# Fatigue Behavior of Prestressed Concrete Beam for Straddle-Type Monorail Tracks
## 3 Finite Element Model  
The three-dimensional finite element model was established based on the commercial finite element program ANSYS. Concrete was simulated using a 3D eight-node solid element, namely SOLID 65. As shown in Figure 9, a model proposed by [PERSON] [33] was applied to describe the compression constitutive relationship of concrete as follows:  
Rising branch (\(\varepsilon\leq\varepsilon_{0}\)):  
\[\sigma=f_{c}[2\frac{\varepsilon}{\varepsilon_{0}}-(\frac{\varepsilon}{ \varepsilon_{0}})^{2}] \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Test** & \multicolumn{3}{c}{**Fatigue Load**} & \multicolumn{8}{c}{**Post-Fatigue Static Load**} \\ \hline grade & min & max & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ Load (kN) & 39.97 & 347.95 & 0 & 47.3 & 142.0 & 236.6 & 331.3 & 425.9 & 473.3 & 236.6 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Test loading value.  
Figure 8: Procedure of the test.  
Figure 7: Schematic of monorail vehicle load (unit: mm).  
Falling branch (\(\varepsilon_{0}\leq\varepsilon\leq\varepsilon_{u}\)):  
\[\sigma=f_{c}[1-0.15\frac{\varepsilon-\varepsilon_{0}}{\varepsilon_{u}- \varepsilon_{0}}] \tag{3}\]  
where \(f_{c}\) is the peak stress, i.e., prism compressive strength of concrete; \(\varepsilon_{0}\) is the strain correspond to peak stress; \(\varepsilon_{u}\) is the ultimate compression strain of concrete.  
The tensile constitutive relationship of the concrete was defined in two parts: the elastic part from zero to the ultimate value and the descending part determined by Equation (4).  
\[\sigma_{t}=\frac{\rho_{t}}{\alpha_{t}(x_{t}-1)^{1.7}+x_{t}}E_{c}\varepsilon_{t} \tag{4}\]  
where \(\rho_{t}=f_{t}/(E_{c}\varepsilon_{tr})\); \(\sigma_{t}\) and \(\varepsilon_{0}\) are the tensile stress and tensile strain respectively; \(f_{t}\) is the tensile strength; \(\varepsilon_{tr}(=0.0001)\) is the strain when the \(\sigma_{t}\) reaches \(f_{t}\); \(\alpha_{t}\) is a parameter related to the shape of descending part of tensile constitutive relationship; \(x_{t}=\varepsilon_{t}/\varepsilon_{tr}\) and \(x_{t}>1\) for the descending part.  
The reinforcement and prestressed tendons were simulated using a 3D two-node link element, namely LINK 180, as shown in Figure 10a,b. The modulus of elasticity was regarded as constant. No relative slippage between concrete and reinforcement was considered, as well as prestressed tendons [34]. The prestressed forces in tendons were applied by defining an initial strain. Tendons had been united with the concrete by establishing reasonable constraint equations between tendons nodes and concrete nodes.  
Due to the symmetry of the specimen, only a half of the structure was modelled, with symmetric constraints on the midspan section, as shown in Figure 10c. A schematic of the symmetric constraints is shown in Figure 11.  
The load applied through the electrohydraulic servo actuator was modeled as a knife edge load. To eliminate the effects of stress concentration, a rigidity block was modeled at the loading points as well as the support position.  
Figure 10: Finite element model based on the commercial finite element program ANSYS: (**a**) Reinforcement; (**b**) prestressed tendons; (**c**) 1/2 beam model.  
Figure 9: Constitutive relation of concrete.