seismic rocking isolation of an asymmetric frame on...

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Seismic Rocking Isolation of an Asymmetric Frame on Spread Footings I. Anastasopoulos 1 ; F. Gelagoti 2 ; A. Spyridaki 3 ; J. Sideri 4 ; and G. Gazetas, M.ASCE 5 Abstract: Rocking isolation is a relatively new design paradigm advocating the intense rocking response of the superstructure as a whole, instead of exural column deformation. This is accomplished through intentionally underdesigning the foundation to guide plastic hinging be- low the ground surface rather than in the columns. A 2-story, 2-bay asymmetric frame is used to explore the effectiveness of this novel design approach. Finite-element dynamic analyses are performed using as seismic excitation idealized pulses and 20 real accelerograms, taking into account material (soil and superstructure) and geometric (uplifting and P-D effects) nonlinearities. A conventionally, Eurocode-designed frame and its foundation are compared to a design featuring the same frame but with substantially underdesigned (unconventional) footings. It is found that the performance of the unconventional system is advantageous, as not only does it escape collapse but it also suffers reparable damage. Despite their reduced width, the residual settlements of the underdesigned footings are comparable to those of the conventional ones. However, the analyses also reveal that residual rotation and differential settlement of the underdesigned footings may be unavoidable and must be critically evaluateda need exaggerated by the asymmetry of the examined frame. Three possible ways of improvement at the foundation level are studied: (1) a single conventional tie beam, monolithically connected to the footings; (2) two separate tie beams hinged at each footing (allowing rotation, but resisting axial deformation); and (3) a hybrid system, comprising a single continuous tie beam connecting the three footings but externally hinged to each of them. The rst solution hardly offers improvement, as it hinders rocking, and the second fails to reduce differential settlements. The hybrid solution provides encouraging results in terms of residual rotation and differential settlement, while it does not hinder the development of benecial rocking isolation mechanisms and fully restrains horizontal differential movements. DOI: 10.1061/(ASCE) GT.1943-5606.0001012. © 2014 American Society of Civil Engineers. Author keywords: Rocking isolation; Soil-structure interaction; Foundation design; Tie beams; Improved foundation; Uplifting; Bearing capacity failure; Differential settlement. Introduction Modern design principles advocate ductility and capacity design, where structural members aboveground are capable of bearing deformations even beyond yielding. The formation of plastic hing- ing is guided to less critical structural members (beams instead of columns) and brittle failure mechanisms (such as shear failure) are avoided (Park and Paulay 1975). To avoid irreparable (and unin- spectable) substantial yielding belowground, the footings are over- designed by the use of overstrength factors to ensure that neither structural yielding of the footing nor bearing-capacity failure mech- anisms develop. However, a new design approach has been under investigation by a growing body of researchers, according to which the foundation is allowed to rock, setting a limit on the inertia loading that may be transmitted onto the superstructure. The potential benets of such rocking isolation have been veried by several studies (e.g., Beck and Skinner 1974; Priestley et al. 1996; Mergos and Kawashima 2005; Kawashima et al. 2007; Deng and Kutter 2012; Deng et al. 2012a, b), and proposed for the retrot of existing structures (ASCE 2000; Martin and Lam 2000). Also, rocking isolation has been applied to a few newly constructed important bridges (Pecker 1998, 2003). A variety of analytical studies is available in the literature, ranging from nite element or nite differences numerical modeling of the entire soil-foundation-structure system (e.g., Paolucci and Pecker 1997; Gazetas et al. 2003; Chatzigogos et al. 2009), to Winkler- based methods (e.g., Chopra and Yim 1985; Apostolou et al. 2007; Kawashima et al. 2007), and comprehensive macroelement modeling (e.g., Paolucci et al. 2008; Chatzigogos et al. 2009; Gajan and Kutter 2009a). Experimental studies can be categorized broadly into large- scale testing (Negro et al. 2000; Faccioli et al. 2001), centrifuge model testing (Kutter et al. 2003; Gajan et al. 2005; Gajan and Kutter 2008, 2009b), and reduced-scale testing (Fukui et al. 2005; Paolucci et al. 2008; Anastasopoulos et al. 2013). This basic idea stems from the fact that the mobilization of soil- foundation bearing-capacity failure under seismic excitation does not necessarily imply failure, thanks to the cyclic and kinematic nature of ground shaking. Intentionally underdesigning the foun- dation (to have a lower moment capacity than the corresponding structural members to which they are attached) could lead to rocking instead of exural response of the superstructure. In other words, 1 Professor, Division of Civil Engineering, Univ. of Dundee, Nethergate, Dundee DD1 4HN, Scotland; formerly, Assistant Professor, School of Civil Engineering, National Technical Univ. of Athens, Athens 10682, Greece (corresponding author). E-mail: [email protected] 2 Postdoctoral Researcher, School of Civil Engineering, National Tech- nical Univ. of Athens, Athens 10682, Greece. 3 Graduate Student, Columbia Univ., New York, NY 10027; formerly, Student, National Technical Univ. of Athens, Athens 10682, Greece. 4 Graduate Student, Columbia Univ., New York, NY 10027; formerly, Student, National Technical Univ. of Athens, Athens 10682, Greece. 5 Professor, School of Civil Engineering, National Technical Univ. of Athens, Athens 10682, Greece. Note. This manuscript was submitted on September 6, 2012; approved on July 24, 2013; published online on July 27, 2013. Discussion period open until June 1, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Geotechnical and Geoenvi- ronmental Engineering, Vol. 140, No. 1, January 1, 2014. ©ASCE, ISSN 1090-0241/2014/1-133151/$25.00. JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2014 / 133 J. Geotech. Geoenviron. Eng. 2014.140:133-151. Downloaded from ascelibrary.org by George Gazetas on 12/24/13. Copyright ASCE. For personal use only; all rights reserved.

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Seismic Rocking Isolation of an Asymmetric Frameon Spread Footings

I. Anastasopoulos1; F. Gelagoti2; A. Spyridaki3; J. Sideri4; and G. Gazetas, M.ASCE5

Abstract: Rocking isolation is a relatively new design paradigm advocating the intense rocking response of the superstructure as a whole,instead of flexural column deformation. This is accomplished through intentionally underdesigning the foundation to guide plastic hinging be-low the ground surface rather than in the columns. A 2-story, 2-bay asymmetric frame is used to explore the effectiveness of this novel designapproach. Finite-element dynamic analyses are performed using as seismic excitation idealized pulses and 20 real accelerograms, taking intoaccountmaterial (soil and superstructure) and geometric (uplifting andP-D effects) nonlinearities. A conventionally, Eurocode-designed frameand its foundation are compared to a design featuring the same frame butwith substantially underdesigned (unconventional) footings. It is foundthat the performance of the unconventional system is advantageous, as not only does it escape collapse but it also suffers reparable damage.Despite their reduced width, the residual settlements of the underdesigned footings are comparable to those of the conventional ones. However,the analyses also reveal that residual rotation and differential settlement of the underdesigned footingsmay be unavoidable andmust be criticallyevaluated—a need exaggerated by the asymmetry of the examined frame. Three possible ways of improvement at the foundation level arestudied: (1) a single conventional tie beam,monolithically connected to the footings; (2) two separate tie beams hinged at each footing (allowingrotation, but resisting axial deformation); and (3) a hybrid system, comprising a single continuous tie beam connecting the three footings butexternally hinged to each of them. The first solution hardly offers improvement, as it hinders rocking, and the second fails to reduce differentialsettlements. The hybrid solution provides encouraging results in terms of residual rotation and differential settlement,while it does not hinder thedevelopment of beneficial rocking isolation mechanisms and fully restrains horizontal differential movements. DOI: 10.1061/(ASCE)GT.1943-5606.0001012. © 2014 American Society of Civil Engineers.

Author keywords:Rocking isolation; Soil-structure interaction; Foundationdesign;Tie beams; Improved foundation;Uplifting;Bearing capacityfailure; Differential settlement.

Introduction

Modern design principles advocate ductility and capacity design,where structural members aboveground are capable of bearingdeformations even beyond yielding. The formation of plastic hing-ing is guided to less critical structural members (beams instead ofcolumns) and brittle failure mechanisms (such as shear failure) areavoided (Park and Paulay 1975). To avoid irreparable (and unin-spectable) substantial yielding belowground, the footings are over-designed by the use of overstrength factors to ensure that neitherstructural yielding of the footing nor bearing-capacity failure mech-anisms develop.

However, a new design approach has been under investigation bya growing body of researchers, according to which the foundation isallowed to rock, setting a limit on the inertia loading that may betransmitted onto the superstructure. The potential benefits of suchrocking isolation have been verified by several studies (e.g., Beckand Skinner 1974; Priestley et al. 1996; Mergos and Kawashima2005; Kawashima et al. 2007; Deng and Kutter 2012; Deng et al.2012a, b), and proposed for the retrofit of existing structures (ASCE2000; Martin and Lam 2000). Also, rocking isolation has beenapplied to a few newly constructed important bridges (Pecker 1998,2003).

A variety of analytical studies is available in the literature, rangingfrom finite element or finite differences numerical modeling of theentire soil-foundation-structure system (e.g., Paolucci and Pecker1997; Gazetas et al. 2003; Chatzigogos et al. 2009), to Winkler-based methods (e.g., Chopra and Yim 1985; Apostolou et al. 2007;Kawashima et al. 2007), and comprehensive macroelement modeling(e.g., Paolucci et al. 2008; Chatzigogos et al. 2009; Gajan and Kutter2009a). Experimental studies can be categorized broadly into large-scale testing (Negro et al. 2000; Faccioli et al. 2001), centrifugemodeltesting (Kutter et al. 2003; Gajan et al. 2005; Gajan and Kutter 2008,2009b), and reduced-scale testing (Fukui et al. 2005; Paolucci et al.2008; Anastasopoulos et al. 2013).

This basic idea stems from the fact that the mobilization of soil-foundation bearing-capacity failure under seismic excitation doesnot necessarily imply failure, thanks to the cyclic and kinematicnature of ground shaking. Intentionally underdesigning the foun-dation (to have a lower moment capacity than the correspondingstructural members to which they are attached) could lead to rockinginstead of flexural response of the superstructure. In other words,

1Professor, Division of Civil Engineering, Univ. of Dundee, Nethergate,Dundee DD1 4HN, Scotland; formerly, Assistant Professor, School of CivilEngineering, National Technical Univ. of Athens, Athens 10682, Greece(corresponding author). E-mail: [email protected]

2Postdoctoral Researcher, School of Civil Engineering, National Tech-nical Univ. of Athens, Athens 10682, Greece.

3Graduate Student, Columbia Univ., New York, NY 10027; formerly,Student, National Technical Univ. of Athens, Athens 10682, Greece.

4Graduate Student, Columbia Univ., New York, NY 10027; formerly,Student, National Technical Univ. of Athens, Athens 10682, Greece.

5Professor, School of Civil Engineering, National Technical Univ. ofAthens, Athens 10682, Greece.

Note. This manuscript was submitted on September 6, 2012; approvedon July 24, 2013; published online on July 27, 2013. Discussion period openuntil June 1, 2014; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Geotechnical and Geoenvi-ronmental Engineering, Vol. 140, No. 1, January 1, 2014. ©ASCE, ISSN1090-0241/2014/1-133–151/$25.00.

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mobilization of soil or soil-footing failure mechanisms averts plas-tification of structural members.

The application of this new design concept has been exploredtheoretically and experimentally for a simple, slender, one–degree offreedom structure, representing a bridge pier (Anastasopoulos et al.2010, 2013), and for a complex 2-story, 2-bay symmetric frame onshallow footings (Gelagoti 2010; Gelagoti et al. 2012a, b). It wasfound that under moderately strong seismic shaking not exceedingthe design motion, the two foundation designs (conventional andunconventional) are practically equivalent, as they both sustainreparable structural damage. But, with very strong seismic shakingin excess of the design limits (hardly an impossible situation), therocking-isolated (unconventional) frameperforms substantially better:damage is restricted to beams and nonstructural elements, leaving theframe columns practically unscathed and, thereby, avoiding collapse.Moreover, despite their reduced width, the underdesigned footings(i.e., intentionally designed to be weaker than the correspondingstructural members) undergo tolerable seismic settlements. This paperextends the exploration of the applicability of rocking isolation to anasymmetric, two-span, 2-story frame—a more realistic structuralconfiguration. The external (overall) dimensions of the studied frameare the same as those of the (symmetric) frame of Gelagoti (2010).

Problem Statement

The studied problem is shown in Fig. 1. In the case of conventionaldesign, the footings are fairly wide and, hence, of larger maximummoment resistance than the bending moment capacity of the cor-responding columns. Therefore, plastic hinging is guided onto thesuperstructure. On the other hand, the footings of the rocking-isolated frame are intentionally underdesigned to be weaker thanthe corresponding columns, guiding plastic hinging at or below thesoil-foundation interface. At the same time, the accelerationstransmitted onto the superstructure are substantially reduced.

The RC frame has been designed in accordance with Eurocode 8[European Committee for Standardization (CEN) 2009], for aneffective design acceleration of 0.36g and a ductility-dependentbehavior factor q5 3:9 (the behavior factor q refers to the overall

ductility of the system, and is used to reduce the design seismicactions, accepting a certain degree of seismic damage). Dead load ofG5 1:5 kN×m2 and live load of Q5 2 kN×m2, typical of residentialbuildings, are adopted. Dimensions and reinforcement details aregiven in Fig. 2. Competent soil conditions are considered, assumingthat the foundation soil consists of stiff (overconsolidated) clay ofundrained shear strengthSu 5 150 kPa and small-strain shearmodulusGo 5 270MPa. The structure is founded on square surface footings ofwidth B.

The conventionally overdesigned footings can mobilize a maxi-mum moment resistance (Mu) from the underlying soil larger thanthe bending moment capacity of the corresponding column (MRD).For static vertical loads, a factor of safety FS $ 3 is required againstbearing-capacity failure. For seismic load combinations, a factor ofsafety FE 5 1 is acceptable. In the latter case, a maximum allowableeccentricity criterion is also enforced: e5M=N#B=3 (where Mand N are the overturning moment and the vertical force of the mostunfavorable load combination). For the investigated soil-structuresystem, the eccentricity criterion was found to be critical, leading tominimum required footingwidthsB5 2:7, 2.5, and 2.4m for the left,middle, and right footings, respectively. Bearing capacities andsafety factors are computed according to the provisions of Euro-code 8 (CEN 2009), which are basically the same as those typicallyused in foundation design practice.

The undersized footings of the rocking isolation design, areweaker than the superstructure, guiding the plastic hinge to or belowthe soil-footing interface instead of the base of the columns. Thesmall width of the footings promotes full mobilization of foundationmoment capacity with substantial uplifting. The eccentricity crite-rion is completely relaxed, while FE , 1 is allowed. FS $ 3 remainsa requirement as a measure against uncertainties regarding soilstrength. Moreover, it turns out that FS $ 5 might be desirable topromote uplifting-dominated response, thereby limiting seismicsettlements (Kutter et al. 2003; Faccioli et al. 2001; Pecker and Pender2000; Kawashima et al. 2007; Chatzigogos et al. 2009; Panagiotidouet al. 2012).

More specifically, applying the methodology that has been out-lined in Gelagoti et al. (2012a), the footings were designed to beadequately small to promote uplifting, but large enough to limit the

Fig. 1. (a) Conventionally designed frame compared to (b) rocking-isolation design

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settlements. Aiming to minimize differential settlements stemmingfrom asymmetry, the three footings were dimensioned in sucha manner so as to have the same FS. Based on the aforementionedcriteria, the resulting footing widths for the rocking-isolated designalternative are B5 1:1, 1.8, and 1.3 m for the left, middle, and rightfootings, respectively (indeed, substantially smaller than those of thecode-based design). Footing dimensions and static factors of safetyagainst vertical loading of the two designs are summarized in Table 1.

Numerical Analysis Methodology

A representative equivalent slice of the soil-foundation-structuresystem is analyzed with the FEM, taking account of material (soiland superstructure) and geometric (uplifting and P-D effects) non-linearities. As depicted in Fig. 3, the soil and the footings aremodeled with quadrilateral plane strain continuum elements, whilebeam elements are used for the superstructure. Special interfaceelements are considered at the soil-foundation interface to realisticallysimulate detachment and sliding. The seismic performance of the

system is analyzed through nonlinear, dynamic, time-history analysis,applying the seismic excitation at the base of the model.

Nonlinear soil behavior is modeled through a simplified kine-matic hardening model with a von Mises failure criterion and as-sociative flow rule. As discussed in detail in Anastasopoulos et al.(2011), the evolution of stresses is defined as

s ¼ s0 þ a (1)

where s0 5 stress at zero plastic strain; and a 5 backstress, de-termining the kinematic evolution of the yield surface in the stressspace. The latter is composed of an isotropic hardening component,which defines the size of the yield surface s0 as a function of plasticdeformation, and of a nonlinear kinematic hardening component,which describes the translation of the yield surface in the stressspace. The evolution of the kinematic component of the yield stressis defined as

_a ¼ C 1s0

ðs2aÞ _ɛpl 2ga_ɛpl

(2)

where C 5 initial kinematic hardening modulus ½C5sy=ɛy5E5 2ð11 vÞGo�; and g 5 parameter determining the rate ofdecrease of the kinematic hardening with increasing plastic de-formation. In the case of clay, the maximum yield stress can bedefined as

sy ¼ffiffiffi

3p

Su (3)

And because sy 5C=g1s0, parameter g can be expressed as

Fig. 2. Geometry and member properties of the idealized asymmetric frame

Table 1. Footing Dimensions and Corresponding Factors of Safety againstVertical Loading for the Seismic Load Combination (G1 0:3Q) for the TwoDesign Alternatives

Conventional design Rocking isolation

Footing B (m) FS B (m) FS

Left 2.7 32.6 1.1 5.4Middle 2.5 10.6 1.8 5.4Right 2.4 18.1 1.3 5.4

Note: Computed following the provisions of Eurocode 8 (CEN 2009).

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g ¼ Cffiffiffi

3p

Su2s0(4)

Parameters0, which controls the initiation of the nonlinear behavior,is defined as a fraction l (typically ranging from 0.1 to 0.3) of theyield stress sy

s0 ¼ lsy (5)

Finally, parameter C corresponds to the Young’s modulus for verysmall strains. If shear wave velocity (Vs) measurements are avail-able, it can be computed directly. Alternatively, it can be estimatedusing empirical correlations (e.g., Hardin 1978; Robertson andCampanella 1983).

The model requires calibration of three parameters only: thesmall-strain elasticity modulus, C; the ultimate strength, sy; andthe yield stress, s0. In the case of clay, the calibration requires thefollowing data: (1) undrained shear strength, Su; (2) Go or Vs

(measured or assessed through the aforementioned empirical cor-relations); and (3) G-g curves to calibrate parameter l. For thepurposes of the current study, model parameters were systematicallycalibrated according to the experimental G-g curves of Vucetic andDobry (1991). The model has been validated thoroughly againstcentrifuge and large-scale model tests, as discussed in detail pre-viously (Anastasopoulos et al. 2011), as well as against reduced-scale tests conducted at the Laboratory of Soil Mechanics of theNational Technical University of Athens (Anastasopoulos et al.2012). It has been shown to be capable of predicting with engi-neering accuracy the experimentalmoment-rotation (M-u) loops andthe settlement-rotation (w-u) response, both in terms of settlementper cycle and total residual settlement.Moreover, themodel has beenvalidated against the failure envelopes of Gourvenec (2007) for

a variety of footing shapes andmoment-to-shear ratios (Gazetas et al.2013); this refers to the ultimate capacity and not the entire range ofnonlinear response.Admittedly, the potential of themodel to reliablysimulate the effects ofmoment-to-shear ratio, embedment, and footingshape on the settlement for any possible combination has to bedemonstrated. While there is a breadth of failure envelopes in theliterature, the experimental data dealingwith cyclic or dynamic loadingare muchmore limited. Specific cases have been tested, and only thesecan be used for validation. Table 2 summarizes the validation of themodel.

The moment-curvature (M-c) response of structural members iscomputed with static cross-sectional analysis using XTRACT 3.0.3.Reasonable assumptions are made for the metaplastic regime (i.e.,c. cu): (1) the residual moment (Mres) is presumed to be 30% of theultimate moment capacity (Mult) (Vintzileou et al. 2007), and (2)Mres

is reached for cmax 5 3cult. A similar kinematic hardening model isemployed, as suggested by Gerolymos et al. (2005), to simulate thenonlinearM-c response of structural members. Model parameters arecalibrated against the XTRACT-computed, M-c relationships. Fora rectangular RCmember of width db and height dh, the strengthsy isdefined as

sy ¼ 4My

db d2h(6)

The small strain modulus C is equal to the Young’s modulus ofRC, while the yield stress is assumed to be

s0 ¼ sy

10(7)

A user subroutine, encoded in ABAQUS 6.9, simulates the meta-plastic response (i.e., the descending branch) of RC cross sections,

Fig. 3. Finite-element model of the soil-foundation-structure system: a typical equivalent slice of the building is analyzed in plane-strain, takingaccount of material (soil and superstructure) and geometric (uplifting and P-D effects) nonlinearities

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as well as stiffness degradation with deformation and cycles ofloading.

Typical results for (monotonic) static pushover loading are pre-sented in Fig. 4(a) for ground floor column 3. Complying withEurocode 8 (CEN 2009) provisions for well-reinforced, concretecross sections, all members are capable of attaining large values ofcurvature ductility (mw � 10). After exhaustion of the availableductility, i.e., for c. cu, the column enters its metaplastic regime(descending branch), reaching its residual state for c5 cmax. Resultsof cyclic response of the structural members, showing the perfor-mance and limitations of the model, have been documented in

Gelagoti et al. (2012b), with due emphasis on the effect of stiffnessdegradation. The M-u response of the corresponding footing 3 isplotted in Fig. 4(b) for the two design alternatives. The momentcapacity of the conventionally designed B5 2:4 m footing issubstantially larger (by a factor of about 1.85) than that of column3: Mult � 370 kN×m.Mc

RD � 200 kN×m. On the contrary, themoment capacity of the rocking-isolated B5 1:3m footing isclearly smaller (by a factor of about 1.5) than that of the column:Mult9 � 130 kN×m,Mc

RD� 200 kN×m. Observe that the reductionof thewidth of the footingB leadsnot only to a reduction of its capacitybut also to reduced toppling rotation (uult).

Table 2. Summary of Model Validation

Reference FS M=Q (m) B (m) Da (m) Soil type Loading type Type/scale

Model effectivenessb

Rotationalstiffness (%)

Momentcapacity (%)

Accumulatedsettlement (%)

Anastasopouloset al. (2011)c

2.8 4.6 2.7 — Clay Cyclic 6 0.01 rad Centrifuge 1:20 1 8 1

2.8 4.6 2.7 — Clay Cyclic 6 0.02 rad Centrifuge 1:20 3 5 12.8 4.6 2.7 — Clay Cyclic 6 0.06 rad Centrifuge 1:20 5 1 6

Anastasopouloset al. (2011)d

3 0.9 1 1 Loose sand Cyclic 6 0.02 rad Real scale 1:1 4 10 5

5 0.9 1 1 Dense sand Cyclic 6 0.02 rad Real scale 1:1 6 7 8Anastasopouloset al. (2012)e

2.1 13 7 — Dense sand Cyclic 6 0.03 rad Reduced 1:20 5 10 14

3.5 13 7 — Dense sand Cyclic 6 0.03 rad Reduced 1:20 12 7 47.3 13 11 — Dense sand Cyclic 6 0.03 rad Reduced 1:20 8 3 7

Gazetas et al. (2013)f 0.01–50 0–10 STg— Clay Monotonic ultimate Analytical 8–14 4–12 —

0.01–50 0–10 SQh— Clay Monotonic ultimate Analytical 9–17 3–8 —

0.01–50 0–10 Ri— Clay Monotonic ultimate Analytical 10–20 6–17 —

0.01–50 0–10 Cj— Clay Monotonic ultimate Analytical 8–17 2–14 —

aEmbedment depth.bError of model prediction compared to experimental or analytical values.cUC Davis centrifuge model tests (Anastasopoulos et al. 2011).dTRISEE large-scale tests (Anastasopoulos et al. 2011).eNTUA reduced-scale tests (Anastasopoulos et al. 2012).fPublished failure envelopes and impedances (Gazetas et al. 2013).gStrip foundations, various widths.hSquare foundations, various widths.iRectangular foundations, 1:3 length-to-width ratio, various widths.jCircular foundations, various diameters.

Fig. 4. (a) M-c response of first-floor column 3 [ductile design according to Eurocode 8 (CEN 2009)]; (b) M-u response of the correspondingfooting 3, illustrating the effect of the reduction of its width B (for combined axial and shear force at a constant lever arm)

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Effectivenessof Rocking Isolation: DynamicAnalysis

The seismic performance of the two design alternatives is in-vestigated through nonlinear dynamic time-history analyses. To thisend, a set of 20 real accelerograms is used as seismic excitation,applied at the base of the soil-foundation-structure model. Asdepicted in Fig. 5 and summarized in Table 3, the selected seismic

records cover a wide range of earthquake characteristics, such aspeak ground acceleration (PGA), pseudo spectral velocity (PSV),maximum spectral acceleration (maxSA), maximum spectral velocity(maxSV), frequency content, number of strong-motion cycles, andduration. Some of these records are well-recognized as bearing thesignature of near source (directivity andfling-step) effects. They rangefrom medium intensity (e.g., El Centro 1940 and Kalamata 1986) to

Fig. 5. Acceleration time histories of the 20 real records used (without scaling) as seismic excitation for the dynamic analysis of the two designalternatives, along with their acceleration and velocity elastic response spectra

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very strong accelerograms (e.g., Takatori 1995 and Tabas 1978). Interms of spectral accelerations (SA), many of the considered accel-erograms surpass the design spectrum of the frame for all periods ofinterest.

Because all the results cannot be presented in detail within thelimits of a single paper, emphasis is placed on elucidating theperformance of the two design alternatives for the case of very strongseismic shaking, using the Takatori record, which substantiallyexceeds the design spectrum of the frame. The performance of the twodesign alternatives for moderate intensity seismic shaking has beenexplored in detail in Gelagoti et al. (2012b). It was shown that, whenthe seismic excitation is within the design limits, the performance ofthe rocking-isolated frame is practically equivalent to the conven-tionally designed one. In both cases, the structure would survive,sustaining acceptable structural damage. With conventional design,the structural damage perhaps could be reparable (flexural cracking ofbeams and columns), but not necessarily within serviceability limits.Incontrast, the rocking-isolated structurewould sufferminor structuraldamage (flexural cracking of beams), and most probably could be inservice immediately after the earthquake. In both cases, the settlementwould be reasonable, with the performance of the rocking-isolatedalternative being slightly worse. Results for the complete ensemble ofmotions are shown only in summary for each design alternative.Subsequently, the idealized Tsang motion (a modulated sinusoid) ofdominant periodT 5 0:5 s and parametrically variablePGA is utilizedfor some of our studies.

Performance in Strong Seismic Shaking

The Takatori record (Fukushima et al. 2000) of theMJMA 7.2, Kobe,Japan, 1995 earthquake, with ground motion beyond the designlimits, constitutes one of the most destructive accelerograms everrecorded, with a PGA� 0:70g, PGV � 169 cm=s, and dominantperiods at about 1.2 and 2 s. It bears the effects of both forwardrupture directivity and soil amplification.

The performance of the two design alternatives is compared inFig. 6. The deformed mesh with superimposed, plastic-strain con-tours of the two alternatives is portrayed in Fig. 6(a). With suchunrelenting seismic shaking, the conventionally designed frame

collapses under its gravity load (due to excessive drift of thestructure, the moments produced by P-D effects cannot be sustainedby the columns, leading to loss of stability and total collapse). Asexpected, plastic hinges firstly develop in the beams and, sub-sequently, at the base of the three columns, while soil under thefootings remains practically elastic. The collapse also is evidencedby the substantial exceedance of the available curvature ductility ofthe columns [Fig. 6(b)]. Conversely, the rocking-isolated framewithstands the shaking, with plastic hinging taking place only in thebeams, leaving the columns almost unscathed (elastic M-c re-sponse). Instead, plastic hinging nowdevelopswithin the underlyingsoil in the form of extended soil plastification [indicated by thered regions under the foundation in Fig. 6(a) and the M-u loops ofFig. 6(c)]. Thanks to the fact that the bendingmoment capacity of thecolumn is larger than that of the footing, damage is guided be-lowground and at the soil-foundation interface, in the form of de-tachment and uplifting [evidenced in Fig. 6(c) by the zero residualrotation upon unloading, unveiling the nonlinear but elastic upliftingcomponent of rotation], instead of aboveground.

In terms of M-u foundation response [Fig. 6(c)], the situationis reversed: conventional footings behave almost elastically, ex-periencing negligible rotation, whereas the underdesigned footingsreach several times their moment capacity and rotate significantly,creating loops indicative of energy dissipation. Nevertheless, aspreviously mentioned, the rotation never exceeds the ultimate ro-tation capacity (uult) of the footing, ensuring safety against collapsedue to rotational instability of the foundation, while residual rotationis minor because of the system’s inherent gravity-induced, self-centering capability. If the footing rotation had reached uult, itsmoment capacity would have been completely lost, and the latterwould perform as a hinge. This could lead to rotational instability ofthe footing and risk of collapse.At this ultimate stage of response, theseismic loads cannot be undertaken through frame action, becausethe beam-column connections have failed already (plastic hinges inbeams have formed already at an earlier stage).

The time histories of interstory drift further elucidate the afore-mentioned behavior of the two design alternatives [Fig. 6(d)]. Whilethe horizontal deformation of the conventionally designed frameincreases uncontrollably until collapse, the residual total drift of the

Table 3. Main Characteristics of the Seismic Records Used for the Analyses

Record Event Year Ms Mechanism RJB (km) PGA (g) PGV (cm=s)

GIC090 Salvador San Salvador, El Salvador 1986 5.8 Strike-slip 4 0.7 80Kalamata Kalamata, Greece 1986 6.2 Normal 7 0.27 24Lefkada Lefkada, Greece 2003 6.4 Strike-slip 8 0.45 34Pacoima Dam 164 San Fernando, CA 1971 6.5 Thrust 3 1.06 112Pacoima Dam 254 San Fernando, CA 1971 6.5 Thrust 3 1.16 -Rinaldi 228 Northridge, CA 1994 6.8 Thrust 0 0.84 148Rinaldi 318 Northridge, CA 1994 6.8 Thrust 0 0.48 65Jensen 292 Northridge, CA 1994 6.8 Thrust 0 0.6 121Sylmar 090 Northridge, CA 1994 6.8 Thrust 3 0.604 74El Centro Imperial Valley, CA 1940 6.9 Strike-slip 8 0.32 29Erzincan Erzincan, Turkey 1992 6.9 Strike-slip 2 0.5 64Treasure Island Loma Prieta 1989 6.9 Strike-slip 77 0.17 33Duzce 180 Duzce, Turkey 1999 7.2 Strike-slip 8 0.35 60Takatori 000 Kobe, Japan 1995 7.2 Strike-slip 3 0.62 127JMA 000 Kobe, Japan 1995 7.2 Strike-slip 1 0.83 81Lucerne 000 Landers, CA 1992 7.3 Strike-slip 3 0.79 32Tabas Tabas, Iran 1978 7.4 Reverse 3 0.84 98Izmit Kocaeli, Turkey 1999 7.6 Strike-slip 4 0.22 30Yarimca Kocaeli, Turkey 1999 7.6 Strike-slip 4 0.26 85TCU-068-east Chi-Chi, Taiwan 1999 7.6 Reverse 1 0.48 260

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Fig. 6. Comparison of the performance of the two design alternatives subjected to very strong seismic shaking (Takatori, Kobe, Japan, 1995):(a) deformed mesh with superimposed plastic strain contours; (b) column bending M-c response; (c) foundation M-u response; (d) time histories ofground floor drift d (flexural dC , and due to foundation rotation dR)

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rocking-isolated frame is limited to 9 cm, corresponding to a driftratio d=h� 2% (where h is the height of a single story). Yet, duringseismic shaking, the maximum total drift reaches 45 cm, whichimplies serious flexural distortion of the superstructure. Althoughnonstructural members (e.g., infill walls) are affected by such lateraldisplacement, columns do not suffer any structural damage, becausethe dominant component of the total drift stems from foundationrotation (dR), while theflexural drift component (dC) isminor [for thetwo drift components, see also Priestley et al. (1996)]. Finally, undersuch severe seismic excitation, when the conventionally designedframe cannot survive, the rocking-isolated frame succeeds inavoiding collapse and even structural damage to its columns, butdamage to nonstructural members and beams (which exhaust theirductility capacity) is inevitable.

The two design alternatives are compared with respect to the w-uresponse of the footings in Fig. 7. As expected, before the initiationof collapse, the conventionally designed footings experience rela-tively small (within serviceability limits) rotations and settlements

[Fig. 7(a)]. Completely different is the w-u response of the rocking-isolated system [Fig. 7(b)]: the footings undergo substantially largerrotations and accumulated settlements but still within tolerablelimits. This marked increase in settlement and rotation, inextricablyconnected to the rocking-induced energy dissipation mechanism, isthe price to pay for the survival of the structure.

Despite the fact that the three footings have been dimensioned tohave the same static factor of safety FS (in an attempt to minimizedifferential settlements exacerbated by asymmetry), the centralfooting settles more than the two side footings, leading to a differ-ential settlement of the order of 3 cm.The difference in the settlementis mainly due to their differences in width. As previously discussed,the central footing was made larger (B5 1:8m compared to 1.1and 1.3 m of the two side footings) to maintain the same FS. Becausethe latter is common for the three footings, if the loading is moreor less the same, their responses should be similar. However, suchequivalence refers to dimensionless quantities not absolute values(seeKourkoulis et al. 2012a). In otherwords,while the three footings

Fig. 7. Performance of the two design alternatives subjected to very strong seismic shaking (Takatori, Kobe, Japan, 1995); comparison of foundationw-u response for (a) the conventionally designed system; (b) the rocking-isolated alternative; (c) axial force time histories of the columns of the rocking-isolated frame

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sustain almost the same dimensionless settlement (w=B), which isroughly equal to 0:025 ð�3 cm=1:2 mÞ for the two side footings and0:033 ð�6 cm=1:8mÞ for the central one, the latter is substantiallylarger in width and, hence, its settlement is larger in absolute terms.Naturally, the three footings are not subjected to exactly the sameloading, something which further complicates the response.

Such differential settlements may inflict additional distress ontothe superstructure and, therefore, are worthy of further investigation.Fig. 7(c) depicts the time histories of axial force N at the base of thethree columns. Observe that the axial force at the central footing doesnot fluctuate as intensely as at the two side footings. More specif-ically, the axial load acting on the left footing (which belongs to thenarrow span) has the lower starting value and exhibits the greatestfluctuation during seismic shaking, leading to complete instantaneousdetachment from the supporting soil (N5 0), rendering the footingprone to sliding. After a complete detachment, the footingmay land ata slightly different position where the soil is in a less disturbed (morevirgin) state, thus limiting its settlement. However, such a landing ata substantially different position also would inflict horizontal differ-ential displacements between the neighboring footings—a potentiallydetrimental kinematic effect.

Apart from this mechanism, the observed differential settlementalso is attributed to the tendency of the central footing to accumulatemore settlement than the two side footings. In fact, this tendency isa direct outcome of the design for a common FS, thanks to which thethree footings tend to accumulate equal dimensionless settlementw=B (see Kourkoulis et al. 2012b). Thus, being the largest in width,the middle footing settles the most.

PerformanceAssessment:SynopsisofAnalysisResults

For all the investigated seismic excitations of Fig. 5, Figs. 8 and 9summarize the performance of the two design alternatives, in terms ofdrift ratio (d=h) and permanent settlement (w). The results are plottedwith respect to the maximum pseudo spectral velocity (maxPSV)—evidently amore representative seismic intensitymeasure for inelasticsystems (Bertero 1976; Garini and Gazetas 2013).

In terms of total (i.e., rotational and flexural) maximum drift ratio(dmax=h), with the exception of the three cases of collapse of theconventionally designed system, the performance of the two designalternatives is practically equivalent [Fig. 8(a)]. Considering the totalresidual drift ratio, dres=h [Fig. 8(b)], the rocking-isolated systemoutperforms the conventionally designed one when maxPSV exceeds150 cm=s. For moderate intensity seismic excitations, not exceedingthe design limits (maxPSV , 100 cm=s), the drift ratio d=h is almostthe same for the two design alternatives. In three extreme seismicscenarios (Takatori, Rinaldi, and JMA), during which the conven-tionally designed system cannot survive, the rocking-isolated systemaccumulates a permanent drift ratio of up to 4% but avoids collapse.The superior performance of the rocking-isolated system becomeseven more evident when examining the flexural residual drift ratio,dc,res=h [Fig. 8(c)], which is a direct indicator of column structuraldamage. In no case does the flexural drift ratio (dc=h) of the rocking-isolated frame exceed 1%: the column response satisfies even strictserviceability limits (Priestley et al. 2007). For moderate seismicshaking (maxPSV , 100 cm=s), the flexural drift ratio, dC=h (whichis an indicator of structural damage), is slightly lower for the rocking-isolated frame, confirming the previously mentioned conclusions(see also Gelagoti et al. 2012b). Conversely, in over 50% of the(intentionally strong) seismic excitations examined herein, the con-ventionally designed frame suffers irrecoverable damage and re-placement is inevitable. In three cases it even reaches the state ofcollapse.

Regarding permanent settlement (w), Fig. 9 shows that the sidefootings (left and right) of the two design alternatives experiencesimilar settlements. For moderate seismic shaking [pseudo spectralvelocity (PSVÞ, 100 cm=s], the two design alternatives are practi-cally equivalent, with the settlement of the central footing reachingroughly 3 cm and that of the side footings not exceeding 2 cm. In thecase of strong seismic shaking, the inherent difference between the twobecomes evident at the central footings, with the underdesignedfooting of the rocking-isolated frame accumulating conspicuouslylarger settlement. Hence, the previously detected problem of differ-ential settlement for the Takatori record proves to be of fairly generalvalidity, and not just a particular case. More specifically, for the(admittedly very strong) seismic excitations used herein, the differ-ential settlement of the rocking-isolated frame ranges from0.3 to 5 cm.(The total seismic settlement of the central footing ranges from 1.5 to7.5 cm.)Although the effectiveness of the newdesign scheme remainsunquestionable in terms of life safety, it would be of interest tominimize differential settlements to improve the performance in termsof serviceability.

The Role of Asymmetry on Differential Settlement

The investigated frame resembles the one analyzed in Gelagoti et al.(2012b) but it is asymmetric. Because the performance of thesymmetric rocking-isolated framewas almost immune to differentialsettlements, it can be inferred that the asymmetry plays a key role inthis respect. Hence, to gain a deeper insight on the role of frameasymmetry, an idealized modulated sinusoidal (Tsang-type) motionwas used next as seismic excitation (Fig. 10). Containing amultitudeof constant-amplitude, strong-motion cycles, it is an extreme butideally symmetric seismic excitation, allowing the role of frameasymmetry to be distinguished. The analysis was conducted fordifferent acceleration amplitudes, ranging from PGA5 0:2 to 1g,aiming to shed light on asymmetry’s effect on the accumulation ofdifferential settlement.

Fig. 10(a) depicts the settlement of the three footings as a functionofPGA. Evidently, the response is asymmetric despite the symmetryof loading. For small to medium intensity seismic excitations,PGA# 0:6g, the differential settlement ranged from 0.5 to 5 cm,being roughly equivalent to what was observed for real seismicexcitations. As the intensity of seismic excitation increased toPGA. 0:6g, the central footing continued to settle at the same oreven increased rate, while the settlements of the two side footingstended to stabilize or even reduce. Hence, the residual differentialsettlement increases, to the detriment of the structural behavior of theframe. For the extreme case of PGA5 1g, the differential settlementreached 18 cm. Interestingly, the settlement of the left footing wasfound to decrease from5 to 1 cmwhen thePGA increased from0.8 to1g; this prompted further investigation of the response for the ex-treme case of PGA5 1g.

Fig. 10(b) illustrates the w-u response of the three footings. Aspreviously noticed with the Takatori excitation, the left footingexperiences sliding and complete detachment from the bearing soil.After each such complete detachment, it tends to land in a differentposition where the soil has not been disturbed with plasticdeformations; thus, its settlement is reduced. The increase of PGAleads to an aggravation of this phenomenon and a correspondingdecrease of its final settlement. At the same time, due to theaforementioned complete loss of contact, its rotation (u) becomesuncontrollable, reaching excessively large values (0.13 rad)compared to the other two footings, which do not experiencefull detachment and barely reach 0.02 rad. At the same time, thecentral footing exhibits a sinking response, accumulating a residual

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settlement of 19 cm (for this motion with a huge number of uniformstrong-motion cycles). As a result, the frame is subjected to ex-cessive differential settlements, in addition to differential hori-zontal displacement (due to the random landing of the left footing ata different location after each cycle), and significant left-footingrotations. This leads to substantial kinematic frame distortion,partly brought about by the differential settlement and partly by thehorizontal divergence of the two footings and their differentialrotations.

To summarize, because of its geometric asymmetry, the rocking-isolated frame may respond unpredictably when subjected to ex-tremely strong seismic excitation. Nevertheless, even under suchextreme multicycle sinusoidal shaking of PGA5 1g, the isolated

structure does not collapse. However, its serviceability is notguaranteed, calling for remedial measures.

Investigation of Remedial Measures

A widely accepted practice, typically entrenched in modern seismiccodes, is the addition of tie beams between the footings (in thelongitudinal and transverse directions). Mobilizing their axial stiff-ness, tie beams act as a diaphragm forcing the footings to maintainthe same horizontal displacement. In addition, thanks to their flex-ural stiffness, they also contribute to reducing differential settle-ments. However, exactly because of their flexural stiffness, the (fully

Fig. 8. Synopsis of the performance of the two design alternatives with respect to the maximum PSV: (a) total maximum drift ratio for the ground floordmax=h (where h is the height of the ground floor); (b) total residual drift ratio for the ground floor dres=h; (c) residual flexural drift ratio dC,res=h; thedamage level is determined with reference to response limit states (Priestley et al. 1996)

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fixed) tie beams also would increase the moment capacity of thefoundation system, and may cancel the rocking isolation. Thus, twoadditional nonconventional solutions are explored.

Frame with a Single Fully Fixed Tie Beam

Monolithically connected to the footings, such a tie beam isdesigned according to current seismic codes to provide adequateaxial stiffness for resisting differential horizontal displacements.According to Eurocode 8 (CEN 2009), for instance, the design axialforce for a given soil category is proportional to: (1) the average,NED, of the axial loads of the two columns connected by this beam;and (2) the effective design ground acceleration, A5ag. ForEurocode 8 (CEN 2009) Soil Category C, appropriate for the sys-tems investigated herein,

FTIE ¼ 60:40a SNED (8)

where a5A=g5 0:40; and S5 1:15. For the studied frame, theresulting tie beams have a 0:303 0:65 m rectangular cross sectionand are reinforced with 6F14—quite typical for structures adheringto the provisions of Eurocode 8 (CEN 2009). Their M-c response iscomputed as describedpreviously usingXTRACT and is of theorder of

150 kN×m. Tsang motions are utilized as seismic excitation, withacceleration amplitudes ranging from 0.2 to 1g and a dominant periodof 0.5 s.

Fig. 11(a) depicts the M-c response of the three ground-floorcolumns for the worst-case scenario, amax 5 1g. As suspected, theresponse of the three columns was highly nonlinear, revealing thatthe addition of the conventional (fully fixed) tie beams practicallycancelled rocking isolation. The middle and the right columnsexperienced a ductility largely exceeding their capacity. The per-formance of the left column was also nonlinear, but its ductilitycapacity was not completely expended. In contrast to the wholerationale of the rocking-isolation concept, exactly owing to theadditionalmoment restraint provided by the fullyfixed tie beams, themoment capacity of the combined foundation system (footings andtie beams) is larger than the capacity of the columns. In other words,their effect is related to the additional resistance they provide in termsof moment capacity of the foundation system. Because the columnmoment capacity, MRD, is of the order of 200 kN×m, eventhe addition of 70 kN×m of moment resistance would be enoughto cancel rocking isolation: the moment capacity, Mult, of theunderdesigned footings is of the order of 130 kN×m [Fig. 4(b)]. Aspreviously mentioned, the simulated tie beams have a momentcapacity of the order of 150 kN×m. As a result, the middle and right

Fig. 9. Synopsis of the performance of the two design alternatives in terms of foundation settlementwwith respect to themaximum PSV of the seismicexcitation: (a) conventionally designed system; (b) rocking-isolated alternative

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footings exhibited an almost elastic response [Fig. 11(b)]. It is notedthat themoment in the footing–tie beam system clearly surpassed theultimate monotonic moment capacity (gray line), which wascomputed for the rocking-isolated frame without tie beams.

Fig. 11(c) illustrates the w-u response of the three footings.Indeed, the tie beams reduce the differential settlement: 12 cm asopposed to 19 cm without tie beams. Regarding the previouslyobserved complete detachment of the left footing and the associatedexcessive rotation and sliding, the addition of conventional tie beamsprovided substantial but not satisfactory improvement: the leftfooting still rotated excessively, partly because of plastic hinging ofthe tie beam itself.

Thus, conventional fully fixed tie beams may eliminate thehorizontal differential displacements, reduce somewhat the differ-ential settlements, and partially hinder the detachment of the leftfooting; but, owing to their monolithic connection to the footings,they also eliminate rocking isolation, rendering the frame perfor-mance almost identical to that of the conventionally designed framewithout tie beams.

Hinged Tie Beams

Aiming to maintain the beneficial effects of rocking isolation whilereducing the differential settlement, hinged tie beams (which would

not restrain the rotation of the footings) are considered. Indeed,thanks to their axial stiffness, the hinged tie beams restrict the lateraldifferential displacements between the footings. Such hinged con-nection can be materialized by placing the reinforcement at thecenter of the concrete cross section, as conceptually shown inFig. 12, or through the addition of prefabricated steel hinges.

The same analysis is repeated, subjecting the frame to the Tsang-type excitation with PGA parametrically varying from 0.2 to 1g.Fig. 12(a) illustrates theM-c response of the three frame columns (attheir bases) for the worst-case scenario, PGA5 1g. As expected,the columns remained elastic, in accordance with the principles ofrocking isolation. The M-u curves of the three footings werenonlinear and consistent with their monotonic curves, which act as(approximate) envelopes for the cyclic loading [Fig. 12(b)]. Owingto their hinged connection to the footings, the tie beams did notincrease the moment capacity of the foundation system, as was thecase with fully fixed tie beams.

However, the performance in terms of differential footing set-tlements was not sufficiently improved [Fig. 12(c)]: the rotation ofthe left footing was reduced by almost 50% (compared to the case ofno tie beams), but the differential settlements still reached 12 cm(compared to 19 cm of the frame without tie beams). Their failure toreduce the differential settlements stems exactly from their hingedconnection to the footings. As schematically illustrated in Fig. 12

Fig. 10. Performance of the rocking-isolated alternative subjected to multicycle Tsang-type pulses (top): (a) settlement of the three footings withrespect to the acceleration amplitude amax; (b) w-u response of the three footings for very strong shaking with amax 5 1g, foundation w-u response; themaximum differential settlement reaches 19 cm

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(top), because of their hinged connection to the footings, the twoseparate tie beams can actually rotate as rigid bodies without de-veloping any substantial resistance in the vertical sense: theirflexural rigidity cannot be mobilized. This means that the centralfooting is free to develop its own settlement, hardly restrained by thesmaller settlement of the two side footings. Naturally, however, theiraxial rigidity is fully mobilized, leading to the reduction of settle-ments and the horizontal differential displacements.

Hybrid Tie Beams

Aiming to combine the advantages of conventional (fully fixed) andhinged tie beams and avoid their distinct disadvantages, a hybridconcept was conceived and investigated. As shown in Fig. 13 (top),it consists of one continuous tie beam (instead of two separate ones)placed behind the columns (i.e., in a second row) and connectedwithexternal hinges (rather than fixed) to the columns. There are various

Fig. 11. Performance of the rocking-isolated alternative equipped with conventional, fully fixed tie beams, subjected to very strong seismic shaking(Tsang-type excitation of amax 5 1g): (a) M-c of the three ground floor columns; (b) M-u response; (c) w-u response of the three footings; themaximum differential settlement is reduced to 12 cm, but rocking isolation is practically canceled

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means to materialize such external hinges, with the simpler onebeing to connect the tie beamwith the columns using centrally placedsteel reinforcement as shown. Alternatively, special prefabricatedsteel components (typically used in bridges) can be utilized.

Thus, it is expected that the flexural rigidity of the tie beams canbe mobilized adequately, homogenizing the settlements of the threefootings (and, hence, reducing their differential settlements), while

at the same time allowing their rocking (and, hence, preservingthe benefits of rocking isolation for the superstructure).

Response to Tsang-Type Excitation

To confirm the aforementioned expectations, the system wassubjected to the same Tsang-type motions, with PGA ranging

Fig. 12. Performance of the rocking-isolated alternative equipped with hinged tie beams, subjected to very strong dynamic shaking (Tsang-typeexcitation of amax 5 1g): (a) M-c of the three ground floor columns; (b) M-u response; (c) w-u response of the three footings; rocking isolation ismaintained, but the reduction of differential settlement still lies within unsatisfactory limits (11.5 cm)

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parametrically from 0.2 to 1g. As shown in Fig. 13(a), whichillustrates theM-c response of the three columns for the worst-casescenario of PGA5 1g, the performance of the frame was con-sistent with the idea of rocking isolation, with all of its columnsexhibiting elastic or nearly elastic response. This is further verifiedby theM-u response of the three footings, which was nonlinear and(with the exception of the left footing) consistent with their monotonicpushover response [Fig. 13(b)]. The left footing exhibited an apparent

overstrength, which is associated with the axial restraint provided bythe hybrid tie beams.

The advantageous performance of the hybrid tie beams becomesevidentwhen examining thew-u response of the footings [Fig. 13(c)].The differential settlement between the middle and the side footingsdecreased impressively to just 2.5 cm, as opposed to 19 cm of thereference case (without tie beams). Given the extreme intensity of theseismic excitation (containing 10 strong-motion cycles of PGA5 1g

Fig. 13. Performance of the rocking-isolated alternative equipped with hybrid tie beams, subjected to very strong dynamic shaking (Tsang-typeexcitation of amax 5 1g): (a) M-c of the three ground floor columns; (b) M-u response; (c) w-u response of the three footings; rocking isolation ismaintained while a spectacular decrease in differential settlement is observed (merely 2.5 cm)

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with a period of 0.5 s), this can be said to constitute a remarkableimprovement in seismic performance.

Response to Recorded Accelerograms

To confirm the effectiveness of the hybrid ties, the system was sub-jected to the previously described set of real seismic motions. Fig. 14compares the performance of the rocking-isolated frame equippedwithhybrid ties to the reference case (rocking-isolated frame without tiebeams), in terms of footing settlement versus peak spectral pseudo-velocity,maxPSV, of the seismic excitation. The addition of the hybridtie beams led to an increase of the settlement of the two side footingsand to a decrease of the settlement of the central footing. As a result,the differential settlement was reduced substantially, by 25–60%,depending on the intensity of the seismic excitation.

To shed more light on the observed performance amelioration,the response to the particular seismic excitation that induces thelargest differential settlement, namely the Tabas record, is presentedin detail in Fig. 15. While in the reference case the differentialsettlement reached 5 cm [Fig. 15(a)], the addition of hybrid tie beamsreduced it to merely 1.9 cm [Fig. 15(b)]. At the same time, theperformance in terms of drift ratio (not shown herein) remainedpractically the same, as the hybrid tie beams do not preclude therocking response of the footings. Therefore, even under such ex-treme seismic shaking, the performance of the frame equipped withhybrid tie beams is excellent, maintaining the advantages of rockingisolation and minimizing the differential settlements.

Conclusions

The presented nonlinear numerical analyses have verified the ef-fectiveness of rocking isolation for an asymmetric 2-story, 2-bayframe. The numerical model employed herein has been validatedthoroughly against centrifuge and large-scale model tests, as well asagainst published failure envelopes for a variety of footing shapesand moment-to-shear ratios. The ability of the model to accuratelysimulate the effects of moment-to-shear ratio, embedment, andfooting shape with respect to the accumulation of settlement havenot yet been demonstrated fully for any possible combination of theabove. Emphasis has been placed on the significance of frame asym-metry, aiming at detecting possible limitations of this new seismicdesign scheme, mainly pertaining to serviceability requirements. Thekey conclusions are summarized as follows:1. In all cases examined, the overall performance of the rocking-

isolated design alternative is superior to that of the convention-ally designed system. While the latter may sustain irreparablestructural damage or even collapse when subjected to verystrong seismic shaking, the rocking-isolated system surviveseven the most extreme seismic excitation with minor flexuraldamage.

2. The asymmetry of the investigated frame complicates theresponse, leading to larger differential settlements than thoseexperienced by the symmetric counterpart. This differencein settlement accumulation stems from the redistribution ofaxial forces during seismic shaking, especially the substantial

Fig. 14. Synopsis of the performance of the two rocking-isolated frames in terms of foundation settlement wwith respect to the maximum PSV of theseismic excitation: (a) no tie beams compared to (b) hybrid tie beams; the differential settlement between the middle and the side footings, evident in allcases, averages around 40%

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unloading of the left footing, inevitably leading to sliding andeven complete detachment from the bearing soil. The ensuinglanding onto the ground surface at a locationwhere the soil has,up to that point, undergone less intense yielding deformation,results in relatively reduced dynamic settlement. Combinedwith the inherent tendency of themiddle footing to accumulatemore settlement, as it carries a larger axial load, the resultingdifferential settlement may lead to additional (kinematicallyinduced) frame distress. Of course, in reality, such footings areusually embedded (to some extent) and such a behavior is lesslikely to be observed.

3. To alleviate the problem of differential displacements, theaddition of tie beams between the footings has been explored.Besides the conventional fully fixed tie beams, two additionalsolutions are conceived and analyzed. The key conclusions areas follows:• Conventional fully fixed tie beams may reduce the differ-

ential settlements and partially hinder the detachment of theleft footing, but prevent rocking isolation as their rotationalresistance is added to that of the footing, rendering the per-formance of the frame almost identical to that of the conven-tionally designed system (with overdesigned foundations).

• In an attempt to maintain their beneficial effects withoutcanceling rocking isolation, two separate hinged tie beamswere placed between the footings. In contrast to conven-tional fully fixed tie beams and owing to their hingedconnection, these beams do not increase the capacity ofthe foundation system, allowing the frame to behaveaccording to its rocking-isolation design. But, the differ-ential settlements are hardly reduced: the hinged tie beamscan rotate as rigid bodies, without mobilizing their flexuralresistance and, hence, without offering substantial resis-tance in the vertical sense. Naturally, their axial rigidity is

fully mobilized, leading to a reduction of the horizontaldifferential displacements.

• Aiming to combine the advantages of conventional (fullyfixed) and hinged tie beams, a hybrid concept has beenconceived and analyzed: a single (continuous) tie beamconnected with external hinges to the three footings. Thishybrid design allowsmobilization of the flexural rigidity ofthe tie beams, leading to homogenization of the settlementsof the three footings and to an impressive reduction ofdifferential settlements, without preventing rocking.

Acknowledgments

The authors are thankful for the financial support provided throughthe research project DARE, by the European Research Council’s(ERC’s) IDEAS Programme, in Support of Frontier Research.Contract No. ERC-2-9-AdG228254-DARE. The authors also ac-knowledge the anonymous reviewers for their very thoughtfulcomments and suggestions.

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