Geometry of reactions in a two-hinged semicircular arch For a two-hinged semicircular arch under variable loading, the locus of the resultant reaction at a support traces which curve in a plane diagram of reactions?

Introduction / Context:
Arches carry loads by combined axial compression and bending; reactions depend on the load position and magnitude. For certain canonical arch geometries, elegant geometric properties emerge. A semicircular two-hinged arch is one such case where the reaction locus exhibits a simple, recognizable curve.

Given Data / Assumptions:

• Arch is semicircular with two supports (hinged at ends).
• Reactions are examined as loads vary in position along the span.
• Plane of analysis; small deflection theory and classical statics apply.

Concept / Approach:
In a semicircular arch, symmetry and constant radius lead to special relationships between horizontal thrust and vertical reaction as load moves. The vectorial sum at a support (resultant reaction) varies in magnitude and direction but maintains a constant relation that plots as a circle in the reaction diagram.

Step-by-Step Solution:
Consider unit and moving loads to generate envelopes of H (horizontal thrust) and V (vertical reaction).Combine H and V vectorially to get the resultant R at a support.Show that as the load position changes, the tip of vector R traces a circle due to the semicircular geometry and constant-radius relationships.

Verification / Alternative check:
Classical arch theory texts present derivations or Mohr diagrams of reactions for circular arches, where the locus of resultant reactions forms a circle under varying loading conditions.

Why Other Options Are Wrong:
Parabola/Hyperbola/Straight line: Do not match the geometric relation arising from the semicircular radius and symmetric boundary conditions.

Common Pitfalls:

• Confusing the locus of the line of thrust within the arch ring with the locus of support reactions in the reaction plane.
• Assuming the same locus shape for noncircular arches; it is specific to semicircular geometry here.

Final Answer:
circle