Need calculate the strain energy due axial

(5 marks)

b) Determine the rotation about x axis of end D by using Castigliano's second theorem.

Answer

To solve this problem using Castigliano's second theorem, we need to calculate the strain energy due to axial, bending, and torsion and then differentiate it with respect to the displacement or rotation of interest. Let's solve each part of the problem step by step:

Where F is the applied force, A is the cross-sectional area of the bar, and ΔL_axial is the change in length due to axial loading. Since the bar is in tension, ΔL_axial can be calculated using Hooke's Law:

ΔL_axial = (F * L) / (A * E)

The strain energy due to torsion (U_torsion) is given by:

U_torsion = (1/2) * (T^2 / (G * J))

δU_total/δδ_y = (δU_axial/δδ_y) + (δU_bending/δδ_y) + (δU_torsion/δδ_y)

We can solve for the displacement using the principle of virtual work:

δU_total/δθ_x = (δU_axial/δθ_x) + (δU_bending/δθ_x) + (δU_torsion/δθ_x)

We can solve for the rotation using the principle of virtual work:

Considering the strain energy components due to axial, bending, and torsion only, neglecting the strain energy due to transverse shear, we can observe the following:

1. Axial loading: Increasing the applied load would result in increased strain energy due to axial loading. However, the rotation about the x-axis is not directly influenced by axial loading, so it would not be affected.