Problem 3: Fixed-fixed 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 u (x=L) = 0 w (x=L) = 0 β (x=L) = 0 |
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
𝗪𝗲 𝗮𝗿𝗲 𝗽𝗿𝗼𝘂𝗱 𝘁𝗼 𝗮𝗻𝗻𝗼𝘂𝗻𝗰𝗲 𝘁𝗵𝗮𝘁 𝗗𝗦𝗣𝗘 𝗣𝗿𝗲𝘀𝗶𝗱𝗲𝗻𝘁 𝗛𝗮𝗻𝘀 𝗞𝗿𝗶𝗸𝗵𝗮𝗮𝗿 𝗵𝗮𝘀 𝗿𝗲𝗰𝗲𝗶𝘃𝗲𝗱 𝗮 𝗚𝗲𝗿𝗮𝗿𝗱 & 𝗔𝗻𝘁𝗼𝗻 𝗛𝗶𝗴𝗵 𝗧𝗲𝗰𝗵 𝗦𝘁𝗮𝗿 𝗔𝘄𝗮𝗿𝗱 𝟮𝟬𝟮𝟲!
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