GATE Civil Engineering (CE) 2027 Test: Determinacy & Indeterminacy -2 Free

4 reactions mean that the system is definitely indeterminate. But stability would depend upon the nature of forces acting on the planar structure.

The degree of indeterminacy = 3 - 1 - 1 = -1 V So structure is deficient.and unstable. It is a mechanism,

For plane truss degree of indeterminacy,
D_s = m + r_e - 2j
r_e = 4; m= 10; j = 5
D_s = 4 + 10 - 2 x 5 = 4

Force method is useful when,
D_s < D_k
Displacement method is useful when,
D_k < D_s

A structure will be statically determinate if the external reactions can be determined from force- equilibrium equations. A structure is stable when the whole or part of the structure is prevented from large displacements on account of loading.
The structure in figure (b) is stable but statically indeterminate to the second degree.
The structure shown in figure (c) has both reaction components coinciding with each other, so the moment equilibrium condition wilt never be satisfied and the structure will not be under equilibrium. In figure'(a), the structure is stable and there are three reaction components which can be determined by two force equilibrium conditions and one moment equilibrium condition.

D_s=D_se+D_si
D_ss= r_E- 3
= 4 - 3 = 1
D_si = 3C - r_R = 3C - ∑(m¹- 1)
= 3 x 2 -(2-1) +(2-1)
= 6 - 2 = 4 
∴ D_s = D_ss + D_si = 5

Degree of indeterminacy,
n = (m + r_E) - 2j
= (9 + 3 ) - 2 x 6 = 0
Since the degree of indeterminacy is zero and the frame is stable so it is a perfect frame.

D_s = 3m + r_e- r_r 3(j + j)
Number of members m=10
Number of external reactions r_e= 12
Number of interna! reaction components released
(r_r) = 2
Number of rigid joints (j) = 8
Number of joints at which releases are located (j') = 2
∴ D_s = 3 x 10 + 12 - 2 - 3 x ( 8 + 2) = 10