1. Introduction
Ultra-high-performance concrete (UHPC) is a new material with ultra-high strength, toughness, and durability, and is considered one of the most sustainable and promising building materials in the future [
1]. Its design theory is the maximum packing density theory. The particles form the closest packing with different particle sizes in the best proportion [
2]. Compared with ordinary cement-based materials [
3,
4], UHPC exhibits better compressive performance [
5], tensile performance [
6,
7], shear performance [
8], seismic performance [
9,
10], and impact resistance [
11,
12]. The incorporation of fiber greatly influences the improvement of the overall strength [
13,
14], and UHPC also shows good durability due to its low water–binder ratio, micro-crack effect, and self-healing effect [
15,
16,
17]. In addition, the excellent bond performance between steel bar and UHPC can optimize the details of steel bar in reinforced concrete members. Therefore, UHPC has broad application prospects in bridge and culvert tunnels, marine structures, explosion-proof projects, long-span structures, and super-high-rise buildings [
1,
18,
19].
The connection modes of steel bar mainly include welding connection, mechanical connection [
20], and lap-splice connection. Compared with welding and mechanical connections, lap-splice connections are steel bar’s most straightforward connections [
21]. The high bond performance between steel bar and UHPC can significantly reduce the lap-splice length of steel bar, which makes UHPC widely used in the connection area of prefabricated components and reduces the onsite wet operation and construction complexity [
22].
The bond performance between steel bar and UHPC directly determines the mechanical performance of the reinforced concrete structure. Many scholars have studied the bond performance between steel bars and UHPC, mainly focusing on the direct pull-out anchorage test [
23,
24,
25,
26,
27,
28]. When the anchorage specimen is directly pulled out, the steel bar is in tension and the surrounding UHPC is under compression, which is inconsistent with the actual structural stress state. The measured bond strength between the steel bar and UHPC is too high, making it difficult to guide practical engineering. On this basis, some scholars studied the direct tensile lap-splice specimens [
29,
30,
31]. This method overcomes the defect that the surrounding UHPC is compressed during the pull-out process of the steel bar, but it is still different from the actual stress state of the structure.
In addition, some scholars have carried out experimental research on lap-spliced UHPC beams. This kind of test can simultaneously obtain specimens’ mechanical properties and bond properties, and the stress state is more in line with the engineering practice. Dagenais et al. [
32,
33] found that UHPC could eliminate splitting failure through monotonic and cyclic loading tests. At the same time, the bond performance between steel bar and UHPC was more than twice that between steel bar and ordinary concrete. Kim et al. [
34] revised the cover thickness of the current UHPC structural design guidelines based on the test results of the lap-spliced beam. Al-Quraishi et al. [
35] concluded that concrete splitting failure was the primary failure mode of lap-spliced UHPC beams through experimental analysis.
There are few reports on the finite element simulation of lap-spliced UHPC beams considering the bond–slip. Li et al. [
36] established a finite element model of steel-skeleton UHPC beam considering the bond–slip for numerical simulation. They selected the appropriate constitutive relationship and damage factor calculation method. The numerical simulation results were in good agreement with the experimental results. Karimipour et al. [
37] carried out the finite element simulation of lap-spliced reinforced concrete beams considering the bond–slip. The load–deflection curve, characteristic load, and failure mode obtained by finite element simulation were in good agreement with the test. Although these numerical simulations consider the bond–slip, they do not quantitatively analyze the bond performance. In order to further understand the mechanical properties and bond properties of lap-spliced UHPC beams, three-dimensional finite element models made in ABAQUS were calibrated with experimental data. The bond–slip between steel bar and UHPC was considered, and the finite element simulation of lap-spliced UHPC beams was carried out. The load–deflection curve, peak load, bond strength, and failure mode obtained from the simulation were compared with the experimental results. Finally, the model was used for parameter analysis to study the influence of various parameters on the mechanical properties and bond properties of lap-spliced UHPC beams.
2. Test Overview
This paper used the corresponding test data [
38] to verify the correctness of the simulation results. The test data included the load–deflection curve, peak load, bond strength, and failure mode. In the test, the section sizes of all beam specimens were the same, with the beam length of 1500 mm, the beam width of 200 mm, and the beam height of 300 mm. The test beam was loaded by a four-point loading device. The loading scheme is shown in
Figure 1. HRB400 steel bars were used for the top steel bars and stirrups of the test beam, and HRB600 steel bars were used for the bottom steel bars of the beam. The geometric dimensions and reinforcement details of specimens are shown in
Figure 2.
The experimental parameters included UHPC age, steel fiber content, cover thickness, lap-splice length, and lap-splice segment stirrup ratio. UA, UB, UC, and UD represent the UHPC age of 3 days, 7 days, 14 days, and 28 days, respectively. S1, S2, S3, and S4 represent the steel fiber content of 1%, 2%, 3%, and 4%, respectively. C1, C1.5, and C2 represent the cover thickness of 16 mm, 24 mm, and 32 mm, respectively. L6, L8, L10, L12, and L15 represent the lap-splice length of 96 mm, 128 mm, 160 mm, 192 mm, and 240 mm, respectively. T0, T2, T3, and T4 represent the lap-splice segment stirrup ratio of 0%, 0.011%, 0.022%, and 0.035%, respectively. Other parameters of the specimen are shown in
Table 1.
3. Finite Element Model of Lap-Spliced UHPC Beam
This study used ABAQUS finite element software to model lap-spliced UHPC beams. The established finite element models have the same size and detailed structure as the test specimens. These models mainly considered the following parameters: UHPC age, steel fiber content, cover thickness, lap-splice length, and lap-splice segment stirrup ratio. The detailed establishment process of the finite element model is introduced in the following sections.
3.1. Material Constitutive Model
3.1.1. UHPC Constitutive Model
ABAQUS provides three concrete constitutive models: concrete damaged plasticity model, concrete smeared cracking model, and concrete brittle cracking model [
39]. The concrete brittle cracking model is generally used for plain concrete or structures with less reinforcement, such as dam engineering. The concrete damaged plasticity model and the concrete smeared cracking model can reasonably simulate concrete structures with normal reinforcement. However, compared with the concrete smeared cracking model, the concrete damaged plasticity model is easy to converge and can significantly improve the calculation efficiency. Therefore, this paper adopted the concrete damaged plasticity model to simulate the material properties of UHPC.
This model defines the compressive strain as negative and the tensile strain as positive. The compression performance of UHPC was simulated by the model proposed by Shan [
40], as shown in
Figure 3a. The compression constitutive relation is expressed as follows:
where
σc is the compressive stress of UHPC;
fc is the axial compressive strength of UHPC prism, which is 0.75
fcu [
41];
εc is the compressive strain of UHPC;
εc0 is the peak compressive strain of UHPC;
Ec is the elastic modulus of UHPC;
A is the parameter of ascending segment, namely, the ratio of initial elastic modulus to peak secant modulus;
B is the descending segment parameter.
The tensile performance of UHPC was simulated by the model proposed by Yang [
42], as shown in
Figure 3b. The expression of tensile constitutive relation is as follows:
where
σt is the tensile stress of UHPC;
ft is the tensile strength of UHPC dog-bone specimen;
εt is the tensile strain of UHPC;
εt0 is the peak tensile strain of UHPC;
a and
b are set to 1.106 and 0.6, respectively;
β is 1.7.
The plastic damage factor was introduced to describe the stiffness degradation of UHPC caused by cracks. At the same time, the monotonic increase of the damage factor should be ensured when calculating the damage factor. The calculation equation of the damage factor [
43] is as follows:
where
k = t,c represents tension and compression, respectively;
dk is the damage factor.
In the structural model, the density and Poisson’s ratio of UHPC were set as 2500 kg/m
3 and 0.2, respectively. The uniaxial constitutive data are calculated by Equations 1–7 and used as constitutive input data of CDP model to form the yield surface and post-yield flow direction in the three-dimensional stress space. The CDP model adopts the yield criterion proposed by Lubliner et al. [
44] and modified by Lee and Fenves [
45]. The shape of the yield surface is mainly determined by the ratio of initial biaxial compressive strength to initial uniaxial compressive strength
fb0/fc0 and the invariant stress ratio
K.
Figure 4a,b show the deviatoric plane yield surface and the plane stress yield surface, respectively. The parameters in the UHPC plastic damage model [
46] are listed in
Table 2.
3.1.2. Steel Bar Constitutive Model
In the finite element model, the linear strengthening constitutive model was used to simulate the steel bar, as shown in
Figure 5. The density of the steel bar is 7800 kg/m
3, the elastic modulus is 200 GPa, and the Poisson’s ratio is 0.3. The material properties of the steel bar are shown in
Table 3.
The following equation calculates the stress–strain relationship of steel bar:
where
σs is the steel bar stress;
εs is the steel bar strain;
Es is the steel bar elastic modulus;
fy is the steel bar yield stress;
εy is the steel bar yield strain;
fu is the steel bar ultimate stress;
εu is the steel bar ultimate strain.
3.2. Bond–Slip Constitutive Model
Scholars worldwide have carried out a lot of research on the bond performance between steel bars and UHPC. According to the test results, the empirical equations were established by regression analyses. The bond strength equations between steel bars and UHPC are listed in
Table 4. The mean (
μ), standard deviation (
σ), and coefficient of variation (
cv) of the ratio between the empirical formula values and the test values of bond strength were compared, as shown in
Figure 6.
Through the above comparison results, the empirical equation of bond strength in reference [
30] is selected:
The bond–slip curve selected in this paper is shown in
Figure 7, and the bond–slip equation [
27] is as follows:
where
τu is the bond strength;
su is the slip value corresponding to bond strength;
p is the ascending segment parameter;
q is the descending segment parameter.
3.3. Boundary Conditions
The simply-supported boundary condition was adopted in the test of the lap-spliced UHPC beam. Therefore, one end constrained the displacement in the X, Y, and Z directions, and the other constrained the displacement in the Y and Z directions. The steel plates were set at the loading position and the support to avoid the stress concentration at the loading position and the support. The displacement-controlled loading method was adopted to avoid the convergence problem. The boundary conditions are shown in
Figure 8. In order to ensure that the steel plate did not deform greatly during the loading process, the elastic modulus of the steel plate was taken as 2000 GPa, and Poisson’s ratio was taken as 0.3.
3.4. Interactions and Constraint Conditions
In this paper, the steel bar and concrete are modeled separately. In general, there are two ways to establish the constraint relationship between steel bar and concrete. One is that the steel bar is embedded in the concrete without considering the bond–slip between the steel bar and concrete. The other uses spring elements to simulate the bond–slip between the steel bar and concrete. The latter can reflect the actual behavior between the steel bar and concrete. Because the relative slip between the steel bar and UHPC was more likely to occur in the lap-splice segment of this model, the bond–slip between the lap-splice segment steel bar and UHPC beam was considered. The non-lap segment steel bar and stirrup were embedded in the UHPC beam. In addition, coupling constraints were used between the reference point and the steel plate. Tie constraints were used between the steel plate and the UHPC beam. The interactions in the model are shown in
Figure 9.
In this study, a steel bar node was used to establish a spring constraint relationship with nine surrounding concrete nodes, as shown in
Figure 10. In the model, the spring direction was three-dimensional, the spring stiffness in the Y and Z directions was infinite, and the stiffness in the X direction was calculated according to the selected bond–slip constitutive relationship.
3.5. Mesh Sensitivity Analysis
In the lap-spliced UHPC beam model, both the UHPC beam and the steel plate adopted the three-dimensional eight-node reduced integral element (C3D8R), which is more accurate in solving the displacement. The element had little influence on the analysis accuracy when the mesh was distorted. The steel bar and stirrup in the beam adopted the three-dimensional two-node truss element (T3D2).
The division of element mesh size significantly impacts the model’s calculation results and efficiency. The mesh sensitivity analysis is shown in
Table 5, and the calculated results are shown in
Figure 11.
According to the mesh sensitivity analysis results, Mesh 2 type was selected. The mesh division results are shown in
Figure 12.