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!

NameSymbolUnitsDescription
Shear forceDNInternal shear force as function of the longitudinal direction (x)
Young’s modulusEN/m2Young’s modulus of the material
Axial forceFNInternal axial force as function of the longitudinal direction (x)
HeightHmThe total length of the beam
LengthLmThe total length of the beam in longitudinal direction
MomentMN/mInternal moment as function of the longitudinal direction (x)
TemperatureΔTKThe biggest temperature difference in the beam
Axial deformationumAxial deformation of the beam as function of the longitudinal direction (x)
DeflectionwmDeflection of the beam as function of the longitudinal direction (x)
CTEαK-1Coefficient of thermal expansion (CTE) of the material
RotationβradRotation of the beam as function of the longitudinal direction (x)
Axial stressσxmAxial 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

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