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Cost optimal design of R/C buildings Alfredo H.-S. Ang a,*, Jae-Chull Lee b a

Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA b Applied Materials, 3320 Scott Blvd., M/S 1170, Santa Clara, CA 95054, USA

Abstract A systematic approach is proposed for evaluating the cost-effectiveness of existing or proposed design criteria from the standpoint of lifecycle cost consideration. A series of alternative designs of a model structure representing a class of R/C buildings would be developed following an existing code procedure, except that the code requirements or parameters will be varied for the alternative designs so that a suite of different structures will be obtained each with a different level of safety or reliability. For each of the designed structures, the probability of exceeding the various damage levels under a given earthquake intensity may be calculated. Aggregating and integrating all the cost components with the damage probability density functions for each of the designed structures, as well as with the probabilities of all possible earthquake intensities over a given life will yield the expected life-cycle costs for the respective structures as a function of structural reliability. From these results, the design with the minimum expected life-cycle cost may then be identi®ed; its underlying safety or reliability can also be determined. The approach is illustrated for a class of reinforced concrete buildings under earthquake loading. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cost-effectiveness; Earthquake-resistant; Structural reliability; Damage probability

1. Introduction The assurance of structural performance under severe earthquakes is the major objective of earthquake-resistant design of building structures. It is implicitly recognized in seismic codes that, even if possible, it would be excessively costly to design for absolute safety and with no damage under all likely earthquake intensities. Therefore, current earthquake-resistant design criteria imply the acceptance of some underlying risk. Given the uncertainties in earthquake loads, structural behavior and performance under a given earthquake loading, risk and probability must be considered when de®ning adequate design criteria. This leads to the formulation of proper design criteria in terms of required target reliabilities. The costs and losses from possible future earthquakes and the dif®culty in repairing post-yielding damage, strongly suggest the need for proper consideration of damage control in the design rather than just for life loss prevention. This can be addressed through the development of design criteria that balances the initial cost of the building with the expected potential losses from future earthquake-induced structural damage. * Corresponding author. E-mail addresses: [email protected] (A.H.-S. Ang), [email protected] (J.-C. Lee).

In developing criteria for performance-based design of structures, one of the key decisions will be the target levels of acceptable risk or reliability. Life safety is obviously essential and important in seismic design and ought to be preserved. However, cost-effectiveness has long been recognized to be also important, even though this issue has thus far not been explicitly included in the development of design requirements. To include cost-effectiveness will require the proper integration of socioeconomics and earthquake engineering technology. The initial cost of each of the designs may then be estimated, covering the costs of design, material, and construction. The potential damage cost from future earthquakes over the life of each of the designs would include the cost of repair and maintenance, the loss of contents, the cost of injury recovery, the cost of life saving, and direct and indirect economic losses. As damage costs are associated with future earthquakes, all such costs must be transformed to present worth. Here, the development of a methodology for determining optimal, cost-effective, earthquake-resistant design criteria is presented and applied to a class of low-rise reinforced concrete structures. Optimal design criteria are determined on the basis of the total expected life-cycle cost and acceptable risk of death from which optimal base shear coef®cients and target reliabilities can be obtained.

0951-8320/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0951-832 0(01)00058-8

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2. Methodology It is necessary to have some means of deciding on the right level of structural reliability or design in which limited resources might be spent to set or improve protection. Even with uncertainties, the quanti®cation of costs and the estimation of corresponding bene®ts of earthquake protection measures are prerequisites in the decision-making process in which the optimal balance between cost and safety may be made. The decision criteria for integrating structural reliability and socioeconomics may be classi®ed into three categories with respective goals based on: (i) minimal lifecycle cost, (ii) balanced risk of death [20], and (iii) acceptable marginal cost of life saving [7,12]. The three decision criteria are linked with each other. It is reasonable that all the criteria should be met. Then, the optimal aseismic design problem involving socioeconomics is stated as follows: minimize ECT ; subject to rF # rF0

1

where ECT is the total expected life-cycle cost in present worth; rF is the expected annual fatality rate for the structure and rF0 is the balanced risk of death. This is a minimum expected life-cycle cost decision criterion with the constraint that the expected annual fatality rate remains below the acceptable balanced risk of death. The results are the optimal design level, which may be de®ned in terms of a base shear coef®cient, the target structural reliability, and the target risk of death for a speci®c building structure at a given site. On this basis, the cost-effectiveness of current criteria for design may be appraised. The proposed methodology and application can be summarized as follows. First, a set of model buildings is designed for different levels of reliability or performance following the procedures of an existing design code, the SEAOC code [16], is illustrated here. For each design, the initial cost of the structure is estimated and the total expected cost of structural damage, including repair cost and other direct and indirect losses, is expressed as a function of the structural reliability or damage probability and in terms of the present worth. The expected risk of death for all designs under all likely earthquake intensities is also expressed as a function of the structural reliability or damage probability. A trade-off between the initial cost of the structure and the expected damage cost is then performed to determine the optimal design level and target reliability that minimizes the expected total life-cycle cost subjected to the constraint of the acceptable risk of death as stated in Eq. (1). 3. Formulation of cost functions Brie¯y, the total expected life-cycle cost function can be expressed as, ECT CI 1 ECD0

2

in which CI is the initial cost of the structure, and CD0 is the cumulative damage cost, in present worth, which includes the direct damage cost and indirect loss under all earthquakes that are likely to occur over the life of the structure. Assuming that the occurrences of earthquakes with a speci®ed minimum intensity constitute a Poisson process, that the occurrences and intensities of earthquakes are statistically independent, and that the structure is repaired every time a signi®cant earthquake occurs, the expected present worth of the cumulative damage cost from future earthquakes over the life L is obtained as follows: t ZL 1 0 ECD ECD n dt lnLECD 3 11q 0 where n is the annual mean occurrence rate of earthquakes with signi®cant intensities; a ln 1 1 q; ECD is the expected current damage cost due to an earthquake, in terms of current dollar value; l is the discount factor equal to {1 2 exp 2aL}= aL; q is the annual discount rate; and n L is the expected number of signi®cant earthquakes during the life L. The damage cost is expressed as a function of the structural damage level x; thus, the expected damage cost ECD in Eq. (3) can be obtained as X Zymax Z1 CDi xfXuY xfY y dx dy 4 ECD i

ymin

0

where X is the structural damage level; Y is the expected maximum ground intensity conditional on the occurrence of an earthquake; CDi x is the cost function for the damage component i; fXuY x is the probability density function of X conditional on Y y; and fY y is the probability density function of Y at the site. Initial cost. The initial cost function is obtained by designing structures using a current seismic design code, e.g. the UBC or the SEAOC code, but with varying base shear coef®cients. For each design, the corresponding cost is estimated and includes the costs of materials, design and construction. The reliability for each design, i.e. the probability of not exceeding a given damage level, is then calculated and the initial cost is expressed as a function of the damage probability, pf ; as CI CI pf : 3.1. Direct loss The direct losses include the cost of repair or replacement of the building, the loss of contents, the cost of injury and the cost of saving lives and is given by CA CR 1 CC 1 CF 1 CJ

5

where CR is the cost of repair or replacement; CC is the cost of contents; CF is the cost of lives and CJ is the cost of injury. Each component of the direct cost will depend on the structural damage level and is described by its expected value. The forms of the cost functions for the various damage components are given below. Repair cost. The repair cost function may be developed

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235

on the basis of available repair cost data from buildings damaged under previous earthquakes. For this purpose, damaged buildings are analyzed to calculate the damage states using the Park±Ang damage model [11] and a reliability analysis procedure such as the one described below. From these results, a regression relation for the normalized repair cost is formulated and expressed in terms of the median global damage index, leading to the following equations for the damage repair cost function:

cipal labor market studies of the implicit value of life in several countries, including the United States. On the basis of his study, Viscusi also determined the central ratio of the cost of a nonfatal injury to the cost of a life saved to be about 7:5 £ 1023 . Based on data from previous earthquakes, it can be estimated that 10% of all injuries may be assumed to be disabling injuries. Assuming that the loss from a disabling injury is equal to the cost of a life saved, the injury cost becomes

CR a1 dm a2 ; 0 # dm # d0

CJ rJ NO 0:9VJ 1 0:1VF

CR CI ;

d m . d0

6

where dm is the median global damage index of the structure and a 1, a 2, and d0 are constants to be evaluated from damage data. Loss of contents. The value of the building contents will, of course, depend on the building usage and is estimated as a percentage of the initial building cost. Here, the content loss is expressed in terms of the Park±Ang damage index in a form analogous to the nonlinear functions in Eq. (6). Cost of life saving and injury. Cost functions for injury and lives saved are determined on the basis of reasonable economic proposals and expected rates of casualty and injury estimated using casualty data for various types of structures damaged under previous earthquakes. The relationship between collapse rate and fatality rate has been proposed by Shiono et al. [15] for 11 construction types. In particular, a model was proposed for R.C. frame structures of good quality, which can be applied for the present study. That model is as follows: rF r0 pfc n

7

where pfc is the collapse probability of the structure; rF is the expected fatality rate, i.e. the number of deaths divided by the number of occupants in the building; r0 is the expected fatality rate for pfc 1 which is estimated at 7:5 £ 1022 for good-quality R.C. frame structures and n 1:6: The ratio of the injury rate to the fatality rate is expressed in terms of the computed median structural global damage index, dm ; on the basis of the data reported in Ref. [1]. Speci®cally, the following equation is used: rJ rF r dm

8

where rJ is the injury rate; rF is the fatality rate given by Eq. (7); and r dm is the ratio of the injury rate to the fatality rate as a function of the median global damage index of the structure. The cost function for life saving is expressed as: CF rF NO VF

9

where NO is the number of occupants in a building which is equal to the total ¯oor area times the number of occupants per unit area and VF is the value of a life saved. The value of life is estimated here using the willingness-to-pay approach for saving a life, which is obtained from the labor market studies conducted by Viscusi [19], compiled from 24 prin-

10

where VJ is the cost of a non-disabling injury. 3.2. Indirect loss Indirect losses are results of the so-called ripple effect on the economy caused by earthquake damage. Usually, indirect losses are evaluated by comparing the post-disaster scenario with a forecast of what the economy would be if the earthquake had not occurred. In order to evaluate the economic impact of earthquakes, it is necessary that certain kinds of economic models need to be developed. In this study, the so-called input±output (I±O) models [9] are used for this purpose. In this study, which is concerned with the cost-effective design of a particular class of buildings, the assessment of indirect loss as a consequence of damage to a single model building is required. Indirect loss items are expressed here in terms of economic surplus, which is the only appropriate measure of bene®ts for a cost/ bene®t analysis [3]. These indirect losses are computed accounting for the reduction in output related speci®cally to the loss of function from the building damage, which is the ®rst-round loss, as well as reduction of the productivity in some industries from the loss of capacity of other industries, which constitutes the second-round loss. For a given global damage level of the building, the ®rst-round loss, CB1, is obtained by: n n X X p tloss loss 1i Yi 1 i Yi n 11 CB1 tIO i i1 i1 where 1i is the economic surplus per unit total output of production sector i in the I±O table; Yip is the total output of sector i without any disaster; tloss is the loss of function measured in time [1]; tIO is the time interval of the I±O model; and ni is the sectoral participation factor, i.e. the ratio of the number of workers in the building participating in sector i to the number of total workers of the regional economy participating in sector i. The input coef®cient matrix of the I±O model, A, in the poster-earthquake economy may be approximated by assuming that the direct input requirements of sector i per unit of output j are reduced in proportion to the reduction in output i [2]. The new level of production is thus estimated as: Yd I 2 Ap 21 I 2 AYp

12

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where Dg is the global damage and < is the union of events. Damage of a critical component or substructure is sometimes computed as a combination of the damage of several components such as, for example, a story damage that is computed as a weighted average of the damage of several story columns and shear walls. 5. Reliability assessment

Fig. 1. Model building.

where A p is the input coef®cient matrix for the post-earthquake economy whose element Apij is equal to Yip =Yip Aij ; and Yip Yip 2 Yiloss : The second-round loss for a given damage level is thus obtained as: CB2

n X i1

1i Yip 2 Yid

13

The production loss for a given sector needed to compute the ®rst round loss is expressed in terms of the median global damage index of the structure. Details of the procedures and algorithms used to compute the indirect losses can be found in Ref. [8]. 4. Structural damage Extensive assessments of structural damage and reliability under various earthquake load intensities are obviously needed to compute the expected life-cycle cost. Damage of a reinforced concrete component, column or girder may be de®ned by the index [11]. D

dm bE 1 0 Q y du du

14

where D is the member damage index; d m is the maximum displacement; Qy is the yielding force; E is the hysteretic energy dissipated; du is the displacement capacity and b 0 is a constant. Global damage of a structure is de®ned as a function of the damages of its constituent elements or components, particularly, the critical components. Denoting damage of a critical component or substructure by Di ; the global damage is de®ned as: Dg . d < Di . d i

15

To properly assess a component damage under random seismic loads, the structure must be adequately modeled and analyzed in order to obtain its response under simulated or recorded earthquake ground motions. Since the structural response under severe and moderate earthquake loads is nonlinear and hysteretic, the calculation of the response statistics under random earthquake loads using appropriate random structural models and capacities becomes an extremely complex task. Here, Monte Carlo simulation is used to compute the desired response statistics. In this regard, the computer program DRAIN-2DX has been modi®ed to perform the desired Monte Carlo simulations, with the critical structural components modeled using the beam-column element in DRAIN-2DX with a tri-linear elasto-plastic hysteresis. Earthquake ground motions used as input for the simulation can be either actual earthquake records or samples of nonstationary, ®ltered Gaussian processes with both frequency and amplitude modulation. Uncertainties in structural properties and capacities and in the critical earthquake loading parameters are modeled as lognormal random variables. The procedure of the simpli®ed Monte Carlo simulation is described in Ref. [8]. 6. Illustrative application The general approach described above is illustrated for a class of ®ve-story R.C. special moment-resisting frame (SMRF) of®ce buildings located in a soft soil site in downtown Los Angeles. A brief description of the model building and the identi®cation of the parameters for some of the damage cost functions described above is presented and followed by a summary of the results obtained. Model building. The structure of each model building consists of nine parallel frames similar to that shown in Fig. 1 and four 4-bay frames in the plane normal to that of Fig. 1. Only the building response in the short direction is considered for this study. Several model buildings were designed according to the SEAOC code for service level seismic base shear coef®cients, C, ranging from 0.04 to 0.13. Currently, C is equal to 0.08 for the design following the SEAOC code. The initial costs are estimated based on the available building cost data [6,14]. A ®nite element model such as the one shown in Fig. 1 is constructed for each model building and used for the nonlinear time-history analyses required for structural damage and reliability assessments. Median global structural damage indices are

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237

Fig. 2. Median global damage index.

shown in Fig. 2 for all design levels considered in this study and for various intensities of the input ground motions. Damage repair cost and loss of contents. A regression relation between the computed median global damage index and the actual or estimated damage repair costs was obtained on the basis of reported damage repair costs for R.C. buildings damaged under previous earthquakes, together with damage assessment analyses. This regression relation is shown in Fig. 3. It is noted that a median global damage of about 0.5 corresponds to the limit of repairable damage, whereas dm 1:0 corresponds to eminent collapse. The value of the contents of the building is assumed to be 40% of the initial building cost, and the loss of contents is assumed to reach a maximum for a median global damage index dm 1:0: Making use of the data in Ref. [5], the loss of contents function for all values of the median structural damage is obtained. Cost of life saving and injury. The life saving and injury cost functions are obtained as described above together with a willingness-to-pay to save a life of seven million dollars. Accordingly, the life saving and injury cost functions for 103 occupants can be formulated, respectively.

Fig. 3. Damage repair cost function.

Fig. 4. Expected life-cycle cost.

Indirect loss function. The indirect cost function is computed using the methodology referred to above in Section 3.2, with the employment population data for the Los Angeles area per economic sector [17] and the I±O table [18]. The computed total indirect loss in terms of economic surplus is expressed as a function of the median damage level. Optimal target reliability and base shear coef®cient. The total expected life-cycle costs for the ®ve designs of the model building are shown in Fig. 4 as a function of the probability of collapse over the 50-year life span of the structure for the seismic hazard of downtown Los

Fig. 5. Expected fatality rate.

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Angeles [4,13] and an average discount rate q 4%: The contributions of the various damage cost items to the total expected life-cycle cost are also shown in Fig. 4. The corresponding expected annual fatality rates, rF, are shown in Fig. 5. Using a voluntary risk of death of 4 £ 1025 for the United States [10], the acceptable annual earthquake-related risk of death, based on 1023 £ voluntary risk [8], is 4 £ 1028 : This threshold is shown in Fig. 5. Using the optimal design criterion as stated in Eq. (1), the optimal base shear coef®cient for the building class would be between 0.08 (current) and 0.10, and its corresponding probability of collapse over the structure life would be around 2 £ 10 24. 7. Conclusions On the basis of the results presented above the following conclusions can be made: Among the various damage cost items, the building damage repair cost contributes the most to the total expected damage cost and the indirect business loss appears to be of the same order as the cost from the loss of contents. Fatality and injury rates for the reinforced concrete SMRF structures are low and, therefore, the costs of life saving and injury do not contribute much to the total expected damage cost. Finally, using the minimum expected life-cycle cost decision criterion with the constraint of the acceptable risk of death, it is noted that for the building class analyzed, the resulting optimal base shear coef®cient appears to be between 0.08 and 0.10, a little higher value than the currently required one, 0.08, in the SEAOC code. The corresponding life-cycle collapse probability would be around 2 £ 1024 : Acknowledgements The developments presented here are based, in part, on a study supported by NSF jointly with the University of Missouri, Rolla. This support is gratefully acknowledged. References [1] Applied Technology Council. Earthquake Damage Evaluation Data for California. Redwood City, CA: ATC-13, 1985.

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