Calculation Method for Punching Shear Capacity of Reinforced UHPC Two-Way Slabs Based on Critical Shear Crack Theory

Abstract

The punching shear capacity of reinforced ultra-high-performance concrete (UHPC) two-way slabs in applications such as floor slabs and bridge decks has attracted increasing attention. However, due to the insufficient consideration of the internal force transmission path and failure mechanism, existing empirical formulas exhibit limited accuracy for predicting the punching shear capacity of reinforced UHPC slabs. Therefore, based on the critical shear crack theory (CSCT), this study proposes a specific theoretical model where the tensile strain-hardening behavior and tensile strength of UHPC, the punching shear-span ratio, and the reinforcement ratio are comprehensively considered. In the proposed model, the steel fiber bridging contribution is derived via the variable engagement method (VEM), for which an equation describing the bond strength between steel fibers and UHPC matrix was developed. The feasibility of the proposed model was validated through an established experimental database. Furthermore, the effects of several key parameters on the punching shear behavior of reinforced UHPC slabs were analyzed. The results show that the proposed models can accurately predict the punching shear capacity and ultimate rotation angle of reinforced UHPC slabs. With increasing slab thickness, UHPC strength, and reinforcement ratio, the punching shear capacity increases, whereas the corresponding ultimate rotation angle and steel fiber contribution ratio decrease. Increasing the fiber volume fraction enhances both the fiber contribution and the punching shear capacity. For slabs with higher UHPC strength, the reinforcing effect of a higher reinforcement ratio is more pronounced.

1. Introduction

Against the backdrop of the growing demand for lightweight and high-durability components in civil engineering structures, reinforced ultra-high-performance concrete (UHPC) slabs are widely used in scenarios such as floor slabs and bridge decks due to their combined advantages of UHPC’s high strength and excellent toughness [1,2,3,4]. Nevertheless, UHPC slabs with thin cross-sections may undergo punching shear failure when subjected to localized concentrated loads [4]. Therefore, an effective evaluation of the punching shear capacity is highly necessary for the design of reinforced UHPC slabs.

Several experimental studies on the punching shear behavior of reinforced UHPC slabs have been documented. Existing findings have confirmed that the bridging effect of steel fibers plays an important role in enhancing the punching shear capacity of UHPC slabs [5,6,7,8,9]. Furthermore, the punching shear resistance of these slabs can be improved to varying degrees, with increases in UHPC’s compressive strength, loading area, steel reinforcement ratio, and slab thickness [4,10,11,12,13]. Fan et al. [10] found that under a concentrated load, UHPC slabs might fail in flexural, punching shear, and flexural-punching shear modes, respectively. Liu et al. [12] reported that a critical thickness of at least 100 mm was identified as the threshold distinguishing punching shear failure from flexural failure; while a higher reinforcement ratio of 3.5% slightly increased the first cracking load, it greatly enhanced the ultimate capacity. Hassan et al. [14] demonstrated that the punching shear capacity of UHPC slabs increased with an increase in punching shear cone angle. Fang et al. [15] reported that both the cone angle and punching shear strength of UHPC slabs decreased with an increase in the punching shear-span ratio within the range of 3 to 7. Liao et al. [16] stated that adding steel fibers could enhance the initial cracking load of the slab, and adding polyethylene fibers could intensify the post-crack ductility response, with both contributing to an enhanced punching shear capacity; the installation of shear reinforcements appeared to be more cost-effective than increasing steel fiber content.

Currently, certain empirical equations have been developed for calculating the punching shear capacity of reinforced UHPC slabs. These empirical equations were established based on the statistical laws pertaining to several parameters. For example, Al-Quraishi et al. [7] empirically approximated the punching shear capacity of UHPC slabs as the sum of three terms related to the contributions of the UHPC matrix, steel fibers, and steel bars, respectively. The formulas provided by the Chinese technical specification DBJ43/T325-2017 [17] and the Japan Society of Civil Engineers (JSCE) [18] adopt a series of empirical coefficients to consider the influences of loading area, slab thickness, and other factors. Nevertheless, in the calculation model suggested by the French code NF P 18-710 [19], the punching shear strength of the slab mainly depends on the tensile strength of UHPC. The incompleteness or inconsistency of parameters in the above empirical formulas stems from their reliance on limited specific experimental databases and thus have limited applicability. This drawback leads to the formulas being either conservative or unsafe when applied to the actual design of UHPC slabs against punching shear failure.

Establishing an effective computational model that can reflect and describe the physical process of punching shear failure is expected to address the shortcomings of the aforementioned empirical formulas. Currently, the critical shear crack theory (CSCT) has been applied to develop calculation models for the punching shear capacity of normal concrete (NC) slabs [20,21,22,23]. This theory directly links the punching shear resistance of reinforced concrete slabs to the critical shear crack width, providing a more reasonable prediction model based on physical mechanisms. However, few studies, if any, have been conducted on a theoretical calculation method for the punching shear capacity of UHPC slabs based on the CSCT. The key unresolved issues regarding the application of CSCT to UHPC slabs lie in the differences in mechanical behavior between UHPC and NC slabs. Firstly, UHPC exhibits a pronounced tensile strain-hardening behavior, leading to a significant difference in the moment-curvature response from that of NC slabs [24,25]. Meanwhile, the tensile strength of UHPC is significant and should be considered, whereas the tensile strength of NC is often ignored. Secondly, the aggregate size of UHPC is smaller than that of NC, leading to a difference in aggregate interlock along the critical shear cracks. Moreover, the punching shear-span ratio has a different effect on the punching shear behavior of UHPC slabs compared to NC slabs [15].

Consequently, the present study aims to develop a mechanism-based theoretical model for calculating the punching shear capacity of reinforced UHPC two-way slabs based on the CSCT. In this model, the tensile strain-hardening behavior and tensile strength of UHPC, as well as the punching shear-span ratio and the reinforcement ratio, were comprehensively considered. Accordingly, the load-rotation curve and the failure criteria were formulated. Moreover, the fiber contribution is calculated by integrating the bridging stress along the punching shear cone, using a newly fitted bond strength equation between steel fibers and UHPC matrix. Based on an experimental database of the punching shear capacity of reinforced UHPC slabs, the feasibility of the proposed model was verified. Subsequently, the effects of several key parameters on the punching shear behavior of UHPC slabs were evaluated.

2. Punching Shear Model for Reinforced UHPC Two-Way Slabs

2.1. Conventional Critical Shear Crack Theory

When a reinforced concrete (RC) slab is subjected to a localized punching load, the shear force is transferred from the column to the inclined compression struts [20]. As the load increases, a critical shear crack forms and propagates, intersecting the compression struts, as presented in Figure 1. Based on the CSCT, the widening of this critical shear crack reduces the shear strength provided by the inclined concrete compressive struts and ultimately leads to punching shear failure. Thus, the punching shear strength decreases with the development of the critical shear crack as the slab rotates.

Muttoni et al. [26] presumed that the width of the critical shear crack (w) was proportional to the rotation angle of the slab (ψ) outside the column region and the effective depth of the cross section (d), as shown in Figure 1. In Vecchio’s study [27], the shear force transferred along the critical shear crack is influenced by the roughness of the punching shear cone surface, which is in turn related to the aggregate size of concrete (d_g). Considering the effects of aggregate size and other critical parameters, Muttoni [20] suggested a failure criterion for RC slabs without transverse reinforcement under punching shear loads, as expressed in Equation (1).

$${\displaystyle \frac{V_{\mathrm{P}.\mathrm{c}}}{b_{0}d\sqrt{f_c}}}={\displaystyle \frac{3/4}{1+15\lambda }}$$

(1)

$$\lambda ={\displaystyle \frac{\psi d}{d_{g0}+d_g}}$$

(1a)

where V_P.c is the punching shear capacity; b₀ is the perimeter of the control cross section at a distance of 0.5d from the column face; and f_c is the axial compressive strength of concrete. d_g0 is the reference aggregate size and is equal to 16 mm [20].

To calculate the punching shear capacity, the load-rotation curve (demand curve) needs to be obtained. At failure, the intersection between the demand curve and the capacity curve (failure criterion) defines the capacity, as depicted in Figure 2.

Figure 3 presents the load-rotation curves of the UHPC slabs tested by Fang et al. [15], where the notations U, h, ρ, and B (followed by numerical values) denote the compressive strength of UHPC, slab thickness, reinforcement ratio, and side length of the loading area, respectively. The intersections between the measured load-rotation curves and the calculated capacity curve lie to the left of the peak point, implying that the failure criterion proposed by Muttoni [20] underestimates the punching shear strength of UHPC slabs. This underestimation may be attributed to the omission of the contribution of steel fibers in Equation (1). Moreover, the aggregate interlock in UHPC slabs is weaker than that in NC slabs because of the smaller aggregate size, a factor also not accounted for in Equation (1).

2.2. Key Assumptions

The distributions of internal forces and deformations of RC slabs are complex and need to be simplified [23]. Compared to NC, both the tensile strain-hardening behavior and the tensile strength of UHPC must be considered. To apply the CSCT to UHPC slabs, several key assumptions were made as follows.

(1)

A geometrical model of an equivalent circular UHPC slab was introduced [20], as shown in Figure 4, where r_c is the column radius; r₀ is the radius of the critical shear crack; r_q is the radius of the circular support line; and r_s is the radius of the slab. In the region within the radius r₀, the intensive tangential cracks are assumed to form, and the radial curvature (χ_r) equals the tangential curvature (χ_t). Outside the radius r₀, the radial curvature decreases quickly and can be ignored. Thus, the corresponding slab portion was assumed to be a cone with a constant rotation. The above simplifications can be expressed as Equation (2), where r denotes the radial distance from the column center.

$${\chi }_{\mathrm{t}}={\chi }_{\mathrm{r}}={\displaystyle \frac{\psi }{r_0}}, \mathrm{for} r\le r_0$$

(2a)

$${\chi }_{\mathrm{t}}={\displaystyle \frac{\psi }{r}} \mathrm{and} {\chi }_{\mathrm{r}}=0, \mathrm{for} r>r_0$$

(2b)

(2)

A rectangular stress block distribution is assumed for both the compressive and tensile zones [28], as shown in Figure 5, where x_c and x_t are the depth of compressive zone and tensile zone, respectively; b is the section width; d is the effective depth; α₁ is the ratio of average compressive stress to f_c; β₁ is the ratio of height of rectangular stress block to x_c; f_y is the yield strength of steel bar; f_t is the axial tensile strength of UHPC; and k is the ratio of average tensile stress to f_t, taken as 0.8 [28].

(3)

Square slabs are converted into equivalent circular slabs based on the virtual work principle and yield line theory, assuming bi-directional uniform reinforcement.

(4)

Aggregate interlock in UHPC is ignored (d_g = 0) because of the small aggregate size (≤5 mm for commercial UHPC).

(5)

The critical shear crack width is proportional to the slab rotation outside the column region and the effective depth (w∝ψd) [21].

(6)

For practical calculation, the steel fiber bridging stress along the punching shear failure surface can be simplified as uniformly distributed [22].

2.3. Load-Rotation Curves of Reinforced UHPC Slabs

Figure 6 shows the moment-curvature curves of different types of RC members [22,25], where m_cr and m_y are the cracking moment and the ultimate moment per unit width, respectively; χ_cr and χ_y are the curvatures at m_cr and m_y, respectively; E_c is the elastic modulus of UHPC; I_g is the gross moment of inertia; and I_cr is the moment of inertia of the cracked section. Compared to traditional fiber-reinforced concrete (FRC) members, UHPC members exhibit a more pronounced tension-hardening response. This is because the denser microstructure of UHPC enhances the bond strength between steel fibers and the matrix, thereby improving the bridging effect. The moment-curvature curves of UHPC members can be divided into three stages, including the uncracked, cracked, and yield stages, respectively.

The cracking curvature (χ_cr) and yielding curvature (χ_y) of UHPC members can be expressed as Equation (3).

$${\chi }_{\mathrm{cr}}={\displaystyle \frac{m_{\mathrm{cr}}}{E_{\mathrm{c}}{I}_{\mathrm{g}}}}$$

(3a)

$${\chi }_{\mathrm{y}}={\displaystyle \frac{m_{\mathrm{y}}-m_{\mathrm{cr}}}{E_{\mathrm{c}}{I}_{\mathrm{cr}}}}+{\displaystyle \frac{m_{\mathrm{cr}}}{E_{\mathrm{c}}{I}_{\mathrm{g}}}}$$

(3b)

Based on the document [25], m_cr, I_g and I_cr of singly reinforced rectangular UHPC member can be expressed as Equations (4) and (5).

$$m_{\mathrm{cr}}={\displaystyle \frac{{\gamma }_{\mathrm{m}}{f}_{\mathrm{t}}{I}_{\mathrm{g}}}{y_{\mathrm{t}}}}$$

(4)

$$I_{\mathrm{g}}={\displaystyle \frac{h^3}{12}}+{\displaystyle \frac{h{\left(2y_{\mathrm{t}}-h\right)}^2}{4}}+A_{\mathrm{s}}\left(n_{\mathrm{s}}-1\right){\left(y_{\mathrm{t}}-{\alpha }_{\mathrm{s}}\right)}^2$$

(4a)

$$y_{\mathrm{t}}={\displaystyle \frac{0.5h^2+{\alpha }_{\mathrm{s}}{A}_{\mathrm{s}}\left(n_{\mathrm{s}}-1\right)}{h+A_{\mathrm{s}}\left(n_{\mathrm{s}}-1\right)}}$$

(4b)

$$I_{\mathrm{cr}}={\displaystyle \frac{d^3}{3}}{\delta }^3+n_{\mathrm{s}}{A}_{\mathrm{s}}{d}^2{\left(1-\delta \right)}^2$$

(5)

$$\delta =\sqrt{2{\rho }_{\mathrm{s}}{n}_{\mathrm{s}}+{\left({\rho }_{\mathrm{s}}{n}_{\mathrm{s}}\right)}^2}-{\rho }_{\mathrm{s}}{n}_{\mathrm{s}}$$

(5a)

where γ_m is the plastic influence coefficient and equal to 1.5 [25]; y_t is the distance from the neutral axis of gross section to the tension face; h is the section depth; A_s is the sectional area of steel bar; n_s is the elastic modulus ratio of steel bar to UHPC; α_s is the distance from the center of steel bars to the tension face; δ is the ratio of the neutral axis depth to the effective depth; ρ_s is the reinforcement ratio.

According to Figure 5 and the moment equilibrium, the expression for calculating the bending capacity m_y of a rectangular UHPC member is derived as Equation (6).

$$m_{\mathrm{y}}=A_{\mathrm{s}}{f}_{\mathrm{y}}\left(d-{\displaystyle \frac{{\beta }_1{x}_{\mathrm{c}}}{2}}\right)+k{f}_{\mathrm{t}}\left(h-x_{\mathrm{c}}\right)\left({\displaystyle \frac{h+x_{\mathrm{c}}-{\beta }_1{x}_{\mathrm{c}}}{2}}\right)$$

(6)

$$x_{\mathrm{c}}={\displaystyle \frac{A_{\mathrm{s}}{f}_{\mathrm{y}}+k{f}_{\mathrm{t}}h}{k{f}_{\mathrm{t}}+{\alpha }_1{f}_{\mathrm{c}}{\beta }_1}}$$

(6a)

$${\alpha }_1=0.75 \mathrm{and} {\beta }_1=0.67 \mathrm{for} \mathrm{slab} \mathrm{with} \mathrm{tensile} \mathrm{failure}$$

(6b)

$${\alpha }_1=0.94 \mathrm{and} {\beta }_1=0.74 \mathrm{for} \mathrm{slab} \mathrm{with} \mathrm{compressive} \mathrm{failure}$$

(6c)

The equation for the theoretical load-rotation curve can be derived by first evaluating the radial moment acting within the region of radius r₀, and then integrating this moment along the tangential direction based on the moment-curvature curve of the UHPC member shown in Figure 6. Thus, the three stages of the moment-curvature curve correspond to three distinct regions within the reinforced UHPC slab. The radii for delimiting each region are calculated by substituting Equation (3) into Equation (2). Specifically, the region extending from the column face to the onset of UHPC cracking is defined by the cracking radius (r_cr). The region spanning from the cracking front to the location of reinforcement yielding is bounded by the yielding radius (r_y).

$$r_{\mathrm{cr}}={\displaystyle \frac{\psi }{{\chi }_{\mathrm{cr}}}}={\displaystyle \frac{\psi E_{\mathrm{c}}{I}_{\mathrm{g}}}{m_{\mathrm{cr}}}}\le r_{\mathrm{s}}$$

(7a)

$$r_{\mathrm{y}}={\displaystyle \frac{\psi }{{\chi }_{\mathrm{y}}}}={\displaystyle \frac{\psi E_{\mathrm{c}}{I}_{\mathrm{cr}}{I}_{\mathrm{g}}}{m_{\mathrm{y}}{I}_{\mathrm{g}}+m_{\mathrm{cr}}\left(I_{\mathrm{cr}}-I_{\mathrm{g}}\right)}}\le r_{\mathrm{s}}$$

(7b)

Figure 7 presents the internal force and stress distributions on the slab portion. Thus, the balance equation is obtained as shown in Equation (8).

$$V{\displaystyle \frac{\Delta \phi }{2\pi }}\left(r_{\mathrm{q}}-r_{\mathrm{c}}\right)=m_{\mathrm{r}}\Delta \phi r_0+\Delta \phi {\displaystyle {\int }_{r_0}^{r_{\mathrm{s}}}{m}_{\mathrm{t}}}d\mathrm{r}$$

(8)

$$r_0=r_{\mathrm{c}}+d/\tan \theta$$

(8a)

$$\theta =\left\{\begin{array}{ccc}{40}^{\circ },& \mathrm{for}& {\lambda }_{\mathrm{p}}\le 3\\ {\displaystyle \frac{160.5}{1+{\lambda }_{\mathrm{p}}}},& \mathrm{for}& 3<{\lambda }_{\mathrm{p}}<7\\ {20}^{\circ },& \mathrm{for}& {\lambda }_{\mathrm{p}}\ge 7\end{array}\right.$$

(8b)

where V is the punching load; Δφ is the angle of the slab portion, and m_r is the radial moment per unit width at r = r₀. To calculate m_r at r₀ from Equation (6), the reinforcement ratio at r₀ is calculated by ρ_s = A_s/(2πdr₀); m_t is the tangential moment per unit width, the value of which is obtained from the moment–curvature relationship shown in Figure 6. θ is the punching shear cone angle of the UHPC slab in degree (°) and is obtained from Ref. [15]; λ_p is the punching shear-span ratio, defined as the ratio of the distance from the column edge to the support line to the effective depth.

Based on Figure 8, λ_p can be calculated using Equation (9), where L₀ and c are the slab span and the side length of the square loading area, respectively.

$${\lambda }_{\mathrm{p}}={\displaystyle \frac{0.5L_0-0.5c}{d}}, \mathrm{for} \mathrm{Square} \mathrm{loading} \mathrm{area}$$

(9a)

$${\lambda }_{\mathrm{p}}={\displaystyle \frac{0.5L_0-r_{\mathrm{c}}}{d}}, \mathrm{for} \mathrm{circular} \mathrm{loading} \mathrm{area}$$

(9b)

With the help of Figure 6, Equations (7) and (8), the expression for the punching load (V) can be derived as Equation (10).

$$V={\displaystyle \frac{2\pi }{r_{\mathrm{q}}-r_{\mathrm{c}}}}\left\{m_{\mathrm{r}}{r}_0+m_{\mathrm{y}}\left[r_{\mathrm{y}}-r_0\right]+m_{\mathrm{cr}}\left[r_{\mathrm{cr}}-r_{\mathrm{y}}\right]+E_{\mathrm{c}}{I}_{\mathrm{cr}}\psi \left[\mathrm{In}\left({\displaystyle \frac{r_{\mathrm{cr}}}{r_{\mathrm{y}}}}\right)\right]+E_{\mathrm{c}}{I}_{\mathrm{g}}\psi \left[\mathrm{In}\left({\displaystyle \frac{r_{\mathrm{s}}}{r_{\mathrm{cr}}}}\right)\right]\right\}$$

(10)

where the operator [x] is x for x > 0 and zero for x < 0.

2.4. Equivalent Method for Square Slabs

Equation (10) is suitable for predicting the load-rotation behavior of circular UHPC slabs with a circular loading area. For practical applications involving square slabs, an equivalent transformation to a circular configuration is therefore required. Referring to the crack patterns documented in Refs. [4,8,11,15], a general yield-line pattern is adopted for the UHPC slabs, as shown in Figure 9, where L is the side length of the square slab; r_s.eq, r_q.eq and r_c.eq are the radii of the equivalent circular slab, the circular support line, and the loading area, respectively.

In this model, the UHPC slab is divided into five parts by the yield lines. For the square slab subjected to a localized load, a unit virtual displacement of part ① is applied. Moreover, part ① is assumed to translate only in the loading direction without rotation. Accordingly, the rotation angle (θ₁₂) between part ① and part ② and the relative rotation angle (θ₂₂) between the two symmetric parts ② are respectively expressed as Equation (11).

$${\theta }_{12}={\displaystyle \frac{1}{\frac{L_0-c}{2}}}={\displaystyle \frac{2}{L_0-c}}$$

(11a)

$${\theta }_{22}=2\times {\displaystyle \frac{1}{\frac{L_0-c}{2}}}\times {\displaystyle \frac{\sqrt{2}}{2}}={\displaystyle \frac{2\sqrt{2}}{L_0-c}}$$

(11b)

Based on the virtual work principle that the internal virtual work equals the external, the ultimate load (P_u) of the square UHPC slab is calculated by Equation (12).

$$P_{\mathrm{u}}=4m_{\mathrm{y}}\left[{\displaystyle \frac{2c}{L_0-c}}+{\displaystyle \frac{2\sqrt{2}}{L_0-c}}\times {\displaystyle \frac{\sqrt{2}\left(L_0-c\right)}{2}}\right]+4{m^{\prime }}_{\mathrm{y}}\left({\displaystyle \frac{2L_0}{L_0-c}}\right)={\displaystyle \frac{8\left(m_{\mathrm{y}}+{m^{\prime }}_{\mathrm{y}}\right)L_0}{L_0-c}}$$

(12)

where m’_y is the negative moment, and m’_y = 0 for the case of a slab simply supported on four edges.

For the circular slab subjected to a localized load, the rotation angle (θ₁₂) between part ① and part ② and the relative rotation angle (θ₂₂) between the two symmetric parts ② are computed via Equation (13).

$${\theta }_{12}={\displaystyle \frac{1}{\frac{\sqrt{2}}{2}{r}_{\mathrm{q}.\mathrm{eq}}-r_{\mathrm{c}.\mathrm{eq}}}}={\displaystyle \frac{2}{\sqrt{2}{r}_{\mathrm{q}.\mathrm{eq}}-2r_{\mathrm{c}.\mathrm{eq}}}}$$

(13a)

$${\theta }_{22}=2\times {\displaystyle \frac{2}{\sqrt{2}{r}_{\mathrm{q}.\mathrm{eq}}-2r_{\mathrm{c}.\mathrm{eq}}}}\times {\displaystyle \frac{\sqrt{2}}{2}}={\displaystyle \frac{2}{r_{\mathrm{q}.\mathrm{eq}}-\sqrt{2}{r}_{\mathrm{c}.\mathrm{eq}}}}$$

(13b)

The ultimate load (P_u) of the circular UHPC slab is calculated by Equation (14).

$$\begin{array}{l}{P}_{\mathrm{u}}=2\pi r_{\mathrm{c}.\mathrm{eq}}{m}_{\mathrm{y}}\left({\displaystyle \frac{2}{\sqrt{2}{r}_{\mathrm{q}.\mathrm{eq}}-2r_{\mathrm{c}.\mathrm{eq}}}}\right)+4m_{\mathrm{y}}\left(r_{\mathrm{q}.\mathrm{eq}}-r_{\mathrm{c}.\mathrm{eq}}\right)\left({\displaystyle \frac{2}{r_{\mathrm{q}.\mathrm{eq}}-\sqrt{2}{r}_{\mathrm{c}.\mathrm{eq}}}}\right)+4{m^{\prime }}_{\mathrm{y}}\left({\displaystyle \frac{2\sqrt{2}{r}_{\mathrm{q}.\mathrm{eq}}}{\sqrt{2}{r}_{\mathrm{q}.\mathrm{eq}}-2r_{\mathrm{c}.\mathrm{eq}}}}\right)\\ ={\displaystyle \frac{8m_{\mathrm{y}}\left[r_{\mathrm{q}.\mathrm{eq}}+\left(\frac{\sqrt{2}\pi }{4}-1\right)r_{\mathrm{c}.\mathrm{eq}}\right]+8{m^{\prime }}_{\mathrm{y}}{r}_{\mathrm{q}.\mathrm{eq}}}{r_{\mathrm{q}.\mathrm{eq}}-\sqrt{2}{r}_{\mathrm{c}.\mathrm{eq}}}}\end{array}$$

(14)

By equating Equations (12)–(14), the equivalent radii of the circular slab and the loading area are calculated by Equation (15).

$$r_{\mathrm{q}.\mathrm{eq}}={\displaystyle \frac{1}{2\pi }}\left[\left(\sqrt{2}\pi -4\right)\left(L_0-c\right)+4\sqrt{2}{L}_0\right]$$

(15a)

$$r_{\mathrm{c}.\mathrm{eq}}={\displaystyle \frac{2c}{\pi }}$$

(15b)

Therefore, the demand curves of square UHPC slabs can be obtained by replacing r_q and r_c in Equation (10) with r_q.eq and r_c.eq, respectively.

2.5. Failure Criterion

According to Ref. [22], the punching shear capacity of the FRC slabs can be estimated as the linear superposition of concrete contribution and fiber contribution. Therefore, the punching shear capacity (V_P) of reinforced UHPC two-way slab can be calculated by Equation (16).

$$V_{\mathrm{P}}=V_{P.\mathrm{c}}+V_{P.\mathrm{f}}$$

(16)

where V_P.c and V_P.f are the contributions of the UHPC matrix and steel fibers to punching shear capacity, respectively.

Fang et al. [15] reported that (1) the punching shear failure surface is smooth except for randomly distributed steel fibers, implying a negligible interlocking effect induced by the aggregates; (2) under the punching shear failure, UHPC slabs present better ductility than NC slabs due to the bridging effect of steel fibers, which allows for more extensive development of inclined shear cracks, thereby further reducing aggregate interlock. Therefore, the value of d_g in Equation (1) can be conservatively taken as zero to neglect the aggregate interlock. For nondimensionalization, a reference aggregate size d_g0 is required. For the commercial UHPC, the maximum aggregate size is often 4 to 5 mm. Thus, d_g0 can be taken as 4 mm. Considering the influences of punching shear-span ratio, reinforcement ratio, and UHPC strength on the punching shear capacity of slab [15], the formula for calculating the contribution of UHPC matrix was proposed as Equation (17).

$${\displaystyle \frac{V_{P.\mathrm{c}}}{b_{0}d{f}_{\mathrm{t}0}}}={\displaystyle \frac{A{\beta }_{\mathsf{\rho }}}{\left(1+{\lambda }_{\mathrm{p}}\right)\left(1+B\psi d/d_{\mathrm{g}0}\right)}}$$

(17)

$${\beta }_{\mathsf{\rho }}={\left(100{\rho }_{\mathrm{s}}\right)}^{0.2}, \mathrm{for} {\rho }_{\mathrm{s}}\le 0.01, {\beta }_{\mathsf{\rho }}=1$$

(17a)

where f_t0 is the elastic limit strength of UHPC, taken as 0.06 times the prism compressive strength [17]; β_ρ is a factor regarding the effect of reinforcement ratio; and b₀ is the perimeter of the control cross section at a distance of 0.5d from the column edge [15].

Based on Figure 8, b₀ is calculated by Equation (18), as follows:

$$b_0=4c+4d, \mathrm{for} \mathrm{Square} \mathrm{loading} \mathrm{area}$$

(18a)

$$b_0=2\pi \left(r_c+0.5d\right), \mathrm{for} \mathrm{circular} \mathrm{loading} \mathrm{area}$$

(18b)

Based on the test results of UHPC slabs without steel fibers in Refs. [9,11], the values of constants A and B were calculated to be 7.3 and 3, respectively, via nonlinear curve fitting. As such, Equation (19) was obtained as

$${\displaystyle \frac{V_{P.\mathrm{c}}}{b_{0}d{f}_{\mathrm{t}0}}}={\displaystyle \frac{7.3{\beta }_{\mathsf{\rho }}}{\left(1+{\lambda }_{\mathrm{p}}\right)\left(1+3\psi d/d_{\mathrm{g}0}\right)}}$$

(19)

The contribution of steel fibers in UHPC to the punching shear strength is a function of the critical shear crack width, which is proportional to the rotation angle and the effective depth of the slab [22]. Based on the kinematic assumption in reference [21], the distribution of the inclined shear crack width along the surface of the punching shear cone is linear, as shown in Figure 10a.

Therefore, the shear crack widths at a vertical distance (ξ) from the top of the slab can be computed by Equation (20).

$$w\left(\psi ,\xi \right)=\kappa \psi \xi , 0\le w\le \kappa \psi d$$

(20)

where w is the critical shear crack width; κ is a coefficient relating the rotation angle to the critical shear crack width. κ = 0.5 is adopted in the present study because this value, as used in the CSCT, provides reasonable predictions for the punching shear strength and deformability of concrete slabs [21].

The distribution of fiber bridging stresses along the failure surface was nonlinear, as presented in Figure 10b, and the stress (σ_tf) was a function of the rotation angle and the positions of steel fibers in the slab [22]. Based on Equation (20), the contribution of steel fibers can be computed by integrating the bridging stress corresponding to each crack width, as shown in Equation (21).

$$V_{\mathrm{P}.\mathrm{f}}={\displaystyle {\int }_{A_{\mathrm{h}}}{\sigma }_{\mathrm{tf}}}\left(w\left(\psi ,\xi \right)\right)d{A}_{\mathrm{h}}={\displaystyle {\int }_{A_{\mathrm{h}}}{\sigma }_{\mathrm{tf}}}\left(\psi ,\xi \right)d{A}_{\mathrm{h}}$$

(21)

where A_h is the horizontally projected area of the punching shear failure surface, which increases with the decrease in punching shear angle (θ).

As indicated by Equation (21), a reasonable method for calculating the bridging stresses of steel fibers in UHPC is crucial. In the variable engagement method (VEM) [29], the tensile behavior of randomly distributed steel fibers can be reasonably assessed, leading to Equation (22) for bridging stress.

$${\sigma }_{\mathrm{tf}}=K_{\mathrm{f}}{\displaystyle \frac{l_{\mathrm{f}}}{d_{\mathrm{f}}}}{V}_{\mathrm{f}}{\tau }_{\mathrm{b}}$$

(22)

where K_f is a coefficient accounting for fiber orientation; l_f, d_f, and V_f are the length, diameter, and volume fraction of steel fibers, respectively; and τ_b is the bond stress between a steel fiber and the UHPC matrix. Voo et al. [29] stated that K_f is a function of critical shear crack width (w), and proposed Equation (23) for factor K_f based on the assumption that randomly distributed steel fibers are pulled out from the matrix rather than fracturing.

$$K_{\mathrm{f}}={\displaystyle \frac{1}{\pi }}\arctan \left({\eta }_{\mathrm{e}}{\displaystyle \frac{w}{d_{\mathrm{f}}}}\right){\left(1-{\displaystyle \frac{2w}{l_{\mathrm{f}}}}\right)}^2$$

(23)

where η_e is the engagement parameter, recommended as 3.5. Voo et al. [29] calibrated η_e by assuming an uniform bond stress along the fiber length and verified its accuracy based on the numerous experimental results.

In the present study, the test results of twenty-five fiber pullout specimens were collected from references [30,31,32,33,34,35] to investigate the bond strengths of steel fibers in UHPC matrix, as shown in Figure 11. A significant linear positive correlation between bond strength and the square root of UHPC compressive strength was obtained. Compared with straight steel fibers, higher bond strengths were obtained for twisted fibers and end-hooked fibers in UHPC.

Based on the linear fitting, Equation (24) was proposed to calculate the bond strength of steel fibers in UHPC:

$${\tau }_{\mathrm{b}}=1.93\sqrt{f_{\mathrm{c}}}-10.98, \mathrm{for} \mathrm{straight} \mathrm{steel} \mathrm{fiber}$$

(24a)

$${\tau }_{\mathrm{b}}=2.29\sqrt{f_{\mathrm{c}}}-10.57, \mathrm{for} \mathrm{twisted} \mathrm{steel} \mathrm{fiber}$$

(24b)

$${\tau }_{\mathrm{b}}=3.66\sqrt{f_{\mathrm{c}}}-24.02, \mathrm{for} \mathrm{end}-\mathrm{hooked} \mathrm{steel} \mathrm{fiber}$$

(24c)

Combining Equations (20)–(24), the steel fiber contributions on punching shear capacity of the UHPC slab can be calculated by Equation (25).

$$V_{\mathrm{P}.\mathrm{f}}={\displaystyle {\int }_{\alpha =0}^{2\pi }{\displaystyle {\int }_{\xi =0}^d\left[\frac{1}{\pi }\arctan \left({\eta }_{\mathrm{e}}\frac{\kappa \psi \xi }{d_{\mathrm{f}}}\right){\left(1-\frac{2\kappa \psi \xi }{l_{\mathrm{f}}}\right)}^2\frac{l_{\mathrm{f}}}{d_{\mathrm{f}}}{V}_{\mathrm{f}}{\tau }_{\mathrm{b}}\right]}}\left(r_{\mathrm{c}}+{\displaystyle \frac{\xi }{\tan \theta }}\right)d\mathsf{\xi }d\mathsf{\alpha }$$

(25)

where α is the angle with reference to the circle radius in the horizontal direction, as depicted in Figure 12.

Nevertheless, Equation (25) is complex for practical application. For FRC slabs or UHPC slabs, the distribution of steel fiber bridging stresses tends to be uniform along the punching shear failure surface [3,22], as depicted in Figure 10. Thus, an average bridging stress was introduced in the present study. The expression for the fiber contribution was proposed as Equation (26).

$$V_{\mathrm{P}.\mathrm{f}}={\displaystyle {\int }_{A_{\mathrm{h}}}{\sigma }_{\mathrm{tf}}}\left(\psi ,\xi \right)d{A}_{\mathrm{h}}={\sigma }_{\mathrm{tf}}\left(\psi ,{\xi }_{\mathrm{c}}\right)A_{\mathrm{h}}$$

(26)

$$A_{\mathrm{h}}=\pi d\left({\displaystyle \frac{d}{{\tan }^2\theta }}+{\displaystyle \frac{2r_{\mathrm{c}}}{\tan \theta }}\right)$$

(26a)

where ξ_c is the control distance from the top of the UHPC slab; A_h is computed based on Figure 12.

The average fiber bridging stress can be taken as the stress at a control distance d/3 from the top of the slab, which is a function of the rotation angle [22]. Hence, Equation (26) can be simplified as Equation (27).

$$V_{\mathrm{P}.\mathrm{f}}={\sigma }_{\mathrm{tf}}\left(w={\displaystyle \frac{\kappa \psi d}{3}}\right)A_{\mathrm{h}}$$

(27)

For ease of application, a term (1 + β_vλ_f) was introduced to comprehensively regard the contribution of steel fibers on the punching shear strength of UHPC slabs [15], where β_v is the enhancement coefficient of steel fibers. This approach significantly reduces the computational effort compared to using Equation (25) or Equation (27). Introducing (1 + β_vλ_f) to Equation (19), a simplified equation for obtaining the capacity curve is given, as shown in Equation (28).

$${\displaystyle \frac{V_P}{b_{0}d{f}_{\mathrm{t}0}}}={\displaystyle \frac{7.3\left(1+{\beta }_{\mathrm{v}}{\lambda }_{\mathrm{f}}\right){\beta }_{\mathsf{\rho }}}{\left(1+{\lambda }_{\mathrm{p}}\right)\left(1+3\psi d/d_{\mathrm{g}0}\right)}}$$

(28)

where λ_f is the characteristic value of steel fiber content, equal to l_f/d_fV_f. Based on Refs. [15,17], the value of β_v is taken as 0.4.

2.6. Calculation Procedure for Reinforced UHPC Slabs

The calculation procedure for determining punching shear capacity of reinforced UHPC slab is depicted in Figure 13. It consists of the following four steps: First, the material and geometric parameters need to be predefined. For square slabs, Equation (15) is used to transform the geometry into an equivalent circular configuration. Next, the load-rotation curve is established. In this step, m_cr, m_y, I_g and I_cr are computed using Equations (4)–(6); r_cr, r_y, r₀ and θ are computed using Equations (7) and (8). The demand curve is then determined by Equation (10). Subsequently, the failure criterion is formulated. Herein, V_P.c is calculated via Equation (19); w, σ_tf, K_f, and A_h are computed using Equation (20), Equation (22), Equation (23), and Equation (26a), respectively; V_P.f is calculated using Equation (25) or Equation (27). The capacity curve is subsequently determined by Equation (16). Finally, the punching shear strength and the corresponding critical rotational angle of the reinforced UHPC slabs are obtained by determining the intersection point of the demand curve and the capacity curve.

3. Verification of Models

3.1. Evaluation of Proposed Models

The demand and capacity curves for the reinforced UHPC slabs tested by Fang et al. [15] were calculated using the proposed equations. The results are shown in Figure 14. The theoretical demand curves agree well with the experimental demand curves. Furthermore, the intersections between the calculated capacity curves and the experimental curves generally coincide with the peak loads from the tests. Therefore, the proposed methods can reasonably predict the punching shear capacity of UHPC slabs.

To further verify the accuracy of the proposed models, the experimental results of forty-three UHPC slabs in the third-party literature were introduced.

Table 1. Configurations of UHPC two-way slabs.

Reference	Specimen Code	Plan Dimensions (mm × mm)	h (mm)	c (mm)	b0 (mm)	rc (mm)	rq (mm)	λp	Vf (%)	d (mm)	fc (MPa)	ρs (%)	Support Conditions
Fang et al. [15]	U120-h80-ρ2.57-B70	900 × 900	80	70	488	44.6	771	7	2	52	119	2.57	simply support
	U150-h80-ρ2.57-B70	900 × 900	80	70	488	44.6	771	7	2	52	139	2.57	simply support
	U150-h60-ρ2.57-B70	900 × 900	60	70	424	44.6	771	10	2	36	139	2.57	simply support
	U150-h100-ρ2.57-B70	900 × 900	100	70	568	44.6	771	5	2	72	139	2.57	simply support
	U150-h80-ρ1.31-B70	900 × 900	80	70	504	44.6	771	7	2	56	139	1.31	simply support
	U150-h80-ρ2.57-B90	900 × 900	80	90	568	57.3	769.6	6.8	2	52	139	2.57	simply support
Liu et al. [12]	U-0-2-3.5	1200 × 1200	50	200	912	127.3	955.8	15	2	28	110	3.50	simply support
	U-10-0-3.5	1200 × 1200	50	200	912	127.3	955.8	15	0	28	91	3.50	simply support
	U-10-1-3.5	1200 × 1200	50	200	912	127.3	955.8	15	1	28	97	3.50	simply support
	U-10-2-0	1200 × 1200	50	200	912	127.3	955.8	15	2	28	106	0	simply support
	U-10-3-3.5	1200 × 1200	50	200	912	127.3	955.8	15	3	28	112	3.50	simply support
Moreillon [8]	BCV-1%-40-0.98	960 × 960	40	40	223	40	440	13	1	20	150	0.98	circular line support
	BCV-1%-40-1.92	960 × 960	40	40	223	40	440	13	1	20	150	1.92	circular line support
	BCV-2%-40-1.92	960 × 960	40	40	223	40	440	13	2	20	150	1.92	circular line support
	BCV-1%-60-0.96	960 × 960	60	40	286	40	440	8.4	1	40	150	0.96	circular line support
	BCV-1%-60-1.96	960 × 960	60	40	286	40	440	8.4	1	40	150	1.96	circular line support
	BCV-2%-60-0.96	960 × 960	60	40	286	40	440	8.4	2	40	150	0.96	circular line support
	BCV-1%-80-1.06	960 × 960	80	40	348	40	440	6.3	1	60	150	1.06	circular line support
	BCV-1%-80-1.31	960 × 960	80	40	348	40	440	6.3	1	60	150	1.31	circular line support
	BCV-1%-80-1.88	960 × 960	80	40	348	40	440	6.3	1	60	150	1.88	circular line support
	BCV-2%-80-1.88	960 × 960	80	40	348	40	440	6.3	2	60	150	1.88	circular line support
Al-Quraishi [7]	G1Ufib0.5	1100 × 1100	100	100	720	63.7	865.9	5.4	0.5	80	198.9	2	simply support
	G1Ufib1.1	1100 × 1100	100	100	720	63.7	865.9	5.4	1.1	80	208.2	2	simply support
	G3Uρ1%	1100 × 1100	100	100	720	63.7	865.9	5.4	0.5	80	198.9	1	simply support
	G4Ut55	1100 × 1100	55	100	576	63.7	865.9	13	0.5	44	199.2	2	simply support
	G5Ufy560	1100 × 1100	100	100	720	63.7	865.9	5.4	0.5	80	198.2	2	simply support
Harris [4]	H2-D1.5	1140 × 1140	55.1	38.1	372.8	24.3	957.5	7.9	2	55.1	218	0	simply support
	H2-D2	1140 × 1140	58.9	50.8	438.8	32.3	956.6	7.3	2	58.9	218	0	simply support
	H2-D1	1140 × 1140	53.8	25.4	316.8	16.2	958.4	8.3	2	53.8	218	0	simply support
	H2.5-D2	1140 × 1140	66.2	50.8	468	32.3	956.6	6.5	2	66.2	218	0	simply support
	H2.5-D1.5	1140 × 1140	64.5	38.1	410.4	24.3	957.5	6.8	2	64.5	218	0	simply support
	H3-D1.5	1140 × 1140	71.8	38.1	439.6	24.3	957.5	6.1	2	71.8	218	0	simply support
	H3-D1	1140 × 1140	76.9	25.4	409.2	16.2	958.4	5.8	2	76.9	218	0	simply support
Park [11]	D40-f1.5	1600 × 1600	40	50	360	31.8	1257.3	14	1.5	40	126.6	0	simply support
	D50-f1.5	1600 × 1600	50	50	400	31.8	1257.3	12	1.5	50	126.6	0	simply support
	D60-f1.5	1600 × 1600	60	50	440	31.8	1257.3	9.6	1.5	60	126.6	0	simply support
	D40-f1.0	1600 × 1600	40	50	360	31.8	1257.3	14	1.0	40	124.2	0	simply support
Joh et al. [13]	PT40-50-75	1600 × 1600	40	50	360	31.8	1257.3	14	2	40	194	0	simply support
	PT40-50-100	1600 × 1600	40	50	360	31.8	1257.3	14	2	40	194	0	simply support
	PT40-50-150	1600 × 1600	40	50	360	31.8	1257.3	13	2	40	194	0	simply support
	PT70-50-50	1600 × 1600	70	50	480	31.8	1257.3	8.2	2	70	194	0	simply support
	PT70-50-100	1600 × 1600	70	50	480	31.8	1257.3	7.8	2	70	194	0	simply support
	PT70-50-125	1600 × 1600	70	50	480	31.8	1257.3	7.6	2	70	194	0	simply support

summarizes the configurations of all specimens, and

Table 2. Punching shear capacities of UHPC two-way slabs.

Reference	Specimen Code	VP.test (kN)	ψu	VP. Equation (25) (kN)	VP. Equation (27) (kN)	VP. Equation (28) (kN)	VP.test /VP. Equation (25)	VP.test /VP. Equation (27)	VP.test /VP. Equation (28)	Cs (%)
Fang et al. [15]	U120-h80-ρ2.57-B70	226.3	0.034	217.6	202.1	202.1	1.04	1.12	1.12	34.2
	U150-h80-ρ2.57-B70	238.8	0.029	231.8	215.1	221.1	1.03	1.11	1.08	30.8
	U150-h60-ρ2.57-B70	140.7	0.053	146.6	125.6	120.3	0.96	1.12	1.17	40.5
	U150-h100-ρ2.57-B70	405.7	0.021	382.7	372.2	329.8	1.06	1.09	1.23	25.7
	U150-h80-ρ1.31-B70	207.8	0.036	197.9	197.9	192.4	1.05	1.05	1.08	35.9
	U150-h80-ρ2.57-B90	262.3	0.028	240.6	232.1	226.1	1.09	1.13	1.16	33.8
Liu et al. [12]	U-0-2-3.5	325.0	0.082	303.7	292.8	298.2	1.07	1.11	1.09	33.8
	U-10-0-3.5	250.0	0.071	255.1	252.5	257.7	0.98	0.99	0.97	0
	U-10-1-3.5	295.0	0.079	273.1	258.8	263.4	1.08	1.14	1.12	28.3
	U-10-2-0	180.0	0.093	160.7	145.2	151.3	1.12	1.24	1.19	43.8
	U-10-3-3.5	355.0	0.089	351.5	366.0	369.8	1.01	0.97	0.96	54.7
Moreillon [8]	BCV-1%-40-0.98	31.0	0.103	24.4	25.4	23.5	1.27	1.22	1.32	27.9
	BCV-1%-40-1.92	49.0	0.088	38.3	34.3	40.5	1.28	1.43	1.21	25.1
	BCV-2%-40-1.92	49.0	0.096	40.5	35.5	35.8	1.21	1.38	1.37	37.7
	BCV-1%-60-0.96	75.0	0.064	81.5	70.8	76.5	0.92	1.06	0.98	22.4
	BCV-1%-60-1.96	130.0	0.051	126.2	135.4	107.4	1.03	0.96	1.21	20.0
	BCV-2%-60-0.96	120.0	0.074	127.7	121.2	105.3	0.94	0.99	1.14	30.5
	BCV-1%-80-1.06	180.0	0.049	169.8	178.2	168.2	1.06	1.01	1.07	19.2
	BCV-1%-80-1.31	220.0	0.042	213.6	205.6	177.4	1.03	1.07	1.24	17.0
	BCV-1%-80-1.88	270.0	0.033	245.5	232.8	238.9	1.10	1.16	1.13	15.7
	BCV-2%-80-1.88	274.0	0.043	279.6	276.8	282.5	0.98	0.99	0.97	23.6
Al-Quraishi [7]	G1Ufib0.5	268.6	0.011	285.7	295.2	271.3	0.94	0.91	0.99	11.5
	G1Ufib1.1	384.5	0.018	349.5	331.5	340.3	1.10	1.16	1.13	22.3
	G3Uρ1%	247.9	0.020	258.2	250.4	258.2	0.96	0.99	0.96	11.3
	G4Ut55	123.9	0.068	113.7	108.7	114.7	1.09	1.14	1.08	12.2
	G5Ufy560	319.8	0.016	371.8	351.4	340.2	0.86	0.91	0.94	13.1
Harris [4]	H2-D1.5	104.0	0.115	102.0	108.3	91.2	1.02	0.96	1.14	30.8
	H2-D2	121.0	0.126	115.2	104.3	100.0	1.05	1.16	1.21	32.4
	H2-D1	101.0	0.134	91.0	82.1	87.8	1.11	1.23	1.15	33.7
	H2.5-D2	147.0	0.090	128.9	136.1	137.4	1.14	1.08	1.07	26.8
	H2.5-D1.5	136.0	0.200	124.8	114.3	120.4	1.09	1.19	1.13	40.8
	H3-D1.5	157.0	0.095	160.2	134.2	141.4	0.98	1.17	1.11	26.8
	H3-D1	148.0	0.171	133.3	127.6	133.3	1.11	1.16	1.11	35.3
Park [11]	D40-f1.5	83.1	0.048	71.6	72.3	79.9	1.16	1.15	1.04	54.9
	D50-f1.5	118.9	0.033	109.1	110.1	97.5	1.09	1.08	1.22	37.5
	D60-f1.5	160.8	0.025	156.1	169.3	147.5	1.03	0.95	1.09	31.7
	D40-f1.0	71.7	0.039	64.0	63.5	69.6	1.12	1.13	1.03	41.0
Joh et al. [13]	PT40-50-75	78.0	0.036	70.3	73.6	65.5	1.11	1.06	1.19	40.5
	PT40-50-100	106.2	0.042	94.0	88.5	94.0	1.13	1.20	1.13	43.8
	PT40-50-150	91.3	0.053	86.1	90.4	86.9	1.06	1.01	1.05	37.9
	PT70-50-50	218.7	0.015	204.4	195.3	183.8	1.07	1.12	1.19	33.8
	PT70-50-100	239.5	0.025	211.9	204.7	193.1	1.13	1.17	1.24	36.6
	PT70-50-125	296.7	0.027	290.9	272.2	277.3	1.02	1.09	1.07	37.3
Average							1.06	1.12	1.13
Standard deviation							0.07	0.10	0.11

shows the corresponding punching shear capacities. Note that for square loading areas and square slabs, r_c and r_q in Table 1 are calculated using Equation (15); λ_p and b₀ are calculated using Equation (9) and Equation (18), respectively. In Table 2, ψ_u and V_P. Equation (25) are the ultimate rotation angle and predicted capacity, respectively, calculated considering the UHPC matrix contribution from Equation (19) and the fiber contribution from Equation (25); V_P. Equation (27) is calculated using Equation (27) for fiber contribution; V_P. Equation (28) is calculated by Equation (28); C_s is the fiber contribution ratio, defined as the ratio of fiber contribution from Equation (25) to V_P. Equation (25).

As shown in Table 2, the direct integration of Equation (25) for the steel fiber contribution gives the best prediction of punching shear capacity, where the mean ratio of measured values to predicted results is 1.06 with a standard deviation of 0.07. Compared to the integration, the application of average fiber bridging stress is more convenient. In so doing, the average ratio of predicted values to test results is 1.12 with a standard deviation of 0.1, indicating an accurate calculation. Equation (28) provided the acceptable prediction with an average ratio of 1.13 and a standard deviation of 0.11. When the fiber volume fraction varied from 0.5% to 3%, the fiber contribution ratio was 16.3% to 54.9%. This range is consistent with the values of 20.5% to 63.8% reported in Refs. [7,8,9,12].

Several cases with notable deviations warrant discussion based on Table 1 and Table 2. First, the model underestimates the capacity for slabs with a thickness of about 40 mm (e.g., an underestimation exceeding 20% for specimen BCV-1%-40-1.92). This may be because the constant factor k is difficult to reflect the law that the tensile contribution of UHPC decreases with the decrease in slab thickness [25]. Second, the model slightly overestimates the capacity when the fiber volume fraction is 0% or 0.5% (e.g., an overestimation of about 10% for specimen G5Ufy560), likely because the weak bridging effect at low fiber contents is not fully captured by the VEM. Based on the analyzed data, the following applicability limits for the proposed models are suggested: (1) steel fiber volume fraction (V_f) varying from 0.5% to 3%; (2) slab thickness (h) varying from 40 mm to 100 mm; (3) UHPC compressive strength (f_c) varying from 100 MPa to 220 MPa; and (4) reinforcement ratio (ρ_s) varying from 0% to 3.5%.

3.2. Assessment of Typical Empirical Formulas

Table 3. Typical equations for calculating punching shear capacity of UHPC slab.

Source	Equations	Notation
Al-Quraishi et al. [7], Equation (29)	$V_{\mathrm{p}}=0.35{\beta }_{\mathrm{h}}\left({\nu }_{\mathrm{fc}}+{\nu }_{\mathrm{fte}}+{\nu }_{\mathsf{\rho }}\right)b_{0\mathrm{f}}d$ where ${\beta }_{\mathrm{h}}={\left(125/d\right)}^{0.87}$; ${\nu }_{\mathrm{fc}}=0.12{\left(f_{\mathrm{c}}\right)}^{0.35}$; ${\nu }_{\mathsf{\rho }}={\left(100{\rho }_{\mathrm{s}}\right)}^{0.21}$$; b_{0\mathrm{f}}=b_{2.5\mathrm{d}}\left(1-KF\right)$$; F=\left(l_{\mathrm{f}}/d_{\mathrm{f}}\right)V_{\mathrm{f}}{D}_{\mathrm{f}}$.	βh: Impact factor of depth of cross-section; νfc and νρ: Contribution of UHPC matrix and steel bars to punching shear strength, respectively; νfte: In total, 14 MPa for a steel fiber volume fraction of 2%; b0f: Modified critical failure perimeter taking into account the fiber effect; b2.5d: Perimeter of critical cross section at 2.5d away from the column edge; K: Non-dimensional constant: 0.45 was used; Df: Bond factor between steel fiber and UHPC: 0.5 for smooth fiber; Vpf and Vpc: Contribution of steel fibers and other factors to capacity, respectively; βr: Impact factor of loading perimeter; fpc: Punching shear strength provided by UHPC; γb: Material partial coefficient, and γb = 1 herein; u: Perimeter of loading area; τmax: Punching strength; klocal: Orientation factor of steel fibers: 1.25 was used in the present study; βs: Ratio of the long side to the short side of the rectangular loading area; αs: Factor to account for column position: 4 for interior position, 3 for edge position, and 2 for corner position; βv: Factor used to account for the effect of steel fibers on punching shear strength: 0.4 was used.
JSCE [18], Equation (30)	$V_{\mathrm{p}}=V_{\mathrm{pc}}+V_{\mathrm{pf}}$ where $V_{\mathrm{pc}}={\beta }_{\mathrm{h}}{\beta }_{\mathsf{\rho }}{\beta }_{\mathrm{r}}{f}_{\mathrm{pc}}{b}_{0}d/{\gamma }_{\mathrm{b}}$; $V_{\mathrm{pf}}=f_{\mathrm{v}}{b}_{0}d/{\gamma }_{\mathrm{b}}$; ${\beta }_{\mathrm{h}}={\left(1000/d\right)}^{0.25}\le 1.5$; ${\beta }_{\mathsf{\rho }}=\sqrt[3]{100{\rho }_{\mathrm{s}}}\le 1.5$; $f_{\mathrm{pc}}=0.2\sqrt{f_{\mathrm{c}}}$; ${\beta }_{\mathrm{r}}=1+1/\left(1+0.25u/d\right)$.
NF P 18-710 [19], Equation (31)	$V_{\mathrm{p}}={\tau }_{\max }{b}_{0}d$ where ${\tau }_{\max }={\displaystyle \frac{0.8}{{\gamma }_{\mathrm{b}}}}\min \left({\displaystyle \frac{f_{\mathrm{t}}}{k_{\mathrm{local}}}};f_{\mathrm{t}0}\right)$.
DBJ43/T325-2017 [17], Equation (32)	$V_{\mathrm{p}}=0.7f_{\mathrm{t}0}(1+{\beta }_{\mathrm{v}}{\lambda }_{\mathrm{f}}){\eta }_{\mathrm{p}}{b}_{0}d$ where ${\eta }_{\mathrm{p}}=\min \left\{0.4+{\displaystyle \frac{1.2}{{\beta }_{\mathrm{s}}}}\right.,\left.0.5+{\displaystyle \frac{{\alpha }_{\mathrm{s}}d}{4b_0}}\right\}$.
Fang et al. [15], Equation (33)	$V_{\mathrm{p}}={\displaystyle \frac{4{\beta }_{\mathsf{\rho }}{f}_{\mathrm{t}0}(1+{\beta }_{\mathrm{v}}{\lambda }_{\mathrm{f}})b_{0}d}{1+{\lambda }_{\mathrm{p}}}}$ where ${\beta }_{\mathsf{\rho }}={\left(100{\rho }_{\mathrm{s}}\right)}^{0.2}$.

lists several typical equations for calculating the punching shear capacity of reinforced UHPC slabs. Al-Quraishi et al. [7] proposed Equation (29) to consider the contributions of UHPC matrix, steel fibers, and steel bars. Equation (30), suggested by JSCE [18], separately accounts for the contributions of steel fibers and other factors. Equation (31), recommended by NF P 18-710 [19], determines the punching shear strength primarily based on the tensile strength of UHPC. In the Chinese specification DBJ43/T325-2017 [17], Equation (32) employs a coefficient (β_v) to holistically account for the positive effects of steel fibers. Equation (33) proposed by Fang et al. [15] incorporates the effect of the punching shear-span ratio on the punching shear capacity.

The punching shear capacities of forty-three UHPC slabs in Refs. [4,7,8,11,12,13,15] were calculated using Equations (29)–(33). The ratios of the test results (V_p.test) to the calculated values (V_p.cal) are presented in Figure 15. Compared to the proposed models, the previous methods suggested by Al-Quraishi et al. [7] and NF P 18-710 [19] exhibit greater scatter and provide less accurate predictions for the punching shear capacity of reinforced UHPC slabs. The equations in JSCE [18] and DBJ43/T325-2017 [17] show relatively reasonable accuracy but do not account for the influence of the punching shear-span ratio. Although Equation (33) proposed by Fang et al. [15] achieves relatively high accuracy, it does not mechanistically incorporate the tensile strain-hardening of UHPC.

4. Parameter Analysis and Discussion

4.1. Influence of Slab Thickness

Figure 16 presents the influence of slab thickness on the punching shear capacity and ultimate rotation angle based on the calculation results (from Equation (25)) of slabs tested by Moreillon [8]. Compared to slabs with a 40 mm thickness, the punching shear capacities of slabs with 60 mm and 80 mm thickness increased by 182% and 487%, respectively, while the corresponding steel fiber contribution ratios decreased by 19.7% and 38.3%, respectively. As the slab thickness increased, the punching shear-span ratio decreased, and the critical failure perimeter increased, resulting in an improvement in the punching shear capacity. Furthermore, the flexural stiffness of the UHPC slabs improved with the increase in slab thickness, which restricted the development of the critical shear crack and reduced the fiber bridging effect [25]. Hence, the fiber contribution ratio decreased.

4.2. Influence of Fiber Volume Fraction

The influence of fiber volume fraction on the punching shear capacity and ultimate rotation angle was investigated based on the predicted results of UHPC slabs tested by Park [11], as depicted in Figure 17. Compared to the slab with a steel fiber volume content of 0.5%, the punching shear capacities of slabs with volume fractions of 1.0% and 1.5% increased by 51.9% and 71.4%, respectively, which results from the enhanced bridging effect of steel fibers. Moreover, a higher fiber volume fraction leads to an increase in the fiber contribution ratio and enhances slab ductility, which consequently improves the ultimate rotation angle.

4.3. Influence of UHPC Compressive Strength

The influence of UHPC compressive strength on the punching shear capacity and ultimate rotation angle was analyzed based on the results of UHPC slabs tested by Fang et al. [15], as shown in Figure 18. Compared to slabs with a UHPC compressive strength of 120 MPa, the punching shear capacities of slabs with a UHPC of 150 MPa and 180 MPa improved by 6.5% and 14.1%, respectively. This improvement is attributed to the increase in tensile strength of UHPC and the shear resistance in the shear-compression zone. Furthermore, the elastic modulus increases with UHPC compressive strength, leading to an improvement in the flexural stiffness of the UHPC slab. Consequently, the development of the critical shear crack was restricted, and the fiber bridging effect decreased. Therefore, the fiber contribution ratio decreased with the increase in UHPC compressive strength.

4.4. Influence of Reinforcement Ratio

The influence of reinforcement ratio on the punching shear capacity and ultimate rotation angle was identified based on the calculation results of slabs tested by Moreillon [8], as shown in Figure 19. Compared to UHPC slabs with a reinforcement ratio of 1.06%, the punching shear capacities of those with reinforcement ratios of 1.31% and 1.88% increased by 25.8% and 44.6%, respectively, while the steel fiber contribution ratios decreased by 11.5% and 18.2%, respectively. This trend results from two primary factors: (1) the dowel action of steel bars improves with increasing reinforcement ratio; and (2) the increased flexural stiffness limits the development of the critical shear crack.

4.5. Discussion

Based on the results in Table 2 and the preceding parametric analysis, a coupling effect exists among several key parameters, including UHPC slab thickness, compressive strength, steel fiber volume fraction, and reinforcement ratio. For instance, as the slab thickness decreases, the contribution of steel fibers becomes more significant, and the influence of the steel fiber volume fraction V_f is consequently amplified. This interaction is quantified by the proposed model: for 40 mm thick slabs (D40-f1.0 and D40-f1.5 in Ref. [11]), a 50% increase in V_f (from 1.0% to 1.5%) results in a 13.9% increase in the fiber contribution ratio C_s. In contrast, for 80 mm thick slabs (BCV-1%-80-1.88 and BCV-2%-80-1.88 in Ref. [8]), a 100% increase in V_f (from 1.0% to 2.0%) yields only a 7.9% increase in C_s. The proposed model effectively captures the interaction effects between different parameters. Therefore, the proposed model provides a mechanism-based foundation for future revisions of UHPC structural design codes pertaining to punching shear design. To further refine the model and broaden its applicability, future work could incorporate a detailed consideration of tensile damage and crack propagation behavior in UHPC [36,37].

5. Conclusions

This study develops a physical–mechanical model for reinforced UHPC slabs based on the critical shear crack theory (CSCT). This model enables the calculation of the punching shear capacity and ultimate rotation angle of reinforced UHPC slabs. The following conclusions are drawn.

1. The contribution of steel fibers is calculated based on the assumption that the critical shear crack width is proportional to the slab rotation angle and effective depth. The proposed model incorporates the tensile strain-hardening behavior and tensile strength of UHPC, the punching shear-span ratio, and the reinforcement ratio, while the aggregate interlock is neglected.

2. The contribution of the UHPC matrix is formulated considering key geometrical and mechanical parameters. The variable engagement method (VEM) can be effectively applied to evaluate the steel fiber contribution to the punching shear capacity of UHPC slabs.

3. A significant linear positive correlation exists between the bond strength of steel fibers in the UHPC matrix and the square root of the UHPC compressive strength. Based on this finding, an equation describing the bond strength between steel fibers and UHPC matrix was developed.

4. Comparisons between experimental results from the literature and model predictions indicate that the proposed model can accurately compute the punching shear capacity and ultimate rotation angle of reinforced UHPC slabs. The models are applicable for the following parameter ranges: steel fiber volume fraction from 0.5% to 3%, slab thickness from 40 mm to 100 mm, UHPC compressive strength from 100 MPa to 220 MPa, and reinforcement ratio from 0% to 3.5%.

5. With increasing slab thickness, UHPC compressive strength, and reinforcement ratio, the punching shear capacity increases, whereas the ultimate rotation angle and fiber contribution ratio decrease. Increasing the fiber volume fraction enhances both the fiber contribution and the ultimate capacity. For slabs with higher UHPC strength, the beneficial effect of a higher reinforcement ratio is more pronounced.