Question

Consider a beam PQ fixed at P, hinged at Q, and subjected to a load \( F \) as shown in the figure (not drawn to scale). The static and kinematic degrees of indeterminacy, respectively, are

• A. 2 and 1
• B. 2 and 0
• C. 1 and 2
• D. 2 and 2

Hint:

The static degree of indeterminacy is determined by comparing the number of reactions to the number of equilibrium equations, while the kinematic degree of indeterminacy is determined by the number of unknown displacements.

Answer 1

The Correct Option is A

This problem involves calculating the static and kinematic degrees of indeterminacy for a beam subjected to a force \( F \), where the beam is fixed at point \( P \) and hinged at point \( Q \). Static Degrees of Indeterminacy:
The static degree of indeterminacy is the number of unknown forces or reactions that cannot be determined by the static equilibrium equations alone. To find the static degree of indeterminacy, we use the following formula: \[ r_s = \text{Number of reactions} - \text{Number of equilibrium equations}. \] In this case, the beam has: - A fixed support at \( P \), which provides 3 reactions (vertical force, horizontal force, and moment). - A hinged support at \( Q \), which provides 2 reactions (vertical and horizontal force). Thus, the total number of reactions is: \[ r_{\text{reactions}} = 3 + 2 = 5. \] Since we have 3 equilibrium equations for static analysis (sum of forces in \( x \)-direction, \( y \)-direction, and sum of moments), the static degree of indeterminacy is: \[ r_s = 5 - 3 = 2. \] Kinematic Degrees of Indeterminacy:
The kinematic degree of indeterminacy refers to the number of unknown displacements that cannot be determined using the kinematic equations. For this beam: - There is one unknown displacement at the point where the force \( F \) is applied (the angular displacement \( \theta \) at point \( Q \)). Thus, the kinematic degree of indeterminacy is 1. Conclusion:
The static degree of indeterminacy is 2, and the kinematic degree of indeterminacy is 1. Therefore, the correct answer is option (A).