OPTIMAL SEISMIC DESIGN METHOD TO INDUCE THE BEAM-HINGING MECHANISM IN REINFORCED CONCRETE FRAMES

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1 10NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 Anchorage, Alaska OPTIMAL SEISMIC DESIGN METHOD TO INDUCE THE BEAM-HINGING MECHANISM IN REINFORCED CONCRETE FRAMES Se Woon Choi 1, Keunhyoung Park 2, Byung Kwan Oh 3, Yousok Kim 4 and Hyo Seon Park 5 ABSTRACT The strong-column weak-beam design criterion is widely used in seismic design procedures to prevent the abrupt collapse of reinforced-concrete (RC) frames and to secure their ductility capacity. However, many studies have demonstrated that the column-hinge collapse mechanism in RC frames can occur even if the strong-column weak-beam design criterion is satisfied. This study presents the automated optimal seismic method to induce the beam-hinging mechanism in RC frames. The proposed optimal seismic method uses on 1) the strengths of the beams and columns, 2) the column-to-beam flexural strength ratio, and 3) the prevention of plastic hinges of columns at the joints. Two objective functions are used to minimize the structural cost and maximize the energy dissipation capacity. The non-dominated genetic algorithm II (NSGA-II) is employed as an optimization tool. A linear static analysis method is employed to evaluate the on the strength of members, and a nonlinear static analysis method is employed to evaluate the energy dissipation capacity and the constraint on the plastic hinge. The multi-core based parallel analysis is adopted to accelerate the optimization process. Finally, a four-story RC moment frame is employed to verify the proposed method. 1 Assistant Professor, Department of Architecture, Catholic University of Daegu, 330, Keumnak-ri, Hayang-eup, Kyeongsan-si, Kyeongbuk , Korea 2 Master Student, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea 3 Ph.D. Student, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea 4 Research Professor, Center for Structural Health Care Technology in Buildings, Yonsei University, 134 Shinchon, Seoul , Korea 5 Professor, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea Choi SW, Park K, Oh BK, Kim Y, Park HS. Optimal seismic design method to induce the beam-hinging mechanism in reinforced concrete frames. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

2 Optimal Seismic Design Method To Induce The Beam-hinging Mechanism In Reinforced Concrete Frames Se Woon Choi 1, Keunhyoung Park 2, Byung Kwan Oh 3, Yousok Kim 4 and Hyo Seon Park 5 ABSTRACT The strong-column weak-beam design criterion is widely used in seismic design procedures to prevent the abrupt collapse of reinforced-concrete (RC) frames and to secure their ductility capacity. However, many studies have demonstrated that the column-hinge collapse mechanism in RC frames can occur even if the strong-column weak-beam design criterion is satisfied. This study presents the automated optimal seismic method to induce the beam-hinging mechanism in RC frames. The proposed optimal seismic method uses on 1) the strengths of the beams and columns, 2) the column-to-beam flexural strength ratio, and 3) the prevention of plastic hinges of columns at the joints. Two objective functions are used to minimize the structural cost and maximize the energy dissipation capacity. The non-dominated genetic algorithm II (NSGA-II) is employed as an optimization tool. A linear static analysis method is employed to evaluate the on the strength of members, and a nonlinear static analysis method is employed to evaluate the energy dissipation capacity and the constraint on the plastic hinge. The multi-core based parallel analysis is adopted to accelerate the optimization process. Finally, a four-story RC moment frame is employed to verify the proposed method. Introduction The devastation of the Northridge (1994) and Kobe (1995) earthquakes led to the development of better seismic engineering and technologies. Interest in and the importance of seismic engineering was further increased because of the significant damage wrought by the Izmit (1999) and Tohoku (2011) earthquakes. A moment frame is an earthquake-resisting structural system composed of beams and columns. This system can provide diversity in architectural planning when compared with other systems, and it is the most widely applied system among earthquake resistance structure systems because of its excellent ductility capacity. Thus, the development of seismic design technologies is important. The inter-story drift ratio is often used to evaluate the structural performance of a moment frame because this ratio can signify the extent of damage or structural deformation. 1 Assistant Professor, Department of Architecture, Catholic University of Daegu, 330, Keumnak-ri, Hayang-eup, Kyeongsan-si, Kyeongbuk , Korea 2 Master Student, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea 3 Ph.D. Student, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea 4 Research Professor, Center for Structural Health Care Technology in Buildings, Yonsei University, 134 Shinchon, Seoul , Korea 5 Professor, Department of Architectural Engineering, Yonsei University, 134 Shinchon, Seoul , Korea Choi SW, Park K, Oh BK, Kim Y, Park HS. Optimal seismic design method to induce the beam-hinging mechanism in reinforced concrete frames. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

3 Therefore, the inter-story drift ratios of moment frames must be reduced to minimize not only the human casualties and but also the economic losses caused by earthquakes. Based on this background, a range of studies on the seismic design method, which evenly distributes the interstory drift ratio by inducing a beam-hinge collapse mechanism, have actively been conducted [1,2]. Under the ACI building requirement, strong-column weak-beam behavior induces the beam-hinge collapse mechanism of moment-resisting frames. However, Dooley and Bracci (2001) [3] and Medina and Krawinkler (2005) [4] demonstrated that stricter conditions compared to the current design standard values are required to induce the beam-hinge collapse mechanism. The current study presents the automated optimal seismic method to induce the beamhinging mechanism in reinforced-concrete (RC) frames. In the proposed optimal seismic method, the on the member strengths of columns and beams, the column-to-beam flexural strength ratio, the prevention of plastic hinge formation of columns at joints are used. By constraining the plastic hinges of columns, the design solution having the beam-hinging mechanism can be obtained. Two objective functions are used to minimize the structural costs and maximize the energy dissipation capacity. The non-dominated genetic algorithm II (NSGA- II) [5] is employed as an optimization tool. A linear static analysis method is employed to evaluate the on the strength of members, and a nonlinear static analysis method is employed to evaluate the energy dissipation capacity and the constraint on the plastic hinge. The multi-core based parallel analysis is adopted to accelerate the optimization process. Finally, a four-story RC moment frame is employed to verify the proposed method. Objective Functions Formulating the Structural Optimization Problem Two objective functions are used in this study. The first objective function minimizes the structural cost of a RC moment-resisting frame. Although other factors, such as the costs of forms and shores, can influence the construction cost of RC frames, in this study, the concrete and rebar costs are only considered based on the assumption that the construction cost of the RC frames is proportional to the total cost of the concrete and rebar. The second objective function maximizes the amount of dissipation energy obtained from the pushover analysis. The amount of dissipation energy is calculated based on the area below the pushover curve. The second objective function is used to improve the ductility capacity of a structure. The first and second objective functions are provided in Eqs. (1) and (2), respectively. Minimize = ( + ) (1) Minimize =1/ (2) where and are the total structural costs of a structure and the reciprocal of the amount of dissipation energy, respectively. The reciprocal of the amount of dissipation energy is minimized to maximize the energy dissipation capacity of a structure, as shown in Eq. (2). and are the costs of the concrete and rebar, respectively, used at the ith member and are calculated based on the volumes of the concrete and rebar. The length of rebar used in a column is equal to the length of the column, whereas the length of rebar used in a beam is calculated separately depending on the end and central portions of the beam. The lengths of

4 the rebar installed at the end and central parts of the beam are assumed to be one-third of the length of the beam. is the area below the pushover curve. Constraints Seven constraint conditions are used in this study. Generally, the strength check of the RC columns utilizes a P-M interaction diagram, as shown in Fig. 1. The required axial strength and required moment strength of the ith column are obtained from the structural analysis; then, the design axial strength and design moment strength are determined. The constraint conditions of axial and flexural column strengths are shown in Eqs. (3) and (4). Axial force D c1 R c1 R c2 D c2 Bending moment Figure 1. P-M interaction diagram. = = 1.0 (3) 1.0 (4) / / The constraint condition on the flexural strength of beams is provided in Eq. (5). = / 1.0 (5) where and are the required flexural strength and design flexural strength of the ith beam member, respectively. The constraint conditions on the shear strength of the columns and beams are provided in Eqs. (6) and (7), respectively. = = 1.0 (6) 1.0 (7) / / where and are the required shear strength and design shear strength, respectively, of the ith column member and and are the required shear strength and design shear strength, respectively, of the ith beam member. The shear-reinforcing bars in both columns and beams are presumed to be arranged under the same conditions (D10@100) for all columns and beams. In this study, the strength constraint conditions of members are evaluated based on strength design equations presented in ACI The constraint condition for the flexural

5 strength ratio at the beam-column joints is provided in Eq. (8). =( ) / 1.0 (8) where and are the sums of the flexural strengths of the beams and columns, respectively, at the ith joint and are calculated according to ACI Eq. (9) is used to induce the beam-hinge collapse mechanism of the RC moment-resisting frame. =, =0 (9) where, is the number of plastic hinges located at the column joints. Designs in which the plastic hinges do not occur at the column joints, except for the joints at the top floor, can be obtained using this constraint. Column joints at the top floor are not considered in the constraint conditions because the plastic hinges at the joints of columns at the top floor have little influence on the overall behavior of the structure. In addition, the constraint condition on the cross sections of vertically continuous columns and the amount of rebar are considered in the constructability of the columns; the size of the cross section at the lower column should be greater than or equal to the cross section of the upper column for any of the two vertically continuous columns. Furthermore, the amount of rebar at the lower column is constrained to be greater than or equal to the amount of rebar at the upper column. To reduce the computational time of the optimization procedure, the column constructability are satisfied by automatically modifying the design variables related to the cross sectional sizes and rebar of the columns during the optimization procedure. Optimal Seismic Design Algorithm NSGA-II [5], which mimics the natural selection process, is employed as an optimization tool in the proposed optimal seismic design algorithm to obtain the optimal solutions to satisfy the given structural optimization problem. The NSGA-II flowchart is shown in Fig. 2. In NSGA-II, the candidate designs are expressed by the individuals that comprise the population. And the genes constituting the individuals represent the design variables of the candidate designs. In this study, each individual in the population is randomly initialized. After the population is initialized, all individuals in the population are evaluated to obtain the values of the considered objective functions and constraint conditions. Then, the ranks and crowding distances, which are used to determine whether the corresponding individual survives in the next generation, are assigned based on the values of the considered objective functions and individual. The individuals in the population evolve through genetic operators, such as the selection, crossover, and mutation. Then, the stopping criteria are evaluated. If the stopping criteria are satisfied, the proposed optimization procedure is terminated. If the stopping criteria are not satisfied, the evolution procedures, as previously mentioned, are repeated until the stopping criteria are satisfied. The structural optimization of RC structures requires extensive computational time for the convergence of solutions because there are more design variables in the optimization procedure for RC structures comprised of concrete and steel rebar than in those for steel structures. To solve this problem, the evaluation process (Fig. 2) is conducted using multi-cores, as shown in Fig. 3. Because the evaluation process includes the structural analysis procedure that

6 assesses the objective functions and of all individuals in the population, the procedure evaluating the individuals accounts for most of the computational time in the structural optimization procedure. In this study, the parallel processing method (based on the multi-cores method instead of the conventional single-core serial processing method) is used to reduce the computational times for the evaluation procedures for the individuals, as shown in Fig. 3. The individuals in the population are distributed to the cores, and then, the evaluation procedure for each individual design is performed at the corresponding core. First, the cross-sectional properties of members (based on the values of the individual design variables) are confirmed, and the constraint on the column constructability is evaluated. If the constraint on the column constructability is not satisfied, the values of the individual design variables are automatically modified to satisfy the constraint on the column constructability. Then, the structure cost, which is the first objective function, is calculated, and the structural modeling and analysis are performed based on the values of the individual design variables. The on the strength conditions of members and the flexural strength ratios at the joints are evaluated using the structural analysis results. If all of the constraint conditions are satisfied, the non-linear static analysis is conducted, and the quantity of energy dissipation (i.e., the second objective function) and the constraint ratios of the plastic hinges are calculated. If any one of the strength conditions or flexural strength ratio conditions is unsatisfied, a non-linear static analysis is not conducted, and a penalty is applied to the energy dissipation value and constraint ratio of the plastic hinge. Figure 2. Flowchart of NSGA-II.

7 Start 1st core 2nd core nth core Modeling of individual Modeling of individual Modeling of individual Linear static analysis Linear static analysis Linear static analysis Calculate cost, strength constraint, strength ratio Calculate cost, strength constraint, strength ratio Calculate cost, strength constraint, strength ratio OK Check the calculated No OK Check the calculated No... OK Check the calculated No Nonlinear static analysis hinge constraint ratio Nonlinear static analysis hinge constraint ratio Nonlinear static analysis hinge constraint ratio Calculate hinge constraint ratio dissipated energy Calculate hinge constraint ratio dissipated energy Calculate hinge constraint ratio dissipated energy Calculate dissipated energy Calculate dissipated energy Calculate dissipated energy No Complete evaluation of individuals Yes End Figure 3. Flowchart of the evaluation procedure for the individual designs using the parallel processing method based on the multi-core analysis. Application The proposed optimal seismic design method, which induces the beam-hinge collapse mechanism of RC moment-resisting frames, is applied in an office example with four stories and four spans, as shown in Fig. 4. The load combinations applied to the structure are 1.4D, 1.2D+1.6L, and 1.4D+1.0E+1.0L (i.e., ASCE 7-05). Based on ASCE 7-05, the dead load and live load are assumed to be 3.5 and 2.4 /, respectively. The site class is B. It is assumed that the structure is located in Los Angeles. This study employs OpenSees [6] for structural analysis. The strength of the concrete and rebar are 35 and 420 MPa, respectively. Linear static analysis is conducted to check the strength condition of the structure, and nonlinear static analysis is also conducted to evaluate the hinge constraint condition of a structure and the quantity of energy dissipation. A nonlinear static analysis of the inverted triangle pattern and displacement control based on the lateral displacement of the top floor is performed. Target displacement is set as lateral drift of top to be 4% of total height of structure. In the modeling process, the elasticbeamcolumn and beamwithhinges element of OpenSees are used. The beamwithhinges element assumes that the plastic behavior of the member only occurs at both ends of the member and uses the fiberbased section. The deformation of the panel zone is not considered; the centerline model is used. The connection of columns and beams is assumed to be fixed. The P-delta effect is considered, whereas degradation of the strength and stiffness of members is not considered. The occurrence of plastic hinges is confirmed by determining whether the tension bars have yielded.

8 C1 C2 C3 C4 B4 B4 B4 B4 B1 C5 C6 C7 C8 C5 C6 C7 C8 C5 C6 C7 C8 B3 B3 B3 B3 B2 B2 B2 B2 B1 B1 B1 C1 C2 C3 C Figure 4. Elevation of an example structure (unit : mm). Fig. 4 presents the number of columns and beams used in this study. The columns are categorized as interior and exterior columns and are divided by each floor into a total of eight groups. The beams are divided by each floor into total of four groups. Table 1 provides the design variables defining the cross section of columns and beams in each group. There are three and four design variables defining the cross section of columns and beams, respectively. Therefore, there are 24 and 16 design variables used for the columns and beams, respectively. The rebar arrangement at the i- and j-ends of beams located on the same story are assumed to be identical, and the compression reinforcement ratio is assumed to be 50% of the tension reinforcement ratio [4]. If the tension reinforcement is an odd number, the compression reinforcement number is given as (original number+1)/2. The compression and tension reinforcement are to be determined within the range that satisfies both the minimum and maximum reinforcement ratio conditions and a minimum reinforcement spacing condition. The column cross section is restricted to a square after the column width is determined. The beam cross section is restricted to a rectangular shape. If the width of the beam is determined, the depth of the beam has a value that is times greater than the determined width. The design variable ranges for the columns and beams are provided in Table 1. Forty optimal solutions are obtained from the proposed optimal seismic design method, as shown in Fig. 5. The quantity of dissipated energy increases with increasing structural cost. These optimal solutions have the beam-hinging collapse mechanism, as demonstrated in Fig. 6, which illustrates the distribution of plastic hinges for the optimal solution with the smallest weight. The plastic hinges did not occur on the beams at the top story because the plastic hinge constraint was applied to the joints, except for those at the top story. In this study, the optimal method includes the stages of initialization, evaluation, sorting, and evaluation for individual designs. Among those stages, the evaluation process for each individual includes a nonlinear analysis and accounts for 99% of the entire run-time. Therefore, a multi-core based parallel analysis is adopted to accelerate the evaluation process; various individuals are simultaneously assessed in the distributed cores instead of progressing in single cores individually. Fig. 7 presents the run-time in accordance with the number of cores. The run-

9 time deceases as the number of cores increases. However, the run-time did not decrease in proportion to the increased number of cores. The efficiency of an increased number of cores decreases. This result is mainly attributed to the waiting time, which means that some cores may complete their analyses earlier than others but should wait until the other analyses finish for further progress. Table 1. Range for the considered design variables. Design variable Column width Rebar diameter for the columns Number of rebars for the columns Beam width Beams depth Diameter of the tension rebars for the beams Number of tension rebars for the beams Range Minimum, 300; maximum, 800; increment, 50 (mm) 19, 22, 25, 29, 32 (mm) 4, 8, 12, 16, 20 Minimum, 300; maximum, 500; increment, 50 (mm) Minimum, 1.5 times the beam width; maximum, 2.5 times the beam width; increment, 50 (mm) 19, 22 (mm) Minimum, 2; maximum, 20 Dissipated engergy (knm) Cost (USD) Figure 5. Distribution of the optimal solutions obtained from the proposed method.

10 Figure 6. Distribution of the plastic hinges for the optimal solution with the lowest weight. Figure 7. Run-time comparison according to the number of cores (unit : sec). Conclusions The optimal seismic design method for ensuring the beam-hinging mechanism in RC momentresisting frames is presented and applied to the seismic design of a RC moment-resisting frame. It is confirmed that the optimal solutions for the beam-hinge collapse modes can be obtained using the proposed optimal seismic design method. By inducing the beam-hinge collapse mode, the abrupt collapse of RC frames can be prevented, and the seismic damage can be reduced. In this study, the multi-core based parallel analysis is adopted to accelerate the evaluation process, and it is confirmed that the run-time deceases with an increasing number of cores. It is believed that the multi-core based parallel analysis can be efficiently utilized for the structural optimization field, including the optimal seismic design of RC frames. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No ). References

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