What is the possible number of independent mechanism in the portal frame given below?

Concept-

When a structure is subjected to a system of loads, it is stable and hence functional until a sufficient number of plastic hinges have been formed to render the structure unstable. As soon as the structure reaches an unstable condition, it is considered to have been failed. The segments of the beam between the plastic hinges are able to move without an increase of load. This condition in a member is called mechanism.

If an indeterminate structure has redundancy r, the insertion of r plastic hinges makes it statically determinate. Any further hinge converts this statically determinate structure into mechanism. Hence for collapse the number of plastic hinges required are r + 1.

Number of independent mechanism-

The number of independent mechanism can be determined as follows,

Let N = Number of possible plastic hinges, r = Number of redundancies

Then number of possible independent mechanism will be N - r.

Given data and Calculation-

Possible location of plastic hinges → 4. (A, B, C, D).

At E, No plastic hinge will form as it is already a hinge joint.

Redundancy of the structure = 2.

No. of independent mechanism = 4 – 2 = 2.

(Beam mechanism & Sway mechanism)