Three-Hinged Arches MCQ with answer and solution

Concept

Bending Moment in Three-Hinged Parabolic Arch

Formula

Horizontal Thrust: H = (w × L²) / (8 × h) Bending Moment: M_x = M_beam - H × y = 0 Arch Profile: y = 4 × h × x × (L - x) / L²

Step-by-step solution

1. Identify given values: w = 1 kN/m, L = 12 m, h = 2 m. 2. Recall that for a three-hinged parabolic arch under a uniform UDL over the full span, the beam moment matches the moment from horizontal thrust (M\_beam = H \* y). 3. Calculate net bending moment (Mx = M\_beam - H \* y = 0), which is zero at all points including the quarter point.

Explanation

When a three-hinged parabolic arch carries a uniformly distributed load over its entire span, the shape of the arch matches the shape of the bending moment diagram. Because of this, the moment created by the horizontal thrust at any point cancels out the moment caused by the load, resulting in a net bending moment of zero throughout the entire arch, including at the quarter point.

Shortcut

Remember: A 3-hinged parabolic arch loaded with UDL over the ENTIRE span has ZERO bending moment everywhere. So, no matter what values of span, rise, or UDL are given, the bending moment at any section (including the quarter point) is always 0.

Common mistake

Students often calculate only the simply supported beam bending moment (such as w·L²/24 = 6 kNm) and select Option C, forgetting that in a 3-hinged parabolic arch under full UDL, the moment caused by horizontal thrust (H·y) completely cancels out the beam moment, making the net bending moment zero everywhere.