performance based earthquake evaluation of reinforced concrete

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Performance based earthquake evaluation of reinforced concrete buildingsusing design of experiments

Mahdi Modirzadeh, Solomon Tesfamariam ⇑, Abbas S. MilaniOkanagan School of Engineering, The University of British Columbia, Canada

a r t i c l e i n f o

Keywords:Building vulnerabilityPerformance based earthquake engineeringDesign of experimentsDecision makingPushover analysis

a b s t r a c t

Seismic resiliency of new buildings has improved over the years due to enhancements in seismic codesand design practices. However, existing buildings designed and built under earlier codes are vulnerableand require a performance-based screening and retrofit prioritization. The performance modifiers consid-ered are soft story, weak story, and the quality of construction, which are collated through a walk downsurvey. The building evaluation is performed through a pushover analysis, and performance objective areobtained through initial stiffness of the pushover curve. Using a design of experiments technique, a reli-able system input–output relation has been identified and used to evaluate the performance criteria atuntried design points (i.e., buildings with different modifier values). The proposed method of perfor-mance based evaluation is illustrated through consideration of the different structural deficiencies ona typical six-storey reinforced concrete building in Vancouver. Through the designed experiments, themain and interaction effects of the performance modifiers have also been studied.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The traditional seismic design practice entails specifying thedesired performance objective, and subsequently the structure isdesigned to meet specific performance levels. Performance-baseddesign is a more general approach in which the criteria areexpressed in terms of achieving a set of performance objectiveswhile the structure is under levels of seismic hazard (Ghobarah,2001). The 1994 Northridge Earthquake, for example, highlightedthe importance of considering performance-based seismic screen-ing criteria (Elms, 2004).

For the performance-based seismic screening, there is a need forreliable building evaluation techniques (Ghobarah, 2001). The cur-rent states of practice for building performance evaluation arecapacity-demand method (ATC, 1996) and linear or nonlinear timehistory analysis (THA) (FEMA, 1996). The nonlinear THA methodprovides the damage initiation and propagation, and also the col-lapse mechanism. The environment under which a building hasbeen constructed (e.g., where there is a potential for lack of exper-tise and quality of construction, often coupled with numerousuncertainties in demand and capacity) should be incorporated inthe performance based evaluation approach. The concept of perfor-mance based design and evaluation is widely used in structuralengineering applications (e.g., Fajfar, 2000; Ghobarah, 2001; Kim& D’Amore, 1999; Porter, 2003). Tesfamariam and Saatcioglu

(2008) summarized prevalence of several building performancemodifiers from various earthquake field reconnaissance reports.Following that work, in this paper three main performance modifi-ers have been selected: soft story index (SSI), weak story index(WSI) and construction quality. The pushover analysis is used forevaluating buildings performance.

Given a structure, for design optimization purposes, often ana-lysts are interested in studying individual and combined effects ofperformance modifiers on overall performance of the structure.This, in turn, can help them concentrate on main aspects of thedesign and also attain some mathematical models for predictingand optimizing the structure response. Design of experiments(DOE) is a technique that helps analysts choose and perform stud-ies of this kind. Different types of objectives can be realized duringa course of DOE (Robinson, 2000). For the first type of objective,one may be interested in a screening procedure in which a smallnumber of factors (called ‘main effects’) are extracted from a largerpool of factors. The second type of objective aims at finding a func-tional description of how factors affect the response (i.e., the in-put–output relation). Eventually, using such a relation the goalcan be to optimize the response surface functions. The third objec-tive is when the experiments are tuned to give an estimation oftesting errors (i.e., the robustness of the solution is of interestrather than its optimality). The fourth objective relies on obtaininga mathematical model for the input–output relation and also esti-mating the typical size and structure of errors.

In this study, a full factorial DOE method with a type-II objec-tive as defined above has been aimed at. More specifically, theinterest is in exploring the main and possible interaction effects

0957-4174/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.eswa.2011.08.153

⇑ Corresponding author. Tel.: +1 250 807 8185.E-mail address: [email protected] (S. Tesfamariam).

Expert Systems with Applications 39 (2012) 2919–2926

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of the performance modifiers (also referred to as design factors) ona main performance indicator (response) of a reinforced concretebuilding under earthquake loads. In doing so, a finite element mod-el of the structure (a typical six-storey RC building in Vancouver)under prescribed load and boundary conditions is employed.Results of the computer experiments are then used to completethe DOE study. It is believed that similar applications of the DOEmethods, especially once combined with computer experiments,can be beneficial for performance-based earthquake engineeringproblems given the high cost of physical experimentations andcomplexity involved in reinforced concrete structures. The conceptof DOE has been previously used in other structural engineeringapplications (e.g., Liel, Haselton, Deierlein, & Baker, 2009; Möller,Foschi, Rubinstein, & Quiroz, 2009; Schotanus, Franchin, Lupoi, &Pinto, 2004; Zhang & Roschi, 2004).

2. Building performance modifiers

The soft story index (SSI) and weak story index (WSI) are encap-sulated under vertical irregularity, and each are quantified throughthe relative storey stiffness and strength, respectively. The problemof soft story was first identified after the San Fernando earthquake(Scarlat, 2000). The softy story is defined by the stiffness of the lat-eral force resisting system in any story being less than 70% of thestiffness in an adjacent story (above or below) or less than 80% ofthe average stiffness of the three stories (above or below) FEMA310 (ASCE, 1998). The relative length between two adjacent floorsis used as a surrogate measure of SSI. The SSI can be quantified as:

SSI ¼ k2

k1¼ L1

L2

� �3

; ð1Þ

where k1 and k2 are stiffnesses of two adjacent stories; and L1 and L2

are column heights of two adjacent floors.The weak story index is defined by lateral force resisting system

strength of any story being less than 80% of the adjacent storystrength (above or below) (ASCE, 1998). The relative strength canbe defined by considering areas of columns, structural walls andpartition walls (Yücemen, Ozcebe, & Pay 2004). For the momentresisting frame building, in this study, only the column areas areconsidered. The WSI is defined as the ration of area of all columnsections of the ground storey to the area of all column sectionsof the first storey:

WSI ¼PðAcolÞ1PðAcolÞ2

; ð2Þ

wherePðAcolÞ1 is the area of the ground storey columns andP

ðAcolÞ2 is the area of the first storey columns. In our case thereare no shear walls so we deal only with columns.

The quality of construction and material used are critical factorsto ensure the intended design protection is in fact in place. Exam-ples of poor construction qualities may be: construction error;improper construction procedures; lack of anchorage of beamand column reinforcement; poor concrete quality. In this study,the compressive concrete strength f =c is used as a surrogate mea-sure of construction quality.

3. Pushover analysis

Pushover analysis is an evaluation technique used to quantifyseismic induced non-linear response of structures. It is a staticanalysis as an approximation of dynamic response of structures,which has extensively been used for seismic performance evalua-tion of buildings (Ghobarah, 2000; Krawinkler & Seneviratna,1998; Kim & D’Amore, 1999). The pushover analysis works onthe premise that response of the structure can be related to the re-

sponse of an equivalent single degree-of-freedom (SDOF) system,so the response is controlled by a single mode, and the shape ofthis mode remains constant throughout the time history response.When the dynamic behavior of a structure is dominated by thefundamental mode of vibration, the result of the pushover analysiswith the load pattern proportional to the shape of the fundamentalmode is accurate. Basically, the pushover analysis is based on avery restrictive assumption that the displacement is time indepen-dent. This makes this method inaccurate when higher mode effectsare significant, i.e., in tall/moderately tall buildings (Fajfar, 2000;Kim & D’Amore, 1999; Mwafy & Elnashai, 2001). In the pushovermethod, there is a considerable correlation between the loadingpattern and observed response. The load pattern can be either‘‘fixed00 or ‘‘variable00 (Tso & Moghadam, 1998). In order to over-come some of the limitations of the method, it has been suggestedto assume two different load patterns and then to envelope theresults at the end (Fajfar, 2000). Fig. 1 shows result of a typical

Fig. 1. A typical performance curve from pushover analysis.

Table 1Comparing various seismic performance ratings (obtained from M Comerio, Univer-sity of California, Berkeley 2000).

FEMA273 SEAOC vision2000 rating

ATC post-earthquakeassessmentdesignation

Rating Performancelevels

Rating Preformanceexpection

Anticipateddamage

Green

S-l Immediateoccupancy

10 Fullyoperational

Negligable

Damagecontrol

9

S-2 8 Operational Light7

S-3 Life safety 6 Life safe Moderat Yellow5

S-4 Limitedsafety

4 Near collpase Severe

S-5 Collapseprevention

3

2 Partialcollapse

Complete Red

2 Partialcollapse-assemblyareas

1 Totalcollapse

2920 M. Modirzadeh et al. / Expert Systems with Applications 39 (2012) 2919–2926


pushover analysis curve, and the performance point is determinedby superimposing the demand curve and finding the intersectionpoint.

Performance of a building can be evaluated based on theexpected damage levels, no damage, minor damage, moderate repair-

able damage, severe damage, and collapse. Furthermore, performanceof the building can be related with the code-specified performancelevels (Table 1). For example, according to FEMA, 1996, the perfor-mance levels can be specified as immediate occupancy, damage con-trol, life safety, limited safety, and collapse prevention. A schematicrelation between these performance levels and corresponding dam-age levels are provided in Fig. 1.

4. Performance objectives

Performance objectives are acceptable level condition that thestructures should be under a given level of earthquake. Often, theexpected performances are specified with regards to the expecteddamage level (e.g. Table 1). Furthermore, in performance baseddesign and evaluation of a building, the performance levels maybe evaluated in terms of its lateral load resistance, global or inter-story drift and potential damage.

Quantification of damage after each earthquake is a dauntingtask. The damage quantification can broadly be classified into threecategories, (i) empirical damage indices are based on observeddamage statistics, (ii) strength-based damage indices that do notconsider response analysis, and (iii) response-based damage indi-ces are based on the analysis of the response parameters of ele-ments (local) or of the whole structure (global) (Ghobarah, 2000).Various response-based damage indices are proposed in the litera-ture. Ghobarah (2000) has proposed initial stiffness of pushovercurve as a response indicator, which has also been adopted in thiswork.

The response-based damage analysis is an attractive approach,however computationally it is time consuming and requires expertknowledge. Therefore, it is desirable to reduce the number of anal-ysis runs through a design of experiments technique. Additionally,it is desired to obtain an input-output relation between the build-ing modifiers and performance evaluative criteria. These possibili-ties are discussed in the following section.

5. Design of experiments: a response surface methodology

In order to perform a DOE study with type II objective defined inthe introduction section, a full factorial design along with a responsesurface methodology can be employed. A brief mathematical back-ground of this methodology is reviewed below and more detailscan be found elsewhere (e.g., Myers & Montgomery, 1995). In the

Fig. 2. Unfactored dead and live loads in a typical internal frame.

Table 2Experimental factors and their levels.

Factor Factor levels

SSI 0.8 0.7 0.6WSI 0.8 0.62 0.47

f =c 25 35 45

Table 3Full factorial combinations and computed experimental results.

Exp. # SSI WSI f =c (MPa) NAB NB-IO NIO-LS NLS-CP K (kN/m)

1 0.8 0.80 25 68 12 4 0 67.202 0.8 0.80 35 67 15 2 0 79.933 0.8 0.80 45 67 15 2 0 91.134 0.8 0.62 25 70 11 3 0 60.055 0.8 0.62 35 68 10 6 0 71.486 0.8 0.62 45 67 12 5 0 81.427 0.8 0.47 25 71 10 3 0 50.358 0.8 0.47 35 71 11 2 0 60.139 0.8 0.47 45 69 12 2 1 68.6610 0.7 0.80 25 68 12 4 0 65.1411 0.7 0.80 35 67 12 4 1 77.6212 0.7 0.80 45 67 12 5 0 88.4513 0.7 0.62 25 69 10 4 1 57.5714 0.7 0.62 35 68 11 4 1 68.7215 0.7 0.62 45 68 10 6 0 78.1416 0.7 0.47 25 72 11 1 0 47.4417 0.7 0.47 35 71 11 2 0 56.7218 0.7 0.47 45 69 12 3 0 64.7819 0.6 0.80 25 67 10 6 1 62.5720 0.6 0.80 35 68 11 5 0 74.6321 0.6 0.80 45 67 12 5 0 84.9322 0.6 0.62 25 70 10 3 1 54.5523 0.6 0.62 35 69 11 3 1 65.1024 0.6 0.62 45 68 11 4 1 74.2425 0.6 0.47 25 71 9 3 1 43.9826 0.6 0.47 35 70 11 3 0 52.6527 0.6 0.47 45 71 12 1 0 60.13

M. Modirzadeh et al. / Expert Systems with Applications 39 (2012) 2919–2926 2921


response methodology approach, the true response of a system(here the building structure) is modeled through a multiple linearregression analysis where main effects and/or interaction effectsbetween design factors can be explicitly accounted for. If the true(normally unknown) model of the system reads y ¼ f ðx; hÞ and theregressed/approximation model (normally a polynomial of ordern) is written as y ¼ f ðx; hÞ, then the approximation error at eachdesign point (i ¼ 1; . . . ;NÞ is written as

ei ¼ yi � yi; ð3Þwhere x is the vector of input (design) variables and the hat signsdenote estimated values. In the case of physical experiments, ei

should be normally responsible for random measurement errors,whereas in the case of computer experiments, it may be attributedto unassigned design variables or their interactions. h is the vectorof unknown coefficients in the model f . To estimate h, a set ofexperimental points from a DOE full factorial table can be used(an example is to follow shortly in the next section) and the leastsquare of prediction error is minimized:XN

i¼1

e2i !Min; ð4Þ

N is the number of experiments to be conducted. For a best lin-ear unbiased estimator, the estimation of unknown coefficientsyields (Myers & Montgomery, 1995):

h ¼ ðxT xÞ�1xT y; ð5Þ

where y is the response vector, ðy1; y2; . . . ; yNÞT , obtained according

to a pre-selected DOE table; x is the augmented design/experimen-tal matrix. Finally, the quality of fit, using the R2-statistic, can beevaluated using

R2 ¼ 1�PN

i¼1e2iPN

i¼1 yi � �yð Þ2

!; ð6Þ

where P is the number of model parameters and y� is the averageresponse over the measured experimental response values.

6. Illustrative example

To illustrate utility of the performance-based seismic evaluationof existing buildings using DOE, a six-storey RC building in Vancou-ver, Canada is considered. The three performance modifiers areconsidered to quantify seismic resiliency of the building.

6.1. The reinforced concrete building model

The six-storey RC building has 3 bays in X direction (2–9 moffice bays and a central 1–3 m corridor) and 7–6 m bays in Ydirection. The interior columns dimensions are 50 � 50 cm while

a b

c

Fig. 3. Performance curves from pushover analysis.



the exterior column dimensions are 45 � 45 cm. The one-way slabfloor system consists of 11 cm thick slab on supporting secondarybeams with dimension of 30 � 35 cm (from top of the slab to theend of the beam). Dimensions of the beams are 40 cmwide � 60 cm deep for the first three stories and 40 � 55 cm forthe three top stories. Further details of this building and load com-binations are provided in (CAC, 2005). In the design and evaluationof the building, plan irregularity is not considered. A two dimen-sional cross-section of the building and design-dead and live loadsare shown Fig. 2. These point loads are transmitted from slabthrough the secondary beams to the main frame beams.

Vancouver is situated in a high seismic hazard area. In congru-ence with the current seismic design practice, the response spec-trum is used to quantify seismic loads. The 5% damped spectralresponse accelerations for Sa(0.2), Sa(0.5), Sa(1.0) and Sa(2.0) are0.94, 0.64, 0.33 and 0.17, respectively (CAC, 2005). Thus, for a funda-mental period of the structure of 1.14 s, the peak spectral accelera-tion is computed as S(1.14) = 0.308. Seismic load was definedaccording to NBCC 2005 and the building importance factor, I, wasset to 1 for this particular office building. Ductility modifier Rd andoverstrength modifier Ro were set to 4 and 1.7, respectively (Mitchellet al. 2003). Once the loads and dimensions are specified, design andanalysis of the building is performed using ETABS (Wilson, 2002).

6.2. Material properties

The 28 day concrete compressive strength considered isf =c = 30 MPa, and corresponding self weight is calculated based ondensity of 24 KN/m3. For normal strength and density of concrete,

modulus of elasticity is calculated as Ec = 4500ffiffiffiffiffif =c

q. Concrete is

assumed to be isotopic with a Poisson’s ratio of 0.3. The steel yieldstress considered is Fy = 400 MPa.

6.3. Performing the DOE: Results of the pushover analysis

Design factors here are the three performance modifiers SSI, WSI,and f =c . The objective function (response) is the value ofK-performance indicator. Accordingly, for a full factorial DOE, basedon the three levels of each factors in Table 2, a total of 33 = 27 com-puter experiments should be performed. The 27 input parametersare summarized in Table 3, and corresponding pushover analysisoutputs are plotted in Fig. 3. Using the pushover analysis results,the initial stiffness of each result shown in Fig. 3 are computed andthe results are also summarized in Table 3.

To identify the (individual) effects of parameters on the response,‘main effect’ plots are constructed as shown in Figs. 4a–c. In Fig. 4a,for example, coordinate of the three points are found by collatingresponse values provided in Table 3 and averaging them over therows where SSI is at its first level (i.e., Experiments 1–8), second level(i.e., Experiments 10–18), and third level (i.e., Experiments 19–27).Similar procedure is applied to arrive at main effect plots for WSIand f =c . As a measure of parameter sensitivity, the variation rate/slope of response with each of the individual factors may indicatethe significance of that factor. Consequently, between WSI and SSIwhere both are dimensionless parameters, it appears that WSI hasa higher effect on the performance of the structure. A more precise

a b

c

Fig. 4. Sensitivity of performance modifiers with respect to (a) soft storey index (SSI) slope = 32, (b) weak storey index (WSI) slope = 66, and (c) construction quality (f’c)slope = 1.

Table 4Coefficients of the prediction model.

Coefficient Value Standard error |t-Ratio|

a0 �30.36 3.60 8.41a1 31.98 4.12 7.74a2 62.44 2.49 25.00a3 1.016 4.12E�02 24.63



(statistical) measure of parameters’ significance may be the t-ratio,as presented at the end of this section (Table 4). The trend ofvariations in Fig. 4a–c suggests that a linear model form may besufficient to capture the response of the structure over the rangeof the proposed factors’ levels.

It is important to note that thus far during the analysis it hasbeen assumed that the design factors are affecting the responseindependent of each other. To verify whether or not this assump-tion is indeed valid, interaction plots for each pair of the factorscan be plotted as shown in Figs. 3a–c. For instance, Fig. 3a showsthat the variation of response with f =c when WSI is set at a givenlevel (0.8 or 0.62 or 0.47). In fact, the interaction plot of f =c -WSIin Fig. 5a suggests that the variation rate of the response with f =cis independent of the value of WSI, and thus no significant interac-tion between these two parameters is detected. Similar results canbe deduced from Figs. 5b and c.

Finally, considering results of both main effect and interactioneffect analyses, a prediction model can be postulated as:

K ¼ a0 þ a1SSI þ a2WSIþ a3f =c ð7Þ

The unknown coefficients ða0; a1; a2; a3Þ of this model are foundby fitting the postulated model to the 27 experimental points inTable 3. With the identified model coefficients (Table 4) the fitquality, measured by R2-statistics, is found to be 99.1%. The abso-lute t-ratio values in Table 4 (i.e., a parameter value divided byits standard error) indicate that WSI has the most significant effecton the response, followed by f =c and SSI. The latter indication is con-sistent with the previous observations from the main effect plots.

a b

c

Fig. 5. Interaction diagrams with respect to K: (a) Interaction between construction quality (f =c ) and weak storey index (WSI); (b) Interaction between weak storey index(WSI) and soft storey index (SSI); (c) Interaction between construction quality (f =c ) and soft storey index (SSI).

Table 5Validation results via nine randomly chosen design points.

Performance modifiers

SSI WSI f =c (MPa) K (analysis) K (predicted)

0.85 0.72 41 82.42 84.040.66 0.79 39 81.15 80.290.76 0.69 33 70.78 71.250.68 0.67 27 62.66 61.190.63 0.64 43 73.45 73.990.78 0.60 36 72.06 69.540.72 0.81 43 87.61 87.330.60 0.76 31 67.44 68.630.58 0.82 42 84.23 82.49

Fig. 6. Comparing the approximated and experimented K values using ninerandomly chosen design points.



6.4. Validation of the prediction model

To validate the prediction model given in Eq. (6), random com-binations were selected within the upper and lower values limitsof the performance modifiers (Table 2). Table 5 summarized therandom combinations of the different performance modifiers,and the estimated and calculated stiffness values. A graphical com-parison of the approximation (via the predictive equation) andanalysis (via computer experiments) is shown also in Fig. 6. Theensuing R2 value is 97.30%.

7. Discussion and conclusions

This study showed the relation between three performancemodifiers for a building and their effect on the performance ofthe structure during seismic activity. Three performance modifiers,soft storey, weak story and construction quality, were chosen tohighlight indicators that have shown to have significant influenceon response of the structure (Tesfamariam & Saatcioglu, 2008).Length of the first story columns is the main factor indicating thepotential for ‘‘soft storey’’ effect. Since the ‘‘softness’’ of a storeyrelates to the stiffness of its columns, and the stiffness is propor-tional to 1/L3, one may choose the ratio of L as a surrogate measureof the Soft Storey Index. The other two performance modifiers arebasically related to the vertical discontinuity of the structurewhich results in a stress concentration and decrease in ‘‘strength’’.Area of the columns of every storey was chosen to resemble thestrength of that storey. Based on this, the A1/A2 may be used toquantify ‘‘weak storey index00. The last modifier is f =c ; which resem-bles the construction quality for the building.

Effects of the three modifiers (design parameters) on the finalresponse of the building structure were investigated by means ofa ‘‘pushover’’ analysis. A total of 27 different runs were performedas for three modifiers with three levels for each. During each run,the stiffness of the structure was calculated as the ultimateresponse of the structure and corresponding prediction model isproposed. Furthermore, the model is further validated with ninerandom combinations of the modifiers, and the prediction foundto be valid with a prediction quality of 97.30%. By increasing theweak and soft storey indices, i.e., with the length and area valuesof the ground floor columns being set close to those of the adjacentstoreys, the final stiffness of the structure was increased. In fact, asexpected, increasing the compressive strength of concreteenhances the stiffness of the structure.

The approximation of K showed that changing WSI has the mostdrastic effect on the final stiffness of the structure. Therefore the

vertical irregularity in the form of area variance can have a greaterrole on how the building responds to earthquake than the height ofground storey. f =c Showed the least effect on the response as com-pared to the other modifiers.

Finally, it should be emphasized that in this study, a static push-over analysis (assuming SDOF) was used for the six-storey building.For taller buildings where higher mode effects are significant, a timehistory dynamic analysis may be performed. It is also to add that formore realistic risk assessment of building structures under earth-quakes, the inclusion of more factors can be required in the analysis(i.e., not only from the structural perspective). As the number of fac-tors increase, the role of design of experiments concept comes to thefront again. In this study, DOE was used to establish an input–outputrelation and also to explore main and interaction effects between themodifiers. More objectives in DOE may be realized, like screeningthe most important factors among a large pool of factors attributingdifferent aspects of risk study in a hierarchical risk assessment struc-ture, or obtaining a mathematical model for predicting underlyingerrors in the response of the structure (e.g., due to uncertainties inthe modifier values). Incorporation of the performance modifiersuncertainty on the estimate of K is illustrated in Fig. 7.

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