Probabilistic Performance Based Design Multi-Objective Optimization for Steel Structures Sanaz Saadat, Ph.D.,1 Charles V. Camp,2 Ph.D., Shahram Pezeshk3 Ph.D., P.E., Christopher M. Foley4, Ph.D., P.E. 1
Gilsanz Murray Steficek LLP, 129 W 27th St., New York, NY 10001, PH (212) 254-0030, FAX: (212)-477-5978, email:
[email protected] 2 Civil Engineering Department, University of Memphis, Engineering Science Building Room 106B PH: (901) 678-3169, FAX: (901) 678-3026, email:
[email protected] 3 Civil Engineering Department, University of Memphis, Engineering Science Building, Room 104A, PH: (901) 678-4727, FAX: (901) 678-3026, email:
[email protected] 4 Department of Civil and Environmental Engineering, Marquette University, 1515 W. Wisconsin Ave. Milwaukee, WI 53233, PH: (414) 288-5741, FAX: (414) 288-7521, email:
[email protected] ABSTRACT Multi-objective optimization for the probabilistic seismic performance based design of an example moment frame steel structure is presented. Direct economic and social losses associated with seismic events, which are of interest in the current recommended frameworks for the performance based design of structures, are considered in the optimization problem defined. Three optimization objectives are selected: the initial construction cost, modeled as the weight of the structural system; expected annual economic loss associated with damage resulting from seismic hazard; and expected annual social loss resulting from seismic hazard induced damage. Hazus recommended procedures are applied in the economic and social loss calculations which include the fragility functions used in the damage analyses and injury event models implemented in the social loss calculations. The multiobjective optimization method uses a non-dominated sorting genetic algorithm strategy. The optimization results for the multiple objectives are presented and discussed in the form of Pareto fronts. Engineering demand parameters implemented for the seismic loss analysis are inter-story drifts and peak floor accelerations and are obtained using inelastic time history analysis for the ground motions associated with various seismic hazard levels. To illustrate the design procedure, loss parameters are calculated for an example steel structure located in Los Angeles, CA.
INTRODUCTION Probabilistic seismic performance-based design (SPBD) of structures provides a framework for expressing design objectives in meaningful terms for different stakeholders. For this purpose, desired performance levels for a specific building and for defined levels of seismic ground motions establish the performance objectives at the beginning of the design process (FEMA 2012, ATC 2007). In other words, SPBD will provide the designer with the better understanding of the risks (economic and social) that may occur during probable future earthquakes. In addition SPBD can enable stakeholders to introduce different measures of building performance for various levels of seismic hazard as the design objectives. SPBD can be used for both the design of new buildings design and the retrofitting of existing buildings. Different aspects of seismic risk are of significant important in several real-state decision making processes such as rehabilitation of existing buildings, design of new structures, earthquake insurances, etc. (Porter et al. 2004). In order to quantify the probable losses due to anticipated seismic events that may occur during the building’s lifecycle, loss parameters can be calculated through the aggregation of the results of the probabilistic seismic hazard analysis, probabilistic seismic demand analysis, probabilistic capacity analysis, and probabilistic loss analysis, using the total probability theorem and implementing the loss assessment framework developed by the PEER center (Krawinkler 2005, Moehle and Deierlein 2004, Hamburger et al. 2004). In this study, seismic loss evaluations are considered to optimize the PBD of an example steel structure located in Los Angeles, CA. A multi-objective optimization method is applied to a SPBD problem that seeks to minimize three objectives: initial construction cost, modeled by the weight of the structural system, expected annual loss value associated with direct economic losses, defined as EAL parameter, and expected annual loss value associated with direct social losses defined as EASL parameter. Inelastic time history analysis is used to evaluate the structural response under different levels of earthquake hazard to determine engineering demand parameters.
PROBLEM DEFINITION The multi-objective optimization problem uses a non-dominated sorting genetic algorithm strategy (Deb et al. 2002) and that minimizes three specified objectives. The optimization problem is expressed as
Minimize (WP , EAL P , EASLP ) Subjected to : ci ≥ 1.0 (i = 1,3)
(1)
where WP, EALP, and EASLP are the penalized values of W , EAL, and EASL, respectively; and ci is the ith constraint that is applied to the optimization problem. W is the weight of the structure. EAL and EASL parameters are calculated as the area under the total economic and social loss curves, respectively (Saadat et al. 2014, Rojas et al. 2011, Foley et al.
2007). The total loss curve is obtained from the loss curves for each hazard level and the hazard curve for the building’s site. Three seismic hazard levels of 2, 10, and 50 percent probability of exceedance (POE) in 50 years are considered in loss calculation. Inelastic time history analysis is performed using suites of ground motions, available from the SAC steel research project (Somerville et al. 1997), for a site located in Los Angeles, CA. EAL is calculated in terms of the percent of building replacement cost (BRC). In calculation of the social losses, casualties caused by an earthquake are modeled by developing a tree of events leading to their occurrence (Hazus-MH 2003b). EASL is presented in terms of %αNo, where α (in $/person) is the comprehensive cost for the first casualty severity level (Hazus-MH 2003b, NSC 2013) and No is the number of building occupants. Damage analysis is performed using the fragility curve parameters and procedures provided in HazusMH (2003a, b) for structural and non-structural members and for different damage states. The constraints for the confidence levels for collapse prevention CLCP and immediate occupancy CLIO are
c1 :
CLCP ≥ 1.0 CLCP,min
(2)
c2 :
CLIO ≥ 1.0 CLIO ,min
(3)
where CLCP,min and CLIO,min are selected as 90 percent and 50 percent, respectively, as recommended by FEMA (FEMA 2000a). The constraint for ensuring the strong column-weak beam criteria of for seismic design, calculated for each connection in the frame is
c3
∑M : ∑M
* pc * pb
≥ 1.0
(4)
where M*pc is the modified flexural strength of the column and M*pb is the modified flexural strength of beam sections. Equation (4) is calculated using the AISC (2011) specifications. Penalty function φ , applied to the optimization objectives, is defined as: 4
ϕ = ∏ϕ i i =1
⎧ϕ i = 2.0 − ci where ⎨ ⎩ϕ i = 1.0
if
ci < 1.0
if
ci ≥ 1.0
(5)
The penalized values WP, EALP and EASLP are calculated as:
WP = ϕ × W
(6)
EALP = ϕ × EAL
(7)
EASLP = ϕ × EASL
(8)
The defined problem is applied to an example steel structure and the optimization results are presented.
EXAMPLE STRUCTURE The multi-objective SPBD optimization is applied to the example 3-story steel structure shown in Figures 1 and 2. The search space includes a list of 60 compact AISC W sections (W10, W12, and W14) for columns and another list of 64 AISC W sections (W18, W21, W24, W27, W30, W33, W36, and W40) for beam elements. The seismic masses are 73.10 kips-sec2/ft for the roof, 67.86 kips-sec2/ft for the 2nd floor, and 69.86 kips-sec2/ft for the 1st floor (the values are for the entire structure) (FEMA 2000b). Masses are lumped at the beam-to-column locations. Moment frame A-F/1 is considered for the design. Lean-on columns are used in the analysis in order to represents the gravity frame system that is tributary to the moment resisting frame.
Figure 1. Plan view of the example structures
Figure 2. Elevation and design variable for the 3-story structure OPTIMIZATION RESULTS In multi-objective optimization problems, since there are more than one objective functions to be optimized simultaneously, due to the usual conflict among different objectives, typically there is not a single solution that is best with respect to all objectives. Instead, there is a set of solutions, called non-dominated solutions or Pareto optimal solutions (Gen and Cheng 2000). Figure 3 shows the results for the multi-objective optimization problem defined. Since three optimization objectives are considered, the results are presented in the form of a three dimensional Pareto front. The presented front shows that a decrease in both economic and social losses is associated with increase in initial cost (modeled as the weight of the structure). Displaying the optimization results in the form of Pareto fronts would provide decision makers with a useful tool to assist in selection of the final design among variety of feasible options. Figure 4 shows the comparison of the seismic loss optimization objective values for the solutions on the Pareto front for the example structure: i.e. EAL (%BRC) vs. EASL (%α No). The results in Figure 4 show the direct relationship between the calculated annual economic and social loss parameters, EAL and EASL, for the example structure located in Los Angeles, CA.
Figure 3. Multi-objective optimization results
Figure 4. Seismic loss optimization objective values for the solutions on the Pareto front for the example structure: EAL (%BRC) vs. EASL (%α No)
Figure 5 shows the distribution of economic loss values between different building components, including structural and non-structural components, for various hazard levels. The comparison of the results is shown for three designs, associated with minimum weight, minimum EAL, and minimum EASL, selected from the Pareto front presented in Figure 3. As designs move along the Pareto front from low weight designs to higher weight designs, which are associated with designs with lower seismic loss values, the contribution of structural (SS) and drift-sensitive non-structural (NSD) components to the total loss value decreases while contribution of acceleration-sensitive non-structural (NSA) components increases. Figure 6 shows the distribution of the economic and social loss values for various hazard levels for the same three designs compared in Figure 5. For both economic and social losses, the results have the similar pattern of higher loss values for events with lower probability of exceedance. A more significant difference between the contributions of different hazard levels is observed for social losses, as compared to the economic losses, for the presented designs.
Figure 5. Distribution of the economic loss values between different building components for various hazard levels
Figure 6. Distribution of economic and social loss values for various hazard levels
SUMMARY AND DISCUSSION Seismic loss values associated with economic and social losses, which are of interest in the current recommended frameworks for the performance based design of structures, are considered in the optimization problem of an example steel structure located in Los Angeles, CA. A multi-objective optimization problem is developed that minimizes three objectives: initial cost, expected annual seismic economic loss parameter EAL, and expected annual seismic social loss parameter EASL. Results of the optimization problem are presented in the form of a three dimensional Pareto front. Decreases in both economic and social losses are associated with an increase in initial cost (modeled as the weight of the structure). Observing results along the Pareto front indicate that as designs increase their structure weight, which is associated with lowering the total seismic loss values, the contribution of structural (SS) and drift-sensitive non-structural (NSD) components to total loss decreases, while the contribution of acceleration-sensitive non-structural (NSA) components increases. Presenting the defined multi-objective optimization results in the form of Pareto fronts would provide decision makers with a useful tool to assist in selection of the final design among variety of feasible options.
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