# Research on the Safety Assessment of Corbel Beams of Reservoir Radial Gates Based on Real States
## 3 Finite Element Simulation and Safety Review of the Corbel Beam of the Radial Gate
### Analysis of Calculation Results and Safety Review  
(1) Safety assessment by theoretical mechanics method  
Through theoretical calculation, it can be seen that the bending moment at the fixed end of the corbel in model 1 is M = 949.34 kN.m; the bending moment at the fixed end of the corbel beam in model 2 is M = 648.72 kN.m. This is because the corbel beam forms a joint force structure with the gate pier, and its internal bending moment is much smaller than that of the corbel cantilever structure. Thus, its ability to bear the supporting force and other loads is much higher than that of the corbel. Therefore, the equivalent coefficient a = 648.72/949.34 = 0.68 can be derived from the fixed end bending moment of model 1 and model 2, and brought into Formula (4) for calculation. The calculation process and results are shown in Table 4. In Table 4, \(0.55f_{kb}bB/(e_{0}/e_{0}+0.20)\) represents the maximum allowable stress, and the safety of the project is determined according to the comparison between the actual stress and this stress. Based on the safety review of the structure, it is evident that the corbel beam structure is in normal working condition and does not require any danger removal or reinforcement. Due to the simple structure of the radial gate corbel beam of small projects, the effect of stress concentration is not obvious, and this method can be directly used for safety review under the premise that no obvious defects are found in the field inspection, which provides the theoretical basis for the next step of the project to de-risk and strengthen. Figure 10: Model of the joint force structure of the corbel beam and gate pier. 2. Structural safety assessment of corbel beams by finite element simulation based on real states The stress state of the joint stressed structure of the corbel beam and gate pier under the real state is more complicated. In addition to the changes in concrete material parameters and stress state, it is also necessary to consider the internal stresses generated by the extrusion and deformation of concrete under the action of self-weight and the support strut thrust. For reinforced concrete, we use the maximum tensile stress to analyze its stress hazard region [39; 40]. Subsequently, detailed post-processing of the equivalent force diagrams for the corbel beam and the gate pier was conducted, as shown in Figure 11.  
From Figure 11, it can be seen that the corbel beam forms a stress concentration at the connection of the gate pier under the action of the support strut thrust and self-weight, and the stress state is symmetrically distributed along the midpoint of the corbel beam. For the gate pier, the stress in the area connecting with the beam is relatively large, and the stress concentration area is mainly in the first half of the thrust side of the beam. For the gate pier, the stress in the area connecting with the beam is relatively large, and the stress concentration area is mainly in the first half of the thrust side of the beam. The remaining portion of the gate pier is solely influenced by gravity, resulting in an even distribution of stress and no safety risk associated with stress concentration. It should be noted that the probe annotation shows that the most dangerous part of the joint stressed structure of the corbel beam and gate pier in this project is at the midpoint of the connection part of the gate pier where the corbel beam is subject to the support strut thrust surface. Once the cracks appear here, deep cracks may be generated with the stress concentration, thus affecting the safety of the overall structure. Combined with the site inspection, the surface of the corbel beam structure appears intact, with no signs of reinforcement corrosion or cracks and defects in the stress concentration area. Thus, the working condition of the corbel beam structure can be judged to be normal. Then, the corbel beam structure was reviewed for safety, and the nodal force at the cross-section (950 mm from the midpoint of the gate pier) where the corbel beam was