Static and kinematic indeterminacy of frame

The frame is a single-storey frame with three columns: left hinged, middle fixed, right on a roller. A continuous girder joins the column tops.

$j = 6,\ m = 5\ (3\ \text{columns} + 2\ \text{beams}),\ r = 2 + 3 + 1 = 6,\ c = 0$

Static indeterminacy

$D_s = 3m + r - 3j - c = 15 + 6 - 18 = 3$

External $= r - 3 = 3$; internal $= 0$ (no closed loop).

Kinematic indeterminacy

• Axial deformation considered: $D_k = 3j - r = 18 - 6 = 12$
• Axial deformation neglected: $D_k = 3j - r - m = 18 - 6 - 5 = 7$

The 7 are: rotations of the three top joints (3), rotation of the hinged base (1), rotation of the roller base (1), horizontal movement of the roller base (1) and sway of the girder (1).

Answer: $D_s = 3$; $D_k = 12$ (7 for inextensible members).