Approach. For rigid (beam/frame) assemblies with no closed rings the degree of static indeterminacy is $\text{DSI}=r-3-c$, where $r$ is the number of reaction components and $c$ the internal condition equations (one per single hinge connecting two members). Where a closed ring is present, $\text{DSI}=3m+r-3n-c$ ($m$ members, $n$ nodes). For pin-jointed trusses $\text{DSI}=m+r-2n$. A negative count, or reactions that are parallel/concurrent, signals a mechanism (unstable) and overrides the count.
(a) Pin + three rollers with two internal hinges under UDL $w$.
(a) Continuous beam, two internal hinges. One pin (2) plus three rollers (1 each) give $r=5$; two single hinges give $c=2$. $\text{DSI}=5-3-2=\boxed{0}$ — statically determinate.
(b) Cranked beam, fixed–roller–fixed with one hinge. A fixed end (3), an intermediate roller (1) and a second fixed end (3) give $r=7$; one internal hinge gives $c=1$. $\text{DSI}=7-3-1=\boxed{3}$ — indeterminate, 3°.
(c) Triangular (A-frame) rigid frame, two pinned feet. Modelling the two legs and the horizontal tie gives $m=5$ members, $n=5$ nodes; two pins give $r=4$; all joints rigid, $c=0$. $\text{DSI}=3(5)+4-3(5)-0=\boxed{4}$ — indeterminate, 4° (one closed ring contributes 3, the redundant reaction adds 1).
(d) Portal frame with a corner hinge. $m=3$, $n=4$; a pinned base (2) and a fixed base (3) give $r=5$; the hinge at the upper-left knee gives $c=1$. $\text{DSI}=3(3)+5-3(4)-1=\boxed{1}$ — indeterminate, 1°.
(e) Five-joint truss with crossing diagonals.
(f) Nine-joint symmetric roof truss.
(e) Truss with crossing diagonals. $m=9$, $n=5$; pin + roller give $r=3$. $\text{DSI}=9+3-2(5)=\boxed{2}$ — indeterminate, 2° (the two crossing diagonals are the redundant members; the equilibrium matrix is full rank, so the truss is stable).
