How to Use Real Work Due to Axial Strains?

Formulate S, A, and E Equations.

We start by identifying \(S\), \(L\), \(A\), and \(E\) per truss member. If there are variations in said variables, one must express these in terms of \(x\) (member's local axis). Our example has constant \(AE\) for all members. So, expressing it in terms of \(x\) is unnecessary.

Afterward, using the method of joints or sections, we solve for internal bar forces \(S\) of each due to the applied loads.

We summarise our results for all bar forces \(S\), \(L\), \(A\), and \(E\) in the table. Positive bar forces are in tension, while negative-sensed bar forces are in compression.

Apply Real Work Equations

The only thing to do is apply the general real work equation. Using the same table, we create a column for \(S^2L/AE\). Afterward, we add the values of this column to obtain the required translation:

\(20^{k N} \times \Delta_{E_v}=\frac{675}{2 E}+\frac{1250}{E}+\frac{1350}{E}+\frac{675}{2 E}+\frac{3125}{E}+\frac{1600}{E}\)

\(\Delta_{E_v}=\frac{400}{E}\)

The positive direction indicates that the vertical translation at \(E\) matches the direction of the \(20 kN\) force, which is downward.