NivaarExam PrepOfficial exam papers ↗

16-Civ-A1 Elementary Structural Analysis · May 2014

Question 7 of 8: Three-hinged frame — reactions and SFD/BMD

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 and compound structures. Sign convention: sagging bending moment positive; upward reactions positive; tension member forces positive (T), compression negative (C).

The paper directs candidates to answer Q1–Q4, then one of Q5/Q6 and one of Q7/Q8. For completeness all eight questions are worked here.

Question 7: Three-hinged frame — reactions and SFD/BMD (22 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.

10 kN/m1 (pin)23 (hinge)45 (pin)4 m6 m7: two columns and two rafters meeting at a hinged apex (node 3); pins at feet 1 and 5; UDL 10 kN/m over the horizontal projection of the rafters.

Given. Nodes $1(3,3),2(3,6),3(7,9),4(13,4.5),5(13,0)$ m; pins at feet 1 and 5; internal hinge at apex 3. UDL $w=10$ kN/m acts vertically over the horizontal projection of the rafters ($x=3\to13$, total 100 kN). Members: column 1–2 (3 m), rafters 2–3 (5 m) and 3–4 (7.5 m), column 4–5 (4.5 m).

Find. Reactions and the SFD/BMD ordinates for each member.

  1. Global + hinge equations. Unknowns $A_x,A_y$ (foot 1) and $E_x,E_y$ (foot 5). $\sum M_1=0$, $\sum F=0$, and $M=0$ at the apex (right free body) give:
  2. $E_y=\boxed{55\ \text{kN}\uparrow}$, $E_x=\boxed{16.7\ \text{kN}\leftarrow}$; $A_y=\boxed{45\ \text{kN}\uparrow}$, $A_x=\boxed{16.7\ \text{kN}\rightarrow}$.
  3. Bending moments. Column 1–2: $0\to50$ kN·m (linear). Rafter 2–3: $50\to0$ at the apex, with a slight reversal ($\approx3$ kN·m). Rafter 3–4: $0\to$ a sagging $+15.3$ ·m at $x\approx8.75$, then to $\boxed{-75\ \text{kN}\cdot\text{m}}$ at node 4. Column 4–5: $75\to0$ (linear). Apex $M=0$ &checkmark.
  4. Shear & axial (member-local). Columns 1–2 and 4–5 carry constant shear $16.7$ kN and axial $45$ kN C, $55$ kN C. Rafter 2–3: shear $-26\to+6$ kN; rafter 3–4: shear $-14\to+34$ kN (zero at the sagging peaks); rafter axials $16\!-\!46$ kN compression.
Member$M$ ordinates (kN·m)Shear (kN)Axial (kN)
Column 1–2$0\to-50$$16.7$$45$ C
Rafter 2–3$-50\to+2.8\to0$$-26\to+6$$40\to16$ C
Rafter 3–4$0\to+15.3\to-75$$-14\to+34$$10\to46$ C
Column 4–5$-75\to0$$16.7$$55$ C

Check (geometry & load): both feet read as pins and the apex as a true hinge (making the frame determinate, 3 hinges); the 10 kN/m is taken as a vertical load per metre of horizontal projection over the rafters (total 100 kN at $x=8$). The rafter rises 3–4–5 (2–3) and falls 6–4.5–7.5 (3–4), both clean triangles.