Question 5 of 8: Three-element truss displacement

Question 5: Three-element truss displacement (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A three-element pin-jointed truss: horizontal member AB (length a), vertical member BD (length b) and diagonal BC, all sharing joint B where the load acts. A and C are pinned, D is a roller; only joint B is free.

Find. the horizontal (u) and vertical (v) displacements of joint B.

Approach. With a single free joint, assemble a 2×2 joint stiffness from the three members' direction cosines and solve for the displacement vector.

1. Member geometry. Taking A(0,0), B(850,0), D(850,−1000), C(0,−1000) mm, the axes from B are AB $(-1,0)$, BD $(0,-1)$, BC $(-0.648,-0.762)$ with $L_{BC}=1312$ mm.
2. Joint stiffness. Summing $(EA/L)\,\hat n\hat n^{\mathsf T}$ over the three members, $$\mathbf K=\begin{bmatrix}94.3&23.7\\23.7&90.9\end{bmatrix}\ \text{kN/mm}.$$
3. Solve. With $\mathbf F=(16500\cos25^\circ,\,16500\sin25^\circ)=(14.95,6.97)$ kN, $$u=\boxed{0.149\ \text{mm}},\qquad v=\boxed{0.0379\ \text{mm}},$$ both positive (rightward and upward).

The diagonal BC provides the coupling term that mixes horizontal and vertical response; without it the two directions would decouple. Because AB is axially very stiff, the horizontal movement stays small despite the larger horizontal load component.