Study on Load–Slip Curve of a PBL Shear Key at a Steel–Concrete Composite Joint

Featured Application

The key link of the steel–concrete composite truss structure, a new type of beam bridge structure, is the joint, and the core area of the joint is mainly transferred via the PBL shear key group. This composite structure puts forward higher requirements for the PBL shear key. Since the calculation results of PBL shear keys in composite joints have not been obtained at this stage, the purpose of this study is to (i) more conveniently and clearly grasp the working mechanism and stress law of PBL shear keys in the core area of new composite structures, such as steel–concrete composite trusses, and (ii) solve the lack of strong theoretical support for the theoretical formula of the load–slip relationship in the entire loading process of a single PBL shear key.

Abstract

The steel–concrete composite truss adopts a new type of steel-concrete composite joint with high rigidity and load-carrying capacity. In order to more conveniently and clearly grasp the working mechanism of Perfobond Leiste (PBL) shear keys in the core area of new composite structures such as steel–concrete composite trusses, the lack of strong theoretical support for the theoretical formula of load–slip relationships in the entire loading process of single PBL shear keys is solved. By proposing a straight–curved–straight three-stage simplified load–slip curve with respect to the PBL shear key, the stress process of the PBL shear key is divided into three stages—the elastic stage, plastic stage, and strengthening stage—based on the compressive yield and failure critical point of tenon concrete in the shear key. With reference to the calculation method of the bearing capacity of the order pile under horizontal loads and by calculating the shear stiffness of the shear key, a theoretical formula suitable for separating the load–slip relationship of a single PBL shear key in the entire loading process of the ear plate composite joint is proposed. The results show that, in the elastic section, the slope of the curve is related to the concrete reaction coefficient and the material parameters of the penetrating steel bar; moreover, in the strengthened section, the coefficient is related to the shear modulus of the penetrating steel bar, and a more uniform length distribution of the penetrating steel bar between the two joint plates will improve the initial stiffness of the PBL shear key to a certain extent. The results of the proposed method are in good agreement with the finite element results and experimental values. This research study’s results can provide a convenient design method for the design of the internal PBL shear keys of new composite structure joints, promoting the promotion and application of new composite structures and advancing the development of the engineering field.

1. Introduction

The emergence and development of composite structures have led to the development of bridge engineering and rail transit. Steel–concrete composite truss structures, a new type of beam bridge structure, have emerged at a historic moment. A new connection method, the steel–concrete hybrid joint, has been adopted between the steel truss and concrete components, and it has the following advantages: high stiffness, high load-bearing capacity, and good comprehensive benefits. The connection between the steel web and concrete bridge deck or chord in steel–concrete composite truss is called a joint, and it has the function of transferring loads and coordinating deformations, which are key links in the design of these structures. The position of the core area of the joint mainly includes two parts: concrete chord and steel joint plate, both of which transfer loads via the PBL shear key group [1,2,3,4,5,6]. The steel–concrete composite truss joint is primarily subjected to shear forces, which are mainly borne by shear connectors. Among the shear connectors, the PBL shear key is a major form of shear connection that can improve the overall performance of the joint. Unlike traditional PBL keys, the PBL key in the steel–concrete composite truss joint has changed boundary support conditions due to the constraints from both steel plates. Therefore, studying the shear bond mechanism of PBL shear keys in steel–concrete composite truss joints can facilitate an improved understanding of the internal stress performance of joints. Many scholars have carried out model tests for different types of composite joints in order to study the mechanical behavior and ultimate bearing capacity of joint specimens. However, due to the dense structural arrangement and complex stress state in the core area of joints, measuring the load of PBL shear keys during tests is difficult, so scholars have researched the force transfer mechanism and load–slip law of PBL shear keys.

Zhang Qinghua et al. [7,8] studied the force transfer mechanism and ultimate bearing capacity of PBL shear connectors based on the load–slip deformation coordination theoretical model and its solution method. The study showed that the essence of the working mechanism of the PBL shear key group is the result of the coordinated deformation of steel, concrete members, and various PBL keys. The study of the PBL shear key group needs a strong basis and method for the study of a single PBL shear key. Xia Song et al. [9] observed that environmental stress, material strength, plate opening, the penetrating steel bar’s size, and other factors influence the ultimate bearing capacity of PBL keys, which can be fitted by linear equations to determine the ultimate bearing capacity of PBL shear keys. Wang Zhenhai [10] studied the breaking characteristics and influencing factors of PBL shear keys in each working stage and proposed the load–slip relation formula of PBL shear keys in the entire process under external load using a normalized treatment coefficient. Xiao Lin et al. [11] used finite element analyses and observed that the failure process of PBL shear keys can be divided into three stages: linear, elastic, and plastic. Via the study of the load–displacement relationship of the PBL shear key, it was concluded that the stress mechanism is as follows: The initial concrete expands outward, the PBL shear key slips with the increase in load, and the concrete slips outward with the shear key. Zhan Haowei [12] designed the static failure tests of 13 groups of 45 specimens based on combinations of parameters such as the diameter of the steel plate’s opening, the penetrating steel bar’s diameter, the steel plate’s thickness, concrete strength, and the number of opening rows. The load–slip relation formulas of single-row shear connectors and double-row shear connectors are fitted.

Based on the calculation method of the PBL shear key’s bearing capacity proposed by predecessors, it was observed that most calculation formulas are only suitable for push-out specimens with unilateral fixed-end bracings and unidirectional loading, some of which are only based on the fitting results of a large number of test data, and the load–slip calculation formula obtained is only a summary of experience and has no specific theoretical support. Some theoretical calculation formulas are too complex, and the prediction of relevant data requires many complex calculations, even requiring calculations using the help of software. The volume of required engineering operations is substantial, more time-consuming, and laborious. There are currently few experimental studies on the PBL shear key inside the composite joint; thus, it is necessary to seek a simple and reliable calculation theory.

In order to provide stronger theoretical support for the theoretical calculation formula of the load–slip relationship of a single PBL shear key, using existing research results, this paper mainly draws lessons from the calculation method of the pile-bearing capacity issued by horizontal load and combines the study of separated ear plate joints in composite truss bridge joints. A theoretical formula suitable for separating the load–slip relationship of a single PBL shear key in the entire loading process of ear plate composite joints is proposed. Because a simplified straight–curved–straight-type load–slip curve is proposed, the critical point between compression yields and the failure of tenon concrete in the shear key is taken as the dividing point of each straight and curved section, which makes the calculation process more convenient; moreover, this provides a convenient design method for the design of PBL shear keys in new composite structure joints.

2. Stress Mechanism of PBL Shear Key

When the composite structure is subjected to external forces, the displacement difference occurs at the junction of the joint slab and concrete, and the PBL shear key will bear this part of the displacement difference and realize the load transfer. Different from headed shear studs, the main load-bearing component is the headed shear studs itself. The PBL shear key is composed of tenon concrete and a penetrating steel bar. When studying the mechanical performance of the PBL shear key, it cannot be analyzed according to the shear connector of a single material, and the interaction between the penetrating steel bar and concrete should be comprehensively considered. With reference to the failure process in previous studies for push-out tests [7] and taking the compressive yield or failure of concrete as the boundary, with an increase in load, the PBL shear key will experience three stages: elastic stage (I), plastic stage (II), and strengthening stage (III). In the elastic stage, the shear stiffness of the shear key is at its highest, and both tenon concrete and the penetrating steel bar in the PBL shear key experience the elastic stage. In the plastic stage, the plastic zone of the concrete at the opening position of the joint plate appears and expands. In the strengthening stage, the concrete loses its bearing capacity and withdraws from the structure, and the steel bar is sheared, yielded, and finally destroyed. These processes are shown in Figure 1.

3. Formula for the Load–Slip Curve of the PBL Shear Key

3.1. Basic Form of the Load–Slip Curve

Although the deformation of the PBL shear key in the joint is similar to that of the PBL shear key in the push-out test, as shown in Figure 2, the boundary support conditions of the joint specimen are different from those of the push-out specimen. The penetrating steel bar of the joint specimen passes through two steel plates, and this results in a state of dual-end constraint for the steel bar between the two plates; moreover, the steel bar on the outside of the two plates experiences fixation at one end and is free on the other end. Additionally, the mechanical performance of the shear key section between the two nodal plates and the shear section at both ends of the joint is different. However, there is no intermediate section with both ends restrained in the push-out specimen. Based on the above considerations, the calculation formula of the push-out test is unsuitable for direct applications relative to the nodal plate.

It was observed that the shape of the load–slip curve of a single shear key in the joint is similar to that obtained by the push-out test. That is, the load–slip curve mainly comprises the elastic–plastic stage with a large slope, the plastic stage near the inflection point of the curve, and the gentle strengthening section of the slope [7]. Without significantly increasing the difficulty of calculations and analyses and ensuring a certain calculation accuracy, we simplify the load–slip curve of PBL shear connectors based on the initial curve. The compressive yield and failure critical point of tenon concrete in the shear key are taken as the dividing points, and two straight lines with different slopes (OA and BC) and a curve (AB) corresponding to the three stress stages of PBL shear keys can be used to replace the curves obtained by tests or finite element analyses, as shown in Figure 3. According to the different mechanical properties of the shear key in the joint, it is divided into two different segments: the two free segments on the outside of the two steel plates and the middle segment sandwiched between the two steel plates—depicted in Figure 4. Because of their different boundary conditions, they are studied.

3.2. Analytical Formula of Load–Slip Curve

When the PBL shear key is loaded, the through steel bar will not only have shear deformations under the shear force but will also be subjected to the reaction force of concrete, which is similar to that of a single pile under horizontal loads. Therefore, the calculation method of the horizontal bearing capacity of a single pile [13,14] is introduced to explore the calculation method of the shear stiffness of shear keys.

3.2.1. Elastic Stage

In the elastic state, the steel bar is equivalent to the elastic foundation beam, and the surrounding concrete is equivalent to the elastic foundation. In order to not significantly increase the workload and difficulty of calculations and analyses, the following assumptions are introduced: (1) The horizontal shear force of the PBL shear key acts directly on the coordinates of the origin rather than the interface position between the joint plate and concrete. (2) The joint plate’s orifice can well restrain the concrete in the hole and the penetrating steel bar, and the interface between the penetrating steel bar and the joint plate does not rotate during the entire process of the PBL shear key’s stress (rotation angle Φ_x=0 ≡ 0). (3) Concrete is a continuous elastic body, and concrete can be regarded as the elastic foundation for penetrating steel bars. (4) Bonding actions between steel and concrete and all vertical actions are not considered. (5) The penetrating steel bar is in accordance with the plane section assumption in the process of deformations.

The calculation diagram of the free segment is shown in Figure 5, where $\overline{p}$ is the unit length reaction force of the steel bar at any point x, and it is a function of deflection y of the steel bar. $\overline{q}\left(x\right)$ indicates the reverse action of steel bars per unit length. Taking element $dx$, as shown in Figure 5, and combining the mechanical analysis of the material, the flexural differential equation for the steel bar is obtained:

$$EI\frac{d^{4}y}{d{x}^4}=-\overline{p}\left(x,y\right)+\overline{q}\left(x\right)$$

(1)

$$p\left(x,y\right)=\frac{\overline{p}\left(x,y\right)}{d_{\mathrm{PBL}}}$$

(2)

Among them, $E$ denotes the elastic modulus (N/mm²), and $I$ denotes the moment of inertia of the cross section of the steel bar (mm⁴).

Considering that the characteristic of concrete is different from that of soil, the back load, $\overline{q}\left(x\right)$, in Formula (1) is 0. With regard to the calculation method of concrete reaction $p\left(x,y\right)$, referring to the foundation reaction’s method of consideration, the elastic foundation reaction method is adopted. By using the bending theory of the beam, the following formula is obtained:

$$p\left(x,y\right)=K\left(x\right)y^n$$

(3)

where $K\left(x\right)$ denotes any function of depth $x$, and $n$ denotes the index of deflection, 0 < n ≤ 1. Because of the characteristics of concrete, $K\left(x\right)$ is constant in the formula, and both the steel bar and concrete are in a stage of linear elasticity; thus, the index is n = 1 in Formula (3).

$$p\left(x,y\right)=ky$$

(4)

Among them, we have the concrete reaction coefficient (N/mm³). In order to calculate the value of k, concrete is regarded as an elastic body, and the formula of concrete reaction coefficient k is obtained with reference to Minderlin [15] and Glick’s [16] solution:

$$k=\frac{22.25E_c\left(1-{\nu }_c\right)}{\left(1+{\nu }_c\right)\left(3-4{\nu }_c\right)\left(4.6{\log }_{10}\frac{2l}{d_{\mathrm{PBL}}}-0.44\right)d_{\mathrm{PBL}}}$$

(5)

In the formula, ${\nu }_c$ denotes the concrete’s Poisson’s ratio; $l$ denotes the calculated length of the free section of the steel bar (mm). By introducing Equation (4) into Equation (1), a fourth-order differential equation with a constant coefficient can be obtained:

$$EI\frac{d^{4}y}{d{x}^4}+d_{\mathrm{PBL}}ky=0$$

(6)

The general solution can be easily obtained as follows:

$$y=e^{\beta x}\left(C_1\cos \beta x+C_2\sin \beta x\right)+e^{-\beta x}\left(C_3\cos \beta x+C_4\sin \beta x\right)$$

(7)

$$\beta =\sqrt[4]{\frac{d_{\mathrm{PBL}}k}{4EI}}$$

(8)

In the formula, $C_1$, $C_2$, $C_3$, and $C_4$ denote integral constants, and $\beta$ denotes the eigenvalue of penetrating steel bars (mm⁻¹). The special point of the force acting on the joint penetrating steel bar, the assumption in this paper, and the boundary conditions can be determined as follows:

$$\{\begin{array}{l}{\phi }_{x=0}=0\\ S_{x=0}=F_s\\ M_{x=l}=0\\ S_{x=l}=0\end{array}$$

(9)

In the formula, $\phi$ denotes the through steel bar section’s corner (rad); $S$ denotes the through steel bar section’s shear force (N); $M$ denotes the through steel bar section’s moment of inertia (N·mm); $F_s$ denotes the horizontal shear force (N). The integral constants can be obtained by combining (7)–(9):

$$\{\begin{array}{l}{C}_1=Cd\left(e^{-2\beta l}+\cos 2\beta l-\sin 2\beta l+2\right)\\ C_2=Cd\left(\cos 2\beta l+\sin 2\beta l-e^{-2\beta l}\right)\\ C_3=Cd\left(\cos 2\beta l+\sin 2\beta l+e^{2\beta l}+2\right)\\ C_4=Cd\left(e^{2\beta l}+\sin 2\beta l-\cos 2\beta l\right)\end{array}$$

(10)

where $Cd=\frac{F_s}{8EI{\beta }^3\left(\sin 2\beta l+\sinh 2\beta l\right)}$.

Because the penetrating steel bar and concrete work together and the deformation is coordinated, the slip of the penetrating steel bar can represent the overall slip of the PBL shear key. Therefore, for a single shear key, the maximum slip is the opening position of the joint plate: that is, the position of $x=0$; thus, the shear stiffness of the free section of the PBL shear key is as follows:

$$K_z=\frac{F_s}{y_{\max }}=\frac{F_s}{y_{x=0}}=\frac{4EI{\beta }^3\left(\sin 2\beta l+\sinh 2\beta l\right)}{\cos 2\beta l+\cosh 2\beta l+2}$$

(11)

The calculation process of the middle section is similar to that of the free section, and the calculation diagram is shown in Figure 6. For the meso-level element, the difference in macroscopic boundary conditions will not make the force type of the element different; thus, under the elastic condition, the flexure-depth differential equation obtained is also consistent with Equation (6). The boundary conditions are changed as follows:

$$\{\begin{array}{l}{\phi }_{x=0}=0\\ S_{x=0}=F_{ms}\\ M_{x=l_m}=0\\ S_{x=l_m}=F_{ms}\end{array}$$

(12)

The integral constants can be obtained by combining three expressions: (7), (8) and (12):

$$\{\begin{array}{l}{C}_{m1}=-C{d}_m\left(e^{\beta l_m}\cos \beta l_m+e^{\beta l_m}\sin \beta l_m+1\right)\\ C_{m2}=C{d}_m\left(e^{\beta l_m}\cos \beta l_m-e^{\beta l_m}\sin \beta l_m+1\right)\\ C_{m3}=C{d}_m\cdot e^{\beta l_m}\left(e^{\beta l_m}+\cos \beta l_m-\sin \beta l_m\right)\\ C_{m4}=C{d}_m\cdot e^{\beta l_m}\left(e^{\beta l_m}+\cos \beta l_m+\sin \beta l_m\right)\end{array}$$

(13)

Among them, $C{d}_m=\frac{F_{sm}}{4EI{\beta }^3\left(e^{2\beta l_m}+2e^{\beta l_m}\cos \beta l_m+1\right)}$; $C_{m1}$, $C_{m2}$, $C_{m3}$, and $C_{m4}$ denote integral constants of the middle section; $l_m$ denotes the calculated length of the middle section of the through the steel bar (mm). In the same manner, the shear stiffness of the PBL shear key in the middle section is as follows:

$$K_m=\frac{F_{sm}}{y_{m\max }}=\frac{F_{sm}}{y_{mx=0}}=\frac{4EI{\beta }^3\left(\cos \beta l_m+\cosh \beta l_m\right)}{\sin \beta l_m-\sinh \beta l_m}$$

(14)

The shear stiffness of the elastic stage of a single PBL shear key in the elastic stage joint is as follows:

$$K_{\mathrm{PBL}}=2K_z+2K_m=\frac{8EI{\beta }^3\left(\sin 2\beta l+\sinh 2\beta l\right)}{\cos 2\beta l+\cosh 2\beta l+2}+\frac{8EI{\beta }^3\left(\cos \beta l_m+\cosh \beta l_m\right)}{\sin \beta l_m-\sinh \beta l_m}$$

It was observed that the shear stiffness is only related to the displacement of the bracing end under the action of the unit shear force. Moreover, considering the displacement continuity of the penetrating steel bar, for the section of the penetrating steel bar that belongs to two sections at the center of the opening of the joint plate, the displacement of the bracing end of the same section is equal; that is, the displacement of the free section of the PBL shear key is equal to that of the middle section at the opening position of the joint plate. Because the penetrating steel bar is all embedded in concrete and combined with the concept of a semi-infinite long pile in foundation engineering and its judgment standard, when the length of the section meets $l_r$ ≥ π/β, the penetrating steel bar in the section is a semi-infinite steel bar.

Combining (8) and $I=\pi d_{\mathrm{PBL}}{}^4/64$, the following was observed:

$$\frac{{\pi }^{5}E}{k{d}_{\mathrm{PBL}}}\le {\left(\frac{l_j}{d_{\mathrm{PBL}}}\right)}^4$$

(15)

In the formula, $l_j$ calculates the length of the through steel bar, with a value of $l$ or $l_m$ (mm). Equation (5) is substituted into (15):

$$\frac{{\pi }^{5}E\left(1+{\nu }_c\right)\left(3-4{\nu }_c\right)\left(4.6{\log }_{10}\frac{2l_j}{d_{\mathrm{PBL}}}-0.44\right)}{22.25E_c\left(1-{\nu }_c\right)}\le {\left(\frac{l_j}{d_{\mathrm{PBL}}}\right)}^4$$

In practical bridge engineering, there is little difference in the elastic modulus corresponding to several types of concrete strength, and the critical aspect ratio of the semi-infinite length steel bar is about 6. Generally speaking, the aspect ratio of the PBL shear key with a reasonable size with respect to the steel bar is much greater than 6; thus, the steel bar in the shear key is semi-infinitely long, and its maximum displacement is always in a support position; that is, the opening position of the nodal plate.

For the PBL shear key in the joint specimen, the position and magnitude of the maximum displacement of the free section and the middle section are the same, and the shear stiffness of the shear key is only related to the maximum displacement of the steel bar; thus, in the follow-up analysis, the shear stiffness of the entire single PBL shear key is four times that of the free section.

3.2.2. Plastic Stage

Considering the continuity of the curve, the endpoint of the elastic stage (the OA section in Figure 3) is the starting point of the plastic stage (the AB section in Figure 3). The calculation formula of the elastic stage of PBL shear key is used to study the starting point of the plastic stage. The calculation diagram is shown in Figure 7a.

Because the internal deformation of the PBL shear key is coordinated, the displacement of the steel bar in the opening position is equal to the deformation of concrete ($u_0$); thus, for this position, there is $y_{x=0}=u_0$. With reference to the calculation method of the plastic deformation of soil with respect to the composite foundation problem [17,18], within the unit length of the shear key in the free section, the width of concrete subjected to compressive stress is larger than the diameter of penetrating steel bar, and the width can be taken as $\kappa \cdot d_{\mathrm{PBL}}$.

$$u_0=\kappa {\epsilon }_0{d}_{\mathrm{PBL}}$$

(16)

In the formula, κ denotes the equivalent width coefficient of the compressive stress of concrete, which is 2.5 here. The horizontal shear force $F_{s_1}$ at the opening position of the joint plate in the PBL shear key is calculated by the combination of Formulas (16) and (7).

$$F_{s_1}=\frac{10{\epsilon }_0{d}_{\mathrm{PBL}}EI{\beta }^3\left(\sin 2\beta l+\sinh 2\beta l\right)}{\cos 2\beta l+\cosh 2\beta l+2}$$

(17)

In the load–slip curve of the PBL shear key, point ($u_0$, $F_{s_1}$) is the end point of the elastic stage and the starting point of the plastic stage. Because the stress condition of the plastic stage is very complex, if the result of the elastic stage method is too large, in order to simplify the calculation and facilitate the application, the power function in the shape of y = ax^b + c is selected as the curve formula of the plastic stage. With reference to the fitting coefficient of the plastic stage of the PBL key [12], the values are as follows:

$$\{\begin{array}{l}a=384.536-16.528t+56.903f_t\left(D^2-d^2\right)-2.165f_{yk}{d}^2\\ b=0.345-0.002t-0.006f_t\left(D^2-d^2\right)+0.001f_{yk}{d}^2\end{array}$$

(18)

Among them, t denotes the thickness of the joint plate; $f_t$ denotes the standard value of the shear strength of tenon concrete; $f_{yk}$ denotes the standard value of tensile strength of the penetrating steel bar; D and d denote the diameter of the opening and the diameter of penetrating steel bar, respectively. The end point of the elastic section is regarded as the starting point of the plastic section, and the c value can be determined according to reference point ($u_0$, $F_{s_1}$); thus, the load–displacement curve formula of the plastic section can be obtained.

As shown in Figure 7b, for the plastic endpoint of the PBL shear key in the free section, the concrete at the position of $x=0$ reaches the ultimate compressive strain. Due to the coordination of the deformation within the PBL shear key, the deflection of the steel bar through the opening position is equal to the deformation of the concrete; thus, we have the following.

$$u_{cu}=2.5{\epsilon }_{cu}{d}_{\mathrm{PBL}}$$

(19)

In the formula, $u_{cu}$ denotes the ultimate deformation of concrete (mm). The combined formulas of (18) and (19) then solves shear force $F_{s_2}$:

$$F_{s_2}=a{u}_{cu}{}^b+c$$

(20)

Point ($u_{cu}$, $F_{s_2}$) obtained by the solution is the plastic end point coordinate of a single PBL shear key on its load–slip curve.

3.2.3. Strengthening Stage

When the PBL shear key is in the strengthening section, the concrete at the opening position of the joint plate reaches the ultimate compressive strain, and it is destroyed and collapses; only the penetrating steel bar slips and bears the shear force.

The process of the shear and final yield of the steel bar is accompanied by the extension of the concrete failure area along its length direction; thus, the interaction between them is more complex at this stage.

In order to not significantly increase the workload of calculations and analyses, it is assumed that the PBL shear key carries all horizontal loads via the steel bar in the entire reinforced section regardless of the bearing capacity of concrete in gradual failure processes, and the calculation diagram shown in Figure 8 is adopted. When the steel bar is separately sheared and yielded, according to the four-strength theory of material mechanics, the following is obtained:

$$\left[{\tau }_J\right]=0.58f_y$$

(21)

In the formula, $f_y$ denotes the steel joint plate’s tensile strength design value (N/mm²); $\left[{\tau }_J\right]$ denotes the steel shear strength design value (N/mm²). The shear force acting on the penetrating steel bar at this time (kN) is as follows:

$$\Delta F_{s3}=A_g\left[{\tau }_J\right]$$

(22)

where $A_g$ denotes the area of the through steel bar’s reinforcement (mm²), which, in this case, is $\pi d_{\mathrm{PBL}}{}^2/4$. Then, the slip of the penetrating steel bar at the opening position of the joint plate is

$$\Delta u_{03}=\frac{k_s\Delta F_{s3}l}{G{A}_g}$$

(23)

$$G=\frac{E}{2\left(1+\nu \right)}$$

(24)

In the formula, $f_y$ denotes the steel joint plate’s tensile strength design value (N/mm²); $\left[{\tau }_J\right]$ denotes steel the shear strength design value (N/mm²); $k_s$ denotes the shear stress distribution’s non-uniform correction coefficient, with a circular section value of 10/9; $\nu$ denotes the Poisson’s ratio of the penetrating steel bar; $G$ denotes the shear modulus of the penetrating steel bar (N/mm²). On the load curve of the shear key of the free segment, point $\left(u_{cu}+\Delta u_{03},F_{s_2}+\Delta F_{s_3}\right)$ is the end point of the strengthening stage. For the PBL shear key’s free segment of the strengthened section, the shear stiffness is as follows:

$$K_{zs}=\frac{\Delta F_{s3}}{\Delta u_{03}}$$

(25)

The overall shear stiffness of a single reinforced PBL shear key section is as follows:

$$K_{sq}=4K_{zq}$$

(26)

In summary, the slip behavior of PBL shear keys is mainly related to the diameter ($d_{\mathrm{PBL}}$) and calculated length ($l_j$) of the through steel bar, as well as the diameter ($D$) and thickness ($t$) of the opening in the node plate. The calculation formula for the load ($F$)–slip ($u$) curve of the PBL key inside the joint of the steel–concrete composite truss is as follows:

$$F=\{\begin{array}{ll}2(K_z+K_m)\cdot u,u\le u_0& \mathrm{Elastic} \mathrm{stage}\\ a{u}^b+c,u_0<u<u_{cu}& \mathrm{Plastic} \mathrm{stage}\\ K_{sq}u,u\ge u_{cu}& \mathrm{Strengthing} \mathrm{stage}\end{array}$$

4. Formula Verification

An ABAQUS finite element model was established based on the PBL shear key design parameters from reference [19], as shown in Figure 9. Concrete was simulated using solid elements with a plastic damage model composed of relationships. The steel plates and penetrating steel bars were also simulated using solid elements, and a bi-linear constitutive model was used for both. The connection relationship between each component is set as follows: The penetrating steel bars and steel plates were embedded in concrete, and the circular holes in the steel plate and the penetrating steel bars were bound by constraints. The comparison between the finite element results and the calculated results is shown in Figure 10, and it is easy to see that the calculated results agree well with experimental values.

The design parameters of PBL shear keys in reference [9] are selected to verify the formula. The experimental values of specimens SB60-22-1 and SB45-12-3 in reference [9] are compared with the calculated results in Figure 11a,b, and their correlation is shown in

Table 1. The correlation between the calculated results and finite element results and experimental values.

Finite Element/Specimen	Related Coefficient (r)
Finite element [19]	0.9983
Specimen SB60-22-1 [9]	0.9411
Specimen SB45-12-3 [9]	0.9612
Specimen PBC-75 [20]	0.9907
Specimen PBC-200 [20]	0.9356

. It can be observed that the results of the formula are in good agreement with the results of the finite element calculation and test. In Figure 10, the experimental values in the elastic stage are slightly higher than the calculated results due to the assumption that the bond between steel and concrete is not considered in the calculation formula. In the actual experiment, the bond effect exists, which results in improved load-carrying capacity in the elastic stage compared to the calculated values. However, the error is small and within an acceptable range.

The design parameters of the PBL shear key in reference [20] are selected to verify the formula. With the exception of different spacings between the two joints, the other parameters of the two specimens—PBC-75 and 200—are the same, and the relevant parameters are included in the formula. The comparison between the test values and the calculated results is shown in Figure 12, and the correlation is shown in Table 1. It can be observed that the test values of PBC-75 and PBC-200 are in good agreement with the calculation results of the formula. The experimental values in the strengthening stage are slightly lower than the calculated values because the calculation formula only considered the main horizontal force for convenience, while the actual failure process is subject to more complex forces, but the error is not significant. According to the results of the formula, it can be observed that the slope of the curve of the two specimens in the elastic section is slightly different, which increases slightly with the increase in plate spacing, but it does not change much; moreover, the load–slip curve after the elastic section has no obvious differences. The difference between the two specimens lies in a single variable, that is, the lengths of the free section of the steel bar, which are 190 mm and 130 mm. This shows that, within a reasonable range, the length distribution of the penetrating steel bar between the two joints is more uniform, which will improve the initial stiffness of the PBL key. The diameter of the penetrating steel bar is 20 mm, and the aspect ratios of the two specimens are all greater than 6. Because the two types of aspect ratios are all within the range of the semi-infinite pile’s length, the influence is not significant.

5. Conclusions

By conducting the analysis of the PBL shear key inside the joint of a steel–concrete composite truss and combining the calculation of the bearing capacity of a single pile, the three stress stages of the load–slip curve of the PBL shear key were studied, and the following conclusions were obtained:

• A simplified straight–curved–straight-type load–slip curve was proposed for the PBL shear key, and it can be easily solved. Based on the compressive yielding and failure critical point of tenon concrete inside the shear key, the curve was divided into three parts: elastic stage, plastic stage, and strengthening stage.
• The theoretical calculation formulas of the load–slip relationship of the PBL shear key in three stages were derived by referring to the calculation method of the bearing capacity of a single pile under horizontal loads. The formulas can be directly used in engineering practice without complex numerical calculations and analyses.
• Comparisons between the calculated values of the proposed load–slip theoretical formula and the finite element calculation results and test values show good agreement, indicating that the formula is highly accurate. In practical engineering, attention should be focused on the detailed design of the steel–concrete composite structure joint—especially the influence of the interface’s bond on structural performance—in order to further improve the performance of the structure.