International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Design of flow isolation systems through multi-objective criteria for the seismic-risk performance Gidaris, Ioannis; Taflanidis, Alexandros A.; Lopez-Garcia, Diego; Mavroeidis, George P. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1Design of Floor Isolation Systems through Multi-Objective Criteria for the Seismic-Risk Performance Ioannis Gidaris Ph.D. Candidate, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN, USA Alexandros A. Taflanidis Associate Professor, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN, USA Diego Lopez-Garcia Associate Professor, Dept. of Civil and Geotechnical Engineering, Pontificia Universidad Catolica de Chile, and National Research Center for Integrated Natural Disaster Management CONICYT/FONDAP/15110017, Santiago, Chile George P. Mavroeidis Assistant Professor, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN, USA ABSTRACT: This paper discusses a probabilistic framework for performance assessment and optimal design of floor isolation systems for the protection of acceleration sensitive contents. A multi-objective formulation is considered for the optimization problem with the two competing objectives corresponding to (i) maximization of the level of protection offered to the sensitive content (acceleration reduction) and (ii) minimization of the demand for appropriate clearance to avoid collision to surrounding objects (floor displacement reduction). Uncertainties are addressed by characterizing these objectives in terms of the associated seismic risk, whereas a surrogate modeling approach is developed to evaluate this risk and support the design optimization. As an illustrative example, the design of a polynomial friction pendulum isolator system is presented. The formulation is demonstrated to efficiently provide design solutions with different performance levels across the considered competing objectives, offering a range of options for selecting the final protection system. 1. INTRODUCTION Floor isolation systems have gained increased attention over the past decade within the earthquake engineering community for the protection of acceleration sensitive contents, such as computer servers or museum artifacts (Jia et al. 2014; Liu and Warn 2012). They operate by using flexible isolators to “decouple” the floor portion containing the group of sensitive contents from the rest of the structure. Proper selection of the properties of these isolators offers enhanced vibration suppression (protection of content), though typically at the expense of a considerable demand for the seismic gap (clearance) between the isolated floor and the surrounding objects to avoid collisions that can have detrimental effects (increase of acceleration demand) on the protection-level offered (Jia et al. 2014). This ultimately leads to two different objectives that need to be considered in the design of floor isolation systems: (a) reduction of the accelerations of the protective content, which is the primary goal of the isolation system, and (b) limitation of the displacements of the isolated floor (avoid collisions to neighboring objects), which is a secondary goal so that excessively large clearances are not required. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2This study considers a multi-objective approach for incorporating the aforementioned two requirements within the design optimization and addresses structural and excitation uncertainties by characterizing these objectives in terms of the associated seismic risk, utilizing a probabilistic framework for quantifying this risk (Jia et al. 2014). The first objective corresponds to maximization of the level of protection offered to the sensitive content, defined as the maximization of the reliability against damages, i.e. the probability that the acceleration response will not exceed acceptable bounds. The second objective is related to the clearance demand around the floor isolation system, and the risk-metric utilized to describe this demand corresponds to minimization of the displacement associated to a specific occurrence probability. An illustrative example is presented that considers the optimization of a polynomial friction pendulum isolator (PFPI) (Lu et al. 2013; Lu et al. 2011) for the protection of a computer server. 2. SYSTEM MODEL 2.1. Structural model and isolation system For computational efficiency an uncoupled analysis approach is adopted; the dynamic interaction between the primary and the secondary system (i.e. the building and the floor isolation system, respectively) is ignored and the two systems are modeled in a decoupled manner.  First the structure’s response due to the ground motion is evaluated and then the total acceleration of the floor that the protected content is located serves as the input excitation of the isolation system (Chen and Soong 1988). Such an assumption of negligible dynamic interaction is well justified when the ratio of the secondary system’s mass over the floor’s mass is small enough (<10%) and the system is detuned, meaning that the fundamental frequency of the secondary system is not close to the fundamental frequency of the primary system, which is the typical case for floor isolation systems (decoupling of dynamic behavior is necessary for facilitating a good isolation level and protection of the contents of interest). Planar analysis is considered, as shown in Figure 1. Let mb be the mass of floor isolation system (without the mass of the protected content) and xb its displacement relative to the displacement of the floor that is located. We will further include in the analysis the dynamics for the vibration of the protected content itself, modeled as a single-degree-of freedom (SDOF) system with stiffness kc, damping coefficient cc and displacement xc relative to the isolation system. The equations of motion of the isolation system, decoupled from the building, are then: ( )( )tb b is c c c c b stc c c c c c c s bm x f c x k x m xm x c x k x m x x               (1) where tsx  is the total acceleration of the floor that the protected content is located and fis corresponds to the isolator forces.  16.6m4.0m 4.6m4x9.1=36.4mxoserverIsolator displacementxbxb,2xb,1ys’(xb)=fis r(xb)/Nk0k1ys’(xb,1)Normalized restoring force (backbone curve)xbIsolation gapNeighboring objectssoftening hardeningmc, kc, ccmbf x f x xis b isrb b( ) ( ) sgn( )= + µ Isolator force (hysterisis loop) Figure 1: Structure equipped with floor isolation system. The isolator forces are also shown. 2.2.  Polynomial friction pendulum isolator The isolation system used in this study, defining ultimately fis, corresponds to a polynomial friction pendulum isolator (Lu et al. 2011). This device is similar to the friction pendulum system (FPS) isolator with the difference that its sliding surface has a variable curvature and the radial cross section of its sliding surface is defined by a 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3polynomial function. This variable curvature allows PFPI to have adaptive isolation stiffness varying continuously along isolator displacement (also shown in Figure 1). This facilitates the dual goal discussed in the introduction (a) long periods for most operational conditions (corresponding to smaller displacements of the isolator) so that the acceleration of the protected content stays below the allowable thresholds, and (b) reduction of the displacement demand for strong earthquakes by a hardening effect that shifts the isolation system to a shorter period when the vibration starts becoming excessive (Lu et al. 2013; Lu et al. 2011), so that pounding between the protection system and neighboring objects can be avoided (when displacement exceeds the isolation gap). Note that alternative approaches have also been proposed to reduce excessive displacements of such isolation systems, such as incorporation of supplemental viscous dampers (Jia et al. 2014) or use of multi-stage pendulum isolators (Morgan and Mahin 2010), though the PFPI provides a more simplistic implementation. The isolator force isf  of the PFPI is described as (Lu et al. 2011): ( ) ( ) ( ) ( ) sgn( )r fis b is b is b s b bf x f x f x Ny x x      (2) where ys(xb) is the geometric function of the sliding surface, μ is the coefficient of friction, Ν=(mc+ mb)g is the average normal force at the isolator and sgn(.)  is the sign function. For avoiding numerical challenges in the solution of the differential equation with discontinuities [as present in Eq. (2) because of the sgn(.) function], the frictional force is further characterized through an equivalent Bouc-Wen model (Constantinou et al. 1990). The geometric function ys(xb) is defined by the following sixth-order polynomial (Lu et al. 2013; Lu et al. 2011) such that the PFPI is able to accommodate the multiple design objectives discussed earlier: y(xb)=1/6axb6+1/4cxb4 +1/2exb2, where a, c and e are three constant coefficients describing the mechanical behavior of the PFPI. These coefficients can be further related to parameters with direct engineering meaning, such as: (a) the initial isolation period Tis = [4π2/(gk0)]0.5, where k0 is the normalized (with respect the floor isolation weight) stiffness at xb=0, (b) the isolator displacement xb,1 indicating transition from softening to hardening behavior of the y’s(xb) - xb relationship of the isolator, and (c) the hardening ratio ah = (ρ4-2ρ2+1)(k0-k1)/k0 between displacements xb,1 and xb,2 = ρ/xb,1. The latter displacement corresponds to onset of significant hardening behavior of the isolator, whereas k1 represents the normalized stiffness at xb,1. Using Tis, xb,1 and ah as the design variables of the problem (in addition to the friction coefficient μ), then a, c and e can be expressed as (Lu et al. 2011): 4,1 0 12,1 0 1 0( ) / 5;2 ( ) / 3;  bba x k kc x k k e k      (3) Ultimately, the restoring force behavior of the PFPI can be fully described through Tis (or k0), xb,1, xb,2 and ah (illustrated also in Figure 1) instead of a, c and e. This requires that the ratio ρ is a-priory selected. 3. EXCITATION MODEL For describing the seismic hazard the same excitation model as in (Gidaris and Taflanidis 2015) is adopted. The broadband (high-frequency component for the excitation is represented through a point source model  based on a parametric description of the temporal envelope and radiation spectrum of the ground motion, both given as a function of the earthquake magnitude, M, and rupture distance, r. Near-fault characteristics are incorporated through the velocity pulse model proposed by Mavroeidis and Papageorgiou (2003) that has as input parameters the pulse period amplitude  Tp, a parameter that controls its amplitude Ap, the oscillatory character (number of half-cycles) γp and its phase vp. For the first two parameters, predictive equations exist that relate them to M and r, and based on recommendations in (Halldórsson et al. 2010; Mavroeidis and Papageorgiou 2003) the following expressions are adopted: log(Ap/0.9) =2.04-0.032r+eA and log(Tp) =-2.9+0.5M+eT, where eA and eT are zero 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4mean Gaussian variables with standard deviations 0.187 and 0.143, respectively. The remaining two pulse-parameters, γp and vp, need to be treated as independent random variables, and based on the recommendations in (Jia et al. 2014)  the probability models for them are chosen as Gaussian with mean 1.8 and standard deviation 0.3 and uniform on the range [0, π], respectively. Since not all near fault excitations exhibit a velocity pulse, the possibility of including such a pulse is considered through the probability model developed by Shahi and Baker (2011) that quantifies the probability of an excitation to include a pulse dependent upon other seismicity characteristics (distance to fault rupture, moment magnitude). 4. RELIABILITY-BASED DESIGN 4.1. Seismic risk quantification Seismic risk is quantified (Taflanidis and Beck 2009) by modeling the uncertainty in the characteristics of the models for the system and for the excitation in terms of probability density functions (PDFs). In this context, let the vector of controllable system parameters (properties of PFPI), referred to herein as design variables, be xnX x  , where X  denotes the admissible design space and let θ  lying inΘ n  , denote the augmented vector of model parameters with PDF denoted as p(θ), where Θ  denotes the space of possible parameter-values. For quantifying the seismic-risk, reliability concepts are adopted, with the latter defined as the probability that a response quantity z (acceleration of the protected content or displacement of isolated floor) will not exceed a prescribed acceptable performance bound zb. The performance measure that is then utilized in the different objective functions is the probability of unacceptable performance for a given seismic event, which is expressed as: [ | ] ( , ) ( ) ( | )pF b F pΘP z z I p P d  x x θ θ θ θ   (4) where εp is a binary (outcomes {yes, no}) random variable describing the probability of pulse existence, P(εp|θ) is the probability model for it (Shahi and Baker 2011), and I(x,θ) is the indicator function for failure, which equals to one if the system that corresponds to (x,θ) fails (i.e. z > zb) and zero if it does not. The probability of failure P[z>zb|x,tlife] over a specified lifetime tlife, will be also needed in the design formulation. This requires an additional assumption pertaining to the occurrences of seismic events. Assuming the common in earthquake engineering Poisson distribution of independent occurrences we have: [ | ][ | , ] 1 life F btli vP z zF b feP z z t e    xx  (5) where v is the expected number of events per year. The latter is related to the assumed probability model for M and will be discussed in more detail later. 4.2. Multi-objective optimal design As discussed in the introduction two different objectives will be considered for the design problem formulation: (a) protection level offered by the floor-isolation system and (b) clearance demand to avoid collision to surrounding objects. The first objective is quantified through the probability that the acceleration cx of the protected content will exceed an acceptable threshold ,c threshx , ,[ | ]F c c threshP x x x  , which is calculated by Eq. (4) with cz x  and ,b c threshz x  . The second objective corresponds to the isolator displacement, denoted xb,thresh, with a specific probability of being exceeded po over an assumed lifespan, representing a design event. This metric is calculated by the inverse problem of Eq. (5) setting z = xb and P[z>zb|x,tlife] = po, and identifying the threshold xb,thresh =zb. Ultimately the mathematical description for the multi-objective design problems is: *, ,,arg min{ [ | ], }such that [ | , ]TF c c thresh b threshXF b b thresh life oP x x xP x x t p  xx xx  (6) This multi-objective formulation leads to a set of points (also known as dominant designs) that form a manifold, the so-called Pareto front. A point belongs to the Pareto front and it is called Pareto optimal point if there is no other point that improves one objective without 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5detriment to the other. The motivation behind this multi-objective formulation is that, once provided, the designer can choose among a range of Pareto optimal floor-isolation configurations that describe different compromises between the considered competing performance criteria. For performing the multi-objective optimization in Eq. (6) the various probabilistic integrals of the form of Eq. (4) are estimated here through stochastic simulation. Using a finite number, Ν, of samples of θ and εp drawn from importance sampling densities q(θ) and Pq(εp) with jθ and εpj denoting the jth sample [more details can be found in (Jia et al. 2014)], an estimate for Eq. (4) is given by:   1( , ) ( ) |1ˆ [ | ] ( )j j j jN F pF b j jj q pI p PP z z N q P  x θ θ θx   θ   (7) The optimization in Eq. (6) is then performed (Gidaris et al. 2014a) by substituting the required probabilistic quantities with the stochastic simulation-based approximations provided by Eq. (7) and by adopting an exterior-sampling approach (Taflanidis and Beck 2008), meaning that the same sample set θj, εpj; j=1,…,N is used throughout the entire optimization to reduce influence of prediction errors associated with the stochastic simulation. Any appropriate elitist genetic algorithm (Kalyanmoy 2001) may be then used to obtain the Pareto front. This approach requires typically a large number of evaluations (for different design choices) of the considered objectives, corresponding in this case to a computationally intensive estimation through stochastic simulation, requiring each time numerical simulation of the system response for N  samples. To reduce this computational burden a kriging surrogate modeling approach is adopted, requiring the numerical simulation of the system response only for a limited number of design configurations. Furthermore the development of the surrogate model simultaneously for the uncertain model parameters and the design variables is adopted (Gidaris et al. 2014a). 4.3. Optimization supported by kriging metamodeling in augmented input space Kriging metamodeling with respect to both x and θ is considered whereas the stochastic characteristics of the excitation model are addressed by assuming that under the influence of the white noise each response quantity of interest zk ( cx  and xb here) follows a lognormal distribution with median kz  and logarithmic standard deviation kz . This leads to the following modification (Gidaris et al. 2014a) for the performance measure (indicator function) involved in Eq. (4): ( , ) ln( / ) / kF k b zI z z   x θ  (8) where [.]  stands for the standard Gaussian cumulative distribution function (CDF). The kriging metamodel is ultimately formulated to provide predictions for the statistical quantities needed for evaluation of Eq. (8), that is, the logarithm of the median response and the associated logarithmic standard deviation. To formalize this concept, let y denote the vector of such quantities to be approximated by the metamodel, and φ=[x θ] the augmented input parameter vector for the kriging formulation. For forming the metamodel initially, a database with nm observations is obtained that provides information for the φ-y pair. For this purpose nm samples for {φj, j=1,…,nm}, also known as support points, are created following a latin hybercube grid over the expected range of values possible for each φi. Stochastic ground motions are then generated according to the excitation model and the structural response is numerically evaluated. The influence of the white noise is addressed by considering nw different samples for each φj and using the statistics under these samples to ultimately quantify the response sample yj. Using this dataset the kriging model is then obtained and is used to calculate the performance within a stochastic simulation setting (different θ samples in Eq. (7)) for each different design configuration (different specific values for x) examined within the optimization algorithm in Eq. (6). More 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6details about the mathematical formulation of the metamodeling approach may be found in (Gidaris et al. 2014b).  5. ILLUSTRATIVE EXAMPLE The design of a floor isolation system located on the second floor of a four-story benchmark building (Goulet et al. 2007) is considered to protect a computer server, with mass mc = 6 tons, modeled as a SDOF (Liu and Warn 2012) with period 0.025 sec and damping ratio 2%. The floor-isolation system has mass mb=25 tons and consists of PFPIs modeled by Eq. (2) with design variables x=[Tis xb,1 ah μ] and ρ taken equal to 1.5. The design space for each of them is defined as Tis [2, 5] (sec), xb,1 [7.5, 20] (cm), ah [1, 1.5625], μ  [0.05, 0.25]. The structure corresponds to design A in the benchmark study presented in (Goulet et al. 2007). Similar model and analysis characteristics as in (Gidaris et al. 2014b) are adopted, whereas all parameters for the structure are taken here to be deterministic. The total weights per story are 7039 kN and 6840 kN for the bottom two, respectively, and 6742 kN for the top two. The hysteretic behavior of the structure is modeled by using the modified Ibarra-Medina-Krawinkler hinge model [more details in (Gidaris et al. 2014b)]. OpenSees is used for the analysis of the structure. Rayleigh damping with damping ratio 5% associated with the first and third mode shapes is assumed. The fundamental period calculated based on the 50% gross section stiffness values (cracked reinforced concrete sections) is equal to 0.99 sec. The mass ratio of the floor isolation system over the second floor mass is equal to 4.32% and, in conjunction with the fact that the fundamental floor isolation period is detuned from the considered support (at least under the optimal designs identified later), it justifies the adoption of the uncoupled analysis approach described in Section 2. The uncertainty in moment magnitude, M , is modeled by the Gutenberg-Richter relationship truncated to the interval [Mmin, Mmax] = [5.5, 8], which leads to the PDF and expected number of events per year given, respectively, by: ( ) ( )M min M maxM b M b Mb MMp M b e e e   and  M M min M M maxa b M a b Mv e e   , with the regional seismicity factors selected as bM=0.9loge(10) and aM=4loge(10), leading to v=0.11. For the uncertainty in the event location, the rupture distance, r, for the earthquake events is assumed to follow a lognormal distribution with median 10 km and c.o.v of 40%. The uncertainty in the remaining model parameters is the one described in Section 3. Ultimately, the group of uncertain model parameters is θ=[M, r, Ap, Tp, γp, vp] when considering excitations with directivity pulses and θ=[M, r] when considering excitations without such pulses. Metamodels are developed separately for each of these excitation cases/model, abbreviated as P and NP, respectively. A total of nm =10,630 and 9,000 support points for P and NP, respectively, and nw =100 white noise sequences are used. Space filling Latin hypercube is selected for the support points, for x within the entire design domain and for θ in the range that is expected to take values based on the defined probability models. The accuracy of the developed surrogate model is evaluated using a leave-one-out cross validation approach. The accuracy established is ultimately high with absolute error less than 5.0% and coefficient of determination over 96% for most approximated response quantities. For the stochastic simulation N = 3000 samples are used. Regarding now the various thresholds and parameters involved in the definition of the multi-objective design in Eq. (6), ,c threshx  corresponds to 0.3g, the probability op  is taken equal to 2%, whereas tlife considered for the isolation system is assumed to be 10 years. The failure probability of the server without the isolation system is estimated very high, 40.80%. Results from the optimal design are shown in Figure 2 and Table 1. Figure 2 presents the Pareto front for the considered multi-objective design. The designer can ultimately select among all these candidates, giving a different priority over the two considered objectives. To further examine the characteristics of the obtained solutions, three optimal designs are distinguished in the Pareto curve. The first two represent extreme designs corresponding to a minimum for each of the two objectives, denoted here as Da 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7and Dd. In this case the other competing objective attains its maximum value over the Pareto front. The third design, denoted as Dm, represents a balanced design around the middle of the Pareto front. The superscripts min, max and m are utilized to represent the performance objectives within these representative designs; min corresponds to the minimum of the objective, max to the maximum over the Pareto front, and m to the balanced design over the Pareto front. Detailed results for these characteristic Pareto optimal solutions are reported in Table 1. The optimal design configurations *x for the isolator as well as the associated performances attained are reported.  0 5 10 15 20 2789101112131415xb,thresh (cm)P x x xF c c thresh b threshmin, ,max[ | ], >( )xP x xF c c thmax,[ >resh b threshx| ],,minx( )F c c thresh b threshP x x x[ | ],m, ,mx>( )P x xF c c thresh[ | ], > x (%)  Figure 2: Pareto front curve  It is evident from the results that the various Pareto optimal solutions lead to significantly different configuration designs. The level of protection provided shows significant variability, with the probability of failure ranging from as low as 0.09% (corresponding to a significant improvement in performance) to as high as 21% (that still represents a good level of protection), with associated displacement demands also varying significantly. This validates the considerations behind the proposed multi-objective design formulation. The competing objectives need to be carefully evaluated before the preferred solution is chosen. For example, if the Da Pareto optimal design is preferred then ,[ | ]F c c threshP x x x   is minimized at the expense, though, of tolerating a high demand for isolation gap xb,thresh, which can be impractical and/or costly due to possible interior space constraints. The optimal configuration for this design corresponds to a flexible isolator (note the high values for Tis and xb,1 in conjunction with the low ah) that is able to greatly reduce the acceleration demand at the expense, though, of high isolator displacements. On the other hand, if the balanced design is chosen, for example Dm, then the optimal configuration corresponds to a less flexible isolation system, leading to a ≈38% reduction in xb,thresh with the trade-off of a considerable increase in ,[ | ]F c c threshP x x x  (6.16%), which still, though, represents an adequate level of protection. It is also interesting to note that as we move along the Pareto curve towards designs with higher priority on reduction of the isolator displacement, the optimal value corresponding to the coefficient of friction increases. This trend is expectable and it is attributed to the fact that friction is the main energy absorption mechanism of the implemented protective system. An interesting extension of these comparisons would be to examine the effectiveness of systems that target the compromise between the two considered objectives through fundamentally different means (for example FPS systems with supplemental dampers or multi-stage pendulum isolators). Such an investigation is topic of future research.  Table 1: Characteristic Pareto optimal solutions Cases Da Dm Dd x* Tis  4.97 2.52 2.83 xb,1  19.78 11.75 7.63 ah 1.08 1.49 1.53 μ 0.15 0.22 0.25 ,[ | ],  /F c c threb shshtrexP x x x  0.09%/ 14.55 cm 6.16%/ 9.06 cm 21.36%/ 7.41 cm 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  86. CONCLUSIONS A multi-objective optimization framework for floor-isolation systems was discussed in this paper for protection of acceleration-sensitive contents. The considered objectives pertained to the level-of protection offered to the sensitive content and the clearance-demand around the floor to avoid collisions to surrounding objects. A risk-based quantification was utilized for these objectives with the system reliability used to represent the seismic-risk, and a stochastic ground motion modeling approach was adopted to describe the seismic hazard. A surrogate modeling approach was adopted for estimating seismic risk and performing the design optimization. As an illustrative example, the design of a polynomial friction pendulum isolator system was presented. Within this example the multi-objective formulation was demonstrated to provide design solutions with different performance levels across the considered objectives, offering a range of options for selecting the final protection system. 7. REFERENCES  Chen, Y., and Soong, T. (1988). "Seismic response of secondary systems." Eng. Str., 10(4), 218-228. Constantinou, M., Mokha, A., and Reinhorn, A. (1990). "Teflon bearings in base isolation II: Modeling." J Str Eng, 116(2), 455-474. Gidaris, I., and Taflanidis, A. A. (2015). "Performance assessment and optimization of fluid viscous dampers through life-cycle cost criteria and comparison to alternative design approaches." B  Earth Eng, 13(4), 1003-1028. Gidaris, I., Taflanidis, A. A., and Mavroeidis, G. P. (2014a). "Multiobjective formulation for the life-cycle cost based design of fluid visocous dampers." Proc., 9th International Conference on Structural Dynamics EURODYN2014, Porto, Portugal. Gidaris, I., Taflanidis, A. A., and Mavroeidis, G. P. (2014b).  "Surrogate modeling implementation for assessment of seismic risk utilizing stochastic ground motion modeling." Proc., 2nd European Conference on Earthquake Engineering and Seismology, Istanbul, Turkey. Goulet, C. A., Haselton, C. B., Mitrani-Reiser, J., Beck, J. L., Deierlein, G., Porter, K. A., and Stewart, J. P. (2007). "Evaluation of the seismic performance of code-conforming reinforced-concrete frame building-From seismic hazard to collapse safety and economic losses." Earth  Eng Str D, 36(13), 1973-1997. Halldórsson, B., Mavroeidis, G. P., and Papageorgiou, A. S. (2010). "Near-Fault and Far-Field Strong Ground-Motion Simulation for Earthquake Engineering Applications Using the Specific Barrier Model." J Str Eng, 137(3), 433-444. Jia, G., Gidaris, I., Taflanidis, A. A., and Mavroeidis, G. P. (2014). "Reliability-based assessment/design of floor isolation systems." Eng. Str., 78(0), 41-56. Kalyanmoy, D. (2001). Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons. Liu, S., and Warn, G. P. (2012). "Seismic performance and sensitivity of floor isolation systems in steel plate shear wall structures." Eng Str, 42, 115-126. Lu, L.-Y., Lee, T.-Y., Juang, S.-Y., and Yeh, S.-W. (2013). "Polynomial friction pendulum isolators (PFPIs) for building floor isolation: An experimental and theoretical study." Eng Str, 56, 970-982. Lu, L. Y., Lee, T. Y., and Yeh, S. W. (2011). "Theory and experimental study for sliding isolators with variable curvature." Eart Eng Str D, 40(14), 1609-1627. Mavroeidis, G. P., and Papageorgiou, A. P. (2003). "A mathematical representation of near-fault ground motions." B Seism Soc A, 93(3), 1099-1131. Morgan, T. A., and Mahin, S. A. (2010). 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