Problem 6: Simple supported-guided-in-x 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) = 0
F (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!

 

Symbols and units
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)