Indeterminacy: frame with hinge and truss

(i) Two-bay, two-storey frame with an internal hinge

Reading of the figure: three column lines over two storeys with floor beams at two levels. Supports: one fixed, one hinged, one roller; one internal hinge.

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

• Static: $D_s = 3(10) + 6 - 3(9) - 1 = 8$ (external $6-3 = 3$; internal $3 \times 2\ \text{closed loops} - 1 = 5$).
• Kinematic, axial deformation allowed: $D_k = 3(9) - 6 + 1 = 22$.
• Kinematic, inextensible members: $D_k = 22 - 10 = 12$.

(ii) Truss with double diagonals

Reading of the figure: four panels with top and bottom chords, five verticals and two diagonals in each panel (X-bracing). Hinge at one end, roller at the other.

$j = 10,\ m = 4 + 4 + 5 + 8 = 21,\ r = 3$

• Static: $D_s = m + r - 2j = 21 + 3 - 20 = 4$ (external 0, internal 4, one for each X-panel).
• Kinematic: $D_k = 2j - r = 20 - 3 = 17$.

Answer: (i) $D_s = 8$, $D_k = 22$ (12 if inextensible); (ii) $D_s = 4$, $D_k = 17$.