## 4 Results and Analysis
### Simulation Results of Arch Culvert under Periodic Vehicle Load
#### 4.2.2 Internal Force of Steel Corrugated Pipe Structure  
Figure 26 shows the circumferential stress and maximum principal stress of the steel corrugated pipe when the wheel load passes 700 times. As shown in Figure 26a,b, a maximum circumferential stress of \(-6\) MPa is observed in the vault, indicating that a large compressive stress is generated when the vault is settled. A maximum principal stress of 3 MPa appears at the arch's waist. Due to the constraint of concrete masonry at the haunch, the deformation at the haunch is smaller than that at the vault. The stress cannot be effectively released, so the stress at the arch waist is the largest. Figure 24: Fitting curve of lateral displacement of arch’s shoulder. Figure 25: Arch’s shoulder (**lower**) deformation curve: (**a**) Settlement; (**b**) Lateral displacement. Figure 27 shows the variation in the bidirectional bending moment and axial force on the vault structure. The Mx represents the bending moment along the \(x\)-axis and the My represents the bending moment along the \(y\)-axis. As shown in Figure 27a, the negative bending moment represents pressure on the structure's inner side of the vault, whereas the positive bending moment represents the opposite. Between PB0 and PB150, the bending moment Mx appears negative at the beginning. The bending moment decreases first and then increases. Finally, the bending moment becomes positive and increases linearly. When the bending moment decreases to a negative value, Mx has a little change. After the inflection point of the negative bending moment appears, CR1-Mx becomes greater than CR5-Mx, and the difference between the two becomes increasingly larger. This is because the CR1 settlement is larger, so the bending moment is larger. In PB700, CR1-Mx and CR5-Mx were 28 kN\(\cdot\)m and 18 kN\(\cdot\)m, respectively, with a difference of approximately 10 kN\(\cdot\)m. Between PB0 and PB150, the bending moment My shows little change. It subsequently begins to grow linearly, but at a much smaller rate than Mx. The difference between CR1-My and CR5-My is small because the steel corrugated pipe vault is uniformly settled along the axis, and My is approximately 8 kN\(\cdot\)m at PB700. As shown in Figure 27b, the axial force represents the pressure at the thin-walled neutral plane of the structure, and negative values indicate compression. The Nx represents the axial force along the \(x\)-axis, and the Ny represents the axial force along the \(y\)-axis. The initial axial force Nx is greater than Ny. Because the structure is circumferential compression  
Figure 27: The variation curve of internal forces in the vault: (**a**) Moment: Mx, My; (**b**) Axial force: Nx, Ny. Figure 26: Structure internal force contour: (**a**) PB700 circumferential stress; (**b**) PB700 maximum principal stress. in the initial state, the force in the circumferential direction is small. Therefore, CR1-Nx is smaller than CR5-Nx. Combining the change in bending moment with the observation that CR1-Mx is greater than CR5-Mx at this point indicates that the deformation of structure CR1 is very large and bending deformation takes precedence, whereas the compression of the arch structure takes precedence at the CR5 position. The axial pressure is not excessively high because CR1 generates an overall settlement, according to the data analysis of CR1-Ny and CR5-Ny. However, CR5, as the edge of the motor vehicle lane and close to the sidewalk, does not generate an overall settlement. Therefore, the axial pressure increment in the \(y\)-direction is considerable. Figure 28 summarizes the variation curves of the axial force around the steel corrugated back end and front end. The solid sign curve shows that the circumferential stress of the back-end pipe decreases rapidly from SP6 to SP1 in the order of SP6 > SP1 > WA1 > CR1 (the position of points can be seen in Figure 11b). It can be observed from Figure 21 that the structure cannot effectively transfer circumferential axial force down at the maximum bending deformation.