Question 8 of 8: Virtual work — truss deflections at L 1 and L 2

Question 8: Virtual work — truss deflections at L₁ and L₂ (22 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 cantilever truss: L₁(0,0), L₂(4.2,0), L₃(7.2,0), U₁(4.2,3), U₂(7.2,3) [pinned to a wall]; L₃ rests on a roller against a vertical face (horizontal reaction only). Loads: 21 kN down at L₁, 14 kN down at U₁. $$AE=3.9\times10^4\text{ kN}$$ throughout.

Find. The vertical deflections at L₁ and L₂.

Approach. Method of joints for the real member forces $$N$$, then virtual work: apply a unit vertical load at L₁ (then separately at L₂) to get the virtual forces $$n$$, and sum $$\Delta=\Sigma nNL/AE$$ over all 7 members.

1. Real member forces (method of joints, working from L₁ toward the supports). $$N_{L1U1}=\boxed{+36.13\text{ kN (T)}},\quad N_{L1L2}=\boxed{-29.4\text{ kN (C)}},\quad N_{L2U1}=\boxed{-35.0\text{ kN (C)}}$$ $$N_{U1U2}=\boxed{+29.4\text{ kN (T)}},\quad N_{L2L3}=\boxed{-64.4\text{ kN (C)}},\quad N_{L2U2}=\boxed{+49.50\text{ kN (T)}},\quad N_{L3U2}=0$$ (Check: $$R_{y,U2}=21+14=35\text{ kN}$$ ✓.)
2. Virtual system 1 — unit load at L₁. Solving the same truss for a 1 kN downward load at L₁ alone gives, member by member, the $$n_1$$ column used below.
3. Virtual system 2 — unit load at L₂. Likewise for a 1 kN downward load at L₂ alone.
4. Sum $$\Delta=\Sigma n N L/AE$$ $$\Delta_{L1}=\boxed{38.0\text{ mm}}\ \downarrow,\qquad \Delta_{L2}=\boxed{12.6\text{ mm}}\ \downarrow$$