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TRANSCRIPT
IAEA TECDOC
On SSI
Johnson, Pecker, Jeremić
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CONTENTS
1. INTRODUCTION.............................................................................................................
1.1 BACKGROUND.................................................................................................1.2 OBJECTIVES......................................................................................................1.3 SCOPE OF THE TECDOC.................................................................................
1.3.1 Design Considerations: General Framework...........................................1.3.2 U.S. Practice............................................................................................1.3.3 Other Member State Practice...................................................................
1.4 STRUCTURE OF TECDOC...............................................................................
2 ELEMENTS OF SSI ANALYSIS...................................................................................
2.1 FREE-FIELD GROUND MOTION..................................................................2.2 MODELING SITE CHARACTERISTICS AND SOIL MATERIAL
BEHAVIOUR....................................................................................................2.2.1 In-situ soil profiles and configurations..................................................2.2.2 Soil material behaviour characterization – Design Basis Earthquake
(DBE) and Beyond Design Basis Earthquake (BDBE).........................2.2.3 Soil material behaviour – construction features....................................
2.3 SITE RESPONSE ANALYSIS.........................................................................2.4 MODELING STRUCTURES AND STRUCTURE FOUNDATIONS............2.5 STRUCTURE MODELING.............................................................................2.6 SSI MODELING...............................................................................................2.7 UNCERTAINTIES............................................................................................
3 SITE CONFIGURATION AND SOIL PROPERTIES...................................................
3.1 SITE CONFIGURATION AND CHARACTERIZATION..............................3.2 CONSTITUTIVE MODELS.............................................................................3.3 EXPERIMENTAL DESCRIPTION OF SOIL BEHAVIOUR.........................
3.3.1 Linear viscoelastic model......................................................................3.3.2 Nonlinear one-dimensional model.........................................................3.3.3 Nonlinear three-dimensional models.....................................................
3.4 PHYSICAL PARAMETERS............................................................................3.5 FIELD AND LABORATORY MEASUREMENTS........................................
3.5.1 Site instrumentation...............................................................................3.5.2 Field investigations................................................................................3.5.3 Laboratory investigations......................................................................3.5.4 Comparison of field and laboratory tests...............................................
3.6 CALIBRATION AND VALIDATION............................................................3.7 UNCERTAINTIES............................................................................................3.8 SPATIAL VARIABILITY................................................................................
4 SEISMIC HAZARD ANALYSIS (SHA).......................................................................
4.1 OVERVIEW......................................................................................................
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4.2 DATA COLLECTION......................................................................................4.3 SEISMIC SOURCES AND SOURCE CHARACTERIZATION....................4.4 GROUND MOTION PREDICTION................................................................4.5 UNIFORM HAZARD RESPONSE SPECTRA...............................................
5 SESIMIC WAVE FIELDS AND FREE FIELD GROUND MOTIONS........................
5.1 SEISMIC WAVE FIELDS................................................................................5.1.1 Perspective and spatial variability of ground motion............................5.1.2 Spatial variability of ground motions....................................................
5.2 FREE FIELD GROUND MOTION ANALYSIS.............................................5.3 RECORDED DATA.........................................................................................
5.3.1 3D (6D) versus 1D Records/Motions....................................................5.3.2 Analytical and numerical (synthetic) earthquake models......................5.3.3 Uncertainties..........................................................................................
5.4 SEISMIC WAVE INCOHERENCE.................................................................5.4.1 Introduction...........................................................................................5.4.2 Incoherence modeling............................................................................5.4.3 Case history - U.S. (in progress - to be completed)...............................
5.5 LIMITATIONS OF TIME AND FREQUENCY DOMAIN METHODS........5.6 REFERENCES..................................................................................................
6 SITE RESPONSE ANALYSIS AND SEISMIC INPUT................................................
6.1 OVERVIEW......................................................................................................6.2 SITE RESPONSE ANALYSIS.........................................................................
6.2.1 Perspective.............................................................................................6.2.2 Foundation Input Response Spectra (FIRS)..........................................
6.3 SITE RESPONSE ANALYSIS APPROACHES..............................................6.3.1 Idealized Site Profile and Wave Propagation Mechanisms...................6.3.2 Non-idealized Site Profile and Wave Propagation Mechanisms...........
6.4 UNCERTAINTIES............................................................................................6.4.1 Analysis Models and Modelling Assumptions......................................
6.5 STANDARD AND SITE SPECIFIC RESPONSE SPECTRA........................6.5.1 Introduction...........................................................................................6.5.2 Standard response spectra......................................................................6.5.3 Site specific response spectra................................................................
6.6 TIME HISTORIES (IN PROGRESS)...............................................................6.7 UNCERTAINTIES AND CONFIDENCE LEVELS (REPEATED FROM
SECTION 6.2.4 – NEED TO DECIDE WHERE IT SHOULD RESIDE).......
7 METHODS AND MODELS FOR SSI ANALYSIS....................................................100
7.1 BASIC STEPS FOR SSI ANALYSIS............................................................1007.2 DIRECT METHODS (JEREMIC)..................................................................104
7.2.1 Linear and nonlinear discrete methods................................................1047.3 SUB-STRUCTURING METHODS................................................................111
7.3.1 Principles.............................................................................................111iii
7.3.2 Rigid or flexible boundary method......................................................1157.3.3 The flexible volume method................................................................1197.3.4 The subtraction method.......................................................................1197.3.5 CLASSI: Soil-Structure Interaction - A Linear Continuum
Mechanics Approach...........................................................................1217.3.6 Discrete methods.................................................................................1277.3.7 Foundation input motion.....................................................................127
7.4 SSI COMPUTATIONAL MODELS..............................................................1297.4.1 Introduction.........................................................................................1297.4.2 Soil/Rock Linear and Nonlinear Modelling........................................1297.4.3 Structural models, linear and nonlinear: shells, plates, walls, beams,
trusses, solids.......................................................................................1357.4.4 Contact Modeling................................................................................1367.4.5 Structures with a base isolation system...............................................1407.4.6 Foundation models..............................................................................1417.4.7 Small Modular Reactors (SMRs).........................................................1437.4.8 Buoyancy Modeling.............................................................................1437.4.9 Domain Boundaries.............................................................................1457.4.10 Seismic Load Input..............................................................................1477.4.11 Liquefaction and Cyclic Mobility Modeling.......................................1507.4.12 Structure-Soil-Structure Interaction.....................................................1517.4.13 Simplified models................................................................................154
7.5 PROBABILISTIC MATERIAL MODELING...............................................1557.5.1 Introduction.........................................................................................155
7.6 PROBABILISTIC RESPONSE ANALYSIS.................................................1577.6.1 Overview..............................................................................................1577.6.2 Simulations of the SSI Phenomena.....................................................158
8 BIBLIOGRAPHY AND REFERENCES......................................................................167
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LIST OF FIGURES
Figure 1-1. Effect of soil stiffness on structure response of a typical nuclear power plant structure; rock (Vs = 6,000 ft/s), stiff soil (Vs = 2,500 ft/s), medium soil (Vs = 1,000 ft/s), and soft soil (Vs = 500 ft/s) (Courtesy of James J. Johnson and Associates). . .4
Figure 1-2. Recorded motions at the Kashiwazaki-Kariwa Nuclear Power Plant in Japan, due to the Niigataken-chuetsu-oki (NCO) earthquake (16 July 2007) in the free-field (top of grade) and in the Unit 7 Reactor Building basement, and third floor (IAEA, KARISMA program [1-3])........................................................................................5
Figure 3-1. Predicted pure shear response of soil at two different shear train amplitudes, from Pisano` and Jeremic (2014).....................................................................................21
Figure 3-2. Cyclic response of soils: (left) no volume change; (middle) compressive response with decrease in stiffness; (right) dilative response with increase in stiffness.........22
Figure 3-3. Idealized stress cycle during an earthquake...............................................................22Figure 3-4. Experimental stress-strain curves under cyclic loading: nonlinear shear behaviour
(left); volumetric behaviour (right)..........................................................................23Figure 3-5. Threshold values for cyclic shear strains...................................................................23Figure 3-6. Shear stress-shear strain curves for constant amplitude cyclic loading.....................24Figure 3-7. One-dimensional viscoelastic model.........................................................................25Figure 3-8. Example of nonlinear characteristics of soil..............................................................27Figure 3-9. Rheological models for Iwan’s nonlinear elastoplastic model..................................30Figure 3-10. Domain of applicability of field and laboratory tests (ASCE 4-16)........................36Figure 3-11. Cyclic stress-strain loops for a soil element with shear strength 65 kPa subjected to
a sinusoidal input seismic motion of 10 s. Predictions produced by the different constitutive models...................................................................................................38
Figure 3-12 Comparison of invasive and non-invasive VS COV values as a function of depth at Mirandola (MIR; a,b), Grenoble (GRE; c) and Cadarache (CAD;d).......................39
Figure 4-1 Steps in the PSHA Process.........................................................................................45Figure 5-1 Propagation of seismic waves in nearly horizontal local geology, with stiffness of
soil/rock layers increasing with depth, and refraction of waves toward the vertical direction. Figure from Jeremic (2016).......................................................................49
Figure 5-2 Propagation of seismic waves in inclined local geology, with stiffness of soil/rock layers increasing through geologic layers, and refraction of waves away from the vertical direction. Figure from Jeremic (2016).........................................................54
Figure 5-3 Acceleration time history LSST07 recorded at SMART-1 Array at Lotung, Taiwan, on May 20th, 1986. This recording was at location FA25. Note the (almost complete) absence of vertical motions. signifying absence of Rayleigh waves. Figure from Tseng et al. (1991).................................................................................................................55
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Figure 5-4 Acceleration time history LSST12 recorded at SMART-1 Array at Lotung, Taiwan, on July 30th, 1986. This recording was at location FA25. Figure from Tseng et al. (1991).........................................................................................................................56
Figure 5-5 Four main sources contributing to the lack of correlation of seismic waves as measured at two observation points. Figure from Jeremic (2016)..................................60
Figure 5-6 Uniform Hazard Response Spectra (UHRS) – annual frequency of exceedance = 10-4
- for 12 rock sites in the U.S. (Courtesy of James J. Johnson and Associates)..........66Figure 6-1 Definition of FIRS for idealized site profiles.............................................................76Figure 7-1: Summary of substructuring methods........................................................................113Figure 7-2 Schematic representation of the elements of soil-structure interaction – sub-structure
method......................................................................................................................113Figure 7-3: Schematic representation of a sub-structure model..................................................114Figure 7-4: 3-step method for soil structure interaction analysis (after Lysmer, 1978).............117Figure 7-5: Substructuring in the subtraction method.................................................................120Figure 7-6 AREVA Finite Element Model of the EPR Nuclear Island (Johnson et al., 2010)
..................................................................................................................................128Figure 7-7 Forces and relative displacements of the element.....................................................137Figure 7-8 Yield surface fs, plastic potential Gs and incremental plastic displacement δup......139Figure 7-9 Description of contact geometry and displacement responses.................................139Figure 7-10 Absorbing boundary consisting of dash pots connected to each degree of freedom of
a boundary node......................................................................................................146Figure 7-11 Large physical domain with the source of load Pe(t) and the local feature (in this
case a soil-structure system...................................................................................148Figure 7-12 Symmetric mode of deformation for two NPPs near each other............................153Figure 7-13 Anti-symmetric mode of deformation for two NPPs near each other....................154Figure 7-14 (a) Left: Transformation relationship between Standard Penetration Test (SPT) N-
value and pressure-meter Young’s modulus, E; (b) Right: Histogram of the residual with respect to the deterministic transformation equation for Young’s modulus, along with fitted probability density function (PDF). From Sett et al. (2011b).....157
Figure 7-15 Mühleberg Nuclear Power Plant – Ensemble of free-field ground motion response spectra – X-direction, 5% damping.......................................................................163
Figure 7-16 Mühleberg Nuclear Power Plant – Ensemble of in-structure response spectra (ISRS) – X-direction, 5% damping – median and 84% NEP.............................................164
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LIST OF TABLES
TABLE 3-1. STRAIN THRESHOLDS AND MODELLING ASSUMPTIONS........................24TABLE 3-2: CHARACTERISTICS OF EQUIVALENT VISCOELASTIC LINEAR
CONSTITUTIVE MODELS.................................................................................26TABLE 4-1 TYPICAL OUTPUT OF PROBABILISTIC SEISMIC HAZARD ANALYSES
[10].........................................................................................................................43TABLE 5-1 ARRAYS USED TO DEVELOP THE COHERENCY MODELS.........................64TABLE 5-2 EARTHQUAKES IN THE ARRAY DATA SETS.................................................64TABLE 7-1 RAYLEIGH WAVE LENGTH AS A FUNCTION OF WAVE SPEED [M/S] AND
WAVE FREQUENCY [HZ].................................................................................153
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1. INTRODUCTION
1.1 BACKGROUND
The response of a structure during an earthquake depends on the characteristics of the ground motion, the surrounding soil, and the structure itself. For structures founded on rock or very stiff soils and subjected to ground motion with frequency characteristics in the low frequency range, i.e., in the frequency range of 2 Hz to 10 Hz, the foundation motion is essentially that which would exist in the rock/soil at the level of the foundation in the absence of the structure and any excavation; this motion is denoted the free-field ground motion. For soft soils, the foundation motion differs from that in the free field due to the coupling of the soil and structure during the earthquake. This soil-structure interaction (SSI) results from the scattering of waves from the foundation and the radiation of energy from the structure due to structural vibrations. Because of these effects, the state of deformation (particle displacements, velocities, and accelerations) in the supporting soil is different from that in the free field. As a result, the dynamic response of a structure supported on soft soil may differ substantially in amplitude and frequency content from the response of an identical structure supported on a very stiff soil or rock. The coupled soil–structure system exhibits a peak structural response at a lower frequency than would an identical rigidly supported structure. Also, the amplitude of structural response is affected by the additional energy dissipation introduced into the system through radiation damping and material damping in the soil. NOTE Boris: I would like to see if we can accommodate the fact that there are always SSI effects, sometimes they are relatively small and sometime they are relatively larger. If SSI effects are always taken into account, that would be a safe option, one can then decide on their own if motions are similar ot not! Main potin is that seismic motions for free field and for SSI will never be the same! Motions might be similar but not the same! We will have an example in the appendix (or in chapter 8 and then more detailed one in the appendix) where we will show that motions are not the same. Differences might be small (will give actuall differences and let readers decide what is small) in some cases, but in some cases differences will be large! Difference will be large for beyond design basis earthquakes, as nonlinear effects of contact zone will start to be felt much more than for small earthquakes. So in that sense, we might postpone statements like this until the end, and provide enough information/examples for readers to decide. I understand that this is a recommendation for doing (or rather, not doing) SSI for last many years, but as such, it would be great if this suggsestions is supported by analysis…
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NOTE JJJ: Not every case requires an SSI analysis which requires software tools that analyze SSI. In ASCE 4-98, we had the following guidance: “A fixed-base support may be assumed in modeling structures for seismic response analysis when the frequency obtained assuming a rigid structure supported on soil springs representing the soil supporting medium, established based on Table 5.1-1 (reference to table of constant soil springs and damping), is more than twice the dominant frequency obtained from a fixed base analysis of the flexible structure representation.”
This provision was not repeated in ASCE 4-16 because for hard rock sites and high frequency ground motion, SSI has been shown to be potentially important, assuming either coherent or, especially incoherent ground motion; almost always in reducing conservatism. So, ASCE 4-16 states: “A fixed-base condition may be assumed for soil-structure systems when the site soil conditions behave in a rock-like manner without significant loss of accuracy in the response calculations. In general, a shear-wave velocity of 8,000 ft/s (2,400 m/s) or greater at a shear strain of 10-4 % or smaller supports a fixed-base analysis without consideration of the SSI effects regardless of the frequency content of the free-field motion.
The assumption of a fixed-base condition needs to be carefully evaluated when considering high-frequency input motions to typical nuclear safety-related structures.”
Consequently, this introduction material should be softened.
Figure 1-1 shows an example of horizontal in-structure response spectra at the top of a typical nuclear power plant structure calculated assuming the structure is founded on four different site conditions ranging from rock to soft soil. It is clear from Figure 1-1 that ignoring SSI in the dynamic response of the structure misrepresents the seismic response of the structure and the seismic input to equipment, components, distribution systems, and supporting sub-structures. NOTE Boris: I moved Figure 1-1 here, it was few pages down, and it is really much more useful if it is in text right after it is mentioned… Colors need to be worked on, yellow does not show up nicely, perhaps we can change to different line types (full, dashed, dash-dot, etc…)NOTE JJJ: I have the original and can modify the colors.
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SSI analysis has evolved significantly over the past five decades.
Initially in the 1960s and early 1970s, SSI was treated with tools developed for calculating the effects of machine vibrations on their foundations, the supporting media, and the machine itself. Foundations were modelled as rigid disks of circular or rectangular shape. Generally, soils were modelled as uniform linear elastic half-spaces. Soil springs were developed from continuum mechanics principles and damping was modelled with dashpots. This approach addressed inertial effects only.
In the 1970s and early 1980s, research activities were initiated. Research centers sprouted up around academic institutions – University of California, Berkeley (Profs. Seed and Lysmer group), University of California, San Diego (Prof. Luco)/University of Southern California (Drs. Wong (NOTE BORIS: I thought that it was Wong and Trifunac together), NOTE JJJ: SSI analysis was Luco and Wong consistently, I have to search H.L. Wong for publications with Prof. Trifunic – maybe on wave propagation.
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Massachusetts Institute of Technology (Profs. Roesset and Kausel, and Dr. John Christian), Dr. John Prof. Wolf (Swiss Federal Institute of Technology).
In addition, nonlinear analyses of soil/rock media subjected to explosive loading conditions led to alternative calculation methods focused on nonlinear material behaviour and short duration, high amplitude loading conditions. Adaptation of these methods to the earthquake problem was attempted with mixed results.
As one element of the U.S. Nuclear Regulatory Commission’s sponsored Seismic Safety Margin Research Program (SSMRP), the state of knowledge of SSI as of 1980 was well documented in a compendium Johnson [1-1] of contributions from key researchers (Luco; Roesset and Kausel; Seed, and Lysmer) and drew upon other researchers and practitioners as well (Veletsos, Chopra). This reference provided a framework for SSI over the 1980s and 1990s.
SSI analysis methodologies evolved over the 1970s and 1980s. Simplified soil spring approaches continued to be used in various contexts. More complete substructure methods emerged, specifically developments by Luco and colleagues (University of California, San Diego) and Roesset, Kausel and colleagues (Massachusetts Institute of Technology). The University of California, Berkeley Team (Seed, Lysmer and colleagues) developed several direct approaches to performing the SSI analyses (LUSH, ALUSH, FLUSH).
In the 1980s, emphasis was placed on the accumulation of substantial data supporting and clarifying the roles of the various elements of the SSI phenomenon. One important activity was the Electric Power Research Institute (EPRI), in cooperation with Taiwan Power Company (TPC), constructed two scale-model reinforced concrete nuclear reactor containment buildings (one quarter and one twelfth scale) within an array of strong motion instruments (SMART-l, Strong Motion Array Taiwan, Number 1) in Lotung, Taiwan. The SMART-I array was sponsored by the U.S. National Science Foundation and maintained by the Institute of Earth Sciences of Academia Sciences of Taiwan. The structures were instrumented to complement the free-field motion instruments of SMART-I.
The expectation was that this highly active seismic area would produce a significant earthquake with strong ground motion. The objectives of the experiment were to measure the responses at instrumented locations due to vibration tests, and due to actual earthquakes. Further, to sponsor a numerical experiment designed to validate analysis procedures and to measure free-field and structure response for further validation of the SSI phenomenon and SSI analysis techniques. These objectives were generally
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accomplished although with some limitations due to the dynamic characteristics of the scale model structure compared to the very soft soil at the Lotung site.
Additional recorded data in Japan and the U.S. served to demonstrate important aspects of the free-field motion and SSI phenomena.
Also, sSignificant progress was made in the development and implementation of SSI analysis techniques, including the release of the SASSI computer program, which continues to be in use today.
In the 1980s, skepticism persisted as to the physical phenomena of spatial variation of free-field motion, i.e., the effect of introducing a free boundary (top of grade) into the free-field system, the effect of construction of a berm for placement of buildings or earthen structures, and other elements. In addition, the lack of understanding of the relationship between SSI analysis “lumped parameter” methods and finite element methods led to a requirement implemented by the U.S. NRC that SSI analysis should be performed by “lumped parameter” and finite element methods and the results enveloped for design.
In the 1990s, some clarity of the issues was obtained. In conjunction with data acquisition and observations, the statements by experts, researchers, and engineering practitioners that the two methods yield the same results for problems that are defined consistently finally prevailed. The EPRI/TPC efforts contributed to this clarity.
The methods implemented during these three decades and continuing to the present are linear or equivalent linear representations of the soil, structure, and interfaces. Although research in nonlinear methods has been performed and tools, such as NRC Real ESSI, have been implemented for verification, validation, and testing on realistic physical situations, adoption in design or validation environments has yet to be done.
In recent years, a significant amount of experience has been gained on the effects of earthquakes on nuclear power plants worldwide. Events affecting plants in high-seismic-hazard areas such as Japan have been documented in International Atomic Energy Agency (IAEA) Safety Reports Series No. 66, “Earthquake Preparedness and Response for Nuclear Power Plants” (2011) [1-2]. In some cases, SSI response characteristics of NPP structures have been documented and studied, in particular the excitation and response of the Kashiwazaki-Kariwa Nuclear Power Plant in Japan, due to the Niigataken-chuetsu-oki (NCO) earthquake (16 July 2007) [1-3]. Figure 1-2 shows the recorded responses of the free-field top of grade measured motion (blue curve) compared to the motions recorded at the Unit 7 Reactor Building basement (red curve) and the third floor (yellow curve). The SSI effects are evident – significantly reduced motions
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from top of grade to the basement level and reduced motions in the structure accompanied by frequency shifts to lower frequencies. These measured motions demonstrate the important aspects of spatial variation of free-field ground motion and soil-structure interaction behaviour.
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Figure 1-1. Effect of soil stiffness on structure response of a typical nuclear power plant structure; rock (Vs = 6,000 ft/s), stiff soil (Vs = 2,500 ft/s), medium soil (Vs = 1,000 ft/s), and soft
soil (Vs = 500 ft/s) (Courtesy of James J. Johnson and Associates)
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Figure 1-2. Recorded motions at the Kashiwazaki-Kariwa Nuclear Power Plant in Japan, due to the Niigataken-chuetsu-oki (NCO) earthquake (16 July 2007) in the free-field (top of grade) and in the Unit 7 Reactor Building basement, and third floor (IAEA, KARISMA program [1-3])
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1.2 OBJECTIVES
This TECDOC provides a detailed treatise on SSI phenomena and analysis methods specifically for nuclear safety related facilities. It is motivated by the perceived need for guidance on the selection and use of the available soil-structure interaction methodologies for the design of nuclear safety-related structures.
The purpose of the task is to review and critically assess the state-of-the-practice regarding soil-structure interaction methods. The emphasis is on the engineering practice, not on methods that are still at the research and development phase. The final goal is to provide practical guidance to the engineering teams performing this kind of analyses in present and in near future. (NOTE BORIS, perhaps we say near future, as if we just review current practice, it will be outdated tomorrow…) NOTE JJJ: Good.The objectives are to:
Describe the physical aspects of site, structure, and earthquake ground motion that lead to important SSI effects on the behaviour of structures, systems, and components (SSCs).
Describe the modelling of elements of SSI analysis that are relevant to calculating the behaviour of SSCs subjected to earthquake ground motion.
Identify the uncertainty associated with the elements of SSI analysis and quantify, if feasible.
Review the state-of-practice for SSI analysis as a function of the site, structures, and ground motion definition of interest. Other important considerations are the purposes of the analyses, i.e., design and/or assessment.
Provide guidance on the selection and use of available SSI analysis methodologies for design and assessment purposes.
Identify sensitivity studies to be performed on a generic basis and a site specific basis to aid in decision-making.
Provide a futuristic view of the SSI analysis field looking to the next five to ten year period.
Document the recommendations observations, recommendations, and conclusions.
The context is facility design, and assessment.
Design includes new nuclear power facility design, such as Reference Designs or Certified Designs of nuclear power plants (NPPs) and the design/qualification of modifications, replacements, upgrades, etc. to an existing facility.
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Assessments encompass evaluations for Beyond Design Basis Earthquake ground motions typically Seismic Margin Assessments (SMAs) or Seismic Probabilistic Risk Assessments (SPRAs). Assessments are necessary for the forensic analysis of facilities that experience significant earthquake ground motions.
This TECDOC is intended for use by SSI analysis practitioners and reviewers (Peer Reviewers and others). This TECDOC is also intended for use by regulatory bodies responsible for establishing regulatory requirements and by operating organizations directly responsible for the execution of the seismic safety assessments and upgrading programmes.
1.3 SCOPE OF THE TECDOC
This TECDOC provides a detailed treatise on SSI phenomena and analysis methods specifically for nuclear safety related facilities. The context is facility design, and assessment.
Design includes new nuclear power facility design, such as Reference Designs or Certified Designs of nuclear power plants (NPPs) and the design/qualification of modifications, replacements, upgrades, etc. to an existing facility.
Assessments encompass evaluations for Beyond Design Basis Earthquake ground motions typically Seismic Margin Assessments (SMAs) or Seismic Probabilistic Risk Assessments (SPRAs). Assessments for the forensic analysis of facilities that experience significant earthquake ground motions.
1.3.1 Design Considerations: General Framework
Design requires a certain amount of conservatism to be introduced into the process. The amount of conservatism is dependent on a performance goal to be established. ASCE 4-16, ASCE 43-05, and U.S. DOE Standards define specific performance goals of structures, systems, and components (SSCs) in terms of design (DBE) and in combination with beyond design basis earthquakes (BDBEs). Performance goals are established dependent on the critical nature of the SSC and the consequences of “failure” to personnel (on-site and public) and the environment.
Performance goals may be defined in probability space. Some Member States, including the U.S. and Canada (NOTE BORIS, perhaps just say members states, not specificailly emphasize USA, not sure if we should make it too USA centric) provide specific risk goals, which become one measure of risk acceptance.
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The US NRC staff identified probabilistic performance goals relative to core damage frequency (CDF) and to large early release frequencies (LERF) in its staff requirements memorandum (SRM) dated June 26, 1990, in response to SECY-90-016. The NRC’s goals are less than 10-4 for mean core damage annual frequency, and less than 10-6 for mean large release annual frequency.
The US Department of Energy (DOE) similarly established probabilistic performance goals to be used as a measure of acceptance of the design of nuclear facilities. The performance goals for DOE nuclear facilities are that confinement should be ensured at an annual frequency of failure of between 10-4 to 10-5 (DOE Order 420.1C and Implementation Guide 420.1-2).
The Canadian Regulatory Document RD-337 specifies three probabilistic performance goals: small release frequencies less than 10-5 per annum; large release frequencies less than 10-6 per annum; and core damage frequencies less than 10-5 per annum.
FRANCE: UK: CHINA: SWITZERLAND: JAPAN: SOUTH AFRICA
Add other Member State performance criteria based on authors’ knowledge and questionnaire to be distributed.
In general, the process of establishing a comprehensive approach to defining and implementing the overall performance goals and, subsequently, tier down these performance goals to SSI (and other elements of the seismic analysis, design, and evaluation process) is as follows:
a. Establish performance goals, or develop a procedure to establish performance goals, for seismic design and beyond design basis earthquake assessments for SSCs.
b. Partition achievement of the performance goal into elements, including SSI.
c. Develop guidance for SSI modelling and analysis to achieve the performance goal.
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Considering that the subject of this TECDOC is SSI, an important element in this process is to establish the levels of conservatism in the current design and evaluation approaches to SSI.
1.3.2 U.S. Practice
One example is U.S. practice. ASCE 4-16[1-4], ASCE 43-05[1-5], and U.S. DOE Standards define specific performance goals of structures, systems, and components (SSCs) in terms of design (DBE) and in combination with beyond design basis earthquakes (BDBEs).
Given the seismic design basis earthquake (DBE), the goal of ASCE/SEI 4-16 is to develop seismic responses with 80% probability of non-exceedance. For probabilistic seismic analyses, the response with 80% probability of non-exceedance is selected.
ASCE/SEI 4-16 is intended to be used together, and be consistent with the revision to ASCE/SEI 43-05. The objective of using ASCE 4 together with ASCE/SEI 43 is to achieve specified target performance goal annual frequencies. To achieve these target performance goals, ASCE/SEI 43 specifies that the seismic demand and structural capacity evaluations have sufficient conservatism to achieve both of the following:
i. Less than about a 1% probability of unacceptable performance for the Design Basis Earthquake Ground Motion, and
ii. Less than about a 10% probability of unacceptable performance for a ground motion equal to 150% of the Design Basis Earthquake Ground Motion.
The performance goals will be met if the demand and capacity calculations are carried out to achieve the following:
i. Demand is determined at about the 80% non-exceedance level for the specified input motion.
ii. Design capacity is calculated at about 98% exceedance probability.
1.3.3 Other Member State Practice
Add other Member State design practice based on authors’ knowledge and questionnaire to be distributed. To add the French experience and practice and other countries
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1.4 STRUCTURE OF TECDOC
Final step.
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2 ELEMENTS OF SSI ANALYSIS
The elements of SSI analysis are:
Free-field ground motion – seismic input Site configuration and modelling of soil properties Site response analysis Modelling of foundation and structure Methods of SSI analysis Uncertainties
Substantial contributions to this chapter are from Johnson [2-1].
2.1 FREE-FIELD GROUND MOTION
The term free-field ground motion denotes the motion that would occur in soil or rock in the absence of the structure or any excavation. Describing the free-field ground motion at a site for SSI analysis purposes entails specifying the point at which the motion is applied (the control point), the amplitude and frequency characteristics of the motion (referred to as the control motion and typically defined in terms of ground response spectra, and/or time histories), the spatial variation of the motion, and, in some cases, duration, magnitude, and other earthquake characteristics.
In terms of SSI, the variation of motion over the depth and width of the foundation is the important aspect. For surface foundations, the variation of motion on the surface of the soil is important; for embedded foundations, the variation of motion over both the embedment depth and the foundation width should be known.
Control motion. Free-field ground motion may be defined by site independent or site dependent ground response spectra.
‒ Site independent ground motion is most often used for performing a new reference design or a Certified Design, which is to be placed on a number of sites with differing characteristics.
‒ Site specific ground motion is most often developed from a probabilistic seismic hazard analysis (PSHA) and is most often used for site specific design or evaluations. The PSHA results may apply directly to top of grade at the site of interest or may be generated at a location within the site profile, such as on hard rock or at a significant interface of soil/rock stiffness. In this latter case, a site
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response analysis is performed to generate the ground motion for input to the SSI analysis.
Control point. The location at which the control motion is applied. Implicit in the control point definition is the assumed boundary condition for control motion application, i.e., actual (or hypothetical outcrop) or in-column motion.
Spatial variation of motion. Typically, defined by an assumed wave propagation mechanism, such as vertically propagating shear and dilatational compressional waves. Also, can be defined by numerical source modelling from fault displacement to site. (NOTE BORIS: I assume that above we talk about P (primary) compressional waves, compressional and S (secondary) shear waves.
NOTE BORIS: Just a brief intro to what we note in chapters 5 and 6, as more and more operators develop motions this way (PG&E for example) ) NOTE JJJ: see below
In addition to the above methods, free field ground motion can be developed using large scale regional models. Large scale regional models require knowledge of local geology and location of causative faults and are thus only possible at a limite number of locations. Large scale regional modelling will be described in some detail in Chapter 5. . Numerical modelling from source to site is most often performed where the seismic hazard is high and the sources are well-defined. Japan is one Member State in which such models are required by the regulator. Such models have been implemented and the results considered in the redefinition of the Ss design basis earthquake ground motion.
Important considerations of defining the free-field ground motion that are discussed later in this TECDOC are:
[a)] Free field ground motions due to wWave types as propagated from source to site without distinct site specific features (such as, slanted layers, topographic effects, etc.) influencing the propagation, i.e., uniform site; for discussion purposes, vertically propagating shear and dilatational compressional waves;
[b)] Free-field ground motion due to natural geological and geotechnical characteristics that differ from a) – slanted layers of soil and rock, hard rock intrusions (dykes), basins, etc.; topographic effects (hills, valleysvalleys, etc.);
a)[c)] Free-field ground motion taking into account man-made features at the site, e.g., excavated soil and rock for construction purposes; construction of berms to support structures; etc.
b)[d)] Relationship between vertical and horizontal ground motion; 14
c)[e)] Effect of high water table on horizontal and vertical ground motion; d)[f)] Incoherence of ground motion, especially for high frequencies; e)[g)] Others to be identified.
These topics are treated in more detail in Chapters 4, 5, and 7.
2.2 MODELING SITE CHARACTERISTICS AND SOIL MATERIAL BEHAVIOUR
2.2.1 In-situ soil profiles and configurations
Field investigations of a site are essential to defining its characteristics. As stated in IAEA SSG-9 [2-2], geological, geophysical, and geotechnical investigations should be performed on four spatial scales. The four spatial areas are regional (encompassing all important features outside the near regional area); near regional (an area composed of not less than a radius about the site of 25 km), site close vicinity (radius of less than 5 km), and site area. The first three scales of investigation lead primarily to progressively more detailed geological and geophysical data and information. The detail of these data is determined by the different spatial scales. The site area investigations are aimed at developing the geotechnical database. The data from these investigations and other sources are to be contained in a geological, geophysical, and geotechnical (GGG) database. To achieve consistency in the presentation of information, whenever possible the data should be compiled in a geographical information system (GIS) with adequate metadata information. The GGG database is complemented by a seismological database. One end result of the data gathering is a geological map.
This data is generally low strain data.
These investigations are feasible for new sites. However, the site area investigations may be difficult for existing sites with operating nuclear facilities.
2.2.2 Soil material behaviour characterization – Design Basis Earthquake (DBE) and Beyond Design Basis Earthquake (BDBE)
The selection of material models for in-situ soil and rock is dependent on numerous issues:
Soil characteristics – hard rock to soft soil. Excitation level. Availability of SSI analysis soil material models (linear, equivalent linear, nonlinear,
elastic-plastic).
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Laboratory tests to define material property parameters (linear and nonlinear); correlation with field investigation results for excitation levels of interest.
Phenomena to be modelled, e.g., dynamic response vs. soil failure. Risk importance of structures to be analyzed. Complexity/importance of the structure or soil-founded component, e.g., singletons in the
systems models are of higher importance to risk and may warrant more sophisticated SSI analyses.
2.2.3 Soil material behaviour – construction features
The selection of material models for construction features of soil and rock is dependent on issues identified in Sub-section 2.2.2, but with the ability to better understand the placement and material characteristics of these man-made features.
Examples are:
Excavation and engineered fill beneath basemats and backfill around structures. Construction of berms, break-waters, channel walls, etc.
Section 2.2 issues are treated in more detail in Chapters 3, 6, and 7.
2.3 SITE RESPONSE ANALYSIS
In the broadest sense, the purpose of site response analysis is to determine the free-field ground motion at one or more locations given the motion at another location. In addition to location, the form of the motion is defined, i.e., in-soil motion or actual/hypothetical outcrop motion. Site response analyses can be performed in the frequency domain or in the time domain.
Frequency domain analyses for earthquake ground motions are linear or equivalent linear. Often, strain-dependent soil material behaviour is simultaneously defined with the definition of the free-field ground motion as seismic input (Chapter 3).
Time domain site response analyses may be linear or nonlinear. Time domain analyses are most often performed to generate the explicit definition of the seismic input motion at all locations along the boundary of the linear or nonlinear SSI model (Chapters 4 and 6).
Investigations of the effects of irregular site profiles can be performed in the frequency domain or the time domain.
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From the perspective of SSI analyses, free-field ground motion may be defined by standard ground response spectra usually at one location in the site profile, or by UHRS derived from a PSHA and defined at a reference location in the site profile. Generally, the site response analysis serves two purposes: (i) to generate strain-dependent soil properties for equivalent linear or nonlinear representation of the soil material behaviour; and (ii) to generate free-field motion at other locations in the site profile, e.g., at structure foundation locations.
2.4 MODELING STRUCTURES AND STRUCTURE FOUNDATIONS
Modelling the structure foundation is an important element in the SSI modelling process. Many simple and sophisticated substructure methods assume the foundation behaves rigidly, which is a reasonable assumption given a multistep seismic response analysis approach (see below) and taking into account the stiffening effects of structural elements supported from the foundation (shear walls and other stiff structural elements).
Structure and foundation modelling are closely coordinated and are dependent on the objectives of the analysis – multistep vs. single step.
Multistep vs. single step seismic structure analyses and structure models. In general, one can categorize seismic structure analysis, and, consequently, the foundation and structure models, into multistep methods and single step methods.
Multistep analyses. In the multistep method, the seismic response analysis is performed in successive steps. In the first step, the overall seismic response (deformations, displacements, accelerations, and forces) of the soil-foundation-structure is determined. The response obtained in this first step is then used as input to other models for subsequent analyses of various portions of the structure. It is common to model the foundation as behaving rigidly in this first step. The subsequent analyses are performed to obtain: (i) seismic loads and stresses for the design and evaluation of portions of a structure; and (ii) seismic motions, such as time histories of acceleration and in-structure response spectra (ISRS), at various locations of the structural system, which can be used as input to seismic analyses of equipment and subsystems. Typically, the structure model of the first step of the multistep analysis represents the overall dynamic behaviour of the structural system but need not be refined to predict stresses in structural elements. The first step model is sufficiently detailed so that the responses calculated for input to subsequent steps or for evaluation of the first model would not change significantly if it was further refined.
A lumped-mass stick model may be used for the first-step model provided that the requirements of ASCE 4-16, Section 3.8.1.3 [2-3] and/or the requirements of SPID Section 6.3.1 [2-4] are met.
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A detailed second-step model that represents the structural configuration in adequate detail to develop the seismic responses necessary for the seismic design of the structure or fragility evaluations is needed. Seismic responses include detailed stress distributions for structure capacity evaluation; detailed kinematic response, such as acceleration, velocity, and displacement time histories, and generated ISRS.
Single step analysis. The objectives of one-step analysis are identical to the multistep method, except that all seismic responses in a structural system are determined in a single analysis. The single step analysis is conducted with the detailed “second-step” model described above. Initially, the single step analysis was most often employed for structures supported on hard rock, with a justified fixed-base foundation condition for analysis purposes. Recently, with the development of additional computing power, single step analyses are performed more frequently.
Decisions concerning foundation modelling should consider the following items:
Purpose of the analysis – DBE design, BDBE evaluation, forensic analysis. Multistep vs. single step analysis (see above discussion). Excitation level. Mat vs. spread/strip footings. Behaving rigidly or flexibly. Surface-or near surface-founded. Embedded foundation with partially embedded structure. Partially embedded (less than all sides). Piles and caissons.
‒ Boundary conditions – baseman slab retains contact with soil/separates from underlying soil.
‒ Pile groups – how to model. Separation of basemat and side walls from soil. Sliding/uplift.
Section 2.3 issues are treated in more detail in Chapters 3, 6, and 7.
2.5 STRUCTURE MODELING
See Section 2.3 for overall discussion of modelling.
Decisions concerning structure modelling should consider the following items:
Purpose of the analysis – DBE design, BDBE evaluation, forensic analysis. 18
Multistep vs. single step analysis (see discussion in Section 2.3). Excitation level. Linear or nonlinear structure behaviour.
‒ Equivalent linear (especially for design) ‒ Nonlinear (especially for BDBE evaluations)
Lumped mass vs. finite element models. Frequency range of interest – especially high frequency considerations (50 Hz, 100 Hz).
Section 2.4 issues are treated in more detail in Chapters 3, 6, and 7.
2.6 SSI MODELING
Decisions concerning SSI modelling should consider the following items:
Purpose of the analysis – DBE design, BDBE evaluation, forensic analysis: ‒ DBE design – conservative ‒ BDBE evaluation – conservative for SMA/realistic for SPRA ‒ Forensic analysis – realistic
Probabilistic/deterministic. Excitation level – equivalent linear or nonlinear soil behaviour. Irregular soil/rock profiles. Partially embedded (less than all sides). High water table. Structure-to-structure interaction. Other issues.
Section 2.1 – 2.62.5 issues are treated in more detail in next Chapters. 6 and 7.
2.7 UNCERTAINTIES
Uncertainties exist in the definition of all elements of soil-structure interaction phenomena and their analyses:
Free-field ground motion – seismic input Site configuration and modelling of soil properties Site response analysis Modelling of foundation and structure Methods of SSI analysis
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In many cases, uncertainties can be explicitly represented by probability distributions of SSI analysis parameters, e.g., soil material properties, structure dynamic properties. In other cases, uncertainties in SSI analysis elements may need to be assessed by sensitivity studies and the results entered in the analysis by combining the weighted results.
In general, uncertainties are categorized into aleatory uncertainty and epistemic uncertainty (ASME/ANS [2-5]) with the following definitions:
“aleatory uncertainty: the uncertainty inherent in a nondeterministic (stochastic, random) phenomenon. Aleatory uncertainty is reflected by modelling the phenomenon in terms of a probabilistic model. In principle, aleatory uncertainty cannot be reduced by the accumulation of more data or additional information. (Aleatory uncertainty is sometimes called “randomness.”)”
“epistemic uncertainty: the uncertainty attributable to incomplete knowledge about a phenomenon that affects our ability to model it. Epistemic uncertainty is reflected in ranges of values for parameters, a range of viable models, the level of model detail, multiple expert interpretations, and statistical confidence. In principle, epistemic uncertainty can be reduced by the accumulation of additional information. (Epistemic uncertainty is sometimes also called “modelling parametric uncertainty.”).”
NOTE BORIS: modelling uncertainty is used in our examples as uncertainty in choosing more or less sophisticated models, on the other hand, parametric uncertainty is used when one decides on model sophistication, but then has do deal with uncertain parameters (range of values, probability distribution of material parameters. In that sense, I replaced term ``modelling uncertainty’’ with ``parametric uncertainty’’. Please let me know if this acceptable!
NOTE JJJ: I e-mailed the following explanation and recommendation to leave the definitions unchanged and, possibly, add “parameter uncertainty” as an additional term:
These definitions are taken directly or paraphrased from the ASME/ANS Standard on PRA (eith Appendix on SMA):
"epistemic uncertainty: the uncertainty attributable toincomplete knowledge about a phenomenon that affectsour ability to model it. Epistemic uncertainty is reflectedin ranges of values for parameters, a range of viablemodels, the level of model detail, multiple expert interpretations,and statistical confidence. In principle, epistemicuncertainty can be reduced by the accumulationof additional information. (Epistemic uncertainty issometimes also called “modeling uncertainty.”)"
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"aleatory uncertainty: the uncertainty inherent in a nondeterministic(stochastic, random) phenomenon. Aleatoryuncertainty is reflected by modeling the phenomenonin terms of a probabilistic model. In principle, aleatoryuncertainty cannot be reduced by the accumulation ofmore data or additional information. (Aleatory uncertaintyis sometimes called “randomness.”)"
These definitions are long-standing going back to the 1970s ...
In a very important example, Diablo Canyon detailed testing of the site conditions reduces epistemic uncertainty by performing the tests - this was emphasized by Nozar Jahangir, DCPP, at the ANS Winter Meeting on the subject of RIDM for extreme external events, which Bob Budnitz and I chaired ...
I view parameter uncertainty more literally, i.e., once I selected a model then the parameters of the model are likely associated with aleatory(ic) and epistemic uncertainty ... also, aleatory uncertainty is the cause of variability in the probability distribution, whereas, epistemic uncertainty is the cause of confidence bounds or limits ...
Finally, we separate the purposes of SSI into "design" and "assessments or evaluations" - in reality, almost all applications today and in the near future are "assessments or evaluations" - those associated with SPRA or SMA are subject to ASME Addendum B (and its revision), so I do not think we should change the terminology ...
Randomness is considered to be associated with variabilities that cannot practically be reduced by further study, such as the exact earthquake time history occurring at the site in each direction and associated response variables, such as random phasing of responses. Uncertainty is generally considered to be variability associated with a lack of knowledge that could be reduced with additional information, data, or models. Aleatory and epistemic uncertainties are often represented by probability distributions assigned to SSI parameters. Further, these probability distributions are typically assumed to be non-negative lognormal distributions (for example lognormal, Weibull, etc.) . An input parameter to the SSI analysis may be represented by a median value (Am) and a double lognormal function (εR and εU) with median values of 1.0 and variability (aleatory and epistemic uncertainty) defined by lognormal standard deviations (βR and βU). (NOTE BORIS: I just expanded to other non-negative distributions that are used…)NOTE JJJ: Good
A = Am εr εu (2.1)
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In some cases, it is advantageous to combine the randomness and modelling parametric uncertainty into a “composite variability” as defined in ASME/ANS, 2013): NOTE JJJ: retain modelling – add “data” as in paragraph below.
“composite variability: the composite variability includes the aleatory (randomness) uncertainty (βR) and the epistemic (modeling and data) uncertainty (βU). The logarithmic standard deviation of composite variability, βC, is expressed as (βR 2 + βU 2)1/2”
The same functional representation of Eqn. 2.1 typically defines the fragility function for SSCs in a SPRA.
All aspects of the SSI analysis process are subject to uncertainties. The issue is how to appropriately address the issue in the context of design and assessments.
Some issues are amenable to modeling probabilistically:
Earthquake ground motion ‒ Control motion (amplitude and phase); ‒ Spatial variation of motion – wave fields generating coherent ground motion; ‒ Random variation of motion – high frequency incoherent ground motion;
Physical material properties (soil, structure dynamic characteristics). Physical soil configurations, e.g., thickness of soil layers). Water table level, including potential buoyancy effects. Others.
Some issues are amenable to sensitivity studies to determine their importance to SSI response:
Linear (equivalent linear) vs. nonlinear soil properties. Sliding/uplift. Non-horizontal layering of soil.
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3 SITE CONFIGURATION AND SOIL PROPERTIES
3.1 SITE CONFIGURATION AND CHARACTERIZATION
The general requirements for site characterization are defined in IAEA Safety Standard NS-R -3 [1]. The size of the region to which a method for establishing the hazards associated with earthquakes is to be applied shall be large enough to include all the features and areas that could be of significance for the determination of the natural induced phenomena under consideration and for the characteristics of the event. For soil-structure interaction (SSI) analyses, description of the soil configuration (layering or stratigraphy) and characterization of the dynamic (material) properties of the subsurface materials, including the uncertainties associated to them, shall be investigated and a soil profile for the site, in a form suitable for design purposes, shall be determined. Determining soil properties to be used in the SSI analysis is the second most uncertain element of the process — the first being specifying the ground motion.
The site shall be characterized from both geological and geotechnical investigations. Attention is drawn on the fact that the incoming seismic waves and therefore the seismic response of soils and structures are not controlled only by the properties of layers in the immediate vicinity of the structures. Therefore, large scale geological investigations are required to define the lateral and in-depth extent of the various strata, the underground topography, the possible existence of basins (large-scale structural formation of rock strata formed by tectonic warping of previously flat-lying strata, structural basins are geological depressions, and are the inverse of domes), the depth to the bedrock, the elevation of the water table, etc. As stated in IAEA SSG-9 [2], geological and geotechnical investigations should be performed at four spatial scales: regional, near regional, close vicinity to the site and site area. The typical scales go from several tens of kilometres to some hundreds of metres. These data should be compiled to form a geographical information system (GIS) and should produce a geological map for the site.
In addition, the geotechnical investigations shall allow assessing the mechanical characteristics required for the seismic studies together with the range of uncertainties and spatial variability associated with each parameter. This is accomplished with cored boreholes, field geophysical investigations, sampling of undisturbed samples for laboratory testing.
The extent of the local site investigations depends on the ground heterogeneities, dimensions of the installation and is typically much larger than for static design. Furthermore, the content shall be defined in connection with the models that are used for the seismic studies and can be guided by parametric studies aiming at identifying the most influential parameters.
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The site characterization phase should allow addressing the following aspects for SSI analyses: is the overall site configuration regular enough to be modelled as a simple 1D geometry? is the above surface and subsurface topography so irregular that the site geometry needs to be represented for the site response analyses with a 2D, or even 3D, model? To help reaching such a decision, the geotechnical engineer may take advantage of his/her own experience, simplified rules proposed in the outcome of the European research project NERA (2014) [3] or be guided by site instrumentation (Section 3.5.1).
NOTE BORIS: above, geotech engineer can use experience, simplified rules or instrumentation… It is not that I do not trust practicing engineers, however, I always find a strong bias toward simplified solutions, even if they are far from being warranted. So in that sense, can we say that we ask practicing engineers to document their choice, especially if it from experience, that is experience is good, however I would like to learn how they make their choice too, would like to see what is experience based on (analysis, case studies, etc.)
Within the framework of the NERA project [3], aggravation factors are defined as the ratios between the spectral acceleration at the ground surface for a 2D configuration to the same quantity for a 1D configuration; aggravation factors different from 1.0 reveal a 2D effect. Pitilakis et al, (2015) [4] tentatively concluded the following based on extensive numerical analyses, involving linear and equivalent linear soil constitutive models:
The aggravation factors are period dependent, The average (over the period range) aggravation factors depend on the geographical
location of the site with respect to the basin edges, The aggrevation factors for locations cClose to the valley edges are smaller than 1.0,
The aggrevations factors for location iIn the centrale part of the valley, they depend on the fundamental period of the valley: they are slightly larger than 1.0 if the basin period is small, typically less than 3.0s, and therefore 2D effect may be considered of minor importance; they may reach high values if the basin period is large.
These results with more precise recommendations, still to be published, combined with seismic instrumentation, constitute helpful guidelines to estimate the potential for 2D effects.
3.2 CONSTITUTIVE MODELS
Soils are known to be highly nonlinear materials as evidenced from both field observations during earthquakes and laboratory experiments on samples. According to Prevost and Popescu, (which reference, year, my (Boris) guess is 1996 from EJGE) during loading, the solid particles which form the soil skeleton undergo irreversible motions such as slips at grain boundaries. In addition, Mindlin and Deresiewicz (1953) show that inelastic, elastic-plastic, irreversible deformation is present for any amount of shear stress in particulate material (soil). When the
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microscopic origin of the phenomena involved are not sought, phenomenological equations are used to provide a description of the behaviour of the various phases which form the soil medium. Considering the particles essentially incompressible, deformation of the granular assembly occurs as the particles translate, slip and/or roll, and either form or break contacts with neighboring particles to define a new microstructure. The result is an uneven distribution of contact forces and particle densities that manifests in the form of complex overall material behaviour such as permanent deformation, anisotropy, and localized instabilities and dilatancy (change of volume during pure shearing deformation) (NOTE BORIS, I thought that dilatancy is probably on the same order of importance as other phenomena and that it needs to be mentioned…). Although very complex when examined on the microscale, soils as many other materials may be idealized at the macroscale as behaving like continua.
In most cases, the soil element is subjected to general three-dimensional state of time-varying stresses. Thus, an adequate modelling of the soil behaviour requires sophisticated constitutive relationships. It is very convenient to split the soil response to any type of loading into two distinct components: the shear behaviour and the volumetric behaviour. In shear, soil response manifests itself in terms of a stiffness reduction and an energy dissipation mechanism, which both take place from very small strains. In reality, soils behave as elastic-plastic materials. Figure3-3Error: Reference source not found shows pure shear response of soil subjected to two different shear strain levels. It is important to note that for very small shear strain cycles (Figure3-3, left) response is almost elastic, with a very small hysteretic loop (almost no energy dissipation). On the other hand, for large shear strain cycles (Figure 3-3, right) there is a significant loss of (tangent) stiffness, as well as a significant energy dissipation (large hysteretic loop).
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Figure 3-3. Predicted pure shear response of soil at two different shear train amplitudes, from Pisano` and Jeremic (2014).
One of the most important features of soil behaviour is the development of volumetric strains even under pure shear, Ishihara (1996) Error: Reference source not found. The result is that during pure shearing deformation, soil can increase in volume (dilatancy) or reduce in volume (contraction). Incorporating volume change information in soil modelling can be very important, especially under undrained behaviour. If models that are used to model soil behaviour do not allow for modelling dilatancy or contraction, potentially significant uncertainty is introduced, and results of seismic soil behaviour might be strongly inaccurate: under undrained conditions, the tendency for volume changes manifests itself in pore water pressure changes and therefore in stiffness reduction/increase and strength degradation/hardening since soil behaviour is governed by effective stresses. shows three responses for no-volume change (left), compressive (middle) and dilative (right) soil with full volume constraint, resulting in changes in stiffness for compressive (reduction in stiffness), and dilative (increase in stiffness).
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Figure 3-4. Cyclic response of soils: (left) no volume change; (middle) compressive response with decrease in stiffness; (right) dilative response with increase in stiffness. (from Jeremic 2016 (CNSC report)
As previously said, the soil element is subjected to general three-dimensional state of time-varying stresses. However, it is frequently assumed that a one-dimensional situation prevails and simpler models can then be used to describe the salient features of the soil response. Therefore, in the following, emphasis is placed on such models, which are used in practice; some indications are nevertheless provided for three-dimensional constitutive models although there is not a single model which can be said to be superior to the others, many of which being still under development today. Chapter 7 provides further description of 3D material models, and references to most commonly used models.
3.3 EXPERIMENTAL DESCRIPTION OF SOIL BEHAVIOUR
As mentioned previously simple one-dimensional models can be used to describe soil behaviour; for instance, it is commonly admitted for site response analyses, or for soil structure interaction problems, to consider that the seismic horizontal motion is caused by the vertical propagation of horizontally polarized shear waves. Under those conditions, a soil element within the soil profile is subjected to shear stress () cycles similar to those presented in Figure 3-5.
Figure 3-5. Idealized stress cycle during an earthquake
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Depending on the amplitude of the induced shear strain, defined as , different types of nonlinearities take place. When the shear strain amplitude is small, typically less than s = 10-5
the behaviour can be assumed to be is essentially linear elastic with no evidence of significant nonlinearity. For increasing shear strain amplitudes, with a threshold typically of the order γv = 1x10-4 to 5x10-4, nonlinearities take place in shear while the volumetric behaviour remains essentially elastic (Figure 3-6, left). It is only for larger than γv strains that significant volumetric strains take place in drained conditions (Figure 3-6 right), or pore water pressures develop under undrained conditions.
These different threshold strains are summarized in Figure 3-7 adapted from Vucetic and Dobry (1991) [8]. Depending on the anticipated strain range applicable to the studied problem, different modelling assumptions, and associated constitutive relationships, may be used (Table 3-1). Accordingly, the number and complexity of constitutive parameters needed to characterize the behaviour will increase with increasing shear strains.
Figure 3-6. Experimental stress-strain curves under cyclic loading: nonlinear shear behaviour (left); volumetric behaviour (right)
0
10
20
30
40
50
60
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Cyclic shear strain
Plas
ticity
Inde
x (%
)
Very smallstrains
Small strains
Moderate to largestrains
s
v
28
Figure 3-7. Threshold values for cyclic shear strains
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TABLE 3-1. STRAIN THRESHOLDS AND MODELLING ASSUMPTIONS
CYCLIC SHEAR STRAIN AMPLITUDE
BEHAVIOURELASTICITY and
PLASTICIY
CYCLIC DEGRADATION in
STAURATED SOILS MODELLING
Very small 0 s Practically linear Practically elastic Non degradable Linear elastic
Small s v Non-linearModerately
elasto-plasticPractically
non-degradableViscoelastic
Equivalent linear
Moderate to large
v Non-linear Elasto-plastic Degradable Non-linear
NOTE BORIS: do we call response nonlinear or elastic-plastic, as they are not exactly the same, nonlinear can be elastic nonlinear (as in hyperbolic model(s)) while elastic-plastic is really full nonlinear model…
3.3.1 Linear viscoelastic model
In the strain range γ ≤ γv, more or less pronounced nonlinearities and energy dissipation become apparent in the shear-strain curve (Figure 3-6 and Figure 3-8).
As a linear viscoelastic model (Figure 3-9) exhibits under harmonic loading hysteresis loops, it is tempting to model the soil behaviour with such models. However, the viscoelastic model lends itself to an energy dissipation mechanism that is frequency dependent, in contradiction with experimental observation.
Figure 3-8. Shear stress-shear strain curves for constant amplitude cyclic loading
For harmonic loading, , the one-dimensional stress-strain relationship (Figure 3-9) can be written:
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(3.1)
G
C =
G
Figure 3-9. One-dimensional viscoelastic model
In equation (3.1), G* is a complex valued modulus formed from the elastic (shear) modulus G
and , called the loss coefficient; this parameter is related to the energy W dissipated during one cycle of loading.
(3.2)
The maximum elastic energy that can be stored in a unit volume of a viscoelastic body is given by:
(3.3)
Then, from equations (3.2) and (3.3):
(3.4)
where is known as the equivalent damping ratio.
Equivalent linear viscoelastic models are defined by a constitutive relationship (for one dimensional loading) of the type given by equation 101\* MERGEFORMAT (.) where the complex valued G* must be defined to yield the same stiffness and damping properties as the actual material. This complex modulus is defined from two experimental parameters (see Section 3.4): G and or )). Several models have been proposed to achieve that purpose. Their characteristics are defined in Table 3-2.
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TABLE 3-2: CHARACTERISTICS OF EQUIVALENT VISCOELASTIC LINEAR CONSTITUTIVE MODELS
COMPLEX MODULUS DISSIPATED ENERGY IN
ONE CYCLE
MODULUS
MATERIAL
MODEL 1
MODEL 2
MODEL 3
The first two models were developed by Seed and his co-workers, Seed et al. (1970) [9]; the third one is due to Dormieux (1990) [10]. Examination of Table 3-2 shows that the first model, which is the simplest one, adequately duplicates the dissipated energy but overestimates the stiffness; the second one duplicates the stiffness but underestimates the dissipated energy and the third one is the only one fulfilling both conditions. In standard practice, the most commonly used model is the second one; this model is implemented in several frequency domain software programs as SHAKE, FLUSH, SASSI (Section 8).
To account for the nonlinear shear behaviour, the parameters entering the definition of the complex modulus, G*, are made strain dependent: the secant shear modulus (Figure 3-8) and the damping ratio are plotted as function of shear strain as depicted in Figure 3-10.
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0
5
10
15
20
25
30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01
Damping ratio (%
)G
/ G
max
Shear strain (%)
Secant shear modulus
Damping ratio
Figure 3-10. Example of nonlinear characteristics of soil
A common mistake in the characterization of the shear behaviour is to measure the G/Gmax curves in the lab under a given confining pressure and to consider that the same curve applies at any depth in the soil profile provided the material does not change. It is well known, however, that not only Gmax but also the shape of the curve depends on the confining pressure (Ishihara, (1996) [7]). To overcome such a difficulty, and to keep the number of tests to a reasonable number, the correct representation is to normalize not only the modulus but also the strain G/Gmax=f( /r), where r is a reference shear strain (Hardin & Drenvich, 1972) [11].
These viscoelastic models are implemented, in frequency domain solutions, with an iterative process in which the soil characteristics (secant shear modulus and equivalent damping ratio) are chosen at variance with the “average” induced shear strain in order to reproduce, at least in an approximate manner, soil nonlinearities. They are typically considered valid for shear strains up to approximately 1x10-3 to 5x10-3. Their main limitation, aside from their range of validity, is their inability to predict permanent deformations.
Extension to three-dimensional situations is straightforward in the framework of viscoelasticity. The constitutive relationship is simply written in terms of complex moduli and writes1:
(3.5)
Where is the tensor of elasticity expressed with complex moduli; for isotropic materials it is defined from the shear modulus G* and bulk modulus B* formed with the physical moduli and 1 The following notations are used: tensors are underlined with a number of lines equal to the order, the symbol ‘:’ is used for the double contraction of tensors (double summation on dummy indices).
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the associated loss coefficients s and p for shear and volumetric strains; usually s is taken equal to p .
3.3.2 Nonlinear one-dimensional model
The definition of one-dimensional inelastic models starts with the definition of the so-called
backbone, which establishes the non-linear stress-strain relationship for monotonic
loading. In this function, the ratio is the secant modulus. In addition, for cyclic deformations, it is necessary to establish an unloading and re-loading rule, which defines the stress-strain path for non-monotonic loading. The most widely used is the so-called Masing’s rule (1926) [12] or extended Masing’s rule (Pyke 1979 [13]; Wang et al 1980 [14]; Vucetic 1990) [15]. Assuming that the backbone curve is anti-symmetric with respect to the strain parameter, which is approximately true in shear when Bauschinger effect is neglected, then the stress-strain relationship during loading-unloading sequences is defined by:
(3.6)
in which (r, r) are the coordinates of the point of last loading reversal. This formulation implies that the shape of the unloading or reloading curve is identical to the shape of the initial loading backbone curve enlarged by a factor of two. This model requires keeping track of the history of all reversal points, so that when an unloading or re-loading sequence intersects a previously taken path, that previous path is followed again as if no unloading or re-loading has taken place before. This is shown schematically in Figure 3-6 (left) where 0a is the initial loading path (backbone curve), ab the unloading path, bc the reloading path which, when reaching point a, resumes the backbone curve.
Several formulations have been proposed for the backbone curve among which the hyperbolic model (Konder 1963 [16], Hardin and Drnevich 1972 [17]) and Ramberg-Osgood (1943) [17] are the most popular ones. In the hyperbolic model, the backbone curve is written:
(3.7)
where max represents the shear strength and r a reference shear strain defined as max/Gmax with Gmax the maximum (small strain) shear modulus (slope of the tangent at the origin to the backbone curve). Some modifications have recently been proposed to eq. (3.7) to better fit the experimental data by raising the term /r in the denominator to a power exponent . It must be pointed out that such modification is not consistent with the behaviour at large strains since /max ~ ( /r)1- except when = 1.0 the strength tends asymptotically towards either 0 ( > 1.0) or towards infinity ( < 1.0).
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The Ramberg-Osgood backbone curve is defined by the expression:
(3.8)
where and R are dimensionless parameters and y an arbitrary reference stress.
For harmonic loading under constant amplitude strains (Figure 3-8), the secant shear modulus and loss coefficient are defined by:
Hyperbolic model
(3.9)
Ramberg-Osgood model
(3.10)
Equations 202\* MERGEFORMAT (.) and 303\* MERGEFORMAT (.) establish the link with the parameters of the viscoelastic equivalent linear models and form the basis for the determination of the constitutive parameters.
Iwan (1967) [18] introduced a class of 1D-models that leads to stress-strain relations which obey Masing’s rule, for both the steady-state and non-steady-state cyclic behaviour once the initial monotonic loading behaviour is known. The concepts of the one-dimensional class of models are extended to three-dimensions and lead to a subsequent generalization of the customary concepts of the incremental theory of plasticity. These models can be conveniently simulated by means of an assembly of an arbitrary number of nonlinear elements, which can be placed (Figure 3-11) either in parallel (elastoplastic springs) or in series (springs and sliders). The advantage of this model is that any experimental backbone curve can be approximated as closely as needed, and the numerical implementation is easy as compared, for instance, to the Ramberg-Osgood model: there is no need to keep track of all the load reversals.
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Figure 3-11. Rheological models for Iwan’s nonlinear elastoplastic model
Several software programs have implemented 1D-nonlinear constitutive models for site response analyses; the most commonly used are DEEPSOIL, DESRA, SUMDES, D-MOD, TESS. One can refer to Stewart et al (2008) [19] for a complete description of these models.
1D-nonlinear models are convenient to describe the nonlinear shear strain-shear stress behaviour but they cannot predict volumetric strains (settlements) that may take place even under pure shear.
3.3.3 Nonlinear three-dimensional models
Unlike the viscoelastic models presented in Section 3.3.1, the extension of 1D-nonlinear models to general 3D states is not straightforward. Usually, such models are based for soils, which are mostly rate-independent materials, on the theory of incremental plasticity. Unlike in 1D-models, coupling exists between shear and volumetric strains; therefore, even in the ideal case of horizontally layered profiles subject to the vertical propagation of shear waves, elastoplastic models will allow calculations of permanent settlements. Nonlinear models can be formulated so as to describe soil behaviour with respect to total or effective stresses. Effective stress analyses allow the modelling of the generation, redistribution, and eventual dissipation of excess pore
pressure during and after earthquake shaking. In these models, the total strain rate is the sum
of an elastic component and of a plastic component . The incremental stress-strain constitutive equation is written:
(3.11)
where C is the tensor of elasticity. The models are fully defined with the yield surface that specifies when plastic deformations take place, the flow rule that defines the amplitude and direction of the incremental plastic deformation and the hardening/softening rule that describes the evolution of the yield surface. A large number of models of different complexity have been proposed in the literature; it is beyond the scope of this document to describe in detail these models. As noted previously they can be viewed as extensions to three dimensional states of
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0G
1G
1
nG
n
0
1G
1
maxG2G
nG
2 n
Iwan’s model. Among all models the commonly used seem to be Prevost’s models (1977) [20], 1985) [21], based on the concept of multiyield surfaces (Mroz, 1967) [22], and Wang et al. [15] (1990) models based on the concept of bounding surface (Dafalias, 1986) [23].
NOTE BORIS: I would like to expand this section a bit and provide a page or two of discussion about current 3D elastic-plastic models. And what requirements are necessary for proper modelling of cyclic behaviour of soils (rotational kimenatic hardening, etc…). I’ll discuss with Alain, and will look to move some of the material from chapter 7 here…
3.4 PHYSICAL PARAMETERS
The essential parameters that need to be determined are the parameters entering the constitutive models described in Section 3.2. In addition, the natural frequency of the soil profile proves to be very important for the analyses: it represents a good indicator for the validation of the site modelling (geometry, properties), at least in the linear range, and is useful to constrain the range of possible variation of the soils characteristics.
The parameters entering the constitutive relationships depend on the adopted model for the soil. For viscoelastic models, elastic (small strain) characteristics and variation of these characteristics with shear strain are required. In view of the large uncertainties involved in soil-structure interaction problems, it is sufficiently accurate to consider an isotropic material, thereby reducing the numbers of moduli to 2: the shear and the bulk moduli, or equivalently the S-wave and P-wave velocities. Energy dissipation is represented by the loss coefficients (or damping ratios) associated to each modulus. Practically, due to the difficulty in damping measurements and to the scatter in the results, a single value of the loss coefficient is usually considered applicable to both moduli. These small strain characteristics depend mainly on the soil density and past and present state of stresses. The nonlinear shear strain-shear stress behaviour is characterized by the variation with shear strain of the secant shear modulus and loss coefficient (Figure 3-10); these curves are mainly influenced by the present state of stresses and the soil plasticity index.
For 1D-nonlinear models, in addition to the previous parameters, the maximum shear strength that can be developed under simple shear loading is required (equation (3.11)); for dry soils or drained conditions, it can be computed from the knowledge of the strength parameters, friction angle ’ and cohesion c’, and of the at rest earth pressure coefficient K0. The latter parameter is the most difficult one to measure directly and is usually estimated from empirical correlation based on the friction angle and overconsolidation ratio of the soil.
(3.12)37
For saturated soils under undrained conditions, the maximum shear strength is the cyclic undrained shear strength of the soil.
For the 3D-nonlinear models the same parameters as for the 1D-nonlinear models are needed but, these models also involve a large number of additional parameters which do not all have a physical meaning; these parameters are “hidden” variables used to calibrate the constitutive model on the experimental data and depend on the formulation of the model. However, such parameters like the dilation angle and the rate of volumetric change under drained conditions, or pore pressure build up under undrained conditions, are essential physical parameters for these models.
3.5 FIELD AND LABORATORY MEASUREMENTS
Measurements of soil parameters are essential to classify the site geometry, to identify the soil strata and to estimate their behaviour under seismic loading. In addition, they should provide the necessary parameters to enter the constitutive models that are used for the soil structure interaction analyses. These parameters can be determined from site instrumentation, field tests and laboratory measurements.
3.5.1 Site instrumentation
Site instrumentation provides very useful and important information to assess potential site effects and their modelling. Two different types of instrumentation may be implemented based on passive measurements of ambient vibrations or active measurements of seismic events.
The profile’s natural frequency can easily be obtained with Ambient Vibration H/V Measurements (AVM). As indicated in Section 3.1, this parameter is a good proxy to decide of the importance of 2D effects in presence of a basin.
This method consists in measuring the ambient noise in continuous mode with velocity meters (not accelerometers) and then computing the ratio between the horizontal and vertical Fourier amplitude spectra (Nakamura, 1989) [24]. Guidelines were produced by the SESAME research programme (SESAME 2004) [25] to implement this technique, which is now reliable and robust. H/V measurements can provide the fundamental frequency of the studied site (but not the associated response amplitude) but can also be used to assess the depth to bedrock and its possible lateral variation when the technique is implemented along profiles. However, in this case, care should be taken when interpreting along the edge of basins, where the bedrock is significantly sloping, because 1-D geometry is assumed in the interpretation of measurements. The knowledge of the soil profile natural frequency is also important to validate the numerical model used for the analyses.
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Active measurement of seismic events is recommended with installation of instruments that allow recording, on site or in the vicinity, ground motions induced by real earthquakes. Based on these free-field records, "site to reference" transfer functions can be determined at various locations across the site. Like the passive H/V measurements, the site to reference transfer functions are useful to detect possible 2D effects and to calibrate the numerical model, at least in the linear range.
3.5.2 Field investigations
The purpose of field investigations is to provide information on the site (stratigraphy, soil properties) at a large scale as opposed to the small scale involved in laboratory tests. They typically rely on borings, which provide information on the spatial distribution of soils (horizontally and with depth) and produce samples for laboratory analyses. However, other techniques, which do not involve borings, may provide essential information to characterize the site stratigraphy: for instance, the depth to the bedrock, which is an important parameter for site response analyses, can be assessed from High Resolution Seismic Reflection Survey (HRSRS) techniques.
In addition to providing information on the site stratigraphy, essential dynamic properties for SSI analyses are measured from field investigations. These include the wave velocity profiles (P-wave and S-wave), which are converted into elastic, or small strain, soil parameters. Various field techniques for measuring in situ, shear and compressional, wave velocities exist (ISSMFE, 1994 [26]; Pecker, 2007 [27]). The most reliable and versatile techniques are invasive techniques based on in-hole measurements; these include the downhole, and crosshole tests but also the suspension logging tests. Use of crosshole tests with three aligned boreholes, one emitter and two receivers, is highly recommended to increase the reliability of the test interpretation: signal processing techniques can be used to identify, almost unambiguously, the time arrival of shear waves. The invasive techniques can be advantageously complemented with non-invasive techniques like the MASW tests or H/V measurements as described in the previous section (Foti et al., 2014 [28]).
Invasive methods are considered more reliable than non-invasive ones because they are based on the interpretation of local measurements of shear-wave travel times and provide good resolution. However, these methods require drilling of at least one borehole, making them quite expensive. Non-invasive techniques provide cost efficient alternatives. In the last decades the methods based on the analysis of surface wave propagation are getting more and more recognition. These methods can be implemented with a low budget without impacting the site. However, they need processing and inversion of the experimental data, which should be carried out carefully. The surface-wave inversion is indeed non-linear, ill-posed and it is affected by solution non-uniqueness. This leads sometimes to strongly erroneous results causing a general lack of confidence in non-invasive methods in the earthquake engineering community.
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The invasive techniques have no theoretical limitations with regards to the depth of investigation; the non-invasive techniques are limited to shallow depth characterization, typically of the order of 20m to 50m. Furthermore, interpretation of MASW measurements implicitly assumes that the site is horizontally layered; therefore, they are not accurate for subsurface sloping layers. However, being less local than the invasive techniques they can provide, at low cost, information on the site heterogeneity and spatial variation of the soil parameters across the site. Furthermore, they may be very useful to constrain the variation of some parameters. For example, in the Pegasos Refinement Project (Renault, 2009 [29]), the dispersion curves measured in MASW tests were used to reject or keep possible alternatives of the soil velocity profiles and thereby allowing to reduce the epistemic uncertainties in this parameter.
In the present state of practice none of the available techniques are adequate for measuring the nonlinear characteristics of the soils; they are limited to the elastic domain and should therefore be complemented with laboratory tests to allow for a complete characterization of the soil behaviour under moderate to large strains that are applicable to seismic loading.
3.5.3 Laboratory investigations
Laboratory tests are essential to measure dynamic soil properties under various stress conditions, such as those that will prevail on site after earthworks and construction of buildings, and to test the materials in the nonlinear strain range. They are therefore used to assess the variation with strain of the soil shear modulus and material damping. Combining field tests and laboratory tests is mandatory to establish a complete description of the material behaviour from the very low strains to the moderate and large strains. However, it is essential that these tests be carried out on truly undisturbed samples as dynamic moduli are highly sensitive to remolding. If sampling of fine cohesive soils can be efficiently performed, retrieving truly undisturbed samples in cohesionless uniform materials is still a challenge. Laboratory tests can be classified in three categories:
Wave propagation tests: shear wave velocities can be measured on laboratory samples using bender elements to measure the travel time of the S wave from one end to the other end of the samples. These tests are limited to elastic strains, like field tests, but unlike those tests they can be performed under various stress conditions; comparison between field measurements and bender tests is a good indicator of the quality of the samples.
Resonant tests: these tests are known as resonant column tests (Wood, 1998 [30]). They are applicable to the measurements of soil properties from very small strains to moderate strains, typically of the order of 5x10-5 to 1x10-4; however, they cannot reach failure conditions. Depending on the vibration mode (longitudinal or torsional) Young’s modulus and shear modulus can be measured. The damping ratio is calculated from the logarithmic decrement in the free vibration phase following the resonant phase. These tests can be performed under various stress conditions and are more accurate than forced
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vibration tests because the moduli are calculated from the knowledge of the sole resonant frequency of the specimen; no displacement or force measurements are involved.
Forced vibration tests: cyclic triaxial tests and simple shear tests belong to that category. Unlike the resonant column tests, measurements of applied force and induced displacement are required to calculate the moduli; therefore, inherent inaccuracies in both measurements immediately translate into errors in the moduli. Due to this limitation and to the classical size of samples (70 to 120mm in diameter) the tests are not accurate for strains below approximately 10-4. These tests can be performed under various stress conditions and can be conducted up to failure, enabling the determination of the sample strength. Damping ratio can be computed from the area of the hysteresis loop or from the phase shift between the applied force and the displacement. In simple shear tests, the shear modulus and the shear strain are directly measured, while in triaxial tests Young’s modulus E and axial strain are the measured parameters. Therefore, determination of the shear modulus from cyclic triaxial tests requires the knowledge of Poisson’s ratio ():
(3.13)
Poisson’s ratio is usually not directly measured but if the tests is conducted on a saturated sample under undrained conditions, = 0.5. In addition to the shear strain – shear stress behaviour, triaxial or simple shear tests are essential to measure the volumetric behaviour of the samples for calibration of 3D nonlinear models or for the prediction, under undrained conditions, of the potential pore pressure build up. Cyclic triaxial tests, which are versatile enough to allow application of various stress paths to the sample, are the main tests available for full calibration of nonlinear constitutive models.
From the range of validity of each test, it appears that a complete description of the soil behaviour from the very small strains up to failure can only be achieved by combining different tests. Furthermore, as previously noted, comparison of bender tests and resonant column tests with field tests is a good indicator of the sample representativity and quality.
3.5.4 Comparison of field and laboratory tests
As mentioned previously field tests are limited to the characterization of the linear behaviour of soils while laboratory tests have the capability of characterizing their nonlinear behaviour. Figure3-12 (from ASCE 4-16 [31]) presents approximate ranges of applicability of tests.
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Figure 3-12. Domain of applicability of field and laboratory tests (ASCE 4-16)
NOTE BORIS: I would argue that SM-EQ as described above (strains induced in strong motion earthquakes) are only accurate for free conditions! When structure is placed on top of soil, strains can reach more than 1% and if failure is considered (strong motions) then strains can reach much more than 1%!!
So in a sense, ASCE4-16 is NOT RIGHT, and I would like us to discuss this!
NOTE JJJ: This is intended to be general guidance for free-field analysis and SSI to emphasize from which in-situ and laboratory tests the data can be derived. Of course, for very local behavior, higher strains could be induced by soil-structure response.
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3.6 CALIBRATION AND VALIDATION
Calibration and validation of the soil constitutive models is an essential step of the analysis process. These steps aim at ensuring that the experimental behaviour of materials under seismic loading is correctly accounted for by the models.
For elastic and viscoelastic linear constitutive models, calibration usually does not pose any problem; the experimental data (elastic characteristics, G/Gmax and damping ratio curves) measured either in situ and/or in the laboratory are directly used as input data to the models. However, validation shall not be overlooked: results should be critically examined since, as indicated previously, those constitutive models are only valid for strains smaller than a given threshold v. If results of analyses indicate larger strains, then the constitutive models should be modified and nonlinear models should be advocated.
Calibration of nonlinear constitutive models is more fastidious and uncertain. For 1D models, the shear strain-shear stress behaviour can be fitted to the experimental data using equations (3.9) and (3.10), but attention must be paid to the energy dissipation: elastoplastic models are known to overestimate this parameter, which means that the damping ratio in equations (3.9) and (3.10) is likely to exceed the experimental data.
For nonlinear 3D models, calibration is even more difficult due to the coupling between shear and volumetric strains. It is stressed that calibration should not rely only on the shear stress-shear strain behaviour but also on the volumetric behaviour. Therefore, laboratory tests providing the required data are warranted. Usually, calibration is best achieved by carrying out numerical experiments duplicating the available experiments. Validation consists in reproducing additional experiments that were not used for calibration; these validations should be performed with the same constitutive parameters and for different stress conditions and stress (or strain) paths.
It is important to note that variability in predictions of constitutive models may be very large. This is illustrated by a recent benchmark carried out within the framework of the SIGMA project (Pecker et al, 2016 [32]). A sample with a prescribed shear strength of 65kPa was subjected to a 10 cycle, quasi harmonic input motion, modulated by a linear amplitude increase, and its behaviour predicted by different models with their associated hysteresis curves, as depicted in Figure 3-13. The full duration of motion leads to very high strain levels (5%), and the stress-strain curves are highly variable from one computation to another. The main differences were attributed to the inability of some models to mimic the prescribed shear strength value and to differences in the way energy dissipation is accounted for by the models. One essential
43
conclusion of the benchmark was that detailed calibration of models is essential and that, in practice, it would be advantageous to use at least two different models for the analyses.
Figure 3-13. Cyclic stress-strain loops for a soil element with shear strength 65 kPa subjected to a sinusoidal input seismic motion of 10 s. Predictions produced by the different constitutive models.
When detailed experimental data are not available for calibration, equations (3.9) and (3.10) can still be used for the definition of the shear stress-shear strain behaviour; liquefaction resistance curves, like those derived from SPT or CPT tests, can then be used to calibrate the volumetric behaviour (Prevost & Popescu, 1996 [5]) but, obviously, this approach is much less accurate since only the global behaviour is predicted and not the detailed evolution of pore pressure (or volumetric strain rate).
3.7 UNCERTAINTIES
NOTE BORIS: Perhaps we can include here a graph that I now have in chapter 7, about uncertainty of varous material parameters, that graph in particular is about youngs modulus (very uncertain) and it is from a paper (well data is from a paper, graph is mine) by KK Phoon and F. Kulhaway… It shows very uncertain (a cloud or points) soil data, with one thin line representing modelling of E, and a large (wide) resuduals… see figure below, taken from page 168 much further down…
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Soil properties in a homogeneous soil layer are affected by a series of uncertainties, such as inherent variability (see Section 3.8), random test errors, systematic test errors, etc. It is common practice to assume, in the absence of precise site specific data, that the elastic shear modulus can vary within a factor of 1.5 around the mean value (ASCE 41-13 [33], ASN2/01 [34], ASCE 4-16 [31]). If additional testing is performed the range could be narrowed to values obtained by multiplying or dividing the mean value by (1+COV) where COV is the coefficient of variation, to be taken larger than 0.5.
In the framework of the SIGMA project (Pecker et al, 2016 [32]), the InterPACIFIC project (Intercomparison of methods for site parameter and velocity profile characterization) compares the non-invasive and invasive methods in order to evaluate the reliability of the results obtained with different techniques. Three sites were chosen in order to evaluate the performance of both invasive and non-invasive techniques in three different subsoil conditions: soft soil, stiff soil and rock. Ten different teams of engineers, geologists and seismologists were invited to take part in the project in order to perform a blind test. The standard deviation of VS values at a given depth is, by and large, higher for non-invasive techniques - coefficients of variation (COV=0.1 to 0.15) than for invasive ones (COV<0.1) as shown in Figure 3-14 for the three tested sites (Garofalo et al, 2015 [35]). These values are significantly less than those recommended by ASCE which would for the modulus imply that the mean value be multiplied by at most 1.152 =1.3; however, it should be considered that 10 highly specialized different teams made their own evaluation and it can be considered that these values are minimum threshold values with no hope of achieving smaller uncertainties.
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Figure 3-14: Comparison of invasive and non-invasive VS COV values as a function of depth at Mirandola (MIR; a,b), Grenoble (GRE; c) and Cadarache (CAD;d)
3.8 SPATIAL VARIABILITY
The necessarily limited number of soil tests and their inherent lack of representability are significant sources of uncertainty in the evaluation of site response analyses, while the uncertainty on the accuracy of analytical or numerical models used for the analysis is in general less significant. Spatial variability may affect the soil properties but also the layer thicknesses. Soil properties in a homogeneous soil layer are affected by a series of uncertainties, such as (Assimaki et al, 2003 [36]): inherent spatial variability, random test errors, systematic test errors (or bias), transformation uncertainty (from index to design soil properties) etc. Since deterministic descriptions of this spatial variability are in general not feasible, the overall characteristics of the spatial variability and the uncertainties involved are mathematically modelled using stochastic (or random) fields. Based on field measurements and empirical correlations, both Gaussian and non-Gaussian stochastic fields are fitted for various soil properties; however, according to Popescu (1995) [37], it is concluded that: (1) most soil properties exhibit skewed, non-Gaussian distributions, and (2) each soil property can follow different probability distributions for various materials and sites, and therefore the statistics and the shape of the distribution function have to be estimated for each case. It is important to realize that the correlation distances widely differ in natural deposits between the vertical and the horizontal directions: typically, in the vertical direction, due to the deposition process the
46
correlation distance is typically of the order of a meter or less, while in the horizontal direction it may reach several meters. Consequently, accurate definition of a site specific distribution can never be achieved in view of the large number of investigation points that would be needed; if for low frequency motions, long wave lengths, a reasonable estimate can be done, description of small scale heterogeneities is clearly out of reach and one has to rely on statistical data collected on various sites.
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4 SEISMIC HAZARD ANALYSIS (SHA)
4.1 OVERVIEW
The basic concept and methodology for seismic hazard evaluation is described in IAEA SSG-9 “Seismic Hazards in Site Evaluation for Nuclear Installations (2009)” [10]. IAEA SSG-9 describes Deterministic Seismic Hazard Analysis (DSHA) and Probabilistic Seismic Hazard Analysis (PSHA). The objective of this section is to describe the probabilistic seismic hazard analysis (PSHA) steps and results that may define the seismic input for the SSI analysis. One method of performing a DSHA is described.
If the performance goal of a facility or structure, system, or component is probabilistically defined, a basic prerequisite is the development of the site specific probabilistically defined seismic hazard associated with that site. The seismic hazard is often termed “seismic hazard curve,” which represents the annual frequency (or rate) of exceedance for different values of a selected ground motion parameter, e.g., PGA or response spectral acceleration at specified spectral frequencies. In the latter case, the PSHA process defines seismic hazard curves for response spectral accelerations over a range of spectral frequencies, e.g., 5-20 discrete natural frequencies ranging from 0.5 Hz to 100 Hz, and for a specified damping value, usually 5%. A uniform hazard response spectrum (UHRS)2 is constructed of spectral ordinates each of which has an equal probability of exceedance. The combination of these discrete response spectral ordinates when interpolated over the complete range of spectral frequencies defines a UHRS.
The seismic hazard curves, one of the end products of the PSHA, are input data to the Seismic Probabilistic Risk Analysis (SPRA). Similarly, the UHRS is an end product of the PSHA and is an essential input to defining the seismic input for SSI analysis for design and assessment.
The elements of the PSHA are: Data collection and developing seismotectonic models; Seismic source characterization; Selection of ground motion prediction equations (GMPEs); Quantification of hazard estimates for the site. Uncertainties propagated and displayed in
the final results; Site Response.
Figure 4-15 shows schematically the PSHA procedure. Each step is briefly discussed in subsequent subsections. Both the aleatory and epistemic uncertainties must be considered when predicting the ground motion. Aleatory uncertainties are represented by probability distributions 2 The terms Uniform Hazard Response Spectrum (UHRS) and Uniform Hazard Spectrum (UHS) are often used interchangeably.
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assigned to uncertain parameters in the seismic hazard integral. Epistemic uncertainties are modelled by fault tree logic. Both are identified in Figure 4-1.
The hazard integral [IAEA, SSG-009] sums over source zones i=1,N, with the integration parameters as shown below and briefly described in subsequent subsections.
ν ( y )=∑i=1
N
λi (m0 )∫m0
mmax
∫r min
rmax
f M i(m )⋅ f Ri
(r ) ⋅P (Y > y∨m ,r )dmdr
where ν ( y ) denotes the annual exceedance rate of ground motions level y, λ i (m0 ) denotes the average annual rate of threshold magnitude exceedance in the seismic source zone i, f Mi (m ) is the probability density distribution of earthquake magnitude within seismic source I, r is distance from the source zone.
P (Y > y )=∫x0
xmax
f X ( x ) ⋅P (Y >x∨x ) dx
The results of a PSHA must be expressed in forms of fractile hazard curves, which display median, mean and other fractile percentile hazard curves, and UHRS.
For many applications, dominant seismic sources are to be identified by providing the deaggregated seismic hazard at the site. Such applications include more realistic site response analyses, generation of ground motion time histories for use in the SSI analyses and other applications, evaluation of the potential for soil failures, such as liquefaction.
TABLE 4-3 [10] lists the typical outputs from the PSHA, including the results of the computation of the mean annual rate of exceedance for the selected ground motion parameter and the associated variability of this rate at a particular site. The variability of the rate is due to uncertainty which can be attributed to both randomness (aleatory uncertainty) and to the lack of knowledge about the earthquake phenomenon affecting the site (epistemic uncertainty).
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TABLE 4-3 TYPICAL OUTPUT OF PROBABILISTIC SEISMIC HAZARD ANALYSES [10]
Output Description Format
Mean hazard curves
Mean annual frequency of exceedance for each ground motion level of interest associated with the suite of epistemic hazard curves generated in the probabilistic seismic hazard analysis.
Mean hazard curves should be reported for each ground motion parameter of interest in tabular as well as graphic format.
Fractile hazard curves
Fractile annual frequency of exceedance for each ground motion level of interest associated with the suite of epistemic hazard curves generated in the probabilistic seismic hazard analysis.
Fractile hazard curves should be reported for each ground motion parameter of interest in tabular as well as graphic format. Unless otherwise specified in the work plan, fractile levels of 0.05, 0.16, 0.50, 0.84 and 0.95 should be reported.
Uniform hazard response spectra
Response spectra whose ordinates have an equal probability of being exceeded, as derived from seismic hazard curves.
Mean and fractile uniform hazard response spectra should be reported in tabular as well as graphic format. Unless otherwise specified in the work plan, the uniform hazard response spectra should be reported for annual frequencies of exceedance of 10−2, 10−3, 10−4, 10−5 and 10−6 and for fractile levels of 0.05, 0.16, 0.50, 0.84 and 0.95.
Magnitude–distance deaggregation
A magnitude–distance (M–D) deaggregation quantifies the relative contribution to the total mean hazard of earthquakes that occur in specified magnitude–distance ranges (i.e. bins).
The M–D deaggregation should be presented for ground motion levels corresponding to selected annual frequencies of exceedance for each ground motion parameter considered in the probabilistic seismic hazard analysis. The deaggregation should be performed for the mean hazard and for the annual frequencies of exceedance to be used in the evaluation or design.
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TABLE 4-1 TYPICAL OUTPUT OF PROBABILISTIC SEISMIC HAZARD ANALYSES[10] (CONTINUED)
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52
Mean and modal magnitude and distance
The M–D deaggregation results provide the relative contribution to the site hazard of earthquakes of different sizes and at different distances. From these distributions, the mean and/or modal magnitudes and the mean and/or modal distances of earthquakes that contribute to the hazard can be determined.
The mean and modal magnitudes and distances should be reported for each ground motion parameter and level for which the M–D deaggregated hazard results are given. Unless otherwise specified in the work plan, these results should be reported for response spectral frequencies of 1, 2.5, 5 and 10 Hz.
Seismic source deaggregation
The seismic hazard at a site is a combination of the hazard from individual seismic sources modelled in the probabilistic seismic hazard analysis. A deaggregation on the basis of seismic sources provides an insight into the possible location and type of future earthquake occurrences.
The seismic source deaggregation should be reported for ground motion levels corresponding to each ground motion parameter considered in the probabilistic seismic hazard analysis. The deaggregation should be performed for the mean hazard and presented as a series of seismic hazard curves.
Aggregated hazard curves
In a probabilistic seismic hazard analysis, often thousands to millions of hazard curves are generated to account for epistemic uncertainty. For use in certain applications (e.g. a seismic probabilistic safety assessment), a smaller, more manageable set of curves is required. Aggregation methods are used to combine like curves that preserve the diversity in shape of the original curves as well as the essential properties of the original set (e.g. the mean hazard).
A group of aggregated discrete hazard curves, each with an assigned probability weight, should be reported in tabular as well as graphic format.
Earthquake time histories
For the purposes of engineering analysis, time histories may be required that are consistent with the results of the probabilistic seismic hazard analysis. The criteria for selecting and/or generating a time history may be specified in the work plan. Example criteria include the selection of time histories that are consistent with the mean and modal magnitudes and distances for a specified ground motion or annual frequency of exceedance.
The format for presenting earthquake time histories will generally be defined in the work plan.
Seismicity Seismic Source Zones Ground Motion Equations(Attenuation)
Recurrence rate (Historical/Geological)Maximum Magnitude
Area sourcesFaults
Inter/intra earthquakesIndex: PGA, SA
Uncertainty
Seismo-tectonic Model
Logic Tree
Alternate ModelsParameter Uncertainty
Relative Likelihood
Probabilistic Seismic Hazard Analysis
Hazard Curves
Hazard Deaggregation-Determining dominant sources
Site response
Soil AmplificationSite-specific Time Histories
Historical, Instrumental earthquake dataPaleoseismology
Geology, Geodesy and Techtonis
Strong Ground Motion records
Expert judgements
Natural period of target structures
Geotechnical data
Uniform Hazard Spectra
Figure 4-15 Steps in the PSHA Process
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4.2 DATA COLLECTION
In conducting a PSHA or DSHA, a comprehensive up-to-date database as described in Chapter 3 and Ref. [10], is to be developed, including geological, geophysical, geotechnical and seismological data at the various geographical layers. Seismological data is derived from instrumental data, historical earthquake data, and paleoseismic data. This data base is often referred to as the GGG data base.
4.3 SEISMIC SOURCES AND SOURCE CHARACTERIZATION
To evaluate the exceedance probability (or frequency) of earthquake ground motions at the site, the PSHA must examine all potential seismic sources. In performing a PSHA, both the aleatory and epistemic uncertainties must be considered in each aspect of the PSHA methodology.
4.4 GROUND MOTION PREDICTION
Given a seismic source, including the definition of all important parameters as probability distributions, ground motion prediction equations (GMPEs) are selected to represent the most appropriate relationships between the source and the site vicinity. GMPEs were formerly referred to as ground motion attenuation relationships. GMPEs model credible mechanisms influencing estimates of vibratory ground motion that can occur at a site given the occurrence of an earthquake of a certain magnitude at a certain location. Three approaches to account for site vicinity behaviour have been used extensively in the recent past: (i) select GMPEs that define the ground motion parameters on a hard rock surface as a hypothetical outcrop motion at the site; local site effects are then introduced through site response analyses (Section 5) to generate the ground motion at the site top of grade or at foundation elevations for the nuclear power plant; (ii) select GMPEs that have been developed for the specific conditions at the site, e.g., deep soil sites of relatively uniform soil; and (iii) generate site specific GMPEs based on recorded ground motions at the site; this approach is dependent on acquiring an adequate number of records to define the GMPE; this approach is most appropriate for benchmarking the high return period portion of the seismic hazard curves.
4.5 UNIFORM HAZARD RESPONSE SPECTRA
If the performance goal of a facility or structure, system, or component is probabilistically defined, a basic prerequisite is the development of the site specific probabilistically defined seismic hazard associated with that site. The seismic hazard is often termed “seismic hazard curve,” which represents the annual frequency (or rate) of exceedance for different values of a selected ground motion parameter, e.g., PGA or response spectral acceleration at specified spectral frequencies. In the latter case, the PSHA process defines seismic hazard curves for
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response spectral accelerations over a range of spectral frequencies, e.g., 5-20 discrete natural frequencies ranging from 0.5 Hz to 100 Hz, and for a specified damping value, usually 5%. A uniform hazard response spectrum (UHRS)3 is constructed of spectral ordinates each of which has an equal probability of exceedance. The combination of these discrete response spectral ordinates when interpolated over the complete range of spectral frequencies defines a UHRS.
Section 6.5.3 discusses the derivation of site specific risk consistent design basis response spectra in detail.
3 The terms Uniform Hazard Response Spectrum (UHRS) and Uniform Hazard Spectrum (UHS) are often used interchangeably.
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5 SESIMIC WAVE FIELDS AND FREE FIELD GROUND MOTIONS
5.1 SEISMIC WAVE FIELDS
5.1.1 Perspective and spatial variability of ground motion
Earthquake motions at the location of interest are affected by a number of factors (Aki and Richards, 2002; Kramer, 1996; Semblat and Pecker, 2009). Three main influences are:
Earthquake Source: the slippage (shear failure) of a fault. Initial slip propagates along the fault and radiates mechanical, earthquake waves. Depending on predominant movement along the failed fault, we distinguish (a) dip slip and (b) strike slip sources. Size and amount of slip on fault will release different amounts of energy, and will control the earthquake magnitude. Depth of source will also control magnitude. Shallow sources will produce much larger motions at the surface, however their effects will be localized (example of recent earthquakes in Po river valley in Italy),
Earthquake Wave Path: mechanical waves propagate from the fault slip zone in all directions. Some of those (body) waves travel upward toward surface, through stiff rock at depth and, close to surface, soil layers. Spatial distribution (deep geology) and stiffness of rock will control wave paths, which will affect amplitude of waves that radiate toward the surface. Body waves are (P) Primary waves (compressional waves, fastest) and (S) Secondary waves (shear waves, slower). Secondary waves that feature particle movements in a vertical plane (polarized vertically) are called SV waves, while secondary waves that feature particle movements in a horizontal plane (polarized horizontally) are called S¿H waves.
Site Response: seismic waves propagating from rock and deep soil layers to the surface create surface seismic waves. Surface waves and shallow body waves are responsible for soil-structure interaction effects. Local site conditions (type and distribution of soil near surface), and local geology (rock basins, inclined rock layers, dykes, etc.) and local topography can have significant influence on seismic motions at the location of interest.
In general, seismic motions at surface and shallow depths consist of (shallow) body waves (P, SH, SV) and surface waves (Rayleigh, Love, etc.). It is possible to analyze SSI effects using a 1D simplification, where a full 3D wave field is replaced with a 1D wave
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field., however iIt is advisable that effects of this simplification on SSI response be carefully assessed.
A usual assumption about propagation of seismic waves (P and S) is based on Snell’s law about wave refraction. The assumption is that Sseismic waves travel from great depth (many kilometers) and as they travel through horizontally layered media (rock and soil layers), where each layer features different wave velocity (stiffness) that decreases toward surface, waves will bend toward vertical (Aki and Richards, 2002). However, even if rock and soil layers are ideally horizontal, and if the earthquake source is very deep, seismic waves will be few degrees off vertical, depending on layering (usually 5-10 degrees off vertical) when they reach the surface. because the stiffness of layers increases with depth. This change in stiffness results in seismic wave refraction, as shown in Figure5-16. This usually small deviation from vertical might not be important for practical purposes. However, such deviation from vertical will produce surface waves, the presence of which can have practical implications for SSI analysis.
Figure 5-16 Propagation of seismic waves in nearly horizontal local geology, with stiffness of soil/rock layers increasing with depth, and refraction of waves toward the vertical direction. Figure from Jeremic (2016).
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Other, a much A more important source of seismic wave deviations from vertical is the fact that rock and soil layers are usually not horizontal. A number of different geologic history effects contribute to non-horizontal distribution of layers. Figure 5-18 shows one such (imaginary but not unrealistic) case where inclined soil/rock layers contribute to bending seismic wave propagation to horizontal direction. Rock basins as well as hard rock protrussions (dykes) are also common and contribute to divation of seismic wave propagationfrom vertical.
Figure 5-17 Propagation of seismic waves in inclined local geology, with stiffness of soil/rock layers increasing through geologic layers, and refraction of waves away from the vertical direction. Figure from Jeremic (2016).
It is important to note that in both horizontal and non-horizontal soil/rock layered case, surface waves are created and propagate/transmit most of the earthquake energy at the surface.
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5.1.2 Spatial variability of ground motions
Spatial variations of ground motion refer to differences in amplitude and/or phase of ground motions with horizontal distance or depth in the free field. As introduced in Section 5.1.1, these spatial variations of ground motion are associated with different types of seismic waves and various wave propagation phenomena, including reflection at the free surface, reflection and refraction at interfaces and boundaries between geological strata having different properties, and diffraction and scattering induced by nonuniform subsurface geological strata and topographic effects along the propagation path of the seismic waves. Chapter 1 summarized the evolution of SSI analyses over the last several decades. Prior to the early 1990s, skepticism existed in some quarters as to the wave propagation behavior of seismic waves in the free-field and their spatial variation with depth in the soil. This skepticism arose from several sources; one of which was the lack of recorded data to provide evidence of a general reduction variability of motion with depth in the soil profile due to wave propagation phenomena. In the 1980s, 1990s, and continuing today, accumulated direct and indirect data verifies the phenomena.
Direct data are measurements of free-field ground motion at depths in the soil. Johnson (2003) [ ] summmarizes the existing data as of 2003. Substantial additional direct data has been accumulated over the intervening decade, especially from Japan where with recordings from K-Net and KiK-Net4. Section 5.3 summarizes important data acquisition over the complete time period.
Indirect data are measurements of response of structures with embedded foundations demonstrating reductions of motion on the foundation compared with free-field ground motions recorded on the free surface. These reductions are due to spatial variation of motion with depth in the soil and due to horizontal and vertical variations of frequency content due to incoherence of ground motions (Johnson (2003); Kim and Stewart (2003)). It should be noted that more recent work suggests that reductions might be in part due to nonlinear effects (seismic energy dissipation) within soil and contact zone adjacent to structural foundations (Orbovic et al 2015).
4 “K-NET (Kyoshin network) is a nation-wide strong-motion seismograph network, which consists of more than 1,000 observation stations distributed every 20 km uniformly covering Japan. K-NET has been operated by the National Research Institute for Earth Science and Disaster Resilience (NIED) since June, 1996. At each K-NET station, a seismograph is installed on the ground surface with standardized observation facilities. KiK-net (Kiban Kyoshin network) is a strong-motion seismograph network, which consists of pairs of seismographs installed in a borehole together with high sensitivity seismographs (Hi-net) as well as on the ground surface, deployed at approximately 700 locations nationwide. The soil condition data explored at K-NET stations and the geological and geophysical data derived from drilling boreholes at KiK-net stations are also available.” (Source http://www.kyoshin.bosai.go.jp/)
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BORIS NOTE: there are other controversies surrounding incoherent motions, (for example the very first paper describing reduction of motions, by Housner, had some errors in processing of data on analog computers… It is likely that observed reductions (mostly reductions) are also, in significant part, due to nonlinearities that are only recently being considered.As introduced in Section 5.1.1, one-dimensional and three-dimensional wave fields consist of particle motion that demonstrates spatial variation of motion with depth in the soil or rock media. For one-dimensional wave propagation, a vertically incident body wave propagating in such a media will include ground motions having identical amplitudes and phase at different points on a horizontal plane. A non-vertically incident plane wave will created a horizontally propagating horizontally (surface) wave at some apparent phase velocity, and will induce ground motion having identical amplitudes but with a shift in phase in the horizontal direction associated with the apparent horizontal propagation velocity of the wave. In either of these ideal cases, the ground motions are considered to be coherent, in that amplitudes (acceleration time histories and their response spectra) do not vary with location in a horizontal plane. Incoherence of ground motion, on the other hand, may result from wave scattering due to inhomogeneities of soil/rock media and topographic effects along the propagation path of the seismic waves. Both of these phenomena are discussed below.
In terms of the SSI phenomenon, variations of the ground motion over the depth and width of the foundation (or foundations for multifoundation systems) are the important aspect. For surface foundations, the variation of motion on the surface of the soil is important; for embedded foundations, the variation of motion on both the embedded depth and foundation width is important. Overall free-field ground motion analysis is discussed next. In Chapter 6, site response analyses are presented.
5.2 FREE FIELD GROUND MOTION ANALYSIS
Free field motions can be developed using two main approaches:
Using empirically developed Ground Motion Prediction Equations (GMPE) (Boore and Atkinson, 2008), (Chapter 4);
Using Large Scale Regional (LSR) simulations (Bao et al., 1998; Bielak et al., 1999; Taborda and Bielak, 2011, Rodgers and Pitarka, etc.)
Both approaches assume elastic and/or anelastic material behaviour for the near field region for which free field motions are developed. That is, from the earthquake source to the neighbourhood of the site of interest, the GMPEs and the numerical simulation
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approaches assume elastic material behaviour with energy dissipation modelled by kappa effects in the former case or by explicit damping modelled in the soil/rock material models.
Site response analyses are then performed to establish the seismic input motions to the SSI analyses taking into account nonlinear behaviour of the local site properties (see Chapter 6).
Site response analyses are currently (usually) performed for the assumptions of one-dimensional wave propagation and horizontally layered soil/rock profiles. as described in Chapter 6. As a minimum, it is becoming increasingly necessary to consider two- or three dimensional site response analysis to generate seismic input to the SSI analyses or, as a minimum, to justify the applicability of one-dimensional site response analysis. This justification applies to the effects of three-dimensional wave fields for sources close to the site and the effects of local geology/site conditions, such as non-horizontal soil layers, hard rock intrusion (dykes), basin effects, and topographic effects (presence of hills, valleys, and sloping ground) (Bielak et al., 2000; Assimaki and Kausel, 2007; Assimaki et al., 2003).
NOTE BORIS: moved this part below to chapter 7 where I talk about material models, although it might actually fit nicely in chapter 3, will see and discuss with Alain.
Material models for site response can be equivalent linear, nonlinear or elastic-plastic.
Equivalent linear models are in fact linear elastic models with adjusted elastic stiffness that represents (certain percentage of a ) secant stiffness of largest shear strain reached for given motions. Determining such linear elastic stiffness requires an iterative process (trial and error). This modeling approach is fairly simple, there is significant experience in professional practice and it works well for 1D analysis and for 1D states of stress and strain. Potential issues with this modeling approach are that it is not taking into account soil volume change (hence it favors total stress analysis, see details about total stress analysis in section 6.4), and it is useable for 3D analysis through deviatoric stress invariant.
Nonlinear material models are representing 1D stress strain (usually in shear, τ −γ) response using nonlinear functions. There are a number of nonlinear elastic models used, for example Ramberg- Osgood (Ueng and Chen, 1992), Hyperbolic (Kramer, 1996), and others. Calibrating modulus reduction and damping curves using nonlinear models is not that hard (Ueng and Chen, 1992), however there is less experience in professional practice with these types of models. Potential issues with this modeling approach are similar (same) as for equivalent linear modeling that it is not taking into account soil volume change as it is essentially based on a nonlinear elastic model,
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Elastic-plastic material models are usually full 3D models that can be used for 1D or 3D analysis. A number of models are available (Prevost and Popescu, 1996; Mr´oz et al., 1979; Elgamal et al., 2002; Pisano` and Jeremi´c, 2014; Dafalias and Manzari, 2004). Use of full 3D material models, if properly calibrated, can work well in 1D as well as in 3D. Potential issues with these models is that calibration usually requires a number of in situ and laboratory tests. In addition, there is far less experience in professional practice with elastic-plastic modeling.
It is important to note that realistic seismic motions always have three-dimensional features. That is, seismic motions feature three translations and three rotations (and three rotations, obtained from differential displacements between closely spaced points (from few meters to few dozen meters to few hundred meters) divided by distance between those points) at each point on the surface and shallow depth. Rotations are present in shallow soil layers due to Rayleigh and Love surface waves, which diminish with depth as a function of their wave length (Aki and Richards, 2002). As noted above, rRotations appear from can also be understood by considering differential vertical (and horizontal) motions at closely spaced points on soil surface and at some depth. Seismic wave traveling effects will produce differences in vertical motions for such closely spaced point, thus producing rotational motions for stiff objects founded on surface (or shallow depth).
For probabilistic site response analysis, it is important not to count uncertainties twice (Abrahamson, 2010). This double counting of uncertainties, stems from accounting for uncertainties in both free field analysis (using GMP equations for soil) and then also adding uncertainties during site response analysis for top soil layers.
Chapter 6 presents site response analysis in more detail.
5.3 RECORDED DATA
There exist a large number of recorded earthquake motions. Most records feature data in three perpendicular directions, East-West (E-W), North-South (N-S) and Up-Down (U-D). Number of recorded strong motions, is (much) smaller. A number of strong motion databases (publicly available) exist, mainly in the east and south of Asia, west coast of North and South America, and Europe. There are regions of the world that are not well covered with recording stations. These same These regions that are not covered with recording stations also happen to be regions are also fairly seismically fairly inactive. However, in some of those regions, return periods of (large) earthquakes are long, and recording of even small events would greatly help gain knowledge about tectonic activity and geology.
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NOTE BORIS: moved this firgure above where it belongs.
Figure 5-18 Propagation of seismic waves in inclined local geology, with stiffness of soil/rock layers increasing through geologic layers, and refraction of waves away from the vertical direction. Figure from Jeremic (2016).
Ergodic Assumption. Development of models for predicting seismic motions based on empirical evidence (recorded motions) reliesy on Ergodic assumption. Ergodic assumption allows statistical data (earthquake recordings) obtained at one (or few) worldwide location(s), over a long period of time, to be used at other specific locations at certain times. This assumption allows for exchange of average of (earthquake) process parameters over statistical ensemble (many realizations, as in many recordings of earthquakes) with an is the same as an average of (earthquake) process parameters over time.
While ergodic assumptions is frequently used, there are issues that need to be addressed when it is applied to earthquake motion records. For example, earthquake records from different geological settings are used to develop GMP equations for specific geologic settings (again, different from those where recordings were made) at the locations of interest.
5.3.1 3D (6D) versus 1D Records/Motions.
Recordings of earthquakes around the world show that earthquakes are almost always featuring all three components (E-W, N-S, U-D). There are very few known recorded events where one of the components was not present or is present in a much smaller magnitude. Presence of two horizontal components (E-W, N-S) of similar amplitude and appearing at about the same time is quite commonexpectable. The four cardinal
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directions (North, East, South and West) which humans use to orient recorded motions have little to do with the mechanics of earthquakess mechanics. The third direction, Up-Down, is different. Presence of the vertical motions before main horizontal motions appear, signify arrival of Primary (P) waves (hence the name). On the other handIn addition, presence of vertical motions at about the same time when horizontal motions appear (these vertical motions appear after first passage of P waves), signifies indicates presence of Rayleigh surface waves. On the other hand, If vertical motions are not present lack (or have very small very reduced magnitudeamplitude) during occurrence of horizontal motionsof vertical motions at about the same time when horizontal motions are present signifies indicates that Rayleigh surface waves are not present. This Lack of Rayleigh (surface) waves is a very rare ocurrenceevent, where a that the combination of source, path and local site conditions produce a plane shear (S) waves that surfaces (almost) vertically. One such (very rare) example (again, very rare) is a recording LSST07 from Lotung recording array in Taiwan (Tseng et al., 1991). Figure 5-19 shows three directional recording of earthquake LSST07 that occurred on May 20th, 1986, at the SMART-1 Array at Lotung, Taiwan.
Figure 5-19 Acceleration time history LSST07 recorded at SMART-1 Array at Lotung, Taiwan, on May 20th, 1986. This recording was at location FA25. Note the (almost complete) absence of vertical motions. signifying absence of Rayleigh waves. Figure from Tseng et al. (1991).
Note almost complete lack of vertical motions at around the time of occurrence of two components of horizontal motions, signifies indicating absence of Rayleigh surface waves. In other words, a plane shear wave front was propagating vertically and surfaced as a plane shear wave front. Other recordings, at locations FA15 and FA35 for event
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LSST07 reveal almost identical earthquake shear wave front surfacing at the same time (Tseng et al., 1991).
On the other hand, recording at the very same location, for a different earthquake (different source, different path) (LSST12, occurring on July 30th 1986) reveals quite different wave field at the surface, as shown in Figure 5-20. Significant vertical and horizontal motions, at the same time, do occur, indicating presence of surface waves.
Figure 5-20 Acceleration time history LSST12 recorded at SMART-1 Array at Lotung, Taiwan, on July 30th, 1986. This recording was at location FA25. Figure from Tseng et al. (1991).
It is important to note that this motion was recorded at the very same location as the previous one, and yet if features all three components. One of possible conclussions would be that local geology alone does not control type of motions that are to be expected. In this case, it is known that at the Lotung SMART-1 array, geology is layered and deep. It would be thus assumed that for a deep soil geology, that is very much layered, and based on Snell’s law, motions would (should) be vertically propagating. Case presented above shows that that is not guaranteed.
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5.3.2 Analytical and numerical (synthetic) earthquake models
In addition to a significant number of recorded earthquakes (in regions with frequent earthquakes and good (dense) instrumentation), analytic and numerical modeling offers another source of high quality seismic motions.
Analytic Earthquake Models
There exist a number of analytic solutions for wave propagation in uniform and layered half space (Wolf, 1988; Kausel, 2006). Analytic solutions do exist for idealized geology, and linear elastic material. While geology is never ideal (uniform or horizontal, elastic layers), these analytic solution provide very useful set of ground motions that can be used for verification and validation. In addition, these analytic solutions can be used to make estimates of behaviour, in cases where geology is close to (ideal) conditions assumed in the analytic solution process. In addition, tThus produced motions can be used to gain better understanding of soil structure interaction response for various types incoming ground motions/wave types (Liang et al., 2013) (P, SH, SV, Rayleigh, Love, etc.).
Numerical Earthquake Model
In recent years, with the rise of high performance computing, it became possible to develop large scale models, that take into account regional geology (Bao et al., 1998; Bielak et al., 1999; Taborda and Bielak, 2011; Cui et al., 2009; Bielak et al., 2010, 2000; Restrepo and Bielak, 2014; Bao et al., 1996; Xu et al., 2002). Large scale regional models that encompass source (fault) and geology in great detail, are currently able to model seismic motions of up to 5Hz. There are currently new efforts projects (US-DOE projects) that will extend modeling frequency to over 10Hz for large scale regions. Improvement in modeling and in ground motion predictions is predicated by fairly detailed knowledge of geology for large scale region, and in particular for the vicinity of location of interest. Free field ground motions obtained using large scale regional models have been validated (Taborda and Bielak, 2013; Dreger et al., 2015; Rodgers et al., 2008; Aagaard et al., 2010; Pitarka et al., 2013, 2015, 2016) and show great promise for development of free field motions.
It is important to note, again, that accurate modeling of ground motions in large scale regions is predicated by knowledge of regional and local geology. Large scale regional models make assumption of anelastic material behaviour, with (seismic quality) factor Q representing attenuation of waves due to viscous (velocity proportional) and material (hysteretic, displacement proportional) effects. With that in mind, effects of softer, surface soil layers are not well represented. In order to account for close to surface soil layers, nonlinear site response analysis has to be performed in 1D or better yet in 3D.
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Moreover, results from large scale regional models can also be used directly in developing seismic motions for SSI models, as described in some details in Chapter 7.
5.3.3 Uncertainties
Earthquakes start at the rupture zone (seismic source), propagates through the rock to the surface soils layers. All three components in this process, the source, the path through the rock and the site response (soil) feature significant uncertainties, which contribute to the uncertainty of ground motions. These uncertainties require use of Probabilistic Seismic Hazard Analysis approach to characterizing uncertainty in earthquake motions (Hanks and Cornell; Bazzurro and Cornell, 2004; Budnitz et al., 1998; Stepp et al., 2001). See Chapter 4.
Uncertain Sources. Seismic source(s) feature a number of uncertainties. Location(s) of the source, magnitude that the earthquake that can be can produced, return period (probability of generating given magnitude of an earthquake in within certain period), are all uncertain (Kramer, 1996), and need to be properly taken into account. Source uncertainty is controlled mainly by the stress drop function (Silva, 1993; Toro et al., 1997).
Uncertain Path (Rock). Seismic waves propagate through uncertain rock (path) to surface layers. Path uncertainty is controlled by the uncertainty in crustal (deep rock) compressional and shear wave velocitiesy, near site anelastic attenuation and crustal damping factor (Silva, 1993; Toro et al., 1997). These parameters are usually assumed to be log normal distributed and are calibrated based on available information/data, site specific measurements and regional seismic information. Both previous uncertainties can be combined into a model that accounts for free field motions (Boore, 2003; Boore et al., 1978).
Uncertain Site (Soil). Once such uncertain seismic motions reach surface layers (soil), they propagate through uncertain soil (Roblee et al., 1996; Silva et al., 1996). Uncertain soil adds additional uncertainty to seismic motions response. Soil material properties can exhibit significant uncertainties and need to be carefully evaluated (Phoon and Kulhawy, 1999a,b; DeGroot and Baecher, 1993; Baecher and Christian, 2003)
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ij
σ"
Effective Stress and Pore Fluid Pressures. Mechanical behaviour of soil depends exclusively on the effective stress (Muir Wood, 1990). Effective stress σ’ij depends on total stress σij (from applied loads, self-weight, etc.) and the pore fluid pressure p:
σ’ij = σij − δij p (4.1)
where δij is Kronecker delta used to distribute pore fluid pressure to diagonal members of stress tensor (δij = 1, when i=j, and δij = 0, when i /= j), Section 6.4.2 provides more information about the use of effective stress principle in modeling.
As described in some detail in section 6.4.11, pore fluid pressure generation and dissipation can (significantly) reduce (for increase in when excess pore fluid pressure increases) and increase (for reduction in when excess pore fluid pressure decreases) stiffness of soil material. Soil can even completely liquefy (becominge a heavy fluid) if pore fluid pressure becomes equal to the mean confining stress (Kramer 1996, Jeremic et al., 2008). Loss of stiffness due to increase in pore fluid pressure can significantly influence results of seismic wave propagation, as described by Taiebat et al. (2010).
FIG.Figure 5.7: One dimensional seismic wave propagation through no-volume change and dilative soil. Please note the (significant) increase in frequency of motions for dilative soil. Left plot is a time history of motions, while the right
plot shows amplitudes at different frequencies.
NOTE BORIS: This will be either moved to appendix or removed
5.4 SEISMIC WAVE INCOHERENCE
5.4.1[5.3.4] Introduction
Seismic motion incoherence is a phenomenon that results in spatial variability of ground motions over small distances. Significant work has been done in researching seismic motion incoherence over the last few decades (Abrahamson et al., 1991; Roblee et al.,
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1996; Abrahamson, 1992a, 2005, 1992b; Zerva and Zervas, 2002; Liao and Zerva, 2006; Zerva, 2009)
The main sources of incoherence lack of spatial correlation (Zerva, 2009) are:
• Attenuation effects, that are responsible for change in amplitude and phase of seismic motions due to the distance between observation points and losses (damping, energy dissipation) that seismic wave experiences along traveling paththat distance. This is a significant source of incoherence lack of correlation for long structures (bridges), however for NPPs it is not of much significance.
• Wave passage effects, contribute to incoherence lack of correlation due to difference in recorded wave field at two location points as the (surface) as wave travels, propagates from first to second point.
• Scattering effects, are responsible to incoherence lack of correlation by creating a scattered wave field. Scattering is due to (unknown or not known enough) subsurface geologic features that contribute to modification of wave field.
• Extended source effects contribute to incoherence lack of correlation by creating a complex wave source field. , Aas the (extended) fault ruptures, rupture propagates and generate seismic sources along the rupturing fault. Seismic energy is thus emitted from different points (along the rupturing fault) and will hasve different travel path and timing as it makes it to observation points.
Figure 5-21 shows an illustration of main sources of lack of correlation.
Figure 5-21 Four main sources contributing to the lack of correlation of seismic waves as measured at two observation points. Figure from Jeremic (2016).
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5.4.2[5.3.5] Incoherence modeling
Early studies concluded that the correlation of motions increases as the separation distance between observation points decreases. In addition to that, correlation increased for decrease in frequency of observed motions. Most theoretical and empirical studies of spatially variable ground motions (SVGM) have focused on the stochastic and deterministic Fourier phase variability expressed in the form of ”lagged coherency” and apparent shear wave propagation velocity, respectively. The mathematical definition of coherency (denoted γ) is given as below in equation:
(5.2)
Coherency is a dimensionless complex-valued number that represents variations in Fourier phase between two signals. Perfectly coherent signals have identical phase angles and a coherency of unity.
Lagged coherency is the amplitude of coherency, and represents the contributions of stochastic processes only (no wave passage). Wave passage effects are typically expressed in the form of an apparent wave propagation velocity.
Lagged coherency does not remove a common wave velocity over all frequencies. Alternatively, plane wave coherency is defined as the real part of complex coherency after removing single plane-wave velocity for all frequencies. Recent simulation methods of SVGM prefer the use of plane-wave coherency as it can be paired with a consistent wave velocity. An additional benefit is that plane-wave coherency captures random variations in the plane-wave while lagged coherency does not. Zerva (2009) has called these variations ’arrival time perturbations’.
Most often, coherency γ is related to the dimensionless ratio of station separation distance ξ to wavelength λ. The functional form most often utilized is exponential (Loh and Yeh, 1988; Oliveira et al., 1991; Harichandran and Vanmarcke, 1986). The second type of functional form relates coherency γ to frequency and distance ξ independently, without assuming they are related through wavelength. This formulation was motivated by the study of ground motion array data from Lotung, Taiwan (SMART-1 and Large Scale Seismic Test, LSST, arrays), from which Abrahamson (1985, 1992a) found that coherency γ at short distances (ξ < 200m) tois not be dependent on wavelength. Wavelength-dependence was found at larger distances (ξ = 400 to 1000m). For SVGM effects over the lateral dimensions of typical structures (e.g., < 200m), non-wavelength dependent models (Abrahamson, 1992a, 2007; Ancheta, 2010) are used.
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Moreover, there is a strong probabilistic nature of this phenomena, as significant uncertainty is present in relation to all four sources of incoherence lack of correlation, mentioned above. A number of excellent references are available on the subject of incoherent seismic motions (Abrahamson et al., 1991; Roblee et al., 1996; Abrahamson, 1992a, 2005, 1992b; Zerva and Zervas, 2002; Liao and Zerva, 2006; Zerva, 2009).
It is very important to note that all current models for modeling incoherent seismic motions make an ergodic assumption. This is very important as all the models assume that a variability of seismic motions at a single site – source combination will be the same as variability in the ground motions from a data set that was collected over different site and source locations. Unfortunately, there does not exist a large enough data set for any particular location of interest (location of a nuclear facility), that can be used to develop site specific incoherence models.
Incoherence in 3D. Empirical SVGM models are developed for surface motions only. This is based on a fact that a vast majority of measured motions are surface motions, and that those motions were used for SVGM model developments. Development of incoherent motions for three dimensional soil/rock volumes creates difficulties.
A 2 dimensional wave-field can be developed, as proposed by Abrahamson (1993), by realizing that all three spatial axes (radial horizontal direction, transverse horizontal direction, and the vertical direction) do exhibit incoherence. Existence of three spatial directions of incoherence requires existence of data in order to develop models for all three directions. Abrahamson (1992a) investigated incoherence of a large set of 3-component motions recorded by the Large Scale Seismic Tests (LSST) array in Taiwan, . and Abrahamson (1992a) concluded that there was little difference in the radial and transverse lagged coherency computed from the LSST array selected events. Therefore, the horizontal coherency models by Abrahamson (1992a) and subsequent models (Abrahamson, 2005, 2007) assumed the horizontal coherency model may apply to any azimuth. Coherency models using the vertical component of array data are independently developed from the horizontal.
Currently, there are no studies of coherency effects with depth (i.e. shallow site response domain). One possible solution is to utilize the simulation method developed by Ancheta et al. (2013) and the incoherence functions for the horizontal and vertical directions developed for a hard rock site by Abrahamson (2005) to create a full 3-D set of incoherent strong motions. In this approach, motions at each depth are assumed independent. This assumes that incoherence functions may apply at any depth within the near surface domain (< 100m). Therefore, by randomizing the energy at each depth, a set of full 3-D incoherent ground motion are created.
Theoretical Assumptions behind SVGM Models. It is very important to note that the use of SVGM models is based on the ergodic assumption. Ergodic assumptions allows statistical data obtained at one (or few) location(s) over a period of time to be used at
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other locations at certain times. For example, data on SVGM obtained from a Lotung site in Taiwan, over long period of time (dozens of years) is developed into a statistical model of SVGM and then used for other locations around the world. Ergodic assumption cannot be proven to be accurate (or to hold) at all, unless more data becomes available. However, ergodic assumption is regularly used for SVGM models.
Very recently, a number of smaller and larger earthquakes in the areas with good instrumentation were used to test the ergodic assumption. As an example, Parkfield, California recordings were used to test ergodic assumption for models developed using data from Lotung and Pinyon Flat measuring stations. Konakli et al. (2013) shows good matching of incoherent data for Parkfield, using models developed at Lotung and Pinyon Flat for nodal separation distances up to 100m. This was one of the first independent validations of family of models developed by Abrahamson and co-workers. This validation gives us confidence that assumed ergodicity of SVGM models does hold for practical purposes of developed SVGM models.
Recent reports by Jeremi’c, et al. (2011); Jeremi´c (2016) present detailed account of incoherence modeling.
[5.3.6] Case history - U.S. (in progress - to be completed)
NOTE BORIS: This set of case histories might be putting a bit too much weight/focus on incoherence. I think that plenty was said about incorence in the above sections and that these cse histories might need to be somehow folded into the upper section…
The treatment of the effects of seismic ground motion incoherence (GMI) on structure response for typical nuclear power plant structures was motivated in part by the development of Uniform Hazard Response Spectra (UHRS) with significant high frequency content, i.e., frequencies greater than 20 Hz. Figure 5-22 shows the UHRS (annual frequency of exceedance = 10-4) at 12 NPP rock sites in the U.S. The peak spectral acceleration for all 12 sites is at 25 Hz.
The evaluation of GMI as it pertains to NPP structures was comprised of the following elements:
Verification of the existence of GMI from recorded motions in the free-field. Process recorded motions to establish ground motion coherency functions.
Abrahamson (2006) investigated and processed recorded motions from 12 sites as listed in TABLE 5-4 for 74 earthquakes as listed in TABLE 5-5.
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TABLE 5-4 ARRAYS USED TO DEVELOP THE COHERENCY MODELS (Abrahamson, 2007)
TABLE 5-5 EARTHQUAKES IN THE ARRAY DATA SETS (Abrahamson, 2007)
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0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.1 1 10 100
Spec
tral
Acc
eler
ation
(g)
Frequency (Hz)
Horizontal Spectra 5% Damping
UHS-Rock-1
UHS-Rock-2
UHS-Rock-3
UHS-Rock-4
UHS-Rock-5
UHS-Rock-6
UHS-Rock-7
UHS-Rock-8
UHS-Rock-9
UHS-Rock-10
UHS-Rock-11
UHS-Rock-12
Figure 5-22 Uniform Hazard Response Spectra (UHRS) – annual frequency of exceedance = 10-4 - for 12 rock sites in the U.S. (Courtesy of James J. Johnson and Associates)
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5.5[5.4] LIMITATIONS OF TIME AND FREQUENCY DOMAIN METHODSFOR FREE FIELD GROUND MOTIONS
Limitations of time and frequency domain methods for Free Filed modeling are twofold. First source of limitations is based on (usual) lack of proper data for (a) deep and shallow geology, surface soil material, and (b) earthquake wave fields. This source of limitations can be overcome by more detailed site and geologic investigation (as recently done for the Diablo Canyon NPP in California NO REFERENCE). The second source of limitations is based on the underlying formulations for both approaches. Time domain methods, with nonlinear modeling, require sophisticated analysis program that is used, sophistication from the analyst, including knowledge of nonlinear solutions methods, elasto-plasticity, etc. Technical limitations on what (sophisticated) modeling can be done are usually with the analyst and with the program that is used (different programs feature different level of modeling and simulation sophistication). On the other hand, frequency domain modeling requires significant sophistication on the analyst side in mathematics. In addition, frequency domain based methods are limited to linear elastic material behaviour (as they rely on the principle of superposition) which limits their usability for seismic events where large nonlinearities are expected.
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5.6[5.5] REFERENCES
Kim, S. and Stewart, J.P. (2003), “Kinematic Soil-Structure Interaction from Strong Motion Recordings,” J. of Geotechnical and Geoenvironmental Engineering, ASCE, April 2003.
Johnson, J.J. (2003), “Soil Structure Interaction,” Earthquake Engineering Handbook, Chap. 10, W-F Chen, C. Scawthorn, eds., CRC Press, New York, NY.
Abrahamson, N. A. (2006), “Program on Technology Innovation: Spatial Coherency Models for Soil Structure Interaction,” EPRI, Palo Alto, CA and the U.S. Department of Energy, Germantown, MD, 1014101, December.
Abrahamson, N. (2005). Spatial Coherency for Soil-Structure Interaction, Electric Power Research Institute, Technical Update Report 1012968, Palo Alto, CA. December.
Abrahamson, N. (2007). Hard Rock Coherency Functions Based on the Pinyon Flat Data, April 2.
EPRI (2006), Program on Technology Innovation: Effect of Seismic Wave Incoherence on Foundation and Building Response, Electric Power Research Institute, Final Report TR-1013504, Palo Alto, CA. November.
Johnson, J.J., Schneider, O., Schuetz, W., Monette, P., Asfura, A.P. (2010), Effects of SSI on EPR In-Structure Response Spectra for a Rock Site, Coherent and Incoherent High Frequency Ground Motion, Proceedings of the ASME 2010 Pressure Vessels & Piping Division / K-PVP Conference, Paper PVP2010-26072, July 2010.
Johnson, J.J., Short, S.A., Hardy, G.S. (2007), Modeling Seismic Incoherence Effects on NPP Structures: Unifying CLASSI and SASSI Approaches, Paper K05-5, 19th Structural Mechanics in Reactor Technology (SMiRT19), Toronto, Canada.
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6 SITE RESPONSE ANALYSIS AND SEISMIC INPUT
6.1 OVERVIEW
Seismic input is the earthquake ground motion that defines the seismic environment to which the soil-structure system is subjected and for which the SSI analyses are performed. Chapter 2 introduces the concept of free-field ground motion, i.e., the motion that would occur in soil or rock in the absence of the structure or any excavation. Chapters 3, 4 and 5 discuss free-field ground motion in more detail.
Two extremely important points are:
The objective of the seismic analysis directly affects the approaches to be implemented for definition of the seismic input, i.e., design or assessment of the facility. The Design Basis Earthquake (DBE) ground motion may be based on standard ground response spectra (Section 6.4) or site specific ground response spectra developed by probabilistic seismic hazard analysis (PSHA) or deterministic seismic hazard analysis (DSHA) (Section 7.2).
Assessments of the facility can be for hypothesized Beyond Design Basis Earthquake (BDBE) ground motions or for actual earthquake events that have occurred and require evaluation. For assessments of the facility subjected to hypothesized BDBE ground motions, PSHA-defined values play an important role, i.e., for seismic margin assessments and seismic probabilistic risk assessments.
The physics of the seismic phenomena dictate that, in terms of SSI, the variation of motion over the dimensional envelope of the foundation (the depth and horizontal dimensions of the foundation) is the essential aspect. The detail of generation of this free-field ground motion is the important factor. For surface foundations, the variation of motion over the surface plane of the soil is important; for embedded foundations, the variation of motion over both the embedment depth and the foundation horizontal dimensions is to be defined.
This chapter discusses various aspects of defining the seismic input for SSI analyses. Seismic input is closely coupled with soil property definition (Chapter 3), free-field ground motion definition (Chapter 5), and SSI analysis methodology (Chapter 7). The
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development of the seismic input for the SSI analysis is closely coordinated with its purpose - design and/or assessment.
Typically, three aspects of free-field ground motion are needed to define the seismic input for SSI analyses: control motion; control point; and spatial variation of motion. Each of these elements contributes to the definition of seismic input for site response analyses and, subsequently, SSI analyses. These elements are discussed in the following subsections. Section 4.2 presents seismic hazard assessment, which is essential to defining the Design Basis Earthquake (DBE) ground motion and the considerations for the Beyond Design Basis Earthquake (BDBE) ground motions.
6.2 SITE RESPONSE ANALYSIS
6.2.1 Perspective
Site response analysis is comprised of many aspects. In the broadest sense, its purpose is to determine the free-field ground motion at one or more locations given the motion at another location. In addition to location, the form of the motion is defined, i.e., in-soil motion or actual/hypothetical outcrop motion. As in the case of SSI analyses, site response analyses can be performed in the frequency domain or in the time domain.
Frequency domain analyses for earthquake ground motions are linear or equivalent linear. Often, strain-dependent soil material behavior is simultaneously defined with the definition of the free-field ground motion as seismic input (Chapter 3).
Time domain site response analyses may be linear or nonlinear. Time domain analyses are most often performed to generate the explicit definition of the seismic input motion at all locations along the boundary of the linear or nonlinear SSI model (Chapters 4 and 6).
Investigations of the effects of irregular site profiles can be performed in the frequency domain or the time domain.
It is important to recognize that specific requirements for seismic input motion may be dependent on the SSI analysis methodology to be used. In some cases, especially for typical sub-structure methods, there are assumptions implemented as to the wave propagation mechanism of the free-field ground motion. These assumptions often are: vertically incident P- and S-waves; non-vertically incident P- and S-waves, and surface waves (Rayleigh or Love waves). Given the assumptions concerning the spatial variation of motion defined by a particular wave propagation mechanism and the compatible soil material property definition, seismic input to the SSI analyses is defined by specifying the
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control motion (normally, three spatial components of free-field ground acceleration) at a point denoted the control point, i.e., location and type of motion (in-soil or outcrop).
6.2.2 Foundation Input Response Spectra (FIRS)
The term Foundation Input Response Spectra (FIRS) was first defined in Refs. [44,45]. Although the definition may be interpreted in a general sense, Refs. [44,45] specifically define FIRS from the site response analysis assuming vertically propagating shear (S) and dilatational (P) waves and semi-infinite horizontal soil layers. The FIRS are defined on a hypothetical or actual outcrop at the foundation level of one or more structures. The important point is that it is a free-field ground motion input to the SSI analysis of a structure.
For a site with many structures of interest and with many different foundation depths (possibly, one foundation depth for each structure), multiple FIRS one at each foundation depth or a single definition at a common location, such as, the free surface at top of grade (TOG) are possible. However, as discussed later in this chapter, there are often additional requirements to which one must adhere, which could introduce additional analysis cases to be performed. One such requirement for design is the requirement to meet a specified minimum input ground motion response spectrum at foundation level.
Background and methodologies for generation of FIRS through the geologic method and full column method are contained in Refs. [44.45]. The primary difference between the geologic method and the full column method lies in the definition of the outcropping motion. Figure 6-1 shows a schematic of the two FIRS definitions. The geologic method requires removal of the strain-compatible soil layers above the foundation level and reanalysis of the soil column to extract the geologic outcrop spectrum. This reanalysis uses the strain compatible soil properties defined in the full column analyses – no additional iterations on soil properties are performed. Needs further clarification with text and adding for example an equation to further explain
The full column method includes the soil layers above the foundation level where the effects of down-coming waves above the foundation level are included in the analysis and the outcrop motion (also referred to as the SHAKE outcrop) assumes that the magnitude of the up-going and down-coming waves are equal at the elevation of the FIRS.
For frequency domain linear or equivalent linear analyses, the full column method is simpler to implement because the outcrop motion at the level of the foundation can be extracted directly from the site response analysis without reanalysis of the iterated soil columns. For linear or nonlinear time domain analyses, the geological method is required.
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In terms of SSI analysis, assuming each structure is analyzed independently of other structures, each structure has a defined Control Point and Control Motion. So for a given site, there may be multiple Control Points and Control Motions. A different point of view is that there is one Control Point and one Control Motion defined at TOG and that defines the input motion for all SSI analyses. Then deconvolution is performed implicitly with a program like SASSI or explicitly with a program like SHAKE.
In all cases when the DBE ground motion is defined at TOG, structures to be analyzed are modeled including embedment, and three deterministic soil profiles are used in the SSI analyses (best estimate, lower bound, and upper bound), the resulting envelope of the three at the TOG shall be verified to be equal to or greater than the TOG DBE ground motion. If not, additional soil profiles or higher FIRS may need to be considered.
6.3 SITE RESPONSE ANALYSIS APPROACHES
Characteristics of site response analysis are discussed next – convolution vs. deconvolution; probabilistic vs. deterministic; response spectra vs. random vibration approach. Convolution analysis may be performed in the frequency domain or in the time domain. Deconvolution analysis is performed in the frequency domain.
Principles of site response analyses:
PSHA results are in terms of seismic hazard curves of spectral acceleration. UHRS values can be used or Frequency domain approach based on time histories or random vibration theory
(RVT); Equivalent linear soil material properties may be generated as part of the process. Randomization
6.3.1 Idealized Site Profile and Wave Propagation Mechanisms
The majority of site response analyses over the last ten years implement techniques based on McGuire, R.K., Silva, W.J., and Costantino, C.J. (2001), “Technical basis for revision of regulatory guidance on design ground motions: Hazard- and risk-consistent ground motion spectra guidelines,” NUREG/CR-6728, U.S. Nuclear Regulatory Commission, Washington, D.C.
Assumptions:
Soil layer stratigraphy (semi-infinite horizontal layers overlying a uniform half-space), variability in layer thickness is modeled;
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Soil material properties (one-dimensional equivalent linear viscoelastic models defined by shear modulus and material damping – median and variability); (Equation 3-1);
Wave propagation mechanism vertically propagating S- and P-waves.
The techniques are designated as Approaches 1, 2A, 2B, 3, and 4; higher numbers associated with the more rigorous approaches specifically with respect to the potential sensitivity of the site amplification functions to magnitude and distance dependency of seismic sources, non-linearity of the soil properties, and consideration of uncertainty in the site profile and dynamic soil properties. The more rigorous of the approaches used extensively is Approach 3 (which is a simplification of Approach 4). Approaches 2B, 2A, and 1 include increasingly simplified assumptions as compared to Approach 3. Approaches 3, 2B, 2A, and 1 are briefly described in the following text.
Convolution. Site response analysis is most often considered to be convolution analysis, i.e., the PSHA defines the UHRS (and all related quantities) at hard rock (9,200 ft/sec) using attenuation laws or GMPEs associated with hard rock. The PSHA intermediate output at hard rock is the seismic hazard curves at a range of discrete natural frequencies - usually 5-20 frequencies, such as 0.1, 2.5, 5, 10, 25, 100 Hz or some other combination - as a function of annual frequencies of exceedance (AFE). The frequencies of calculation are referred to as “conditioning frequencies” in some applications. To define a UHRS at AFE, the discrete values of acceleration (or spectral acceleration) at the requested AFE of each of the natural frequency hazard curves is selected and the values are connected by segmented lines or fit with a curve. This becomes the UHRS at AFE at the location of interest and form of interest (in-soil or outcrop).
The next step then is to propagate that hard rock motion from the actual or hypothetical hard rock location at the site to locations within the soil profile - usually, to generate FIRS at foundation levels (locations) that correspond to structures of interest to design or evaluations. This is especially the case now as the new seismic hazard values for each site in the U.S. have been produced on hard rock; similarly, for many other Member States. This means generating site specific UHRS at requested locations (requested by the Team performing the SSI analyses), which are then used to define the seismic input to the SSI analyses (Chapter 6).
Frequency domain – either using time histories or random vibration theory (RVT) methods.
All approaches begin with the PSHA-defined seismic hazard curves at bedrock as an outcrop motion. The objective is to generate Site Amplification Functions (SAFs) between the bedrock outcrop motion and the point(s) of interest in the site profile.
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The approaches McGuire et al. to generate the amplification factors are:
Description of Approach 3, 2B, 2A, and 1 in progress.
These approaches are applied to the UHRS at hard rock to generate UHRS at TOG and at locations of interest in the soil media between the hard rock and the TOG. Given seismic hazard curves at 5-20 natural frequencies, these approaches may be customized to apply to different natural frequency ranges [low frequency (about 1 Hz) and high frequency (about 10 Hz)] as a function of the deaggregated seismic hazard of the PSHA. In the end, UHRS are developed at the location of interest, i.e., a UHRS (FIRS) at a given location and a selected annual frequency of exceedance (AFE).
These site response analyses (convolution) are most often probabilistic, accounting for strain dependency of the soil between hard rock and the FIRS locations and/or TOG, and randomness and uncertainty in the soil configuration and soil material properties.5 Normally, at least over the last five or more years, these probabilistic analyses are performed for a series of 60 earthquake simulations.
The convolution analysis (deterministic or probabilistic) is not tied to PSHA results only, e.g., a site with a significant mismatch of impedance values where a soft soil overlies a stiff soil or rock, site independent ground response spectra may be defined on a hypothetical outcrop at the interface and then site response analysis is performed to calculate TOG and/or FIRS for locations of foundations between impedance mis-match and foundation location.
Deconvolution. In other cases, the PSHA results (UHRS) are generated at a site location, usually TOG, using attenuation laws or GMPEs somewhat tailored to the site properties, in some cases through a Vs30 term. This case is most often implemented for relatively uniform soil properties. In this case, the UHRS can be input directly to surface-founded structures. They are the FIRS. Generally, however, some estimates or calculated soil properties adjusted for strain levels induced in the free-field by the ground motion are generated for SSI analyses. These free-field soil properties could be generated probabilistically or deterministically. For structures with embedded foundations, FIRS could be generated by deconvolution or, if the SSI analysis program permits exciting the SSI system with the TOG motion, then FIRS are not required for the SSI analysis, but FIRS may be required for checks against limits in the reduction of free-field ground motion at foundation level.
5 Ground motion response spectra peak and valley variability is explicitly treated in the PSHA. Consequently, the only ground motion randomness treated in the site response analyses and in the SSI analyses is randomness of phase.
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6.3.2 Non-idealized Site Profile and Wave Propagation Mechanisms
To be added. Determine what This is now section 6.2.5, few paragraphs to introduce to 6.2.5
6.4 UNCERTAINTIES
Epistemic (E) and aleatoric (A) uncertainties are to be included in the probabilistic analyses.
Ground motion – phase variability between the three spatial components (A), vertical vs. horizontal components (E,A), rock seismic hazard curves/UHRS (E,A), etc.
Stratigraphy Idealized soil profile – thickness of layers (E), correlation of soil
properties between layers (E), variation of soil properties in discretized layers (A), etc.
Non-idealized soil profile – geometry (E,A) Soil behaviour
Material model (linear/equivalent linear) Stiffness Energy dissipation
Wave propagation mechanism (E,A)
Figure 6-23 Definition of FIRS for idealized site profiles
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6.4.1 Analysis Models and Modelling Assumptions
Development of analysis models for free field ground motions and site response analysis require an analyst to make modeling assumptions. Assumptions must be made about the treatment of 3D or 2D or 1D seismic wave field (see sections 6.2.1, 6.2.2), treatment of uncertainties (see section 6.2.3), material modeling for soil (see section 6.3), possible spatial variability of seismic motions (see section 4.4).
This section is used to address aspects of modeling assumptions. The idea is not to cover all possible (more or less problematic) modeling assumptions (simplification), rather to point to and analyze some commonly made modeling assumptions. It is assumed that the analyst will have proper expertise to address all modeling assumptions that are made and that introduce modeling uncertainty (inaccuracies) in final results.
Addressed in this section are issues related to free field modeling assumptions. Firstly, a brief description is given of modeling in 1D and in 3D. Then, addressed is the use of 1D seismic motions assumptions, in light of full 3D seismic motions (Abell et al., 2016). Next, an assumption of adequate propagation of high frequencies through models (finite element mesh size/resolution) is addressed as well (Watanabe et al., 2016). There are number of other issues that can influence results (for example, nonlinear SSI of NPPs Orbovi´c et al. (2015)) however they will not be elaborated upon here.
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1D Models.
One dimensional (1D) wave propagation of seismic motions is a commonly made assumption in free ground motion and site response modeling. This assumptions stems from an assumption that horizontal soil layers are located on top of (also) horizontally layered rock layers. Snell’s law (Aki and Richards, 2002) can then be used to show that refraction of any incident ways will produce (almost) vertical seismic waves at the surface. Using Snell’s law will not produce vertical waves, however, for a large number of horizontal layers, with stiffness of layers increasing with depth, waves will be slightly off vertical (less than 10o in ideal, horizontal layered systems with deep sources). It is noted that with this ideal setup, with many horizontal layers of soil and rock and with the increase of stiffness with depth. waves will indeed become more and more vertical as they propagate toward surface (but never fully vertical!). However, seismic waves will rarely (never really) become fully 1D. Rather, seismic wave will feature particle motions in a horizontal plane, and there will not be a single plane to which these particle motions will be restricted. In addition, even if it is assumed that seismic waves propagating from great depth, are almost vertical at the surface, incoherence of waves at the surface at close distances, really questions the vertical wave propagation assumption. Nonetheless, much of the free and site response analysis is done in 1D.
It is worth reviewing 1D site response procedures as they are currently done in professional practice and in research. Currently most frequently used procedures for analyzing 1D wave propagation are based on the so called Equivalent Linear Analysis (ELA). Such procedures require input data in the form of shear wave velocity profile (1D), density of soil, stiffness reduction (G/Gmax) and damping (D) curves for each layer. The ELA procedures are in reality linear elastic computations where linear (so called strain compatible) elastic constants are set to secant values of stiffness for (a ratio of) highest achieved strain value for given seismic wave input. This means that for different seismic input, different material (elastic stiffness) parameters need to be calibrated from G/Gmax and D curves. In addition, damping is really modeled as viscous damping, whereas most damping in soil is frictional (material) damping (Ostadan et al., 2004). Due to secant choice of (linear elastic) stiffness for soil layers, the ELA procedures are also known to bias estimates of site amplification (Rathje and Kottke, 2008). This bias becomes more significant with stronger seismic motions. In addition, with the ELA procedures, there is no volume change modeling, which is quite important to response of constrained soil systems (di Prisco et al., 2012).
Instead of Equivalent Linear Analysis approach, a 1D nonlinear approach is also possible. Here, 1D nonlinear material models are used (described in section 3.2). Nonlinear 1D models of soil (nonlinear elastic, with a switch to account for unloading-
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reloading cycles) are used for this modeling. Nonlinear 1D models overcome biased estimates of site amplification, as they model stiffness in a more accurate wave. While modeling with nonlinear models is better than with ELA procedures, it is noted, that volume change information is still missing. It is also noted that 1D models used here only work for 1D, that is, extension to 2D or 3D is rather impossible to be done in a consistent way.
3D Models.
In reality, seismic motions are always three dimensional, featuring body and surface waves (see more in section 4.2). However, development of input, free field motions for a 3D analysis is not an easy task. Recent large scale, regional models (Bao et al., 1998; Bielak et al., 1999; Taborda and Bielak, 2011; Cui et al., 2009; Bielak et al., 2010, 2000; Restrepo and Bielak, 2014; Bao et al., 1996; Xu et al., 2002; Taborda and Bielak, 2013; Dreger et al., 2015; Rodgers et al., 2008; Aagaard et al., 2010; Pitarka et al., 2013, 2015, 2016) have shown great promise in developing (very) realistic free field ground motions. What is necessary for these models to be successfully used is the detailed knowledge about the deep and shallow geology as well as a local site conditions (nonlinear soil properties in 3D). This data is unfortunately usually not available.
When the data is available, a better understanding of dynamic response of an NPP can be developed. Developed nonlinear, 3D response will not suffer from numerous modeling uncertainties (1D vs. 3D motions, elastic vs. nonlinear/inelastic soil in 3D, soil volumetric response during shearing, influence of pore fluid, etc.). Recent US-NRC, CNSC and US-DOE projects have developed and are developing a number of nonlinear, 3D earthquake soil structure interaction procedures, that rely on full 3D seismic wave fields (free field and site response) and it is anticipated that this trend will only accelerate as benefits of (a more) accurate modeling become understood.
3D versus 1D Seismic Models
We start by pointing out one of the biggest simplifying assumptions made, that of a presence and use of 1D seismic waves. As pointed out in section 4.2.1 above, worldwide records do not show evidence of 1D seismic waves. It must be noted, that an assumption of neglecting full 3D seismic wave field and replacing it with a 1D wave field can sometimes be appropriate. However, such assumption should be carefully made, taking into account possible intended and unintended consequences.
An example is developed that illustrates differences between use 3D seismic motions and 1D seismic motions. Example is available in the appendix.
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NOTE BORIS: moving this section on 3D vs 1D waves to either chapter 8 or to the appendix with examples. Currenlty is will be just a separate section at the end in the Appendix
Present below is a simple study that emphasizes differences between 3D and 1D wave fields, and points out how this assumption (3D to 1D) affects response of a generic model NPP. It is noted that a more detailed analysis of 3D vs. 3×1D vs. 1D seismic wave effects on SSI response of an NPP is provided by (Abell et al., 2016).
Assume that a small scale regional models is developed in two dimensions (2D). Model consists of three layers with stiffness increasing with depth. Model extends for 2000m in the horizontal direction and 750m in depth. A point source is at the depth of 400m, slightly off center to the left. Location of interest (location of an NPP) is at the surface, slightly to the right of center.
Figure 6.10 shows a snapshot of a full wave field, resulting from a small scale regional simulation, from a point source (simplified), propagating P and S waves through layers. Wave field is 2D in this case, however all the conclusion will apply to the 3D case as well,
It should be noted that regional simulation model shown in Figure 4.10 is rather simple, consisting of a point source at shallow depth in a 3 layer elastic media. Waves propagate, refract at layer boundaries (turn more ”vertical”) and, upon hitting the surface, create surface waves (in this case, Rayleigh waves). In our case (as shown), out of plane translations and out of plane rotations are not developed, however this simplification will not affect conclusions that will be drawn. A seismic wave field with full 3 translations and 3 rotations (6D) will only emphasize differences that will be shown later.
Assume now that a developed wave field, which in this case is a 2D wave field, with horizontal and vertical translations and in plane rotations, are only recorded in one horizontal direction. From recorded 1D motions, one can develop a vertically propagating shear wave in 1D, that exactly models 1D recorded motion. This is usually done using de-convolution Kramer (1996).
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FIG.Figure 6.9: Snapshot of a full 3D wave field
Figure 6.9: (Left) Snapshot of a wave field, with body and surface waves, resulting from a point source at 45o off the point of interest, marked with a vertical line, down-left. This is a regional scale model of a (simplified) point source (fault) with three soil layers. (Right) is a zoom in to a location of interest. Figures are linked to animation of full wave propagation.
Figure 6.10 illustrates the idea of using a full 3D seismic wave field to develop a reduced, 1D wave field.
Figure 6.10: Illustration of the idea of using a full wave field (in this case 2D) to develop a 1D seismic wave field.
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Two seismic wave fields, the original wave field and a subset 1D wave field now exist. The original wave field includes body and surface waves, and features translational and rotational motions. On the other hand, subset 1D wave field only has one component of motions, in this case an SV component (vertically polarized component of S (Secondary) body waves).
Figure 6.11 show local free field model with 3D and 1D wave fields respectively.
Please note that seismic motions are input in an exact way, using the Domain Reduction Method (Bielak et al., 2003; Yoshimura et al., 2003) (described in chapter 6.4.10) and how there are no waves leaving the model out of DRM element layer (4th layer from side and lower boundaries). It is also important to note that horizontal motions in one direction at the location of interest (in the middle of the model) are exactly the same for both 2D case and for the 1D case.
Figure 6.12 shows a snapshot of an animation (available through a link within a figure) of difference in response of an NPP excited with full seismic wave field (in this case 2D), and a response of the same NPP to 1D seismic wave field.
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FIG.Figure 6.11: Left: Snapshot of a full 3D wave field, and Right: Snapshot of a reduced 1D wave field. Wave fields are at the location of interest. Motions from a large scale model were input using DRM, described in chapter 6 (6.4.10). Note body and surface waves in the full wave field (left) and just body waves (vertically propagating) on the right. Also note that horizontal component in both wave fields (left, 2D and right 1D), are exactly the same. Each figure is linked to an animation of the wave propagation.
FIG.Figure 6.12: Snapshot of a 3D vs 1D response of an NPP, upper left side is the response of the NPP to full 3D wave field, lower right side is a response of an NPP to 1D wave field.
Figures 6.13 and 6.14 show displacement and acceleration response on top of containment building for both 3D and 1D seismic wave fields.
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A number of remarks can be made:
Accelerations and displacements (motions, NPP response) of 6D and 1D cases are quite different. In some cases 1D case gives bigger influences, while in other, 6D case gives bigger influences.
Differences are particularly obvious in vertical direction, which are much bigger in 6D case.
Some accelerations of 6D case are larger thanthat those of a 1D case. On the other hand, some displacements of 1D case are larger than those of a 6D case. This just happens to be the case for given source motions (a Ricker wavelet), for given geologic layering and for a given wave speed (and length). There might (will) be cases (combinations of model parameters) where 1D motions model will produce larger influences than 6D motions model, however motions will certainly again be quite different. There will also be cases where 6D motions will produce larger influences than 1D motions. These differences will have to be analyzed on a case by case basis.
In conclusion, response of an NPP will be quite different when realistic 3D (6D) seismic motions are used, as opposed to a case when 1D, simplified seismic motions are used. Recent paper, by Abell et al. (2016) shows differences in dynamic behaviour of NPPs same wave fields is used in full 3D, 1D and 3×1D configurations.
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FIG.Figure 6.13: Displacements response on top of a containment building for 3D and 1D seismic input.
FIG.Figure 6.14: Acceleration response on top of a containment building for 3D and 1D seismic input.
Propagation of Higher Frequency Seismic Motions
Seismic waves of different frequencies need to be accurately propagated through the model/mesh. This is particularly true when it is required that higher frequencies of seismic motions be propagated. An illustrative example is used to analyse propagation of seismic waves of different frequencies throught finite element mesh. Resulting damping
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of higher frequencies is clearly observable, and should be taken into account when finite element models are designed, and decisions about mesh quality (finite element size) are made during model development. Recent paper by Watanabe et al. 2016 presents an in depth analysis of wave propagation through different mesh sizes, and for elastic as well as for nonlinear (elastic-plastic) materials.
NOTE BORIS, moving this analysis of waves of different frequencies propagation back to the appendix, will only leave a brief intro here…
It is known that mesh size can have significant effect on propagating seismic waves (Argyris and Mlejnek, 1991; Lysmer and Kuhlemeyer, 1969; Watanabe et al., 2016; JeremicJeremi´c, 2016). Finite element model mesh (nodes and element interpolation functions) needs to be able to approximate displacement/wave field with certain required accuracy without discarding (filtering out) higher frequencies. For a given wave length λ that is modeled, it is suggested/required to have at least 10 linear interpolation finite elements (8 node bricks in 3D, where representative element size is ∆hLE ≤ λ/10) or at least 2 quadratic interpolation finite elements (27 node bricks in 3D, where representative element size is ∆hQE ≤ λ/2) for modeling wave propagation.
Since wave length λ is directly proportional to the wave velocity v and inversely proportional to the frequency f , λ = v/f . we can devise a simple rule for appropriate size of finite elements for wave propagation problems:
Linear interpolation finite elements (1D 2-node truss, 2D 4-node quad, 3D 8-node brick) the representative finite element size needs to satisfy the following condition:
Quadratic interpolation finite elements (1D 3-node truss, 2D 9-node quad, 3D 27-node brick) the representative finite element size needs to satisfy the following condition:
It is noted that while the rule for number of elements (or element size ∆h) can be used to delineate models with proper and improper meshing, in reality having bigger finite element sizes than required by the above rule will not filter out higher frequencies at once, rather they will slowly degrade with increase in frequency content.
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Simple analysis can be used to illustrate above rules (Jeremi´c, 2016). Example analysis is presented in the appendix for linear elastic material model. When material becomes nonlinear (elastic-plastic), stiffness of the material is reduced, and thus the finite element size is reduced as well. Cases with nonlinear material are described by Watanabe et al. (2016).
NOTE BORIS, Moved to the appendix
One dimensional column model is used for this purpose, as shown in figure 4.16.
Total height of the model is 1000m. Two models are built with element height/length of 20 m and 50 m, and each model has two different shear wave velocities (100 m/s and 1000 m/s). Density is set to ρ = 2000 kg/m3, and Poisson’s ratio is ν = 0.3. Various cases are set and tested as shown in table 4.1. Both 8 node and 27 node brick elements are used for all models. Thus, total 24 parametric study cases are inspected. Linear elastic elements are used for all analyses. All analyses are performed in time domain with Newmark dynamic integrator without any numerical damping (γ = 0.5, and β = 0.25, no numerical damping, unconditionally stable).
Ormsby wavelet (Ryan, 1994) is used as an input motion and imposed at the bottom of the model.
Ormsby wavelet, given by the equation 4.3 below:
features a controllable flat frequency content. Here, f1 and f2 define the lower range frequency band, f3 and f4 define the higher range frequency band, A is the amplitude of the function, and ts is the time at which the maximum amplitude is occurring, and sinc(x) = sin(x)/x. Figure 6.17 shows an example of Ormsby wavelet with flat frequency content from 5 Hz to 20 Hz.
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FIG.Figure 6.16: One dimensional column test model to address the mesh size effect.
FIG.Figure 6.17: Ormsby wavelet in time and frequency domain with flat frequency content from 5 Hz to 20 Hz.
As an example of such an analysis, consider three cases, as shown in Figures 6.18, 6.19, and 4.20m with cutoff frequencies of Ormsby wavelets set as 3Hz, 8Hz, and 15Hz. As
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shown in figure 5.18, case 1 and 7 (analysis using Ormsby wavelet with 3 Hz cutoff frequency) predict exactly identical results to the analytic solution in both time and frequency domain. In this case, the number of nodes per wavelength for both cases is over 10 and presented very accurate results (in both time and frequency domain) are expected. Increasing cutoff frequency to 8Hz induces numerical errors, as shown in figure 6.19.
In frequency domain, both 10 m and 20 m element height model with 27 node brick element predict very accurate results. However, in time domain, asymmetric shape of time history displacements are observed. Observations from top of 8 node brick element models show more numerical error in both time and frequency domain due to the decreasing number of nodes per wavelength. Figure 4.20 shows analysis results with 15Hz cutoff frequency. Results from 27 node brick elements are accurate in frequency domain but asymmetric shapes are observed in time domain. Decreasing amplitudes in frequency domain along increasing frequencies are observed from 8 node brick element cases.
Table 6.1: Analysis cases to determine a mesh size
Case Vs Cutoff Element Max. propagation
1 1000 3 10 102 1000 8 10 103 1000 15 10 104 100 3 10 15 100 8 10 16 100 15 10 17 1000 3 20 58 1000 8 20 59 1000 15 20 510 100 3 20 0.511 100 8 20 0.512 100 15 20 0.5
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Material Modeling and Assumptions
Material models that are used for site response need to be chosen to have appropriate level of sophistication in order to model important aspects of response. For example, for site where it is certain that 1D waves will model all important aspects of response, and that motions will not be large enough to excite fully nonlinear response of soil, and where volumetric response of soil is not important (soil does not feature volume change during shearing), 1D equivalent linear models can be used. On the other hand, for sites where full 3D wave fields are expected to provide important aspects of response (3D wave fields develop due to irregular geology, topography, seismic source characteristics/size, etc.), and where it is expected that seismic motions will trigger full nonlinear/inelastic response of soil, full 3D elastic-plastic material models need to be used. More details about material models that are used for site response analysis are described in some detail in section 3.2, and in section 4.3, above.
3D vs 3×1D vs 1D Material BehaviourBehavior and Wave Propagation Models
In general behaviour of soil is three dimensional (3D) and very nonlinear/inelastic. In some cases, simplifying assumptions can be made and soil response can be modeled in 2D or even in 1D. Modeling soil response in 1D makes one important assumption, that volume of soil during shearing will not change (there will be no dilation or compression). Usually this is only possible if soil (sand) is at the so called critical state (Muir Wood, 1990) or if soil is a fully saturated clay, with low permeability, hence there is no volume change (see more about modeling such soil in section 6.4.2).
If soils will be excited to feature a full nonlinear/inelastic response, full 3D analysis and full 3D material models need to be used. This is true since for a full nonlinear/inelastic response it is not appropriate to perform superposition, so superimposing 3×1D analysis, is not right.
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FIG.
Figure 6.18: Comparison between (a) case 1 (top, Vs = 1000 m/s,
3 Hz, element size = 10m) and (b) case 7 (bottom, Vs = 1000 m/s,
3 Hz, element size = 20m)
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FIG.Figure 6.19: Comparison between (a) case 2 (top, Vs = 1000 m/s, 8 Hz, element size = 10m) and (b) case 8 (bottom, Vs = 1000 m/s, 8 Hz, element size = 20m)
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FIG.
Figure 6.20: Comparison between (a) case 3 (top, Vs = 1000 m/s, 15 Hz, element size = 10m) and (b) case 9 (bottom, Vs = 1000 m/s, 15 Hz, element size = 20m)
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Material Modeling and Assumptions
Material models that are used for site response need to be chosen to have appropriate level of sophistication in order to model important aspects of response. For example, for site where it is certain that 1D waves will model all important aspects of response, and that motions will not be large enough to excite fully nonlinear response of soil, and where volumetric response of soil is not important (soil does not feature volume change during shearing), 1D equivalent linear models can be used. On the other hand, for sites where full 3D wave fields are expected to provide important aspects of response (3D wave fields develop due to irregular geology, topography, seismic source characteristics/size, etc.), and where it is expected that seismic motions will trigger full nonlinear/inelastic response of soil, full 3D elastic-plastic material models need to be used. More details about material models that are used for site response analysis are described in some detail in section 3.2, and in section 4.3, above.
3D vs 3×1D vs 1D Material BehaviourBehavior and Wave Propagation Models
In general behaviour of soil is three dimensional (3D) and very nonlinear/inelastic. In some cases, simplifying assumptions can be made and soil response can be modeled in 2D or even in 1D. Modeling soil response in 1D makes one important assumption, that volume of soil during shearing will not change (there will be no dilation or compression). Usually this is only possible if soil (sand) is at the so called critical state (Muir Wood, 1990) or if soil is a fully saturated clay, with low permeability, hence there is no volume change (see more about modeling such soil in section 6.4.2).
If soils will be excited to feature a full nonlinear/inelastic response, full 3D analysis and full 3D material models need to be used. This is true since for a full nonlinear/inelastic response it is not appropriate to perform superposition, so superimposing 3×1D analysis, is not right.
Recent increase in use of 3×1D models (1D material models for 3D wave propagation) requires further comments. Such models might be appropriate for seismic motions and behaviour of soil that is predominantly linear elastic, with small amount of nonlinearities. In addition, it should be noted that vertical motions recorded on soil surface are usually a result of surface waves (Rayleigh). Only very early vertical motions/wave arrivals are due to compressional, primary (P) waves. Modeling of P waves as 1D vertically propagating waves is appropriate. However, modeling of vertical components of surface (Rayleigh) waves as vertically propagating 1D
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waves is not appropriate. Recent work by (Elgamal and He, 2004) provides nice description of vertical wave/motions modeling problems.
NEED to Bring Reference
6.5 STANDARD AND SITE SPECIFIC RESPONSE SPECTRA
6.5.1 Introduction
The amplitude and frequency characteristics of the free-field ground motion are one of the most important elements of the SSI analyses. Generally, the free-field ground motion is defined by ground response spectra. The ground response spectra may be site independent, i.e., uncorrelated or weakly correlated with site specific conditions, or site specific, i.e., the end product of a seismic hazard analysis (deterministically or probabilistically derived) as discussed in Sections 5.2 and 5.3. These cases are discussed in this section.
6.5.2 Standard response spectra
For nuclear power plants, depending on the vintage of the plant and the site soil conditions, the majority of the design ground response spectra have been relatively broad-banded standard spectra representing a combination of earthquakes of different magnitudes and distances from the site. Construction of such design spectra is usually based on a statistical analysis of recorded motions and frequently targeted to a 50% or 84% NEP. Three points are important relative to these broad-banded spectra. First, earthquakes of different magnitudes and distances control different frequency ranges of the spectra. Small magnitude earthquakes contribute more to the high frequency range than to the low frequency range and so forth. Second, it is highly unlikely that a single earthquake will have frequency content matching the design ground response spectra. Hence, a significant degree of conservatism is added when broad-banded response spectra define the control motion. Third, a single earthquake can have frequency content that exceeds the design ground response spectra in selected frequency ranges. The likelihood of the exceedance depends on the NEP of the design spectra.
Currently, standard or site independent ground response spectra are most often used in the design process for a new reference design or a Certified Design. Such designs are intended to be easily licensed for a large number of sites in many different seismic environments and site conditions. Therefore, the Design Basis Earthquake (DBE) ground motion is defined by broad-banded standard ground response spectra that are site-independent or weakly site-dependent and termed Certified Seismic Design Response Spectra (CSDRS).
Another application of broad-banded standard ground response spectra is the verification that the Foundation Input Response Spectra (FIRS) satisfies a minimum specified design basis ground motion:
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Minimum of 0.1g PGA, International Atomic Energy Agency (IAEA) Safety Standards Series No. NS-G-1.6, “Seismic Design and Qualification for Nuclear Power Plants, Safety Guide,” 2003;
Appendix S to 10 CFR Part 50, the minimum PGA for the horizontal component of the SSE at the foundation level in the free-field should be 0.1g or higher. The response spectrum associated with this minimum PGA should be a smooth broadband response spectrum (e.g., RG 1.60, or other appropriate shaped spectra, if justified) and is defined as outcrop response spectra at the free-field foundation level.
Figure 6-3 provides examples of standard ground response spectra for the horizontal direction:
U.S. NRC Regulatory Guide 1.60 (Rev. 2, 2014) U.S. NRC Regulatory Guide 1.60 enhanced in the high frequency range, which is the
CSDRS for the Westinghouse AP1000 Pressurized Water Reactor; European Utility Requirements EUR) - three design spectra corresponding to three broad
site conditions, i.e., hard rock, medium soil, and soft soil; Advanced CANDU Reactor (ACR-1000) – two reference design response spectra
corresponding to two broad site conditions, i.e., rock and soil.
6.5.3 Site specific response spectra
For this discussion, assume a PSHA has been performed and UHRS are developed for the range of annual frequencies of exceedance of the ground motion. This range of annual frequencies of exceedance is from about 1e-02 to 1e-07. Section 1.3.2 introduces one example of performance goals, which based on the design of SSCs to achieve less than a 1% probability of unacceptable behaviour at the DBE level and less than a 10% probability of unacceptable behaviour for a ground motion equal to 150% of the DBE level. Unacceptable behaviour is tied to the Seismic Design Category (SDC) and its probabilistically defined acceptance criteria.
The concept is based on performance goals. Develop a ground motion response spectra that, when coupled with seismic response procedures and seismic design procedures, will confidently achieve the probabilistically defined performance goal.
Given this context, ASCE 43 and U.S. NRC Regulatory Guide 1.208 establish a performance-based approach to developing the DBE (termed the ground motion response spectra - GMRS) that achieves these goals. The concept is to assume the above performance goal (1% and 10%) is achieved for an SSC of interest (or the plant) and associate it with a lognormal probability distribution with hypothesized lognormal standard deviations (based on previously performed studies of SSC and plant performance).
For high hazard facilities, focus on a performance goal of mean annual probability of unacceptable behaviour of Pf = 1e-05. Consider two UHRS – mean 1e-04 and mean 1e-05.
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Develop the relationships between the individual spectral accelerations (at the discrete frequencies of the calculated seismic hazard curves) at mean annual UHRS of 1e-04 and 1e-05. Develop scale factors to be applied to the spectral accelerations (SAs) of the mean 1e-05 to obtain risk consistent GMRS. The scale factors (less than or equal to 1.0) are based on previously performed studies of the integration of seismic hazard curves over the lognormal probability distribution of performance as introduced above. Scale factors are applied to the mean 1e-05 seismic hazard curves and the GMRS is constructed by connecting the SAs at these discrete natural frequencies.
The end result is the risk consistent definition of the site specific DBE (GMRS) associated with the performance criteria of less than a 1% probability of unacceptable behaviour at the DBE and less than a 10% probability of unacceptable behaviour at ground motion 150% times the DBE. The GMRS concept is to be applied consistently to the free-field ground motion that serves as the seismic input to the SSI analyses.
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0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1 1 10 100
Spec
tral
Acc
eler
atio
n (g
)
Frequency (Hz)
Horizontal Spectra 5% Damping
RG 1.60
US EPR, EUR Hard
US EPR, EUR Medium
US EPR, EUR Soft
AP1000 RG 1.60
ACR, CSA Soil
ACR, CSA Rock
FIG. 6-3 Examples of site independent Design Basis Earthquake (DBE) ground motion response spectra for standard design or reference design nuclear power plants
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6.6 TIME HISTORIES (IN PROGRESS)
Throughout Chapter 6, the discussion has been focused on ground motion response spectra and scalar descriptors of the ground motion – peak ground acceleration, velocity, and displacement. SSI analyses are performed for seismic input defined by acceleration time histories. This section discusses the selection and generation of acceleration time histories.
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6.7 UNCERTAINTIES AND CONFIDENCE LEVELS (REPEATED FROM SECTION 6.2.4 – NEED TO DECIDE WHERE IT SHOULD RESIDE)
Epistemic (E) and aleatoric (A) uncertainties are to be included in the probabilistic analyses.
Ground motion – phase variability between the three spatial components (A), vertical vs. horizontal components (E,A), rock seismic hazard curves/UHRS (E,A), etc.
Stratigraphy Idealized soil profile – thickness of layers (A), correlation of soil
properties between layers (A), variation of soil properties in discretized layers (A), etc.
Non-idealized soil profile – geometry (E,A) Soil behaviour
Material model (linear/equivalent linear) Stiffness Energy dissipation
Wave propagation mechanism (E,A)
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7 METHODS AND MODELS FOR SSI ANALYSIS
7.1 BASIC STEPS FOR SSI ANALYSIS
To identify candidate SSI models, model parameters, and analysis procedures, assess:
Scoping the problem, identify all relevant phenomena that will be simulated (1D vs 3D seismic wave field, linear or nonlinear response in 1D or 3D for soil, contacts and structure, dry (single phase), or saturated soil (effective stress, fully coupled analysis or total stress analysis), etc. Determine relative importance of phenomena to be modelled. It is noted that determination of relative importance can only be properly estimated using numerical experiments and a numerical tool that can properly model said phenomena and that can also model simplified model, neglecting (completely or partially) phenomena of interest.
Determine what numerical code (finite element, finite difference, or some other approach) possesses all (or most) required (identified above) modelling capabilities.
Confirm availability of a verification suite for a chosen numerical code. It should be required that chosen numerical code features an extensive verification suite, at least for modelling and simulation components that will be required for current analysis. Verification suite should be available (published) and be accessible to general public. Estimation of numerical errors for code components (solution advancement algorithms, elements, material models, etc.) should also be provided within verification suite.
Confirm availability of validation suite for material models (for materials) that are to be used in modelling and simulation
Perform a sensitivity study for all chosen modelling and simulation parameters in order to determine sensitivity of solution(s) to parameters. For example, seinsitivity study for material parameters should determine relative importance (sensitivity to changes) of each material parameter. Furthemore, sensivity to simulation parameters (for example values of β and γ parameters for Newmark time integration algorithm) should also be peformed for all simulation parameters.
Develop a set of simplified solutions, using simplified models (with understanding that these models do introduce modelling uncertainty, however these models are eaiser to manage and offer an efficient way to start a hierarchy, from less sophisticated to more sophisticated, of models.
For a specific site, a number of activities are advised in order to decide level of sophistication of a model
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Determine the purposes of the SSI analysis (design and/or assessment):
– Seismic response of structure for design or evaluation (forces, moments,
stresses or deformations, such as story drift, number of cycles of response)
– Input to the seismic design, qualification, evaluation of subsystems supported in the structure
(in-structure response spectra ISRS; relative displacements, number of cycles)
– Base-mat response for base-mat design
– Soil pressures for embedded wall designs
1. Determine the characteristics of the subject ground motion (seismic input motion):
a. Magnitude (excitation level) and frequency content (low vs. high frequency)
∗ Low frequency content (2 Hz to 10 Hz) affects structure and subsystem design/capacity; high frequency content (> 20 Hz) only affects operation of mechanical/electrical equipment and components;
b. Incoherence of ground motion;
c. Are ground motions 3D? Are vertical motions coming from P or S
(surface) waves. If from S and surface waves, full 3D motions.
Appropariate analysis procedures are given in Chapters 3, 4, and 5 for
free-field ground motion and seismic input.
2. Determine the characteristics of the site:
a. Assess the site profile from field investigation data
a) Determine site stratigraphy for selecting site profile model; b) Assess the need to perform sensitivity studies to determine two- or three-dimensional modelling for the site;
b. Idealized site profile is applicable
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∗ Linear or equivalent linear soil material model applicable (visco- elastic model parameters assigned) ∗ Nonlinear (inelastic, elastic-plastic) material model necessary?
c. Non-idealized site profile necessary?
d. Sensitivity studies to be performed to clarify model requirements for site
characteristics (complex site stratigraphy, inelastic modelling, etc.)?
3. Determine the characteristics of a structure of interest (identify safety related structures, structures housing safety related equipment and large components located in the yard for which SSI is important):
a. Identify function to be performed during and after earthquake shaking, e.g., leak tightness, i.e., pressure or liquid containment; structural support to subsystems (equipment, components, distribution systems); prevention of failure causing failure of safety related SSCs; b. Identify load bearing systems for modelling purposes, e.g., shear wall
structures, steel frame structures;
c. Expected behaviour of structure (linear or nonlinear);
d. Based on initial linear model of the structure, perform preliminary seismic response analyses (response spectrum analyses) to determine stress levels in structure elements;
e. If significant cracking or deformations possible (occur) such that portions of the structure behave nonlinearly, refine model either approximately introducing cracked properties or model portions of the structure with nonlinear elements;
f. For expected structure behaviour, assign material damping values;
4. Detarmine the foundation characteristics:
a. Effective stiffness is rigid due to base-mat stiffness and added stiffness due to
structure being anchored to base-mat, e.g., honey-combed shear walls
anchored to base-mat;
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b. Effective stiffness is flexible, e.g. if additional stiffening by the structure is
not enough to claim rigid; or for strip footings;
There are a number of other good practice activities that are suggested to be performed.
This is particulatly important as the level of sophisitication of analysis and models
increase, and as our capability to intuitively (using engineering judgement) check model
results diminishes.
Before initial results are available, make estimates of what type of behaviour is expected.
Develop a review process where developed results (at different levels of sophistication)
will be presented to and defended in front of independent reviewers.
Model development requires a hierarchical set of models with gradual increase in sophistication. As hierarchy of models is developed, each model needs to be verified and be capable to (properly, accurately) model phenomena of interest.
Model verification is used to verify that mechanical features that are of interest are indeed properly modeled. In other words, model verification is required to prove that results obtained for a given (developed) model are accurately modeling features of interest. For example, if propagation of higher frequency motions is required (analyzed), it is necessary to verify that developed (used) model is capable of propagating waves of certain wave lengths and frequency. Model verification is different than code and solution verification and validation, described in some details in Chapter 9 and also in much more detail in a number of references (Roache, 1998; Oberkampf et al., 2002; Oberkampf, 2003; Oden et al., 2005; Babuˇska and Oden, 2004; Oden et al., 2010a,b; Roy and Oberkampf, 2011).. Model verification has to be performed for each developed model in order to gain confidence that modeling results are usable for design, licensing, etc.
Modelling sequence should be:
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5. Linear elastic, model components first then slowly complete the model:
– soil only, (self weight in all three directions (vertical and both horizontal in
order to verify model), static loads, point, self-weight, eigen analysis, etc.);
dynamic loads (point loads, etc.); free field ground motions
– Components of structural model only (for example containment only, internal
structure only. Aux building, and in general all structures on the nuclear
island (EPR has 9 separate sub-structures), start with a fixed base model for
all, etc.), and then full structural model (just the structure, no soil), static loads
(self weight in three directions to verify model and load paths, point loads to
verify model and load paths); then dynamic loads (point loads, and seismic
loads) (Visual verification of results Is important, visual verification of
element shapes
– complete structure and foundation, (apply same load scenarios as above)
– complete structure, foundation, soil system, (apply same load scenarios as above)
6. Equivalent linear modeling, and observe changes in response, to determine possible
plastification effects. It is very important to note that it is still an elastic analysis,
with reduced (equivalent) linear stiffness. Reduction in secant stiffness really steams
from plastification, although plastification is not explicitly modeled, hence an idea
can be obtained of possible effects of reduction of stiffness.
One has to be very careful with observing these effects, and focus more on verification of model (for example wave propagation through softer soil, frequencies will be damped, etc...).
7. Nonlinear/inelastic modelling, slowly introduce nonlinearities to test models,
convergence and stability, in all the components as above.
8. Investigate sensitivities for both linear elastic and nonlinear/inelastic simulations!
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9. Develop documentation according to project requirements and procedures on
modelling, choices/assumptions, uncertainties (how are they dealt with) results, etc.
It is highly recommendable and strongly suggested to do regular, independent peer
reviews of project, with contininous involvement during project, managed by a (an
independent) technical integrator.
7.1.1 General guidance on soil structure interaction modelling and analysis
NOTE: There is a bit of this in Chapter 2 and then in 7.1 so either move to 7.,1 or combine somehow
7.2 DIRECT METHODS
7.2.1 Linear and nonlinear discrete methods
Linear and nonlinear mechanics of solids and structures relies on equilibrium of external and internal forces/stresses (Bathe (1996) and Zienkieqicz and Taylor (1991). . Such equilibrium can be expressed as σij,j = fi − uρ ¨i (7.1)where σij,j is a small deformation (Cauchy) stress tensor, fi are external (body (fi
B) and surface (fi
S) ) forces, ρ is material density and u¨i are accelerations. Inertial forces uρ ¨i
follow from d’Alembert’s principle (D’Alembert, 1758).The equilibrium above equation forms a basis for both Finite Element Method (FEM) and Finite Difference Method (FDM). Above equation can be pre-multiplied with virtual
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displacements uδ i and then integrated by parts to obtain the weak form, as further elaborated below in section 6.2.1. This equation can also be directly solved using finite differences, as noted in section 6.2.1.It is important to note that equilibrium equation 7.1 is usually not directly satisfied in either FEM or FDM. Rather it is satisfied in an approximate fashion, with a smaller or large deviation, depending on type of FEM or FDM used.
Finite Element Method
Finite element governing equiations are Equilibrium Equations Development of finite element equations is efficiently done by using through use of principle of virtual displacements. This principle states that the equilibrium of the body requires that for any compatible, small virtual displacements, which satisfy displacement boundary conditions imposed onto the body, the total internal virtual work is equal to the total external virtual work. Dynamic equiations of motions can thus be developed
Finite Element Equations After some manipulations (Zienkiewicz and Taylor, 1991a,b), we can write the finite element equations as:
MPQ u¨¯P + CPQ u¯˙P + KPQ u¯P = FQP,Q = 1,2,...,(#ofDOFs)N (7.2)
where MPQ is a mass matrix, CPQ is a damping matrix, KPQ is a stiffness matrix and FQ is a force vector. Damping matrix CPQ cannot be directly developed from a formulation for a single phase solid or structure. In other words, viscous damping is a results of interaction of fluid and solid/structure and is not part of this formulation (Argyris and Mlejnek, 1991). Viscous damping can, however, be added through viscoelastic constitutive material models and through Rayleigh damping, or a more general, Caughey damping. Viscous damping can also be added through viscoelastic constitutive material models .
In general Caughey damping is defined as (Semblat, 1997) (Semblat and Pecker 2009):
m−1
C = [M] X aj([M]−1[K])j(7.3)
j=0
where the order used depends on number of modes to be considered for damping in the problem. The second order Caughey damping , is also known as Rayleigh damping, with j = 1 in Equation (7.3).
In reality, damping matrix (more precisely, damping resulting from viscous effects) results from an interaction of soils and/or structures with surrounding fluids (Argyris and
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Mlejnek, 1991) (Ostadan et al 2004). For porous solid with pore space filled with fluid, a direct derivation of damping matrix is possible (Jeremic, et al., 2008).
Stiffness matrix KPQ can be linear (elastic) or nonlinear, elastic-plastic.
Finite element analysis comprises a discretization of a solid and/or structure into an assemblage of discrete finite elements. Finite elements are connected at nodal points.
It is very important to note that the finite element method is an approximate method. Generalized displacement solutions at nodes are approximate solutions. A number of factors control the quality of such approximate solutions. For example it can be shown (Zienkiewicz and Taylor, 1991a,b; Hughes, 1987; Argyris and Mlejnek, 1991) that an increase in a number of nodes, finite elements (refinement of discretization) and a reduction of increments (loads steps or time step size) will lead to a more accurate solution. However, this refinement in mesh discretization and reduction of step size, will lead to longer run times. A fine balance needs to be achieved between accuracy of the solution and run time. This is where verification procedures (described in some details in section 8 on verification and validation.) become essential. Verification procedures provide us magnitudes of errors that we can expect from our finite element (approximate) solutions. Results from verification procedures should thus be used to decide appropriate discretization (in space (mesh) and load/time) to achieve desired accuracy in solution.
Finite Elements
There exist different types of finite elements. They can be broadly classified into:
• Solid elements (3D brick, 2D quads etc.)
• Structural elements (truss, beam, plate, shell, etc.)
• Special Elements (contacts, etc.)
Solid finite elements usually feature displacement unknowns in nodes, 3 displacements for 3D elements, and 2 unknowns displacements for 2D elements. The most commonly used 3D solid finite elements are bricks, that can have 8, 20, and 27 nodes. In 3D, tetrahedral elements (4 and 10 nodes) are also popular due to their ability to be meshed into any volume, while solid brick elements sometimes can have problems with meshing. In 2D most common are quads, with 4, 8 or 9 nodes (Zienkiewicz and Taylor, 1991a,b; Bathe, 1996a). Triangular elements are also popular (3, 6 and 10 nodes), due to the same reason, that is triangles can be meshed in any plane shape, unlike quads. Two dimensional finite elements can approximate plane stress, plane strain or axisymmetric
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continuum. It is important to note that 3 node triangular elements feature constant strain field, and thus lead to discontinuous strains, and possibility of mesh locking.
Solid finite elements are also used to model coupled problems where porous solid (soil skeleton) is coupled with pore fluid (water), as described by Zienkiewicz and Shiomi (1984); Zienkiewicz et al. (1990, 1999). These elements and the underlying formulation will be described in some detail in section 7.4.11.
Structural finite elements use integrated section stress to develop section generalized forces (normal, transversal and moments). Truss elements can have 2 or 3 nodes. Beams usually have 2 nodes, although 3 node beam elements are also used (Bathe, 1996a). Most beam elements are based on a Euler-Bernoulli beam theory, which means that they do not take into account shear deformation, and thus should only be used for slender beams, where the ratio of beam length to (larger) beam cross section dimension is more than 10 (some authors lower this number to 5) (Bathe, 1996a). For beams that are not slender, Timoshenko beam element is recommended (Challamel, 2006), as it explicitly takes into account shear deformation.
Plate, wall and shell elements are usually quads or triangles. Plate finite elements model plate bending without taking into account forces in the plane of the plane plate. Main unknowns are transversal displacement and two bending (in plane) moments.
In plane forcing and deformation is modeled using wall elements that are very similar to plane stress 2D elements noted above. In plane nodal rotations are usually not taken into account. If possible, it is beneficial to include rotational (drilling) degree of freedom (Bergan and Felippa, 1985), so that wall elements have three degrees of freedom per node (two in plane displacements and out of plane rotation). Shell element is obtained by combining plate bending and wall elements.
Special elements are used for modeling contacts, base isolation and dissipation devices and other special structural and contact mechanics components of an NPP soil-structure system (Wriggers, 2002).
Finite Difference Method
Finite different methods (FDM) operate directly on dynamic equilibrium equation 6.1, when it is converted into dynamic equations of motion. The FDM represents differentials in a discrete form. It is best used for elasto-dynamics problems where stiffness remains constant. In addition, it works best for simple geometries (Semblat and Pecker, 2009), as finite difference method requires special treatment boundary conditions, even for straight boundaries that are aligned with coordinate axes.
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Finite Difference Solution Technique The FDM solves dynamic equations of motion directly to obtain displacements or velocities or accelerations, depending on the problem formulation. Within the context of the elasto-dynamic equations, on which FDM is based, elastic-plastic calculations are performed by changes to the stiffness matrix, in each step of the time domain solution.
Nonlinear discrete methods
Nonlinear problems can be separated into (Felippa, 1993; Crisfield, 1991, 1997; Bathe, 1996a)
• Geometric nonlinear problems, involving smooth nonlinearities (large deformations, large strains), and
• Material nonlinear problems, involving rough nonlinearities (elasto-plasticity, damage)
Main interest in modeling of soil structure interaction is with material nonlinear problems. Geometric nonlinear problems involve large deformations and large strains and are not of much interest here.
It should be noted that sometimes contact problems where gapping occurs (opening and closing of gaps) are called geometric nonlinear problems. For problems of interest here, namely, gap opening and closing between foundation and soil/rock, these problems are not geometrically nonlinear. They are not geometric nonlinear problems in the sense of large deformation and large strains. for cases of interest here, namely, gap opening and closing between foundations. Problems where (small) gap opens and closes are material nonlinear problems where material stiffness (and internal forces) vary between very small values (zeros in most formulations) when the gap is opened, and large forces when the gap is closed.
Material nonlinear problems problems for soil can be modeled using
• Linear elastic models, where linear elastic stiffness is the initial stiffness or the equivalent elastic stiffness (Kramer, 1996; Semblat and Pecker, 2009; Lade, 1988; Lade and Kim, 1995).
– Initial stiffness uses highest elastic stiffness of a soil material for modeling. It is usually used for modeling small amplitude vibrations. These models can be used for 3D modeling.
– Equivalent elastic models use secant stiffness for the average high estimated strain (typically. 65% of maximum strain) achieved in a given layer of soil. Eventual modeling is linear elastic, with stiffness reduced
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from initial to approximate secant. These models should really be only used for 1D modeling.
Nonlinear 1D models, that comprise variants of hyperbolic models (described in section 3.2), utilize a predefined stress-strain response in 1D (usually shear stress
τ versus shear strain γ) to produce stress for a given strain.
There are other nonlinear elastic models also, that define stiffness change as a function of stress and/or strain changes (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988)
These models can successfully model 1D monotonic behaviour of soil in some cases. However, these models cannot be used in 3D. In addition, special algorithmic measures (tricks) must be used to make these models work with cyclic loads.
Elastic-Plastic material modeling can be quite successfully used for frys frys both monotonic, and cyclic loading conditions (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and Whittle, 1995). Elastic plastic modeling can also be used for limit analysis (de Borst and Vermeer, 1984).
Material nonlinear problems for concrete can be modelled on two levels
Solid concrete level, where concrete is modelled using solid models and 2D or 3D elastic-damage-plastic material models (Willam et al. de Borst et al, Fenstra et al., Fenves and Kim, etc.)
Beam and/or plate/wall cross section, where 1D fibers are used to model nonlinear normal stress behaviour (concrete and reinforcement) in cross section (Fillipou et al, Spacone et al. Scott et al,, etc.). It is important to note that in this type of modelling influence of shear stress is neglected, that is pure bending is assumed.
Inelasticity, Elasto-Plasticity
Inelastic, elastic-plastic modeling relies on incremental theory of elasto-plasticity to solve elastic-plastic constitutive equations, with appropriate/chosen material model. Most solutions are strain driven, while there exist techniques to exert stress and mixed control (Bardet and Choucair, 1991). There are two levels of nonlinear/inelastic modeling when elasto-plasticity is employed:
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Constitutive level, where nonlinear constitutive equations with appropriate material models are solved for stress and stiffness (tangent or consistent) given strain increment
Global, finite element level, where nonlinear dynamic finite element equations are solved for given dynamic loads and current (elastic-plastic) stiffness (tangent or consistent).
Material Models for Dynamic Modeling At the constitutive level, general 3D strain increments (incremental strain tensor, or in other words, increments in all six independent components of strain, normal (σxx, σyy, and σyy) and shear (σxy, σyz, and σzx)) is driving the nonlinear constitutive solution.
Proper elastic-plastic material models must be chosen to obtain accurate results. Elastic-plastic material models, consist of four main components:
Elasticity, that governs the elastic response, before material yields. Yield function, a function in stress and internal variables (shear strength,
friction angle, back-stress, etc.) space, that separates elastic region from the elastic-plastic region.
Plastic flow directions, that provide directions of plastic strain, once material plastifies. Magnitude of plastic strain is obtained from the solution of constitutive equations.
Hardening/softening rules that control evolution of yield surface and plastic flow direction, during plastic deformation. There are four main types of hardening/softening rules, that can be combined between each other (for example isotropic and kinematic hardening models can be combined):– Perfect plastic material behaviour, where yield function and plastic
flow directions do not change during plastic deformation. There is no internal variable for this type of hardening/softening.
– Isotropic hardening/softening material behaviour, where yield function and plastic flow directions change isotropically (proportionally). This type of hardening/softening is only good for monotonic loading and should not be used for cyclic loading. Internal variables are of scalar type, for example friction angle, shear strength, maximum isotropic confinement, etc.
– Kinematic hardening where yield function and plastic flow direction either translate (works well for metals and total stress analysis of undrained, soft clays), or rotate (works well for soils, concrete, rock and other pressure sensitive materials). This type of hardening
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(usually it is only used for hardening, there is no softening) is good for cyclic loading. Internal variables are the back stress.
– Distortional hardening where yield function and plastic flow direction can have a general change in stress and internal variable space. This type of hardening/softening is the most general case and contains all the previous hardening/softening cases, however it is rarely used, as it requires a large number of tests.
Dynamic modeling, where stresses and strain cyclically change requires models that feature kinematic hardening. In case of pressure sensitive materials, like soil, concrete and rock, rotational kinematic hardening is used. For materials that do not have pressure sensitivity (metals, and saturated clays when modeled with a total stress approach (as opposed to effective stress approach (Jeremic, et al., 2008)), translational kinematic hardening is used.
There exist a number of models developed recently that can produce satisfactory modeling of dynamic response of geomaterials (Dafalias and Manzari, 2004; Taiebat and Dafalias, 2008; Dafalias et al., 2006; Mr´oz et al., 1979; Mroz and Norris, 1982; Prevost and Popescu, 1996). Of particular importance is availability of calibration tests, and addressing the issue of uncertainty and sensitivity of material response to changes in parameters.
It is also important to address the issue of spatial variability and uncertainty in material parameters for soils, as the ensuing response can also be quite uncertain. The issue of spatial variability and uncertainty in material modeling will be addressed in more detail in section will be addressed in section 6.5.
Nonlinear Dynamics Solution Techniques On the global, finite element level, finite element equations are solved using time marching algorithms. Most often used are Newmark algorithm (Newmark, 1959) and Hilber-Hughes-Taylor (HHT) α algorithm (Hilber et al., 1977). Other algorithms (Wilson θ, l’Hermite, etc.) also do exist Argyris and Mlejnek (1991); Hughes (1987); Bathe and Wilson (1976), however they are used less frequently. Both Newmark and HHT algorithm allow for numerical damping (through appropriate choice of parameters, and γ for Newmark method and γ for HHT method) to be included in order to damp out higher frequencies that are introduced artificially into FEM models by discretization of continua into discrete finite elements. These numerical damping terms are dependent on the choice of the integration parameters.
Solution to the dynamic equations of motion can be done by either enforcing or not enforcing convergence to equilibrium. Enforcing the equilibrium usually requires use of Newton or quasi Newton methods to satisfy equilibrium within some tolerance. This
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results in a (much) longer running times, however, provided that the convergence tolerance is small enough, analyst is assured that his/her solution is within proper material response and equilibrium. Solutions without enforced equilibrium are faster, and if they are done using explicit solvers, there is a requirement of small time step, which can then slow down the solution process.
7.3 SUB-STRUCTURING METHODS
7.3.1 Principles
As described in Section 1.1, the natural progression for addressing soil-structure interaction (SSI) of structures, such as nuclear power plant structures, was to implement tools developed for machine vibrations. This was an initial attempt at sub-structuring methods. It took into account the dynamic behaviour of the semi-infinite half-space, i.e., the force-displacement behaviour defined by impedance functions that are complex-valued and frequency dependent. For a foundation assumed to behave rigidly, these impedances are uniquely defined by 6x6 frequency-dependent matrices of complex-valued impedances relating foundation forces and moments to six rigid body degrees of freedom. For foundations that behave flexibly, a sufficient number of flexible impedance matrices (frequency-dependent and complex-valued) are developed relating forces and displacements.
The desire to break the complicated soil-structure problem into more manageable parts has lead to the sub-structure methods.
Generally, sub-structuring methods applied to the SSI problem are considered to be a linear process, i.e., each step in the analysis is solved separately and then the separate parts are combined through super-position to solve the complete problem. Most sub-structuring methods solve the SSI problem in the frequency domain: the seismic input is defined by acceleration time histories, which are transformed into the frequency domain for the analysis; scattering functions that define the foundation input motion are frequency dependent; impedance functions that define the force-dispacement relationships between foundation node points (or the rigid foundation) are complex-valued, frequency-dependent functions; and the dynamic behavior of the structure is frequency-dependent (as demonstrated by eigen-system extraction into normal modes).
Conceptually, sub-structuring methods can be classified into four types depending on the methodology of treating the interface between the soil and structure, i.e., how the soil and structure interface degrees-of-freedom are treated. Figure 7-24 shows the four types. These four types are: 1) the rigid boundary method, where the term “rigid” refers to the
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boundary between the foundation6 and the soil; 2) the flexible boundary methods; 3) the flexible volume method or the direct method; and 4) the substructure subtraction method.
It is noted that the Domain Reduction Method (DRM), described in section 7.4.10, is a direct method as it solves the (nonlinear) problem of earthquake soil structure interaction (ESSI) using a single model. However, the DRM can also be considreted as a multistep method if one considers modelling of an earthquake process from the source (causative fault) to the NPP site. Regional model taking into account the fault, can be described as the first step, while ESSI modelling is then a second step. However, large scale, regional model does not have to be modelled, input motions for DRM input can be developed using other, simplified methods, for example a directo deconvolution of surface motions. The DRM is decribed in some detail in section 7.4.10, while original reference is by Bielak et al. (2003).
Unlike direct methods of analyses in which all the elements involved in a soil-structure model are gathered in a single model and the dynamic equilibrium equations are solved in one step, sub-structuring methods replace the one-step analysis by a multi-step approach: each step in the process is analyzed by the most appropriate tools available. the global model is replaced by several sub-models of reduced sizes, which are solved independently and successively. To express that the individual models belong to the same global problem, compatibility conditions are enforced between the sub-models: these compatibility conditions simply state that at the shared nodes of two sub-models displacements must be equal and forces must have the same amplitude but opposite signs. The sub-structure approach is schematically depicted in Figure 7-26.
6 The term “foundation” is used herein to denote the foundation of the structure (base mat) and partially embedded structure elements, such as exterior walls in contact with the soil.
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Figure 7-24: Summary of substructuring methods
Figure 7-25 Schematic representation of the elements of soil-structure interaction – sub-structure method
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A
B
Sub-structure 1
Sub-structure 2
FB
FA
Figure 7-26: Schematic representation of a sub-structure model
At nodes A and B, which are at the same location in space, the displacement vectors satisfy UA = UB and the force vectors FA = –FB. Note that here the partitioning between the two subsystems is made along the boundary separating the structure and the soil; however, as it will be briefly mentioned below, other possible alternatives. The boundary considered in Figure 7-26 does not need to be rigid but the size of the problem to handle is significantly reduced when this assumption holds.
Several approaches are proposed to handle the sub-structure methods. They differ essentially by the way in which the interaction between the two sub-structures is handled. One possibility is to define the interface between the two sub-systems along the external boundary of the structure which can be either a plane (2D model) or a surface (3D model); the approach is known as the (rigid or flexible) boundary method and is due to Kausel and Roësset (1974) and Luco and Wong (1971). Other sub-structuring methods have been introduced more recently; they are known as the flexible volume method (Tabatabaie-Raissi, 1982) and the subtraction method (Chin, 1998).
The seismic SSI sub-problems that these three types of substructuring methods are required to solve to obtain the final solution are compared in Figure 7-24 (SASSI manual). As shown in this figure, the solution for the site response problem is required by all four methods. This is, therefore, common to all methods. The analysis of the structural response problem is also required and involves essentially the same effort for all methods. The necessity and effort required for solving the scattering and impedance problems, however, differ significantly among the different methods. For the rigid boundary method, the scattering and impedance problems are typically solved simultaneously (Luco and Wong (1971), Johnson et al. (2007)) once Green’s functions
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have been calculated or when the flexible impedance matrix is calculated. The flexible boundary model requires calculation of the scattering and impedance matrices separately. The flexible volume method and the substructure subtraction method, because of the unique substructuring technique, require only one impedance analysis and the scattering analysis is eliminated. Furthermore, the substructuring in the subtraction method often requires a much smaller impedance analysis than the flexible volume method. The latter two methods are those implemented in SASSI.
Note also that the domain reduction method (DRM) described in section 7.4.10 6.4.10 can be viewed as a substructure method in which the substructure includes not only the structure but part of the soil. However, reference to substructure methods is usually limited in practice to one of the four methods described above.
Actually, the substructure methods apply only to linear systems. However, the nonlinear behaviour of the soil may be accounted for by using strain compatible properties derived from site response analyses carried out with the viscoelastic equivalent linear model.
7.3.2 Rigid or flexible boundary method
The multi-step approach involved in sub-structuring is based on the so-called superposition theorem established by Kausel and Roësset (1974). It is schematically depicted in Figure 7-25 and Figure 7-27 (Lysmer, 1978) and can be described as follows. Note that Figure 7-27 has been drawn for the special case described below of a 2D problem and a rigid foundation.
The solution to the global SSI problem is obtained by solving successively:
The scattering problem, which aims at determining the kinematic foundation motions, which represent, at each foundation node, the three (or six) components of motion of the massless foundation with its actual stiffness subjected to the incident wave field of the global SSI problem. Note, these motions differ from the free-field motions except for surface foundations subjected to coherent vertically propagating waves. In this case, scattering functions are real and equal to unity in the direction of the excitation of interest and zero in other excitation directions. For embedded foundations, scattering functions vary with location of points on the interface between the soil and the foundation. This is due to the spatial variation of ground motion with the depth and width of the foundation and the scattering of waves by the same.
For foundations modelled as behaving rigidly, the scattering matrix is a (6x3) matrix relating rigid body displacements and rotations to the three components of coherent free-field ground motion (two horizontal and the vertical direction) (see Chapter 6 for further discussion of the free-field ground motion). Foundation
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input motion (FIM) is defined as the result of multiplying the scattering matrix times the free-field ground motion. .
The impedance problem; the impedance function expresses the relationship between a unit harmonic force applied in any direction and at any location along the soil-foundation interface and the induced displacements at any node in any direction along the same interface. In calculating the impedance function the foundation is massless and modelled with its actual rigidity, which is not necessarily infinite. The size of the impedance matrix is kn by kn where n is the number of nodes along the boundary and k the number of degrees of freedom at each node. In the most general case k=6 and the stiffness matrix is full with 6n by 6n non-zero terms.
When the foundation can be considered rigid, its kinematics can be described by the displacement of a single point7, for instance the geometric centre of the basemat, and the size of the frequency-dependent complex valued impedance matrix is 6 by 6 for each frequency of calculation for the analysis.
The structural analysis problem which establishes the response of the original structure connected to a support through the impedance functions and subjected to the kinematic foundation motions.
It can be proved that, provided each of the previous steps are solved rigorously, the solution to third problem (step 3 of the analysis) is strictly equivalent to the solution of the original global SSI problem.
Figure 7-27: 3-step method for soil structure interaction analysis (after Lysmer, 1978)
7 For foundations assumed to behave rigidly analyzed by CLASSI, he point about which the scattering and impedance matrices are calculated is named the “Foundation Reference Point (FRP).” This also the point about which the SSI solution is generated.
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Sub-structure methods can be sub-divided into methods that efficiently analyze structures whose foundation can be modelled as behaving rigidly or flexibly. This categorization is often tied to the multi-step vs. single step analyses in conjunction with the required SSI analysis results.
Foundations that behave flexibly are treated by the flexible boundary method, flexible volume method (direct method), and the subtraction method – described later in this chapter.
Foundations behaving rigidly. Two programs that are extensively used to analyzeanalyse linear elastic soil-structure systems when the foundation of the structure can be treated as behaving rigidly are CLASSI (Soil-Structure Interaction: A Linear Continuum Mechanics Approach) (Luco and Wong, 1980) and SUPELM (Kausel, xxxx). In the same spirit as the evolution of SASSI, CLASSI has evolved over the last decades such that it is a staple tool in analyst’s tool boxes. Section 7.3.5 describes the principle features of CLASSI. Section 7.3.5 also describes the hybrid method of CLASSI-SASSI, in which embedded foundations are modelled with SASSI and the resulting SASSI flexible impedance matrix is derived and condensed to six degrees-of-freedom for CLASSI analyses. The modules of CLASSI and SUPELM were used extensively in the SASSI V&V program to benchmark individual modules of the SASSI program.
In general, all linear and nonlinear SSI methods of analysis rely on numerical techniques continuum mechanics, finite elements, finite differences, or boundary element methods. As introduced above, sub-structuring methods are most efficiently implemented in the frequency domain, which requires that theoretically solutions must be established for each frequency of the Fourier decomposition of the seismic input. In the past, this would represent a formidable task and, in practice, only less than the total number of Fourier transform frequencies are analyzed explicitly (typically, 50 – 200 frequencies). Interpolation schemes have been developed and implemented to fill in the functions at missing frequencies, e.g., Tajirian, 1981. With the advent of High Performance Computing (HPC), this has beome less of an issue.
The main steps of this implementation are briefly described below for finite element modelling with reference to Figure 7-27; for more details in the derivation of the following equations, the reader can refer to Kausel and Roësset (1974).
(NOTE by BORIS, I would suggest that all the equation below be replaced with a text descriptions (brief) of a methodology, with benefits and drawbacks itemize, and with references to the actual method development and equations!)
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Noting with a superscript ~ the Fourier transform of any quantity, the impedance
matrix S() relates the force vector applied to the foundation to its
displacement vector by (solution of step 2, Figure 7-27 (b))
(7.4)
If is the solution of the scattering problem (Figure 7-27 (a)), it can then be shown that the dynamic equilibrium equation for the structure can be written in the frequency domain
(7.5)
In the above equation the whole system has been partitioned in degrees of freedom belonging to the structure (indices bb) and degrees of freedom belonging to the structure-foundation interface (indices fb). The damping terms (viscous damping if any) has been omitted for simplicity and the energy dissipation term is represented only by the imaginary part of the complex stiffness terms K (viscoelastic equivalent linear model, see chapter 3).
Equation (7.5) represents the solution to step 3 (Figure 7-27 (c)). If the foundation is infinitely rigid, the displacement vector can be expressed as a function of the displacement at a particular point through a transformation matrix T:
(7.6)
and eq.Error: Reference source not found then becomes
(7.7)
In eq.(7.7), the superscript T represents the transpose of the transformation matrix and the second line has been left multiplied by TT to render the system symmetric.
represents the impedance matrix of the rigid foundation (dimensions at
most 6 by 6). Noting that , eq.Error: Reference source not found represents the motion of the structure connected to a support by the impedance
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matrix T and subject to a support motion which is the kinematic foundation input motion, solution of the scattering problem.
SUGGESTION by Boris, remove above section, and replace with a list of benefits and detriments of the method, while making a reference to papers/reports where equations can be found!
Once the solution is determined for a set of discrete frequencies and interpolated for the missing frequencies, an inverse Fourier transform provides the time domain solution. It must again be stressed that, as superposition is involved in the frequency domain solution, the substructure methods are limited to linear systems, even though nonlinearities can partially be accounted for in the soil by using strain compatible properties.
7.3.3 The flexible volume method
The flexible volume substructuring method (Tabatabaie-Raissi, 1982) is based on the concept of partitioning the total soil-structure system into three substructure systems. Substructure I consists of the free-field site, substructure II consists of the excavated soil volume, and substructure III consists of the structure, of which the foundation replaces the excavated soil volume. The substructures I, II and III, when combined together, form the original SSI system. The flexible volume method presumes that the free-field site and the excavated soil volume interact both at the boundary of the excavated soil volume and within its body, in addition to interaction between the substructures at the boundary of the foundation of the structure. The SASSI program implements the flexible volume method denoting it the “direct method”.
7.3.4 The subtraction method
The substructure subtraction method (Chin, 1998) is basically based on the same substructuring concept as the flexible volume method. The subtraction method partitions the total soil structure system into three substructure systems (Error: Reference source notfound). Substructure I consists of the free-field site, substructure II consists of the excavated soil volume, and substructure III consists of the structure. The substructures I, II and III, when combined together, form the original SSI system. However, the subtraction method recognizes, as opposed to the flexible volume method, that soil structure interaction occurs only at the common boundary of the substructures, that is, at the boundary of the foundation of the structure. This often leads to a smaller impedance analysis than the flexible volume method.
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NOTE by Boris, similar to the above note, I would suggest removing specifics of the method below (that is actdually very specific about SASSI, and it is (somewhat) controversial), just list pros and cons and give references. The way this is given, TECDOC becomes very heavy on theory for SASSI, which is available elsewhere!
Note that the subtraction method in SASSI is an approximate method to the flexible volume method (direct method). Frequently, the combination of soil properties and geometry of the excavated soil have led to spurious results (amplified frequencies associated with the excavated soil mass). To alleviate this anomaly, the subtraction method is modified by adding interaction nodes to the excavated soil further restraining the free body frequencies so that any spurious results occur at frequencies above the maximum frequency of interest in the SSI problem being solved. The subtraction method with these additions is called the extended (modified) subtraction method. When implementing any of the subtraction methods, validation that no spurious results have been introduced is necessary. (ASCE 4-16)
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Figure 7-28: Substructuring in the subtraction method.
Noting C the matrix K–2M and S the impedance matrix, and identifying the various degrees of freedom with the fllowing subscripts:
b nodes at the boundary of the total system i nodes at the boundary between the ground and the structure w nodes within the excavated soil volume g nodes at the remaining part of the freefield s nodes at the remaining part of the structure
The equation of motion is partitioned as follows:
(7.8)
Where the superscripts refer to the substructures. The complex frequency dependent dynamic stiffness matrix on the left of eq. Error: Reference source notfound simply indicates the stated partitioning according to which the stiffness and mass of the excavated soil volume are subtracted from the dynamic stiffness of the free-field site and the structure. As previously the freefield motions, associated with nodes i, are noted with a * superscript; they represent the load vector as in eq.Error:Reference source not found.
7.3.5 CLASSI: Soil-Structure Interaction - A Linear Continuum Mechanics Approach
Overview
Key elements of the CLASSI approach (Luco and Wong, 1980) are:
The site is modeled as semi-infinite viscoelastic horizontal layers overlying a semi-infinite viscoelastic halfspace.
1. Complex-valued, frequency-dependent Green’s functions for horizontal and vertical point loads are used in the generation of the foundation input motion (scattering functions) and the foundation impedances.
In the standard version of CLASSI, Green’s functions are generated from continuum mechanics principles. The Green’s functions are integrated over the
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discretized foundation sub-region areas to calculate a resultant set of forces and displacements at sub-region centroids. The foundation impedances are developed by imposing rigid body constraints to the sub-region centroidal displacements corresponding to unit displacements at a defined foundation reference point (FRP). The scattering matrix is developed in a second stage where the free-field ground motions are imposed on the assumed rigid, massless foundation yielding the foundation input motion and, thereby, the scattering matrix.
2. In the Hybrid Method, the advantages of CLASSI and SASSI are combined for generation of scattering functions and foundation impedances. In the same manner as described above, the Green’s functions from SASSI are integrated over the sub-regions creating the unconstrained flexible impedance matrix. Applying the constraints of rigid body motion yields the impedance and scattering matrices from the SASSI flexible impedance matrix for embedded foundations of arbitrary shape. In the hybrid method, any soil profile modelled in SASSI can be treated in CLASSI-SASSI.
3. The foundation geometry is discretized for generating the scattering functions and foundation impedances by applying rigid body constraints. The results consist of complex-valued, frequency-dependent scattering functions and impedances. Multiple rigid foundations may be modeled including through soil-coupling.
4. The structures are idealized by simple or very detailed three-dimensional finite element models. The dynamic characteristics of the structure are represented by the fixed-base eigensystem. For the SSI analysis, these dynamic properties are projected onto the foundation, i.e., the fixed-base dynamic characteristics of the structure are represented exactly by its mode shapes, frequencies and modal damping values projected onto the foundation. An example of a detailed three-dimensional finite element structure model as analyzed by CLASSI is shown in Figure 7-29. Key parameters of the model are: number of nodes = 70,366; number of degrees-of-freedom = 422,196; and number of fixed-base modes included in the SSI analyses = 3004 (frequcies ranging from 4.33 Hz to 61.05 Hz) (Johnson et al., 2010).
Multiple structures on a single foundation may be modeled.
5. Material damping is introduced by the use of complex moduli in the soil and modal damping in the structure.
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6. The seismic environment may consist of an arbitrary three-dimensional superposition of inclined body and surface waves. Also, a version of CLASSI treats incoherent ground motion.
7. The earthquake excitation is defined by time histories of free-field accelerations, called control motion. The control motion is applied at the control point which may be defined on the soil free surface or at a point within the soil column.
8. The Fast Fourier Transform technique is used.
9. The solution of the complete SSI problem is performed in stages: (i) the SSI response of the foundation is calculated including the effects of the structure (fixed-base modes), foundation (mass), and supporting soil (impedance functions) when subjected to the foundation input motion; and (ii) the dynamic response of the structure degrees of freedom, when subjected to the foundation SSI response, are calculated.
Figure 7-25 shows schematically the steps in the CLASSI methodology.
Important Equations
NOTE By BORIS; similar to the above, recommend removing detailed equations, they are
available elsewhere and are making this section too equation heavy! Probably best approach would be to develop a list of pros and cons of the methodology, and include reference for the details of the methodology (Luco and Wong papers/reports…)
The SSI problem is solved in the frequency domain, frequency-by-frequency for the frequencies of the Fourier transformed input motion. The equations of motion are written about the foundation reference point (FRP), in terms of the unknown foundation response. Equation 1 shows the equations of motion.
The equations of motion have been re-arranged with the unknown foundation response U(ω) on the left hand side. The first term represents the inertial loading conditions (forces and moments) of the foundation and the structure applied at the foundation reference point (FRP). The foundation loads are a result of the foundation mass being treated as a rigid body, [Mo]. The structure loads are comprised of two parts. The first part is due to the loads induced at the FRP due to
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the rigid body motion of the foundation (foundation translations and rotations). The second part is the vibrational response of the structure when excited by the foundation motion, i.e., a base excitation comprised of the foundation translations and rotations. The two additional terms represent loads due to the resistance of the supporting soil. The foundation impedances act on the differential motion between the foundation input motion U*(ω) and the resulting foundation response (see Eqn. 2). In all cases, the foundation response U(ω) is that motion of the foundation as a result of SSI, including kinematic and inertial interaction. The vibrational response of the structure is calculated using the fixed-base modes of the structure. The following equations assume M independent structures supported by the foundation.
M - ω2 [Mo] + ∑ [MB(ω)]i + [Ks(ω)] U(ω)
i=1
= [Ks(ω)] U*(ω) (7.9) Resistance of the supporting soil:
[Ks(ω)] U(ω) - U*(ω) (7.10)
[Mo] = the N x N mass matrix of the rigid foundation mat composed of the mass and mass moments of inertia about the FRP; N is the number of foundation degrees of freedom (N ≤ 6).
[MB(ω)]i = the frequency dependent equivalent mass matrix of the ith structure supported on the foundation mat.
[Ks(ω)] = the frequency dependent impedance matrix of the foundation-soil system; the real part of [Ks(ω)] represents the stiffness and inertia of the soil; the imaginary part of [Ks(ω)], when divided by the frequency, corresponds to the radiation and material damping of the soil.
U*(ω) = up to three N x 1 vectors describing the Foundation Input Motion for each of the three directions of excitation of the free-field, i.e., the response of the massless, rigid foundation subjected to the free-field ground motion.
U(ω) = unknown frequency-dependent response of the foundation mat, including all aspects of SSI (kinematic and inertial interaction).
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[MB(ω)]i = [MBO]i + [Г]iT [D(ω)]i [Г]i (7.11)
where
[MBO]i = the 6 x 6 rigid body mass matrix of the ith structure supported on the foundation mat;
[MBO]i = [α]iT [M]i [α]i
[M]i = nodal mass matrix of the ith structure; assume q x q where q is the number of structure dynamic degrees of freedom;
[α]i = structure geometry matrix; a q by 6 rigid body transformation matrix of the ith structure about the FRP; [α]i is calculated as:
The second term on the right hand side of Eqn. 7.11 represents the dynamic response of the structure when subjected to the base excitation of translations and rotations. The dynamic response characteristics of the structure are represented by its fixed-base modes, frequencies ωi and mode shapes [Φ]. The modal participation factor matrix for a given mode r with mass normalized mode shape φr can be calculated as:
<Гr> = φrT [M] [α] (7.12)
Where <Гr> is the modal participation factors for translation and rotations (1 x 6), φrT is the transpose of the rth fixed-base mass normalized mode shape (1 x q), [M] is the structure mass matrix (q x q), and [α] is the rigid body transformation matrix of the structure about the FRP. The matrix [Г] is comprised of the modal participation factors for all fixed-base modes considered in the structure model.
The matrix [D(ω)] is a diagonal matrix comprised of the dynamic amplification factors for each of the fixed-base modes:
[ D ]=[ (ω /ωr )2
(1−ω2
ωr2 )+2iβ r (ω/ωr) ]
(7.13)
Where ωr is the frequency (rad/sec) and βr is the modal damping (fraction of critical) for mode r.
The solution process proceeds by solving for the foundation response U(ω) and then solving the dynamic response problem for the structure subjected to this foundation motion.
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SSI Parameters By The Hybrid Approach
The general formulation follows that of Luco et al., 1980 and Johnson et al., 2010 (PVP2010-25350).
Foundation Impedance Matrix
The foundation impedance matrix for the rigid, massless foundation, [Ks(ω)], is derived from the SASSI flexible impedance matrix as follows.
For each frequency of interest, assume the unconstrained SASSI flexible impedance matrix is generated. Figure 7-25 shows the foundation and free-field schematically.
NOTE by Boris: similar note, develop a list of pros and cons and reference related equations/methodology development!
The unconstrained SASSI flexible impedance matrix, [k], relates the displacements, u, occurring at the n degrees of freedom on the foundation to the forces, f, being applied to them. This relationship is only dependent on the properties of the soil and the geometry of the n degrees of freedom.
f = [k]u (7.14)
For a rigid foundation the impedance functions can be obtained by enforcing the constraint that all displacements, u, are related to each other by rigid-body motion.
u = [a]U (7.15)
where [a] is a rigid-body transformation matrix, which relates the displacements at each degree of freedom, u, to the six degrees of freedom, U, at the foundation reference point (FRP).
The resultant force vector at the FRP, F, is
F = [a]Tf (7.16)
= [a]T[k]u
= [a]T[k][a]U
Define [Ks(ω)] = [a]T[k][a] and
F(ω) = [Ks(ω)] U(ω) (7.17)
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[Ks(ω)] is the impedance matrix of the rigid, massless foundation.
Foundation Input Motions (FIMs)
NOTE by Boris: similar to the above, it would be good to develop a list of pros and cons and then reference method development and equations from papers and reports.
Let u' be the vector of the unconstrained foundation input motions (FIMs), defined at the n degrees of freedom of the foundation. The unconstrained FIMs are defined by the given free-field motion, U0, i.e., the control motion (amplitude and frequency content at the control point) and the spatial variation of motion over the depth and width of the embedded foundation. At this stage, this assumes the foundation node points are unconstrained by any foundation stiffness. The vector u' is the same as the free-field motion at the node points defining the foundation. If the amplitude of U0 is unity, then u' can be thought of as the unconstrained scattering vector, s'.
Let U* be the unknown FIMs at the FRP, assuming the foundation behaves rigidly. Let u* be the rigid FIMs transformed to the degrees of freedom of the flexible foundation.
u* = [a]U* (7.18)
where [a] is defined above.
The difference between the displacements of the unconstrained flexible foundation and the assumed rigid foundation is:
du = u’ - u* (7.19)
The force pattern required to produce du is df where
df = [k]du (7.20)
This force pattern must be self-equilibrating and therefore its resultant at the FRP dF must equal zero
dF = [a]T df= [a]T [k]u’ - u* = 0 (7.21)
Therefore:
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[a]T[k][a]U* = [a]T [k]u’
and
U* = [Ks(ω)]-1[a]T [k]u’ (7.22)
For a unit amplitude free-field motion at frequency ω, u' = s', and U*(ω) is the scattering vector, S(ω).
7.3.6 Discrete methods
(finite elements and finite difference)
7.3.7 Foundation input motion
(Reference to section 4.5 and 5.1)
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Figure 7-29 AREVA Finite Element Model of the EPR Nuclear Island (Johnson et al., 2010)
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7.4 SSI COMPUTATIONAL MODELS
7.4.1 Introduction
Soil structure interaction computational models are developed with a focus on three components of the problem:
Earthquake input motions, encompassing development of 1D or 2D or 3D motions, and their effective input in the SSI model,
Soil/rock adjacent to structural foundations, with important geological (deep) and site (shallow) conditions near structure, contact zone between foundations and the soil/rock, and
Structure, including structural foundations, embedded walls, and the superstructure
It is advisable to develop models that will provide enough detail and accuracy to be able to address all the important issues. For example, for modeling higher frequencies of earthquake motions, analyst needs to develop finite element mesh that will be capable to propagate those frequencies and to document influence of numerical/mesh induced dissipation/damping of frequencies. Modeling steps described in section 7.1 present good guidance for effective modelling of all three components above.
(Note, multi-step and single step models are introduced previously in Chapter 2 and Section 7.1 and should be referenced here.)
(NOTE: ALL EQUATION NUMBERS NEED TO BE REVISED WHEN TEXT IS FINALIZED)
[7.4.2] Soil/Rock Linear and Nonlinear Modelling
Effective and Total Stress Analysis
Soil and rock adjacent to structural foundations can be either dry or fully (or partially) saturated (Zienkiewicz et al., 1990; Lu and Likos, 2004).
Dry Soil:. In the case of dry soil, without pore fluid pressures, it is appropriate to use models that are only dependent on single phase stress, that is, a stress that is obtained from applying all the loads (static and/or dynamic) without any consideration of pore fluid pressures.
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Partially Saturated Soil:. For partially saturated soil, effective stress principle (see equation 7.4 below) must also the include influence of gas (air) present in pore of soils. There are a number of different methods to do that (Zienkiewicz et al., 1999; Lu and Likos, 2004), however computational frameworks that incorporate those methods are not yet well developed. Main approaches to modeling of soil behaviour within a partially saturated zone of soil (a zone where water rises due to capillary effects) are dependent on two main types of partial saturation
Voids of soil fully saturated with fluid mixed with air bubbles, water in pores is fully connected and can move and pressure in the mixture of water and air can propagate, with reduced bulk stiffness of water-air mixture.
This type of partial saturation can be modeled using fully saturated approaches, given in section 6.4.11 noted in section 7.4.11 below. It is noted that bulk modulus of fluid-air mixture is (much) lower than that of fluid alone, and to be tested for. Therefore, only methods that assume fluid to be compressible should be used (u − p − U, u − U, see section 6.4.11 for details). In addition, permeability will change from a case of just fluid seeping through the soil, and additional testing for permeability of water-air mixture is warranted. It is also noted, that since this partial saturation is usually found above water table, (capillary rise), hydrostatic pore pressure can be suction.
Voids of soil are full of air, with water covering thin contact zone between particles, creating water menisci, and contributing to the apparent cohesion of cohesionless soil material (think of wet sand at the beach, there is an apparent cohesion, until sand dries up).
This type of partial saturation can be modeled using dry (unsaturated) modeling, where elasticplastic material models used are extended to include additional cohesion that arises from thin water menisci connecting soil particles.
Saturated Soil. In the case of full saturate, effective stress principle (Terzaghi et al., 1996) has to be applied. This is essential as for porous material (soil, rock, and sometimes concrete) mechanical behaviour is controlled by the effective stresses. Effective stress is obtained from total stress acting on material (σij), with reductions due to the pore fluid pressure:
(7.4)
where is effective stress tensor, σij is total stress tensor, δij is Kronecker delta (a diagonal matrix with numbers 1 on a diagonal and numbers 0 on non-diagonal positions, that is δij = 1, when i=j, and δij = 0, when i 6= j), and p is the pore fluid pressure. We use
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standard mechanics of materials convention that tensile components of stress are positive, and so the pore fluid pressure p is negative when in compression (Zienkiewicz et al., 1999). All the mechanical behaviour of soils and rock is a function of the effective stress σ’ij, which is affected by a full coupling with the pore fluid, through a pore fluid pressure p.
A note on clays:. Clay particles (platelets) are so small that their interaction with water is quite different from silt, sand and gravel. Clays feature chemically bonded water layer that surrounds clay platelets. Such water does not move freely and stays connected to clay platelets under working loads.
Usually, clays are modeled as fully saturated soil material. In addition, clays feature very small permeability, so that, while the effective stress principle (from Equation 7.4) applies, pore fluid pressure does not change during fast (earthquake) loading. Hence clays should be analyzed using total stress analysis, where the initial total stress is a stress that is obtained from an effective stress calculation that takes into account hydrostatic pore fluid pressure. In other words, clays are modeled using undrained, total stress analysis, using effective stress (total stress reduced by the pore fluid pressure) for initializing total stress at the beginning of loading.
Drained and Undrained Modeling
Depending on the permeability of the soil, on relative rate of loading and seepage, and on boundary conditions (Atkinson, 1993), a decision needs to be made if analysis will be performed using drained or undrained behaviour. Permeability of soil (k) can range from k > 10−2m/s for gravel, 10−2m/s > k > 10−5m/s for sand, 10−5m/s > k > 10−8m/s for silt, to k < 10−8m/s for clay. If we assume a unit hydraulic gradient (reduction of pore fluid pressure/head of 1m over the seepage path length of 1m), then for a dynamic loading of 10 − 30 seconds (earthquake), and for a semi-permeable silt with k = 10−6m/s, water can travel few millimeters. However, pore fluid pressure will propagate (much) faster (further) and will affect mechanical behaviour of soil skeleton. This is due to high bulk modulus of water (Kw = 2.15×105 kN/m2), which results in high speed of pressure waves in saturated soils. Thus a simple rule is that for earthquake loading, for gravel, sand and permeable silt, relative rate of loading and seepage requires use of drained analysis. For clays, and impermeable silt, it might be appropriate to use undrained analysis for such short loading.
Drained Analysis: Drained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions allow for full movement of pore fluid and
pore fluid pressures during loading event. As noted above, use of the effective stress for the analysis is essential, as is modeling of full coupling of pore fluid pressure with the
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mechanical behaviour of soil skeleton. This is usually done using theory of mixtures (Green and Naghdi, 1965; Eringen and Ingram, 1965; Ingram and Eringen, 1967; Zienkiewicz and Shiomi, 1984; Zienkiewicz et al., 1999) and will be elaborated upon in some detail in section 6.4.11. During loading events, pore fluid pressures will dynamically change (pore fluid and pore fluid pressures will displace) and will affect the soil skeleton, through effective stress principle. All nonlinear (inelastic) material
modeling applies to the effective stresses ( ). Appropriate inelastic material models that are used for modeling of soil (as noted in section 6.4.2) should be used.
Undrained Analysis Undrained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions do not allow movement of pore fluid and pore fluid pressures during loading event. This is usually the case for clays and for low permeability silt. There are three main approaches to undrained analysis:
Total stress approach, where there is no generation of excess pore fluid pressure (pore fluid pressure in addition to the hydraulic pressure), and soil is practically impermeable (clays and low permeability silt). In this case hydrostatic pore fluid (water) pressures are calculated prior to analysis, and effective stress is established for the soil. This approach assumes no change in pore fluid pressure. This usually happens for clays and low permeability silt, and due to very low permeability of such soils, a total stress analysis is warranted, using initial stress that is calculate based on an effective stress principle and known hydrostatic pore fluid pressure. Since pore fluid pressure does not affect shear strength (Muir Wood, 1990), for very low permeability soils (impermeable for all practical purposes), it is convenient to perform elastic-plastic analysis using undrained shear strength (cu) within a total strain setup. Since only shear strength is used, and all the change in mean stress is taken by the pore fluid, material models using von Mises yield criteria can be used.
Locally undrained analysis where excess pore fluid pressure (change from hydrostatic pore pressure) can be created. Excess pore fluid pressures can be created, due to compression effects on low permeability soil (usually silt). On the other hand, pore fluid suction can also be created due to dilatancy effects within granular material (silt). Due to very low permeability, pore fluid and pore fluid pressure does not move during loading, and therefor pore fluid pressure increase or decrease at one location will not affect nearby locations. However, pore fluid pressure change (increase or decrease) will affect effective stress. Effective stress will change, and will affect constitutive behaviour of soil. Analysis is essentially undrained, however, pore fluid pressure can and will change locally due to compression or dilatancy effects in granular soil.Due to very low permeability,
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pore fluid and pore fluid pressure does not move during loading, and hence, effective stress will change, and will affect constitutive behaviour of soil. Analysis is essentially undrained, however, pore fluid pressure can and will change locally due to compression or dilatancy effects in granular soil. Appropriate inelastic (elastic-plastic) material models that are used for modeling of soil (as noted in section 6.4.2) should be used, while constitutive integration should take into account local undrained effects and convert any change in voids into excess pore fluid pressure change (excess pore pressure). This is still a single phase analysis, as all the pore fluid pressure changes are taken into account on local, constitutive level, and there is no two phase material (pore fluid and porous solid) where pore fluid pressure is able to propagate.
Very low permeability soils, that can, but to not have to develop excess pore fluid pressure do to constitutive level volume change of soil can also be analyzed as fully drained continuum, while using very low, realistic permeability. In this case, although analysis is officially drained analysis, results will be very similar if not the same as for undrained behaviour (one of two approaches above) due to use of very low, realistic permeability. Effective stress analysis is used, with explicit modeling of pore fluid pressure and a potential for pore fluid to displace and pore fluid pressure to move. However, due to very low permeability, and fast application of load (earthquake) no fluid will displace and no pore fluid pressure will propagate.
This approach can be used for both cases noted above (total stress approach and locally undrained approach). While this approach is actually explicitly allowing for modeling of pore fluid movement, results for pore fluid displacement should show no movement. In that sense, this approach is modeling more variables than needed, as some results are known before simulations (there will be no movement of water nor pore fluid pressure). However, this approach can be used to verify modeling using the first two undrained approaches, as it is more general.
It is noted that globally undrained problems, where for example soil is permeable, but boundary conditions prevent water from moving, should be treated as drained problems, while appropriate boundary conditions should prevent water from moving across impermeable boundaries.
Linear and Nonlinear Elastic Models
NOTE BORIS: adding this from chapter 5, will still have to work on it a bit to fit nicely here…
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Material modelmodelling for soil s for site response can be equivalent linear, nonlinear or elastic-plastic.
Equivalent linear models are in fact linear elastic models with adjusted elastic stiffness that represents (certain percentage of a ) secant stiffness of largest shear strain reached for given motions. Determining such linear elastic stiffness requires an iterative process (trial and error). This modeling approach is fairly simple, there is significant experience in professional practice and it works well for 1D analysis and for 1D states of stress and strain. Potential issues with this modeling approach are that it is not taking into account soil volume change (hence it favors total stress analysis, see details about total stress analysis in section 6.4), and it is useable for 3D analysis through deviatoric stress invariant.
Nonlinear material models are representing 1D stress strain (usually in shear, τ −γ) response using nonlinear functions. There are a number of nonlinear elastic models used, for example Ramberg- Osgood (Ueng and Chen, 1992), Hyperbolic (Kramer, 1996), and others. Calibrating modulus reduction and damping curves using nonlinear models is not that hard (Ueng and Chen, 1992), however there is less experience in professional practice with these types of models. Potential issues with this modeling approach are similar (same) as for equivalent linear modeling that it is not taking into account soil volume change as it is essentially based on a nonlinear elastic model,
Elastic-plastic material models are usually full 3D models that can be used for 1D or 3D analysis. A number of models are available (Prevost and Popescu, 1996; Mr´oz et al., 1979; Elgamal et al., 2002; Pisano` and Jeremi´c, 2014; Dafalias and Manzari, 2004). Use of full 3D material models, if properly calibrated, can work well in 1D as well as in 3D. Potential issues with these models is that calibration usually requires a number of in situ and laboratory tests. In addition, there is far less experience in professional practice with elastic-plastic modeling.
Linear and nonlinear elastic models are used for soil, rock and structural components. Linear elastic model that are used are usually isotropic, and are controlled by two constants, the Young’s modulus E and the Poisson’s ratio ν, or alternatively by the shear modulus G and the bulk modulus K.
Nonlinear elastic models are used mostly in soil mechanics, There are a number of models proposed over years, tend to produce initial stiffness of a soil for given confinement of over-consolidation ratio (OCR) (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988).
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Anisotropic material models are mostly used for modeling of usually anisotropic rock material (Amadei and Goodman, 1982; Amadei, 1983).
Equivalent Linear Elastic Models Equivalent elastic models are linear elastic models where the elastic constants were determined from nonlinear elastic models, for a fixed shear strain value. They are secant stiffness 1D models and usually give relationship between shear stress ( τ = σxz) and shear strain ( ). Determination of secant shear stiffness is done iteratively, by performing 1D wave propagation simulations, and recording average high estimated strain (65% of maximum strain) for each level/depth. Such representative shear strain is then used to determine reduction of stiffness using modulus reduction curves (G/Gmax and the analysis is re-run. Stable secant stiffness values are usually reached after few iterations, typically 5-8. It is important to emphasize that equivalent elastic modeling is still essentially linear elastic modeling, with changed stiffness. More details are available in sections 3.2 and 4.5.
Elastic-Plastic Models
Elastic plastic modeling can be used in 1D, 2D and full 3D. A number of material models have been developed over years for both monotonic and cyclic modeling of materials. Material models for soil (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and Whittle, 1995; Prevost and Popescu, 1996; Mroz and Norris, 1982), rock (Lade and Kim, 1995; Hoek et al., 2002; Vorobiev, 2008) have been developed over last many years.
It should be noted that 3D elastic plastic modeling is the most general approach to material modeling of soils and rock. If proper models are used (see section 6.2.1) it is possible to achieve modeling that is done using simplified modeling approaches described above (linear elastic, equivalent linear elastic, modulus reduction curves, etc.). However, calibration of models that can achieve such modeling sophistication requires expertise. The payoff is that important material response effects, that are usually neglected if simplified models are used, can be taken into account and properly modeled. As an example, soil volume change during shearing is a first order effects, however it is not taken into account if modulus reduction curves are used.
A Note on Constitutive Level and Global Level Equilibrium
There are two main types of numerical algorithms for constitutive integrations:
Explicit or Forward Euler, is an algorithm that produces tangent stiffness tensor on the constitutive level. This algorithm does not enforce equilibrium and error in constitutive integrations (drift from the yield surface) is accumulated. This
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algorithm is simpler and faster than the implicit algorithm (next item) and is implemented and used in most (all) computer programs.
Implicit or Backward Euler) is an algorithm that produces algorithmic (consistent) stiffness tensor (matrix) that can produce very fast convergence (quadratic for Newton scheme) on the global, finite element equilibrium iterations. This algorithm is iterative and does enforce equilibrium (within user specified tolerance). It is usually slower than the explicit algorithm (see above) and implementation can be quite complicated, particularly for elastic plastic material models for soil and concrete (Crisfield, 1987; Jeremicand Sture, 1997; Jeremi´c, 2001).
On the global, finite element level, there are two main types of solution advancement algorithmsways to advance the solutions:
Solution advancement without enforcing the equilibrium. In this case, solutions is produced using current tangent stiffness matrix (relying on the tangent stiffness tensor, developed on the constitutive level, as noted above). For each step of loading (static or dynamic) difference between applied loads and internal loads (stresses) is not checked for. This means that error in unbalanced forces is accumulating as computations progress. Usual remedy is to make steps small enough so that error is also reduced. However this reduction in step size (or time step size) can significantly increase computational times. In one specific instance, if lumped mass matrix is used, instead of a consistent mass matrix (which is theoretically more accurate), solution of a large system of equations can be completely circumvented. For particular explicit dynamic computations, only inverse of a diagonal mass matrix is required, which is trivial to obtain.
Solution advancement with enforcement of the equilibrium. In this case, equilibrium is explicitly checked for, and if unbalanced forces are not balanced within certain (user specified) tolerance, an iterative scheme is used until equilibrium is achieved (within tolerance). Alternative method for ending iterations (instead of achieving equilibrium) is to check for iterative displacements and place a low limit below which iterations are not worth while any more and therefore end them. The most commonly used iteration methods are based on Newton iterative scheme Crisfield (1984). This approach is computationally demanding, however it does benefit the solution as it yields (close to) equilibrium solutions. In addition, if consistent (algorithmic) stiffness is used on the constitutive level (see Implicit constitutive algorithm above), a fast convergence (sometimes even close to quadratic) is achieved.
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Concluding note for both constitutive and global level solution advancement is that simpler methods (explicit, no equilibrium check) will lead to accumulating error (unbalance stress and force) and will thus render solutions that are not in equilibrium and are possibly quite wrong. This can be remedied by reducing step size (time increment), however computational times are then becoming long On the other hand, methods that enforce equilibrium (within tolerance) are (much) more complicated to develop, implement and execute, yet they enforce equilibrium (again within tolerance).
7.4.2[7.4.3] Structural models, linear and nonlinear: shells, plates, walls, beams, trusses, solids
Linear and nonlinear structural models are not used as much in the industry for modeling and simulation of behaviour of nuclear power plants (NPP). One of the main reasons is that NPPs are required to remain, effectively elastic during earthquakes (design basis earthquake ground motion). Nevertheless modeling of nonlinear effects in structures remains a viable proposition. It is important to note that inelastic behaviour of structural components (trusses, beams, walls, plates, shells) features a localization of deformation (Rudnicki and Rice, 1975). While localization of deformation is also present in soils, soils are more ductile medium (unless they are very dense) and so inelastic treatment of deformation in soils, with possible localization, is more benign than treatment of localization of deformation in brittle concrete. Significant work has been done in modeling of nonlinear effects in mass concrete and concrete beams, plates, walls and shells (Feenstra, 1993; Feenstra and de Borst, 1995; de Borst and Feenstra, 1990; de Borst, 1987, 1986; de Borst, 1987; de Borst et al., 1993; Bi´cani´c et al., 1993; Kang and Willam, 1996; Rizzi et al., 1996; Menetrey and Willam, 1995; Carol and Willam, 1997; Willam, 1989; Willam and Warnke, 1974; Etse and Willam, 1993; Scott et al., 2004, 2008; Spacone et al., 1996a,b; Scott and Fenves, 2006).
The main issue is still that concrete structural elements still develop plastic hinges (localized deformation zones). Finite element results with localized deformation are known to be mesh dependent (change of mesh will change the result), and as such are hard to verify. Recent work on rectifying this problem (Larsson and Runesson, 1993) shows promises, however these methods are still not widely accepted. One possible, rather successful solution relies on classical developments of Cosserat continua (Cosserat, 1909), where results looked very promising (Dietsche and Willam, 1992), however sophistication required by such analysis and lack of programs makes this approach still very exotic.
7.4.3[7.4.4] Contact Modeling
In all soil-structure systems, there exist interfaces between structural foundations and the soil or rock beneath. There are two main modes of behaviour of these interfaces, contacts:
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Normal contact where foundation and the soil/rock beneath interact in a normal stress mode. This mode of interaction comprises normal compressive stress, however it can also comprise gap opening, as it is assumed that contact zone has zero tensile strength.
Shear contact where foundation and the soil/rock beneath can develop frictional slip.
Contact description provided here is based on recent work by Jeremic (2016) and Jeremic et al. (1989-2016).
Modeling of contact is done using contact finite elements. Simplest contact elements are based on a two node elements, the so called joint elements which were initially developed for modeling of rock joints. Typically normal and tangential stiffness were used to model the pressure and friction at the interface (Wriggers, 2002; Haraldsson and Wriggers, 2000; Desai and Siriwardane, 1984).
The study of two dimensional and axisymmetric benchmark examples have been done by Olukoko et al. (1993) for linear elastic and isotropic contact problems. Study was done considering Coulomb’s law for frictional behaviour at the interface. In many cases the interaction of soil and structure is involved with frictional sliding of the contact surfaces, separation, and re-closure of the surfaces. These cases depend on the loading procedure and frictional parameters. Wriggers (2002) discussed how frictional contact is important for structural foundations under loading, pile foundations, soil anchors, and retaining walls.
Two-dimensional frictional polynomial to segment contact elements are developed by Haraldsson and Wriggers (2000) based on non-associated frictional law and elastic-plastic tangential slip decomposition. Several benchmarks are presented by Konter (2005) in order to verify the the results of the finite element analyses performed on 2D and 3D modelings. In all proposed benchmarks the results were approximated pretty well with a 2D or an axisymmetric solutions. In addition, 3D analyses were performed and the results were compared with the 2D solutions.
Contact Modeling Formulation
The formulation for contact is represented by a discretization which establishes constraint equations and contact interface constitutive equations on a purely nodal basis. Such a formulation is called node-tonode contact (Wriggers, 2002). The variables adopted to formulate the model are shown in Figure 7-30: the force (F) and displacement vectors (u).
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Figure 7-30 Forces and relative displacements of the element.
Each vector is composed of three terms: the first one acts along the longitudinal direction (nlocal) whereas the other two components lie on the orthogonal plane (mlocal and llocal). The total relative displacement additively decomposed into elastic and plastic components (6.6).
F = [p ; t]T ; u = [v; gs]T (7.5)
el h
el eliT
pl h
pl pliT
el pl (7.6)u = v ; gs ; u = v ; gs ; u = u + u
Elastic Bbehaviour:. The elastic behaviour (contact and no slip) is defined by the relation:
KN(p) 0 0
el dF = E · du ; E = 0 KT
0 0 0 KT (7.7)
The normal displacement-normal force relationship, valid for loading and unloading conditions, that can be either
Constant contact stiffness (penalty stiffness, hard contact). One simple rule of thumb in choosing this stiffness is to prescribe contact penalty stiffness K = 1000EA/h, where E is a stiffness modulus of one of the materials adjacent to contact zone (hence there is a possibility that this stiffness might be coming from soil (relatively low stiffness) or from concrete (relatively large stiffness), A is a tributary area for that contact element, and h is a thickness of a contact zone (usually a small number, on the order of few centimeters). It is important to note that above recommended penalty stiffness can vary orders of magnitude, and that
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numerical experiments need to be performed in order to test contact element performance with chosen stiffness, or
Nonlinear function (soft contact). Functional relationship that works well for concrete – soil contact is:
,if ux < 0(6.8)
,if ux < 0
where kn is a constant and ux is a relative displacement of two contact nodes. Force–displacement equation given in equation 6.8 is a parabola that has a non-zero tangent at ux = 0. The value of stiffness 0.0001kn is chosen as stiffness at 1/10 of millimeter (0.0001m) of penetration.
The tangential stiffness KT is assumed to be constant. Alternative contact stiffness functions were proposed by Gens et al. (1988, 1990) for a more stiff contact between two rock (or concrete) surfaces.
Plastic model: The plastic model is defined in terms of yield surface and plastic potential surface, as shown in Figure 7.2. Yield surface is a frictional cone (with friction coefficient µ = tanφ), with no cohesion. Plastic potential, that defines plastic flow directions, preserves volume, that is, there is no dilation of compression due to plastic slip.
Figure 7-31 Yield surface fs, plastic potential Gs and incremental plastic displacement δup.
Geometry description
Figure 7-32 shows geometry of the two node contact element and its main deformations modes. It is important to note the importance of properly numbering element nodes, consistently with the definition of normal x1. Node I is the first node, node J is the second
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node and normal goes from node I toward node J. If reversed, elements behaves like a hook.
Definition of local axes
Nodes in contact Movement in normal direction
Movement in tangent
direction
Figure 7-32 Description of contact geometry and displacement responses.
7.4.4[7.4.5] Structures with a base isolation system
Base isolation system are used to change dynamic characteristics of seismic motions that excite structure and also to dissipate seismic energy before it excites structure. Therefore there are two main types of devices:
Base Isolators (Kelly, 1991a,b; Toopchi-Nezhad et al., 2008; Huang et al., 2010; Vassiliou et al., 2013) are usually made of low damping (energy dissipation) elastomers and are primarily meant to change (reduce) frequencies of motions that are transferred to the structural system. They are not designed nor modeled as energy dissipators.
NOTE for Alain: what I meant is that base isolators will modify frequency of motions that to into the structure, I changed above item to reflect that.However if it is still not clear, I can add more explanation...
Base Dissipators Kelly and Hodder (1982); Fadi and Constantinou (2010); Kumar et al. (2014) are developed to dissipate seismic energy before it excites the structure. There two main types of such dissipators:
– Elastomers made of high dissipation rubber, and
– Frictional pendulum dissipators
Both isolators and dissipators are usually developed to work in two horizontal dimensions, while motions in vertical direction are not isolated or dissipated. This can create potential problems, and need to be carefully modeled.
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J
I
J
I
J
I
contact plane
lz
ly
lxJ
I
z
y
x
systemcoordinateGlobal
Modeling of base isolation and dissipation system is done using two node finite elements of relatively short length.
Base Isolation Systems are modeled using linear or nonlinear elastic elements. Stiffness is provided from either tests on a full sized base isolators, or from material characterization of rubber (and steel plates if used in a sandwich isolator construction). Depending on rubber used, a number of models can be used to develop stiffness of the device (Ogden, 1984; Simo and Miehe, 1992; Simo and Pister, 1984).
Particularly important is to properly account for vertical stiffness as vertical motions can be amplified depending on characteristics of seismic motions, structure and stiffness of the isolators Hijikata et al. (2012); Araki et al. (2009). It is also important to note that assumption of small deformation is used in most cases. In other words, stability of the isolator, for example overturning or rolling is not modeled. It is assumed that elastic stiffness will not suddenly change if isolator becomes unstable (rolls or overturns).
Base Dissipator Systems are modeled using inelastic (nonlinear) two node elements. There are three basic types of dissipator models used:
High damping rubber dissipators Rubber dissipators with lead core Frictional pendulum (double or triple) dissipators
Each one is calibrated using tests done on a full dissipator. It is important to be able to take into account influence of (changing) ambient temperature and increase in temperatures due to energy dissipation (friction) on resulting behaviour. Ambient temperature can have significant variation, depending on geographic location of installed devices and such variation will affects base dissipator system response. In addition, energy dissipation results in heating of devices, and resulting increase in temperature will influences base dissipators response as well.
7.4.5[7.4.6] Foundation models
Foundation modeling can be done using variable level of sophistication. Earlier models assumed rigid foundation slabs. This was dictated by the use of modeling methods that rely on analytic solution, which in turn have to rely on simplifying assumptions in order to be solved. For example, soil and rock beneath and adjacent to foundations was usually assumed to be an elastic half space.
Foundation response plays an important role in overall soil-structure interaction (SSI) response. Major energy dissipation happens in soil and contact zone beneath the foundation. Buoyant forces (pressures) act on foundation if water table is above bottom of the lowest foundation level.
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Shallow and Embedded Slab Foundations Foundation slabs and walls can be are modelled as flexible or rigid. Their thickness can range from 3 − 5 meters, however, their horizontal but they extentd can be for up to 100 meters. Containment, and shield and auxiliary buildings are rigidly connected to foundation slabs, and will significantly contribute to stiffness of slab/building system stiffen it up. In addition, auxiliary buildings, will also stiffen up foundation slabs. However, even with all these stiffening effects, slabs and walls should be modeled using flexible models Depending on relative stiffness of building/slab systems they can be modelled as rigid systems, however influence if such (simplified) modelling needs to be analysed. If possible, buildings (containment, shield and auxiliary) and slab systems should be modelled using realistic stiffness of all components. (plates and walls). Using realistic stiffness models, influence of rigid or no rigid assumptions on results can be analysed. Moreover, if inelastic (nonlinear) response of concrete plates and walls is expected (for example for beyond design basis seismic motions) use of realistic stiffness, nonlinear models is recommended.
Flexible modeling of foundation slabs is best done using either shell elements (plate bending and in plane wall) or solids.
For shell element models, it is important to bridge over half slab or wall thickness to the adjacent soil. This is important as shell elements are geometrically representing plane in the middle of a solid (slab or wall) with a finite thickness, so connection over half thickness of the slab or wall is needed. It is best if shell elements with drilling degrees of freedom are used (Ibrahimbegović´c et al., 1990; Militello and Felippa, 1991; Alvin et al., 1992; Felippa and Militello, 1992; Felippa and Alexander, 1992), as they properly take into account all degrees of freedom (three translations, two bending rotations and a drilling rotation).
For solid element models, it is important to use proper number of solids so that they represent properly bending stiffness. For example, a single layer of regular 8 node bricks will over-predict bending stiffness over 2 times (200%). Hence at least 4 layers of 8 node bricks are needed for proper bending stiffness. If 27 node brick elements are used, a single layer is predicting bending stiffness within 4% of analytic solution.
Piles and Shaft Foundations For nuclear facilities built on problematic soils, piles and shafts are usually used for foundation system. Piles can carry loads at the bottom end, and in addition to that, can also carry loads by skin friction. Shafts usually carry majority of load at the bottom end.
Piles (including pile groups) and shafts have been modeled using three main approaches:
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Analytic approach (Sanchez-Salinero and Roesset, 1982; Sanchez-Salinero et al., 1983; Myionakis and Gazetas, 1999; Law and Lam, 2001; Sastry and Meyerhof, 1999; Abedzadeh and Pak, 2004; Mei Hsiung et al., 2006; Sun, 1994) where a main assumptions is that of a linear elastic behaviour of a pile and the soil represented by a half space, while contact zone is fully connected, and no slip or gap is allowed. More recently there are some analytic solution where mild nonlinear assumptions are introduced (Mei Hsiung, 2003), however models are still far from realistic behaviour. These approaches are very valuable for small vibrations of piles, pile groups and shafts as well as for all modes of deformation where elasto-plasticity will be very mild if it exists at all.
P-Y and T-Z approach where experimentally measured response of piles in lateral direction (P-Y) and vertical direction (T-Z) is used to construct nonlinear springs that are then used to replace soil (Stevens and Audibert, 1979; Brown et al., 1988; Bransby, 1996, 1999; Tower Wang and Reese, 1998; Reese et al., 2000; Georgiadis, 1983). This approach is very popular with practicing engineers. However this approach does make some simplifying assumptions that make its use questionable for use with cases where calibrations of P-Y and T-Z curves do not exist. For example, for layered soils, of for piles in pile groups, this approach can produce problematic results. Moreover, in dynamic applications, dynamics of soils surrounding piles and piles groups is poorly approximated using springs.
Nonlinear finite element models in 3D have been developed recently for treatment of piles, pile groups and shafts, in both dry and liquefiable soils (Brown and Shie, 1990, 1991; Yang and Jeremic, 2003; Yang and Jeremi´c, 2005a,b; McGann et al., 2011; S.S.Rajashree and T.G.Sitharam, 2001; Wakai et al., 1999). In these models, elastic-plastic behaviour of soil is taken into account, as well as inelastic contact zone within pile-soil interface. Layered soils are easily modeled, while proper modeling of contact (see section 6.4.4) resolves both horizontal and vertical shear (slip) behaviour. Moreover, with proper modeling, effects of piles in liquefiable soils can be evaluated as well (Cheng and Jeremi´c, 2009).
Deeply Embedded Foundations In case of Small Modular Reactors, foundations are deeply embedded, and the foundation walls, in addition to the base slab, contribute significantly more to supporting structure for static and dynamic loads. Main issues are related to proper modeling of contact (see more in sections 76.4.4 and 6.4.7), as well inelastic behaviour of soil adjacent to the slab and walls. Of particular importance for deeply embedded foundations is proper modeling of buoyant stresses (forces) as it is likely that ground water table will be above base slab.
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Foundation Flexibility and Base Isolator/Dissipator Systems. There are special cases of foundations where base isolators and dissipators are used. In this case there are two layers of foundations slabs, one at the bottom, in contact with soil and one above isolators/dissipators, beneath the actual structure. Those two base slabs are connected with dissipators/isolators. It is important to properly (accurately) model stiffness of both slabs as their relative stiffness will control how effective will isolators and dissipators be during earthquakes.
[7.4.7] Deeply Embedded StructuresSmall Modular Reactors (SMRs)
Deeply Embedded Structures (DES) such as Small Modular Reactors (SMR) (example describing analysis of an SMR is given in the appendix) requires special considerations. For modeling of DES, it is important to note extensive contact zone of deeply embedded DES walls and base slab with surrounding soil. This brings forward a number of modeling and simulation issues for DESs. listed below, In addition, noted are suggested modeling approaches for each listed issue.
Seismic Motions: Seismic motions will be quite variable along the depth and in horizontal direction. This variability of motions is a results of mechanics of inclined seismic wave propagation, inherent variability (incoherence) and the interaction of body waves (SH, SV and P) with the surface, where surface waves are developed. Surface waves do extend somewhat into depth (about two wave lengths at most (Aki and Richards, 2002)). This will result in different seismic motion wave lengths (frequencies, depending on soil/rock stiffness), propagating in a different way at the surface and at depth of a DES. As a results, a DES will experience very different motions at the surface, at the base and in between.
Due to a number of complex issues related to seismic motions variability, as noted above, it is recommended that a full wave fields be developed and applied to SSI models of DESs.
– In the case of 1D wave propagation modeling, vertically propagating shear waves are to be developed (deconvolution and/or convolution) and applied to SSI models.
– For 3D wave fields, there are two main options:
∗ use of incoherence functions to develop 3D seismic wave fields. This option has a limitation as incoherent functions in the vertical direction are not well developed.
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∗ develop a full 3D seismic wave field from a wave propagation modeling using for example SW4. This option requires knowledge of local geology and may require modeling on a regional scale, encompassing causative faults, while another option is to perform stress testing using a series of sources/faults (Abell et al., 2015).
Nonlinear/Inelastic Contact: Large contact zone of a DES concrete walls and foundation slab, with surrounding soil, with its nonlinear/inelastic behaviour will have significant effect on dynamic response of a DES.
Use of appropriate contact model that can model frictional contact as well as possible gap opening and closing (most likely in the near surface region) is recommended. In the case of presence of water table above DES foundation base, effective stresses approach needs to be used, as well as modeling of (possibly dynamically changing) buoyant forces, as described in section 6.4.8 and also below.
Buoyant Forces: With deep embedment, and (a possible) presence of underground water (water table that is within depth of embedment), water pressure on walls of DES will create buoyant forces. During earthquake shaking, those forces will change dynamically, with possibility of cyclic mobility and liquefaction, even for dense soil, due to water pumping during shaking (Allmond and Kutter, 2014).
Modeling of buoyant forces can be done using two approaches, namely static and dynamic buoyant force modeling, as described in section 6.4.8.
Nonlinear/Inelastic Soil Behaviour: With deep embedment, dynamic behaviour an DES is significantly influenced by the nonlinear/inelastic behaviour of soil adjacent to adjacent SMR walls and foundation slab.
Use of appropriate inelastic (elastic-plastic) 3D soil models is recommended. Of particular importance is proper modeling of soil behaviour in 3D as well as proper modeling of volume change due to shearing (dilatancy). One dimensional equivalent elastic models, used for 1D wave propagation are not recommended for use, as they do not model properly 3D effects and lack modeling of volume change.
Uncertainty in Motions and Material: Due to large contact area and significant embedment, significant uncertainty and variability (incoherence) in seismic motions will be present. Moreover, uncertainties in properties of soil material surrounding DES will add to uncertainty of the response.
Uncertainty in seismic motions and material behaviour can be modeled using two approaches, as described in section 7.5. One approach is to rely on varying input motions and material parameters using Monte Carlo approach, and its variants (Latin Hypercube, etc.). This approach is very computationally demanding and not too accurate. Second
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approach is to use analytic stochastic solutions for components or the full problem. For example, stochastic finite element method, with extension to stochastic elasto-plasticity with random loading. More details are given in section 7.5.
Figure 7.4 illustrates modeling issues on a simple, generic DES (an SMR) finite element model (vertical cut through middle of a full model is shown). Illustrative example using this SMR is provided in the appendix.
FIG. 7.4: Four main issues for realistic modeling of Earthquake Soil Structure Interaction of SMRs: variable weave field at depth and surface, inelastic behaviour of contact and adjacent soil, dynamic buoyant forces, and uncertain seismic motions and material.
It is important to develop models with enough fidelity to address above issues. It is possible that some of the issues noted above will not be as important to influence results in any significant way, however the only way to determine importance (influence) of above phenomena on seismic response of an SMR is through modeling.
A modelling and simulation example for an SMR is presented in the appendix and is used to illustrate challenges and modelling and simualion methodology.
Add a link to SMR section in chapter 8 and make a bit of intro here….
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Jim: introduce why this has special characteristics and why does in need special treatment than other currently described…
[7.4.8] Buoyancy Modeling
For NPPs (and related) structures for which lowest foundation level is below the water table, there exist a buoyant pressure/forces on foundation base and all walls, creating symmetric buoyant forces. In addition, for some NPP related structures, like intake structures, there exist different pore fluid pressures in the soil on diferent sides, creating a nonsymmetric buoyant forces. For static loads, symmetric buoyant force B can be calculated using Archimedes principle: ”Any object immersed in water is buoyed up by a force equal to the weight of water displaced by the object”, B = ρwgV where ρw = 999.972 kg/m3 (for salty sea/ocean water values of density are higher ρw = 1020.0 − −1029.0 kg/m3) is the mass density of water (at temperature of +4oC with small changes of less than 1% up to +40oC), g = 9.81 m/s2 is the gravitational acceleration, and V is the volume of displaced fluid (volume of foundation under water table). Buoyant force can be applied as a single force or a small number of resultant forces directed upward around the stiff center of foundation. This is strictly applicable if foundation is rigid, but it can probably work in most cases for static loads. For nonsymmetric buoyant forces, static forces on all (pore flujid pressures) walls and base slab have to be taken into account separately.
During dynamic loading, buoyant force (buoyant pressures) can dynamically change, as a results of a dynamic change of pore fluid pressures in soil adjacent to the foundation concrete. This is particularly true for soils that are dense, where shearing will lead increase of inter-granular void space (dilatancy), and reduction in buoyant pressures or for soils that are loose, where shearing will lead to reduction of inter-granular void space (compression), and increase in buoyant pressures.
For strong shaking, it also expected that gaps will open between soil and foundation walls and even foundation slab. This will lead to pore water being sucked into the opening gap and pumped back into soil when gap closes. This ”``pumping” of water will lead to large, dynamic changes of buoyant pressures.
Different dynamic scenarios, described above, create conditions for dynamic, nonlinear changes in buoyant force.
Dynamic Buoyant Stress/Force Modeling. Fully coupled finite elements (u-p or u-p-U or u-U, as described in section 76.4.11) are used for modeling saturated soil adjacent to foundation walls and base. Modeling of contact between soil and the foundation concrete needs to take into account effects of pore fluid pressure – buoyant stress within the contact zone, on order to properly model normal stress for frictional contact. This modeling can be done using:
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coupled contact elements, that explicitly model water displacements and pressures and allows for explicit gap opening, filling of gap with water, slipping (frictional) when the gap is closed, and pumping of water as gap opens and closes. This contact element takes into account the pore water pressure information from saturated soil finite elements, as well as the information about the displacement (movement) of pore water within a gap. It is based on a dry version of the contact element and incorporates effective contact (normal) stress, based in the effective stress principle (see equation 6.4). Potential for modeling of pumping of water during gap opening and closing is important as it might influence dynamic response. Noted pumping of water, can also lead to liquefying of even dense soil, in the zone where gap opening happens (Allmond and Kutter, 2014).
coupled finite elements in contact with impermeable concrete finite elements (solids or shells/plates). In this case, impermeable concrete finite elements create a natural barrier for water flow, thus allowing generation of excess pore fluid pressure in the contact zone during dynamic shaking. This approach precludes formation of gaps and pumping of water.
7.4.6[7.4.9] Domain Boundaries
JIM: motivation, intake structure is not embedded on all sides, therefor it receives pressures from sides that are embedded only, so this is a structure which will be important to model using this approach… Introduce a complexity of the issue, set a context (intake structure it is beleiverd to be a conservative approach however do a sensitivity study and show if it is indeed or not (conservative).
One of the biggest problems in dynamic ESSI in infinite media is related to the modeling of domain boundaries. Because of limited computational resources the computational domain needs to be kept small enough so that it can be analyzed in a reasonable amount of time. By limiting the domain however an artificial boundary is introduced. As an accurate representation of the soil-structure system this boundary has to absorb all outgoing waves and reflect no waves back into the computational domain.
The most commonly used types of domain boundaries are presented in the following:
• Fixed or free
By fixing all degrees of freedom on the domain boundaries any radiation of energy away from the structure is made impossible. Waves are fully reflected and resonance frequencies can appear that don’t exist in reality. The same happens if the degrees of freedom on a boundary are left ’free’, as at the surface of the soil.
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Combination of free and fully fixed boundaries should be chosen only if the entire model is large enough and if material damping of the soil prevents reflected waves to propagate back to the structure.
For cases where compressional and/or shear waves travel very fast, in this case boundaries have to be placed very far, thus significantly increasing the size of models.
• Absorbing Lysmer Boundaries
A possible Way solution to eliminate waves propagating outward from the structure is to use Lysmer boundaries. This method is relatively easy to implement in a finite element code as it consists of simply connecting dash pots to all degrees of freedom of the boundary nodes and fixing them on the other end (Figure 7-33).
Lysmer boundaries are derived for an elastic wave propagation problem in a one-dimensional semi-infinite bar. It can be shown that in this case a dash pot specified appropriately has the same dynamic properties as the bar extending to infinity (Wolf, 1988). The damping coefficient C of the dash pot equals
C = A cρ (7.9)
where A is the section of the bar, ρ is the mass density and c the wave velocity that has to be selected according to the type of wave that has to be absorbed (shear wave velocity cs or compressional wave velocity cp).
Figure 7-33 Absorbing boundary consisting of dash pots connected to each degree of freedom of a boundary node
In a 3d or 2d model the angle of incidence of a wave reaching a boundary can vary from almost 0 up to nearly 180. The Lysmer boundary is able to absorb completely only those under an angle of incidence of 90. Even with this type of absorbing boundary a large number of reflected waves are still present in the
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domain. By increasing the size of the computational domain the angles of incidence on the boundary can be brought closer to 90 and the amount of energy reflected can be reduced.
• Infinite elements can also be used (Zienkiewicz and Taylor, 1991) however their use does not guaranty full absorbtion of outgoing waves.
More sophisticated boundaries modeling wave propagation toward infinity (boundary elements). However, use of boundary elements for outside FEM models destgroys sparsity of the resulting stiffness matrix a full matrix, and thust places high computational burden on FEM solvers.
For a spherical cavity involving only waves propagating in radial direction a closed form solution for radiation toward infinity, analogous to the Lysmer boundary for wave propagation in a prismatic rod, exists (Sections 3.1.2 and 3.1.3 in Wolf (1988)). Since this solution, in contrast to the Lysmer boundary, includes radiation damping it can be thought of as an efficient way of eliminating reflections on a semi-spherical boundary surrounding the computational domain.
More generality in terms of absorption properties and geometry of the boundary are provided by the various boundary element methods (BEM) available in the literature.
[7.4.10] Seismic Load Input
MOVE this up and combine with section 7.3 which has different methods for seismic onut (SASSSI, CLASSI and so on…)
A number of methods are used to input seismic motions into finite element model. Most of them are based on simple intuitive approaches, and as such are not based on rational mechanics. Most of those currently still widely used methods cannot properly model all three components of body waves as well as always present surface waves. There exist a method that is based on rational mechanics and can model both body and surface seismic waves input into finite element models with high accuracy. That method is called the Domain Reduction Method (DRM) and was developed fairly recently Bielak et al. (2003); Yoshimura et al. (2003)). It is a modular, two-step dynamic procedure aimed at reducing the large computational domain to a more manageable size. The method was developed with earthquake ground motions in mind, with the main idea to replace the force couples at the fault with their counterpart acting on a continuous surface surrounding local feature of interest. The local feature can be any geologic or manmade object that constitutes a difference from the simplified large domain for which displacements and accelerations are easier to obtain. The DRM is applicable to a much wider range of problems. It is essentially a variant of global–local set of methods and as
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formulated can be used for any problems where the local feature can be bounded by a continuous surface (that can be closed or not). The local feature in general can represent a soil–foundation–structure system (bridge, building, dam, tunnel...), or it can be a crack in large domain, or some other type of inhomogeneity that is fairly small compared to the size of domain where it is found.
The Domain Reduction Method
A large physical domain is to be analyzed for dynamic behaviour. The source of disturbance is a known time history of a force field Pe(t). That source of loading is far away from a local feature which is dynamically excited by Pe(t) (see Figure 7-34).
The system can be quite large, for example earthquake hypocenter can be many kilometers away from the local feature of interest. Similarly, the small local feature in a machine part can be many centimeters away from the source of dynamic loading which influences this local feature. In this sense the term large domain is relative to the size of the local feature and the distance to the dynamic forcing source.
It would be beneficial not to analyze the complete system, as we are only interested in the behaviour of the local feature and its immediate surrounding, and can almost neglect the domain outside of some relatively close boundaries. In order to do this, we need to somehow transfer the loading from the source
Figure 7-34 Large physical domain with the source of load Pe(t) and the local feature (in this case a soil-structure system.
to the immediate vicinity of the local feature. For example we can try to reduce the size of the domain to a much smaller model bounded by surface Γ as shown in Figure 6.6. In doing so we must ensure that the dynamic forces Pe(t) are appropriately propagated to the much smaller model boundaries Γ.
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Seismic source
Ω
Ω+
Local feature
Large scale domain
Γ
eu
buiu
)t(eP
It can be shown (Bielak et al., 2003) that the consistent dynamic replacement for the dynamic source forces Pe is a so called effective force, Peff:
(6.10)
eff
Pi 0
Peff = Pbeff = − MbeΩ+u¨e0 − KbeΩ+u0e (6.10)
Peeff MebΩ+u¨0b + KebΩ+u0b
where MbeΩ+ and Meb
Ω+ are off-diagonal components of a mass matrix, connecting boundary (b) and external (e) nodes, Kbe
Ω+ and KebΩ+ are off-diagonal component of a
stiffness matrix, connecting boundary (b) and external (e) nodes, u¨0e and u¨0
b are free field accelerations of external (e) and boundary (b) nodes, respectively and, and u0
b are free field displacements of external (e) nodes, and boundary (b) nodes, respectively and. The effective force Peff consistently replace forces from the seismic source with a set of forces in a single layer of finite elements surrounding the SSI model. The DRM is quite powerful and has a number of features that makes an excellent choice for SSI modeling:
Single Layer of Elements used for Peff. Effective nodal forces Peff involve only the submatrices Mbe, Kbe, Meb, Keb. These matrices vanish everywhere except the single layer of finite elements in domain Ω+ adjacent to Γ. The significance of this is that the only wave-field (displacements and accelerations) needed to determine effective forces Peff is that obtained from the simplified (auxiliary) problem at the nodes that lie on and between boundaries Γ and Γe
Only residual waves outgoing. Solution to the DRM problem produces accurate seismic displacements inside and on the DRM boundary. On the other hand, the solution for the domain outside the DRM layer represents only the residual displacement field. This residual displacement field is measured relative to the reference free field displacements. Residual wave field has low energy when compared to the full seismic wave field, as it is a results of oscillations of the structure only. It is thus fairly easy to be damped out. This means that DRM can very accurately model radiation damping.
This is significant for two more reasons:
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Large models can be reduced in size to encompass just a few layers of elements outside DRM boundary,
Residual unknown field can be monitored and analyzed for information about the dynamic characteristics of the soil structure system
Inside of DRM boundary can be nonlinear/inelastic. This is a very important conclusion, based on a fact that only change of variables was employed in DRM development, and solution does not rely on superposition.
All types of realistic seismic waves are modeled. Since the effective forcing Peff consistently replaces the effects of the seismic source, all appropriate (real) seismic waves are properly (analytically) modeled, including body (SV, SH, P) and surface (Rayleigh, Love, etc...) waves.
A Note on Free Field Input Motions for DRM. Seismic motions (free field) that are used for input into a DRM model need to be consistent. In other words, a free field seismic wave that is used needs to fully satisfy equations of motion. For example, if free field motions are developed using a tool (SHAKE, or EDT or SW4, or fk, &c.) using time step ∆t = 0.01s and then you decide that you want to run your analysis with a time step of ∆t = 0.001s, simple interpolation (10 additional steps for each of the original steps) might create problems. Simple linear interpolation actually might (will) not satisfy wave propagation equations and if used will introduce additional, high frequency motions into the model. It is a very good idea to generate free field motions with the same time step as it will be used in ESSI simulation.
Similar problem might occur if spacial interpolation is done, that is if location of free field model nodes is not very close to the actual DRM nodes used in ESSI model. Spatial interpolation problems are actually a bit less acute, however one still has to pay attention and test the ESSI model for free conditions and only then add the structure(s) on top.
7.4.7[7.4.11] Liquefaction and Cyclic Mobility Modeling
JIM will send examples of liquefaction modelling so that I can motivate this a bit more. Needs to be rewritten to reflect motivation and perhaps shorten already short description…
Introduction
Saturated soils should be analyzed using information about pore fluid pressure. Modeling of saturated soils, where pore fluid is able to move during loading (see note on effective and total stress analysis conditions in section 76.4.2) is best done using an effective stress approach.
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Effective stress analysis, where stress in soil skeleton and pore fluid pressures are treated separately is appropriate for both drained and undrained (low permeability) cases. This approach is necessary if permeability of soil is such that one can expect movement of pore fluid and pore fluid pressures during dynamic event duration. For example, saturated sandy soils, will feature a fully coupled behaviour of pore fluid and soil skeleton. It is expected that pore fluid and pore fluid pressure will move and, through the effective stress principle, such pore fluid pressure change will affect (significantly) response of soil skeleton. There are three main approaches to modeling fully coupled pore fluid – solid skeleton systems (Zienkiewicz and Shiomi, 1984):
u − p − U, where the main unknowns are displacements of porous solid skeleton (ui), pore fluid pressures (p), and displacements of pore fluid (Ui),
u−U where the main unknowns are displacements of porous solid skeleton (ui), and displacements of pore fluid (Ui),
u−p where the main unknowns are displacements of porous solid skeleton (ui), pore fluid pressures (p).
Zienkiewicz and Shiomi (1984) notes pros and cons of each approach, and concludes that the u−p−U is the most flexible and accurate for dynamic events where there is a significant rate of change in pore fluid pressures and pore fluid and solid skeleton displacements, as is a case for soil structure interaction problems. In the case of u − U formulation, stumbling block is the high bulk stiffness of the fluid, which can create numerical problems. Those numerical problems are elegantly resolved in the u−p−U formulation through the inclusion of pore fluid pressure into the unknown field (although pore fluid pressures and pore fluid displacements are dependent variables). The u − p formulation is the simplest one, but has issues in treating problems with high rate of change of pore fluid pressure in space and time.
Liquefaction Modeling Details and Discussion
Modeling of liquefaction and its effects requires availability of computer programs that model behaviour of fully saturated, fully coupled pore fluid – porous solid materials (soils). Chapter 8 list some of the available programs. Modeling and simulation of liquefaction requires significant test data for soil. Both laboratory and in situ test data is required. Recent publications (Peng et al., 2004; Elgamal et al., 2002, 2009; Arduino and Macari, 1996, 2001; Jeremic et al., 2008; Shahir et al., 2012; Taiebat et al., 2010b,a; Cheng and Jeremic, 2009; Taiebat et al., 2010a) describe various possibilities in modeling of cyclic mobility and liquefaction effects. It is important to properly verify and validate modeling tools for coupled (liquefaction) modeling, as multi–physics modeling of these problems can be difficult and results interpretation requires full confidence in simulation programs (Tasiopoulou et al., 2015a,b).
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7.4.8[7.4.12] Structure-Soil-Structure Interaction
(in phase, out phase, distance between structures relative to the surface wave length, etc.)
Soil-Structure-Soil-Structure Interaction (SSSSI) sometimes needs to be taken into account, as it might change levels of seismic excitation for adjacent NPPs. There are a number of approaches to model SSSSI.
• Direct Models. The simplest and most accurate approach is to develop a direct model all (two or more) structures on subsurface soil and rock, develop input seismic motions and analyze results. While this approach is the most involved, it is also the most accurate, as it allows for proper modeling of all the structure, foundation and soil/rock geometries and material without making any unnecessary simplifying assumptions.
The main issue to be addressed with this approach is development of seismic motions to be used for input. Possible approach to developing seismic motions is to use incoherent motions with appropriate separation distance. Alternatively, regional seismic wave modeling can be used to develop realistic seismic motions and use those as input through, for example the Domain
• Reduction Method (see section 76.4.10).
• Symmetry and Anti-Symmetry Models. These models are sometimes used in order to reduce complexity and sophistication of the direct model (see recent paper by Roy et al. (2013) for example). However, there are a number of concerns regarding simplifying assumptions that need to be made in order for these models to work. These models have to make an assumption of a vertically propagating shear waves and as such do not take into account input surface waves (Rayleigh, Love, etc). Surface waves will additionally excite NPP for rocking and twisting motions, which will then be transferred to adjacent NPP by means of additional, induced surface waves. If only vertically propagating waves are used for input (as is the case for symmetry and antisymmetry models) energy of input surface waves is neglected. It is noted that depending on the surface wave length and the distance between adjacent structures, a simple analysis can be performed to determine if particular surface waves, emitted/radiated from one structure toward the other one (and in the opposite direction) can influence adjacent structures. It is noted that the wave length can be determined using a classical equation λ = v/f where λ is the length of the (surface) wave, v is the wave speed8 and f is the wave frequency of interest. Table6.1 below gives Rayleigh wave lengths for four
8 For Rayleigh surface waves, a wave speed is just slightly below the shear wave speed (within 10%, depending on
elastic properties of material), so a shear wave speed can be used for making Rayleigh wave length estimates.173
different wave frequencies (1,5,10,20 Hz and for three different Rayleigh (very close to shear) wave velocities (300,1000,2500 m/s):
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TABLE 7-6 RAYLEIGH WAVE LENGTH AS A FUNCTION OF WAVE SPEED [M/S] AND WAVE FREQUENCY [HZ].
1.0Hz 5.0Hz
10.0Hz
20Hz
300m/s 300m 60m 30m 15m
1000m/s
1000m
200m 100m 50m
2500m/s
2500m
500m
250m 125m
It is apparent that for given separation between NPP buildings, different surface wave (frequencies) will be differently transmitted with different effects. For example, for an NPP building that has a basic linear dimension (length along the main rocking direction) of 100m, the low frequencies surface wave (1Hz) in soft soil (vs ≈ 300m/s) will be able to encompass a complete building within a single wave length, while for the same soil stiffness, the high frequency (20Hz) will produce waves that are too short to efficiently propagate through such NPP structure. On the other hand, for higher rock stiffness (vs ≈ 2500m/s), waves with frequencies up to approximately 5Hz, can easily affect a building with 100m dimension.
Further comments on are in order for making symmetric and antisymmetric assumptions (that is modeling a single building with one boundary having symmetric or antisymmetric boundary condition so as to represent a duplicate model, on the other side of such boundary):
Symmetry: motions of two NPPs are out phase and this is only achievable, if the wave length of surface wave created by one NPP (radiating toward the other NPP) is so large that half wave length encompasses both NPPs. This type of motions (symmetry) is illustrated in Figure 7-35 below
Figure 7-35 Symmetric mode of deformation for two NPPs near each other.
Antisymmetry: motions of two NPPs are in phase. This is achievable if distance between two NPPs is perfectly matching wave lengths of the radiated wave from
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one NPP toward the other one, and if the dimension of NPPs is not affecting radiated waves. This type of SSSSI is illustrated in Figure 7-36 below
Figure 7-36 Anti-symmetric mode of deformation for two NPPs near each other.
Both symmetry and antisymmetry assumptions place very special requirements on wave lengths that are transmitted/radiated and as such do not model general waves (various frequencies) that can be affecting adjacent NPPs.
7.4.9[7.4.13] Simplified models
Simplified Models
Simplified models are used for fast prototyping and for parametric studies, as they have relatively low computational demand. It is very important to note that a significant expertise is required from analyst in order to develop appropriate modeling simplifications that retain mechanical behaviour of interests, while simplifying out model components that are not important (for particular analysis).
Simplified, Discrete Soil and Structural Models
Using discrete models can provide useful results for some aspects of mechanical behaviour while requiring relatively small computational effort. Discrete, simplified models can be used for analyzing many aspects of SSI behaviour of NPPs, provided that simplifications are made in a consistent way and that thus developed models can be verified for required modeling purposes.
P-Y and T-Z Springs. Simplified models using P-Y and T-Z springs (Bransby, 1999; Allen, 1985; Georgiadis, 1983; Stevens and Audibert, 1979; Bransby, 1996) have been used to model response of piles, pile groups and other types of foundations in elastic-plastic soil. The ideas is based on Winkler foundation springs that are now defined as nonlinear springs. Radial (transversal) behaviour is modeled using P-Y springs, while axial (longitudinal) behaviour is modeled using T-Z spings. It is important to note that calibration of P-Y and T-Z springs is done in full scale tests, for a given pile or foundation stiffness and for a given soil type. In that sense, P-Y and T-Z springs can be understood as post-processing (recording) of the actual response, that is then transferred to nonlinear springs behaviour, through a combination of nonlinear springs, dashpots,
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sliders and other components that are combined in order to mimic recorded P-Y and T-Z behaviour.
While P-Y and T-Z springs have been successfully used for modeling monotonic loading of single piles in uniform soils (sand and clay), their use for loading in different directions, in layered soils and for dynamics applications cannot be fully verified. Particularly problematic is P-Y behaviour in layered soils and for large pile groups Yang and Jeremic (2002, 2003); Yang and Jeremic (2005a).
Simplified, Continuum Soil Models
While modeling of SSI is best done using continuum models, there are number of simplifications that can be done as well.
Linear Elastic. One of the most commonly made simplification is to use linear elastic models for modeling of soil. While this might be appropriate for small seismic excitations, it is likely that models with more significant seismic load levels, where plastification will have larger effects, cannot be validated and that these models will miss significant aspects of behaviour that are important for understanding response.
Stiffness Reduction (G/Gmax) and Damping Curve Models. Plastification (not significant) can be taken into account using modulus reduction and damping curves. However, as noted in section 4.5, such results have to be carefully used as this material modeling simplification introduces a number of artifacts. For example amplification factors for 1D models using this approximations are known to be biased (Rathje and Kottke, 2008).
7.5 PROBABILISTIC MATERIAL MODELING
NOTE: This needs to be moved (at least in part to Chapter \3 where Alain talks about material modeling…
This part is also what has been mentioned/described
[7.5.1] Introduction
Uncertainty is present in all phases of modeling and simulation of an earthquake soil structure interaction problem for nuclear facilities. There are two main sources of uncertainty:
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• Modeling Uncertainty is introduced when simplifying modelling assumptions are made. In general, only simplifying modelling assumptions that do not introduce significant uncertainties, and inaccuracies in results should be used. However, that is not always the case. Significance of the influence of modeling uncertainties on results can only be assessed if higher sophistication models can be simulated and results compared with lower sophistication models (where modeling uncertainties were introduced).
• Parametric uncertainty is introduced when there are uncertainties in
Material modeling parameters,
and
Loads (earthquakes).
Uncertainty in material modelling parameters are a result of measuring errors and transformation errors (Phoon and Kulhawy, 1999a,b; Fenton, 1999; Baecher and Christian, 2003), as well as spatial variability that contributes to uncertainties as averaging of material volume has to be done in order to determine material properties.
Parametric (material modeling) uncertainties can be significant. For example Figure 6.9 shows experimental data for a relationship between the Standard Penetration Test (SPT) and elastic modulus (Ohya et al., 1982). In addition, shown is a mean trend, as well as a histogram of a scatter with respect to the mean. It is important to note that significant uncertainty will be introduced in results if a mean relationship is used alone. For example, if one uses the above relationship, and has a soil with a blow-count of 10, elastic stiffness will be (according to the mean equation/relationship) approximately 8,000 kPa, while test data shows that values can be as low as 1,000 kPa and as high as 33,000 kPa. Hence, assuming deterministic material property (elastic stiffness in this case), analyst will not model (probable) very soft or very stiff material response. In other words, parametric uncertainty has to be taken into account in order to provide full information about (probabilities of) response.
There are a number of approaches to take probabilistic response into account.
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5 10 15 20 25 30 35
SPT N Value Residual (w.r.t Mean) Young’s Modulus (kPa)
Figure 7-37 (a) Left: Transformation relationship between Standard Penetration Test (SPT) N-value and pressure-meter Young’s modulus, E; (b) Right: Histogram of the residual with respect to the deterministic transformation equation for Young’s modulus, along with fitted probability density function (PDF). From Sett et al. (2011b).
7.6 PROBABILISTIC RESPONSE ANALYSIS
7.6.1 Overview
In general, probabilistic response analysis is compatible with the definition of a performance goal for design and input requirements for assessments, such as seismic margin assessment (SMA) and seismic probabilistic risk analysis (SPRA). Section 1.3 discusses these needs in the context of performance goals defined probabilistically.
For design, one definition is to calculate seismic demand on structures, systems, and components (SSCs) as 80% non-exceedance probability (NEP) values conditional on the design basis earthquake ground motion (DBE), as specified in ASCE 4. Deterministic response analysis approaches specified in ASCE 4 are developed to approximate the 80% NEP response level based on sensitivity studies and judgment. The preferred approach is to perform probabilistic response analysis and generate the 80% NEP directly.
For BDBE assessments, SPRA or SMA methods are implemented.
• For SPRA assessments, the full probability distribution of seismic demand conditional on the ground motion (UHRS or GMRS) is required. The median values (50% NEP) and estimates of the aleatory and epistemic uncertainties (or estimates of composite uncertainty) at the appropriate risk important frequency of occurrence are needed. The preferred method of development is by probabilistic response analyses as described in succeeding sections. An alternative is to rely on generic studies that have been performed to generate approximate values.
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Young’s Modulus, E (kPa)
• Two approaches to SMA assessments are used, i.e., the conservative deterministic failure margin (CDFM) method and the fragility analysis (FA) method. For the CDFM, the seismic demand is defined as the 80% NEP conditional on the Review Level Earthquake (RLE). The procedures of ASCE 4 are acceptable to do so. For the FA method, the full probability distribution is required. In both cases, the 80% NEP values are needed to apply the screening tables of EPRI NP-6041 (1991), which provide screening values for high capacity SSCs. Most applications of SMA use the CDFM method.
Soil-structure interaction includes the two most significant sources of uncertainty in the overall seismic response analysis process, i.e., the definition of the ground motion and characterization of the in situ soil geometry and its material behaviour. Throughout this document, aleatory and epistemic uncertainties are identified in the site response and SSI models and analyses. Implementing probabilistic response analysis permits the analysts to explicitly take into account some of these uncertainties.
Commentary. In addition to the above described applications of probabilistic analysis methods, to have a rational consistent approach to applying risk-informed techniques to decision-making, requires that the seismic demand on SSCs be quantified probabilistically so that the industry (Licensee and Regulator) knows the likelihood of exceedance of a given loading condition (excitation) applied to individual SSCs and that the likelihood of exceedance when combined with the seismic design and fragility parameters leads to a balanced design.
In today’s world, the framework requires that each SSC be individually designed to meet the code, with nothing in the way of cognizance of how that SSC fits into the overall seismic risk profile, beyond the overall classification of an SSC as either “safety-related” or not. Finally, because the design codes have been developed by different code committees (ASME, ASCE, IEEE, ANS, ACI, etc.) at different times using different ideas, there is little in the way of consistency in the margins that these design codes introduce at or above the design basis, and little in the way of “working together” to produce a rational performance-based design.
7.6.2 Simulations of the SSI Phenomena
Generally, some type of simulation is performed to calculate probabilistic responses of SSCs. The seismic analysis and design methodology chain is comprised of: (i) ground motion definition (amplitude, frequency content, primary earthquakes contributing to its definition - deaggregation of the seismic hazard, incoherence, location of the motion, site response analysis); (ii) SSI phenomena; (iii) soil properties (stratigraphy, material properties – low strain and strain-dependent; (iv) structure dynamic characteristics (all
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aspects); and (v) equipment, components, distribution systems dynamic behaviour (all aspects).
An important aspect of the elements of the seismic response process is that all elements are subject to uncertainties.
SMACS
Probabilistic methods to analyze or reanalyze SSCs to develop seismic demands for input to SPRA evaluations were pioneered in the late 1970s in the U.S. NRC Seismic Safety Margin Research Program (SSMRP). A family of computer programs called the Seismic Methodology Analysis Chain with Statistics (SMACS) was developed and implemented in the SPRA of the Zion nuclear power plant in the U.S. (Johnson et al., 1981). This method is often called the “SSMRP” method. The method is still used in the development of seismic response of SSCs of interest to the SPRA so-called Seismic Equipment List (SEL) items (Nakaki et al., 2010).
The SMACS methodology is based on analyzing NPP SSCs for simulations of earthquakes defined by acceleration time histories at appropriate locations within the NPP site. Modeling, analysis procedures, and parameter values are treated as best estimate with uncertainty explicitly introduced. For each simulation, a new set of soil, structure, and subsystem properties are selected and analyzed to account for variability in the dynamic properties of the soil/structure/subsystems.
For the SSI portion of the seismic analysis chain, the substructure approach is used and the basis for the SMACS SSI analysis is the CLASSI (Continuum Linear Analysis for Soil Structure Interaction) family of computer programs (Luco and Wong, 1980). CLASSI implements the substructure approach to SSI (see Section 7.3).
The basic steps of the SMACS methodology are:
• Seismic Input • The seismic input is defined by sets of earthquake ground motion
acceleration time histories at a location in the soil profile defined in the PSHA or the site response analyses (Chapter 5). Three spatial components of motion, two horizontal and the vertical, comprise a set.
• The number of earthquake simulations to be performed for the probabilistic analyses is defined. Thirty simulations generally provide a good representation of the mean and variability of the input motion and the structure response. Although recent studies have suggested that a greater number of simulations improves the definition of the probability distributions of the response quantities of interest.
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• Often, it is advantageous to use the deaggregated seismic hazard as the basis for the definition of the ground motion acceleration time histories as discussed in Chapter 5.
• Development of the median soil/rock properties and variability.
• Chapter 3 describes site investigations to develop the low strain soil profiles – stratigraphy and other soil properties. Chapter 3, also, describes laboratory testing of soil samples the results of which form the basis for defining the strain-dependent soil material properties. For the idealized case (semi-infinite horizontal layers overlying a half-space), Chapter 5 describes site response procedures to develop an ensemble of probabilistically defined equivalent linear viscoelastic soil profiles – individual profiles from the analyses and their derived median-centered values and their variability. These viscoelastic properties are defined by equivalent linear shear moduli and material damping - and Poisson’s ratios.
• The end result of the process is used to define the best estimate and variability of the soil profiles to be sampled in the SMACS analyses.
• Structure model development.
The structure models are general finite element models. The input to the SMACS SSI analyses is the model geometry, mass matrix, and fixed-base eigen-system. Adequate detail is included in the model to permit the generation of structure seismic forces for structure fragility evaluation and in-structure response spectra (ISRS) for fragility evaluation of supported equipment and subsystems.
The structure model is developed to be median-centered for a reference excitation corresponding to an important seismic hazard level for the SPRA or SMA. Stiffness properties are adjusted to account for anticipated cracking in reinforced concrete elements and modal damping is selected compatible with this anticipated stress level.
• SSI parameters for SMACS:
• Foundation input motion. The term “foundation input motion” refers to the result of kinematic interaction of the foundation with the free-field ground motion. The foundation input motion differs from the free-field ground motion in all cases, except for surface foundations subjected to vertically incident waves. The motions
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differ for two reasons. First, the free-field motion varies with soil depth. Second, the soil-foundation interface scatters waves because points on the foundation are constrained to move according to its geometry and stiffness. The foundation input motion is related to the free-field ground motion by means of a transformation defined by a scattering matrix.
• Foundation impedances. Foundation impedances describe the force-displacement characteristics of the soil/rock. They depend on the soil/rock configuration and material behaviour, the frequency of the excitation, and the geometry of the foundation. In general, for a linear elastic or viscoelastic material and a uniform or horizontally stratified soil deposit, each element of the impedance matrix is complex-valued and frequency dependent. For a rigid foundation, the impedance matrix is 6 x 6, which relates a resultant set of forces and moments to the six rigid body degrees-of-freedom.
• The standard CLASSI methodology is based on continuum mechanics principles and is most applicable to structural systems supported by surface foundations assumed to behave rigidly. Hybrid approaches exist to calculate the scattering matrices and impedances for embedded foundations with the computational efficiency of CLASSI. Median-centered foundation impedances and scattering functions are calculated using the embedment geometry and the median soil profile.
• Statistical sampling and Latin Hypercube experimental design.
Input to this step in the SMACS analyses are the number of simulations (N), the variability of soil/rock stiffness and damping, and the variability of the structure’s dynamic behaviour (structure frequencies and damping). As presented above, the median-centered properties of the soil and structure are derived in initial pre-SMACS activities. The probability distributions of the soil stiffness, soil material damping, structure frequency, and structure damping are represented by scale factors with median values of 1.0 and associated coefficients of variation. Stratified sampling is used to sample each of the probability distributions for the defined parameters. Latin Hypercube experimental design is used to create the combinations of samples for the simulations. These sets of N combinations of parameters when coupled with the ground motion time histories provide the complete probabilistic input to the SMACS analyses.
• SMACS analyses.
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Using the free-field time histories, median-centered structural model, median-centered SSI parameters, and experimental design obtained above, perform N SSI analyses. For each simulation, time histories of seismic responses are calculated for quantities of interest: (i) internal forces, moments, and stresses for structure elements from which peak values are derived; (ii) acceleration time histories at in-structure locations, which provide input for subsystem analyses, i.e., equipment, components, distribution systems, in the form of time histories, peak values, and in-structure response spectra (ISRS).
On completion of the SMACS analysis of N simulations, the N values of a quantity of interest, e.g., peak values of an internal force, ISRS at subsystem support locations, are processed to derive their median value, i.e., 50% non-exceedance probability (NEP), and other NEP values, e.g., the 84% NEP. This then permits fitting probability distributions to these responses for combination with probability distributions of capacity thereby creating families of fragility functions for risk quantification. The specific requirements of the SPRA, guide the identification of these probabilistically-defined in-structure response quantities based on structure and equipment fragility needs.
• Example results
Nakaki et al. (2010) present the SMACS probabilistic seismic response analyses of selected Muhleberg Nuclear Power Plant structures for the purposes of the SPRA.
• For the Muhleberg site, the ensemble of X-direction seismic input motions displayed as response spectra (5% damping) is presented in Figure 7-38. The input motion variability is apparent.
• For the Muhleberg NPP Reactor Building, a representative ensemble of ISRS for the X-direction at Elevation 8m (approximately 20m above the basmat) is shown in Figure 6.5-2.
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Figure 7-38 Mühleberg Nuclear Power Plant – Ensemble of free-field ground motion response spectra – X-direction, 5% damping
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Figure 7-39 Mühleberg Nuclear Power Plant – Ensemble of in-structure response spectra (ISRS) – X-direction, 5% damping – median and 84% NEP
Monte Carlo
Monte Carlo approach is used to estimate probabilistic site response, when both input motions (rock motions at the bottom) and the material properties are uncertain. For a simplified approach, using equivalent linear (EqL) approach (strain compatible soil properties with (viscous) damping, a large number of combinations (statistically significant) of equivalent linear (elastic) stiffness for each soil layer are analyzed in a deterministic way. In addition, input loading can also be developed into large number (statistically significant) rock motions. Large number of result surface motions, spectra, etc. can then be used to develop statistics (mean, mode, variance, sensitivity, etc.) of site response. Methodology is fairly simple as it utilizes already existing EqL site response modeling, repeated large number of times. There lies a problem, actually, as for proper (stable) statistics, a very large number of simulations need to be performed, which makes this method very computationally intensive.
While Monte Carlo method can sometimes be applied to a 1D EqL site response analysis, any use for 2D or 3D analysis (even linear elastic) creates an insurmountable number of (now more involved, not 1D any more) of simulations that cannot be performed in reasonable time even on large national supercomputers. Problems becomes even more
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overwhelming if instead of linear elastic (equivalent linear) material models, elastic-plastic models are used, as they feature more independent (or somewhat dependent) material parameters that need to be varied using Monte Carlo approach.
Random Vibration Theory
Random Vibration Theory (RVT) is used for evaluating probabilistic site response. Instead of performing a (statistically significant) large number of deterministic simulations of site response (all still in 1D), as described above, RVT approach can be used Rathje and Kottke (2008). RVT uses Fourier Amplitude Spectrum (FAS) of rock motions to develop FAS of surface motions. Developed FAS of surface motions can them be used to develop peak ground acceleration and spectral acceleration at the surface. However, time histories cannot be developed, as phase angles are missing.
Stochastic Finite Element Method
Instead of using Monte Carlo repetitive computations (with high computational cost), uncertainties in material parameters (left hand side (LHS) and the loads (right hand side, RHS), can be directly taken into account using stochastic finite element method (SFEM). There are a number of different approaches to SFEM computations. Perturbation approach was popular in early formulations of SFEM (Kleiber and Hien, 1992; Der Kiureghian and Ke, 1988; Mellah et al., 2000; Gutierrez and De Borst, 1999), while spectral stochastic finite element method (SSFEM) proved to be more versatile (Ghanem and Spanos, 1991; Keese and Matthies, 2002; Xiu and Karniadakis, 2003; Debusschere et al., 2003). Review of various issues with different SFEM approaches were recently provided by Matthies et al. (1997); Stefanou (2009); Deb et al. (2001); Babuska and Chatzipantelidis (2002); Ghanem (1999); Soize and Ghanem (2009). Most previous approaches pertained to linear elastic uncertain material. Recently, (Sett et al., 2011a) developed Stochastic Elastic Plastic Finite Element Method (SEPFEM), that can be used for modeling of seismic wave propagation through inelastic (elastic-plastic) stochastic material (soil).
Both SFEM and the SEPFEM rely on discretization of finite element equations in both stochastic and spatial dimensions. In particular, (a) Karhunen–Lo`eve (KL) expansion (Karhunen, 1947; Lo`eve, 1948; Ghanem and Spanos, 1991) is used to discretize the input material properties random field into a number of independent basic random variables, (b) Polynomial chaos (PC) expansion (Wiener, 1938; Ghanem and Spanos, 1991) is used to discretize degrees of freedom into stochastic space(s), and (c) (classical) shape functions (Zienkiewicz and Taylor, 2000; Bathe, 1996b; Ghanem and Spanos, 1991) are used to discretize the spatial components of displacements into the (above) polynomial chaos expansion.
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In addition to the above, SEPFEM relies on Probabilistic Elasto Plasticity Jeremic et al. (2007); Sett et al. (2007); Jeremic and Sett (2009); Sett and Jeremic (2010); Sett et al. (2011b,a); Karapiperis et al. (2016) to solve for probabilistic elastic-plastic problem and supply probability distributions of stress and stiffness to finite element computations.
Both SFEM and SEPFEM provide very accurate results in terms of full probability density functions (PDFs) of main unknowns (Degrees of Freedom, DoFs) and stress (forces). Of particular importance is the very accurate calculation of full PDF, which supplies accurate tails of PDF, so that Cumulative Distribution Functions (CDFs, or fragilities) can be accurately obtained. However, while SFEM and SEPFEM are (can be) extremely powerful, and can provide very useful, full probabilistic results (generalized displacements, stress/forces), it requires significant expertise form the analyst. In addition, significant site characterization data is needed in order for uncertain (stochastic) characterization of material properties.
If such data is not available, one can of source resort to (non-site specific) data available in literature (Baecher and Christian, 2003; Phoon and Kulhawy, 1999a,b). However, use of non-site specific data significantly increases uncertainties (tails of material properties distributions become very`` ”thick”) as data is now obtained from a number of different, non-local sites, and is averaged, a process which usually increases variability.
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1.5 REFERENCES
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[2] INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA), “Earthquake Preparedness and Response for Nuclear Power Plants,” Safety Reports Series No. 66, Vienna, 2011.
[3] INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA), “Review of Seismic Evaluation Methodologies for Nuclear Power Plants Based on a Benchmark Exercise,” IAEA-TECDOC-1722, Vienna, 2013.
[4] American Society of Civil Engineers (ASCE), “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” Standard ASCE 4-16, 2016. (currently in publication)
[5] American Society of Civil Engineers (ASCE), “Seismic Design Criteria for Structures, Systems and Components in Nuclear Facilities,” ASCE/SEI 43-05 (Revision in progress).
2.8 REFERENCES
[1] Johnson, J.J., “Soil-Structure Interaction,” Chapter 10, Earthquake Engineering Handbook, CRC Press, 2002.
[2] INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA), “Seismic Hazards in Site Evaluation for Nuclear Installations,” Specific Safety Guide SSG-9, Vienna, 2010.
[3] American Society of Civil Engineers (ASCE), “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” Standard ASCE 4-16, 2016. (currently in publication)
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[4] Electric Power Research Institute, “Seismic Evaluation Guidance: Screening, Prioritization and Implementation Details (SPID) for the Resolution of Fukushima Near-Term Task Force Recommendation 2.1: Seismic,” EPRI 1025287, Electric Power Research Institute, Palo Alto, CA, 2012.
[5] American Society of Mechanical Engineers/American Nuclear Society (ASME/ANS), “Standard for Level 1/Large Early Release Frequency Probabilistic Risk Assessment for Nuclear Power Plant Applications,” so-called ”Addendum B,” Standard ASME/ANS RA-Sb-2012, 2012.
3.8 REFERENCES
[1] IAEA (2003). Site Evaluation for Nuclear Installation. Safety Standard Series n° NS-R -3, Vienna, 28 pages.
[2] IAEA (2010). Seismic Hazard in Site Evaluation for Nuclear Installations. Specific Safety Guide SSG-9.
[3] Woods, R.D. (1978). Measurement of Dynamic Soil Properties, Proc. ASCE Specialty Conference on Earthquake Engineering and Soil Dynamics, vol. 1, Pasadena, CA, 91–178.
[4] Vucetic, M., Dobry, R. (1991). Effect of soil plasticity on cyclic response, Journal of Geotechnical Engineering, ASCE, Vol. 117, n°1, 89-107.
[5] Seed, H.B., Idriss, I.M. (1970). Soil moduli and damping factors for dynamic response analysis, Report EERC 70-10. Earthquake Engineering Research Center.
[6] Dormieux, L., Canou, J. (1990). Determination of Dynamic Characteristics of a Soil based on Cyclic Pressumeter Test, 3rd Int. Symposium on Pressumeters, Oxford, 159-168.
[7] Ishihara K. (1996). Soil Behaviour in Earthquake Geotechnics. Oxford Engineering Science Series, Oxford University Press, UK, 342pp
[8] Masing, G. (1926). “Eigenspannungen und Verfestigung beim Messing”, Proc. 2nd International Congress of Applied Mechanics, Zurich, pp. 332-335.
[9] Konder, R. L. (1963). Hyperbolic stress-strain response: cohesive soils. Journal of the Soil Mechanics and Foundations Division (ASCE). Vol. 89, No. SM1, pp. 115-143.
[10] Hardin, B.O., Drnevich, V. P. (1972). Shear modulus and damping in soils: design equations and curves. Journal of the Soil Mechanics and Foundations Division (ASCE). Vol. 98, No. SM7, pp. 667 to 691.
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[11] Ramberg, W., Osgood, W.R. (1943). “Description of stress-strain curves by three parameters”, Technical note 902, National Advisory Committee for Aeronautics, Washington D.C.
[12] Iwan, W.D. (1967). On a class of models for the yielding behavior of continuous composite systems. Journal of Applied Mechanics, 34(E3), 612-617.
[13] Pyke, R.M. (1979). Nonlinear soil models for irregular cyclic loadings, J. Geotech. Eng., ASCE, 105(GT6), 715-726.
[14] Vucetic, M. (1990). Normalized behavior of clay under irregular cyclic loading, Canadian Geotech. J., 27, 29-46.
[15] Stewart, J.P., Hashash, Y.M.A., Matasovic, N., Pyke, R., Wang, Z., Yang, Z. (2008). Benchmarking of Nonlinear Geotechnical Ground Response Analysis Procedures, PEER report 2008/04, Berkeley, CA.
[16] Prevost, J.H. (1978). Plasticity theory for soil stress-strain behavior, J. Eng. Mech., 104 (EM5), 1177-1194.
[17] Prevost, J.H. (1985). A simple plasticity theory for frictional cohesionless soils, Soil Dyn. Earthquake Eng., 4(1), 9-17.
[18] Wang, Z.L., Y.F. Dafalias, C.K. Shen (1990). Bounding surface hypoplasticity model for sand, J. Eng. Mech., ASCE, 116 (5), 983-1001.
[19] Mroz, Z. (1967). On the Description of Antisotropic Workhardening, J. Mech. Phys. Solids, 15, 163-175.
[20] Dafalias, Y.F. (1986). Bounding Surface Plasticity: Mathematical Foundation and Hypo-plasticity, J.Eng. Mech., ASCE, 112(9), 966-987.
[21] Nakamura Y., (1989). A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface, Quarterly Report Railway Tech. Res. Inst., 30-1, 25-33, Tokyo, Japan.
[22] NERA: Network of European Research Infrastructures for Earthquake Risk Assessment and Mitigation (2014). Deliverable D11.5, Code cross-check, computed models and list of available results. http://www.orfeus-eu.org/organization/projects/NERA/Deliverables/
[23] Pitilakis K., Riga E., Anastasiadis A., Makra K. (2015). New elastic spectra, site amplification factors and aggravation factors for complex subsurface geometry towards
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the improvement of EC8; 6th International Conference on Earthquake Geotechnical Engineering, Christchurch, New Zealand
[24] SESAME (2004). European research project WP12, Guidelines for the implementation of the H/V spectral ratio technique on ambient vibrations –measurements, processing and interpretation– Deliverable D23.12. European Commission – Research General Directorate Project No. EVG1-CT-2000-00026 SESAME.
[25] Renault P. (2009) PEGASOS / PRP Overview. Joint ICTP/IAEA Advanced Workshop on Earthquake Engineering for Nuclear Facilities. Abdus Salam International Center for Theoretical Physics, Trieste, Italy.
[26] ISSMFE Technical committee #10 (1994). Geophysical characterization of sites, R.D. Woods edt., XIII ICSMFE, New Delhi, India.
[27] Pecker A. (2007). Determination of soil characteristics, in Advanced Earthquake Engineering Analysis, Chapter 2, Edt A. Pecker, CISM N°494, Springer.
[28] Foti S., Lai C.G., Rix G.J., Strobbia C. (2014). Surface Wave Methods for Near-Surface Site Characterization. CRC Press, Boca Raton, Florida (USA), 487 pp.
[29] Woods R.D. (1998). Measurement of dynamic soil properties, Proceedings of the Conference on Earthquake Engineering and Soil Dynamics, ASCE, Pasadena.
[30] Pecker A., Faccioli E., Gurpinar A., Martin C., Renault P. (2016). Overview & Lessons Learned from a Probabilistic Seismic Hazard Assessment for France and Italy. SIGMA project, to be published.
[31] Prevost J.H., Popescu R. (1996). Constitutive relations for soil materials, Electronic Journal of Geotechnical Engineering-ASCE, 1-39.
[32] ASCE/SEI 41-13 (2014). Seismic Evaluation and Retrofit of Existing Buildings.
[33] ASN/2/01 (2006). Prise en compte du risque sismique à la conception des ouvrages de génie civil d'installations nucléaires de base à 1'exception des stockages à long terme des déchets radioactifs.
[34] American Society of Civil Engineers (ASCE), “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” Standard ASCE 4-98, 1998. (ASCE 4-16 currently in publication)
[35] Garofalo F., Foti S., Hollender F., Bard P.Y., Cornou C., Cox B.R., Dechamp A., Ohrnberger M., Perron V., Sicilia D., Teague D., Vergniault C. (2016), InterPACIFIC
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project: comparison of invasive and non-invasive methods for seismic site characterization, Part II: inter-comparison between surface-wave and borehole methods, Soil Dyn. and Earthq. Eng., 82, 241-254.
[36] Assimaki D., Pecker A., Popescu R., Prevost J.H. (2003). Effects of Spatial Variability of Soil Properties on Surface Ground Motion, Journ. Earthq. Eng. Vol7, special issue 1, 1-44.
[37] Popescu R. (1995). Stochastic variability of soil properties: data analysis, digital simulation, effects on system behavior, PhD thesis, Princeton University, Princeton, New Jersey.
5.7 REFERENCES
B. T. Aagaard, R. W. Graves, A. Rodgers, T. M. Brocher, R. W. Simpson, D. Dreger, N. A. Petersson, S. C. Larsen, S. Ma, and R. C. Jachens. Ground-motion modeling of hayward fault scenario earth- quakes, part II: Simulation of long-period and broadband ground motions. Bulletin of the Seismological Society of America, 100(6):2945–2977, December 2010.
C. Larsen, S. Ma, and R. C. Jachens. Ground-motion modeling of hayward fault scenario earth- quakes, part II: Simulation of long-period and broadband ground motions. Bulletin of the Seismological Society of America, 100(6):2945–2977, December 2010.
J. A. Abell, N. Orbovi´c, D. B. McCallen, and B. Jeremi´c. Earthquake soil structure interaction of nuclear power plants, differences in response to 3-d, 3×1-d, and 1-d excitations. Earthquake Engineering and Structural Dynamics, In Review, 2016.
N. Abrahamson. Spatial variation of earthquake ground motion for application to soil-structure interaction. EPRI Report No. TR-100463, March., 1992a.
N. Abrahamson. Generation of spatially incoherent strong motion time histories. In A. Bernal, editor, Earthquake Engineering, Tenth World Conference, pages 845–850. Balkema, Rotterdam, 1992b. ISBN 90 5410 060 5.
N. Abrahamson. Spatial variation of multiple support inputs. In Proceedings of the First U.S. Seminar, Seismic Evaluation and Retrofit of Steel Bridges, San Francisco, CA, October 1993. UCB and CalTrans.
N. Abrahamson. Updated coherency model. Report Prepared for Bechtel Corporation, April, 2005.
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N. Abrahamson. Sigma components: Notation & initial action items. In Proceedings of NGA-East “Sigma” Workshop, University of California, Berkeley, February 2010. Pacific Earthquake Engineering Research Center. (http://peer.berkeley.edu/ngaeast/2010/02/sigma-workshop/).
N. A. Abrahamson. Estimation of seismic wave coherency and rupture velocity using the Smart 1 strong- motion array recordings. Technical Report EERC-85-02, Earthquake Engineering Research Center, University of California, Berkeley, 1985.
N. A. Abrahamson. Hard rock coherency functions based on Pinyon Flat data. unpublished data report, 2007.
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APPENDIX, EXAMPLES
Section title
Three Dimensional Seismic Motions and Their Use for 3D and 1D SSI Modeling
Present below is a simple study that emphasizes differences between 3D and 1D wave fields, and points out how this assumption (3D to 1D) affects response of a generic model NPP. It is noted that a more detailed analysis of 3D vs. 3×1D vs. 1D seismic wave effects on SSI response of an NPP is provided by (Abell et al., 2016).
Assume that a small scale regional models is developed in two dimensions (2D). Model consists of three layers with stiffness increasing with depth. Model extends for 2000m in the horizontal direction and 750m in depth. A point source is at the depth of 400m, slightly off center to the left. Location of interest (location of an NPP) is at the surface, slightly to the right of center.
Figure 6.10 shows a snapshot of a full wave field, resulting from a small scale regional simulation, from a point source (simplified), propagating P and S waves through layers. Wave field is 2D in this case, however all the conclusion will apply to the 3D case as well,
It should be noted that regional simulation model shown in Figure 4.10 is rather simple, consisting of a point source at shallow depth in a 3 layer elastic media. Waves propagate, refract at layer boundaries (turn more ”vertical”) and, upon hitting the surface, create surface waves (in this case, Rayleigh waves). In our case (as shown), out of plane translations and out of plane rotations are not developed, however this simplification will not affect conclusions that will be drawn. A seismic wave field with full 3 translations and 3 rotations (6D) will only emphasize differences that will be shown later.
Assume now that a developed wave field, which in this case is a 2D wave field, with horizontal and vertical translations and in plane rotations, are only recorded in one horizontal direction. From recorded 1D motions, one can develop a vertically propagating shear wave in 1D, that exactly models 1D recorded motion. This is usually done using de-convolution Kramer (1996).
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FIG.Figure 6.9: Snapshot of a full 3D wave field
Figure 6.9: (Left) Snapshot of a wave field, with body and surface waves, resulting from a point source at 45o off the point of interest, marked with a vertical line, down-left. This is a regional scale model of a (simplified) point source (fault) with three soil layers. (Right) is a zoom in to a location of interest. Figures are linked to animation of full wave propagation.
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Figure 6.10: Illustration of the idea of using a full wave field (in this case 2D) to develop a 1D seismic wave field.
Two seismic wave fields, the original wave field and a subset 1D wave field now exist. The original wave field includes body and surface waves, and features translational and rotational motions. On the other hand, subset 1D wave field only has one component of motions, in this case an SV component (vertically polarized component of S (Secondary) body waves).
Please note that seismic motions are input in an exact way, using the Domain Reduction Method (Bielak et al., 2003; Yoshimura et al., 2003) (described in chapter 6.4.10) and how there are no waves leaving the model out of DRM element layer (4th layer from side and lower boundaries). It is also important to note that horizontal motions in one
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direction at the location of interest (in the middle of the model) are exactly the same for both 2D case and for the 1D case.
Figure 6.12 shows a snapshot of an animation (available through a link within a figure) of difference in response of an NPP excited with full seismic wave field (in this case 2D), and a response of the same NPP to 1D seismic wave field.
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FIG.Figure 6.11: Left: Snapshot of a full 3D wave field, and Right: Snapshot of a reduced 1D wave field. Wave fields are at the location of interest. Motions from a large scale model were input using DRM, described in chapter 6 (6.4.10). Note body and surface waves in the full wave field (left) and just body waves (vertically propagating) on the right. Also note that horizontal component in both wave fields (left, 2D and right 1D), are exactly the same. Each figure is linked to an animation of the wave propagation.
FIG.Figure 6.12: Snapshot of a 3D vs 1D response of an NPP, upper left side is the response of the NPP to full 3D wave field, lower right side is a response of an NPP to 1D wave field.
Figures 6.13 and 6.14 show displacement and acceleration response on top of containment building for both 3D and 1D seismic wave fields.
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A number of remarks can be made:
Accelerations and displacements (motions, NPP response) of 6D and 1D cases are quite different. In some cases 1D case gives bigger influences, while in other, 6D case gives bigger influences.
Differences are particularly obvious in vertical direction, which are much bigger in 6D case.
Some accelerations of 6D case are larger thanthat those of a 1D case. On the other hand, some displacements of 1D case are larger than those of a 6D case. This just happens to be the case for given source motions (a Ricker wavelet), for given geologic layering and for a given wave speed (and length). There might (will) be cases (combinations of model parameters) where 1D motions model will produce larger influences than 6D motions model, however motions will certainly again be quite different. There will also be cases where 6D motions will produce larger influences than 1D motions. These differences will have to be analyzed on a case by case basis.
In conclusion, response of an NPP will be quite different when realistic 3D (6D) seismic motions are used, as opposed to a case when 1D, simplified seismic motions are used. Recent paper, by Abell et al. (2016) shows differences in dynamic behaviour of NPPs same wave fields is used in full 3D, 1D and 3×1D configurations.
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FIG.Figure 6.13: Displacements response on top of a containment building for 3D and 1D seismic input.
FIG.Figure 6.14: Acceleration response on top of a containment building for 3D and 1D seismic input.
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Seismic wave propagation and damping out high frequencies of motions
One dimensional column model is used for this purpose, as shown in figure 4.16.
Total height of the model is 1000m. Two models are built with element height/length of 20 m and 50 m, and each model has two different shear wave velocities (100 m/s and 1000 m/s). Density is set to ρ = 2000 kg/m3, and Poisson’s ratio is ν = 0.3. Various cases are set and tested as shown in table 4.1. Both 8 node and 27 node brick elements are used for all models. Thus, total 24 parametric study cases are inspected. Linear elastic elements are used for all analyses. All analyses are performed in time domain with Newmark dynamic integrator without any numerical damping (γ = 0.5, and β = 0.25, no numerical damping, unconditionally stable).
Ormsby wavelet (Ryan, 1994) is used as an input motion and imposed at the bottom of the model.
Ormsby wavelet, given by the equation 4.3 below:
features a controllable flat frequency content. Here, f1 and f2 define the lower range frequency band, f3 and f4 define the higher range frequency band, A is the amplitude of the function, and ts is the time at which the maximum amplitude is occurring, and sinc(x)
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= sin(x)/x. Figure 6.17 shows an example of Ormsby wavelet with flat frequency content from 5 Hz to 20 Hz.
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FIG.Figure 6.16: One dimensional column test model to address the mesh size effect.
FIG.Figure 6.17: Ormsby wavelet in time and frequency domain with flat frequency content from 5 Hz to 20 Hz.
As an example of such an analysis, consider three cases, as shown in Figures 6.18, 6.19, and 4.20m with cutoff frequencies of Ormsby wavelets set as 3Hz, 8Hz, and 15Hz. As shown in figure 5.18, case 1 and 7 (analysis using Ormsby wavelet with 3 Hz cutoff frequency) predict exactly identical results to the analytic solution in both time and
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frequency domain. In this case, the number of nodes per wavelength for both cases is over 10 and presented very accurate results (in both time and frequency domain) are expected. Increasing cutoff frequency to 8Hz induces numerical errors, as shown in figure 6.19.
In frequency domain, both 10 m and 20 m element height model with 27 node brick element predict very accurate results. However, in time domain, asymmetric shape of time history displacements are observed. Observations from top of 8 node brick element models show more numerical error in both time and frequency domain due to the decreasing number of nodes per wavelength. Figure 4.20 shows analysis results with 15Hz cutoff frequency. Results from 27 node brick elements are accurate in frequency domain but asymmetric shapes are observed in time domain. Decreasing amplitudes in frequency domain along increasing frequencies are observed from 8 node brick element cases.
TABLE 6.1: ANALYSIS CASES TO DETERMINE A MESH SIZE
CASE VS CUTOFF ELEMENT
MAX. PROPAGATION
1 1000 3 10 10
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FIG.
Figure 6.18: Comparison between (a) case 1 (top, Vs = 1000 m/s, 3 Hz, element size = 10m) and (b) case 7 (bottom, Vs = 1000 m/s,
3 Hz, element size = 20m)
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FIG.Figure 6.19: Comparison between (a) case 2 (top, Vs = 1000 m/s, 8 Hz, element size = 10m) and (b) case 8 (bottom, Vs = 1000 m/s, 8 Hz, element size = 20m)
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FIG.
Figure 6.20: Comparison between (a) case 3 (top, Vs = 1000 m/s, 15 Hz, element size = 10m) and (b) case 9 (bottom, Vs = 1000 m/s, 15 Hz, element size = 20m)
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