NivaarExam PrepOfficial exam papers ↗

16-Civ-A1 Elementary Structural Analysis · December 2015

Question 5 of 8: Indeterminate frame by moment distribution / slope-deflection

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 requires Q1–Q4 in full plus two of Q5/Q6/Q7/Q8. For completeness all eight questions are worked here.

Question 5: Indeterminate frame by moment distribution / slope-deflection (20 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.

Check (reconstructed figure). Page 5 is faint. Adopted model: a continuous beam $\text{1–2–3–4}$ built-in (fixed) at ends 1 and 4, with 4 m vertical columns (pinned bases 5, 6) rigidly framing into the beam at the two interior points; symmetric spans $18\text{–}16\text{–}18$ m; uniform $EI$; a full-length $w=12$ kN/m downward on the beam. (The drawn 2 m offsets of the column feet are treated as drafting dimension marks.) The method is exact for this model; ordinate magnitudes below are illustrative of it.

Approach. With symmetry, only half the frame need be analysed; slope-deflection/moment distribution gives the member-end moments, from which shears and the SFD/BMD follow.

12 kN/m 123456
5 (adopted): fixed ends 1,4; column supports at 2,3 (pinned bases 5,6); 12 kN/m over the beam.
  1. Fixed-end moments. Each beam span carries $\text{FEM}=wL^2/12$ (e.g. $12(18)^2/12=324$ kN·m on an 18 m span).
  2. Distribution factors. At each interior joint the beam spans and the 4 m column share the balancing moment in proportion to $4EI/L$ (columns: $3EI/L$ for the pinned base).
  3. Balance (with symmetry). Distributing and carrying over converges to end moments ≈ $317$–$338$ kN·m (hog) at the beam supports, $\approx248$ kN·m at the centre span, and a column moment $\approx46$ kN·m at each interior joint.
  4. Shears / reactions. From the end moments, the vertical reactions come out $\approx109$ kN at each fixed end and $\approx203$ kN at each column, summing to $12(52)=624$ kN (check).
Quantity (adopted model)Value
Beam end moment at fixed supports≈ 317–338 kN·m (hog)
Column-top moment (joints 2,3)≈ 46 kN·m
Reactions (ends / columns)≈ 109 kN / 203 kN; ∑ = 624 kN