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Static and kinematic indeterminacy of frame

Short answer

Static indeterminacy = 8. Kinematic indeterminacy = 22, or 12 if members are axially inextensible.

Determine degrees of static and kinematic indeterminacies for the frame shown below. [Figure: two-storey, two-bay frame with an internal hinge at the top-left; supports: fixed at the left, hinged in the middle (with a fixed base), roller/hinge at the right as drawn.]

Reading of the figure: two-storey, two-bay frame with three column lines, a hinge at the top-left joint, and supports fixed (left), hinged (middle) and roller (right).

j=9, m=10 (6 columns+4 beams), r=3+2+1=6, c=1j = 9,\ m = 10\ (6\ \text{columns} + 4\ \text{beams}),\ r = 3 + 2 + 1 = 6,\ c = 1

Static indeterminacy

Ds=3m+r−3j−c=30+6−27−1=8D_s = 3m + r - 3j - c = 30 + 6 - 27 - 1 = 8

Kinematic indeterminacy

  • Axial deformation considered: Dk=3j−r+c′=27−6+1=22D_k = 3j - r + c' = 27 - 6 + 1 = 22
  • Axial deformation neglected: Dk=22−m=22−10=12D_k = 22 - m = 22 - 10 = 12

Answer: Ds=8D_s = 8; Dk=22D_k = 22 (12 for inextensible members).

✓ Answer checked by Claude Opus 5.5 (high effort)

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