Simplified model for estimating the punching load and deformation of RC flat plate based on big data mining

Abstract

In the previous paper, based on the big data of gathered crack image and load-deformation relationship of laboratory experiments, the bending-shearing critical crack method (BSCCM) has been developed to give the mechanical explanation of the punching shear phenomenon in interior slab-column connections without shear reinforcement and the analytical method could analysis the punching shear strength and deformation capacity simultaneously. Based on the big data of the gathered experiments, this paper firstly validates the accuracy of the BSCCM for the standard specimen with 45 tests data, and secondly deduces the equivalent transformation formulas for non-standard specimen through the yield line theory and verifies the applicability of the BSCCM for the non-standard specimens with 98 tests data. And on this basis, through data mining and factor analysis, this paper studies the factors that affecting the punching shear strength and deformation and proposes the simplified calculation formulas for engineering practice. At last, this paper fully validates the reasonability and accuracy of the formulas with 432 tests data. Results show that the BSCCM is applied to the calculation of punching shear strength and deformation for both standard and non-standard specimens. The simplified formulas which considered concrete strength, yield strength of steel/bar, reinforcement ratio, punching span ratio and perimeter thickness ratio of critical section can accurately and rapidly predict the punching shear strength and deformation of the slab-column connection. The conclusion can be available for engineering application and be used as the basis of the performance design of RC flat plate structure.

Keywords

Structural engineering big data data mining punching design formula

1 Introduction

The traditional reinforced concrete (RC) flat plate structure and the hollow floor structure (as shown in Fig. 2, which is more widely applied currently) have proven to be the most appropriate ceiling system for structural and industrial engineering, because there are no visible beams, column capitals, nor drop panels, making the simplicity both for construction (less formwork, simple reinforcement) and use (easy placement of equipment and pipeline installation), meanwhile making the highly space utilization (decreasing storey height significantly, arranging partition freely).

However, the flat plate structure is susceptible to punching shear failure at slab-column connection under bending-shearing composite forces. Once punching shear failure happened, it makes gravity redistribution, worse still, the structure may progressive collapse [2, 33]. Thus, the flat plate structure and the hollow floor structure are cautiously used in high-rise and ultrahigh-rise buildings, let alone in the performance-based seismic design. The bottleneck is that there is no theory-based quantitative analysis method of the punching shear strength and deformation capacity of the slab-column connection.

The previous paper [18] have developed the bending-shearing critical crack method (BSCCM) which could theoretically and quantificationally analysis the punching shear strength and deformation capacity simultaneously. In a slab subjected to a concentrated load with the boundary supported, the flexure cracks will appear from the column edge to the support gradually. Each flexure crack corresponds to the specific shearing-compression zone and the specific shear strength and each flexure crack might become the critical crack. Figure 1 shows the solution of the BSCCM, the shear strength and deformation of each crack are determined at the intersection (1, 2, k …) of the capacity curve (which decreases with the deformation) and the demand curve (which increases with the deformation). The flexure crack section where the shear strength is minimum is regarded as the critical section. The punching shear strength (V_pun) and deformation (ψ_pun) of the slab-column connection are defined as the shear strength and deformation of the critical section, respectively. In the previous study [18], the biaxial strength criteria of concrete, the refined layer model of shearing-compression zone, the distribution of the curvature, the judgment of the critical section and the consideration of size effect and dowel action were described, all these fundamental elements constituted the BSCCM.

Fig.1

The solution of BSCCM.

Fig.2

Hollow floor.

This paper mainly validates the accuracy of BSCCM through the collected standard specimen tests data, and promotes BSCCM to the non-standard specimens. By analyzing the factors controlling the punching behavior of the slab-column connection, this paper proposes the simplified formulas for engineering practice and validates the design formulas with a large number of tests data.

2 Invalidation of the standard specimens

2.1 Test data overview

“Standard specimen” refers to:

as shown in Fig. 3(a): square slab on inner supports arrayed in circular pattern and column loaded (or column supported and square slab loaded in circular pattern), the column should be circular or square;

as shown in Fig. 3(b): circle slab supported on edge and column loaded (or column supported and circle slab loaded in circular edge), the column should be circular or square.

Fig.3

Test specimen.

The deformation and crack of standard specimens are centrally symmetric, which is consistent with the premise and assumption of BSCCM proposed in the previous paper [18]. This paper collects a total of 45 standard specimens from 7 literatures [15 , 43] which report both the failure deflection (or rotation) and the failure load.

The distribution of parameters in the 45 standard specimens are described in Fig. 4. Where h is slab thickness (40 mm∼240 mm), ρ is reinforcement ratio (0.331% ∼2.292%), f_c is concrete strength (9.395MPa∼108.821MPa, φ150 × H300 cylinder compressive strength), λ is punching span ratio (4.324∼6.667, λ = (r_q - r_c)/d, r_q and r_c are the load radius and column radius respectively, d is the effective thickness of the slab), f_y is yield strength of steel (420MPa∼650MPa), K is comprehensive coefficient (0.025∼0.641, K = ρf_y/f_c). According to the former researches [8 , 42], K is a parameter describing the effect of concrete strength (f_c) and the ratio (ρ) and the yield stress (f_y) of the flexural reinforcement.

Fig.4

Distribution of parameters in 45 standard specimens.

The 45 standard specimens basically cover the common situations and even include the situations of high-strength concrete and high-strength reinforcing bar.

2.2 Calculation results

Figure 5 presents the calculation results of two specimens with different reinforcement ratio according to BSCCM. Each flexure crack is possible to develop to be the critical crack and each flexure crack corresponds to a shearing-compression zone and a capacity curve and a demand curve. The minimum value among all the failure loads (i. e. the control point C in Fig. 5) is taken as the actual punching shear load of the specimen. The average rotation ψ which corresponding to the punching shear load is the punching rotation inclination. Where ψ = δ/(r_q - r_c), δ is the deflection of slab end, r_q and r_c are the load radius and column radius respectively, as shown in Fig. 3(a, b), r_q = B₁/2 and r_c = 2b/π for square column with side length b. In the follow-up analysis, to simplify, the final punching load V_pun and rotation ψ_pun are given directly.

Fig.5

Calculation result of capacity curve and demand curve.

Figure 6(a, b) are the scatter diagrams comparing between the calculation values and the test values about the punching load and punching rotation, respectively. As shown in Fig. 6, the relative error between the test values and the calculation values of the punching load is less than 15%, and the relative error of the punching rotation is less than 20%. It can be seen from Fig. 5 that in the peak part of the test curve which closed to failure, the curve changes dramatically. Especially for the specimens with low reinforcement ratio, the slab has good ductility, and the rotation changes significantly, that is the rotation is more sensitive. As a result, the discreteness of the calculation value of the punching rotation is larger than that of the punching load, which can also be reflected from Table 1. The table shows the relative ratio between test values and the calculation results by BSCCM.

Fig.6

Comparison between calculation value and test value ——standard specimens.

Table 1

Relative ratio between the test values and the calculation results of standard specimens

Reference	No. of Slabs	V / (test/cal)		ψ / (test/cal)
		AVG	COV	AVG	COV
[15]	5	0.978	0.151	1.060	0.186
[20]	12	0.942	0.122	1.034	0.167
[23]	5	0.876	0.072	0.946	0.070
[36]	4	1.078	0.077	0.988	0.066
[39]	12	1.081	0.078	0.951	0.158
[40]	6	0.960	0.140	0.934	0.139
[43]	1	1.136	–	0.960	–
total	45	0.995	0.125	0.987	0.149

Note: AVG = average value; COV = coefficient of Variation.

3 Validation of the non-standard specimens

“Non-standard specimen” refers to:

as shown in Fig. 3(c): square slab on inner supports arrayed in square pattern and column loaded (or column supported and square slab loaded in square pattern), the column should be circular or square;

as shown in Fig. 3(d): square slab supported on edge and column loaded (or column supported and square slab loaded in square edge), the column should be circular or square.

The deformation and crack of non-standard specimen are not centrally symmetric, which is not fully consistent with the premise and assumption of BSCCM. This paper collects a total of 98 non-standard specimens from 20 literatures [3 , 50] which report both the failure deflection (or rotation) and the failure load.

3.1 Equivalent transformation

The BSCCM [18] is based on the sector model proposed by Kinnunen and Nylander [20]. For the standard specimen (as shown in Fig. 7(b), both the plate and column are circular), the boundary, deformation and crack are symmetrical about the column, the sector model takes the representative sector block to analysis. For square plate which is more common, as shown in Figs. 3(c, d) and 7(a), the crack develops in a different way from that of standard specimen. As shown in Fig. 7, to promote BSCCM to wider application scope, it is necessary to propose the equivalent conversion principle transforming the non-standard specimens to the standard specimens. Here, this paper deduces the equivalent conversion principle according to the plastic yield line theory [37] and virtual work principle.

Fig.7

Equivalent conversion of the non-standard specimen.

Figure 8(a) shows a triangle isolation plate with orthogonal reinforcement, and assuming m_x and m_y are the unit width ultimate bending moment in x- and y-, respectively. The angle between the yield line and horizontal axis is α. As for isotropic slab with orthogonal reinforcement, m_x = m_y = m_u. As shown in Fig. 8(b), according to the symmetry, isolating the quarter part to analysis. The yield lines AB, BC, AC divide the slab into three portions $\underline{A}$ , $\underline{B}$ , $\underline{C}$ , $\underline{A}$ and $\underline{C}$ rotate about the column edge, $\underline{B}$ rotates about the yield line BC. As shown in Fig. 8(b), the acute angle between yield line and column side is θ, supposing there is a virtual rotation β with $\underline{A}$ , and the rotation of $\underline{C}$ will be β, the rotation of $\underline{B}$ will be $\sqrt{2} \cdot (1 + tan θ) β$ , so, the virtual work of punching load is: $W_{e} = V_{R} β (b_{q} - b / 2)$ (1)

Fig.8

Plastic yield line of square slab.

Where V_R is the punching load, the virtual work of internal force is: $\begin{matrix} W_{i} = 8 m_{u} β [(\frac{b}{2} + \frac{B - b}{2} tan θ) + \frac{\bar{BC}}{2} \cdot \frac{\sqrt{2}}{1 + tan θ}] \\ = 8 β m_{u} (\frac{B - b}{2} \cdot \frac{1 + {tan}^{2} θ}{1 + tan θ} + \frac{b}{2}) \end{matrix}$ (2)

By W_e = W_i: $\frac{V_{R}}{m_{u}} = \frac{8}{b_{q} - b / 2} [\frac{B - b}{2} \cdot \frac{1 + {tan}^{2} θ}{1 + tan θ} + \frac{b}{2}]$ (3)

When θ= 22.5deg, Equation (3) has the minimum value: $\frac{V_{R}}{m_{u}} = \frac{8}{b_{q} - b / 2} [(B - b) \cdot (\sqrt{2} - 1) + \frac{b}{2}]$ (4)

For the circular plate, supposing that the virtual rotation of slab about column side is β, and the vertical deflection of slab end is (r_q - r_c) β.

The virtual work of punching load is: $W_{e} = V_{R} β (r_{q} - r_{c})$ (5)

The virtual work of internal force is: $W_{i} = m_{u} \cdot β \cdot 2 π \cdot r_{s}$ (6)

By W_e = W_i: $\frac{V_{R}}{m_{u}} = 2 π \cdot \frac{r_{s}}{r_{q} - r_{c}}$ (7)

From Equations (4 and 7), and r_q - r_c = b_q - b/2, the radius of equivalent circular slab is: $r_{s, eq} = \frac{2}{π} [2 (B - b) \cdot (\sqrt{2} - 1) + b]$ (8)

Where B and b are the side length of square slab and column, respectively. For square column, it can be equivalently transformed to circular column by the principle of consistent perimeter, the equivalent radius of circular column is: $r_{c, eq} = \frac{2 b}{π}$ (9)

When the square slab loaded by line load or slab supported by outer sides, as shown in Fig. 7(a), the equivalent load radius is: $r_{q, eq} = b_{q}$ (10)

When the square slab loaded by symmetrical concentrated loads, as shown in Fig. 7(a), the equivalent load radius is: $r_{q, e q} = b_{q}^{'}$ (11)

3.2 Test data overview

The basic information of 98 specimens collected in the paper are shown in Fig. 9. Where h = 55mm∼300mm, ρ= 0.370% ∼3.810%, f_c= 12.750MPa∼50.530 MPa, K = 0.025∼0.641, λ= 3.441∼13.423. Where λ = (r_q.eq - r_c.eq)/d, r_q.eq and r_c.eq are the equivalent load radius and column radius respectively, calculated by Equations (9 to 11).

Fig.9

Distribution of parameters in 98 non-standard specimens.

3.3 Calculation results

The calculation results of non-standard specimens by BSCCM are shown in Fig. 10 and Table 2. Since the crack development of non-standard specimen is not as symmetrical as that of standard specimen and the stress concentration exists in the slab corner and column corner, the computational accuracy is poorer than that of standard specimen. However, as shown in Fig. 10, the relative error between the test values and the calculation values of the punching load and the punching rotation are still less than 20%. Both the geometric parameters and material parameters of the collected test specimens are in the common scope of practical engineering, indicating the good practicability of the theoretical method.

Fig.10

Comparison between calculation value and test value——non-standard specimens.

Table 2

Relative ratio between the test values and the calculation results of non-standard specimens

Reference	No. of Slabs	V / (test/cal)		ψ / (test/cal)
		AVG	COV	AVG	COV
[3]	1	1.050	–	0.890	–
[4]	3	0.829	0.004	0.903	0.060
[9]	2	1.038	0	0.971	0.260
[11]	21	0.987	0.131	0.923	0.175
[14]	1	0.729	–	0.820	–
[17]	4	1.031	0.175	1.164	0.144
[24]	9	1.033	0.165	1.042	0.123
[27]	6	1.219	0.054	0.917	0.177
[28]	2	1.058	0.194	1.092	0.075
[29]	2	0.924	0.063	1.221	0.031
[30]	1	0.848	–	0.861	–
[31]	8	1.128	0.070	0.890	0.060
[35]	4	1.071	0.116	1.022	0.127
[38]	8	1.051	0.095	1.008	0.156
[40]	4	1.255	0.020	0.855	0.169
[44]	1	1.114	–	1.131	–
[45]	5	0.943	0.063	0.885	0.152
[46]	6	0.987	0.149	1.080	0.095
[48]	2	0.958	0.105	1.069	0.173
[50]	8	0.959	0.074	1.157	0.060
total	98	1.026	0.137	0.989	0.161

Note: AVG = average value; COV = coefficient of Variation.

4 Factor analysis of punching shear strength and deformation capacity

The above work has validated the overall computational accuracy of BSCCM. To further explore the influence of various factors on punching shear strength and punching deformation, this section will analyze the single factors. As shown in Fig. 11, the tests with a certain investigated factor changing and other factors unchanged are selected. The test results and calculation results by BSCCM are presented. At the same time, to fully reflect the impact of the factors on punching shear strength and punching deformation, the supplementary examples which designed according to the test specimens are also shown in Fig. 11. When there are small differences of “unchanged factors” among the test specimens, the average value would be adopted to design the supplementary example. And in Fig. 11:

Fig.11

Influence factor analysis of punching capacity and deformation.

ψ · λ · d represents the slab end deflection when punching occurred, where ψ = δ/(r_q - r_c) is the average rotation of the slab, λ = (r_q - r_c)/d is the punching span ratio, d is the effective thickness of the slab.

$V / (f_{c}^{1 / 2} \cdot b_{0} \cdot d)$ represents the regular nominal punching shear strength, where V is the punching load, $f_{c}^{1 / 2}$ is the concrete item (as shown in Table 3, referring to the standards of different countries), b₀ is the perimeter of the critical section, calculated according to the round corner section which is 0.5d from the column edge, shown as in Fig. 12(a, c).

Fig.12

Position and shape of the critical section.

Table 3

Specifications of different Codes for punching shear strength^a

Design code	Critical section (u)	Effective height (d)^c	Concrete strength^d	Size effect(S_e)	Other factors (η)^e	Punching shear strength (V)
GB 50010-2010 [13]	Fig. 12 (a)(b)	(d_x + d_y)/2	f _t	1.0-(d-800)/12000; 0.9 ≤ S_e ≤ 1.0	η₁ = f₁ (c₁/c₂) ; η₂ = f₂ (d/u) ; η₁ = min(η₁, η₂)	0.7S_ef_tηud
ACI318-14 [1]	Fig. 12 (a)(b)	(d_x + d_y)/2		–	η₁ = f₃ (c₁/c₂) ; η₂ = f₄ (d/u) ; η = min(η₁, η₂, 0.33)
CEB-FIP MC2010 [7]	Fig. 12 (a)(c)	(d_x + d_y)/2	$f_{ck}^{1 / 2}$	–	η = f₅ (E_s, f_y, d_g, r_s)	$f_{c}^{1 / 2} η ud$
BS 8110-97 [5]	Fig. 12 (d)(e)	(d_x + d_y)/2	$f_{cu}^{1 / 3}$	(400/d)^1/4; S_e ≥ 1.0	η = f₆ (ρ) ; ρ ≤ 0.03	0.79S_e (f_cu/25) ^1/3ηud^b
CAN/CSA A23.3-04 [6]	Fig. 12 (a)(b)	(d_x + d_y)/2		1300/(1000+d); d ≥ 300	η₁ = f₇ (c₁/c₂) ; η₂ = f₈ (d/u) ; η = min(η₁, η₂, 0.38)
EN 1992-1-1:2004 [12]	Fig. 12 (g)(h)	(d_x + d_y)/2	$f_{ck}^{1 / 3}$	1+(200/d)^1/2; S_e ≤ 2.0	η = f₆ (ρ) ; ρ ≤ 0.02	$0.18 S_{e} f_{ck}^{1 / 3} η ud$
DIN 1045-1:2008 [10]	Fig. 12 (d)(f)	(d_x + d_y)/2	$f_{ck}^{1 / 3}$	1+(200/d)^1/2; S_e ≤ 2.0	η = f₆ (ρ) ; ρ ≤ 0.02	$0.21 S_{e} f_{ck}^{1 / 3} η ud$
JSCE 2012 [19]	Fig. 12 (a)(b)	(d_x + d_y)/2		(1000/d)^1/4; S_e ≤ 1.5	η₁ = f₆ (ρ) ; η₂ = f₉ (c₁, c₂, d) ; η₁ ≤ 1.5, η = η₁ × η₂

^aThe units in Table 3 are MPa and mm; ^bWhen f_cu < 25MPa, f_cu= 25; and f_cu≤40MPa; ^cd_x and d_y are the effective height in two orthogonal directions; ^df_t, $f_{c}^{'}$ , f_ck, f_cu are the axial tension strength, cylinder compressive strength, eigenvalue of compressive strength, cubic (150mm×150mm×150mm) compressive strength, respectively; ^ec₁ and c₂ are the long side and short side of rectangular column, respectively, when c₁/c₂ < 2, c₁/c₂ = 2, and c₁/c₂ = 2 for the round column; the functions in Table 3: f₁ = 0.4 + 1.2/(c₁/c₂); f₂ = 0.5 + 0.25α_sd/u; f₃ = 0.166 + 0.332/(c₁/c₂); f₄ = 0.166 + 0.083α_sd/u; f₅ = 1/[1.5 + 43.2r_sf_y/(16 + d_g)/E_s]; f₆ = (100ρ) ^1/3; f₇ = 0.19 + 0.38/(c₁/c₂); f₈ = 0.19 + 0.1α_sd/u; f₉ = 1 +1 [1 +0.5 (c₁ + c₂)/d]; where α_s is 40 for interior columns, 30 for edge columns, 20 for corner columns; ρ is the reinforcement ratio; r_s denotes the position where the radial bending moment is zero with respect to the support axis; f_y is the yield strength of steel; d_g is the maximum size of the aggregate; E_s is the elastic modulus of steel.

Figure 11(a, b) show the impacts of reinforcement ratio (ρ) on punching deformation and punching shear strength, respectively. The figures present the test values of three specimens (a-32, a-25 and c-30) from Kinnunen and Nylander [20] and the calculation results according to BSCCM [18]. Among the three specimens and the supplementary examples, only the reinforcement ratio (ρ) is different. As shown in Table 3, the design specifications in China (GB 50010-2010), Canada (CAN/CSA A23.3-04) and the USA (ACI318-14) did not consider the effects of reinforcement ratio. However, as shown in Figure 11, the influence on punching deformation and punching shear strength are significant. By increasing the reinforcement ratio, the punching deformation reduces significantly and the punching shear strength increases significantly. When the reinforcement ratio increases gradually, the slab failure type also transits from flexure failure to punching failure. In other words, increasing the reinforcement ratio could increase the punching shear strength, but strongly decrease the deformation capacity.

Figure 11(c, d) show the impacts of steel yield strength (f_y) on punching deformation and punching shear strength, respectively. The figures present the test values of three specimens (H1, S1-60 and S1-70) from Moe [31] and the calculation results according to BSCCM [18]. Among the three specimens and the supplementary examples, only the yield strength of steel (f_y) is different. The specifications (as shown in Table 3) and former researchers have not regarded steel yield strength as the key factor for investigating the punching capacity so far, while, it can be found from Fig. 11(c, d) that, with increasing the yield strength of steel, slab deformation reduces significantly and the punching shear strength increases gradually. Especially when using the high-strength steel, the depth of shearing-compression zone is larger than that of regular strength steel under the same reinforcement ratio, at the same time, the development of crack is re-strained. Thus, high-strength steel would increase the punching shear strength, however weaken the energy consumption capacity.

Figure 11(e, f) shows the impacts of concrete strength (f_c) on punching deformation and punching shear strength, respectively. The figures present the test values of four specimens (A-1(a), (b), (c) and (e)) from Elstner and Hognestad [11] and the calculation results according to BSCCM [18]. Among the four specimens and the supplementary examples, only the concrete strength (f_c) is different. According to the figures, concrete strength is a core factor affecting punching deformation and punching shear strength, which could also be convinced from the fact that the concrete strength theory is the foundation of BSCCM [18]. The concrete strength controls the shear resistance level not only when the bending-shearing crack develops inside the slab (where $τ_{u i} = \sqrt{f_{t i}^{'} (f_{t i}^{'} - σ_{u i})}$ ), but also when the bending-shearing crack reaches the column edge (where $τ_{ui} = \sqrt{f_{c} (f_{c} - σ_{ui})}$ , and the influence of concrete strength on the shear resistance in this area is more direct and significant).

Figure 11(g, h) show the impacts of a comprehensive factor (λb₀/d) on punching deformation and punching shear strength, respectively. The figures present the test values of four specimens (6, 7, 10 and 11) in Ladner [23] and the calculation results according to BSCCM [18]. Among the four specimens and the supplementary examples, only the comprehensive factor (λb₀/d) is different. The selected comprehensive factor λb₀/d (where λ is the punching span ratio λ = (r_q.eq - r_c.eq)/d, d is the effective thickness of the slab section; b₀ is the perimeter of the critical section) considers that both λ and b₀/d can affect punching deformation and punching shear strength, but those two factors are also affected by slab size, column width and effective thickness of slab. Therefore, λb₀/d is used to comprehensively express the influence of the punching span ratio (λ) and perimeter thickness ratio of the critical section (b₀/d) on punching deformation and punching shear strength. As shown in Fig. 11(g, h), with increasing λb₀/d, the punching deformation capacity increases gradually and the punching shear strength reduces gradually.

As mentioned above, the punching shear strength and the punching deformation of slab-column connection restrict each other. For the slab-column connections without shear reinforcement, the punching shear strength increases at the expense of reducing the punching deformation. To increase the deformation ability, it is bound to weaken the punching shear strength.

5 Simplified calculation method based on BSCCM

Although the BSCCM proposed previously had good calculation accuracy and wide application scope through the above analysis, it still should be simplified for engineering design [21]. This section will propose the simplified calculation method of punching shear strength and punching deformation through the collected test data and a large number of examples supplemented according to the BSCCM.

5.1 Simplified calculation formulas

Through the parameter analysis in the previous section, the factors that affect punching shear strength and punching deformation include: reinforcement ratio (ρ), concrete strength (f_c), yield strength of steel (f_y), punching span ratio (λ) and perimeter thickness ratio of critical section (b₀/d). As shown in Fig. 11, ρ and f_y are negatively correlated with punching deformation and positively correlated with punching shear strength. In contrast, f_c, λ and b₀/d are positively correlated with punching deformation and negatively correlated with punching shear strength. Therefore, this paper constructs the comprehensive parameter K₀ (as shown in Equation (12)) which represents the comprehensive influence of above factors on punching shear strength and punching deformation. $K_{0} = \frac{ρ f_{y}}{\sqrt{f_{c}} λ b_{0} / d}$ (12)

Figure 13(a) shows the overall scatterplot of K₀ - ψλd, and Fig. 13(b) shows the fitted curve of K₀ - ψλd. Due to the limited test points, 157 supplementary examples are designed based on BSCCM. As shown in Fig. 13(a), the supplementary points and the test points demonstrate the good correlation between K₀ and ψλd. Here, as shown in Fig. 13(b), to improve the fitting accuracy, the paper intercepts the key area (i.e. the practical area in engineering) for curve fitting.

Fig.13

Fitted curve of slab punching deformation.

Supposing the column width span ratio (c/L, where c is column width, L is the axis-to-axis spacing of the columns) is α₁, slab thickness span ratio (h/L, where h is slab thickness) is α₂, and the distance between the point of slab inflection and column axis is 0.22L (i. e. r_s = 0.22L, according to a linear-elastic estimate) [32], thus, the punching span ratio (λ) will be (0.22L - 0.5α₁L)/α₂L, and the perimeter thickness ratio (b₀/d) of critical section will be (4α₁L + πα₂L)/α₂L. The comprehensive parameter K₀ can be expressed: $K_{0} = \frac{ρ f_{y}}{\sqrt{f_{c}} \frac{0.44 - α_{1}}{2 α_{2}} \cdot (4 \frac{α_{1}}{α_{2}} + π)}$ (13)

In engineering practice, 0.5% ≤ ρ ≤ 2 %, 15MPa ≤ f_c ≤ 50MPa, 300MPa ≤ f_y ≤ 50MPa, α₂ ≥ 1/30, and α₁ ≈ 0.1, there will be 0.003 ≤ K₀ ≤ 0.05, thus, select this region as the key area. As shown in Fig. 13(b), according to the test points and supplementary points, when K₀ - ψλd correlation which shows in Equation (14 -1) is adopted, the fitted curve has the smallest residual sum of squares. At the same time, Fig. 13(b) also shows the upper limit curve and lower limit curve with 95% reliability and the calculation formulas are as shown in Equations (14 -2) and (14 -3), respectively.

$ψ λ d = 6.613 + 46.469 \cdot e^{- K_{0} / 0.00603} (14 - 1)$ $ψ λ d = 7.471 + 50.321 \cdot e^{- K_{0} / 0.00674} (14 - 2)$ $ψ λ d = 5.755 + 42.617 \cdot e^{- K_{0} / 0.00531} (14 - 3)$ $v_{0} = \frac{V}{\sqrt{f_{c}} b_{0} d}$ (14)

Figure 14(a) shows the overall scatterplot of ψλd - v₀, and Fig. 14(b) shows the fitted curve of ψλd - v₀ (the calculation of the regular nominal shear stress v₀ is shown in Equation (15)). Due to the limited test, 124 supplementary examples are designed based on BSCCM. As shown in Fig. 14(a), the supplementary points and the test points demonstrate the good correlation between ψλd and v₀. Here, as shown in Fig. 14(b), to improve the fitting accuracy, the paper intercepts the key area (i.e. the practical area in engineering) for curve fitting.

Fig.14

Fitted curve of punching load-punching deformation.

As previously mentioned, in engineering practice, when 0.003 ≤ K₀ ≤ 0.05, ψλd will not larger than 50; and when regular nominal shear stress v₀ = 0.7, it is far beyond the practical possible range. Thus, as shown in Fig. 14(a), in the present study, the area with ψλd ≤ 50 and v₀ ≤ 0.7 is intercepted as the key area for curve fitting. As shown in Fig. 14(b), according to the test points and supplementary points, when ψλd - v₀ correlation which is shown in Equation (16 -1) is adopted, the fitted curve has the smallest residual sum of squares. At the same time, Fig. 14(b) also shows the upper limit curve and lower limit curve with 95% reliability and the calculation formulas are as shown in Equation (16 -2) and (16 -3 ), respectively. $100 \cdot ν_{0} = 15.498 + 62.780 \cdot e^{- ψ λ d / 16.913} (16 - 1)$ $100 \cdot ν_{0} = 11.545 + 59.328 \cdot e^{- ψ λ d / 13.493} (16 - 2)$ $100 \cdot ν_{0} = 19.450 + 66.232 \cdot e^{- ψ λ d / 20.332} (16 - 3)$

5.2 Calculation and verification

Ospina and Birkle [34] developed a databank of two-way RC slabs without shear reinforcement at interior supports. The databank makes a detailed statistic of 519 specimens. Based on the series of Data Acceptance Criteria (DAC), this paper collects 148 specimens from 11 literatures [3 , 49–51], which written in Chinese and Japanese and not collected by the databank [34] (two parts constitute the database contain a total of 667 specimens). This paper selects two kinds of tests, namely, C-C group (as shown in Fig. 3(a, b), $\underline{C}$ ircular boundary loaded or supported, $\underline{C}$ ylinder column or square column) and S-S group (as shown in Fig. 3(c, d), $\underline{S}$ quare boundary loaded or supported, $\underline{S}$ quare column), to make the detailed discussion. There are two reasons for extracting those two-group data: firstly, the specimen shape and the loading pattern of the two groups are the most regular and the closest to the theoretical method built previously; secondly, most researchers have chosen those two ways to test, being 136 and 324, respectively, accounting for 67.85% of the database, so the analysis results of those two groups are the most representative. Since key data missing, 22 specimens in C-C group and 6 specimens in S-S group are removed, and there remains 114 and 318 test data in C-C group and S-S group, respectively.

Figures 15 and 16 show the overall parameters information of the test specimens in C-C group and S-S group, respectively. Each parameter is divided into different grades, the first grade and last grade are uncommon cases and the middle grades are common cases in engineering. It also calculates the proportion of each grade in the group. It should be noted that the international units (MPa, mm, N) are adopted in the proposed simplified calculation formulas (Equations (14 and 16)), and f_c is the φ150×H300 cylinder compressive strength. Conversion of different concrete compressive strengths into f_c (φ150×H300) is adopted by Reineck et al. [41].

Fig.15

Distribution of parameters in C-C group.

Fig.16

Distribution of parameters in S-S group.

Here, the test specimens both in C-C group and S-S group are calculated by the simplified calculation methods proposed in this paper (Equations (14 and 16)) and proposed by Muttoni [32] which served as a basis for the fib Model Code for Concrete Structures of 2010. The r_c,eq and r_q,eq of S-S group are calculated by Equations (9 to 11). To make better comparison with the specifications, China’s concrete structure design code (GB50010-2010), American concrete institute code (ACI318-14), British standard (BS8110-97), Canadian standards association (Design of concrete structures CAN/CSA A23.3-04), European concrete structure design code (EN 1992-1-4:2004), Germany standard for concrete structure (DIN 1045-1:2008) and Japanese concrete institute specification (JSCE2012) are introduced for comparison (the detailed contents are shown in Table 3).

The calculation results are shown in Table 4. The statistical results in the table are the ratios of the test values to the calculated values of punching shear load. It can be seen from the table that the simplified calculation method proposed in the paper (Equations (14 and 16)) has the best accuracy. The calculation results of the seven design codes have great deviations, in which JSCE code has the best calculation results; the calculation results of China’s concrete code are better than those of ACI but poorer than European code and JSCE. JSCE considers the influence of size effect, concrete strength, reinforcement ratio and perimeter thickness ratio of critical section on punching shear strength, thus it has the best calculation effect. Besides, the method suggested in this paper still considers the influence of punching span ratio.

Table 4

Calculation results of punching shear load

Calculation method	C-C Group		S-S Group
	AVG	COV	AVG	COV
Simplified BSCCM	1.00	0.188	1.075	0.173
Muttoni	1.075	0.232	1.446	0.200
GB50010-2010	1.491	0.341	1.285	0.330
ACI318-14	1.676	0.332	1.455	0.329
BS 8110-97	1.244	0.244	1.115	0.316
CAN/CSA A23.3-04	1.456	0.331	1.265	0.328
EN 1992-1-4:2004	1.240	0.264	1.289	0.321
DIN 1045-1:2008	1.280	0.256	1.309	0.316
JSCE2012	1.119	0.269	1.117	0.315

Note: AVG = average value; COV = coefficient of Variation.

It can be found that the simplified method proposed in this paper has high efficiency and accuracy. Since it needs to calculate the punching deformation ψλd according to Equation (14) when calculating the punching shear strength according to Equation (16), it indirectly indicates that the method also has high accuracy in deformation calculation.

6 Conclusion and prospect

Based on the previous research [18], this paper validates the calculation results of BSCCM, deduces the equivalent transformation method of non-standard specimens, and based on factor analysis, the paper finally proposes the simplified calculation formulas of punching shear strength and punching deformation for engineering practice. Results show that the previous theoretical method and the simplified calculation formulas proposed in this paper have high calculation efficiency and accuracy.

Based on the research results of the previous paper [18] and this paper, it is possible to accurately quantify the punching behavior of the slab-column structure and reasonably consider the restriction between the punching shear strength and punching deformation of slab-column connections.

It also facilitates the evaluation on the capacity and deformation of connections and provides a new idea for the design method of the slab-column structure based on performance. When flat plate structure bears the eccentric loading, the column rotation will increase the rotation of the slab due to concentric gravity load ψ_v at one half of the slab and decrease it at the opposite half. Punching failure occurs when the sum of the rotations reaches the ultimate rotation of the slab, ψ_u, which is associated with concentric punching failure and could be calculated by BSCCM. However, it is necessary to make in-depth researches in the future to further validate the feasibility of BSCCM through the full-size laboratory tests and the non-linear analysis. Moreover, it is necessary to further study the situation while the slab has the punching reinforcement.

Footnotes

Acknowledgments

This research was financially supported by the Doctoral Program Foundation of Zunyi Normal College (BS[2015]17#), Youth Scientific Talents Project of Education Department of Guizhou Province of China (Qian Jiao He KY Zi [2016]257), United Foundation of Science and Technology Department of Guizhou Province of China (Qian Ke He LH Zi [2017]7083), Cooperative Education Project of Ministry of Education of China (No. 201701074015, No. 201701061006).

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