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Indeterminacy and stability of truss-frames

Short answer

X-braced truss with cantilever: static indeterminacy 2, kinematic 11. Braced single-storey frame: static 5, kinematic 5 (inextensible). Both are stable.

Determine the external/internal static indeterminacy and kinematic indeterminacy of the structures shown in figure below. Are they stable or unstable? [Figure: (i) a truss-like frame with X-bracing in two panels, hinged at one support and roller at the other end, plus a cantilever arm; (ii) a braced frame with diagonals and three supports (fixed, hinged, roller).]

(i) Truss with X-bracing in two panels and a cantilever arm

Reading: top and bottom chords with three verticals and two X-braced panels, hinge at one end and roller at the other, plus a cantilever arm of two bars carrying a joint.

j=6+1=7, m=4+3+4+2=13, r=3j = 6 + 1 = 7,\ m = 4 + 3 + 4 + 2 = 13,\ r = 3

  • Ds=m+r−2j=13+3−14=2D_s = m + r - 2j = 13 + 3 - 14 = 2 (internal 2, one per X-panel; external 0)
  • Dk=2j−r=14−3=11D_k = 2j - r = 14 - 3 = 11
  • Stable: triangulated panels, the cantilever arm is attached to the rigid truss by two bars, and the reactions are non-concurrent.

(ii) Braced single-storey frame

Reading: three columns (fixed, hinged, roller), two beams and two pin-ended diagonals.

j=6, m=7, r=3+2+1=6, c=4 (bar ends)j = 6,\ m = 7,\ r = 3 + 2 + 1 = 6,\ c = 4\ \text{(bar ends)}

  • Ds=3(7)+6−18−4=5D_s = 3(7) + 6 - 18 - 4 = 5 (external 3, internal 2 from the two closed braced bays)
  • Dk=18−6−7=5D_k = 18 - 6 - 7 = 5 (inextensible)
  • Stable: braced panels are rigid and reactions prevent all rigid-body motion.

Answer: (i) Ds=2, Dk=11D_s = 2,\ D_k = 11; (ii) Ds=5, Dk=5D_s = 5,\ D_k = 5. Both stable.

✓ Answer checked by Claude Opus 5.5 (high effort)

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