The locus of reaction of a two hinged semi-circular arch, is

Solution (By Examveda Team)

1. Locus of Reactions:
The reaction locus refers to the path traced by the reaction forces at the hinge supports as the load moves along the arch.

2. Two-Hinged Semi-Circular Arch Behavior:
A two-hinged arch is statically indeterminate to one degree, meaning it requires additional equations (beyond equilibrium equations) to determine its reactions.
The horizontal thrust at the hinges plays a crucial role in defining the locus of reactions.

3. Mathematical Justification:
When deriving the equation of the reaction locus, it is found to follow the equation of a hyperbola rather than a circle.
This happens because the reactions depend on the horizontal thrust, which varies with the load position in a hyperbolic manner.

4. Why Not a Circle?
If the locus were a circle, it would imply a uniform reaction distribution independent of the load position.
However, in a two-hinged arch, the reactions change non-linearly, leading to a hyperbolic relationship rather than a circular one.

Conclusion:
The correct answer is Hyperbola (Option D), as it accurately represents the mathematical and structural behavior of the reaction locus in a two-hinged semi-circular arch.