Mechanics of cables and arches

A cable can carry only tension. Under its own weight it hangs in a catenary; under a load spread uniformly along the horizontal, as in a suspension bridge deck, it takes the shape of a parabola. Because a cable cannot resist bending, its shape adjusts to the load so that the tension alone keeps every point in equilibrium. That is why cables are so efficient: the whole section works at nearly the same stress.

For a parabolic cable with span L and central sag d under a uniform load w per horizontal metre, taking moments about the lowest point gives the horizontal component of tension, H, equal to w L squared over 8 d. The horizontal component is the same everywhere along the cable; the vertical component grows towards the supports, so the maximum tension occurs at the supports. A small sag means a very large horizontal pull, which the anchorages or towers must resist.

An arch is the inverted cable. If an arch follows the funicular shape of its load, a parabola for a uniform load, it carries that load in pure compression with no bending. A three-hinged arch is statically determinate: the extra hinge at the crown gives the extra equation needed to find the horizontal thrust, which equals w L squared over 8 h for a parabolic arch of rise h. Two-hinged and fixed arches are indeterminate and develop bending under non-funicular loads such as a moving vehicle.

In practice loads are never exactly funicular, so arches carry some bending and cables need stiffening girders to limit deflection under moving traffic. Both forms push or pull horizontally on their supports, so foundations, abutments and anchor blocks are a large part of the design. Temperature and creep change the sag or rise, which changes the thrust.