Analysis of Elastic-Stage Mechanical Behavior of PBL Shear Connector in UHPC

Abstract

This paper investigates the mechanical behavior of PBL shear connectors in UHPC during the elastic stage, utilizing push-out experiments and numerical simulation. This study simplifies the mechanical behavior of PBL shear connectors in UHPC under normal service conditions as a plane strain problem for the UHPC dowel and a Winkler’s Elastic foundation beam theory for the transverse reinforcement. The UHPC dowel is a thick-walled cylindrical shell subjected to non-axisymmetric loads inside and outside simultaneously in the plane-strain state. The stress solution is derived by assuming the contact stress distribution function and using the Airy stress function. The displacement solution is subsequently determined from the stresses by differentiating between elastic and rigid body displacements. By modeling the transverse reinforcement as an infinitely long elastic foundation beam, its displacement solution and stress solution are obtained. We obtain the load–slip curve calculation method by superimposing the displacement of UHPC with the transverse reinforcement in the direction of shear action. The proposed analytical solutions for stress and slip, as well as the method for calculating load–slip, are shown to be reliable by comparing them to the numerical simulation analysis results.

1. Introduction

The steel-UHPC composite structure offers enhanced mechanical performance and economic efficiency, as it fully exploits the superior tensile resistance of steel and exceptional compressive resistance of UHPC [1,2,3,4]. The application of UHPC material can effectively reduce the size of structural components and enhance the spanning capability. In addition, the reduction in the amount of material used during construction, as well as the excellent durability of UHPC, is conducive to achieving the goal of energy conservation and emission reduction. Therefore, it has been widely used in various engineering projects [5,6,7,8]. The shear connectors play a crucial role in composite construction as they are the key component for synergistic interaction. Among the various types of shear connectors in steel-concrete composite constructions, the Perfobond strip (PBL) connector is widely used due to its superior stiffness, ductility, and fatigue performance [9,10,11].

In recent years, many scholars have investigated the mechanical behavior of PBL shear connectors in steel-UHPC composite bridges. By combining experiments and finite element analysis, Wu et al. [12] have investigated the differences in PBL shear connection mechanical performance between steel-UHPC bridges and ordinary steel-concrete composite bridges. The results reveal that UHPC demonstrates slower crack development, fewer cracks, smaller crack widths, and a higher degree of concrete integrity compared with ordinary concrete specimens. Xiao et al. [13,14] performed a series of twenty-four push-out tests to investigate the effects of the load transfer mechanisms and the perforated plate thickness on failure modes. The results showed that concrete dowels initially provided the primary shear resistance of PBL shear connectors under relatively low loads. However, once the concrete dowels failed, the shear resistance was then transferred to the perforated rebar. Cao et al. [15] conducted monotonic loading push-out tests on four PBL shear connectors embedded in UHPC and one reference specimen cast in normal concrete, employing a parametric experimental analysis to evaluate the effects of concrete types, embedded depth, and transverse reinforcement diameter on structural performance. He et al. [16] performed twelve push-out tests to investigate the failure mechanisms, ductility characteristics, and constituent contributions to the ultimate shear capacity of PBL. They developed a theoretically derived analytical model and predictive equations to predict the ultimate resistance of PBL connectors under concrete dowel shear failure modes by interpreting concrete dowel shear failure patterns. Tu et al. [17] tested twelve specimens to investigate the shear performance of PBL connectors, investigating the effects of concrete type, embedment depth, and transverse reinforcement diameter. And proposed an equation for accurately predicting the ultimate shear capacity of steel-UHPC PBL connectors.

The aforementioned experimental investigations have elucidated the mechanical behavior of PBL shear connectors. For conventional concrete-embedded PBL connectors, it is difficult to study their mechanical behavior by theoretical methods due to the inherent heterogeneity and premature cracking propensity of the concrete matrices [18,19,20]. In contrast, UHPC is a type of advanced cementitious composite that has an optimized mixture of cement gel, silica fume, quartz sand, water-reducing admixture steel fibers, and other materials. It exhibits approximately homogeneous material properties on the macroscopic scale of the structure [21,22,23]. This intrinsic continuity enables reliable theoretical modeling of PBL connector behavior in UHPC systems through continuum mechanics approaches. Zhang et al. [24] established a nonlinear mechanical model for characterizing force transfer mechanisms within the PBL shear connector group. This methodology systematically accounts for equivalent stiffness degradation of critical constituent elements, including individual connectors, perforated steel plates, and concrete substrates. Duan et al. [25] established a full-range analytical formula for the load–slip behavior of UHPC single-hole PBL shear connector by employing the elastic foundation beam model based on Timoshenko beam theory. The derived formulation demonstrated good agreement with the experimental results. Zhang et al. [26,27] proposed a coordinated load–slip calculation method for PBL shear connector groups, integrating the force characteristics of individual connectors with the relationship between steel-concrete interfacial slip and layer-wise connector load distribution. Ma et al. [28] established a theoretical model to predict the shear bearing capacity of T-Perfobond shear connectors, incorporating the effects of end concrete bearing pressure and the reduction coefficient for welding. The model shows close agreement with both experimental and numerical results. Liao et al. [29] conducted eight samples with different arrangements of PBL shear connectors. The tests identified the major contributions from concrete end bearing and the PBL bar. The authors proposed a new model for estimating the shear capacity of PBL connectors in DSTMs, which demonstrated a high level of accuracy.

The above studies explain the internal force transfer mode of the PBL shear connector from the theoretical perspective and provide references and means for further theoretical derivation of the PBL shear connector. However, existing research efforts present several limitations. A portion of studies are constrained to empirical generalizations derived from experimental observations [30,31,32], while others attempting mechanistic analyses frequently employ oversimplified assumptions that inadequately represent actual service conditions. In addition, current investigations predominantly focus on macroscopic load–slip curve characterizations, and lack in-depth research on the stress and displacement distribution inside the UHPC matrices. The stress response of PBL shear connectors in UHPC interactions between the UHPC dowel mechanism, penetrating reinforcement, and perforated steel plate components [33,34,35]. Therefore, it is difficult to quantitatively elucidate the microscopic force-transfer phenomenon at the mechanistic level by only studying from a single level or summarizing the law from the experimental phenomenon.

This paper aims to comprehensively explain the mechanical behavior of PBL shear connectors in UHPC. This investigation employs the linear relationship between load and slip as well as reinforcement strain in the elastic stage by three sets of PBL single connector push-out specimens. We propose a simplified mechanical model of the elastic stage. In this model, the force behavior of the PBL shear connector in UHPC is simplified to the plane strain problem of the UHPC dowel and the Winkler elastic foundation beam problem of the penetrating reinforcement. This paper obtains the analytical solutions of the stress and displacement distributions of the UHPC dowel and the penetrating reinforcement and derives the computation method of the load–slip curves. The study ultimately verifies the accuracy and reliability of the proposed analytical method through systematic comparison of the derived solutions with both finite element simulation results and experimental datasets.

2. Push-Out Test of PBL Shear Connector in UHPC

2.1. Experiment Setting

This paper investigates the mechanical characteristics of the PBL shear connector in UHPC through static push-out tests to provide a reference for establishing its mechanical model. To minimize the influence of the eccentric moment on the test results in the standard push-out test, the push-out specimens are designed with a “sandwich” structure, as shown in Figure 1a. There are 3 specimens in total, denoted by S1, S2, and S3, respectively, where “S” stands for static test. Each of the specimens contains a PBL shear connector. The compressive strength of the specimen UHPC is 127.2 MPa, the tensile strength is 6.5 MPa, and the elastic modulus is 42319 MPa; the perforated steel plate is Q355D with 24 mm thickness and a perforated with 60 mm diameter; the transverse rebar grade is HRB335 with 20 mm diameter. The structural dimensions are shown in Figure 1b. The width and height of the loading steel plate are 300 mm and 175 mm, respectively. The perforated steel plate width and height are 550 mm and 300 mm, respectively. The foam boards placed at the bottom of the perforated steel plate have a thickness of 24 mm, a width of 400 mm and a height of 50 mm, respectively. The stirrups in all specimens have a diameter of 10 mm, and the stirrup reinforcement ratio is 3.4%.

This study employs the formula for the shear capacity of the PBL connector in UHPC as proposed in Reference [36].

$$P_{Rd}=({\beta }_1\xi (t/D)A_c{f}_c+{\beta }_2{A}_{tr}{f}_y)/{\gamma }_v$$

(1)

where A_c and A_tr denote the cross-sectional area of UHPC dowel and transverse rebar, respectively; f_c and f_y represent the design strength of UHPC dowel and transverse rebar, respectively; t and D correspond to the perforated steel plate thickness and the hole diameter. The coefficients β₁, β₂, and ξ account for the influence of the concrete dowel, transverse rebar, and hoop reinforcement. The partial safety factor γ_v is specified as 1.1.

During the preloading process, the load is directly applied to 0.2 P_Rd and maintained for 3 min. This procedure is repeated three times to eliminate system compliance and stabilize interfacial contact conditions. During the formal loading, stepwise force-controlled loading at 20 kN increments until reaching the design capacity P_Rd (413 kN), holding each loading level for 5 min. And then unloaded to zero in 50 kN steps. Subsequently, displacement-controlled loading was initiated with displacement increments of 0.2 mm at all levels and a constant loading rate of 0.2 mm/min until the specimen failed.

The applied load, strain in the transverse rebar and UHPC, and relative slip are recorded for five minutes during each loading phase. The test equipment reads the applied load, and a micrometer measures the relative slip. The arrangement of these measurement points is illustrated in Figure 1c. Figure 1d shows strain measurement points for both the transverse rebar and UHPC. Specifically, there are four symmetrically placed measuring locations on the left, right, up, and downsides of each transverse rebar, and the stress measurement points are situated at 32 mm on the left and right sides of the center. Resistance strain gauges are used to evaluate the comparatively deformed reinforcement, which is welded with angle steel to provide a measuring point. The arrangement of strain measurement points is shown in Figure 2, where C and T represent the strain gauges on the upper and lower surfaces of the transverse rebar respectively.

2.2. Test Result

2.2.1. Failure Mode

The failure process of the three push-out specimens is similar. At the design load, the relative slip of the specimen is less than 0.6 mm, and no cracks were observed on the UHPC surface. When the relative slip is about 1 mm under continuous displacement loading, small vertical cracks began to appear on the side of the specimen, and the measured load increment decreased. Further loading caused the specimen to emit a muffled sound, indicating the UHPC dowel’s shear failure. Figure 3a shows the morphology of the UHPC dowel after separation of the open steel plate and UHPC. The relative slip began to increase dramatically, while the load increment was relatively small. The load and relative slip increment showed a gradual slowdown with further loading. Finally, the transverse rebar was a shear failure with a loud noise. Figure 3b shows the cross-section of the transverse bar following a vertical cut of the material. All three specimens exhibited maximum relative slips greater than 20 mm.

2.2.2. Load–Strain Curve

During the test, the measured stress level in UHPC was relatively low. This paper mainly discusses the development of strain in the transverse rebar. The arrangement direction of the strain gauge is consistent with the axial direction of the transverse rebar. The load–strain curves are shown in Figure 4. There is no axial strain in the transverse rebar before the load exceeds 250 kN. As the loading continued to the design load (413 kN), the measurement points on the upper and lower sides of the transverse rebar began to exhibit axial strains in opposite directions but with similar magnitudes. This indicates that bending strains began to appear at the measurement points at this time, while the strain level was low. As the load increases, the UHPC dowel undergoes shear failure, transferring the load borne originally by the UHPC to the through transverse rebar. The bending strain of the transverse rebar rapidly increases to 2000 με, indicating that the penetrating bar has already yielded. As the load continued to increase, the strain of the penetrating bar kept growing and entered the strain-hardening stage (which was not measured due to the range limitation of the strain gauge), and finally shear failure.

2.2.3. Load–Slip Curve

The load–slip curves of all specimens are shown in Figure 5, where the relative slip values represent the average of all relative slip measurement points. The inset illustrates the linear fit (R² = 0.95) applied to the average load–slip curve of the three specimens during the initial elastic stage. Based on the load–slip curves and the failure processes, the mechanical behavior of the PBL shear connector in the UHPC can be divided into three stages. At the Elastic stage, the load is transferred from the UHPC dowel to the transverse rebar with both components sharing the load. During this stage, the relative slip between steel and UHPC remains approximately 0.2 mm, and the strain in the transverse rebar does not exceed 100 με. The load–slip curve and load–strain curve exhibit a nearly linear relationship, as shown in Figure 6 (the strain and slip are average values of the three specimens), indicating that both the transverse rebar and UHPC remain in elasticity. At the Plastic stage, the relative slip between steel and UHPC accelerates with increasing load. The UHPC dowel enters into a plastic state due to excessive shear deformation. However, the surrounding UHPC remains in a multi-axial stress state due to being constrained by axial restraint, and there is no failure. As the load further increases to the peak, the UHPC dowel shear fails. At the Ductile stage, the load is completely transferred to the transverse rebar after the shear failure of UHPC. It results in a rapid reduction in the specimen’s shear stiffness. At the large shear load, the steel bar has yielded and entered the strain-hardening stage. Due to the excellent ductility of the steel, the specimen undergoes significant slip until the transverse rebar reaches ultimate strength and ultimately fails by shear fracture.

3. Mechanical Model and Analysis of PBL Shear Key in UHPC in the Elastic Stage

3.1. Mechanical Model

The code [37] recommends 0.2 mm slip as the serviceability limit state threshold for shear connectors. According to the experimental results in Section 2.2, when the relative slip is less than 0.2 mm, UHPC, the perforated steel plates and penetrating steel bars of PBL shear connectors remain in the elastic stage. The loading-bearing capacity and shear stiffness are jointly provided by the UHPC dowel and transverse rebar.

From a geometric perspective, the UHPC dowel is wrapped around the transverse rebar and embedded in the perforated steel plate. The UHPC plate thickness significantly exceeds that of the perforated steel plate, combined with the confining effect of the stirrups, the UHPC dowel within the steel plate’s thickness range is constrained by the UHPC plate on the lateral side of the steel plate, existing in the plane strain state. In the vertical plane shown in Figure 7, the outer side of the UHPC dowel is subject to compressive stress from the perforated steel plate’s hole walls within the plate’s thickness range, while the transverse rebar supports the inner side. This structure forms a thick-walled cylindrical shell that simultaneously resists non-axisymmetric loading from both internal and external. Therefore, the UHPC dowel in the range of the thickness of the perforated steel plate can be regarded as a thick-walled cylindrical shell with non-axisymmetric loads both inside and outside the circle under plane strain state.

The transverse rebar of the PBL shear connector is embedded in the UHPC dowel and is subjected to vertical shear within the thickness range of the steel plate, while there is no relative displacement with the UHPC outside the steel plate thickness. As shown in Figure 8, its deformation is slight during the elastic stage. Given the limited capacity of this deformation to diffuse to both ends, it can be considered as a Winkler foundation beam supported by UHPC.

3.2. Mechanical Analysis of UHPC Dowel

The problem of plane strain in thick-walled cylindrical shells subjected to axisymmetric loads can be solved using the well-known Lame equations. However, if non-axisymmetric loads are present, the Lame equations cannot provide a solution [38]. Yiannopoulos [39,40] investigated the plane strain problem of a circular ring subjected to two opposing concentrated or distributed forces on its outer wall, which provides ideas for solving the plane strain in a thick-walled cylindrical shell under non-axisymmetric loads both inside and outside the ring.

3.2.1. Basic Equations for the UHPC Dowel

The mechanical model of the UHPC dowel in Figure 7 is symmetric about the x-axis, defining the positive x-axis direction as the initial direction. The stress sign follows the regulations of elastic mechanics.

The distributed force perpendicular and pointing to the inner surface is q₁(θ) with the distribution range of [−β₁, β₁], and the resultant force is P₁. Similarly, the distributed force perpendicular and pointing to the outer surface is q₂(θ) with the distribution range of [−β₂, β₂], and the resultant force is P₂. Considering the force equilibrium of the dowel has:

$$P_1={\displaystyle {\int }_{-{\beta }_1}^{{\beta }_1}{q}_1(\theta )}\cos \left(\theta \right)at\mathrm{d}\theta =P_2={\displaystyle {\int }_{-{\beta }_2}^{{\beta }_2}{q}_2(\theta )}\cos \left(\theta \right)bt\mathrm{d}\theta$$

(2)

where a is the inner radius, b is the outer radius, and t is the steel plate thickness.

The solution to the plane strain elasticity problem can be derived using the Airy stress function Φ. The compatibility equation expressed in polar coordinates (r,θ) is:

$$\left({\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\theta }^2}}\right)\left({\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial {\theta }^2}}\right)=0$$

(3)

The stress boundary conditions are:

$$\left\{\begin{array}{c}\begin{array}{l}{\left.{\sigma }_r\right|}_{r=a,-{\beta }_1\le \theta \le {\beta }_1}=q_1\left(\theta \right)\\ {\left.{\sigma }_r\right|}_{r=a,-\pi \le \theta \le -{\beta }_1}={\left.{\sigma }_r\right|}_{r=a,{\beta }_1\le \theta \le \pi }=0\\ {\left.{\tau }_{r\theta }\right|}_{r=a}=0\\ {\left.{\sigma }_r\right|}_{r=b,-{\beta }_2\le \theta \le {\beta }_2}=q_2\left(\theta \right)\end{array}\\ \begin{array}{l} {\left.{\sigma }_r\right|}_{r=b,-\pi \le \theta \le -{\beta }_2}={\left.{\sigma }_r\right|}_{r=b,{\beta }_2\le \theta \le \pi }=0 \\ {\left.{\tau }_{r\theta }\right|}_{r=b}=0\end{array}\end{array}\right.$$

(4)

In Equation (4), σ_r, τ_rθ represent the radial stress and shear stress in polar coordinates of the UHPC dowel, respectively.

The functions q₁(θ) and q₂(θ) are symmetric about the x-axis and both are even functions. They can be expanded as Fourier series:

$$\left\{\begin{array}{c}{q}_1(\theta )={\displaystyle \frac{A_0}{2}}+{\displaystyle \sum _{n=1}^{\infty }{A}_n\cos \left(n\theta \right)}\\ q_2(\theta )={\displaystyle \frac{B_0}{2}}+{\displaystyle \sum _{n=1}^{\infty }{B}_n\cos \left(n\theta \right)}\end{array}\right.$$

(5)

where A₀, A_n, B₀, B_n are undetermined coefficients, and there are:

$$\left\{\begin{array}{c}{A}_i={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }{q}_1(\theta )\cos (i\theta )\mathrm{d}\theta }\\ B_i={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }{q}_2(\theta )\cos (i\theta )\mathrm{d}\theta }\end{array}\right.\hspace{1em}i=0,1,2\dots$$

(6)

Assuming that q₁(θ) and q₂(θ) are parabolic distributions, they can be expressed as:

$$\left\{\begin{array}{c}{q}_1(\theta )=q_{10}\left[1-{\left({\displaystyle \frac{\theta }{{\beta }_1}}\right)}^2\right],\hspace{1em}-{\beta }_1\le \theta \le {\beta }_1\\ q_2(\theta )=q_{20}\left[1-{\left({\displaystyle \frac{\theta }{{\beta }_2}}\right)}^2\right],\hspace{1em}-{\beta }_2\le \theta \le {\beta }_2\end{array}\right.$$

(7)

where q₁₀ and q₂₀ are the peak values of the distributed loads. From Equations (2) and (7), the relationship between p with q₁₀ and q₂₀ can be derived as follows:

$$\left\{\begin{array}{c}{p}_1={\displaystyle \frac{4q_{10}at}{{{\beta }_1}^2}}(\sin {\beta }_1-{\beta }_1\cos {\beta }_1)\\ p_2={\displaystyle \frac{4q_{20}bt}{{{\beta }_2}^2}}(\sin {\beta }_2-{\beta }_2\cos {\beta }_2)\end{array}\right.$$

(8)

The Fourier coefficients can be calculated by substituting Equation (7) into Equation (6).

3.2.2. Stress Solution

The general solution of Equation (3) is:

$$\begin{array}{ll}\hfill \mathrm{\Phi }=& a_0\ln r+b_0{r}^2+{\displaystyle \frac{a_1}{2}}r\theta \sin \theta -{\displaystyle \frac{c_1}{2}}r\theta \cos \theta \hfill \\ \hfill & +{\displaystyle \sum _{n=2}^{\infty }\left[a_n{r}^n+b_n{r}^{n+2}+a_n\prime r^{-n}+b_n\prime r^{2-n}\right]}\cos (n\theta )\hfill \\ \hfill & +{\displaystyle \sum _{n=2}^{\infty }\left[c_n{r}^n+d_n{r}^{n+2}+c_n\prime r^{-n}+d_n\prime r^{2-n}\right]}\sin (n\theta )\hfill \\ \hfill & +(b_1{r}^3+{\displaystyle \frac{a_1\prime }{r}}+b_1\prime r\ln r)\cos \theta \hfill \\ \hfill & +(d_1{r}^3+{\displaystyle \frac{c_1\prime }{r}}+d_1\prime r\ln r)\sin \theta \hfill \end{array}$$

(9)

where a₀, b₀, a₁, b₁, c₁, d₁, a₁′, b₁′, c₁′, d₁′, a_n, b_n, c_n, d_n, a_n′ and b_n′, c_n′, d_n′ are all undetermined constants. The displacement of the annulus should be single-valued. In order to satisfy this condition, the following requirements must hold:

$$b_1\prime =-{\displaystyle \frac{a_1(1-v)}{4}},\hspace{1em}{d}_1\prime =-{\displaystyle \frac{c_1(1-v)}{4}}$$

(10)

The stress components expressed using stress functions are:

$$\begin{array}{l}{\sigma }_r={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial {\theta }^2}}, {\sigma }_{\theta }={\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial r^2}} \\ {\tau }_{r\theta }=-{\displaystyle \frac{\partial }{\partial r}}({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial \theta }}), {\sigma }_z=v({\sigma }_r+{\sigma }_{\theta })\end{array}$$

(11)

where v is the Poisson’s ratio.

Substituting the stress function Equation (9) into the stress Equation (11), the stress components can be expressed by undetermined constants. By combining the Fourier series Equation (5) and the stress boundary condition Equation (4), along with the displacement single-value Equation (8) and the requirement of equal coefficients for corresponding cosine term, all undetermined constants can be expressed by the Fourier coefficients as follows:

$$\begin{gathered} a_0={\displaystyle \frac{1}{2}}{\displaystyle \frac{a^2{b}^2\left(B_0-A_0\right)}{a^2-b^2}},\hspace{1em}{b}_0={\displaystyle \frac{1}{4}}{\displaystyle \frac{A_0{a}^2-B_0{b}^2}{a^2-b^2}},\hspace{1em} \\ {a}_1=A_{1}a,\hspace{1em} \\ {b}_1={\displaystyle \frac{3A_{1}a{b}^2+A_{1}a{b}^{2}v-4B_1{b}^3+A_1{a}^3-A_1{a}^{3}v}{8\left(a^4-b^4\right)}},\hspace{1em} \\ {c}_1=0,\hspace{1em}{d}_1=0,\hspace{1em} \\ \begin{array}{ll}\hfill a_n=& -{\displaystyle \frac{ a^2{b}^2}{2}}(n{B}_n{a}^{2n}{b}^{2+n}+A_n{a}^{3n+2}-n{A}_n{a}^n{b}^{2n+2}\hfill \\ & +n{A}_n{a}^{n+2}{b}^{2n}-A_n{a}^{n+2}{b}^{2n}-n{B}_n{a}^{2n+2}{b}^n\hfill \\ & -B_n{a}^{2n}{b}^{n+2}+B_n{b}^{3n+2})/\left[-n^2{a}^{2n}{b}^{2n}{\left(a^2-b^2\right)}^2\right.\hfill \\ & \left.+a^2{b}^2{\left(a^{2n}-b^{2n}\right)}^2\right]/\left(n-1\right)\hfill \end{array} \\ \begin{array}{ll}\hfill b_n=& {\displaystyle \frac{1}{2}}(n{B}_n{a}^{ 2n}{b}^{n+4}+n{A}_n{a}^{n+4}{b}^{2n}-n{B}_n{a}^{2n+2}{b}^{n+2}\hfill \\ & -n{A}_n{a}^{n+2}{b}^{2n+2}-A_n{a}^{n+2}{b}^{2n+2}-B_n{a}^{2n+2}{b}^{n+2}\hfill \\ & +B_n{a}^2{b}^{3n+2}+A_n{a}^{3n+2}{b}^2)/\left[a^2{b}^2{\left(a^{2n}-b^{2n}\right)}^2\right.\hfill \\ & \left.-n^2{a}^{2n}{b}^{2n}{\left(a^2-b^2\right)}^2\right]/\left(n+1\right)\end{array} \\ \begin{array}{ll}\hfill a_n\prime =& -{\displaystyle \frac{ a^{n+2}{b}^{n+2}}{2}}(-B_n{a}^{3n}{b}^2+n{A}_n{a}^{2n+2}{b}^n-A_n{a}^2{b}^{3n}\hfill \\ & -n{A}_n{a}^{2n}{b}^{n+2}-n{B}_n{a}^{n+2}{b}^{2n}+n{B}_n{a}^n{b}^{2n+2}\hfill \\ & +B_n{a}^n{b}^{2n+2}+A_n{a}^{2n+2}{b}^n)/\left[a^2{b}^2{\left(a^{2n}-b^{2n}\right)}^2\right.\hfill \\ & -\left.n^2{a}^{2n}{b}^{2n}{\left(a^2-b^2\right)}^2\right]/\left(n+1\right)\hfill \end{array} \\ \begin{array}{ll}\hfill b_n\prime =& {\displaystyle \frac{ a^n{b}^n}{2(n-1)}}(-n{A}_n{a}^{2n+2}{b}^{n+2}-n{B}_n{a}^{n+2}{b}^{2n+2}\hfill \\ & +n{B}_n{a}^n{b}^{2n+4}+n{A}_n{a}^{2n+4}{b}^n+A_n{a}^{2n+2}{b}^{n+2}\hfill \\ & -B_n{a}^{3n+2}{b}^2+B_n{a}^{n+2}{b}^{2n+2}-A_n{a}^2{b}^{3n+2})\hfill \\ & /\left[a^2{b}^2{\left(a^{2n}-b^{2n}\right)}^2-n^2{a}^{2n}{b}^{2n}{\left(a^2-b^2\right)}^2\right]\hfill \end{array} \\ a_1\prime ={\displaystyle \frac{a^3{b}^2\left(3A_1{a}^2+A_1{a}^{2}v-4a{B}_{1}b+A_1{b}^2-v{A}_1{b}^2\right)}{8\left(a^4-b^4\right)}}, \\ \begin{array}{c}{b}_1\prime ={\displaystyle \frac{ A_{1}a\left(v-1\right)}{4}},\hspace{1em}{c}_1\prime =0,\hspace{1em}{d}_1\prime =0,\hspace{1em}{c}_n=0,\hspace{1em}{d}_n=0\\ c_n\prime =0,\hspace{1em}{d}_n\prime =0\end{array} \end{gathered}$$

(12)

After the undetermined constants are determined, substitute them back into Equation (9). Then, substituting the obtained stress function into Equation (11) to obtain the required stress component as follows:

$$\begin{gathered} \begin{array}{ll}\hfill {\sigma }_r=& {\displaystyle \frac{1}{r}}\{{\displaystyle \frac{a_0}{r}}+2b_{0}r+{\displaystyle \frac{a_1}{2}}\theta \sin \theta \hfill \\ & +{\displaystyle \sum _{n=2}^{\infty }[n{a}_n}{r}^{n-1}+(n+2)b_n{r}^{n+1}-n{a}_n\prime r^{-n-1}\hfill \\ & +(2-n)b_n\prime r^{1-n}]\cos (n\theta )\hfill \\ & +(3b_1{r}^2-{\displaystyle \frac{a_1\prime }{r^2}}+b_1\prime \ln r+b_1\prime )\cos \theta \}\hfill \\ & +{\displaystyle \frac{1}{r^2}}\{{\displaystyle \frac{a_1}{2}}r(2\cos \theta -\theta \sin \theta )\hfill \\ & -{\displaystyle \sum _{n=2}^{\infty }[a_n}{r}^n+b_n{r}^{n+2}+a_n\prime r^{-n}+b_n\prime r^{2-n}]n^2\cos (n\theta )\hfill \\ & -(b_1{r}^3+{\displaystyle \frac{a_1\prime }{r}}+b_1\prime r\ln r)\cos \theta \}\hfill \end{array} \\ \begin{array}{ll}\hfill {\sigma }_{\theta }=& -{\displaystyle \frac{ a_0}{r^2}}+2b_0+{\displaystyle \sum _{n=2}^{\infty }[n(n-1)a_n}{r}^{n-2}\hfill \\ & +(n+2)(n+1)b_n{r}^n+n(n+1)a_n\prime r^{-n-2}\hfill \\ & +(2-n)(1-n)b_n\prime r^{-n}]\cos (n\theta )\hfill \\ & +{(6\mathrm{b}}_{1}r+{\displaystyle \frac{2a_1\prime }{r^3}}+{\displaystyle \frac{b_1\prime }{r}})\cos \theta \hfill \end{array} \end{gathered}$$

(13)

3.2.3. Displacement Solution

Once the stress components are determined, the displacement and strain components can be obtained using the constitutive Equation (14) and geometric Equation (15) of the plane strain problem in elasticity:

$$\left\{\begin{array}{l}{\epsilon }_r={\displaystyle \frac{1-v^2}{E}}({\sigma }_r-{\displaystyle \frac{v}{1-v}}{\sigma }_{\theta }),\\ {\epsilon }_{\theta }={\displaystyle \frac{1-v^2}{E}}({\sigma }_{\theta }-{\displaystyle \frac{v}{1-v}}{\sigma }_r),\\ {\gamma }_{r\theta }={\displaystyle \frac{2(1+v)}{E}}{\tau }_{r\theta }\end{array}\right.$$

(14)

$$\left\{\begin{array}{l}{\epsilon }_r={\displaystyle \frac{\partial u_r}{\partial r}},\\ {\epsilon }_{\theta }={\displaystyle \frac{u_r}{r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_{\theta }}{\partial \theta }},\\ {\gamma }_{r\theta }={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_r}{\partial \theta }}+{\displaystyle \frac{\partial u_{\theta }}{\partial r}}-{\displaystyle \frac{u_{\theta }}{r}}\end{array}\right.$$

(15)

The displacement of an elastic body consists of two parts: “deformation-related displacement” and “deformation-independent displacement”. Deformation-related displacements (denoted as u_r1 and u_θ1) are directly determined from the strain components. Deformation-independent displacements (denoted as u_r2 and u_θ2) are directly determined from the strain components. Thus, the total displacement can be expressed as:

$$u_r=u_{r1}+u_{r2},\hspace{1em}{u}_{\theta }=u_{\theta 1}+u_{\theta 2}$$

(16)

u_r1 and u_θ1 are obtained by integrating Equation (15) with:

$$\left\{\begin{array}{l}{u}_{r1}={\displaystyle \int {\epsilon }_r\mathrm{d}r}\\ u_{\theta 1}={\displaystyle \int (r{\epsilon }_{\theta }-{\displaystyle \int {\epsilon }_r\mathrm{d}r})\mathrm{d}\theta }\end{array}\right.$$

(17)

The displacement expression for the rigid body displacement of an elastic body in a rectangular coordinate system is:

$$\left\{\begin{array}{l}{u}_x=k_x+k_{2}z-k_{3}y\\ u_y=k_y+k_{3}x-k_{1}z\\ u_z=k_z+k_{1}y-k_{2}x\end{array}\right.$$

(18)

where (k_x, k_y, k_z) represents the translation and (k₁, k₂, k₃) represents the rotation.

For plane problems in rectangular coordinates, the expression is simplified to:

$$\left\{\begin{array}{l}{u}_x=k_x-k_{3}y\\ u_y=k_y+k_{3}x\end{array}\right.$$

(19)

The rigid body displacement Equation (19) is transformed into a polar coordinate expression:

$$\left\{\begin{array}{l}{u}_{r2}=k_y\sin \theta +k_x\cos \theta \\ u_{\theta 2}=k_y\cos \theta +k_{3}r-k_x\sin \theta \end{array}\right.$$

(20)

Assuming that the radial displacement and circumferential displacement of the dowel are constrained at the polar coordinate position (a, 0) on the inner surface, and the circumferential displacement of the dowel is constrained at the polar coordinate position (a, π) on the inner surface. Then, the displacement boundary conditions are as follows:

$$\left\{\begin{array}{c}{\left.u_r\right|}_{r=a,\theta =0}=0,{\left.u_{\theta }\right|}_{r=a,\theta =0}=0\\ {\left.u_{\theta }\right|}_{r=a,\theta =\pi }=0\end{array}\right.$$

(21)

Considering the symmetry of the geometry shape, the stress boundary conditions, and the displacement boundary conditions of the UHPC dowel, it can be seen that u_r2 is an even function and u_θ2 is an odd function. Therefore, it concludes k_y = 0, k₃ = 0.

Thus, the rigid body displacement expression can be further simplified as:

$$\left\{\begin{array}{l}{u}_{r2}=k_x\cos \theta \\ u_{\theta 2}=-k_x\sin \theta \end{array}\right.$$

(22)

The displacement expression consisting of “deformation-related displacements” and “deformation-independent displacements” is ultimately given by:

$$\left\{\begin{array}{l}{u}_r=u_{r1}+u_{r2}={\displaystyle \int {\epsilon }_r\mathrm{d}r}+k_x\cos \theta \\ u_{\theta }=u_{\theta 1}+u_{\theta 2}={\displaystyle \int (r{\epsilon }_{\theta }-{\displaystyle \int {\epsilon }_r\mathrm{d}r})\mathrm{d}\theta }-k_x\sin \theta \end{array}\right.$$

(23)

where k_x can be obtained from the displacement boundary Equation (21) with u_r = 0 when r = a and θ = 0, i.e.:

$$k_x=-\int \frac{1-{\mu }^2}{E}({\sigma }_r-\frac{\mu }{1-\mu }{\sigma }_{\theta })\mathrm{d}r$$

(24)

3.3. Mechanical Analysis of Transverse Rebar

3.3.1. Basic Equations for the Transverse Rebar

The basic assumption of the Winkler foundation beam is that the intensity of the foundation reaction force at any point is directly proportional to the beam’s deflection at that point.

According to this assumption, the intensity of the foundation reaction force f(x) at any point on the beam is expressed as:

$$f\left(x\right)=-kw\left(x\right)$$

(25)

where w(x) is the beam deflection at any point (units: mm); k is the stiffness of the foundation (units: N/mm³). The f(x) represents the foundation reaction pressure (units: N/mm²).

If a beam is subjected to a uniformly distributed load q(x), based on the relationship between the distributed load and the bending moment, as well as the approximate differential equation for the deflection curve, the following can be derived:

$$EJ{\displaystyle \frac{{\mathrm{d}}^{4}w}{\mathrm{d}{x}^4}}=-q(x)-bkw$$

(26)

where EJ is the bending stiffness of the beam, b is the width of the beam, which can be taken as the diameter for a circular cross-section.

The support reaction force is defined as positive upward. This is a fourth-order linear constant coefficient equation, assuming:

$$\xi =\sqrt[4]{{\displaystyle \frac{kb}{4EJ}}}$$

(27)

Then Equation (26) can be rewritten as:

$${\displaystyle \frac{{\mathrm{d}}^{4}w}{\mathrm{d}{x}^4}}+4{\xi }^{4}w=-{\displaystyle \frac{q(x)}{EJ}}$$

(28)

When q(x) does not exceed the third power of x, the generalized solution of Equation (28) is:

$$\begin{array}{ll}\hfill w=& -e^{-\xi x}(C_1\cos (\xi x)+C_2\sin (\xi x))\hfill \\ & +e^{\xi x}(C_3\cos (\xi x)+C_4\sin (\xi x))-{\displaystyle \frac{q(x)}{k}}\hfill \end{array}$$

(29)

where C₁, C₂, C₃, and C₄ are undetermined constants.

3.3.2. Displacement and Stress Solutions

For an infinitely long beam subjected to concentrated force, with the load application point set as the coordinate origin. It can utilize structural symmetry and analyze the right half of the load application point (excluding the area around the coordinate origin), the distributed load q(x) = 0. The deflection and internal forces of the beam should tend to zero at points far away from the load application location. Thus, the coefficients C₃ and C₄ must be zero.

Due to the symmetry of the applied loads and foundation reaction forces, the tangent to the deflection curve at the load application point should be horizontal, i.e., when x = 0, there is:

$${\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}=-\xi e^{-\xi x}\left[(C_1+C_2)\sin \xi x+(C_1-C_2)\cos \xi x\right]=0$$

(30)

From Equation (30), it can be derived that C₁ = C₂.

The shear force at a point on the beam is denoted as Q. At x = 0, the shear force Q = −P/2. Substituting this condition into the expression for Q, it can be obtained that C₁ = −P/8EJξ³.

$$Q=EJ{\displaystyle \frac{{\mathrm{d}}^{3}w}{\mathrm{d}{x}^3}}=4C_{1}EJ{\xi }^3{e}^{-\xi x}\cos \xi x$$

(31)

From these, the equations for the deflection, rotation, bending moment, and shear force of the beam at any section x can be obtained as:

$$\left\{\begin{array}{l}w=-{\displaystyle \frac{P}{8EJ{\xi }^3}}{e}^{-\xi x}(\sin \xi x+\cos \xi x) \\ \theta ={\displaystyle \frac{\mathrm{d}w}{\mathrm{d}x}}={\displaystyle \frac{P}{4EJ{\xi }^2}}{e}^{-\xi x}\sin \xi x\\ M=EJ{\displaystyle \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{x}^2}}={\displaystyle \frac{P}{4\xi }}{e}^{-\xi x}(\cos \xi x-\sin \xi x)\\ Q=EJ{\displaystyle \frac{{\mathrm{d}}^{3}w}{\mathrm{d}{x}^3}}=-{\displaystyle \frac{P}{2}}{e}^{-\xi x}\cos \xi x\end{array}\right.$$

(32)

When x = 0, Equation (32) simplifies as:

$$w=-{\displaystyle \frac{P}{8EJ{\xi }^3}},\hspace{1em}\theta =0,\hspace{1em}M={\displaystyle \frac{P}{4\xi }},\hspace{1em}Q=-{\displaystyle \frac{P}{2}}$$

(33)

Then, the maximum bending stress and maximum shear stress of the transverse rebar can be calculated as follows:

$${\sigma }_{\max }={\displaystyle \frac{P}{\xi \pi a^3}},\hspace{1em}{\tau }_{\max }={\displaystyle \frac{2P}{3\pi a^2}}$$

(34)

3.4. Load–Slip Calculation Method

The relative slip between the steel plate and UHPC can be expressed by the relative displacement between the center of the steel plate hole and the center of the penetrating rebar. During the elastic stage, the perforated steel plates exhibit negligible plastic deformation (the hole remains circular). The radial compressive stress of the penetrating rebar is minimal, resulting in basically no radial compression deformation. Additionally, the compressive stress level in the UHPC outside the perforated steel plates is also very low, causing almost no displacement in the shear direction. Therefore, the relative slip between components is primarily caused by the compression deformation of the UHPC dowel along the shear direction and the deflection of the penetrating rebar.

The displacement of the UHPC dowel in the rectangular coordinate system is as follows:

$$\left\{\begin{array}{c}{u}_x=u_r\cos \theta -u_{\theta }\sin \theta \\ u_y=u_r\sin \theta +u_{\theta }\cos \theta \end{array}\right.$$

(35)

By substituting the stress-strain relationship (14) into the displacement Equation (23) of the UHPC dowel in the polar coordinate system, and then substituting the result into the displacement Equation (35) in the rectangular coordinate system, the vertical displacement of the UHPC dowel can be derived:

$$\begin{array}{ll}\hfill u_x& ={\displaystyle \int [}{\displaystyle \frac{1-v^2}{E}}({\sigma }_r-{\displaystyle \frac{v}{1-v}}{\sigma }_{\theta })]dr\cdot \cos \theta \hfill \\ & -{\displaystyle \int \{r\cdot \frac{1-v^2}{E}({\sigma }_{\theta }-}{\displaystyle \frac{v}{1-v}}{\sigma }_r)\hfill \\ & -{\displaystyle \int [\frac{1-v^2}{E}({\sigma }_r-\frac{v}{1-v}{\sigma }_{\theta })]}dr\}d\theta \cdot \sin \theta +k_x\hfill \end{array}$$

(36)

By superimposing the vertical displacement result of the UHPC dowel (Equation (35)), and the deflection of the transverse rebar directly beneath the dowel (Equation (33)). The vertical slip of the PBL shear connector under applied load is determined. That is as follows:

$$\begin{array}{ll}\hfill u=& u_x+w\hfill \\ \hfill =& {\displaystyle \int [}{\displaystyle \frac{1-v^2}{E}}({\sigma }_r-{\displaystyle \frac{v}{1-v}}{\sigma }_{\theta })]dr\cdot \cos \theta \hfill \\ & -{\displaystyle \int [r\cdot \frac{1-v^2}{E}({\sigma }_{\theta }-}{\displaystyle \frac{v}{1-v}}{\sigma }_r)\\ & -{\displaystyle \int \frac{1-v^2}{E}({\sigma }_r-\frac{v}{1-v}{\sigma }_{\theta })}dr]d\theta \cdot \sin \theta \hfill \\ & -{\displaystyle \int (\frac{1-v^2}{E}}{q}_{10})dr-{\displaystyle \frac{p}{8EJ{\xi }^3}}\hfill \end{array}$$

(37)

The above expression for load–slip is relatively cumbersome. For the convenience of calculation, the calculation steps are given in the form of a flow chart, as illustrated in Figure 9.

4. Verification of Analytical Solution

The push-out test struggles to accurately measure the stress and displacement state of the PBL shear connector, while the numerical simulation analysis demonstrates sufficient accuracy in the elastic stage. Therefore, this section combines the numerical simulation analysis results of the push-out test to verify the validity of the analytical stress and displacement solutions derived in the previous section.

4.1. Finite Element Analysis Model

The numerical model of the push-out specimen was established using ABAQUS. In the model, the stirrups were modeled using three-dimensional truss elements (T3D2), while the perforated steel plate, the transverse rebar, and the UHPC were simulated with solid elements (C3D8R). The Q355D steel plate was simulated by a tri-linear strengthened constitutive model, while the HRB335 adopts an ideal elastic-plastic model. The elastic modulus (E_s) and Poisson’s ratio (v_s) of the steel were taken as 206,000 MPa and 0.3, respectively. The damage plasticity model which considers cracking and crushing, was used to model the concrete material. The modulus of elasticity (E_c) and Poisson’s ratio (v_c) of the steel were taken as 42,319 MPa and 0.2, respectively. Other material and geometric parameters of each component were the same as those of the push-out specimen.

According to the symmetry of the push-out specimen, a 1/2 model was established for computational analysis. The interaction relationship between the steel plate and the UHPC block adopts surface-to-surface contact, with a frictionless tangential behavior and hard contact normal behavior. Similarly, surface-to-surface contacts were applied between the steel plate and the UHPC dowel, as well as between the UHPC dowel and the transverse rebar. These interfaces adopted a penalty friction tangential behavior with friction coefficients of 0.3 and hard contact normal behavior. The stirrups were embedded within the UHPC block. Symmetrical constraints were applied on the symmetry plane of the 1/2 model. The bottom face of the model adopted hinge constraints, while the top surface of the steel plate carried a vertical downward displacement load. The analysis employed the dynamic implicit method for computation. The finite element model is shown in Figure 10.

Extracting the vertical reaction force at the reference point on the top surface of the steel plate and the vertical displacement at the central node of the transverse rebar cross section to plot the load–slip curve, thereby validating the accuracy of the FEA results, Figure 11 compares the load–slip curves obtained from the push-out test and the finite element analysis (FEA), which indicates the FEA results show a load–slip curve highly consistent with the experimental results, exhibiting three distinct stages. During the elastic stage, the slip displacement remains minimal, and the load increases nearly linearly with the slip. During the plastic stage, the slip displacement begins to accelerate under increasing load. During the ductile stage, the slip displacement grows rapidly while the development of the load stabilizes with negligible variation. The FEA results demonstrate close consistency with the test in the first two stages.

Figure 12 shows the principal tensile stress in UHPC and the Von. Mises stress in the perforated steel plate and the transverse rebar at the relative slip of 0.2 mm, corresponding to a load of 235.6 kN. The results indicate that the UHPC dowel experiences compression at its upper part and tension at its lower part. The maximum principal tensile stress (9.75 MPa) is located on the lower surface of the UHPC dowel, while the corresponding maximum compressive stress (106.4 MPa) is located on its upper surface. The perforated steel plate exhibits its maximum stress (355.1 MPa) at the upper edge of the hole. Within the thickness range of the steel plate, the transverse rebar shows approximate yield stress levels at its lower part and the shear surface of the two sides.

4.2. Verification of Stress Analysis Results

A finite element model with the same dimensions and material parameters as described above was used to verify the analytical solution. We take the vertical force applied to the reference point of the perforated steel plate (finite element model) as the concentrated force P exerted on the penetrating reinforcement (analytical model). Assuming that the contact stress distribution ranges between [−π/2, π/2] for both the UHPC dowel-the steel plate and the transverse rebar, i.e., β₁ = β₂ = π/2. The surrounding UHPC is a continuous elastic material, then the support stiffness per unit length can be defined as k = CE_c, where C is the undetermined constant. Through trial calculations, the foundation modulus k = 423.19 N/mm³ and the flexibility characteristic value ξ = 0.034 mm-1 is calculated by Equation (27).

Since the load corresponding to the slip of 0.2 mm is about 235.6 kN, the UHPC dowel almost fails and exhibits nonlinear characteristics. Since the analytical solution can only consider linear elastic materials, we have analyzed the stress of the UHPC dowel under a small load (100 kN), as shown in Figure 13a. Figure 13b shows the radial stress curves of the analytical solution and numerical solution at points in the 0° and 45° directions of the UHPC dowel under the load of 100 kN (the corresponding slip is about 0.05 mm at this time), where tension stress is defined as positive. The figure shows that the radial stress decreases with the increase of radius in both 0° and 45° directions, and the rate of decrease gradually slows down. The maximum radial stress occurs at r = 10 mm in the 0° direction, where the analytical solution is 86.23 MPa, the numerical solution is 81.48 MPa, resulting in the maximum error of 5.5%. The agreement between the two solutions is satisfactory in other points.

4.3. Verification of Displacement Analysis Results

Figure 14 shows the radial displacement curves of the analytical solution and numerical solution at points in the 0° and 45° directions of the UHPC dowel under a load of 100 kN, where the radial displacement direction along the radius outward is defined as positive. The figure shows that the radial displacement increases with the increase in the radius. As the angle increases, the tendency of radial displacement to rise with increasing radius gradually diminishes. While discrepancies between the curves are relatively significant at smaller radii, the agreement improves with larger radii. The peak radial displacement occurs at the position r = 30 mm in the 0° direction. The reason behind this discrepancy can be attributed to the approximation of the contact algorithm. In the numerical model, the normal and tangential constraints at the contact interface between the inner surface of the UHPC dowel and the penetrating reinforcement fail to replicate the real mechanical behavior fully, thus inducing displacement deviations at the contact interface.

Table 1. Comparisons between numerical and analytic solutions.

	${\mathit{\sigma }}_{\mathit{r}}/\mathrm{MPa}$				${\mathit{u}}_{\mathit{r}}/\mathrm{mm}$				A/N
	Numerical Solution		Analytic Solution		Numerical Solution		Analytic Solution		${\mathit{\sigma }}_{\mathit{r}}$		${\mathit{u}}_{\mathit{r}}$
r/mm	0°	45°	0°	45°	0°	45°	0°	45°	0°	45°	0°	45°
10	−81.49	−68.40	−86.23	−68.52	−0.025	−0.019	−0.031	−0.021	1.06	1.00	1.25	1.11
12	−73.38	−55.89	−72.38	−55.01	−0.028	−0.021	−0.032	−0.022	0.99	0.98	1.16	1.04
14	−63.05	−47.21	−60.81	−46.49	−0.030	−0.023	−0.033	−0.023	0.96	0.98	1.12	1.00
16	−55.16	−40.77	−52.92	−40.53	−0.032	−0.025	−0.035	−0.024	0.96	0.99	1.09	0.97
18	−48.95	−35.71	−47.19	−36.03	−0.033	−0.026	−0.035	−0.025	0.96	1.01	1.07	0.96
20	−43.97	−31.75	−42.83	−32.50	−0.035	−0.027	−0.036	−0.026	0.97	1.02	1.05	0.94
22	−39.29	−28.78	−39.41	−29.70	−0.036	−0.028	−0.037	−0.026	1.00	1.03	1.04	0.93
24	−36.73	−26.59	−36.67	−27.51	−0.037	−0.029	−0.038	−0.027	1.00	1.03	1.03	0.92
26	−34.74	−24.91	−34.41	−25.63	−0.038	−0.030	−0.039	−0.027	0.99	1.03	1.02	0.91
28	−31.96	−23.49	−32.44	−24.10	−0.039	−0.030	−0.039	−0.028	1.01	1.03	1.01	0.91
30	−30.98	−22.88	−30.48	−23.25	−0.039	−0.031	−0.040	−0.028	0.98	1.02	1.01	0.90
Average value									0.99	1.01	1.08	0.96

Note: A/N: The ratio of the analytical solution to the numerical solution.

presents a comparison of the numerical analytic solutions at selected points in the 0° and 45° directions of the UHPC dowel under a load of 100 kN. The average ratios between the solutions were 0.99, 1.01, 1.08, and 0.96, respectively, demonstrating that the proposed analytical method can accurately calculate stress and displacement in the elastic stage of UHPC dowel.

4.4. Verification of Load–Slip Analysis Results

Figure 15 compares the load–slip curve in the elastic stage, including push-out experiment, analysis solution, and finite element numerical solution. Taking the downward displacement defined as positive, the analytical displacement equals the sum of the transverse rebar displacement and the vertical displacement of the UHPC dowel at r = 30 mm in the 0° direction. As shown in Figure 15, the numerical solution exhibits an agreement with the experimental value. When the load is below 175 kN, the analytical solution is slightly larger than the numerical solution, with the maximum discrepancy of 0.008 mm occurring at 111 kN. For the load exceeding 200 kN, the numerical solution remains consistent with experimental data, the analytical solution begins to diverge from both curves. Beyond this threshold, the displacement of the numerical solution increases dramatically, indicating the structural entry into the nonlinear stage. While the analytical solution continues to exhibit linear behavior, the analytical model is no longer applicable.

These discrepancies can be attributed to several factors: the analytical solution neglects subtle interface effects such as microscopic gaps and friction. Although the finite element model can simulate more complex contact conditions and nonlinear material responses, it still relies on idealized boundary conditions. The experiments may involve loading imperfections, which contribute to the deviations among the analytical, numerical, and experimental results. However, the number of test samples in this study was relatively small, so additional tests and further assessments on the rigor of the proposed model are still required in the future.

5. Conclusions

This study establishes a mechanical model in the elastic phase based on the mechanical behavior observed in the push-out test of the PBL shear connector in UHPC. The model derives the analytical solutions for stresses and displacements in the UHPC dowel and transverse rebar, as well as a calculation method for load–slip curves. The reliability of the proposed solutions for stress, displacement, and load–slip results is verified by finite element simulation analysis and experimental results from model tests. The main conclusions are as follows:

(1)

The mechanical behavior of PBL shear connectors in UHPC can be divided into elastic, plastic, and ductile stages. During the elastic stage, the load is jointly carried by the UHPC dowel and transverse rebar. Based on their respective mechanical characteristics in this stage, the system can be idealized as a plane strain problem for the UHPC dowel and the Winkler elastic foundation beam model for the transverse rebar.

(2)

The UHPC dowel can be simplified as a plane strain problem of a thick-walled cylindrical shell subjected to non-axisymmetric loads on both inside and outside surfaces. The stress and displacement analytical solutions of the UHPC dowel are derived by analyzing the Airy stress function expressed in polar coordinates. For the Winkler elastic foundation beam problem of transverse rebar, establish analytical solutions for its deflection, rotation, bending moment, shear force, and the corresponding stress calculation formulas. The relative slip associated with the shear force can be obtained by superimposing the elastic displacement of the UHPC dowel with the rigid body displacement of the transverse rebar.

(3)

The reliability of the proposed stress, displacement, and load–slip curve calculation method is verified by comparison with the finite element analysis and push-out test results of PBL shear keys in UHPC in the elastic stage. However, these conclusions are based on a limited sample size, and further validation through more experiments is necessary. Additionally, the presented analytical method is primarily applicable to the elastic stage, within which it provides a reliable and computationally efficient means of determining the initial shear stiffness.