*4.1. Current Models*

Previously, variety formulas for the ultimate bond strength of anchoring or lapping rebar were provided in GB 50010-2010 [19], AS-3600 [25], ACI 318-05 [26], and Wu [27], as shown in Equations (3)–(6).

$$
\tau\_{\rm u} = (0.82 + 0.9 \frac{d}{l\_a})(1.6 + 0.7 \frac{c}{d} + 20 \rho\_{sv})f\_t \tag{3}
$$

$$
\tau\_u = 0.265(\frac{c}{d} + 0.5)\sqrt{f\_u} \tag{4}
$$

$$
\tau\_u = 0.083(1.2 + 3\frac{c}{d} + 50\frac{d}{l\_a})\sqrt{f\_u} \tag{5}
$$

$$
\pi\_u = (0.36 + 30.81 \frac{d}{l\_s})(2.48 - 6.2 \frac{d}{D} + 46.9 \rho\_{sv})f\_t \tag{6}
$$

where *τ*<sup>u</sup> is the ultimate bond strength, *d* is the rebar diameter, *l*<sup>a</sup> is the anchorage length, *l*<sup>s</sup> is the lapped length, *c* is the concrete cover thickness, *ρ*sv is the spiral hoop ratio, *f* <sup>u</sup> is the ultimate compressive strength of concrete or grout mortar, and *f* t is the ultimate tensile strength of concrete or grout mortar, which can be established from Equation (7) [28].

$$f\_t = 0.26f\_u^{2/3} \tag{7}$$

The comparison of the ultimate bond strength between various models and test data is shown in Figure 11. As the high-strength grouted mortar and the spiral hoop effect is not considered in AS-3600 [25] and ACI 318-05 [26], the estimated values from these two specifications are much smaller than the tested values, and the maximum error reaches 20%. Wu's model [27] shows obvious difference with the test data as his model is developed based on the experiment of the long-lapped-rebar splices. GB 50010-2010 [19] is a little underestimated as the maximum error reaches −23%. For predicting the behavior of the short-lapped-rebar splice, it is necessary to develop a more accurate model for the ultimate bond strength.

**Figure 11.** *Cont*.

**Figure 11.** Comparison of ultimate bond strength between various models and test data. (**a**) GB50010- 2010 [19] (**b**) AS-3600 [25] (**c**) ACI 318-05 [26] (**d**) Wu [27].

#### *4.2. A Semi-Empirical Model for the Ultimate Bond Strength*

Figures 12–14 illustrate the mechanical mechanism of the short-lapped-rebar splice. Similar to conventional splices, the shear force is carried by the friction force, mechanical interlocking force, and chemical cemented force. However, in conventional splices, cracks easily appear on the concrete near the zone of the connection. In the short-lapped-rebar splice, the concrete is constrained by the spiral hoops, which can effectively limit the development of cracks and increase the ultimate bond strength. As illustrated in Figure 12, the shear force is transferred from the anchored rebar to the embedded rebar through grout, metal duct, and concrete.

**Figure 12.** Bond mechanism of the short-lapped-rebar splices.

Figure 13 illustrates the stress distribution of the interaction surface between the anchored rebar and the grout mortar along the longitudinal direction. Under the action of external anchorage force *F*, the interaction surface between the rebar rib and grout mortar is subjected to extrusion stress *p* and friction stress *μp*. On the basis of equivalent conditions, the stress in the horizontal and circumferential direction can be obtained, as shown in Equation (8).

$$\begin{cases} \tau = p \sin \alpha + \mu p \cos \alpha \\ \quad q = p \cos \alpha - \mu p \sin \alpha \end{cases} \tag{8}$$

where *τ* is the bond strength between the rebar and grout mortar; *q* is the circumferential compressive stress; *p* is the extrusion stress between the rebar and the grout mortar; *μ* is

the friction coefficient between concrete and rebar, which is 0.3; and *α* is the inclined angle between rebar rib and grout mortar, which is 45 degrees [19,29].

**Figure 13.** Stress distribution of the interaction surface.

**Figure 14.** Stress distributed on the cross section.

The stress distributed on the cross section of the short-lapped-rebar splice is shown in Figure 14. Note that the assumption that the stress and cracks only spread to the outer bound of the spiral hoop is made. The constraining effect of the spiral hoop can be considered as a thick-walled cylinder with two rebar subjected to uniformly distributed stress, as shown in Figure 14. Based on the thick-walled cylinder theory [29], the circumferential tensile stress of the grout mortar or concrete at a certain point can be obtained from Equation (9).

$$\begin{cases} \begin{array}{c} q\pi d = \frac{1}{2} q\_1 \pi D \\ \sigma\_\theta = \frac{q\_1 \left(D/2\right)^2}{\left(c + D/2\right)^2 - \left(D/2\right)^2} [1 + \frac{\left(c + D/2\right)^2}{r^2}] \end{array} \tag{9}$$

where *q*<sup>1</sup> is the compressive stress of the grout mortar, *c* is the protective layer of concrete, *D* is the inner diameter of the spiral hoop, and *r* is the distance from the certain point to the center of specimen, *σ*<sup>θ</sup> is the circumferential tensile stress at the certain position.

Similar with Xu's method [30], a constraint coefficient, *β*, is utilized to consider the constraint effect of the spiral hoop, as shown below.

$$
\sigma\_{\emptyset}|\_{r=D/2} = \pounds f\_{\emptyset} \tag{10}
$$

where *β* is the constraint coefficient of the spiral hoop, and it is suggested to be 1.2, according to Xu's test [30].

On the basis of Equations (7)–(10), the ultimate bond strength can be obtained, as shown below.

$$\pi\_{\rm u} = 1.12 \frac{D}{d} \frac{\left(c + D/2\right)^2 - \left(D/2\right)^2}{\left(c + D/2\right)^2 + \left(D/2\right)^2} f\_l \tag{11}$$

As shown in Figure 15 and Table 5, there is a significant error between theoretical and tested values. The reason for that is that the influences of the lapped length, the rebar diameter, and the protective layer are not fully considered. As a result, another coefficient for considering the effects of the lapped length, the rebar diameter, and the protective layer is proposed, as shown below.

$$
\pi\_u' = \eta \,\pi\_u \tag{12}
$$

where *τ<sup>u</sup>* is the ultimate bond strength, and *η* is the affecting coefficient considering the effects of the lapped length, the rebar diameter, and the protective layer.

**Figure 15.** Comparison of the ultimate bond strength between estimated values and test data (**a**) Theoretical model (**b**) Semi-empirical model.

From the results of the test, the affecting coefficient has good relations with the rebar diameter, the ratio of the rebar diameter to the lapped length, and the ratio of the protective layer to the rebar diameter, as shown in Equation (13).

$$\eta = \begin{cases} \begin{pmatrix} 0.08\frac{d}{l\_l} + 0.007\frac{c}{d} + 0.02 \end{pmatrix} d & \frac{c}{d} \le 5.0\\ \begin{pmatrix} 0.08\frac{d}{l\_l} + 0.055 \end{pmatrix} d & \frac{c}{d} > 5.0 \end{cases} \tag{13}$$

where *τ*<sup>u</sup> is the ultimate bond strength, *d* is the anchored rebar diameter, *l*<sup>l</sup> is the lap length, and *c* is the concrete cover thickness.

Based on Equations (7)–(13), the ultimate bond strength of the short-lapped-rebar splice can be obtained. The comparison of the ultimate bond strength between the proposed models and the tested values is shown in Figure 15 and Table 5. The average error estimated from the proposed model is only 4.49%, and the maximum error varies from −15% to 15%. Clearly, the proposed model is more accurate than the existing models.


**Table 5.** Comparison of the ultimate bond strength between theoretical and semi-empirical values and test data.

Note: *τ*exp represents the values of tested data, *τ<sup>u</sup>* represents the values of theoretical values, *τ*'*u*represents the values of semi-empirical model.

#### **5. Conclusions**

In this study, a new type of connection (the short-lapped-rebar splices) in precast concrete structure was developed, and the failure modes, strain distribution, bond slip behavior, and bond strength of the connection were experimentally investigated. A semiempirical model was proposed to predict the ultimate bond strength of the short-lappedrebar splices. The following conclusions are made:

(1) Two different types of failure modes for the short-lapped-rebar splices, namely, the fracture of rebar and the pull-out failure of rebar, are found.

(2) The short-lapped-rebar splices have higher ultimate strength or ultimate bond strength than that of the conventional lapped splices.

(3) The stress of the anchored rebar in the short-lapped-rebar splices is distributed symmetrically along the longitudinal direction, and the maximum bond stress is approximately twice that of the conventional specimens.

(4) A semi-empirical model considering the effect of the spiral hoop is developed to predict the ultimate bond strength of the short-lapped-rebar splices, which shows good agreement with test data.

Note that the cyclic behavior of the short-lapped-rebar splices and structural reliability of the short-lapped-rebar splices are not included in this study; the determination of the lapped length of the rebar should be further studied before the design of the short-lappedrebar splices.

**Author Contributions:** Conceptualization, Q.L. and X.L.; methodology, Q.L.; validation, Q.L., X.L. and R.C.; formal analysis, Q.L.; investigation, Q.L.; resources, X.L.; data curation, R.C.; writing original draft preparation, Q.L.; writing—review and editing, Z.K.; visualization, T.X.; supervision, Z.K.; project administration, Q.L.; funding acquisition, Q.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by (1) [Anhui University of Technology Foundation] grant number [QZ202015]; (2) [National Natural Science Foundation of China] grant number [52078042].

**Data Availability Statement:** The data presented in this study are available upon request from the corresponding author.

**Acknowledgments:** All the authors are very grateful to the anonymous reviewers and editors for their careful review and critical comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
