Shear strength of steel plate reinforced concrete shear wall

Abstract

This article investigates the shear strength of steel plate reinforced concrete shear wall under cyclic loads. A nonlinear three-dimensional finite element model in ABAQUS was developed and validated against published experimental results. Then, a parametric study was conducted to evaluate the effects of the parameters on the lateral capacity of composite shear wall, including shear span ratio, concrete strength, axial load ratio, steel plate ratio and transverse reinforcement ratio of the web. Furthermore, a modified formula of shear strength of composite shear wall was proposed. Regression analyses were used to obtain the contribution coefficients of different parts from 720 finite element models. Finally, the shear strengths of specimens from published tests were compared with design strengths calculated using the proposed formula, American Institute of Steel Construction Provisions and Chinese Code. It was found that the Chinese Code well predicts the shear strength of composite shear wall of a steel plate ratio of less than 5%, while unsafely predicting that of a higher steel plate ratio. The American Institute of Steel Construction Provisions predictions are quite conservative because the contribution of the reinforced concrete is neglected. The modified formula safely predicts the shear strength of composite shear wall.

Keywords

design code finite element model regression analyses shear strength steel plate reinforced concrete shear wall

Introduction

In high-rise buildings, shear walls are generally designed as the main lateral resistance of structures. Steel–concrete composite shear wall can provide sufficient stiffness and strength with better capacity of energy dissipation than conventional reinforced concrete (RC) shear walls. Steel plate reinforced concrete (SPRC) shear wall consists of steel plate in the RC wall. The co-working of the steel plate and RC makes full use of the advantages of both steel and concrete. RC on both sides of the steel plate confines and prevents it from failure, that is, local buckling. Shear yielding mechanism of the steel plate will result in more stable hysteretic loops than for RC walls (American Institute of Steel Construction (AISC), 2016).

Tests of SPRC shear walls have been carried out by many researchers. Zhao and Astaneh-Asl (2004) tested a series of composite shear walls and found them rather promising with better capacity of energy dissipation. Cao et al. (2013) conducted pushing out tests of SPRC shear walls with arrayed studs under monotonic loading and discussed the effects of different configurations of shear studs on the shear capacity of SPRC shear walls. It was observed that the shear capacity is obviously improved using smaller diameter studs or decreasing the distance between studs when the total area of studs is kept unchanged. Several researchers (Cao et al., 2012; Gan et al., 2008; Ji et al., 2017; Wang et al., 2018; Wu, 2014; Xiao et al., 2012; Zhu, 2015) have conducted experimental study on lateral behaviour of SPRC shear walls. The effects of parameters, including shear span ratio, reinforcement ratio, thickness of the concrete wall and steel plate and so on, on the cyclic behaviour of SPRC shear walls were investigated. Ji et al. (2017) compared the shear strengths calculated using Chinese Code for design of composite structures and AISC Seismic provisions for structural steel buildings. It was found that the Chinese Code conservatively predicts the shear strengths of most specimens, while the AISC Provisions significantly underestimates them.

Numerical modelling has been carried out by researchers to better understand the behaviour of SPRC shear walls. Rahnavard et al. (2016) conducted numerical analysis on different types of composite steel–concrete shear walls to study their failure modes and energy dissipation capacity. Jin et al. (2016) simulated the behaviour of steel plate shear walls with inclined-slots. Both simulations found out that the concrete on two sides of steel plate can completely prevent it from local buckling. A numerical model by OpenSees was developed by Wang et al. (2017) to study the cyclic behaviour of composite shear walls. It was noted that the bond-slip behaviour can be ignored in composite shear walls with enough shear studs to coordinate the deformation between steel plate and concrete.

Current experimental studies on SPRC shear walls are still focused on a limited range of parameters. Few finite element studies on the cyclic loading of SPRC shear walls have been performed. The main objective of this study is to develop an efficient nonlinear three-dimensional (3D) finite element model to investigate the shear strength of SPRC shear wall under cyclic loads. The effects of concrete confinement were carefully considered. The finite element model was validated against published test results. Parametric study was conducted to investigate the effects of the main parameters on the shear strength of composite shear walls under cyclic loads, including shear span ratio, concrete strength, axial load ratio, steel plate ratio and transverse reinforcement ratio. In addition, 720 models with different parameters were developed and analysed. Based on a linear combination of shear contributions of different parts of composite shear wall, regression analyses were conducted to propose a modified formula of the shear strength of composite shear wall. Finally, the shear strengths of composite shear walls calculated using Chinese Code, AISC Provisions and proposed formula were compared with those of the published tests.

Finite element modelling

Finite element software ABAQUS is used to simulate the behaviour of SPRC shear walls. The main modelling approach is described as follows.

Modelling approach

The SPRC composite shear wall studied in this article is composed of four parts, namely concrete, structural steel, steel plate, longitudinal and transverse reinforcement bars in wall web and boundary columns. The concrete part is divided into three zones, which are highly confined concrete, partially confined concrete and unconfined concrete with different states of confinement. The highly confined concrete is taken from the web of the structural steel to the mid-width of each flange outstand. The unconfined concrete is taken from the centreline of the longitudinal reinforcement to the concrete cover. The partially confined concrete is the remaining part, which is confined by steel wires that pass through the reserved holes in steel plate as shown in Figure 1(a) (Ellobody and Young, 2006a). The modelling of SPRC shear walls is shown in Figure 2.

Figure 1.

(a) Typical section of SPRC shear wall, (b) simplified confinement zones of concrete and (c) actual confinement zones of concrete.

Figure 2.

Modelling of steel plate reinforced concrete shear walls: (a) steel and reinforcement, (b) concrete and (c) finite element model.

Finite element type, mesh, modelling of interfaces and boundary conditions

The concrete is modelled by the 3D 8-node solid elements with reduced integration C3D8R. The steel plate is modelled by the 4-node doubly curved shell elements with reduced integration S4R. The 2-node linear displacement truss elements are used to model the steel rebars. An average mesh size of 50 mm (depth) ×25 mm (width) × 50 mm (height) was applied to most of the concrete elements. The mesh size of steel plate and reinforcement is approximately 50 mm. It was found that the mesh can provide both accurate results and less computational time. Well-distributed studs welded to the steel plate can almost assure the co-working of concrete and steel plate. Moreover, high-strength steel wires that pass through the reserved holes in steel plate tie to the transverse rebars in concrete wall web to strengthen the connect between concrete and steel plate. As the tests show, these configurations can well control the concrete and steel plate to deform together. Thus, the possible sliding between concrete and steel plate is ignored. The steel plate, structural steel and reinforcements are embedded into the concrete using the *Embedded Constraint option in ABAQUS, assuming a perfect bond. The bottom surface of the lower loading beam was restrained against all degrees of freedom. The side surface of the upper loading beam was coupled to a reference point located at its centre. The displacement loading was applied to the reference point. The reference point is free to displace in other directions.

Material modelling of concrete

Concrete was modelled using the damaged plasticity model for concrete (concrete damaged plasticity (CDP) model) implemented in ABAQUS. The concrete strength and ductility under the confinement of steel and reinforcements is considerably improved. As illustrated by Ellobody and Young (2006b), the CDP model may not be able to accurately simulate the compressive behaviour of concrete under high level of confinement. In this case, a confined concrete model needs to be developed. The input concrete compressive strength f_c will be taken as the unconfined concrete cylinder compressive strength f_c0, while the corresponding concrete strain ε_c will be taken as the confined strain ε_cc. According to Yu et al. (2010), parameters that control the yield surface and flow rule are altered for partially and highly confined concrete, respectively. For high confined concrete, K_c = 0.725 and ψ = 56°. For partially confined concrete, K_c = 0.667 and ψ = 42°. To improve the convergence speed, a viscosity parameter μ equal to 0.001 is introduced.

(1) Unconfined concrete

The concrete compressive strain ε_c is taken as ε_c0, which is the unconfined concrete strain corresponding to the unconfined concrete strength f_c0, taken as 0.003 as recommended by fib model code for concrete structures (fib, 2013).

(2) Partially confined concrete

The compressive strain ε_c of confined concrete can be determined by equation (1), proposed by Mander et al. (1988) and improved by Denavit et al. (2011)

ε_{c} = ε_{c 0} (1 + 5 (K - 1))

(1)

K = 1 + A f_{l} (0.1 + \frac{0.9}{1 + B f_{l}})

(2)

A = 6.8886 - (0.6069 + 17.275 r) e^{- 4.989 r}

(3)

B = \frac{4.5 A}{5 (0.9849 - 0.6306 e^{- 3.8939 r}) - 0.1 A} - 5

(4)

f_{l} = \frac{f_{l 1} + f_{l 2}}{2 f_{c 0}}

(5)

r = \frac{f_{l 1}}{f_{l 2}}, f_{l 1} \leq f_{l 2}

(6)

where f_l is the equivalent lateral confining pressure, f_l1 and f_l2 are the lateral confining pressures imposed by the reinforcement bars with different directions, respectively, as given by Mander (1988).

(3)Highly confined concrete

High confinement of concrete is provided by the steel and reinforcements. The lateral confining pressure (f′_ly) imposed by the flange of the steel is given by Huang et al. (2017)

f'_{ly} = K_{e} {f ″}_{ly}

(7)

{f ″}_{ly} = \frac{f_{ys} t^{2}}{\sqrt{9 l^{4} + 3 l^{2} t^{2}}}

(8)

K_{e} = \frac{(L - h) (H - 2 t) - (1 / 3) {(H - 2 t)}^{2}}{(L - h) (H - 2 t)}

(9)

where f_ys is the yield strength of the steel, L is the flange length, H is the web length and t is the flange thickness of the steel. The confined strain ε_c of the highly confined concrete can be determined by the lateral confining pressure imposed by the reinforcements and steel in different directions.

The model proposed by Popovics (1973) and Collins (1993) was adopted to simulate the concrete response under uniaxial compression. The initial Young’s modulus of concrete (E_c) is taken as $4730 f_{c}^{0.5}$ given by the ACI Specification (2008). The Poisson’s ratio of concrete is taken as 0.2. Under uniaxial tension, the stress–strain response follows a linear elastic relationship until reaching the value of the failure stress. The tensile failure stress was assumed to be $0.395 f_{cu}^{0.55}$ where f_cu is the cubic compressive strength of concrete. The softening stress strain response, past the maximum tensile stress, was represented by a linear line defined by the fracture energy G_f and crack opening w. The fracture energy in N/mm was taken as $73 f_{c}^{0.18}$ as recommended by the fib code (fib, 2013).

The CDP model assumes that the reduction of the elastic modulus is given in terms of a scalar degradation variable d described as E = (1 − d) × E₀, where E₀ is the initial modulus. The stiffness degradation variable d is a function of the stress state and the uniaxial damage variables d_t and d_c. According to Birtel and Mark (2006), the uniaxial damage variables d_t and d_c can be calculated by equation (10). It is assumed that the plastic strain ε_c(t)^pl is proportional to the inelastic strain ε_c(t)ⁱⁿ, that is ε_c(t)^pl = b_c(t)ε_c(t)ⁱⁿ

d_{c (t)} = 1 - \frac{σ_{c (t)} E_{0}^{- 1}}{ε_{c (t)}^{in} (1 - b_{c (t)}) + σ_{c (t)} E_{0}^{- 1}}

(10)

where b_c = 0.7 and b_t = 0.1, suggested by Birtel and Mark (2006). ε_c(t)ⁱⁿ is the compressive (tensile) inelastic strain of concrete. The crack opening w is a product of the inelastic strain and an internal length parameter l_t which is assumed to be the cubic root of the volume between integration points for a solid element.

Material modelling of steel and reinforcement

A bilinear kinematic hardening model with Von Mises yield criterion is adopted for the structural steel, steel plate and reinforcements. It is defined by the experimental measured yield strength (f_ys) and a hardening modulus E = 0.01E₀, where E₀ is the experimentally measured initial Young’s modulus. Poisson’s ratio equals 0.3. The material behaviour is provided by ABAQUS using the *PLASTIC option.

Validation of the finite element model

Based on the modelling method above, finite element analyses (FEAs) of 23 SPRC shear walls with different shear span ratios were performed (Wu, 2014; Xiao et al., 2012; Zhu, 2015), as shown in Table 1.

Table 1.

Parameter of test specimens.

References	Test no.	h_w × l_w × t_w (mm)	n	λ	f_c (Mpa)	Web			Boundary member
						f _yt (MPa), ρ_t (%)	f _yl (MPa), ρ_l (%)	f _yp (MPa), ρ_p (%)	f _ys (MPa), ρ_s (%)	f _ya (MPa), ρ_a (%)
Xiao et al. (2012)	SPRCW1	2100 × 800 × 150	0.42	2.7	69	291, 0.67	298, 0.38	310, 3.33	440, 0.68	334, 1.26
	SPRCW2	2100 × 800 × 150	0.5	2.7	69	291, 0.67	298, 0.38	310, 3.33	440, 0.68	334, 1.26
	SPRCW3	2100 × 800 × 150	0.58	2.7	69	291, 0.67	298, 0.38	310, 3.33	440, 0.68	334, 1.26
Zhu (2015)	GB2-3-1	1500 × 1000 × 125	0.25	2.0	65	576, 0.59	576, 0.59	330, 2.5	475, 2.18	353, 5.70
	GB2-3-2	1500 × 1000 × 125	0.3	2.0	60	576, 0.59	576, 0.59	330, 2.5	475, 2.18	353, 5.70
	GB2-3-3	1500 × 1000 × 125	0.3	2.0	60	498, 0.59	498, 0.59	330, 2.5	475, 2.18	353, 5.70
	GB2-3-4	1500 × 1000 × 125	0.3	2.0	68	576, 0.38	576, 0.38	330, 2.5	475, 2.18	353, 5.70
	GB2-5-1	1500 × 1000 × 125	0.25	2.0	66	576, 0.59	576, 0.59	353, 4.2	475, 2.18	353, 5.70
	GB2-5-2	1500 × 1000 × 125	0.3	2.0	69	576, 0.59	576, 0.59	353, 4.2	475, 2.18	353, 5.70
	GB2-5-3	1500 × 1000 × 125	0.3	2.0	68	498, 0.59	498, 0.59	353, 4.2	475, 2.18	353, 5.70
	GB2-5-4	1500 × 1000 × 125	0.3	2.0	65	576, 0.38	576, 0.38	353, 4.2	475, 2.18	353, 2.18
Wu (2014)	W1-a	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 2.5	409, 9.24	416, 3.14
	W1-b	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 1.0	409, 9.24	416, 3.14
	W1-c	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 1.5	409, 9.24	416, 3.14
	W1-d	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 2.0	409, 9.24	416, 3.14
	W2-a	1200 × 800 × 60	0.5	1.5	62	409, 1.26	409, 0.47	416, 1.0	409, 9.24	416, 3.14
	W2-b	1200 × 800 × 70	0.5	1.5	62	409, 1.08	409, 0.40	416, 1.0	409, 9.24	416, 3.14
	W3-a	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 1.0	409, 9.24	416, 6.28
	W3-b	1200 × 800 × 80	0.5	1.5	62	409, 0.94	409, 0.35	416, 1.5	409, 9.24	416, 6.28
	W4-a	1200 × 800 × 80	0.5	1.5	62	409, 0.63	409, 0.35	416, 1.0	409, 9.24	416, 3.14
	W4-b	1200 × 800 × 80	0.5	1.5	62	409, 1.96	409, 0.35	416, 1.0	409, 9.24	416, 3.14
Zhu (2015)	GB1-3-5	1000 × 1000 × 125	0.3	1.0	62	494, 0.59	494, 0.59	409, 2.5	689, 8.73	401, 5.7
Zhu (2015)	GB1-3-6	1000 × 1000 × 125	0.3	1.0	60	576, 0.38	576, 0.38	330, 2.5	576, 8.73	353, 5.7

h _w, l_w and t_w are the height, length and thickness of the composite shear wall; the concrete cylinder compressive strength f_c is taken as 0.76f_cu for C50, 0.82f_cu for C80 and linear interpolation for concrete grade between; n is the axial load ratio; λ is the shear span ratio; f_yt, f_yl, f_yp, ρ_t, ρ_l and ρ_p are the yield strengths and ratios of the transverse, longitudinal reinforcement and steel plate of the web; f_ys, f_ya, ρ_s and ρ_a are the yield strengths and ratios of the longitudinal reinforcement and structural steel of the boundary members.

The predicted load–displacement cyclic and skeleton curves are compared with those of the tests in Figure 3. The results of FEA reflect the pinching effect and damage conditions under cyclic load, as well as the decline of strength after peak load.

Figure 3.

Comparison of predicted and test load–displacement cyclic and skeleton curves.

Two failure modes were observed in the test and confirmed in the FEA as summarized in Table 2, that is flexural failure (F) and flexural–shear failure (FS). SPRC shear walls with shear span ratio of 2.0 and 2.7 failed in a typical flexural mode, while shear walls with shear span ratio of 1.5 and 1.0 failed in both shear and bending. Figure 4(a)–(c) demonstrates the predicted and test concrete damage of specimen GB2-3-1 which shows a typical flexural failure. As shown in Figure 4(a), bottom concrete of boundary columns suffered the most severe compressive damage, which can be observed in the test (concrete crushing) as shown in Figure 4(c). Specimen W1-a failed in a flexural and shear failure mode. Obvious shear crack developed along the height of the shear wall. The bottom concrete crushed under flexural and shear effects and the web concrete crushed under shear, which can be found in both test and numerical results. Concrete crack developed along the height of boundary column and upper height of wall web as shown in Figure 4(f), which can be found in tensile damage contour in Figure 4(e). The corresponding yield loads and peak loads are listed in Table 2. The yield point is determined by equivalent energy method. It is found that good agreement has been achieved between test results and most of the FEAs.

Table 2.

Comparison between tests and finite element results.

References	Test no.	Yield load (kN)		Q_TEST/Q_FEM	Peak load (kN)		P_TEST/P_FEM	Failure mode
		Q _TEST	Q _FEM		P _TEST	P _FEM		Test	FEM
Xiao et al. (2012)	SPRCW1	542.5	552.3	0.98	638.8	604.8	1.06	F	F
	SPRCW2	555.2	555.9	1.00	659.7	633.9	1.04	F	F
	SPRCW3	570.7	566.7	1.00	687.7	651.8	1.06	F	F
Zhu (2015)	GB2-3-1	607.0	630.1	0.96	828.0	840.1	0.99	F	F
	GB2-3-2	613.0	633.2	0.97	841.0	844.2	1.00	F	F
	GB2-3-3	612.0	631.0	0.97	822.0	841.3	0.98	F	F
	GB2-3-4	612.0	648.4	0.94	843.0	864.6	0.98	F	F
	GB2-5-1	625.0	726.5	0.86	891.0	968.7	0.92	F	F
	GB2-5-2	742.0	742.3	1.00	1003.0	989.4	1.01	F	F
	GB2-5-3	708.0	745.3	0.95	963.0	993.7	0.97	F	F
	GB2-5-4	667.0	712.1	0.94	939.0	949.5	0.99	F	F
Wu (2014)	W1-a	1520.0	1509.3	1.01	1730.0	1687.0	1.02	FS	FS
	W1-b	1120.0	1178.4	0.95	1450.0	1376.0	1.05	FS	FS
	W1-c	1214.0	1248.9	0.97	1515.0	1476.0	1.03	FS	FS
	W1-d	1340.0	1334.4	1.00	1703.0	1586.6	1.07	FS	FS
	W2-a	978.0	1145.1	0.85	1341.0	1311.4	1.00	FS	FS
	W2-b	1015.0	1163.4	0.87	1431.0	1339.6	1.06	FS	FS
	W3-a	1306.0	1489.7	0.88	1685.0	1594.1	1.06	FS	FS
	W3-b	1367.0	1549.4	0.88	2001.0	1714.0	1.17	FS	FS
	W4-a	977.0	1032.4	0.95	1382.0	1372.2	1.01	FS	FS
	W4-b	1188.0	1193.3	0.99	1533.0	1390.6	1.10	FS	FS
Zhu (2015)	GB1-3-5	1098.0	1236.0	0.89	1675.0	1651.3	1.01	FS	FS
Zhu (2015)	GB1-3-6	1385.0	1450.6	0.96	1921.6	1944.8	0.99	FS	FS
Mean		–	–	0.95	–	–	1.02
Standard variation, σ		–	–	0.05	–	–	0.05

FEM: finite element modelling; F denotes flexural failure; FS denotes flexural and shear failure.

Figure 4.

Predicted and test concrete damage: (a)–(c) GB2-3-1 and (d)–(f) W1-a. (a) Concrete compressive damage, (b) concrete tensile damage, (c) test failure, (d) concrete compressive damage, (e) concrete tensile damage and (f) test failure.

Parametric study and discussions

A parametric study of SPRC shear walls was performed to investigate the effect of the main parameters on the behaviour of composite shear wall, including shear span ratio, concrete strength, axial load ratio, steel plate ratio and web reinforcement ratio. A basic specimen was designed based on Zhu’s test (2015) and the Chinese code for design of composite structures, the typical section as shown in Figure 1(a). Shear span ratios of 1.5, 2.5 and 3.5 are considered. Concrete grade of C80, axial load ratio of 0.2, steel plate ratio of 3% and web reinforcement ratio of 0.6% are taken as the base values of the basic specimen.

Steel plate ratio

Steel plate ratios of 1%, 3%, 5% and 7% were studied. Specimens are named as SPRCW-LmRSn, where m and n represent the shear span ratio and steel plate ratio, respectively. The load–displacement skeleton curves of specimens of different shear span ratios are shown in Figure 5. The shear force contributions of different parts at peak point of the skeleton curves are shown in Figure 6. It can be found that the amount of shear forces by concrete and structural steel are kept stable as steel plate ratio increases. The increase of its capacity is mostly contributed by the steel plate. As shear span increases, the capacity ratio of the steel plate tends to decrease, which can be explained by the shear mechanism of the steel plate.

Figure 5.

Load–displacement skeleton curves of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Figure 6.

Shear contributions of different parts of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Concrete strength

Generally, the concrete grade of composite shear walls in high-rise buildings ranges from C50 to C80. Concrete grades of C50, C60, C70 and C80 were studied. Specimens are named as SPRCW-LmCn, where n represents the concrete grade. The load–displacement skeleton curves are shown in Figure 7. The shear force contributions of different parts at peak point are shown in Figure 8. As concrete grade increases, the peak force gets bigger, but the capacity decreases faster. For shear walls of high shear span ratio, the force ratio of steel plate tends to decrease when concrete grade increases. Measures shall be taken to improve the ductility of the composite shear wall when concrete of high grade is used.

Figure 7.

Load–displacement skeleton curves of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Figure 8.

Shear contributions of different parts of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Axial load ratio

Axial load ratios of 0.1, 0.2, 0.3, 0.4 and 0.5 were studied. Specimens are named as SPRCW-LmNn, where n represents the axial load ratio. The load–displacement skeleton curves are shown in Figure 9. The shear force contributions of different parts at peak point are shown in Figure 10. As axial load ratio increases, the ductility of the specimen keeps decreasing. And its capacity starts to decrease when the axial load ratio reaches 0.3. The shear force amount and ratio of steel plate decreases as axial load ratio increases.

Figure 9.

Load–displacement skeleton curves of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Figure 10.

Shear contributions of different parts of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Transverse reinforcement ratio of wall web

Web transverse reinforcement ratios of 0.3%, 0.4%, 0.6%, 0.8% and 1.0% were studied. Specimens are named as SPRCW-LmRRn, where n represents the transverse reinforcement ratio. The longitudinal reinforcement ratio is taken as the same with transverse reinforcement ratio. The load–displacement skeleton curves are shown in Figure 11. The force ratios of different parts at peak point are shown in Figure 12. As transverse reinforcement ratio of the web increases, the peak force and ductility of the specimen increases. Higher transverse reinforcement ratio can provide more confinement to the concrete, resulting in slight increase of force ratio of RC.

Figure 11.

Load–displacement skeleton curves of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Figure 12.

Shear contributions of different parts of SPRC shear walls: (a) λ = 1.5, (b) λ = 2.5 and (c) λ = 3.5.

Shear strength of SPRC shear walls

As illustrated by Zhu’s tests, since the shear mechanism of steel plate in the concrete wall, typical shear failure was hardly observed in SPRC shear wall. The failure modes of SPRC shear walls with lower shear span ratio are generally controlled by both shear and bending. Loss of capacity was mostly caused by the buckling of structural steel after its cover concrete crushed. Considering the difficulty in designing SPRC shear wall with pure shear failure, the potential flexural–shear failure is also considered to be shear failure in this case. Thus, numerical analyses of SPRC shear walls of shear span ratio between 1.0 and 1.5 was used to study the shear capacity of SPRC composite shear walls.

Chinese Code for design of composite structures

The calculation of the shear strength of SPRC shear wall proposed by Chinese code for design of composite structures is based on a linear combination of shear contributions of four parts, namely concrete, transverse reinforcement, structural steel and steel plate

\begin{matrix} V = \frac{1}{λ - 0.5} (0.5 f_{t} b_{w} h_{w 0} + 0.13 N \frac{A_{w}}{A}) \\ + f_{yh} \frac{A_{sh}}{s} h_{w 0} + \frac{0.3}{λ} f_{a} A_{a 1} + \frac{0.6}{λ - 0.5} f_{p} A_{p} \end{matrix}

(11)

where N is the designed axial force, and N is taken as 0.2f_cb_wh_w when N ≥ 0.2f_cb_wh_w; A is the section area of the wall; A_w is the section area of the web of T-shape or I-shape wall, taken as A for rectangular shear wall; λ is the shear span ratio of the calculated section, equal to M/Vh_w0, taken as 1.5 when λ ≤ 1.5 and taken as 2.2 when λ ≥ 2.2; h_w0 is the effective section height of the shear wall; A_a1 is the smaller section area of structural steel of one-side boundary member and A_p is the section area of the steel plate.

AISC seismic provisions for structural steel buildings

The AISC Provisions limit the shear strength of the walls to the yield stress of the steel plate because there is insufficient basis to develop design rules for combining the yield stress of the steel plate and the RC. Only the contribution from steel plate is considered (AISC, 2016)

V = 0.6 f_{p} A_{p}

(12)

Proposed formula for shear strength of SPRC shear wall

The shear strength of SPRC composite shear wall is assumed to be linearly combined by the contributions of concrete, transverse reinforcement, structural steel and steel plate. Coupling of different parts are ignored. Based on the Chinese Code for design of composite structures, a modified formula to calculate the shear strength of SPRC shear wall is proposed

\begin{matrix} V = \frac{1}{λ - 0.5} (α_{1} f_{t} b_{w} h_{w 0} + α_{2} N \frac{A_{w}}{A}) \\ + \frac{β}{λ} f_{yh} \frac{A_{sh}}{s} h_{w 0} + \frac{0.3}{λ} f_{a} A_{a 1} + \frac{a + b ρ_{p}}{λ - 0.5} f_{p} A_{p} \end{matrix}

(13)

where the steel plate ratio ρ_p = t_p/t_w; t_p and t_w are the thickness of the steel plate and wall web. α₁, α₂, β, a and b are the combination coefficients of different parts.

A similar parametric study of 720 steel plate reinforced shear walls was conducted. Main variables of these composite shear walls are listed in Table 3. FEAs of all specimens under cyclic loads were carried out. The shear strength V_FEM is taken as the peak load from load–displacement curves.

Table 3.

Main variables of the parametric study.

Parameter	Values
Shear span ratio	1.0, 1.2, 1.5
Concrete cylinder compressive strength (MPa)	32.4, 38.5, 44.5, 50.2
Axial load ratio	0.1, 0.2, 0.3
Transverse reinforcement ratio	0.3%, 0.4%, 0.6%, 0.8%, 1.0%
Steel plate ratio	1%, 3%, 5%, 7%

The combination coefficients α₁, α₂, β, a and b were obtained by regression analysis with the corresponding coefficient of determination of 72.8%. Then, the shear capacity of SPRC shear wall was obtained

\begin{matrix} V = \frac{1}{λ - 0.5} (1.52 f_{t} b_{w} h_{w 0} + 0.26 N \frac{A_{w}}{A}) + \frac{0.55}{λ} f_{yh} \frac{A_{sh}}{s} h_{w 0} \\ + \frac{0.3}{λ} f_{a} A_{a 1} + \frac{0.37 - 1.35 ρ_{p}}{λ - 0.5} f_{p} A_{p} \end{matrix}

(14)

The shear strength V_pred calculated by the proposed formula is compared with that V_FEM from numerical analyses in Figure 13. The mean value of V_pred/V_FEM is 1.01 with the corresponding covariance (COV) of 12.39%. It is shown that the proposed formula well predicts the shear strength of SPRC shear walls. To prevent shear wall with lower shear span ratio from shear failure, a safer calculation of its shear capacity is achieved by taking the shear span ratio λ as 1.5 when λ ≤ 1.5 in equation (14) that explains the difference between shear capacity predictions in Figure 13(a).

Figure 13.

Comparison of FEA and proposed formula: (a) different shear span ratios and (b) different steel plate ratios.

Comparison of proposed formula with design guides

For design proposal, a certain amount of redundancy of the strength shall be assured. About 15% reduction of the concrete contribution is considered, then a formula for design purpose is obtained

\begin{matrix} V = \frac{1}{λ - 0.5} (1.2 f_{t} b_{w} h_{w 0} + 0.22 N \frac{A_{w}}{A}) + \frac{0.5}{λ} f_{yh} \frac{A_{sh}}{s} h_{w 0} \\ + \frac{0.3}{λ} f_{a} A_{a 1} + \frac{0.32 - 1.2 ρ_{p}}{λ - 0.5} f_{p} A_{p} \end{matrix}

(15)

Tests of SPRC shear walls with shear span ratio between 1.0 and 1.5 are selected, as listed in Tables 1 and 4. The shear strengths of the test are compared with that calculated using proposed equation (15), the Chinese Code and AISC Provisions, separately, as shown in Table 5. The predictions of shear walls with different shear span ratios and steel plate ratios by the AISC Provisions, Chinese code and proposed formula are shown in Figures 14 and 15, respectively.

Table 4.

Parameter of additional test specimens.

References	Test No.	h_w × l_w × t_w (mm)	n	λ	f_c (Mpa)	Web			Boundary member
						f_yt (MPa), ρ_t (%)	f_yl (MPa), ρ_l (%)	f_yp (MPa), ρ_p (%)	f_ys (MPa), ρ_s (%)	f_ya (MPa), ρ_a (%)
Zhu (2015)	GB1-3-1	1000 × 1000 × 125	0.25	1.0	75	598, 0.59	598, 0.59	407, 2.5	531, 8.73	384, 5.7
	GB1-3-2	1000 × 1000 × 125	0.3	1.0	70	598, 0.59	598, 0.59	407, 2.5	531, 8.73	384, 5.7
	GB1-3-3	1000 × 1000 × 125	0.2	1.0	70	598, 0.59	598, 0.59	407, 2.5	531, 8.73	384, 5.7
	GB1-3-4	1000 × 1000 × 125	0.34	1.0	72	598, 0.59	598, 0.59	407, 2.5	531, 8.73	384, 5.7
	GB1-3-5	1000 × 1000 × 125	0.3	1.0	62	494, 0.59	494, 0.59	409, 2.5	689, 8.73	401, 5.7
	GB1-3-6	1000 × 1000 × 125	0.3	1.0	60	576, 0.38	576, 0.38	330, 2.5	576, 8.73	353, 5.7
	GB1-5-1	1000 × 1000 × 125	0.25	1.0	67	598, 0.59	598, 0.59	384, 4.2	531, 8.73	384, 5.7
	GB1-5-2	1000 × 1000 × 125	0.3	1.0	65	598, 0.59	598, 0.59	384, 4.2	531, 8.73	384, 5.7
	GB1-5-3	1000 × 1000 × 125	0.3	1.0	62	598, 0.59	598, 0.59	384, 4.2	531, 8.73	384, 5.7
	GB1-5-4	1000 × 1000 × 125	0.34	1.0	68	598, 0.59	598, 0.59	384, 4.2	531, 8.73	384, 5.7
	GB1-5-5	1000 × 1000 × 125	0.3	1.0	64	498, 0.59	498, 0.59	353, 4.2	576, 8.73	353, 5.7
	GB1-5-6	1000 × 1000 × 125	0.3	1.0	65	576, 0.38	576, 0.38	353, 4.2	576, 8.73	353, 5.7
	GB1-7-1	1000 × 1000 × 125	0.25	1.0	66	689, 0.59	689, 0.59	394, 6.0	689, 8.73	401, 5.7
	GB1-7-2	1000 × 1000 × 125	0.3	1.0	64	689, 0.59	689, 0.59	394, 6.0	689, 8.73	401, 5.7
	GB1-7-3	1000 × 1000 × 125	0.25	1.0	63	689, 0.59	689, 0.59	394, 6.0	494, 8.73	401, 5.7
	GB1-7-4	1000 × 1000 × 125	0.3	1.0	65	689, 0.59	689, 0.59	394, 6.0	494, 8.73	401, 5.7
	GB1-7-5	1000 × 1000 × 125	0.3	1.0	60	494, 0.59	494, 0.59	394, 6.0	689, 8.73	401, 5.7
	GB1-7-6	1000 × 1000 × 125	0.3	1.0	68	576, 0.38	576, 0.38	369, 6.0	576, 8.73	353, 5.7
Cao et al. (2012)	SWA-1	1110 × 740 × 100	0.3	1.5	35	669, 0.63	669, 0.63	460, 2	278, 5.61	536, 2.0
	SWA-2	1110 × 740 × 100	0.3	1.5	37	669, 0.63	669, 0.63	460, 4	278, 5.61	536, 2.0
	SWA-3	1110 × 740 × 100	0.3	1.5	37	669, 0.63	669, 0.63	460, 6	278, 5.61	536, 2.0
	SWA-4	1110 × 740 × 100	0.3	1.5	35	669, 0.63	669, 0.63	460, 4	278, 5.61	536, 2.0

Table 5.

Comparison of shear strengths from tests, design guides and proposed formula.

References	Test no.	V _TEST (kN)	V _ASIC341 (kN)	V _JGJ138 (kN)	V _pred (kN)	V _ASIC341/V_TEST	V _JGJ138/V_TEST	V_pred/V_TEST
Zhu (2015)	GB1-3-1	1898	767	1735	1745	0.40	0.91	0.92
	GB1-3-2	1940	767	1707	1680	0.40	0.88	0.87
	GB1-3-3	1862	767	1708	1683	0.41	0.92	0.90
	GB1-3-4	1864	767	1719	1708	0.41	0.92	0.92
	GB1-3-5	1661	547	1377	1481	0.33	0.83	0.89
	GB1-3-6	1821	481	1212	1395	0.26	0.67	0.77
	GB1-5-1	1897	1067	1994	1751	0.56	1.05	0.92
	GB1-5-2	1948	1067	1982	1721	0.55	1.02	0.88
	GB1-5-3	2108	1067	1970	1693	0.51	0.93	0.80
	GB1-5-4	2037	1067	2001	1766	0.52	0.98	0.87
	GB1-5-5	1982	990	1825	1658	0.50	0.92	0.84
	GB1-5-6	2089	990	1746	1642	0.47	0.84	0.79
	GB1-7-1	2100	1409	2395	1863	0.67	1.14	0.89
	GB1-7-2	2100	1409	2385	1840	0.67	1.14	0.88
	GB1-7-3	2246	1409	2381	1831	0.63	1.06	0.82
	GB1-7-4	2231	1409	2389	1850	0.63	1.07	0.83
	GB1-7-5	2160	1409	2227	1745	0.65	1.03	0.81
	GB1-7-6	2147	1392	2162	1800	0.65	1.01	0.84
	GB2-3-1	828	481	998	933	0.58	1.20	1.13
	GB2-3-2	841	481	984	901	0.57	1.17	1.07
	GB2-3-3	822	481	923	880	0.58	1.12	1.07
	GB2-3-4	843	481	867	924	0.57	1.03	1.10
	GB2-5-1	891	990	1314	1056	1.11	1.47	1.19
	GB2-5-2	1003	990	1323	1077	0.99	1.32	1.07
	GB2-5-3	963	990	1262	1056	1.03	1.31	1.10
	GB2-5-4	939	990	1171	1019	1.05	1.25	1.09
Wu (2014)	W1-a	1730	562	1175	1403	0.32	0.68	0.81
	W1-b	1450	225	838	1275	0.15	0.58	0.88
	W1-c	1515	337	950	1320	0.22	0.63	0.87
	W1-d	1703	449	1063	1364	0.26	0.62	0.80
	W2-a	1341	225	809	1200	0.17	0.60	0.90
	W2-b	1431	225	824	1238	0.16	0.58	0.87
	W3-a	1685	225	942	1380	0.13	0.56	0.82
	W3-b	2001	337	1055	1427	0.17	0.53	0.71
	W4-a	1382	225	926	1310	0.16	0.67	0.95
	W4-b	1533	225	838	1280	0.15	0.55	0.83
	W4-c	1492	225	838	1279	0.15	0.56	0.86
Cao et al. (2012)	SWA-1	706	123	623	607	0.17	0.88	0.86
	SWA-2	782	302	806	682	0.39	1.03	0.87
	SWA-3	772	460	964	732	0.60	1.25	0.95
	SWA-4	763	302	803	674	0.40	1.05	0.88

Figure 14.

Comparison of different shear span ratios: (a) test and AISC Provisions, (b) tests and Chinese code and (c) tests and proposed formula.

Figure 15.

Comparison of different steel plate ratios: (a) test and AISC Provisions, (b) tests and Chinese code and (c) tests and proposed formula.

The AISC Provisions significantly underestimate the shear strengths of SPRC shear walls; the mean value of V_AISC341/V_Test is 0.40, with the corresponding COV of 45.0%. The Chinese Code predicts the shear strengths of SPRC shear walls with steel plate ratio less than 5% with some redundancy but tends to overestimate those with steel plate ratio more than 5%, which is unsafe for design purposes. The mean value of V_JGJ138/V_Test is 0.85, with the corresponding COV of 20.7%. The proposed formula predicts the shear strengths of SPRC shear walls conservatively and safely. The mean value of V_pred/V_Test is 0.85, with the corresponding COV of 8.4%.

Conclusion

The article presents a finite element model for analysis of steel plate RC shear walls under cyclic loads. The effect of concrete confinement was carefully considered. The comparison between numerical results and test results showed good agreement in predicting the strength and hysterical behaviour of SPRC shear wall.

A parametric study was conducted to investigate the effects of the main parameters of SPRC shear wall on its cyclic behaviour. It was shown that steel plate ratio remarkably improves the strength of the composite shear wall. The axial load ratio of SPRC shear wall is suggested to be within 0.3 since significant degradation of the ductility was found as well as slight reduction of the strength for shear walls under axial load larger than 0.3. Higher grade of concrete may result in a more brittle behaviour of shear walls. Thus, measures that can help improve the ductility of shear walls shall be made in such cases, such as a proper setup of the transverse reinforcement to provide confinement to the core concrete.

A formula to calculate the shear strength of steel plate RC shear wall in a linear form of combination of the shear contributions of different parts was proposed based on another parametric study. Shear strengths from published tests were compared with that calculated using design guides and the proposed formula. It was found that the prediction using Chinese code for SPRC shear wall with a steel plate ratio of more than 5% is unsafe. The AISC predictions are quite conservative because the contribution of the concrete is neglected. The modified formula will lead to a more rational design of shear strength of steel plate RC shear wall.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors acknowledge with thanks the support from (a) the National Key Research and Development Programme of China (No. 2017YFC1500701), (b) the National Natural Science Foundation of China (Project 51638016 and Project 51608406) and (c) the Open Foundation of Hubei Key Laboratory of Theory and Application of Advanced Materials Mechanics (Wuhan University of Technology; No. TMA201810).

ORCID iD

Zhi Zhou

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