The effects of rehabilitation objectives on near optimal trade-off relation between minimum weight and maximum drift of 2D steel X-braced frames considering soil-structure interaction using a cluster-based NSGA II

Abstract

A cluster-based non-dominated sorting genetic algorithm (NSGA) II has been considered to investigate the effects of rehabilitation objectives on multi-objective design optimization of two-dimensional (2D) steel X-braced frames in the presence of soil-structure interaction. The substructure elasto-perfect plastic model has been adopted for modeling of the soil-structure interaction and the nonlinear pushover analysis is used to evaluate the performance level of the frames for a specified hazard level. Cross-sections of grouped elements of the frames are considered to be discontinuous design variables of the problem. Via implementing some of the constraints, which are independent of doing the time-consuming nonlinear analysis, input population of the optimization technique has been clustered. By using the nonlinear analysis technique in conjunction with the cluster-based NSGA II, near optimal trade-off relation between minimum weight and maximum story drifts of the frames are obtained. The allowable rotations, geometry, and resistance constraints of the structural elements are considered in the optimization design of the frames. The effects of the enhanced basic safety and limited selective rehabilitation objectives on optimum design of the frame are studied. The results show differences between the optimum results of the three mentioned rehabilitation objectives and effects of soil types.

Appendices

Appendix 1: Nonlinear static analysis method (pushovers)

Nonlinear static analysis method can be used to check the performance level of a structure (FEMA 356 2000). In this analysis, the lateral load gradually increases and displacement of a certain or control point of the structure exceeds to reach a target displacement. By increasing the lateral load, the deformations and internal forces of the structural elements are continuously checked. The control point of the structure should properly represent the actual structural changes. In this paper, the center of mass of the roof is selected as the control point. The target displacement of a rigid diaphragm of a structural system can be determined via employing the presented coefficient method given in FEMA 356 (2000). In this method, first, a nonlinear static analysis must be performed to obtain the pushover curve. Considering this curve, target displacement of the structure can be computed as follows.

$$ {\delta}_t={C}_0{C}_1{C}_2{C}_3{S}_a\frac{T_e^2}{4{\uppi}^2}g $$

(A-1)

$$ {T}_e={T}_i\sqrt{\frac{k_i}{k_e}} $$

(A-2)

where δ_t is the target displacement; C₀, C₁, C₂, and C₃ are the modification factors provided by FEMA 356 (2000); S_a is the response spectrum acceleration at effective fundamental period; T_e and T_i are the effective and elastic fundamental periods, respectively; k_e and k_i are the effective and elastic lateral stiffness of the building, and g is the gravity acceleration. The response spectrum acceleration can be considered using the following relations.

$$ {S}_a^i=\left\{\begin{array}{c}\ {F}_a{S}_s^i\left(0.4+3\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{${T}_0$}\right.\right)\kern5.5em \mathrm{if}\kern0.5em 0<\mathrm{T}\le 0.2{T}_0^i\kern2.25em \\ {}{F}_v{S}_s^i\kern11.25em \mathrm{if}\kern0.5em 0.2{T}_0^i<\mathrm{T}\le {T}_0^i\kern0.75em \\ {}\kern0.75em {F}_v\raisebox{1ex}{${S}_1^i$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\kern9.75em \mathrm{if}\kern0.5em \mathrm{T}>{T}_0^i\kern4em \end{array}\right.\kern1.75em i=20\%/50\ \mathrm{years},\mathrm{BSE}-1,\mathrm{BSE}-2\kern2em $$

(A-3)

$$ {T}_0^i=\frac{F_v{S}_1^i}{F_a{S}_s^i} $$

(A-4)

The recommended values of F_v, F_a, S₁, and S_s for three different hazard levels are considered same as those of the Gholizadeh and Poorhoseini (2016) which are tabulated in Table 9.

Table 9 The recommended values for F_v, F_a, S₁, and S_s

Also, to investigate the simultaneous effect of gravity and earthquake loads in the nonlinear static analysis, the following upper and lower gravity loads combinations should be chosen.

$$ {Q}_G=1.1\left[{Q}_D+{Q}_L\right] $$

(A-5)

$$ {Q}_G=0.9\ {Q}_D $$

(A-6)

where Q_D and Q_L are the dead and live loads, respectively. The lateral load distribution is considered to be proportional to the mass of stories, and is assumed constant through the analysis and design process of the X-braced steel frames as follows.

$$ {F}_i=\frac{W_i\ {h}_i^k}{\sum_{j=1}^n{W}_j\ {h}_j^k}\kern0.5em V $$

(A-7)

where F_i is the lateral applied force on the ith story; W_i is the effective seismic weight of ith story; h_i is the height of the ith story from the base; V is the base shear; and k can be obtained from the following equation.

$$ \mathrm{k}=\left\{\begin{array}{c}1\\ {}\left(\frac{T+1.5\ }{2}\right)\\ {}2\end{array}\right.{\displaystyle \begin{array}{c}\ T\le 0.5\\ {}\kern0.75em 0.5<T\kern0.5em \le \kern0.5em 2.5\\ {}T\kern0.5em >2.5\end{array}} $$

(A-8)

Appendix 2: Recommended soil-structure interaction by FEMA356 and ASCE 41-17

A series of Winkler springs are recommended to implement soil-structure interaction model for analysis of structures by refs. (FEMA 356 2000; ASCE 41-17 2017). The characteristics of the springs should be as follows.

• 1. The characteristics of these springs are obtained in a static state and are discarded from their dependence in the vibrational frequency, which is an appropriate approximation in the usual frequency domain.

• 2. Nonlinear elasto-plastic behavior of soil has been used in the analyses, Fig. 18.

Fig. 18

Elasto-plastic load-deformation behavior of soil (FEMA 356 2000; ASCE 41-17 2017)

where Q_c is equal to the vertical load bearing capacity of the soil and can be obtained from the following equation.

$$ {Q}_c=3{q}_a $$

(B-1)

where q_a is the load capacity of the soil.

For taking into account the flexural stiffness effect of a rigid foundation (ASCE 41-17 2017), stiffness of the springs at the two ends of foundation is considered to differ from its middle part, Fig. 19. The per unit length stiffness of the springs can be computed by using (B-2) through (B-4)

$$ {k}_{\mathrm{end}}=6.83\frac{G}{1-\upupsilon} $$

(B-2)

$$ {k}_{\mathrm{mid}}=0.73\frac{G}{1-\upupsilon} $$

(B-3)

$$ {K}_i={l}_i{k}_j\kern2em j=\mathrm{mid}\ \mathrm{and}\ \mathrm{end} $$

(B-4)

where G, υ, l_i, and K_i are the shear modulus, the Poisson’s ratio, effective length, and the stiffness of the foundation, respectively.

• 3. The movement of the foundation in the horizontal direction is assumed to be tight.

• 4. The tensile strength of the soil has been completely ignored.

Fig. 19

Vertical stiffness modeling for shallow bearing footings (FEMA 356 2000; ASCE 41-17 2017)