Three-Hinged Arches MCQ with answer and solution

Concept

For a suspension cable subjected to a uniform load per unit horizontal length, the bending moment at any section is zero, resulting in a parabolic profile where the horizontal component of cable tension is constant.

Formula

H = wL² / (8d)

Step-by-step solution

1. Consider a suspension cable of span L and central dip d subjected to a uniform load w per unit horizontal length. 2. Taking moments about the crown (center of the cable) where the vertical shear is zero: M_crown = (wL/2) × (L/4) - (w × L/2 × L/4) / 2 - H × d = 0. 3. Solving for horizontal reaction H yields H = wL² / (8d).

Explanation

A suspension cable carrying a uniformly distributed load per unit horizontal length takes the shape of a parabola. Taking moments about the lowest point (crown) or using equilibrium equations, the horizontal reaction at the supports is given by H = wL² / (8d), where L is the span and d is the central dip.

Why the other options are wrong

A: wL²/4d overestimates the horizontal reaction. C: wL²/16d underestimates the horizontal reaction. D: wL²/24d is incorrect for this configuration.

Shortcut

Remember that for parabolic cables/arches with udl over horizontal span, H = wL²/(8d), whereas for circular/parabolic arches with udl over the span along the arch, the formula slightly differs. For horizontal span udl, it is always 8d.

Common mistake

Confusing the denominator 8d with 4d (which applies to maximum bending moment in simply supported beams or central deflection formulas) or 16d.