Problem 7: Simple supported beam
The equations for all 3 thermal load cases are shown below, the colors represent temperature field (white is high and black is low). The equations give the axial deformation u of the beam, the deflection w of the beam, and the axial stresses in the beam.
![]() | Boundary Conditions: u (x=0) = 0 w (x=0) = 0 M (x=0) = MT u (x=L) = 0 w (x=L) = 0 M (x=L) = MT |

The figure of the deformed beam illustrates the deflection resulting from load case C. Colors represent the axial stress field (green is zero, red is positive and blue is negative). Note that the boundary effects as shown in the figure is not included in the equations!
Name | Symbol | Units | Description |
Shear force | D | N | Internal shear force as function of the longitudinal direction (x) |
Young’s modulus | E | N/m2 | Young’s modulus of the material |
Axial force | F | N | Internal axial force as function of the longitudinal direction (x) |
Height | H | m | The total length of the beam |
Length | L | m | The total length of the beam in longitudinal direction |
Moment | M | N/m | Internal moment as function of the longitudinal direction (x) |
Temperature | ΔT | K | The biggest temperature difference in the beam |
Axial deformation | u | m | Axial deformation of the beam as function of the longitudinal direction (x) |
Deflection | w | m | Deflection of the beam as function of the longitudinal direction (x) |
CTE | α | K-1 | Coefficient of thermal expansion (CTE) of the material |
Rotation | β | rad | Rotation of the beam as function of the longitudinal direction (x) |
Axial stress | σx | m | Axial stress of the beam, i.e. combined thermal and mechanical stress, as function of the longitudinal direction and the direction perpendicular (x and y) |
References

This website was developed by a group of people, passionate about Thermo-Mechanics. They contributed their free time to writing and reviewing the content and building the website. If there are...
Read more
From left to right: Frank Sperling (Chairman sessions 6-7 of the DSPE Conference 2016)Marc Vermeulen (Chairman of the Jury Best Contribution Awards)Ad VermeerStefan LampaertErik Hennes Best Presentation Award 2016 ...
Read more
This edition of Mikroniek aims to honour the memory of Wim van der Hoek with articles about designs in his vein and reflections that build on his ideas. Wim van der...
Read more