Degree of kinematic indeterminacy of a pin jointed plane frame is given by .............

1. Solution to the MCQ

Question: Degree of kinematic indeterminacy of a pin-jointed plane frame is given by...

• Correct Option: A) $2j - r$

Brief Justification:

In a pin-jointed plane frame (2D Truss), each joint ($j$) has two degrees of freedom: translation in the $X$ and $Y$ directions. Rotation is not considered because the members are pin-connected. Therefore, the total degrees of freedom are $2j$. To find the kinematic indeterminacy ($D_k$), we subtract the number of external constraints or reactions ($r$) provided by the supports. Hence, $D_k = 2j - r$.

2 & 3. High-Yield MCQs (Structural Analysis)

The following questions are curated based on standard patterns found in GATE, ESE, and SSC JE exams.

Q1. What is the degree of kinematic indeterminacy ($D_k$) for a rigid-jointed plane frame?

• A) $2j - r$

• B) $3j - r$

• C) $6j - r$

• D) $3j + r$

• Answer: B) $3j - r$ (Rigid joints in 2D have two translations and one rotation).

Q2. For a space truss (pin-jointed space frame), the degree of kinematic indeterminacy is:

• A) $2j - r$

• B) $3j - r$

• C) $6j - r$

• D) $j - r$

• Answer: B) $3j - r$ (Each joint in 3D space has three translational degrees of freedom).

Q3. If the members of a rigid-jointed plane frame are assumed to be axially inextensible, and '$m$' is the number of members, the kinematic indeterminacy becomes:

• A) $3j - r - m$

• B) $3j - r + m$

• C) $2j - r - m$

• D) $3j - r$

• Answer: A) $3j - r - m$ (Each inextensible member removes one translational degree of freedom).

Q4. The number of independent equilibrium equations available for a single joint in a pin-jointed plane frame is:

• A) 1

• B) 2

• C) 3

• D) 6

• Answer: B) 2 ($\sum F_x = 0$ and $\sum F_y = 0$).

Q5. Kinematic indeterminacy is also known as:

• A) Redundancy

• B) Degrees of freedom

• C) Compatibility

• D) Determinacy

• Answer: B) Degrees of freedom

4. Core Theoretical Concepts: Kinematic Indeterminacy

• Definition: Kinematic Indeterminacy ($D_k$) refers to the number of independent joint displacements (translations and rotations) that are necessary to describe the deformed shape of a structure.

• Pin-Jointed Plane Frame (2D Truss):

  • Each joint can move in $X$ and $Y$ directions.

  • No internal rotation resistance exists at the joints.

  • Formula: $D_k = 2j - r$

• Rigid-Jointed Plane Frame (2D Frame):

  • Each joint has 3 degrees of freedom: $\Delta_x$, $\Delta_y$, and $\theta_z$.

  • Formula: $D_k = 3j - r$

• Space Structures (3D):

  • Pin-jointed (Truss): $D_k = 3j - r$ (Translations in $X, Y, Z$).

  • Rigid-jointed (Frame): $D_k = 6j - r$ (3 Translations, 3 Rotations).

• Effect of Inextensibility: If members are "rigid" or "axially inextensible," they do not change length. Each such member reduces the $D_k$ by 1.

  • Adjusted Formula: $D_k = (Degrees\ of\ Freedom \times j) - r - m$

• Significance in Analysis: $D_k$ is the fundamental starting point for the Stiffness Method (Displacement Method) of structural analysis, where joint displacements are treated as the primary unknowns.