Pin-Jointed Frame: Stability and Indeterminacy Analysis

This question relates to the analysis of pin-jointed plane frames, specifically focusing on their stability and degree of static indeterminacy. We need to determine the nature of the frame given a specific relationship between its components.

Understanding Frame Classification

In structural analysis, frames are classified based on their stability and how their internal forces and reactions are determined:

• Statically Determinate Frame: A frame whose reactions and internal forces can be determined solely by the equations of static equilibrium ($ \sum F_x = 0 $, $ \sum F_y = 0 $, $ \sum M = 0 $).
• Statically Indeterminate Frame: A frame that cannot be fully analyzed using only the static equilibrium equations. It possesses more unknown forces and reactions than available equilibrium equations. This requires additional equations, usually derived from deformation compatibility, to solve.
• Stable Frame: A frame that maintains its shape and equilibrium under applied loads without collapsing.
• Unstable Frame: A frame that lacks rigidity and may collapse under load, even if static equilibrium equations are satisfied.

Degree of Static Indeterminacy (DOS)

For a pin-jointed plane frame, the Degree of Static Indeterminacy (DOS) is calculated using the following formula:

DOS = m + r - 2j

Where:

• m = number of members in the frame
• r = number of reaction components at the supports
• j = number of joints in the frame

Applying the Given Condition

The question provides the condition:

(m + r) > 2j

To find the DOS, we can rearrange this inequality:

m + r - 2j > 0

This directly corresponds to the DOS formula. Therefore, the condition implies:

DOS > 0

Interpreting the Result

A frame with DOS > 0 is classified as statically indeterminate.

Regarding stability, the condition m + r > 2j (or DOS > 0) generally indicates that the frame has sufficient members and reactions to provide stability, assuming a proper geometric configuration of members and supports. While specific geometric arrangements can lead to instability even when DOS > 0, in the absence of such information, this condition is conventionally associated with stability.

Therefore, a pin-jointed plane frame satisfying (m + r) > 2j is considered stable and statically indeterminate.

Conclusion

Based on the analysis, the frame is stable because the condition generally ensures sufficient constraints, and it is statically indeterminate because the Degree of Static Indeterminacy is greater than zero.