Analysis of Cables and Suspension Systems

Core Idea

Cables that support uniformly distributed loads (like their own weight) hang in a parabolic shape and can be analyzed using distributed load concepts. Cables supporting point loads hang in straight segments between loads. The horizontal tension in a cable is constant throughout, while vertical components vary depending on the load distribution and geometry.

How It's Best Learned

Analyze symmetric cable configurations first (like suspension bridges with uniform loading), then progress to asymmetric cases. Use moment equations about a point to relate geometry and load to tension.

Explainer

The key to cable analysis is recognizing what is constant and what varies along the cable. Because a flexible cable can transmit no bending moment or shear — it can only pull — every internal force along the cable is purely tensile and directed along the cable's tangent. This constraint, combined with your equilibrium skills, produces a powerful result: the horizontal component of tension is the same at every point along the cable. Think of it as the "throughput" of horizontal force that must be consistent end-to-end for equilibrium.

The shape a cable takes depends on how the load is distributed. When a cable supports a load that is uniform per unit horizontal distance (like the deck of a suspension bridge, where hangers are evenly spaced horizontally), the cable hangs in a parabolic shape. You can derive this by applying the distributed-load analysis you already know: cut the cable at position x, replace the distributed load with its resultant, and write ΣFx = 0 and ΣFy = 0 for the free-body diagram. The result is a second-order differential equation whose solution is a parabola y = x²·w/(2T₀), where w is load per unit length and T₀ is the horizontal tension. When a cable supports only its own weight — distributed uniformly along its arc length rather than horizontally — the true shape is a catenary, a hyperbolic cosine curve. For many engineering problems the parabola is a sufficient approximation when the sag is small relative to the span.

For cables loaded by discrete point loads, the geometry is even simpler: the cable forms a series of straight segments connecting the load application points. Between loads, there is no distributed force, so each segment must be straight. Your approach here uses moment equations: write equilibrium for the entire cable, find support reactions, then cut at each joint and apply ΣFx = 0, ΣFy = 0. The angles of the segments set the geometry, and the horizontal tension threads consistently through every segment.

The practical payoff of constant horizontal tension is computational leverage. Once you determine T₀ — typically from the geometry at one known point, such as the lowest point of a symmetric cable — you have a fixed quantity that connects every other calculation. The total tension at any point is T = T₀/cos(θ), where θ is the local slope angle. Maximum tension always occurs at the supports, where the cable is steepest. This hierarchy — find T₀ from geometry, derive everything else from T₀ — is the standard solution pathway for all cable problems, whether parabolic, catenary, or piecewise-linear.