The analysis of suspension bridges is a complex and critical process that involves multiple factors, including the shape and tension of the main cables. The main cables of suspension bridges are subjected to distributed loads, which result in changes in tensile forces and deformation of the cable. These loads must be carefully considered to ensure that the bridge is safe and capable of supporting the weight of different loads.

To analyze a cable subjected to distributed loads, consider a small segment of the cable and draw a free-body diagram showing the distributed load and changes in tensile force. A set of equations is obtained by applying equilibrium equations and substituting the horizontal and vertical forces and moments acting on the segment. Dividing each equation by delta Δx and taking each limit tending to zero yields a new set of three equations. The first equation obtained from this analysis helps determine the tensile force's horizontal component at any point along the cable.

It also allows calculating the cable's tension at any point along its length. The expression for the slope of the cable is obtained by dividing the second equation, which is the vertical component of tensile force, with the component of horizontal tensile force and substituting the value of tan θ.

Finally, performing a second integration on the derived equation determines the curve of the cable.

Thus, the process helps us understand the cable's behavior when subjected to distributed loads. It is also useful in designing the bridge's support system and ensuring that the cables can handle the weight of traffic and other stresses.