Question 1 of 8: Classify each structure — unstable, statically determinate, or statically indeterminate

Question 1: Classify each structure — unstable, statically determinate, or statically 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.

Given. Six structures (a)–(f): (a) a beam with pin + internal hinge + roller + roller; (b) two inclined beams meeting at a pinned apex, each also propped by a roller; (c) a trapezoidal portal with two fixed bases and hinges at both top corners; (d) an L-shaped beam–column chain with a pin, an internal hinge/pin at the foot of the leg, and an end roller; (e) a wall-mounted, two-panel X-braced truss; (f) a wall-mounted Warren-type truss with two separate pin supports.

Find. The classification (unstable / determinate / indeterminate, with degree) of each.

[Figure not reproduced: Structures (a)–(f) — supports and hinge locations as printed on the exam sheet. See the official exam paper.]

Approach. For a beam/frame, count reactions r and subtract 3 (global equilibrium) and one condition equation per internal hinge c that joins exactly two members: $$i=r-3-c$$ For a pin-jointed truss with m members, r reactions and n joints: $$i=m+r-2n$$ i=0 is determinate, i>0 is indeterminate to that degree, i<0 is a mechanism (unstable).

1. (a) Beam — pin, hinge, roller, roller. Reactions: pin (2) + roller (1) + roller (1) $$r=4$$. One hinge joining 2 members, $$c=1$$. $$i=4-3-1=\boxed{0}$$ Statically determinate.
2. (b) Two inclined beams to a pinned apex. Each inclined member is propped by its own roller at the top; the apex where the two members meet is itself a pin support. Reactions: roller + roller + pin $$r=1+1+2=4$$. The apex is a hinge joining the 2 inclined members, $$c=1$$. $$i=4-3-1=\boxed{0}$$ Statically determinate — each rafter is simply a propped cantilever back to the shared apex pin.
3. (c) Trapezoidal portal, two fixed bases, hinges at both top corners. Reactions: fixed + fixed $$r=3+3=6$$. Two hinges, each joining 2 members, $$c=2$$. $$i=6-3-2=\boxed{1}$$ Statically indeterminate to the 1st degree.
4. (d) L-shaped chain — pin, hinge/pin at the knee, roller. Reactions: pin (2, top) + pin (2, at the hinged knee support) + roller (1) $$r=5$$. One hinge (at the knee) joining 2 members, $$c=1$$. $$i=5-3-1=\boxed{1}$$ Statically indeterminate to the 1st degree.
5. (e) Wall-mounted X-braced truss. Members: 2 top chord + 2 bottom chord + 1 end vertical + 4 diagonals (2 crossing pairs, unconnected) $$m=9$$. Joints $$n=6$$. The truss is pinned to the wall at 2 separate points, $$r=2+2=4$$. $$i=m+r-2n=9+4-12=\boxed{1}$$ Statically indeterminate to the 1st degree — both diagonals of each panel are present, one more brace than a minimally-stable panel needs.
6. (f) Wall-mounted Warren truss, two separate pins. Members: 2 top chord + 2 bottom chord + 5 zig-zag diagonals + 1 support diagonal to the lower pin $$m=10$$. Joints $$n=7$$ (including the extra lower-pin joint). Reactions: pin + pin $$r=4$$. $$i=10+4-14=\boxed{0}$$ Statically determinate.