a simplified method of evaluating the seismic performance of buildings

Vol.3, No.2 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION December, 2004

Article ID: 1671-3664(2004)02-0223-14

A simplified methd of evaluating the seismic performance of buildings

Ashutosh Bagchi*

Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Quebec, H3G I MS, Canada

A b s t r a c t : This paper presents a simplified method of evaluating the seismic performance of buildings. The proposed method is based on the transformation of a multiple degree of freedom (MDOF) system to an equivalent single degree of freedom (SDOF) system using a simple and intuitive process. The proposed method is intended for evaluating the seismic performance of the buildings at the intermediate stages in design, while a rigorous method would be applied to the final design. The performance of the method is evaluated using a series of buildings which are assumed to be located in Victoria in western Canada, and designed based on the upcoming version of the National Building Code of Canada which is due to be published in 2005. To resist lateral loads, some of these buildings contain reintbrced concrete moment resisting frames, while others contain reinforced concrete shear walls. Each building model has been subjected to a set of site-specific seismic spectrum compatible ground motion records, and the response has been determined using the proposed method and the general method for MDOF systems. The results from the study indicate that the proposed method can serve as a useful tool for evaluation of seismic perfoNnance of buildings, and carrying out performance based design.

Keywords: seismic hazard; modal analysis; static pushover analysis; dynamic time history analysis; performance-based design

1 Introduction

The seismic design of a structure is performed with the anticipation that a severe earthquake would cause some damage, but that it would be contained to an acceptable limit. The extent of damage sustained by a building would determine the level of seismic performance it achieved. The goal of performance- based design is to analyze the performance objectives of a structure to specified levels of hazard, and design the structure to achieve such performance. Evaluation of the seismic performance of a building often involves a number of analytical techniques including modal analysis, static pushover analysis, and inelastic dynamic time history analysis. The design of a structure typically undergoes a number of revisions based on the capacity and performance requirements, and an evaluation of tile seismic performance corresponding to each design revision could be quite cumbersome and time consuming. To address this concern, many researchers (Fajfar and Gaspersic, 1996; Mazzolani and Piluso, 1997; Ghobarah et al., 1997; Chopra and Goel, 2001)

Correspondence to: Ashutosh Bagchi, Department of Building, Civil and Environmental Engineering Concordia University 1455 de Maisonneuve Blvd. West, BE-341, Montreal, Quebec, H3G [ MS, Canada Tel: (514) 848-3213; Fax: (514) 848-7965 E-mail: [email protected]

tAssistant Professor Received date: 2003-I1-04; Accepted date: 2004-10-15

have developed simplified methods for calculating the dynamic response of buildings. A simplified method, by nature, has limitations in terms of accuracy. However, it could be very useful for a quick evaluation of a building's performance in the intermediate stages of design. Such a method need not be very complex and computationally intensive. With that objective in mind, a simplified method for the evaluation of the seismic performance of buildings is developed in this paper. The method is based on an equivalent single degree of freedom system (SDOF) of a multi degree of freedom (MDOF) system. The equivalent SDOF system is derived from the load-deformation curve obtained from static pushover analysis of the MDOF system. The peak response of the SDOF system is obtained from dynamic or response spectrum analysis. The pushover results of the MDOF system could be used to derive a relationship between the roof displacement and the maximum story drift, and this relationship could be used for interpreting the SDOF response to obtain an estimate of the response of the actual structure.

Fajfar and Gaspersic, (1996) suggested a method called the N2 method for seismic performance evaluation. The method is based on an equivalent SDOF system, derived using the static displacement shape of the original structure produced by an assumed distribution of the lateral forces and a two stage pushover analysis is necessary to estimate the MDOF response. The simplified analysis approach suggested by Mazzolani and Piluso (1997), is based on the tri- linear representation of the load-displacement curve and

224 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.3

elastic and rigid-plastic analysis of the equivalent SDOF system. Relationships between roof-displacement and the damage index have been established in Ghobarah et al. (1997) for the seismic performance evaluation. Such relationships could be used for interpreting the response of the SDOF structure and evaluating the performance of the building based on the SDOF response. Chopra and Goel (2001) have combined the strengths of various simplified methods and developed a method based on modal pushover analysis.

Although there are a number of simplified methods are currently available, they are still complicated for common use. In the method suggested here, an equivalent SDOF system is derived based on the bilinear idealization of the pushover curve of the original structure. The peak response of the SDOF system for a given hazard level can be obtained either by using inelastic response spectra or performing inelastic dynamic time history analysis using appropriate ground motion records. The peak SDOF response is then translated to the performance of the MDOF system using the results of the pushover analysis performed earlier.

The proposed method bears some resemblance to the N2 method mentioned above. However, the procedures for deriving the equivalent SDOF system and the method of estimating the MDOF response from the SDOF response are quite different. Unlike the N2 method, the static displacement shape of the original structure is not used in the derivation of the equivalent SDOF system and a second stage pushover analysis is not required.

The proposed method is illustrated with a number of reinforced concrete buildings designed according to the National Building Code of Canada (NBCC). The National Building Code of Canada is currently undergoing major revisions to incorporate the location based seismic hazard spectra and the latest knowledge of earthquake engineering. While the proposed method is quite general, the case studies are based on the upcoming version of NBCC and they are discussed with reference to NBCC seismic provisions.

2 Evaluation of the seismic performance

Earthquake resistant design of buildings is based on the concept of acceptable levels of damage under one or more events of specified intensity. The acceptable level of damage is related to the performance objective of the building. For example, the objective may be specified in the form of a requirement that the building is fully operational with little or no damage during an earthquake that has a 50% probability of exceedance (PE) in 50 years; but may have moderate damage during an earthquake with a 10% PE in 50 years.

The Geological Survey of Canada (GSC) has recently developed site-specific spectra corresponding to various levels of seismic hazard for a number of locations in Canada (Adams et al., 1999). These spectra are called the uniform hazard spectra (UHS). Figure 1 (a)

shows the spectra for Victoria. An earthquake event with a return period of 475 years (10% PE in 50 years) would be designated as UHS-500 for easy reference in the text. Similarly, a 970-year event (10% PE in 100 years), and a 2500-year event (2% PE in 50 years) would be designated as UHS-1000 and UHS-2500, respectively.

The Vision 2000 committee (1995) of the Structural Engineers Association of California (SEAOC) has suggested performance objectives for buildings of different types. A fully operational performance would mean that the building has suffered minor damage, and a repair may not be necessary; an operational performance means that the building has sustained some damage, but it is safe for post-earthquake occupancy, and minor repair may be necessary; a life-safe performance level means that the building has sustained extensive damage, and it is not safe for post-earthquake occupancy without major repair work; a near collapse performance means the lateral load carrying capacity of the building has diminished to a great extent, and the building is not repairable. Similar definitions are adopted by the upcoming version ofNBCC (Humar, 2000).

2.1 NBCC seismic provisions

The reinforced concrete buildings considered here designed according to the seismic provisions of the upcoming version of the National Building Code of Canada. Thus a brief discussion of the seismic provisions and the design philosophy of the new version of NBCC would be of interest.

According to NBCC 1995, buildings are required to be life-safe under an earthquake with a return period of 475 years. However, such buildings have considerable reserve strength to sustain earthquakes of higher intensity. Suppose that a building has sufficient over-strength to sustain twice the design shear without collapse. From the hazard spectral shapes, it could be observed that a building in western Canada would be able to sustain a 2400-year earthquake although it was designed to sustain a 475-year earthquake, while a building in eastern Canada will be able to sustain an earthquake with a return period of about 1600 years (Humar, 2000). Consequently, the two buildings have different levels of protection. Thus, to design the two buildings to have the same levels of protection, the design base shear should be derived for, say a 2500-year earthquake and then reduced by a factor to account for the reserve strength. This approach is adopted in the new version of NBCC. Calculation of the design base- shear according to the new version on NBCC is briefly discussed below.

The elastic base shear, Vo for a single degree-of- freedom building can be obtained from the spectral acceleration, S(T) corresponding to the period of the building, T and weight of the building, IV.

vo= s (:r) w (1)

No.2 Ashutosh Bagchi: A simplified methd of evaluating the seismic performance of buildings 225

The design spectral acceleration, S(T) value is calculated based on the site-specific values of the spectral

acceleration So(T) available from the Geological Survey of Canada (Adams et al., 1999).

S(T)=

-FaS~ (0.2) for T < 0.2s

FvSa(0.5 ) or F.S~(0.5)

whichever is less for T =0.5s

F~S (1.O) for T=l .Os

FvS(2 .0 ) for T : Z . 0 s

F~S (2.0)/2 for T > 4.0s (2)

where Fa and F~ are the foundation factors. The design base shear is calculated using the following expression.

v - s(ra)MvZw >_ s(2.o)Mj w

RoR RoRa (3)

where, My accounts for higher mode effect, Ic is the importance factor, Rd and Ro account for ductility and over-strength, respectively.

2.2 Damage parameters and performance metrics

Seismic performance of a building is generally expressed in qualitative terms. However, for engineering evaluation of performance, a quantitative measure of damage is calculated. It is often necessary to correlate the quantitative measures of the damage parameters to the overall performance of the building in qualitative terms. The selection of appropriate damage parameters is very important for performance evaluation. Overall lateral deflection ductility demand, inter-story drift, etc., are commonly used damage parameters. The damage index developed by Park and Ang (! 985) is regarded as a good representation for structural damage as it accounts for the damage caused by cyclic deformations into the post-yield level.

Obtaining a qualitative measure of performance

from the quantitative values of the damage parameters is quite difficult. Vision 2000 (1995) provides some guidelines for evaluating the performance of a building based on the drift values (Table 1 ).

2.3 Analysis procedures

The evaluation of seismic performance of any structure requires the estimation of its capacity, dynamic characteristics and the prediction of its response to the ground motion to which it could be subjected during its service life. The commonly used methods for evaluating the seismic performance of a building are, (1) static pushover analysis, (2) modal analysis, and (3) dynamic time history analysis.

DRAIN-2DX (Prakash et al., 1993), a general purpose program for dynamic analysis of building frames, is used for the analysis in the present study.

Selection of appropriate earthquake records is very important for dynamic time history analysis. If actual earthquake records are used, only those records whose spectra closely match the uniform hazard spectrum corresponding to the design earthquake level for the site of interest can be used. As an alternative, the UHS- compatible ground motion time histories developed by Atkinson and Beresnev (1998) could be used.

Atkinson and Beresnev (1998) have produced physically realistic time history records, which not only match the hazard spectrum, but also are representative of motions for specified magnitude distance scenarios in the regions of interest. The UHS of a given location can be represented by the composite two ground motions; one corresponding to a short duration, low magnitude earthquake at a very short distance and the other to a long duration, high magnitude earthquake far away from the site (Atkinson and Beresnev, 1998). The characteristics of these artificial ground motion records are summarized in Bagchi (2001). Figures 1 (b) and (c) show a set of records corresponding to UHS-500. There are sets of four records for UHS-500 and UHS-1000, whi!e a set of eight records for UHS-2500 available for this study.

3 Details of the proposed method

Static pushover analysis is performed to construct the base shear - top displacement curve for the frame. A curve representing the relationship between the roof- displacement and the maximum inter-story drift (let us

Table 1 Permissible structural damage at various performance levels (Vision 2000, 1995)

Damage Performance level

parameter Fully o p e r . Operational Life-safe Near collapse Collapse

Drift

(a) Transient < 0.2% < 0.5% < 1.5% < 2.5% > 2.5% (b) Permanent Negligible Negligible < 0.5% < 2.5% > 2.5%


Fig. 1 Uniform hazard spectra and sample ground motion records for Victoria

call it the drift curve) is also constructed. The pushover curve is then idealized as a bilinear force-displacement relationship as shown in Fig. 2 (a), where Fy and Dy are the effective yield force and yield displacement, respectively; and Fr is the force corresponding to a reference displacement beyond yield. The idealized bilinear curve is constructed based on the approximate equal energy criteria. The equivalent stiffness of the single degree of freedom system (SDOF) is obtained from the following expressions

k = Fy/De (4)

where k is the stiffness of the equivalent SDOF system. The period of the SDOF system is given by

T = 2rt ( m x / ~ (5)

where m is the mass of the equivalent system. The period T is taken to be equal to the fundamental period of the structure as obtained from the modal analysis of the MDOF model. Equation 5 thus allows the determination of mass m of the equivalent system.

A bilinear behaviour is assumed for the SDOF system. IfFr is the base shear and Dr is the inter-story drift

No.2 Ashutosh Bagchi: A simplified mcthd of evaluating the seismic performance of buildings 227

~D

m

Push-over curve F~ / (MDOF)

-F/ - ~ ~.~.......~..:...=..=...:..= .............. / r Bilinea.r approximation of

the push-over curve

i ~ w ~ l l ~ . 1 . . . . . . . . . . . . . . . . . . . . . . . .

, ' I L i

Dy Roof displacement Dr

(a) Bilinear idealization of the pushover curve

O O

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

UHs:

Maximum inter-storey drift

(b) A drift curve representing the relation between the maximum inter-story drift and roof displacement

Fig. 2 Derivation of an equivalent SDOF model

corresponding to a reference level of roof displacement, an equivalent strain hardening ratio could be calculated from Fig. 2(a). The bilinear hysteretic behaviour of the equivalent SDOF system can be derived from the loading curve shown in Fig. 2(a) with a diagonally symmetric unloading curve. No strength or stiffness degradation is assumed in modeling the hysteretic behaviour of the SDOF system.

Seismic response of the equivalent SDOF system is obtained through a series of inelastic dynamic time history analyses for the UHS-based ground motion records corresponding to all three levels of seismic hazard. The envelope of the SDOF response for a given level of hazard is used as the peak response. Alternatively, the peak displacement response could be obtained using the inelastic response spectra for a given level of seismic hazard, if available. As opposed to many other simplified methods, such as the N2 method, the SDOF model derived here does not utilize any algebraic transformation based on the modal or assumed shape vectors. Thus, SDOF response corresponding to a given level of seismic hazard can be taken to be equal to the peak roof-displacement of the MDOF building frame, which is then used for predicting the maximum inter- story drift in the MDOF structure using the drift curve.

4 Description of the building models

Two types of lateral load resisting systems are considered here: concrete moment resisting frame, and concrete shear wall frame. The buildings are assumed to be located in Victoria in western Canada. The buildings with moment resisting frames are designed to be six and twelve-stories high, while the shear wall buildings are designed to be twelve and twenty-stories high. The geometric details of the buildings considered here are shown in Figs. 3 and 4. The plan layout is common to all the buildings. It has several 6 m bays in the N-S direction and 3 bays in the E-W direction. The E-W bays consist of two 9 m office bays and a central 6 rn corridor bay. The story height is 4.85 m tbr the first story and

3.65 m for all the other stories. The yield stress, fy for reinforcing steel, and the 28-day concrete compressive stress, .s are assumed to be 400 MPa and 30 MPa, respectively. The following gravity loads are used in the design. Dead load is assumed to be 3.5 kN/m 2 on the roof and 5.0 kN/m 2 on all other floors. Live load is assumed to be 2.2 kN/m 2 on the roof and 2.4 kN/m 2 on all other floors. The seismic lateral forces are obtained using the new UHS based methodology (Eqs. 1 through (3)). The base shear is distributed across the height of the frame, using the procedure suggested by NBCC 1995 to obtain the floor level forces (triangular distribution). All transverse frames are designed to be ductile lateral load resisting frames.

For moment resisting frame systems, a bare frame model, and a masonry infilled model have been considered. The masonry infill has been considered to account for the effect of non-structural elements in the building and to match the fundamental period of the model with the building code specified value (normally the period of a bare frame model is much higher than the code specified value). The masonry infilled panels are modelled as equivalent struts developed by Stafford- Smith (1966). Clay masonry with a thickness of 100 ram, and crushing strength of 8.6 MPa has been used in the study. The CMRF frame models are shown in Fig. 5. For the six-story building, every alternate frame is considered to be infilled such that the period of the structure is close to the NBCC 1995 values. To construct the analytical model, an infilled frame and a bare frame are connected together using rigid links. For the twelve- story building, all frames are considered to be infilled. For a shear wall frame system, a single wall model, and a model consisting of the shear wall and the associated frame have been considered. The reinforcement details are not shown here to conserve space. The building models as described in Table 2 are considered here to demonstrate the effectiveness of the simplified method of seismic performance evaluation of buildings.

A preliminary analysis of the seismic performance


At..

A B C D L 9m I 6m _l_ 9m _l 1~ ~ i l l ~i ~ ~I

2 : : : : : : �9 : - - - 2 - 1 . . . . . .

t - - I - - - : : 2 - I - : : : - I - : : - - - ]

�9 : : : - i t 2 . = : : : : : : ; _ - ; : _- - . -

- C : % : % -% I : C : C : - ~ l I --- -% : % : :

Plan

j A

Fig. 3 Details of the CMRF buildings

Elevation-six storey

Elevation-twelve storey

q ' 5

r162

4.85m

t t 3 ' , D

- - ~ r ~ 5 m

@ I_ 9m ~., 6m _L 9m _[

r',,, 4

. . . . . . . -1 . . . . . - I - . . . . . .

�9 : : - : : : : : : : : Z . : : - I : : : : : : :

Plan

A A

9m _1~ 6m _L 9m ~ 1 ~ ~ r ~

:.: . :->: : .b : .5>: . : .>:

r . - , - . ' . ' . ' , ' . ' . ' . ' . ' . ' . ' . ' . ' . l

r . ' . - . . . - : . . ' = .w . ' . ' . ' . ' . l

: ' :-5:.:-: : ' : ' > : ' : ' : ' : ' : ' : t

rs . . ' . , = . . = . . ' . ' . ' . ' . ' . :

::::::::::::::::::::::::::::::

, . , =., . . . . . . . . . . : . K

r . . . , . . : . : . . . . . , 1....1

Elevation

L

4.85m

Fig. 4 Details of the CSWF buildings

of these buildings has been reported in Bagchi (1999), and Humar and Bagchi (2000). A more detailed study on the seismic performance of these buildings using the rigorous analysis of the MDOF models has been

reported in Bagchi (2001). The performance of the simplified method would be evaluated by comparing the response parameters obtained using this method with those reported in Bagchi (2001).


Rigid links Frame A ~ Frame B

L I . . . .

(a) CMRF-V6FB

, ' ! - ' , ' , ' ,

Frame A: Frame with infill panels Infill panel Frame B: Neighbouring bare frame

(b) CMRF-V6FP

' ' ' - - "

. m m m ,

(c) CMRF-V 12FB (d) CMRF-V12FP

Fig. 5 Bare and infill frame models of the CMRF buildings

Table 2 Description of the building models

Model Description

CMRF-V6FB

CMRF-V6FP

CMRF-VI 2FB

CMRF-V 12FP

CSWF-VI2FW

CSWF-VI2FF

CSWF-V20FW

CSWF-V20FF

Six story CMRF building: bare frame model

Six story CMRF building: infilled flame model

Twelve story CMRF building: bare frame model

Twelve story CMRF building: infilled frame model

Twelve story CSWF building: wall model

Twelve story CSWF building: wall-frame model

Twenty story CSWF building: wall model

Twenty story CSWF building: wall-flame model


5 Performance of the simplified method

The idealized bilinear load-deformation curves for the building models considered here are obtained from the respective pushover curves as shown in Fig. 6. The bilinear curves are derived from these curves. The details of the models are given in Table 3, where H is the height, W is the weight, Fy is the yield base shear, Dy is the lateral drift at yield, Fr is the base shear corresponding to a reference roof-displacement, and T is the fundamental period. The reference roof-

displacement is arbitrarily chosen from the push over curves to get a good representation of the bilinear curve, and in the present work it is chosen to be 2% for CMRF buildings and 3% for the shear wall buildings. The equivalent SDOF models can be derived from this information. The properties of the equivalent SDOF models are given in Table 4.

The drift curves for the models are obtained from the static pushover analysis. However, These curves are not shown here to conserve space. Tables 5 through 7 show the response of the SDOF models and how they

Table 3 Details of the building models

Model H(m) W(kN) Fy/W Dy(%H) F/W T (s)

CMRF-V6FB 22.8 3592 0.160 0.55 O. 180 1.47

CMRF-V6FP 22.8 3592 0.310 0.45 0.330 0.77

CM RF-V 12FB 44.7 7372 0.070 0.58 0.075 2.91

CMRF-VI2FP 44.7 7372 0.I 50 0.43 0.155 1.29

CSWF-V 12FW 44.7 7372 0.072 0.40 0.095 2.05

CSWF-V 12FF 44.7 7372 0.11 0.40 0.180 1.80

CSWF-V20FW 73.9 12436 0,040 0.48 0.065 3.57

CSWF-V20FF 73.9 12436 0.062 0.48 0.095 2.70

Table 4 Details of the equivalent SDOF models

Model H (m) Stiffness, k (kN/m) Mass, m (t)

CMRF-V6FB 22.8 4583.1 250.9

CMRF-V6FP 22.8 10853.0 163.0

CMRF-V 12FB 44.7 1990.4 426.9

CM RF-V 12FP 44.7 5829.8 245.7

CSWF-V12FW 44.7 2968.6 316.0

CSWF-V 12FF 44.7 4535.4 372.2

CSWF-V20FW 73.9 1402.3 452.7

CSWF-V20FF 73.9 2173.6 401.4

Table 5 Response under UHS-500 events

Model SDOF response, D~r (% H) Estimated drift, D~ (% h) MDOF drift, Dsm (%h) Ds~/D~m

CMRF-V6FB 0.40 0.57 0.60 0.95

CMRF-V6FP 0.35 0.48 0.50 0.96

CMRF-VI 2FB 0.50 0.90 1.00 0.90

CMRF-V 12FP 0.30 0.42 0.45 0.93

CSWF-V 12FW 0.70 0.94 1.10 0.86

CSWF-V12FF 0.45 0.65 0.75 0.87

CSWF-V20FW 0.45 0.62 0.60 1.03

CSWF-V20FF 0.35 0.50 0.55 0.91


compare to the response of the corresponding MDOF models reported in Bagchi (2001). In these tables, the SDOF response, D~ is expressed as a percentage of the building height, H, the estimated inter-story drift, D~ is expressed as a percentage of the story height, h, and the corresponding inter-story drift using the MDOF model is D~.

In Tables 5 through 7, the comparison is made between D~ and D~m, representing the dynamic response calculated using the simplified method, and the MODF time-history analysis, respectively.

It is observed from the results presented here that the simplified method gives a good estimate of the

maximum story drift corresponding to UHS-500 and UHS-1000. However, in some cases with UHS-2500, the estimated story drift deviates as much as 11% from the MDOF response. This implies that when displacement demand is very high and the structure is well in the yield zone, the response by simplified method tends to be less accurate. In most of the cases, inter-story drift calculated using the simplified method is lower than the actual values.

A summary of the single degree of freedom response, D~ under various levels of seismic hazard is compiled from Tables 5 through 7, and the values are shown on the idealized bilinear load-displacement curves in Fig. 6 as

0.5

0.4

0.3 E

~E

0.2

r 0.1

0.0 0.0

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0,00 0.0

. . . . . CMRF-V6FB ......... CMRF-V6FP

( 0 . 3 1 ~ .......... ~ . . . . . . . . . . . . .

. . . . . . . . .

3

0.4 0.8 1.2 1.6 2.0

Roof displacement (% H)

(a) CMRF six story

",,2

r

0.20

0.16

0.12

0.08

0.04

0.00 0.0

. . . . . CMRF-VI2FB

......... CMRF-VI2FP

(0.15, 0.4~." _m"" .......... "'-.-~ .. ~ , ~ : $ " . , ' -

/ r (0.07, 0.~0)~,..~.-....-..:...'....'."~" . . . . . . " - , ,

3

!, �9 ]

n i n t

0.4 0.8 1.2 1.6 2.0


(b) CMRF twelve story

. . . . . ' c s w i - v l 2 F W �9

......... CSWF-V 12FF ........ 7 '

(0. 1 ,0 .40) . , f~ . , /

�9 " (0.072, 0.40)

I | ! ! I

0.5 1.0 1.5 2.0 2.5 3.0


(c) CSWF twelve story

0.2

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00 0.0

. . . . . CSWF-V20FW

......... CSWF-V20FF

(0.062, 0.48) 3 o . , ~ . ~ " ~ : ~ ' ' ' ' w ~ ' ~ - - ~ ' ' '

"2 ( J 3

(0.04, 0.40)

I I I I n

0.5 1.0 1.5 2.0 2.5 3.0


(d) CSWF twenty story

Fig. 6 Pushover curves of the building models



Model SDOF response, Dsr(%H) Estimated drift, Dss(%h) MDOF drift, D~m(%h) DJD~m

CMRF-V6FB 0.45 0.65 0.70 0.93

CMRF-V6FP 0.42 0.62 0.60 1.03

CMRF-V 12FB 0.70 1.25 1.20 1.04

CMRF-V 12FP 0.35 0.50 0.55 0.91

CSWF-VI2FW 0.80 1.10 1.20 0.92

CSWF-V12FF 0.55 0.75 0.80 0.94

CSWF-V20FW 0.55 0.70 0.70 1.00

CSWF-V20FF 0.40 0.55 0.60 0.92


Model SDOF response, Dsr (%H) Estimated drift, Ds~(%h ) MDOF drift, Dsm (%h) DssrDsm

CMRF-V6FB 1.10 1.95 2.20 0.89

CMRF-V6FP 0.80 1.52 1.70 0.89

CMRF-V12FB 1.32 2.50 2.50 1.00

CMRF-V12FP 0.75 1.65 1.50 1.10

CSWF-V 12FW 1.81 2.60 2.80 0.93

CSWF-VI2FF 1.33 1.82 2.00 0.91

CSWF-V20FW 1.55 2.20 2.40 0.92

CSWF-V20FF 1.05 1.33 1.50 0.89

points 1, 2 and 3 corresponding to UHS-500, UHS-1000 and UHS-2500, respectively. It can be noted from Fig.6, that the values of Dsr for UHS-500 and UHS-1000 are well below the reference roof-displacement and some of the values are even below the yield displacement. This is indicative of negligible or minor damage as determined from the MDOF analysis. Although the SDOF responses due to UHS-2500 events are beyond the yield displacements, they are still lower than the reference roof displacements. One may argue that a smaller value for the reference displacement could be taken to obtain the idealized bilinear curves. As indicated earlier in the paper that the reference roof displacement is chosen arbitrarily to match the push-over curve as closely as possible. Thus it is possible to choose a different value for the reference roof displacement which may alter the strain hardening segment of the idealized bilinear curve, and the response may change to some extent. A sensitivity analysis could be carried out to quantify the extent of change in the SDOF response due to different values of the reference roof displacement. To the author's opinion, the reference roof displacement should be high enough to capture the entire pushover curve and the post-yield behaviour of the structure before collapse, which will be useful for estimating the reserve deformation capacity of the structure beyond the UHS- 2500 level of seismic hazard at which it is expected to have life-safety performance.

5.1 Account ing for the effects o f higher modes

The distribution of lateral loads used in the pushover analysis is based on the NBCC 1995 guidelines (triangular distribution). Pushover analysis using the NBCC 1995 distribution of lateral loads may fail to capture the effects of higher modes and the pushover results may not provide an accurate prediction of the dynamic behaviour of the structure. Humar and Rahgozar (2000), and Humar and Mahgoub (2000) have shown that the effect of higher modes on the base-shear depends on structural types, modal periods and spectral shapes. The higher mode effect is found to be more pronounced for flexural wall systems than for shear flame structures.

To evaluate the effect of higher modes in the present analysis, the following shear wall models are considered: CSWF-V12FW and CSWF-V20FW (refer Table 2). For a building model, the contribution of each mode to the base shear is computed using the method suggested in Humar and Rahgozar (2000). Components of the base shear corresponding to the individual modes are calculated based on the modal contribution ratios. Each component is distributed along the height of the building according to the corresponding modal shape. The modal distributions of lateral loads are then combined together using square root of sum squares (SRSS) technique and scaled to the design base shear. Instead of using the


e~

t~ ~

>

o

120

100

80

60

40

20

0 0.0 1.0 2.0 3.0 4.0

Period (s)

(a) Spectral ordinates for CSWF-V12FW model

GSC spectrum, S (UHS-2500)

I .......... Design spectrum, S(T)

l ,.o \ l_ i . . . . . . . . . _ x ........

,0.45 : 2.05 ................................

I I I I I I I I I

5.0

e~ t~

120

100

80

60

40

20

i ilo:ol ....... , :

r ...... : . . . . . .

0 0.0 1.0 2.0 3.0 4.0 5.0

Period (s)

(b) Spectral ordinates for CSWF-V20FW model

12

10

%~%~*%.%.% /

%

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NBCCdist I~// ! ) 1 st mode dist ........

4 ml . . . . ~ -:r ~ I~// /lll m 2 ~

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contributions

-50 0 50 100 Lateral force (kN)

(c) Lateral load distribution in CSWF-V12FW model

0 -100

20

15

10 NBCC d i s t [ i I i l l i t : ~

1 st mode dist ....... ml . . . .

m2 ---- f 7 t r , , , Scaled SRSS ~ - ~L./r

m 1,m2 are modal contributions

0 , t -50 0 50 100

Lateral force (kN)

(d) Lateral load distribution in CSWF-V20FW model

-100

Fig. 7 Accounting for higher modes

triangular distribution, this SRSS modal distribution of lateral forces could be used in the pushover analysis. The proposed simplified method is then used to evaluate the seismic response. Only the first two modes are considered in the present study.

Figures 7 (a) and (b) show the design spectral acceleration curve (based on UHS-2500) for Victoria. The first two modal periods for the CSWF-V12FW model are 2.05 s and 0.45 s. Corresponding values of effective modal weight are 0.68 W and 0.18 W, W being the weight of the structure. The spectral acceleration corresponding to each mode is obtained from Fig. 7 (a). The values of spectral acceleration corresponding to the

first mode and the second mode are 0.19g and 0.88g, respectively (g is the acceleration due to gravity). By multiplying the spectral acceleration with the modal weight, the modal base shear is obtained. These values are tabulated in Table 8. The base shear contribution of the second mode is also quite considerable in the twenty- story shear wall building model, CSWF-V20FW.

Figures 7 (c) and (d) show the distribution of modal contributions to lateral loads. For the purpose of comparison, the NBCC 1995 distribution and the distribution of lateral loads according to the first mode only are also shown. It is observed that the NBCC distribution follows the first mode only distribution more


Fig. 8 Effect of higher modes

Table 8 Higher mode effect on the base-shear

Effective modal Spectral Modal base-shear Ratio of modal Model Period (s) weight acceleration factor base-shear

CSWF-V 12FW 2.05 0.68 W 0.19g O. 129 0.45

0.45 0. l 8 W 0.88g 0.158 0.55

CSWF-V2OFW 3.57 0.64 W 0. I 0g 0.134 0.47

0.58 0.20W 0.75~ 0.150 0.53

Table 9 Estimated story-drift by the simplified method

Model Method UHS-500 UHS- 1000 UHS-2500

CSWF-VI2FW NBCC-1995 dist. 0.94 1.10 2.60

SRSS-modal dist. 1.05 1.16 2.76

MDOF response 1.10 1.20 2.80

CSWF-V20FW NBCC-1995 dist. 0.62 0.70 1.33

SRSS-modal dist. 0.63 0.71 1.46

MDOF response 0.60 0.70 1.50


closely than the SRSS modal distribution. To account for the higher order modes, the NBCC distribution assigns higher magnitude of lateral force at the top story.

Pushover curves using the SRSS modal distribution of lateral forces, and the corresponding drift curves are shown in Fig. 8. It is observed from Fig. 8 that there is some difference between the pushover response of a building using the NBCC distribution and the SRSS modal distribution when first two modes are considered.

Table 9 shows the values of the maximum inter-story drift estimated by the proposed simplified method using the NBCC and SRSS modal distribution of the lateral forces. The results presented in Table 9 show some improvement in estimated response. The contribution of the higher modes could be significant as observed by Chopra and Goel (2001). The estimated response based on the simplified method presented here still provides a valuable guidance on the seismic performance. It should be noted that the pushover analysis here has been carried out using the combined distribution of lateral loads, not individual modal distribution, and the reference roof displacement is assumed to be 2% or 3% as in the case of NBCC distribution.

6 Discussion and conclusions

From the results presented above, it is observed that the accuracy of the simplified method in predicting the maximum inter-story drift is reasonably good in some cases. Given the nature of approximation involved in the proposed method, its performance is acceptable. From the examples presented by Fajfar and Gaspersic (1997), the N2 method seems to achieve a similar level of accuracy.

Based on the estimated maximum values of inter- story drift obtained using the proposed simplified method, the following observations could be made on the performance of the buildings studied here.

�9 The six-story CMRF building achieves a performance level of operational under UHS-500 and UHS-1000, and life-safe under UHS-2500 when the bare frame model is considered. When the infill frame model is considered, its performance levels are fully operational under UHS-500, operational under UHS- 1000 and life-safe under UHS-2500.

�9 The twelve-story CMRF building achieves a performance level of operational under UHS-500 and UHS-1000, and near collapse under UHS-2500 when the bare frame model is considered. With infill panels, its performance levels are fully operational under UHS- 500, operational under UHS-1000 and life-safe under UHS-2500.

�9 The twelve-story CSWF building achieves a performance level of operational under both UHS-500 and UHS-1000 and life-safe under UHS-2500 when the wall model is considered. With the wall-frame model, the pertbrmance level is operational under both UHS- 500 and UHS-1000 and life-safe under UHS-2500.

�9 The twenty-story CSWF building in Victoria achieves a performance level of operational under both UHS-500 and UHS-1000 and life-safe under UHS-2500 for both wall and wall-frame models.

In most cases, the qualitative performance levels mentioned above are consistent with the performance levels evaluated through a rigorous study reported in Bagchi (2001 ).

Simplified analysis as described above could form an important intermediate step in the earthquake resistant design of buildings. The designer can quickly estimate the possible performance level of the structure and modify the design based on the results. Once the design is finalized, a detailed analysis could be performed to evaluate the seismic performance of the structure.

The simplified method presented here has some limitations. For cases where the effect of higher order modes is significant, this technique may not very effective in estimating the maximum inter-story drift. Pushover analysis using the SRSS modal distribution of the lateral loads improves the solution to some extent. However, the method could be used for an easy and approximate estimate of the seismic pertbrmance of a building for given level of hazard.

The proposed method may be criticized for being overly simple and approximate. Designers must be aware of its limitations while using it for practical design. The simplified method presented here would, however, be of assistance to designers in carrying out a perfbrmance- based design. The estimated response using the method described here would allow the designers to get an approximate measure of the qualitative performance of the structure and could help them in revising the design through a number of iterations in order to achieve the desired performance characteristics, strength and deformation capacity. A detailed MDOF analysis must be carried out in the final stage for estimating the seismic response more accurately.

Acknowledgement

The work presented here forms a part of the author's Doctoral thesis. The author would like to thank his thesis supervisor Professor Jag Humar at Carleton University, Ottawa, Canada, for his guidance, help and encouragement.

References

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a simplified method of evaluating the seismic performance of buildings

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