Structural Analysis Fundamentals / Arches
The cable–arch analogy
Why an arch is an upside-down cable, and what a funicular shape means.
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Read the lesson register and outstanding review notesWhy this matters
Masonry has limited tensile capacity. Arches, vaults and domes can carry suitable loading mainly through compression, but the thrust line, support conditions and material resistance still need checking.
By the end of this lesson you should be able to
- State the cable–arch analogy
- Define a funicular shape
- Explain why an arch thrusts outwards
What you should already know
- Cables and the constant horizontal component (Module 5)
- Equilibrium and taking moments (Module 2)
A hanging chain finds an equilibrium shape under its applied loading. Invert that geometry and reverse the internal force signs while retaining corresponding downward loads: the same equilibrium can be carried by compression. This is the cable–arch analogy. It establishes an ideal funicular shape, not its resistance to buckling, material failure or support movement.
A shape is funicular for a loading when it carries that loading through axial force without bending. Proportional increases in the same load pattern retain the funicular shape in first-order analysis. A change in the distribution may introduce bending.
For example, a parabola is funicular for a vertical load uniform per horizontal metre. A circular axis generally develops bending under that loading.
Predict first
A parabolic arch spans 40 m with a rise of 8 m and carries 15 kN/m. A cable of the same span and sag carries the same load. How do their horizontal thrusts compare?
Practice
A parabolic arch spans 30 m with a rise of 6.0 m and carries a uniform load of 20 kN/m over the full span. What is the horizontal thrust at each abutment, in kN?
Practice
For the same arch, what is the vertical reaction at each springing, in kN?
Practice
The same arch is rebuilt with the rise increased to 12 m. What is the new horizontal thrust, in kN?
Summary
- An arch is an inverted cable: same equilibrium, compression instead of tension
- A funicular shape carries one particular loading with no bending
- The arch thrusts outwards, and the supports must resist that
- Depart from the funicular shape and bending appears