1.4 Thermo-mechanical beam equations ("vergeet-mij-nietjes")

The equations for a single beam are widely used for simple hand calculation in the design of mechanical structures. Similarly, equations can be derived to describe the deflection of, and stresses in beams. In this section, the beam equations are given for three thermal load cases and seven combinations of boundary conditions.

Three thermal load cases are considered**:

  1. Uniform temperature change,
  2. Temperature gradient in longitudinal direction, and
  3. Temperature gradient over the cross section.

Each of these thermal load cases is assessed for seven different boundary conditions, which are shown below* (clickable)

 Problem 1: fixed-free beam
 Problem 2: fixed-guided-in-y beam
 Problem 3: fixed-fixed
 Problem 4: fixed-guided-in-x beam
 Problem 5: fixed-simple supported beam
 Problem 6: simple supported-guided-in-x beam
 Problem 7: simple supported beam

* Assumptions:

  • The Bernoulli-Euler beam theory;
  • The equations do not include non-linear effects, like buckling;
  • Rectangular cross section only, so the thermal load case C is valid.

** Colours represent the temperature field (white represente positive as black represents negative and orange denotes the undeformed temperature T0)