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16-Civ-A1 Elementary Structural Analysis · May 2017

Question 1 of 8: Classify each structure — unstable / determinate / indeterminate

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

Notes on this paper

Paper: National Exams — May 2017, 16-Civ-A1 Elementary Structural Analysis (closed book, 3 h). 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), 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 beams, indeterminate frames.

Sign convention: upward reactions positive; sagging bending moment positive (tension on the underside); member tension positive (T), compression negative (C).

These A1 papers are defined entirely by their figures, so every structure has been read from the printed figure and redrawn to scale below.

Question 1: Classify each structure — unstable / determinate / indeterminate (6 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.

Approach. For beam/frame assemblies use the degree of static indeterminacy $\text{DSI}=3m+r-3n-c$ (equivalently $r-3-c$ for a load path with no closed ring), where $m$ = members, $n$ = joints, $r$ = reaction components, $c$ = internal condition (release) equations, one per internal hinge. For trusses use $\text{DSI}=m+r-2n$. A positive result is the degree of indeterminacy; zero is determinate; negative (or a bad reaction geometry) is a mechanism.

wP
(a)
PP
(b)
2PP
(c)
90°PP
(d)
2PP
(e)
PP
(f)
  1. (a) Continuous beam: wall (fixed) + three rollers + two internal hinges. $r=3+1+1+1=6$, hinges give $c=2$. $\text{DSI}=6-3-2=\boxed{1}$ — indeterminate, 1°.
  2. (b) Two cantilevers built into opposite walls, joined by an internal hinge that also sits on a roller. $r=3+3+1=7$; the hinge connecting the two beams gives $c=1$. $\text{DSI}=7-3-1=\boxed{3}$ — indeterminate, 3° (a fixed–fixed beam is already 3°; the added roller and hinge cancel).
  3. (c) Open (tree) rigid frame on pin + roller + pin. No closed ring, so $\text{DSI}=r-3=(2+1+2)-3=\boxed{2}$ — indeterminate, 2°.
  4. (d) Closed triangular rigid frame (one ring) on three rollers. $m=3,\;n=3,\;r=1+1+1=3$: $\text{DSI}=3(3)+3-3(3)=\boxed{3}$ — a single closed ring is 3° internally, and the three non-concurrent roller reactions make it externally just-rigid — indeterminate, 3°.
  5. (e) Truss, pin + roller, crossing diagonals unconnected. $m=5,\;r=3,\;n=4$: $\text{DSI}=5+3-2(4)=\boxed{0}$ — statically determinate and stable.
  6. (f) Cantilever truss off a wall on pin + two rollers. $m=13,\;r=2+1+1=4,\;n=8$: $\text{DSI}=13+4-2(8)=\boxed{1}$ — indeterminate, 1°.
Structure(a)(b)(c)(d)(e)(f)
ClassificationIndet. 1°Indet. 3°Indet. 2°Indet. 3°DeterminateIndet. 1°
Check (figure reading): (b) is read as two wall cantilevers joined by a hinge that also rests on the roller; if instead the two beams were independent (right beam free at the hinge) the count would be 1°. (c)’s centre support and (f)’s three wall connections were read as roller / pin / roller from the drawn wheels — the counts above follow that reading.
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