Indeterminacy and stability of truss and frame

Axial deformation is neglected, so every member is axially rigid.

(i) Double-diagonal truss with hinged supports

Reading: four panels, chords, five verticals, X-diagonals, both supports hinged.

$j = 10,\ m = 21,\ r = 4$

• Static: $D_s = m + r - 2j = 21 + 4 - 20 = 5$ (external 1, internal 4)
• Kinematic: with rigid bars and a stable arrangement no joint can move, so $D_k = 0$.
• Stability: every panel is triangulated and four reactions (non-concurrent) prevent rigid-body motion. The truss is stable.

(ii) Two-bay braced frame with an internal hinge

Reading: two storeys, three column lines, X-bracing in all four panels, one internal hinge, three base supports (fixed, hinged, roller).

$j = 9,\ m_{frame} = 10,\ \text{braces} = 8,\ r = 6,\ c = 1\ \text{(hinge)} + 16\ \text{(bar ends)}$

• Static: $D_s = 3(18) + 6 - 3(9) - 17 = 16$ (frame alone 8, plus 8 braces)
• Kinematic: rigid X-braces stop all joint translations, so only rotations remain: six upper joints (6), hinged base (1), roller base (1), and the extra rotation at the hinge (1). $D_k = 9$.
• Stability: braced panels and well-placed reactions give a stable frame.

Answer: (i) $D_s = 5$, $D_k = 0$, stable; (ii) $D_s = 16$, $D_k = 9$, stable.