Explain indeterminacies; degrees of frame

Static indeterminacy

A structure is statically indeterminate when the equilibrium equations ($\Sigma F_x = 0,\ \Sigma F_y = 0,\ \Sigma M = 0$) are not enough to find all reactions and member forces. The excess number of unknowns is the degree of static indeterminacy $D_s$, the number of redundants. For plane frames $D_s = (3m + r) - (3j + c)$. It has an external part ($r - 3 - c$) and an internal part (from closed loops).

Kinematic indeterminacy

It is the number of independent joint displacements (rotations and translations) that are unknown. $D_k = 3j - r + c'$, and $D_k = 3j - r - m + c'$ if axial deformation is neglected. It decides the work of the displacement method.

Frame in the figure

Reading: two-storey, two-bay frame, bases fixed, hinged and roller, no internal hinge.

$j = 9,\ m = 10,\ r = 3 + 2 + 1 = 6$

• $D_s = 3(10) + 6 - 3(9) = 9$ (external 3, internal $3 \times 2 = 6$)
• Members are inextensible: $D_k = 3(9) - 6 - 10 = 11$

Answer: $D_s = 9$, $D_k = 11$.