Kinematic Indeterminacy MCQ with answer and solution

Concept

Kinematic Indeterminacy of Beams

Formula

Dk = 3j - r - m

Step-by-step solution

1. Identify the parameters: joints j = 2, reactions r = 3 (2 at hinge + 1 at roller), inextensible members m = 0. 2. Substitute into the formula: Dk = 3(2) - 3 - 0 = 3.

Explanation

A simply supported beam has two supports: a hinged support and a roller support. At the hinged support, vertical and horizontal movements are restricted, leaving 1 degree of freedom (rotation). At the roller support, only vertical movement is restricted; since the member is extensible, it can move horizontally and rotate, giving 2 degrees of freedom. Adding these together gives a total kinematic indeterminacy of 3 (option d).

Shortcut

Use the formula Dk = 3j - r. For a simply supported beam: 2 joints (j = 2), 3 reactions (r = 3: 2 at hinge, 1 at roller). Extensible means m = 0, so Dk = 3(2) - 3 = 3. Shortcut tip: Count the unconstrained degrees of freedom directly—1 rotation at the hinge, plus 1 rotation and 1 horizontal displacement at the roller = 3.

Common mistake

Many students mistakenly assume the member is inextensible (m = 1) instead of extensible (m = 0), leading them to subtract axial deformation and incorrectly select 2 instead of 3. Another common mistake is neglecting horizontal displacement at the roller support and considering only rotational degrees of freedom.