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Integrated optimum design of viscoelastically damped structural systems
Engineering Structures 26 (2004) 581–591 www.elsevier.com/locate/engstruct
Integrated optimum design of viscoelastically damped structural systems Kwan-Soon Park , Hyun-Moo Koh, Daegi Hahm School of Civil, Urban and Geosystem Engineering, Seoul National University, San 56-1 Shilim-dong, Gwanak-gu, Seoul 151-742, South Korea Received 4 June 2003; received in revised form 8 December 2003; accepted 10 December 2003
Abstract This paper describes a new approach for an integrated optimum design of a viscoelastically damped structural system. The criterion selected for the optimization is minimization of the life-cycle cost that mainly consists of the initial construction cost and the damage cost estimated by failure probability over its entire lifetime. Considering the structure and dampers as a combined or an integrated system, the characteristics of the system and the design constraints can be accounted for from the design step. Optimization problem is also formulated by adopting structural sizing variables, locations and the amount of the viscoelastic damper as design variables. A genetic algorithm is used as a numerical searching technique in order to simultaneously find the optimum parameters of the integrated system. The proposed design method was verified with a numerical example of an eight-story building. From comparative results, it is found that the integrated design approach can improve the seismic performance of the structural system while it maintains low life-cycle cost. # 2003 Elsevier Ltd. All rights reserved. Keywords: Integrated optimum design; Viscoelastic damper; Life-cycle cost; Genetic algorithm; Seismic performance
1. Introduction Civil structures must be protected from natural hazards such as strong winds or earthquakes and structural response control has become more important in ensuring not only living comfort but also structural safety [1]. Over the last decades, a number of energy absorption and damping devices have been proposed for civil engineering applications to reduce large vibration that sometimes occurs in structures [2–4]. Among such devices, viscoelastic dampers have been shown to be effective in reducing seismic responses of steel and reinforced concrete structures as well as windinduced responses [5–7]. Although the effectiveness of these devices for reducing seismic and wind responses has been confirmed through analytical and experimental studies, the development of methods for the optimal utilization of these devices continues to be an Corresponding author. Tel.: +82-2-880-8325; fax: +82-2-8723325. E-mail address: [email protected] (K.-S. Park).
0141-0296/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2003.12.004
important research issue [8]. The damping capacity of a device and specifically where and how many are placed on a structure will have a significant effect on its ability to reduce a response and to achieve the desired design objectives. In the literature, researchers have suggested different practical schemes for the optimal design of dampers. Zhang and Soong used a sequential optimization procedure to determine the optimal location of viscoelastic dampers in multi-story building structures [9,10]. Under such approaches, the design problem is mainly focused only on the damper system. The structure is designed first, and damper systems are then optimally placed. When the selection of the geometry, cross-sectional areas of the members, and materials are completed for a structure, structural frequencies and mode shapes serve as input to the design of the dampers. This can cause a lack of design integration that typically requires several redesigns to obtain a satisfactorily performing system. Occasionally the desired performance of the system cannot be achieved. The cross-sectional areas of structural elements influence the structural frequencies and their distribution that
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affects the damper location. Therefore, it would be more appropriate to integrate the two steps and to determine the optimum values for the design variables in both systems. In this study, an integrated optimum design approach for a viscoelastically damped structural system is described with the goal of obtaining improved seismic performance. By considering a structure and viscoelastic damper system as a combined or an integrated system, the characteristics and design constraints of the system can be accounted for from the design step. Thus, optimization of the combined system can be achieved simultaneously. In the design of earthquake resistant structures, optimal design variables may be based on a tradeoff between the costs of protection versus potential future losses caused by earthquakes [11]. Therefore, regarding the underlying concept of economics, this study is concerned with the problem of optimal engineering design, in which initial construction cost and expected damage costs are balanced. Combining the initial cost and the expected damage costs yields the expected life-cycle cost function for a given seismicity, and minimization of the lifecycle cost constitutes an optimization problem. In the estimation of failure probability that will be used for the calculation of the expected damage cost, spectral analysis and crossing theory are used. As a numerical tool for searching for optimum solutions, a genetic algorithm is adopted. One of the major advantages of a genetic algorithm in the context of our integrated optimal design problem is that it is particularly well suitable to the problems with objective functions that are discrete or non-differentiable, and their values change drastically over the range of the design variables and as a result, they have several local optima. As an example to illustrate the application of the proposed approach, an earthquake excited eight-story building is used and a discussion of the numerical results is presented.
order to consider the effects of the damper, we adopt the modal strain energy method. Previous studies have shown that the equivalent damping ratio of a viscoelastically damped structure might be accurately predicted by this method [7,12,13]. In this approach, when the inherent damping of the structure is small, damping ratios of the viscoelastically damped structure can be represented as, " # gv /Ti K0 /i 1 T ð2Þ ni ¼ 2 /i K/i
2. System modeling
kd;j ¼
The equations of motion for a building structure with added viscoelastic damping devices subjected to earthquake excitation can be written as,
where ad, h and G0 are the shear area, thickness and shear storage modulus of the viscoelastic material, respectively. In this study, viscoelastically damped building is modeled as a shear building structure. The simplified shear building model which has one degree-of-freedom (DOF) of horizontal direction per node, cannot express the effect of vertical displacement and/or rotational deformation in each node. However, the failures of earthquake excited building structures are mainly governed by story drifts due to horizontal motion. Generally, the modeling method to describe a structural system depends on the design stage and/or what the
M€ x þ Cx_ þ Kx ¼ M½1€ xg
ð1Þ
where x is the n 1 relative displacement vector with €g is the earthquake ground respect to the ground; x T motion; ½1 ½1; . . . ;1 ; M, C and K represent, respectively, the n n mass, damping and stiffness matrices of a viscoelastically damped structure. It is generally known that adding a viscoelastic damper not only increases the stiffness of the structure but also provides a significant amount of damping. In
where ni is the equivalent damping ratio for i-th mode; /i is i-th vibration mode shape of the viscoelastically damped structure; K0 is the stiffness matrix of the structure without a damper; and gv is the loss factor which is often used as a measure of energy dissipation capacity of the viscoelastic material. The loss factor is defined as follows [14]: gv ¼
G 00 G0
ð3Þ
where G00 is defined as the shear loss modulus, which gives a measure of the energy dissipated per cycle and G0 is defined as the shear storage modulus of the viscoelastic material, which is a measure of the energy stored and recovered per cycle, respectively. If the change in vibration mode shapes due to the addition of viscoelastic dampers can be neglected, Eq. (2) can be reduced to the following: 2 g x ni ¼ v 1 i2 ð4Þ 2 xi i and xi are i-th natural frequencies of the where x structure without and with added dampers, respectively. In addition, for a viscoelastic material with a known shear storage modulus at the design frequency and temperature, added stiffness for j-th story, kd,j, due to viscoelastic damper can be determined as, ad G 0 h
ð5Þ
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analysis used for. Therefore, the shear building model can be an economic simple alternative especially in the preliminary design stage which only needs basic information on the design variables, i.e. the column stiffness and the amount of viscoelastic materials, although more general structural models such as 2-D model of three DOF per node or 3-D model of six DOF per node can also be applied with ease to the optimal design procedures proposed by this study.
3. Estimation of failure probability For the evaluation of the probabilistic characteristics of an earthquake excited structure, it is possible to estimate the probabilistic distribution of a structure by Monte Carlo simulation, a statistical estimation of the structural response from a sufficient number of deterministic analyses based on artificially generated time histories [15]. A huge number of time history analyses need to be performed to obtain reliable results. In the spectral analysis, on the other hand, one power spectral density function (PSDF), rather than a collection of time histories, can be used as an input model for a stochastic analysis in the frequency domain. Although it cannot consider the nonlinear structural behavior, the failure probability of the structure can be estimated relatively easily. Therefore, the frequency domain approach is more appropriate for this study because a simple method is needed for the numerous, iterated computations in the process of life-cycle cost minimization. However, it should be noted that this method is an approximate way to deal with failure probability of structures, and more accurate methods for dynamic analysis are required in the stage of design verification that needs to account for the structural nonlinearity. This section briefly describes the procedures used for the estimation of failure probability. 3.1. Modeling of input ground motions In general, seismic performance of the structural system highly depends on the magnitudes and the frequency characteristics of ground motion. In the design of earthquake resistant structures or systems, therefore, adopting an appropriate excitation model, capable of reflecting the specific characteristics of the construction site, makes it possible to obtain an improved seismic performance. To account for the available site characteristics, in this study, power spectral density functions that are compatible with the response spectra of the 1997 Uniform Building Code (UBC-97) for combinations of various seismic intensities and soil profile types are generated using the method developed by Koh et al. [16]. The site-dependent input ground motion can be obtained by iteratively improving a
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PSDF such that the empirical response spectrum based on the PSDF matches the target response spectrum of the site. The procedures are as follows. First, the target response spectrum with the acceleration coefficient and site coefficient given for the specific region is constructed. According to UBC-97, an elastic design response spectrum can be constructed by the combinations of the seismic zone factor (Z) representing the regional seismic intensity and the soil profile type. The seismic zone factors with their approximately correlated maximum ground accelerations and the soil profile types with their generic descriptions are shown in Tables 1 and 2, respectively, [17,18]. Secondly, an empirical response spectrum is constructed from the results of numerous dynamic analyses of single DOF structures under artificial ground accelerations. The artificial ground acceleration can be simulated by the spectral representation method [19] with the initially assumed white noise ( ) N x 1 X aðkÞ Bl exp½iðlDxÞðjDtÞ ; g ðjDtÞ ¼ Re l¼0 2
ði ¼ 1; j ¼ 0;1; . . . ;Nx 1Þ pffiffiffi ðkÞ Bl ¼ 2Al exp½i/l ; ðl ¼ 0;1; . . . ;Nx 1Þ Al ¼ ð2Sg ðlDxÞDxÞ1=2 ðkÞ where ag ðtÞ ðkÞ iteration; /l
Dx ¼ xu =Nx
ð6Þ
is the acceleration time history at k-th
is the random variable vector between 0 and 2p used for k-th iteration; Sg is the PSDF generating time histories; Dt is time interval; Dx is frequency spacing; xu is an upper cut-off frequency; and Nx is the
Table 1 Seismic zone factors specified in UBC-97 Zone
Seismic zone factor (Z) Maximum acceleration
0 1 2A 2B 3 (not near a great fault) 4 (near a great fault)
0.04 0.075 0.15 0.2 0.3
0.040 g 0.075 g 0.150 g 0.200 g 0.300 g
0.4
0.400 g
Table 2 Soil profile types specified in UBC-97 Soil profile type
Soil profile name/generic description
SA SB SC SD SE SF
Hard rock Rock Very dense soil and soft rock Stiff soil profile Soft soil profile Soil requiring site-specific evaluation
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number of frequency spacing, respectively. The current PSDF, Sg(x), is then improved based on the comparison between the two response spectra as follows:
RT ðxÞ 2 Sg ðxÞ ! Sg ðxÞ ð7Þ RE ðxÞ where RT(x) is the target design response spectrum for the specific region and RE(x) is the empirical response spectrum generated from the current PSDF, Sg(x), respectively. By repeating these steps until the empirical response spectrum of the current PSDF converges to the target design response spectrum, the PSDF compatible with the site-depend response spectra can be constructed. Fig. 1(a) shows that the generated PSDFs are proportional to the seismic zone factor, which confirms that this spectral density model is capable of reflecting the magnitude of ground motion acceleration represented by the seismic zone factor. From Fig. 1(b), it
can be seen that the peaks of the PSDFs move to the long-period region and that their magnitudes increase as the site condition becomes softer. This implies that soft-soil layers amplify the magnitudes, resulting in a longer period. From these results, it can be confirmed that the power spectral density model used in this study properly reflects the properties of the input ground motion defined by the seismic zone factor and the soil profile type in the code. 3.2. Definition of limit state To estimate the failure probability of a structure, limit states of the structure are required to be predefined. The structural limit states vary according to performance requirements and load types [20]. The failure of a building structure subjected to horizontal ground motion is largely caused by excessive shear force or bending moment. In general, shear stress and bending stress of structural elements of each floor should be governed by story drift. Yun [21] utilized the drift ratio to investigate and analyze the structural capacity for the incipient collapse of a steel moment frame building. Wen and Kang [22] reported on the performance and damage level of a building structure in terms of story drift. Hence, we also adopt the story drift for defining the limit states of a building structure subjected to a horizontal ground motion. The limitations of inelastic story drift are provided in the UBC97 provisions to control inelastic deformations and to prevent potential instabilities in both structural and nonstructural elements. By using the UBC-97, the story drift limit can be defined as follows: 8 0:025hs > > for Tn 0:7 sec < D Dlim ¼ 0:7R ð8Þ > > : D D ¼ 0:020hs for Tn > 0:7 sec lim 0:7R where D is the story drift obtained from elastic dynamic analysis, Dlim is the story drift limit, R is the response modification factor which is determined by the structural material and type used, hs is the story height and Tn is fundamental period of the building, respectively. On the other hand, an excessive shear deformation of viscoelastic material leads to shear failure of the damper. The shear deformation limit of the damper can be defined as follows by using the relative displacement of a viscoelastic material dd < dlim ¼ clim h
Fig. 1. Power spectral density functions with seismic zone factors and soil profile types (a) seismic zone 3; (b) soil profile type SB.
ð9Þ
where dd is the relative displacement of a viscoelastic material, clim is the shear strain limit (usually 60%) and h is the thickness of the viscoelastic material, respectively.
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3.3. Failure probability For the estimation of the probability that the system response stays within a safe region, within a specified interval of time, failure of the building is considered to have occurred if story drift of any floors exceeds the preselected limit value of in a given earthquake duration. The determination of such a probability is generally known as the first-passage problem [23]. Thus, it is required to find the probability Pf; j that story drift of j-th floor (Dj) exceeds the limit state (Dlim) at least once in a given interval of time. In the calculation of the probability, we assume mean zero stationary Gaussian random process of strong ground motion with an interval of time td. Then, Pf,j can be obtained as follows. Firstly, the average rate of positive slope crossings of the level Dj ¼ Dlim is found as Eq. (10) by using Rice’s formula [24] ð1 D_ j pðDj ¼ Dlim ; D_ j ÞdD_ j mþ ¼ Dj 0 ð1 D_ j pðDlim ; D_ j ÞdD_ j ¼ ð10Þ 0
where mþ Dj is the single-sided crossing rates of the story drift at j-th floor, and pðD ; D_ Þ is the joint probability j
j
density function for Dj, and D_ j . Since the input ground motion is Gaussian, it is obvious that the story drift (Dj) and its derivative ðD_ j Þ are also mean zero Gaussian random processes. Therefore, the joint probability density function in Eq. (10) is written as, h i 1 pðDlim ; D_ j Þ ¼ pffiffiffiffiffiffi exp D2lim =2r2Dj 2prDj h 2 i 1 pffiffiffiffiffiffi exp D_ j =2r2D_ ð11Þ j 2prD_ j
where td is the time duration of strong ground motion, and mDj ð ¼ 2mþ Dj Þ is the double-sided crossing rate, respectively. Although exact analytical solutions for Pf, j are generally unavailable, it is known that Eq. (13) provides an approximate, upper-bound estimate of Pf, j which can be used for design purposes from a practical viewpoint [23]. Now, the failure probability, Pf, that story drift of any floors exceeds the preselected limit value due to earthquake excitation can be calculated as, Pf ¼ 1
Assuming that the occurrence of exceeding the limit state is subjected to a Poisson distribution, the probability Pf, j that the story drift exceeds the limit states at least once within an earthquake duration is written as, Pf; j ¼ 1 expð mDj td Þ
ð13Þ
n Y ð1 Pf; j Þ
ð14Þ
j¼1
It should be noted that mþ Dj of Eq. (12) requires the calculation of the standard deviations for story drift at j-th floor and its time derivative. These can be simply calculated in frequency domain as follows. From Eq. (1), the transfer matrix, hg(x), whose components are the transfer function of relative displacement with respect to the ground at j-th floor, hg, j (x), is introduced as, T hg ðxÞ ¼ hg;1 ðxÞ; hg;1 ðxÞ; . . . ;hg;n ðxÞ 1 ¼ x2 I ixM1 C M1 K ½1 ð15Þ where i2 ¼ 1. By using hg, j (x), the transfer function of the story drift at j-th floor, hj(x), is written as, hg;1 ðxÞ; for j ¼ 1 hj ðxÞ ¼ ð16Þ hg; j ðxÞ hg; j1 ðxÞ; for j ¼ 2;3; . . . ;n The standard deviations for story drift at j-th floor and its time derivative can be estimated by the following integration in frequency domain ð1 hj ðxÞ2 Sg ðxÞdx; j ¼ 1;2; . . . ;n ð17Þ r2Dj ¼ 1
r2D_ ¼ j
where rDj and rD_ j are the standard deviations of story drift at j-th floor and its time derivative, respectively. Substituting Eq. (11) into Eq. (10) gives, h i 1 mþ exp D2lim =2r2Dj Dj ¼ pffiffiffiffiffiffi 2prDj ð1 h 2 i 1 pffiffiffiffiffiffi exp D_ j =2r2D_ D_ j dD_ j j 2prD_ j 0 h i rD_ j ¼ exp D2lim =2r2Dj ð12Þ prDj
585
ð1
2 x2 hj ðxÞ Sg ðxÞdx;
j ¼ 1;2; . . . ;n
ð18Þ
1
where Sg(x) is the PSDF of input ground motion, which is obtained from the Eq. (7).
4. Life-cycle cost minimization In the design of earthquake resistant structures, several studies concerning the problem of balancing the expected costs and the reliability of structural system have been reported [25–28]. The use of viscoelastic dampers or larger amounts of structural members may decrease the probability of failure and the expected damage cost due to possible earthquake events although it may increase the initial construction cost. Therefore, optimum design problem can be formulated based on a tradeoff between the costs of protection versus potential future losses caused by earthquakes
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explicitly with risk or the reliability of viscoelastically damped structural system are considered, while other cost items that do not directly influence the risk or reliability of the system are not included. The expected value of the life-cycle cost function can then be defined by using the design variables of column stiffness vector (kc) and damper area vector (ad) as follows: E ½CT ðkc ;ad Þ ¼ Cs Vs ðkc Þ þ Cd Vd ðad Þ m þCD Pf ðkc ;ad Þ ð1 ektlife Þ k T T kc ¼ kc;1 ; . . . ;kc;n ; ad ¼ ad;1 ; . . . ;ad;n
Fig. 2. Life-cycle cost minimization concept.
(Fig. 2). Regarding the underlying concept of economics, this study is concerned with the problem of an optimal engineering design of a viscoelastically damped structural system, while minimizing life-cycle cost that implies total expected cost during the structure’s lifetime. Life-cycle cost can be defined as the sum of the initial construction cost and the expected damage costs. The initial construction cost include the material cost, fabrication labor cost, erection labor cost and other cost items including outside services other than erection, shop drawings, etc. Although life-cycle cost is an objective function to be minimized, the main concern of this study is to obtain cost-effective design alternatives, which vary the combination of each design variable, rather than precise estimation of life-cycle cost itself. In order to state the optimization problem properly, therefore, it is important to define the lifecycle cost function for viscoelastically damped structures in terms of appropriate design variables which can cause large differences between initial construction cost and expected damage cost. This will help designers to make decisions on the use of viscoelastically damped structural system. Column members and viscoelastic damper materials, which control the volume of columns and viscoelastic damper, respectively, are the main variables that impact the initial construction cost. In addition, the expected damage cost is also dependent on these two variables since seismic responses of the system can be controlled by changing the amount of damper used or stiffness of the structure. Therefore, in defining the lifecycle cost function, only the material costs of the column members and viscoelastic dampers that vary
ð19Þ
where E½CT ð Þ denotes the expected value of life-cycle cost, Vs and Vd are the total volume of columns and viscoelastic material; Cs and Cd are the initial construction cost of column and viscoelastic damper per each unit volume; CD is the damage cost caused by failure of the structural system; Pf is the failure probability of the structural system; m, k and tlife are earthquake occurrence rate per 1-year, discount rate and lifetime of structure; ke, j and ad, j are column stiffness and viscoelastic damper material area at j-th story, respectively. On the other hand, the amount of the actual construction cost is likely to vary with the actual regional situations. Therefore, to describe the general life-cycle cost function, we introduce the normalized life-cycle cost function that is divided by the initial cost per unit volume of the column. With these normalization procedures, the cost of any category can always be measured in terms of the volume of the column. The normalized life-cycle cost is defined as, ~ T ðkc ;ad Þ ¼ Vs ðkc Þ þ rd=c Vd ðad Þ E½C m þ Vf Pf ðkc ;ad Þ ð1 ektlife Þ k
ð20Þ
~ T ð Þ denotes the expected value of life-cycle where E½C cost, normalized to the cost of the volume of column member; rd/c is the cost ratio of the viscoelastic damper to column member per unit volume; Vf is the damage scale, which varies with the importance of the building structure. Now, an integrated optimal design problem minimizing the life-cycle cost of viscoelastically damped structural system can be formulated as, ~ T ðkc ;ad Þ Find kc and ad ; which minimize E C subject to kc;min kc; j kc;max
ð21Þ
0 ad; j ad;max ðj ¼ 1; . . . ;nÞ where kc,min, kc,max and ad,max are the design constraints of column stiffness and viscoelastic dampers, respectively.
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5. Integrated design by genetic algorithm Determining optimal design parameters for an integrated system is a very complex problem because the varying dynamic properties of the system due to interactions between the structure and the dampers should be considered. Moreover, in practical applications, design variables such as column stiffness, damper capacity, and dampers locations may not be a continuous function because of commercial and manufacturing constraints. Thus, the optimal design of such an integrated structural system can be viewed as a combinatorial optimization problem since the design space is discrete. In principle, the optimal solution for such a discrete problem can be found by an exhaustive of enumerative search of every possible combination of column and damper size with the damper locations. However, the practical implementation of this search is not at all effective due to the large number of feasible design combinations. Searching each possible combination is obviously a daunting task even for currently available computing facilities and, therefore, a more systematic and an efficient approach must be used. In order to effectively address this problem, a genetic algorithm (GA) is adopted as a numerical searching technique. The GA is an optimization technique simulating the evolutionary process based on the principles of natural biological evolution where the stronger individuals are likely to be winners in a competing environment. For the past few years, the GA introduced by Holland [29] has been successfully applied to a wide range of engineering applications and proved to be very effective in solving such problems, for its features of evolutionary, multi-point, direct and parallel searching [30–32]. The GA works with a coding of the parameter set, not the parameters themselves. Binary code with 0’s and 1’s is employed in this study to represent the design variables of kc, j and ad, j. The design variables can be encoded by mapping in a certain range using an
Fig. 3.
587
n-bit of binary unsigned integer and the GA starts with an initial population, which comprises Ngn randomly created binary strings as shown in Fig. 3. The following equation can be used to decode a string bk;j ; 2n 1 ba;j ¼ ad;min þ ðad;max ad;min Þ n 2 1
kc; j ¼ kc;min þ ðkc;max kc;min Þ ad; j
ð22Þ
where bk, j and ba, j are the decimal integer values of binary string for kc, j and ad, j, respectively. The design strings are searched for their optimum values through three basic operations, i.e. selection, crossover and mutation. Reproduction is a selection process to determine the survival potential of a string according to the string’s fitness value. Strings with higher fitness values have a higher probability of proceeding to subsequent operation. In the present study, the expression for fitness value of a string is defined as follows, using the life-cycle cost function of Eq. (20)
~ T þ min E½C ~ T E½C ~ T fi ¼ max E½C ð23Þ i i i i¼1;...;Ngn
i¼1;...;Ngn
where fi is the fitness value of i-th individual string.
Crossover is a recombination operator which produces two new offspring strings, with crossover probability pc, by randomly exchanging the corresponding bits between two parent strings. It follows the reproduction procedures. The newly produced strings have partially the genetic information of their parents. Crossover operator makes it possible to have more chance to obtain better offspring strings. Mutation provides diversity in a population by occasional random change in bit values of strings. In mutation operator, the randomly chosen bit values of a small fraction of strings are changed from 0 to 1, or vice versa. The mutation happens with probability pm at randomly selected positions of the strings. Mutation operator can prevent falling all solutions in populations into a local optimum.
Binary code description of design variables.
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namely, smaller life-cycle cost. The designs of viscoelastically damped structural system will be gradually improved as the evolutionary cycle is repeated.
6. Numerical example
Fig. 4. An eight-story example building.
For an evolutionary process, the individual strings of a population are manipulated cyclically through three basic operations and replacement to produce a new population that tends to have a larger fitness value,
Fig. 5.
As an example to demonstrate the proposed design approach, an eight-story moment-resisting steel frame building is used, as shown in Fig. 4. The mass of each floor is 5 104 kg and damping of the fundamental mode without viscoelastic damper is assumed to be 2.0%. For the integrated design under consideration in this study, we also assumed the followings: the thickness of the viscoelastic material, h, is 1.8 cm; the maximum values for column stiffness kc,max and damper area ad,max are 5 104 kN=m and 4:0 101 m2 , v respectively; the ambient temperature is 21 C; the duration of strong motion is 6.6 s (td ¼ 6:6); earthquake occurrence rate per 1 year m is 0.1; the discount rate k is 5.0%; the lifetime of the structure tlife is 50 years; the cost ratio of a viscoelastic damper to a column member per unit volume, rd/c, is 4.0; and considering a general purpose office building [33], the damage scale Vf is assumed to be 6.13. Firstly, the failure probabilities with various combinations of column stiffness and added viscoelastic dampers were investigated (Fig. 5). In the calculation of the failure probability in Fig. 5, we assume that column stiffness and damper area of each floor are equally distributed. It is clear that reliability of the structural system is sensitively affected by stiffness of structural members and amount of dampers. The failure probability decreases as the column stiffness and/or the damper area increase. This implies that the two variables, floor stiffness and damper area, can be main
Failure probability of a viscoelastically damped structure (soil profile type SE, seismic zone 3).
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design variables which control not only the initial cost but also the expected damage cost. The proposed design procedures were applied to the example structure and results are summarized in Figs. 6–8 and Table 3. Fig. 6 shows an example of the generation history of the optimization process. In the beginning of the optimization process, a lower initial construction cost is used, and the expected damage cost is relatively large. However, the expected damage cost rapidly decreases as the initial construction cost increases slightly for the first 50 generations, and the total life-cycle cost converges into its optimal value after about 200 generations. The solution seems to converge to practically the best solution in 300 generations. In the following,
the result obtained in the 700th generation is shown as the converged optimal result. In the application of GA, one population consists of 50 chromosomes (N gn ¼ 50), and each chromosome of the population is a string of size 16n. Selection is based on roulette wheel selection, the crossover operation is performed with a crossover probability (pc) of 0.85 and mutation operates with a mutation probability (pm) of 0.01. Fig. 7 depicts the optimal distribution of the stiffness and the damper. It is noteworthy that the optimal values for column stiffness of the low floor levels are much larger than those of the high floor levels, while the dampers are more concentrated at the middle floor levels rather than at the top or base floor levels. The r.m.s. displacement responses are compared with those obtained by the uniform design case, in which structural members and viscoelastic dampers are equally distributed to each floor (Fig. 8). For the sake of comparison, the two design cases have the same amount of floor stiffness and viscoelastic damper, i.e. the identical initial cost is used. It can be seen that the proposed integrated design method exhibits a better seismic performance than uniform design case. The r.m.s. displacement at the top floor is decreased by about 7% with compared to the uniform design case. In Table 3, the optimized costs for various site conditions are summarized. Comparison of the estimated costs between the two design cases reveals that the expected damage cost due to seismic hazard can be considerably reduced by the proposed design approach. This effect is more easily reported especially in a region of soft site condition rather than stiff site condition. In this example, therefore, the integrated optimum design
Fig. 7. Optimal distribution of floor stiffnesses and viscoelastic dampers (soil profile type SB, seismic zone 3).
Fig. 8. The r.m.s. displacement responses of each floor (soil profile type SB, seismic zone 3).
Fig. 6. Generation history of the normalized costs (soil profile type SB, seismic zone 3).
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Table 3 Results of optimized costs for various soil profile types and seismic intensities Soil profile type
Stiff (SB)
Seismic zone (Z)
Moderate (0.15)
Strong (0.30)
Design method
Uniform
Integrated
Uniform
Structural cost Damper cost Initial cost Expected damage cost Total life-cycle cost
3255 27 3283 5025 8556
3255 27 3283 294 3577
3348 133 3481 4260 8168
Flexible (SE) Moderate (0.15)
Strong (0.30)
Integrated
Uniform
Integrated
Uniform
Integrated
3348 133 3481 473 3955
3885 256 4141 12,479 16,953
3885 256 4141 443 4585
4894 249 5144 57,045 62,571
4894 249 5144 553 5697
Unit: cm3.
approach is highly effective especially in the site of soft soil condition. On the other hand, comparing the costs of the integrated designs, it is also found that the expected damage cost and the total life-cycle cost increase as the seismic intensity and soil conditions at the site become stronger and more flexible, respectively. In the case of stiff soil profile type, the damper cost of a region of strong seismicity is about 5 times higher than that of a region of moderate seismicity while the structural cost is almost the same. For a structural system with a high seismic performance, therefore, increasing the amount of viscoelastic damper is much more cost-effective than increasing structural stiffness. However, in flexible soil conditions, the structural cost in a region of strong seismicity is considerably larger (about 26%) than that of a region of moderate seismicity while the damper cost is nearly unchanged. In this case, it would be more cost-effective to increase the structural stiffness than to use a larger amount of the damper.
coelastic damper is more cost-effective rather than increasing structural stiffness, especially in a region of stiff soil conditions. From comparative results, it is also found that the integrated design approach can improve the seismic performance of the structural system while it maintains low life-cycle cost. Thus, it is concluded that the proposed method has the advantages, not only from the viewpoint of seismic performance but also economic aspects. Acknowledgements This work was supported by the Brain Korea 21 Project and by the Korea Science and Engineering Foundation (KOSEF) through the Korea Earthquake Engineering Research Center at Seoul National University. References
7. Conclusions An integrated optimum design method for a viscoelastically damped structural system is presented. In the method, the size of structural members, the amount and the location of viscoelastic dampers are considered as design variables, i.e. a simultaneous optimization problem is formulated by considering the structure and damper system as a combined or an integrated system. The expected life-cycle cost, which consists of the initial construction cost and potential losses over the structure’s lifetime due to seismic hazard, is adopted as an objective function to be minimized. Optimum values of design variables are successfully determined by a genetic algorithm, which was used in this study as a numerical searching technique. The proposed design method was verified with a numerical example of an eight-story building. Optimum designs for various seismic intensities and soil profile types were performed. In the sense of an integrated optimal design of viscoelastically damped structures, increasing the amount of vis-
[1] Kobori T. Proceedings of the Third World Conference on Structural Control, Como, Italy. Past, present and future in seismic response control of civil engineering structures, vol. 1. 2002, p. 9– 14. [2] Samali B, Kwok KCS. Use of viscoelastic dampers in reducing wind- and earthquake induced motion of building structures. Engineering Structures 1995;17(9):639–54. [3] Dargush GF, Soong TT. Behavior of metallic plate dampers in seismic passive energy dissipation systems. Earthquake Spectra 1995;11(4):545–68. [4] Makris N, Dargush GF, Constantinou MC. Dynamic analysis of viscoelastic fluid dampers. Journal of Engineering Mechanics (ASME) 1995;121(10):1114–21. [5] Mahmoodi P, Keel CJ. Performance of structural dampers for the Columbia Center Building. In: Building motion in wind. New York; 1986. p. 83–106. [6] Shen KL, Soong TT, Chang KC, Lai ML. Seismic behaviour of reinforced concrete frame with added viscoelastic dampers. Engineering Structures 1995;17(5):372–80. [7] Chang KC, Soong TT, Oh S-T, Lai ML. Seismic behavior of steel frame with added viscoelastic dampers. Journal of Structural Engineering 1995;121(10):1418–26. [8] Singh MP, Moreschi LM. Optimal placement of dampers for passive response control. Earthquake Engineering and Structural Dynamics 2002;31:955–76.
K.-S. Park et al. / Engineering Structures 26 (2004) 581–591 [9] Zhang RH, Soong TT. Seismic design of viscoelastic dampers for structural applications. Journal of Structural Engineering 1992;118(5):1375–92. [10] Garcia DL, Soong TT. Proceedings of the Third World Conference on Structural Control, Como, Italy. Efficiency of a simple approach for the design of optimal damper configurations in MDOF structures, vol. 3. 2002, p. 431–6. [11] Ang AH-S, Leon DD. Determination of optimal target reliabilities for design and upgrading of structures. Structural Safety 1997;19(1):91–103. [12] Johnson CD, Keinholz DA. Finite element prediction for damping in structures with constrained viscoelastic layers. American Institute of Aeronautics and Astronautics Journal 1982;20(9):1284–90. [13] Chang KC, Soong TT, Oh S-T, Lai ML. Ambient temperature on a viscoelastically damped structure. Journal of Structural Engineering 1992;118(7):1955–73. [14] Soong TT, Dargush GF. Passive Energy Dissipation Systems in Structural Engineering. West Sussex, England: John Wiley & Sons; 1997. [15] Soong TT, Grigoriu M. Random Vibration of Mechanical and Structural Systems. New Jersey: PRT Prentice-Hall; 1993. [16] Koh HM. Proceedings of the Third World Conference on Structural Control, Como, Italy. Random vibration of mechanical and structural systems, vol. 1. 2002, p. 69–83. [17] Lindeburg MR, Baradar M. Seismic Design of Building Structures. Belmont, CA: Professional Publications, Inc; 2001. [18] International Conference of Building Officials, 1997. Uniform Building Code—Volume 2. Whittier, CA: ICBO; 1997. [19] Shinozuka M, Deodatis G. Simulation of stochastic processes by spectral representation. Applied Mechanics Reviews 1991;44(4):191–203. [20] Wen YK, Kang YJ. Minimum building life-cycle cost design criteria. I: Methodology. Journal of Structural Engineering 2001;127(3):330–7.
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[21] Yun SY. Investigation on performance, prediction and evaluation of low ductility moment frames. PhD Thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 2000. [22] Wen YK, Kang YJ. Minimum building life-cycle cost design criteria. II: Applications. Journal of Structural Engineering 2001;127(3):338–46. [23] Roberts JB, Spanos PD. Random Vibration and Statistical Linearization. West Sussex, England: John Wiley & Sons; 1990. [24] Rice SO. Mathematical analysis of random noise. Selected Papers on Noise and Stochastic Processes, Dover, NY 1954:133–294. [25] Montes-Iturrizaga R, Heredia-Zavoni E, Esteva E. Optimal maintenance strategies for structures in seismic zones. Earthquake Engineering and Structural Dynamics 2003;32:245–64. [26] Liu SC, Neghabat F. A cost optimization model for seismic design of structures. The Bell System Technical Journal 1972;51(10):2209–25. [27] Rosenblueth E. Optimum design for infrequent disturbances. Journal of the Structural Division (ASCE) 1976;102(ST9):1807– 25. [28] Enevoldsen I, Sorensen JD. Reliability-based optimization in structural engineering. Structural Safety 1994;15:169–96. [29] Holland JH. Adaptation in Natural and Artificial Systems. Ann Arbor, MI: University of Michigan Press; 1975. [30] Furuya H, Haftka RT. Placing actuators on space structures by genetic algorithms and effectiveness indices. Structural Optimization 1995;9:69–75. [31] Furuya O, Hamazaki H, Fujita S. Proper placement of energy absorbing devices for reduction of wind-induced vibration caused in high-rise buildings. Journal of Wind Engineering and Industrial Aerodynamics 1998;74–76:931–42. [32] Kim Y, Ghaboussi J. A new method of reduced order feedback control using genetic algorithms. Earthquake Engineering and Structural Dynamics 1999;28(3):193–212. [33] ATC-13. Earthquake Damage Evaluation Data for California. Redwood City, CA: Applied Technology Council; 1986.
Integrated optimum design of viscoelastically damped structural systems Kwan-Soon Park , Hyun-Moo Koh, Daegi Hahm School of Civil, Urban and Geosystem Engineering, Seoul National University, San 56-1 Shilim-dong, Gwanak-gu, Seoul 151-742, South Korea Received 4 June 2003; received in revised form 8 December 2003; accepted 10 December 2003
Abstract This paper describes a new approach for an integrated optimum design of a viscoelastically damped structural system. The criterion selected for the optimization is minimization of the life-cycle cost that mainly consists of the initial construction cost and the damage cost estimated by failure probability over its entire lifetime. Considering the structure and dampers as a combined or an integrated system, the characteristics of the system and the design constraints can be accounted for from the design step. Optimization problem is also formulated by adopting structural sizing variables, locations and the amount of the viscoelastic damper as design variables. A genetic algorithm is used as a numerical searching technique in order to simultaneously find the optimum parameters of the integrated system. The proposed design method was verified with a numerical example of an eight-story building. From comparative results, it is found that the integrated design approach can improve the seismic performance of the structural system while it maintains low life-cycle cost. # 2003 Elsevier Ltd. All rights reserved. Keywords: Integrated optimum design; Viscoelastic damper; Life-cycle cost; Genetic algorithm; Seismic performance
1. Introduction Civil structures must be protected from natural hazards such as strong winds or earthquakes and structural response control has become more important in ensuring not only living comfort but also structural safety [1]. Over the last decades, a number of energy absorption and damping devices have been proposed for civil engineering applications to reduce large vibration that sometimes occurs in structures [2–4]. Among such devices, viscoelastic dampers have been shown to be effective in reducing seismic responses of steel and reinforced concrete structures as well as windinduced responses [5–7]. Although the effectiveness of these devices for reducing seismic and wind responses has been confirmed through analytical and experimental studies, the development of methods for the optimal utilization of these devices continues to be an Corresponding author. Tel.: +82-2-880-8325; fax: +82-2-8723325. E-mail address: [email protected] (K.-S. Park).
0141-0296/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2003.12.004
important research issue [8]. The damping capacity of a device and specifically where and how many are placed on a structure will have a significant effect on its ability to reduce a response and to achieve the desired design objectives. In the literature, researchers have suggested different practical schemes for the optimal design of dampers. Zhang and Soong used a sequential optimization procedure to determine the optimal location of viscoelastic dampers in multi-story building structures [9,10]. Under such approaches, the design problem is mainly focused only on the damper system. The structure is designed first, and damper systems are then optimally placed. When the selection of the geometry, cross-sectional areas of the members, and materials are completed for a structure, structural frequencies and mode shapes serve as input to the design of the dampers. This can cause a lack of design integration that typically requires several redesigns to obtain a satisfactorily performing system. Occasionally the desired performance of the system cannot be achieved. The cross-sectional areas of structural elements influence the structural frequencies and their distribution that
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affects the damper location. Therefore, it would be more appropriate to integrate the two steps and to determine the optimum values for the design variables in both systems. In this study, an integrated optimum design approach for a viscoelastically damped structural system is described with the goal of obtaining improved seismic performance. By considering a structure and viscoelastic damper system as a combined or an integrated system, the characteristics and design constraints of the system can be accounted for from the design step. Thus, optimization of the combined system can be achieved simultaneously. In the design of earthquake resistant structures, optimal design variables may be based on a tradeoff between the costs of protection versus potential future losses caused by earthquakes [11]. Therefore, regarding the underlying concept of economics, this study is concerned with the problem of optimal engineering design, in which initial construction cost and expected damage costs are balanced. Combining the initial cost and the expected damage costs yields the expected life-cycle cost function for a given seismicity, and minimization of the lifecycle cost constitutes an optimization problem. In the estimation of failure probability that will be used for the calculation of the expected damage cost, spectral analysis and crossing theory are used. As a numerical tool for searching for optimum solutions, a genetic algorithm is adopted. One of the major advantages of a genetic algorithm in the context of our integrated optimal design problem is that it is particularly well suitable to the problems with objective functions that are discrete or non-differentiable, and their values change drastically over the range of the design variables and as a result, they have several local optima. As an example to illustrate the application of the proposed approach, an earthquake excited eight-story building is used and a discussion of the numerical results is presented.
order to consider the effects of the damper, we adopt the modal strain energy method. Previous studies have shown that the equivalent damping ratio of a viscoelastically damped structure might be accurately predicted by this method [7,12,13]. In this approach, when the inherent damping of the structure is small, damping ratios of the viscoelastically damped structure can be represented as, " # gv /Ti K0 /i 1 T ð2Þ ni ¼ 2 /i K/i
2. System modeling
kd;j ¼
The equations of motion for a building structure with added viscoelastic damping devices subjected to earthquake excitation can be written as,
where ad, h and G0 are the shear area, thickness and shear storage modulus of the viscoelastic material, respectively. In this study, viscoelastically damped building is modeled as a shear building structure. The simplified shear building model which has one degree-of-freedom (DOF) of horizontal direction per node, cannot express the effect of vertical displacement and/or rotational deformation in each node. However, the failures of earthquake excited building structures are mainly governed by story drifts due to horizontal motion. Generally, the modeling method to describe a structural system depends on the design stage and/or what the
M€ x þ Cx_ þ Kx ¼ M½1€ xg
ð1Þ
where x is the n 1 relative displacement vector with €g is the earthquake ground respect to the ground; x T motion; ½1 ½1; . . . ;1 ; M, C and K represent, respectively, the n n mass, damping and stiffness matrices of a viscoelastically damped structure. It is generally known that adding a viscoelastic damper not only increases the stiffness of the structure but also provides a significant amount of damping. In
where ni is the equivalent damping ratio for i-th mode; /i is i-th vibration mode shape of the viscoelastically damped structure; K0 is the stiffness matrix of the structure without a damper; and gv is the loss factor which is often used as a measure of energy dissipation capacity of the viscoelastic material. The loss factor is defined as follows [14]: gv ¼
G 00 G0
ð3Þ
where G00 is defined as the shear loss modulus, which gives a measure of the energy dissipated per cycle and G0 is defined as the shear storage modulus of the viscoelastic material, which is a measure of the energy stored and recovered per cycle, respectively. If the change in vibration mode shapes due to the addition of viscoelastic dampers can be neglected, Eq. (2) can be reduced to the following: 2 g x ni ¼ v 1 i2 ð4Þ 2 xi i and xi are i-th natural frequencies of the where x structure without and with added dampers, respectively. In addition, for a viscoelastic material with a known shear storage modulus at the design frequency and temperature, added stiffness for j-th story, kd,j, due to viscoelastic damper can be determined as, ad G 0 h
ð5Þ
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analysis used for. Therefore, the shear building model can be an economic simple alternative especially in the preliminary design stage which only needs basic information on the design variables, i.e. the column stiffness and the amount of viscoelastic materials, although more general structural models such as 2-D model of three DOF per node or 3-D model of six DOF per node can also be applied with ease to the optimal design procedures proposed by this study.
3. Estimation of failure probability For the evaluation of the probabilistic characteristics of an earthquake excited structure, it is possible to estimate the probabilistic distribution of a structure by Monte Carlo simulation, a statistical estimation of the structural response from a sufficient number of deterministic analyses based on artificially generated time histories [15]. A huge number of time history analyses need to be performed to obtain reliable results. In the spectral analysis, on the other hand, one power spectral density function (PSDF), rather than a collection of time histories, can be used as an input model for a stochastic analysis in the frequency domain. Although it cannot consider the nonlinear structural behavior, the failure probability of the structure can be estimated relatively easily. Therefore, the frequency domain approach is more appropriate for this study because a simple method is needed for the numerous, iterated computations in the process of life-cycle cost minimization. However, it should be noted that this method is an approximate way to deal with failure probability of structures, and more accurate methods for dynamic analysis are required in the stage of design verification that needs to account for the structural nonlinearity. This section briefly describes the procedures used for the estimation of failure probability. 3.1. Modeling of input ground motions In general, seismic performance of the structural system highly depends on the magnitudes and the frequency characteristics of ground motion. In the design of earthquake resistant structures or systems, therefore, adopting an appropriate excitation model, capable of reflecting the specific characteristics of the construction site, makes it possible to obtain an improved seismic performance. To account for the available site characteristics, in this study, power spectral density functions that are compatible with the response spectra of the 1997 Uniform Building Code (UBC-97) for combinations of various seismic intensities and soil profile types are generated using the method developed by Koh et al. [16]. The site-dependent input ground motion can be obtained by iteratively improving a
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PSDF such that the empirical response spectrum based on the PSDF matches the target response spectrum of the site. The procedures are as follows. First, the target response spectrum with the acceleration coefficient and site coefficient given for the specific region is constructed. According to UBC-97, an elastic design response spectrum can be constructed by the combinations of the seismic zone factor (Z) representing the regional seismic intensity and the soil profile type. The seismic zone factors with their approximately correlated maximum ground accelerations and the soil profile types with their generic descriptions are shown in Tables 1 and 2, respectively, [17,18]. Secondly, an empirical response spectrum is constructed from the results of numerous dynamic analyses of single DOF structures under artificial ground accelerations. The artificial ground acceleration can be simulated by the spectral representation method [19] with the initially assumed white noise ( ) N x 1 X aðkÞ Bl exp½iðlDxÞðjDtÞ ; g ðjDtÞ ¼ Re l¼0 2
ði ¼ 1; j ¼ 0;1; . . . ;Nx 1Þ pffiffiffi ðkÞ Bl ¼ 2Al exp½i/l ; ðl ¼ 0;1; . . . ;Nx 1Þ Al ¼ ð2Sg ðlDxÞDxÞ1=2 ðkÞ where ag ðtÞ ðkÞ iteration; /l
Dx ¼ xu =Nx
ð6Þ
is the acceleration time history at k-th
is the random variable vector between 0 and 2p used for k-th iteration; Sg is the PSDF generating time histories; Dt is time interval; Dx is frequency spacing; xu is an upper cut-off frequency; and Nx is the
Table 1 Seismic zone factors specified in UBC-97 Zone
Seismic zone factor (Z) Maximum acceleration
0 1 2A 2B 3 (not near a great fault) 4 (near a great fault)
0.04 0.075 0.15 0.2 0.3
0.040 g 0.075 g 0.150 g 0.200 g 0.300 g
0.4
0.400 g
Table 2 Soil profile types specified in UBC-97 Soil profile type
Soil profile name/generic description
SA SB SC SD SE SF
Hard rock Rock Very dense soil and soft rock Stiff soil profile Soft soil profile Soil requiring site-specific evaluation
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number of frequency spacing, respectively. The current PSDF, Sg(x), is then improved based on the comparison between the two response spectra as follows:
RT ðxÞ 2 Sg ðxÞ ! Sg ðxÞ ð7Þ RE ðxÞ where RT(x) is the target design response spectrum for the specific region and RE(x) is the empirical response spectrum generated from the current PSDF, Sg(x), respectively. By repeating these steps until the empirical response spectrum of the current PSDF converges to the target design response spectrum, the PSDF compatible with the site-depend response spectra can be constructed. Fig. 1(a) shows that the generated PSDFs are proportional to the seismic zone factor, which confirms that this spectral density model is capable of reflecting the magnitude of ground motion acceleration represented by the seismic zone factor. From Fig. 1(b), it
can be seen that the peaks of the PSDFs move to the long-period region and that their magnitudes increase as the site condition becomes softer. This implies that soft-soil layers amplify the magnitudes, resulting in a longer period. From these results, it can be confirmed that the power spectral density model used in this study properly reflects the properties of the input ground motion defined by the seismic zone factor and the soil profile type in the code. 3.2. Definition of limit state To estimate the failure probability of a structure, limit states of the structure are required to be predefined. The structural limit states vary according to performance requirements and load types [20]. The failure of a building structure subjected to horizontal ground motion is largely caused by excessive shear force or bending moment. In general, shear stress and bending stress of structural elements of each floor should be governed by story drift. Yun [21] utilized the drift ratio to investigate and analyze the structural capacity for the incipient collapse of a steel moment frame building. Wen and Kang [22] reported on the performance and damage level of a building structure in terms of story drift. Hence, we also adopt the story drift for defining the limit states of a building structure subjected to a horizontal ground motion. The limitations of inelastic story drift are provided in the UBC97 provisions to control inelastic deformations and to prevent potential instabilities in both structural and nonstructural elements. By using the UBC-97, the story drift limit can be defined as follows: 8 0:025hs > > for Tn 0:7 sec < D Dlim ¼ 0:7R ð8Þ > > : D D ¼ 0:020hs for Tn > 0:7 sec lim 0:7R where D is the story drift obtained from elastic dynamic analysis, Dlim is the story drift limit, R is the response modification factor which is determined by the structural material and type used, hs is the story height and Tn is fundamental period of the building, respectively. On the other hand, an excessive shear deformation of viscoelastic material leads to shear failure of the damper. The shear deformation limit of the damper can be defined as follows by using the relative displacement of a viscoelastic material dd < dlim ¼ clim h
Fig. 1. Power spectral density functions with seismic zone factors and soil profile types (a) seismic zone 3; (b) soil profile type SB.
ð9Þ
where dd is the relative displacement of a viscoelastic material, clim is the shear strain limit (usually 60%) and h is the thickness of the viscoelastic material, respectively.
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3.3. Failure probability For the estimation of the probability that the system response stays within a safe region, within a specified interval of time, failure of the building is considered to have occurred if story drift of any floors exceeds the preselected limit value of in a given earthquake duration. The determination of such a probability is generally known as the first-passage problem [23]. Thus, it is required to find the probability Pf; j that story drift of j-th floor (Dj) exceeds the limit state (Dlim) at least once in a given interval of time. In the calculation of the probability, we assume mean zero stationary Gaussian random process of strong ground motion with an interval of time td. Then, Pf,j can be obtained as follows. Firstly, the average rate of positive slope crossings of the level Dj ¼ Dlim is found as Eq. (10) by using Rice’s formula [24] ð1 D_ j pðDj ¼ Dlim ; D_ j ÞdD_ j mþ ¼ Dj 0 ð1 D_ j pðDlim ; D_ j ÞdD_ j ¼ ð10Þ 0
where mþ Dj is the single-sided crossing rates of the story drift at j-th floor, and pðD ; D_ Þ is the joint probability j
j
density function for Dj, and D_ j . Since the input ground motion is Gaussian, it is obvious that the story drift (Dj) and its derivative ðD_ j Þ are also mean zero Gaussian random processes. Therefore, the joint probability density function in Eq. (10) is written as, h i 1 pðDlim ; D_ j Þ ¼ pffiffiffiffiffiffi exp D2lim =2r2Dj 2prDj h 2 i 1 pffiffiffiffiffiffi exp D_ j =2r2D_ ð11Þ j 2prD_ j
where td is the time duration of strong ground motion, and mDj ð ¼ 2mþ Dj Þ is the double-sided crossing rate, respectively. Although exact analytical solutions for Pf, j are generally unavailable, it is known that Eq. (13) provides an approximate, upper-bound estimate of Pf, j which can be used for design purposes from a practical viewpoint [23]. Now, the failure probability, Pf, that story drift of any floors exceeds the preselected limit value due to earthquake excitation can be calculated as, Pf ¼ 1
Assuming that the occurrence of exceeding the limit state is subjected to a Poisson distribution, the probability Pf, j that the story drift exceeds the limit states at least once within an earthquake duration is written as, Pf; j ¼ 1 expð mDj td Þ
ð13Þ
n Y ð1 Pf; j Þ
ð14Þ
j¼1
It should be noted that mþ Dj of Eq. (12) requires the calculation of the standard deviations for story drift at j-th floor and its time derivative. These can be simply calculated in frequency domain as follows. From Eq. (1), the transfer matrix, hg(x), whose components are the transfer function of relative displacement with respect to the ground at j-th floor, hg, j (x), is introduced as, T hg ðxÞ ¼ hg;1 ðxÞ; hg;1 ðxÞ; . . . ;hg;n ðxÞ 1 ¼ x2 I ixM1 C M1 K ½1 ð15Þ where i2 ¼ 1. By using hg, j (x), the transfer function of the story drift at j-th floor, hj(x), is written as, hg;1 ðxÞ; for j ¼ 1 hj ðxÞ ¼ ð16Þ hg; j ðxÞ hg; j1 ðxÞ; for j ¼ 2;3; . . . ;n The standard deviations for story drift at j-th floor and its time derivative can be estimated by the following integration in frequency domain ð1 hj ðxÞ2 Sg ðxÞdx; j ¼ 1;2; . . . ;n ð17Þ r2Dj ¼ 1
r2D_ ¼ j
where rDj and rD_ j are the standard deviations of story drift at j-th floor and its time derivative, respectively. Substituting Eq. (11) into Eq. (10) gives, h i 1 mþ exp D2lim =2r2Dj Dj ¼ pffiffiffiffiffiffi 2prDj ð1 h 2 i 1 pffiffiffiffiffiffi exp D_ j =2r2D_ D_ j dD_ j j 2prD_ j 0 h i rD_ j ¼ exp D2lim =2r2Dj ð12Þ prDj
585
ð1
2 x2 hj ðxÞ Sg ðxÞdx;
j ¼ 1;2; . . . ;n
ð18Þ
1
where Sg(x) is the PSDF of input ground motion, which is obtained from the Eq. (7).
4. Life-cycle cost minimization In the design of earthquake resistant structures, several studies concerning the problem of balancing the expected costs and the reliability of structural system have been reported [25–28]. The use of viscoelastic dampers or larger amounts of structural members may decrease the probability of failure and the expected damage cost due to possible earthquake events although it may increase the initial construction cost. Therefore, optimum design problem can be formulated based on a tradeoff between the costs of protection versus potential future losses caused by earthquakes
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explicitly with risk or the reliability of viscoelastically damped structural system are considered, while other cost items that do not directly influence the risk or reliability of the system are not included. The expected value of the life-cycle cost function can then be defined by using the design variables of column stiffness vector (kc) and damper area vector (ad) as follows: E ½CT ðkc ;ad Þ ¼ Cs Vs ðkc Þ þ Cd Vd ðad Þ m þCD Pf ðkc ;ad Þ ð1 ektlife Þ k T T kc ¼ kc;1 ; . . . ;kc;n ; ad ¼ ad;1 ; . . . ;ad;n
Fig. 2. Life-cycle cost minimization concept.
(Fig. 2). Regarding the underlying concept of economics, this study is concerned with the problem of an optimal engineering design of a viscoelastically damped structural system, while minimizing life-cycle cost that implies total expected cost during the structure’s lifetime. Life-cycle cost can be defined as the sum of the initial construction cost and the expected damage costs. The initial construction cost include the material cost, fabrication labor cost, erection labor cost and other cost items including outside services other than erection, shop drawings, etc. Although life-cycle cost is an objective function to be minimized, the main concern of this study is to obtain cost-effective design alternatives, which vary the combination of each design variable, rather than precise estimation of life-cycle cost itself. In order to state the optimization problem properly, therefore, it is important to define the lifecycle cost function for viscoelastically damped structures in terms of appropriate design variables which can cause large differences between initial construction cost and expected damage cost. This will help designers to make decisions on the use of viscoelastically damped structural system. Column members and viscoelastic damper materials, which control the volume of columns and viscoelastic damper, respectively, are the main variables that impact the initial construction cost. In addition, the expected damage cost is also dependent on these two variables since seismic responses of the system can be controlled by changing the amount of damper used or stiffness of the structure. Therefore, in defining the lifecycle cost function, only the material costs of the column members and viscoelastic dampers that vary
ð19Þ
where E½CT ð Þ denotes the expected value of life-cycle cost, Vs and Vd are the total volume of columns and viscoelastic material; Cs and Cd are the initial construction cost of column and viscoelastic damper per each unit volume; CD is the damage cost caused by failure of the structural system; Pf is the failure probability of the structural system; m, k and tlife are earthquake occurrence rate per 1-year, discount rate and lifetime of structure; ke, j and ad, j are column stiffness and viscoelastic damper material area at j-th story, respectively. On the other hand, the amount of the actual construction cost is likely to vary with the actual regional situations. Therefore, to describe the general life-cycle cost function, we introduce the normalized life-cycle cost function that is divided by the initial cost per unit volume of the column. With these normalization procedures, the cost of any category can always be measured in terms of the volume of the column. The normalized life-cycle cost is defined as, ~ T ðkc ;ad Þ ¼ Vs ðkc Þ þ rd=c Vd ðad Þ E½C m þ Vf Pf ðkc ;ad Þ ð1 ektlife Þ k
ð20Þ
~ T ð Þ denotes the expected value of life-cycle where E½C cost, normalized to the cost of the volume of column member; rd/c is the cost ratio of the viscoelastic damper to column member per unit volume; Vf is the damage scale, which varies with the importance of the building structure. Now, an integrated optimal design problem minimizing the life-cycle cost of viscoelastically damped structural system can be formulated as, ~ T ðkc ;ad Þ Find kc and ad ; which minimize E C subject to kc;min kc; j kc;max
ð21Þ
0 ad; j ad;max ðj ¼ 1; . . . ;nÞ where kc,min, kc,max and ad,max are the design constraints of column stiffness and viscoelastic dampers, respectively.
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5. Integrated design by genetic algorithm Determining optimal design parameters for an integrated system is a very complex problem because the varying dynamic properties of the system due to interactions between the structure and the dampers should be considered. Moreover, in practical applications, design variables such as column stiffness, damper capacity, and dampers locations may not be a continuous function because of commercial and manufacturing constraints. Thus, the optimal design of such an integrated structural system can be viewed as a combinatorial optimization problem since the design space is discrete. In principle, the optimal solution for such a discrete problem can be found by an exhaustive of enumerative search of every possible combination of column and damper size with the damper locations. However, the practical implementation of this search is not at all effective due to the large number of feasible design combinations. Searching each possible combination is obviously a daunting task even for currently available computing facilities and, therefore, a more systematic and an efficient approach must be used. In order to effectively address this problem, a genetic algorithm (GA) is adopted as a numerical searching technique. The GA is an optimization technique simulating the evolutionary process based on the principles of natural biological evolution where the stronger individuals are likely to be winners in a competing environment. For the past few years, the GA introduced by Holland [29] has been successfully applied to a wide range of engineering applications and proved to be very effective in solving such problems, for its features of evolutionary, multi-point, direct and parallel searching [30–32]. The GA works with a coding of the parameter set, not the parameters themselves. Binary code with 0’s and 1’s is employed in this study to represent the design variables of kc, j and ad, j. The design variables can be encoded by mapping in a certain range using an
Fig. 3.
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n-bit of binary unsigned integer and the GA starts with an initial population, which comprises Ngn randomly created binary strings as shown in Fig. 3. The following equation can be used to decode a string bk;j ; 2n 1 ba;j ¼ ad;min þ ðad;max ad;min Þ n 2 1
kc; j ¼ kc;min þ ðkc;max kc;min Þ ad; j
ð22Þ
where bk, j and ba, j are the decimal integer values of binary string for kc, j and ad, j, respectively. The design strings are searched for their optimum values through three basic operations, i.e. selection, crossover and mutation. Reproduction is a selection process to determine the survival potential of a string according to the string’s fitness value. Strings with higher fitness values have a higher probability of proceeding to subsequent operation. In the present study, the expression for fitness value of a string is defined as follows, using the life-cycle cost function of Eq. (20)
~ T þ min E½C ~ T E½C ~ T fi ¼ max E½C ð23Þ i i i i¼1;...;Ngn
i¼1;...;Ngn
where fi is the fitness value of i-th individual string.
Crossover is a recombination operator which produces two new offspring strings, with crossover probability pc, by randomly exchanging the corresponding bits between two parent strings. It follows the reproduction procedures. The newly produced strings have partially the genetic information of their parents. Crossover operator makes it possible to have more chance to obtain better offspring strings. Mutation provides diversity in a population by occasional random change in bit values of strings. In mutation operator, the randomly chosen bit values of a small fraction of strings are changed from 0 to 1, or vice versa. The mutation happens with probability pm at randomly selected positions of the strings. Mutation operator can prevent falling all solutions in populations into a local optimum.
Binary code description of design variables.
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namely, smaller life-cycle cost. The designs of viscoelastically damped structural system will be gradually improved as the evolutionary cycle is repeated.
6. Numerical example
Fig. 4. An eight-story example building.
For an evolutionary process, the individual strings of a population are manipulated cyclically through three basic operations and replacement to produce a new population that tends to have a larger fitness value,
Fig. 5.
As an example to demonstrate the proposed design approach, an eight-story moment-resisting steel frame building is used, as shown in Fig. 4. The mass of each floor is 5 104 kg and damping of the fundamental mode without viscoelastic damper is assumed to be 2.0%. For the integrated design under consideration in this study, we also assumed the followings: the thickness of the viscoelastic material, h, is 1.8 cm; the maximum values for column stiffness kc,max and damper area ad,max are 5 104 kN=m and 4:0 101 m2 , v respectively; the ambient temperature is 21 C; the duration of strong motion is 6.6 s (td ¼ 6:6); earthquake occurrence rate per 1 year m is 0.1; the discount rate k is 5.0%; the lifetime of the structure tlife is 50 years; the cost ratio of a viscoelastic damper to a column member per unit volume, rd/c, is 4.0; and considering a general purpose office building [33], the damage scale Vf is assumed to be 6.13. Firstly, the failure probabilities with various combinations of column stiffness and added viscoelastic dampers were investigated (Fig. 5). In the calculation of the failure probability in Fig. 5, we assume that column stiffness and damper area of each floor are equally distributed. It is clear that reliability of the structural system is sensitively affected by stiffness of structural members and amount of dampers. The failure probability decreases as the column stiffness and/or the damper area increase. This implies that the two variables, floor stiffness and damper area, can be main
Failure probability of a viscoelastically damped structure (soil profile type SE, seismic zone 3).
K.-S. Park et al. / Engineering Structures 26 (2004) 581–591
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design variables which control not only the initial cost but also the expected damage cost. The proposed design procedures were applied to the example structure and results are summarized in Figs. 6–8 and Table 3. Fig. 6 shows an example of the generation history of the optimization process. In the beginning of the optimization process, a lower initial construction cost is used, and the expected damage cost is relatively large. However, the expected damage cost rapidly decreases as the initial construction cost increases slightly for the first 50 generations, and the total life-cycle cost converges into its optimal value after about 200 generations. The solution seems to converge to practically the best solution in 300 generations. In the following,
the result obtained in the 700th generation is shown as the converged optimal result. In the application of GA, one population consists of 50 chromosomes (N gn ¼ 50), and each chromosome of the population is a string of size 16n. Selection is based on roulette wheel selection, the crossover operation is performed with a crossover probability (pc) of 0.85 and mutation operates with a mutation probability (pm) of 0.01. Fig. 7 depicts the optimal distribution of the stiffness and the damper. It is noteworthy that the optimal values for column stiffness of the low floor levels are much larger than those of the high floor levels, while the dampers are more concentrated at the middle floor levels rather than at the top or base floor levels. The r.m.s. displacement responses are compared with those obtained by the uniform design case, in which structural members and viscoelastic dampers are equally distributed to each floor (Fig. 8). For the sake of comparison, the two design cases have the same amount of floor stiffness and viscoelastic damper, i.e. the identical initial cost is used. It can be seen that the proposed integrated design method exhibits a better seismic performance than uniform design case. The r.m.s. displacement at the top floor is decreased by about 7% with compared to the uniform design case. In Table 3, the optimized costs for various site conditions are summarized. Comparison of the estimated costs between the two design cases reveals that the expected damage cost due to seismic hazard can be considerably reduced by the proposed design approach. This effect is more easily reported especially in a region of soft site condition rather than stiff site condition. In this example, therefore, the integrated optimum design
Fig. 7. Optimal distribution of floor stiffnesses and viscoelastic dampers (soil profile type SB, seismic zone 3).
Fig. 8. The r.m.s. displacement responses of each floor (soil profile type SB, seismic zone 3).
Fig. 6. Generation history of the normalized costs (soil profile type SB, seismic zone 3).
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Table 3 Results of optimized costs for various soil profile types and seismic intensities Soil profile type
Stiff (SB)
Seismic zone (Z)
Moderate (0.15)
Strong (0.30)
Design method
Uniform
Integrated
Uniform
Structural cost Damper cost Initial cost Expected damage cost Total life-cycle cost
3255 27 3283 5025 8556
3255 27 3283 294 3577
3348 133 3481 4260 8168
Flexible (SE) Moderate (0.15)
Strong (0.30)
Integrated
Uniform
Integrated
Uniform
Integrated
3348 133 3481 473 3955
3885 256 4141 12,479 16,953
3885 256 4141 443 4585
4894 249 5144 57,045 62,571
4894 249 5144 553 5697
Unit: cm3.
approach is highly effective especially in the site of soft soil condition. On the other hand, comparing the costs of the integrated designs, it is also found that the expected damage cost and the total life-cycle cost increase as the seismic intensity and soil conditions at the site become stronger and more flexible, respectively. In the case of stiff soil profile type, the damper cost of a region of strong seismicity is about 5 times higher than that of a region of moderate seismicity while the structural cost is almost the same. For a structural system with a high seismic performance, therefore, increasing the amount of viscoelastic damper is much more cost-effective than increasing structural stiffness. However, in flexible soil conditions, the structural cost in a region of strong seismicity is considerably larger (about 26%) than that of a region of moderate seismicity while the damper cost is nearly unchanged. In this case, it would be more cost-effective to increase the structural stiffness than to use a larger amount of the damper.
coelastic damper is more cost-effective rather than increasing structural stiffness, especially in a region of stiff soil conditions. From comparative results, it is also found that the integrated design approach can improve the seismic performance of the structural system while it maintains low life-cycle cost. Thus, it is concluded that the proposed method has the advantages, not only from the viewpoint of seismic performance but also economic aspects. Acknowledgements This work was supported by the Brain Korea 21 Project and by the Korea Science and Engineering Foundation (KOSEF) through the Korea Earthquake Engineering Research Center at Seoul National University. References
7. Conclusions An integrated optimum design method for a viscoelastically damped structural system is presented. In the method, the size of structural members, the amount and the location of viscoelastic dampers are considered as design variables, i.e. a simultaneous optimization problem is formulated by considering the structure and damper system as a combined or an integrated system. The expected life-cycle cost, which consists of the initial construction cost and potential losses over the structure’s lifetime due to seismic hazard, is adopted as an objective function to be minimized. Optimum values of design variables are successfully determined by a genetic algorithm, which was used in this study as a numerical searching technique. The proposed design method was verified with a numerical example of an eight-story building. Optimum designs for various seismic intensities and soil profile types were performed. In the sense of an integrated optimal design of viscoelastically damped structures, increasing the amount of vis-
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