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16-Civ-A1 Elementary Structural Analysis · May 2018

Question 3 of 8: Vertical deflection at B of a non-prismatic beam

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

Notes on this paper

Paper: National Exams — May 2018, 16-Civ-A1 Elementary Structural Analysis (closed book, 3 h; Casio/Sharp calculator permitted). Six questions constitute a complete paper: answer Q1–Q5 and one of Q6/Q7/Q8. For completeness all eight questions are worked here.

Reference texts: R. C. Hibbeler, Structural Analysis (10th ed., Pearson) — determinacy (Ch. 2), method of joints & sections (Ch. 3), shear & moment diagrams (Ch. 4), influence lines (Ch. 6), virtual-work deflections (Ch. 8–9), moment distribution / slope–deflection (Ch. 11–12); A. Kassimali, Structural Analysis (6th ed., Cengage) — internal hinges, compound (Gerber) beams, indeterminate frames.

Sign convention. Upward reactions positive; sagging bending moment positive (tension on the underside), hogging negative; member axial force tension positive (T), compression negative (C).

These A1 papers are defined entirely by their figures, so every structure is redrawn to scale below.

Question 3: Vertical deflection at B of a non-prismatic beam (16 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. Simple span $A(0)\!-\!D(9\text{ m})$, pin at $A$, roller at $D$; segment stiffnesses $A\!-\!B=EI$, $B\!-\!C=3EI$, $C\!-\!D=EI$ (each $3\text{ m}$); point loads $81\text{ kN}$ at $B$ and $72\text{ kN}$ at $C$. Find. $\delta_B$ (vertical).

81 kN72 kNABCDEI3EIEI3 m3 m3 m
Q3 — non-prismatic simply-supported beam.
  1. Reactions (real system). $\Sigma M_A=0:\;9R_D=81(3)+72(6)\Rightarrow R_D=75\text{ kN}$, $R_A=78\text{ kN}$.
  2. Real moment $M$. $M=78x$ ($0\!-\!3$); $M=243-3x$ ($3\!-\!6$); $M=675-75x$ ($6\!-\!9$). Peak $M_B=234\text{ kN}\cdot\text{m}$.
  3. Virtual system. Unit downward load at $B$: $r_D=\tfrac13,\;r_A=\tfrac23$, so $m=\tfrac23x$ ($0\!-\!3$) and $m=3-\tfrac13x$ ($3\!-\!9$).
  4. Virtual work with variable $EI$. $\displaystyle \delta_B=\int_0^9\frac{M\,m}{EI(x)}\,dx=\frac{1}{EI}\!\int_0^3 Mm\,dx+\frac{1}{3EI}\!\int_3^6 Mm\,dx+\frac{1}{EI}\!\int_6^9 Mm\,dx=\frac{468+345+225}{EI}=\frac{1038}{EI}.$
  5. Evaluate. $\displaystyle \delta_B=\frac{1038}{3.0\times10^{4}}=0.03460\text{ m}.$ $$\boxed{\delta_B = 34.6\text{ mm}\ \downarrow}$$