Each cable of the Golden Gate Bridge supports a load w=11.1...

Each cable of the Golden Gate Bridge supports a load $w=11.1$ kips fft
along the horizontal. Knowing that the span $L$ is $4150 \mathrm{ft}$ and that the sag $h$ is $464 \mathrm{ft}$, determine (a) the maximum tension in each cable, $(b)$ the length of each cable.

Key Concepts

Sag-to-Span Relationship

The sag-to-span relationship is a key geometrical factor in determining the shape and behavior of a suspended cable. It involves the ratio between the maximum vertical deflection (sag) and the horizontal span. This ratio influences both the curvature of the cable and the magnitude of the internal forces, and it is critical for the accurate analysis and design of cable-supported structures.

Cable Length Calculation

Determining the length of a cable that follows a parabolic shape involves integrating the curve’s arc length. This process takes into account the cable’s horizontal span and its sag. The computed length is necessary for material estimation and in further analyses of the cable's mechanical properties, such as the distribution of internal forces along its length.

Uniformly Distributed Load

A uniformly distributed load is one where the force is spread evenly over the horizontal projection of the cable. In the context of suspension bridges or similar structures, this kind of loading leads to a predictable and continuous distribution of force along the span, which is fundamental to calculating both the cable's geometry and the internal forces that develop within it.

Parabolic Cable Approximation

When a cable is subjected to a uniformly distributed load along the horizontal, its shape can be well approximated by a parabola. This approximation greatly simplifies the calculation of key quantities such as maximum tension and the arc length of the cable. It is particularly useful when the sag is small relative to the span, although even with significant sag, the parabolic model can yield reasonably accurate results.

Maximum Tension in a Cable

The maximum tension in a cable under a uniformly distributed load typically occurs at the supports. This happens because the combination of the horizontal force components and the accumulation of vertical loads along the span creates the highest resultant force at these points. Understanding how to compute maximum tension is essential for ensuring the structural integrity and safe design of suspension systems.

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