Question 6 of 8: Truss forces and virtual-work deflection

Question 6: Truss forces and virtual-work deflection (9 + 13 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. $L_1(0,0)\,L_2(6,0)\,L_3(12,0)$; $M_1(3,4)\,M_2(9,4)$; apex $U_1(6,8)$; pin $L_1$, roller $L_3$; horizontal $90\text{ kN}\rightarrow$ at each of $U_1,M_1,M_2$. Find. three member forces and $\delta_{U_1,\text{horiz}}$.

1. Reactions. $\sum F_x:\;L_{1x}=-270\text{ kN}$ (the three $90\text{ kN}$ loads); $\sum M_{L_1}:$ a couple $L_{1y}=-120,\;R_{L_3y}=+120\text{ kN}$.
2. Diamond diagonals. Joint resolution gives $L_2M_1=\boxed{-75\text{ kN (C)}}$ and $L_2M_2=\boxed{+75\text{ kN (T)}}$ (equal and opposite, as the horizontal loading is antisymmetric about the vertical axis through $L_2$).
3. Horizontal tie. At joint $L_2$ the two inclined diagonals’ vertical components cancel and their horizontal components are balanced by the chords, leaving $M_1M_2=\boxed{0}$.
4. Deflection (part b). Apply a unit horizontal load at $U_1$, get virtual forces $n$, then $\displaystyle \delta_{U_1}=\sum\frac{N\,n\,L}{AE}=\frac{2685}{1.5\times10^{5}}=\boxed{0.0179\text{ m}=17.9\text{ mm}}$ (rightward, i.e. in the direction of the loads).

Every one of the nine members must appear in the summation, since a member that happens to be zero in the real system can still carry a virtual force (and vice versa). Here the tie $M_1M_2$ is zero in both systems, so it contributes nothing; the four sloping chord members ($L_1M_1$, $M_1U_1$, $U_1M_2$, $M_2L_3$) do most of the work because they are both heavily loaded and steeply inclined to the horizontal unit load. Note also that the drawing shows a faint vertical line from $U_1$ down to $L_2$: it is a construction axis, not a member — including it as a bar would make the truss one degree indeterminate and the method-of-joints solution (and this virtual-work calculation) would no longer apply. The determinacy check $m+r-2n=9+3-12=0$ confirms the correct reading.