Temperature Change

The beam is fixed at end B and free at end A. It is loaded at end A with the concentrated force 4F. At the same time there is a temperature change ΔT. Calculate the length change Δx. (Modulus of elasticity E, Cross section area A_S , Temperature coefficient.)

Solution

There are two causes for the length change.

The first cause for the length change is force 4F. The area of the cross section of the beam is A_S so the stress is σ = 4F / A_S . The change in length is Δx1 and the deformation of the beam is ε1= Δx1/x. The Hook law gives :

σ = E * ε1 -> 4F / A_s = E * Dx1 / x -> Dx1 = 4F*x / E*A_S

The second cause for the length change is the temperature change. 

Δx2 = a * x * ΔΤ

When the bean is simultaneously loaded by the force 4F and the temperature change ΔΤ the length change is: 

Δx = Δx1 + Δx2 = 4F * x / E * A_S + a * x * Δx