# Design Approach for Composite Pavement Structure Incorporating Reflective Crack Considerations
## 3 Derivation of Calculation Formula for Temperature Warping Stress  
### Calculation Model for Temperature Warping Stress  
To analyze the impact of the board warping phenomenon on the bottom position of the asphalt layer, the finite element method was employed in this study. This approach facilitates the consideration of the foundation reaction and the self-weight constraint of the board on warping stress [21; 22]. When developing the temperature warping stress model, the following assumptions were made:  
1. Each structural layer of the road surface was a uniformly distributed and isotropic elastic body;
2. Assuming that the temperature within the pavement structure remains constant laterally, heat transfer occurs only in the vertical (depth) direction of the pavement;
3. The structural layers of the road surface were in close contact and exhibited continuous heat transfer. The finite element model is employed to calculate the thermal stresses, and the specific implementation steps are as follows: (1) Determine the material properties of each layer in the pavement structure. (2) Assign various parameters for the materials, as shown in Table 1. (3) Apply the material properties to the model and then discretize the model by meshing. (4) Apply temperature loads according to the temperature gradient. The positive temperature gradient is set up, meaning the surface temperature of the slab is higher than the bottom, causing the edges of the slab to curl upwards, resulting in tensile stress at the bottom of the slab. (5) Solve the model and extract the thermal warping stresses at position 1 on the bottom of the asphalt layer and the interlayer shear stresses. Figure 2: Design flow chart of composite pavement structure. Furthermore, relevant research [23] indicated that the distribution of cross-sectional stress along the thickness in both three-dimensional and planar models showed good consistency under temperature gradients. Any variations between the two models were minimal and mainly existed in regions with high stress concentrations. Considering the need for computational efficiency, this study adopted a planar model. Different temperature gradients cause thermal contraction deformation in the concrete slab, which alters the stress at position 1 on the bottom of the asphalt layer. Given that this position is the most critical stress point within the pavement structure, The stress was calculated at position 1 of the asphalt layer bottom under six temperature gradients: \(-\)50 \({}^{\circ}\)C/m, \(-\)40 \({}^{\circ}\)C/m, \(-\)30 \({}^{\circ}\)C/m, 30 \({}^{\circ}\)C/m, 40 \({}^{\circ}\)C/m, and 50 \({}^{\circ}\)C/m. The calculation results are shown in Figure 3.  
Based on Figure 3, it can be observed that, as the temperature gradient increased, the warping stress also increased. This was because the temperature difference across the plate increased with a higher temperature gradient, resulting in elevated temperature warping stress. For the concrete slab, under the same temperature gradient, the effect of a negative temperature gradient on its temperature warping stress was slightly smaller compared to a positive temperature gradient. Previous research [3] has indicated that positive temperature gradients in pavement structures were generally much larger than negative temperature gradients. Therefore, the subsequent analysis focused on the influence of various structural parameters on temperature stress when considering a temperature gradient of 50 \({}^{\circ}\)C/m [23].