# Design Approach for Composite Pavement Structure Incorporating Reflective Crack Considerations
## 3 Derivation of Calculation Formula for Temperature Warping Stress
### Formula for Calculating Temperature Warping Stress  
In Equation (9), \(N_{t}\) is as shown in Equations (11) and (12):  
\[N_{t}=\alpha_{c}E_{1}T_{g}h_{1}h_{2}\varphi_{N} \tag{11}\]  
\[\varphi_{N}=Ag(E_{1}/E_{2})g(h_{1})g(h_{2})g(k) \tag{12}\]  
where \(\alpha_{c}\) depicts the shrinkage coefficient, taken as 0.00001; \(T_{g}\) is the temperature gradient, taken as 50 \({}^{\circ}\)C/m; \(\varphi_{N}\) represents the correction factor for the axial force of the section; \(A,g(E_{1}/E_{2}),g(h_{1}),g(h_{2}),g(k)\) is the fitted parameter or formula.  
In Equation (9), \(M_{t}\) is as shown in Equations (13) and (14):  
\[M_{t}=\alpha_{c}E_{1}T_{g}h_{1}{}^{2}h_{2}\varphi_{M} \tag{13}\]  
\[\varphi_{M}=Bf(E_{1}/E_{2})f(h_{1})f(h_{2})f(k) \tag{14}\]  
where \(\varphi_{M}\) is the section bending moment correction factor; \(B,f(E_{1}/E_{2}),f(h_{1}),f(h_{2}),f(k)\) is the fitted parameter or formula.  
Using the Levenberg-Marquardt algorithm, a program was written to fit a total of 22 parameters after 185 iterations. The correlation coefficient between the simulated and measured values was determined to be 0.99013, indicating a strong correlation. Furthermore, the root mean square error was calculated to be 0.08838, indicating a small deviation between the fitted values and the actual values. The fitted value for \(\sigma_{t0}\) can be found in Equations (15)-(19).  
\[\sigma_{t0}=\left(\frac{N_{t}}{h_{1}}+\frac{6M_{t}}{h_{1}^{2}}\right) \tag{15}\]  
\[\begin{array}{l}N_{t}=\alpha_{c}E_{1}T_{g}h_{1}h_{2}Ag\left(\frac{E_{1}}{E_{ 2}}\right)g(h_{1})g(h_{2})g(k)\\ =0.00001\times 50E_{1}h_{1}h_{2}Ag\left(\frac{E_{1}}{E_{2}}\right)g(h_{1})g(h_{2} )g(k)\end{array} \tag{16}\]  
Figure 9: Temperature stress at different k values.  
\[\left\{\begin{array}{c}A=0.8119\\ g(E_{1}/E_{2})=0.9344+0.8849e^{0.9449(E_{1}/E_{2})}\\ g(h_{1})=0.9162+0.9037e^{0.9874h_{1}}\\ g(h_{2})=0.8119h_{2}{}^{1.1796}\\ g(k)=0.8119k^{-0.0726}\end{array}\right. \tag{17}\]  
where \(A\) is a constant parameter; \(g(E_{1}/E_{2})\) represents the influence of the ratio between the asphalt layer modulus and the cement slab modulus on \(N_{t}\); \(g(h_{1})\) represents the influence of the thickness of the asphalt layer on \(N_{t}\); \(g(h_{2})\) represents the influence of the thickness of the cement slab on \(N_{t}\); \(g(k)\) represents the influence of the subgrade reaction modulus on \(N_{t}\).  
\[\begin{array}{c}M_{t}=\alpha_{c}E_{1}T_{g}h_{1}{}^{2}h_{2}Bf(E_{1}/E_{2})f(h _{1})f(h_{2})f(k)\\ =0.00001\times 50E_{1}h_{1}{}^{2}h_{2}Bf(E_{1}/E_{2})f(h_{1})f(h_{2})f(k) \\ \left\{\begin{array}{c}B=0.8887\\ f(E_{1}/E_{2})=1.0147+0.8679e^{0.8304(E_{1}/E_{2})}\\ f(h_{1})=0.9589+0.9291e^{0.9691h_{1}}\\ f(h_{2})=0.8887h_{2}{}^{0.9844}\\ f(k)=0.8887k^{-0.2117}\end{array}\right.\end{array} \tag{18}\]  
where \(B\) is a constant parameter; \(f(E_{1}/E_{2})\) represents the influence of the ratio between the asphalt layer modulus and the cement slab modulus on \(M_{t}\); \(f(h_{1})\) represents the influence of the thickness of the asphalt layer on \(M_{t}\); \(f(h_{2})\) represents the influence of the thickness of the cement slab on \(M_{t}\); \(f(k)\) represents the influence of the subgrade reaction modulus on \(M_{t}\).