Approach. For beam/rigid-frame assemblies, degree of static indeterminacy $\text{DSI}=3m+r-3n-c$ ($m$ members, $n$ rigid nodes, $r$ reaction components, $c$ condition equations = one released moment per single hinge). For pin-jointed trusses, $\text{DSI}=m+r-2j$. A negative count, or reactions that are all parallel or concurrent, means a mechanism (unstable) regardless of the arithmetic.
(a) Continuous beam with two internal hinges. Supports read (left→right) roller, roller, roller, pin ⇒ $r=1{+}1{+}1{+}2=5$; two single hinges ⇒ $c=2$. $\text{DSI}=r-3-c=5-3-2=\boxed{0}$ — statically determinate (and stable: the pin supplies horizontal restraint, each rigid link is held by two support/hinge points).
(b) Two-bay, part two-storey rigid frame. Members $m=13$, rigid nodes $n=11$; supports = fixed (3) + pin (2) + fixed (3) ⇒ $r=8$; hinges at the two top outer corners ⇒ $c=2$. $\text{DSI}=3(13)+8-3(11)-2=\boxed{12}$ — indeterminate, 12° (three closed panels give $3\times3=9$, the three fixed/pin bases add $8-3-2=3$).
(c) Portal-type bent with an internal panel. $m=5$, $n=5$, two fixed bases $r=6$, no hinge $c=0$. $\text{DSI}=3(5)+6-3(5)=\boxed{6}$ — indeterminate, 6° (one closed cell $=3$, plus $6-3=3$ from the two fixed supports).
(d) Beam on two inclined links and a central roller. Each inclined member is pinned to a wall and hinged to the beam — a two-force member delivering one force along its own line. The beam therefore sees three reaction components (two link axials + one roller), non-parallel and non-concurrent: $\text{DSI}=r-3=3-3=\boxed{0}$ — statically determinate and stable.
(e) “Wheel” hexagonal truss. Six rim joints + one hub $=j=7$; six rim members + six spokes $=m=12$; pin + roller $\Rightarrow r=3$. $\text{DSI}=m+r-2j=12+3-14=\boxed{1}$ — internally indeterminate, 1° (a fully-triangulated hexagon carries one redundant member).
(f) Triangle with its medial (inner) triangle. $j=6$, $m=9$; supports pin + roller + pin $\Rightarrow r=5$. $\text{DSI}=9+5-12=\boxed{2}$ — indeterminate, 2° (the extra pin adds two external redundants to an internally-determinate truss).
Structure
Classification
(a)
Determinate (DSI = 0)
(b)
Indeterminate, 12°
(c)
Indeterminate, 6°
(d)
Determinate (DSI = 0)
(e)
Indeterminate, 1°
(f)
Indeterminate, 2°
Reading each drawing carefully is decisive: two of these “beam-type” cases (a and d) look indeterminate at a glance yet are determinate once the internal hinges and the two-force links are recognised, while the innocuous-looking multi-storey frame (b) is highly redundant because every enclosed panel locks in three extra unknowns. The rule of thumb is to count reactions and releases first, then confirm the reaction lines are not all parallel or concurrent before trusting a zero result.
