Deflection of a trapezoidal bent (flexural, virtual work): 16-Civ-A1 Elementary Structural Analysis, December 2015, Q3

Question 3: Deflection of a trapezoidal bent (flexural, virtual work) (18 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.

Joint 1 pin $(0,0)$; joint 4 roller $(12,0)$	Joint 2 $(3,4)$; joint 3 $(9,4)$
Members 1–2, 2–3 (top), 3–4	$EI=28.8\times10^{3}$ kN·m²; 32 kN ↓ at 2

Find. $\delta_{v3}$ (part a); $\delta_{v2}$ for the moved load (part b).

3: pinned–roller trapezoidal bent, members 1–2–3–4.

Approach. Unit-load (virtual work) with only flexural strain: $\delta=\sum\int \dfrac{M\,m}{EI}\,dx$, where $M$ is the real bending moment (32 kN at 2) and $m$ from a unit vertical load at 3. Carried out numerically with a flexure-only stiffness model (axial rigidity made large so only bending contributes).

Real system. Reactions for 32 kN at joint 2 (pin at 1, roller at 4) give the bending diagram over the three members.
Virtual system. Unit vertical load at joint 3 gives $m(x)$.
Integrate. $\displaystyle\delta_{v3}=\sum\int\frac{Mm}{EI}dx=\boxed{20.0\text{ mm}\ \downarrow}$.
Part (b) — Maxwell’s reciprocal theorem. The deflection at 2 due to a load at 3 equals the deflection at 3 due to the same load at 2. Hence $\delta_{v2}=\boxed{20.0\text{ mm}\ \downarrow}$ — no re-analysis needed.

Part	Result
(a) $\delta_{v3}$, load at 2	20.0 mm ↓
(b) $\delta_{v2}$, load at 3	20.0 mm ↓ (= part a, Maxwell–Betti)

Question 3 of 8: Deflection of a trapezoidal bent (flexural, virtual work)

Question 3: Deflection of a trapezoidal bent (flexural, virtual work) (18 marks)