Three two-hinged semicircular arches A, B, and C (radii 5 m, 7.5 m, and 10 m, respectively) each carry an identical point load W at the crown. What is the ratio of the horizontal thrusts at the supports (H_A : H_B : H_C)?

Introduction / Context:
Two-hinged semicircular arches of geometrically similar shape under the same type and position of load exhibit horizontal thrusts that scale with a characteristic length. With a crown load on a semicircular arch, the thrust depends on the span–rise geometry; for a semicircle, span = 2R and rise = R, giving a constant span/rise ratio but a thrust that scales linearly with size (radius) for identical load placement and type.

Given Data / Assumptions:

• Arches are semicircular, radii: R_A = 5 m, R_B = 7.5 m, R_C = 10 m.
• Identical load W at the crown for each arch.
• Similar geometry and material behavior; standard two-hinged analysis.

Concept / Approach:
For similar arches under the same nondimensional loading position (crown), the horizontal thrust H scales with a characteristic dimension (here, R) when the load magnitude W is the same. Therefore, H ∝ R for the crown point load case in similar semicircular arches, and the ratio of H values follows the ratio of radii.

Step-by-Step Solution:
1) Recognize geometric similarity: semicircular shape with identical load position.2) For the given loading, adopt proportionality H ∝ R (same W).3) Compute ratio using radii: 5 : 7.5 : 10 → divide by 5 → 1 : 1.5 : 2.4) Map to options: 1 : 1 1/2 : 2.

Verification / Alternative check:
Dimensionless analysis with forces normalized by W and lengths by R shows thrusts scale with R for similar arches under identical load types and positions, confirming the 1 : 1.5 : 2 ratio.

Why Other Options Are Wrong:

• 2 : 1 1/2 : 1 and 1 : 1 : 2: Do not follow the linear scaling with radius.
• None of these: Incorrect because 1 : 1 1/2 : 2 fits exactly.

Common Pitfalls:

• Assuming H is independent of size for similar shapes—this neglects the geometric scaling.
• Confusing span scaling with rise scaling; for a semicircle both scale with R.

Final Answer:
1 : 1 1/2 : 2.