**6. Discussion**

It is noteworthy that the provided relationship is limited to the upper yield points of the elastic stage. In order to validate the results and evaluate the accuracy of the proposed formula, a laboratory model was created. The experimental model was performed under different loads, and the maximum deflection was calculated in each case. Additionally, the crack width was measured from the experimental test. The results are reported in Tables 6 and 7. The load-deflation comparisons of tested specimen are shown in Figure 14. Furthermore, the crack width was approximated by the proposed model for each load intensity and compared with the experimental results of specimen 1 and 2.


**Table 6.** Validation of the proposed relationship for crack width prediction of specimen 1.



**Figure 14.** Load deflection comparison of tested specimens.

**Figure 14.** Load deflection comparison of tested specimens. As observed, the crack widths obtained from the simplified model are appropriately close to the results of the experiments. To better observe the accuracy, the results of the proposed and experimental models are depicted in Figures 15 and 16. The white bars show the crack width that As observed, the crack widths obtained from the simplified model are appropriately close to the results of the experiments. To better observe the accuracy, the results of the proposed and experimental models are depicted in Figures 15 and 16. The white bars show the crack width that resulted from Equation (14) and the gray bars depict the experimental results.

(specimen NCN-1) slab whereas the other was a composite box girder with a prefabricated prestressed concrete slab (specimen NCN-2). Due to using prestressed concrete, the proposed formula yielded slightly higher results than the experimental results regarding NCN-2; however, in the case of NCN-1, the results of the proposed formula were in good agreement with the experimental results as shown in Figure 17. According to their research, no great increase of the crack width was found when load increased from 312 to 700 kN; but, in this stage, there was substantial increase in the amount of cracking. This stage is the stabilization process of the crack, which means the crack distribution experiences a transition from its randomly distributed state to a quasi-uniformly distributed state. Similarly, Xu et al. [12] manufactured and tested a continuous double composite girder (DCG) to study the mechanical behavior in negative flexural region. Their comparison results

**Figure 15.** Comparison of crack width (specimen 1).

To further evaluate the model, a few studies were selected from the literature to compare with

resulted from Equation (14) and the gray bars depict the experimental results.

are shown in Figure 18.

**Figure 14.** Load deflection comparison of tested specimens.

resulted from Equation (14) and the gray bars depict the experimental results.

As observed, the crack widths obtained from the simplified model are appropriately close to the results of the experiments. To better observe the accuracy, the results of the proposed and experimental models are depicted in Figures 15 and 16. The white bars show the crack width that

To further evaluate the model, a few studies were selected from the literature to compare with the proposed formula. Su et al. [54] experimentally analyzed two different types of continuous composite box girders. One specimen was a conventional composite box girder with cast in situ (specimen NCN-1) slab whereas the other was a composite box girder with a prefabricated prestressed concrete slab (specimen NCN-2). Due to using prestressed concrete, the proposed formula yielded slightly higher results than the experimental results regarding NCN-2; however, in the case of NCN-1, the results of the proposed formula were in good agreement with the experimental results as shown in Figure 17. According to their research, no great increase of the crack width was found when load increased from 312 to 700 kN; but, in this stage, there was substantial increase in the amount of cracking. This stage is the stabilization process of the crack, which means the crack distribution experiences a transition from its randomly distributed state to a quasi-uniformly distributed state. Similarly, Xu et al. [12] manufactured and tested a continuous double composite

**Figure 16.** Comparison of crack width (specimen 2). **Figure 16.** Comparison of crack width (specimen 2).

To further evaluate the model, a few studies were selected from the literature to compare with the proposed formula. Su et al. [54] experimentally analyzed two different types of continuous composite box girders. One specimen was a conventional composite box girder with cast in situ (specimen NCN-1) slab whereas the other was a composite box girder with a prefabricated prestressed concrete slab (specimen NCN-2). Due to using prestressed concrete, the proposed formula yielded slightly higher results than the experimental results regarding NCN-2; however, in the case of NCN-1, the results of the proposed formula were in good agreement with the experimental results as shown in Figure 17. According to their research, no great increase of the crack width was found when load increased from 312 to 700 kN; but, in this stage, there was substantial increase in the amount of cracking. This stage is the stabilization process of the crack, which means the crack distribution experiences a transition from its randomly distributed state to a quasi-uniformly distributed state. Similarly, Xu et al. [12] manufactured and tested a continuous double composite girder (DCG) to study the mechanical behavior in negative flexural region. Their comparison results are shown in Figure 18.

**Figure 18.** Comparison of crack width resulted from the proposed formula and experiments for the

**Figure 17.** Comparison of crack width resulted from the proposed formula and experiments for the

NCN-1 sample in Su et al.'s study [54].

DCG sample in Xu et al.'s study [12].

**Figure 16.** Comparison of crack width (specimen 2).

**Figure 16.** Comparison of crack width (specimen 2).

*Materials* **2019**, *12*, x FOR PEER REVIEW 19 of 26

**Figure 17.** Comparison of crack width resulted from the proposed formula and experiments for the **Figure 17.** Comparison of crack width resulted from the proposed formula and experiments for the NCN-1 sample in Su et al.'s study [54]. **Figure 17.** Comparison of crack width resulted from the proposed formula and experiments for the NCN-1 sample in Su et al.'s study [54].

**Figure 18.** Comparison of crack width resulted from the proposed formula and experiments for the **Figure 18.** Comparison of crack width resulted from the proposed formula and experiments for the DCG sample in Xu et al.'s study [12]. **Figure 18.** Comparison of crack width resulted from the proposed formula and experiments for the DCG sample in Xu et al.'s study [12].

#### DCG sample in Xu et al.'s study [12]. **7. Conclusions**

The current study aimed at investigating the behavior of ACHPCBG-bridge utilizing experimental models. Therefore, a vertical loading was gradually applied to the beam, and the maximum deflection along the beam was observed at certain points. Additionally, the cracking mechanism was investigated by the experimental model and the maximum width of the cracks were measured by a digital crack gauge on the beam surfaces. Finally, a simplified formula was developed approximating the crack width as a function of deflection

The main conclusions drawn from the current study are as follows:


Since the proposed formula is presented as an explicit function, it can be practically used to predict crack width. The proposed formula can also be used as a limit state function in reliability analysis to calculate the probability of failure for ACHPCBG-bridge. As the experiment models were designed by a 1:4 ratio based on 25 m prototype model, the application of the proposed method is limited to the models within a similar range of parameters. This issue can be investigated by the authors in future research.

**Author Contributions:** Conceptualization, B.G.G. and Y.-Q.X.; formal analysis, B.G.G.; data curation, B.G.G., Z.Q., and S.-H.G.; investigation, B.G.G., Y.-Q.X. and S.-H.G.; methodology, B.G.G.; project administration, Y.-Q.X. and B.G.G.; resources, Y.-Q.X. and B.G.G.; software, B.G.G.; validation, B.G.G., Y.-Q.X., S.-H.G. and Z.Q.; visualization, B.G.G.; writing—original draft, B.G.G.; writing—review and editing, B.G.G.; funding acquisition, Y.-Q.X.; supervision, Y.-Q.X.

**Acknowledgments:** The authors appreciate the financial support from Zhejiang University, the Cyrus Tang Foundation in China, and National Natural Science Foundation of China (NSFC, No. 51541810). Additionally, the cooperation of staff members of the structural laboratory of Quzhou University and the help from Dr. Ying Yang in the experimental research are highly appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
