## 6 Finite Element Analysis of Prestressed Hollow Slab
### Establishment of the Finite Element Model for the Prestressed Hollow Slab Reinforced Components
#### 6.3.2 Theoretical Calculation Model of the Elastic Moduli of the Materials  
Xiao [41] performed a freeze-thaw test on concrete by controlling related influencing factors. The constitutive law for the dynamic modulus of elasticity of the concrete was analyzed under freezing and thawing conditions, and an equation for the change in this dynamic modulus of elasticity with the number of freeze-thaw cycles was derived. \[E_{c}(\widetilde{n})=C_{2}E_{c}(0)e^{\lambda_{2}\widetilde{n}} \tag{2}\]  
Figure 18: Finite element model of the prestressed hollow slab reinforced component. In the formula, \(E_{c}(\widetilde{n})\) indicates the residual dynamic elastic modulus of the concrete as a function of the freeze-thaw cycles. \(E_{c}(0)\) indicates the initial elastic modulus of the concrete. \(C_{2}\) and \(\lambda_{2}\) are the material characteristic parameters. In addition, the dynamic and static elastic moduli of concrete can be interconverted, and the conversion formula was presented in [42]. \[\hat{E}_{c}(\widetilde{n})=-0.52+0.77E_{c}(\widetilde{n}) \tag{3}\]  
where \(\hat{E}_{c}(\widetilde{n})\) and \(E_{c}(\widetilde{n})\) indicate the static and dynamic elastic moduli of the concrete, respectively. 3.3 Finite Element Analysis of Fatigue Characteristics of Prestressed Reinforcement Components under Different Freeze-Thaw Cycles  
A simulation analysis of the fatigue characteristics of the finite element model for the reinforced prestressed hollow slabs subjected to freezing and thawing corrosion was performed according to the theoretical decay model for the strength and elasticity of the concrete materials, which were primarily subjected to freezing and thawing. The fatigue properties were simulated and analyzed by changing the material parameters of the concrete materials under freeze-thaw erosion. The initial compressive strength of the concrete obtained from the experiment was 31.6 MPa. After 50, 100, 150, and 200 freeze-thaw cycles, the residual compressive strengths of the concrete were calculated to be 28.63 MPa, 27.23 MPa, 25.91 MPa, and 24.64 MPa, respectively, using Formula (1). The modulus of elasticity of the concrete was set based on the theoretical decay model for the concrete materials in the freeze-thaw cycle. The residual dynamic modulus of the concrete after the freeze-thaw cycle was calculated using Formula (2) and converted into a static elastic modulus using Formula (3). The initial elastic modulus was 34,500 MPa, and the residual static elastic moduli of the concrete after 50, 100, 150, and 200 freeze-thaw cycles were 20,577 MPa, 18,084 MPa, 15,582 MPa, and 13,412 MPa, respectively. In addition, the material degradation in the steel bars under the action of freeze-thaw cycles was minute; therefore, the effects of freezing and thawing on the mechanical properties of the steel bars were not considered. In the fatigue cyclic loading simulation, 5.0 \(\times\) 10\({}^{5}\) cycles of fatigue loading were applied for different freeze-thaw cycles at a 0.8 stress level. The vertical displacement program of the reinforced components is shown in Figure 19.  
Figure 19 shows that the reinforced specimens primarily produced large displacement deformations in the middle of the span under the double effects of freeze-thaw cycles and fatigue loads. Thus, the midspan node 576 was selected as the reference node, and the deflection values at this reference node for different freeze-thaw cycles and 20.0 \(\times\) 10\({}^{5}\) fatigue cycles of loading were extracted, as shown in Figure 20. It was shown that the midspan deflection in the reinforced component that had not undergone freeze-thaw cycle corrosion was 1.12 mm after 20.0 \(\times\) 10\({}^{5}\) cycles of fatigue loading. The deflection values of the reinforced components after 50, 100, 150, and 200 freeze-thaw cycles and fatigue loading  
Figure 19: Vertical displacement program of reinforced components with different freeze–thaw cycles: (**a**) 100 freeze–thaw cycles; (**b**) 150 freeze–thaw cycles.