Indeterminacy and stability of frame and truss

(i) Braced two-bay frame

Reading: one storey, three columns (two fixed, one pinned), two beams, one pin-ended diagonal in each bay, and two internal hinges.

$j = 6,\ m = 5 + 2 = 7,\ r = 3 + 3 + 2 = 8,\ c = 2\ \text{hinges} + 4\ \text{bar ends} = 6$

• $D_s = 3(7) + 8 - 3(6) - 6 = 5$. External: $r - 3 = 8 - 3 = 5$; internal: each brace adds 1 and each hinge releases 1, so $2 - 2 = 0$. Check: the frame without braces gives $15 + 8 - 18 - 2 = 3$, and each brace adds 1, giving 5.
• Kinematic (inextensible members): the columns stop vertical movement of the top joints, the beams tie them together, and the braces stop sway, so no joint translation remains. The unknowns are the rotations of the three top joints (3), the rotation of the pinned base (1) and the extra rotations at the two hinges (2): $D_k = 6$. (The shortcut $3j - r + c' - m = 5$ undercounts here, because the two braces give two constraints against only one sway.) With axial deformation, $D_k = 3j - r + c' = 18 - 8 + 2 = 12$.
• Stability: reactions are non-concurrent and non-parallel, and the bays are closed by braces. The frame is geometrically stable.

(ii) Double-diagonal truss

Reading: four panels, chords, five verticals, both diagonals in each panel, hinge and roller.

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

• $D_s = m + r - 2j = 21 + 3 - 20 = 4$ (internal)
• $D_k = 2j - r = 17$
• Stability: $m + r = 24 > 2j = 20$, all panels are braced triangles, and the three reactions are neither parallel nor concurrent. The truss is stable.

Answer: (i) $D_s = 5$, $D_k = 6$ for inextensible members (12 if extensible), stable; (ii) $D_s = 4$, $D_k = 17$, stable.