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16-Civ-A1 Elementary Structural Analysis · May 2016

Question 4 of 8: Truss member forces

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

Notes on this paper

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, compound (Gerber) beams and three-hinged frames. Sign convention: sagging bending moment positive; upward reactions positive; member tension positive (T), compression negative (C).

The paper directs candidates to answer Q1–Q5 and then one of Q6/Q7/Q8. For completeness all eight questions are worked here.

Question 4: Truss member forces (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.

4(a)

Given. Pin at $L_1(0,3)$, roller at $L_5(16,3)$; loaded bottom chord $L_2(4,0),L_3(8,0),L_4(12,0)$ carrying $24,48,72\text{ kN}$; apex $U_2(8,9)$, with $U_1(4,6),U_3(12,6)$.

Find. $L_1\!-\!L_2$, $U_3\!-\!L_4$ and $L_3\!-\!L_4$ (tension/compression). Highlighted red below.

244872 kNL1L2L3L4L5U1U2U3
4(a) Requested members in red.
  1. Reactions. $\sum M_{L_1}:\;V_{L_5}(16)=24(4)+48(8)+72(12)=1344\Rightarrow V_{L_5}=84\text{ kN}$; $V_{L_1}=144-84=60\text{ kN}$; $H_{L_1}=0$.
  2. Joint $L_1$. Members $L_1\!-\!L_2$ (slope $\downarrow$ $3\!:\!4$) and $L_1\!-\!U_1$ (slope $\uparrow$ $3\!:\!4$). $\sum F_x\Rightarrow F_{L_1U_1}=-F_{L_1L_2}$; $\sum F_y:\;60-\tfrac{6}{5}F_{L_1L_2}=0\Rightarrow F_{L_1L_2}=\boxed{50\text{ kN (T)}}$.
  3. Joint $L_4$. Members $L_3\!-\!L_4$, $L_4\!-\!L_5$ ($4\!:\!3$) and vertical $U_3\!-\!L_4$, under the $72\text{ kN}$ load. Solving the joint gives $F_{U_3L_4}=\boxed{30\text{ kN (T)}}$ and $F_{L_3L_4}=\boxed{56\text{ kN (T)}}$.

4(b)

Given. Top chord $U_1\dots U_6$ at $y=6$ (spacing $4\text{ m}$); pinned supports at $L_1(0,0)$ and $L_5(20,3)$; interior lower nodes $L_2(4,3),L_3(12,3),L_4(16,3)$; three $30\text{ kN}$ loads at $U_3,U_4,U_5$.

Find. $U_3\!-\!L_3$, $L_4\!-\!U_6$ and $L_3\!-\!L_4$ (tension/compression).

Note on this truss. Nodes $L_2$ and $L_3$ are joined only through the apex $U_3$ (there is no $L_2\!-\!L_3$ chord), so both feet are pinned; the second pin removes the otherwise-present internal mechanism, and $m+r-2n=18+4-22=0$ — the truss is stable and determinate (the equilibrium matrix has full rank).
303030 kNU1U2U3U4U5U6L1L2L3L4L5
4(b) Both feet pinned; requested members in red.
  1. Reactions. Global equilibrium with two pins gives $V_{L_1}=V_{L_5}=45\text{ kN}$ and $H_{L_1}=+60\text{ kN}$, $H_{L_5}=-60\text{ kN}$ (the horizontal thrust is demanded by the two-triangle geometry). Check $\sum M_{L_1}=-1080+45(20)+(-60)(-3)=0$.
  2. Member forces (whole-structure joint solution, tension +): $F_{U_3L_3}=\boxed{25\text{ kN (T)}}$, $F_{L_4U_6}=\boxed{75\text{ kN (T)}}$, and $F_{L_3L_4}=\boxed{0}$ (zero-force member for this loading).
MemberForceSense
4(a) $L_1\!-\!L_2$$50$ kNTension
4(a) $U_3\!-\!L_4$$30$ kNTension
4(a) $L_3\!-\!L_4$$56$ kNTension
4(b) $U_3\!-\!L_3$$25$ kNTension
4(b) $L_4\!-\!U_6$$75$ kNTension
4(b) $L_3\!-\!L_4$$0$—