Interrelation between static and kinematic degree of indeterminacy of structure

Interrelation between static and kinematic degree of a structure:

    The static and kinematic degree of indeterminacy can be related to each other and make a single equation. This equation can be obtained by adding algebraic expression of both quantities. We can categorise this process in two cases:

  

    Case 1: (When given structure is a beam, or a frame or an arch) 
The summation of static (Ds) and kinematic (Dk) degree of indeterminacy for beams, frames and

arches is equals to the product of degree of freedom (f) times the summation of number of joints (j), number of loops (l) and negative of number of parts (n).

Ds + Dk = f ( j + l - n)

    For 2D structure the degree of freedom is three (movement along x and y axis; and rotation about z-axis) and for 3D structure the degree of freedom is six (movement along and rotation about x, y and z axis).

Proof:

Dse = r – f * n (external degree of indeterminacy of the structure)

Dsi = f * l (internal degree of indeterminacy of the structure)

Ds = Dse + Dsi

= ( r – f * n ) + ( f * l )

= r – f ( l – n )

Dk = f * j - r

Ds + Dk = f ( j + l – n )

Case 2: (When given structure is a truss) 

The summation of static (Ds) and kinematic (Dk) degree of indeterminacy for a truss structure is equals to the number members (m).

Ds + Dk = m

Proof:

Ds = ( m + r ) – a*j ( for 2D trusses a = 2, for 3D trusses a = 3)

Dk = a*j - r

Ds + Dk = m