Locations of zero bending moment in a three-hinged arch For a three-hinged arch with hinges at both springings and at the crown, at which hinge locations is the internal bending moment identically zero under any loading?

Introduction / Context:
Three-hinged arches are widely used in bridges and long-span roofs. Hinges (pins) release moment at their locations, which has direct consequences for internal force distributions and influence lines.

Given Data / Assumptions:

• Hinges exist at both springings (supports) and one hinge at the crown.
• Arbitrary static loading within the plane of the arch.
• Rigid arch rib between hinges.

Concept / Approach:
By definition, an ideal hinge cannot transmit bending moment. Therefore, at each hinge of a three-hinged arch, the internal bending moment is zero regardless of loading. The crown hinge also divides the arch into two determinate segments, each with zero moment at the hinge location.

Step-by-Step Solution:
Identify hinge locations: left springing, crown, right springing.Apply the hinge property: M_hinge = 0 at each hinge.Conclude that bending moment vanishes at all three hinges under any loading.

Verification / Alternative check:
Influence lines for moment in a three-hinged arch always pass through zero at each hinge; structural analysis textbooks and codes rely on this fact for determinate solution procedures.

Why Other Options Are Wrong:
Options A, B, and C restrict zero moment to fewer hinges, contradicting the fundamental behavior of an ideal hinge.

Common Pitfalls:
Confusing hinges with fixed joints; a fixed joint can transfer moment, but a hinge cannot.

Final Answer:
at all the three hinges