16-Civ-A1 Elementary Structural Analysis · Undated paper

Question 1 of 8: Classification of six structures

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

Notes on this paper

Paper format. Engineers Canada / EGBC national examination 16-Civ-A1 Elementary Structural Analysis, 3 hours, CLOSED BOOK. 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 and stability (Ch. 2), method of joints and sections (Ch. 3), shear and bending-moment diagrams (Ch. 4), influence lines and moving loads (Ch. 6), virtual-work deflections (Ch. 8–9), slope-deflection and moment distribution (Ch. 10–11); A. Kassimali, Structural Analysis (6th ed., Cengage) — internal hinges, compound (Gerber) beams, three-hinged frames, Maxwell’s law of reciprocal deflections.

Sign convention. Upward reactions positive; sagging bending moment positive (tension on the underside of a beam), hogging negative; member axial force tension positive (T), compression negative (C). Shear is positive when the resultant of the forces on the left of a section acts upward.

Check: figure reading. All geometry, loads and support symbols used below were read directly from the printed figures. Where a support symbol governs the answer it is called out explicitly in the Given.

Question 1: Classification of six structures (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.

Given. Six sketches. Reading the support symbols from the drawing — a plain hatched ground line with the member built into it is a fixed support (3 restraints), a triangle on hatching is a pin (2), a triangle on wheels is a roller (1), and a small open circle drawn on a member is an internal hinge (one moment release) — the member, joint, reaction and release counts are:

Case	Type	Members m	Joints j	Reactions r	Releases c
(a)	beam, UDL, one internal hinge	1 chain	–	4 (pin + 2 rollers)	1
(b)	single-bay two-storey rigid frame	6	6	4 (2 pins)	0
(c)	gable, rigid apex	2	3	5 (fixed + pin)	0
(d)	beam on 2 rollers with 2 inclined props	5	6	6 (2 rollers + 2 pins)	0
(e)	X-braced tower truss (diagonals connected at the crossings)	14	8	4 (2 pins)	–
(f)	Warren truss, two cross-braced panels	19	10	3 (pin + roller)	–

Find. For each structure: unstable, statically determinate, or statically indeterminate to degree n.

[Figure not reproduced: Q1 — the six structures, redrawn from the printed figure. Note the fixed base on the left leg of (c) (plain hatched ground line, no triangle) and the two pinned feet of the tower truss (e). See the official exam paper.]

Approach. Count restraints against available equations: for beams and rigid frames use the member/joint form of the degree of static indeterminacy, for pin-jointed trusses use the bar-count form, then confirm that the restraints are also correctly arranged (a positive count is worthless if the structure is a mechanism).

State the two counting rules. For a planar assembly of rigid (beam-type) members with m members, j joints (including supports and free ends), r support reaction components and c internal releases, $$\text{DSI} = 3m + r - 3j - c$$ For a pin-jointed planar truss with m bars, j pins and r reactions, $$\text{DSI} = m + r - 2j$$ In either case DSI > 0 is indeterminate to that degree, DSI = 0 is determinate provided the restraints are properly arranged, and DSI < 0 is a mechanism.
(a) Continuous beam with one internal hinge. A single, unbranched beam is most quickly handled with the reduced form $\text{DSI} = r - (3 + c)$. Here the left end is a pin (2), and there are two rollers (1 each), so $r = 4$, with $c = 1$ for the hinge: $$\text{DSI} = 4 - (3 + 1) = \boxed{0}$$ The arrangement is sound: the right-hand segment is held vertically by its two rollers and horizontally through the hinge back to the pin, so the beam is statically determinate.
(b) Single-bay two-storey rigid frame on pinned feet. Two columns each split at the intermediate floor give 4 column members plus 2 beams, so $m = 6$, $j = 6$ and $r = 2 + 2 = 4$: $$\text{DSI} = 3(6) + 4 - 3(6) = \boxed{4}$$ The same answer follows from the closed-loop rule: two closed rectangular loops are $3 \times 2 = 6$ redundants for fixed feet, less 2 for replacing each fixed base by a pin. Indeterminate to the 4th degree.
(c) Two-member gable with a rigid apex. The critical reading is the left base: the member runs straight into a hatched ground line with no support triangle and no hinge circle, which is a fixed support, while the right base carries the circle-on-triangle of a pin. With $m = 2$, $j = 3$ and $r = 3 + 2 = 5$: $$\text{DSI} = 3(2) + 5 - 3(3) = \boxed{2}$$ Indeterminate to the 2nd degree. (Had the apex carried a hinge the answer would drop to 1.)
(d) Beam on two rollers, propped by two inclined struts. The struts frame rigidly into the beam and are pinned to the ground, so the beam is divided into three members and there are two struts: $m = 5$, $j = 6$ (two beam ends, two strut-to-beam joints, two ground pins) and $r = 1 + 1 + 2 + 2 = 6$: $$\text{DSI} = 3(5) + 6 - 3(6) = \boxed{3}$$ The two ground pins supply the horizontal restraint that the rollers cannot, so the frame is stable: indeterminate to the 3rd degree.
(e) X-braced tower truss. The diagonals of both panels are connected where they cross, so each crossing is a real joint and each diagonal counts as two bars. Chords 2, columns 4, and 4 + 4 diagonal halves give $m = 14$; the joints are the six panel corners plus the two crossings, $j = 8$; both feet are pins, $r = 4$: $$\text{DSI} = 14 + 4 - 2(8) = \boxed{2}$$ Indeterminate to the 2nd degree.
(f) Warren truss with two cross-braced panels. Here the question states that the crossing diagonals are not connected, so each cross-brace is two full-length bars and the crossing point is not a joint. Four top-chord bars, four bottom-chord bars, three verticals, four end diagonals and the two pairs of cross-diagonals give $m = 19$, with $j = 10$ and $r = 3$ (pin plus roller): $$\text{DSI} = 19 + 3 - 2(10) = \boxed{2}$$ The two redundant diagonals are exactly the two extra bars in the two braced panels: indeterminate to the 2nd degree.

Structure	Count	Classification
(a)	r − 3 − c = 4 − 4	Statically determinate
(b)	3m + r − 3j = 18 + 4 − 18	Indeterminate, degree 4
(c)	3m + r − 3j = 6 + 5 − 9	Indeterminate, degree 2
(d)	3m + r − 3j = 15 + 6 − 18	Indeterminate, degree 3
(e)	m + r − 2j = 14 + 4 − 16	Indeterminate, degree 2
(f)	m + r − 2j = 19 + 3 − 20	Indeterminate, degree 2

Topic. Static determinacy and stability of planar beams, frames and trusses.

Key relations. Rigid-member form $\text{DSI} = 3m + r - 3j - c$; truss form $\text{DSI} = m + r - 2j$; single-beam shortcut $\text{DSI} = r - 3 - c$; closed-loop form $\text{DSI} = 3(\text{loops}) - (\text{releases}) + (r - 3)$.

Why this works. Each rigid planar member carries three unknown internal actions (axial, shear, moment) at a cut, and each planar joint supplies three equilibrium equations. Counting the unknowns (member actions plus reactions) against the equations (joint equilibrium plus the extra condition each release contributes) gives the surplus of unknowns, which is exactly the number of redundants that a compatibility condition must supply. For a pin-jointed truss the members carry axial force only, so each bar is one unknown and each pin gives two equations, which collapses the same argument to $m + r - 2j$.

Common pitfall. Treating the count as sufficient. A structure with DSI = 0 can still be a mechanism if the reactions are parallel or concurrent, or if a truss panel has no bracing; conversely a structure can be locally over-braced and globally unstable at the same time. The second pitfall here is the support symbols: in (c) the left base is fixed, not pinned, and reading it as a pin gives 1 instead of 2, while in (e) both feet are pins, not pin plus roller.

Source. Hibbeler, Structural Analysis, 10th ed., §2.4–2.5 (beams and frames) and §3.2 (trusses); Kassimali, Structural Analysis, 6th ed., Ch. 3.

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