Performance-based multiobjective optimum design of steel structures ...

Struct Multidisc Optim (2006) 32: 1–11 DOI 10.1007/s00158-006-0009-y

R E S E A R C H PA P E R

Michalis Fragiadakis · Nikos D. Lagaros · Manolis Papadrakakis

Performance-based multiobjective optimum design of steel structures considering life-cycle cost

Received: 18 May 2005 / Revised manuscript received: 7 October 2005 / Published online: 27 April 2006 © Springer-Verlag 2006

Abstract A new methodology for the performance-based optimum design of steel structures subjected to seismic loading considering inelastic behavior is proposed. The importance of considering life-cycle cost as an additional objective to the initial structural cost objective function in the context of multiobjective optimization is also investigated. Life-cycle cost is considered to take into account during the design phase the impact of future earthquakes. For the solution of the multiobjective optimization problem, Evolutionary Algorithms and in particular an algorithm based on Evolution Strategies, specifically tailored to meet the characteristics of the problem at hand, are implemented. The constraints of the optimization problem are based on the provisions of European design codes, while additional constraints are imposed by means of pushover analysis to control the load and deformation capacity of the structure.

Keywords Structural optimization · Life-cycle cost · Earthquake · Pushover analysis · Evolution Strategies · Multiobjective optimization

1 Introduction Severe damages caused by recent earthquakes in the United States and Japan made the engineering community to question the effectiveness of seismic design codes. Although the number of losses in terms of human lives was low, the economic loss was significant. Given that the primary goal of contemporary seismic design is the protection of human life, it is evident that additional performance targets should be considered. Recently, the concept of performance-based design for structures subjected to seismic loading conditions Michalis Fragiadakis · Nikos D. Lagaros · Manolis Papadrakakis (B) Institute of Structural Analysis & Seismic Research, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, 15780, Athens, Greece e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

was introduced. The progress that took place in the last two decades in the fields of computational mechanics and hardware technology made possible to employ more realistic design procedures based on nonlinear analysis in place of procedures based on linear analysis. The use of structural optimization methods provides the computational environment to replace the traditional “trial and error” process with an automatic procedure. Therefore, more elaborate, nonlinear analysis methods can be easily introduced for the design of new structures when a structural optimization environment is present, as opposed to the current practice where these methods are used primarily for the assessment of existing structures. Performance-based design concepts have been introduced over the last 10 years by various guidelines [ATC-40 (1996); FEMA-356 (2000); SEAOC Vision 2000 (1995)] for the assessment and rehabilitation of existing structures and the analysis and design of new ones. The main objective is to increase the safety against natural hazards and in the case of earthquakes to make them having a predictable and reliable performance. In other words, the structures should be able to resist earthquakes in a quantifiable manner and to preset levels of desired possible damage. A typical limit-statebased design can be viewed as a two-level approach where the serviceability and the ultimate limit-states are considered. This approach is adopted by seismic design codes such as Eurocode 3 (EC3) (1992). On the other hand, PerformanceBased Earthquake Engineering (PBEE) is a multi-level design approach where various levels of structural performance are considered. For example, FEMA-356 (2000) suggests the following performance levels: operational level, immediate occupancy, life safety, and collapse prevention. For assessing the structural performance, the guidelines suggest the use of various types of analysis methods: linear static, nonlinear static, linear dynamic, and nonlinear dynamic. The most commonly used approach is the nonlinear static analysis known as pushover analysis. Pushover analysis allows for the direct evaluation of the performance of the structure at each limit-state as opposed to conventional static analysis-based design procedures, such as that of EC3, where the structure is designed for the ultimate limit-state. In the latter case, a num-

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ber of deflection checks are performed at the end of the design process to check the serviceability limit-state requirements. Perhaps the most appropriate objective function in a structural optimization problem is to minimize the total cost of the structure during its lifetime, which is defined as the sum of the initial and the life-cycle cost. Usually, life-cycle cost refers to the deterioration of the structural components’ capacity over time due to phenomena such as corrosion or deterioration of the joints or the bearings [Frangopol et al. (1997)]. However, life-cycle cost may also refer to the risk related to natural hazards, such as wind or earthquake. In this case, the lifecycle cost is related to the possible losses due to unsatisfactory performance of the structure under loading with random occurrence and intensity during its life. The design process should consider both direct economic and human life losses within a given social context [Sanchez-Silva and Rackwitz (2004)]. The objective of this work is the performance-based optimum design of steel moment-resisting structures with respect to their initial and their life-cycle cost. A number of studies have been published in the past where the concept of performance-based design was adopted. Among others, Ganzerli et al. (2000) proposed a performance-based optimization procedure of RC frames using convex optimum design models and a mathematical optimizer, while Fragiadakis et al. (2006) proposed a performance-based optimization procedure for steel moment-resisting frames in the probabilistic framework of FEMA-350. Structural optimization with life-cycle cost considerations has also received attention in several recent studies. Liu et al. (2005) proposed a Genetic Algorithm-based multiobjective structural optimization procedure for steel frames considering four objective functions; weight, maximum interstorey drift for two performance levels, and design complexity criteria. Khajehpour and Grierson (2003) investigated the trade-off between life-cycle profitability and load-path safety for an example high-rise office building project. Using a multicriterion Genetic Algorithm, they proposed alternative building layouts for highrise buildings to increase safety. Sarma and Adeli (2002) addressed the life-cycle optimization of steel structures, solving a four-criterion optimization problem. Their study is based on a detailed breakdown of all factors that influence the life-cycle cost of steel structures. Li and Cheng (2003) introduced a damage-reduction-based technique in the framework of structural optimization and showed that the proposed damage-reduction design concept leads to designs with better seismic performance in terms of both life-cycle cost and maximum interstorey drift criteria. Life-cycle cost, as considered in this study, represents the cost of the expected damages caused by earthquake events that are expected to occur during the design life of the structure. Other parameters that influence life-cycle cost such as operating costs and inspection costs are not accounted. Furthermore, the proposed methodology is based on the performance-based design concept, where pushover analysis is performed to determine the capacity of each candidate design. It is shown that an optimum design with respect to the

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minimum initial cost is far from being optimum with respect to the total cost corresponding to the lifetime of the structure. Therefore, life-cycle cost serves as an additional objective function of the optimization problem to take into account the level of damage caused by future earthquakes. In the methodology proposed, for the first time, a design procedure based on pushover analysis is incorporated into the analysis phase of a multiobjective structural optimization algorithm. For the solution of the two-criterion optimization problem at hand, an Evolutionary Algorithm (EA), and in particular an algorithm based on Evolution Strategies (ES), is employed. To meet the demands of the multicriterion optimization problem, a number of modifications to the ES algorithm are implemented (Papadrakakis et al. (2002)). Each design is initially checked against gravity loading and the Eurocode provisions, and then pushover analysis is carried out to determine if the performance targets at three limit-states are met. 2 Seismic optimum design procedure 2.1 Formulation of the multiobjective optimization problem In general, the mathematical formulation of a multiobjective optimization problem includes a set of n design variables, a set of m objective functions, and a set of k constraint functions. The formulation of the optimization problem at hand can be stated as follows: mins∈F [C I C (s), C LC (s)]T subject to g j (s) ≤ 0 j = 1,...,k ,

(1)

where CIC and CLC are the initial and the life-cycle cost objective functions, while s represents the design vector that corresponds to the cross sections of each member of the structure. F is the feasible set in the discrete design space Rn , defined as the set of design variables that satisfy the constraint functions gj (s):  F = s ∈ R n g j (s) ≤ 0 j = 1,...,k .

(2)

In a multiobjective optimization problem, there is no unique point that would represent the optimum in terms of either minimum CIC or CLC . Thus, a set of optimum designs that correspond to the Pareto front of the problem is sought. A design vector s*∈F is a Pareto optimum for the problem of (1) if and only if there exists no other design vector s∈F such that: C I C (s) ≤ C I C s∗




with C LC (s) < C LC s∗




or

(3)

C LC (s) ≤ C LC s . ∗




with C I C (s) < C I C s∗




Performance-based multiobjective optimum design of steel structures considering life-cycle cost

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A candidate design is checked whether it belongs to the feasible set of (2) at every design step of the optimization process (Fig. 1). The design step consists primarily of three substeps. Initially, Eurocode checks that do not require any analysis are performed and subsequently, the structure is verified against gravity loads. Pushover analysis is carried out where constraints on maximum interstorey drift ∆ are imposed.

2.2 Seismic loading and optimization constraints A number of constraints must be considered in the framework of a structural optimization problem to ensure that the optimum design will meet the design requirements. Each candidate design is assessed against these constraints, while care should be taken during the optimization process to perform the checks that require less computational effort first. Therefore, before any analysis, the column to beam strength ratio is calculated and also a check is performed on whether the sections chosen are of class 1, as EC3 suggests. The latter check is necessary to ensure that the members have the capacity to develop their full plastic moment and rotational ductility, while the former is necessary to have a design consistent with the ‘strong column–weak beam’ design philosophy. Subsequently, all EC3 checks must be satisfied for the gravity loads obtained from the following load combination: X

Sd = 1.35

G k j + 1.50

X

j

Q ki ,

(4)

i

where “+” implies “to be combined with”, the summation symbol 6 implies “the combined effect of”, Gkj denotes the characteristic value of the permanent action j, and Qki refers to the characteristic value of the variable action i. If the above constraints are satisfied, pushover analysis is performed where earthquake loading is considered using the following load combination: Sd =

X j

G k j + Ed +

X

ψ2i Q ki ,

(5)

i

where Ed is the design value of the seismic action and ψ 2i is the combination coefficient for quasi permanent value of the variable action i, here taken equal to 0.30. The load combination of (5) implies that the seismic loads are applied incrementally, while the structure is also loaded with constant gravity loads Gkj and Qki . The proposed design procedure allows the use of higher level design checks as opposed to the prescriptive building codes such as Eurocodes (EC3) (1992) and EC8 (1994) which include a great number of provisions to ensure adequate strength of structural members and indirectly of the overall structural strength. Design codes do not provide acceptable levels of building performance for actions that correspond to earthquake of with different return periods.

Fig. 1 Flowchart of the design step

2.3 Pushover analysis The purpose of pushover analysis is to assess the structural performance in terms of strength and deformation capacity globally as well as at the element level. The structural model is “pushed” using a predefined fixed lateral load pattern, usually proportional to the fundamental mode. The method can provide the sequence of member yielding and identify the regions where inelastic deformations are expected to be high. Pushover analysis is based on the assumption that the response of the structure is related to the response of an equivalent single degree of freedom system with properties propor-

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tional to the first mode of the structure, although extensions to account for higher mode effects have been proposed [e.g., Chintanapakdee and Chopra (2003)]. The pushover analysis step is initiated as soon as the structure has satisfied the initial static analysis step. Gravity loads are present and follow the load combination of (5). Geometric nonlinearities are also considered. A lateral load distribution that follows the fundamental mode is adopted, while the analysis is terminated as soon as a target displacement that corresponds to the 2% in 50 (2/50) years earthquake is reached or earlier if the algorithm fails to converge. The target displacement is obtained from the FEMA-356 (2000) formula: δt = C0 C1 C2 C3 Sa

Te2 g, 4π 2

(6)

where C0 , C1 , C2 , and C3 are modification factors. C0 relates the spectral displacement to the likely building roof displacement. C1 relates the expected maximum inelastic displacements to the displacements calculated for linear elastic response. C2 represents the effect of the hysteresis shape on the maximum displacement response, and C3 accounts for second order effects. For the test case considered in this study, C1 and C3 are both equal to 1.0 while C2 takes the values of 1.0, 1.1, and 1.2 depending on the performance level considered. Te is the effective fundamental period of the building in the direction under consideration. Sa is the response spectrum acceleration corresponding to the Te period, normalized by g. The pushover curve is converted to a bilinear curve with a horizontal post-yield branch that balances the area above and below the curve. The constraints considered for the pushover analysis step are related to the maximum interstorey drift ratio ∆, which is the largest value of the height-wise peak interstorey drift ratios at each intensity level. This is an excellent measure of both structural and nonstructural damage because of its close relationship to plastic rotation demands on individual beam–column connection assemblies. In this study, three intensity levels are considered that correspond to the 50, 10 and 2% probabilities of exceedance in 50 years. The allowable maximum interstorey drift ratio ∆ for each limit-state was considered to be 0.6, 1.5, and 3%, respectively. The ∆ values of the three intensity levels are also used for the calculation of life-cycle cost, as will be described in the next section.

factors. Other parameters that affect the initial cost, like the cost of nonstructural components, the cost of coating against fire or corrosion, the cost of heating or electrical installments or any sort of aesthetic components, are not considered. With the term life-cycle cost, we refer to the potential damage cost from earthquakes that may occur during the life of the structure. It accounts for the cost of repair after an earthquake, the cost of loss of contents, the cost of injury recovery or human fatality, and other direct or indirect economic losses. Other expenses, not related to earthquake damages, but may arise during the life of the structure, such as maintenance costs, are omitted. The quantification of the losses in economical terms depends on several socioeconomic parameters. For example, there is a number of approaches for the calculation of the cost of the loss of a human life, ranging from purely economic approaches to more sensitive ones that consider the loss of a person irreplaceable [Warszawski et al. (1996)]. Therefore, the estimate of the cost of exceeding the collapse prevention damage state will vary according to which approach is adopted. For the purpose of this work, the cost of exceedance of a damage state is obtained as a percentage of the initial cost from the table of ATC-13 (1985) (Table 1, column 2). The life-cycle cost function used in this study is based on the work of Wen and Kang (2001): N

C LC =


X ν 1 − e−λt Ci Pi , λ i=1

(7)

where N is the total number of limit-states considered, P i =
Pi (∆ > ∆i ) − Pi+1 (∆ > ∆i+1 ) and Pi (∆ > ∆i ) = −1 t   · ln 1 − Pi (∆ > ∆i ) . Pi is the probability of the ith damage state being violated given the earthquake occurrence and Ci is the corresponding cost (Table 1, column 4); Pi (∆ > ∆i ) is the annual exceedance probability of the maximum interstorey drift value ∆i ; ν the annual occurrence rate of significant earthquakes, modeled as a Poisson process, and t is the service life of a new structure or the remaining life of a retrofitted structure. The first component of (7) that contains the exponential term is used to express CLC in present worth, where λ is the annual momentary discount rate considered to be constant and equal to 5%. It is assumed that after an earthquake occurs, the structure is fully restored to its initial state. Table 1 Damage states, maximum interstorey drift limits, and cost

2.4 Initial material and life-cycle cost objective functions Initial material cost of a new structure refers to the cost of the structure during the construction stage. For steel-framed structures, it is usually considered to be proportional to the total weight of its components. The initial cost may also be influenced by other parameters such as the cost, of connections, or other detailing that influences the performance. All these parameters may be accounted by the use of appropriate

Performance level 1 2 3 4 5 6 7

Damage state None Slight Light Moderate Heavy Major Destroyed

Interstorey drift (%) ∆