Flexible vs rigid design; Westergaard temperature stresses

Flexible pavement design vs rigid pavement design

Westergaard's concept of temperature stresses

Westergaard (1927) treated the slab as a plate on a dense-liquid foundation and analysed curling due to a temperature difference between top and bottom.

• A temperature gradient $\Delta T$ across the thickness $h$ makes the top and bottom expand or contract differently, so the slab tends to curve. In the daytime the top is warmer and the slab edges curl down; at night the edges curl up. The slab weight and the subgrade reaction resist the curvature, creating stresses.
• If the slab were completely restrained from curving, the stress would be $\sigma=\dfrac{E\alpha\Delta T}{2(1-\mu)}$.
• For a finite slab free to curl, the stress is reduced by coefficients that depend on the ratio of slab length to radius of relative stiffness ($L/l$). Bradbury's version of Westergaard's result for the edge is

$$\sigma_{te}=\frac{C\,E\,\alpha\,\Delta T}{2}$$

and for the interior $\sigma_{ti}=\dfrac{E\alpha\Delta T}{2}\left(\dfrac{C_x+\mu C_y}{1-\mu^2}\right)$.

• For short slabs ($L/l$ below about 4) $C$ is small, so closely spaced joints reduce warping stress; for slabs about 8-9 $l$ long $C$ reaches a maximum (about 1.08), and then decreases slightly.
• The stress is tension at the bottom in the daytime and tension at the top at night; it is added to the load stress for the critical combination.