Finite Element Analysis of Reinforced Concrete Slab-Rectangular Column Connections Using ABAQUS

Abstract

Rectangular columns used in flat-slab structures run the risk of punching shear damage due to stress concentrations, especially when bending moments and vertical forces act together at the connections. Using the finite element analysis method, the existing experiments are numerically simulated using the 3D modeling software ABAQUS, and describe the cracking behavior of concrete using the concrete damaged plasticity model. The accuracy of the numerical simulation was calibrated by load–displacement curves and crack patterns using the experimental results. The study of the model was set up with different moment-to-shear ratios and outputs the trend of the average shear stress on the eccentric force side of the slab. The moment transfer coefficients are derived through the equation of ACI-318 and compared with the code values. A safe range of side length ratios is proposed to reduce the risk of punching shear damage from the use of rectangular columns. This provides a reference for practical design, but more experiments are needed to support the proposed recommendations.

1 Introduction

Flat-slab structures supported by rectangular columns are very common. Due to the high stresses near the columns, the slab-column connection may cause punching shear damage at the joint. Unlike square columns, the stresses around columns are uniformly distributed when the nodes are subjected to concentric forces. However, in the case of rectangular columns, the larger the column side length ratio (${\text{C}}_{{{\text{max}}}} /{\text{C}}_{{{\text{min}}}}$), the more the stresses are concentrated near the short side of the column, as stated by Sagaseta et al. [1] and Oliveira et al. [2]. When the slab-column connection suffers the bending moment, the transmission of the moment in the slab increases the shear stress in the vicinity of the column. This will increase the risk of punching shear damage to the connection and make the slab less resistant to punching shear. This may result in the collapse of the structure, threatening human life and property.

In ACI-318 [3], when the connection is subjected to the combined action of vertical force and bending moment, the formula for calculating the shear stress on the critical section is Eq. (1). The critical section is specified as the perimeter position at one-half of the effective thickness of the slab from the surface of the column.

$${\text{V}}_{{\text{u}}} = \frac{{\text{V}}}{{{\text{b}}_{0} {\text{d}}}} + \frac{{\upgamma _{{\text{v}}} {\text{Mx}}}}{{{\text{J}}_{{\text{c}}} }}$$

(1)

where, V is the vertical shear force. ${\text{b}}_{0}$ is perimeter of the critical section, d is the effective depth of slab. M is the unbalanced bending moment, x is the distance from the centroid to the critical sections, and ${\text{J}}_{{\text{c}}}$ is the polar moment of inertia. ${\upgamma }_{{\text{v}}}$ is the ratio of the bending moment transferred by the eccentric shear force to the total unbalanced bending moment, and calculated by Eq. (2). ${\text{c}}_{1}$ and ${\text{c}}_{2}$ are the side length of the column, as shown in Fig. 1.

$$\upgamma _{{\text{v}}} = 1 - \frac{1}{{1 + \frac{2}{3}\sqrt {\frac{{{\text{c}}_{1} + {\text{d}}}}{{{\text{c}}_{2} + {\text{d}}}}} }}$$

(2)

Fig. 1

Assumed distribution of shear stress from ACI-318

In this paper, numerical simulation of the experiment of Teng et al. [4] with a slab- rectangular column connection was carried out using the finite element method. Three-dimensional modeling was performed using the commercial finite element analysis program ABAQUS. Taking the moment transfer coefficient specified in ACI-318 as the safe value, the safe range of the side-length ratio of the rectangular column was explored by changing the moment-to-shear ratio (M/V). According to the simulation results, it can provide a reference for the actual structural design, to reduce the risk of punching shear failure of rectangular column flat slab structure.

2 Test Specimens

Teng et al. [4] tested slab-rectangular column connections with three different column side length ratios, 1:1, 1:3, and 1:5 respectively. The center of the slab is fixed and the jack applies vertical forces on all four sides of the plate. As shown in Fig. 2b. The height of the column for all specimens is 200 mm. The concrete cover is 20 mm. The arrangement of all specimens with compressive reinforcement is ${\Phi }10@260$. The geometry of the three slabs is shown in Fig. 2a. The specific information of specimens is shown in Table 1.

Fig. 2

Experimental settings of Teng S et al. test slabs

Table 1 Characteristics of the Teng S et al. test slabs

3 Finite Element Simulations

3.1 Modeling in ABAQUS

According to the experimental setup, the four sides of the slab are applied with vertical force by four jacks through the spreader beam. Therefore, eight shims are set on the slab through TIE, and the force application points are set on them by coupling. Figure 3 shows the boundary conditions for the connection of the column with an aspect ratio of 1. U1 = 0 means that the degree of freedom in the x-direction is restricted, UR1 = 0 represents that the degree of freedom of rotation in the x-direction is restricted, and 2 and 3 indicate the y and z-directions [5].

Fig. 3

Load and boundary conditions of the Teng S et al. specimen in ABAQUS

The concrete is simulated with an 8-node hexahedral linear reduction integral element (C3D8R) and the reinforcement is simulated with a 2-node truss unit (T3D2). The combination of the two is embedded, i.e., there is no relative slip. One of the most important steps in the finite element is to divide the mesh. To prevent hourglassing problems, the maximum size of the mesh is one-quarter of the slab thickness [5]. And the mesh size of this model is 37.5 (Fig. 6).

The experimental setup is to apply the concentric force to the specimens, while the purpose of this paper is to study the shear stress of the plate on the critical section of the rectangular column for different moment-to-shear ratios. Therefore, different M/V can be achieved by controlling the displacements applied at the force application points on the four sides of the slab.

3.2 Concrete Damage Plasticity Model

The ABAQUS finite element software used in this paper permits the user to use the concrete damage plasticity (CDP) model which can simulate the nonlinear behavior of concrete relatively accurately. The CDP model was first put forward by Lubliner et al. [6] and revised by Lee and Fenves [7]. The yield function, which describes the size of the yield surface shape, is represented by Eq. (3).

$${\text{F}} = \frac{1}{{1 - \upalpha }}\left( {{\overline{\text{q}}} - 3\upalpha {\overline{\text{p}}} +\upbeta \left( {\upvarepsilon ^{{{\text{pl}}}} } \right)\left\langle {{\widehat{\overline{\upsigma }}}_{{{\text{max}}}}} \right\rangle -\upgamma - \left\langle {{\widehat{\overline{\upsigma }}}_{{{\text{max}}}}} \right\rangle } \right) - \overline{\upsigma }_{{\text{c}}} \left( {\upvarepsilon _{{\text{c}}}^{{{\text{pl}}}} } \right)$$

(3)

where,

$$\upalpha = \frac{{\left( {\upsigma _{{{\text{b}}0}} /\upsigma _{{{\text{c}}0}} } \right) - 1}}{{2\left( {\upsigma _{{{\text{b}}0}} /\upsigma _{{{\text{c}}0}} } \right) - 1}},\quad 0 \le \upalpha \le 0.5$$

(4)

$$\upbeta \left( {\upvarepsilon ^{{{\text{pl}}}} } \right) = \frac{{\overline{\upsigma }_{{\text{c}}} \left( {\upvarepsilon _{{\text{c}}}^{{{\text{pl}}}} } \right)}}{{\overline{\upsigma }_{{\text{t}}} \left( {\upvarepsilon _{{\text{t}}}^{{{\text{pl}}}} } \right)}}\left( {1 - \upalpha } \right) - \left( {1 + \upalpha } \right)$$

(5)

$$\upgamma = \frac{{3\left( {1 - {\text{K}}_{{\text{c}}} } \right)}}{{2{\text{K}}_{{\text{c}}} - 1}},\quad 0.5 < {\text{K}}_{{\text{c}}} \le 1$$

(6)

In Eq. (3), ${\overline{\text{q}}}$ is the Mises equivalent effective stress, and ${\overline{\text{p}}}$ is the hydrostatic pressure stress. ${\upalpha }$ is given by Eq. (4), where $\left( {\upsigma _{{{\text{b}}0}} /\upsigma _{{{\text{c}}0}} } \right)$ is the ratio of the concrete biaxial compressive strength and the uniaxial compressive strength. This ratio of different concrete types and strengths is different. The ratio made by Shang et al. [8] at the laboratory of the Dalian University of Technology in China is probably in the range of 1.06–1.48. ABAQUS set it to the default value of 1.16. The function $\upbeta \left( {\upvarepsilon ^{{{\text{pl}}}} } \right)$ is expressed by Eq. (5), which exists when the algebraically maximum principal effective stress ($\widehat{{\overline{\upsigma }}}_{{{\text{max}}}}$) is positive. In the CDP model, the effective stress is $\overline{\upsigma }_{{\text{c}}} = {\text{E}}_{0} \gamma \left( {\upvarepsilon -\upvarepsilon ^{{{\text{pl}}}} } \right)$. The Macauley bracket 〈∙〉 is calculated as ${\text{x}} = \frac{1}{2}\left( {\left| {\text{x}} \right| + {\text{x}}} \right)$. In this equation, $\overline{\sigma }_{{\text{c}}} \left( {\upvarepsilon _{{\text{c}}}^{{{\text{pl}}}} } \right)$ and $\overline{\upsigma }_{{\text{t}}} \left( {\upvarepsilon _{{\text{t}}}^{{{\text{pl}}}} } \right)$ is the effective compressive and tensile cohesion stresses, respectively. The parameter $\upgamma$ is influence the shape of the yield surface. It appears when the maximum principal effective stress ($\widehat{{\overline{\upsigma }}}_{{{\text{max}}}}$) is negative. ${\text{K}}_{{\text{c}}}$ defines the shape of the yield surface in the deviatory plane and it is the ratio of the tensile to the compressive meridian. The difference in its value has little effect on the simulation results as verified by Genikomsou [9]. ABAQUS set it to the default value of 2/3.

The flow rule expresses the direction of the plastic strain increment. In the CDP model, the flow potential function is defined by Eq. (7).

$${\text{G(}}\upsigma {)} = \sqrt {\left( {\upvarepsilon \upsigma _{{{\text{t}}0}} \tan \,\uppsi } \right)^{2} + {\overline{\text{q}}}^{2} } - {\overline{\text{p}}}\,\tan \,\uppsi$$

(7)

where $\upvarepsilon$ represents the ratio of the plastic potential function close to the asymptote, that is, the eccentricity. It defaults to 0.1 in ABAQUS. The $\upsigma _{{{\text{t}}0}}$ is uniaxial tensile stress in concrete. $\uppsi$ is the dilation angle that expresses the direction of the plastic strain increment vector. The larger the dilation angle, the more brittle the concrete. The recommended values in the ABAQUS user manual are 30–45°. The value of the dilation angle in this paper needs trial calculation and was finally determined to be 40°, the calibration results are presented in Sect. 4. ${\overline{\text{q}}}$ and ${\overline{\text{p}}}$ are introduced in Eq. (3).

4 Finite Element Analyses Results

4.1 Load–Deflection Response

Figure 4 shows the load–displacement curves of the results of the three simulated specimens compared with the test results. The finite element model (FEM) results match the experimental values, both for stiffness and ultimate load. The experimental values of ultimate loads for the three samples S11-139, S13-143, and S15-143 are 454 kN, 718 kN, and 776 kN, respectively, and the numerical simulation results are 463 kN, 712 kN, and 757 kN, respectively. The ratios of experimental to simulated values are 0.98, 1.00, and 1.09, respectively. Therefore, the numerical simulation results are desirable.

Fig. 4

Load–displacement response for experimental versus FEM of Teng S et al. slabs

4.2 Crack Pattern

Figure 5 shows the ABAQUS post-processed crack images and experimental photos. Since the numerical simulation is an ideal state, the cracks are symmetrical patterns when the same size of the load is applied to all four sides of the slab. However, the experimental specimens may show the randomness of cracks due to errors in equipment or inhomogeneity of concrete. Therefore, the comparison of the results will not match exactly, but the direction of crack development is the same for all.

Fig. 5

Comparison of cracks patterns of S11-139, S13-143 and S15-143 between experiment and ABAQUS cloud image of Teng S et al. slabs

Fig. 6

Location of the output shear stress in ABAQUS

From the comparison results of the load–displacement curves with the crack patterns, it can be concluded that the accuracy of this numerical simulation is satisfactory. Therefore, the magnitude of the load applied to the four sides of the plate can be varied so that the study of the shear stress of the plate under different moment-to-shear ratios can be carried out.

4.3 Average Shear Stress in the Critical Section

The shear stress used in ACI-318 is in the critical section. For a clear view, the red part of Fig. 6 shows the location of the critical section, i.e., half the effective plate thickness from the column (green part). Figure 7 displays the distribution of shear stress in the critical section. For a more complete comparison, a simulation of the slab with a column side length ratio of 1:2 is added (Fig. 7b). From Fig. 7, it can be seen that the larger the ratio of the column side length (${\text{C}}_{{{\text{max}}}} /{\text{C}}_{{{\text{min}}}}$), the more the stress is concentrated near the short side, which again confirms the results of previous studies.

Fig. 7

Shear stress in the critical section of the slabs by Teng et al.

Relatively large loads are applied to one side of the plate and small loads of the same size are applied to the other three sides to achieve different M/V effects. The shear stress on the critical section on the eccentric force side is recorded, resulting in the trend diagram shown in Fig. 8. When M//${\text{C}}_{{{\text{min}}}}$, the shear stress is positively correlated with M/V. When ${\text{M}} {\bot} {\text{C}}_{{{\text{min}}}}$, the shear stress has a slightly decreasing trend with the increase of M/V. The average shear stress on the critical section of the slab-square column connection on the eccentric force side has an approximately constant trend.

Fig. 8

Average shear stress versus M/V for columns with different side length ratios

4.4 Coefficient of the Bending Moment Transferred by Eccentric Shear

The case of different column side length ratio

Substituting the shear stress in Fig. 8 into Eq. (1) and deriving the value of ${\upgamma }_{{\text{v}}}$, and summarizing it into Fig. 9. When ${\text{M}} {\bot} {\text{C}}_{{{\text{min}}}}$, the moment transfer coefficients of slabs with different column side length ratios have a similar trend with the increase of M/V. The code values calculated from Eq. (2) of ACI-318 for the cases of column side length ratios of 2, 3, and 5 are 0.34, 0.31, and 0.26, respectively. Therefore, it can be seen from Fig. 9a that none of the simulation results exceed the code values, so in the actual design, the moment transfer coefficient in the case where the bending moment is perpendicular to the short side of the column can follow the value calculated by the code formula.

Fig. 9

M/V versus moment transfer coefficient for different column side length ratio slabs

When M//${\text{C}}_{{{\text{min}}}}$, as can be seen in Fig. 9b, they all show an increasing trend with the increase of M/V. The dashed lines indicate the code values, and the dashed lines of the same color correspond to the dots for the same case. Thus only the slab with a column side length ratio of 2 does not exceed the code value. The simulation results for the case of the column’s ${\text{C}}_{{{\text{max}}}} /{\text{C}}_{{{\text{min}}}} = 5$ are out of range and therefore not shown here.

The case of different reinforcement ratio of S13

Because when M/V is relatively small, the ${\upgamma }_{{\text{v}}}$ of the slab with column side length ratio of 3 and reinforcement ratio of 1.43% are within the specification value, as in Fig. 9b. Therefore, the reinforcement ratio of the S13 slab is changed, resulting in the results shown in Fig. 10. Except for the slab with $\rho = 1.6\%$ which is a supplementary model to this paper, the other four S13 slabs with different reinforcement ratios are the experiment specimens of Teng et al. The moment transfer coefficient trends were derived by simulating these five slabs at different M/V. For the case of column side ratio of 3, the ACI formula (Eq. (2)) calculates $\gamma_{v}$ as 0.5. Therefore, the gray part in Fig. 10 is the region where the code value is exceeded. It can be seen that only the slabs with reinforcement ratios less than 0.5% are within the specification values at any M/V.

Fig. 10

M/V versus moment transfer coefficient for S13 slabs with different reinforcement ratios

5 Conclusions

The finite element model created in ABAQUS was used to analyze the punching shear resistance of the slab-rectangular column connection based on the setup of the existing experiments. The concrete damage plasticity model was used to simulate the nonlinear characteristics of concrete. The accuracy of the simulation can be demonstrated by comparing the experimental and simulated load–displacement curves and crack patterns. The shear stress on the critical section is output by post-processing with ABAQUS to obtain the distribution law of shear stress when the plate is subjected to concentric forces. As the ratio of the long and short sides of the column increases, the more the shear stress is concentrated near the short side of the column. By controlling the loads on the four sides of the slab to achieve different M/V, the results calculated using the equation of ACI-318 summarize the trend of shear stress on the eccentric force side, and the magnitude of the moment transfer coefficient for comparison with the code value.

According to the results, as shown in Sect. 4.4, for the designers’ reference, the side length ratio of the columns used should preferably not exceed 2. When columns with a side length ratio of 3 have to be used due to practical conditions, the reinforcement ratio of the slab should not exceed 0.5%. Since only one experiment is simulated in this paper, more experiments are needed to support this proposal. The experiments on the rectangular column in the database are yet to be supplemented, and more experiments on it should be designed in the future to improve the study of the performance of rectangular columns.