*3.4. Formulas*

The characteristic bond load and the average characteristic bond strength of specimens are shown in Table 5. The formulas for the average bond stress of the four factors were established by statistical regression analysis. They can be expressed as follows.

$$\overline{\tau}\_{s} = (\frac{-0.686C\_{sv}}{d} + \frac{0.020L\_{\varepsilon}}{d} + 2.506\rho\_{sv} + 0.067)f\_{l} \tag{15}$$

$$\overline{\tau}\_{\text{\tiny\text{\tiny\text{\tiny}}}} = (\frac{-0.335\text{C}\_{\text{sv}}}{d} + \frac{0.015\text{L}\_{\text{\tiny\text{\tiny\text{\tiny}}}}}{d} + 0.718\rho\_{\text{sv}} + 0.683)f\_t \tag{16}$$

$$\overline{\pi}\_{r} = (\frac{-0.493 \text{C}\_{\text{ss}}}{d} + \frac{0.006 \text{L}\_{\text{e}}}{d} - 0.842 \rho\_{\text{sv}} + 0.590) f\_{\text{l}} \tag{17}$$

where τ*s* is average initial bond strength; τ*u* is average ultimate bond strength; τ*r* is average residual bond strength.

In order to verify the reliability of the formulas, the comparison was performed between the calculation of the formulas and the experiment data from this test, as well as using data from Yin et al. [35], Chen et al. [24], and Chen et al. [36]. The results are shown in Table 6.

Table 6 indicates that the average ultimate bond strength and the average residual bond strength fit well, but the fitting result of the average initial bond strength has a certain error. One of the reasons for this error is the different values of initial load between the man-made and instrument methods. In addition, Equation (2) by Xiao et al. [34] for the tensile strength of RAC was used in this study, but the rest of the articles adopted ordinary concrete formulas. The tensile strength of RAC under the same compressive strength is higher than in this paper.


SRRC-7 0.848 0.972 0.872 1.382 1.457 0.949 0.900 0.847 1.062 SRRC-8 1.065 1.140 0.934 1.444 1.472 0.981 0.973 1.097 0.887 SRRC-9 0.731 0.602 1.213 1.442 1.448 0.996 0.853 0.755 1.129 Chen et al. SRRAC-11 1.438 1.670 0.861 1.719 1.963 0.876 1.258 1.404 0.896 Chen et al. SRAC-5 1.318 0.930 1.417 1.821 1.510 1.206 1.238 0.990 1.251 SRRC-34 0.860 0.501 1.717 1.213 0.905 1.340 0.801 0.412 1.944 SRRC-35 0.807 0.349 2.311 1.125 1.139 0.988 0.798 0.629 1.269 SRRC-36 0.379 0.492 0.770 1.000 1.038 0.964 0.573 0.637 0.899

Yin et al.

## **4. Numerical Simulation**

The simulation of interfacial bond stress is a di fficult point in the simulated process of bond slip between section steel and RAC. Nonlinear spring units were utilized to solve the problem in this study, which included two aspects: one was the preparation of the nonlinear syntax for the inp file, and the other was the determination of the constitutive relationship of the spring element.

#### *4.1. Finite Element Model*

#### 4.1.1. Element and Material

The solid element C3D8R for section steel and RAC was used in this study, which is an 8-node hexahedron reduction integral element. This element has more accurate results and saves calculation time, and is also suitable for meshing refinement. The linear three-dimensional truss element T3D2 was adopted for steel and stirrups, which has two nodes, each with three degrees of freedom. The steel and stirrups were assigned as truss elements when meshing.

The experimental materials adopted in this study were described as shown in Table 7. The properties of the steel used in the tests were determined in accordance with the "Code for Design of Concrete Structures" [37]. The elastic properties of the second-class coarse aggregate for RAC are shown in Table 8. The plastic damage model of Abaqus was selected for the plastic part of the RAC.


**Table 7.** Properties of steel.


Properties of concrete.

**Table 8.**

#### 4.1.2. Analysis Step and Constraint

The initial incremental step was 0.02, the minimum incremental step was the default 0.00001, and the maximum incremental step was 1000. These were set to meet the calculation requirements in this simulation. In the interaction module, the embedding relationship was defined between the RAC and reinforcement cage, which was made of stirrups and steel. The reference point above the surface of the section steel acting on the displacement load was provided. The loading speed was 0.3 mm/min, and the loading frequency was set as the amplitude. The binding constraints between the bottom slab and the RAC were defined, and the boundary conditions at the bottom of the slab were set as fixed. The specimen adding constraint is shown in Figure 9.

**Figure 9.** Specimen containing section steel and RAC.
