Cables and Arches - Theory & Concepts

Learning Objectives

  • Understand the fundamental characteristics of structural cables and arches.
  • Learn the key assumptions for analyzing cables.
  • Analyze cables subjected to concentrated and distributed loads.
  • Differentiate between parabolic and catenary cable profiles and apply their respective formulas.
  • Understand the function of a three-hinged arch and the procedure for determining its support reactions.

This lesson explores the mechanics of cables and arches, two highly efficient structural forms used for spanning large distances. By resisting loads primarily through axial tension and compression, these elements minimize bending moments, making them essential in the design of suspension bridges, stadium roofs, and other long-span structures.

Introduction to Cables and Arches

Cables and arches are specialized structural elements that carry loads primarily through axial forces. Because they minimize bending moments, they are highly efficient over long spans and are extensively used in modern civil engineering structures, such as suspension bridges and stadium roofs. A fundamental principle of both forms is that their shape is closely tied to the applied loading.

Cable

A structural member that is perfectly flexible and can only resist axial tension. It offers zero resistance to bending or compression.

Assumptions for Cable Analysis

The analysis of cables relies on a few fundamental assumptions:

  • Perfect Flexibility: The cable has no bending stiffness. As a result, the internal tension force is always perfectly tangent to the curve of the cable at any given point.
  • Inextensibility: The change in length of the cable due to the applied tension is usually small enough to be neglected.
  • Form Follows Loading: The final shape the cable assumes depends entirely on the type and distribution of the load it supports.

Inextensibility Assumption Limitations

While the inextensibility assumption is valid for materials like steel cables under typical working loads, it fails for highly elastic materials (like rubber) or exceptionally long spans where axial deformation significantly alters the cable's sag and shape. In such cases, non-linear geometric analysis is required.

Cables Subjected to Concentrated Loads

When a cable of negligible weight supports a series of discrete, vertical concentrated loads, it assumes the shape of a series of straight-line segments, forming a polygon.

  • The horizontal component of the cable tension (TxT_x) remains constant throughout the entire length of the cable.
  • The maximum tension occurs in the segment with the steepest slope, which is typically near the supports.
  • The shape of the cable and internal forces can be found by applying equilibrium equations to sections of the cable or individual joints (nodes).

Cables Subjected to Distributed Loads

If a cable is subjected to a continuous distributed load, it takes the form of a smooth curve rather than discrete linear segments. There are two primary categories depending on how the load is distributed: parabolic cables and catenary cables.

Parabolic Cable

A cable that takes the shape of a parabola, which occurs when a uniformly distributed load is applied continuously along the horizontal span projection (e.g., the deck of a suspension bridge).

Parabolic Cable Equation

The mathematical equation describing the shape of a cable subjected to a uniform horizontal load.

y=w02T0x2y = \frac{w_0}{2T_0} x^2

Variables

SymbolDescriptionUnit
yyThe vertical dip or sag at a horizontal distance x from the lowest pointm
xxThe horizontal distance from the lowest point of the cablem
w0w_0The uniform horizontal load intensityN/m
T0T_0The constant horizontal tension at the lowest point of the cableN

Maximum Tension in Parabolic Cables

The maximum tension force occurring at the highest support points of the parabolic cable.

Tmax=T02+(w0x)2T_{\mathrm{max}} = \sqrt{T_0^2 + (w_0 x)^2}

Variables

SymbolDescriptionUnit
TmaxT_{\mathrm{max}}The maximum tension in the cableN
T0T_0The constant horizontal tensionN
w0w_0The uniform horizontal load intensityN/m
xxThe horizontal distance from the lowest point to the highest supportm

Cable Length for Parabolic Cables

For a parabolic cable with a uniform horizontal load, the exact length SS spanning between two supports at the same elevation, separated by distance LL and with a maximum sag hh, can be determined by integration. For practical engineering purposes where the sag-to-span ratio (h/Lh/L) is relatively small, the length is often approximated using a series expansion formula.

Parabolic Cable Length Approximation

The series expansion used to approximate the un-tensioned length of a parabolic cable with a small sag-to-span ratio.

S≈L[1+83(hL)2−325(hL)4]S \approx L \left[ 1 + \frac{8}{3}\left(\frac{h}{L}\right)^2 - \frac{32}{5}\left(\frac{h}{L}\right)^4 \right]

Variables

SymbolDescriptionUnit
SSThe approximate total length of the cablem
LLThe horizontal distance between the supports (span)m
hhThe maximum sag at the center of the spanm

Catenary Cable

A cable that takes the shape of a catenary curve, which occurs when a continuous load is distributed uniformly along the length of the cable itself (e.g., a free-hanging power transmission line supporting its own self-weight).

Catenary Cable Equation

The mathematical equation involving hyperbolic functions that describes the shape of a hanging cable under its own uniform weight.

y=T0w0(cosh⁡(w0T0x)−1)y = \frac{T_0}{w_0} \left( \cosh \left( \frac{w_0}{T_0} x \right) - 1 \right)

Variables

SymbolDescriptionUnit
yyThe vertical dip or sag at a horizontal distance x from the lowest pointm
xxThe horizontal distance from the lowest pointm
T0T_0The constant horizontal tension at the lowest pointN
w0w_0The uniform weight per unit length of the cableN/m

Interactive Simulation

Explore how different types of loading (Concentrated, Parabolic, Catenary) change the shape and the internal tension forces of a cable spanning a distance. Note how increasing the sag dramatically decreases the tension forces in the cable. Interact with the simulation below to visualize these concepts.

Interactive Physics Simulation

Suspension Cable Force & Shape Simulator

Compare structural cable configurations. Analyze how concentrated point loads or uniform distributed loads yield parabolic vs. catenary curves.

12 kN/m
5.0 m
Cable Mechanics Governing Equations
Minimum Cable Tension (at lowest point):
T0=fracwcdotL28cdothT_0 = \\frac{w \\cdot L^2}{8 \\cdot h}
Maximum Cable Tension (at support anchors):
Tmax=sqrtT02+left(fracwcdotL2right)2T_{max} = \\sqrt{T_0^2 + \\left(\\frac{w \\cdot L}{2}\\right)^2}
Span L: 20 meters
Minimum Cable Tension (T0)
120.0 kN
Maximum Anchor Tension (Tmax)
169.7 kN

Arch

A curved structural member that spans an opening and carries load primarily in pure axial compression, functioning essentially as an inverted cable.

Characteristics of Arches

While a cable carries loads purely in tension, an ideally shaped arch carries loads purely in compression. However, because practical arches must support moving and variable loads, they are designed to resist some bending and shear forces as well. Arches direct loads down to their supports, creating outward horizontal thrusts that must be resisted by strong foundations or tie rods.

Three-Hinged Arch

A type of statically determinate arch structure composed of two arch segments pinned together at a central crown hinge and pinned at both base supports.

Three-Hinged Arch Analysis Procedure

  • Step 1: Overall Free-Body Diagram. Draw the free-body diagram of the entire arch. There are 4 unknown reaction components in total (two orthogonal force components at each pin support).
  • Step 2: Global Equilibrium. Apply the global equilibrium equations ∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, and take moments about one of the supports (∑M=0\sum M = 0). This provides three equations, which is not enough to solve for all 4 unknowns.
  • Step 3: Crown Hinge Condition. The internal bending moment at the central crown hinge is zero (M=0M = 0). Make an imaginary cut at the crown hinge and draw the free-body diagram of either the left or the right arch segment.
  • Step 4: Solve for Unknown Reactions. Take the sum of moments about the crown hinge for the isolated segment. This provides the 4th independent equation needed to solve for the remaining support reactions.
  • Step 5: Internal Forces. Once all reactions are known, use the method of sections at any point along the arch to find the internal normal force, shear force, and bending moment.
Key Takeaways
  • Cables can only resist tension. They are perfectly flexible and assume a shape entirely dictated by their loading.
  • For vertically loaded cables, the horizontal component of tension (T0T_0 or TxT_x) remains constant throughout the length of the cable.
  • A uniform load distributed horizontally (e.g., a bridge deck) creates a parabolic cable profile.
  • A uniform load distributed along the cable length (e.g., self-weight) creates a catenary cable profile.
  • The maximum tension in a cable always occurs at the highest support point where the slope is steepest.
  • A three-hinged arch is statically determinate. The zero-moment condition at the central crown hinge provides the critical extra equation needed to solve for the four support reactions.
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