4.1.3. Meshing

The mesh generation of the specimen was performed after the assembly of the components and the constraint settings were completed, which a ffected the establishment of subsequent nonlinear springs. The intersection interface for the spring element arrangemen<sup>t</sup> is most important, and it is located between the section steel and the RAC. The mesh for the interface between the section steel and the RAC was divided in order to successfully arrange the subsequent spring elements. The section steel was cut according to the geometrical axis, as shown in Figure 10, and the section steel and RAC parts were set as independent.

**Figure 10.** Mesh division for specimen containing section steel and RAC.

The node set of the interface for the section steel and the RAC was established after the mesh was divided, as shown in Figure 11. All nodes in the node set were exported in post-processing and were numbered using the "VLOOKUP" function in excel software.

**Figure 11.** Node set: (**a**) section steel; (**b**) RAC.

Each specimen in this simulation has 1140 nodes. The VLOOKUP function of a corresponding node is

=VLOOKUP(C4&D4&E4,CHOOSE({1,2,3,4},\$G\$4:\$G\$1143&\$H\$4:\$H\$1143&\$I\$4:\$I\$1143,\$F\$4: \$F\$1143),2,0).

SRRC-1 is used as an example in Table 9.


**Table 9.** Relationship of node for SRRC-1.

Note: Table cells are expressed in absolute form; C4, D4, and E4 are the columns where the x, y, and z coordinates of Section 1 are located. G4: G1143 represents the query columns, which correspond to the C4 column of Section 1; the rest can be deduced by analogy. F4: F1143 is the column for the output formula.

#### 4.1.4. Plastic Damage Model

The premise for studying the plastic damage model for RAC is reflected in the constitutive relationship. The constitutive relationship between the compression and tension in RAC is basically the same as that of ordinary concrete. The specific di fferences are reflected in several coe fficients related to the replacement rate. The equation proposed by Xiao et al. [38] was used for calculation in this study. The stress and strain at di fferent strength levels are shown in Table 10.


**Table 10.** Characteristic parameters of RAC.

*Appl. Sci.* **2020**, *10*, 887

The plastic damage model of concrete is mainly used to provide a universal analysis model for analyzing the structure of concrete under cyclic and dynamic loads. This model is based on plastic and isotropic failure assumptions, and it can be used in unidirectional loading, cyclic loading, and other functions [39].

The evolution of the yield or failure surface is controlled by ε*inc* and ε*int* , where ε*inc* represents a compressive inelastic strain and ε*int*represents a tensile inelastic strain.

$$
\overline{\varepsilon}\_{\varepsilon}^{\rm in} = \varepsilon\_{\varepsilon} - \varepsilon\_{0\varepsilon}^{\rm el} \tag{18}
$$

$$
\varepsilon\_{0\varepsilon}^{cl} = \sigma\_{\varepsilon} / E\_0 \tag{19}
$$

Let the plastic strain in the inelastic strain be ε*pl c* , then the proportion of the inelastic strain is β*<sup>c</sup>*, according to the Abaqus user manual:

$$
\overrightarrow{\varepsilon}\_{\varepsilon}^{pl} = \overrightarrow{\varepsilon}\_{\varepsilon}^{in} - \frac{d\_c}{1 - d\_c} \times \frac{\sigma\_c}{E\_0} \tag{20}
$$

$$d\_c = \frac{(1 - \beta\_C)\vec{\varepsilon}\_c^{in} E\_0}{\sigma\_c + (1 - \beta\_c)\vec{\varepsilon}\_c^{in} E\_0} \tag{21}$$

$$d\_{t} = \frac{(1 - \beta\_{t})\overline{\varepsilon}\_{t}^{in} E\_{0}}{\sigma\_{t} + (1 - \beta\_{t})\overline{\varepsilon}\_{t}^{in} E\_{0}} \tag{22}$$

where *dc* is the concrete compression damage factor; *dt* is the concrete tensile damage factor; σ*c* is the peak compressive stress of RAC; σ*t* is the peak tensile stress of RAC; β*c* is 0.6; β*t* is 0.8; ε*inc* is the inelastic compressive strain of RAC; ε*int*is the inelastic tensile strain of RAC.

#### 4.1.5. F–D Relation of Nonlinear Spring Unit

The constitutive relationship of the spring unit was determined before the establishment of the nonlinear spring unit, and was determined according to the constitutive relationship of the average bond strength at the loading end of the specimen. The interface between the section steel and RAC had three directions, namely longitudinal tangential, normal, and transverse tangential directions. The experiments proved that in the case of structural failure: first, the normal and transverse tangential deformation were much smaller than the longitudinal tangential deformation; second, the longitudinal tangential interaction was characterized by the bond slip phenomenon in the section steel and RAC. The Force-Displacement curve (F–D curve) consistent with the longitudinal tangential direction was employed by the spring element constitutive relationship, because the transverse tangential assumption was consistent with the longitudinal tangential interaction. The law-up interaction was set to a spring element with infinite stiffness because it was subjected to pressure and had high stiffness. The spring element F–D relationship is calculated by:

$$\mathbf{F} = \boldsymbol{\pi} \times \mathbf{A} \tag{23}$$

where A is the area occupied the spring connection surface, with the calculation diagram shown in Figure 12.

The F–D relationship of the springs under each node can be calculated through the bond slip constitutive relation, which was obtained from the test. However, the calculated F–D relationship did not pass through the coordinate origin and did not satisfy the input requirements of Abaqus, so it was processed. It was completely symmetrically processed in its negative direction in order to complete the F–D curve. The F–D relationship is shown in Figure 13 (taking the intermediate node of the spring at SRRC-1 as an example).

**Figure 12.** Schematic diagram of the force area calculation for the spring element.

**Figure 13.** F–D curve for intermediate node spring element of SRRC-1.

#### 4.1.6. Rewriting of Inp File

In this paper, the specific method used in addition of the nonlinear spring unit was as follows: the linear spring was first added to the simulated specimen, then the keywords in the inp file were found, and finally the linear spring was rewritten. The precautions were as follows: first, the rewritten inp file could not be imported into Computer Aided Engineering(CAE) and was applied by Abaqus command operation; second, the added maximum spring force was greater than the maximum force balanced with it; third, the nonlinear stiffness was symmetrically defined in the case of convenience and without affecting the result.

Taking SRRC-1 as an example, the nonlinear spring unit of the rewritten inp file is as follows:

\*Spring, elset=Springs/Dashpots-1-spring, nonlinear 1, 1 Nonlinear constitutive relation \*Element, type=Spring2, elset=Springs/Dashpots-1-spring Serial number, part one, node one, part two, node two

Nonlinear constitutive relations and corresponding nodes were replaced in order to reduce the length of the article, as shown above. It should be noted that "nonlinear" was required as a keyword after the spring set, indicating that all springs in the set were nonlinear. Two points should be noted when adding nonlinear constitutive relations: one is that the F–D curve needs to pass the coordinate origin, and the other is that the force is connected by "," in the middle of the displacement. For example, in 570, 0.00005, 570 means the force is 570N, and 0.00005 means 0.00005 m. When adding multiple sets of spring sets, the serial number of the corresponding node should be accumulated at once, otherwise the nonlinear stiffness will be overwritten by subsequent coverage, resulting in the failure of multiple sets for spring addition.

In this paper, each set of springs had a total of 1140 nodes and each specimen contained three sets of springs, which were longitudinal tangential, transverse tangential, and normal. Nine sets of specimens were simulated and the results were as outlined in the following subsections.
