## 2. Thermal Stresses in Clamped Beam

An aluminum beam of length L=1m, height H=0.05m, and thickness t=0.01m is clamped on both sides and subjected to differential temperature distribution. The beam is modeled by quadratic 2D solid elements (Q8 and Q9), the model being the same as the clamped beam example, with the difference that the beam is clamped on both sides, producing stresses but no deformations. The analysis is performed on a common mesh in 2 steps:

• Heat analysis: Calculate temperatures at mesh nodes. Prescribed temperatures at the top are T=100 and at the bottom T=-100.

• Deformation analysis: With temperatures from above step, calculate deformations and stresses.

A simple analytical solution for the maximum longitudinal stress σxx of a clamped beam subjected to a uniform temperature variation T at the top to -T at the bottom can be found in Roark, "Formulas for Stress and Strain":

σxx = +- α 2 T E J / h

The figure below displays discrete values of σxx (at the Gauss or 'sampling' points), the temperature being the same as in the previous example. Figure 93. σxx at Gauss points: Values at +-0.7746 H/2 of half height of beam (blue and red dots).