*Article* **Experimental and Numerical Study on Interface Bond Strength and Anchorage Performance of Steel Bars within Prefabricated Concrete**

**Zhijian Hu <sup>1</sup> , Yasir Ibrahim Shah 1,\* and Pengfei Yao <sup>2</sup>**


**Abstract:** This study investigates the interface bond strength and anchorage performance of steel bars within prefabricated concrete. Twenty-two specimens were designed and manufactured to study the interface bond behavior of deformed and plain steel bars under a larger cover thickness. Diameter of steel bars, strength grade of concrete, and anchorage length were considered influential factors. The finite element method (ABAQUS) was used for the validation of experimental results. The interface bond's failure mechanism and the anchorage length in the prefabricated concrete under different concrete strength levels were explored and compared to national and international codes. A suitable value of the basic anchoring length for the prefabricated structure was recommended. The results show that the interface bond strength of prefabricated bridge members is directly proportional to the strength grade of the concrete, inversely proportional to the reinforcement diameter, and less related to anchorage length. The effect of the cover thickness of the surrounding concrete is negligible. Conversely, the bearing capacity of prefabricated bridge members depends on the strength of the concrete, the diameter of the steel bar, and the anchorage length. Furthermore, it is concluded that the mechanical bond strength accounts for 88% of the bond strength within prefabricated concrete.

**Keywords:** bond strength; prefabricated concrete structure; anchorage performance; mechanical bond strength

## **1. Introduction**

In recent years, prefabricated reinforced concrete structures are widely used to construct commercial buildings, temporary safety protection structures, and large and mediumsize bridges. The prefabricated structure has the advantages of a short construction periods, industrialized production, high dimensional accuracy, and less environmental pollution. The sufficient bond strength and anchorage performance of steel bars within prefabricated concrete are the key to ensure the service performance of the structure [1–3].

At present, the design of the anchorage length of the connecting reinforcement within the prefabricated concrete structure is usually considered according to the relevant provisions of the cast-in-place structure. However, due to the different characteristics of the prefabricated connection, the thickness of the protective layer of the connecting reinforcement is generally more than 50 mm larger than that of the cast-in-place structure and so the influence of the thickness of the protective layer can be ignored in the calculation of the anchorage length and interface bond strength. Presently, the existing relevant specifications of the assembled concrete structure still utilizes the relevant provisions of the cast-in-place structure for the anchorage length of the connecting steel bars, which is not suitable for calculating the bond strength and anchorage performance of the prefabricated assembled bridge with large protective layer thickness.

**Citation:** Hu, Z.; Shah, Y.I.; Yao, P. Experimental and Numerical Study on Interface Bond Strength and Anchorage Performance of Steel Bars within Prefabricated Concrete. *Materials* **2021**, *14*, 3713. https:// doi.org/10.3390/ma14133713

Academic Editor: Eva O. L. Lantsoght

Received: 23 May 2021 Accepted: 14 June 2021 Published: 2 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

There are only a few publications for the case of prefabricated concrete that have been published; this is the motivation for conducting the presented research. For the general case of a bond between reinforcement and concrete, however, much work has been conducted and reported in publications; the most relevant is mentioned here and shortly discussed. Steel bars have a significant effect on the mechanical properties and bond strength of concrete [4]. Li et al. established the formula of the ultimate bond strength by an experimental study on bond anchorage performance of 1860-grade high-strength prestressed steel strands and lightweight aggregate concrete [5].

The experimental study analyzes the compressive bond anchorage properties of 500 MPa steel bars in concrete. Five influence factors, including concrete strength, the steel bar's diameter, concrete cover, embedment length, and transverse reinforcement, were considered. The result shows that the influence of the surrounded concrete cover thickness on compressive bond strength is more than the steel bar's diameter [6]. Saeed et al. concluded from an experimental study that the anchor strength and stiffnesses are directly proportional to the bond length; the cross-sectional area ratio of Carbon Fibre-Reinforced Polymer (CFRP) rods to anchor borehole affects the stiffness and bonding capacity of the anchor [7]. Dang et al. proposed the standard test to investigate the bond performance of 18 mm prestressing strands used in precast/prestressed concrete applications. The pull-out resistance of steel can be improved by controlling the crack growth inside the concrete [8]. Hayashi et al. used the three-dimensional discrete model to analyze reinforced concrete (RC) anchorage performance. Results indicate that concrete strength is reduced if reinforcement spacing between the column and the embedded is very close because of a non-homogeneous behavior of concrete anchorage performance in a multidirectional arrangement of reinforcement bars [9].

Bond performance between concrete and steel bars was examined under corrosion level and temperature and it was found, from the study, that bond strength was influenced by temperature and corrosion [10]. John et al. revised how the anchorage's contribution was calculated and recognized the contribution of end bearing to laps and anchorages of compression bars. Bond influences the width and spacing of transverse cracks, tension stiffening, and flexural curvature. At the ultimate limit state, the bond is responsible for the strength of end anchorages and lapped joints of reinforcement and influences the rotation capacity of plastic hinge regions [11]. The hysteretic behavior of the anchorage slip is examined in reinforced concrete structures. Reinforced concrete columns subjected to axial compression and inelastic lateral deformation reversals develop significant rotations due to anchorage slip [12].

Anchorage and pull-out behavior depend on the geometry of steel fibers and is also related to the characteristics of the matrix [13]. The form of wet connection is to weld, lap, or mechanically connect the reserved connecting steel bar or connecting rod at the connection part, anchor the steel bar through post cast concrete or other grouting materials, and to connect the different prefabricated components. The dry connection is to embed the steel connecting parts in the prefabricated concrete components and then to connect them into one through bolt connection or welding a holistic approach [14]. The failure mode of reinforced concrete central pull-out specimen and the whole failure process of the reinforced concrete bond interface is divided into two stages: the elastic stage without crack and the working stage with crack. Based on the experimental results, the corresponding calculation methods of bond interface energy in different stages are obtained. The process of bond failure of reinforced concrete was analyzed from the perspective of energy [15].

Much numerical research has been carried out on the stochastic character of concrete [16–21]. Concrete is a primary construction material composed of cement and aggregates and the geometry and distribution of aggregates significantly affect the interface bond performance of the concrete structure. Jonak et al. used a contact interface between concrete and undercut anchor by using the finite element method (ABAQUS) and studied the cone failure occurring in the pull-out test. The result shows that the break-out angle of the undercut anchor was considerably less than the concrete capacity design method [18,19]. Ombres

et al. conducted direct single-lap shear tests on 20 specimens in order to study the bond behavior of steel reinforce grout to concrete joints. Experimental results were compared with finite element simulation and spectated to be in good agreement [21]. Funari et al. proposed a moving mesh numerical model using interface elements to calculate debonding mechanisms, crack opening, and cracks propagation in fiber-reinforced polymer (FRP) concrete beams [22].

It is concluded from the literature review that research on the bond strength and anchorage performance of reinforcement-concrete used in prefabricated bridges is still limited in number. There are still some differences in the design standards of the assembly type concrete and anchoring. However, due to the differences in test conditions and test design set by different scholars, the current test results are relatively discrete and the conclusions are quite different. Moreover, the influence of the failure mode on the interface bond strength calculation has not been clearly distinguished in the existing experimental studies. The complete interface failure method was used to calculate the bond strength, leading to the smaller calculated value of bond strength between steel and prefabricated concrete. A high value of the anchorage length is not suitable for the design and construction of prefabricated bridges. Existing codes for prefabricated structures stipulate the anchorage length of a post-cast straight anchor connecting steel bars of precast concrete members (JGJ 145-2004) (GB 50010-20118) [23,24]. The mechanical characteristics of reinforced concrete bonding interface under sufficient cover thickness are an essential issue in the design and construction of prefabricated bridges.

This study investigates the influential factors that affect the interface bond strength and the anchorage performance of steel bars within prefabricated concrete. Twenty-two specimens were manufactured for the pull-out test by utilizing a larger cover thickness. Based on experimental results and finite element simulation, the failure mode, ultimate load, the load-displacement curves, and the effect of the different influential factors on interface bond strength and anchorage performance of steel bars within prefabricated concrete were analyzed.

Furthermore, the bond strength calculation formulas and anchorage length for the steel bars within prefabricated concrete were fitted and derived. The national and international codes for the anchorage length and interface bond strength of steel bars within prefabricated concrete under different concrete strength levels were compared and analyzed. Recommended values of bond strength and the anchorage length of steel bars in the design of prefabricated bridges are given.

### **2. Experimental Program and Analysis of Test Results**

### *2.1. Specimen Design and Fabrication*

Twenty-two reinforced concrete specimens were designed and manufactured for pullout tests using larger cover thicknesses are shown in Figure 1. The section size of each specimen is 300 mm × 300 mm. The concrete strength grades were C30 and C50, the anchorage lengths of reinforcement were 200 mm and 300 mm, the diameters of the ribbed steel bar were 16 mm and 20 mm, and the diameters of plain steel bars were 8 mm and 20 mm.

### *2.2. Materials Test and Properties*

Hot-rolled Ribbed Bar (HRB400) [25] with the diameter of 12 mm and 16 mm was used in this study. Elastic modulus and compressive strength of C50 and C30 were calculated according to relevant specifications [26]. Three groups of standard cubes 150 mm × 150 mm × 150 mm were made for the compression test of C50 and C30 (each group has three specimens). An elastic modulus test was carried out on a prism block of the size 150 mm × 150 mm × 300 mm; three groups of specimens were designed with three test specimens in each group. The mix proportions of C30 and C50 are given in Table 1.

**Figure 1.** Reinforced concrete test specimens. **Figure 1.** Reinforced concrete test specimens.

*2.2. Materials Test and Properties* **Table 1.** Mix proportion of C30 and C50.


results of concrete and steel bars are shown in Table 2. **Table 1.** Mix proportion of C30 and C50. The tensile test was carried out according to the Chinese code [27,28] out on HRB400 with a diameter of 12 mm and 16 mm. The average value of material performance test results of concrete and steel bars are shown in Table 2.



C30 32521 31.2

### Steel bar Yield strength (MPa) Ultimate strength (MPa) HRB400Φ12 525 645 *2.3. Test Instruments*

middle.

HRB400Φ16 605 705 *2.3. Test Instruments* A bolt comprehensive parameter tester was used as a loading device. An intelligent digital pressure gauge indicates the load's value and the loading end displacement was measured by electronic displacement, as shown in Figure 2. At the first stage of loading, A bolt comprehensive parameter tester was used as a loading device. An intelligent digital pressure gauge indicates the load's value and the loading end displacement was measured by electronic displacement, as shown in Figure 2. At the first stage of loading, the load increment was 2 kN–5 kN and, at the second stage of loading, the load increment was 5 kN–10 kN. Electronic displacement meters were simultaneously set at the loading end and free end, respectively, with the spacing of 70 mm at the end and 100 mm in the middle.

the load increment was 2 kN–5 kN and, at the second stage of loading, the load increment was 5 kN–10 kN. Electronic displacement meters were simultaneously set at the loading end and free end, respectively, with the spacing of 70 mm at the end and 100 mm in the

**Figure 2.** Schematic diagram of the loading device. **Figure 2.** Schematic diagram of the loading device.

### *2.4. Pull-Out Test Results 2.4. Pull-Out Test Results*

The failure mode and ultimate load of each specimen in the pull-out test are shown in Table 3. The specimens in Table 3 are numbered according to the concrete strength grade-reinforcement diameter-anchorage length, such as C30-12-300. The failure mode and ultimate load of each specimen in the pull-out test are shown in Table 3. The specimens in Table 3 are numbered according to the concrete strength grade-reinforcement diameter-anchorage length, such as C30-12-300.

**Table 3.** Failure modes of reinforced concrete pull-out specimens. **Table 3.** Failure modes of reinforced concrete pull-out specimens.


C30-16-300 Reinforcement failure 106.82 C50-16-300 Reinforcement failure <sup>128</sup> C50-ø8-300 Pull-out of a plain bar 11.48 C50-16-400 Reinforcement failure 119.7 C50-ø20-300 Pull-out of a plain bar 18.1 C50-20-300 Reinforcement failure 153.77 Test results of all the specimens show the tensile failure of reinforcement. The ultimate load difference of C50 and C30 indicates that, for prefabricated concrete members with larger cover thicknesses, the concrete strength grade and reinforcement diameter significantly influences the ultimate load of the bond interface between reinforcement and concrete. The ultimate load of the ribbed steel bar specimen C50-20-300 is 8.5 times higher than that of the plain steel bar specimen C50-ø20-300.

Test results of all the specimens show the tensile failure of reinforcement. The ultimate load difference of C50 and C30 indicates that, for prefabricated concrete members with larger cover thicknesses, the concrete strength grade and reinforcement diameter significantly influences the ultimate load of the bond interface between reinforcement and concrete. The ultimate load of the ribbed steel bar specimen C50-20-300 is 8.5 times higher It can be estimated that the mechanical interlocking force accounts for about 88% of the bond strength between deformed steel bars and concrete. In addition, by comparing the failure modes of different specimens, it is concluded that, with the decrease in concrete strength and the increase in steel bar diameter, the failure mode of specimens gradually changes from the tensile failure of reinforcement to pull-out failure.

### than that of the plain steel bar specimen C50-ø20-300. *2.5. Load-Displacement Curves*

It can be estimated that the mechanical interlocking force accounts for about 88% of the bond strength between deformed steel bars and concrete. In addition, by comparing the failure modes of different specimens, it is concluded that, with the decrease in concrete The load-displacement curve from the experimental results is drawn and demonstrates the tensile failure of the specimen, as shown in Figure 3. Specimen C50-12-300 is taken as an example.

changes from the tensile failure of reinforcement to pull-out failure.

*2.5. Load-Displacement Curves*

taken as an example.

**Figure 3.** Load-displacement curve-tensile failure of reinforcement. **Figure 3.** Load-displacement curve-tensile failure of reinforcement.

Test specimens were loaded three times. At the first stage of loading, the specimen's ductility was large and there is a yield strengthening stage before the steel bars yield. In the second stage of loading, when the load value reaches its maximum value of the first load the steel bars begin to yield and there is a longer yield deformation stage. At third stage of loading, when the load reaches the yield value the steel bars break and there is no obvious post-yield strengthening stage. The diameter of the reinforcement determines the ultimate bearing capacity of the specimen and it possesses a positive linear correlation with the cross-sectional area of the reinforcement. Test specimens were loaded three times. At the first stage of loading, the specimen's ductility was large and there is a yield strengthening stage before the steel bars yield. In the second stage of loading, when the load value reaches its maximum value of the first load the steel bars begin to yield and there is a longer yield deformation stage. At third stage of loading, when the load reaches the yield value the steel bars break and there is no obvious post-yield strengthening stage. The diameter of the reinforcement determines the ultimate bearing capacity of the specimen and it possesses a positive linear correlation with the cross-sectional area of the reinforcement.

strength and the increase in steel bar diameter, the failure mode of specimens gradually

The load-displacement curve from the experimental results is drawn and demonstrates the tensile failure of the specimen, as shown in Figure 3. Specimen C50-12-300 is

### *2.6. Failure Mode 2.6. Failure Mode*

There were two main failure modes of the steel bar pull-out specimens in prefabricated concrete structures: reinforcement pull-out failure and tensile failure of the steel bar. There were two main failure modes of the steel bar pull-out specimens in prefabricated concrete structures: reinforcement pull-out failure and tensile failure of the steel bar.

(1) Pull-out failure of reinforcement. (1) Pull-out failure of reinforcement.

When the anchorage is insufficient, the pull-out force was greater than the bond interface bearing capacity of reinforced concrete and the failure of the bond interface occurs in the specimens. It can be seen from Figure 4a when the ribbed bar is pulled out, the concrete at the loading end was damaged in a cone shaped manner and there was no obvious necking of the bar. It can be seen from Figure 4b that the transverse rib of the reinforcement at the bond between the pulled-out reinforcement and concrete was intact. Under the pull-out load, the reinforcement was pulled out together with some intercostal concrete. The residual intercostal concrete accounts for about 50% of the spacing between the transverse ribs of the reinforcement. When the anchorage is insufficient, the pull-out force was greater than the bond interface bearing capacity of reinforced concrete and the failure of the bond interface occurs in the specimens. It can be seen from Figure 4a when the ribbed bar is pulled out, the concrete at the loading end was damaged in a cone shaped manner and there was no obvious necking of the bar. It can be seen from Figure 4b that the transverse rib of the reinforcement at the bond between the pulled-out reinforcement and concrete was intact. Under the pull-out load, the reinforcement was pulled out together with some intercostal concrete. The residual intercostal concrete accounts for about 50% of the spacing between the transverse ribs of the reinforcement.

(2) Tensile failure of the steel bar.

As shown in Figure 4c, when the reinforcement was sufficiently anchored, the bearing capacity of the reinforced concrete interface was more significant compared to the tensile bearing capacity of the reinforcement. The tensile failure of the reinforcement occurs in the specimens with a strength grade of C50.

### *2.7. Analysis of Plain and Ribbed Steel Bar Diameter*

Figure 5 shows the load-displacement curve of plain and ribbed reinforcement with different diameters. It can be observed from the figure that the pull-out failure of steel

bars occurs in both groups of the test specimens. With the increase in steel bar diameter, the ultimate load of the specimens increases gradually. The ultimate load of C50-G8-300 and C50-G20-300 is 11.48 kN and 18.1 kN, respectively. The analysis shows that, due to the Poisson's ratio effect of steel, the reinforcement will produce radial deformations under the pull-out force. If the diameters of steel bars were large, there would be more obvious shrinkage. *Materials* **2021**, *14*, x FOR PEER REVIEW 7 of 24

**Figure 4.** (**a**) Pull-out failure of reinforcement at loading end. (**b**) Cone-shaped failure after pulling out. (**c**) Tensile failure of reinforcement. **Figure 4.** (**a**) Pull-out failure of reinforcement at loading end. (**b**) Cone-shaped failure after pulling out. (**c**) Tensile failure of reinforcement. *Materials* **2021**, *14*, x FOR PEER REVIEW 8 of 24

(2) Tensile failure of the steel bar.

forced concrete, the mechanical interlocking force of bond interface is small or negligible, the bearing capacity of the bond interface between the plain steel bar and concrete is weak, **Figure 5.** Load-displacement curve of plain and ribbed reinforcement. **Figure 5.** Load-displacement curve of plain and ribbed reinforcement.

and the ultimate load is far less than that of the deformed steel bar. **3. Modelling** *3.1. Specimen Design* The finite element model (ABAQUS) of reinforced concrete pull-out specimens was established to simulate the interface bond and anchorage characteristics of steel bars within prefabricated concrete. In order to ensure the accuracy of the model and improve the calculation efficiency, the bilinear axisymmetric quadrilateral reduced integral element CGAX4R was used for both reinforcement and concrete. The mesh density of the bonding interface, reinforcement axis, and concrete edges were 0.5 mm, 0.5 mm, and 2 The bond interface between reinforcement-concrete was to be debonding and bond stress decreases between plain reinforcement and concrete with the increase in reinforcement diameter. Comparing the ultimate load of C50-8-300 and C50-G8-300 specimens with C50-20-300 and C50-G20-300, it is found that the interface bond strengths of plain steel bar are mainly composed of chemical bond strength and friction force between reinforced concrete, the mechanical interlocking force of bond interface is small or negligible, the bearing capacity of the bond interface between the plain steel bar and concrete is weak, and the ultimate load is far less than that of the deformed steel bar.

mm, respectively. By establishing the surface size of the transverse rib, the influences of the artificial definition of interface elements on the analysis results were reduced. The

The detailed characteristics of the bond interface between steel and concrete were considered. The normal behavior of the contact surface was simulated by "Hard Contact"

**Figure 6.** Mesh generation of the finite element model.

and implemented by the Classical Lagrange Multiplier Method.

*3.2. Contact Problem Simulation*

the model was 0.4 mm.

### **3. Modelling** *3.1. Specimen Design*

**3. Modelling**

0

20

40

60

80

100

Load (kN)

120

140

160

180

### *3.1. Specimen Design* The finite element model (ABAQUS) of reinforced concrete pull-out specimens was

The finite element model (ABAQUS) of reinforced concrete pull-out specimens was established to simulate the interface bond and anchorage characteristics of steel bars within prefabricated concrete. In order to ensure the accuracy of the model and improve the calculation efficiency, the bilinear axisymmetric quadrilateral reduced integral element CGAX4R was used for both reinforcement and concrete. The mesh density of the bonding interface, reinforcement axis, and concrete edges were 0.5 mm, 0.5 mm, and 2 mm, respectively. By establishing the surface size of the transverse rib, the influences of the artificial definition of interface elements on the analysis results were reduced. The model diagram and mesh generation are shown in Figure 6. The minimum mesh size of the model was 0.4 mm. established to simulate the interface bond and anchorage characteristics of steel bars within prefabricated concrete. In order to ensure the accuracy of the model and improve the calculation efficiency, the bilinear axisymmetric quadrilateral reduced integral element CGAX4R was used for both reinforcement and concrete. The mesh density of the bonding interface, reinforcement axis, and concrete edges were 0.5 mm, 0.5 mm, and 2 mm, respectively. By establishing the surface size of the transverse rib, the influences of the artificial definition of interface elements on the analysis results were reduced. The model diagram and mesh generation are shown in Figure 6. The minimum mesh size of the model was 0.4 mm.

C50-8-300 C50-20-300 C50-G8-300 C50-G20-300

**Figure 5.** Load-displacement curve of plain and ribbed reinforcement.

0 10 20 30 40 50 60 70

Displacement (mm)

*Materials* **2021**, *14*, x FOR PEER REVIEW 8 of 24

**Figure 6.** Mesh generation of the finite element model. **Figure 6.** Mesh generation of the finite element model.

### *3.2. Contact Problem Simulation 3.2. Contact Problem Simulation*

The detailed characteristics of the bond interface between steel and concrete were considered. The normal behavior of the contact surface was simulated by "Hard Contact" and implemented by the Classical Lagrange Multiplier Method. The detailed characteristics of the bond interface between steel and concrete were considered. The normal behavior of the contact surface was simulated by "Hard Contact" and implemented by the Classical Lagrange Multiplier Method.

The transmitted compressive stress between the contact surfaces was unlimited. When the pressure on the contact surface becomes negative or zero, the two contact surfaces were separated. The tangential behavior was simulated by "Penalty Friction." In the Classical Coulomb Friction Model, the critical friction stress depends on the contact pressure and the elastic slip was allowed on the contact surface. Assuming that the friction coefficient µ between the contact surfaces was the same 0.1 [29], axial symmetry was adopted for boundary conditions.

### *3.3. Parameter Selection*

In order to study the interface bond strength and anchorage performance between the reinforcement and the concrete of the prefabricated bridge, the larger thickness of the protective layer was adopted according to the structural characteristics. The cross-sectional dimensions of the specimens were 300 mm × 300 mm according to the prefabricated structure specification, as per experimental design. The test parameters include the diameter of reinforcement, concrete strength, and anchorage length. The concrete strength grade was C30 and C50. The reinforcement diameter was 12 mm, 16 mm, and 20 mm and the anchorage length was 150 mm, 200 mm, and 300 mm, respectively.

### *3.4. Material Constitutive Model*

In order to simulate the failure and crack development of the bond interface of reinforced concrete, the elastic-plastic damage constitutive model was adopted for concrete, which can be used to observe and analyze the development law of cracks by using the cloud diagram of concrete tensile damage (DAMAGET). The pull-out phenomenon of rein-

forced concrete's bonding interface was observed through concrete compression damage (DAMAGEC) [30]. The constitutive model of concrete compression damage and tension damage is shown in Figure 7.

**Figure 7.** Plastic damage constitutive curve of concrete. (**a**) Compression damage constitutive curve of concrete. (**b**)Tensile damage constitutive curve of concrete. **Figure 7.** Plastic damage constitutive curve of concrete. (**a**) Compression damage constitutive curve of concrete. (**b**)Tensile damage constitutive curve of concrete.

The constitutive model can be expressed in the following:

$$
\sigma\_{t,\mathcal{c}} = (1 - d\_{t,\mathcal{c}}) E\_0(\varepsilon\_{t,\mathcal{c}} - \hat{\varepsilon}\_{t,\mathcal{c}}^{pl}) \tag{1}
$$

where *dt*,*<sup>c</sup>* is the damage factor of concrete, *E*<sup>0</sup> is the elastic modulus of concrete, *εt*,*<sup>c</sup>* is the concrete strain, and <sup>e</sup>*<sup>ε</sup> pl t*,*c* is the equivalent plastic strain of concrete.

In the simulation of reinforced concrete structure, the interface effect between reinforcement and concrete (such as bond-slip and locking behavior) was simulated by introducing "Tensile Hardening" into the concrete model. The tensile hardening data were defined according to the cracking strain <sup>e</sup>*<sup>ε</sup> pl t* . The relationship between the equivalent plastic strain <sup>e</sup>*<sup>ε</sup> ck t* and the cracking strain <sup>e</sup>*<sup>ε</sup> pl t* in the model is described as follows <sup>e</sup>*<sup>ε</sup> ck t* .

$$
\hat{\varepsilon}\_t^{pl} = \hat{\varepsilon}\_t^{ck} - \frac{d\_t}{(1 - d\_t)} \frac{\sigma\_t}{E\_0} \tag{2}
$$

According to the definition of compression hardening, the hardening data are defined according to the inelastic strain <sup>e</sup>*<sup>ε</sup> inl c* . The relationship between the equivalent plastic strain e*ε pl <sup>c</sup>* and the inelastic strain <sup>e</sup>*<sup>ε</sup> inl c* in the model is as follows:

$$
\hat{\mathfrak{e}}\_{\mathcal{c}}^{pl} = \hat{\mathfrak{e}}\_{\mathcal{c}}^{inl} - \frac{d\_{\mathcal{c}}}{(1 - d\_{\mathcal{c}})} \frac{\sigma\_{\mathcal{c}}}{E\_0} \tag{3}
$$

$$
\sigma\_s = \begin{cases}
 \ E\_s \varepsilon\_s & \varepsilon\_s \le \varepsilon\_y \\
 \ f\_y & \varepsilon\_s > \varepsilon\_y
\end{cases}
\tag{4}
$$

where *σ<sup>s</sup>* is the stress of reinforcement, *E<sup>s</sup>* is the elastic modulus of reinforcement, *f<sup>y</sup>* is the yield strength of reinforcement, *ε<sup>s</sup>* is the strain of reinforcement, and *ε<sup>y</sup>* is the yield strain of reinforcement. Properties of concrete and reinforcement are indicated in Table 4.
