Cast-Iron Arch Bridge

What is modelled

A three-hinged parabolic arch with hinges at both banks and at the crown, carrying the nominal bridge mass as a uniform line load plus an added deck load. Because three hinges make the structure statically determinate, the reactions follow from equilibrium alone. A uniform load on a parabolic arch produces no bending at all; loading only part of the span does, and the thrust line moves away from the arch. The span, mass, five ribs and the full-load formulae are those of the 3D page; the rise ratio, partial load and stresses are added here.

Equations

• Vertical reaction V = q L / 2 and thrust H = q L2 / (8 f) for full uniform load q (the 3D page uses f = L/2).
• Arch shape y = 4 f x (L - x) / L2; bending moment M(x) = VA x - H y - ∫ q (x - s) ds; thrust line y + M/H.
• Half-span load: Mmax = q L2 / 64 at a quarter span.
• Rib stress σ = N/A + M/Z with N = √(V² + H²) shared over 5 ribs and Z = 0.45 A d.

Checks run in the page

• Element-by-element statics against q L²/8f and q L/2.
• Zero bending under a full uniform load.
• Half-span moment against q L²/64 and zero moment at the crown hinge.
• Thrust doubling when the rise halves.

Honest simplifications

The ribs are treated as one equivalent arch with the load shared equally, the section modulus is a rule-of-thumb fraction of area times depth, and joints, casting flaws, abutment movement, temperature and the wind are ignored. The real Iron Bridge has a rise of about half the span only in the idealised description used by the 3D page. These numbers are for teaching and are not a safety assessment of any bridge.

• Choose Half-span traffic and watch the thrust line leave the arch and the moment peak near a quarter span.
• Choose Shallow arch: the thrust grows and the abutments must resist much more sideways push.
• Tick the sweep box to move the load across the span and see the rib stress rise and fall.