Numerical Simulation of Lap-Spliced Ultra-High-Performance Concrete Beam Based on Bond–Slip

Abstract

In this paper, 3D finite element simulations were conducted for lap-spliced ultra-high-performance concrete (UHPC) beams using ABAQUS software. Based on the concrete damaged plasticity (CDP) model, the plastic damage factor was introduced to simulate the material properties of UHPC. The nonlinear characteristics of the steel bar and UHPC were considered, and the bond–slip constitutive relationship was selected to evaluate the bond–slip between the lap-spliced steel bar and UHPC. The simulated load–deflection curve, peak load, bond strength, and failure mode were in good agreement with the experimental results. The verified finite element model was used to analyze the parameters of the lap-spliced UHPC beam. The effects of lap-spliced steel bar diameter, stirrup spacing of non-lap segment, and shear span ratio on the mechanical properties and bond properties of the lap-spliced UHPC beam were studied. This study can provide a reference for the future simulation and design of lap-spliced UHPC beams.

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. Details of test specimens.

Test Variable	Specimen Abbreviation	fcu (MPa)	ft (MPa)	d (mm)	Vf (%)	c (mm)	ls (mm)	ρst (%)
UA-S2C1-L10-T0	UA	98.34	6.84	16	2	16	160	0
UB-S2C1-L10-T0	UB	109.64	7.31	16	2	16	160	0
UC-S2C1-L10-T0	UC	116.10	8.23	16	2	16	160	0
UD-S2C1-L10-T0	UD	139.82	9.46	16	2	16	160	0
UD-S1C1-L10-T0	S1	125.18	8.46	16	1	16	160	0
UD-S3C1-L10-T0	S3	144.97	9.76	16	3	16	160	0
UD-S4C1-L10-T0	S4	147.98	10.11	16	4	16	160	0
UD-S2C1.5-L10-T0	C1.5	140.06	9.69	16	2	24	160	0
UD-S2C2-L10-T0	C2	138.61	9.15	16	2	32	160	0
UD-S2C1-L6-T0	L6	136.12	9.42	16	2	16	96	0
UD-S2C1-L8-T0	L8	140.42	9.12	16	2	16	128	0
UD-S2C1-L12-T0	L12	138.91	9.44	16	2	16	192	0
UD-S2C1-L15-T0	L15	137.54	9.07	16	2	16	240	0
UD-S2C1-L10-T2	T2	137.78	9.19	16	2	16	160	0.011
UD-S2C1-L10-T3	T3	141.15	9.52	16	2	16	160	0.022
UD-S2C1-L10-T4	T4	138.87	9.39	16	2	16	160	0.035

Note: f_cu is the compressive strength of UHPC cube; f_t is the tensile strength of UHPC dog-bone specimen; d is the steel bar diameter; V_f is the fiber volume content; c is the cover thickness; l_s is the lap-splice length of steel bar; ρ_st is the lap-splice segment stirrup ratio.

.

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:

$$y_c=\{\begin{array}{l}A{x}_c+(6-5A)x_c{}^5+(4A-5)x_c{}^6 0\le x_c\le 1\\ \frac{x_c}{B{(x_c-1)}^2+x_c} x_c\ge 1\end{array}$$

(1)

$$y_c=\frac{{\sigma }_c}{f_c}$$

(2)

$$x_c=\frac{{\epsilon }_c}{{\epsilon }_{c0}}$$

(3)

$$A=\frac{E_c{\epsilon }_{c0}}{f_c}$$

(4)

where σ_c is the compressive stress of UHPC; f_c is the axial compressive strength of UHPC prism, which is 0.75f_cu [41]; ε_c is the compressive strain of UHPC; ε_c0 is the peak compressive strain of UHPC; E_c 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:

$$y_t=\{\begin{array}{l}a{x}_t+(3-2a)x_t{}^2+(a-2)x_t{}^3 0\le x_t\le 1\\ \frac{x_t}{b{(x_t-1)}^{\beta }+x_t} x_t\ge 1\end{array}$$

(5)

$$y_t=\frac{{\sigma }_t}{f_t}$$

(6)

$$x_t=\frac{{\epsilon }_t}{{\epsilon }_{t0}}$$

(7)

where σ_t is the tensile stress of UHPC; f_t is the tensile strength of UHPC dog-bone specimen; ε_t is the tensile strain of UHPC; ε_t0 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:

$$d_k=1-\sqrt{\frac{{\sigma }_k}{E_c{\epsilon }_k}} ,(k=t,c)$$

(8)

where k = t,c represents tension and compression, respectively; d_k is the damage factor.

In the structural model, the density and Poisson’s ratio of UHPC were set as 2500 kg/m³ 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 f_b0/f_c0 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. Plastic damage parameters of UHPC.

φ	e0	fb0/fc0	K	μ
30°	0.1	1.16	0.6667	0.0001

Note: φ is the dilation angle of concrete; e₀ is the eccentricity of flow potential function; f_b0/f_c0 is the ratio of initial biaxial compressive strength to initial uniaxial compressive strength; K is the invariant stress ratio; μ is viscosity coefficient.

.

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/m³, 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. Material properties of steel bar.

Steel Bar Grade	Steel Bar Diameter (mm)	Yield Strength (MPa)	Ultimate Strength (MPa)
HRB400	6	483	617
	10	478	613
HRB600	16	680	861

.

The following equation calculates the stress–strain relationship of steel bar:

$${\sigma }_s=\{\begin{array}{l}{E}_s{\epsilon }_s 0\le {\epsilon }_s\le {\epsilon }_y\\ f_y+\frac{f_u-f_y}{{\epsilon }_u-{\epsilon }_y}({\epsilon }_s-{\epsilon }_y) {\epsilon }_y\le {\epsilon }_s\le {\epsilon }_u \end{array}$$

(9)

where σ_s is the steel bar stress; ε_s is the steel bar strain; E_s is the steel bar elastic modulus; f_y is the steel bar yield stress; ε_y is the steel bar yield strain; f_u is the steel bar ultimate stress; ε_u 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. Equations of bond strength between steel bar and UHPC.

Author	Equation
Alkaysi [24]	${\tau }_u=1.1\sqrt{{f_c}^{\prime }}$
Roy [47]	${\tau }_u=(0.45c/d+38.5/l_s+0.23V_f)f_t$
Sturm [48]	${\tau }_u=(0.0018c+0.186){f_c}^{\prime }$
Marchand [23]	${\tau }_u=0.875(c/d)\sqrt{{f_c}^{\prime }}$
Ma [49]	${\tau }_u=(0.36+2.02d/l_s)(0.86+0.57c/d)(0.83+2\sqrt{{\rho }_{sv}}+11.65V_f)\sqrt{{f_c}^{\prime }}$
Fang [30]	${\tau }_u=\alpha (1+0.55d/l_s)(1+0.22c/d+0.22{\rho }_s{}_v)\sqrt{{f_c}^{\prime }}$

Note: f_c’ is the compressive strength of UHPC; f_t is the tensile strength of UHPC; c is the cover thickness; d is the steel bar diameter; l_s is the anchorage (lap) length of steel bar; V_f is fiber volume content; λ_f is the fiber characteristic parameter; ρ_sv is the stirrup ratio; α is the lap coefficient, which is 1 for non-contact lap and 0.9 for contact lap.

. The mean (μ), standard deviation (σ), and coefficient of variation (c_v) 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:

$${\tau }_u=\alpha (1+0.55d/l_s)(1+0.22c/d+0.22{\rho }_v)\sqrt{{f_c}^{\prime }}$$

(10)

The bond–slip curve selected in this paper is shown in Figure 7, and the bond–slip equation [27] is as follows:

$$\tau =\{\begin{array}{l}{\tau }_u{(\frac{s}{s_u})}^p s\le s_u\\ {\tau }_u{e}^{-q(s/s_u-1)} s\ge s_u\end{array}$$

(11)

where τ_u is the bond strength; s_u 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. Mesh size analysis.

Mesh Type	Mesh 1	Mesh 2	Mesh 3
Mesh size of lap segment (mm)	10 × 18 × 18	10 × 15 × 15	10 × 12 × 12
Mesh size of non-lap segment (mm)	18 × 18 × 18	15 × 15 × 15	12 × 12 × 12
Total number of solid elements	21,302	30,356	52,032
Total number of truss elements	1074	1294	1612
Computation time (s)	3108	3624	10,403
Peak load (kN)	418.43	416.34	422.30
Displacement at the peak load (mm)	3.56	3.58	3.68
Bond strength (MPa)	13.42	13.69	13.65

, 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.

4. Finite Element Model Validation

The load–deflection curve, peak load, bond strength, and failure mode of lap-spliced UHPC beam obtained by finite element simulation were compared with the test results to verify the correctness of the established finite element model.

4.1. Load–Deflection Curve

The comparison of the load–deflection curves of the lap-spliced UHPC beam obtained from the test and the simulation is shown in Figure 13. The test results of Bae et al. [50] were also used further to prove the universality of the finite element model, as shown in Figure 14. The initial stiffness, peak load, and descending part of the simulation curves agree well with the test curves.

4.2. Peak Load

The peak load of the lap-spliced UHPC beam was extracted from the finite element simulation results and compared with the test results, as shown in

Table 6. Summary of peak load and bond strength.

Specimen	Fu (kN)			τu (MPa)
	FEA	EXP	Δ (%)	FEA	EXP	Δ (%)
UA-S2C1-L10-T0	357.14	389	8.19	10.44	10.62	1.69
UB-S2C1-L10-T0	381.90	402	5.00	11.44	11.74	2.56
UC-S2C1-L10-T0	407.11	405	0.52	12.38	12.80	3.28
UD-S2C1-L10-T0	416.34	410	1.55	13.69	14.18	3.46
UD-S1C1-L10-T0	272.60	245	11.27	13.15	11.75	11.91
UD-S3C1-L10-T0	449.24	441	1.87	13.92	15.65	11.05
UD-S4C1-L10-T0	464.13	451	2.91	14.01	14.30	2.03
UD-S2C1.5-L10-T0	408.66	407	0.41	14.72	15.78	6.72
UD-S2C2-L10-T0	425.89	402	5.94	15.60	17.00	8.24
UD-S2C1-L6-T0	333.10	305	9.21	13.76	15.36	10.42
UD-S2C1-L8-T0	349.32	345	1.25	13.25	13.28	0.23
UD-S2C1-L12-T0	428.47	430	0.36	13.21	13.62	3.01
UD-S2C1-L15-T0	486.29	463	5.03	11.81	11.33	4.24
UD-S2C1-L10-T2	429.68	412	4.29	14.02	16.62	15.64
UD-S2C1-L10-T3	431.01	416	3.61	14.50	16.85	13.95
UD-S2C1-L10-T4	435.18	425	2.40	14.97	16.50	9.27
120-5db-V1.5	219.78	205	7.21	25.38	26.50	4.23
120-10db-V1.5	276.09	270	2.26	12.65	13.25	4.53
180-10db-V1	243.51	242	0.62	12.83	13.25	3.17
180-10db-V2	323.36	319	1.37	12.94	13.25	2.34

Note: F_u is the peak load; τ_u is the bond strength; Δ = |FEA-EXP|/EXP × 100%.

. Except for the 11.27% difference between the simulated value and the test value of the specimen UD-S1C1-L10-T0, the difference between the simulated value and the test value of the other specimens was within 10%. Figure 15 shows the comparison of simulated peak load and test peak load. The mean (μ), standard deviation (σ), and coefficient of variation (c_v) of the ratio of simulated peak load to test peak load were 1.024, 0.043, and 0.042, respectively. The simulated values are in good agreement with the test values.

4.3. Bond Strength

The stress of the steel bar at the end of the lap-splice segment was further extracted from the finite element simulation results to verify the correctness of the established finite element model. The bond strength between the lap-splice segment steel bar and UHPC was calculated by Equation (12), and the bond strength was compared with that obtained from the test. The results are shown in Table 6. The difference between the simulated and test bond strength is within the acceptable range. Figure 16 shows the comparison between simulated bond strength and test bond strength. The mean (μ), standard deviation (σ), and coefficient of variation (c_v) of the ratio of simulated bond strength to test bond strength were 0.955, 0.060, and 0.063, respectively. The simulated bond strength is less than the test bond strength, indicating that the bond strength obtained by finite element simulation is relatively conservative.

$${\tau }_{u,FEM}=\frac{f_s{A}_s}{\pi d{l}_s}=\frac{f_{s}d}{4l_s}$$

(12)

where τ_u,FEM is the simulated bond strength; f_s is the steel bar stress at the end of the lap-splice segment, which is obtained from the finite element model; A_s is the cross-section area of the steel bar; d is the steel bar diameter; l_s is the lap-splice length of the steel bar.

4.4. Failure Mode

Taking the specimen UD-S2C1-L10-T0 as an example, Figure 17 shows the typical damage mode of the lap-spliced UHPC beam obtained by finite element simulation and the final failure mode of the test. The test results show that an apparent main crack appears when the lap-spliced UHPC beam fails, the other cracks are thin and short, and the distribution is dense. These phenomena are close to the finite element simulation results. In the finite element simulation results, the damage on the right side of the lap-spliced UHPC beam is more serious, and the main crack is formed. The height of the main crack is consistent with the height of the main crack in the test, as shown in Figure 17. The damage value in other regions is small. The primary failure mode of the lap-spliced UHPC beam obtained by simulation is in good agreement with the experimental phenomena.

5. Parameter Analysis

The simulated load–deflection curve, peak load, bond strength, and failure mode are in good agreement with the test. Through these comparisons, the applicability of the finite element model established in this paper is verified. The established finite element model is used for parameter analysis to study the influence of main variables on the performance of lap-spliced UHPC beams. The specimen UD-S2C1-L10-T0 was selected as the control group. The research parameters included the lap-spliced steel bar diameter, the stirrup spacing of non-lap segment, and the shear span ratio.

5.1. Lap-Spliced Steel Bar Diameter

When other conditions remain unchanged, the selected lap-spliced steel bar diameter is 14 mm, 16 mm, 18 mm, and 20 mm. Figure 18 shows the influence of the lap-spliced steel bar diameter change on the load–deflection curve, peak load, and bond strength of the lap-spliced UHPC beam. It can be seen from Figure 18a that when the lap-spliced steel bar diameter increases from 14 mm to 20 mm, the initial stiffness and peak load of the lap-spliced UHPC beam gradually increase at the curve rising stage. The descending part of the curves are the same, and the lap-spliced steel bar diameter does not affect the trend of the descending segment.

Figure 18b shows the influence of the lap-spliced steel bar diameter on the peak load. The peak load increases linearly with the increase of the lap-spliced steel bar diameter. When the lap-spliced steel bar diameter increases from 14 mm to 16 mm, 18 mm, and 20 mm, the peak load increases by 6.0%, 13.6%, and 18.7%, respectively.

Figure 18c shows the influence of the lap-spliced steel bar diameter on the bond strength. The bond strength shows a linearly decreasing trend with the increased lap-spliced steel bar diameter. When the lap-spliced steel bar diameter increases from 14 mm to 16 mm, 18 mm, and 20 mm, the bond strength decreases by 3.8%, 6.1%, and 7.8%, respectively.

5.2. Stirrup Spacing of Non-Lap Segment

This section analyzes the influence of the stirrup spacing in the non-lap segment on the performance of the lap-spliced UHPC beam. The selected non-lap segment stirrup spacing is 40 mm, 80 mm, 120 mm, 160 mm, and 200 mm. Figure 19 shows the influence of non-lap segment stirrup spacing on the load–deflection curve, peak load, and bond strength. It can be found from Figure 19a that with the increase in the non-lap segment stirrup spacing (the stirrup ratio decreases), the specimens’ initial stiffness, peak load, and ductility remain unchanged.

Figure 19b shows the influence of the stirrup spacing of the non-lap segment on the peak load. With the increase of the stirrup spacing in the non-lap segment, the variation law of the peak load is insignificant. Compared with lap-spliced UHPC beam specimen with the stirrup spacing of 40 mm, when the stirrup spacing increases to 80 mm, the peak load decreases slightly (within 0.5%), and when the stirrup spacing increases to 120 mm, 160 mm, and 200 mm, the peak load increases to varying degrees (within 2.5%). Therefore, the rise in the stirrup spacing in the non-lap segment (the decrease of stirrup ratio) does not lead to reducing the bearing capacity of the lap-spliced UHPC beam. In the design of the lap-spliced UHPC beam or UHPC member, the stirrup ratio can be reduced to avoid steel bar congestion.

Figure 19c shows the influence of the stirrup spacing of the non-lap segment on the bond strength. With the increased stirrup spacing in the non-lap segment, the bond strength between the lap-spliced steel bar and UHPC remains unchanged.

5.3. Shear Span Ratio

When other conditions remain unchanged, the shear span ratio is selected as 1.268, 1.812, 2.355, 2.899, and 3.442, and the corresponding beam lengths are 1200 mm, 1500 mm, 1800 mm, 2100 mm, and 2400 mm, respectively, as shown in Figure 20. To analyze the influence of the shear span ratio on the performance of lap-spliced UHPC beams, Figure 21 shows the influence of the shear span ratio on the load–deflection curve, initial stiffness, peak load, and bond strength. It can be found from Figure 21a that with the increase of shear span ratio, the initial stiffness and peak load gradually decrease, the deflection corresponding to the peak load increases significantly, and the downward trend of the curve becomes slow.

Figure 21b shows the influence of the shear span ratio on the initial stiffness, and Figure 21c shows the influence of the shear span ratio on the peak load. With the increase of the shear span ratio, the initial stiffness and peak load show a hyperbolic downward trend. When the shear span ratio increases from 1.268 to 1.812, 2.355, 2.899, and 3.442, the initial stiffness decreases by 44.2%, 65.9%, 79.5%, and 86.6%, respectively, and the peak load decreases by 29.9%, 46.0%, 55.4%, and 63.1%, respectively.

Figure 21d shows the influence of the shear span ratio on the bond strength. With the increase of the shear span ratio, the bond strength remains unchanged, indicating that the shear span ratio does not affect the bond performance between the lap-spliced steel bar and UHPC.

6. Conclusions

In this paper, the three-dimensional finite element models of lap-spliced UHPC beams made in ABAQUS were calibrated with experimental data. Through the finite element model validation and parameter analysis, the following main conclusions are drawn:

(1)

The load–deflection curve, peak load, bond strength, and failure mode obtained by simulation agree well with the test results. The mean (μ), standard deviation (σ), and coefficient of variation (c_v) of the ratio of simulated peak load to test peak load are 1.024, 0.043, and 0.042, respectively. The mean (μ), standard deviation (σ), and coefficient of variation (c_v) of the ratio of simulated bond strength to test bond strength are 0.955, 0.060, and 0.063, respectively.

(2)

The parametric analysis results show that with the lap-spliced steel bar diameter increase, the bearing capacity of the lap-spliced UHPC beam increases linearly, and the bond strength decreases linearly.

(3)

The stirrup spacing of the non-lap segment has no apparent effect on the bearing capacity and bond strength of the lap-spliced UHPC beam. Increasing the stirrup spacing of the non-lap segment will not cause a decline in structural performance.

(4)

With the increase of the shear span ratio, the initial stiffness and bearing capacity of the lap-spliced UHPC beam load–deflection curve show a hyperbolic decrease, and the bond strength remains unchanged.