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

Question 1 — Classify each structure: unstable, determinate or indeterminate [6]

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 $D = (3m + r) - (3n + c)$ ($m$ members, $n$ joints, $r$ reaction components, $c$ released condition equations — one per internal hinge in a beam line). For a beam line this reduces to $D = r - (3 + c)$. For a pin-jointed truss use $D = (m + r) - 2n$. Then $D=0$ & stable ⇒ determinate; $D>0$ ⇒ indeterminate to degree $D$; a reaction/member set that cannot resist every equilibrium mode is unstable regardless of the count.

Structure (a) — Statically determinate

Supports: two rollers ($1+1$) and a fixed end ($3$), so $r=5$. There are two internal hinges (both circle symbols — the “typical hinge” note labels the symbol), giving $c=2$ condition equations. $D=r-(3+c)=5-3-2=\boxed{0}$: determinate. One horizontal restraint (the fixed end) plus five reactions and two hinges leave the beam just-rigid.

Structure (b) — Indeterminate — degree 5

Rigid frame: $m=6$ members, $n=6$ joints; supports pin ($2$) + fixed ($3$) ⇒ $r=5$, no internal hinge ($c=0$). $D=3m+r-3n-c=18+5-18-0=\boxed{5}$. Equivalently the closed upper cell is internally $3^{\circ}$ indeterminate and the reactions carry $5-3=2$ external redundants ($3+2=5$).

Structure (c) — Indeterminate — degree 1

One rigid bent body; two rollers ($1+1$) under the beam and a pin ($2$) at the top of the right riser give $r=4$, no internal hinge. $D=r-3=4-3=\boxed{1}$. The pin supplies horizontal restraint, so the bent is stable with one redundant.

Structure (d) — Indeterminate — degree 2

Members $m=4$, joints $n=5$; supports pin ($2$) + roller ($1$) + fixed ($3$) ⇒ $r=6$, one internal hinge ⇒ $c=1$. $D=3m+r-3n-c=12+6-15-1=\boxed{2}$.

Structure (e) — Statically determinate

Counting all bars (four top chords, two bottom chords, three verticals and four fan diagonals from the two supports down to the lower nodes) gives $m=13$; joints $n=8$; $r=3$. $D=m+r-2n=13+3-16=\boxed{0}$ — determinate, with complete triangulation (no mechanism).

Structure (f) — Indeterminate — degree 1

Roof triangle plus a square panel carrying both diagonals: $m=8$ ($2$ rafters, top tie, two sides, bottom chord, two crossing diagonals), $n=5$, $r=3$. $D=m+r-2n=8+3-10=\boxed{1}$ — the second panel diagonal is the internal redundant.