*3.5. Finite Element Model Validation*

1

Taking C50-12-300 as an example, the finite element model was established. According to the results of finite element analysis, the load-displacement curve of the specimen was extracted and compared with the experimental result curve, as shown in Figure 8. It can be observed that the rising stage and ultimate load of the finite element model are consistent with that of the test specimen. The errors between the limit load of the finite element model and the experimental value were 2.4% and 0.8%, respectively.


element model and the experimental value were 2.4% and 0.8%, respectively.

C50 0.2 34429 53.8 C30 0.2 32521 31.2

Steel bar / / Ultimate strength (MPa)

**Table 4.** Material properties of concrete and reinforcement. HRB400Φ12 0.3 206000 645

**Table 4.** Material properties of concrete and reinforcement.

**Specimen Poisson's Ratio Young's Modulus**

*Materials* **2021**, *14*, x FOR PEER REVIEW 10 of 24

<sup>−</sup>

According to the definition of compression hardening, the hardening data are de-

<sup>−</sup>

where is the stress of reinforcement, is the elastic modulus of reinforcement, is the yield strength of reinforcement, is the strain of reinforcement, and is the yield strain of reinforcement. Properties of concrete and reinforcement are indicated in Table 4.

(1 − )

in the model is as follows:

(1 − )

**(MPa)**

 ≤ >

̃ = ̃ 

̃ = ̃ 

=

. The relationship between the equivalent plas-

. The relationship between the equivalent plastic

.

**Compressive Strength (MPa)**

(2)

(3)

(4)

in the model is described as follows

 

> 

defined according to the cracking strain ̃

fined according to the inelastic strain

and the inelastic strain ̃

and the cracking strain ̃

tic strain ̃

strain ̃  **Figure 8.** Test finite element comparison.

### *3.6. Analysis of Interface Bond Failure Process*

Under the pull-out load, the failure mode of the bond interface of reinforced concrete was the same as that of the experimental test results. The development of cracks cannot be directly observed in the test, but the development law of cracks can be observed and analyzed by using the cloud picture of concrete tensile damage (DAMAGET) in the finite element results. The failure appearance of the bonding interface of reinforced concrete can be observed through concrete compression damage (DAMAGEC) and the main failure mode is shown in Figure 9.

The bond strength between reinforcement and concrete was determined by the properties of the interface between them, mainly including three factors: (1) the chemical bond force between the concrete substrate and the surface coating of reinforcement; (2) the relative sliding friction resistance between reinforcement and concrete along with the interface; (3) the mechanical interlocking force caused by the unevenness of the interface between reinforcement and concrete [30–33]. The test results show that the failure modes of reinforced concrete interface bond specimens under pull-out load are mainly divided into the yield fracture of reinforcement and interface concrete failure. The bond strength of ribbed steel bar and concrete was composed of chemical bond force, friction force, and mechanical interlocking force.

Under the pull-out load, the failure mode of the bond interface of reinforced concrete was the same as that of the experimental test results. The development of cracks cannot be directly observed in the test, but the development law of cracks can be observed and analyzed by using the cloud picture of concrete tensile damage (DAMAGET) in the finite element results. The failure appearance of the bonding interface of reinforced concrete can be observed through concrete compression damage (DAMAGEC) and the main failure

**Figure 8.** Test finite element comparison.

mode is shown in Figure 9.

*3.6. Analysis of Interface Bond Failure Process*

**Figure 9.** Failure process of the bonding interface. (**a**) Concrete crack development. (**b**) Interface concrete failure. **Figure 9.** Failure process of the bonding interface. (**a**) Concrete crack development. (**b**) Interface concrete failure.

The bond strength between reinforcement and concrete was determined by the properties of the interface between them, mainly including three factors: (1) the chemical bond At the initial loading stage, the bonding interface's slip resistance was assumed by the chemical bonding force. In contrast, the mechanical interlocking force and friction force do not play a role temporarily. With the increase in the load, the chemical bond force fails, the bond interface slips relatively, the mechanical interlocking force and friction force begin to play a role, and the interface slip resistance is then provided by the oblique extrusion force between the transverse rib and the concrete. The axial component of the oblique extrusion force renders the concrete between the ribs subject to bending and shearing as a cantilever beam. The radial component of the oblique extrusion force results in the concrete around the reinforcement producing circumferential tensile stress. At this time, the concrete around the reinforcement was in a three-phase stress state. As shown in Figure 9a, the concrete behind the transverse rib of the steel bar was pulled by the oblique extrusion force, while the concrete in front of the rib was pressed. With the increase in the load, the radial cracks

first appear behind the rib and develop along the direction of 60◦ , with the axial direction of the steel bar (the inclination angle of the transverse rib of the reinforcing bar). The radial crack depths of the cracks were approximately equal to the spacing between the transverse ribs of the reinforcing bar. As shown in Figure 9b, the radial failure depth was about two times the transverse rib height of the reinforcement. This failure process is called the shear bond failure of ribbed bars.

### *3.7. Load-Displacement Curve*

According to the results of the finite element analysis, the load-displacement curve of the specimen was extracted, the stress characteristics and failure mechanism of different stages were analyzed, and the influences of concrete strength grade, reinforcement diameter, and anchorage length on the load-displacement curve were compared.

As shown in Figure 10, the constitutive interface model can be divided into an initial linear elastic stage and failure stage. The failure process can be divided into the micro-slip stage, internal crack slip-stage, and pull-out fluctuation decline stage. In the micro-slip stage, the slip at the loading end is minimal and there is no slip at the free end, which shows a linear phase on the load slip curve. It can be considered that the bond force gradually transfers from the near end to the far end and the adhesive force was complete during the internal crack sliding stage. When the load continues to increase, the bond force was transferred to the far end, a small amount of slip begins to appear at the far end, and the adhesive force disappears. The interface bond was mainly maintained by the friction force and mechanical interlocking forces between the concrete and reinforcement ribs. In the pull-out stage, the interface concrete was damaged, the load suddenly decreases, and the reinforcement is gradually pulled-out. *Materials* **2021**, *14*, x FOR PEER REVIEW 13 of 24

**Figure 10.** Load-displacement curve. **Figure 10.** Load-displacement curve.

### **4. Parametric Study**

**[kN]**

**Scheme Failure Mode Ultimate Load** 

out

C30-20-200 Reinforcement pull-

### **4. Parametric Study** *4.1. Scope of Investigation*

*4.1. Scope of Investigation* Parametric analysis was carried out with concrete strength, reinforcement diameter, and anchorage length as a variable. The parameters, failure mode, ultimate load, and bond strength of each specimen are shown in Table 5. The specimens in the Table are numbered according to the concrete strength grade—reinforcement diameter—anchorage length, Parametric analysis was carried out with concrete strength, reinforcement diameter, and anchorage length as a variable. The parameters, failure mode, ultimate load, and bond strength of each specimen are shown in Table 5. The specimens in the Table are numbered according to the concrete strength grade—reinforcement diameter—anchorage length, such as C30-12-300. The calculation formula of bond stress in the Table is as follows:

$$
\pi = \frac{F}{\pi dl} \tag{5}
$$

$$
\text{If } \pi \text{ is a constant, then } \pi \text{ is a constant.}
\text{ This means that } \pi \text{ is a constant.}
$$

**[kN]**

**Bond Strength /MPa**

 (5) where is the bond strength of reinforced concrete, *F* is the pull-out load, *d* is the diamwhere *τ* is the bond strength of reinforced concrete, *F* is the pull-out load, *d* is the diameter of reinforcement, and *l* is the effective bond length.

99.97 10.83 C50-20-200 Reinforcement failure 124.27 13.86

The load-displacement curves of the experiment and FEM are compared, which are in good agreement and shown in Figure 11. Load-displacement comparison shows that with the increase in the concrete strength, the steel bar's pull-out failure changes into the tensile failure of the steel bar. The ultimate load-bearing capacity of the specimen increases significantly. It is concluded that in engineering practice, high-strength concrete can improve the bond strength of steel bars and reduce the requirements for anchorage

**ber Failure Mode Ultimate Load** 

C30-16-200 Reinforcement failure 75.96 11.53 C50-12-300 Reinforcement failure 44.71 15.82

C30-16-300 Reinforcement failure 74.4 11.81 C50-16-200 Reinforcement failure 75.74 14.57

*4.2. Comparison of Load-Displacement Curves of FEM and Test Result*

**Specimen Num-**

**Table 5.** Analysis results.

eter of reinforcement, and is the effective bond length.

**Bond Strength /MPa**

length of steel bars to a certain extent.


**Table 5.** Analysis results.

### *4.2. Comparison of Load-Displacement Curves of FEM and Test Result*

The load-displacement curves of the experiment and FEM are compared, which are in good agreement and shown in Figure 11. Load-displacement comparison shows that with the increase in the concrete strength, the steel bar's pull-out failure changes into the tensile failure of the steel bar. The ultimate load-bearing capacity of the specimen increases significantly. It is concluded that in engineering practice, high-strength concrete can improve the bond strength of steel bars and reduce the requirements for anchorage length of steel bars to a certain extent. *Materials* **2021**, *14*, x FOR PEER REVIEW 14 of 24

**Figure 11.** Comparison of load-displacement curves. **Figure 11.** Comparison of load-displacement curves.

### *4.3. Influence of the Concrete Strength Grade 4.3. Influence of the Concrete Strength Grade*

The bond strength and ultimate load of specimens with different concrete strengths were compared in Figure 12. The d16-100 series in Figure 12 represent five specimens with different concrete strengths with a reinforcement diameter of 16 mm and anchorage length of 100 mm. The bond strength and ultimate load of specimens with different concrete strengths were compared in Figure 12. The d16-100 series in Figure 12 represent five specimens with different concrete strengths with a reinforcement diameter of 16 mm and anchorage length of 100 mm.

According to Figure 12a, the bond strength between reinforcement and concrete increases with concrete strength. Compared with C30, C40, C50, and C60 specimens, the bond strength of C80 specimens increases by 44.2%, 34.7%, 16.4%, and 5.8%, respectively. According to Figure 12a, the bond strength between reinforcement and concrete increases with concrete strength. Compared with C30, C40, C50, and C60 specimens, the bond strength of C80 specimens increases by 44.2%, 34.7%, 16.4%, and 5.8%, respectively.

d16-100 series d16-150 series 55 60 65 70 75 80 Utimate load (kN) d16-100 series d16-150 series Steel bar pull-out Steel bar failure At the same time, it can be observed from Figure 12a that the correlation between the bond strength and the anchorage length of the specimens with different anchorage lengths was small for the connection reinforcement of prefabricated assembled concrete structure with larger cover thickness. The influence of anchorage length can be ignored in the calculation of bond strength. It can be observed from Figure 12b that with the increase in concrete strength, the failure mode of the specimen changes from the pull-out failure to the tensile failure of the reinforcement and the ultimate bearing capacity was significantly increased. The results show that when the pull-out failure occurs, the anchorage length was insufficient and the concrete strength possesses a significant impact on the ultimate bearing capacity of the specimens. However, when the steel bar was failing, the anchorage length

At the same time, it can be observed from Figure 12a that the correlation between the bond strength and the anchorage length of the specimens with different anchorage lengths was small for the connection reinforcement of prefabricated assembled concrete structure with larger cover thickness. The influence of anchorage length can be ignored in the calculation of bond strength. It can be observed from Figure 12b that with the increase in concrete strength, the failure mode of the specimen changes from the pull-out failure to

30 40 50 60 70 80

d16-200 series d16-300 series

Concrete strength (MPa)

(**a**) (**b**) **Figure 12.** Influence of concrete strength grade. (**a**) Comparison of bond strength and concrete strength. (**b**) Ultimate load

40 45 50

concrete strength comparison.

30 40 50 60 70 80

d16-200 series d16-300 series

Concrete strength (MPa)

Bond strength (MPa)

**Figure 11.** Comparison of load-displacement curves.

0 10 20 30 40 50

Displacement (mm)

*4.3. Influence of the Concrete Strength Grade*

length of 100 mm.

0

10

20

30

40

Load (kN)

50

60

70

80

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

was sufficient, such as the d16-200 series and d16-300 series, and the concrete strength had little influence. creases with concrete strength. Compared with C30, C40, C50, and C60 specimens, the bond strength of C80 specimens increases by 44.2%, 34.7%, 16.4%, and 5.8%, respectively.

The bond strength and ultimate load of specimens with different concrete strengths were compared in Figure 12. The d16-100 series in Figure 12 represent five specimens with different concrete strengths with a reinforcement diameter of 16 mm and anchorage

C30-16-200 FEM C50-16-200 FEM

According to Figure 12a, the bond strength between reinforcement and concrete in-

**Figure 12.** Influence of concrete strength grade. (**a**) Comparison of bond strength and concrete strength. (**b**) Ultimate load concrete strength comparison. **Figure 12.** Influence of concrete strength grade. (**a**) Comparison of bond strength and concrete strength. (**b**) Ultimate load concrete strength comparison.

### At the same time, it can be observed from Figure 12a that the correlation between the *4.4. Influence of the Reinforcement Diameter*

bond strength and the anchorage length of the specimens with different anchorage lengths was small for the connection reinforcement of prefabricated assembled concrete structure with larger cover thickness. The influence of anchorage length can be ignored in the calculation of bond strength. It can be observed from Figure 12b that with the increase in concrete strength, the failure mode of the specimen changes from the pull-out failure to The load-displacement curves of specimens with different reinforcement diameters were compared, as shown in Figure 13a. When the anchorage length and concrete strength grades were the same, the specimens' bearing capacity gradually increased with the increase in the reinforcement diameter. However, the failure mode of the specimens changed from the tensile failure (C30-12-200 and C30-16-200) to the pull-out failure (C30- 20-200 and C30-25-200), indicating that the specimens did not pull out and the anchorage length required for failure was significantly increased; that is, the anchorage length closely related to the diameter of reinforcement.

From the previous analysis, it can be seen that the interfacial bond strength between the reinforcement and the concrete was mainly composed of the mechanical interlocking force. Shape parameters determine the mechanical interlocking force, the transverse rib's height, and the spacing between the transverse ribs. The relative rib area (the ratio of the projected area of the transverse rib on the surface of the reinforcement to the reinforcement's surface area) is taken as the index to evaluate the bond performance.

When the relative rib area was larger, the bond performance should be improved. According to (GB 1499.2-2018) [25], the relative rib area of reinforcement decreases with the increase in the diameter of steel bars and the bond performance between the connecting reinforcement and the concrete also decreases, as shown in Figure 13b Compared with the diameter of the specimen of 8 mm, 12 mm, 16 mm, and 20 mm, the bond strength of the 25 mm steel bar decreases by 33.9%, 28.1%, 15.6%, and 11.1%. It can be seen from Figure 13c that the ultimate load is linearly related to the cross-sectional area of the reinforcement. As the reinforcement's diameter increases, the specimen's failure mode changes from tensile failure to pull-out failure. The diameter of the reinforcement is no longer the main influencing factor of the ultimate load (such as the C30-200 series).

When the diameter of reinforcement increases, the bond area and mechanical interlocking depth of reinforcement concrete increases. The increment of interface bearing capacity caused by the increase in bond area was more significant than the decrease caused by the decrease in bond strength under the same conditions. It was concluded that the ultimate load still increases with the increase in reinforcement diameter.

**Figure 13.** (**a**) Comparison of load-displacement curves of specimens with different reinforcement diameters. Influence of steel bar diameter: (**b**) Bond strength and steel bars diameter; (**c**) Ultimate load and steel bars diameter. **Figure 13.** (**a**) Comparison of load-displacement curves of specimens with different reinforcement diameters. Influence of steel bar diameter: (**b**) Bond strength and steel bars diameter; (**c**) Ultimate load and steel bars diameter.

### From the previous analysis, it can be seen that the interfacial bond strength between *4.5. Influence of Anchorage Length*

strength had little influence.

*4.4. Influence of the Reinforcement Diameter*

related to the diameter of reinforcement.

the reinforcement and the concrete was mainly composed of the mechanical interlocking It can be observed from the C50-16-100 specimen in Figure 14a that when the steel bars were not sufficiently anchored, the pull-out failure of the specimen occurs. Before the steel bars were pulled out, the interface stiffness decreases rapidly with the load increase and the bond interface between reinforcement and concrete fails. When the reinforcement is fully anchored (C50-16-200 and C50-16-300 specimens in Figure 14a and the interface stiffness decreases with the increase in load before the reinforcement yields, the change amplitude was small. It can be observed from Figure 14b that the anchorage length has little effect on the bond strength between the connecting steel bar and the concrete.

the tensile failure of the reinforcement and the ultimate bearing capacity was significantly increased. The results show that when the pull-out failure occurs, the anchorage length was insufficient and the concrete strength possesses a significant impact on the ultimate bearing capacity of the specimens. However, when the steel bar was failing, the anchorage length was sufficient, such as the d16-200 series and d16-300 series, and the concrete

The load-displacement curves of specimens with different reinforcement diameters were compared, as shown in Figure 13a. When the anchorage length and concrete strength grades were the same, the specimens' bearing capacity gradually increased with the increase in the reinforcement diameter. However, the failure mode of the specimens changed from the tensile failure (C30-12-200 and C30-16-200) to the pull-out failure (C30- 20-200 and C30-25-200), indicating that the specimens did not pull out and the anchorage length required for failure was significantly increased; that is, the anchorage length closely

By comparing the ultimate load of different specimens in Figure 14c, it was found that the ultimate load of specimens with an anchorage length of 200 mm in the C30-16 series increases by 21.5% and 64.2%, respectively, compared with specimens with anchorage lengths of 150 mm and 100 mm. It was concluded that when the anchorage length was not sufficient, the ultimate load increases with the increase in anchorage length and the failure mode of the specimen will change from pull-out failure to tensile failure.

force. Shape parameters determine the mechanical interlocking force, the transverse rib's height, and the spacing between the transverse ribs. The relative rib area (the ratio of the projected area of the transverse rib on the surface of the reinforcement to the reinforce-

When the relative rib area was larger, the bond performance should be improved. According to (GB 1499.2-2018) [25], the relative rib area of reinforcement decreases with the increase in the diameter of steel bars and the bond performance between the connecting reinforcement and the concrete also decreases, as shown in Figure 13b Compared with the diameter of the specimen of 8 mm, 12 mm, 16 mm, and 20 mm, the bond strength of the 25 mm steel bar decreases by 33.9%, 28.1%, 15.6%, and 11.1%. It can be seen from Figure 13c that the ultimate load is linearly related to the cross-sectional area of the reinforcement. As the reinforcement's diameter increases, the specimen's failure mode changes from tensile failure to pull-out failure. The diameter of the reinforcement is no longer the

When the diameter of reinforcement increases, the bond area and mechanical interlocking depth of reinforcement concrete increases. The increment of interface bearing capacity caused by the increase in bond area was more significant than the decrease caused by the decrease in bond strength under the same conditions. It was concluded that the

It can be observed from the C50-16-100 specimen in Figure 14a that when the steel bars were not sufficiently anchored, the pull-out failure of the specimen occurs. Before the steel bars were pulled out, the interface stiffness decreases rapidly with the load increase and the bond interface between reinforcement and concrete fails. When the reinforcement is fully anchored (C50-16-200 and C50-16-300 specimens in Figure 14a and the interface stiffness decreases with the increase in load before the reinforcement yields, the change amplitude was small. It can be observed from Figure 14b that the anchorage length has little effect on the bond strength between the connecting steel bar and the concrete.

ment's surface area) is taken as the index to evaluate the bond performance.

main influencing factor of the ultimate load (such as the C30-200 series).

ultimate load still increases with the increase in reinforcement diameter.

*4.5. Influence of Anchorage Length*

**Figure 14.** Influence of anchorage length. (**a**) Load-displacement curves for anchorage lengths. (**b**) Bond strength and anchorage length. (**c**) Relationship b/w ultimate load and anchorage length. **Figure 14.** Influence of anchorage length. (**a**) Load-displacement curves for anchorage lengths. (**b**) Bond strength and anchorage length. (**c**) Relationship b/w ultimate load and anchorage length.

### By comparing the ultimate load of different specimens in Figure 14c, it was found **5. Simplified Calculation Formula**

index of bond strength calculation.

connecting steel bar can be analyzed.

control.

### that the ultimate load of specimens with an anchorage length of 200 mm in the C30-16 *5.1. Calculation Formula for the Bond Strength*

series increases by 21.5% and 64.2%, respectively, compared with specimens with anchorage lengths of 150 mm and 100 mm. It was concluded that when the anchorage length was not sufficient, the ultimate load increases with the increase in anchorage length and the failure mode of the specimen will change from pull-out failure to tensile failure. **5. Simplified Calculation Formula** *5.1. Calculation Formula for the Bond Strength* For calculating the bond strength of steel bars within the concrete, scholars and codes have given the corresponding semi-empirical and semi-theoretical calculation formulas [23,24,33,34] by comprehensively considering different factors. The relevant formulas consider the concrete strength, protective layer thickness, anchorage length, and reinforce-For calculating the bond strength of steel bars within the concrete, scholars and codes have given the corresponding semi-empirical and semi-theoretical calculation formulas [23,24,33,34] by comprehensively considering different factors. The relevant formulas consider the concrete strength, protective layer thickness, anchorage length, and reinforcement diameter. The typical calculation formula of bond strength is shown in Table 6. The concrete structure design code [24] only calculates the bond strength between steel bar and concrete from the perspective of concrete tensile strength, without considering the influence factors such as steel bar type, steel bar diameter, anchorage length, and cover thickness. The Australian code [33] and the American code [34] considered the concrete strength, which is the ratio of concrete cover thickness to steel bar diameter, as the key index of bond strength calculation.

ment diameter. The typical calculation formula of bond strength is shown in Table 6. The concrete structure design code [24] only calculates the bond strength between steel bar and concrete from the perspective of concrete tensile strength, without considering the influence factors such as steel bar type, steel bar diameter, anchorage length, and cover

strength, which is the ratio of concrete cover thickness to steel bar diameter, as the key

Although the concrete cover in the fabricated structure is sufficient, the bond strength of the connecting steel bar will not be affected due to the too-small cover thickness. Therefore, the existing formula for calculating the bond strength of reinforcement is not suitable for the calculation of the connecting reinforcement of prefabricated bridges, which will lead to the length of the reserved connecting reinforcement of prefabricated components and will have an adverse impact on the construction difficulty and assembly accuracy

Therefore, the existing formula of bond strength of steel bars is not suitable for calculating the connection reinforcement of prefabricated bridges, which will lead to the long reserved connection reinforcement of prefabricated members and will have an adverse effect on the construction difficulty and accuracy control of the assembly. According to the above analysis, the main factors influencing the bond strength of the prefabricated and assembled concrete structure connecting steel bars are the diameter of the reinforcement and the concrete's strength. Therefore, the calculation formula of the bond strength of the


**Table 6.** Calculation formula of bond strength.

Although the concrete cover in the fabricated structure is sufficient, the bond strength of the connecting steel bar will not be affected due to the too-small cover thickness. Therefore, the existing formula for calculating the bond strength of reinforcement is not suitable for the calculation of the connecting reinforcement of prefabricated bridges, which will lead to the length of the reserved connecting reinforcement of prefabricated components and will have an adverse impact on the construction difficulty and assembly accuracy control.

Therefore, the existing formula of bond strength of steel bars is not suitable for calculating the connection reinforcement of prefabricated bridges, which will lead to the long reserved connection reinforcement of prefabricated members and will have an adverse effect on the construction difficulty and accuracy control of the assembly. According to the above analysis, the main factors influencing the bond strength of the prefabricated and assembled concrete structure connecting steel bars are the diameter of the reinforcement and the concrete's strength. Therefore, the calculation formula of the bond strength of the connecting steel bar can be analyzed.

In the Table 6, *τ<sup>u</sup>* is the interface bonding strength of reinforced concrete interface, *fcu* is the standard value of concrete compressive strength, *ft*,*<sup>r</sup>* is the characteristic value of concrete tensile strength, and *f<sup>t</sup>* is the splitting strength of concrete. *d* is the diameter of reinforcement and *c* is the thickness of the protective layer. The anchorage length is *l* and *ρsv* is the reinforcement ratio. Multiple linear regression analysis was used to determine the influence proportion of each factor of the above formulas. The results are shown in Figure 15. It can be seen that the bond strength of the interface between reinforcement and concrete is positively correlated with the strength of concrete and negatively correlated with the diameter of reinforcement. The formula for calculating the bond strength of connecting bars in prefabricated concrete structure can be fitted as follows:

$$
\pi\_{\rm u} = 0.1f\_{\rm cu} - 0.35d + 14.9 \tag{6}
$$

where *τ<sup>u</sup>* is the bond strength of the interface between steel bar and concrete, *fcu* is the standard value of concrete compressive strength, and *d* is the diameter of the steel bar.

The minimum error between the fitting formula and the test value in literature is 2%, the maximum error is 11%, and the overall error is 7%. The minimum error is 0.6%, the maximum error is 7%, and the overall error is 4%. The average ratio of the fitting value to the test value is 1.04, and the standard deviation and coefficient of variation of the ratio are 0.06 and 5.77%, respectively. The results show that the fitting values are in good agreement with the experimental values and the dispersion is low.

According to the principle of the complete failure of interface, the formula (6) was compared with this paper's test results and the references [35,36]. The comparison results are shown in Figure 16. It was found that the average ratio of the bond strength fitting value to the test value is 1.04, the standard deviation and coefficient of variation of the ratio are 0.06 and 5.77%, respectively, and the goodness of fit R<sup>2</sup> between the fitting value and the measured value is 0.87, which indicates that the fitting value is in good agreement with the test value and that the dispersion is low.

**Table 6.** Calculation formula of bond strength.

**Standard Calculation Formula of Bond Strength**

In the Table 6, is the interface bonding strength of reinforced concrete interface, is the standard value of concrete compressive strength, , is the characteristic value of concrete tensile strength, and is the splitting strength of concrete. *d* is the diameter of reinforcement and *c* is the thickness of the protective layer. The anchorage length is and is the reinforcement ratio. Multiple linear regression analysis was used to determine the influence proportion of each factor of the above formulas. The results are shown in Figure 15. It can be seen that the bond strength of the interface between reinforcement and concrete is positively correlated with the strength of concrete and negatively correlated with the diameter of reinforcement. The formula for calculating the bond strength

= 0.1 − 0.35 + 14.9 (6)

Australia AS3600 [33] = 0.3 × (0.5 + /) America ACI318-11 [34] = 0.08 × (1.2 + 3/ + 50/) Literature [35] = (1.6 + 0.7/ + 20)(0.82 + 0.9/)

of connecting bars in prefabricated concrete structure can be fitted as follows:

where is the bond strength of the interface between steel bar and concrete, is the standard value of concrete compressive strength, and *d* is the diameter of the steel bar.

**Figure 15.** Multiple linear regression curve. **Figure 15.** Multiple linear regression curve.

**Figure 16.** Comparison curve between fitting value and test value of bond strength.

Taking the diameter of a steel bar, 25 mm, and the thickness of the protective layer 85 mm commonly used in the connection of prefabricated concrete members as an example, the differences of bond strength calculation between domestic and foreign codes and the fitting formula in this paper are shown in Figure 17. It can be observed that the bond strength calculation value of ACI 318-11 [34] is 25% and 15.9% higher than that of AS3600 [33] and JTG 3362-2018 [27], respectively, and this indicates that there are some differences in bond strength calculation between domestic and foreign codes. Compared with AS3600 [33], JTG 3362-2018 [27], and ACI318-11 [34], the bond strength of the fitting formula (6) was increased by 62.7%, 50.9%, and 30.25%, respectively, and this indicates that the calculation results of formula (6) in China and abroad are smaller without considering the influence of cover thickness, which is not suitable for the calculation of bond strength between reinforcement and concrete of prefabricated concrete members. literature formula **Figure 16.** Comparison curve between fitting value and test value of bond strength. Taking the diameter of a steel bar, 25 mm, and the thickness of the protective layer 85 mm commonly used in the connection of prefabricated concrete members as an example, the differences of bond strength calculation between domestic and foreign codes and the fitting formula in this paper are shown in Figure 17. It can be observed that the bond strength calculation value of ACI 318-11 [34] is 25% and 15.9% higher than that of AS3600 [33] and JTG 3362-2018 [27], respectively, and this indicates that there are some differences in bond strength calculation between domestic and foreign codes. Compared with AS3600 [33], JTG 3362-2018 [27], and ACI318-11 [34], the bond strength of the fitting formula (6) was increased by 62.7%, 50.9%, and 30.25%, respectively, and this indicates that the calculation results of formula (6) in China and abroad are smaller without considering the influence of cover thickness, which is not suitable for the calculation of bond strength between reinforcement and concrete of prefabricated concrete members.

fitting formula AS3600

ACI318-11 JTG3362-2018

*5.2. Calculation Formula for the Anchorage Length*

2

4

6

8

10

Bond strength (MPa)

12

14

16

**Figure 17.** Comparison of calculation formulas of bond strength in China and abroad.

30 35 40 45 50 55 60

Concrete strength (MPa)

tween reinforcement and concrete of prefabricated concrete members.

**Figure 16.** Comparison curve between fitting value and test value of bond strength.

0 5 10 15 20 25

Formula(6)Fitting

Experimental values

Literature test value

value

Specimen number

4

6

8

10

12

Bond strength (MPa)

14

16

18

Taking the diameter of a steel bar, 25 mm, and the thickness of the protective layer 85 mm commonly used in the connection of prefabricated concrete members as an example, the differences of bond strength calculation between domestic and foreign codes and the fitting formula in this paper are shown in Figure 17. It can be observed that the bond strength calculation value of ACI 318-11 [34] is 25% and 15.9% higher than that of AS3600 [33] and JTG 3362-2018 [27], respectively, and this indicates that there are some differences in bond strength calculation between domestic and foreign codes. Compared with AS3600 [33], JTG 3362-2018 [27], and ACI318-11 [34], the bond strength of the fitting formula (6) was increased by 62.7%, 50.9%, and 30.25%, respectively, and this indicates that the calculation results of formula (6) in China and abroad are smaller without considering the influence of cover thickness, which is not suitable for the calculation of bond strength be-

**Figure 17.** Comparison of calculation formulas of bond strength in China and abroad. **Figure 17.** Comparison of calculation formulas of bond strength in China and abroad.

### *5.2. Calculation Formula for the Anchorage Length 5.2. Calculation Formula for the Anchorage Length*

When the bond strength between reinforcement and concrete was constant, the failure mode of the specimen was determined by the anchorage length. There was a critical length between the pull-out failure and the tensile failure of the reinforcement, which is called the critical anchorage length *lcr*. When the anchorage length was less than the critical anchorage length, the pull-out failure occurs.

When the anchorage length was greater than the critical anchorage length, the bearing capacity of the bonding interface was greater than the ultimate tensile load of the reinforcement and the tensile failure of the reinforcement occurs. For the calculation of the critical anchorage length of the connection reinforcement in the prefabricated concrete structure, the calculation formula of the critical anchorage length can be given according to the bond strength calculation formula as follows:

$$l\_{cr} = \frac{P\_{\rm{\tiny{\tiny{\tiny{\tiny{\tiny{\pi}}}}}}}{\pi d \tau\_{\rm{\tiny{\tiny{\pi}}}}} = \frac{\sigma\_{\rm{\tiny{\text{s}}}} d}{4 \tau\_{\rm{\tiny{\pi}}}} = \frac{\sigma\_{\rm{\tiny{\text{s}}}} d}{4(0.10 f\_{\rm{\tiny{\text{\tiny{\text{s}}}}}} - 0.35 d + 14.9)}\tag{7}$$

where *P<sup>u</sup>* is the ultimate tensile load of reinforcement, *σ<sup>s</sup>* is the ultimate strength of reinforcement, *d* is the diameter of reinforcement, and *τ<sup>u</sup>* is the bond strength.

The basic anchorage length *L<sup>a</sup>* of reinforcement specified in the code is generally determined based on the critical anchorage length calculated by the corresponding bond strength and multiplied by the corresponding safety factor. The calculation formula of the basic anchorage length of reinforcement, the ratio *La*/*Lcr* of the basic anchorage length, and the critical anchorage length in domestic and foreign codes are shown in Table 7.

The basic anchorage length of the tensile reinforcement is *la*, *n* is the concrete strength coefficient, *f<sup>y</sup>* is the design value of the tensile strength of the reinforcement, *fcu* is the standard value of the concrete compressive strength, and *f<sup>t</sup>* is the design value of the axial tensile strength of the concrete. The diameter of the anchored reinforcement is *d*, α is the shape coefficient of the anchored reinforcement, and *c* is the thickness of the protective reinforcement layer. *ψ<sup>t</sup>* , *ψ<sup>e</sup>* , and *ψ<sup>s</sup>* are the reinforcement location coefficient, coating coefficient, and variety coefficient. *λ* is a concrete variety coefficient, *Ktr* is the reinforcement coefficient, *k*<sup>1</sup> and *k*<sup>2</sup> are reinforcement location coefficient of AS3600 [33] specification, and *A* is a cross-sectional area of anchorage reinforcement. By comparison

*Materials* 

of critical anchorage length between domestic and foreign codes (Figure 18) it can be observed that the critical anchorage length corresponding to formula (7) is 0.66 times, 0.61 times, and 0.79 times of the critical anchorage length of JTG3362-2018 [27], ACI318-11 [34] and AS3600 [33] respectively, which indicates that the anchorage length requirement of the precast assembled concrete structure is far less than the standard value and special consideration should be given to it in the design and calculation.

**Table 7.** Calculation formula of basic anchorage length (domestic and foreign codes).


**Figure 18.** Comparison of critical anchorage length calculation between domestic and foreign **Figure 18.** Comparison of critical anchorage length calculation between domestic and foreign codes.

Compared with the provisions of domestic and foreign codes for the basic anchorage length of reinforcement (Figure 19), it can be observed that the basic anchorage length of JTG3362-2018 [27] is 1.61 times and 2.14 times of that of ACI318-11 [34] and AS3600 [33], respectively, compared with Figure 18 and, in Figure 18, the relationship between the critical anchorage length and the basic anchorage length in Chinese and foreign codes can be obtained. The basic anchorage length La of JTG3362-2018 [27], ACI318-11 [34], and AS3600 [33] was 1.14, 1.2, and 1.0 times of the critical anchorage length Lcr, respectively, which indicates that the Chinese code is more conservative than the foreign code. By considering 1.7 times of safety factor, the basic anchorage length of connecting reinforcement of prefabricated concrete members can be obtained. Compared with the basic anchorage length of domestic and foreign codes, the basic anchorage length of the connecting reinforcement of prefabricated bridge is 0.49 times, 0.79 times, and 1.05 times of that of JTG3362-2018 [27], ACI318-11 [34] and AS3600 [33], respectively. In Table 8, the comparison of anchor-Compared with the provisions of domestic and foreign codes for the basic anchorage length of reinforcement (Figure 19), it can be observed that the basic anchorage length of JTG3362-2018 [27] is 1.61 times and 2.14 times of that of ACI318-11 [34] and AS3600 [33], respectively, compared with Figure 18 and, in Figure 18, the relationship between the critical anchorage length and the basic anchorage length in Chinese and foreign codes can be obtained. The basic anchorage length La of JTG3362-2018 [27], ACI318-11 [34], and AS3600 [33] was 1.14, 1.2, and 1.0 times of the critical anchorage length Lcr, respectively, which indicates that the Chinese code is more conservative than the foreign code. By considering 1.7 times of safety factor, the basic anchorage length of connecting reinforcement of prefabricated concrete members can be obtained. Compared with the basic anchorage length of domestic and foreign codes, the basic anchorage length of the connecting reinforcement of prefabricated bridge is 0.49 times, 0.79 times, and 1.05 times of that of JTG3362-2018 [27], ACI318-11 [34] and AS3600 [33], respectively. In Table 8, the comparison of anchorage length between domestic and foreign codes and formula (7).

age length between domestic and foreign codes and formula (7).

**Table 8.** Comparison of anchorage length between domestic and foreign codes.

JTG-3362-2018 [27] 0.32 1.14 1.0 0.49 ACI-318-11 [33] 0.56 1.22 1.61 0.79 AS3600 [34] 0.79 1.0 2.14 1.05

codes.

**Figure 19.** Comparison of calculation of basic anchorage length in domestic and foreign codes. **Figure 19.** Comparison of calculation of basic anchorage length in domestic and foreign codes.


In short, if the current domestic codes of prefabricated concrete will still be followed **Table 8.** Comparison of anchorage length between domestic and foreign codes.

**6. Discussion** Calculation of the interface bond strength and anchorage length of steel bar within prefabricated concrete are usually considered according to the relevant provisions of the cast-in-place structure. However, the cover thickness is generally more than 50 mm larger than that of the cast-in-place structure. The complete interface failure method was used to calculate the bond strength, which may lead to the smaller calculated value of bond In short, if the current domestic codes of prefabricated concrete will still be followed for the design of the anchorage length of prefabricated members, the size of prefabricated components will be too large, which will increase the construction cost and construction difficulty. Therefore, it is necessary to modify the anchorage length of the existing specifications. By considering the safety factor of 1.7 times, it is suggested to take it as 0.5 times the design value of the existing JTG3362-2018 [27].

### strength and high values of the anchorage length are not suitable for the design and construction of prefabricated bridges. Combined with experimental research and numerical **6. Discussion**

analysis methods, this paper calculates the formula for interface bond strength and anchorage length by considering the main influencing factors and compares it with national and international codes. It is recommended that the basic anchorage length should be 18 d when the concrete strength grade is C35 or less and 15 d when the strength grade of the concrete is C40 or more. **7. Conclusions** From experimental research and numerical analysis methods, the following calculations are made. 1. It is concluded that the effect of cover thickness of the surrounding concrete is negligible for calculating interface bond strength within prefabricated structures. Calculation of the interface bond strength and anchorage length of steel bar within prefabricated concrete are usually considered according to the relevant provisions of the cast-in-place structure. However, the cover thickness is generally more than 50 mm larger than that of the cast-in-place structure. The complete interface failure method was used to calculate the bond strength, which may lead to the smaller calculated value of bond strength and high values of the anchorage length are not suitable for the design and construction of prefabricated bridges. Combined with experimental research and numerical analysis methods, this paper calculates the formula for interface bond strength and anchorage length by considering the main influencing factors and compares it with national and international codes. It is recommended that the basic anchorage length should be 18 d when the concrete strength grade is C35 or less and 15 d when the strength grade of the concrete is C40 or more.

### 16-400 increased by 85.7% and 49.9%, respectively, and 19.8% and 20.8%, respec-**7. Conclusions**

tively, when compared with C30-16-300 and C30-16-400, which indicates that for precast concrete members with larger cover thicknesses, the concrete strength grade and From experimental research and numerical analysis methods, the following calculations are made.

reinforcement diameter possess a significant influence on the ultimate load of the

2. Compared with C50-12-300 and C50-12-400, the ultimate load of C50-16-300 and C50-

bond interface between reinforcement and concrete.


**Author Contributions:** Z.H., Supervision, Investigation, Funding acquisition. Y.I.S., Methodology, Visualization, Writing-review &editing. P.Y., Formal analysis, Data curation, Software. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was financially supported by the Project of National Key Research & Development (2017YFC0806000) and the "5511" Innovation Driven Project of Jiangxi Province (20165ABC28001).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Some or all data, models, or codes generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions.

**Conflicts of Interest:** The authors declare that they have no known conflict competing for financial interest or personal relationships that could have appeared to influence the work reported in this paper.

## **References**

