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

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 count reaction components $r$ and internal condition equations $c$ (one per single hinge joining two members). With no closed ring the degree of static indeterminacy is $\text{DSI}=r-3-c$; when a closed ring is present use $\text{DSI}=3m+r-3n-c$ ($m$ members, $n$ joints). For pin-jointed trusses $\text{DSI}=m+r-2n$. A negative count — or reactions that are parallel or concurrent — means a mechanism (unstable) and overrides the arithmetic.

[Figure not reproduced: Q1 — the six structures (a)–(f), redrawn from the exam figure with the classification result under each. See the official exam paper.]

1. (a) Continuous beam, wall + three rollers + one internal hinge. Reactions $r=3\;(\text{fixed})+1+1+1=6$; one hinge gives $c=1$. $\text{DSI}=6-3-1=\boxed{2}$ — indeterminate, 2°.
2. (b) Upper cantilever resting through a roller link on a lower cantilever, both built into the wall. Treat as two rigid bodies. External reactions $r=3+3+1(\text{roller})=7$; the internal roller between the two beams transmits one force. Unknowns $=7+1=8$; equations $=2\times3=6$. $\text{DSI}=8-6=\boxed{2}$ — indeterminate, 2° (each beam is a propped cantilever, $1^\circ$ each).
3. (c) Stepped rigid frame on three pinned bases. No closed ring (tree). $r=2+2+2=6$, $\text{DSI}=6-3=\boxed{3}$ — indeterminate, 3°.
4. (d) Closed rigid frame (one ring) on three pinned bases. $m=8,\;n=8,\;r=6$: $\text{DSI}=3(8)+6-3(8)=\boxed{6}$ — equivalently $3$ (one closed ring) $+\,(6-3)$ external $=6$ — indeterminate, 6°.
5. (e) Truss, pin + roller. $m=11,\;r=3,\;n=7$: $\text{DSI}=11+3-2(7)=\boxed{0}$ — statically determinate (and stable: fully triangulated).
6. (f) Cantilever truss on three pinned feet. $m=9,\;r=6,\;n=7$: $\text{DSI}=9+6-2(7)=\boxed{1}$ — indeterminate, 1°.