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The Over-Strength Coefficient of Masonry-Infilled RC Frame Structures under Bidirectional Ground Motions

The Over-Strength Coefficient of Masonry-Infilled RC Frame Structures under Bidirectional Ground... buildings Article The Over-Strength Coefficient of Masonry-Infilled RC Frame Structures under Bidirectional Ground Motions 1 , 2 3 3 , 1 , 2 3 , 4 Xiaomin Wang , Yuhan Su , Jingchang Kong * , Maosheng Gong and Chunhui Liu Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, China School of Civil Engineering, Yantai University, Yantai 264005, China Department of Civil & Environmental Engineering, University of Alberta, Edmonton, AB T6G 2R3, Canada * Correspondence: kjch8811@126.com Abstract: The over-strength coefficient is one of the key factors for the seismic safety of a structure. For RC frames, the infill wall may improve the lateral bearing capacity, while the seismic demand increases as well, which leads to the unexpected seismic performance of an infilled RC frame in past earthquakes. Therefore, it is necessary to systematically study the over-strength effect of the infilled RC frames from the point of seismic capacity and demand. In this paper, 36 RC frame structures with/without infill walls are designed, and the corresponding finite element modelings, considering the in-plane and out-of-plane performance coupling effect of infill walls, are established to conduct incremental dynamic analyses (IDA). The seismic capacity values of over-strength coefficients are calculated, utilizing the IDA results under bidirectional ground motions. The effects of seismic precautionary intensity and number of stories on the over-strength coefficient of the RC frame with/without infill walls are discussed. The over-strength coefficient capacity value of the infilled frame is apparently higher than that of the bare frame, due to the contribution of infill walls. However, Citation: Wang, X.; Su, Y.; Kong, J.; the seismic demand analysis of the over-strength coefficient shows that the capacity–demand ratio Gong, M.; Liu, C. The Over-Strength Coefficient of Masonry-Infilled RC of masonry-infilled RC frame structures is greatly reduced, especially for the bottom soft-story Frame Structures under Bidirectional infilled frame. Ground Motions. Buildings 2022, 12, 1290. https://doi.org/10.3390/ Keywords: infilled RC frame; over-strength coefficient; bidirectional ground motions; precautionary buildings12091290 intensity; number of stories Academic Editors: Jiulin Bai, Wei Guo and Junxian Zhao Received: 16 July 2022 1. Introduction Accepted: 11 August 2022 In the FEMA-P695 report of the project ATC-63 [1], the overall seismic performance co- Published: 23 August 2022 efficient is a general term for the response modification coefficient, the system over-strength Publisher’s Note: MDPI stays neutral coefficient and the displacement amplification coefficient. The system over-strength coeffi- with regard to jurisdictional claims in cient refers to the ratio of the actual strength to the design strength of the structure, which published maps and institutional affil- reflects the structural strength reserve. The effect of system over-strength has been con- iations. firmed in previous earthquake damage investigations and shaking table tests of reinforced concrete structures and steel structures [2–6]. The source of the system over-strength is quite complicated, which can be attributed to three aspects: design over-strength, material over-strength and structural over-strength [7,8]. Copyright: © 2022 by the authors. At present, the values of system over-strength coefficients in seismic design codes Licensee MDPI, Basel, Switzerland. or standards of different countries are determined according to engineering experience, This article is an open access article and the values vary greatly. Therefore, it has attracted extensive attention worldwide, distributed under the terms and and a lot of research work on the over-strength coefficient has been carried out. Osteraas conditions of the Creative Commons and Krawinkler [9] adopted a pushover analysis method to systematically study the over- Attribution (CC BY) license (https:// strength coefficient of steel frames designed according to the allowable stress design method. creativecommons.org/licenses/by/ The results show that the over-strength coefficient of the flexural steel frame is between 8.0 4.0/). Buildings 2022, 12, 1290. https://doi.org/10.3390/buildings12091290 https://www.mdpi.com/journal/buildings Buildings 2022, 12, 1290 2 of 17 and 2.1, and the over-strength coefficient of the center-supported steel frame is between 2.8 and 2.2. Rahgozar and Humar [10] studied the over-strength coefficient of flexural steel frames with different heights. The structural strength reserve (i.e., the structural over-strength effect) decreases with the increase in the height, and the P-D effect makes the structural reserve strength further decrease. Calderoni et al. [11] studied the seismic performance of steel frames under different precautionary intensities and pointed out that the existence of the over-strength ensured that the structure had a certain anti-collapse ability. Balendra and Huang [12] analyzed the behavior of 3, 6 and 10 stories of flexural steel frames designed by British standard BS5950 by using pushover analysis. This shows that the steel frame has good over-strength effect and ductility, and the over-strength coefficient for the steel frame structures with X-bracing and Y-bracing is further increased. Stefano et al. [13] studied the effect of the over-strength effect of the element cross-section on the performance of the irregular structure. Rossi and Lombardo [14] studied the effect of node connection on over-strength of eccentric braced frame structures with different stories. In the specifications of different countries, the provisions of the over-strength co- efficient are different. Wei et al. [15] compared the relevant provisions on the system over-strength in the design codes of various countries. Zhai and Xie [16] quantitatively analyzed the over-strength coefficient of RC frame structures designed according to the Chinese seismic code. Zhou et al. [17] analyzed the influence of various factors on the over-strength capability of the RC frame structure. Ma and Cao [18] studied the influence of the number of spans and the precautionary intensity on the system over-strength coeffi- cient of the frame structures. Xin et al. [19] discussed the reasonable value of the system over-strength coefficient in the displacement-based seismic design method. Kuylmaz and Topkaya [20] studied the over-strength effect of eccentrically braced steel frames, indicating that the over-strength effect is mainly affected by the length and span of the bracing, and secondly by the precautionary intensity and the height of the structure. Elnashai and Mwafy [21] discussed the influence of periodic variation on the over-strength coefficient and the relationship between the over-strength coefficient and the response modification coefficient. Dickof et al. [22] studied the over-strength coefficient and ductility coefficient of wood–steel composite structures. It is recommended that the over-strength coefficient be 1.25 and the ductility coefficient be 2.5. Cui et al. [23] studied the effect of infill walls and floors on the system over-strength of RC frame structures. Li et al. [24] analyzed the over-strength coefficient of infill wall RC frame structures by using adaptive POA and IDA. Johnson and Dowell [25] evaluated the over-strength coefficient for nonstructural component anchorage into concrete via dynamic shaking table tests. Xin [26] studied the over-strength coefficient of the steel grid box frame structure under different pushover methods and IDA method. The importance of the over-strength coefficient is much larger than that of the structural ductility reduction coefficient. Abraik and Youssef [8] evaluated the ductility and over-strength of shape–memory-alloy reinforced concrete shear walls. It can be seen that there are few studies focusing on the over-strength coefficient of RC frame structures with infill walls. In the most recent earthquake events, masonry-infilled RC frame structures exhibited unfavorable failure phenomenon, although the infill wall can improve the bearing capacity of the structure [27]. On the contrary, the research indicates that the seismic demand of the infilled RC frame increases compared to the bare frame [28]. Therefore, it is necessary to systematically study the over-strength effect of the infilled RC frames from the point of seismic capacity and demand. In this paper, 36 representative masonry-infilled RC frame structures are designed according to Chinese seismic design codes, and a simplified finite element modeling technique, considering in-plane and out-of-plane performance coupling effect of infill walls, is presented. The corresponding finite element modelings for the designed structures are built in OpenSees [29]. Then, 20 pairs of ground motions are selected, the averaged response spectrum of which is in agreement with the designed response spectrum and is utilized to carry out incremental dynamic analyses under bidirectional ground motions [30,31]. The capacity values of over-strength coefficients are calculated to analyze the effects of seismic Buildings 2022, 12, x FOR PEER REVIEW 3 of 19 built in OpenSees [29]. Then, 20 pairs of ground motions are selected, the averaged re- sponse spectrum of which is in agreement with the designed response spectrum and is utilized to carry out incremental dynamic analyses under bidirectional ground motions Buildings 2022, 12, 1290 3 of 17 [30,31]. The capacity values of over-strength coefficients are calculated to analyze the ef- fects of seismic precautionary intensity and number of stories on the over-strength coeffi- cient of RC frame with/without infill walls. The capacity–demand ratio of the over- precautionary intensity and number of stories on the over-strength coefficient of RC frame strength coefficient for the 36 structures is analyzed as well. with/without infill walls. The capacity–demand ratio of the over-strength coefficient for the 36 structures is analyzed as well. 2. Design of Masonry-Infilled RC Frame Structures In this section, in accordance with the Chinese seismic code [32], a series of repre- 2. Design of Masonry-Infilled RC Frame Structures sentative masonry-infilled RC frame structures with 3, 6, 9 and 12 stories and in the level In this section, in accordance with the Chinese seismic code [32], a series of represen- VI (0.05 g), VII (0.1 g), and VIII (0.2 g) of seismic precautionary intensity regions are de- tative masonry-infilled RC frame structures with 3, 6, 9 and 12 stories and in the level VI signed, respectively. The seismic precautionary intensity means that the structure is ex- (0.05 g), VII (0.1 g), and VIII (0.2 g) of seismic precautionary intensity regions are designed, pected to have recoverable performance when it is subjected to the corresponding PGA respectively. The seismic precautionary intensity means that the structure is expected to value [33]. have recoverable performance when it is subjected to the corresponding PGA value [33]. The plan of all structures is the same, as shown in Figure 1a. The elevations of the The plan of all structures is the same, as shown in Figure 1a. The elevations of the different structures with/without infill walls are shown in Figure 1b–d, in which the height different structures with/without infill walls are shown in Figure 1b–d, in which the height of the ground story is 3.9 m, and the height of other stories is 3.3 m. The design information of the ground story is 3.9 m, and the height of other stories is 3.3 m. The design information includes: basic wind pressure is 0.55 kN/m , ground roughness belongs to Class C, basic includes: basic wind pressure is 0.55 kN/m , ground roughness belongs to Class C, basic 2 2 snow pressure is 0.30 kN/m , de 2 ad load of floor is 5.0 kN/m , and 2 live load of floor is 2.5 snow pressure is 0.30 kN/m , dead load of floor is 5.0 kN/m , and live load of floor is kN/m . Moreover, an inaccessible roof is adopted, and the live load of the roof is 0.5 2.5 kN/m . Moreover, an inaccessible roof is adopted, and the live load of the roof is kN/m . Site classification belongs to Site II, and the site characteristic period is 0.35 s. 0.5 kN/m . Site classification belongs to Site II, and the site characteristic period is 0.35 s. 6000 2400 6000 6000 2400 6000 7200 7200 7200 7200 (a) (b) 6000 2400 6000 6000 2400 6000 (c) (d) Figure 1. Plan and elevation views of RC frame structures with/without infill walls: (a) plan; Figure 1. Plan and elevation views of RC frame structures with/without infill walls: (a) plan; (b) el (b evat ) elevation ion of ba of re bar fram e frame; e; (c) el (cevat ) elevation ion of ful of lfully y infil infilled led fram frame; e; (d) ( d el ) eva elevation tion ofof the the bot bottom tom so soft-story ft-story infilled frame. infilled frame. The compressive strength of the concrete of the columns and beams is 30 MPa, the The compressive strength of the concrete of the columns and beams is 30 MPa, the strength of longitudinal reinforcement is 400 MPa, and the hooping strength is 235 MPa. strength of longitudinal reinforcement is 400 MPa, and the hooping strength is 235 MPa. The details of the reinforcement can be seen in Reference [28]. The dimensions of the beams The details of the reinforcement can be seen in Reference [28]. The dimensions of the and columns are listed in Table 1. The thickness of the floor is 100 mm, and the thickness of beams and columns are listed in Table 1. The thickness of the floor is 100 mm, and the the infill wall is 200 mm. The concrete hollow block, a widely used construction material in China, is used for infill wall, and the compressive strength of the block is 2.8 MPa. (n 2)×3300 3900 3300 6000 2400 6000 3900 3300 (n 2)×3300 (n 2)×3300 3900 3300 (n-2)×3300 3900 3300 Buildings 2022, 12, 1290 4 of 17 Table 1. Dimensions of beams and columns in structures. The Section of Columns The Section of Beams Story Number of Stories b (mm) h (mm) b (mm) h (mm) c c b b 3 1~3 500 500 250 500 5 1~5 600 600 300 600 8 1~8 650 650 300 600 1 700 700 350 700 2~8 650 650 300 600 9~10 600 600 300 600 1 700 700 350 700 2~10 650 650 300 700 11~12 600 600 300 600 To evaluate the effect of infill walls on the seismic performance of frame structures, three different infill wall arrangements were considered: (a) The effects of stiffness and load bearing capacity of the infill wall are not considered, and the structure is regarded as a bare frame with only the mass of the infill wall imposed on the beam-column joints. (b) The effects of stiffness and load bearing capacity of the infill wall are considered, the infill wall is in full layout, and it is simulated by a simplified analysis model. (c) The frame with a soft-story at the bottom story, a usual arrangement for infill walls, is also analyzed. These three types of frame structures are represented by F, IF, and IFE, respectively, and the three seismic precautionary intensity levels are represented by 6, 7 and 8, respectively. On this basis, these structures can be numbered by combining the number of stories (e.g., F3-6 represents a three-story bare frame structure for seismic precautionary intensity VI). In summary, 36 RC frame structures were analyzed in this paper. 3. Finite element Modeling Technology 3.1. Modeling of Frames and Infill Walls Three-dimensional finite element modelings for these 36 RC frame structures are built and simulated in OpenSees. The beam and column elements are modeled by using displacement-based beam-column elements, and the element sections are composed of fiber sections. The fiber section model is close to the actual mechanical performance in the modeling of RC frame structures [34]. For the concrete, the uniaxial material model Concrete02 is used. In order to consider the improvement of concrete strength caused by the lateral restraint of the stirrups, the core area of the section is composed of confined concrete. The improved characteristic parameters for the confined concrete is determined by the Kent–Scott–Park model [35]. For the steel, the uniaxial material model Steel01 is adopted, which is widely used due to its high computational efficiency. The infill wall is simulated by a simplified five-element model constituted of four diagonal beam elements with rigid behavior, a central horizontal beam element representing the in-plane and out-of-plane non-linear hysteretic behavior of the infill wall, and the two nodes at the ends of the central element with the wall mass, as shown in Figure 2. More specifically, when the infill wall is subjected to in-plane loads, the central element section is subjected to tension and compression to represent the performance of the infill wall; when the infill wall is subjected to out-of-plane loads, the central element is subjected to bending moments, and half of the fibers on the section are in tension and half in compression, as shown in Figure 3. Since the two fibers in the section have a certain distance from the center of the wall, the model can better reflect the arch mechanism in out of plane of the infill wall. Buildings 2022, 12, x FOR PEER REVIEW 5 of 19 from the center of the wall, the model can better reflect the arch mechanism in out of plane of the infill wall. Buildings 2022, 12, 1290 5 of 17 Buildings 2022, 12, x FOR PEER REVIEW 5 of 19 from the center of the wall, the model can better reflect the arch mechanism in out of plane Diagonal strut of the infill wall. Non-linear element Diagonal strut Central nodes with Non-linear element OOP nodes Central nodes with OOP nodes Figure 2. Simplified five-element model for infill walls. Figure 2. Simplified five-element model for infill walls. Figure 2. Simplified five-element model for infill walls. tinf/2 d/2 d/2 t /2 inf tinf/2 t /2 F inf d/2 d/2 t /2 inf (a) (b) Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. The material constitutive model of the central element adopts the Pinching4 model, The material constitutive model of the central element adopts the Pinching4 model, as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight points points as shown in the solid line, and the unloading and reloading path is shown in the t /2 inf as shown in the solid line, and the unloading and reloading path is shown in the dotted dotted line. Compared with other material models such as Hysteretic [36,37], Concrete01 in line. Compared with other material models such as Hysteretic [36,37], Concrete01 in OpenSees, Pinching4 material can simulate the strength degradation, stiffness degradation OpenSees, Pinching4 material can simulate the strength degradation, stiffness degrada- and increase l in deformation caused by loading and unloading, which gives Pinching4 tion and material increa as gr e eat in de advantage formation in simulating caused by in-plane loading and and out-of-plane unloading, damage which g interactions ives Pinching in 4 the infill wall [38,39]. material a great advantage in simulating in-plane and out-of-plane damage interactions in the infill wall [38,39]. (a) (b) Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. The material constitutive model of the central element adopts the Pinching4 model, as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight points as shown in the solid line, and the unloading and reloading path is shown in the dotted line. Compared with other material models such as Hysteretic [36,37], Concrete01 in OpenSees, Pinching4 material can simulate the strength degradation, stiffness degrada- tion and increase in deformation caused by loading and unloading, which gives Pinching4 material a great advantage in simulating in-plane and out-of-plane damage interactions in the infill wall [38,39]. Buildings 2022, 12, x FOR PEER REVIEW 6 of 19 Buildings 2022, 12, 1290 6 of 17 Load (dmax,f(dmax)) (ePd3,ePf3) (ePd2,ePf2) (ePd1,ePf1) (ePd4,ePf4) (*,uForceP·ePf3) (rDispP·dmin,rForceP·f(dmin)) Deformation (*,uForceN·eNf3) (rDispN·dmin,rForceN·f(dmin)) (eNd4,eNf4) (eNd1,eNf1) (eNd2,eNf2) (eNd3,eNf3) (d f(d )) min, min (a) Force (kN) 𝑚𝑎𝑥 𝑑 𝑑 𝑑 𝑑 𝑐 𝑦 𝐹 𝑢 Inter-story drift (%) 𝑚𝑎𝑥 (b) Figure 4. Pingching 4 constitutive model; (a) hysteretic behavior; (b) skeleton curve. Figure 4. Pingching 4 constitutive model; (a) hysteretic behavior; (b) skeleton curve. The determination of the mechanical performance of the infill wall needs to first The determination of the mechanical performance of the infill wall needs to first de- determine the skeleton curve. The tension and compression parts of the skeleton curve used termine the skeleton curve. The tension and compression parts of the skeleton curve used in this paper are the same, and the load and displacement values for the four characteristic in this paper are the same, and the load and displacement values for the four characteristic points are shown is Figure 4b and are determined as follows [40,41]: points are shown is Figure 4b and are determined as follows [40,41]: * The ratio of the cracking strength F to the peak strength F is taken as 0.55, and c max * The ratio of the cracking strength 𝐹 to the peak strength 𝐹 is taken as 0.55, and 𝑐 𝑎𝑥𝑚 the cracking inter-story drift ratio d is between the values 0.075% and 0.12%. the cracking inter-story drift ratio 𝑑 is between the values 0.075% and 0.12%. * The yielding strength F and inter-story drift ratio d are the intermediate values y y * The yielding strength 𝐹 and inter-story drift ratio 𝑑 are the intermediate values 𝑦 𝑦 between the coordinate values of the cracking and peak points, that is, F = (F + F )/2, y max c between the coordinate values of the cracking and peak points, that is, 𝐹 = (𝐹 + 𝐹 )/2 𝑦 𝑎𝑥𝑚 𝑐 and d = (d + d )/2. y F c max , and d = (𝑑 + 𝑑 )/2. y 𝐹 𝑐 𝑚𝑎𝑥 Buildings 2022, 12, x FOR PEER REVIEW 7 of 19 * The in-plane peak strength of the infill wall generally occurs when the inter-story Buildings 2022, 12, 1290 7 of 17 drift ratio is between 0.25% and 0.5%. The peak strength 𝐹 can be calculated by the 𝑎𝑥𝑚 following formula: F = 0.818L t f C * The in-plane peak strength of the infill wall generally occurs when the inter-story (1) max inf inf tp drift ratio is between 0.25% and 0.5%. The peak strength F can be calculated by the max following formula: C = 1+ C +1 / C (2) F = 0.818L t f C (1) ( 11 ) max in f in f t p C = 1 + C + 1 /C (2) inf C = 1.925 (3) in f inf C = 1.925 (3) in f where 𝑓 is the masonry cracking strength, which can be taken as 1/9 to 1/10 of the ma- 𝑡𝑝 where f is the masonry cracking strength, which can be taken as 1/9 to 1/10 of the t p sonry compressive strength 𝑓 , and 𝑓 can be easily obtained through experimental tests, 𝑐 𝑐 masonry compressive strength f , and f can be easily obtained through experimental tests, c c 𝐿 is the length of the infill wall, 𝑡 is the thickness of the infill wall, and ℎ is the L is the length of the infill wall, t is the thickness of the infill wall, and h is the in f in f in f height of the infill wall. height of the infill wall. * The residual strength 𝐹 is adopted as 20% of the 𝐹 , and the corresponding re- 𝑢 𝑎𝑥𝑚 * The residual strength F is adopted as 20% of the F , and the corresponding u max sidual inter-story drift ratio 𝑑 is five times the 𝑑 . 𝑢 𝐹 𝑚𝑎𝑥 residual inter-story drift ratio d is five times the d . max In addition, the hysteresis rule of the pinching4 model takes into account the stiffness In addition, the hysteresis rule of the pinching4 model takes into account the stiffness degradation, strength degradation and pinching effect of the infill wall through three pa- degradation, strength degradation and pinching effect of the infill wall through three rameters α, β and γ. parameters , and . 3.2. Verification of Finite Element Modeling 3.2. Verification of Finite Element Modeling A shaking table test of an infilled RC frame specimen was adopted to verify the ac- A shaking table test of an infilled RC frame specimen was adopted to verify the curacy of the simplified model in simulating the in-plane and out-of-plane interaction ef- accuracy of the simplified model in simulating the in-plane and out-of-plane interaction fects of infill walls. The test was performed by Hashemi and Mosalam [42]. The test spec- effects of infill walls. The test was performed by Hashemi and Mosalam [42]. The test imen is a substructure of a five-story multi-span infilled frame structure, and the ground specimen is a substructure of a five-story multi-span infilled frame structure, and the motion is applied to the specimen in a direction perpendicular to the infill wall. ground motion is applied to the specimen in a direction perpendicular to the infill wall. The relationship between the base shear force and the top displacement of the test The relationship between the base shear force and the top displacement of the test specimen under the action of the two ground motions is shown in Figure 5, the time his- specimen under the action of the two ground motions is shown in Figure 5, the time history tory curve of the base shear force for ground motion Tar6 case is shown in Figure 6, and curve of the base shear force for ground motion Tar6 case is shown in Figure 6, and the the relative error is listed in Table 2. It can be seen that the finite element simulation results relative error is listed in Table 2. It can be seen that the finite element simulation results are are in good agreement with the test results, and the pinch effect and loading and unload- in good agreement with the test results, and the pinch effect and loading and unloading ing trends are roughly the same. The relative error between the numerical values and the trends are roughly the same. The relative error between the numerical values and the test values in terms of bearing capacity and stiffness is small, the peak bearing capacity is test values in terms of bearing capacity and stiffness is small, the peak bearing capacity with is within in 20% 20%, , and and the the initi initial al stiff stif nefness ss is with is within in 10% 10%. . Gen Generally erally, th , the e simp simplified lified mo model del can can sisimulate mulate the the perform performance ance of i ofn infill fill wa walls lls under the under the act action ion of of ground ground mo motion tion wel well. l. (a) (b) Figure 5. Base shear force-top displacement curves: (a) Tar6 case; (b) Duz7 case. Figure 5. Base shear force-top displacement curves: (a) Tar6 case; (b) Duz7 case. 𝑖𝑛𝑓 𝑖𝑛𝑓 𝑖𝑛𝑓 Buildings 2022, 12, x FOR PEER REVIEW 8 of 19 Buildings 2022, 12, 1290 8 of 17 Figure 6. Time history curve of the base shear force for the Tar6 case. Figure 6. Time history curve of the base shear force for the Tar6 case. Table 2. Comparison of numerical and test results. Table 2. Comparison of numerical and test results. Tar6 Duz7 Initial Residual Initial Residual Peak Load Peak Load Tar6 Duz7 Stiffness Stiffness Stiffness Stiffness (kN) (kN) (kN/mm) (kN/mm) (kN/mm) (kN/mm) Initial Stiff- Residual Initial Stiff- Residual Peak Load Peak Load Test 63.77 50.63 588.96 48.70 28.03 763.35 ness Stiffness ness Stiffness Numerical 68.29 47.97 691.93 53.51 21.72 687.39 (kN) (kN) Error 7.09% 5.26% 17.48% 9.86% 22.51% 9.95% (kN/mm) (kN/mm) (kN/mm) (kN/mm) Test 63.77 50.63 588.96 48.70 28.03 763.35 4. Ground Motions Numerical 68.29 47.97 691.93 53.51 21.72 687.39 In this paper, 20 pairs of ground motions recommended by the report FEMA-P695 [43] Error 7.09% 5.26% 17.48% 9.86% 22.51% 9.95% were adopted for incremental dynamic analysis (IDA) and were further used for structural over-strength coefficient analysis. The 20 pairs of ground motions include ten pairs of far-field ground motions, five pairs of near-field pulsed ground motions, and five pairs 4. Ground Motions of near-field non-pulsed ground motions. Table 3 lists the related information of these ground motions, where PGA ranges from 0.15 to 1.2 g, and PGA/PGV ranges from 0.3 to In this paper, 20 pairs of ground motions recommended by the report FEMA-P695 2.9 g/(cm/s). [43] were adopted for incremental dynamic analysis (IDA) and were further used for Table 3. Ground motion records. structural over-strength coefficient analysis. The 20 pairs of ground motions include ten PGV PGD PGA/PGV pairs of far-field ground motions, five pairs of near-field pulsed ground motions, and five No. Ground Motions PGA (g) (cm/s) (cm) (g/(cm/s)) pairs of near-field non-pulsed ground motions. Table 3 lists the related information of RSN126_GAZLI_GAZ090 0.8639 67.62 20.71 0.0128 GM-1 RSN126_GAZLI_GAZ000 0.7017 66.18 27.32 0.0106 these ground motions, where PGA ranges from 0.15 to 1.2 g, and PGA/PGV ranges from RSN160_IMPVALL.H_H-BCR230 0.7769 44.92 15.09 0.0173 GM-2 0.3 to 2.9 gRSN160_IMPV /(cm/s). ALL.H_H-BCR140 0.5987 46.73 20.21 0.0128 RSN165_IMPVALL.H_H-CHI012 0.2699 24.79 9.29 0.0109 GM-3 RSN165_IMPVALL.H_H-CHI282 0.2542 29.89 7.65 0.0085 RSN495_NAHANNI_S1010 1.1079 43.90 6.80 0.0252 Table 3. Ground motion records. GM-4 RSN495_NAHANNI_S1280 1.2007 40.61 10.20 0.0296 RSN741_LOMAP_BRN000 0.4564 51.36 8.11 0.0089 GM-5 RSN741_LOMAP_BRN090 0.5023 44.47 5.05 PGV 0.0113 PGD PGA/PGV No. Ground Motions PGA(g) RSN181_IMPVALL.H_H-E06140 0.4473 66.99 27.88 0.0067 GM-6 (cm/s) (cm) (g/(cm/s)) RSN181_IMPVALL.H_H-E06230 0.4490 113.50 72.85 0.0040 RSN182_IMPVALL.H_H-E07140 0.3408 51.65 27.98 0.0066 GM-7 RSN126_GAZLI_GAZ090 0.8639 67.62 20.71 0.0128 RSN182_IMPVALL.H_H-E07230 0.4691 113.08 46.92 0.0041 GM-1 RSN292_ITALY_A-STU270 0.3205 71.92 29.31 0.0045 GM-8 RSN126_GAZLI_GAZ000 0.7017 66.18 27.32 0.0106 RSN292_ITALY_A-STU000 0.2267 36.96 13.11 0.0061 RSN802_LOMAP_STG000 0.5145 41.56 16.32 0.0124 RSN160_IMPVALL.H_H- GM-9 RSN802_LOMAP_STG090 0.3262 45.95 33.31 0.0071 0.7769 44.92 15.09 0.0173 BCR230 GM-2 RSN160_IMPVALL.H_H- 0.5987 46.73 20.21 0.0128 BCR140 RSN165_IMPVALL.H_H-CHI012 0.2699 24.79 9.29 0.0109 GM-3 RSN165_IMPVALL.H_H-CHI282 0.2542 29.89 7.65 0.0085 RSN495_NAHANNI_S1010 1.1079 43.90 6.80 0.0252 GM-4 RSN495_NAHANNI_S1280 1.2007 40.61 10.20 0.0296 RSN741_LOMAP_BRN000 0.4564 51.36 8.11 0.0089 GM-5 RSN741_LOMAP_BRN090 0.5023 44.47 5.05 0.0113 RSN181_IMPVALL.H_H-E06140 0.4473 66.99 27.88 0.0067 GM-6 RSN181_IMPVALL.H_H-E06230 0.4490 113.50 72.85 0.0040 RSN182_IMPVALL.H_H-E07140 0.3408 51.65 27.98 0.0066 GM-7 RSN182_IMPVALL.H_H-E07230 0.4691 113.08 46.92 0.0041 Buildings 2022, 12, x FOR PEER REVIEW 9 of 19 RSN292_ITALY_A-STU270 0.3205 71.92 29.31 0.0045 GM-8 RSN292_ITALY_A-STU000 0.2267 36.96 13.11 0.0061 RSN802_LOMAP_STG000 0.5145 41.56 16.32 0.0124 GM-9 RSN802_LOMAP_STG090 0.3262 45.95 33.31 0.0071 RSN828_CAPEMEND_PET000 0.5908 49.30 16.59 0.0120 GM-10 RSN828_CAPEMEND_PET090 0.6616 88.47 33.20 0.0075 RSN68_SFERN_PEL090 0.2248 21.71 15.91 0.0104 GM-11 Buildings 2022, 12, 1290 9 of 17 RSN68_SFERN_PEL180 0.1949 16.93 12.87 0.0115 RSN125_FRIULI.A_A-TMZ000 0.3571 22.84 4.59 0.0156 GM-12 RSN125_FRIULI.A_A-TMZ270 0.3151 30.50 5.21 0.0103 Table 3. Cont. RSN169_IMPVALL.H_H- 0.2357 26.31 14.69 0.0090 DLT262 PGV PGD PGA/PGV GM-13 No. Ground Motions PGA (g) RSN169_IMPVALL.H_H- (cm/s) (cm) (g/(cm/s)) 0.3497 32.98 20.17 0.0106 DLT352 RSN828_CAPEMEND_PET000 0.5908 49.30 16.59 0.0120 GM-10 RSN828_CAPEMEND_PET090 0.6616 88.47 33.20 0.0075 RSN174_IMPVALL.H_H-E11140 0.3668 36.00 25.08 0.0102 GM-14 RSN68_SFERN_PEL090 0.2248 21.71 15.91 0.0104 GM-11 RSN174_IMPVALL.H_H-E11230 0.3794 44.59 21.31 0.0085 RSN68_SFERN_PEL180 0.1949 16.93 12.87 0.0115 RSN125_FRIULI.A_A-TMZ000 0.3571 22.84 4.59 0.0156 RSN721_SUPER.B_B-ICC000 0.3573 48.05 19.27 0.0074 GM-12 GM-15 RSN125_FRIULI.A_A-TMZ270 0.3151 30.50 5.21 0.0103 RSN721_SUPER.B_B-ICC090 0.2595 41.77 21.85 0.0062 RSN169_IMPVALL.H_H-DLT262 0.2357 26.31 14.69 0.0090 GM-13 RSN169_IMPVALL.H_H-DLT352 0.3497 32.98 20.17 0.0106 RSN725_SUPER.B_B-POE270 0.4750 41.15 7.73 0.0115 GM-16 RSN174_IMPVALL.H_H-E11140 0.3668 36.00 25.08 0.0102 RSN725_SUPER.B_B-POE360 0.2862 29.00 11.36 0.0099 GM-14 RSN174_IMPVALL.H_H-E11230 0.3794 44.59 21.31 0.0085 RSN752_LOMAP_CAP000 0.5111 38.01 7.06 0.0134 RSN721_SUPER.B_B-ICC000 0.3573 48.05 19.27 0.0074 GM GM-15 -17 RSN721_SUPER.B_B-ICC090 0.2595 41.77 21.85 0.0062 RSN752_LOMAP_CAP090 0.4386 29.60 4.91 0.0148 RSN725_SUPER.B_B-POE270 0.4750 41.15 7.73 0.0115 GM-16 RSN767_LOMAP_G03000 0.5591 36.29 10.84 0.0154 RSN725_SUPER.B_B-POE360 0.2862 29.00 11.36 0.0099 GM-18 RSN752_LOMAP_CAP000 0.5111 38.01 7.06 0.0134 RSN767_LOMAP_G03090 0.3682 45.40 24.09 0.0081 GM-17 RSN752_LOMAP_CAP090 0.4386 29.60 4.91 0.0148 RSN900_LANDERS_YER270 0.2445 51.10 41.69 0.0048 RSN767_LOMAP_G03000 0.5591 36.29 10.84 0.0154 GM-19 GM-18 RSN9 RSN767_LOMAP_G03090 00_LANDERS_YER360 0.3682 0.1518 29.08 45.40 23.13 24.09 0.005 0.00812 RSN900_LANDERS_YER270 0.2445 51.10 41.69 0.0048 RSN953_NORTHR_MUL009 0.4434 59.27 15.47 0.0075 GM-19 RSN900_LANDERS_YER360 0.1518 29.08 23.13 0.0052 GM-20 RSN953_NORTHR_MUL279 0.4880 66.68 12.17 0.0073 RSN953_NORTHR_MUL009 0.4434 59.27 15.47 0.0075 GM-20 RSN953_NORTHR_MUL279 0.4880 66.68 12.17 0.0073 Taking the seismic precautionary intensity VI as an example, the peak acceleration value of each ground motion is adjusted to 0.035 g for the more frequent case, 0.1 g for the Taking the seismic precautionary intensity VI as an example, the peak acceleration precautionary case, 0.22 g for the rare case, and 0.32 g for the extremely rare case [44,45]. value of each ground motion is adjusted to 0.035 g for the more frequent case, 0.1 g for the After calculating the response spectrum of each ground motion, the average response precautionary case, 0.22 g for the rare case, and 0.32 g for the extremely rare case [44,45]. spectrum and the seismic design response spectrum of four different intensities are plot- After calculating the response spectrum of each ground motion, the average response ted in Figure 7. It can be seen that the averaged response spectrum is in good agreement spectrum and the seismic design response spectrum of four different intensities are plotted with the designed response spectrum in the short period, which meets the analysis re- in Figure 7. It can be seen that the averaged response spectrum is in good agreement with quirement the designed s. response spectrum in the short period, which meets the analysis requirements. Figure 7. Average response spectrum and seismic design response spectrum for seismic precautionary intensity VI. 5. Over-Strength Coefficient Capacity Analysis 5.1. Introduction of Over-Strength Coefficient Capacity Analysis In earthquake damage investigations, it has been shown that the actual seismic capacity of the structure is generally higher than the designed seismic capacity, which is referred to as the system over-strength effect. The existence of system over-strength is an important factor that prevents the structure from collapsing when subjected to strong earthquakes greater than the design seismic force [35,40,46]. The system over-strength coefficient R can be defined as the ratio of the actual yielding strength of the structure to the design Buildings 2022, 12, 1290 10 of 17 seismic force, which reflects the strength reserve of the structure. It can be calculated by the following formula: R = V /V (4) s y where V is the yielding strength of the structure, and V is the design seismic force. y d The process of system over-strength coefficient analysis is as follow: (1) Perform incremental dynamic analysis (IDA) of the model under one pair of ground motion, obtain the maximum inter-story drift ratio under different ground motion levels, and plot the IDA curve of the model under this pair of ground motion. (2) Obtain the structural dynamic capacity curve in the form of maximum base shear force (V ) and maximum top displacement (D ), then transform the dynamic max max capacity curve into a double-linear elastic–plastic curve by using the equal energy principle, and solve the equivalent yielding strength V of the structure. (3) Calculate the design seismic force V of the structure by using the bottom shear method or the mode-superposition response spectrum method. (4) Calculate the system over-strength coefficient R by its definition as shown in Formula (4). (5) Repeat the above steps to obtain the system over-strength coefficient under each pair of ground motion. (6) Take the median value as the system over-strength coefficient capacity value. 5.2. Over-Strength Coefficient Capacity Analysis of Infilled RC Frame Structure According to the process described above, the incremental dynamic analysis is first Buildings 2022, 12, x FOR PEER REVIEW 11 of 19 performed. For the sake of brevity and the limitation of text space, only the IDA curves of the three structural models are given, as shown in Figure 8. (a) (b) (c) Figure 8. The IDA curves of 20 pairs of ground motions for (a) Model F6-6; (b) Model IF6-6; (c) Model IFE6-6. Figure 8. The IDA curves of 20 pairs of ground motions for (a) Model F6-6; (b) Model IF6-6; (c) Model IFE6-6. Then, structural dynamic capacity curves are obtained, and the median values of over-strength coefficients of 36 RC frame structures under bidirectional ground motion action are calculated, which are shown in Table 4 and Figure 9. It can be seen that the system over-strength coefficient capacity values of the bare frame structures designed ac- cording to different precautionary intensities of VI, VII and VIII are between 9.7–19.69, 5.24–9.01, 3.31–4.57, which are greater than 9.5, 5.0, 3.0, respectively. The over-strength coefficient capacity values of the fully infilled frame structures for different precautionary intensities are between 21.45–33.62, 11.52–20.51, 6.05–9.83, reaching 21.0, 11.5, and 6.0, re- spectively. The over-strength coefficient capacity values of the infilled frame structures with a bottom soft-story for different precautionary intensities are between 15.39–26.79, 9.76–18.41, 3.63–6.46, reaching 15.0, 9.5, 3.5, respectively. Table 4. Over-strength coefficient values of 36 RC frame structures. Number of Stories 3 6 9 12 precautionary inten- 6 7 8 6 7 8 6 7 8 6 7 8 sity bare frame 9.7 5.24 3.31 14.5 8.1 3.9 16.95 8.9 4.1 19.69 9.01 4.57 Buildings 2022, 12, 1290 11 of 17 Then, structural dynamic capacity curves are obtained, and the median values of over-strength coefficients of 36 RC frame structures under bidirectional ground motion action are calculated, which are shown in Table 4 and Figure 9. It can be seen that the system over-strength coefficient capacity values of the bare frame structures designed according to different precautionary intensities of VI, VII and VIII are between 9.7–19.69, 5.24–9.01, 3.31–4.57, which are greater than 9.5, 5.0, 3.0, respectively. The over-strength coefficient capacity values of the fully infilled frame structures for different precautionary intensities are between 21.45–33.62, 11.52–20.51, 6.05–9.83, reaching 21.0, 11.5, and 6.0, respectively. The over-strength coefficient capacity values of the infilled frame structures with a bottom soft-story for different precautionary intensities are between 15.39–26.79, 9.76–18.41, 3.63–6.46, reaching 15.0, 9.5, 3.5, respectively. Buildings 2022, 12, x FOR PEER REVIEW 12 of 19 Table 4. Over-strength coefficient values of 36 RC frame structures. fully infilled frame 21.45 11.52 6.05 32 20.51 6.22 33.62 17 9.83 29.5 13.5 6.15 Number of Stories 3 6 9 12 bottom soft-story precautionary intensity 6 7 8 6 7 8 6 7 8 6 7 8 15.39 9.76 3.63 22 12.2 6.2 26.79 18.41 5 24 12.5 6.46 frame bar e frame 9.7 5.24 3.31 14.5 8.1 3.9 16.95 8.9 4.1 19.69 9.01 4.57 fully infilled frame 21.45 11.52 6.05 32 20.51 6.22 33.62 17 9.83 29.5 13.5 6.15 bottom soft-story frame 15.39 9.76 3.63 22 12.2 6.2 26.79 18.41 5 24 12.5 6.46 (a) (b) (c) Figure 9. Over-strength coefficient values of 36 RC frame structures: (a) bare frame; (b) fully infilled Figure 9. Over-strength coefficient values of 36 RC frame structures: (a) bare frame; (b) fully infilled frame; (c) bottom soft-story infilled frame. frame; (c) bottom soft-story infilled frame. Specifically, with the increase in the precautionary intensity, the over-strength coef- Specifically, with the increase in the precautionary intensity, the over-strength coeffi- ficient shows a significant decrease trend regardless of the existence of the infill wall. As cient shows a significant decrease trend regardless of the existence of the infill wall. As the the intensity increases, the increase trend of the yielding base shear force 𝑉 of the struc- intensity increases, the increase trend of the yielding base shear force V of the structure is ture is obviously smaller than that of the design seismic force 𝑉 . This may be because ythe higher obviously the precau smaller tionary than intensity, that of the the higher design the se seismic ismic forc for e ce that Vth .e This struct may ure may be be be cause the higher subjected to, and the more difficult it is to realize the seismic capacity of the structure. the precautionary intensity, the higher the seismic force that the structure may be subjected Therefore, the strength reserve of the structure decreases with the increase in the precau- to, and the more difficult it is to realize the seismic capacity of the structure. Therefore, the tionary intensity, which is expressed as the over-strength coefficients that are significantly strength reserve of the structure decreases with the increase in the precautionary intensity, decreasing. which is expressed as the over-strength coefficients that are significantly decreasing. For structures designed according to the same precautionary intensity, the over- strength coefficient of the bare frame structures increases with the increase in the number of stories, and the over-strength coefficients of the fully infilled frame and the bottom soft- story infilled frame structures first increase and then decrease with the increase in the number of stories. For the bare frame structures, as the number of stories increases, the yielding strength 𝑉 increases, and the system over-strength coefficient increases with the increase in the number of stories, although the design seismic force 𝑉 also increases. 𝑑 Buildings 2022, 12, 1290 12 of 17 For structures designed according to the same precautionary intensity, the over- strength coefficient of the bare frame structures increases with the increase in the number of stories, and the over-strength coefficients of the fully infilled frame and the bottom soft-story infilled frame structures first increase and then decrease with the increase in the number of stories. For the bare frame structures, as the number of stories increases, the yielding strength V increases, and the system over-strength coefficient increases with Buildings 2022, 12, x FOR PEER REVIEW the incr ease in the number of stories, although the design seismic force13 V of also 19 increases. However, this growth trend gradually slows down. In general, the natural vibration period is larger for a taller structure, and the seismic influence coefficient used for calculating However, this growth trend gradually slows down. In general, the natural vibration pe- design seismic force is smaller, while under practical earthquake action, due to the influence riod is larger for a taller structure, and the seismic influence coefficient used for calculating of higher-order mode, the yielding force V of the structure is relatively higher. Therefore, design seismic force is smaller, while under practical earthquake action, due to the influ- the strength reserves of high-rise structures are higher. Furthermore, the design of high-rise ence of higher-order mode, the yielding force 𝑉 of the structure is relatively higher. structures in this paper may be more cautious and conservative, which also strengthens Therefore, the strength reserves of high-rise structures are higher. Furthermore, the design the result. For the fully infilled frame and the bottom soft-story infilled frame structures, of high-rise structures in this paper may be more cautious and conservative, which also although strengthens thethe infill result wall . Fohas r the high fully infil rigidity led fr,ame its deformation and the bottomability soft-story is poor infille ,d and frame seri ous brittle structures, although the infill wall has high rigidity, its deformation ability is poor, and damage often occurs. In a high-rise structure, the displacement response is often larger; serious brittle damage often occurs. In a high-rise structure, the displacement response is thus, the damage of infill walls is more serious. As a result, the system over-strength often larger; thus, the damage of infill walls is more serious. As a result, the system over- coefficient decreases in the high-rise structure. strength coefficient decreases in the high-rise structure. To exhibit the effect of infill walls on the over-strength coefficient of the frame structure, To exhibit the effect of infill walls on the over-strength coefficient of the frame struc- the over-strength coefficient capacity values of the three types of structures are plotted ture, the over-strength coefficient capacity values of the three types of structures are plot- together, as shown in Figure 10. It can be seen that the over-strength coefficient of the fully ted together, as shown in Figure 10. It can be seen that the over-strength coefficient of the infilled frame structure is 93.5% higher than that of the bare frame structure on average; the fully infilled frame structure is 93.5% higher than that of the bare frame structure on av- erage; the over-strength coefficient of the bottom soft-story infilled frame structure is over-strength coefficient of the bottom soft-story infilled frame structure is 50.5% higher 50.5% higher than that of the bare frame structure on average; the over-strength coefficient than that of the bare frame structure on average; the over-strength coefficient of the fully of the fully infilled frame structure is increased by 31.8% on average compared with the infilled frame structure is increased by 31.8% on average compared with the result of the result of the bottom soft-story infilled frame structure. bottom soft-story infilled frame structure. (a) (b) (c) Figure 10. Over-strength coefficient values for different precautionary intensities: (a) intensity Ⅵ; Figure 10. Over-strength coefficient values for different precautionary intensities: (a) intensity VI; (b) intensity Ⅶ; (c) intensity Ⅷ. (b) intensity VII; (c) intensity VIII. Buildings 2022, 12, 1290 13 of 17 The over-strength coefficient is a reflection of the seismic capacity of the structure. The existence of the infill wall improves the rigidity and strength of the structure, and the damage of the infill wall will consume a part of the energy so that the yielding strength V of the structure is significantly increased, and the seismic performance of the structure is greatly improved. However, this strengthening effect of infill walls is gradually reduced as the number of stories increases. This may be due to the fact that infill walls are more severely damaged in the high-rise structure. With the increase in the precautionary intensity, the over-strength coefficient of the structure shows a significant decreasing trend regardless of whether the effect of the infill wall is considered. As a result, the effect of the infill wall on the over-strength coefficient has no significant relevance with the precautionary intensity. 6. The Assessment of Over-Strength Coefficient In order to evaluate the rationality of the system over-strength coefficient value, the capacity–demand ratio of the over-strength coefficient is adopted, and the calculation process is as follows: (1) Develop the structural dynamic capacity curve of the model under one pair of ground motion; then, convert the structural capacity curve into the spectral acceleration- displacement curve, that is, the capacity spectrum curve. (2) Convert the seismic response spectrum with 5% damping into the spectral displacement– acceleration format as well, that is, the demand spectrum curve. (3) Solve the intersection of the capacity spectrum and demand spectrum curves by using the dynamic capability spectrum method; then, obtain the target vertex displacement De and the corresponding base shear force V under different strength spectra. (4) Knowing the seismic design force V of the structure, according to Formula (4), calcu- late the demand value of the over-strength coefficient at different strength spectra. (5) Calculate the over-strength coefficient capacity–demand ratio of the structure accord- ing to the following formula. l = R /R (5) Rs sC sD (6) Repeat the above steps to obtain the demand values and capacity–demand ratios of the system over-strength coefficient under each pair of ground motions. (7) Take the median values as the demand value and capacity–demand ratio of the system over-strength coefficient. Using the dynamic capability spectrum method, the over-strength coefficient demand values of 36 structures for the more frequent, precautionary, rare, and extremely rare ground motion demand spectra are calculated. The results are shown in Figure 11. It can be seen from Figure 11a that the demand values of the over-strength coeffi- cients of the bare frame structures under different precautionary intensities are 2.66~2.11, 2.24~1.78, and 2.02~1.59 for the precautionary case and reached 2.1, 1.7, and 1.5, respec- tively. For the rare case, the values are 5.58~3.80, 4.02~3.07, 3.32~2.32 and reached 3.8, 3.0, and 2.3, respectively. In Figure 11b, for the precautionary case, the demand values of fully infilled frame structures are 7.1~2.31, 6.02~2.29, and 3.72~1.54, which are above 2.3, 3.2, and 1.5, respectively. For the rare case, the values are 13.79~5.00, 8.72~4.07, and 5.5~2.47, which are above 5.0, 4.0, and 2.4, respectively. In Figure 11c, for the precautionary case, the demand values of bottom soft-story infilled frame structures are 7.33~2.42, 5.11~1.93, and 3.52~1.44, which are above 2.7, 1.9, and 1.4, respectively. For the rare case, the values are 14.07~4.34, 7.97~3.47, and 5.39~2.66, which are above 4.3, 3.4, and 2.6, respectively. In general, the over-strength coefficient demand value of the three types of frame structures increases with the increase in the seismic demand spectrum intensity, decreases with the increase in the precautionary intensity, and increases with the increase in the number of stories. The fully infilled frame has the largest over-strength coefficient demand value, while the bare frame has the smallest over-strength coefficient demand value. Buildings 2022, 12, 1290 14 of 17 Buildings 2022, 12, x FOR PEER REVIEW 15 of 19 (a) (b) (c) Figure 11. Demand analysis based on dynamic capacity spectra: (a) bare frame; (b) gully infilled Figure 11. Demand analysis based on dynamic capacity spectra: (a) bare frame; (b) gully infilled frame; (c) bottom soft-story infilled frame. frame; (c) bottom soft-story infilled frame. It can be seen from Figure 11a that the demand values of the over-strength coeffi- cients of the bare frame structures under different precautionary intensities are 2.66~2.11, The over-strength coefficient capacity–demand ratio values are obtained by dividing Buildings 2022, 12, x FOR PEER REVIEW 16 of 19 2.24~1.78, and 2.02~1.59 for the precautionary case and reached 2.1, 1.7, and 1.5, respec- the over-strength coefficient capability value by the demand value in the rare case, as tively. For the rare case, the values are 5.58~3.80, 4.02~3.07, 3.32~2.32 and reached 3.8, 3.0, shown in Figure 12. and 2.3, respectively. In Figure 11b, for the precautionary case, the demand values of fully infilled frame structures are 7.1~2.31, 6.02~2.29, and 3.72~1.54, which are above 2.3, 3.2, and 1.5, respectively. For the rare case, the values are 13.79~5.00, 8.72~4.07, and 5.5~2.47, which are above 5.0, 4.0, and 2.4, respectively. In Figure 11c, for the precautionary case, the demand values of bottom soft-story infilled frame structures are 7.33~2.42, 5.11~1.93, and 3.52~1.44, which are above 2.7, 1.9, and 1.4, respectively. For the rare case, the values are 14.07~4.34, 7.97~3.47, and 5.39~2.66, which are above 4.3, 3.4, and 2.6, respectively. In general, the over-strength coefficient demand value of the three types of frame structures increases with the increase in the seismic demand spectrum intensity, decreases with the increase in the precautionary intensity, and increases with the increase in the number of stories. The fully infilled frame has the largest over-strength coefficient demand value, while the bare frame has the smallest over-strength coefficient demand value. The over-strength coefficient capacity–demand ratio values are obtained by dividing the over-strength coefficient capability value by the demand value in the rare case, as shown in Figure 12. (a) (b) (c) Figure 12. The structural capacity–demand ratio of over-strength coefficient: (a) rare frame; (b) fully Figure 12. The structural capacity–demand ratio of over-strength coefficient: (a) rare frame; (b) fully infilled frame; (c) bottom soft-story infilled frame. infilled frame; (c) bottom soft-story infilled frame. It can be seen that for the bare frame, the capacity–demand ratio is between 1.15 and 2.98; for the fully infilled frame, the ratio is between 1.00 and 3.05; they are all greater than or equal to 1. However, for the bottom soft-story infilled frame, the minimum capacity– demand ratio value is only 0.90. Therefore, for the structure designed according to the seismic code, the bare frame has a good strength reserve under the action of a rare earth- quake; however, when the infill wall is considered, although the infill wall can improve the bearing capacity of the structure, the structural stiffness and vibration model have been changed, especially when the bottom infill wall of the structure is missing, the over- strength coefficient capacity–demand ratio is greatly reduced, which indicates that the structure is at risk under the action of rare earthquakes. Therefore, in the design of the masonry-infilled RC frame structure, the arrangement of the soft-story at the bottom should be avoided, or some strengthen technologies should be adopted. 7. Conclusions In this paper, a total of 36 bare frame, fully infilled frame, and bottom soft-story in- filled frame structures with 3, 6, 9, and 12 stories and for seismic precautionary intensities VI, VII and VIII are designed, and three-dimensional finite element modelings are built by using an improved five-element infill wall model in OpenSees. Then, the system over-strength coefficient capacity value of the structures under bi- directional ground motion action is obtained by using the IDA method, and the effect of Buildings 2022, 12, 1290 15 of 17 It can be seen that for the bare frame, the capacity–demand ratio is between 1.15 and 2.98; for the fully infilled frame, the ratio is between 1.00 and 3.05; they are all greater than or equal to 1. However, for the bottom soft-story infilled frame, the minimum capacity–demand ratio value is only 0.90. Therefore, for the structure designed according to the seismic code, the bare frame has a good strength reserve under the action of a rare earthquake; however, when the infill wall is considered, although the infill wall can improve the bearing capacity of the structure, the structural stiffness and vibration model have been changed, especially when the bottom infill wall of the structure is missing, the over-strength coefficient capacity–demand ratio is greatly reduced, which indicates that the structure is at risk under the action of rare earthquakes. Therefore, in the design of the masonry-infilled RC frame structure, the arrangement of the soft-story at the bottom should be avoided, or some strengthen technologies should be adopted. 7. Conclusions In this paper, a total of 36 bare frame, fully infilled frame, and bottom soft-story infilled frame structures with 3, 6, 9, and 12 stories and for seismic precautionary intensities VI, VII and VIII are designed, and three-dimensional finite element modelings are built by using an improved five-element infill wall model in OpenSees. Then, the system over-strength coefficient capacity value of the structures under bidi- rectional ground motion action is obtained by using the IDA method, and the effect of the infill wall on the over-strength coefficient is discussed. It can be concluded that the system over-strength coefficients of the three types of frame structures all decrease with the increase in the precautionary intensity, and the values of over-strength coefficients for different intensities are quite different. For the bare frame structure, the over-strength coef- ficient increases with the increase in the number of stories; for the fully infilled frame and the bottom soft-story infilled frame structures, the over-strength coefficient first increases and then decreases with the increase in the number of stories. The inclusion of infill walls significantly enhances the performance of frame structures; therefore, the over-strength coefficient of the infilled frame structure is significantly higher than that of the bare frame structure, and the coefficient of the fully infilled frame structure is greater than that of the bottom soft-story infilled frame structure. As the number of stories increases, the effect of the infill wall is gradually reduced. However, for different intensities, it does not show a clear rule for the increase in the over-strength coefficient of the infilled frame structure. In addition, the capacity–demand ratio of the over-strength coefficient is analyzed to evaluate the strength reserve of three types of frame structures with and without infill walls. It has shown that the bare frame structure has a good strength reserve, while the seismic performance of masonry-infilled RC frame structures is deteriorated, although the infill wall can improve the bearing capacity of the structure, and the design of the bottom soft-story infilled frame structure requires some additional strengthening treatments. Author Contributions: Conceptualization, J.K.; methodology, X.W.; software, Y.S. and X.W.; formal analysis, Y.S.; investigation, M.G., C.L. and X.W.; resources, J.K., X.W. and Y.S.; writing—original draft preparation, J.K., writing—review and editing, X.W.; visualization, J.K. and X.W.; supervision, M.G. and C.L.; project administration, X.W.; funding acquisition, X.W., M.G., C.L. and J.K. All authors have read and agreed to the published version of the manuscript. Funding: The research is supported by the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (grant nos. 2018B07, 2019A01, and 2021EEEVL0301), the National Nature Science Foundation of China (grant nos. 51808478 and 51908484), and the Natural Science Foundation of Shandong (grant no. ZR2019QEE033). This support is greatly appreciated. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Buildings 2022, 12, 1290 16 of 17 Acknowledgments: Grateful acknowledgment is given to Liang Cui and Bo Liu at the School of Civil Engineering, Yantai University. Conflicts of Interest: The authors declare no conflict of interest. References 1. Applied Technology Council. 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The Over-Strength Coefficient of Masonry-Infilled RC Frame Structures under Bidirectional Ground Motions

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buildings Article The Over-Strength Coefficient of Masonry-Infilled RC Frame Structures under Bidirectional Ground Motions 1 , 2 3 3 , 1 , 2 3 , 4 Xiaomin Wang , Yuhan Su , Jingchang Kong * , Maosheng Gong and Chunhui Liu Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, China School of Civil Engineering, Yantai University, Yantai 264005, China Department of Civil & Environmental Engineering, University of Alberta, Edmonton, AB T6G 2R3, Canada * Correspondence: kjch8811@126.com Abstract: The over-strength coefficient is one of the key factors for the seismic safety of a structure. For RC frames, the infill wall may improve the lateral bearing capacity, while the seismic demand increases as well, which leads to the unexpected seismic performance of an infilled RC frame in past earthquakes. Therefore, it is necessary to systematically study the over-strength effect of the infilled RC frames from the point of seismic capacity and demand. In this paper, 36 RC frame structures with/without infill walls are designed, and the corresponding finite element modelings, considering the in-plane and out-of-plane performance coupling effect of infill walls, are established to conduct incremental dynamic analyses (IDA). The seismic capacity values of over-strength coefficients are calculated, utilizing the IDA results under bidirectional ground motions. The effects of seismic precautionary intensity and number of stories on the over-strength coefficient of the RC frame with/without infill walls are discussed. The over-strength coefficient capacity value of the infilled frame is apparently higher than that of the bare frame, due to the contribution of infill walls. However, Citation: Wang, X.; Su, Y.; Kong, J.; the seismic demand analysis of the over-strength coefficient shows that the capacity–demand ratio Gong, M.; Liu, C. The Over-Strength Coefficient of Masonry-Infilled RC of masonry-infilled RC frame structures is greatly reduced, especially for the bottom soft-story Frame Structures under Bidirectional infilled frame. Ground Motions. Buildings 2022, 12, 1290. https://doi.org/10.3390/ Keywords: infilled RC frame; over-strength coefficient; bidirectional ground motions; precautionary buildings12091290 intensity; number of stories Academic Editors: Jiulin Bai, Wei Guo and Junxian Zhao Received: 16 July 2022 1. Introduction Accepted: 11 August 2022 In the FEMA-P695 report of the project ATC-63 [1], the overall seismic performance co- Published: 23 August 2022 efficient is a general term for the response modification coefficient, the system over-strength Publisher’s Note: MDPI stays neutral coefficient and the displacement amplification coefficient. The system over-strength coeffi- with regard to jurisdictional claims in cient refers to the ratio of the actual strength to the design strength of the structure, which published maps and institutional affil- reflects the structural strength reserve. The effect of system over-strength has been con- iations. firmed in previous earthquake damage investigations and shaking table tests of reinforced concrete structures and steel structures [2–6]. The source of the system over-strength is quite complicated, which can be attributed to three aspects: design over-strength, material over-strength and structural over-strength [7,8]. Copyright: © 2022 by the authors. At present, the values of system over-strength coefficients in seismic design codes Licensee MDPI, Basel, Switzerland. or standards of different countries are determined according to engineering experience, This article is an open access article and the values vary greatly. Therefore, it has attracted extensive attention worldwide, distributed under the terms and and a lot of research work on the over-strength coefficient has been carried out. Osteraas conditions of the Creative Commons and Krawinkler [9] adopted a pushover analysis method to systematically study the over- Attribution (CC BY) license (https:// strength coefficient of steel frames designed according to the allowable stress design method. creativecommons.org/licenses/by/ The results show that the over-strength coefficient of the flexural steel frame is between 8.0 4.0/). Buildings 2022, 12, 1290. https://doi.org/10.3390/buildings12091290 https://www.mdpi.com/journal/buildings Buildings 2022, 12, 1290 2 of 17 and 2.1, and the over-strength coefficient of the center-supported steel frame is between 2.8 and 2.2. Rahgozar and Humar [10] studied the over-strength coefficient of flexural steel frames with different heights. The structural strength reserve (i.e., the structural over-strength effect) decreases with the increase in the height, and the P-D effect makes the structural reserve strength further decrease. Calderoni et al. [11] studied the seismic performance of steel frames under different precautionary intensities and pointed out that the existence of the over-strength ensured that the structure had a certain anti-collapse ability. Balendra and Huang [12] analyzed the behavior of 3, 6 and 10 stories of flexural steel frames designed by British standard BS5950 by using pushover analysis. This shows that the steel frame has good over-strength effect and ductility, and the over-strength coefficient for the steel frame structures with X-bracing and Y-bracing is further increased. Stefano et al. [13] studied the effect of the over-strength effect of the element cross-section on the performance of the irregular structure. Rossi and Lombardo [14] studied the effect of node connection on over-strength of eccentric braced frame structures with different stories. In the specifications of different countries, the provisions of the over-strength co- efficient are different. Wei et al. [15] compared the relevant provisions on the system over-strength in the design codes of various countries. Zhai and Xie [16] quantitatively analyzed the over-strength coefficient of RC frame structures designed according to the Chinese seismic code. Zhou et al. [17] analyzed the influence of various factors on the over-strength capability of the RC frame structure. Ma and Cao [18] studied the influence of the number of spans and the precautionary intensity on the system over-strength coeffi- cient of the frame structures. Xin et al. [19] discussed the reasonable value of the system over-strength coefficient in the displacement-based seismic design method. Kuylmaz and Topkaya [20] studied the over-strength effect of eccentrically braced steel frames, indicating that the over-strength effect is mainly affected by the length and span of the bracing, and secondly by the precautionary intensity and the height of the structure. Elnashai and Mwafy [21] discussed the influence of periodic variation on the over-strength coefficient and the relationship between the over-strength coefficient and the response modification coefficient. Dickof et al. [22] studied the over-strength coefficient and ductility coefficient of wood–steel composite structures. It is recommended that the over-strength coefficient be 1.25 and the ductility coefficient be 2.5. Cui et al. [23] studied the effect of infill walls and floors on the system over-strength of RC frame structures. Li et al. [24] analyzed the over-strength coefficient of infill wall RC frame structures by using adaptive POA and IDA. Johnson and Dowell [25] evaluated the over-strength coefficient for nonstructural component anchorage into concrete via dynamic shaking table tests. Xin [26] studied the over-strength coefficient of the steel grid box frame structure under different pushover methods and IDA method. The importance of the over-strength coefficient is much larger than that of the structural ductility reduction coefficient. Abraik and Youssef [8] evaluated the ductility and over-strength of shape–memory-alloy reinforced concrete shear walls. It can be seen that there are few studies focusing on the over-strength coefficient of RC frame structures with infill walls. In the most recent earthquake events, masonry-infilled RC frame structures exhibited unfavorable failure phenomenon, although the infill wall can improve the bearing capacity of the structure [27]. On the contrary, the research indicates that the seismic demand of the infilled RC frame increases compared to the bare frame [28]. Therefore, it is necessary to systematically study the over-strength effect of the infilled RC frames from the point of seismic capacity and demand. In this paper, 36 representative masonry-infilled RC frame structures are designed according to Chinese seismic design codes, and a simplified finite element modeling technique, considering in-plane and out-of-plane performance coupling effect of infill walls, is presented. The corresponding finite element modelings for the designed structures are built in OpenSees [29]. Then, 20 pairs of ground motions are selected, the averaged response spectrum of which is in agreement with the designed response spectrum and is utilized to carry out incremental dynamic analyses under bidirectional ground motions [30,31]. The capacity values of over-strength coefficients are calculated to analyze the effects of seismic Buildings 2022, 12, x FOR PEER REVIEW 3 of 19 built in OpenSees [29]. Then, 20 pairs of ground motions are selected, the averaged re- sponse spectrum of which is in agreement with the designed response spectrum and is utilized to carry out incremental dynamic analyses under bidirectional ground motions Buildings 2022, 12, 1290 3 of 17 [30,31]. The capacity values of over-strength coefficients are calculated to analyze the ef- fects of seismic precautionary intensity and number of stories on the over-strength coeffi- cient of RC frame with/without infill walls. The capacity–demand ratio of the over- precautionary intensity and number of stories on the over-strength coefficient of RC frame strength coefficient for the 36 structures is analyzed as well. with/without infill walls. The capacity–demand ratio of the over-strength coefficient for the 36 structures is analyzed as well. 2. Design of Masonry-Infilled RC Frame Structures In this section, in accordance with the Chinese seismic code [32], a series of repre- 2. Design of Masonry-Infilled RC Frame Structures sentative masonry-infilled RC frame structures with 3, 6, 9 and 12 stories and in the level In this section, in accordance with the Chinese seismic code [32], a series of represen- VI (0.05 g), VII (0.1 g), and VIII (0.2 g) of seismic precautionary intensity regions are de- tative masonry-infilled RC frame structures with 3, 6, 9 and 12 stories and in the level VI signed, respectively. The seismic precautionary intensity means that the structure is ex- (0.05 g), VII (0.1 g), and VIII (0.2 g) of seismic precautionary intensity regions are designed, pected to have recoverable performance when it is subjected to the corresponding PGA respectively. The seismic precautionary intensity means that the structure is expected to value [33]. have recoverable performance when it is subjected to the corresponding PGA value [33]. The plan of all structures is the same, as shown in Figure 1a. The elevations of the The plan of all structures is the same, as shown in Figure 1a. The elevations of the different structures with/without infill walls are shown in Figure 1b–d, in which the height different structures with/without infill walls are shown in Figure 1b–d, in which the height of the ground story is 3.9 m, and the height of other stories is 3.3 m. The design information of the ground story is 3.9 m, and the height of other stories is 3.3 m. The design information includes: basic wind pressure is 0.55 kN/m , ground roughness belongs to Class C, basic includes: basic wind pressure is 0.55 kN/m , ground roughness belongs to Class C, basic 2 2 snow pressure is 0.30 kN/m , de 2 ad load of floor is 5.0 kN/m , and 2 live load of floor is 2.5 snow pressure is 0.30 kN/m , dead load of floor is 5.0 kN/m , and live load of floor is kN/m . Moreover, an inaccessible roof is adopted, and the live load of the roof is 0.5 2.5 kN/m . Moreover, an inaccessible roof is adopted, and the live load of the roof is kN/m . Site classification belongs to Site II, and the site characteristic period is 0.35 s. 0.5 kN/m . Site classification belongs to Site II, and the site characteristic period is 0.35 s. 6000 2400 6000 6000 2400 6000 7200 7200 7200 7200 (a) (b) 6000 2400 6000 6000 2400 6000 (c) (d) Figure 1. Plan and elevation views of RC frame structures with/without infill walls: (a) plan; Figure 1. Plan and elevation views of RC frame structures with/without infill walls: (a) plan; (b) el (b evat ) elevation ion of ba of re bar fram e frame; e; (c) el (cevat ) elevation ion of ful of lfully y infil infilled led fram frame; e; (d) ( d el ) eva elevation tion ofof the the bot bottom tom so soft-story ft-story infilled frame. infilled frame. The compressive strength of the concrete of the columns and beams is 30 MPa, the The compressive strength of the concrete of the columns and beams is 30 MPa, the strength of longitudinal reinforcement is 400 MPa, and the hooping strength is 235 MPa. strength of longitudinal reinforcement is 400 MPa, and the hooping strength is 235 MPa. The details of the reinforcement can be seen in Reference [28]. The dimensions of the beams The details of the reinforcement can be seen in Reference [28]. The dimensions of the and columns are listed in Table 1. The thickness of the floor is 100 mm, and the thickness of beams and columns are listed in Table 1. The thickness of the floor is 100 mm, and the the infill wall is 200 mm. The concrete hollow block, a widely used construction material in China, is used for infill wall, and the compressive strength of the block is 2.8 MPa. (n 2)×3300 3900 3300 6000 2400 6000 3900 3300 (n 2)×3300 (n 2)×3300 3900 3300 (n-2)×3300 3900 3300 Buildings 2022, 12, 1290 4 of 17 Table 1. Dimensions of beams and columns in structures. The Section of Columns The Section of Beams Story Number of Stories b (mm) h (mm) b (mm) h (mm) c c b b 3 1~3 500 500 250 500 5 1~5 600 600 300 600 8 1~8 650 650 300 600 1 700 700 350 700 2~8 650 650 300 600 9~10 600 600 300 600 1 700 700 350 700 2~10 650 650 300 700 11~12 600 600 300 600 To evaluate the effect of infill walls on the seismic performance of frame structures, three different infill wall arrangements were considered: (a) The effects of stiffness and load bearing capacity of the infill wall are not considered, and the structure is regarded as a bare frame with only the mass of the infill wall imposed on the beam-column joints. (b) The effects of stiffness and load bearing capacity of the infill wall are considered, the infill wall is in full layout, and it is simulated by a simplified analysis model. (c) The frame with a soft-story at the bottom story, a usual arrangement for infill walls, is also analyzed. These three types of frame structures are represented by F, IF, and IFE, respectively, and the three seismic precautionary intensity levels are represented by 6, 7 and 8, respectively. On this basis, these structures can be numbered by combining the number of stories (e.g., F3-6 represents a three-story bare frame structure for seismic precautionary intensity VI). In summary, 36 RC frame structures were analyzed in this paper. 3. Finite element Modeling Technology 3.1. Modeling of Frames and Infill Walls Three-dimensional finite element modelings for these 36 RC frame structures are built and simulated in OpenSees. The beam and column elements are modeled by using displacement-based beam-column elements, and the element sections are composed of fiber sections. The fiber section model is close to the actual mechanical performance in the modeling of RC frame structures [34]. For the concrete, the uniaxial material model Concrete02 is used. In order to consider the improvement of concrete strength caused by the lateral restraint of the stirrups, the core area of the section is composed of confined concrete. The improved characteristic parameters for the confined concrete is determined by the Kent–Scott–Park model [35]. For the steel, the uniaxial material model Steel01 is adopted, which is widely used due to its high computational efficiency. The infill wall is simulated by a simplified five-element model constituted of four diagonal beam elements with rigid behavior, a central horizontal beam element representing the in-plane and out-of-plane non-linear hysteretic behavior of the infill wall, and the two nodes at the ends of the central element with the wall mass, as shown in Figure 2. More specifically, when the infill wall is subjected to in-plane loads, the central element section is subjected to tension and compression to represent the performance of the infill wall; when the infill wall is subjected to out-of-plane loads, the central element is subjected to bending moments, and half of the fibers on the section are in tension and half in compression, as shown in Figure 3. Since the two fibers in the section have a certain distance from the center of the wall, the model can better reflect the arch mechanism in out of plane of the infill wall. Buildings 2022, 12, x FOR PEER REVIEW 5 of 19 from the center of the wall, the model can better reflect the arch mechanism in out of plane of the infill wall. Buildings 2022, 12, 1290 5 of 17 Buildings 2022, 12, x FOR PEER REVIEW 5 of 19 from the center of the wall, the model can better reflect the arch mechanism in out of plane Diagonal strut of the infill wall. Non-linear element Diagonal strut Central nodes with Non-linear element OOP nodes Central nodes with OOP nodes Figure 2. Simplified five-element model for infill walls. Figure 2. Simplified five-element model for infill walls. Figure 2. Simplified five-element model for infill walls. tinf/2 d/2 d/2 t /2 inf tinf/2 t /2 F inf d/2 d/2 t /2 inf (a) (b) Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. The material constitutive model of the central element adopts the Pinching4 model, The material constitutive model of the central element adopts the Pinching4 model, as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight points points as shown in the solid line, and the unloading and reloading path is shown in the t /2 inf as shown in the solid line, and the unloading and reloading path is shown in the dotted dotted line. Compared with other material models such as Hysteretic [36,37], Concrete01 in line. Compared with other material models such as Hysteretic [36,37], Concrete01 in OpenSees, Pinching4 material can simulate the strength degradation, stiffness degradation OpenSees, Pinching4 material can simulate the strength degradation, stiffness degrada- and increase l in deformation caused by loading and unloading, which gives Pinching4 tion and material increa as gr e eat in de advantage formation in simulating caused by in-plane loading and and out-of-plane unloading, damage which g interactions ives Pinching in 4 the infill wall [38,39]. material a great advantage in simulating in-plane and out-of-plane damage interactions in the infill wall [38,39]. (a) (b) Figure 3. Details of the central element: (a) set of two fiber section; (b) analysis of section force. The material constitutive model of the central element adopts the Pinching4 model, as shown in Figure 4. The skeleton curve of the Pinching4 model is defined by eight points as shown in the solid line, and the unloading and reloading path is shown in the dotted line. Compared with other material models such as Hysteretic [36,37], Concrete01 in OpenSees, Pinching4 material can simulate the strength degradation, stiffness degrada- tion and increase in deformation caused by loading and unloading, which gives Pinching4 material a great advantage in simulating in-plane and out-of-plane damage interactions in the infill wall [38,39]. Buildings 2022, 12, x FOR PEER REVIEW 6 of 19 Buildings 2022, 12, 1290 6 of 17 Load (dmax,f(dmax)) (ePd3,ePf3) (ePd2,ePf2) (ePd1,ePf1) (ePd4,ePf4) (*,uForceP·ePf3) (rDispP·dmin,rForceP·f(dmin)) Deformation (*,uForceN·eNf3) (rDispN·dmin,rForceN·f(dmin)) (eNd4,eNf4) (eNd1,eNf1) (eNd2,eNf2) (eNd3,eNf3) (d f(d )) min, min (a) Force (kN) 𝑚𝑎𝑥 𝑑 𝑑 𝑑 𝑑 𝑐 𝑦 𝐹 𝑢 Inter-story drift (%) 𝑚𝑎𝑥 (b) Figure 4. Pingching 4 constitutive model; (a) hysteretic behavior; (b) skeleton curve. Figure 4. Pingching 4 constitutive model; (a) hysteretic behavior; (b) skeleton curve. The determination of the mechanical performance of the infill wall needs to first The determination of the mechanical performance of the infill wall needs to first de- determine the skeleton curve. The tension and compression parts of the skeleton curve used termine the skeleton curve. The tension and compression parts of the skeleton curve used in this paper are the same, and the load and displacement values for the four characteristic in this paper are the same, and the load and displacement values for the four characteristic points are shown is Figure 4b and are determined as follows [40,41]: points are shown is Figure 4b and are determined as follows [40,41]: * The ratio of the cracking strength F to the peak strength F is taken as 0.55, and c max * The ratio of the cracking strength 𝐹 to the peak strength 𝐹 is taken as 0.55, and 𝑐 𝑎𝑥𝑚 the cracking inter-story drift ratio d is between the values 0.075% and 0.12%. the cracking inter-story drift ratio 𝑑 is between the values 0.075% and 0.12%. * The yielding strength F and inter-story drift ratio d are the intermediate values y y * The yielding strength 𝐹 and inter-story drift ratio 𝑑 are the intermediate values 𝑦 𝑦 between the coordinate values of the cracking and peak points, that is, F = (F + F )/2, y max c between the coordinate values of the cracking and peak points, that is, 𝐹 = (𝐹 + 𝐹 )/2 𝑦 𝑎𝑥𝑚 𝑐 and d = (d + d )/2. y F c max , and d = (𝑑 + 𝑑 )/2. y 𝐹 𝑐 𝑚𝑎𝑥 Buildings 2022, 12, x FOR PEER REVIEW 7 of 19 * The in-plane peak strength of the infill wall generally occurs when the inter-story Buildings 2022, 12, 1290 7 of 17 drift ratio is between 0.25% and 0.5%. The peak strength 𝐹 can be calculated by the 𝑎𝑥𝑚 following formula: F = 0.818L t f C * The in-plane peak strength of the infill wall generally occurs when the inter-story (1) max inf inf tp drift ratio is between 0.25% and 0.5%. The peak strength F can be calculated by the max following formula: C = 1+ C +1 / C (2) F = 0.818L t f C (1) ( 11 ) max in f in f t p C = 1 + C + 1 /C (2) inf C = 1.925 (3) in f inf C = 1.925 (3) in f where 𝑓 is the masonry cracking strength, which can be taken as 1/9 to 1/10 of the ma- 𝑡𝑝 where f is the masonry cracking strength, which can be taken as 1/9 to 1/10 of the t p sonry compressive strength 𝑓 , and 𝑓 can be easily obtained through experimental tests, 𝑐 𝑐 masonry compressive strength f , and f can be easily obtained through experimental tests, c c 𝐿 is the length of the infill wall, 𝑡 is the thickness of the infill wall, and ℎ is the L is the length of the infill wall, t is the thickness of the infill wall, and h is the in f in f in f height of the infill wall. height of the infill wall. * The residual strength 𝐹 is adopted as 20% of the 𝐹 , and the corresponding re- 𝑢 𝑎𝑥𝑚 * The residual strength F is adopted as 20% of the F , and the corresponding u max sidual inter-story drift ratio 𝑑 is five times the 𝑑 . 𝑢 𝐹 𝑚𝑎𝑥 residual inter-story drift ratio d is five times the d . max In addition, the hysteresis rule of the pinching4 model takes into account the stiffness In addition, the hysteresis rule of the pinching4 model takes into account the stiffness degradation, strength degradation and pinching effect of the infill wall through three pa- degradation, strength degradation and pinching effect of the infill wall through three rameters α, β and γ. parameters , and . 3.2. Verification of Finite Element Modeling 3.2. Verification of Finite Element Modeling A shaking table test of an infilled RC frame specimen was adopted to verify the ac- A shaking table test of an infilled RC frame specimen was adopted to verify the curacy of the simplified model in simulating the in-plane and out-of-plane interaction ef- accuracy of the simplified model in simulating the in-plane and out-of-plane interaction fects of infill walls. The test was performed by Hashemi and Mosalam [42]. The test spec- effects of infill walls. The test was performed by Hashemi and Mosalam [42]. The test imen is a substructure of a five-story multi-span infilled frame structure, and the ground specimen is a substructure of a five-story multi-span infilled frame structure, and the motion is applied to the specimen in a direction perpendicular to the infill wall. ground motion is applied to the specimen in a direction perpendicular to the infill wall. The relationship between the base shear force and the top displacement of the test The relationship between the base shear force and the top displacement of the test specimen under the action of the two ground motions is shown in Figure 5, the time his- specimen under the action of the two ground motions is shown in Figure 5, the time history tory curve of the base shear force for ground motion Tar6 case is shown in Figure 6, and curve of the base shear force for ground motion Tar6 case is shown in Figure 6, and the the relative error is listed in Table 2. It can be seen that the finite element simulation results relative error is listed in Table 2. It can be seen that the finite element simulation results are are in good agreement with the test results, and the pinch effect and loading and unload- in good agreement with the test results, and the pinch effect and loading and unloading ing trends are roughly the same. The relative error between the numerical values and the trends are roughly the same. The relative error between the numerical values and the test values in terms of bearing capacity and stiffness is small, the peak bearing capacity is test values in terms of bearing capacity and stiffness is small, the peak bearing capacity with is within in 20% 20%, , and and the the initi initial al stiff stif nefness ss is with is within in 10% 10%. . Gen Generally erally, th , the e simp simplified lified mo model del can can sisimulate mulate the the perform performance ance of i ofn infill fill wa walls lls under the under the act action ion of of ground ground mo motion tion wel well. l. (a) (b) Figure 5. Base shear force-top displacement curves: (a) Tar6 case; (b) Duz7 case. Figure 5. Base shear force-top displacement curves: (a) Tar6 case; (b) Duz7 case. 𝑖𝑛𝑓 𝑖𝑛𝑓 𝑖𝑛𝑓 Buildings 2022, 12, x FOR PEER REVIEW 8 of 19 Buildings 2022, 12, 1290 8 of 17 Figure 6. Time history curve of the base shear force for the Tar6 case. Figure 6. Time history curve of the base shear force for the Tar6 case. Table 2. Comparison of numerical and test results. Table 2. Comparison of numerical and test results. Tar6 Duz7 Initial Residual Initial Residual Peak Load Peak Load Tar6 Duz7 Stiffness Stiffness Stiffness Stiffness (kN) (kN) (kN/mm) (kN/mm) (kN/mm) (kN/mm) Initial Stiff- Residual Initial Stiff- Residual Peak Load Peak Load Test 63.77 50.63 588.96 48.70 28.03 763.35 ness Stiffness ness Stiffness Numerical 68.29 47.97 691.93 53.51 21.72 687.39 (kN) (kN) Error 7.09% 5.26% 17.48% 9.86% 22.51% 9.95% (kN/mm) (kN/mm) (kN/mm) (kN/mm) Test 63.77 50.63 588.96 48.70 28.03 763.35 4. Ground Motions Numerical 68.29 47.97 691.93 53.51 21.72 687.39 In this paper, 20 pairs of ground motions recommended by the report FEMA-P695 [43] Error 7.09% 5.26% 17.48% 9.86% 22.51% 9.95% were adopted for incremental dynamic analysis (IDA) and were further used for structural over-strength coefficient analysis. The 20 pairs of ground motions include ten pairs of far-field ground motions, five pairs of near-field pulsed ground motions, and five pairs 4. Ground Motions of near-field non-pulsed ground motions. Table 3 lists the related information of these ground motions, where PGA ranges from 0.15 to 1.2 g, and PGA/PGV ranges from 0.3 to In this paper, 20 pairs of ground motions recommended by the report FEMA-P695 2.9 g/(cm/s). [43] were adopted for incremental dynamic analysis (IDA) and were further used for Table 3. Ground motion records. structural over-strength coefficient analysis. The 20 pairs of ground motions include ten PGV PGD PGA/PGV pairs of far-field ground motions, five pairs of near-field pulsed ground motions, and five No. Ground Motions PGA (g) (cm/s) (cm) (g/(cm/s)) pairs of near-field non-pulsed ground motions. Table 3 lists the related information of RSN126_GAZLI_GAZ090 0.8639 67.62 20.71 0.0128 GM-1 RSN126_GAZLI_GAZ000 0.7017 66.18 27.32 0.0106 these ground motions, where PGA ranges from 0.15 to 1.2 g, and PGA/PGV ranges from RSN160_IMPVALL.H_H-BCR230 0.7769 44.92 15.09 0.0173 GM-2 0.3 to 2.9 gRSN160_IMPV /(cm/s). ALL.H_H-BCR140 0.5987 46.73 20.21 0.0128 RSN165_IMPVALL.H_H-CHI012 0.2699 24.79 9.29 0.0109 GM-3 RSN165_IMPVALL.H_H-CHI282 0.2542 29.89 7.65 0.0085 RSN495_NAHANNI_S1010 1.1079 43.90 6.80 0.0252 Table 3. Ground motion records. GM-4 RSN495_NAHANNI_S1280 1.2007 40.61 10.20 0.0296 RSN741_LOMAP_BRN000 0.4564 51.36 8.11 0.0089 GM-5 RSN741_LOMAP_BRN090 0.5023 44.47 5.05 PGV 0.0113 PGD PGA/PGV No. Ground Motions PGA(g) RSN181_IMPVALL.H_H-E06140 0.4473 66.99 27.88 0.0067 GM-6 (cm/s) (cm) (g/(cm/s)) RSN181_IMPVALL.H_H-E06230 0.4490 113.50 72.85 0.0040 RSN182_IMPVALL.H_H-E07140 0.3408 51.65 27.98 0.0066 GM-7 RSN126_GAZLI_GAZ090 0.8639 67.62 20.71 0.0128 RSN182_IMPVALL.H_H-E07230 0.4691 113.08 46.92 0.0041 GM-1 RSN292_ITALY_A-STU270 0.3205 71.92 29.31 0.0045 GM-8 RSN126_GAZLI_GAZ000 0.7017 66.18 27.32 0.0106 RSN292_ITALY_A-STU000 0.2267 36.96 13.11 0.0061 RSN802_LOMAP_STG000 0.5145 41.56 16.32 0.0124 RSN160_IMPVALL.H_H- GM-9 RSN802_LOMAP_STG090 0.3262 45.95 33.31 0.0071 0.7769 44.92 15.09 0.0173 BCR230 GM-2 RSN160_IMPVALL.H_H- 0.5987 46.73 20.21 0.0128 BCR140 RSN165_IMPVALL.H_H-CHI012 0.2699 24.79 9.29 0.0109 GM-3 RSN165_IMPVALL.H_H-CHI282 0.2542 29.89 7.65 0.0085 RSN495_NAHANNI_S1010 1.1079 43.90 6.80 0.0252 GM-4 RSN495_NAHANNI_S1280 1.2007 40.61 10.20 0.0296 RSN741_LOMAP_BRN000 0.4564 51.36 8.11 0.0089 GM-5 RSN741_LOMAP_BRN090 0.5023 44.47 5.05 0.0113 RSN181_IMPVALL.H_H-E06140 0.4473 66.99 27.88 0.0067 GM-6 RSN181_IMPVALL.H_H-E06230 0.4490 113.50 72.85 0.0040 RSN182_IMPVALL.H_H-E07140 0.3408 51.65 27.98 0.0066 GM-7 RSN182_IMPVALL.H_H-E07230 0.4691 113.08 46.92 0.0041 Buildings 2022, 12, x FOR PEER REVIEW 9 of 19 RSN292_ITALY_A-STU270 0.3205 71.92 29.31 0.0045 GM-8 RSN292_ITALY_A-STU000 0.2267 36.96 13.11 0.0061 RSN802_LOMAP_STG000 0.5145 41.56 16.32 0.0124 GM-9 RSN802_LOMAP_STG090 0.3262 45.95 33.31 0.0071 RSN828_CAPEMEND_PET000 0.5908 49.30 16.59 0.0120 GM-10 RSN828_CAPEMEND_PET090 0.6616 88.47 33.20 0.0075 RSN68_SFERN_PEL090 0.2248 21.71 15.91 0.0104 GM-11 Buildings 2022, 12, 1290 9 of 17 RSN68_SFERN_PEL180 0.1949 16.93 12.87 0.0115 RSN125_FRIULI.A_A-TMZ000 0.3571 22.84 4.59 0.0156 GM-12 RSN125_FRIULI.A_A-TMZ270 0.3151 30.50 5.21 0.0103 Table 3. Cont. RSN169_IMPVALL.H_H- 0.2357 26.31 14.69 0.0090 DLT262 PGV PGD PGA/PGV GM-13 No. Ground Motions PGA (g) RSN169_IMPVALL.H_H- (cm/s) (cm) (g/(cm/s)) 0.3497 32.98 20.17 0.0106 DLT352 RSN828_CAPEMEND_PET000 0.5908 49.30 16.59 0.0120 GM-10 RSN828_CAPEMEND_PET090 0.6616 88.47 33.20 0.0075 RSN174_IMPVALL.H_H-E11140 0.3668 36.00 25.08 0.0102 GM-14 RSN68_SFERN_PEL090 0.2248 21.71 15.91 0.0104 GM-11 RSN174_IMPVALL.H_H-E11230 0.3794 44.59 21.31 0.0085 RSN68_SFERN_PEL180 0.1949 16.93 12.87 0.0115 RSN125_FRIULI.A_A-TMZ000 0.3571 22.84 4.59 0.0156 RSN721_SUPER.B_B-ICC000 0.3573 48.05 19.27 0.0074 GM-12 GM-15 RSN125_FRIULI.A_A-TMZ270 0.3151 30.50 5.21 0.0103 RSN721_SUPER.B_B-ICC090 0.2595 41.77 21.85 0.0062 RSN169_IMPVALL.H_H-DLT262 0.2357 26.31 14.69 0.0090 GM-13 RSN169_IMPVALL.H_H-DLT352 0.3497 32.98 20.17 0.0106 RSN725_SUPER.B_B-POE270 0.4750 41.15 7.73 0.0115 GM-16 RSN174_IMPVALL.H_H-E11140 0.3668 36.00 25.08 0.0102 RSN725_SUPER.B_B-POE360 0.2862 29.00 11.36 0.0099 GM-14 RSN174_IMPVALL.H_H-E11230 0.3794 44.59 21.31 0.0085 RSN752_LOMAP_CAP000 0.5111 38.01 7.06 0.0134 RSN721_SUPER.B_B-ICC000 0.3573 48.05 19.27 0.0074 GM GM-15 -17 RSN721_SUPER.B_B-ICC090 0.2595 41.77 21.85 0.0062 RSN752_LOMAP_CAP090 0.4386 29.60 4.91 0.0148 RSN725_SUPER.B_B-POE270 0.4750 41.15 7.73 0.0115 GM-16 RSN767_LOMAP_G03000 0.5591 36.29 10.84 0.0154 RSN725_SUPER.B_B-POE360 0.2862 29.00 11.36 0.0099 GM-18 RSN752_LOMAP_CAP000 0.5111 38.01 7.06 0.0134 RSN767_LOMAP_G03090 0.3682 45.40 24.09 0.0081 GM-17 RSN752_LOMAP_CAP090 0.4386 29.60 4.91 0.0148 RSN900_LANDERS_YER270 0.2445 51.10 41.69 0.0048 RSN767_LOMAP_G03000 0.5591 36.29 10.84 0.0154 GM-19 GM-18 RSN9 RSN767_LOMAP_G03090 00_LANDERS_YER360 0.3682 0.1518 29.08 45.40 23.13 24.09 0.005 0.00812 RSN900_LANDERS_YER270 0.2445 51.10 41.69 0.0048 RSN953_NORTHR_MUL009 0.4434 59.27 15.47 0.0075 GM-19 RSN900_LANDERS_YER360 0.1518 29.08 23.13 0.0052 GM-20 RSN953_NORTHR_MUL279 0.4880 66.68 12.17 0.0073 RSN953_NORTHR_MUL009 0.4434 59.27 15.47 0.0075 GM-20 RSN953_NORTHR_MUL279 0.4880 66.68 12.17 0.0073 Taking the seismic precautionary intensity VI as an example, the peak acceleration value of each ground motion is adjusted to 0.035 g for the more frequent case, 0.1 g for the Taking the seismic precautionary intensity VI as an example, the peak acceleration precautionary case, 0.22 g for the rare case, and 0.32 g for the extremely rare case [44,45]. value of each ground motion is adjusted to 0.035 g for the more frequent case, 0.1 g for the After calculating the response spectrum of each ground motion, the average response precautionary case, 0.22 g for the rare case, and 0.32 g for the extremely rare case [44,45]. spectrum and the seismic design response spectrum of four different intensities are plot- After calculating the response spectrum of each ground motion, the average response ted in Figure 7. It can be seen that the averaged response spectrum is in good agreement spectrum and the seismic design response spectrum of four different intensities are plotted with the designed response spectrum in the short period, which meets the analysis re- in Figure 7. It can be seen that the averaged response spectrum is in good agreement with quirement the designed s. response spectrum in the short period, which meets the analysis requirements. Figure 7. Average response spectrum and seismic design response spectrum for seismic precautionary intensity VI. 5. Over-Strength Coefficient Capacity Analysis 5.1. Introduction of Over-Strength Coefficient Capacity Analysis In earthquake damage investigations, it has been shown that the actual seismic capacity of the structure is generally higher than the designed seismic capacity, which is referred to as the system over-strength effect. The existence of system over-strength is an important factor that prevents the structure from collapsing when subjected to strong earthquakes greater than the design seismic force [35,40,46]. The system over-strength coefficient R can be defined as the ratio of the actual yielding strength of the structure to the design Buildings 2022, 12, 1290 10 of 17 seismic force, which reflects the strength reserve of the structure. It can be calculated by the following formula: R = V /V (4) s y where V is the yielding strength of the structure, and V is the design seismic force. y d The process of system over-strength coefficient analysis is as follow: (1) Perform incremental dynamic analysis (IDA) of the model under one pair of ground motion, obtain the maximum inter-story drift ratio under different ground motion levels, and plot the IDA curve of the model under this pair of ground motion. (2) Obtain the structural dynamic capacity curve in the form of maximum base shear force (V ) and maximum top displacement (D ), then transform the dynamic max max capacity curve into a double-linear elastic–plastic curve by using the equal energy principle, and solve the equivalent yielding strength V of the structure. (3) Calculate the design seismic force V of the structure by using the bottom shear method or the mode-superposition response spectrum method. (4) Calculate the system over-strength coefficient R by its definition as shown in Formula (4). (5) Repeat the above steps to obtain the system over-strength coefficient under each pair of ground motion. (6) Take the median value as the system over-strength coefficient capacity value. 5.2. Over-Strength Coefficient Capacity Analysis of Infilled RC Frame Structure According to the process described above, the incremental dynamic analysis is first Buildings 2022, 12, x FOR PEER REVIEW 11 of 19 performed. For the sake of brevity and the limitation of text space, only the IDA curves of the three structural models are given, as shown in Figure 8. (a) (b) (c) Figure 8. The IDA curves of 20 pairs of ground motions for (a) Model F6-6; (b) Model IF6-6; (c) Model IFE6-6. Figure 8. The IDA curves of 20 pairs of ground motions for (a) Model F6-6; (b) Model IF6-6; (c) Model IFE6-6. Then, structural dynamic capacity curves are obtained, and the median values of over-strength coefficients of 36 RC frame structures under bidirectional ground motion action are calculated, which are shown in Table 4 and Figure 9. It can be seen that the system over-strength coefficient capacity values of the bare frame structures designed ac- cording to different precautionary intensities of VI, VII and VIII are between 9.7–19.69, 5.24–9.01, 3.31–4.57, which are greater than 9.5, 5.0, 3.0, respectively. The over-strength coefficient capacity values of the fully infilled frame structures for different precautionary intensities are between 21.45–33.62, 11.52–20.51, 6.05–9.83, reaching 21.0, 11.5, and 6.0, re- spectively. The over-strength coefficient capacity values of the infilled frame structures with a bottom soft-story for different precautionary intensities are between 15.39–26.79, 9.76–18.41, 3.63–6.46, reaching 15.0, 9.5, 3.5, respectively. Table 4. Over-strength coefficient values of 36 RC frame structures. Number of Stories 3 6 9 12 precautionary inten- 6 7 8 6 7 8 6 7 8 6 7 8 sity bare frame 9.7 5.24 3.31 14.5 8.1 3.9 16.95 8.9 4.1 19.69 9.01 4.57 Buildings 2022, 12, 1290 11 of 17 Then, structural dynamic capacity curves are obtained, and the median values of over-strength coefficients of 36 RC frame structures under bidirectional ground motion action are calculated, which are shown in Table 4 and Figure 9. It can be seen that the system over-strength coefficient capacity values of the bare frame structures designed according to different precautionary intensities of VI, VII and VIII are between 9.7–19.69, 5.24–9.01, 3.31–4.57, which are greater than 9.5, 5.0, 3.0, respectively. The over-strength coefficient capacity values of the fully infilled frame structures for different precautionary intensities are between 21.45–33.62, 11.52–20.51, 6.05–9.83, reaching 21.0, 11.5, and 6.0, respectively. The over-strength coefficient capacity values of the infilled frame structures with a bottom soft-story for different precautionary intensities are between 15.39–26.79, 9.76–18.41, 3.63–6.46, reaching 15.0, 9.5, 3.5, respectively. Buildings 2022, 12, x FOR PEER REVIEW 12 of 19 Table 4. Over-strength coefficient values of 36 RC frame structures. fully infilled frame 21.45 11.52 6.05 32 20.51 6.22 33.62 17 9.83 29.5 13.5 6.15 Number of Stories 3 6 9 12 bottom soft-story precautionary intensity 6 7 8 6 7 8 6 7 8 6 7 8 15.39 9.76 3.63 22 12.2 6.2 26.79 18.41 5 24 12.5 6.46 frame bar e frame 9.7 5.24 3.31 14.5 8.1 3.9 16.95 8.9 4.1 19.69 9.01 4.57 fully infilled frame 21.45 11.52 6.05 32 20.51 6.22 33.62 17 9.83 29.5 13.5 6.15 bottom soft-story frame 15.39 9.76 3.63 22 12.2 6.2 26.79 18.41 5 24 12.5 6.46 (a) (b) (c) Figure 9. Over-strength coefficient values of 36 RC frame structures: (a) bare frame; (b) fully infilled Figure 9. Over-strength coefficient values of 36 RC frame structures: (a) bare frame; (b) fully infilled frame; (c) bottom soft-story infilled frame. frame; (c) bottom soft-story infilled frame. Specifically, with the increase in the precautionary intensity, the over-strength coef- Specifically, with the increase in the precautionary intensity, the over-strength coeffi- ficient shows a significant decrease trend regardless of the existence of the infill wall. As cient shows a significant decrease trend regardless of the existence of the infill wall. As the the intensity increases, the increase trend of the yielding base shear force 𝑉 of the struc- intensity increases, the increase trend of the yielding base shear force V of the structure is ture is obviously smaller than that of the design seismic force 𝑉 . This may be because ythe higher obviously the precau smaller tionary than intensity, that of the the higher design the se seismic ismic forc for e ce that Vth .e This struct may ure may be be be cause the higher subjected to, and the more difficult it is to realize the seismic capacity of the structure. the precautionary intensity, the higher the seismic force that the structure may be subjected Therefore, the strength reserve of the structure decreases with the increase in the precau- to, and the more difficult it is to realize the seismic capacity of the structure. Therefore, the tionary intensity, which is expressed as the over-strength coefficients that are significantly strength reserve of the structure decreases with the increase in the precautionary intensity, decreasing. which is expressed as the over-strength coefficients that are significantly decreasing. For structures designed according to the same precautionary intensity, the over- strength coefficient of the bare frame structures increases with the increase in the number of stories, and the over-strength coefficients of the fully infilled frame and the bottom soft- story infilled frame structures first increase and then decrease with the increase in the number of stories. For the bare frame structures, as the number of stories increases, the yielding strength 𝑉 increases, and the system over-strength coefficient increases with the increase in the number of stories, although the design seismic force 𝑉 also increases. 𝑑 Buildings 2022, 12, 1290 12 of 17 For structures designed according to the same precautionary intensity, the over- strength coefficient of the bare frame structures increases with the increase in the number of stories, and the over-strength coefficients of the fully infilled frame and the bottom soft-story infilled frame structures first increase and then decrease with the increase in the number of stories. For the bare frame structures, as the number of stories increases, the yielding strength V increases, and the system over-strength coefficient increases with Buildings 2022, 12, x FOR PEER REVIEW the incr ease in the number of stories, although the design seismic force13 V of also 19 increases. However, this growth trend gradually slows down. In general, the natural vibration period is larger for a taller structure, and the seismic influence coefficient used for calculating However, this growth trend gradually slows down. In general, the natural vibration pe- design seismic force is smaller, while under practical earthquake action, due to the influence riod is larger for a taller structure, and the seismic influence coefficient used for calculating of higher-order mode, the yielding force V of the structure is relatively higher. Therefore, design seismic force is smaller, while under practical earthquake action, due to the influ- the strength reserves of high-rise structures are higher. Furthermore, the design of high-rise ence of higher-order mode, the yielding force 𝑉 of the structure is relatively higher. structures in this paper may be more cautious and conservative, which also strengthens Therefore, the strength reserves of high-rise structures are higher. Furthermore, the design the result. For the fully infilled frame and the bottom soft-story infilled frame structures, of high-rise structures in this paper may be more cautious and conservative, which also although strengthens thethe infill result wall . Fohas r the high fully infil rigidity led fr,ame its deformation and the bottomability soft-story is poor infille ,d and frame seri ous brittle structures, although the infill wall has high rigidity, its deformation ability is poor, and damage often occurs. In a high-rise structure, the displacement response is often larger; serious brittle damage often occurs. In a high-rise structure, the displacement response is thus, the damage of infill walls is more serious. As a result, the system over-strength often larger; thus, the damage of infill walls is more serious. As a result, the system over- coefficient decreases in the high-rise structure. strength coefficient decreases in the high-rise structure. To exhibit the effect of infill walls on the over-strength coefficient of the frame structure, To exhibit the effect of infill walls on the over-strength coefficient of the frame struc- the over-strength coefficient capacity values of the three types of structures are plotted ture, the over-strength coefficient capacity values of the three types of structures are plot- together, as shown in Figure 10. It can be seen that the over-strength coefficient of the fully ted together, as shown in Figure 10. It can be seen that the over-strength coefficient of the infilled frame structure is 93.5% higher than that of the bare frame structure on average; the fully infilled frame structure is 93.5% higher than that of the bare frame structure on av- erage; the over-strength coefficient of the bottom soft-story infilled frame structure is over-strength coefficient of the bottom soft-story infilled frame structure is 50.5% higher 50.5% higher than that of the bare frame structure on average; the over-strength coefficient than that of the bare frame structure on average; the over-strength coefficient of the fully of the fully infilled frame structure is increased by 31.8% on average compared with the infilled frame structure is increased by 31.8% on average compared with the result of the result of the bottom soft-story infilled frame structure. bottom soft-story infilled frame structure. (a) (b) (c) Figure 10. Over-strength coefficient values for different precautionary intensities: (a) intensity Ⅵ; Figure 10. Over-strength coefficient values for different precautionary intensities: (a) intensity VI; (b) intensity Ⅶ; (c) intensity Ⅷ. (b) intensity VII; (c) intensity VIII. Buildings 2022, 12, 1290 13 of 17 The over-strength coefficient is a reflection of the seismic capacity of the structure. The existence of the infill wall improves the rigidity and strength of the structure, and the damage of the infill wall will consume a part of the energy so that the yielding strength V of the structure is significantly increased, and the seismic performance of the structure is greatly improved. However, this strengthening effect of infill walls is gradually reduced as the number of stories increases. This may be due to the fact that infill walls are more severely damaged in the high-rise structure. With the increase in the precautionary intensity, the over-strength coefficient of the structure shows a significant decreasing trend regardless of whether the effect of the infill wall is considered. As a result, the effect of the infill wall on the over-strength coefficient has no significant relevance with the precautionary intensity. 6. The Assessment of Over-Strength Coefficient In order to evaluate the rationality of the system over-strength coefficient value, the capacity–demand ratio of the over-strength coefficient is adopted, and the calculation process is as follows: (1) Develop the structural dynamic capacity curve of the model under one pair of ground motion; then, convert the structural capacity curve into the spectral acceleration- displacement curve, that is, the capacity spectrum curve. (2) Convert the seismic response spectrum with 5% damping into the spectral displacement– acceleration format as well, that is, the demand spectrum curve. (3) Solve the intersection of the capacity spectrum and demand spectrum curves by using the dynamic capability spectrum method; then, obtain the target vertex displacement De and the corresponding base shear force V under different strength spectra. (4) Knowing the seismic design force V of the structure, according to Formula (4), calcu- late the demand value of the over-strength coefficient at different strength spectra. (5) Calculate the over-strength coefficient capacity–demand ratio of the structure accord- ing to the following formula. l = R /R (5) Rs sC sD (6) Repeat the above steps to obtain the demand values and capacity–demand ratios of the system over-strength coefficient under each pair of ground motions. (7) Take the median values as the demand value and capacity–demand ratio of the system over-strength coefficient. Using the dynamic capability spectrum method, the over-strength coefficient demand values of 36 structures for the more frequent, precautionary, rare, and extremely rare ground motion demand spectra are calculated. The results are shown in Figure 11. It can be seen from Figure 11a that the demand values of the over-strength coeffi- cients of the bare frame structures under different precautionary intensities are 2.66~2.11, 2.24~1.78, and 2.02~1.59 for the precautionary case and reached 2.1, 1.7, and 1.5, respec- tively. For the rare case, the values are 5.58~3.80, 4.02~3.07, 3.32~2.32 and reached 3.8, 3.0, and 2.3, respectively. In Figure 11b, for the precautionary case, the demand values of fully infilled frame structures are 7.1~2.31, 6.02~2.29, and 3.72~1.54, which are above 2.3, 3.2, and 1.5, respectively. For the rare case, the values are 13.79~5.00, 8.72~4.07, and 5.5~2.47, which are above 5.0, 4.0, and 2.4, respectively. In Figure 11c, for the precautionary case, the demand values of bottom soft-story infilled frame structures are 7.33~2.42, 5.11~1.93, and 3.52~1.44, which are above 2.7, 1.9, and 1.4, respectively. For the rare case, the values are 14.07~4.34, 7.97~3.47, and 5.39~2.66, which are above 4.3, 3.4, and 2.6, respectively. In general, the over-strength coefficient demand value of the three types of frame structures increases with the increase in the seismic demand spectrum intensity, decreases with the increase in the precautionary intensity, and increases with the increase in the number of stories. The fully infilled frame has the largest over-strength coefficient demand value, while the bare frame has the smallest over-strength coefficient demand value. Buildings 2022, 12, 1290 14 of 17 Buildings 2022, 12, x FOR PEER REVIEW 15 of 19 (a) (b) (c) Figure 11. Demand analysis based on dynamic capacity spectra: (a) bare frame; (b) gully infilled Figure 11. Demand analysis based on dynamic capacity spectra: (a) bare frame; (b) gully infilled frame; (c) bottom soft-story infilled frame. frame; (c) bottom soft-story infilled frame. It can be seen from Figure 11a that the demand values of the over-strength coeffi- cients of the bare frame structures under different precautionary intensities are 2.66~2.11, The over-strength coefficient capacity–demand ratio values are obtained by dividing Buildings 2022, 12, x FOR PEER REVIEW 16 of 19 2.24~1.78, and 2.02~1.59 for the precautionary case and reached 2.1, 1.7, and 1.5, respec- the over-strength coefficient capability value by the demand value in the rare case, as tively. For the rare case, the values are 5.58~3.80, 4.02~3.07, 3.32~2.32 and reached 3.8, 3.0, shown in Figure 12. and 2.3, respectively. In Figure 11b, for the precautionary case, the demand values of fully infilled frame structures are 7.1~2.31, 6.02~2.29, and 3.72~1.54, which are above 2.3, 3.2, and 1.5, respectively. For the rare case, the values are 13.79~5.00, 8.72~4.07, and 5.5~2.47, which are above 5.0, 4.0, and 2.4, respectively. In Figure 11c, for the precautionary case, the demand values of bottom soft-story infilled frame structures are 7.33~2.42, 5.11~1.93, and 3.52~1.44, which are above 2.7, 1.9, and 1.4, respectively. For the rare case, the values are 14.07~4.34, 7.97~3.47, and 5.39~2.66, which are above 4.3, 3.4, and 2.6, respectively. In general, the over-strength coefficient demand value of the three types of frame structures increases with the increase in the seismic demand spectrum intensity, decreases with the increase in the precautionary intensity, and increases with the increase in the number of stories. The fully infilled frame has the largest over-strength coefficient demand value, while the bare frame has the smallest over-strength coefficient demand value. The over-strength coefficient capacity–demand ratio values are obtained by dividing the over-strength coefficient capability value by the demand value in the rare case, as shown in Figure 12. (a) (b) (c) Figure 12. The structural capacity–demand ratio of over-strength coefficient: (a) rare frame; (b) fully Figure 12. The structural capacity–demand ratio of over-strength coefficient: (a) rare frame; (b) fully infilled frame; (c) bottom soft-story infilled frame. infilled frame; (c) bottom soft-story infilled frame. It can be seen that for the bare frame, the capacity–demand ratio is between 1.15 and 2.98; for the fully infilled frame, the ratio is between 1.00 and 3.05; they are all greater than or equal to 1. However, for the bottom soft-story infilled frame, the minimum capacity– demand ratio value is only 0.90. Therefore, for the structure designed according to the seismic code, the bare frame has a good strength reserve under the action of a rare earth- quake; however, when the infill wall is considered, although the infill wall can improve the bearing capacity of the structure, the structural stiffness and vibration model have been changed, especially when the bottom infill wall of the structure is missing, the over- strength coefficient capacity–demand ratio is greatly reduced, which indicates that the structure is at risk under the action of rare earthquakes. Therefore, in the design of the masonry-infilled RC frame structure, the arrangement of the soft-story at the bottom should be avoided, or some strengthen technologies should be adopted. 7. Conclusions In this paper, a total of 36 bare frame, fully infilled frame, and bottom soft-story in- filled frame structures with 3, 6, 9, and 12 stories and for seismic precautionary intensities VI, VII and VIII are designed, and three-dimensional finite element modelings are built by using an improved five-element infill wall model in OpenSees. Then, the system over-strength coefficient capacity value of the structures under bi- directional ground motion action is obtained by using the IDA method, and the effect of Buildings 2022, 12, 1290 15 of 17 It can be seen that for the bare frame, the capacity–demand ratio is between 1.15 and 2.98; for the fully infilled frame, the ratio is between 1.00 and 3.05; they are all greater than or equal to 1. However, for the bottom soft-story infilled frame, the minimum capacity–demand ratio value is only 0.90. Therefore, for the structure designed according to the seismic code, the bare frame has a good strength reserve under the action of a rare earthquake; however, when the infill wall is considered, although the infill wall can improve the bearing capacity of the structure, the structural stiffness and vibration model have been changed, especially when the bottom infill wall of the structure is missing, the over-strength coefficient capacity–demand ratio is greatly reduced, which indicates that the structure is at risk under the action of rare earthquakes. Therefore, in the design of the masonry-infilled RC frame structure, the arrangement of the soft-story at the bottom should be avoided, or some strengthen technologies should be adopted. 7. Conclusions In this paper, a total of 36 bare frame, fully infilled frame, and bottom soft-story infilled frame structures with 3, 6, 9, and 12 stories and for seismic precautionary intensities VI, VII and VIII are designed, and three-dimensional finite element modelings are built by using an improved five-element infill wall model in OpenSees. Then, the system over-strength coefficient capacity value of the structures under bidi- rectional ground motion action is obtained by using the IDA method, and the effect of the infill wall on the over-strength coefficient is discussed. It can be concluded that the system over-strength coefficients of the three types of frame structures all decrease with the increase in the precautionary intensity, and the values of over-strength coefficients for different intensities are quite different. For the bare frame structure, the over-strength coef- ficient increases with the increase in the number of stories; for the fully infilled frame and the bottom soft-story infilled frame structures, the over-strength coefficient first increases and then decreases with the increase in the number of stories. The inclusion of infill walls significantly enhances the performance of frame structures; therefore, the over-strength coefficient of the infilled frame structure is significantly higher than that of the bare frame structure, and the coefficient of the fully infilled frame structure is greater than that of the bottom soft-story infilled frame structure. As the number of stories increases, the effect of the infill wall is gradually reduced. However, for different intensities, it does not show a clear rule for the increase in the over-strength coefficient of the infilled frame structure. In addition, the capacity–demand ratio of the over-strength coefficient is analyzed to evaluate the strength reserve of three types of frame structures with and without infill walls. It has shown that the bare frame structure has a good strength reserve, while the seismic performance of masonry-infilled RC frame structures is deteriorated, although the infill wall can improve the bearing capacity of the structure, and the design of the bottom soft-story infilled frame structure requires some additional strengthening treatments. Author Contributions: Conceptualization, J.K.; methodology, X.W.; software, Y.S. and X.W.; formal analysis, Y.S.; investigation, M.G., C.L. and X.W.; resources, J.K., X.W. and Y.S.; writing—original draft preparation, J.K., writing—review and editing, X.W.; visualization, J.K. and X.W.; supervision, M.G. and C.L.; project administration, X.W.; funding acquisition, X.W., M.G., C.L. and J.K. All authors have read and agreed to the published version of the manuscript. Funding: The research is supported by the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (grant nos. 2018B07, 2019A01, and 2021EEEVL0301), the National Nature Science Foundation of China (grant nos. 51808478 and 51908484), and the Natural Science Foundation of Shandong (grant no. ZR2019QEE033). This support is greatly appreciated. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Buildings 2022, 12, 1290 16 of 17 Acknowledgments: Grateful acknowledgment is given to Liang Cui and Bo Liu at the School of Civil Engineering, Yantai University. Conflicts of Interest: The authors declare no conflict of interest. References 1. Applied Technology Council. 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Journal

BuildingsMultidisciplinary Digital Publishing Institute

Published: Aug 23, 2022

Keywords: infilled RC frame; over-strength coefficient; bidirectional ground motions; precautionary intensity; number of stories

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