Numerical and analytical investigation on the moment and deformation capacity of interior slab-column connections without shear reinforcement

Abstract

In this study, nonlinear finite element analyses of reinforced concrete interior slab-column connections were performed to investigate their punching shear behavior under gravity and lateral loads without shear reinforcement. A numerical model based on the concrete damaged plasticity (CDP) model in ABAQUS was developed with suitable constitutive models for concrete and reinforcements using eight-node brick elements with reduced integration and three-dimensional truss elements, respectively. The model was calibrated in comparison with the test results of an interior slab-column connection without shear reinforcement. Moreover, the validity of the calibrated model was verified using other test results. Then, a parametric study was conducted to examine the influence of different design variables on the unbalanced moment and deformation capacity of slab-column connections. Finally, based on the existing experimental data and finite element analysis results obtained in this work, formulas for calculating the ultimate unbalanced moment and drift ratio were proposed using the nonlinear fitting method. The proposed formulas were compared with the existing methods provided by other national codes and researchers, verifying the rationality of the proposed formulas.

Keywords

punching shear slab-column connections finite element method concrete damaged plasticity flat slab

Introduction

The problem of punching shear failure occurs in flat slab-column connections that are not typically reinforced with shear reinforcement. Many researchers have studied the punching shear behavior of slab-column connections without shear reinforcement (Drakatos et al., 2016; Inácio et al., 2020; Pan and Moehle, 1989; Robertson and Johnson, 2006). However, there is no agreement on the punching shear failure mechanism. Therefore, this work aims to conduct further research on the failure mechanism and mechanic behavior of slab-column connections without shear reinforcement.

Existing experimental results show that the main factors affecting the punching failure of slab-column connections are the gravity load level and flexural reinforcement ratio. The gravity load level is usually represented using the gravity shear ratio V_g/V_c, where V_g is the shear force acting on a critical perimeter due to the gravity load and V_c is the punching shear capacity of concrete calculated according to ACI318-19 (2019). Pan and Moehle (1989) tested two isolated slab-column connections with gravity shear ratios of 0.18 and 0.37. Moreover, Robertson and Durrani (1992) and Robertson and Johnson (2006) reported three single-story structures with two-span slab-column frames and three isolated slab-column connections with gravity shear ratios ranging from 0.18 to 0.54. Tang et al. (2019) reported four isolated slab-column connections with gravity shear ratios ranging from 0.39 to 0.72. The above test results showed that increasing the gravity shear ratio reduces the unbalanced moment capacity, deformation capacity, and lateral stiffness of slab-column connections. Pan and Moehle (1989) believed that the gravity shear ratio must not exceed 0.4 to ensure a connection drift capacity of 1.5%, while Robertson and Durrani (1992) suggested that the value of 0.4 should be reduced to 0.35. Generally, increasing the slab flexural reinforcement ratio could improve the unbalanced moment capacity and lateral stiffness of slab-column connections but reduce the lateral deformation capacity. This trend was confirmed by the test results presented by Choi et al. (2007), Ghali et al. (1976), and Robertson and Johnson (2006), wherein the gravity shear ratios were between 0.23 and 0.30. However, Drakatos et al. (2016) found that when the gravity shear ratio was approximately 0.6, increasing the slab longitudinal reinforcement ratio had little effect on the unbalanced moment capacity of slab-column connections that experienced punching shear failure, which was consistent with the test results of the specimens that experienced flexural failure with low gravity shear ratios of less than 0.05 (Morrison et al., 1983). When the gravity shear ratio was approximately 0.9, the specimens with reinforcement ratios of 0.8% and 1.6% showed little difference in terms of lateral deformation ability. Drakatos et al. (2016) and Morrison et al. (1983) showed inconsistent test results based on specimens with moderate gravity shear ratios ranging from 0.23 to 0.30 for plate slabs of 114–152 mm. These discrepancies may arise from the fact that Morrison et al. (1983) used low gravity shear ratios of less than 0.05 in the case of thin plate slabs of 76 mm, whereas Drakatos et al. (2016) used high gravity shear ratios of more than 0.59 in the case of thick plate slabs of 250 mm. Furthermore, the test results of Tang et al. (2019) showed that at a gravity shear ratio of 0.39, increasing the reinforcement ratio could not only improve the unbalanced moment capacity and lateral stiffness but also enhance the lateral deformation capacity of slab-column connections. Moreover, Inácio et al. (2020) found that the lateral deformation capacity of slab-column connections could be significantly improved using high-strength concrete in local areas of connections.

Several mechanical models have been proposed by Choi et al. (2014a, 2014b), Deifalla (2020a, 2020b, 2021a, 2021b), Fan et al. (2021), Kinnunen and Nylander (1960), Muttoni (2008), Park and Islam (1976), Stasio and Buren (1960), and others, in addition to extensive experimental research on punching shear behavior. Some of them served as the foundation for the design formulas used in code provisions.

With the development of computers and computing technologies, the nonlinear finite element method has become a powerful tool for studying the performance of concrete structures. Many existing studies have been conducted on the finite element analysis (FEA) of the mechanical performance of slab-column connections under gravity loads (Bompa and Elghazouli, 2017; Eder et al., 2010; Lapi et al., 2020; Navarro et al., 2018; Shu et al., 2016; Vidosa et al., 1988; Wörle, 2014; Xiao and Chin 2007). However, there are relatively few studies on mechanical performance analyses under combined gravity and horizontal loads. Polak (2005) utilized three-dimensional (3D) layer shell elements to study the punching shear transfer mechanisms of slabs and discussed the influence of material parameters (cracked shear modulus of concrete, tension stiffening factors, and dowel action coefficients) on the numerical results. Guan (2009) also used layer shell elements to simulate the punching shear performance, and the focus of this research was to examine the impacts of the openings and column aspect ratio on slab-edge column connections. The failure criteria for the shell modeling approaches used by Polak (2005) and Guan (2009) are tension cracking or concrete crushing. Genikomsou and Polak (2015) established a 3D model with brick elements using the ABAQUS software, and the parameters were calibrated using the test results. Additionally, the rationality of the FEA model was verified by comparing the numerical results with the test results in terms of the load–deformation curve and cracking pattern. Setiawan et al. (2019) used a 3D nonlinear finite element analysis (NLFEA) using the software ATENA to investigate the effect of the cyclic degradation mechanism on the unbalanced moment capacity, lateral stiffness, and lateral deformation capacity of slab-column connections under lateral cyclic loading.

The existing experimental studies show that the gravity shear ratio, reinforcement ratio, and concrete strength have a great influence on the mechanical performance of slab-column connections without shear reinforcement under the combination of gravity and horizontal loads. In most concrete structure design codes, simplified empirical formulas are used to calculate the shear stress on the critical section of slab-column connections (ACI318-19, 2019; EC2-04, 2004). However, there are not enough experimental data and numerical analysis results for fully supporting the design calculation method. Therefore, the nonlinear FEA method was adopted in this work to study the strength and deformation performance of slab-column connections under the combination of gravity and horizontal loads and provide a reference for revising relevant codes. First, the model parameters were calibrated based on the specimen (RC1) tested by Tang et al. (2019), and the rationality of the calibrated model was further verified by other specimens (RC2, RC3, and RC4) tested by Tang et al. (2019) and (L0.5, G0.5, and G1.0), which were tested by Tian et al. (2008). Then, a parameter analysis was completed based on the calibrated model. The influence of the gravity shear ratio, reinforcement ratio, and concrete strength on the punching shear performance of the slab-column connections was studied. Finally, based on the experimental and FEA results, formulas for calculating the ultimate unbalanced moment and ultimate drift ratio were proposed, and the rationality of the proposed formulas was verified by comparing them with the existing standard calculation methods and the formulas proposed by other scholars.

Finite element model

Description of the finite element model

The FEA software ABAQUS was used for analysis in this article. Eight-node three-dimensional brick elements with reduced integration (C3D8R) were used to model the concrete, while the reinforcement was modeled using two-node three-dimensional truss elements (T3D2). The rebars were embedded into the concrete, so a perfect bond was assumed between the rebars and the concrete. According to the mesh sensitivity study of Deng (2018), six brick elements were used through the depth of the 150-mm slab. To save computational effort, a refined mesh size of 25 mm × 25 mm × 25 mm was used for the concrete in the slab width (c + 3h) centered on the column, while the mesh size of 25 mm × 50 mm × 50 mm was used for the remaining concrete. Here, c is the column size and h is the slab thickness. Further, a mesh size of 50 mm was used for all the rebars. The geometry and boundary conditions of the specimen (RC1), which were used for calibration, are presented in Figure 1. A reference point (RP) was established at the center of each support surface. A coupling constraint was adopted to couple the motion of a collection of nodes on the support surface to the motion of the RP. It was assumed that the column remained elastic to provide a simple approach, which was also used by other researchers (Genikomsou and Polak, 2016; Liu et al., 2015). The concrete damaged plasticity (CDP) model in ABAQUS was adopted to model the concrete behavior. The CDP model is suitable for concrete members under low confining pressure. Considering the damage effect, it is more suitable for simulating the crack development, crack closure, and stiffness recovery of concrete structures under cyclic loading (Hibbitt et al., 1998).

Figure 1.

Geometry and boundary conditions of the model.

Main parameters

The CDP model has five main parameters, which are the dilation angle (ψ), eccentricity (ε), yield stress ratio (σ_b0/σ_c0), yield surface shape factor (K_c), and viscosity parameter (μ). Some of their default values are usually assumed. The eccentricity was considered to be 0.1, the yield stress ratio was assumed to be 1.16, and the yield surface shape factor was assumed to be 0.667.

The stress–strain relationship of the concrete under uniaxial compression was represented based on Hognestad parabola (Genikomsou and Polak, 2015), which can be subdivided into three parts (Figure 2). The first part O-A represents the linear elastic branch, assumed until $0.4 f_{c}^{'}$ and adopting the initial elasticity modulus, $E_{c} = 5000 \sqrt{f_{c}^{'}}$ (Laguta, 2020; Milligan, 2018; Stoner, 2015). The second segment A-B shows the nonlinear ascending branch with the modulus of elasticity $E_{t} = 5500 \sqrt{f_{c}^{'}}$ , which is tangent to the parabola at the origin. The strain corresponding to the peak compressive stress $f_{c}^{'}$ was determined to be $ε_{0} = 2 f_{c}^{'} / E_{t}$ . The third section B-C indicates the descending branch from the peak stress to the ultimate strain $ε_{u}$ , and the expression of the assumed stress–strain curve is expressed as

σ = {\begin{cases} E_{c} ε_{c}, i f ε_{c} \leq 0.4 f_{c}^{'} / E_{c} \\ f_{c}^{'} [2 (\frac{ε_{c}}{ε_{0}}) - {(\frac{ε_{c}}{ε_{0}})}^{2}], i f ε_{c} > 0.4 f_{c}^{'} / E_{c} \end{cases}

(1)

Figure 2.

Compressive stress–strain relationship for concrete.

The uniaxial tensile stress–strain behavior of concrete was considered linear elastic until the peak tensile stress, $f_{t}^{'}$ , which was determined as

f_{t}^{'} = 0.33 \sqrt{f_{c}^{'}}

(2)

After cracking, an exponential tension-softening branch, which was proposed by Hordijk (1991), was selected for the tensile stress–crack width relationship to reduce the mesh sensitivity of the numerical results (Figure 3).

\frac{σ}{f_{t}^{'}} = [1 + {(\frac{c_{1} w}{w_{c}})}^{3}] \cdot e^{(- c_{2} w / w_{c})} - \frac{w}{w_{c}} \cdot (1 + c_{1}^{3}) \cdot e^{(- c_{2})}

(3)

w_{c} = 5.14 \frac{G_{f}}{f_{t}^{'}}

(4)

where c₁ and c₂ are material constants equal to 3 and 6.93, respectively. The fracture energy of concrete G_f was defined as the energy required to propagate a tensile crack of a certain unit area, and it represents the area under the σ_t–w curve. In this study, three different G_f expressions are adopted according to the Model Code 1990 Comité Euro-International du Béton (1993), Model Code 2010 Comité Euro-International du Béton (2012), and Lapi et al. (2020), which are expressed using Equations (5)–(7): respectively

G_{f} = {\begin{cases} G_{f 0} \cdot {(f_{c m} / f_{c m 0})}^{0.7}, f_{c m} \leq 80 MPa \\ 4.3 \cdot G_{f 0}, f_{c m} > 80 MPa \end{cases}

(5)

G_{f} = 73 \cdot f_{c m}^{0.18}

(6)

G_{f} = 64 \cdot f_{c m}^{0.11}

(7)

where G_f0 is the base fracture value, which depends on the maximum aggregate size d_max (assumed to be 16 mm), f_cm is the mean compressive strength of concrete, which was calculated using the characteristic concrete strength, f_ck, plus 8 MPa, and f_cm0 equals 10 MPa. To minimize fracture localization, the tensile stress–crack width relationship was converted into a tensile stress–strain relationship (Figure 3). A characteristic length of the element (l_c) was adopted for the 3D elements, and it is equal to the cubic root of the element volume (Genikomsou and Polak, 2015).

Figure 3.

(a) Tensile stress–crack width relationship and (b) tensile stress–strain relationship for concrete.

Two damage variables (d_c and d_t) were used for the concrete under compression and tension, respectively. Under compression, the damage was introduced after reaching the assumed elastic limit (σ_cA), as shown in Figure 2, and the tensile damage occurred when the tensile strength $(f_{t}^{'})$ was reached. The damage parameters were determined according to the energy equivalence principle of Cordebois and Sidoroff (1982), which assumed that the damaged complementary energy was obtained by substituting the stress with the effective stress under undamaged complementary energy. Then, a formulation applicable for calculating the compression and tensile damage parameters was derived, as in equation (8). The maximum value for the damage parameters in both tension and compression was assumed to be 0.9, which corresponded to the value used by Genikomsou and Polak (2015) and Wosatko et al. (2015).

d = 1 - \sqrt{\frac{σ}{E_{0} ε}}

(8)

Meanwhile, the steel reinforcement stress–strain relationship was assumed to be elastically perfectly plastic, with the elastic modulus equal to 200,000 MPa according to ACI318-19. The Poisson’s ratio for reinforcement was usually set to 0.3 (Genikomsou and Polak, 2015; Laguta, 2020; Milligan, 2018).

Description of the test specimens

The experiments conducted by Tang et al. (2019) and Tian et al. (2008) were adopted to verify the proposed numerical model. The specimens (RC1, RC2, RC3, and RC4) were tested by Tang et al. (2019). Each of them consisted of a slab measuring 2900-mm square and 150-mm thick and 350-mm square column extending 1175 mm beyond the slab top surface and 925 mm beyond the bottom surface. The test variable of RC1, RC2, and RC3 was the gravity shear ratio (Table 1), which was defined as the ratio of the punching shear force V_g on a critical section to the punching shear capacity V_c given by ACI318-19, where

V_{c} = 0.33 \sqrt{f_{c}^{'}} A_{c} (MPa)

. Here, A_c is the critical section area at d/2 from the column surface and it is equal to 4d (c + d), d is the effective thickness of the slab, and c is the column width. The longitudinal reinforcement ratio of the RC1 and RC4 specimens was different (Table 1). The specimens (L0.5, G0.5, and G1.0), which were tested by Tian et al. (2008), consisted of a slab measuring 4267-mm square and 152-mm thick and 406-mm square column extending 1397 mm beyond the slab top surface and 1016 mm beyond the bottom surface. The gravity shear ratio was 0.23 for specimen L0.5 (Table 1). The specimens G0.5 and G1.0, which showed different reinforcement ratios, were only applied to the vertical load.

Table 1.

The test parameters and results of the specimens.

ID	Loading type	c (mm)	d (mm)	f_c^' (MPa)	f_y (MPa)	ρ (%)		M_u (kN·m)	V_g (kN)	V_g/V_c	e/c	FM
ID	Loading type	c (mm)	d (mm)	f_c^' (MPa)	f_y (MPa)	Top	Bottom	M_u (kN·m)	V_g (kN)	V_g/V_c	e/c	FM
RC1	V + C	350	123	31.9	421	0.81	0.80	147.1	171.0	0.39	2.46	P
RC2	V + C	350	123	34.2	421	0.81	0.80	120.7	254.0	0.56	1.36	P
RC3	V + C	350	123	31.5	421	0.81	0.80	76.7	315.5	0.72	0.69	P
RC4	V + C	350	123	33.0	421	0.61	0.61	103.1	174.1	0.39	1.69	P
L0.5	V + C	406	127	25.6	441/469	0.50	0.25	123.6	118.0	0.23	2.58	P
G0.5	V	406	127	31.3	407/421	0.50	0.25	0	312	0.62	0.00	P
G1.0	V	406	127	28.0	407/421	1.00	0.50	0	404	0.85	0.00	P

V + C refers to applying the gravity load and horizontal cyclic load and V refers to only applying the gravity load.

f_y is the yield strength of flexural reinforcement and ρ is the reinforcement ratio within a slab width of (c + 3h) centered on the column. P refers to the punching shear failure.

The punching shear failure occurred in all specimens, with e/c values less than 2.6 (Table 1). Here, e is the eccentricity equal to M_u/V_g, M_u is the ultimate unbalanced moment, and c is the column size. The following sections present information on the comparison analysis between test and simulation results.

Finite element analysis

Parameter calibration of the finite element model

The test results of RC1 were used to calibrate the parameters of the FEA model. It was assumed that the envelope (skeleton curve) of the hysteretic curve for the specimen is the same as the monotonic loading curve. In other words, the influence of the cyclic loads on the skeleton curve was ignored. In this study, the displacement-controlled mode was used to apply a horizontal monotonic load during the simulation test. According to the loading rate of conventional seismic tests, such tests could be regarded as quasistatic problems. The implicit (ABAQUS/Standard) algorithm was used to solve the model. The numerical integration was performed using the Newton–Raphson method. Under the premise of no obvious loss of calculation accuracy and efficiency, the convergence rate of the model in the softening segment could be improved by taking a smaller value of the viscosity parameter. In this article, the viscosity coefficient was set as 0.00001, which is consistent with the research results of Genikomsou and Polak (2015) and Navarro et al. (2018). The analysis showed that when the slab thickness to element size ratio is greater than or equal to six, the numerical simulation accuracy of the punching failure is good and does not change significantly with the refined mesh size (Deng 2018). To determine the refined mesh size in this study, it was reasonable to use the ratio of the slab thickness to the element size to be equal to six (i.e., six layers of elements were divided along the thickness of the slab).

The concrete volume change due to the inelastic strains is called dilatancy, and it could be modeled by adopting the dilation angle in the CDP model. According to Genikomsou and Polak (2015), the dilation angle (ψ) should range from 31° to 42°. In the calibration process, four different values of the dilation angle (i.e., ψ = 20°, 30°, 38°, and 42°) were examined (see Figure 4(a)). In the figure, the unbalanced moment was determined from the measured lateral load and effective column height (H_c) and the drift ratio was defined as the ratio of the column top displacement to the effective column height. Figure 4(a) shows that a dilation angle of 38° provided better prediction results in terms of the ultimate unbalanced moment and drift ratio. Lapi et al. (2020) summarized the dilation angle used in the existing FEA and found that it ranged from 35° to 40°, so the value of the dilation angle in this article is also consistent with the statistical results. Therefore, the dilation angle was selected as 38° for all the subsequent specimens.

Figure 4.

Unbalanced moment–drift ratio response of RC1: (a) for different dilation angle values and (b) for different fracture energy values.

The fracture energy of concrete is defined as the energy required to produce a tensile crack per unit area (Figure 3). According to Equations (5)–(7), the fracture energy can be associated with the concrete strength. Three different values derived from Equations (5)–(7) were investigated, and it could be seen that the influence of the concrete tensile behavior on the slab response is notable (Figure 4(b)). The results obtained from either Equation (5) or (6) present an underestimation or overestimation of the ultimate unbalanced moment. However, the most accurate results could be acquired using Equation (7). Thus, Equation (7) was considered to determine the fracture energy values for all the simulated specimens.

According to the above-calibrated results, the static analysis method was adopted. The viscosity coefficient, refined mesh size, and dilation angle were 0.00001, 25 mm, and 38°, respectively. The fracture energy was determined according to Equation (7). Finally, the simulated unbalanced moment–drift ratio response was obtained and then compared with the test results. The results show that the simulation response of the connection under monotonic loading is in good agreement with the skeleton curve of the tested hysteresis response for the positive horizontal loading displacement (Figure 5). The FEA overestimated the specimen’s stiffness possibly because the adopted concrete model did not realistically simulate the punching shear behavior owing to its complexity.

Figure 5.

The test and simulation response of the unbalanced moment–drift ratio for RC1.

The CDP model assumes that when the maximum principal plastic strain is positive, the concrete cracks and crack direction are perpendicular to the maximum principal plastic strain direction (Genikomsou and Polak, 2015). Therefore, the maximum plastic principal strain (PE, max. Principal) can be used to approximate the failure mode of the slab-column connections. In the event of failure, Figure 6 depicts the tested cracks as well as the maximum principal plastic strain contour plots of the specimen RC1. By comparing of the test and FEA results in Figure 6, it could be seen that the maximum principal plastic strain calculated based on ABAQUS could well simulate the cracking pattern of the slab-column connections.

Figure 6.

Crack patterns of specimen RC1: (a) top slab and (b) cross section.

Verification of the calibrated finite element model

The calibrated FEA model was further verified using the specimens RC2, RC3, and RC4 in the study by Tang et al. (2019) and the specimens L0.5, G0.5, and G1.0 in the study by Tian et al. (2008). For the specimens RC2, RC3, RC4, and L0.5, which were subjected to gravity and lateral loads, the ultimate unbalanced moment–drift ratio responses obtained using the FEA simulation were in good agreement with the test results (Figures 7(a)–(d)). For the specimens G0.5 and G1.0, which were subjected to vertical loads, the vertical load-deflection responses obtained using simulations showed good agreement with the experimental results (Figures 7(e) and (f)). According to Table 2, the maximum relative error between FEA and test results was within 15% in all specimens, which is an acceptable difference. Figure 8 shows a comparison of the tested crack patterns and the maximum principal plastic strain contour plots of all the specimens at failure. The main characteristics of tested cracks at failure, which are concentrated near the column and developed to the slab boundaries, can be captured realistically using NLFEA. The simulations all showed brittle punching shear failure, similar to the test. The slab top reinforcements passing through the column reached the yield strain under the vertical load in the case of simulated specimen RC2 in Figure 8(a). When the horizontal load was applied, the slab bars near the column gradually yielded, whereas the other slab bars far away from the column remained elastic. When the peak load was reached, the slab concrete near the column could no longer support the horizontal load due to the large tensile strain and the connection failed because of brittle punching failure. This is consistent with the experimental results. For the slab top reinforcements passing through the column under a vertical load, a strain of nearly half of the yield strain was achieved for simulated specimen L0.5 in Figure 8(d). The yielding of top bars extended to 530 mm outside the column face in the lateral loading direction when the peak lateral load was reached at 1.75%. Owing to the large plastic strain, the slab concrete around the column was severely damaged after the peak load was reached. The sudden drop of combined vertical and lateral loads revealed a brittle punching shear failure. The FEA results agreed well with the experimental results. The numerical simulations also confirmed differences in crack distribution between specimens RC2 and L0.5, which could be attributed to different reinforcement arrangements.

Figure 7.

Unbalanced moment–drift ratio response or gravity load-deflection response for specimens: (a) RC2, (b) RC3, (c) RC4, (d) L0.5, (e) G0.5, and (f) G1.0.

Table 2.

Test and FEA results.

ID	Gravity shear ratio	ρ_t (%)	Test results		FEA results		Test/FEA results
ID	Gravity shear ratio	ρ_t (%)	M_u (kN·m)/V_u (kN)	θ_u (%)/w_u (mm)	M_u (kN·m)/V_u (kN)	θ_u (%)/w_u (mm)	(M_u)_test/(M_u)_FEA or (V_u)_test/(V_u)_FEA	(θ_u)_test/(θ_u)_FEA or (w_u)_test/(w_u)_FEA
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8) = (4)/(6)	(9) = (5)/(7)
RC1	0.39	0.81	147.1	2.00	143.3	1.94	1.03	1.03
RC2	0.56	0.81	120.7	1.25	126.2	1.37	0.96	0.91
RC3	0.72	0.81	76.7	0.75	84.0	0.75	0.91	1.00
RC4	0.39	0.61	103.1	1.50	111.4	1.71	0.93	0.88
L0.5	0.23	0.50	123.6	2.00	129.3	2.25	0.96	0.89
G0.5	0.62	0.50	312.0	25	328.0	26.4	0.95	0.95
G1.0	0.85	1.00	404.0	24	420.3	25.2	0.96	0.95
Mean							0.96	0.94
St Dev							0.04	0.06
COV							0.04	0.06

ρ_t is the reinforcement ratio within a slab width of (c + 3h) centered on the column, M_u is the ultimate unbalanced moment, V_u is the ultimate vertical load, θ_u is the ultimate drift ratio, w_u is the ultimate deflection, G0.5 and G1.0 are only subjected to vertical loads, and the test and FEA results are the ultimate vertical load (V_u) and ultimate deflection (w_u), respectively.

Figure 8.

Crack patterns at failure on the top surface of connections only under gravity loads: (a) RC2, (b) RC3, (c) RC4, (d) L0.5, (e) G0.5, and (f) G1.0.

Parametric study

A parametric study for the RC slab-column connections without shear reinforcement was conducted using the calibrated CDP model. The geometrical size and loading position of the slab-column connections were the same as those of RC1. There are three main parameters: the gravity shear ratio (V_g/V_c), reinforcement ratio (ρ_s) for flexural reinforcement within a slab width of (c + 3h) centered on the column, and compressive strength of concrete slab $(f_{c}^{'})$ . The reinforcements at the top and bottom of slabs were reinforced equally, which is also the case of tests by Mahroug et al. (2014), Morrison et al. (1983), Rubaye et al. (2020), Sadraie et al. (2019), and Tang et al. (2019). Each variable had four different values. The gravity loads were 65.7, 153.4, 241.0, and 328.7 kN, and the gravity shear ratio (V_g/V_c) was adopted to measure the magnitudes of the gravity load. As the influence of the concrete strength in V_c was considered, the gravity shear ratio would vary with the concrete strength. The gravity shear ratios corresponding to the concrete strength of 32 MPa in the basic series were 0.15, 0.35, 0.55, and 0.75, respectively. The reinforcement ratios were 0.50%, 0.80%, 1.20%, and 1.60%, and the concrete strengths were 32 MPa, 52 MPa, 72 MPa, and 92 MPa, respectively. Therefore, 64 connections (4 × 4 × 4) were designed considering the combination of different values for the investigated variables.

Figure 9 shows the ultimate unbalanced moment with varying reinforcement ratios and gravity shear ratios. The ultimate unbalanced moment of the slab-column connections decreases with an increase in the gravity shear ratio and increases with an increase in the reinforcement ratio or concrete strength. However, the increment in the ultimate unbalanced moment is minor when the reinforcement ratio is more than 1.2%. The reason is that when the reinforcement ratio reaches a certain level, the concrete strength plays a dominant role in determining the unbalanced moment capacity. In this case, increasing the reinforcement ratio has no significant effect on the unbalanced moment capacity. Figure 10 depicts the ultimate drift ratio with varying reinforcement and gravity shear ratios. The ultimate drift ratio decreases with an increase in the gravity shear ratio and increases with an increase in the concrete strength. When the gravity shear ratio is less than a threshold value (approximately 0.23), ductile flexural failure often occurs at the connections with a small reinforcement ratio. Alternatively, increasing reinforcement ratio, which leads to the brittle punching failure of the connections, reduces the lateral deformation capacity. However, if the gravity shear ratio is greater than this threshold value, there is a critical reinforcement ratio of nearly 0.8%. For low reinforcement ratios up to the critical value, increasing the reinforcement ratio reduces the cracking of slab concrete under the vertical load, which improves the lateral deformation capacity of the slab-column connections subsequently subjected to the horizontal load. Moreover, if the reinforcement ratio is greater than the critical value, the reinforcement fails to yield or yields partially before failure occurs. In this case, increasing the reinforcement ratio reduces the lateral deformation capacity of the slab-column connections. For the same gravity shear ratio, the connection with a higher concrete strength is subject to greater gravity load, but exhibits a larger unbalanced moment capacity and improved lateral deformation capacity.

Figure 9.

Curves of the ultimate unbalanced moment with the changes in the reinforcement and gravity shear ratios: (a) f_c^' = 32 MPa, (b) f_c^' = 52 MPa, (c) f_c^' = 72 MPa, and (d) f_c^' = 92 MPa.

Figure 10.

Curves of the drift ratio with the changes in the reinforcement and gravity shear ratios: (a) f_c^' = 32 MPa, (b) f_c^' = 52 MPa, (c) f_c^' = 72 MPa, and (d) f_c^' = 92 MPa.

Validation of the proposed formulas

When calculating the unbalanced moment capacity, both ACI318-19 (2019) and EC2-04 (2004) adopted the eccentric shear stress model. ACI318-19 assumed that the shear stress was linearly distributed on the critical section, while EC2-04 assumed that the shear stress was uniformly distributed on the critical half-section. Moreover, the formulas of the punching shear capacity in addition to the positions and shapes of the critical section were different. ACI318-19 considered the influence of the gravity shear ratio and provided the formula of the ultimate drift ratio, while EC2-04 did not provide the corresponding formula.

For the connections with square columns, ACI318-19 assumed that 40% of the unbalanced moments was transferred by the linear shear stress on the critical section of concrete and that the remaining 60% was transferred by the flexural reinforcement in the (c + 3h) slab width. The unbalanced moment capacity M_ACI of the reinforced concrete slab-column connections with the square columns according to ACI318-19 can be calculated using Equation (9):

M_{A C I} = \min (\frac{5 (V_{c} - V_{g}) J_{c}}{(c + d) A_{c}}, \frac{M_{f, 1} + M_{f, 2}}{0.6})

(9)

where V_c is the punching shear capacity calculated using Equation (10), V_g is the shear force acting on a critical perimeter due to the gravity load, J_c is the polar moment of inertia of the critical perimeter calculated using Equation (11), c is the column width, d is the effective slab depth, A_c is the area of the critical section located at a distance of 0.5 days from the column perimeter, M_f,1 is the slab flexural capacity for the negative bending evaluated in the slab width of (c + 3h), and M_f,2 is the slab flexural capacity for the positive bending evaluated in the slab width (c + 3h).

V_{c} = 0.33 \sqrt{f_{c}^{'}} A_{c}

(10)

J_{c} = \frac{2}{3} d {(c + d)}^{3} + \frac{1}{6} (c + d) d^{3}

(11)

For a given shear force V_g and moment M, the shear stress v_Ed acting on the critical section of the slab-column connection according to EC2-04 is

v_{E d} = \frac{V_{g}}{u_{1} \cdot d} + \frac{k \cdot M}{W_{1} \cdot d}

(12)

where u₁ is the perimeter of the critical section located at a distance of 2d from the column faces; d is the effective depth of the slabs; k is a coefficient dependent on the column’s section dimensions and its value is a function of the portion of unbalanced moment transmitted by uneven shear, bending, and torsion; and W₁ is akin to the plastic modulus of the control perimeter along the axis of the bending acting through the centroid of the control perimeter.

The maximum permissible shear stress v_Rd acting on the critical section is

v_{R d} = 0.18 ζ {(100 ρ_{1} f_{c}^{'})}^{1 / 3}

(13)

where ζ is the size effect factor, which is equal to a maximum of

1 + \sqrt{200 / d}

and 2, ρ₁ is the average tension reinforcement ratio, and

f_{c}^{'}

is the compressive strength of the concrete cylinder. For a given shear force V_g, the ultimate unbalanced moment can be obtained by substituting v_Rd with v_Ed in Equation (12).

Herein, 81 test data and 64 FEA data of the interior slab-column connections were collected (Appendix A). Based on these test and FEA data, the effect of the gravity shear ratio on the moment ratio was investigated (Figure 11). The ordinate moment ratio is the ratio of the tested unbalanced moment to the calculated value based on ACI318-19, EC2-04, and the proposed formula. The abscissa γ_G,ACI is equal to V_g/V_c,ACI, where V_c,ACI is the punching shear capacity calculated using ACI318-19. The abscissa γ_G,EC is equal to V_g/V_c,EC, where V_c,EC is the punching shear capacity calculated using EC2-04. Notably, the moment ratio calculated using ACI318-19 did not change significantly with the gravity shear ratio but exhibited a large dispersion (Figure 11(a)). The moment ratio calculated using EC2-04 showed a small dispersion; however, the calculated results significantly deviated from the test results when the gravity shear was relatively large (Figure 11(b)). Therefore, based on EC2-04, a modified formula (14) considering the gravity shear ratio was proposed to predict the ultimate unbalanced moment with improved accuracy.

M_{p} = α \cdot M_{0}

(14)

M_{0} = \frac{5}{3} v_{R d} \cdot W_{1} \cdot d

(15)

α = (1 - γ_{G, E C}) (0.9 γ_{G, E C}^{2} + 0.4 γ_{G, E C} + 0.8)

(16)

where M_p is the ultimate unbalanced moment recommended in this study. Figure 11(c) shows the moment ratio calculated using the proposed formula. This figure indicates that the results calculated using the proposed formula were in good agreement with the experimental results. Considering the nonlinear relationship between the unbalanced moment and gravity shear ratio, the proposed formula is more accurate than ACI318-19 and EC2-04.

Figure 11.

Relationship between the moment ratio and gravity shear ratio based on different methods: (a) ACI318-19, (b) EC2-04, and (c) proposed formula.

When the ultimate drift ratio of the slab-column connections was determined, the corresponding lateral force resistance system could be designed. If necessary, the shear reinforcement could be provided to avoid the brittle punching failure of the slab-column connections. Considering the gravity load influence, ACI318-19 (2019), Hueste and Wight (1999) and Megally and Ghali (1994) proposed different formulas for calculating the ultimate drift ratio θ_u, which are expressed using Equations (17)–(19), respectively.

θ_{u} (%) = \max {3.5 - 5 γ_{G, A C I}, 0.5}

(17)

θ_{u} (%) = \max {- 12.5 γ_{G, A C I} + 6.5, - 1.67 γ_{G, A C I} + 2.17} f o r γ_{G, A C I} \geq 0.2

(18)

θ_{u} (%) = 0.5 / γ_{G, A C I}^{0.85} f o r γ_{G, A C I} > 0

(19)

Figure 12 shows the relationship between the ultimate drift ratio and gravity shear ratio as well as the calculation curves of the ultimate drift ratio. In Figure 12, “Proformula” is the abbreviation of the proposed formula. It could be seen from the figure that the ultimate drift ratio generally showed a decreasing trend with the increase in the gravity shear ratio. The ACI318-19 formula roughly gave the lower limit of the ultimate drift ratio, but there were some overestimated data. The calculation formula proposed by Hueste and Wight (1999) was only applicable to the case in which the gravity shear ratio was greater than 0.2 and roughly yielded the average predicted value of the ultimate drift ratio. The calculation formula provided by Megally and Ghali (1994) is not applicable to the case where the gravity shear ratio is relatively small. It is obviously unreasonable that the ultimate drift ratio tended to infinity as the gravity shear ratio approached zero. In the slab-column-wall structures, the slab-column frame mainly shared the gravity load, while the horizontal load was mainly carried by the shear wall. Hence, it is very important to ensure that the slab-column frame has a good lateral deformation capacity to work with the shear wall. Therefore, considering the influence of the gravity load, a relatively conservative formula (20) was obtained based on the test and FEA results. Compared with the other formulas, the proposed formula is more reasonable in agreement with the lower limit of the ultimate drift. In addition, γ_G,EC in Equation (20) was replaced by γ_G,ACI. Figure 12(b) indicates that this formula can also well fit the lower limit of the relationship between the ultimate drift ratio and gravity shear ratio calculated using ACI318-19.

θ_{u} = \frac{1}{4 γ_{G, E C}^{2} + 0.3}

(20)

Figure 12.

Relationship between the ultimate drift ratio and gravity shear ratio based on different codes: (a) EC2-04 and (b) ACI318-19.

Conclusions

In this study, FEA was performed to investigate the punching shear behavior of slab-column connections without shear reinforcement. A numerical model was proposed based on the CDP model in ABAQUS, whose major parameters were calibrated by comparing the test and numerical results. This calibrated model was applied to a parametric study considering the effects of various variables. The primary conclusions are as follows.

An FEA model was established based on the CDP model in ABAQUS. The parameters of the model, such as the viscosity coefficient, mesh size, and fracture energy, were calibrated using a tested slab-column connection. Moreover, the principle of each parameter value was presented and the calibrated model was used for analyzing other tested connections. The analytical results were in good agreement with the experimental results.

Based on the calibrated FEA model, the effects of the gravity shear ratio, reinforcement ratio, and concrete strength on the punching shear performance of the reinforced slab-column connections without shear reinforcement were investigated. The results showed that the ultimate unbalanced moment of the slab-column connections decreased with an increase in the gravity shear ratio and increased with an increase in the reinforcement ratio or concrete strength. Moreover, the ultimate drift ratio decreased with an increase in the gravity shear ratio and increased with an increase in the concrete strength. When the gravity shear ratio was less than the threshold value (nearly 0.23), the lateral deformation capacity of the connection decreased with an increase in the reinforcement ratio. However, if the gravity shear ratio was greater than this threshold value, the lateral deformation capacity of the connection increased first and then decreased with an increase in the reinforcement ratio. For the same gravity shear ratio, the connection with a higher concrete strength showed a larger unbalanced moment capacity and improved lateral deformation capacity.

According to the experimental and FEA results, the formulas for calculating the ultimate unbalanced moment in ACI318-19 and EC2-04 were evaluated. Based on the formula for EC2-04, the correction coefficient of the formula for calculating the ultimate unbalanced moment was proposed and the accuracy of the corrected formula showed obvious improvements. To ensure the lateral deformation capacity of the connection, a formula for calculating the ultimate drift ratio was also proposed, which exhibited a higher guarantee rate than the other formulas.

The proposed methods for predicting the unbalanced moment capacity and ultimate drift ratio are limited to the test and numerical database used in this study and to interior slab-column connections with square columns, unless other evidence exists.

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 would like to acknowledge the financial support from the Natural Science Foundation of China (No. 51178175), which made the research possible.

ORCID iDs

Ming Tang

Hui Chen

Appendix A

References

ACI Committee 318 (2019) Building Code Requirements for Structural Concrete (ACI318-19) and Commentary. Farmington Hills: American Concrete Institute.

Ali

Alexander

(2002) Behaviour of Slab-Column Connections with Partially Debonded Reinforcement under Cyclic Lateral Loading: SER243. Edmonton: University of Alberta, Department of Civil Engineering.

Almeida

AFO

Inácio

MMG

Lúcio

VJG

, et al. (2016) Punching behaviour of RC flat slabs under reversed horizontal cyclic loading. Engineering Structures 117: 204–219.

Bompa

Elghazouli

(2017) Numerical modelling and parametric assessment of hybrid flat slabs with steel shear heads. Engineering Structures 142: 67–83.

Cao

(1993) Seismic Design of Slab-Column Connections. Alberta: University of Calgary.

Choi

Shin

Park

(2014a) Shear strength model for slab-column connections subjected to unbalanced moment. ACI Structural Journal 111(3): 491–550.

Choi

Shin

Park

(2014b) Modification of the ACI 318 design method for slab-column connections subjected to unbalanced moment. Advances in Structural Engineering 17(10): 1469–1480.

Choi

Cho

Han

, et al. (2007) Experimental study of interior slab-column connections subjected to vertical shear and cyclic moment transfer. Key Engineering Materials 348–349: 641–644. DOI: 10.4028/www.scientific.net/kem.

Comité Euro-International du Béton (1993) CEB-FIP Model Code 1990. London: Thomas Telford.

10.

Comité Euro-International du Béton (2012) Model Code 2010-Volume 1, Bulletin 6. Lausanne, switzerland: Internation Federation for Structural Concrete (fib).

11.

Cordebois

Sidoroff

(1982) Damage induced elastic Anisotropy. In: Boehler

(eds) Comportment Méchanique des Solides Anisotropes. Dordrecht: Springer, 761–774.

12.

Deifalla

(2020a) Strength and ductility of lightweight reinforced concrete slabs under punching shear. Structures 27: 2329–2345.

13.

Deifalla

(2020b) Design of lightweight concrete slabs under two-way shear without shear reinforcements: a comparative study and a new formula. Engineering Structures 222: 111076.

14.

Deifalla

(2021a) A mechanical model for concrete slabs subjected to combined punching shear and in-plane tensile forces. Engineering Structures 231: 111787.

15.

Deifalla

(2021b) A strength and deformation model for prestressed lightweight concrete slabs under two-way shear. Advances in Structural Engineering. DOI: 10.1177/13694332211020408.

16.

Deng

(2018) Study on Punching Shear Behavior and Capacity of Reinforced Concrete Slab-Column Connections. Changsha: Hunan University.(in Chinese).

17.

Drakatos

Muttoni

Beyer

(2016) Internal slab-column connections under monotonic and cyclic imposed rotations. Engineering Structures 123: 501–516.

18.

Durrani

Luo

(1995) Seismic resistance of nonductile slab-column connections in existing flat-slab buildings. ACI Structural Journal 92(4): 479–487.

19.

Eder

Vollum

Elghazouli

, et al. (2010) Modelling and experimental assessment of punching shear in flat slabs with shearheads. Engineering Structures 32: 3911–3924.

20.

Emam

Marzou

Hilal

(1997) Seismic response of slab-column connections constructed with high-strength concrete. ACI Structural Journal 94(2): 197–205.

21.

Eurocode 2 (2004) Design of Concrete Structures-Part 1-1: General Rules and Rules for buildings(EC2-04). Brussels, Belgium: European Committee of Standardization.

22.

Fan

, et al. (2021) Critical shear crack theory-based punching shear model for FRP-reinforced concrete slabs. Advances in Structural Engineering 24(6): 1208–1220.

23.

Genikomsou

Polak

(2015) Finite element analysis of punching shear of concrete slabs using damaged plasticity model in ABAQUS. Engineering Structures 98: 38–48.

24.

Genikomsou

Polak

(2016) Finite-element analysis of reinforced concrete slabs with punching shear reinforcement. Journal of Structural Engineering 142(12): 04016129.

25.

Ghali

Elmasri

Dilger

(1976) Punching of flat plates under static and dynamic horizontal forces. ACI Structural Journal 10(73): 566–572.

26.

Guan

(2009) Prediction of punching shear failure behaviour of slab-edge column connections with varying opening and column parameters. Advances in Structural Engineering 12(1): 19–36.

27.

Hanson

(1968) Shear and moment transfer between concrete slabs and columns. Journal of the PCA Research and Development Laboratories 10(1): 2–16.

28.

Hibbitt

Karlsson

Sorensen

(1998) ABAQUS/Standard User Subroutines Reference Manual. USA: The Pennsylvania State University.

29.

Hordijk

(1991) Local Approach in Fatigue of Concrete. Delft: Delft University.

30.

Hueste

Wight

(1999) Nonlinear punching shear failure model for interior slab-column connections. Journal of Structural Engineering 125(9): 997–1008.

31.

Inácio

Isufi

Lapi

, et al. (2020) Rational use of high-strength concrete in flat slab-column connect--ions under seismic loading. ACI Structural Journal 117(6): 297–310.

32.

Islam

Park

(1976) Tests on slab-column connections with shear and unbalanced flexure. Journal of the Structural Division 102(3): 549–568.

33.

Kang

Park

Kim

(2013) Lattice-reinforced slab-column connections under cyclic lateral loading. ACI Structural Journal 110(6): 929–939.

34.

Kang

Wallace

(2008) Seismic performance of reinforced concrete slab column connections with thin plate stirrups. ACI Structural Journal 105(5): 617–625.

35.

Kinnunen

Nylander

(1960) Punching of Concrete Slabs without Shear Reinforcement. Stockholm: Royal Institute of Technology. Transaction No. 158.

36.

Laguta

(2020) Effect of Unbalanced Moment on Punching Shear Strength of Slab-Column Joints. Waterloo: University of Waterloo.

37.

Lapi

Secci

Teoni

, et al. (2020) A hybrid method for the calibration of finite element models of punching-shear in R/C flat slabs. Computers & Structures 238: 1–14.

38.

Liu

Tian

Orton

(2015) Resistance of flat-plate buildings against progressive collapse. II: system response. Journal of Structural Engineering 141(12): 04015054.

39.

Luo

Durrani

Conte

(1994) Equivalent frame analysis of flat plate buildings for seismic loading. Journal of Structural Engineering 120(7): 2137–2155.

40.

Mahroug

MEM

Ashour

Lam

(2014) Experimental response and code modelling of continuous concrete slabs reinforced with BFRP bars. Composite Structures 107: 664–674.

41.

Marzouk

Osman

Hussein

(2001) Cyclic loading of high-strength lightweight concrete slabs. ACI Structural Journal 98(2): 207–214.

42.

Megally

Ghali

(1994) Design considerations for slab-column connections in seismic zones. ACI Structural Journal 91(3): 303–314.

43.

Milligan

(2018) Nonlinear Finite Element Analysis of Punching Shear of Reinforced Concrete Slabs Supported on Rectangular Columns. Waterloo: University of Waterloo.

44.

Morrison

Hirasawa

Sozen

(1983) Lateral-load tests of R/C slab-column connections. Journal of Structural Engineering 109(11): 2698–2714.

45.

Muttoni

(2008) Punching shear strength of reinforced concrete slabs without transverse reinforcement. ACI Structural Journal 105(4): 440–450.

46.

Navarro

Ivorra

Varona

(2018) Parametric computational analysis for punching shear in RC slabs. Engineering Structures 165: 254–263.

47.

Pan

Moehle

(1989) Lateral displacement ductility of reinforced concrete flat plates. ACI Structural Journal 86(3): 250–258.

48.

Park

Kim

Song

, et al. (2012) Lattice shear reinforcement for enhancement of slab-column connections. Journal of Structural Engineering 138(3): 425–437.

49.

Park

Islam

(1976) Strength of slab-column connections with shear and unbalanced flexure. Journal of the Structural Division 102(9): 1879–1901.

50.

Polak

(2005) Shell finite element analysis of RC plates supported on columns for punching shear and flexure. Engineering Computations 22(4): 409–428.

51.

Rha

Kang

Shin

, et al. (2014) Gravity and lateral load-carrying capacities of reinforced concrete flat plate systems. ACI Structural Journal 111(4): 753–764.

52.

Robertson

(1990) Seismic Response of Connections in Indeterminate Flat-Slab Subassemblies. Houston: Rice University.

53.

Robertson

Durrani

(1992) Gravity load effect on seismic behavior of interior slab-column connections. ACI Structural Journal 89(1): 37–45.

54.

Robertson

Kawai

Lee

, et al. (2002) Cyclic testing of slab-column connections with shear reinforcement. ACI Structural Journal 99(5): 605–613.

55.

Robertson

Johnson

(2006) Cyclic lateral loading of nonductile slab-column connections. ACI Structural Journal 103(3): 356–364.

56.

Rubaye

Manalo

Alajarmeh

, et al. (2020) Flexural behaviour of concrete slabs reinforced with GFRP bars and hollow composite reinforcing systems. Composite Structures 236: 111836.

57.

Sadraie

Khaloo

Soltani

(2019) Dynamic performance of concrete slabs reinforced with steel and GFRP bars under impact loading. Engineering Structures 191: 62–81.

58.

Setiawan

Vollum

Macorini

(2019) Numerical and analytical investigation of internal slab-column connections subject to cyclic loading. Engineering Structures 184: 535–554.

59.

Shu

Plos

Zandi

, et al. (2016) Prediction of punching behaviour of RC slabs using continuum non-linear FE analysis. Engineering Structures 125: 15–25.

60.

Song

Kim

Song

, et al. (2012) Effective punching shear and moment capacity of flat plate-column connection with shear reinforcements for lateral loading. International Journal of Concrete Structures and Materials 6(1): 19–29.

61.

Stark

Binici

Bayrak

(2005) Seismic upgrade of reinforced concrete slab-column connections using carbon fiber-reinforced polymers. ACI Structural Journal 102(2): 324–333.

62.

Stasio

Buren

MPV

(1960) Transfer of bending moment between flat plate floor and column. ACI Journal Proceedings 57(9): 299–314.

63.

Stoner

(2015) Finite Element Modelling of GFRP Reinforced Concrete Beams. Waterloo: University of Waterloo.

64.

Tang

Liu

(2019) Investigation on seismic performance of interior slab-column connections subjected to reversed cyclic loading. Earthquake Engineering and Engineering Vibration 39(3): 109–121 (in Chinese).

65.

Tian

Jirsa

Bayrak

, et al. (2008) Behavior of slab-column connections of existing flat-plate structures. ACI Structural Journal 105(5): 561–569.

66.

Vidosa

Kotsovos

Pavlovic

(1988) Symmetrical punching of reinforced concrete slabs: an analytical investigation based on nonlinear finite element modeling. ACI Structural Journal 85(3): 241–250.

67.

Wörle

(2014) Enhanced shear punching capacity by the use of post installed concrete screws. Engineering Structures 60: 41–51.

68.

Wosatko

Pamin

Polak

(2015) Application of damage-plasticity models in finite element analysis of punching shear. Computers & Structures 151: 73–85.

69.

Xiao

Chin

(2007) Flat slabs at slab-column connection: nonlinear finite element modelling and punching shear capacity design criterion. Advances in Structural Engineering 10(5): 567–579.

70.

Zee

Moehle

(1984) Behavior of Interior and Exterior Flat Plate Connections Subjected to Inelastic Load Reversals: UCB/EERC-84/07[R]. Berkeley: University of California, Earthquake Engineering Research Center.