Three-hinged arch in structural analysis In classical structural analysis and bridge engineering, how is a three-hinged arch classified with respect to static determinacy? Provide the most accurate description used in design and exam standards.

Introduction / Context:
A three-hinged arch is a fundamental form used in bridge and roof structures. Knowing whether it is statically determinate or indeterminate is critical because it dictates the analysis method, sensitivity to support settlements, and thermal effects.

Given Data / Assumptions:

• An arch with hinges at both springings (supports) and one hinge at the crown (third hinge).
• Loads are arbitrary, supports are ideal pins, and the arch is a rigid body between hinges.
• Linear elastic behavior is assumed for internal force interpretation.

Concept / Approach:
Each pin (hinge) releases a moment. With hinges at both ends and at the crown, the arch has just enough internal releases to remove redundancy. Therefore, all reaction components can be obtained using only the three static equilibrium equations in two dimensions: ΣFx = 0, ΣFy = 0, and ΣM = 0.

Step-by-Step Solution:
Count unknown reactions: two at each springing (Ax, Ay and Bx, By) → four unknowns.Add the internal crown hinge condition: the bending moment at the crown hinge is zero → this gives an additional relationship (used via sectional equilibrium).Use global and sectional equilibrium to solve for all unknowns without invoking compatibility → structure is determinate.

Verification / Alternative check:
Three-hinged arches are immune to thermal stress buildup and support settlement–induced indeterminacy because the crown hinge accommodates rotations, allowing solution purely by statics.

Why Other Options Are Wrong:
Statically indeterminate requires more unknowns than equilibrium equations; not the case here. Geometrically unstable is false because three hinges provide stability with proper geometry. Structurally sound but indeterminate contradicts determinate nature.

Common Pitfalls:
Confusing two-hinged arches (indeterminate to degree 1) with three-hinged arches (determinate). Also, misusing the crown hinge condition.

Final Answer:
statically determinate