## 3 Test Results and Discussion
### Barrier Maximum Moment Associated with the Diagonal Tension Crack in the Deck Slab Overhang  
Upon the cracking of a reinforced concrete joint, the joint behaves as a composite structure composed of both concrete and reinforcement, making its analysis significantly more complex than that of homogeneous materials [69]. The failure at the joint may occur due to diagonal tension cracks, which arise when tensile stresses from external flexural moments are not adequately countered by the reinforcement, and when appropriate reinforcement configurations are lacking. Figure 14a illustrates the diagonal tensile crack in the deck overhang under the barrier wall due to the influence of a transverse load, \(F_{t}\), while Figure 14b depicts a free-body diagram of the associated forces that develop the diagonal crack. Figure 14c depicts the stress distribution at the corner joint of the barrier wall resulting from the applied moments in both the wall and the deck slab. The bending stress, \(\sigma_{x}\), reveals a high tensile stress at the inner corner of the joint, suggesting the potential for crack initiation at this location under minimal loading conditions. Conversely, the stresses perpendicular to the bending stresses, \(\sigma_{y}\), generate tensile stresses that lead to diagonal cracking across the corner joint. Such diagonal tension cracks can precipitate abrupt failure unless suitable reinforcement arrangements are implemented in these critical areas [69]. As shown in Figure 14a, the applied transverse load, \(F_{t}\), produces a moment at the centerline of the deck overhang (i.e., mid-depth of the deck overhang), \(M_{b}\), which is equated with the deck overhang internal couple as follows:  
\[M_{b}=F_{t}\left(H_{t}+0.5t_{d}\right)=\left(C_{d}-0.5F_{t}\right).\ 0.9d_{d} \tag{2}\]  
where \(H_{t}\) is the height of the applied transverse vehicle impact load from the top surface of the deck slab overhang, \(C_{d}\) is the resultant compressive force in the deck section at the  
Figure 14: Forces and stresses due to diagonal tension failure at the barrier wall deck slab junction. (**a**) Internal forces; (**b**) Free-body diagram; (**c**) Stress distribution along and normal to the diagonal crack indicating tensile stresses (positive sign) and compressive stresses (negative sign); (**d**) Diagonal crack length measurement. inner side of the barrier wall, \(\rm t_{d}\) is the deck overhang thickness, \(\rm d_{d}\) is the effective depth of the deck overhang, and the distance between the resultant compressive and tensile forces is approximately taken as \(\rm 0.9d_{d}\). Analyzing forces parallel to the diagonal crack in Figure 14b leads to the following equation. \[C_{\rm b}.\rm sin\alpha=(C_{d}-0.5F_{t}).\rm cos\alpha \tag{3}\]  
where \(\alpha\) is the inclination of the diagonal tension crack and \(\rm C_{b}\) is the resultant compressive force in the barrier wall. To calculate the diagonal crack angle between \(\rm C_{b}\) and \(\rm(C_{d}-0.5F_{t})\) forces, it is conversative to assume their locations are approximately 0.9 times the barrier thickness at the barrier-deck interface and 0.9 times the deck slab thickness at the inner face of the barrier wall, respectively, as depicted in Figure 14d. Equation (3) is rearranged as follows. \[C_{b}=\frac{C_{d}-0.5F_{t}}{\tan\alpha} \tag{4}\]  
By analyzing forces normal to the diagonal crack in Figure 14b, the tensile force creating the diagonal crack, T, is calculated as follows:  
\[\rm T=C_{b}.\rm cos\alpha+(C_{d}-0.5F_{t}).\rm sin\alpha \tag{5}\]  
Substituting Equation (4) in Equation (5) yields the following equation. \[(C_{d}-0.5F_{t})=T.\rm sin\alpha \tag{6}\]  
The modulus of rupture of concrete, \(\rm f_{r}\), is calculated as follows [70]. \[f_{r}=0.6\lambda\sqrt{f^{\prime}_{c}} \tag{7}\]  
where \(\lambda=\) concrete density factor and \(f^{\prime}_{c}\) = concrete compressive strength.