Indeterminacy of truss and beam-cable structure

All members are extensible, so $D_k$ counts all joint displacements.

(i) Truss with hinge and roller supports and internal hinges

Reading: a three-panel truss with top and bottom chords, four verticals and three diagonals (no redundant member). The internal hinges are pin joints of the truss and do not change the count.

$j = 8,\ m = 3 + 3 + 4 + 3 = 13,\ r = 3$

• $D_s = m + r - 2j = 13 + 3 - 16 = 0$ (determinate)
• $D_k = 2j - r = 16 - 3 = 13$

(ii) Beam supported by cables

Reading: a horizontal beam fixed at one end and held by two cables (tension-only bars) anchored above, attached at two points along the beam.

Unknowns: 3 fixed-end reactions + 2 cable tensions = 5. Equations = 3.

$D_s = 5 - 3 = 2$

Free beam joints: two cable-attachment points and the free end = 3 joints, each with 2 translations and 1 rotation, with the cable anchors fixed:

$D_k = 3 \times 3 = 9$

Answer: (i) $D_s = 0$, $D_k = 13$; (ii) $D_s = 2$, $D_k = 9$.