Seismic optimization design for uniform damage of reinforced concrete moment-resisting frames using consecutive modal pushover analysis

Abstract

In this study, a practical optimization procedure is developed for uniform damage design of reinforced concrete moment-resisting frames using consecutive modal pushover analysis based on the framework of performance-based earthquake engineering. Consecutive modal pushover, which can capture the higher-mode effect well, is employed to derive the inelastic seismic demands of structures subjected to considered seismic hazards. Furthermore, the optimization problems are formulated using the profile of inter-story drift ratios and component hinge rotations to redistribute the steel reinforcements from components with low damage to ones that experience high damage under the constant-cost constraint, until a state of uniform damage distribution prevails. By applying the proposed design procedure to two prototype frames with five and eight stories designed using the Chinese Seismic Design Code (GB 50011-2010), the efficiency of the method as well as the variation of steel reinforcements is demonstrated. The seismic performances of final solutions and prototype frames are compared through nonlinear dynamic analysis for scenarios of 22 ground motions matching well with the code-compliant spectrum. The results show that the proposed method allows for a significant reduction of maximum story drifts combined with more evenly distributed story drifts and component hinge rotations, indicating better seismic performance for the optimized structures.

Keywords

component hinge rotation consecutive modal pushover analysis inter-story drift ratio reinforced concrete moment frames structural optimization uniform damage design

Introduction

Conventional seismic design in earthquake-prone regions is performed to provide the structure with sufficient strength, stiffness, and ductility to resist seismic actions. The applied lateral inertial forces excited by strong ground motions are taken into account by equivalent lateral force (ELF) procedure (GB 50011-2010, 2010; ICC, 2006; ICBO, 1997). A basic issue of ELF procedure is to determine the height-wise distribution of lateral forces. Current code-conforming lateral force patterns are generally based on the dynamic behavior of elastic structural systems (Hart, 2000; Park, 2007). As a consequence, when structures are expected to experience inelastic deformation under severe earthquakes, the use of code-conforming lateral forces may not lead to optimum structural properties and accurate predictions of force and deformation demands, causing structures to respond in a rather undesirable and unpredictable manner (Chao et al., 2007; Moghaddam and Hajirasouliha, 2006).

The need for obtaining the optimum structural behavior has given rise to the development of many structural optimization design approaches. Due to the limit of huge computational requirements and difficulty in calculating the approximate expressions for optimization constraints, conventional optimization design methods generally can obtain the optimum solution of elastic structural systems under static loads (Hajirasouliha et al., 2012; Li et al., 2011). Using the optimization result of elastic structures could not achieve the desired nonlinear seismic responses (Chao et al., 2007). The recently formulated optimization design methods could take the structural nonlinear behaviors (e.g. the inter-story drift ratio (IDR) and component damage level) into the consideration. Ganzerli et al. (2000) implemented the performance-based constraints in terms of plastic rotations of the beams and columns into the design process and proposed the structural optimization design procedure considering the structural nonlinear response and structural cost. They took the uncertainties of structural period and earthquake excitation into account using the convex models. To establish the performance-acceptance criteria and parameters for seismic design, Esteva et al. (2002) developed a life-cycle optimization formulation and estimated the peak dynamic responses of the system by first-order second-moment probabilistic criteria. Liu et al. (2003) treated life-cycle cost as two separate objectives of initial material costs and lifetime seismic damage costs and adopted genetic algorithm to obtain the optimum solution of steel moment frames and then developed the multiobjective optimization procedure for performance-based seismic design (Liu et al., 2005). Fragiadakis and Papadrakakis (2008) proposed the optimum seismic methodology based on nonlinear response history analysis. They treated the optimum problems using a deterministic manner and reliability-based optimization approach, and a number of limit states from serviceability to collapse prevention were considered. Application to reinforced concrete (RC) frames illustrated that the proposed methodology can achieve a significant reduction of the direct construction cost and improved seismic performances. Zou et al. (2007) and Li et al. (2012) developed the optimization technique for minimizing the life-cycle cost using the nonlinear static pushover analysis.

It should be seen that the above-mentioned optimization methods do not take the deformation distribution into account. Current studies indicate that the structural deformation demand subjected to strong earthquakes is not uniformly distributed (Chao et al., 2007; Chopra, 2001). Therefore, the structural material is not fully exploited along the building height since some parts of the structure do not reach the allowable level of seismic capacity. If the material shifts from strong parts to weak parts, a status of uniform deformation can be eventually obtained (Moghaddam and Hajirasouliha, 2006). It is expected in such a condition, not only the material capacity is fully exploited, but also the seismic response can be potentially reduced since the energy-dissipating capacity of all stories is maximized. This is considered as the concept of uniform distribution of deformation or uniform damage design (Hajirasouliha et al., 2012; Leelataviwat et al., 2002; Moghaddam and Hajirasouliha, 2004, 2009; Takewaki, 1997), which is the basis of this study.

Due to the simplicity and efficiency, nonlinear static pushover analysis has been widely used to evaluate the seismic demands (ATC-58, 2012; FEMA 445, 2006; GB 50011-2010, 2010). Using nonlinear static pushover analysis as a optimization tool has been widely investigated (e.g. Ganzerli et al., 2000; Li et al., 2012; Liu et al., 2003, 2005; Zou and Chan, 2005; Zou et al., 2007) since it can predict the structural seismic demands with acceptable accuracy and easy operation, as well as can avoid the confusion of selecting suitable ground motion as the seismic input for structural nonlinear optimization because different ground motion inputs can result in different final optimum solutions. Nevertheless, there is no investigation on the inelastic structural optimization for uniform damage design. In order to bridge the gap, this article presented a nonlinear static pushover-based method in the framework of performance-based seismic design for obtaining the uniform damage of RC moment frame structures subjected to strong earthquakes. Optimization problems were formulated using consecutive modal pushover (CMP) procedure (Poursha et al., 2009), which could take into account higher-mode effects and estimate the seismic demands with high accuracy. Furthermore, the IDR demands and component rotations were considered into the design process. Two prototype RC frame structures with five and eight stories, designed using the Chinese Code (GB 50011-2010), were examined, and the effectiveness of the developed method was verified through extensive time history analyses.

CMP-based seismic optimization design

CMP analysis procedure

CMP procedure, initially developed by Poursha et al. (2009), utilizes the multi-stage and single-stage pushover analyses to estimate the seismic demands which are determined by enveloping the results of single-stage and multi-stage pushover analyses. The single-stage pushover analysis uses an invariant inverted triangular force distribution for medium-rise buildings and an invariant uniform load pattern for high-rise buildings, to mainly capture the first-mode response. In the multi-stage pushover analyses, the lateral force distribution using mode shapes for up to three modes is applied consecutively in stages in a single pushover analysis. The CMP has the advantage that the modal nonlinear interactions are explicitly modeled in a way that may cause different inelastic mechanism to form, and the capacity limits on demand are inherently represented in the analyses where the member forces are consistent with the capacity limits (Fragiadakis et al., 2014).

The required number of modes (i.e. the number of stages) in the CMP procedure depends on the fundamental period T of the structures. When the fundamental period is less than 2.2 s, only single-stage and two-stage pushover analyses will be performed, while single-stage, two-stage, and three-stage pushover analyses will be carried out when fundamental period T equals 2.2 s or higher. The step-by-step analytical procedure for the CMP procedure can be summarized as follows:

Conduct the modal analysis of elastic structural system to calculate the natural period of vibration T_n, and mode shapes $ϕ_{n}$ for the three modes. The mode shapes $ϕ_{n}$ are normalized by the roof component ( $ϕ_{rn} = 1$ ).

Calculate $s_{n}^{*} = m ϕ_{n}$ , where m is the mass matrix of the structure, and $s_{n}^{*}$ is the lateral force distribution for the nth stage of multi-stage pushover analysis.

The CMP procedure consists of independent analyses: single-stage and multi-stage pushover procedure. After applying the gravity loads, the following sub-steps will be performed:

(a) Conduct the single-stage pushover analysis using the inverted triangular load pattern or uniform load pattern, pre-determined according to the fundamental period of the building. Specifically, the total target displacement at the roof u_t will be determined based on the capacity curve of this single-stage analysis. N2 method is used to establish the target displacement. R_y − µ − T relation proposed by Vidic et al. (1994) was used to construct the inelastic spectra

\begin{matrix} S_{a} = \frac{S_{a}^{e} (T_{e}, ξ)}{R_{y}} \\ S_{d} = μ_{s} S_{a} {(\frac{T_{e}}{2 π})}^{2} \end{matrix}

(1)

where S_a and S_d are acceleration spectrum and the displacement spectrum, respectively; $S_{a}^{e} (T_{e}, ξ)$ is the elastic acceleration spectrum corresponding to the period T_e and a fixed viscous damping ratio ξ, µ_s is the displacement ductility factor and R_y is the strength reduction factor. Of course, other inelastic spectrum, such as the Newmark and Hall (1982) equation, can also be used to reduce the elastic spectrum.

(b) Conduct the two-stage pushover analysis. In the first stage, pushover analysis using the lateral force profile $s_{1}^{*} = m ϕ_{1}$ will be performed until the roof displacement reaches $u_{r 1} = α_{1} u_{t}$ , where α₁ is the first-mode effective modal participating mass ratio defined as follows

α_{n} = \frac{M_{n}^{*}}{M^{*}} = \frac{{(ϕ_{n}^{T} ml)}^{2} / ϕ_{n}^{T} m ϕ_{n}}{\sum_{j = 1}^{N} m_{j}}

(2)

where $M_{n}^{*}$ is the effective modal mass, $M^{*}$ is the total mass of the structure, m_j is the lumped mass of jth floor, l is the unit vector, and N is the number of stories.

Then, the second stage pushover analysis using lateral force distribution $s_{2}^{*} = m ϕ_{2}$ will be implemented consecutively until the roof displacement increment reaches $u_{r 2} = (1 - α_{1}) u_{t}$ . As a consequence, the structure is monotonically pushed to the state that the roof displacement equals u_t through two-stage pushover analysis. It should be noted that the initial state of second stage pushover analysis is same as the final state of the first stage pushover analysis.

(c) Conduct the three-stage pushover analysis. First, carry out the pushover analysis using the lateral forces $s_{1}^{*} = m ϕ_{1}$ until the roof displacement reaches $u_{r 1} = α_{1} u_{t}$ . Then, continue the pushover analysis using lateral forces $s_{2}^{*} = m ϕ_{2}$ until the incremental roof displacement reaches $u_{r 2} = α_{2} u_{t}$ , where α₂ is the second-mode effective modal participating mass ratio calculated by equation (2). Finally, perform the third stage pushover analysis using lateral forces $s_{3}^{*} = m ϕ_{3}$ , and the roof incremental displacement is set as $u_{r 3} = (1 - α_{1} - α_{2}) u_{t}$ .

4. Calculate the seismic demands of the structure R, by enveloping the desired peak response of interest, such as the floor displacements, story drifts, and component rotation

R = {\begin{matrix} \max (R_{1}, R_{2}) if T < 2.2 s \\ \max (R_{1}, R_{2}, R_{3}) if T ⩾ 2.2 s \end{matrix}

(3)

where R₁, R₂, and R₃ are the peak values of desired response for single-stage, two-stage, and three-stage pushover analyses, respectively.

Based on the above CMP procedure, the seismic demands can be easily obtained, which accounts for the load path using consecutive modal lateral forces in the order of modes to push the structure to the intended roof displacement.

Seismic optimization design for uniform story drift

In current code-based design systems (e.g. the Chinese Seismic Design Code, GB 50011-2010 (2010)), the structural seismic design is carried out based on the principle that structures should resist minor but frequently occurring earthquakes without damage, should maintain function with repairable damage when subjected to moderate earthquakes, and should not collapse when subjected to rare but severe earthquakes. As a consequence, the structures are expected to remain elastic in minor earthquakes and experience nonlinear deformation under moderate-to-severe earthquakes. Usually, the story drift demands under earthquake loads are not uniformly distributed along the height of the structures, which could not fully exploit the material behavior, as mentioned before. Therefore, it is necessary to optimize the structural design to achieve the uniform drift demands for maximizing the seismic performance.

Objective functions

To obtain the uniformly distributed story drift, the objective function can be

Minimize : f = \frac{co v_{IDR}}{co v_{IDR, Original}}

(4)

where $co v_{IDR}$ is the coefficient of variation (COV) of IDR for the proceeded optimized structures, and the $co v_{IDR, Original}$ is the COV of IDR for original structure. It should be noted that the COV usually refers to a statistic term, and employing it here is to present the degree of uniform IDR for different stories during the optimization process and it will reduce to zero when structures have same story drifts. IDR is selected mainly due to its direct relation to the story local deformation which is deemed as an indicator for the potential collapse mechanism. For structural design purpose, the induced story drift ratio should be uniformly distributed along the height, rather than concentrated at one or some specified levels. Reasonably, the structures are expected to experience less damage if the structures exhibit near-uniform drift under earthquake shakings, and the seismic performance is enhanced accordingly.

Optimization variables

The seismic performance of RC moment frames is mainly influenced by the geometry configuration, material properties (including steel reinforcements and concrete), component sizes, and section reinforcements. Usually, the geometry configuration is pre-determined based on the building occupancy and space constraints, while the material properties are selected based on the availability. Therefore, only the component sizes and section reinforcements will be determined to achieve the desired seismic performance.

The height-wise distribution of component sizes plays a significant role in providing the elastic lateral stiffness, and therefore, the elastic drift demands are mainly controlled by the sectional sizes. Although the component sizes can affect the initial material cost of RC buildings, however, in design practice, the dimensions of beams and columns are originally determined mainly to satisfy the building requirements, such as the elastic drift limit, serviceability limit states for beams, axial compression ratio for columns, and so on. Of course, the member sizes can also be optimized based on the code-specified elastic story drift ratio limit to minimize the structural initial cost (Chan and Zou, 2004). While in the inelastic range, both the component sizes and section reinforcements have effects on the inelastic seismic response. Since the component sizes have originally been determined to meet the elastic drift demand and serviceability limit states, only the sectional reinforcements are assumed to be the design variables to control the inelastic drift demands in this study. Because the symmetrical column sections are used, the cross-sectional area of the steel in columns is the identical design variable. While for beams, the top flexure reinforcements and bottom reinforcements are independently determined to resist the maximum negative moments and positive moments due to gravity and seismic loadings. For simplicity in the optimizing process, the bottom reinforcements are assumed to be proportional to the top reinforcements and the ratio is approximately set as 0.5, which reduces the design variables to achieve the single design variable for a specific beam component. In addition, two other assumptions must be made: (1) adequate shear confinement reinforcements are provided for each component, which are roughly proportional to the amount of flexural reinforcements (Hajirasouliha et al., 2012); (2) the beam–column joints are rigidly connected and no failure for joints will occur. These two assumptions are the criteria of “strong-joints weak-components” and “strong-shear weak-flexure” in structural design philosophy.

Constraint conditions

The component longitudinal reinforcements must satisfy the minimum and maximum reinforcement constraints

\begin{matrix} ρ_{col mim} ⩽ ρ_{col} ⩽ ρ_{col max} \\ ρ_{bt min} ⩽ ρ_{bt} ⩽ ρ_{bt max} \\ ρ_{bb min} ⩽ ρ_{bb} ⩽ ρ_{bb max} \end{matrix}

(5)

where ρ_col, ρ_bt, and ρ_bb are the reinforcement ratios for a column, a beam top, and beam bottom, respectively; ρ_{col min}, ρ_{bt min}, and ρ_{bb min} are the corresponding minimum requirement specified in code provisions; and ρ_{col max}, ρ_{bt max}, and ρ_{bb max} are the maximum allowable values. The minimum and maximum reinforcement requirements are mainly used to avoid the brittle failure due to the absence of enough flexure reinforcements and concrete crushing without steel yielding, respectively. In this study, the specific value of the minimum and maximum values can be obtained from the GB 50011-2010 (2010).

To achieve the enhanced seismic performance under the design level earthquake, the structural initial cost, mainly due to the material cost, is set as unchanged in the optimizing process. Since the component section dimensions are set as the fixed value, the total reinforcement will keep unchanged to achieve the same material cost before and after optimization

V_{i} = V_{0}

(6)

where V₀ is the initial reinforcement volume and V_i is the volume for the ith iteration. To achieve this, the total reinforcements of columns and beams after each iteration will be scaled to the original condition, using a same scaling factor λ for all columns and beams where λ is the ratio of total reinforcement volume before optimization to that of after optimization.

Optimization design procedure

Optimization procedure

In the study, the seismic demands are obtained using CMP analysis. The engineering demand parameters under the design earthquake level, such as the IDR and peak component rotation, are captured. Usually, the story drift ratio distribution is not uniform. To achieve the uniformly distributed drift ratio, the story reinforcements will be shifted from stories with lower story drift to the stories which experienced higher story drift by adopting the following equation

[(A_{drift})_{i}]_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \cdot [(A_{drift})_{i}]_{k}

(7)

where k denotes the kth iteration; (A_drift)_i is the drift-related story reinforcements of ith story; Δ_i and Δ_mean are the story drift ratio of ith story and the mean value of story drift ratios, respectively; α is the converging parameter and 0 < α < 1. It should be noted that equation (7) is the expression of so-called optimality criteria method in structural optimization design (William and Keith, 2009). Since the lateral drift for a specific story is not only related to the properties of columns and beams of this story, for example, the ith story, but also related to the columns of the adjacent stories, that is, the (i − 1)th story and the (i + 1)th story, and the beams beneath this story, that is, the (i − 1)th story. Of course, the properties of other stories will also have influences to the drift of this story. Note that the influences from other stories are small and therefore are neglected. Meanwhile, at this specific story, the component local deformation measures, such as the rotation, plastic rotation, damage index, and so on, usually are not the same amount, which potentially results in excessive local damage for some components to develop the local failure mode. In order to develop the uniformly distributed story drift along the height and component local damage along the span, the following equations are used to derive the beam and column new reinforcements

\begin{matrix} {[(A_{scol, 1})_{i + 1}^{j}]}_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \cdot {(\frac{(θ_{col})_{i + 1}^{j}}{{(\bar{θ_{col}})}_{i + 1}})}^{γ} \\ \cdot {[(A_{scol, 1})_{i + 1}^{j}]}_{k} \\ {[(A_{scol, 2})_{i}^{j}]}_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \cdot {(\frac{(θ_{col})_{i}^{j}}{{(\bar{θ_{col}})}_{i}})}^{γ} \cdot {[(A_{scol, 2})_{i}^{j}]}_{k} \\ {[(A_{scol, 3})_{i - 1}^{j}]}_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \cdot {(\frac{(θ_{col})_{i - 1}^{j}}{{(\bar{θ_{col}})}_{i - 1}})}^{γ} \\ \cdot {[(A_{scol, 3})_{i - 1}^{j}]}_{k} \end{matrix}

(8)

\begin{matrix} {[(A_{sbeam, 1})_{i}^{j}]}_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \cdot {(\frac{(θ_{beam})_{i}^{j}}{{(\bar{θ_{beam}})}_{i}})}^{γ} \cdot {[(A_{sbeam, 1})_{i}^{j}]}_{k} \\ {[(A_{sbeam, 2})_{i - 1}^{j}]}_{k + 1} = {(\frac{Δ_{i}}{Δ_{mean}})}^{α} \\ \cdot {(\frac{(θ_{beam})_{i - 1}^{j}}{{(\bar{θ_{beam}})}_{i - 1}})}^{γ} \cdot {[(A_{sbeam, 2})_{i - 1}^{j}]}_{k} \end{matrix}

(9)

where $(A_{scol, 1})_{i + 1}^{j}$ , $(A_{scol, 2})_{i}^{j}$ , and $(A_{scol, 3})_{i - 1}^{j}$ are the column reinforcements of the jth column along the span for the (i + 1)th, ith, and (i + 1)th story, respectively; $(θ_{col})_{i + 1}^{j}$ , $(θ_{col})_{i}^{j}$ , and $(θ_{col})_{i - 1}^{j}$ are the component maximum rotation demand of the jth column for the (i + 1)th, ith, and (i + 1)th story, respectively; $(\bar{θ_{col}})_{i + 1}$ , $(\bar{θ_{col}})_{i + 1}$ , and $(\bar{θ_{col}})_{i + 1}$ are the mean rotation of all columns for the (i + 1)th, ith, and (i + 1)th story, respectively; $(A_{sbeam, 1})_{i}^{j}$ and $(A_{sbeam, 1})_{i - 1}^{j}$ are beam top reinforcements of the jth beam for the ith and (i − 1)th story, respectively; $(θ_{beam})_{i}^{j}$ and $(θ_{beam})_{i - 1}^{j}$ are the beam maximum rotation of jth beam for the ith and (i − 1)th story, respectively; $(\bar{θ_{beam}})_{i}$ and $(\bar{θ_{beam}})_{i - 1}$ are the mean rotation of all beams for the ith and (i − 1)th story, respectively; γ is the converging parameter and 0 < γ < 1. Note that in equation (8), the subscripts “1,”“2,” and “3” denote the new column reinforcements derived by the drift demands of adjacent lower story, this story, and the adjacent upper story, respectively, while in equation (9) the subscripts “1” and “2” denote the new beam reinforcements derived by the drift demands of this story and the adjacent upper story, respectively. It should be noted that in this study, the component rotation is selected as the deformation measure because it not only can reflect the component deformation level, but also be a continuous deformation measure from elastic condition to inelastic condition which make the new reinforcement derivation (equations (8) and (9)) can be realized when beams or columns remain in elastic range.

Based on the above derivation, the column new reinforcements for a specific story are determined by three parts: this story’s drift-derived reinforcements and those derived by the two adjacent stories, while the beams are controlled by two parts of reinforcements: this story and the upper story. As a consequence, the new beam reinforcements can be formulated as follows

\begin{matrix} [(A_{sbeam})_{i}^{j}]_{k + 1} = (1 - β) \cdot [(A_{sbeam, 1})_{i}^{j}]_{k + 1} + β \\ \cdot [(A_{sbeam, 2})_{i}^{j}]_{k + 1} \end{matrix}

(10)

\begin{matrix} [(A_{scol})_{i}^{j}]_{k + 1} = ω_{i}^{1} [(A_{scol, 1})_{i}^{j}]_{k + 1} + ω_{i}^{2} [(A_{scol, 2})_{i}^{j}]_{k + 1} \\ + ω_{i}^{3} [(A_{scol, 3})_{i}^{j}]_{k + 1} \end{matrix}

(11)

where $(A_{sbeam})_{i}^{j}$ and $(A_{scol})_{i}^{j}$ are the new beam and column reinforcements for the jth component for the ith story, respectively; β, $ω_{i}^{1}$ , $ω_{i}^{2}$ , and $ω_{i}^{3}$ are the reinforcement contribution factors. In this study, β = 0.5 for all stories except the roof beams and β = 0 for roof beams. It must be pointed out that the selection of β = 0.5 is to consider the beams of this story and the upper story has approximately the same contributions to the drift demand of this story. For columns, the column stiffness are employed to construct the drift contribution, and $ω_{i}^{1} = k_{i - 1}^{1} / (k_{i - 1}^{1} + 2 k_{i}^{2} + k_{i + 1}^{3})$ , $ω_{i}^{2} = 2 k_{i}^{2} / (k_{i - 1}^{1} + 2 k_{i}^{2} + k_{i + 1}^{3})$ , and $ω_{i + 1}^{3} = k_{i - 1}^{3} / (k_{i - 1}^{1} + 2 k_{i}^{2} + k_{i + 1}^{3})$ where $k_{i - 1}^{1}$ , $k_{i}^{2}$ , and $k_{i + 1}^{3}$ are the summation of linear stiffness of all columns for the(i − 1)th, ith, and (i + 1)th story, respectively. Note that for the first story $k_{i - 1}^{1} = 0$ and for the roof story $k_{i + 1}^{3} = 0$ .

Factored gravity load check

Apart from the reinforcement constraints shown in equation (5), beams must keep elastic condition under factored gravity loads. Therefore, the structural analytical model must be established by applying only the factored gravity load and the stress–strain state of beam reinforcement will be monitored. When a beam yields under the factored gravity, the beam reinforcements will be updated based on the axial strain of the steel reinforcements using the following equation

{[(A_{sbeam})_{i}^{j}]}_{1} = \frac{(ε_{steel})_{i}^{j}}{ε_{y}} \cdot {[(A_{sbeam})_{i}^{j}]}_{0}

(12)

where $ε_{y}$ and $(ε_{steel})_{i}^{j}$ are the yield strain and strain of beam reinforcements, respectively; ${[(A_{sbeam})_{i}^{j}]}_{0}$ and ${[(A_{sbeam})_{i}^{j}]}_{1}$ are the beam reinforcements before and after factored gravity check for yielded beams, respectively.

Design flowchart

The flowchart of the proposed optimization design is shown in Figure 1. When the target function reaches the minimum within 30 steps, the whole iterated process achieves the stop criteria, while if not, the procedure will terminate at 30 steps. It should be noted that the present optimization design procedure has similar properties with other published nonlinear optimization procedures in obtaining the nonlinear seismic demands and constructing the new component section information, and many available software tools can be used to perform the optimization design since only the distribution of IDR and component rotation demands are required.

Figure 1.

Flowchart of the proposed optimization design.

Illustrated examples

To illustrate the effectiveness of the CMP-based optimization design procedure to achieve the uniform drift demands, two prototype RC moment-resisting frames, with five story and eight story (as shown in Figure 2), were examined. The structural elevation view and component configurations are also shown in Figure 2. These two buildings were designed using the Chinese Seismic Design Code GB 50011-2010. GB50011-2010, similar to UBC-97 and Euro code, performs the structural design using the equivalent static force procedure, where a structure is designed for strength and checked for drift. Also, the capacity design method is employed to amplify the column capacity to achieve the desired “strong-column weak-beam” mechanism. The section reinforcements of components are derived from the sectional internal forces and must be consistent with the maximum/minimum reinforcement requirement.

Figure 2.

Structural elevation and beam and column configurations of five- and eight-story RC frames (cm).

The five- and eight-story buildings have three bays (3@5.0 m) and four bays (4@6.0 m), respectively. HRB400 and C30 were selected as the concrete and reinforcement material, respectively, for both structures. The floor (roof) dead load and live load were taken as 6.0 and 2.0 N/mm², respectively, for both structures. The two structures were assumed to locate at the region with seismic intensity of 8 and characteristic period of 0.35 s. It should be pointed out that the two structures are employed to validate the efficiency and effectiveness of the proposed optimization procedure for structures with different stories and bays. Furthermore, the main purpose of the proposed optimization design procedure is to make nonlinear seismic demands of code-conforming structures uniformly distributed under severe earthquakes. Therefore, the code-based design results of the two structures are taken as the initial optimization variables.

OpenSees (2013) platform was used to develop the analytical model of the structures. Columns and beams were modeled using beamWithHinges elements where the nonlinearity was modeled using lumped plastic hinges at the ends with the middle region modeled with linear-elastic properties. Fiber section model was employed to model the nonlinear behavior of the hinges. Uniaxial materials, Concrete01 and Steel02, were used to model the concrete and steel reinforcement, respectively. To account for the component stiffness reduction due to concrete cracking, reduction factors of 0.5 and 0.7 were selected to model the elastic portion of beams and columns, respectively (FEMA 356, 2000). In addition, P-Delta effects were considered in all nonlinear static and dynamic analyses. Soil–structure interaction (SSI) effects were not considered. Rayleigh damping of 5% was assigned at the first and third mode periods.

To validate the seismic performance of structures before and after optimization, 22 ground motion records were selected from the PEER Next Generation Attenuation (NGA) database (PEER, 2005). The detailed information of the ground motions can be available from Bai (2015). The ground motions were selected such that the mean spectra of the selected ground motions matched the code design spectrum for the seismic hazard level within the period range of 0.1–6 s with the same weight factors for each period. Figure 3 shows the median of the scaled spectra match reasonably well with the target code spectrum in the whole period range.

Figure 3.

Comparison between the selected spectra and target code spectrum (5% damped).

Results and discussion

To investigate the influence of converging parameters α and γ (presented in equations (8) and (9)) on the convergence of final near optimal results, sensitive analyses were carried out by changing α and γ from 0.1, 0.2, 0.5 to 1.0 for more efficient seismic design of the five- and eight-story frames. Figure 4 compares the variation of target function from code-based design to final solution for different α and γ values. As can be seen, the convergence speed was rapid during the first few steps for most cases, and the convergence speed increased with the increase in α and γ from 0.1 to 0.5 without any fluctuation for both structures. However, for α and γ equaled 1.0, unstable solution was observed for the five-story frame.

Figure 4.

Influence of different converging parameters α and γ on the final solution: (a) five-story frame and (b) eight-story frame.

Nevertheless, different values of α and γ can result in nearly the same final solution. For considerations of stability and convergence speed, it can be concluded that using the converging parameters α and γ from 0.2 to 0.5 can achieve the acceptable convergence solution for the proposed CMP-based seismic optimization procedure for uniform drift. For the present two frames, the selection of α and γ equaling 0.2 was used. It should be noted that α and γ are theoretically independent values ranging from 0 to 1, and their near optimum solutions can be obtained through a large number of parametric analyses. However, the near optimum solutions are not fixed values for different structural systems and for the simplification and easy operation, α and γ are set as the same values in this study.

To investigate the change of IDR using the proposed optimization method, Figure 5 shows the comparison of story drifts for conventional code-based design and optimized design for the five- and eight-story structures using the CMP method. As demonstrated in the figure, both optimized frames experienced more uniform story drift. The COVs of story drifts of the five-story frame were 26.5% and 7.6% for original code-based design and optimized design, respectively, while the corresponding values of eight-story frame were 33.2% and 10.8%, respectively. Meanwhile, the optimization design procedure can achieve the maximum story drift reduction, with 6.3% and 2.5% for five- and eight-story frames, respectively. Although the maximum drift reduction was slight, as will be shown later, the design procedure can effectively reduce the maximum story drift ratio under the real ground motions.

Figure 5.

Story drift ratio for conventional code-based design and near optimum design using CMP method: (a) five-story frame and (b) eight-story frame.

Figure 6 illustrates the component reinforcement variation of five-story frame during the optimization process. As evidenced in Figure 6(a), the structural total reinforcements remained unchanged, while the total beam reinforcement and total column reinforcements also stayed almost unchanged. This indicated the reinforcements of beams shifted mainly from beams experienced low damage to beams with high damage, which was the same for columns. Figure 6(b) and (c) shows the variation of column reinforcements and beam reinforcements for each story, respectively. For columns, the reinforcements shifted from first, fourth, and fifth stories to second and third stories (Figure 6(b)). Beam reinforcements of first and fourth story shifted to second and third story, while the beam reinforcements of fifth story kept unchanged since they are controlled by the factored gravity loads. Similar results can be obtained for the eight-story building. The steel reinforcement ratio of beams and columns of five- and eight-story structures is summarized in Tables 1 and 2, respectively, for the conventional code-based design and proposed optimized design. It must be pointed out that in Table 1 although the reinforcement ratio of some optimized columns is less than 1%, these results are consistent with the minimum acceptable values prescribed in the code (GB50011-2010).

Figure 6.

Variation of component reinforcements from code-based design to final design for five-story frame: (a) all beams, all columns, and total reinforcements, (b) column reinforcements, and (c) beam reinforcement.

Table 1.

Comparison of longitudinal steel reinforcement ratio for five-story code-based design and near optimum design.

Story level	Code-based design (GB50011-2010)				Near optimum design
	Columns (%)		Beams (%)		Columns (%)		Beams (%)
	Side columns	Inner columns	Outer bays	Central bay	Side columns	Inner columns	Outer bays	Central bay
1	2.55	2.35	1.96	1.71	2.00	1.90	1.85	1.53
2	1.12	1.73	2.09	1.85	0.94	2.96	2.47	2.21
3	1.74	2.48	1.95	1.76	1.04	4.40	2.10	1.71
4	2.38	2.96	1.80	1.61	1.16	3.34	1.43	1.29
5	1.96	2.20	1.06	1.02	0.92	2.48	1.09	0.96

Table 2.

Comparison of longitudinal steel reinforcement ratio for eight-story code-based design and near optimum design.

Story level	Code-based design(GB50011-2010)					Near optimum design
	Columns (%)			Beams (%)		Columns (%)			Beams (%)
	Side columns(axis:①/⑤)	Inner columns(axis:②/④)	Middlecolumns(axis:③)	Outerbays	Centralbays	Side columns(axis:①/⑤)	Innercolumns(axis:②/④)	Middlecolumns(axis:③)	Outerbays	Centralbays
1	2.47	2.16	2.16	1.53	1.44	1.11	1.10	1.10	0.98	0.97
2	1.42	1.42	1.42	1.85	1.75	1.44	1.39	1.16	2.51	2.36
3	1.43	1.63	1.61	1.80	1.70	1.10	3.02	2.86	2.14	2.04
4	1.43	1.63	1.61	1.67	1.58	1.12	2.44	2.41	1.49	1.46
5	1.42	2.03	2.03	1.75	1.70	1.10	2.62	2.66	1.82	1.84
6	1.42	1.90	2.01	1.49	1.45	1.10	2.82	2.78	1.77	1.65
7	1.81	2.18	2.28	1.17	1.20	1.11	2.05	2.22	0.83	0.87
8	1.53	1.56	1.54	0.88	0.87	1.11	1.18	1.28	0.81	0.80

To validate the efficiency of the proposed design procedure, nonlinear time history analyses using the selected 22 ground motions were carried out for the structures before and after optimization. The story drift ratio distribution of the two structures is shown in Figures 7 and 8, where the median value and median plus standard deviation (Stdev) are also plotted. The results indicated that the structures designed using the optimization procedure can achieve more uniform drift demands, as well as lower maximum story drift ratio. For the five-story building (Figure 7), the COVs of maximum median story drift ratio were 22.5% and 6.4% for code-based design and optimization design, respectively, while the corresponding maximum median story drift ratios were 1.48% and 1.31%, respectively, with a reduction of 11.7%. When it comes to the eight-story frame (Figure 8), the optimization design made the maximum median story drift reducing 17.2%, from 1.55% for code-based design to 1.28% for proposed design, and resulted in the COV of maximum median story drifts reduced from 30.7% to 12.8%. These indicate the proposed optimization design not only can realize the uniform damage distribution, but also can reduce the structural maximum drift demands effectively, which agreed with the concept of uniform damage demands emphasized in this study.

Figure 7.

Comparison of story drift ratio of five-story frame for code-based and optimization design under 22 ground motions:(a) code-based design, (b) optimization design, and (c) comparison of median value.

Figure 8.

Comparison of story drift ratio of eight-story frame for code-based and optimization design under 22 ground motions: (a) code-based design, (b) optimization design, and (c) comparison of median value.

In order to achieve a quantitative perception of component local damage, the column and beam rotation were recorded and compared. The distribution of median beams’ maximum rotations and columns’ rotation is shown in Figures 9 and 10 for five- and eight-story frame, respectively. As can be seen from both figures, the optimization design procedure leaded to more uniformly distributed beam rotation demand for structures with different stories and the beams of optimized structures experienced less rotations at their ends. In addition, the beam rotations of all bays for each story were uniformly distributed. Note that the columns’ rotation were in relatively low level, which indicated columns remained in elastic condition or near elastic condition. This implied the “strong column-weak beam” design criterion can be guaranteed. In particular, the optimization design structures can achieve somewhat smaller column rotation than code-based design structures, for columns except the base columns. It should be noted that the optimized structures can achieve the potential uniform drift demands for all stories, which was the product of global yield mechanism where all beams and column bases yield to dissipate the seismic energy. According to the global yield mechanism, the column bases have the same rotations with the beam ends, which can explain why the column bases of optimized structures have larger rotations than code-based design results, but comparative values with those of beam ends.

Figure 9.

Comparison of median beam and column maximum rotation distribution for five-story frame: (a) columns’ rotation and (b) beams’ rotation.

Figure 10.

Comparison of median beam and column maximum rotation distribution for eight-story frame: (a) columns’ rotation and (b) beams’ rotation.

The conventional code-based structural seismic design in earthquake-prone zones is performed using the applied lateral seismic loads which play an important role in determining the strength and stiffness distribution. As a consequence, the inelastic seismic response is highly dependent on the applied lateral force patterns. Figure 11 compared the normalized story shear of code-based design and optimized design with the code-defined patterns: UBC pattern (International Conference of Building Officials (ICBO), 1997), GB 50011-2010 (2010) pattern, and IBC pattern (International Code Council (ICC), 2006). The results showed that there was a discrepancy between the shear forces of code-based structures and the forces determined by code definition. This indicated using the code-defined force pattern to design structures cannot obtain the code-conforming story shear demands. This is due to fact that code-defined patterns established mainly based on the dynamic behavior of elastic structural systems could not predict the inelastic structural response, for example, the story shears (Chao et al., 2007; Hajirasouliha and Moghaddam, 2009; Park, 2007). The results also implied the story shears of optimized structures were not consistent with the code-defined patterns. It is necessary to point out the objective of this section is to demonstrate the discrepancy between applied lateral force profiles and the earthquake-induced story shears, to emphasize the need to develop new design lateral forces considering the inelastic state of structures (Chao et al., 2007) and uniform distribution of deformation (Hajirasouliha and Moghaddam, 2009), and assess the seismic performance of structures designed by the new load patterns and update the design load patterns based on their seismic demands. Although the present optimization approach can achieve the uniform damage of structures using the iterative methods, if a new advanced design lateral force patterns have been developed, no iterations are required to realize the uniform damage, and this will be our future work.

Figure 11.

Comparison of normalized story shear with code lateral load patterns: (a) five stories and (b) eight stories.

It should be noted that in this study, the two case structures could represent low- and medium-rise structures, and higher structures were not investigated. Basically, if the nonlinear static pushover analysis can precisely predict the structural seismic demands, the proposed design procedure can be extended to high-rise buildings. However, due to the inherent shortcomings of pushover analysis, the extension of the proposed design procedure to high-rise buildings should be more investigative.

Conclusion

An efficient performance-based optimization technique, using the CMP analysis to estimate the inelastic seismic demands of RC moment frames under the considered seismic hazard, has been developed in this article, which provides an effective way to realize the uniform damage design. Some conclusions can be drawn from the analytical results:

The concept of uniform damage distribution can be employed as the target design under the constraints of constant cost to maximize the seismic performance of structures subjected to gravity load and seismic actions.

The story drift distribution and component rotation can be used as the damage measures to construct the optimization procedure to redistribute the steel reinforcements from components with low damage to ones that experience high damage.

Extensive nonlinear dynamic analyses of the illustrated examples of five- and eight-story RC frames show that the developed optimization procedure can achieve the maximum story drift reduction with 11.7% and 17.2%, respectively, as well as more uniform IDR distribution. Moreover, the column and beam rotation demands reveal the optimized structures tend to form the potential global yield mechanism to more effectively dissipate the seismic energy.

It is necessary to develop new advanced design lateral force patterns to realize the uniform damage design to avoid any iterations. Nevertheless, the presented optimization approach provides a good basis for uniform damage design of RC frame structures.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 51261120376), Project No. 0903005203376 supported by the Fundamental Research Funds for the Central Universities and Scholarship Award for Excellent Doctoral Student granted by Ministry of Education. The financial support is gratefully acknowledged.

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