A multi-objective design method for seismic retrofitting of existing reinforced concrete frames using pin-supported rocking walls

Abstract

Over the past several decades, a variety of technical ways have been developed in seismic retrofitting of existing reinforced concrete frames (RFs). Among them, pin-supported rocking walls (PWs) have received much attentions to researchers recently. However, it is still a challenge that how to determine the stiffness demand of PWs and assign the value of the drift concentration factor ( DCF) for entire systems rationally and efficiently. In this paper, a design method has been exploited for seismic retrofitting of existing RFs using PWs (RF-PWs) via a multi-objective evolutionary algorithm. Then, the method has been investigated and verified through a practical project. Finally, a parametric analysis was executed to exhibit the strengths and working mechanism of the multi-objective design method. To sum up, the findings of this investigation show that the method furnished in this paper is feasible, functional and can provide adequate information for determining the stiffness demand and the value of the DCFfor PWs. Furthermore, it can be applied for the preliminary design of these kinds of structures.

Graphical abstract

Keywords

pin-supported rocking wall / reinforced concrete frame / seismic retrofit / stiffness demand / drift concentration factor / multi-objective design / genetic algorithm / Pareto optimal solution

1 Introduction

In the last few decades, reinforced concrete frames (RFs) have been utilized widely in seismic areas all over the world. It is well-known that an RF should possess enough energy dissipation capacity and ductility in order to reduce seismic responses [1,2]. Consequently, it often causes large residual drifts and momentous structural damage under severe seismic excitations. On account of this, severe economic losses will be incurred and rebuilding or repairing necessities will be needed after a serious earthquake. Furthermore, the idea of “Strong Column Weak Beam” has been adopted in leading seismic design codes in many countries for several decades [3]. However, weak-story failures and inter-story drift concentrations have emerged in some high-rise frames designed on the basis of the existing codes. Besides, many existing RFs have structural deficiencies owing to the inadequacy of old design provisions, e.g., some regions with high seismic risks now were not diagnosed as they are or their levels of seismic hazards were underestimated before the upgraded codes [4].

In the light of the reasons depicted above, the seismic retrofitting of existing RFs is considered to be increasingly drawing the attention in earthquake engineering field currently. Numerous techniques have been developed for seismic retrofitting of existing RFs and can be employed in combination or alternatively to improve the seismic performance of structures, such as: Buckling Restrained Braces (BRBs) [5–9], Fiber-Reinforced Polymer (FRP) [4,10], Base Isolation [11–17], Precast concrete spreader-wall [18], Dissipative steel exoskeletons [19] and reinforced concrete rocking walls [1,20–24], etc.

Among them, RC rocking walls have been exploited to mitigate the seismic vulnerability of existing structures in the last decade [21,25]. Generally speaking, reinforced concrete rocking walls can be classified into two categories, viz. pin-supported and stepping rocking walls (see Fig.1). Their major difference is that the former rocks around only one fulcrum, while the latter has fulcrums on both sides of the bottom of the rocking wall [26]. In this paper, the pin-supported rocking walls (PWs) is concentrated on and investigated both conceptually and theoretically.

Moreover, the existing body of research on retrofitting of existing RFs via PWs suggests that the deformation pattern of RFs can be controlled effectively using PWs so as to avoid weak story failures and inter-story drift concentrations [1,27]. The effectiveness of PWs in reducing inter-story drift concentrations can be indicated viz. the drift concentration factor (DCF) [1,21], i.e., DCF can be employed to assess the uniformity of Inter-Drift Ratios (IDRs) of structures. The expression of DCF can be defined as Eq. (1).

(1)DCF=\frac{\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{d_i}{h_i}}}{\text{Δ}/\sum _i{h}_i},

where d_i, h_i, and ∆ mean the inter-story displacement, the story height, and the roof displacement of structures, respectively. Specifically, the closer to 1.0 the value of DCF is, the better deformation pattern of the structure is controlled. However, a uniform DCF would be an ideal case while it would demand an impractical and infinite PW. Furthermore, a major problem in applying DCF to reflect PW’s effect, which has been found in most previous research with regard to the stiffness demand, is difficult to assign a design target value, i.e., the determination of the DCF is blind and its physical meaning would be obscure. Therefore, it is worth studying that how to choose the appropriate DCF value in the light of the actual engineering requirements and structural performance demands.

In this study, a design method by a multi-objective evolutionary optimization (MEO) is explored for seismic retrofitting of existing RFs using PWs (RF-PWs). Specifically, based on genetic algorithm (GA) and the theories of RFs retrofitted with PWs, a group of Pareto Optimal Solutions (POSs) will be furnished, which not only meets the performance requirements of the retrofitted structures but also possesses a characteristic of diversity. Meantime, the physical meaning of DCFs and the performance of the retrofitted structures can be displayed clearly and intuitively. Afterwards, the best one can be selected by designers through comparing the differences of individuals in the POSs. The multi-objective design method may boost the preliminary design of this kind of structures.

2 Multi-objective evolutionary optimization via genetic algorithm

It is well-known that MEO methods have been the significant techniques to tackle practical problems in the last decade [28]. In engineering fields, most real-world problems get involved in optimizing multi-objectives simultaneously which have tradeoff relationship normally. In general, the commonly used form of the MEO problem can be stated as Eq. (2).

(2)\begin{array}{r}\text{Minimize/maximize}{F}_m(\mathit{\text{z}}),m=1,2,\dots ,M;\\ G_j(\mathit{\text{z}})⩾0,j=1,2,\dots ,J\text{;}\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}{H}_k(\mathit{\text{z}})=0,k=1,2,\dots ,K\text{;}\\ C_r(\mathit{\text{z}})⩽0,r=1,2,\dots ,R\text{;}\\ D_s(\mathit{\text{z}})=0,s=1,2,\dots ,S\text{;}\\ {\mathit{\text{z}}}_{(l)i}⩽{\mathit{\text{z}}}_i⩽{\mathit{\text{z}}}_{(u)i},i=1,2,\dots ,n.\end{array}\},

where z is the design variables’ vector: \mathit{\text{z}}=({\mathit{\text{z}}}_1,{\mathit{\text{z}}}_2,\dots ,{\mathit{\text{z}}}_n{)}^{\text{T}}; F_m(\mathit{\text{z}}) is the objective functions’ vector: F_m(\mathit{\text{z}})=(f_1(\mathit{\text{z}}),f_2(\mathit{\text{z}}),\dots ,f_M(\mathit{\text{z}}){)}^{\text{T}}; G_j(\mathit{\text{z}}) means the constraints’ vector of linear inequality: G_j(\mathit{\text{z}})={\left(g_1(\mathit{\text{z}}),g_2(\mathit{\text{z}}),\dots ,g_J(\mathit{\text{z}})\right)}^{\text{T}}; H_k(\mathit{\text{z}}) denotes the constraints’ vector of linear equality: H_k(\mathit{\text{z}})={\left(h_1(\mathit{\text{z}}),h_2(\mathit{\text{z}}),\dots ,h_K(\mathit{\text{z}})\right)}^{\text{T}}; C_r(\mathit{\text{z}}) stands for the constraints’ vector of nonlinear inequality: C_r(\mathit{\text{z}})=(c_1(\mathit{\text{z}}),c_2(\mathit{\text{z}}),\dots ,c_R(\mathit{\text{z}}){)}^{\text{T}}; D_s(\mathit{\text{z}}) symbolizes the constraints’ vector of nonlinear equality: D_s(\mathit{\text{z}})={\left(d_1(\mathit{\text{z}}),d_2(\mathit{\text{z}}),\dots ,d_S(\mathit{\text{z}})\right)}^{\text{T}}; {\mathit{\text{z}}}_{(l)i},{\mathit{\text{z}}}_{(u)i} represent the lower and upper bounds on {\mathit{\text{z}}}_i, respectively.

In this study, GA was suggested to address this problem [29,30]. Meantime, the schematic of the EMO using GA is presented in Fig.2.

Moreover, NSGA-II approach in which ranking is based sequentially on: (1) constraint satisfaction, (2) domination, and (3) crowding distance was utilized to address the multiobjective optimization [31,32]. The EMO using GA was executed through the GA Optimization Toolbox in Matlab [33]. Tab.1 shows the parameters adopted by EMO using GA in this study.

3 Parametric model for seismic retrofitting of existing reinforced concrete frames using pin-supported rocking walls

In this study, the model for RF-PWs is displayed in Fig.3. Fig.3(a)–3(e) represent the lateral load, RF, PW with a rotational spring, RF-PW and parametric model for RF-PWs, respectively. Specifically, the RF-PW is simulated as a combination of and one flexural beam one shear beam linked with rigid links at the location of the floors (see Fig.3(e)). The flexural beam and the shear beam symbolize the PW and the RF, respectively.

In Fig.3(e), p(x) is the lateral load, which is assigned along the structure’s height whereas p_{\text{F}}(x) represents the distributed internal force between the RF and the PW, where x means the height from the bottom to a certain position of the structure; H stands for the entire structure’s height; E_{\text{w}}{I}_{\text{w}} is the bending stiffness of the PW; K with a unit of Newtons denotes the shear distributed stiffness of the RF, which equals the shear force when a unit IDR is applied. K_{\text{r}} is the rotational spring’s stiffness on the bottom of the PW [26,34,35].

According to the Refs. [26,34], the differential equations and the corresponding solutions can be derived (see Tab.2 and Tab.3) under four different lateral load models. Accordingly, the bending moment M_{\text{w}} and shear force V_{\text{w}} of the flexural beam can be obtained as Eqs. (3) and (4), respectively.

(3)M_{\text{w}}=\frac{E_{\text{w}}{I}_{\text{w}}}{H^2}\frac{{\text{d}}^{\text{2}}y}{\text{d}{\xi }^2},

(4)V_{\text{w}}=-\frac{E_{\text{w}}{I}_{\text{w}}}{H^3}\frac{{\text{d}}^{3}y}{\text{d}{\xi }^3}.

4 Case study

4.1 Model of multi-objective evolutionary optimization via genetic algorithm

The problem of an RF-PW can be stated as follows:

One Design Variable:

Find

E_{\text{w}}{I}_{\text{w}}=E_{\text{w}}\cdot I_{\text{w}},

where E_{\text{w}}{I}_{\text{w}} is the bending stiffness of the cross-section of the PW; E_{\text{w}} is the elasticity modulus of concrete of the PW; I_{\text{w}} means the section moment of inertia of the PW, I_{\text{w}}={\displaystyle \frac{t_{\text{w}}{b}_{\text{w}}^3}{12}}; where t_{\text{w}} is the thickness of the PW (unit: m); b_{\text{w}}^{} is the width of the PW (unit: m).

Two Trade-off Objective Functions:

(1) \eta and (2) DC{F}^{\text{PW}},

where \eta ={\displaystyle \frac{1}{\lambda }}·\eta, the reciprocal of \lambda, is a nondimensional parameter representing the stiffness ratio between the flexural beam and shear beam; and DC{F}^{\text{PW}} is the DCF of the PW.

Constraints:

Subjected to:

(1) the range of the design variable, E_{\text{w}}{I}_{\text{w}}.

1.0\times {10}^7⩽E_{\text{w}}{I}_{\text{w}}⩽1.0\times {10}^9,

where in order to determine the range of the design variable, t_{\text{w}}=0.6 and are assumed, respectively.

(2) the range of the maximum IDR of the RF-PW, ID{R}_{max}^{\text{PW}}.

For the efficiency of convergence of the optimization program, the restraint of ID{R}_{max}^{\text{PW}} is set between 0 and ID{R}_{max}^{\text{RF}}, where the ID{R}_{max}^{\text{RF}} denotes the maximum IDR of the RF and equals 1/1289 in this case (see Tab.4).

(3) the range of the DCF of the RF-PW, DC{F}^{\text{PW}}.

where the DC{F}^{\text{RF}} denotes the DCF of the RF and equals 1.62 in this case.

4.2 Case employed in this study

Fig.4 presents an existing two-dimensional five-story RF [26]. In this case, the bending stiffness of the frame beams is assumed to be infinite; K=1.86\times {10}^6\text{kN}; the uniformly distributed load is adopted as the lateral load acting on the structure, i.e., p(\xi )=q, where q=100\text{kN}/\text{m}; K_r=0, i.e., no rotational spring is set. Furthermore, the basic information of the RF is given in Tab.5.

It is very easy to obtain the lateral displacement and IDRs of the RF by any kind of finite element software. The lateral displacements and ID{R}^{\text{RF}} of the RF are furnished in Tab.4 and Fig.5, respectively. As shown in Tab.4 and Fig.5, the maximum lateral displacement and ID{R}_{max}^{\text{RF}} of the structure are 7.17 mm and 1/1289, respectively. Furthermore, the DC{F}^{\text{RF}} of the structure equals 1.62 according to Eq. (1), where ID{R}_{max}^{\text{RF}} and DC{F}^{\text{RF}} represent the maximum IDR and the DCF of the RF, respectively.

5 Results analysis

According to Tab.1, the number of POSs will be 50 in this case, for Population size and Pareto front population fraction are 100 and 0.5, respectively. Tab.6 provides the 50 individuals of POSs at Generation 50. Fig.6 and Fig.7 show the bar diagrams of two objective functions, viz. \eta and DCF, of 50 POSs at Generation 50, respectively. From Tab.6 and Fig.6–Fig.7, we can see that 0.169⩽\eta ⩽1.514, and 1.026⩽DC{F}^{\text{PW}}⩽1.494. It is apparent that the values of DC{F}^{\text{PW}} are all smaller than the one of DC{F}^{\text{RF}}, viz. 1.62 (the red bar in right side of Fig.7), which means the effectiveness of the PW in reducing inter-story drift concentrations of the RF.

Fig.8 displays the scatter diagram between two objective functions, viz. \eta and DC{F}^{\text{PW}}, at Generation 50. Looking at Fig.8, it is clear that a trade-off correlation is demonstrated between \eta and DC{F}^{\text{PW}}, i.e., specifically, with the gradual increase of \eta, DC{F}^{\text{PW}} will continue to decrease, and vice versa. i.e., with the rise of E_{\text{w}}{I}_{\text{w}}, the deformation control effect of the PW on the RF will be increasingly obvious.

Fig.9 and Fig.10 present the 3D and plan diagrams of the quantity distribution of objective function values of 50 POSs at Generation 50. From Fig.9 and Fig.10, we can see that some values of the objective functions among POSs are very close, which can be considered as equal individuals. e.g., when , 0.31 and , , there are 6 and 4 similar POSs, respectively. In addition, as shown in Fig.9 and Fig.10, there are 18 kinds of POSs, which displays a relatively uniform scatter. Also, no local convergence appears. As a consequence, the solutions of multi-objective optimization have the characteristic of diversity in contrast to the only one of single objective optimization.

Fig.11 and Fig.12 display the scatter and bar diagrams of the distribution of the design variables, viz. the bending stiffness of the PW, E_{\text{w}}{I}_{\text{w}}, of 50 POSs at Generation 50, respectively. Fig.13 presents the diagram of the relationship between b_{\text{w}} and DC{F}^{\text{PW}}. From Fig.11 and Fig.12, we can see that the distribution of the design variables has good discreteness and diversity. From Fig.13, it is evident that there is an inverse correlation between b_{\text{w}} and DC{F}^{\text{PW}}. Moreover, the width of the PW, viz. b_{\text{w}} can be acquired when t_{\text{w}} is set as 0.6 m and E_{\text{w}}=E_{\text{f}}, respectively (see Fig.14). It can be seen from Tab.6 and Fig.14 that 2.0\text{m}⩽b_{\text{w}}⩽8.6\text{m}, which means there are a variety of options when designers try to determine the value of b_{\text{w}}.

On the other hand, it is apparent that the distribution of individuals in POSs is also a little biased from Fig.11. Specifically, more individuals cluster in the end with lower E_{\text{w}}{I}_{\text{w}} value. In other words, there are more individuals with lower b_{\text{w}} and higher DC{F}^{\text{PW}} in terms of Fig.14. The reason of this phenomenon may be linked with the constraints of the optimization problem, e.g., due to 1.0⩽DC{F}^{\text{PW}}⩽1.62, the closer to the lower E_{\text{w}}{I}_{\text{w}} value the individuals are (i.e., the DC{F}^{\text{PW}} nears to 1.0 and b_{\text{w}} approaches to 8.6 m), the less feasible the design schemes are. For as aforementioned, when the value of DCF closer to 1.0, the corresponding scheme will be impractical due to the demand of the infinite stiffness of the PW. In contrast, when DC{F}^{\text{PW}}=1.62, the scheme corresponds to the RF, which is an existing structure, i.e., the RF-PW with a value of DCF close to 1.62, it needs a lower E_{\text{w}}{I}_{\text{w}} and easy to be realized. From the perspective of probability, these kinds of solutions have a large probability of occurrence so as to cause the above-mentioned bias phenomenon.

\lambda has been utilized as a significant parameter in the previous studies on RF-PWs, which is the reciprocal of \eta as mentioned in Subsection 4.1. For the sake of comparison with the existing findings, the values and diagram of \lambda are provided in Tab.6 and Fig.15, respectively. As shown in Tab.6 and Fig.15, 0.66⩽\lambda ⩽5.905.

Fig.16 and Fig.17 illustrate the bar diagrams of the distribution of ID{R}_{max}^{\text{PW}} and \zetaof 50 POSs at Generation 50, respectively. Fig.18 provides the diagram of the relationship between \eta and \zeta. From Tab.6 and Fig.16, we can know that {\displaystyle \frac{1}{1978}}⩽ID{R}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\text{PW}}⩽{\displaystyle \frac{1}{1359}}. It is clear that ID{R}_{max}^{\text{PW}} are all less than ID{R}_{max}^{\text{RF}}={\displaystyle \frac{1}{1289}} which reflects the effect of the RF after retrofitted by the PW. Furthermore, from Tab.6 and Fig.16, 0.652⩽\zeta ⩽0.948, which means the ratios of ID{R}_{max}^{\text{PW}} over ID{R}_{max}^{\text{RF}} are from 65.2% to 94.8%, respectively.

Moreover, from Fig.18, it is apparent that the correlation between \eta and \zeta is inverse. That is to say, with the increase of the E_{\text{w}}{I}_{\text{w}}, the lateral global stiffness of the RF-PW increasingly ascends, so as to lead to the decrease of the ID{R}_{max}^{\text{PW}}.

Fig.19 presents the correlation between \zeta and ID{R}_{max}^{\text{PW}}. From Fig.19, it is apparent that \zetais linearly and positively proportional to ID{R}_{max}^{\text{PW}}, i.e., the larger \zeta of the structure is, the bigger the corresponding ID{R}_{max}^{\text{PW}} is, and vice versa. In addition, the values of \zeta and ID{R}_{max}^{\text{PW}} are distributed in a reasonable scope, which meets the optimization constraints, such as: and 1.0⩽DC{F}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\text{PW}}⩽1.62.

Fig.20 and Fig.21 demonstrate the bar diagrams of the distribution of V_{\text{wb}} and \delta of 50 POSs at Generation 50, respectively. Fig.22 presents the diagram of the relationship between \eta and \delta. From Tab.6 and Fig.20, we can know that 252.4\text{kN}⩽V_{\text{wb}}⩽723.2\text{kN}, which means the PW shares the lateral load with the RF. Also, From Tab.6 and Fig.21, the ratios of V_{\text{wb}} over the resultant force of the lateral load q\cdot H, viz. \delta are from 15.2% to 43.4%, respectively, which reveals the efficiency provided by the PW to the RF. Furthermore, according to Fig.22, it is clear that with the increase of \eta, \delta ascends gradually, and vice versa. Meanwhile, when \eta ⩽0.4, there is an approximately linear relationship between them. By contrast, when \eta >0.4, the growth rate of \delta begins to decline gradually and inclines to zero finally, which shows the maximum V_{\text{wb}} shared by the PW will not exceed 45% of the resultant force of the lateral load q\cdot H in this case.

Lastly, returning to the aim posed at the beginning of this paper, it is now possible to demonstrate how to apply this multi-objective design method. For instance, (1) if designers want to determine the scheme with the minimum ID{R}_{max}^{\text{PW}}, individual No. 3 would be the best choice on the basis of Tab.6. Meantime, other performance indexes relating to individual No. 3 can be found in Tab.6 as well; (2) Assuming according to the characteristics of structural plane layout, b_{\text{w}}=4.0\text{m} is suitable to it. As a result, individual No. 21 can be chosen as the best one in the light of Tab.6; (3) if a design target value of DC{F}^{\text{PW}} is assigned as 1.320, individual No. 42 would be the second-one choice. The corresponding performance indexes of the 3 chosen individuals can be found in Tab.7.

6 Conclusions

The following conclusions can be drawn from the present study.

1) A multi-objective design method has been developed for seismic retrofitting of existing RFs using PWs. Then, it has been investigated and verified via a real structure. The results demonstrated that this method is feasible, practical and can be employed for the preliminary design of these kinds of structures.

2) MEO is integrated to produce POSs with a certain number of structural performance indexes. After that, the most suitable one can be picked out based on the requirements of practical projects so that the blindness in determining the target value of DCF can be addressed.

3) Owing to POSs, designers can compare and weigh the pros and cons of them, a best one can be determined so that the idea of performance-based design can be realized through this method.

Limitation and future studies:

1) In this paper, the research about the proposed design procedure of PWs as retrofitting system is based on elastic theory. The nonlinear seismic analysis are necessary to investigate the actual behavior of the RF-PWs.

2) The effects of the high modes are getting obvious with the increase of structure height. Therefore, the effects of the high modes should be considered when analyzing the structural performance.

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