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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**

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