5.1. Funicular Polygons

A cable is a slender flexible member with zero or negligible flexural stiffness, thus it can only transmit tensile forces.

This statement introduces what a cable is, emphasizing its key physical properties. Unlike rigid structures that can resist bending, cables can only pull or stretch because they lack the stiffness to bend under load. This flexibility allows cables to efficiently carry loads but also to change shape significantly.

Imagine a rubber band; it can stretch and hold loads but cannot resist being bent. Like the rubber band, cables function best under tension rather than compression or bending.

The tensile force at any point acts in the direction of the tangent to the cable (as any other component will cause bending).

Tensile forces in cables always pull along the length of the cable. Because cables cannot resist bending, any force acting perpendicular to the cable would cause it to bend, which is not possible. Therefore, it is essential that the force aligns with the tangent to the cable to ensure that it only experiences tension.

Think of a tightrope walker. The rope can only support their weight if it is tight and straight. If there are any forces that pull down at an angle, the rope will sag and cannot hold the weight efficiently.

Cables resist vertical forces by undergoing sag (h) and thus developing tensile forces. The horizontal component of this force (H) is called thrust.

When a cable is loaded vertically, it will sag downwards. This sag creates a tension that allows the cable to respond to the load. The horizontal component of the force generated due to this tension is referred to as 'thrust,' and is crucial for understanding how cables carry loads effectively.

Consider a hammock. When you sit in it, the fabric sags under your weight, and the sagging creates tension in the ropes that can help support you. The horizontal force acting to keep the ropes taut is similar to 'thrust' in a cable.

The distance between the cable supports is called the chord. The sag to span ratio is denoted by $r = \frac{h}{l}$.

The 'chord' refers to the straight line distance between the two points where the cable is supported. The sag to span ratio gives a numerical relationship between the height sagging down (h) and the distance between the supports (l). This ratio helps in designing cables for specific loads and spans.

If you were to hang a rope between two trees, the distance between the trees is the 'chord.' If the rope bounces downwards in the middle, the height it drops down is the 'sag.' The ratio of how far it dips compared to how far apart the trees are helps determine how sturdy your rope setup is.

When a set of concentrated loads is applied to a cable of negligible weight, then the cable deflects into a series of linear segments and the resulting shape is called the funicular polygon.

The funicular polygon is the shape formed by a cable when it supports several concentrated loads. Instead of a smooth curve, the cable forms straight lines between these points of load due to its nature of only being able to hold tensile forces. This provides engineers a method to visualize load paths on cables.

Picture a bunch of keys hanging from a string. As you add more keys, the string will adjust into straight lines connecting each key. The overall shape formed by the string and the keys showcases how the load distribution affects the string’s shape, resembling a funicular polygon.

If a cable supports vertical forces only, then the horizontal component H of the cable tension T remains constant.

When only vertical loads are applied, the horizontal component of the tension does not vary. This constancy simplifies calculations and helps predict how the cable will behave under different loads, as the horizontal tension can be treated as a fixed value in engineering design.

Think about the wires holding up a sign. If the sign were to sway only in the vertical direction with wind, the wires would remain tight horizontally, keeping the sign securely in place. The tension in the wire along the horizontal direction doesn't change even as the sign moves.

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