Module 6 · Lesson 6.1
The cable–arch analogy
Why an arch is an upside-down cable, and what a funicular shape means.
Why this matters
Masonry is strong in compression and almost useless in tension, which is why every great masonry structure is an arch, a vault or a dome. Understanding the funicular idea explains why those shapes are what they are — and why a badly shaped arch cracks.
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)
Hang a chain between two points and it finds a shape in which every link is in pure tension. Now freeze that shape, turn it upside down and build it in stone. Every force reverses direction, so what was tension becomes compression — and the structure still carries the load with no bending at all.
That is the cable–arch analogy, and it is not just a metaphor. The equilibrium equations are identical; only the sign changes. Robert Hooke put it in 1675: as hangs the flexible line, so but inverted stands the rigid arch.
A shape that carries a particular loading in pure axial force is called funicular for that loading, or a linear arch. The important word is particular. A shape can only be funicular for one pattern of load. Change the load and the same arch is no longer funicular, so bending moments appear.
This is why a parabolic arch is right for a uniformly distributed load, but a circular arch under the same load carries bending — its shape is not the funicular one.
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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint