The options show frames consisting of rigid bars connected by pin joints. Which one of the frames is non-rigid?

The correct option is the third frame, which is represented by the image showing a three-panel truss where the middle panel lacks any diagonal bracing.

To determine if a plane truss/frame consisting of rigid bars connected by pin joints is rigid (stable) or non-rigid (unstable/mechanism), we analyze both the mathematical criteria for stability and the arrangement of its members.

1. General Mathematical Condition for Rigidity:
For a plane frame to be internally stable and statically determinate, the relationship between the number of members (m) and the number of joints (j) must satisfy:

$$m\ge 2j-3$$

However, satisfying this equation is a necessary condition but not a sufficient one. The members must also be distributed properly so that no part of the structure can behave as a mechanism (undergo deformation without straining the members).

2. Detailed Analysis of the Correct Option (Third Frame):
Let us count the joints and members for the third frame:
• Joints (j): There are 4 joints along the top chord, 4 joints along the bottom chord, and 1 joint at the center of the rightmost panel where the diagonal members intersect. This gives:

$j=9$

• Members (m):
- 3 top horizontal members
- 3 bottom horizontal members
- 4 vertical members
- 1 diagonal member in the leftmost panel
- 4 half-diagonal members connecting the corners of the rightmost panel to its central joint
This gives a total of:

$m=3+3+4+1+4=15$

Using the stability formula:

$$2j-3=2\left(9\right)-3=15$$

Although $m=2j-3$ is satisfied, the members are not distributed properly.
The middle panel of this frame consists of 4 pin-connected joints forming a rectangle with no diagonal bracing member. A four-bar rectangular link with pinned joints behaves as a kinematic mechanism with one degree of freedom. It cannot resist shear forces and will undergo large deformations (shearing into a parallelogram shape) when subjected to an external load (such as a vertical force P applied at the top chord).
Because this middle panel can deform freely without any resistance from the axial members, the entire frame is non-rigid.