direct generation of floor design spectra (fds) from ......draft direct generation of floor design...

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Direct generation of Floor Design Spectra (FDS) from Uniform Hazard Spectra (UHS) - Part II: Extension and

Application of the method

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2018-0151.R3

Manuscript Type: Article

Date Submitted by the Author: 08-Dec-2019

Complete List of Authors: Asgarian, Amin; McGill University, Department of Civil Engineering and Applied MechanicsMcClure, Ghyslaine; McGill University, Department of Civil Engineering and Applied Mechanics

Keyword:Spectrum-to-spectrum method, Operational and Functional Components (OFCs), Secondary systems, Earthquake Engineering, Seismic Assessment and Design

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjce-pubs

Canadian Journal of Civil Engineering

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Direct generation of Floor Design Spectra (FDS) from

Uniform Hazard Spectra (UHS) - Part II: Extension and

Application of the method

Amin Asgarian 1, Ghyslaine McClure 1

1 Civil Engineering and Applied Mechanics, McGill University

Corresponding author:

Name: Amin Asgarian

Address:

Civil Engineering Department, McGill University

Macdonald Engineering Building

817 Sherbrooke Street West

Montreal, Quebec H3A 0C3

Tel: +1 514-398-6860

Email address: [email protected]

Word count: 7702

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Abstract

This paper extends the methodology presented in the companion paper (Asgarian and McClure,

2019) to study the effects of non-structural components’ (NSCs) damping ratio and their

location in the building on the Pseudo Acceleration Floor Response Spectra (PA-FRS) of

reinforced concrete buildings, and propose equations to derive Floor Acceleration Design

spectra (FDS) directly from the Uniform Hazard Design spectra (UHS) for Montreal, Canada.

The buildings used in the study are 27 existing reinforced concrete structures with braced

frames and shear walls as their lateral load resisting systems: 12 are low-rise (up to 3 stories

above ground), 10 are medium-rise (4 to 7 stories) and 5 are high-rise (10 to 18 stories). Based

on statistical and regression analysis of floor accelerations spectra generated from linear

dynamic analysis of coupled building-NSC systems, two sets of modification factors are

proposed to account for floor elevation and NSC damping, applicable to the experimentally-

derived FDS for roof level and 5% NSC damping. Modification factor equations could be

derived only for the low-rise and medium-rise building categories, as insufficient correlation

in trends could be obtained for high-rises given their low number. The approach is illustrated

in details for two typical buildings of the database, one low-rise (Building #4) and one medium-

rise (Building#18), where the proposed FDS/UHS results show agreement with those obtained

from detailed dynamic analysis. The work is presented in the context of a more general

methodology to show its potential general applicability to other building types and locations.

Keywords: Spectrum-to-spectrum method; Operational and Functional Components (OFCs);

Secondary systems; Earthquake Engineering; Seismic Assessment and Design.

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1. Introduction

In response to the current increasing demand for high performance structures, careful attention

must be paid to seismic design and assessment of the operational and functional components

of buildings (OFCs), generally referred to as Non-Structural Components (NSCs) in earthquake

structural engineering literature. As experienced in past earthquakes, many buildings have

failed meeting their performance objectives solely due to failure/malfunction of their NSCs

while the structural elements and systems have performed satisfactorily as per design. Damage

to NSCs is often ensued by some undesired aftereffects which are mainly associated with: a)

life-safety hazards (i.e. fatalities/injuries caused by falling/overturning NSCs and etc. (Adham

and Ballif 1985; Taly 1988; Reitherman et al. 1995), b) property loss due to direct/indirect

damage costs (e.g. major part of approximate economic loss of 25 billion dollar in 2010 Maule,

Chile earthquake (Miranda et al. 2012) and 2 billion dollar in 2001 Nisqually (Seattle)

earthquake (Filiatrault et al. 2001), and c)- loss of building functionality (e.g. impairment or

complete shut-down of 130 hospitals in 2010 Maule, Chile earthquake (Miranda et al. 2012)

and of 32 commercial data processing centers in 1989 Loma Prieta earthquake (Ding et al.

1990). A comprehensive description of these consequences accompanied with several

examples can be found in FEMA E-74, “Reducing the Risk of Nonstructural Earthquake

Damage – A Partial Guide” (FEMA E-74 2011).

These observations clearly demonstrate the essential need for a reliable approach to properly

quantify the two main Engineering Demand Parameters (EDP) needed for seismic

design/analysis of NSCs. Depending on the type of NSCs, two EDPs are required: 1- story

drift/displacement demand needed for drift-sensitive components, and 2- component

acceleration demand needed for acceleration-sensitive components. After the introduction of

the displacement-based design approach, firstly in 1993 in New Zealand (Priestley 1993), many

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studies (Priestley et al. 2007; Sullivan 2011; Calvi 2014; and Welch et al. 2014, to name a few)

have focused on quantifying story drift demand and as a result a range of reliable approaches

have been developed to estimate building displacement demand, and drift-sensitive NSC

design can then proceed using the inter-story drift limits corresponding to their required target

performance levels. However, there still appears to be a need for a simplified method that can

properly quantify acceleration demand on NSCs (i.e. acceleration-sensitive components). In

this regard, most of the current building codes contain empirical equations to estimate NSC

acceleration demand but they remain incapable of considering several key factors such as the

effects of building higher frequency modes and torsional modes and the effect of NSCs internal

damping. These shortcomings have been exposed in several studies such as in Mondal and Jain

2005; Sullivan et al. 2013; Wieser et al. 2013; and Petrone et al. 2014. In particular, it has been

shown that the peak NSC accelerations derived from first-mode building response predictions

are inaccurate, especially at roof or top floor level; they are typically largely underestimated in

the building short-period range, which is of concern considering the large building stocks of

low-rise and medium-rise constructions in most densely populated areas. The fact remains that

floor acceleration responses may vary considerably depending on elevation, and NSC response

may vary depending on the actual NSC location on the floor, away from the center of rigidity

or mass, especially when torsion is present. Moreover, for NSC having ξNSC < 5%, ignoring

NSC damping properties may lead to considerable underestimation of the NSC acceleration

demand if based on the nominal 5% first-mode damping approach used for building design.

In an attempt to introduce alternative approaches capable of resolving these issues, there have

been many efforts (see Biggs 1972; Singh 1980; Villaverde 2004; Sullivan et al. 2013; Calvi

and Sullivan 2014; among others) to develop Floor Response Spectra (FRS), which provide

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the acceleration demand on NSCs as a function of their fundamental period so each component

can be assessed exclusively.

A step forward to these improvements is to develop Floor Design Spectra (FDS) that can be

used for seismic assessment of NSCs in a similar way as Design Response Spectra (DRS) are

for structural elements. In a companion paper (Asgarian and McClure 2019) the authors have

introduced the methodology to generate Pseudo Acceleration Floor Response Spectrum (PA-

FRS) using data extracted from Ambient Vibration Measurement (AVM) records; the

methodology has been applied to a database of 27 existing Reinforced Concrete (RC) frame

buildings, all located in Montreal Canada and designated as post-disaster buildings. PA-FRS

were derived for every floor of the buildings considering four different NSC critical damping

ratios. The generated PA-FRS have been statistically analysed and a set of equations proposed

to generate FDS for low and medium-rise RC frame buildings directly from the corresponding

5% damped Uniform Hazard Spectra (UHS). At that stage, the proposed method was limited

to generate the FDS only at the roof level and for NSCs with 5% viscous damping ratio

(ξNSC=5%). This paper examines the effect of NSCs location in the building (using the Z/H

ratio where Z is the NSC-supporting floor elevation, and H is the roof height) and internal

damping ratios (ξNSC) on the FDS. It uses statistical and regression analysis and extends the

application of the method to cover any selection of floor level (0.0≤Z/H≤1.0) and NSCs’

damping ratios (1%≤ξNSC≤20%). A wide range of underdamped ratios was considered to

provide a larger perspective of the effect of this parameter on the FDS, also reflecting the large

variability in connection details for various NSCs and noting that NSC damping can be

measured on site. Two additional sets of modification factors are therefore introduced and

added to the equations derived in the companion paper to account for the NSCs’ floor location

(Z/H) and damping (ξNSC). In the following sections, the extended methodology is described in

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details and its application through two complete case-studies, Building#4 (low-rise) and

Building#18 (medium-rise) of the database, is presented. These two examples are deemed

representative of the reinforced concrete buildings of their respective height category, for

which AVM data were collected (See Table 2 of the companion paper Asgarian and McClure

2019 for detailed information on the building database.)

The goal and significance of the research is to provide a methodology to derive FDS for NSC

seismic design or vulnerability assessment in new or existing buildings, directly from the UHS

specified for the building site. For any particular existing building, AVM tests or other types

of dynamic measurements will provide direct measures of the dynamic interactions between

the NSC and the building structure.

2. Code provisions for acceleration-sensitive NSCs

Acceleration-sensitive NSCs are those components which are designed against seismically

induced acceleration/force while building drift is not a controlling factor for them. Current

building codes address these components by establishing a set of design requirements in the

form of empirical equations to calculate the equivalent static seismic force that the components

and their connections must resist. Table 1 briefly summarizes the recommendation of three

well-known building codes in the versions available at the time of the study: 1- Canadian

(NBCC 2015 (National Research Council of Canada (NRC) 2015)), 2- American (ASCE/SEI-

07-16 (American Society of Civil Engineers 2016)), and European (Eurocode 8, EN. 1988. 1.

2004. (Comité Européen de Normalization 2004)). Detailed descriptions of the parameters and

their recommended values can be found in (Comite Europeen de Normalization 2004; National

Research Council of Canada (NRC) 2015; American Society of Civil Engineers 2016). All

these code equations are conceptually similar in the sense that they compute the seismic force

demand as a multiplication of component weight by the component peak acceleration. They

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also have similar shortcomings as they do not consider the effects of higher frequency and

torsional modes of the primary system and the effects of NSC damping. Except for Eurocode

8, the other two also disregard the tuning effect (i.e. matching the natural period of NSCs with

one of the fundamental periods of supporting system which causes resonance). One could argue

that tuning/detuning effects are adequately covered by the simplified code procedures that

prescribe a component force modification factor (typically an Ar value of 2.5 in the National

Building Code of Canada – Art. 4.1.8.18.) for all flexible or flexibly-connected elements.

However, this approach essentially considers that all flexible acceleration-sensitive elements

will enter resonance during the design earthquake, irrelevant of their natural frequencies, and

that if there is indeed resonance the amplification is capped by a fixed factor. This may partly

explain why in many cases, the code estimation of acceleration demand on NSCs is of limited

accuracy and reliability.

3. Methodology to extend the proposed approach

Figure 1 schematically explains the steps of the global research methodology, which had to be

applied to generate the results presented in the both companion papers to generate floor

acceleration design spectra directly from uniform hazard spectra. 20 compatible seismic record

with the NBCC 2015 UHS for Montreal (National Research Council of Canada (NRC) 2015)

were obtained based on the works by Atkinson et al. (Atkinson and Beresnev 1998; Adams and

Atkinson 2003; Atkinson 2009) and used as input excitations. The floor response histories of

every floor of the 27 buildings of the database were derived in terms of relative displacement

(results not discussed here) and absolute acceleration. A program has been implemented in

MATLAB (The MathWorks Inc. 2014) which takes the floor response histories as input and

performs a linear dynamic analysis of the NSC system, where the NSC is modeled as a

viscously damped, single-degree-of-freedom oscillator. The MATLAB analysis is repeated for

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each floor of the 27 buildings of the database and an ensemble of 20 synthetic seismic records

compatible with the NBC 2015 UHS for Montreal (National Research Council of Canada

(NRC) 2015), according to the methodology established by Atkinson et al. (Atkinson and

Beresnev 1998; Adams and Atkinson 2003; Atkinson 2009). The NSC time history acceleration

responses are used to generate the NSC pseudo-acceleration spectra (PA-FRS) for any selection

of floor level, NSC damping ratio (ξNSC), and NSC period range of interest (TNSC). The

horizontal axis of the PA-FRS is normalized with respect to the fundamental period of the

building (T1-B) to provide a proper parameter used to reflect tuning effects. A detailed

description of the MATLAB code and its validation can be found in Asgarian et al. 2014 and

Asgarian and McClure 2019. In the study reported here, four NSC damping ratios have been

considered (2, 5, 10, and 20% critical viscous damping) and approximately 132,000 PA-FRS

have been generated. Statistical analysis of these normalized PA-FRS has been used to develop

an approach for generation of FDS directly from UHS. At the first step in Part 1 (Asgarian and

McClure 2019), a set of equations were recommended to generate FDS directly from UHS for

each building exclusively using its natural frequency (T1-B) and its design spectral acceleration

(UHS(T1-B)), while the method was limited to produce FDS for roof level and 5% NSC

damping. This paper extends the proposed methodology to generate FDS for NSCs located at

any floor level and having any viscous damping ratio. The effect of NSCs’ location/elevation

in the building (i.e. Z/H) is quantified through statistical analysis of the generated PA-FRS for

different floor levels of the various buildings. Similarly, the effect of NSCs damping ratios is

obtained from statistical and regression analysis of the dynamic analysis results corresponding

to various NSCs damping ratios. Finally, a set of complete equations is recommended to

develop FDS directly from UHS for any selection of floor level (0.0≤Z/H≤1.0) and NSCs’

damping ratio (1%≤ξNSC≤20%). These equations define normalized NSC acceleration spectra

with respect to UHS as a function of the period ratio, TNSC/T1-B. While the proposed method

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can be applied to a larger database of existing buildings to provide code recommendations for

FDS appropriate to various building categories of seismic load resisting systems and heights,

the FDS equations derived next are valid only for medium and low-rise reinforced concrete

buildings of the type included in the database.

4. Effect of NSCs’ elevation on FDS

The seismic acceleration demand on NSCs varies depending on the location/elevation of the

component in the building. Moving up from ground level to roof level, the NSC acceleration

response will, in general, increase and be filtered by the dynamic characteristics of the building

(i.e. the response corresponding to dominant building frequencies is stronger) rather than the

frequency content of the ground motion. The generation of FDS for roof level of RC buildings

was described in the companion paper (Asgarian and McClure 2019) and as mentioned above

this second part extends the application to any selected floor level. To do so, the effect of NSCs’

location along the height of the building on the FDS, referred to as the relative height effect

(denoted Z/H hereafter), is addressed through statistical analysis of the generated PA-FRS for

different floor levels. With a view to identify trends in the response to develop more general

FDS models, the spectral domain is divided into three distinct ranges, namely short-period

(0<TNSC/T1-B≤0.7), building fundamental-period (0.7≤ TNSC/T1-B ≤1.0), and long-period region

(TNSC/T1-B ≥ 1). The generated PA-FRS for different floor levels of each building were plotted

on the same graph and the horizontal axis (i.e. T1-NCS) was normalized by the natural period of

the corresponding building (T1-B), which causes most of the peaks to be located in the building

fundamental-period region (0.7≤ TNSC/T1-B ≤1.0). To illustrate with a typical example for a

medium-rise building, Figure 2.a depicts the PA-FRS of all five floors (including roof and

excluding ground floor) of Building#18. As the FDS generation has been already formulated

for the roof level and ξNSC=5%, the roof response was taken as the reference of the analysis and

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the PA-FRS of all floor levels in each building (Sa NSC-Fl) are normalized with respect to the

roof level spectral acceleration values (Sa NSC-Roof), as shown in Figure 2.b as to emphasize the

variations in floor responses compared to the roof level response. Note that the PA-FRS of the

ground floor is excluded from the analysis as it is influenced only by the dynamic

characteristics of the seismic excitation. Next, the median+σ of Sa NSC-Fl/ Sa NSC-Roof ratio over

each selected spectral regions (i.e. short, fundamental, and long period regions) is calculated at

each floor level and the scatter of these ratios vs their corresponding relative height (Z/H) is

plotted and a curve is fitted to the data points in each of the three spectral regions. The

dispersion of the results obtained from the effects of the 20 selected ground motions was

quantified in the companion paper (Asgarian and McClure 2019) where the median plus one

standard deviation (σ) results were shown to provide a rational basis for design

recommendations.

The effects of relative height in low and medium rise buildings were observed to be different,

so each building category is addressed separately but using the same methodology, as discussed

next.

4.1. Low-rise buildings

The scatter of the (SaNSC-Fl / Sa NSC-Roof) ratio vs relative height (Z/H) for the short and

fundamental period regions (0.0< TNSC/T1-B ≤1.0) and the long period region (1.0≤ TNSC/T1-B

≤5.0) is illustrated in Figure 3.a and Figure 3.b, respectively. A linear trend of the PA-FRS is

assumed and a linear function is fitted to the data points. As the effect of relative height over

the short and fundamental period regions was found very similar, the data points for these two

spectral regions are combined and one set of recommendations is made for both regions.

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However, the long period range presents a different behaviour and was studied separately. The

equation of the fitted line accompanied with is coefficient of correlation (R2 value) can be seen

on the graphs. As could been seen in Figure 2.b, and quantified in Figure 3, lower floors show

close acceleration to roof level acceleration over the long period regions and the linear

regression provides a modification factor of 0.80 + 0.2 Z/H (regression equations shown in

Figure 3.b). However, for the more rigid NSC ranges (short and fundamental period regions)

the effect of floor elevation is more important and regression provides a modification factor of

0.33 + 0.67 Z/H (regression equations shown in Figure 3.a). These findings are significant and

clearly confirm the benefits of relocating rigid/rigidly connected acceleration-sensitive NSCs

at lower building levels inasmuch as possible unless the components are adequately isolated

for transverse vibrations. Using these regression formulas, the previously produced FDS for

roof level for ξNSC=5% can be now modified for the lower floor levels. The effect of NSC

damping is addressed in the next section and incorporated as well. The final FDS

recommendations for RC low-rises are illustrated for Building#4 in Section 6.

4.2 Medium-rise buildings

The same analysis as above is performed for the medium-rise buildings of the database. The

data points of the short and fundamental period regions are combined again due to their similar

trends while the long period region is again addressed separately. In medium-rise buildings,

the points are distributed more evenly along the horizontal axis compared to the clustered

pattern observed in the low rises: This is essentially caused by the larger number of floor levels

in medium rises. The results of the short and fundamental period and long period regions are

illustrated in Figures 4.a and 4.b, respectively. The modification factor in the large period ratio

range is the same as for low-rise buildings (0.8 + 0.2 Z/H) (regression equations shown in

Figure 4.a), while the effect is less important in the pre-resonance and resonance ranges, with

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a modification factor of 0.64 + 0.36 Z/H (regression equations shown in Figure 4.b). As can

be seen, the recommendations for medium and low rises are different while the building codes

do not make that distinction. Illustration of the final recommendations for RC medium-rise

buildings is shown for Building#18 in Section 6.

5. Effect of NSCs’ internal damping on FDS

The effect of NSCs’ damping on FDS is quantified using the same statistical methodology

adopted for the study of relative height effect. To measure this effect, the generated PA-FRS

for different NSCs damping ratios (2, 5, 10 and 20 % of viscous critical) at each floor level

were plotted on the same graph and the horizontal axis (i.e. TNCS) was normalized by the natural

period of the corresponding building (T1-B), which brings most of the peak responses in the

fundamental-period region (0.7≤ TNSC/T1-B ≤1.0). Figure 5.a depicts the PA-FRS for the roof

of Building#18 considering the selected four NSCs damping ratios. The PA-FRS for ξNSC=5%

at each floor level is taken as the reference for the analysis and the PA-FRS values obtained at

each floor are normalized with respect to the PA-FRS values for ξNSC=5% of the corresponding

floor and presented as a function of TNSC/T1-B. Figure 5.b illustrates the normalized PA-FRS

for different NSCs damping ratios of all floors and roof of Building#18 (Sa ξ / Sa ξ=5% vs

TNSC/T1-B). It was observed that the results of all floors are close and consistent with each other

in each mid-rise building, confirming that the relative effect of NSC damping is independent

of the NSC location along building height. Ratio over each selected spectral region (i.e. short,

fundamental, and long period regions) is calculated. The scatter of these normalized PA-FRS

values (Sa ξ / Sa ξ=5%) is studied in each spectral region with a linear regression. After

reviewing the results obtained for all the low-rise and medium-rise buildings of the database,

it was observed that the effect of NSCs damping throughout the spectral range was similar and,

hence, their data points are combined and studied together next.

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The scatters of the (Sa ξ / Sa ξ=5%) ratio vs NSCs damping ratio (ξNSC) for the short (0.0< TNSC/T1-

B ≤0.7), fundamental (0.7< TNSC/T1-B ≤1.0), and long period regions (1.0< TNSC/T1-B ≤5.0 are

illustrated in Figure 6.a, 6.b and 6.c, respectively. Different exponential and rational functions

have been tested for data fitting, and the rational function in the form of Y=(a.X+b)/(X+c) was

found to best represent the data. As the effects of NSC damping over different spectral regions

were found to be different, separate analyses have been done in each segment. The equation of

the fitted curves accompanied with its R2 value can be seen on the graphs of Figure 6. Using

these equations, the previously produced FDS for ξNSC=5% is modified for the other ξNSC

values. The results of Figure 6 clearly indicate that the amount of NSC damping has an

important influence on the magnitude of the acceleration response, and the effect differ

depending on the fundamental period ratio of the NSC and the building. For lower damping

(say 2%) the values of the accelerations (normalized to the values obtained with ξNSC=5%) are

amplified by a factor of the order of 1.1 to 1.3 in the lower range, by as much as 1.3 to 1.5 in

the resonance range, and by 1.3 in the higher range. As one would expect, larger damping (say

20%) significantly reduces the NSC acceleration response and the reduction factors are of the

order of 0.8, 0.4-0.6, and 0.6, in the three regions, respectively, and more scatter is observed in

the resonant range.

This section presents a set of equations recommended to develop FDS directly from UHS for

any selection of floor level (0.0≤Z/H≤1.0) and NSCs’ damping ratio (1%≤ξNSC≤20%) in both

low and medium rise buildings. In the companion paper (Asgarian and McClure 2019a) the

FDS generation for roof level and 5% NSC damping was presented. Figure 7 schematically

shows how the spectral acceleration is idealized in each spectral region for both low and

medium rises, and the final recommendations are described in sections 5.1 and 5.2,

respectively. Essentially, the regression functions obtained in Figures 3 and 4 (to account for

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floor elevation effects - Z/H) and Figure 6 (to account for ξNSC) multiply the spectral

acceleration values obtained at roof level (Z/H = 1) for ξNSC = 5%. The complete sets of

equations are presented next.

5.1. Low-rise buildings

As illustrated in Figure 7, the recommended FDS has a linear variation in the short-period

region (point “a” to point “b”), a constant value in fundamental-period region (points “b” to

“c”), and decays according to a rational function in the long-period region (points “c” to “d”).

The following equations describe how the FDS values are calculated in each spectral region

for RC low-rise buildings. It should be mentioned that in all the recommended equations, the

first bracket is to calculate the FDS values at roof level given 5% NSC damping (discussed in

the companion paper Asgarian and McClure 2019), the second bracket is the modification

factor which accounts for relative height effect (0.0≤Z/H≤1.0), and the third bracket is the

modification factor that accounts for NSCs’ damping effect (1%≤ξNSC≤20%).

In the short-period region, the FDS values are increased linearly from point “a” at TNSC/T1-B =

0.0 to point “b” at TNSC/T1-B = 0.7. Values of point “a” and “b” can be calculated according to

Equation 1:

𝑆𝑎𝑁𝑆𝐶

𝑈𝐻𝑆(𝑇1 ― 𝐵) = {[2.0] × [0.33 + 0.67(𝑍𝐻)] × [0.69 × 𝜉𝑁𝑆𝐶 + 3.33

𝜉𝑁𝑆𝐶 + 1.78 ] @ "a", 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵= 0.0

[10.5] × [0.33 + 0.67(𝑍𝐻)] × [0.14 × 𝜉𝑁𝑆𝐶 + 7.36

𝜉𝑁𝑆𝐶 + 3.06 ] @ "b", 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵= 0.7

Equation 1

In the fundamental-period region, the FDS has a constant value determined at point “b” using

Equation 1, between points “b” at TNSC/T1-B = 0.7 and “c” at TNSC/T1-B = 1.0. In the long-period

region, the value of FDS is calculated according to Equation 2:

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𝑈𝐻𝑆(𝑇1 ― 𝐵) = 𝑚𝑖𝑛{ [10.5] × [0.33 + 0.67(𝑍𝐻)] × [0.14 × 𝜉𝑁𝑆𝐶 + 7.36

𝜉𝑁𝑆𝐶 + 3.06 ][ 1.89

( 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵) ― 0.82] × [0.8 + 0.2(𝑍

𝐻)] × [0.3 × 𝜉𝑁𝑆𝐶 + 8.3𝜉𝑁𝑆𝐶 + 4.8 ]} : 1.0 ≤

𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵≤ 5.0

Equatio

n 2

The FDS is taken as the minimum of the two proposed formulas. It is seen that the first

corresponds to the value of point “c” for a period ratio of 1.0, as obtained from Equation 1,

while the second is the rational function corresponding to the long-period region. It has been

observed in the companion paper that the first term in the square bracket of the lower part of

Equation 2 that gives the reference spectral value at rooftop does, in some cases, overestimate

the FDS values in the vicinity of TNSC/T1-B = 1.0, so the value is capped at point “c”. If the FDS

is required to be extended for a longer range, 5.0 ≤ TNSC/T1-B ≤10.0, a conservative and simple

approach is proposed where the SaNSC/UHS(T1-B) is decreased linearly from its value at

TNSC/T1-B = 5.0 to half of that at TNSC/T1-B =10.0.

5.2. Medium-rise buildings

For RC medium-rise buildings, FDS is generated using the same methodology as described for

low-rise buildings (See Section 5.1) but using a different set of equations are described below,

where the modification factors are obtained from the appropriate regression functions presented

in Section 5.

In the short-period region, the FDS values are increased linearly from point “a” at TNSC/T1-B =

0.0 to point “b” at TNSC/T1-B = 0.7. Values of point “a” and “b” can be calculated according to

Equation 3.

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𝑈𝐻𝑆(𝑇1 ― 𝐵) = {[3.0] × [0.2 + 0.8(𝑍𝐻)] × [0.69 × 𝜉𝑁𝑆𝐶 + 3.33

𝜉𝑁𝑆𝐶 + 1.78 ] @ "a", 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵= 0.0

[12.0] × [0.2 + 0.8(𝑍𝐻)] × [0.14 × 𝜉𝑁𝑆𝐶 + 7.36

𝜉𝑁𝑆𝐶 + 3.06 ] @ "b", 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵= 0.7

Equation 3

In the fundamental-period region, the FDS takes the constant value determined at point “b”

using Equation 3, between points “b” at TNSC/T1-B = 0.7 and “c” at TNSC/T1-B = 1.0. In the long-

period region, the value of FDS is calculated according to Equation 4:


𝑈𝐻𝑆(𝑇1 ― 𝐵) = 𝑚𝑖𝑛{ [12.0] × [0.2 + 0.8(𝑍𝐻)] × [0.14 × 𝜉𝑁𝑆𝐶 + 7.36

𝜉𝑁𝑆𝐶 + 3.06 ][ 1.68

( 𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵) ― 0.86] × [0.64 + 0.36(𝑍

𝐻)] × [0.3 × 𝜉𝑁𝑆𝐶 + 8.3𝜉𝑁𝑆𝐶 + 4.8 ]} : 1.0 ≤

𝑇𝑁𝑆𝐶

𝑇1 ― 𝐵≤ 5.0

Equatio

n 4

As explained previously for low-rise buildings, the FDS is taken as the minimum of the two

expressions in Equation 4 as the spectral values need to be capped at resonance (for the period

ratio of 1.0). Likewise, if the FDS is required to be extended for 5.0 ≤ TNSC/T1-B ≤10.0, the

same approach as indicated for lower-rise buildings can be used.

Equations 1 to 4 provide a general formulation to determine the FDS of low-rise and medium-

rise reinforced concrete buildings within the range of building characteristics of the database

described in the companion paper, located in Montreal. These equations can be used directly

for new buildings and existing buildings for which the dynamic characteristics are not known

from field testing (AVM or other methods). The full advantages of the formulation, however,

are obtained when the modal characteristics of the buildings are determined experimentally, as

was done for the database, and the general methodology proposed in Figure 1 is applied, in

which case the floor responses are determined more accurately. The full-fledged method, which

does not require a detailed structural analysis model of the building, is ideally suited for the

risk assessment of existing NSCs in buildings, as a more realistic seismic response can be

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predicted based on the measured dynamic characteristics of the building and NSC system,

therefore accounting for any special floor distribution effects (higher modes and vertical

irregularities), torsional effects (horizontal irregularities) and tuning effects. Furthermore, it is

a promising tool for the rapid assessment of seismic functionality in buildings for which

detailed structural drawings are not readily available or incomplete.

The process of generating FDS for both low and medium-rise buildings according to the above

equations was coded in the MATLAB program (The MathWorks Inc. 2014). The extended

code requires four inputs: the fundamental period of the building (T1-B), its corresponding

uniform hazard design spectral acceleration (UHS(T1-B)), the number of floors and their

corresponding heights, and the category of the building (either low-rise or medium-rise).

6. Illustration of FDS results using the proposed methodology

Here, the application of the proposed method is presented through generation of FDS for two

typical examples of RC buildings of the database: Building#4, as a low-rise example, and

Building#18, as a medium-rise. The proposed FDS for all floors of both buildings considering

four different NSC damping ratios (2, 5, 10, and 20% of critical viscous damping) are generated

using the MATLAB code (The MathWorks Inc. 2014) and compared with the corresponding

PA-FRS derived from the dynamic analysis. A summary of the building information

accompanied with the corresponding results are presented next.

6.1. Building#4 (Low-rise building)

Building#4 (label is referring to the database described in Asgarian and McClure 2019 is a

reinforced concrete moment frame (RCFM) low-rise building with three stories above ground.

The general information of the building, typical plan view, elevation view, and the AVM results

for the first three modes are summarized in Table 2. The building was constructed in 1957 and

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the structural drawings available were of relatively poor visual quality. The mode shapes are

illustrated schematically where the blue color shapes show the building at rest and the green

color represents the deformed modal shape corresponding to the extracted natural frequency.

Figure 8 shows the proposed FDS (Solid lines) for all three floors of Building#4 given four

different NSC damping ratios compared with the corresponding PA-FRS (dashed lines) derived

from the dynamic analysis of the building. The comparison shows that the proposed

methodology provides a good match with the detailed analysis results and confirms the

reliability the generated FDS to estimate the seismic acceleration demand on NSCs with any

damping ratio and located at any floor level.

6.2. Building#18 (Medium-rise building)

Building#18 (label is referring to the database described in Asgarian and McClure 2019) is a

medium-rise reinforced concrete shear wall (RCSW) building with five stories above ground.

The general information of the building, typical plan view, elevation view, and the AVM results

for the first three modes are summarized in Table 3. The mode shapes are illustrated

schematically where the blue color shapes show the building at rest and the green color

represents the deformed modal shape corresponding to the extracted natural frequency.

Figure 9 shows the proposed FDS (Solid lines) for all five floors of Building#18 given four

different NSC damping ratios compared with the corresponding PA-FRS (dashed lines) derived

from the dynamic analysis of the building. Again, a good consistency can be seen between the

proposed FDS and the PA-FRS obtained from analysis, which indicates that the proposed

method is capable of reliably estimating the seismic acceleration demand on NSCs.

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7. Summary and Conclusion

The work presented builds on the companion paper (Asgarian and McClure 2019) that

proposed an original approach to generate Floor Design Spectra (FDS) at roof level of

Reinforced Concrete (RC) low and medium- rise buildings given 5% NSC internal damping.

The recommendations were formulated for each building category (i.e. RC low and medium

rises) to produce FDS directly from the 5% damped design response spectrum specified in

building codes (Uniform Hazard Spectrum (UHS) of NBCC 2015 for Montreal to be specific).

The FDS is produced for three distinct spectral regions (i.e. short, fundamental, and large

period regions).

To describe the major outputs of this part of the study, the proposed FDS equations for roof

level were extended to produce FDS for any floor level and NSC damping ratio. To achieve

this result, the Pseudo Acceleration Floor Response Spectra (PA-FRS) have been derived for

every floor of the buildings in the database (12 low-rise, 10 medium-rise, and 5 high-rise)

considering four different NSC damping ratios (2, 5, 10, and 20 % viscous damping).

Approximately 132,000 PA-FRS have been generated for statistical analysis. The effects of

NSCs damping ratio (ξNSC) and their location along the building height (Z/H) on the derived

PA-FRS have been quantified through statistical analysis and a height and a damping

modification factors have been introduced in the form of regression functions in which the

period ratio of the NSC and the fundamental period of the building (TNSC/TB-1) is the

independent variable. Two sets of extended equations are recommended, one for low-rise

(Equations 1 and 2) and one for medium-rise buildings (Equations 3 and 4). No FDS/UHS

equations were proposed for the high-rise building category because the small number of cases

in the database for Montreal could not provide sufficient correlation in the regression functions.

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The recommended equations for low-rise and medium-rise buildings have been coded in the

MATLAB program (The MathWorks Inc. 2014) and then applied over the database, for 12-

low rise and 10 medium-rise buildings. The FDS were generated for every floor of these

selected buildings given four different NSC damping ratios and compared with the

corresponding PA-FRS derived from dynamic analysis. The comparison was illustrated for the

case of low-rise Building#4 and medium-rise Building#18 and showed consistency between

the results which attests the validity reliability of the proposed FDS/UHS equations 1 to 4,

within the limits of their derivation:

The UHS used is for Montreal, Canada. As such, the results could also be directly

applicable to the Ottawa region that has comparable seismicity characteristics. A similar

study could be conducted using a database of buildings located in a seismicity area with

different tectonic characteristics, for example in the Canadian West Coast cities of

Victoria and Vancouver where seismic risk is high and more severe shaking is exposed

in the low-period range.

The buildings are not located or on sensitive or possibly liquefiable soil, as such

conditions require special attention to properly account for soil-structure interaction

effects on the building response.

The buildings are not seismically isolated.

The building types are reinforced concrete shear walls and moment resisting frames. The

low-rise equations are valid for buildings with no more than 3 stories above ground

while the mid-rise equations are applicable to cases of 4 to 7 stories. Testing of more

buildings of various structural types and taller buildings would allow to derive equations

that would further extend the applicability of the tool.

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The NSC is assumed to behave as a linear single-degree-of-freedom system in the

analysis. The FDS will provide the force demand necessary to design/assess the

connection of the NSC to the supporting floor, assuming this connection remains linear

elastic under seismic shaking. In future work, dynamic analysis could be expanded to

include nonlinear NSC response.

The full advantages of the proposed FDS/UHS formulation, however, are obtained when the

modal characteristics of the buildings are determined experimentally, as was done for the

database, and the general methodology proposed in Figure 1 is applied, in which case the floor

responses are determined more accurately. The full-fledged method, which does not require a

detailed structural analysis model of the building, is ideally suited for the risk assessment of

existing NSCs in buildings, as a more realistic seismic response can be predicted based on the

measured dynamic characteristics of the building and NSC system, therefore directly

accounting for any special floor distribution effects (higher modes and vertical irregularities),

torsional effects (horizontal irregularities) and tuning effects. Furthermore, it is a promising

tool for the rapid assessment of seismic functionality in buildings for which detailed structural

drawings are not readily available or incomplete. The authors have found it particularly useful

to assess the seismic functionality of acceleration-sensitive NSCs in existing post-critical

buildings that have to remain operational (in whole or in part) during and after a design level

seismic event.

8. Acknowledgment

This work has received financial support from the Natural Sciences and Engineering Research

Council of Canada, strategic grant project STPGP-396464, and McGill University’s MEDA

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program for the first author in the form of a doctoral fellowship. The detailed results of the

study are available by request to the first author.

9. References

Asgarian, A., Mirshafiei, F., and McClure, G. 2014. Experimental floor response

spectra for seismic evaluation of operational and functional components of building. CSCE

2014, 4th International Structural Specialty Conference. Halifax, NS, May 28 to 31: 10.

Asgarian, A., and McClure, G. 2019. "Generation of experimental floor response

spectra for seismic assessment of Non-Structural Components (NSCs) based on ambient

vibration measurements." Manuscript submitted for publication: 22.

Adams, J., and Atkinson, G. 2003. "Development of seismic hazard maps for the

proposed 2005 edition of the National Building Code of Canada." Canadian Journal of Civil

Engineering 30(2): 255-271.

Adham, S. A., and Ballif, B. 1985. "The Borah Peak, Idaho earthquake of October 28,

1983—buildings and schools." Earthquake spectra 2(1): 169-182.

American Society of Civil Engineers, 2016. ASCE Standard 7-16: Minimum Design

Loads for Buildings and Other Structures. Reston, VA, American Society of Civil Engineers.

Asgarian, A., and McClure, G. 2019. "Direct generation of Floor Design Spectra (FDS)

from Uniform Hazard Spectra (UHS) - Part I: Formulation of the method." Manuscript

submitted for publication: 19.

Atkinson, G. M. 2009. "Earthquake time histories compatible with the 2005 National

building code of Canada uniform hazard spectrum." Canadian Journal of Civil Engineering

36(6): 991-1000.

Atkinson, G. M., and Beresnev, I. A. 1998. "Compatible ground-motion time histories

for new national seismic hazard maps." Canadian Journal of Civil Engineering 25(2): 305-318.

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Biggs, J. M. 1972. Seismic response spectra for equipment design in nuclear power

plants. Proceedings of the First International Conference on Structural Mechanics in Reactor

Technology. Berlin, Germany. 5 (PT.K): 329-343.

Petrone. C., Magliulo, G., Cimmino, M., and Manfredi, G. 2014. Evaluation of the

seismic demand on acceleration sensitive nonstructural components in RC frame structures.

Second European conference on earthquake engineering and seismology. Instanbul, Turkey.

Calvi, P. M. 2014. "Relative displacement floor spectra for seismic design of non

structural elements." Journal of Earthquake Engineering 18(7): 1037-1059.

Calvi, P. M., and Sullivan, T. J. 2014. "Estimating floor spectra in multiple degree of

freedom systems." Earthquakes and Structures 7(1): 17-38.

Comite Europeen de Normalization, 2004. Eurocode 8–Design of Structures for

earthquake resistance–Part 1: General rules, seismic actions and rules for buildings. EN-1998-

1. Brussels, Belgium. 1.

Ding, D., Arnold, C., Lagorio, H., Tobriner, S., Rihal, S., Mangum, R., Hezmalhalch,

G., Green, M., Watson, A., and Mah, D. 1990. "Architecture, building contents, and building

systems." Earthquake Spectra 6(S1): 339-377.

FEMA E-74, 2011. Reducing the Risks of Nonstructural Earthquake Damage – A

Practical Guide. Washington, D.C., FEDERAL EMERGENCY MANAGEMENT AGENCY.

Filiatrault, A., Uang, C.M., Folz, B., Chrstopoulos, C., and Gatto, K. 2001.

"Reconnaissance report of the February 28, 2001 nisqually (seattle-olympia) earthquake."

SSRP: 67.

Miranda, E., Mosqueda, G., Retamales, R., and Pekcan, G. 2012. "Performance of

nonstructural components during the 27 February 2010 Chile earthquake." Earthquake Spectra

28(S1): S453-S471.

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Mondal, G., and Jain, S. K. 2005. "Design of non-structural elements for buildings: A

review of codal provisions." Indian concrete journal 79(8): 22-28.

National Research Council of Canada (NRC), Institute for Research in Construction

(IRC), 2015. National Building Code of Canada (NBCC). Ottawa, ON, Canada.

Priestley, M., Calvi, G., and Kowalsky, M. 2007. Direct displacement-based seismic

design of structures. 2007 NZSEE conference.

Priestley, M. N. 1993. "Myths and fallacies in earthquake engineering—conflicts

between design and reality." Bulletin of the New Zealand National Society for Earthquake

Engineering 26(3): 329-341.

Reitherman, B., Sabol, T., Bachman, R., Bellet, D., Bogen, R., Cheu, D., Coleman, P.,

Denney, J. , Durkin, M., and Fitch, C. 1995. "Nonstructural damage." Earthquake Spectra

11(S2): 453-514.

Singh, M. P. 1980. "Seismic design input for secondary systems." Journal of the

Structural Division 106(2): 505-517.

Sullivan, T. J. 2011. "An energy-factor method for the displacement-based seismic

design of RC wall structures." Journal of Earthquake Engineering 15(7): 1083-1116.

Sullivan, T. J., Calvi, P. M., and Nascimbene, R. 2013. "Towards improved floor

spectra estimates for seismic design." Earthquake and Structures 4(1): 109-132.

Taly, N. 1988. "The Whittier Narrows, California Earthquake of October 1, 1987-

Performance of Buildings at California State University, Los Angeles." Earthquake Spectra

4(2): 277-317.

The MathWorks Inc. 2014. MATLAB. Natick, MA.

Villaverde, R. 2004. Seismic Analysis and Design of Nonstructural Elements.

Earthquake engineering: From engineering seismology to performance-based engineering,

CRC Press.

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Welch, D., Sullivan, T., and Filiatrault, A. 2014. Equivalent Structural Damping of

Drift Sensitive Nonstructural Building Components. Proceedings of the 10th US National

Conference on Earthquake Engineering.

Wieser, J., Pekcan, G., Zaghi, A.E., Itani, A., and Maragakis, M. 2013. "Floor

accelerations in yielding special moment resisting frame structures." Earthquake Spectra 29(3):

987-1002.

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List of Figures

Figure 1 – Flow-chart of the global research methodology .....................................................31

Figure 2– Building#18, ξNSC=5%: a) PA-FRS for all floors of the building; b) Normalized PA-

FRS of all floors with respect to the roof. ................................................................................32

Figure 3 – Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for low-rises

given ξNSC=5%: a) median+σ of the ratios over short & fundamental period regions; b)

median+σ of the ratios over long period region.......................................................................33

Figure 4 - Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for medium-rise

buildings given ξNSC=5%: a) median+σ of the ratios over short and fundamental period regions;

b) median+σ of the ratios over long period region...................................................................34

Figure 5 - Building #18: a) Roof PA-FRS given ξNSC = 2, 5, 10, 20 % vs (TNSC / T1-B); b)

Normalized PA-FRS at all floors, for all ξNSC with respect to the values for ξNSC =5% at

corresponding floor. .................................................................................................................35

Figure 6 - Scatter of (Sa ξ / Sa ξ=5%) ratio vs NSC damping ratio (ξNSC) for low and medium-

rise buildings given: a) median+σ of the ratios over short period regions; b) median+σ of the

ratios over fundamental period region; c) median+σ of the ratios over long period region. ...36

Figure 7 – Schematic of the proposed FDS and idealization of spectral acceleration for NSCs

..................................................................................................................................................37

Figure 8 – Illustration and comparison of the proposed FDS and the real PA-FRS generated for

all floors of Building#4 considering NSCs damping ratios of 2, 5, 10, and 20 %...................38


all floors of Building#18 considering NSCs damping ratios of 2, 5, 10, and 20 %.................39

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List of Tables

Table 1 - Code provisions for acceleration-sensitive NSCs...................................................277

Table 2 - Structural information and AVM results of Building #4..........................................28

Table 3 - Structural information and AVM results of Building #18 …………………………29

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Table 1 - Code provisions for acceleration-sensitive NSCs§

NBC 2015

(Division B-Part 4) ; , 𝑽𝑷 = 𝟎.𝟑𝑭𝒂𝑺𝒂(𝟎.𝟐)𝑰𝑬𝑺𝑷𝑾𝑷 0.7 ≤ 𝑆𝑃 =

𝐶𝑃𝐴𝑟𝐴𝑥

𝑅𝑃≤ 4.0 𝐴𝑥 = 1 + 2

ℎ𝑥

ℎ𝑛

ASCE/SEI-07-16

(Chapter 13)

; 𝑭𝑷 = 𝟎.𝟒𝒂𝑷𝑺𝑫𝑺

(𝑹𝑷𝑰𝑷) × (𝟏 + 𝟐

𝒁𝑯) × 𝑾𝑷 1.6𝑆𝐷𝑆𝐼𝑃𝑊𝑃 ≤ 𝐹𝑃 ≤ 0.3𝑆𝐷𝑆𝐼𝑃𝑊𝑃

Eurocode 8

(Part 4.3.5) ; 𝑭𝒂 =

(𝑺𝒂𝑾𝒂𝜸𝒂)𝒒𝒂

𝑆𝑎 = 𝛼.𝑆.[ 3(1 +𝑍𝐻)

1 + (1 ―𝑇𝑎𝑇1)2 ― 0.5] ≥ 𝛼𝑆

§ Symbols are defined in Table 1 of the companion paper, Asgarian and McClure (2019).

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Table 2 - Structural information and AVM results of Building#4

Building # 4LLRS type RCMF Construction year 1957HA / HB (m) 11.0 / 0.0 Typical plan dimension (m)

NA / NB 3 / 0 L = 53.7 W = 11Typical plan view Elevation view

Modal properties extracted from AVMMode 1-Translation in X dir. Mode 2-Translation in Y dir. Mode 3-Torsionf = 5.42 Hz ξ = 1.5 % f = 5.69 Hz ξ = 1.3 % f = 10.0 Hz ξ = 2.0 %

Nomenclature: HA=Height above ground level [m], HB=Height below ground level [m], NA=Number of floors above ground level, NB=Number of floors below ground level, ξ=Modal viscous damping ratio (percentage).

N N N

Y

N

11

m

53.7

m

X

X

Z

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Table 3 - Structural information and AVM results of Building#18

Building # 18LLRS type RCSW Construction year 1988HA / HB (m) 19.6 / 3.6 Typical plan dimension (m)

NA / NB 5 / 1 L = 33.7 W = 24.6Typical plan view Elevation view

37.6 m

24.6

m

N

Modal properties extracted from AVMMode 1-Translation in Y dir. Mode 2-Translation in X dir. Mode 3-Torsionf = 2.52 Hz ξ = 2.32 % f = 2.76 Hz ξ = 1.66 % f = 3.56 Hz ξ = 2.76 %

Table nomenclature is the same as in Table 2.

N

N NY

X X

Z

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List of Figures

Figure 1 – Flow-chart of the global research methodology .....................................................31

Figure 2– Building#18, ξNSC=5%: a) PA-FRS for all floors of the building; b) Normalized PA-

FRS of all floors with respect to the roof. ................................................................................32

Figure 3 – Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for low-rises

given ξNSC=5%: a) median+σ of the ratios over short & fundamental period regions; b)

median+σ of the ratios over long period region.......................................................................33

Figure 4 - Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for medium-rise

buildings given ξNSC=5%: a) median+σ of the ratios over short and fundamental period regions;

b) median+σ of the ratios over long period region...................................................................34


Normalized PA-FRS at all floors, for all ξNSC with respect to the values for ξNSC =5% at

corresponding floor. .................................................................................................................35

Figure 6 - Scatter of (Sa ξ / Sa ξ=5%) ratio vs NSC damping ratio (ξNSC) for low and medium-

rise buildings given: a) median+σ of the ratios over short period regions; b) median+σ of the

ratios over fundamental period region; c) median+σ of the ratios over long period region. ...36


..................................................................................................................................................37


all floors of Building#4 considering NSCs damping ratios of 2, 5, 10, and 20 %...................38


all floors of Building#18 considering NSCs damping ratios of 2, 5, 10, and 20 %.................39

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Figure 1 – Flowchart of the global research methodology

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33

a)

TNSC/T1-B

0 1 2 3 4 5

SaN

SC [g

]

0

1

2

3

4

5Floor#1Floor#2Floor#3Floor#4Roof

b)

TNSC/T1-B

0 1 2 3 4 5

SaN

SC-F

l / Sa

NSC

-Roo

f

0

0.5

1

1.5

Floor#1Floor#2Floor#3Floor#4Roof

ShortPeriod

(I)

LongPeriod

(III)

FundamentalPeriod

(II)

Figure 2– Building#18, ξNSC=5%: a) PA-FRS for all floors of the building; b) Normalized PA-FRS

of all floors with respect to the roof.

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a)

Z/H0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SaN

SC-F

l / Sa

NSC

-Roo

f

0

0.5

1

1.5

Y = 0.33 + 0.67*XR2 = 0.95

b)

Z/H0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SaN

SC-F

l / Sa

NSC

-Roo

f

0

0.5

1

1.5

Y = 0.8 + 0.2 *XR2 = 0.90

Figure 3 – Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for low-rises given

ξNSC=5%: a) median+σ of the ratios over short & fundamental period regions; b) median+σ of the

ratios over long period region.

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a)

Z/H0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SaN

SC-F

l / Sa

NSC

-Roo

f

0

0.5

1

1.5

Y = 0.2 + 0.8*XR2 = 0.92

b)

Z/H0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SaN

SC-F

l / Sa

NSC

-Roo

f

0

0.5

1

1.5

Y = 0.64 + 0.36*XR2 = 0.92

Figure 4 - Scatter of (SaNSC-Fl / SaNSC-Roof) ratio vs relative height (Z/H) for medium-rise buildings

given ξNSC=5%: a) median+σ of the ratios over short and fundamental period regions; b) median+σ of

the ratios over long period region.

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a)

TNSC/T1-B

0 1 2 3 4 5

SaN

SC[g

]

0

1

2

3

4

5

6

7

8

9

10

NSC =2%

NSC =5%

NSC =10%

NSC =20%

b)

TNSC/T1-B

0 1 2 3 4 5

Sa /

Sa =

5%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Floor#1Floor#2Floor#3Floor#4Roof

NSC = 10% NSC = 20%

NSC = 2%NSC = 5%


Normalized PA-FRS at all floors, for all ξNSC with respect to the values for ξNSC =5% at corresponding

floor.

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a)

b)

c)

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Figure 6 - Scatter of (Sa ξ / Sa ξ=5%) ratio vs NSC damping ratio (ξNSC) for low and medium-rise

buildings given: a) median+σ of the ratios over short period regions; b) median+σ of the ratios over

fundamental period region; c) median+σ of the ratios over long period region.


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00.5

11.5

Pseudo Acceleration response spectrum [g]

0246810NS

C=2%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Roo

fPA

-FRS

- Roo

f

00.5

11.5

0246810NS

C=5%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Roo

fPA

-FRS

- Roo

f

T NSC [s

]0

0.51

1.50246810

NSC=1

0%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Roo

fPA

-FRS

- Roo

f

00.5

11.5

0246810NS

C=20%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Roo

fPA

-FRS

- Roo

f

Figure 8 – Illustration and comparison of the proposed FDS and the real PA-FRS generated for all

floors of Building#4 considering NSCs damping ratios of 2, 5, 10, and 20 %

Page 39 of 40



Draft

40

00.5

11.5

Pseudo Acceleration response spectrum [g]

02468NS

C=2%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Floo

r#3PA

-FRS

- Floo

r#3FD

S - Fl

oor#4

PA-F

RS - F

loor#4

FDS -

Roo

fPA

-FRS

- Roo

f

00.5

11.5

02468NS

C=5%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Floo

r#3PA

-FRS

- Floo

r#3FD

S - Fl

oor#4

PA-F

RS - F

loor#4

FDS -

Roo

fPA

-FRS

- Roo

f

T NSC [s

]0

0.51

1.502468

NSC=1

0%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Floo

r#3PA

-FRS

- Floo

r#3FD

S - Fl

oor#4

PA-F

RS - F

loor#4

FDS -

Roo

fPA

-FRS

- Roo

f

00.5

11.5

02468NS

C=20%

FDS -

Floo

r#1PA

-FRS

- Floo

r#1FD

S - Fl

oor#2

PA-F

RS - F

loor#2

FDS -

Floo

r#3PA

-FRS

- Floo

r#3FD

S - Fl

oor#4

PA-F

RS - F

loor#4

FDS -

Roo

fPA

-FRS

- Roo

f

Figure 9 – Illustration and comparison of the proposed FDS and the real PA-FRS generated for all

floors of Building#18 considering NSCs damping ratios of 2, 5, 10, and 20 %

Page 40 of 40



direct generation of floor design spectra (fds) from ......draft direct generation of floor design...

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