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 rigid (beam/frame) assemblies with no closed rings, degree of static indeterminacy $\text{DSI}=r-3-c$, where $r$ = number of reaction components and $c$ = internal condition equations (one per single hinge joining two members). Where closed rings exist, $\text{DSI}=3\,(\text{rings})+(r-3)$. For pin-jointed trusses, $\text{DSI}=m+r-2j$ ($m$ members, $j$ joints). A negative count or a set of parallel/concurrent reactions signals a mechanism (unstable), which always overrides the count.

1. (a) Propped beam with one internal hinge. Two rollers (1 each) plus a fixed end (3) give $r=5$; one hinge gives $c=1$. $\text{DSI}=5-3-1=\boxed{1}$ — statically indeterminate, 1°. (The fixed end supplies the only horizontal restraint; with no horizontal load this is consistent and stable.)
2. (b) Rigid L-bent. A pin at the foot of the leg (2) and a pin under the beam (2) give $r=4$; the 90° corner is rigid, $c=0$. $\text{DSI}=4-3=\boxed{1}$ — indeterminate, 1°.
3. (c) Suspended (hung) lower beam. The upper beam is fixed + roller; the lower beam is carried only by a vertical two-force link (pinned top and bottom) at its right end and a vertical roller near mid-length. Both of these restraints are vertical and parallel, so the lower beam has no horizontal restraint and is free to translate as a rigid body — a mechanism. The structure is unstable (geometrically, by parallel reactions), even though the naive count $r-3-c=(3{+}1{+}1)-3-2=0$ would read “determinate.”
4. (d) Two-storey rigid frame on two pins. The frame encloses two closed rectangular panels ($2\times3=6$ internal redundants) and the two pins give $r=4$, i.e. one redundant reaction $(4-3)$. $\text{DSI}=3(2)+(4-3)=\boxed{7}$ — indeterminate, 7°.
5. (e) Parallel-chord truss with one X-panel. $m=18$, $j=10$, $r=3$ (pin + roller). $\text{DSI}=18+3-2(10)=\boxed{1}$ — indeterminate, 1° (the extra crossing diagonal is the single redundant; the equilibrium matrix is full-rank, so it is stable).
6. (f) Braced tower on two pins. $m=10$, $j=6$, $r=4$ (two pins). $\text{DSI}=10+4-2(6)=\boxed{2}$ — indeterminate, 2° (both X-braced panels are over-braced by one diagonal each).