Cables and Arches

Analysis of statically determinate cables subjected to concentrated and distributed loads, and the structural behavior of three-hinged arches.

Introduction to Cables

Cables and Funicular Shapes

Cables are flexible structural members capable of supporting only tensile forces. Because they cannot resist shear or bending, a cable's geometry changes perfectly to align with the applied loads. This shape is known as the funicular shape.
  • Funicular Polygon: The shape assumed by a cable subjected to a series of discrete, concentrated vertical loads. The cable forms a series of straight-line segments.
  • Funicular Curve: The shape assumed by a cable subjected to a continuous distributed load. It forms a smooth curve (e.g., a parabola or a catenary).

Cables Subjected to Concentrated Loads

When a cable supports a series of vertical concentrated loads, it takes the shape of a series of straight-line segments.

Assumptions

  • The cable is perfectly flexible.
  • The cable is inextensible (no stretching under load).
  • The weight of the cable is negligible compared to the applied concentrated loads.

Analysis Procedure

Procedure

  1. Treat the entire cable as a single rigid body and apply global equilibrium equations (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, ∑M=0\sum M = 0) to find support reactions. Note that the horizontal component of tension (Tx=FHT_x = F_H) is constant throughout the cable.
  2. If the sag at a specific point is known, cut the cable at that point and analyze one portion to find the horizontal tension FHF_H.
  3. Use joint equilibrium (Method of Joints) at each load point to find the tension in each segment and its vertical geometry (sag).
  4. The maximum tension always occurs in the steepest segment, which is typically adjacent to one of the supports.

Cables Subjected to Distributed Loads

When a cable supports a continuous distributed load, it takes the shape of a continuous curve. The two most common cases are uniform loads along the horizontal projection and uniform loads along the length of the cable itself.

Uniform Load Along Horizontal Projection (Parabolic Cable)

This approximates the loading on a suspension bridge where the heavy deck represents a uniform load w0w_0 per horizontal meter.

Procedure

  • Equation of the Curve: The cable takes the shape of a parabola.
y=w0x22FH y = \frac{w_0 x^2}{2F_H}
Where FHF_H is the constant horizontal tension, and the origin is at the lowest point of the cable.
  • Maximum Tension: The maximum tension occurs at the supports where the slope is steepest:
Tmax=FH2+(w0xmax)2 T_{max} = \sqrt{F_H^2 + (w_0 x_{max})^2}

Uniform Load Along the Cable Length (Catenary Cable)

This occurs when a cable hangs under its own weight, such as electrical transmission lines.

Procedure

  • Equation of the Curve: The cable takes the shape of a catenary.
y=FHw0[cosh⁡(w0xFH)−1] y = \frac{F_H}{w_0} \left[ \cosh \left( \frac{w_0 x}{F_H} \right) - 1 \right]
Where w0w_0 is the weight per unit length of the cable.

Three-Hinged Arches

Arches

An arch is a curved structure designed to carry loads primarily through axial compression. However, unlike a cable, an arch also carries shear and bending moments. A three-hinged arch is formed by placing hinges at both supports and a third hinge usually at the crown (the highest point).

Characteristics of a Three-Hinged Arch

  • It is a statically determinate structure.
  • It has four support reactions (two at each pinned support).
  • The three global equilibrium equations plus one condition equation (bending moment at the internal hinge is zero) provide the four equations needed to solve for the four reactions.

Analysis Procedure

Procedure

  1. Apply global equilibrium (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, ∑M=0\sum M = 0) to the entire arch.
  2. Separate the arch into two halves at the internal crown hinge.
  3. Apply the condition that the sum of moments about the crown hinge for either half of the arch is zero (∑Mcrown=0\sum M_{crown} = 0). This allows you to solve for the horizontal thrust forces.
  4. Once the reactions and crown hinge forces are known, use the method of sections to find the internal axial force, shear force, and bending moment at any cross-section along the arch.

Interactive Simulation: Cables and Arches

Observe the difference between parabolic and catenary cable shapes under load, and analyze forces in a three-hinged arch.

Cable Loading Simulation

Notice how the perfectly flexible cable forms a funicular shape of two straight segments under a single point load.

Tied Arches vs. Fixed Arches

Arch Support Types

Arches develop horizontal thrusts at their supports. To resist this thrust, different support mechanisms are used:
  • Fixed Arches: The abutments provide full resistance to horizontal thrust, vertical reaction, and moment. They are heavily statically indeterminate.
  • Tied Arches: A horizontal tie (cable or steel rod) connects the two ends of the arch, carrying the horizontal thrust internally. The external reactions are purely vertical, making the tied arch statically determinate externally (if supported by a pin and a roller).

The Funicular Shape

Funicular Polygons

The funicular shape is the natural shape a cable assumes under a specific loading condition. An arch designed to match the inverted funicular shape of its loading will experience pure axial compression with zero bending moment. For example, a parabolic arch experiences no bending under a uniform horizontal load.
Key Takeaways
  • Cables support loads exclusively through tension and adjust their shape based on the loading.
  • Cables carrying concentrated loads form straight-line segments; those carrying horizontally distributed loads form parabolas; and those carrying their own weight form catenaries.
  • The horizontal component of tension is constant throughout any given cable.
  • Three-hinged arches are statically determinate structures that carry loads primarily through compression, though they also experience shear and bending.
  • The internal hinge in a three-hinged arch provides an essential condition equation (∑Mcrown=0\sum M_{crown} = 0) to determine the horizontal support reactions.
  • Cables carry loads purely through axial tension and assume a shape dictated by the loading (funicular curve).
  • Cables under uniform horizontal loads take a parabolic shape, while cables under self-weight take a catenary shape.
  • Arches carry loads primarily through axial compression but often develop bending moments under non-funicular loading.
  • Three-hinged arches are statically determinate and rely on a central internal hinge to release bending moments.
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