A flexible cable is anchored at two supports A and B, which are at the same elevation and $10\text{ m}$ apart. A single concentrated load of $100\text{ kN}$ is hung exactly in the middle of the span ($x = 5\text{ m}$). The sag at the center is measured to be $2\text{ m}$. Determine the maximum tension in the cable. Neglect the weight of the cable.

A suspension bridge cable has a span of $100\text{ m}$ and a maximum sag of $10\text{ m}$. It supports a uniformly distributed deck load of $w = 20\text{ kN/m}$ (measured horizontally). Determine the equation of the cable shape and the maximum tension.

A high-voltage transmission line is strung between two towers $200\text{ m}$ apart. The cable itself weighs $10\text{ N/m}$ along its own length. If the minimum tension in the cable (at the lowest point) is $5000\text{ N}$, find the maximum sag.

A flexible cable is supported at $A$ and $B$, which are at the same elevation and $12\text{ m}$ apart. The cable is subjected to two concentrated loads: $P_1 = 40\text{ kN}$ at $x = 4\text{ m}$ from $A$, and $P_2 = 60\text{ kN}$ at $x = 8\text{ m}$ from $A$. If the sag at the first load $P_1$ is $1.5\text{ m}$, determine the sag at the second load $P_2$ and the maximum tension in the cable.

A three-hinged semicircular arch has a radius of $10\text{ m}$ and is supported by pin connections at $A$ and $B$, which are at the same elevation. A hinge is located at the crown $C$. A concentrated load of $50\text{ kN}$ is applied vertically at a point $D$, which is located on the arch at a horizontal distance of $5\text{ m}$ from the left support $A$. Determine the horizontal thrust at the supports and the vertical reactions.

A three-hinged parabolic arch has a span of $40\text{ m}$ and a rise (height at the crown) of $8\text{ m}$. The hinges are located at the two supports, $A$ and $B$, which are at the same elevation, and at the crown $C$. The arch carries a uniformly distributed load of $30\text{ kN/m}$ over its entire horizontal span. Determine the horizontal thrust and the maximum bending moment in the arch.

A cable is attached to two supports $A$ and $B$. Support $B$ is $10\text{ m}$ higher than support $A$. The horizontal distance between the supports is $50\text{ m}$. The cable carries a uniform load of $20\text{ kN/m}$ along the horizontal span. The lowest point of the cable is $2\text{ m}$ below support $A$. Determine the horizontal tension in the cable and the distance from support $A$ to the lowest point.

A uniform cable weighing $15\text{ N/m}$ is suspended between two points $A$ and $B$, which are on the same horizontal line and $60\text{ m}$ apart. The horizontal tension in the cable is known to be $1200\text{ N}$. Determine the maximum sag and the total length of the cable.

The main cables of the Golden Gate Bridge are often assumed to be parabolas, while the cables of high-voltage transmission lines are modeled as catenaries. What is the fundamental physical difference in the loading that causes these two distinct mathematical shapes?

The Gateway Arch in St. Louis is designed in the shape of an inverted weighted catenary. How does the concept of a "funicular shape" relate arches to hanging cables, and why is this shape structurally ideal?

When designing a large span arch bridge, an engineer must choose between a three-hinged arch and a two-hinged arch. What is the fundamental difference in their static determinacy, and how does this affect their response to temperature changes and foundation settlement?

Many modern arch bridges, such as the Sydney Harbour Bridge or various river crossings, use a "tied-arch" or "bowstring arch" design. In classic arches, the horizontal thrust must be resisted by massive abutments built into the earth. How does a tied-arch bridge resolve this issue, and what structural member acts analogously to the tension cable in a suspension bridge?