# Design Approach for Composite Pavement Structure Incorporating Reflective Crack Considerations
## 3 Derivation of Calculation Formula for Temperature Warping Stress
### The Effect of Structural Parameters on Temperature Warping Stress  
In previous research [23], several factors were identified as the main influencers of temperature warping stress in asphalt layers, including the elastic modulus of the asphalt layer (\(E_{1}\)), the thickness of the asphalt layer (\(h_{1}\)), the elastic modulus of the cement concrete slab (\(E_{2}\)), the thickness of the cement concrete slab (\(h_{2}\)), the length of the cement concrete slab (\(L\)), and the foundation reaction modulus (\(k\)). The stress indicators analyzed were the temperature warping stress and interlayer shear stress at position 1, which is the bottom of the asphalt layer under a temperature gradient.  
(1) The influence of \(E_{1}\)  
As depicted in Figure 4, an increase in \(E_{1}\) led to a gradual increase in both temperature stress and interlayer shear stress. Specifically, for each increment in modulus from 500 MPa, the temperature warping stress approximately doubled. However, the increase in interlayer shear stress was not significant. When E\({}_{1}\) reached 4000 MPa, the shear stress was only 0.00651 MPa, and when E\({}_{1}\) increased to 10,000 MPa, the shear stress was 0.0124 MPa.  
(2) The influence of \(h_{1}\)  
Figure 5 illustrates that, with an increase in h\({}_{1}\), both the temperature warping stress and interlayer shear stress exhibited a gradual decrease. Specifically, for every 4 cm increment in h\({}_{1}\), the reduction in warping stress under a temperature gradient ranged from approximately 1.3% to 7.8%. On the other hand, the decrease in interlayer shear stress was minimal, with the minimum interlayer shear stress recorded at 0.00299 MPa and the maximum interlayer shear stress at 0.00358 MPa.  
Figure 4: Temperature stress at different \(E_{1}\) values.  
(3) The influence of \(E_{2}\)  
In Figure 6, it is evident that the temperature warping stress and interlayer shear stress gradually increased with an increase in E\({}_{2}\). Specifically, under the influence of a temperature gradient, for every 500 MPa increased in E\({}_{2}\), the warping stress exhibited an approximate 3% increment. On the other hand, the increase in interlayer shear stress was minimal, with the minimum interlayer shear stress recorded at 0.00233 MPa and the maximum interlayer shear stress at 0.00412 MPa.  
(4) The influence of \(h_{2}\)  
According to the findings presented in Figure 7, it is observed that an increase in h\({}_{2}\) corresponded to a gradual increase in both temperature warping stress and interlayer shear stress. When subjected to a temperature gradient, each 0.02 m increment in h\({}_{2}\) led to an approximate increase of 1.7-6.1% in the warping stress. However, the impact on the interlayer shear stress was negligible, with the minimum value recorded at 0.00248 MPa and the maximum at 0.00478 MPa.  
Figure 5: Temperature stress at different \(h_{1}\) values.  
Figure 6: Temperature stress at different \(E_{2}\) values.  
(5) The influence of L  
As shown in Figure 8, as L increased, the temperature warping stress was found to gradually decrease. Under the effect of a temperature gradient, L increased from 3.8 m to 5.2 m, and the decrease in the temperature warping stress was noted to be about 1.4%. The interlayer shear stress remained basically unchanged, always at 0.00348 MPa.  
(6) The influence of k  
As shown in Figure 9, an increment in k caused a gradual decrease in the temperature warping stress. Under the effect of a temperature gradient, the decrease in the temperature warping stress was about 10-15% for every doubling of k. The decrease in the interlayer shear stress was negligible, with a minimum interlayer shear stress of 0.00335 MPa and a maximum interlayer shear stress of 0.00348 MPa.  
Figure 8: Temperature stress at different \(L\) values.  
Figure 7: Temperature stress at different \(h_{2}\) values.