Clumped beam with parabolic thermal load

A beam fixed at both ends is assumed to be subjected to one-step parabolic thermal load:

T = 2(T_1+T_2-2T_3)\frac{y^2}{h^2}+(T_1-T_2)\frac{y}{h}+T_3

with T₁ = -20 ^oC, T₂= 40 ^oC and T₃ = 0 ^oC.

Plane stress is assumed and the physical properties of the beam are: E = 10⁶ N/m², v = 0.3, α = 10^-7 ^oC^-1.

The analitical solution for the stress is:

T = -E\alpha[2(T_1+T_2-2T_3)\frac{y^2}{h^2}+(T_1-T_2)\frac{y}{h}]

Numerical results and analytical solution are compared in the following Figure.

Reference

Hemayati & Karami (2001) A BOUNDARY ELEMENTS AND PARTICULAR INTEGRALS IMPLEMENTATION FOR
THERMOELASTIC STRESS ANALYSIS.