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Degrees of indeterminacy of frame with hinges

Short answer

External static indeterminacy = 5, internal = 0, total = 5. Kinematic indeterminacy = 16, or 9 if members are considered axially inextensible.

Determine the external and internal degree of static indeterminacy and also the kinematic indeterminacy of the structure given below. [Figure: two-bay frame with internal hinges, supports as drawn (two hinged/roller bases and two fixed bases).]

The figure is read as a single-storey frame on four columns (two hinged bases, two fixed bases) joined by a continuous girder having two internal hinges: j=8, m=7, r=2(2)+2(3)=10, c=2j = 8,\ m = 7,\ r = 2(2) + 2(3) = 10,\ c = 2.

Static indeterminacy

  • External: Dse=r−3−c=10−3−2=5D_{se} = r - 3 - c = 10 - 3 - 2 = 5 (the two hinges give two extra equations of condition for the reactions).
  • Internal: the frame has no closed loop, so Dsi=0D_{si} = 0.
  • Total: Ds=3m+r−3j−c=21+10−24−2=5D_s = 3m + r - 3j - c = 21 + 10 - 24 - 2 = 5.

Kinematic indeterminacy

Each internal hinge adds one independent rotation (c′=2c' = 2).

  • With axial deformation: Dk=3j−r+c′=24−10+2=16D_k = 3j - r + c' = 24 - 10 + 2 = 16
  • Neglecting axial deformation: Dk=16−m=16−7=9D_k = 16 - m = 16 - 7 = 9

Answer: external Dse=5D_{se} = 5, internal Dsi=0D_{si} = 0 (total 5); kinematic Dk=16D_k = 16 (9 if members are inextensible).

✓ Answer checked by Claude Opus 5.5 (high effort)

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