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Indeterminacy of frame with internal hinges

Short answer

External static indeterminacy is 2, internal 1 (total 3). Kinematic indeterminacy is 15 considering axial deformation, or 9 if axial deformation is neglected.

Determine the static indeterminacy (external/internal) and kinematic indeterminacy for the structure as shown. [Figure: trapezoidal two-storey frame with two internal hinges (one on each of two levels), left base hinged, right base fixed.]

Reading of the figure: trapezoidal two-storey frame with two inclined columns on each side, a floor beam and a top beam forming one closed loop, one internal hinge at each of two levels, left base hinged, right base fixed.

j=6, m=6, r=2+3=5, c=2j = 6,\ m = 6,\ r = 2 + 3 = 5,\ c = 2

Static indeterminacy

  • External: r−3=2r - 3 = 2
  • Internal: 3×1 loop−2=13 \times 1\ \text{loop} - 2 = 1
  • Total: Ds=3m+r−3j−c=18+5−18−2=3D_s = 3m + r - 3j - c = 18 + 5 - 18 - 2 = 3

Kinematic indeterminacy

  • Axial deformation considered: Dk=3j−r+c′=18−5+2=15D_k = 3j - r + c' = 18 - 5 + 2 = 15
  • Axial deformation neglected: Dk=15−6=9D_k = 15 - 6 = 9

Answer: external 2, internal 1 (total 3); Dk=15D_k = 15 (9 if inextensible).

✓ Answer checked by Claude Opus 5.5 (high effort)

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