Punching of reinforced concrete slab without shear reinforcement: Standard models and new proposal

Abstract

Reinforced concrete (RC) slabs are characterized by reduced construction time, versatility, and easier space partitioning. Their structural behavior is not straightforward and, specifically, punching shear strength is a current research topic. In this study an experimental database of 113 RC slabs without shear reinforcement under punching loads was compiled using data available in the literature. A sensitivity analysis of the parameters involved in the punching shear strength assessment was conducted, which highlighted the importance of the flexural reinforcement that are not typically considered for punching shear strength. After a discussion of the current international standards, a new proposed model for punching shear strength and rotation of RC slabs without shear reinforcement was discussed. It was based on a simplified load-rotation curve and new failure criteria that takes into account the flexural reinforcement effects. This experimental database was used to validate the approaches of the current international standards as well as the new proposed model. The latter proved to be a potentially useful design tool.

Keywords

punching shear strength / reinforced concrete / slabs / reinforcement ratio

Introduction

The use of reinforced concrete (RC) slabs has a number of advantages, including reduced and simpler formwork, versatility, and easier space partitioning. They represent economical and efficient structural systems, but however common their use may be, their structural behavior is not straightforward and has been analyzed over numerous years [¹]. It is the subject of current studies that are considering its response under impulsive and blast loading [²–⁵].

Therefore, the analysis of punching in RC slab is topical for a number of reasons. Punching collapse can be dangerous as it develops with a brittle mechanism and affects the integrity of the structure. A localized punching collapse can result in a general structural failure. A very interesting state of the art report is presented in Ref. [⁶].

International standards (Eurocode 2 [⁷], ACI 318 [⁸], and Model Code 2010 [⁹]) present different punching models and prescriptions that yield different structural design results. In addition, if a comparison between theoretical and experimental punching shear strengths [¹⁰–²⁰] is conducted, different safety coefficients are obtained.

In Model Code 2010 (MC10) [⁹], the punching shear strength model is based on the critical shear crack theory (CSCT) [²¹], whereas ACI 318 [⁸] and Eurocode 2 (EC2) [⁷] refer to empirical models based on the maximum shear stress. The CSCT also provides an estimate of the slab rotation from punching; however, the other two models do not provide this information.

Comparisons between the theoretical models and the experimental results can be found in Refs. [¹¹,¹³,¹⁹, ²⁰,²²–²⁴], but they either do not consider all of the standard models or their experimental benchmark databases are limited. Sigrist et al. [²⁵] point out that shear resistance understanding is incomplete and provide a very interesting discussion on the advantage of physical models over empirical formulae.

Over the past few years, a number of researchers have analyzed the effects of fiber reinforcement on RC structures and on RC slabs subject to punching loads [²⁶–²⁸] with the aim of assessing the performance of new materials and innovative technologies. In particular [²⁷], presents an innovative approach for the design of punching shear reinforcement for RC slabs using fiber reinforced polymer bars and stirrups. An interesting experimental analysis is developed in Ref. [²⁸] where the punching performance of RC slabs strengthened with ultra-high performance fiber reinforced cement-based composite is described.

Interesting results were obtained with reinforced recycled concrete slabs [¹⁴–¹⁵,¹⁸,²⁹–³²], where the low mechanical performance of the materials corresponded to unexpectedly good structural performance of the whole structural element. As shown in Ref. [¹⁴], the increasing replacement percentage of recycled aggregates with respect to the natural ones produced a reduction of compressive strength and elastic modulus of concrete, but RC slab specimens realized with recycled aggregates present the same punching resistance of slab realized with natural aggregates.

Concrete elements with plastic void formers [³³] and lightweight aggregate were also tested for punching shear strength, and Ref. [³⁴] reports on both experimental results and design equations considering slabs with small reinforcement ratios.

In RC slabs without shear reinforcement (see Refs. [³⁴–³⁵]), brittle punching failure may be a critical factor in the design, and specific attention should be paid to this to ensure adequate deformation and redistribution of bending moments.

The empirical approach, adopted in a number of international codes, is sufficient for simple design practice. In addition, further limitations in construction details lead to very conservative approaches. However, to develop efficient construction systems, it is essential to understand the physical phenomena and to justify the behavior of existing slabs whenever they do not fall within the typical limits and conditions.

In this study the punching shear strength models adopted by the abovementioned international standards (ACI 318, EC2, and MC10) are discussed in Section 2 in order to highlight the characteristics, limits, and reliability of each model. A comprehensive experimental database was developed using the results reported in the literature, and direct comparisons with the theoretical models, and their analysis are presented in Section 3. The considered database is representative of the entire population of RC slabs under punching load, as the frequency distribution of the key parameters considered in the theoretical models is sufficiently uniform. A sensitivity analysis is developed and is explained in Section 4; this analysis identifies the primary geometric and mechanical parameters which influence the experimental punching shear strength. The proposal of a new punching shear strength model based on CSCT is presented in Section 5. This model takes into account the rotation capacity of the slab and it is designed specifically for elements without shear reinforcement. It is characterized by a parameter representing the flexural reinforcement in the collapse criterion formulation. Thus, it can highlight the flexural reinforcement role in RC slab punching. The conclusions drawn are presented in Section 6.

Standard theoretical models for punching shear strength

ACI 318

The model presented in ACI 318 [⁸] assesses the punching shear strength of slabs without specific shear reinforcement but does not provide any information regarding their rotation capability. In this case, the punching load V_theo,ACI (N) is obtained as the minimum value between Eqs. (1a), (1b), and (1c)

(1a)V_{\mathrm{c}}=0.17\cdot (1+\frac{2}{\beta })\cdot \lambda \cdot \sqrt{f_{\mathrm{c}}}\cdot b_0\cdot d,

(1b)V_{\mathrm{c}}=0.083\cdot \left(\frac{a_{\mathrm{s}}\cdot d}{b_0}+2\right)\cdot \lambda \cdot \sqrt{f_{\mathrm{c}}}\cdot b_0\cdot d,

(1c)V_{\mathrm{c}}=0.33\cdot \lambda \cdot \sqrt{f_{\mathrm{c}}}\cdot b_0\cdot d,

where b is the ratio between the minimum and maximum dimensions of column cross sections (or loading surface); a_s = 40 for an interior column, 30 for a side column, and 20 for a corner column; l= 1 for normal weight concrete and 0.75 for lightweight concrete, and is determined considering volumetric proportions of lightweight and normal weight (normal weight concretes are characterized by a density varying between 2155–2560 kg/m³) aggregates (see ACI 318 [⁸]); f_c (MPa) is the cylindrical compressive strength of concrete, b₀ (mm) is the perimeter of the critical section located at a distance d/2 from the edges of the loaded area (in the critical section curved corners are not taken into consideration); and d (mm) is the effective depth (the distance from extreme compression fiber to the centroid of the longitudinal flexural reinforcement of the cross section), see Fig. 1.

This model does not consider the contribution of flexural reinforcement to punching shear strength, which is typically determined by Eq. (1c). However, experimental tests have shown that thin slabs [¹⁰] and elongated rectangular load areas [¹²] exhibit significant resistance reductions, and that Eq. (1c) is not conservative. Specifically, if the greatest load surface size is considerably greater than the thickness of the slab, the appropriate equations are Eqs. (1a) and (1b). In addition, for square columns, Eq. (1b) is valid when the side of the load surface is approximately four times greater than the thickness of the slab. In the case of significantly elongated rectangular load surfaces (height/base ratio>3), the experimental results indicate that the optimum equation to be used is Eq. (1a).

The ACI 318 punching model can be beneficial to the early design of slab dimensions, as it allows determination of the slab thickness knowing the punching load, surface load, and concrete mechanical characteristics.

Eurocode 2

The Eurocode 2 (EC2) [⁷] punching model is capable of assessing the punching shear strength of slabs without shear reinforcement, but does not provide any information concerning rotation capability. The theoretical punching strength V_theo,EC2 is given by Eq. (2).

V_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o},\mathrm{E}\mathrm{C}2}=\left[\frac{0.18}{{\gamma }_{\mathrm{c}}}k\cdot {\left({\rho }_{1}100f_{\mathrm{c}\mathrm{k}}\right)}^{\frac{1}{3}}+k_1{\sigma }_{\mathrm{c}\mathrm{p}}\right]\cdot b_{0}d

(2)\ge \left(v_{\min }+k_1{\sigma }_{\mathrm{c}\mathrm{p}}\right)\cdot b_{0}d,

where g_c = 1.5; k=1+\sqrt{\frac{200}{d}}\le 2.00; d in mm; {\rho }_1= \sqrt{{\rho }_{1x}\cdot {\rho }_{1y}}\le 0.02; r_lx and r_ly are the reinforcement ratios of the flexural reinforcement in the x- and y-directions, respectively;

v_{\min }=0.035\cdot k^{3/2}\cdot f_{\mathrm{c}\mathrm{k}}^{1/2}

; s_cp represents the prestressing stress, k₁ = 0.1; and b₀ is the perimeter of the critical section, as can be seen in Fig. 2.

The EC2 allows the determination of the slab thickness given the surface load sizes, concrete compressive strength, and reinforcement ratio. However, these parameters can be unknown in the early stages of structural design.

Model Code 2010

According to MC10 [⁹], the theoretical punching shear strength and the corresponding rotation of the slab can be determined by the intersection of the load-rotation curve and the punching failure criterion.

The MC10 load-rotation curve representing the flexural behavior of the slab without shear reinforcement is based on the CSCT proposed by Muttoni [²¹] based on the works carried out in Switzerland in the 80s.

Actually, MC10 provides four different approximation levels (from I to IV) to estimate the rotation \psi. In this work the authors consider MC10 level II approach that is more accurate than level I without the computational cost of the nonlinear analysis required in the level IV case. Finally, level III is recommended for irregular slabs with particular geometric characteristics.

The MC10 level II approach is based on a re-distribution of the bending action expressed by the following equation:

(3)\psi =1.5\frac{r_{\mathrm{s}}}{d}\cdot \frac{f_{\mathrm{y}}}{E_{\mathrm{s}}}\cdot {\left(\frac{m_{\mathrm{E}\mathrm{d}}}{m_{\mathrm{R}\mathrm{d}}}\right)}^{1.5},

where E_s is the steel elastic modulus (typically 200 GPa); f_y is the steel yielding strength; r_s can be expressed as 0.22 L_x and 0.22 L_y for the x- and y-directions, respectively, in the case of regular slab with span ratios (L_x/L_y) ranging from 0.5 to 2; m_Ed is the bending moment from loads per unit length used for reinforcement design in a given direction; m_Rd is the bending moment capacity per unit length for a given direction. In evaluating m_Rd, the following assumptions are made: plane sections remain plane; the strain in bonded reinforcement, whether under tension or compression, is the same as that in the surrounding concrete; the tensile strength of the concrete is ignored; stress block distribution is as for concrete under compression; and perfect elastic-plastic behavior for steel reinforcement under tension and compression.

The collapse scenario hypotheses are as follows: concrete ultimate compressive strain equal to 3.5‰; and yielded flexural steel reinforcement. The compressive reinforcement is not taken into account.

(4)m_{\mathrm{Rd}}=\rho d^2{f}_{\mathrm{y}}(1-\frac{\rho }{2}\cdot \frac{f_{\mathrm{y}}}{f_{\mathrm{c}}}),

where r is the tensile reinforcement ratio; and, f_c is the concrete compressive strength.

For internal columns without eccentricity of shear-force V_Ed, it is typically taken such that m_{\mathrm{E}\mathrm{d}}=V_{\mathrm{E}\mathrm{d}}/8. Therefore, considering Eqs. (3) and (4), it is possible to obtain a simplified load-rotation equation:

(5)V=8\cdot m_{\mathrm{R}\mathrm{d}}\cdot {\left[\frac{\psi E_{\mathrm{s}}d}{1.5r_{\mathrm{s}}{f}_{\mathrm{y}}}\right]}^{2/3}\mathrm{.}

The intersection between the load-rotation Eq. (5) and the failure criterion Eq. (6) provides the theoretical punching shear load.

(6)V_{\mathrm{R},\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h}}=k_{\psi }\cdot \frac{\sqrt{f_{\mathrm{c}}}}{{\gamma }_{\mathrm{c}}}\cdot b_{0}d,

where d is the effective depth of the slab; b₀ is the critical perimeter at a distance d/2 from the actual load area; and the parameter k_ψ is related to the slab rotation y by the following equation:

(7)k_{\psi }=\frac{1}{1.5+0.9\cdot k_{d_{\mathrm{g}}}\cdot \psi \cdot d}\le \mathrm{0.6.}

If the aggregate maximum size is greater than 16 mm, then k_{d_{\mathrm{g}}}= 1, and if it is smaller than 16 mm,

(8)k_{d_{\mathrm{g}}}=\frac{32}{16+d_g}\ge 0.75.

For high-strength concrete lightweight concrete, the aggregate particles could break, resulting in a reduced aggregate interlock contribution. Therefore, it is conservative to assume k_{d_{\mathrm{g}}}= 2 (in which case, d_g = 0).

Figure 3 shows the load-rotation and failure criterion curves. The intersection point is highlighted, and represents the theoretical punching load V_theo,MC10 and the corresponding rotation y_theo,MC10.

The exact punching strength can be obtained from the MC10 model using a numerical iterative process to determine the intersection point (y_theo,MC10,V_theo,MC10) of Eqs. (5) and (6). If V_theo,MC10 is smaller than V_Ed the slab characteristics must be modified. However, a simple design check can be performed calculating the slab rotation y_d corresponding to the shear force V_d using Eq. (5). Then, the corresponding punching shear strength can be obtained plugging y_d in the failure criterion (Eq. (6)), see Ref. [²¹].

Standard models discussion

The three models (ACI 318, EC2, and MC10) are currently the predominant models available for punching load assessment.

ACI 318 is based on a purely empirical model and does not consider the contribution of the flexural reinforcement. EC2 is based on the comb model integrated and tuned by several experimental parameters.

The MC10 model is based on CSCT, as above mentioned. It is notable that its load-rotation curve (Eq. (5)) depends on the flexural reinforcement, while the failure criterion Eq. (6) does not consider this parameter. In contrast with the ACI 318 and EC2 models, but in agreement with Walraven [³⁶], Vecchio and Collins [³⁷], and Muttoni [²¹], the MC10 punching shear model considers a direct relationship between the punching shear strength and the maximum aggregate size. The roughness of the critical crack and its capacity to resist shear forces is proportional to the maximum aggregate size. Also, other recent studies addressed the influence of the maximum aggregate size to the shear strength, see Refs. [³⁸–⁴⁰].

It is of interest to develop a reliability analysis based on experimental data which can be found in literature using the abovementioned models. In this manner it is possible to assess and compare their performance in punching shear strength evaluation for RC slabs without shear reinforcement.

Experimental database

From a literature search the authors developed a database of 113 experimental punching test results. The rectangular or circular RC slabs are simply supported along the edges, do not have specific shear reinforcement, and they are characterized by slenderness, reinforcement ratio, concrete strength, and maximum aggregate size varying over a wide range in order to represent the bulk of actual cases.

Figure 4 presents the experimental punching load and rotation (V-y)_punch,exp data gathered from the literature. In addition, in Table 1 the reference, sample number, publication year, primary focus of the research, and the method used to assess the punching rotation of database are presented. The y_punch,_exp was directly assessed in a number of cases and obtained by indirect measurement, and as the ratio between the deflection and the span length in other cases (Table 1: D = direct measurement; and I = indirect measurement).

Figure 4 clearly shows that the punching rotation is inversely proportional to the punching load. Therefore, it is important to understand the geometric and mechanical parameters which influence this phenomenon.

The geometrical and mechanical parameters of the slab database are presented in Table 2. In addition, Fig. 5 presents the frequency distribution of the abovementioned parameters, where it can be seen that 75% of the sample exhibits a slenderness ratio h/L = 0.10. This value ensures that the slabs flexural behavior is acceptable and conform to international standards. The distributions of other parameters are approximately uniform. The authors consider the sample to be representative of the typical RC slab population, and suitable for testing the theoretical models presented in the previous section.

Given the high variance of the abovementioned parameters it is essential to consider a dimensionless punching load. In agreement with ACI 318 and MC10, the experimental punching load V_punch,exp is transformed into a dimensionless load:

(9)V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\exp ,\mathrm{a}\mathrm{d}\mathrm{m}}=\frac{V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\exp }}{d{b}_0\sqrt{f_{\mathrm{c}}}},

where b₀ is the critical perimeter at a distance d/2 from the actual load area. The correlation diagrams between the dimensionless punching load and the other parameters are presented in Figs. 6–11. The applicable correlation coefficients r are given in each figure:

(10)r=\left|\frac{{\displaystyle \sum _i^n\left(x_i-x\right)}\left(y_i-y\right)}{\sqrt{{\displaystyle \sum _i^n{\left(x_i-\overline{x}\right)}^2}}\sqrt{{\displaystyle \sum _i^n{\left(y_i-\overline{y}\right)}^2}}}\right|,

where x_i and y_i are the investigated variables; and \overline{x} and \overline{y}are the average values of the variables, respectively.

Figure 6 shows a clear correlation between the dimensionless punching shear strength and the punching rotation. The value of r is significant (0.515) and it can be observed that greater values of y correspond to smaller load values.

As can be seen in Fig. 7, the slenderness correlation is not clear. As discussed previously, the experimental parameter was not investigated across a wide range of this variable, and it is difficult to observe its influence on the punching shear strength.

There is significant variance in the a/L-V_{punch,exp,adm} and D_aggr,max/h-V_{punch,exp,adm} correlations, and there is no clear trend in Figs. 8 and 9. However, a clear trend can be observed in Fig. 10 where the flexural reinforcement ratio has the greatest value of the correlation coefficient r (0.702), and it can be concluded that the greater the value of r, the greater the punching shear strength. Finally, the ratio between top and bottom reinforcement ratio r′/r also exhibits a strong correlation with the punching load. In this case the variance is significant, but it is apparent that the greater the ratio, the smaller the punching shear strength.

Figure 12, extracted from Ref. [²¹], presents the load-rotation curves obtained by Kinnunen and Nylander [²²]. A strong correlation can be clearly identified between the dimensionless punching load, the rotation y, and the flexural reinforcement ratio r.

Muttoni [²¹] reported that the dimensionless punching load V_{punch,exp,adm} and the ultimate rotation y_punch,exp (Fig. 12, rectangular spots) are highly dependent on the flexural reinforcement ratio r. Greater values of y_punch,exp correspond to smaller values of r and V_{punch,exp,adm}, and a smaller y_punch,exp corresponds to greater values of r and V_{punch,exp,adm}.

The authors developed a database characterized by a greater variation range of the flexural reinforcement ratio r, which enabled an expansion of what was reported in Refs. [²¹,²²]. In Fig. 13, it can be seen that for smaller values of flexural reinforcement (0.50%<r<0.60%) there is a significant data variance. In particular, when r is small (0.20%<r<0.30%), there is a poor correlation between the ultimate rotation and punching shear strength. However, with mid-sized (0.70%<r<1.20%) and greater (2%<r<4%) values of r, the flexural reinforcement ratio trend reported by Refs. [²²,³⁵] is clearly observed.

International standards models for punching shear strength

ACI 318 punching model

Figure 14 shows a comparison between the database experimental punching loads V_punch,exp and the theoretical values obtained using ACI 318 (V_{punch,ACI 318}) approach, discussed in Section 2.1. The complete experimental data set is shown in Fig. 14(a), and the range 0–1000 kN is shown in Fig. 14(b). The values of V_{punch,ACI 318} were estimated using Eqs. (1)–(3) with f_c = f_cm.

Eurocode 2 punching model

The experimental punching loads V_punch,exp, and the theoretical values obtained using the EC2 model (V_punch,EC2), discussed in Section 2.2, are shown in Fig. 15. The complete experimental data set is shown in Fig. 15(a), and the range 0–1000 kN is shown in Fig. 15(b). V_punch,EC2 was estimated using Eq. (2) with g_c = 1 and f_ck = f_cm.

Model Code 2010 simplified punching model (level II)

Figure 16 presents a comparison between the experimental punching loads V_punch,exp and the theoretical values (V_punch,MC10) obtained using the simplified MC10 approach presented in Section 2.3. The complete experimental data set is shown in Fig. 16(a), and the range 0–1000 kN is shown in Fig. 16(b). The values of V_punch,MC10 were estimated using Eqs. (5) and (6) with g_c = 1 and f_c = f_cm. This approach is based on an effective bending moment re-distribution. The rotations around the supported area are calculated with level II approximation (see Ref. [⁹]).

Figure 17 shows the comparison between the experimental and theoretical punching rotation y_punch values. Figure 17(a) shows the complete data set, and Fig. 17(b) the range 0–0.04 rad.

The MC10, ACI 318, and EC2 punching models were found to be accurate in determining the punching shear load, and the average ratio between the experimental and theoretical values is marginally greater than one in each case. Only the MC10 model could determine the ultimate rotation y_punch,theo.

The average, maximum, and minimum values of the ratio V_punch,exp/V_punch,theo with the corresponding coefficient of variation (CoV, see Eq. (11)) are presented in Table 3. For the MC10 case the same analysis is presented for the ratio y_punch,exp/y_punch,theo.

(11)CoV=\frac{\mathrm{standard\; deviation}}{\mathrm{average\; value}}=\frac{\frac{\sqrt{{\sum }_{i=1}^n{\left(x_i-x\right)}^2}}{n-1}}{x}

where x_i=\frac{V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\mathrm{e}\mathrm{x}\mathrm{p}}}{V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}}; and \overline{x}={\displaystyle \sum _{i=1}^n\frac{x_i}{n}}.

From Table 3 it can be seen that the EC2 model yields the best results for punching load determination.

The maximum and average values of the V_punch,exp/V_punch,theo ratio of the EC2 approach are the closest to one, which indicates that excessive oversizing will not occur. The minimum value of the V_punch,exp/V_punch,theo ratio is smaller than one for all models, which could result in non-conservative designs. In the case of the EC2 model (V_punch,exp/V_punch,theo)_min = 0.74 by using Eq. (2) with g_c = 1 and f_c = f_cm. However, if g_c = 1.5 and f_c = f_ck had been assumed, the theoretical punching load could have been greater than the experimental one. In this specific case (V_punch,exp/V_punch,theo)_min increases from 0.74 to 1.23. Therefore, the former approach ensures a safe design.

The MC10 model is more conservative, as it allows an estimation of the ultimate rotation of the slabs, however, there is a significant data scattering. Actually, the collapse criterion of the MC10 is designed to assess the characteristic punching strength and this can explain its conservative results.

The ACI 318 model could be non-conservative with (V_punch,exp/V_punch,theo)_min = 0.59. Equations (1), (2), and (3) do not contain safety coefficients, and the use of f_ck in place of f_cm determines a theoretical punching value greater than the experimental one in a number of cases.

New proposal for punching load model

The three models (ACI 318, EC2, and MC10) are currently the predominant models available for punching load assessment. ACI 318 is based on a purely empirical model and does not take into account the contribution of the flexural reinforcement. EC2 model is based on the comb model modified by several experimental parameters.

Empirical or conservative simplified approaches (like those presented in ACI 318 or EC2) can be sufficient for early punching design. But in order to develop a more effective design it is necessary to truly understand the physical phenomena. Figure 12 clearly shows as higher rotation y corresponds to lower punching strength V. Consequently, the correlation between these two parameters seems very important. For this reason, the authors believe that any physical models of punching should consider also the rotation of the slab. The MC10 approach is based on CSCT, takes into account slab rotations capability and presents different levels of approximation. An improved close form formulation of CSCT has been recently proposed by Simões et al. [⁴¹]. In the same document the authors state that there is no consensus on the mechanics governing the punching phenomenon.

These considerations and the results presented in Section 4 indicate that there is a need for an improvement in the estimation of punching shear performance of RC slabs without shear reinforcement.

In this study the authors propose a model which allows the punching shear strength and the punching rotation to be estimated with a safe and reliable approach.

The assessment of the punching shear strength and the ultimate rotation necessarily implies prior knowledge of the flexural behavior of the RC slab without shear reinforcement.

To simplify the MC10 approach, the authors propose: a new parabolic load-rotation curve based on the CSCT with smaller computational costs; and a new failure criterion which takes the correlation analyses into account.

The correlation analysis between the geometrical and mechanical characteristics of the slabs, and the ultimate punching load and rotation (see Figs. 6–11), indicate that the flexural reinforcement ratio is a key-parameter with the greatest correlation coefficient r = 0.702, as can be seen in Fig. 10. On the contrary, the maximum aggregate size does not present significant values of the correlation coefficient, r = 0.041. In addition, the compressive reinforcement ratio r'/r proved to have a relatively significant correlation coefficient, as can be seen in Fig. 11. However, this new model does not take the compressive reinforcement ratio into account for three primary reasons.

1) The load-rotation curve does not significantly change its shape when considering the compressive reinforcement r'. However, the computational costs of a model taking r' into account increases, while disregarding its contribution is on the conservative side.

2) The slabs with compressive reinforcement in the experimental database are limited (56 out of 113 samples).

3) Given the geometrical and mechanical characteristics of the considered slabs, the compressive reinforcement can also be in the tensile regime at collapse, which in conjunction with the abovementioned issues, indicate the complexities when assessing the actual influence of compressive reinforcement on the punching load.

Considering these issues, a new failure criterion, based on the MC10 model (see Eq. (6)), is proposed.

The theoretical punching load V_punc,theo and the corresponding ultimate rotation y_punc,theo can be determined by the intersection between the failure criterion and a simplified load-rotation curve.

Simplified load-rotation relationship

A parabolic relationship between the punching load and rotation has been assumed on the horizontal axis. This approach is based on the consideration that the experimental V-y relationships can be represented by a simple parabolic curve. Its vertex is located at the origin of the reference system and it comprises the limit point (y_R,flex, V_R,flex), where V_R,flex and y_R,flex can be determined according to Eqs. (A19) and (A20) in Appendix A. The load-rotation and the corresponding rotation-load curve equations are as follows:

(12)V=V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}\sqrt{\frac{\psi }{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}},

(13)\psi =\frac{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{V}^2\mathrm{.}

New failure criterion

The new proposal for failure criterion is defined as follows:

(14)V_{\mathrm{R},\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h}}=k_{\psi }\frac{\sqrt{f_{\mathrm{c}}}}{{\gamma }_{\mathrm{c}}}{b}_{0}d,

where the effective depth is d; and b₀ is the critical perimeter located at d/2 from the loading surface.

The sensitivity analysis developed in Section 3 highlighted that the flexural reinforcement ratio r has a significant influence on the punching strength, while the maximum aggregate size is less correlated. Therefore, the parameter k_y has been expressed as a function of slab rotation y, effective depth d, and flexural reinforcement ratio r in Eq. (15). The influence of the maximum aggregate size d_g has been neglected.

(15)k_{\psi }=\frac{a_1\cdot {\rho }^2+a_2\rho +a_3}{{\beta }_1+{\beta }_2\cdot \psi \cdot d}\le 0.6,

where α₁, α₂, α₃, β₁, and β₂ are numerical parameters.

It is critical to emphasize that the intersection point between the simplified load-rotation curve and this failure criterion can be assessed in closed-form. The value of the ultimate rotation can be expressed by the following equation which represents the only real solution of the 3rd order polynomial expression representing the above-mentioned intersection:

(16)\psi =\frac{1}{6d}\left(-\frac{4{\beta }_1}{{\beta }_2}+\frac{2^{\frac{4}{3}}{\beta }_1^2}{A}+\frac{2^{\frac{2}{3}}A}{{\beta }_2^2}\right),

(17)A=\frac{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{\left(\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\hfill 27a_3{b}_0^2{\beta }_2^2{d}^3{f}_{\mathrm{c}}+2{\beta }_1^2{\beta }_2^2\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}+54a_2{a}_3{b}_0^2{\beta }_2^4{d}^3{f}_{\mathrm{c}}\rho +27a_2{b}_0^2{\beta }_2^4{d}^3{f}_{\mathrm{c}}{\rho }^2+\hfill \\ \hfill 54a_1^2{a}_3{b}_0^2{\beta }_2^4{d}^3{f}_{\mathrm{c}}{\rho }^2+54a_1{a}_2{b}_0^2{\beta }_2^4{d}^3{f}_{\mathrm{c}}{\rho }^3+27a_1^2{b}_0^2{\beta }_2^2{d}^3{f}_{\mathrm{c}}{\rho }^4+\hfill \end{array}\hfill \end{array}\hfill \\ \hfill 3\sqrt{3}{b}_0{d}^{\frac{3}{2}}{\left({\beta }_2^7{f}_{\mathrm{c}}\left(\begin{array}{c}\hfill \begin{array}{c}\hfill 27a_3{b}_0^2{\beta }_2{d}^3{f}_{\mathrm{c}}+4{\beta }_1^3\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}+\hfill \\ \hfill 54a_3{b}_0^2{\beta }_2{d}^3{f}_{\mathrm{c}}\rho (a_2+a_1\rho )+\hfill \end{array}\hfill \\ \hfill 27b_0^2{\beta }_2{d}^3{f}_{\mathrm{c}}{\rho }^2{(a_2+a_1\rho )}^2\hfill \end{array}\right)\right)}^{\frac{1}{2}}\cdot |a_3+\rho (a_2+a_1\rho )|\hfill \end{array}\right)}^{\frac{1}{3}}\mathrm{.}

The numerical parameters α₁, α₂, α₃, β₁, and β₂ have been tuned using a numerical approach based on the simulated annealing algorithm [⁴²] developed in the MATLAB environment. Using the experimental database, it was possible to tune the failure criterion expressed in Eq. (14) in order to reduce the difference between the theoretical and experimental values. The values of the unknown parameters were found by minimising e={\displaystyle \sum _i^{113}{\left(\frac{V_{\exp }^i-V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}^i}{V_{\exp }^i}\right)}^2}.

The numerical values of the parameter are

{\alpha }_1=-9997.5787;{\alpha }_2=-9988.9091;{\alpha }_3=-4005.3062;

(18){\beta }_1=-6937.0915;{\beta }_2=-1751.8665.

By taking into account the results presented in Refs. [¹¹,²¹,³⁶], and the MC10 [⁹] approach which considers the influence of the maximum aggregate size d_g on the punching strength, a further definition of k_y, also taking d_g into account, is proposed

(19)k_{\psi ,d_{\mathrm{g}}}=\frac{{\delta }_1{\rho }^2+{\delta }_2\rho +{\delta }_3}{1.5+0.9k_{d_{\mathrm{g}}}\psi d}\le 0.6,

(20)k_{d_{\mathrm{g}}}=\frac{32}{16+d_{\mathrm{g}}}\ge \mathrm{0.75.}

The values of the numerical parameters tuned using the abovementioned numerical approach are as follows

{\delta }_1=-181.5905\mathrm{;}\mathrm{}{\delta }_2=8.69371556\mathrm{;}

(21){\delta }_3=\mathrm{1.12853373.}

In this case it is also possible to obtain a closed-form expression for the intersection point between the simplified load rotation curve and this failure criterion:

(22)\psi =\frac{5^{\frac{4}{3}}{d}^2{\left(\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}\right)}^2{k}_{d_{\mathrm{g}}}^2-10d\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}{k}_{d_{\mathrm{g}}}{(C+B)}^{\frac{1}{3}}+5^{\frac{2}{3}}{(C+B)}^{\frac{2}{3}}}{9d^2\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}{k}_{d_{\mathrm{g}}}^2{(C+B)}^{\frac{1}{3}}},

(23)B=5d^3{\left(\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}\right)}^3{k}_{d_{\mathrm{g}}}^3+18b_0^2{d}^6{f}_{\mathrm{c}}{\left(\frac{V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}\right)}^2{k}_{d_{\mathrm{g}}}^2{({\delta }_3+\rho ({\delta }_2+{\delta }_1\rho ))}^2,

(24)C=6\sqrt{b_0^2{d}^9{f}_{\mathrm{c}}{\left(\frac{v_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}\right)}^4{k}_{d_{\mathrm{g}}}^7{({\delta }_3+\rho ({\delta }_2+{\delta }_1\rho ))}^2(5\frac{v_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}^2}{{\psi }_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}}+9b_0^2{d}^3{f}_{\mathrm{c}}{k}_{d_{\mathrm{g}}}{({\delta }_3+\rho ({\delta }_2+{\delta }_1\rho ))}^2)}\mathrm{.}

We now compare the performance of the new proposed model in assessing the punching strength V_{\mathrm{R},\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}} based on the failure criterion characterized by k_{\psi } (without d_g), and V_{\mathrm{R},\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{d}_{\mathrm{g}}} based on the failure criterion characterized by k_{\psi ,d_{\mathrm{g}}} (with d_g).

New model results

Figure 18 presents the load-rotation curves of the MC10 simplified approach and the new proposed model, as well as the above-mentioned failure criteria, the MC10 criterion, and the new failure criterion proposed by the authors (Eqs. (14) and (15)). Three experimental cases were considered when investigating the influence of the flexural reinforcement ratio r on the new model effectiveness: small value (r = 0.2%) in Fig. 18(a) [¹¹], average value (r = 0.5%) in Fig. 18(b) [¹³], and large value (r = 2.5%) in Fig. 18(c) [¹³]. The performance of the new model without the contribution of maximum aggregate size is presented in Figs. 19 and 20. The theoretical punching load has been obtained from Eqs. (12) and (14) with g_c = 1 and f_c= f_cm, and is compared to the experimental value in Fig. 19. The theoretical and experimental ultimate rotations are plotted in Fig. 20.

Figures 21 and 22 present the results obtained with the new proposed model V_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h},\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{d}_{\mathrm{g}}} which also takes the maximum size of aggregate d_g into account (Eq. (19)).

Table 4 presents the average, minimum, and maximum values, as well the CoV of the V_punch,exp/V_punch,theo and y_punch,exp/y_punch,theo ratios for the new proposed and standards models.

The failure criterion with the new simplified load rotation curve exhibited highly satisfactory performance. This approach allows the punching performance of RC slabs to be designed in compliance with an advanced and reliable physical model without the need for oversizing. The average value of the V_punch,exp/V_punch,theo ratio of 1.06 is close to one. From Table 4 it can be seen that the average of the new models is smaller than the 1.17 of the EC2 model. The maximum value of 1.46 of the new proposed models is smaller than the 1.71 of the EC2 model, which ensures a design which limits oversizing. The minimum value of (V_punch,exp/V_punch,theo)_min is smaller than the values of the new proposal, which could result in an non-conservative design. This value was obtained assuming g_c = 1 and f_c = f_cm and using Eqs. (12) and (14). If g_c = 1.5 and f_c = f_ck had been considered (V_punch,exp/V_punch,theo)_min = 1.1, which is a safer value. Using the latter parameters, in the case of the EC2 model (V_punch,exp/V_punch,theo)_min increased from 0.74 to 1.71. Therefore, it is possible to have a conservative design and, even in this case, to limit the oversizing. The CoV is also smaller. The new proposal with d_g exhibited poorer results than the proposed model which neglected the effect of the maximum aggregate size.

With reference to the rotation estimation of slabs (Table 4), the comparison could only be conducted with the simplified MC10 model. The new and MC10 models obtained the same average and minimum values of the ratio y_punch,exp/y_punch,theo. However, looking at the maximum values there are critical differences, as it is 8.68 for the new model with failure criterion 2, and 6.81 for the MC10 model. This significant difference could signify a different deformation properties estimation which requires further investigation.

However, it is important to highlight that the proposed model is based on the fitting of a quite large, but limited, database. For this reason, its accuracy has been proved only inside this database whose main characteristics are reported in Table 2 and not in every other case. In addition, an accurate study of the ultimate rotation cannot be developed on the basis of the available data, and it can be seen from Table 1 that the rotations were not all evaluated in a similar fashion. In approximately 50% of the cases they were evaluated using a non-direct approach. Therefore, further experimental studies are essential.

Conclusions

In this study an experimental database of 113 RC slabs under punching loads was compiled from data available in the literature. A sensitivity analysis of the parameters involved in the punching strength assessment was conducted, which indicated to the authors the importance of the flexural reinforcement.

A new proposal for punching strength and ultimate rotation estimation model for RC slabs was then described. This approach can be seen as a modification of the MC10 simplified model (level II approximation) in the presence of significant redistribution of the bending moment. It is based on a simplified load-rotation curve and on new failure criteria that takes into account flexural reinforcement effects.

The experimental database was used to evaluate the approaches of current international standards, ACI 318, EC2, and MC10, as well as the new proposed model. The latter could be an effective design tool with a relatively simple load-rotation curve definition which would allow the collapse point to be obtained without requiring an iterative process.

Further developments are expected using a computational mechanics approach, similar to those presented in [⁴³–⁴⁶], in order to build a nonlinear numerical model tuned on a number of experimental benchmarks capable of improving the sensitivity analysis presented in this study.

In addition, a new experimental campaign was planned by the authors with the aim of improving the literature database. Specifically, the influence of different flexural reinforcement ratio on the punching shear strength and the direct measurement of the slab rotation y are the primary aims of future experimental studies.

Appendix A—Critical shear crack theory

The critical shear crack theory (CSCT) allows the determination of the load-rotation curve which, in the case of an axisymmetric condition, can be obtained by the numerical integration of the moment-curvature relationship. A synthesis of the known CSCT formulation is reported in this section.

The CSCT model identifies a critical inclined crack, whose projection on the extrados surface can be approximated by a circumference of radius r₀ centered at the resultant load application point. The extrados of the crack portion has a truncated cone shape and is characterized by radial cracks which generate sectors of amplitude Δϕ. During collapse, these sectors rigidly rotates describing an angle y with respect to the initial horizontal configuration (see Fig. A1). This angle increases with increasing shear load V.

The plate can be schematised by a central core delimited by the critical crack and a series of sectors, of amplitude Δϕ, delimited by radial cracks. The individual sectors are connected in the compressed area of the slab intrados, close to the loading surface. The inclined component of the load V on the single sector is applied at point O (see Fig. A1).

The equation of the rotational equilibrium of the single sector of slab with the pole at point O is as follows (see Fig. A1):

(A1)V\cdot \frac{\mathrm{\Delta }\varphi }{2\mathrm{\pi }}(r_{\mathrm{q}}-r_{\mathrm{c}})-m_{\mathrm{r}}\cdot \mathrm{\Delta }\varphi \cdot r_0-\mathrm{\Delta }\varphi {\int }_{r_0}^{r_{\mathrm{s}}}{m}_{\mathrm{t}}\cdot \mathrm{d}r=0,

where r_q is the distance between the reaction of the support and the loading point; and r_c is the radius of the loaded surface.

This model can also represent circular or square plates with uniform or discrete support at the edges, with uniform distribution of reinforcement in both directions.

The radial moment for units of length m_r and the tangential moment for units of length m_t can be expressed as a function of the geometry of the element, the presence of the reinforcement, the mechanical characteristics of the steel and concrete, and the curvature and rotationy produced by the applied load.

The curvature in the tangential direction q_t, is expressed as a function of rotation y:

(A2){\theta }_{\mathrm{t}}=\frac{\psi }{r}\mathrm{,}\mathrm{}\mathrm{f}\mathrm{o}\mathrm{r}r>r_0\mathrm{.}

Equation (A2) indicates that tangential curvature q_t is inversely proportional to the radius.

Inside the critical shear crack the radial and tangential curvature can be considered to be equal:

(A3){\theta }_{\mathrm{t}}={\theta }_{\mathrm{r}}=\frac{\psi }{r_0}\mathrm{,}\mathrm{}\mathrm{f}\mathrm{o}\mathrm{r}r\le r_0\mathrm{.}

The radius r₀ (see Fig. A1) identifies the position of the circumferential crack, and is equal to (r_c + d/2) according to the MC. However, in the Muttoni model [²¹] r₀ = r_c + d, where d is the effective depth of the slab.

The curvature-bending moment relationship can be expressed by a quadrilateral curve:

(A4)m=E_{\mathrm{c}}\cdot I_0\theta \mathrm{,}\mathrm{}\mathrm{f}\mathrm{o}\mathrm{r}\theta \le {\theta }_{\mathrm{c}\mathrm{r}},

(A5)

(A6)

(A7)m=m_{\mathrm{R}}\mathrm{,}\mathrm{}\mathrm{f}\mathrm{o}\mathrm{r}\theta >{\theta }_{\mathrm{y}}\mathrm{.}

The concrete elastic modulus is E_{\mathrm{c}}=10000\cdot {}^3\sqrt{f_{\mathrm{c}}} (MPa) and its tensile strength is E_{\mathrm{c}}=10000\cdot {}^3\sqrt{f_{\mathrm{c}}} (MPa). The steel elastic modulus is E_s = 200000 MPa.

The resistant cross section has total height h, effective depth d, and a flexural reinforcement ratio r.

For a unitary width b, with an uncracked cross section, it is possible to obtain:

(A8)I_0=\frac{h^3}{12}\mathrm{,}\mathrm{}

(A9)m_{\mathrm{c}\mathrm{r}}=f_{\mathrm{c}\mathrm{t}}\frac{h^2}{6},

(A10){\theta }_{\mathrm{c}\mathrm{r}}=\frac{m_{\mathrm{c}\mathrm{r}}}{E_{\mathrm{c}}{I}_0}\mathrm{.}

For a cracked cross section, without compressive reinforcement, using the hypothesis of conservation of the plane sections, perfect adherence of concrete and steel, no traction strength for concrete, and Hooke's law assumptions, the neutral axis depth x, and moment of inertia I₁ are obtained:

(A11)x=\frac{E_{\mathrm{s}}}{E_{\mathrm{c}}}\rho d(-1+\sqrt{1+\frac{2E_{\mathrm{c}}}{\rho E_{\mathrm{s}}}})\mathrm{,}\mathrm{}

(A12)I_1=\frac{x^3}{3}+\frac{E_{\mathrm{s}}}{E_{\mathrm{c}}}\rho d{(d-x)}^2\mathrm{.}

The contribution of the tensile concrete between two consecutive cracks, tension stiffening, is to be considered constant and produces a reduction of the curvature equal to q_TS:

(A13){\theta }_{\mathrm{T}\mathrm{S}}=\frac{f_{\mathrm{c}\mathrm{t}}}{\rho E_{\mathrm{s}}}\cdot \frac{1}{6h}\mathrm{.}

Upon reaching the first cracking bending moment m_cr, the bending action remains constant while the curvature increases up to the value q₁:

(A14){\theta }_1=\frac{m_{\mathrm{c}\mathrm{r}}}{E_{\mathrm{c}}{I}_1}-{\theta }_{\mathrm{T}\mathrm{S}}\mathrm{.}

The ultimate bending moment m_R is obtained with the following collapse scenario: ultimate compressive strain of concrete equal to 0.00035; yielded reinforcement; stress block for concrete; and ideal elastic-plastic steel as constitutive laws. The presence of compressive reinforcement is disregarded.

(A15)m_{\mathrm{R}}=\rho \cdot d^2\cdot f_{\mathrm{y}}\cdot (1-\frac{\rho \cdot f_{\mathrm{y}}}{2\cdot f_{\mathrm{c}}})\mathrm{.}

The yielding curvature q_y is:

(A16){\theta }_{\mathrm{y}}=\frac{m_{\mathrm{R}}}{E_{\mathrm{c}}{I}_1}-{\theta }_{\mathrm{T}\mathrm{S}}\mathrm{.}

To develop Eq. (A1) it is necessary to use the radial and tangential bending moment (m_r and m_t) definitions, as in Eqs. (A4–A7). It is assumed that:

(A17)\mathrm{f}\mathrm{o}\mathrm{r}r\le r_0,{\theta }_{\mathrm{r}}={\theta }_{\mathrm{t}}=\frac{\psi }{r_0},

(A18)\mathrm{f}\mathrm{o}\mathrm{r}r>r_0,{\theta }_{\mathrm{r}}=0,{\theta }_{\mathrm{t}}=\frac{\psi }{r_0}\mathrm{.}

The slab failure criterion is satisfied when the circumferential flexural reinforcement, characterized by radius r_s have yielded. In this condition, it is possible, using Eq. (A1), to calculate V_R,flex:

(A19)V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}=2\varphi \cdot m_{\mathrm{R}}\cdot \frac{r_{\mathrm{s}}}{(r_{\mathrm{q}}-r_{\mathrm{c}})}\mathrm{.}

By using Eq. (A18) in the limit condition r=r_{\mathrm{y}}=r_{\mathrm{s}}\mathrm{a}\mathrm{n}\mathrm{d}{\theta }_{\mathrm{t}}={\theta }_{\mathrm{y}}, it is possible to obtain:

(A20)V_{\mathrm{R},\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}}={\theta }_{\mathrm{y}}\cdot r_{\mathrm{s}}\mathrm{.}

To obtain the full load-rotation curve it is necessary to develop an iterative process: for the given value of y (varying between 0 and y_R,flex), using Eq. (A1), it is possible to obtain the punching load V using the quadrilateral bending moment-curvature relationship (Eqs. (A4)–(A7)).

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