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13 Earthquake Engineering and Engineering Seismology Earthquake Engineering and Engineering Seismology 13 Volume 1, Number 1, September 1999, pp. 13–26 Practical Seismic Design Criteria and Life-Cycle Optimization for Structures with Hysteretic Energy-Dissipating Devices Luis Esteva1) Orlando Díaz1) Jaime García1) 1) Professor, Institute of Engineering, National University of Mexico. ABSTRACT A review is presented of several topics related to seismic design criteria, response and damage estimation methods and life-cycle optimization analysis of structural frames provided with hysteretic energy dissipating devices. A ductility-based design method is proposed, adequate for practical applications. Stiffness-and-strength degrading hysteretic moment-rotation models are used to represent the behavior of critical sections of reinforced concrete beams and columns. These models are used to study the process of damage accumulation, as well as the influence of some seismic-design parameters on the performance and reliability of multistory frames. INTRODUCTION ever, many of the concepts and criteria presented are directly applicable to One approach of modern earthquake systems provided with other types or engineering to the control of structural devices, or may be easily adapted to their damage produced by earthquakes is the specific conditions. use of energy-dissipating devices (EDD’s). Up to now, a large portion of the These are elements capable of undergoing applications of EDD’s in earthquake a large number of high amplitude engineering problems have been ad- deformation cycles without experiencing dressed to the strengthening or retro- any significant degradation in the fitting of existing structures, including strengths and stiffnesses of their cons- both cases, when the system considered titutive functions (stress-strain or force- is in an undamaged state and when it deformation curves). Our discussion will shows some initial damage. The latter be oriented to those devices that base may arise as a result of previous earth- their energy-dissipation capability on the quakes or of other perturbations, such as hysteretic behavior of the materials or differential settlements or excessive elements that constitute them. How- stresses produced by gravitational loads. 14 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 Here we devote attention to this type of The need for a simple and realistic, applications, as well as to those when the performance-based, seismic design cri- use of EDD’s is considered at the design terion is obvious. The approach pro- stage of a new construction as a means posed here is an attempt to contribute to for attaining an efficient structural satisfy this need. It is based on repre- system from a life-cycle perspective. For senting the EDD’s as conventional this purpose, the mentioned devices must structural members, characterized by be considered as ordinary structural their constitutive functions for alternating elements, described in terms of their load cycles. The main frame system is cyclic constitutive functions. In addi- described also in terms of its constitutive tion, the decisions regarding design functions, expressed in terms of either the values and repair and maintenance base shear vs. top displacement curves or policies must be made in terms of a cost- of the shear vs. drift at each story. In benefit analysis that accounts for the both cases, these curves correspond to manners in which the initial costs and monotonic load application, and are those expected to be generated in the modified by simplified rules that account future may be affected by those decisions. for stiffness and strength degradation In order to permit the comparison of resulting from damage accumulation. initial costs with those that may be In order to establish optimum design generated at unknown times in the criteria for systems of the type described future, the latter must be transformed above, it is necessary to formulate the into their equivalent values at the mo- problem within a life-cycle optimization ment for which initial costs are estimated. framework that permits the comparison The main benefits of the use of EDD’s of alternatives regarding design require- in structures exposed to severe earth- ments and maintenance policies. This quakes arise from (a) their capability to creates the need for damage accumu- control damage on the structural and lation models and indices for a given non-structural elements of the main system, as well as of simple relations that system, and (b) the possibility of being can be readily used to transform those easily replaced when damage accu- indices into the corresponding variations mulated on them reaches unacceptable in the mechanical properties and relia- levels. However, these objectives are not bility functions of the system. These explicitly considered in quantitative terms problems are the subject of the last in the conventional practice of earth- sections of this paper. quake resistant design, which empha- sizes the additional damping supplied by EDD’s and the local ductility demands DUCTILITY-BASED DESIGN experienced by them, as well as by the members of the main structural system at The design criterion that follows is their critical sections. But talking about applicable to cases where the energy- additional equivalent damping ratio when dissipation system (EDS) is assumed to dealing with hysteretic dampers may be act in parallel with the conventional frame inadequate, or at least ambiguous, since (CF) at each story. Thus, the design that value is a function of the response parameters are the contributions of both amplitudes and frequencies, which are in systems to the story strengths and turn correlated. stiffnesses (Rd , Rc , Kd and Kc , res- Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 15 pectively). These are conveniently expressed in terms of the resulting story strengths and stiffnesses (R and K), and of the ratios a = Kd / Kc and b = Rd / Rc . The design conditions are expressed in terms of md*, mc* and d*, the allowable values of the story ductility demands md and mc , for the energy-dissipating and conventional systems, and of the story drift d, respectively. As a first approximation for preliminary design, a set of values K, R, a Fig. 1 Idealized shear vs. drift function and b are chosen so that the combined for a story of a building frame system (CS) satisfies the conditions mentioned above for the set of nonlinear response spectra of the design earth- the ordinates of the acceleration response quake for different ductility values. For spectra are assumed to be S(T, m). The this purpose, the global properties of the ratio S(T, 1) /S(T, m) will be denoted by CS (K and R at each story, as well as the Q, and the ratio Q / m will be denoted by g. allowable ductility value, m* ) are ex- Design proceeds in accordance with the pressed in terms of the parameters following steps: defined above. For an elasto-plastic 1. Target values of mc* and md* are system, the following relations are established. In some cases, these obtained from Fig. 1: values are established independently of each other; that is, there are no d d d restrictions to the possible values of e. mc = , md = , m= d yc d yd dy In many cases, either Kc / Rc or (1a, 1b, 1c) Kd / Rd , or both, can only adopt values contained within narrow Here, dyc = Rc / Kc , dyd = Rd / Kd and dy = intervals, which imposes restrictions R / K are the yield deformations of the CF, to the possible values of e. Under these conditions, a new set of the EDS and the CS, respectively. It is allowable ductility values mc¢ < mc*, and convenient to define a new parameter e = md¢ < md* is adopted, such the e lies a / b, which is equal to md* / mc* = dyc / dyd . within the range of its acceptable From the foregoing concepts it is easy values. to show the following relation between the 2. In any of the previous cases, the ductility ratios developed by the CS and design is started, for instance, by the CF: assuming reasonable values for Kc 1+ a and Kd . This determines a and T. m = mc (2) 3. Once a practically feasible value of e 1+b has been adopted, b is taken equal to The design earthquake is supposed a / e. to be defined by a set of elasto-plastic 4. Taking into account Eq. (2), the target response spectra for different ductility ductility value for the CS is equal to m¢ ratios. For the linear-behavior natural = mc¢ (1 + a ) / (1 + b ). The required period of the system to be designed, base-shear ratio is determined from 16 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 the ordinate of the design response different ductility levels, by a system with spectrum corresponding to this an equivalent ductile capacity; second, ductility value. This determines the the strength and stiffness degradation required lateral strength R at each properties of members of the CF differ story. from those typical of the EDD’s. 5. The peak story displacement d is The calibration proposed is underway. obtained as m¢ R / K, and compared It has consisted in comparing the ratios of with d*. If d < d*, the design has been actual ductility demands (computed by completed at the level of global system means of step-by-step integration) to their parameters; otherwise, a new set of target values, for both, conventional tentative parameters must be inves- frames and systems provided with EDD’s, tigated. assuming they are all designed on the basis of allowable values of story ductility In ordinary multi-story frames, EDD’s demands. are connected to the beams or to the Figures 2 and 3 show ductility spectra beam-column joints by means of diagonal for systems designed in accordance with members. The axial forces acting on the the algorithm described above for columns will be the result of the story prescribed values of the allowable shear forces taken by both, the CF and ductilities, mc¢ and md¢, for different values the EDS. of a and e. The case a = 0 corresponds to the ordinary frame without EDD’s. These curves are representative of a CALIBRATION OF DESIGN number of curves that were obtained for CRITERION systems with different values of those parameters. Each curve corresponds to a The design criterion sketched above is consistent with modern ideas about performance-based design of con- ventional systems. Because the EDD’s are explicitly dealt with as structural elements with known force-deflection functions, peak story drifts and ductility demands on those elements and on the CF are estimated with the same tools applied in the design of conventional systems. However, before adopting this criterion for routine practical design, it seems convenient to calibrate it, comparing the safety levels attained through its use with those implicitly accepted for conventional systems according to current design procedures. The need to do this arises from two main considerations. First, an approximation is implied in the replacement of the CS, Fig. 2 Ductility demand for 1dof con- made of elements capable of developing ventional system Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 17 ductility m¢ that were studied, the ductility demands on the CF are largest for aT = 0.4 and bT = 0.33. Here, aT and bT are the parameters of Takeda’s model, such that aT = 0 means that the stiffness of the unloading branch is not reduced, while b = 1.0 corresponds to the case where the re-loading branch reaches the original force-deflection curve at the yield-point for the first load cycle. For m¢ = 2, the use of EDD’s reduces in 44 percent the maximum ductility demand, which occurs for a natural period equal to 2.5s. Also, for both values of m¢ the use of EDD’s causes an increase of the response for systems with aT = 0, regardless of the value of bT , for natural periods shorter than 1.5s. Thus, the change in dynamic properties resulting from the use of EDD’s is not always beneficial to the expected behavior of a structure. More systematic studies are necessary to clarify this point. The results discussed in the foregoing paragraphs correspond to deterministic systems subjected to random earthquake Fig. 3 Ductility demand for 1dof system ground motion of a given intensity. In with EDD; a = 0.5, e = 2.0 reality, the design conditions are established in terms of nominal values of value of m¢, the target ductility demand for the design variables (loads, strengths, the CS. The ordinates show the expected stiffnesses, safety factors, etc.). The values of the ductility demands on the CF expected or the most likely value of the for systems subjected to a sample of base-shear lateral capacity of a given simulated acceleration time histories with structure are usually significantly greater statistical properties equal to those of than the value implied by the nominal the EW component of the SCT (Mexico base-shear coefficient used in design. City) record of September 19, 1985 Therefore, before using graphs similar to (SCT850919EW). The behavior of the those included in Figs. 2 and 3 for the EDD’s was assumed to be elasto-plastic, purpose of estimating expected ductility while the CF was assumed to behave in demands of actual systems, adequate accordance with a Takeda model, with criteria must be developed to transform parameters representative of those the nominal values of lateral strengths obtained in laboratory tests of ductile into their expected or most likely values. moment-resisting reinforced concrete Table 1 gives an idea of the order of frame components [1]. magnitude of the ratios between expected A comparison of Figs. 2 and 3 shows and nominal lateral strength coefficients that, for the two values of the target for some typical multistory frames 18 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 designed in accordance with the Mexico concentrated at the critical sections City seismic regulations of 1993. The located at their member ends. For each variables considered are N, the number of of these sections, the plastic component stories, and the nominal value of the of the curvature varies with the acting base-shear coefficient used in design. internal moment as shown in Fig. 4. The The results show that the ratios between constitutive law shown [2] was derived expected and nominal values of lateral starting from one previously proposed by capacities, measured by base-shear Wang and Shah [3], which appeared to coefficients, are ordinarily greater than 2, predict excessive stiffness-and-strength and can be very high for low values of the degradation. In its present version, nominal design coefficient. moments vary within two enveloping straight lines that intersect the vertical axis at the values of the corresponding Table 1 Comparison of nominal and yield moments. A damage index D is expected values of base shear defined, which is associated with the ratios (c , c ) residual stiffness and capacity of the critical section in each loading N c c /c direction. This index is obtained as a 0.05 3.88 conventional fatigue index: D = S (q /qF), 5 0.10 2.42 where q and qF are respectively the 0.15 2.07 plastic-hinge rotation and the rotation at 0.05 3.10 failure (the point where the lateral load 10 0.10 2.37 reaches its maximum) under monotonic 0.15 2.31 loading conditions. The reloading 0.05 2.52 branch in a given direction is a straight 15 line that intersects the vertical 0.10 2.32 corresponding to the maximum rotation 0.05 2.46 20 amplitude previously reached in that 0.10 2.38 direction at an ordinate M ¢ smaller than DAMAGE ACCUMULATION MODELS The behavior models described in the following paragraphs were used in the study on ductility and damage distri- bution in multistory frames presented below. They are representative of cases frequently found in engineering practice. Behavior Models for Members of Conventional Frames The nonlinear behavior of the Fig. 4 Stiffness-degrading functions for conventional frames is assumed to be plastic-hinge rotation Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 19 that associated with the virgin moment- the initial damage values on both groups rotation curve for monotonic load of elements. Thus, it was deemed application (M). The original and the convenient to obtain curves displaying reduced values are related by the the expected final value of the damage equation M ¢ = M (1 – l), where l = 1 – index Dk as a function of the initial value, exp (– k D). Moment-rotation curves of for different intensities [9]. Figures 5 this form are used in this paper. A value and 6 show some of those curves for of k equal to 0.0671 was adopted, after systems with natural periods equal fitting the mathematical models to some respectively to 1.0 and 1.5s, built on the experimental results reported by Ma, et soft soil area in Mexico City, and designed al., [4], Wang and Shah [3], Townsend according to the current seismic code for and Hanson [5], Scribner and Wight [6] a nominal base shear ratio of 0.10. The and Uzumeri [7]. stiffness and strength ratios, a and b, are both equal to 1.0. The intensity of the Behavior Models for EDD’s excitation is represented by its normalized The constitutive functions proposed for EDD’s are based on some derived from the results of laboratory tests carried out by Aguirre and Sánchez [8] in U-shaped hysteretic EDD’s. A non-degrading bi- linear moment-rotation curve is assumed. Failure of one of these elements is assumed to occur when the fatigue index DF exceeds unity, where DF = S Ni–1, and Ni is the expected number of constant- amplitude deformation cycles that lead to fatigue failure. This number is given by the equation Ni = exp (121 (x –0.02 – 1) ), where xi is the ratio of the amplitude of the Fig. 5 Cumulative damage functions for deformation cycle to the deformation at frame with EDD; T = 1.0s, rk = 0.5 failure under monotonic load. Story Damage Index At any individual story, the damage index Dk in a given load direction is defined as the ratio (K0 – Ks) / K0, where Ks is the secant stiffness associated with the maximum story drift in that direction, and K0 is the corresponding initial tangent stiffness for small deformations. The life-cycle optimization formulation presented at the end of this paper makes use of conditional probability distri- butions of the damage on both, the CF and the EDD’s, at the end of an Fig. 6 Cumulative damage functions for earthquake, in terms of the intensity and frame with EDD; T = 1.5s, rk = 0.5 20 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 value, y / ym , with respect to that of Table 2 Properties of systems studied SCT850919EW. Use was made of the modified Wang and Shah model, with Stories System with T (s) rk y m r strength-degradation parameters ob- EDD’s tained from the results published by them. a None 1.41 ¾ ¾ 4 ¾ Because these parameters seem to b all 1.33 0.25 0.50 4 ¾ predict excessively high degradation c all 1.38 0.25 0.50 5 ¾ values, the damage indexes shown in Figs. d all 1.37 0.50 0.50 4 ¾ 5 and 6 may be too high. e all 1.45 0.50 0.50 5 ¾ f all 1.34 0.50 1.00 5 ¾ g DUCTILITY AND DAMAGE 1~4 1.36 0.50 1.00 5 1.0 h 1~4 1.34 0.50 1.00 5 1.1 DISTRIBUTION IN MULTISTORY i 1~4 1.34 0.50 1.00 5 1.2 FRAMES A study on the spatial distributions of members of the lowest stories (see over- ductility demands and damage indexes strength factor r, in the last column in was carried out in a group of several Table 2). In all cases, the excitation was building frames, including one that did represented by a sample of simulated not have EDD’s (see Table 2, taken from acceleration time histories with statistical Campos [10]). As in the sdof systems properties equal to those of studied in the previous section, the SCT850919EW [11], normalized to the responses computed for the system same spectral intensity I : without EDD’s serve as a basis of comparison for the responses of the other 1 T1 cases. All the frames studied in this I = 2p ò 0 T × S a (T , z) dT (3) series were fourteen-story high, with their fundamental periods of vibration ranging Figure 7 shows the average ordinates between 1.34 and 1.41s. The variables of the elasto-plastic response spectra for studied were the nominal value m of the different values of the ductility value. target design ductility for the combined The cyclic behavior of the EDD’s was system, the ratio y of the yield deflection assumed to be bilinear, with a post- of the EDD to that of the CF at each story, yielding stiffness equal to 3.2 percent of and the ratio rk between the stiffnesses of that corresponding to the linear range. the EDD and the CS at each story. The The nonlinear behavior of the conven- last three cases in Table 2 correspond to tional frame was assumed to be con- systems where EDD’s were placed only in centrated at plastic hinges forming at its the first four stories, with the objective of member ends, which were represented by concentrating all energy dissipation in stiffness-and-strength-degrading elements those stories. In case g, the members in following the model proposed by Shah the stories above the first four were and Wang [3], modified by Esteva and designed for the same load factor used for Díaz [2], and summarized above. The the lower ones. In cases h and i the load results of the analysis are summarized in factors used for the design of the upper Figs. 8 and 9, which represent the mean stories were respectively 1.1 and 1.2 values of the story damage indexes and of times those applied for the design of the the story ductility demands, respectively. Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 21 Fig. 7 Mean response spectra for simu- Fig. 8 Story damage indexes in multi- lated accelerograms story frames The results show that story ductility demands never exceeded significantly of 3.0, and were usually substantially smaller, in spite of the fact that the target design ductilities for the CS were equal to 4.0 or 5.0. This is the result of two effects that tend to compensate each other. On one hand, the degrading behavior of the CF tends to produce larger ductility demands than expected for the non-degrading bilinear model; on the other, nominal values of design loads and member capacities lie on the conservative side of their most probable values. This leads to larger expected ductility de- mands than their nominal values assumed in design. The beneficial influence of the non- Fig. 9 Story ductility demand in multi- story frames degrading behavior of the EDD’s, as compared with the degrading behavior of the CF, is clearly shown in Figs. 6 and 7. keeping target ductility values for the CS Thus, a comparison of curves corres- equal to 4.0. By means of curves a, c, e ponding to systems a, b and d show that and f a comparison is made of the mean story damage indexes and ductility behavior of conventional frames designed demands are significantly reduced when for a target ductility value of 4.0 with that fractions of story stiffnesses equal to 0.25 of systems with EDD’s that had been and 0.50 are provided by EDD’s, while designed for a system ductility of 5.0. It 22 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 can be appreciated that damage indexes stand for expected value and standard and ductility demands are lower along a deviation, respectively. For frame a, bC large portion of the structure’s height for was equal to 3.18. The largest value the systems with EDD’s than for the adopted by any of the systems with EDD’s conventional frame system. corresponded to systems d, f and i (4.76, Frames g, h and i were studied in 7.88 and 5.81, respectively), while the order to obtain some information about lowest values corresponded to systems e the efficiency of EDD’s used only in the and h (3.62 and 3.27, respectively). lowest stories of a frame. The design values of the lateral shear forces acting on the stories of the CS were derived from the design spectrum corresponding to a DAMAGE ACCUMULATION nominal target ductility of 5.0. However, UNDER RANDOM an additional load factor greater than EARTHQUAKE SEQUENCES unity was applied to the design of the structural members in stories above the For simplicity, the discussion that fourth one for systems h and i. Thus, the follows refers to a single-story system stories having EDD’s were conceived as with one EDD. Just after the occurrence subsystems providing partial base of the j-th earthquake, the damage isolation. The figures show that for the accumulated on that element is equal to lowest four stories the damage indexes Ddj , while that affecting the conventional and ductility demands were lower for the structural frame is equal to Dcj . After the systems with EDD’s than for the (j + 1)-th event, these values become conventional frame designed for a respectively Dd (j+1) = Ddj + dd (j+1) and Dc (j+1) nominal target ductility value of 4.0. At =Dcj + dc (j+1), where dd (j+1) and dc (j+1) are the the upper stories, ductility demands in systems g and h were significantly larger corresponding damage increments. If Dcj than for frame a, but remained in general exceeds a given threshold, designated below the maximum value experienced by here as Drc , the frame is repaired in such the conventional frame anywhere along a manner as to eliminate the damage its height. accumulated, thus restoring its initial strength and stiffness, Rc and Kc . It is The response studies of the systems assumed that the damage level on the in Table 2 were also used to assess the frame can be assessed from observation influence of the EDD’s on the structural of the evidences of physical deterioration, reliability levels. For this purpose, the while that on the EDD’s is inferred from reliability for the family of ground motion records used to represent the random the estimated value of the low-cycle- excitation for a given intensity was fatigue index. This information is used measured by means of index bC (Cornell’s to implement the preventing strategy of beta). This was defined in terms of the replacing the EDD after the occurrence of random variable Z, equal to the natural a number of high-intensity earthquakes, logarithm of the minimum value, along on the basis of a threshold value Drd , the system’s height, of the ratio of the defined within the framework of a life- story available ductility to the cycle optimization approach. corresponding ductility demand that Whether the process of occurrence of results from the dynamic response earthquake ground motions with different analysis: bC = E(Z) / sZ , where E( ) and s characteristics involves some kind of Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 23 correlation with previous history or is objective function must be minimized: independent from it, the levels of damage é¥ ù accumulated Dcj and Ddj , j = 1, ¼, ¥, at U = C + E êå L-q Ti ú i (4) the end of the j-th earthquake occur as ëi =1 û events of a Markov process. The Here, E[ × ] stands for expected value, and transition probabilities from (Dcj, Ddj ) to q is an adequate discount rate. (Dc (j+1), Dd(j+1)) are obtained from the proba- bility density functions of dc (j+1) and dd (j+1), which depend on Dcj and Ddj, as well as on the probability density function of Yj +1, CASE STUDY the intensity of the (j + 1)-th event. Esteva, et al., [9] applied the optimi- In order to determine the conditional zation analysis described above to three probability density functions of Dc (j+1) and reinforced concrete buildings: five, ten Dd (j+1), given the values corresponding to and fifteen-story high, respectively. The the end of the j-th earthquake, it is following paragraphs make a brief necessary both, to calculate the joint description of the method used and probability density function of the present the main results for the tallest intensity of the (j + 1)-th and the waiting building. It has a square plan area of time to its occurrence, and to determine 346m2, and a fundamental vibration the damage states Dci¢ and Ddi¢ of the period of 1.5s. It is assumed that it will system’s components after carrying out be built at a soft soil site in the Valley of the operations of repairing the Mexico, where local soil conditions are conventional frame members and/or similar to those of the recording site of the replacing the EDD’s. The conditional accelerogram SCT850919EW mentioned probability functions obtained in this above. The seismic hazard at the site manner are integrated recursively in was expressed by an intensity-recurrence order to obtain the marginal probability curve previously obtained. distributions of all Dcj and Ddj . Details The behavior of the reinforced are given by Díaz and Esteva [2]. In the concrete members was represented by life-cycle optimization studies reported in Wang and Shah’s model [3]. The this paper, the direct approach described modifications proposed by Díaz and above was replaced by a Monte Carlo Esteva [2] were not included. Therefore, simulation procedure described by Esteva, according to the comments presented in et al. [9]. the section on damage accumulation models, it is thought that the predicted values of the stiffness and strength LIFE-CYCLE OPTIMIZATION degradation of those members are excessively large, thus leading to Let Ci be the initial construction cost excessively large responses and damage of a system of interest, Ti , i = 1, ¼, ¥ the levels. The case study is presented, (random) times of occurrence of however, because of the value of the earthquakes that may affect it, and Li , i = qualitative conclusions extracted from its 1, ¼, ¥ the losses associated with those results. earthquakes; they include damage and On the basis of some approximate failure consequences, as well as repair estimates of initial construction costs in and maintenance actions. The following terms of the seismic design coefficient, c, 24 Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1 it was concluded that the initial con- The estimation process starts by struction cost C of the frame system simulating a number of seismic histories, varies with that coefficient according to each consisting of the intensities and the the equation C = C0 (1 + 0.14 × Nc1.5 ) 0.4, times of occurrence of the corresponding where N is the number of stories and C0 is events. Damage levels on the CF and the the value of C for c = 0. The cost of EDS at the end of each event are obtained installing an EDD of the type considered, by Monte Carlo simulation, taking into with a lateral yield strength P, is equal to account the intensity of the ground 6.17 P 0.68, where P is expressed in kilo- motion and the damage levels on both grams and the cost is expressed in US elements at the beginning of the dollars. earthquake. Repair or replacements The repair-cost of the structural and actions are taken, if adequate, after non-structural elements at a given story comparing the damage levels with the was assumed to vary linearly with the pre-established threshold values. The damage index Dk at that story. initial damage conditions for the next The variables considered in the study earthquake are established. The of the building selected were the seismic corresponding costs are then obtained design coefficient and the pre-established and used to calculate the value of the threshold values of the story damage term inside the parenthesis in Eq. (4). levels whose exceedance would lead to After doing this for a sufficiently large repair and/or replacement actions (Drc sample of seismic histories, the expected and Drd , for elements of the CF and of the value appearing in that equation can be EDS, respectively). At each story, the obtained. The results for the system contribution of the EDS, both to the considered are shown in Table 3. There, lateral stiffness and to the lateral strength C is the initial construction cost for the of the CS, is equal to 75 percent. This structure; it is a function of the seismic value is higher than those found in typical design coefficient c. C1 is the initial practical cases. construction cost for a reference system, The mechanical properties of the which in this case was a conventional structure and the gravitational loads frame, without EDD’s, designed for c = 0.1. acting on it were taken equal to their The figures reported in the table are expected values. Once these values were values of U / C1. The minimum values of derived from the nominal design para- this variable are shown in bold type. It is meters and the assumed statistical mo- easy to see that the normalized utility is dels and safety factors, an equivalent sdof not very sensitive to the repair and system was assumed to represent the real replacement thresholds. It is interesting system for the purpose of performing a to see that the optimum Drc is very high, life-cycle optimization analysis. The free which may be a consequence of the fact variables are the seismic design that the contribution of the CF to the coefficient c and the threshold values for lateral stiffness and strength of the repair and replacement of the CF and the system is relatively low (0.25). EDS: Drc and Drd . For each combination This example is only intended to show of these variables an estimate is made of an approach to the life-cycle optimization the negative utility U, given by Eq. (4). of a system with EDD’s on the basis of The estimate is obtained by Monte Carlo expected utilities. Another approach simulation. deserving attention is that of decision Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization 25 making on the basis of tolerable risk Because the reductions in response criteria. and damage achieved by the use of EDD’s have a cost, it may happen that a stronger conventional frame, designed for the Table 3 Influence of c, Drd on C/C1 and same tolerable lateral drifts as one with U/C1 EDD’s, has a cost lower than that of the c Drc latter. Optimum decisions in these Drd C/C1 0.2 0.4 0.6 0.8 cases have to be made under a life-cycle 0.05 0.2 1.470 1.493 1.508 1.507 framework that accounts for the process 1.168 0.4 1.472 1.493 1.510 1.501 of damage accumulation. 0.6 1.470 1.489 1.500 1.502 0.10 0.2 1.280 1.282 1.287 1.285 1.194 0.4 1.281 1.288 1.286 1.290 0.6 1.281 1.283 1.285 1.289 ACKNOWLEDGEMENTS 0.15 0.2 1.257 1.252 1.250 1.252 1.2259 0.4 1.255 1.252 1.251 1.250 The authors express their gratitude to 0.6 1.255 1.252 1.251 1.250 Danto Campos, Miguel Ruiz and Leonardo Veras, for their valuable colla- boration in some of the studies reported here. CONCLUDING REMARKS Significant reductions in the expected seismic response of conventional struc- REFERENCES tural frames may be obtained by replacing a portion of their lateral strength and 1. Bertero, V.V. and Popov, E.P. (1975). stiffness with that provided by hysteretic “Hysteretic behavior of ductile energy-dissipating devices. This is a moment-resisting reinforced concrete consequence of the capacity of the latter frame components,” Earthquake Engi- elements to sustain large numbers of neering Research Center, University of large-amplitude deformation cycles with- California, Berkeley, Report EERC 75- out suffering significant degradation of 16. their mechanical properties. The best 2. Díaz, O.J. and Esteva, L. (1993). way to estimate their possible influence “Seismic damage indexes in decisions on the response of specific nonlinear related to structural safety,” Proc. 7th degrading systems is by dealing with both, IFIP WG7.5 Working Conference, EDD’s and conventional frame members, University of Colorada, Boulder. as with nonlinear elements, each charac- 3. Wang, M.L. and Shah, S.P. 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(1998). “Optimization analysis in the (1977). “Reinforced concrete connection earthquake-resistant design of struc- hysteresis loops,” ACI Special Publi- tures with energy-dissipating devices,” cation SP53-13, pp. 351–370. draft of a paper reporting work in 6. Scribner, C.F. and Wight, J.K. (1978). progress at the Institute of Engineering, “Delaying shear-strength decay in National University of Mexico. 10. Campos, D. (1998). “Optimization reinforced concrete flexural members criteria for the design of buildings with and large load reversals,” Report UMEE hysteretic energy-dissipating devices,” 78R2, Department of Civil Engineering, doctoral thesis in progress, National University of Michigan. University of Mexico. 7. Uzumeri, S.M. (1977). “Strength and 11. Alamilla, J., Esteva, L., García, J. and ductility of cast-in-place beam-column Díaz, O.J. 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