KSCE Journal of Civil Engineering (2013) 17(5):1051-1063 DOI 10.1007/s12205-013-0334-9
Structural Engineering
www.springer.com/12205
Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique Shinyoung Kwag* and Seung-Yong Ok** Received November 30, 2010/Revised February 21, 2012/Accepted September 2, 2012
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Abstract This study proposes a robust optimal design approach of seismic isolation system for bridges against uncertainties in the system such as model parameters of bridge and isolation bearing, for which a constrained multi-objective optimization problem is formulated by using the bi-objective and constraint functions. The two objective functions are defined as the mutually-conflicting failure probabilities of the bridge pier and the seismic isolator, and the constraint function is constructed by using a robustness index which is defined as the average value of the perturbed amount of the failure probabilities of the bridge so that it can be interpreted as the quantitative measure of the seismic performance degradation of the bridge system caused by the uncertainties. The nonlinear random vibration analyses that estimate the stochastic responses of the seismic isolation bridge system used in the bi-objective and constraint functions are efficiently handled by the stochastic linearization method without performing numerous nonlinear timehistory analyses. In order to demonstrate the effectiveness of the proposed approach, the constrained multi-objective optimization is performed on an example bridge. For comparison purposes, the unconstrained multi-objective optimization with no uncertainties in the system parameters is performed as well. The comparative results verify that the robust seismic isolation system obtained by the proposed design approach can balance the failure probabilities of both pier and isolator in a reasonable level while guaranteeing the bridge performance to remain insensitive to changes in the bridge parameters as well as the design variables of the isolator. The parametric investigation of the robust performance between two different systems also confirms guarantee of the robustness of the proposed method despite the uncertain characteristics of the stochastic ground motions. Keywords: robust optimal design, seismic isolation system, constrained multi-objective optimization technique, stochastic linearization method, stochastic response, uncertainty ··································································································································································································································
1. Introduction In recent decades, many attempts have been made to protect the bridge against damage-causing seismic events, either by providing the bridge with supplementary strength and ductility or by isolating the bridge from severe seismic ground motions. Especially, the seismic isolated bridge has gained greater popularity due to its inherent stability and economical benefit (Kelly, 1986; Buckle and Mayes, 1989). However, if the natural period of the seismic isolated brige is excessively prolonged by using an overly flexible bearing, the responses between the superstructure and the substructure exceed the allowable limits on deformation of the bearings or spacing of the abutments. As a result, an unexpected damage of the seismic isolator or the unseating of superstructure may take place. On the other hand, if less flexible or relatively stiff bearings are used, the pier can experience some damages and hardly survive. In this regard, the design of seismic isolation system becomes an optimization problem to decide a compromising solution between mutually conflicting responses
of the superstructure and the substructure in a reasonable level. Many studies on optimal design of the seismic isolation system can be found in literature. As summarized in Kunde and Jangid (2003), the studies attempt to investigate the parametric behaviors of seismically isolated bridges and then identify their optimum parameters for the most favorable performance. For example, Ghobarah (1988) considered the highway bridge with lead-rubber bearings modeled as a bi-linear spring, and investigated the influence of isolator stiffness and pier stiffness on the effectiveness of seismic isolation. Additionally, Ghobarah and Ali (1988) demonstrated that the two- or three-span highway bridges could be designed for base isolation under the assumption of the rigid behavior of bridge deck in a horizontal direction without significant loss of accuracy. Adachi et al. (1998) conducted a parametric study on seismic behavior of reinforced concrete bridge with seismic isolator using a two-degree-of-freedom system in which two different bilinear hysteresis models are used for seismic isolator and bridge pier, respectively. Koh et al. (2000) investigated the cost-effectiveness of seismically isolated bridges
*Senior Researcher, Research Reactor Engineering Division, Korea Atomic Energy Research Institute, Daejeon 305-353, Korea (E-mail: kwagsy@ kaeri.re.kr) **Member, Assistant Professor, Dept. of Safety Engineering, Hankyong National University, Anseong 456-749, Korea (Corresponding Author, E-mail:
[email protected]) − 1051 −
Shinyoung Kwag and Seung-Yong Ok
in low-to-moderate seismicity based on a minimum life cycle cost concept and identified the optimal parameters of the seismic isolation bearings. Further, many studies in literature (Pagnini and Solari, 1999; Ates et al., 2005; Marano and Sgobba, 2007; Jangid, 2008a; Jangid, 2008b) employed nonlinear random vibration analysis by use of a stochastic linearization method (Wen, 1980; Roberts and Spanos, 1990) in order to compute the responses of the seismic isolated bridges. This is due to the fact that a nonlinear time-history analysis by the direct integration in time domain is time-consuming and often impractical especially for the parametric investigation on the optimal seismic isolation system. The optimal design method proposed in this study also adopts the nonlinear stochastic dynamic analysis by the linearization approximation for the response assessment of the bridge system. As described in the first paragraph, the design of seismic isolation system corresponds to the reasonable selection of the seismic bearing to compromise the mutually conflicting responses of the superstructure and the substructure. These multiple competing objectives are converted into a weighted single objective function in a typical single-objective optimization approach, by using arbitrary weighting factors. The solution mainly depends on the selection of the weighting factors. If the weighing factor is put more on the pier response than on the isolator response, the optimally obtained solution is more likely to reduce the pier response so that the pier will have a higher probability of surviving. On the contrary, a greater weighting factor on the isolator will guarantee a higher likelihood of the isolator’s safety against the uncertain earthquake excitations. As pointed out by Ok et al. (2008a), however, a weighting-based single-objective optimization approach has the following pitfall. In most engineering problems, the relative importance of multiple objectives is practically unknown prior to an optimization. In order to choose reasonable weighting factors, one needs to perform the single-objective optimization repeatedly. However, the selection of reasonable weighting factors requires an expensive computational costs, especially for a problem where the optimal solutions are sensitive to the change in importance weights. In order to efficiently deal with multiple objective functions for the design of the seismic isolation system, we employ a multiobjective optimization approach (Fonseca and Fleming, 1993; Horn et al., 1994; Srinivas and Deb, 1994) that requires no arbitrary weighting factors and obtains a set of Pareto optimal solutions in a single run of the optimization process. Lead rubber bearing (LRB) is considered as the seismic bearing in this study. It has been widely recognized that the model properties of LRB are highly dependent on its surrounding temperature. Kang and Lee (2002) experimentally demonstrated that variations in temperature in the range of -30oC to +50oC can considerably change the effective stiffness and damping ratio of the LRB, eventually causing serious degradation of bearing performance. Furthermore, the field testing result of the seismic isolated bridge (Gilani et al., 1995; Koh et al., 2005) showed that there could exist significant difference from the original and insitu dynamic properties of the bridge system such as the effective
stiffness and damping ratio of the pier and isolator. It has been said that such difference could be incurred by fabrication errors or field conditions, e.g., change of atmospheric temperature, friction of the deteriorated bearing, etc. Therefore, if the isolation system is simply designed without considering possible variations of the system parameters, it may sometimes lead to the nonrobust optimal isolation system; the seismic performance may be optimal in original conditions but may be degraded significantly in perturbed conditions caused by the uncertainties in the system parameters. Based on these observations, this study proposes a robust design method of the seismic isolation bridge system against the system uncertainties, for which a constrained multiobjective optimization problem is formulated by using the biobjective and constraint functions. In order to demonstrate the effectiveness of the proposed robust optimal design method, the constrained multi-objective optimization is performed on an example bridge. For comparison purposes, the unconstrained multi-objective optimization with no uncertainties in the system parameters is also performed. The validity of the proposed approach is discussed by comparing the robust performances of both solutions.
2. Stochastic Dynamic Analysis of Seismically Isolated Bridge 2.1 Equation of Motion Figure 1 illustrates a multi-span continuous bridge with seismic isolation bearings mounted between superstructure and piers. For the design purpose, the seismically isolated bridge system can be represented by the two-degree-of-freedom model in Fig. 2 (Ghobarah, 1998; Ghobarah and Ali, 1988; Adachi et
Fig. 1. Multi-span Continuous Seismic Isolated Bridge
Fig. 2. Two-degree-of-freedom Model for Nonlinear Isolation Bridge System
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Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique
al., 1998; Marano and Sgobba, 2007; Chang and Liu, 1997) where the nonlinear behaviors of pier and isolation bearing are characterized by the nonlinear hysteretic Bouc-Wen model (Wen, 1976). The restoring force of the Bouc-Wen model is defined as: d f ( t ) = c ∆x· ( t ) + αk ∆x ( t ) + ( 1 – α )ky( t )
(1)
where c is the damping coefficient; k is the initial elastic stiffness; α is the post-yield stiffness ratio or the post-elastic stiffness divided by the initial elastic stiffness; ∆x( t ) and ∆x· ( t ) represent the relative displacement and velocity of the oscillator; and y ( t ) is the internal variable to account for the hysteretic restoring force. Note that the first two terms correspond to the linear viscouselastic contribution and the third denotes the nonlinear hysteretic contribution. The differential equation governing the variable y ( t ) is given by: n–1 n y· ( t ) = –γ ∆x· ( t ) ⋅ y( t ) ⋅ y ( t ) – β ∆x· ( t ) ⋅ y( t ) (2) + A ∆x· ( t ) where the operator [ · ] is the first-order derivative with regard to
time; the operator | · | represents the absolute value of the quantity; β and γ are the parameters that control the shape of the hysteretic behavior (Barber and Wen, 1981); the parameter A controls the scale of the hysteresis loop; and the parameter n determines the sharpness of the hysteresis cycle on the boundary between the elastic and inelastic zones. For example, if n goes to infinity, the Bouc-Wen model corresponds to the bilinear model. For a random vibration analysis of the bridge system, a stationary filtered white-noise process defined by the KanaiTajimi power spectral density function (Tajimi, 1960) is used to describe the random earthquake acceleration in this study. This model has found wide applications in random vibration analyses of structures by simply describing the ground motions with a predominant frequency ω g , bandwidth ζg , and intensity Φ0 . Then the absolute ground acceleration x··g ( t ) can be expressed by: r r 2 r x··g ( t ) + 2ζg ω g x· g ( t ) + ω g xg ( t ) = w ( t )
(3a)
r r 2 r x··g ( t ) = x··g ( t ) – w ( t ) = – 2ζg ω g x· g ( t ) – ω g xg ( t )
(3b)
where x ( t ) is the ground displacement relative to the base; and w ( t ) is a white noise process with intensity Φ0 . Finally, the equations of motion for the isolation bridge system in Fig. 2 are described by five ordinary differential equations such that: m1 x··1 + ( c1 + c2 )x· 1 –c2 x· 2 + ( α1 k1 + α2 k2 )x1 – α2 k2 x2 (4a) r 2 r + ( 1 – α1 )k1 y1 – ( 1 – α2 )k2 y2 – 2m1 ζg ω g x· g –m1 ω g xg = 0 r g
m2 x··2 – c2 x· 1 + c2 x· 2 – α2 k2 x1 + α2 k2 x2 + ( 1 – α2 )k2 y2 r 2 r – 2m2 ζg ω g x· g – m2 ω g xg = 0
(4b)
n–1 n y· 1 = –γ1 x· 1 y1 y1 – β1 x· 1 y1 + Ax· 1
(4c)
n–1 n y· 2 = – γ2 x· 2 – x· 1 y2 y2 – β2 ( x· 2 – x· 1 ) y2 + A ( x· 2 – x· 1 )
(4d)
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r r 2 r x··g + 2ζg ω g xg + ω g xg = w
(4e)
where the subscript i denotes the pier ( i = 1 ) and the superstructure ( i = 2 ), respectively; xi , x· i and x··i are the displacement, velocity and acceleration of the pier and the superstructure, respectively; yi is the displacement of the hysteretic restoring force acting on the pier ( i = 1 ) and isolator ( i = 2 ); mi, ci and ki are the mass, damping coefficient and initial stiffness of the pier and the superstructure; αi is the corresponding post-yield stiffness ratio, and βi and γi are the parameters that control the shape of the hysteretic behavior of the pier ( i = 1 ) and isolator ( i = 2 ). For simplicity, the time ‘(t)’ is omitted in the equations. Note that the two equations (Eqs. (4c) and (4d)) are nonlinear while the others are linear. These nonlinear differential equations will be transformed, in a stochastic sense, into equivalently linearized differential equations. 2.2 Estimation of Stochastic Response of Seismic Isolation Bridge System For numerical efficiency, this study employs a stochastic linearization method (Wen, 1980) that facilitates reasonable estimation of the probabilistic seismic responses of the nonlinear structural system without repetitive nonlinear time-history analyses. It enables the realization of the linearized system equivalent to the originally nonlinear hysteretic system by minimizing the mean-squared error of the responses between the nonlinear system and the equivalent linear system. Given the nonlinear system in the form of Eq. (2) (consequently Eqs. (4c) & (4d)), the equivalent linear system is represented by: eq eq y· = –C ∆x· – K y
(5)
If the input ground motions are assumed to follow zero-mean stationary Gaussian process, the equivalent linear coefficients Ceq and Keq are defined, for the most common case of n = 1, by (Atalik and Utku, 1976): γE [ ∆x· y ] eq C = --2- ⎛ -------------------- + βσy⎞ – A ⎝ ⎠ σ∆x· π
(6a)
βE [ ∆x· y ] eq K = --2- ⎛ γσ∆x· + ---------------------⎞ ⎝ σy ⎠ π
(6b)
where σ∆x· and σy are the standard deviation of the two responses, ∆x· and y; and E [ ∆x· y ] corresponds to the covariance or the statistical second moments of the two quantities. Note that eq eq the equivalent coefficients C and K are functions of the statistical second moments of the Gaussian responses, i.e., E [ ∆x· y ] , σ ∆x· and σy . These statistical second moments are acquired by solving an equation of a covariance matrix, often called the Lyapunov equation (Nigam, 1983; Lin, 1967). In order to construct the Lyapunov equation, we transform Eq. (4) into a state-space equation where the nonlinear differential equations of Eqs. (4c) and (4d) are replaced by the equivalent linear differential equation of Eq. (5) with the coefficients in Eq. (6). By introducing a state vector,
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Shinyoung Kwag and Seung-Yong Ok r r T z = [ x1 x2 x· 1 x· 2 y1 y2 xg x· g ]
(7)
algebraic Lyapunov equation for the stationary input process: T
0 = As S + SAs + Q
the final state-space representation is given by: z· = As z + Bw w
(8)
where As and Bw are defined as:
(10)
where the last component of the Q matrix, Q88 = 2π ⋅ Φ0 ; and the remaining components are all zero (Song, 2004). Here, Φ0 corresponds to the intensity of the white noise process in Eq. (3). In addition, the i-th-row and j-th-column element of the covariance matrix S is: 2
⎧ E [ z i zj ] = σ z =⎨ ⎩ E [ z i zj ]
Sij lim
i&j ∈ { 1, …, 8 }
(9a) T
(9b) Bw = {0 0 0 0 0 0 0 1 } T Then, the covariance matrix S = E [ zz ] satisfies the following
i
for
i=j
for
i≠j
(11)
Given As and Q, the solution to this equation (Bartels and Stewart, 1972) provides the second moments of the responses. Since the system matrix As involves the equivalent linear eq eq coefficients C and K as in Eq. (9a) and the coefficients are again functions of the second moments as in Eq. (6), an iterative scheme such as the parameter-sweeping approach (Ni et al., 2001) is required to obtain the solution to Eq. (10). The detailed procedure of the stochastic linearization method is summarized in Fig. 3. Note that the Root-Mean-Squared (RMS) responses of the system under the zero-mean stationary Gaussian input process can be calculated from the second moments of the responses.
3. Robust Design Approach by use of Constrained Multi-objective Optimization 3.1 Formulation of Multi-objective Optimization Problem The main goal of the proposed robust design approach is to find optimal and robust seismic isolation bearing for the bridge structure in the presence of uncertainty in the system parameters. In this context, the optimal solution means that it reduces both incompatible responses of the superstructure and substructure of the bridge as much as possible, and the robust solution means that its reductoin effect on both responses is not diminished significantly over the variations in the system parameters. In other words, its seismic performance is insensitive or not degraded significantly under the changes of the uncertain properties that may exist in the system. In order to guarantee the optimal and robust seismic isolation system, the optimization problem is formulated into the constrained multi-objective optimization problem such that: pier
⎧ P f ( x ;p ) Minimize f ( x ;p ) = ⎨ isol ⎩ Pf ( x ;p ) subject to g ( x, δx ;p, δp ) = η ( x, δx ;p, δp ) – ηt arg et ≤ 0 –
+
⎧ δx ≤ δx ≤ δx for ⎨ – + ⎩ δp ≤ δp ≤ δp
Fig. 3. Flowchart of Stochastic Linearization Method
(12)
where x is a vector of design variables. In this study, design variables are considered as the model properties of the isolation bearing that are the initial elastic stiffness, the post-yield stiffness − 1054 −
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Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique
ratio, and the yield displacement of the isolator; p is a vector of the model parameters given, e.g., dynamic properties of the bridge such as initial elastic stiffness, the post-yield stiffness ratio, and the yield displacement of the pier. More details on these vectors will be described later in the example section; δx and δp respectively denote the perturbation of the design – + – + variables and the model parameters; ( δx , δx ) and ( δp , δp ) pier represent the variation bounds on x and p, respectively; Pf ( x ;p ) isol and Pf ( x;p ) are two objective functions to be minimized and correspond to the failure probabilities of the pier and isolator. Note that, in this paper, the bridge performance is characterized by these failure probabilities in a stochastic sense rather than the magnitudes of the structural responses; for the robust design, robustness index η is introduced as a function of the parameter variations, i.e., x, p, δx and δp; ηtarget is the target robustness index of the solution prescribed by the designer; and g ( x, δx ;p, δp ) is a constraint function used to maintain the system robustness below the given target robustness. It should be mentioned that we express the constraint function g in bold type. It implies that the constraint function can also be used in a vector form. The proposed framework allows multiple constraints to be used, which will be shown in numerical example. In what follows, we describe more details on the bi-objective and the constraint functions. First, in the bi-objective functions, the failure probabilities of the seismic isolator and bridge pier can be computed by the marginal first-passage probability (Song and Der Kiureghian, 2006). The failure probability of the pier is defined as the probability that the displacement of the top of the pier exceeds the limit of the inelastic deformation, and the failure probability of the isolation bearing is defined as the probability that the peak displacement of the isolator exceeds the allowable limit on the isolator deformation. Let these limits on the the pier and bearing displacements be thresholds a. Under the stochastic load, the probability of a stochastic process X ( t ) going beyond a given critical threshold a during a certain period of time τ can be approximated by Vanmarcke’s formula (Vanmarcke, 1975; Song and Der Kiureghian, 2006), as follows: τ
P ( a ;τ ) ≅ 1 – B exp –∫ η ( a ;t ) dt 0
normalized by the standard deviation of the stochastic process σX ; the mean crossing rate of the envelope process η ( a ;t ) becomes constant in time for the zero-mean stationary Gaussian process; ν ( a ) is the unconditional mean crossing rate of X ( t ) over a, 2 and derived by Rice (1945) as ν ( a ) = σX· ⁄ ( πσX ) × exp ( –r ⁄ 2 ) ; and δ is a shape factor which features the bandwidth of the process. The corresponding formula is: 2
λ1 δ = 1 – --------λ0 λ2
(15)
where, λm ( m = 0, 1, 2, … ) is the m-th spectral moments defined by: λm =
∞
m ∫ –∞ ω SXX ( ω ) dω
(16)
2
where, SXX ( ω ) = H ( ω ) is the power spectral density function defined as the square of the frequency response function H ( ω ) of the system as in Eqs. (8) and (9). Here, the system corresponds to the equivalently linearized system by the aforementioned stochastic linearization method. For more details, see the references (Song and Der Kiureghian, 2006; Ok et al., 2008b). The thresholds on the allowable displacements of the pier and isolator are respectively defined as the product of yield displacement and ductility factor and the product of the thickness of the rubber in LRB and the maximum allowable shear strain in the lateral direction such that: x1 = µ × u y
(17a)
x2 = γs × trubber
(17b)
lim
pier
lim
Next, in the constraint function in Eq. (12), the robustness index η is defined as the average of the perturbed amount of the objective functions between the original and perturbed conditions, normalized by the original objective function, as follows: Nk
f ( x + δxk ;p + δpk ) – f ( x ;p ) 1 η = ----- ∑ --------------------------------------------------------------Nk f ( x ;p )
(18)
k=1
(13)
In the case that the stochastic process X(t) follows a zero-mean stationary Gaussian process, the two quantities B and η ( a ;t ) can be derived by:
where Nk represents the number of perturbation point set, { δx , δp }; and the operator is the 2-norm of the vector and geometrically denotes the distance between two points. Therefore, the robustness index η denotes the averaged value of
2
r B = 1 – exp ⎛ – ----⎞ ⎝ 2⎠
(14a)
⎧ ⎫ 1.2 ⎪ exp ⎛ – π --- δ r⎞ ⎪ ⎝ 2 ⎠⎪ ⎪ η ( a ;τ ) = η ( a ) = ν ( a ) ⎨ 1 – ---------------------------------⎬ 2 ⎪ ⎛ – r----⎞ ⎪ 1 – exp ⎪ ⎝ 2⎠ ⎪ ⎩ ⎭
(14b)
where, r = a ⁄ σX is the ratio of the given critical threshold a Vol. 17, No. 5 / July 2013
Fig. 4. Graphical Representation of Robustness Index in Failureprobabilty Space
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Shinyoung Kwag and Seung-Yong Ok
the difference between the original and perturbed failure probabilities, normalized by its original condition point, as illutrated in Fig. 4. It is noteworthy that the design variable with smaller η represents the more robust design. In order to evaluate the constraint function, the perturbation of the parameters, δx and δp need to be generated arbitrarily (Deb and Gupta, 2005). There can be a number of ways of simulating the perturbed conditions of the system parameters. For numerical efficiency, the Latin Hypercube sampling (LHS) method (Stein, 1987; Pebesma and Heuvelink, 1999) is employed to generate the neighboring points in the vicinity of the design variables x and model parameters p. By solving this constrained multi-objective optimization problem, we can obtain the optimal and robust seismic isolation system that can guarantee the minimized failure probabilities of the pier and isolation bearing while maintaining the performance degradation of the solution below a prescribed criterion, i.e.,
preventing the solution from being sensitive to changes in the parameters of the bridge pier as well as the isolator. 3.2 Constrained Multi-objective Optimization Method A solution to this constrained multi-objective optimization problem can be also obtained by the same algorithm used in Ok et al. (2008a; 2008b) except for using the following constraintdominance principle instead of simple dominance principle (Deb et al., 2001): A solution x1 is said to constrained-dominate a solution x2, if any of the following conditions is true: (1) Solution x1 is feasible and solution x2 is not. (2) The solutions x1 and x2 are both feasible, but solution x1 dominates x2. (3) The solutions x1 and x2 are both infeasible, but solution x1 has a smaller constraint violation than the solution x2. The expression “solution is feasible” means that the solution
Fig. 5. Flowchart of Constrained Multi-objective Optimization − 1056 −
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Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique
satisfies the constraint condition, and the expression “one solution dominates another solution” means that one solution is superior to another solution in terms of the multi-objective functions. According to this principle, we can assess the superiority or nondominances of two solutions and subsequently sort all solutions based on their non-dominations of the vectorized multiple objectives. As a result, we can assign the fitness values consecutively to all solutions. In addition to this fitness assignment, another measure, so-called crowding distance, is assigned to the solutions as well in order to maintain population diversity in the current non-dominated solution set. Note that computing the crowding distance only applies to the individuals with the same rank since it is only used to decide the precedence of the selection for the individuals in the same rank group. Please refer to Ok et al. (2008a; 2008b) for more details on the fitness and crowding distance. Given the rank/fitness and the crowding distance of the individuals, a crowded comparison operator is used to guide the selection operator towards a uniformly distributed Paretooptimal solution front in the multi-objective space. The basic principle is as follows: A solution x1 is selected prior to a solution x2, if any of the following conditions is satisfied: (1) Solution x1 has a lower rank than solution x2. (2) The solutions x1 and x2 have the same rank, but solution x1 has a larger crowding distance than solution x2. In summary, the overall procedure of the constrained multiobjective optimization approach is illustrated in Fig. 5. First, N sets of design variable vectors x, each with size n, are randomly generated in design space. Second, perturbation points ( δx, δp ) with perturbation range ±δ(%) are generated using the LHS method (Stein, 1987; Pebesma and Heuvelink, 1999). The number of perturbed points Nk is closely related to both numerical efficiency and evaluation accuracy of robust measure. As more points are used, the evaluated robustness of the function becomes more accurate, but more computation time is also required. Since the LHS method ensures a full coverage of each variable with a small number of sampling points, the robustness performance of the system can be evaluated in an efficient manner. Using the generated design variables and perturbation points, the objective functions and constraint conditions can be evaluated following the aforementioned procedure. Next, the superior individuals are chosen by use of the crowded comparison operator based on the rank and crowding distance values of the individuals. The superior individuals that have the first ranks and larger crowding distances are preferentially preserved at each iteration by the elitism operator. The GA repeatedly modifies a population of individual solutions. At each step, the GA selects individuals at random from the current population to be parents and uses them to produce the offsprings for the next generation through the crossover and mutation operators. Since the superior individuals constitute the current population, the population is able to evolve toward Pareto-optimal solutions over successive generations. All the above processes are repeated until the stopping criteria are satisfied, e.g., up to the specified maximum number of generation. Vol. 17, No. 5 / July 2013
In this study, we can finally obtain the Pareto-optimal solutions that minimize the two failure probabilities and are insensitive to the change in the system parameters caused by potential uncertainties in the system.
4. Illustrative Example To demonstrate the proposed robust optimal design method of the seismic isolation system, Nam-Han river bridge shown in Fig. 6 is chosen as an example bridge. The dynamic properties of the bridge are listed in Table 1. The Kanai-Tajimi model with the 2 3 parameters Φ0 = 0.0218 m ⁄ s , ζg = 0.6 and ω g = 5π rad ⁄ s is used to simulate the earthquake motions. In Eq. (17), ductility pier factor µ is assumed to be 5, the yield displacement of the pier u y is given in Table 1, the maximum allowable shear strain of the bearing γs is set to 250%, and the thickness of the rubber in LRB is trubber = 9.6 cm . The failure of the unseating of the superstructure at the abutment is not considered in this study since the lateral spacing of the abutment specified by the AASHTO standard lim specifications was calculated as 40.74 cm ( » x2 = 24 cm ) for this bridge. In engineering aspects, using the model parameters to directly shape the nonlinear hysteretic curve of the isolation bearing is more practical than using the equivalent stiffness and damping ratio. Marano and Sgobba (2007) said that only three parameters are indispensable in this regard. They are initial stiffness, postyield stiffness ratio and yield displacement, and are selected as the design variables in this study. Under this fact, the yield displacement of the isolator is directly associated with BoucWen model parameters which contribute to the shape of hysteretic behavior. Marano and Sgobba (2007) provided the relationship between the Bouc-Wen model parameters and yield displacement of the isolator. First, the parameters A = 1 and n = 1 are used in this study which were identified in experimental studies (Skinner et al., 1980; Constantinou and Tadjbakhsh,
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Fig. 6. Nam-han River Bridge Table 1. Dynamic Properties of Nam-han River Bridge Properties Effective mass of deck (m2) Effective mass of pier (m1) Initial stiffness of pier (k1) Post-yield stiffness ratio of pier (α1) Damping ratio of pier Damping ratio of isolator pier Yield displacement of pier ( uy )
Values 544.4 ton 101.5 ton 2.16×108 N/m 0.01 0.02 0.1 1.11 cm
Shinyoung Kwag and Seung-Yong Ok
1985). Next, the condition β = γ can be established if the unloading stiffness is equal to the elastic stiffness (it happened in many cases). According to Wong et al. (1994), the yield displacement can be represented by the parameters of the BoucWen model such that: 1---
A n uy = ⎛ ----------⎞ ⎝ β + γ⎠
(19)
As a result, the two model parameters reduce to β = γ = 1 ⁄ 2uy . In the multi-objective optimization process, the search domains for the design variables are set to [10%~50%] of the pier initial elastic stiffness k1 for the initial elastic stiffness k2, [1%~30%] for the post-yield stiffness ratio α2, and [1 mm~30 mm] for the yield displacement uy. All system parameters such as initial stiffness, post-yield stiffness ratio and yield displacement of the pier and pier isol isolator (k1, α1, uy , k2, α2, uy ) are considered as random variables. For simplicity, their distributions are assumed to follow the standard normal distribution. The perturbation ranges of the random variables are equally considered in a range of ± 30%, and a total of 50 perturbation points are randomly generated within the range by the LHS method. The target robustness index ηtarget is set to 0.4.
Fig. 7. Distribution of Pareto-optimal Solutions in Objective Space
4.1 Pareto-optimal Solutions by Unconstrained Multiobjective Optimization For illustration purposes, the typical multi-objective optimization method in a seismic isolation system has been performed without using the constraint conditions on the robustness of the solution (Kwag et al., 2010). In the GA optimization process, one
Fig. 8. Distribution of Design Variables: (a) Initial Stiffness Ratios of Pareto-optimal Isolators to Pier, (b) Post-yield Stiffness ratios of Pareto-optimal Isolators, (c) Yield Displacements of Pareto-optimal Isolators
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Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique
population in each generation consists of 100 individuals, and the searching operation is terminated when the number of the generation reaches 1,000. The optimization results are depicted in objective-function space in Fig. 7. The horizontal and vertical axes correspond to the failure probabilities of the pier and isolator, respectively. We can see that the Pareto-optimal solutions form a uniformly distributed surface, so-called a Pareto front, which demonstrates that this method has successfully maintained the population diversity in the searching process. In addition, an obvious conflicting relationship exists in the solutions in the front: the failure probability of the pier increases as the failure probability of the isolator decreases, and vice versa. In Fig. 8, a total of 100 Pareto optimal solutions are reordered in an increasing order of the failure probability of the pier and a decreasing order of the failure probability of the isolator. Thus, the first solution represents one with the largest failure probability of the isolator and the smallest failure probability of the pier, while the last solution corresponds to the opposite. The vertical axis in Fig. 8(a) represents the ratio of the initial elastic stiffness of the isolator to that of the pier ( k2 ⁄ k1 ). Fig. 8(b) shows the distribution of the post-yield stiffness ratio (α2), that is, the post-yield stiffness divided by the intial stiffness of the Paretooptimal isolators. These two design variables show similar patterns of variations in their magnitudes. In consideration of the sorting order, this indicates that the increase in the initial and post-yield stiffness of the seismic isolator contributes to the increase in the failure probability of the pier. As already mentioned in the introduction, it is self-evident in that the use of the flexible isolator increases the displacement of the superstructure but the use of the stiff isolator produces the opposite result, i.e., the increase in the transmitted inertia force from the superstructure and consequently a higher probability of the pier failure. Unlike the two design variables, the yield displacement of the isolator shown in Fig. 8(c) converges to 0.36 cm and remains constant for all different failure probabilities of the pier and isolator. Therefore, the yield displacement has little effect on the variations of the failure probabilities of the pier, although an optimal value exists. In what follows, the robust Pareto-optimal solutions are explored using the proposed constrained multi-objective optimization approach. The robustness of the solutions obtained by the unconstrained and constrained multi-objective optimization approaches will be compared to each other. 4.2 Robust Pareto-optimal Solutions by Constrained Multiobjective Optimization In the constrained multi-objective GA, we use 20 individuals for each population of the design variables to improve the numerical efficiency. The 1,000 generations are repeated as in the previous optimization. Since we use only 20 individuals instead of 100 individuals, the Pareto-optimal solutions are likely to be distributed more sparsely in the objective space. Therefore, in order to maintain the acceptable distribution density of the Pareto-optimal solutions in the objective space, two additional Vol. 17, No. 5 / July 2013
Fig. 9. Comparison of Distributions of Nominal and Robust Paretooptimal Solutions in Objective Space; ∗ 100 Nominal Solutions without Considering Constraints on Solution Robustness, 20 Robust Solutions
constraints on the failure probabilities of the pier and isolator are instead added by limiting the searching space as follows: pier
pier
isol
isol
g2 ( x ;p ) = P f ( x ;p ) – P f, t arg et ≤ 0
(20a)
g3 ( x ;p ) = P f ( x ;p ) – P f, t arg et ≤ 0
(20b) pier f, t arg et
isol f, t arg et
where the two target failure probabilities P and P are chosen to be 1.35×10−3, corresponding to the generalized –1 reliability index of 3 such that β = –Φ [ P f, t arg et ] = 3 . Φ[ ⋅ ] is the Cumulative Distribution Function (CDF) of the standard normal distribution. By an engineering decision, a reliability index of 3 is reasonable because a solution with reliability less than 3 (and consequently a higher corresponding failure probability) may not be accepted for the seismic isolation bridge system. Hence, the use of constrained multi-objective GA enables a user to maintain the solution space as dense as in unconstrained approach while improving the numerical efficiency. Figure 9 illustrates the comparison between the 100 Paretooptimal solutions obtained without considering the robustness index and the 20 Pareto-optimal solutions obtained by considering the robustness index, in the objective space. For convenience, the 100 Pareto-optimal solutions are termed as nominal Pareto-optimal solutions, and the 20 Pareto-optimal solutions are termed as robust Pareto-optimal solutions. Unlike the nominal Pareto-optimal solutions, the robust Pareto-optimal solutions show a rapid change in their distribution diverging at a certain point from the nominal solution front. In other words, a part of the nominal solution front coincides exactly with the robust solution front and indicates a similar level of robust performance between the two fronts. The other part of the nominal solution front, however, differs greatly from the robust solution front, and has no guarantee of the prescribed robustness of the solutions. Thus, the robustness index defined in Eq. (18) is also applied to the nominal Pareto-optimal solutions obtained by the
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Fig. 10. Comparison of Robustness between Two Pareto-optimal Solution Sets: (a) Nominal Pareto-optimal Solutions, (b) Robust Paretooptimal Solutions
Fig. 11. Comparison of Distribution of Nominal and Robust Design Variables in conjunction with Two Objectives: (a) Initial Stiffness Ratios of Isolators to Pier, (b) Post-yield Stiffness Ratios of Isolators, (c) Yield Displacements of Isolators; ∗ 100 Nominal Solutions, 20 Robust Solutions
unconstrained multi-objective GA in order to examine the robustness of the solutions. Fig. 10 displays the comparative results of the robust indices between the nominal and robust Pareto-optimal solutions. As seen in Fig. 10, the robustness of the nominal solutions covers a wide range approximately from 0.21 to 0.60, while the robust solutions maintain the robust performance in the vicinity of the prescribed level of target robustness. This indicates that the nominal solutions that are relatively high in the failure probability of the pier and low in that of the isolator are somewhat sensitive to the changes in the
system parameters. On the other hand, as can be seen in Fig. 10, the solutions in the upper-left side of the coincident region in Fig. 9 have robustness index values much lower than the target robustness index of 0.4. These solutions, however, are not captured by the proposed approach because of the additional constraints that set the target failure probabilities of the pier and isolator to 1.35×10−3 or smaller. Thus, the first solution in the robust solution front has an isolator failure probability very close to the target failure probability, and the following solutions are distributed coincidently with the nominal solution front up to the
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Robust Design of Seismic Isolation System using Constrained Multi-objective Optimization Technique
Fig. 12. Comparison of Robust Performances between Nominal and Robust Pareto-optimal Solutions with respect to Variations in: (a) Seismic Intensity, (b) Do-minant Frequency, (c) Bandwidth. ∗ Nominal Solution, Robust Solution
bifurcation point. At the bifurcation point, the robust solution front leaves the nominal Pareto front rapidly because of the significant degradation of the robust performances of the nominal solutions in this region, as can be found in Fig. 10. On the contrary, the robust solutions evolve in a robust direction at the sacrifice of the failure probability of the pier in nominal or unperturbed conditions. It is noteworthy, however, that even if the failure probability of the pier is increased as the robust front leaves the nominal front, the maximum failure probability of the solutions is still acceptable, i.e., 1.59×10−5. Such robust solutions cannot be captured by the unconstrained multi-objective optimization approach. In this regard, unless the optimization method takes into account the variations in the system parameters in an appropriate manner, the optimally designed seismic isolation system can show some degraded seismic performance under perturbed conditions of the system parameters. Furthermore, if we have further information or design requirements on the system (e.g., Pf 1.35×10−3), the proposed constrained multiobjective optimization approach enables us to reduce the searching solution space, which could not only improve numerical efficiency but also help make a decision on choosing a reasonable solution. Figure 11 illustrates the comparative results of the distribution of the design variables for nominal and robust solutions. The two horizontal axes represent the failure probabilities of the pier and Vol. 17, No. 5 / July 2013
isolator, and the vertical axis represents the three design variables of the isolator such as the ratio of the initial stiffness to that of pier, the post-yield stiffness ratio, and the yield displacement. It can be observed from the comparative result with the nominal isolation system that a smaller initial stiffness (k2), and greater post-yield stiffness ratio (α2) and yield displacement (uy) of the isolator contribute to the establishment of the robust isolation system. In order to confirm guarantee of robustness of the proposed method, the bridge system with robustly designed seismic isolator has been further simulated under various seismic events. For comparison purposes, the bridge system with optimally designed seismic isolator without considering its robustness (nominal solution) is also taken into account. Two solutions are chosen to have the same pier failure probability but distinct robustness, which are denoted by a marker ‘□’ in Figs. 9 and 10. The pier failure probabilities of the nominal and robust solutions are 4.77×10−6, whereas their robustness indices are 0.58 and 0.40, respectively. Fig. 12 represents variations in the robustness index defined in Eq. (18) of the two seismic isolation bridge systems (nominal and robust solutions) with respect to the variations in intensity (Φ0), dominant frequency (ωg) and bandwidth (ζg) of the seismic excitations. Fig. 12(a) is the assessment result of robustness index according to the change of the seismic intensity. It is observed that the robustness index is
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increased proportionally to the increase of the seismic load. This implies that the system responses are more sensitively changed against uncertainty for the larger seismic load, so the failure probability of the system used to define the robustness index is also sensitively changed over the increase of the seismic load. Besides, the comparative results show that the proposed robust system exhibits the more enhanced robust performance than the nominal system for a wide range of seismic intensity. Fig. 12(b) is the variation of robustness index with respect to the change of the dominant frequency of the earthquake excitations. As for the equivalent linearized systems to the nominal and robust seismic isolated bridges, the first two modal frequencies are computed as 0.81 Hz and 7.59 Hz for the nominal solution, and 0.65 Hz and 7.50 Hz for the robust solution, respectively. As shown in Fig. 12(b), the robustness indices are increased as the dominant frequencies of the excitation approach to the first and second modal frequencies of the system. Therefore, Fig. 12(b) indicates that both responses of the nominal and robust systems are sensitively amplified in the vicinity of their first and second modal frequencies. Again, the comparative results demonstrate the robustness improvement of the proposed robust system over the conventional nominal system for a wide range of excitation frequency. Similar results are also observed in Fig. 12(c), where the robust performances are compared for the range of the bandwidth parameter ζg from 0.1 to 6.0. A high value of the parameter indicates a wide-band process. In summary, the proposed optimal design approach systematically finds an optimal isolation system with the enhanced robustness despite the uncertainty in the stochastic ground motion model.
5. Conclusions This study proposes a robust optimal design method of the seismic isolation system for bridges against uncertainties in the system parameters. The method employs a constrained multiobjective optimization approach and a stochastic linearization method. These have three major advantages. First, the multiobjective optimization approach can handle the mutually conflicting and equally important objectives without relying on the arbitrarily weighted single-objective function. Accordingly, it produces a set of optimal solutions in a single run, whereas a conventional single-objective optimization approach needs to be applied iteratively. This helps choose reasonable design alternatives in the decision making process. Second, the use of constraint conditions on the solution enables us to reduce the searching space, which could improve the numerical efficiency in the optimization process. If we have any further information or design requirements on the system, it rapidly facilitates the decision making process for choosing a reasonable solution. Third, the stochastic linearization method helps a fast estimation of the stochastic responses of the seismic isolation bridge system. As a result, the optimal design process can avoid numerous nonlinear time-history analyses. The numerical example of exploring the robust optimal
isolation systems for Nam-han river bridge demonstrates that the proposed approach can systematically achieve the most desirable seismic performance in terms of the failure probabilities of the pier and isolator, and simultaneously maintain the prescribed robust performance in the presence of uncertainty in the system parameters. Moreover, the parametric investigation of the robust performance between nominal and robust solutions confirms that the robustly designed seismic isolation bridge system exhibits the more enhanced robust performance than the nominal system for a wide range of uncertainty in the seismic characteristics. These results finally demonstrate that the proposed method successfully guarantees the robust performance of the seismic isolation bridge system despite the uncertain characteristics of the stochastic ground motions as well as variations in the system parameters.
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