Degree of kinematic indeterminacy of a pin jointed plane frame refers to the number of independent displacements the frame can undergo without violating the equilibrium conditions. It is denoted by "d" and can be calculated using the formula:

d = 2j - r

where:
j = number of joints in the frame
r = number of reaction components

Let's understand this formula in detail:

1. Number of Joints (j):
Joints in a frame are the points where two or more members meet and are connected by pin joints. Each joint allows for two degrees of freedom, one for translation (either in the horizontal or vertical direction) and one for rotation. So, the total number of degrees of freedom contributed by the joints is 2j.

2. Number of Reaction Components (r):
Reaction components refer to the forces and moments exerted by the supports or foundations on the frame. These can be calculated using the equations of equilibrium. The number of reaction components depends on the type of supports and constraints applied to the frame.

3. Calculation of Degree of Kinematic Indeterminacy (d):
The formula for calculating the degree of kinematic indeterminacy takes into account the contributions from both joints and reaction components. By subtracting the number of reaction components from twice the number of joints, we obtain the degree of kinematic indeterminacy.

4. Interpretation of the Formula:
The formula 2j - r signifies that each joint contributes two degrees of freedom, while the reaction components restrict the overall degrees of freedom. Therefore, the degree of kinematic indeterminacy represents the remaining number of independent displacements that can occur in the frame.

5. Example:
Suppose we have a pin jointed frame with 5 joints (j = 5) and 3 reaction components (r = 3). Substituting these values into the formula, we get:

d = 2j - r
d = 2(5) - 3
d = 10 - 3
d = 7

So, the degree of kinematic indeterminacy for this frame is 7, meaning it can undergo 7 independent displacements without violating the equilibrium conditions.

To summarize, the formula for calculating the degree of kinematic indeterminacy of a pin jointed plane frame is given by 2j - r, where j represents the number of joints and r represents the number of reaction components. The formula takes into account the contributions from both joints and reaction components and provides a measure of the frame's ability to undergo independent displacements while maintaining equilibrium.