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16-Civ-A1 Elementary Structural Analysis · December 2013

Question 3 of 8: Deflection of a non-prismatic beam by virtual work

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams, December 2013 — 98-Civ-A1 Elementary Structural Analysis. Three-hour, closed-book examination (approved Sharp/Casio calculator only). Format: six questions constitute a complete paper — answer all of #1–#5 and any one of #6, #7 or #8. All eight questions are solved below for completeness.

Reference texts: R.C. Hibbeler, Structural Analysis (10th ed., Pearson) — determinacy, method of joints/sections, virtual-work deflections, moment distribution, slope–deflection, influence lines; A. Kassimali, Structural Analysis (6th ed., Cengage) — internal hinges and compound structures. Sign convention: sagging bending moment positive; upward reactions positive; tension member forces positive (T), compression negative (C).

Question 3: Deflection of a non-prismatic beam by 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.

12 kN 8 kN A B C D EI 3EI 3EI 6 m 6 m 3 m
Beam 3: pin A, roller C at 12 m, overhang to D; 12 kN at B (6 m), 8 kN at D (15 m); flexural rigidity EI over AB, 3EI over BC and CD.

Given. Simply supported beam, pin A (x = 0), roller C (x = 12 m), free overhang to D (x = 15 m). Point loads 12 kN at B (x = 6 m) and 8 kN at D. Rigidity $EI$ over segment AB, $3EI$ over BC and CD, with $EI = 8.0\times10^{3}$ kN·m².

Find. The vertical deflection at B.

SegmentRange (m)Real moment $M(x)$ (kN·m)Unit moment $m(x)$Rigidity
AB0–6$4x$$0.5x$$EI$
BC6–12$-8x+72$$-0.5x+6$$3EI$
CD12–15$8x-120$$0$$3EI$
  1. Real reactions. $\sum M_A = 0$: $R_C(12) = 12(6)+8(15) \Rightarrow R_C = 16$ kN; $R_A = 20-16 = 4$ kN.
  2. Unit (virtual) system. Apply a 1 kN downward dummy load at B: $r_A = r_C = 0.5$. Beyond C the overhang carries no unit moment, so $m=0$ on CD.
  3. Virtual-work integral. $\displaystyle \delta_B = \int \frac{M\,m}{EI(x)}\,dx$. Over AB: $\int_0^6 \frac{(4x)(0.5x)}{EI}dx = \frac{144}{EI}$. Over BC: $\int_6^{12} \frac{(-8x+72)(-0.5x+6)}{3EI}dx = \frac{72}{3EI} = \frac{24}{EI}$. Over CD: $0$.
  4. Total. $\delta_B = \dfrac{144+24}{EI} = \dfrac{168}{8.0\times10^{3}} = \boxed{0.0210\ \text{m} = 21.0\ \text{mm}\ \downarrow}$.
QuantityValue
Vertical deflection at B21.0 mm downward