Approach. Count external reaction components r and released internal conditions c (an internal hinge in a flexural chain releases one moment, c = 1; a pin joining k members releases k−1). For beams/frames without closed rings, DSI = r − 3 − c; for pin-jointed trusses DSI = m + r − 2j. A body held by fewer independent restraints than its rigid-body freedoms is a mechanism ⇒ unstable.
(a) Propped/continuous beam: fixed wall at the left, an internal hinge just to its right, then two roller supports, free right end. $r = 3(\text{fixed})+1+1 = 5$; one hinge $c=1$. $DSI = 5-3-1 = \boxed{1}$ — indeterminate, degree 1.
(b) Loaded beam resting on the apex of an A-frame through a single crown hinge, beam ends free. The beam is connected to the rest of the structure at exactly one pin, so it is free to rotate about that pin — a rigid body on one hinge has an unrestrained rotation. $\Rightarrow$ unstable (internal mechanism), regardless of the A-frame beneath.
(c) Loaded beam with two rigidly-attached columns fixed at their bases, plus a pin-ended inclined strut to a pinned foot at each end. Treating beam + two columns as one rigid body: restraints $= 3+3$ (two fixed feet) $+1+1$ (two axial struts) $=8$; one rigid body gives 3 equations. $DSI = 8-3 = \boxed{5}$ — indeterminate, degree 5.
(d) Two stacked simply-supported beams (upper beam on a pin + roller that bear on the lower beam; lower beam on a ground pin + roller). Two rigid bodies (6 equations); reactions $r=3$ (pin+roller to ground); internal transfer forces $=2+1$ (pin + roller between beams). $DSI = (3+3)-6 = \boxed{0}$ — statically determinate.
(e) Loaded horizontal bar hung from two inclined two-force members to two pinned supports. The bar is a rigid body restrained only by the two axial links (2 constraints) but needs 3; the two link axes meet at a point about which the bar can rotate. $\Rightarrow$ unstable (one-degree mechanism).
(f) Trapezoidal truss, pin + roller feet, one panel with crossed diagonals. $m=10,\ r=3,\ j=6$. $DSI = m+r-2j = 10+3-12 = \boxed{1}$ — indeterminate, degree 1 (one redundant diagonal in the X-panel).
(g) Rectangular-panel truss with two pinned supports on the left edge and a crossed-diagonal panel. $m=9,\ r=4$ (two pins), $j=6$. $DSI = 9+4-12 = \boxed{1}$ — indeterminate, degree 1 (the internal truss is just-rigid, so the extra reaction is the single redundancy).
Structure
Classification
(a) beam: fixed + hinge + 2 rollers
Statically indeterminate — degree 1
(b) beam on single crown hinge
Unstable (mechanism)
(c) beam + 2 fixed columns + 2 struts
Statically indeterminate — degree 5
(d) two stacked simple beams
Statically determinate
(e) bar on two inclined links
Unstable (mechanism)
(f) trapezoidal X-panel truss
Statically indeterminate — degree 1
(g) rectangular X-panel truss, 2 pins
Statically indeterminate — degree 1
Check (figure interpretation): (b) and (e) are read as the beam/bar being connected to the rest of the structure only through pin(s), giving a rotational mechanism — if any of those circles were a rigid (moment) connection the member would instead be indeterminate. (g) is read with both left supports as pins and no right support (the right bottom node carries only the load P); a right roller instead would give degree 2.
