Analytical study to evaluate the effect of higher modes of reinforced concrete moment-resisting frames with thin steel shear wall under simple pulse

Abstract

The response of flexible structures with long period to near-fault earthquakes shows an imposed demand on these structures which exceeds their capacity. Also, the relationship between frequency content of earthquake and the main frequency of structure is a significant parameter to the response of structure. Therefore, the sensitivity of the response of structure to period of pulse and the lack of enough records for near-fault earthquakes with different amounts of period of pulse is the most important challenge of structural analysis. Of all methods for this analysis, proposed model by Agraval was used in this study. To achieve this goal, various ratios of period of pulse to main period of structure (T_p/T₁) were considered, where the effect of higher modes on estimating displacement demands was assessed. Meanwhile, the distribution of shear forces for 6-, 12-, and 24-story reinforced concrete moment frames with steel shear wall was evaluated. The results showed that maximum displacement and force demands were obtained for different structures with T_p/T₁ = 1. Meanwhile, by increasing the number of stories, the effect of higher modes decreased and structures tended to fluctuate in first mode. Furthermore, the most effect of higher modes was obtained for shear force at the roof (V_roof) and then base shear force (V_base), where displacement of roof (U_roof) did not have any effect on the period of models.

Keywords

concrete frame equivalent pulse higher modes near-fault earthquake steel shear wall

Introduction

Studies on the near-fault earthquakes date back to more than half a century ago. The first near-fault earthquake was reported by Beniof in 1955 when the earthquakes were categorized as near-fault and far-fault earthquakes. These definitions changed gradually by Hazner et al. in 1967 (Soleimani Amiri et al., 2013). The phrase of near-fault velocity pulse was first discussed by Bolt (1975). Later on, extensive studies were conducted on the effects of structures near to fault, which showed that imposed demand on structures with long period exceeds their capacity (Alavi and Krawinkler, 2001). In addition, the relationship between frequency content of earthquake and the main frequency of structure is a significant parameter to the response of structure (Choi et al., 2005). Bertero et al. (1978) evaluated structural damages due to pulse-like nature of earthquake near to fault of San Fernando earthquake. Using earthquake-shock transmission, Westergaard (1933) demonstrated that by approaching period of pulse continuity to main period of the structure, collision of sweep waves in middle floors imposes remarkable demands on the structure. Biggs (1964) showed that the amplitude of simplified pulse and the duration of continuity of pulse, relative to the period of single-degree-of-freedom (SDOF) system, are two main factors for controlling maximum resilient response.

Hall et al. (1995) demonstrated that displacement pulses in the vicinity of fault impose higher displacement demands for tall buildings. Anderson et al. (1999) evaluated the performance of some tall concrete frames with shear walls and some steel frames, subjected to severe pulse-type ground motions. They found that an increase in stiffness by increasing shear walls is not workable for long-period building structures, subjected to severe pulse-like motions. The reason is that this way causes to reduce the period of structure and subsequently leads to higher spectral acceleration. Chai and Loh (1999) demonstrated that demand for resistance is dependent on the period of pulse and the ratio of structural period to period of pulse.

Studies by Alavi and Krawinkler (2001) showed that by keeping the ratio of period of structure to dominant period of velocity pulse (T_p/T₁) in the range of 0.375–3 s, the model of simplified pulse with real record can be used instead of real earthquake record. Also, by obtaining major structure responses for multi-degree-of-freedom (MDOF) in the first mode, the simplified pulse with real record has acceptable results (Mavroeidis et al., 2004). In addition, the results, obtained by Sehhati et al. (2011), showed that by remaining the ratio of period of pulse (T_p) to period of structure (T_s) in the range of 0.5–2.5 s, equivalent pulse model can predict properly structural response for near-fault ground motions. Gerami et al. (2015) used from the equation, proposed by Agraval for pulse-like model, to model multi-story steel moment frame structures. They concluded that maximum displacement and force are generated in different frames for T_p/T₁ = 1.0, while the effect of higher modes decreased by increasing the number of stories.

Fardis evaluated the effects of earthquake near to fault on the lateral displacement of reinforced concrete structures with the ratio of period of structure to dominant period of pulse (T_p/T₁) (Liossatou and Fardis, 2016). Hosseini Vaez and Jahan Abadi (2017) performed some studies on equivalent pulse of pulse-like motions for estimating response reinforced concrete moment frames. They concluded that by increasing the number of stories and approaching period of pulse to main period of the structure, the response of structure can be more accurate.

To ensure the seismic performance of steel shear walls, a number of experimental research have been conducted in recent years. Goodsir et al. (1983) investigated the inelastic seismic response of reinforced concrete coupled frame-shear wall structures. Driver et al. (1996, 1998) performed some studies on four-story shear wall under cyclic loading. An experimental study by Rezai (1999) showed that energy absorption in steel shear walls occurred in the first story and upper stories rotate around the first story, as an axis, similar to solid object. Behbahanifard et al. (2003) conducted cyclic test on a four-story structure with steel shear wall under lateral quasi-static loading in the presence of gravity loads, applied to three upper stories. Furthermore, Veladi et al. (2007) investigated analytically and experimentally the cyclic behavior of non-reinforced and reinforced shear walls. Chen and Jhang (2006) evaluated cyclic behavior of steel shear wall with low yield point steel.

Sabouri-Ghomi and Gholhaki (2008) evaluated the ductility of two thin steel plate shear walls under cyclic loading. They concluded that thin steel plate shear wall had high ductility and strength and also better energy dissipation capacity. Choi and Park (2010) conducted an experimental study on cyclic behavior of reinforced concrete walls with thin steel plates. They showed that their specimens had remarkable strength, deformation capacity, and energy dissipation capacity similar to steel shear wall with boundary elements. Ozkok et al. (2010) studied different rehabilitation methods, including thin steel plate application. They found that the cyclic behavior of specimens including pinching and energy dissipation capacities was found to be quite different for different strengthening methods. Marco Valente (2012) investigated an innovative method based on low-yield steel plate shear walls for seismic retrofitting of existing reinforced concrete structures. Borello and Fahnestock (2017) conducted a full-scale experimental study on cyclic behavior of steel shear wall.

Wen-Yang Liu et al. (2017) investigated experimentally on reinforced concrete frames with two-side-connected buckling-restrained steel plate shear walls. They showed that the presence of buckling-restrained steel plate shear wall can not only enhance the stiffness and load-bearing capacity but also improve the ductility and energy dissipation capacity of reinforced concrete frame structures. Two failure modes are found for reinforced concrete frames with buckling-restrained steel plate shear walls.

According to the previous studies, since there is not any study about the effect of simplified pulse on elastic and inelastic behaviors of reinforced frame with steel shear wall, this system has been investigated in the range of T_p/T₁ in present study. To achieve this goal, 6-, 12-, and 24-story reinforced concrete frames with steel shear wall under simplified sinusoidal pulse were assessed and their effects on the displacement demands and distribution of story shear force, shear at the roof and shear base were evaluated. Meanwhile, the effect of the number of modes using modal spectral dynamic analysis and specially the effect of higher modes to estimate elastic demands for various amounts of T_p/T₁ were investigated. It is noteworthy that the types of pulses were simplified models including rectangular, triangular, and sinusoid.

Verification of finite element model

To verify two-dimensional (2D) numerical model employing OpenSees, an experimental study on one-span and three-story reinforced concrete moment frame with steel plate infilled wall (SPIW1), performed by Choi and Park (2010), was used. Geometric dimensions of reinforcement in the concrete frame are shown in Figure 1. In addition, properties of reinforced concrete frame with SPIW1 are presented in Table 1. The ratio of length to height of wall is equal to 1.5. As proposed by Park et al. (2007), cyclic load has been applied to the specimen to achieve target displacement.

Figure 1.

Configuration and geometric details of reinforcement in concrete frame with SPIW1 (Choi and Park, 2010).

Table 1.

Properties of reinforced concrete frame with SPIW1 (Choi and Park, 2010).

Concrete compressive strength (MPa)			26.4
Steel plate	Thickness (mm)		2
Steel plate	Yield stress (MPa)		302
Column	Longitudinal reinforcement	Area (mm²)	3336^a
	Longitudinal reinforcement	Reinforcement ratio (%)	3.7
	Transverse reinforcement^b	Spacing (mm)	50
	Column dimensions	Width × depth (mm)	300 × 300
Ends Beam	Longitudinal reinforcement	Area (mm²)	794^c
	Longitudinal reinforcement	Reinforcement ratio (%)	0.66
	Transverse reinforcement^b	Spacing (mm)	60
	End beam dimensions	Width × depth (mm)	300 × 400
Middle beam	Longitudinal reinforcement	Area (mm²)	794^c
	Longitudinal reinforcement	Reinforcement ratio (%)	1.3
	Transverse reinforcement^b	Spacing (mm)	60
	Middle beam dimensions	Width × depth (mm)	300 × 200

SPIW1: steel plate infilled wall.

6-D22 (A_b = 387.1 mm², f_y = 430 MPa) and 2-D25 (A_b = 506.7 mm², f_y = 443 MPa).

D10 (A_b = 71.3 mm², f_y = 486 MPa).

4-D16 (A_b = 198.6 mm², f_y = 471 MPa).

Roberts and Sabouri-Ghomi (1992) proposed the method based on the interaction between the steel plate and boundary frame for designing and analyzing of different types of steel shear walls including thin and thick steel plate shear walls, stiffened steel shear wall, and steel shear wall with opening. Thorburn et al. (1983) used the inclined truss member instead of the infill plate in the steel shear wall where was accepted in appendix of Canadian steel design code (CAN/CSA S16-01, 2001). Berman and Bruneau (2003) improved the proposed relationships of Thorburn et al. by determining the ultimate shear capacity of steel shear wall for pinned and rigid beam-to-column connections. Since the analyze time in the proposed method by Roberts and Sabouri-Ghomi is long, in this study, the inclined truss member method presented by Thorburn et al. was used for numerical modeling. This method recognizes the post-buckling strength in thin steel plate shear wall. In thin steel plates, most capacity of plate is in the range of the post-buckling strength owing to the diagonal tension field. As shown in Figure 2, to achieve to the more suitable modeling of the plate in the post-buckling condition, inclined tension strips were used. To model reinforced concrete moment frame, nonlinear beam–column element by controlling deformation was used. This element is capable of applying P-Δ effect and large deformation, which can help to consider nonlinear geometric effects on the model. For plasticity modeling of elements in program, the section of each element divided into a number of fiber and segments, along its length.

Figure 2.

Inclined truss member modeling according to the experimental study by Choi and Park (2010).

The behavioral model of Uniaxial Material Concrete01 was assumed to model concrete materials where tensile strength was equal to zero. Furthermore, the behavioral model of Uniaxial Material Steel01, as a bilinear kinematic hardening model, was used to model rebars. In addition, the behavioral model of Uniaxial Material Hysteretic was considered for simulating plate material which was capable of modeling the behavior of steel in the form of a trilinear curve for tension and compression. The push plot of analytical model is compared with that of experimental study as shown in Figure 3(a), and there is a good compatibility between experimental and numerical initial stiffnesses. However, first, the experimental model existed from linear region and the yielding occurred in it. This is due to the occurrence of cracks in concrete that are impossible to model them in software. Therefore, the yielding and nonlinear behavior of numerical model happened subsequently. In addition, the ultimate capacity of numerical model was very close to experimental model. Generally, the results demonstrated that there is a good compatibility between experimental and numerical results and this is a proper proof for verifying the inclined tension strip model in OpenSees software. According to characteristics of the experimental specimen, the model was designed against lateral loads based on the shear behavior. It is quite evident in the behavior of numerical model. Figures of numerical model and the failure mode for experimental and numerical specimens under lateral loads were indicated in Figure 3(b) and (c), respectively. Evaluating the failure mechanism in Figure 3(b) and (c) clarified that the tension field and the plastic hinge were occurred in the steel plate of shear wall for all stories. Furthermore, plastic hinges were formed in the base of column in the first story and also the end of beams in first and second stories.

Figure 3.

(a) Comparative push plots for experimental specimen and numerical model, (b) failure mechanism for experimental specimen, and (c) failure mechanism for numerical model.

By comparing the results of Table 2, the energy dissipation of finite element model has good compatibility with that of the experimental specimen. So, this demonstrates that the inclined truss member method in dynamic researches can present and predict appropriate energy absorption for analyses.

Table 2.

Comparison results.

Specimen	Maximum load (P_max), kN		Energy absorption (kN m)
Specimen	Experimental	Finite element	Experimental	Finite element
SPIW1	886	903	323.98	349.23

SPIW1: steel plate infilled wall.

Introduction of numerical models

In this study, numerical models were categorized three types including short-height building (H/B ≤ 1.57), middle-height building (1.57 ≤ H/B ≤ 3.14), and tall building (3.14 ≤ H/B ≤ 4.71) with 6, 12, and 24 stories, respectively, where H and B were the height and width of building, respectively. The height of each story was equal to 4 m, and the number of span was five, where the span width was equal to 6 m. As shown in figure 4, these models in second and fourth spans had steel shear walls. The frames and steel shear walls were modeled according to reinforced concrete special moment frame system and inclined truss member method, respectively. Based on Iranian National Building Code, part 6 (2013a) and Iranian Seismic Code (2800 Code (ver. 4), BHRC-2800, 2014), seismic and gravity loadings were calculated for designing models. The dead and live loads were equal to 600 and 200 kg/m², respectively. The kind of structures was considered as residential buildings where building importance factor (I) was 1. In addition, basic design acceleration (A) was equal to 0.35g and the ground was classified as the type III. Since behavioral factor for lateral load resisting system of reinforced concrete moment frames with steel shear wall is not offered, the behavioral factor of 7 was assumed for designing models logically, based on experimental results of concrete frame with steel shear wall (Choi and Park, 2010) and ASCE/SEI 7-10 (2010).

Figure 4.

Geometric properties of two-dimensional frames with 6, 12, and 24 stories: (a) plan and (b) perspective.

To design and analyze the models, ETABS program (Computers and Structures, Inc., 2015) was employed based on American Institute of Steel Construction (AISC, 2010) and American Concrete Institute (ACI, 2015). Then, the results were controlled with Iranian National Building Codes, parts 9 and 10. Furthermore, OpenSees (Mazzoni et al., 2006) was used for finite element modeling based on Iranian National Building Code, part 9 (2013b) where compressive strength of concrete was 25 MPa and class of reinforcement AIII was assumed for numerical modeling. In addition, the type of steel for equivalent brace was A992Fy50.

To model reinforced concrete moment frames, nonlinear beam–column element by controlling deformation was used. This element is capable of applying P-Δ effect and large deformation, which can help to consider nonlinear geometric effects of the model. For plasticity modeling of elements in program, the section of each element divided into a number of fiber and segments, along its length. By considering the fact that the type of steel was A992Fy50 in this study, yield stress, ultimate stress, and modulus of elasticity were equal to 3515, 4569, and $2.03 \times 10^{6} kg / c m^{2}$ , respectively, and the slope of inelastic region was considered 2%. Plot of strain–stress for the rebar, the plate of steel shear walls, and concrete (column and beam) is presented in Figure 5. Meanwhile, inclined truss member method was used for designing steel shear wall. Design process of steel shear wall is indicated in Figure 6.

Figure 5.

Plot of strain–stress: (a) rebar, (b) the plate of steel shear walls, and (c) concrete.

Figure 6.

Design process of steel shear wall using inclined truss member method.

Simplified equivalent pulse model

To define of near-fault earthquake and the effects of pulse-like motion under the progressive condition, real records and simplified pulses can be used. One of the basic limitations for analyzing these types of structures subjected to the earthquake is the lack of access to considerable number of records for near-fault earthquakes. So, different researchers have introduced synthetic pulses for simulating the acceleration of near-fault earthquakes. Simplified pulses can be calculated using various methods, and these researchers have proposed different mathematical forms for these pulses. Simplified equivalent pulses can be calculated using various methods, and different mathematical forms have been proposed by researchers. Agrawal and He (2002) offered a closed-form to define mapping velocity for producing three types of pulses, including A, B, and D. Equation (1) is proposed for pulse D

\begin{matrix} {\overset{\cdot\cdot}{u}}_{g} = V_{p} e^{at} [a \sin (bt) + b \sin (bt)] \\ a = - ξ_{p} ω_{p}, b = ω_{p} \sqrt{1 - ξ_{p}^{2}} \end{matrix}

(1)

where T_p is the period of dominant pulse; ω_p is the period of dominant frequency; V_p and ξ_p are the period of dominant velocity and the damping soil, respectively.

In this study, pulse D was used for analysis where V_p and ξ_p were assumed to be 140 m/s and 5%, respectively. Also, entry time delay was equal to 5 s. The main issue regarding equivalent pulse is that the frequency content for earthquake near to fault cannot be introduced properly.

The pulse-like model of the Landers earthquake with main records in the Lakren station is indicated in Figure 7, where pulse D has enough accuracy compared to pulses A and B (Siapolo, 2015).

Figure 7.

Comparison of synthetic pulses with main pulse for Landers earthquake in the Lakren station (Siapolo, 2015).

For producing simplified pulse, the ratio of T_p/T₁ was assumed to be 0.5, 0.8, 1, 1.2, and 1.5. To generate inelastic response spectrum for SDOF system for various amounts of T_p/T₁, SeismoSignal program (Antoniou and Pihno, 2012) was used. The plot of acceleration time history of pulse D for each ratio of T_p/T₁ in six-story model is shown in Figure 8.

Figure 8.

Plot of acceleration time history of pulse D for each ratio of T_p/T₁ in six-story model.

Evaluation of the effect of simplified pulse on the displacement demands and shear force of stories

To evaluate the effect of simplified pulse on the displacement demands and shear force of stories, all models with different amounts of T_p/T₁ were analyzed using modal spectral dynamic analysis. Displacement demands including maximum displacement, drift angle of structure, and relative drift angle of stories were assessed. In addition, the distribution of story shear forces along the height of structure with different amounts of T_p/T₁ was investigated. Maximum and average displacement demands are indicated in Figure 9. Furthermore, maximum and average story shear force, distributed along the height of structure with different amounts of T_p/T₁, are shown in Figure 10.

Figure 9.

Maximum and average displacement demands for different amounts of T_p/T₁: (a) maximum displacement, (b) maximum floor drift angle (%), and (c) relative drift angle (%).

Figure 10.

Maximum and average story shear forces, distributed along the height of structure with different amounts of T_p/T₁: (a) 6 stories, (b) 12 stories, and (c) 24 stories.

Displacement demands are as significant and effective items for higher modes in the models. In six-story model, difference among maximum relative drift angles of stories is negligible, which shows that the model tends to fluctuate in the main mode. In this mode, relative drift angle is distributed among stories uniformly, which causes maximum relative drift angle among stories to be close to each other. By increasing height of structure, not only is distribution of maximum relative drift angle among stories affected by higher modes but also this distribution is varied for different amounts of T_p/T₁. In 12- and 24-story models, the effect of higher modes causes that displacement demands tend to collect upper stories where this effect is enhanced by increasing the amounts of T_p/T₁. As shown in Figure 10, by increasing and closing the amount of T_p/T₁ to 1, maximum relative drift angle of stories is obtained. For all models, maximum shear force is occurred in lower stories; meanwhile, by increasing the number of stories, shear forces of upper stories for various amounts of T_p/T₁ are close to each other and the higher modes do not have any special effect on distribution of shear forces.

Investigation of simplified pulse on displacement demands and shear force at the roof

The modal spectral dynamic analysis is commonly used in a number of codes including IBC, UBC, and FEMA. In this method, MDOF structure was separated into the number of SDOF structures by means of the principle of the separation of variables. The period of each separated SDOF structure is the representative of period of each mode of the MDOF structure. The merit of equivalency type is the simplification of the calculation of force and displacement demands for each one of the SDOF structures. The notable point in modal analysis is the use of a limited number of modes can lead to reducing volume of analysis for calculating ultimate the response of structure. This matter is a positive property of modal analysis. It is obvious that the number of modes is dependent on the shape of loading, response spectrum, and properties of structure. After estimating response of each SDOF system, reflection spectrums can be combined using conventional methods including complete quadratic combination (CQC) and square root of the sum of the squares (SRSS). Iranian Seismic Code (BHRC-2800, 2014) offers three criteria for determining the number of modes:

At least three first modes.

All modes with period more than 0.4 s.

All modes which sum of their effective mass is more than 90% of total weight of structure.

The effect of the number of modes on shear force at the roof (V_roof), base shear force (V_base), and displacement of roof (U_roof) has been evaluated.

The maximum number of selected modes was considered in such way to provide 100% reflection spectrum; therefore, modal collinearity factor (MCF) can be calculated using equation (2)

MCF = \frac{R_{N}}{R_{N max}} \times 100

(2)

where R_N and R_N_max are corresponding reflection with N mode and corresponding response with maximum number of mode which produces 100% reflection, respectively. The modal combination method of SRSS was used for all models and their results were drawn for various amounts of T_p/T₁.

Figures 11 to 13 show the effect of higher modes on V_roof, V_base, and U_roof, respectively, for models with various amounts of T_p/T₁.

Figure 11.

Effect of higher modes on V_roof for models with various amounts of T_p/T₁: (a) 6 stories, (b) 12 stories, and (c) 24 stories.

Figure 12.

Effect of higher modes on V_base for models with various amounts of T_p/T₁: (a) 6 stories, (b) 12 stories, and (c) 24 stories.

Figure 13.

Effect of higher modes on U_roof for models with various amounts of T_p/T₁: (a) 6 stories, (b) 12 stories, and (c) 24 stories.

As shown in Figure 11, in six-story model with T_p/T₁ = 0.5, the modal participation for first mode is equal to 13.84% of total reflection (six modes). Meanwhile, by increasing the amount of T_p/T₁, the modal participation increases in all structures. The highest effect of first mode for all structures is for T_p/T₁ = 1, where the models tend to fluctuate in first mode. In other words, if T_p is far less than T₁, an increase in the number of stories leads to a decrease in interfering of first mode in total response; therefore, the effect of higher modes on V_roof for low ratio of T_p/T₁ is more. This process can be shown for two other reflections. In 6-, 12-, and 12-story models with T_p/T₁ = 0.5, modal participations for first mode in total response of V_base are equal to 78%, 72.19%, and 37.27%. Concerning V_base, by increasing the amount of T_p/T₁, the modal participation increases as shown in Figure 12. For instance, the effect of V_roof and V_base for 12-story model with T_p/T₁ = 0.8 is calculated 16.22% and 97.9%, respectively. As indicated in Figure 13, the lowest effect of the number of mode on reflection of structure is for U_roof, where by increasing the amount of T_p/T₁, U_roof is estimated acceptably with the use of one mode.

By comparing the results of Figures 11 to 13, it can be concluded that the effect of higher modes on response of V_roof has the highest significance; thereafter, they affect V_base, and finally, the effect of higher modes on response of U_roof does not have so much importance. By increasing and closing the amount of T_p/T₁ to 1 at all reflections shown in Figures 11 to 13, the importance of first mode increases. In other words, the use of first mode for T_p/T₁ between 0.8 and 1.2 can lead to the acceptable results.

The parameter of $α_{R}^{el}$ can be calculated using equation (3) for evaluating the effect of higher modes

α_{R}^{el} = \frac{R_{N} - R_{1}}{R_{N}} \times 100

(3)

where R is representative of various reflections including V_roof, V_base, and U_roof. Meanwhile, $α_{R}^{el}$ is the error percentage of removal of higher Modes for calculating R reflection from N-story model which has been obtained by inelastic modal analysis, R₁; and R_N and R_N_max are reflection with one and N modes, respectively.

The amounts of $α_{R}^{el}$ for 6-, 12-, and 24-story models with various ratios of T_p/T₁ are presented in Table 3. For better understanding the results of each ratio of T_p/T₁, the amounts of $α_{R}^{el}$ with main period of structures are shown in Figure 14.

Table 3.

Removal of higher modes for calculating various reflections in models.

No. of stories	T₁ (s)	T_p/T₁	$α_{R}^{el}$
No. of stories	T₁ (s)	T_p/T₁	V_roof	V_base	U_roof
6	0.93	0.5	86.15	21.77	0.85
		0.8	52.86	1.93	0.05
		1	30.57	0.52	0.01
		1.2	46.85	1.1	0.03
		1.5	50.69	1.28	0.03
12	1.59	0.5	91.87	27.8	1.26
		0.8	83.78	4.23	0.12
		1	61.41	1.06	0.03
		1.2	68.75	1.17	0.03
		1.5	77.59	2.03	0.05
24	2.89	0.5	99.22	62.73	5.27
		0.8	94.83	8.68	0.19
		1	90.05	2.01	0.04
		1.2	95.25	5.1	0.09
		1.5	94.53	4.22	0.07

Figure 14.

Effect of higher modes on $α_{R}^{el}$ for models with various amounts of T_p/T_1.

As shown in Figure 14, the maximum amount of $α_{R}^{el}$ for various amounts of T_p/T₁ is for V_roof and then V_base and finally U_roof. This matter shows that U_roof does not have any effect on the period of models. In addition, by increasing the number of stories and subsequently an increase in period of structure, V_roof increases. As shown in Figure 14, by increasing the amount of T_p/T₁, $α_{R}^{el}$ decreases. In other words, by increasing the amount of T_p/T₁, the effect of higher modes decreases for various amounts of periods.

Effect of higher modes in maximum drift angle of stories

If stiffness and strength of MDOF under a loading pattern are distributed in a way that the shape of inelastic deformation is a straight line, all stories are yielded simultaneously and the results of analysis show that drift angle between stories are the same and can be predicted equal to global drift ratio $(δ_{t} / h_{t})$ , while the amounts of drift angle between stories, obtained from dynamic analysis, vary remarkably along the height. However, if the structure is affected by higher modes not just first mode, the amounts of drift angle between stories are different from the case of the drift angle, calculated based on all modes of structure. For determining this effect, the parameter of $α_{HME}^{d, el}$ , as a criterion for the effect of higher modes in inelastic story drift, can be defined using equation (4)

α_{HME}^{d, el} = \frac{δ_{IDR, max}^{N} - δ_{IDR, max}^{1}}{δ_{IDR, max}^{N}} \times 100

(4)

where $δ_{IDR, max}$ is the maximum drift angle between stories, which can be calculated under the effect of one and N modes. The distribution of $α_{HME}^{d, el}$ for different stories of 6-, 12-, and 24-story models is shown in Figure 15, where the distribution of $α_{HME}^{d, el}$ along the height is drawn for various ratios of T_p/T₁ in each plot. As shown in plots of Figure 15, maximum amount of $α_{HME}^{d, el}$ occurred in the lower stories for all models. In addition, maximum amount of $α_{HME}^{d, el}$ is obtained for T_p/T₁ = 0.5. By increasing the amount of T_p/T₁ from 0.5, $α_{HME}^{d, el}$ decreases sharply. From 0.83% height of all stories to higher, $α_{HME}^{d, el}$ converges to zero, while the amount of $α_{HME}^{d, el}$ for T_p/T₁ = 0.5 is occurred after 0.83% height. This equivalent point is a turning point of the effect of higher modes on the enhancement of drift between upper stories of tall building.

Figure 15.

Rate of change of $α_{HME}^{d, el}$ for models with various amounts of T_p/T₁: (a) 6 stories, (b) 12 stories, and (c) 24 stories.

Conclusion

In this study, the ability of equivalent pulse for estimating response of 6-, 12-, and 24-story reinforced concrete moment frames with thin steel plate shear wall, near to the fault with the effect of higher modes, was evaluated using the mathematical form proposed by Agraval. Furthermore, displacements and shear force demands with the various ratios of T_p/T₁ were investigated using modal spectral dynamic analysis. The results showed that by increasing height of structure not only is distribution of the maximum relative drift angle among stories affected by higher modes but also this distribution is varied for different amounts of T_p/T₁. The difference among maximum relative drift angles of stories is negligible which shows that the model tends to fluctuate in the main mode. In this mode, the relative drift angle is distributed among stories uniformly. However, by increasing and closing the amount of T_p/T₁ to 1, maximum relative drift angle of stories is obtained, and the higher modes do not have any special effect on distribution of shear forces. For all modes, the highest effect of first mode is for T_p/T₁ = 1; as if, the structure tends to fluctuate in first mode completely. The effect of equivalent pulse on displacements, the base shear and the roof shear demands demonstrated that the effect of higher modes on V_roof for lower ratios of T_p/T₁ is obvious. This manner can be shown in two other reflections (V_base and U_roof). In addition, the effect of higher modes has the highest significance to response of V_roof and then V_base and finally the effect of higher modes on response of U_roof does not have so much importance. To evaluate the effect of higher modes on various reflections of structure, the maximum amount of $α_{R}^{el}$ for various amounts of T_p/T₁ is for V_roof and then V_base and finally U_roof. This matter shows that U_roof does not have any effect on the period of models. In addition, by increasing the number of stories and subsequently an increase in period of structure, V_roof increases. In the end of study, the effect of higher modes on the elastic drift of story was studied, and the results showed that maximum value of $α_{HME}^{d, el}$ is occurred in the lower stories for all models where this value is obtained for T_p/T₁ = 0.5. In this condition, the probability of collinearity of the second and third modes is increased for determining the reflections of structure. In addition, if the period of the second or third modes approaches to T_p, ignoring the higher modes can lead to unrealistic results. Following this, the factor of $α_{HME}^{d, el}$ is sharply increased due to higher modes effects in upper stories. In addition, minimum amount of $α_{HME}^{d, el}$ is obtained for T_p/T₁ = 1; because, the major structure response is in the first mode and also the use of one mode can be obtained an acceptable estimation for the structure response.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Agrawal

(2002) A close-form approximation of near-fault ground motion pulses for flexible structures. In: ASCE engineering mechanics conference, New York, 2–5 June.

Alavi

Krawinkler

(2001) Effects of Near-Fault Ground Motions on Frame Structures. Stanford, CA: John A. Blume Earthquake Engineering Center.

American Concrete Institute (ACI) (2015) Building Code Requirements for Structural Concrete (ACI 318-14), an ACI Standard: Commentary on Building Code Requirements for Structural Concrete (ACI 318R-14), an ACI Report. Farmington Hills, MI: ACI.

American Institute of Steel Construction (AISC) (2010) Specification for Structural Steel Buildings (ANSI/AISC 360-10). Chicago, IL: AISC.

Anderson

Bertero

(1999) Performance Improvement of Long Period Building Structures Subjected to Severe Pulse-type Ground Motions. Berkeley, CA: Pacific Earthquake Engineering Research Center, College of Engineering, University of California.

Antoniou

Pihno

(2012) SeismoSignal 5.0. Pavia: Seismosoft srl.

ASCE/SEI 7-10 (2010) Minimum design loads for buildings and other structures. Available at: https://dx-doi-org.web.bisu.edu.cn/10.1061/9780784412916

Behbahanifard

Grondin

Elwi

(2003) Experimental and numerical investigation of steel plate shear walls. Available at: https://era.library.ualberta.ca/items/d3a1bbad-4280-43e9-af85-55e7b4d77a4a/view/70a025b9-731c-4020-9e00-b928f4a51b2f/SER254.pdf

Berman

Bruneau

(2003) Plastic analysis and design of steel plate shear walls. Journal of Structural Engineering 129(11): 1448–1456.

10.

Bertero

Mahin

Herrera

(1978) Aseismic design implications of near-fault San Fernando earthquake records. Earthquake Engineering & Structural Dynamics 6(1): 31–42.

11.

BHRC-2800 (2014) Iranian Code of Practice for Seismic Resistant Design of Buildings: Standard No. 2800. 4th ed.Tehran, Iran: Ministry of Roads & Urban Development (in Persian).

12.

Biggs

(1964) Introduction to Structural Dynamics. New York: McGraw-Hill College.

13.

Bolt

(1975) The San Fernando earthquake, 1971. Magnitudes, aftershocks, and fault dynamics. In: Bulletin 196. Sacramento, CA: California Division of Mines and Geology.

14.

Borello

Fahnestock

(2017) Large-scale cyclic testing of steel-plate shear walls with coupling. Journal of Structural Engineering 143(10): 04017133.

15.

CAN/CSA S16-01 (2001) Limit states design of steel structures.

16.

Chai

Loh

(1999) Near-fault ground motion and its effect on civil structures. In: International workshop on mitigation of seismic effects on transportation structures, Taipei, Taiwan, July 12–14, pp. 70–81. Taipei, Taiwan: National Center for Research on Earthquake Engineering.

17.

Chen

S-J

Jhang

(2006) Cyclic behavior of low yield point steel shear walls. Thin-Walled Structures 44(7): 730–738.

18.

Choi

I-K

Kim

Choun

Y-S

et al . (2005) Shaking table test of steel frame structures subjected to scenario earthquakes. Nuclear Engineering and Technology 37(2): 191–200.

19.

Choi

I-R

Park

H-G

(2010) Cyclic loading test for reinforced concrete frame with thin steel infill plate. Journal of Structural Engineering 137(6): 654–664.

20.

Computers and Structures, Inc. (2015) ETABS 15.0. Berkeley, CA: Computers and Structures, Inc.

21.

Driver

Kulak

Kennedy

et al . (1996) Seismic performance of steel plate shear walls based on a large-scale multi-storey test. In: Proceedings of the 11th world conference on earthquake engineering, Acapulco, Mexico, June 1996, p. 8.

22.

Driver

Kulak

Kennedy

et al . (1998) cyclic test of four-story steel plate shear wall. Journal of Structural Engineering 124(2): 112–120.

23.

Goodsir

Paulay

Carr

(1983) A study of the inelastic seismic response of reinforced concrete coupled frame-shear wall structures. Bulletin of the New Zealand National Society 16: 185–200.

24.

Hall

Heaton

Halling

et al . (1995) Near-source ground motion and its effects on flexible buildings. Earthquake Spectra 11(4): 569–605.

25.

Hosseini Vaez

Jahan Abadi

(2017) Evaluation equivalent pulse of pulse-like ground motion to estimate the response of RC moment-resisting frames. Journal of Structural and Construction Engineering 4(2): 47–67.

26.

Iranian National Building Code (2013a) Applied Loads on Buildings, Part 6. Tehran, Iran: Ministry of Roads & Urban Development (in Persian).

27.

Iranian National Building Code (2013b) Design and Implement of Concrete Buildings, Part 9. Tehran, Iran: Ministry of Roads & Urban Development (in Persian).

28.

Liossatou

Fardis

(2016) Near-fault effects on residual displacements of RC structures. Earthquake Engineering & Structural Dynamics 45(9): 1391–1409.

29.

Liu

W-Y

G-Q

Jiang

(2017) Experimental study on reinforced concrete frames with two-side connected buckling-restrained steel plate shear walls. Advances in Structural Engineering 21: 460–473.

30.

Mavroeidis

Dong

Papageorgiou

(2004) Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthquake Engineering & Structural Dynamics 33(9): 1023–1049.

31.

Mazzoni

McKenna

Scott

et al . (2006) The Open System for Earthquake Engineering Simulation (OpenSees) user command-language manual. Available at: http://opensees.berkeley.edu/OpenSees/manuals/usermanual/

32.

Ozkok

Ozcelik

Binici

(2010) Performance comparisons of external strengthening methods for deficient RC frames. In: Improving the seismic performance of existing buildings and other structures, San Francisco, California, USA, 9–11 December 2009, pp. 926–936.

33.

Park

H-G

Kwack

J-H

Jeon

S-W

et al . (2007) Framed steel plate wall behavior under cyclic lateral loading. Journal of Structural Engineering 133(3): 378–388.

34.

Rezai

(1999) Seismic Behaviour of Steel Plate Shear Walls by Shake Table Testing. Vancouver, BC, Canada: University of British Columbia.

35.

Roberts

Sabouri-Ghomi

(1992) Hysteretic characteristics of unstiffened perforated steel plate shear panels. Thin-Walled Structures 14(2): 139–151.

36.

Sabouri-Ghomi

Ghol

(2008) Ductility of thin steel plate shear walls. Asian Journal of Civil Engineering (Building and Housing) 9(2): 153–166.

37.

Sehhati

Rodriguez-Marek

ElGawady

et al . (2011) Effects of near-fault ground motions and equivalent pulses on multi-story structures. Engineering Structures 33(3): 767–779.

38.

Gerami

Siapolo

(2015) The Effect of Near-Field Earthquake on Seismic Demand of SMRFs with MDOF’s Considerations. Semnan, Iran: Semnan University (in Persian).

39.

Soleimani Amiri

Ghodrati Amiri

Razeghi

(2013) Estimation of seismic demands of steel frames subjected to near-fault earthquakes having forward directivity and comparing with pushover analysis results. The Structural Design of Tall and Special Buildings 22(13): 975–988.

40.

Thorburn

Kulak

Montgomery

(1983) Analysis of steel plate shear walls. Structural engineering report no.107. Edmonton, AB, Canada: Department of Civil Engineering, The University of Alberta.

41.

Valente

(2012) Seismic performance of existing R/C frames strengthened by steel plate shear walls. Applied Mechanics and Materials 193–194: 1470–1475.

42.

Veladi

Armaghani

Davaran

(2007) Experimental investigation on cyclic behavior of steel shear walls. Asian Journal of Civil Engineering (Building and Housing) 8(1): 63–75.

43.

Westergaard

(1933) Earthquake-shock transmission in tall buildings. Engineering News-Record 111(22): 654–656.