Problem 1: Fixed-free 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 β (x=0) = 0 F (x=L) = 0 M (x=L) = MT D (x=L) = dM/dx |
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
The W.M. Keck Observatory telescopes on Hawaii are receiving upgrades that include an Adaptive Secondary Mirror (ASM) and the Keck Wide Field Imager (KWFI).
Read more
In the rapidly evolving field of photonics, the integrity of photonic wafers is paramount to the performance and reliability of optical devices.
Read more
In the area of precision engineering – where tolerances are measured in microns or even nanometers – additive manufacturing (AM) offers a new level of possibility.
Read more