Truss joint deflections by virtual work: 16-Civ-A1 Elementary Structural Analysis, May 2018, Q8

Question 8: Truss joint deflections by virtual work (22 marks)

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Given. Cantilever truss anchored to a wall on the right (pin at $U_a$, horizontal roller at $L_3$); $L_1(0,0)\!-\!L_2(4.2,0)\!-\!L_3(7.2,0)$ along the bottom, $U_1(4.2,1.75)$ on the raking top chord to $U_a(7.2,3)$; loads $21\text{ kN}\downarrow$ at $L_1$ and $14\text{ kN}\downarrow$ at $U_1$. Find. $\delta_{L_1},\,\delta_{L_2}$ (vertical).

Q8 — cantilever truss; deflections sought at $L_1$ and $L_2$.

Real bar forces $N$. Method of joints from the free tip gives, e.g., $L_1U_1=+54.6$, $L_1L_2=-50.4$, $L_2L_3=-64.4$, $L_2U_1=-14.0$, $L_2U_a=+19.8\text{ kN}$ (T +). The wall reactions are $U_{a}=(64.4,\,35)\text{ kN}$, $L_{3x}=-64.4\text{ kN}$.
Virtual bar forces $n$. Apply a unit vertical load at $L_1$ (then, separately, at $L_2$) with all real loads removed, and recover the corresponding $n_1$, $n_2$.
Virtual work. $\displaystyle \delta=\sum \frac{N\,n\,L}{AE}$ over the seven members, with $AE=3.9\times10^{4}\text{ kN}$: $$\boxed{\delta_{L_1}=53.3\text{ mm}\ \downarrow,\qquad \delta_{L_2}=8.0\text{ mm}\ \downarrow}$$

Joint	Vertical deflection
$L_1$ (cantilever tip)	$53.3\text{ mm}\ \downarrow$
$L_2$	$8.0\text{ mm}\ \downarrow$

Topic. Truss joint deflection by the unit-load (virtual-work) method.

Key relations. $\delta=\displaystyle\sum_{\text{members}}\frac{N\,n\,L}{AE}$, where $N$ are the real bar forces and $n$ the bar forces under a unit load at, and in the direction of, the sought deflection.

Why this works. Applying a dummy unit load and equating external virtual work $1\cdot\delta$ to internal virtual work $\sum n\,(NL/AE)$ isolates the wanted displacement. With a single $AE$ common to every bar it factors out, so the sum reduces to $\tfrac{1}{AE}\sum N n L$.

Common pitfall. Mixing tension/compression signs of $N$ or $n$, or using each bar’s projected (not true) length; a two-force member on the load path (here $L_2\!-\!U_1$ carries the $14\text{ kN}$ directly) is an easy self-check.

Source. Hibbeler 10e, Ch. 8 (trusses by virtual work); Kassimali 6e, Ch. 7.

Question 8 of 8: Truss joint deflections by virtual work

Question 8: Truss joint deflections by virtual work (22 marks)