Cables and Arches - Theory & Concepts

Cables and Arches - Theory & Concepts

Cables and arches are specialized structural elements that carry loads primarily through axial tension (cables) or axial compression (arches). Because they transfer loads efficiently without relying primarily on bending or shear resistance, they can span large distances with relatively little material.

Analysis of Cables

Cables are perfectly flexible structures, meaning they have no flexural (bending) stiffness and cannot resist shear forces or bending moments. They resist external loads entirely by developing internal axial tensile forces. Their shape changes dramatically depending on how they are loaded.

Key Assumptions for Cable Analysis

When analyzing cables in structural theory, three fundamental assumptions are typically made:

Perfectly Flexible: The cable offers no resistance to bending ( $M = 0$ everywhere).
Inextensible: The length of the cable remains constant under load (axial deformation is considered negligible).
Self-Weight Ignored (Usually): Unless specified, the weight of the cable itself is assumed negligible compared to the large external loads it carries. If self-weight is considered, it forms a catenary curve.

General Cable Theorem

The general cable theorem connects the shape of a loaded cable to the bending moment of a simply supported beam. It states that at any point on a cable supporting vertical loads, the product of the horizontal tension (

H

) and the vertical sag (

y

) equals the bending moment (

M

) that would exist at that same point in a simply supported beam spanning the same distance and carrying the identical loads.

H \cdot y_x = M_x

This theorem provides an incredibly fast way to find the sag at any point along a cable subjected to complex loading without solving the entire cable geometry from scratch.

Cables Subjected to Concentrated Loads

When a perfectly flexible cable is subjected to a series of vertical concentrated point loads, it forms a funicular shape—a series of straight-line segments connecting the load application points. The internal tensile force is constant within any given straight segment between two loads.

Cable Loading Simulation

Load Position (horizontal)

Load Magnitude (affects sag)

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

Important

Since the bending moment is zero everywhere in the cable, you can take a section cut at any point, split the cable into two free-body diagrams, and sum the moments about the cut section to zero (

\sum M_{cut} = 0

). This is the primary equation used to find the sag at unknown points or the horizontal tension.

Cables Subjected to Uniform Distributed Loads

When a cable carries a uniformly distributed load along its horizontal projection (e.g., the deck of a suspension bridge), the cable assumes the shape of a parabola. This differs from a catenary curve, which occurs when a cable hangs under its own weight (a load uniformly distributed along the length of the cable itself).

Parabolic vs. Catenary Cables

When analyzing parabolic cables, the horizontal component of the tension (

H

) is constant throughout the length. The maximum tension (

T_{max}

) occurs at the supports. The relationship between the uniform load

w

per unit horizontal length, the span

L

, and the maximum sag

h

is given by:

H = \frac{w L^2}{8h}

The maximum tension

T_{max}

at the support is calculated using the Pythagorean theorem:

T_{max} = \sqrt{H^2 + \left(\frac{wL}{2}\right)^2}

The shape a perfectly flexible cable assumes depends entirely on the distribution of the applied load. The two most common and critical cable shapes in structural engineering are parabolic and catenary.

Parabolic Cable

A cable assumes a parabolic shape when subjected to a load that is uniformly distributed along its horizontal projection (

w_0

). Suspension bridge cables are approximated as parabolas because the weight of the flat bridge deck is the dominant load.

The equation for the curve is

y = \frac{w_0 x^2}{2 H}

, where

H

is the constant horizontal tension. The maximum tension (

T_{max}

) occurs at the highest support and is calculated as

T_{max} = \sqrt{H^2 + (w_0 L/2)^2}

, where

L

is the total horizontal span.

The approximate total length (

S

) of a parabolic cable with a uniform horizontal load, spanning a distance

L

with a maximum sag

h

, can be calculated using the series expansion formula:

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

Catenary Cable

A cable assumes a catenary shape when it sags purely under its own self-weight, meaning the load is uniformly distributed along the length of the cable itself (

s

), not horizontally. Power lines and ski-lift cables form catenaries.

The equation for the curve involves hyperbolic cosines:

y = \frac{H}{w_0} \left[ \cosh \left(\frac{w_0 x}{H}\right) - 1 \right]

, where

H

is horizontal tension, and

w_0

is weight per unit length of the cable. The total length

s

of the catenary curve between the lowest point and point

(x,y)

is given by

s = \frac{H}{w_0} \sinh\left(\frac{w_0 x}{H}\right)

Note

The maximum tension in a parabolic cable occurs at the supports and is given by

T_{max} = \sqrt{H^2 + (\frac{wL}{2})^2}

, where

L

is the total horizontal span.

Analysis of Arches

An arch is essentially an inverted cable. While a cable under vertical loads develops purely tensile forces, an arch with the exact same geometry (inverted) under the same loads will develop purely compressive forces. However, because arches are made of rigid materials (like concrete, masonry, or steel), they also possess flexural stiffness and can resist bending moments and shear forces, allowing them to carry varying loads without changing their shape.

Classification of Arches

Arches are classified by their determinacy and support conditions, dictating how they handle thermal expansion and support settlement.

Types of Arches

Three-Hinged Arch: Supported by two pins at the base and a third internal hinge at the crown. It is a statically determinate structure. Thermal changes or minor support settlements do not induce secondary internal stresses, making them ideal for unstable soils.
Two-Hinged Arch: Supported by two pins at the base but continuous (no hinge) at the crown. It is statically indeterminate to the first degree ( $DOI=1$ ). It is stiffer than a three-hinged arch but susceptible to thermal stresses.
Fixed Arch: Rigidly fixed at both supports. It is statically indeterminate to the third degree ( $DOI=3$ ). It offers the highest stiffness and material efficiency but requires extremely solid, unyielding foundations (like solid bedrock) because any support settlement will induce massive internal moments.

Three-Hinged Arch

An arch consisting of two solid segments connected by a hinge at the crown and supported by hinges at its two springing points. It is a statically determinate structure, making it easier to analyze than fixed or two-hinged arches.

Tied Arch

A type of arch where the horizontal outward thrust at the supports is counteracted by a tension tie (often the bridge deck or a separate cable) connecting the two ends of the arch. This effectively converts the arch's horizontal reactions into internal tension within the tie, allowing the arch to be supported vertically like a simply supported beam without needing massive abutments.

Thermal Effects on Arches

Temperature changes cause materials to expand or contract. The type of arch dictates how this strain translates into internal forces.

Thermal Stresses by Arch Type

Three-Hinged Arches: Being statically determinate, a uniform temperature increase simply causes the two segments to lengthen. The central crown hinge allows the arch to rise slightly without generating any internal thermal stresses or bending moments.
Two-Hinged and Fixed Arches: These are statically indeterminate. When temperature increases, the arch attempts to expand, but the rigid supports prevent outward translation. This "locked-in" deformation induces a significant horizontal thrust reaction at the supports, which in turn creates a massive bending moment throughout the entire arch rib. This is a critical design consideration for indeterminate arches.

Analyzing a Three-Hinged Arch

The analysis of a three-hinged arch involves finding the support reactions. Because there are four unknown reaction components (two at each pinned support) and only three global equations of equilibrium, a fourth equation is needed.

The internal hinge at the crown provides this extra condition: the internal bending moment at the hinge is always zero.

Step 1: Consider the entire arch as a free-body diagram and apply global equilibrium equations ( $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ ).
Step 2: Disconnect the arch at the internal crown hinge, creating two separate free-body diagrams (left half and right half).
Step 3: Sum moments about the crown hinge for either the left or right half and set the sum to zero ( $\sum M_{hinge} = 0$ ). This allows you to solve for the remaining unknown support reactions.
Step 4: Once all reactions are known, you can use the method of sections to find internal normal forces, shear forces, and bending moments at any point along the arch.

Arch Shape Optimization

The shape of an arch can be optimized so that it carries loads purely in axial compression, eliminating bending moments entirely. This ideal shape is called the funicular polygon or line of pressure for a specific set of loads.

The Funicular Shape

Funicular Arches: For a specific loading, an arch can be designed to match the inverted shape of a cable hanging under the exact same loads. For example:
- If a cable carrying a uniform distributed load (like a suspension bridge deck) forms a parabola, an arch designed as an inverted parabola under that same load will experience zero bending.
- If a cable carrying its own weight forms a catenary, an arch carrying its own weight must be shaped as an inverted catenary to carry the load purely in compression.
Sensitivity: Because the funicular shape is perfectly optimized for a single load case, any live loads or wind loads that alter the loading pattern will immediately induce bending moments. Real arches must therefore be designed with enough flexural strength to handle these variations, making true pure-compression funicular arches practical only for loads that are massive and unvarying (like huge masonry structures).

The Three-Hinged Arch Procedure

A three-hinged arch is the most common statically determinate arch type. It consists of two curved members connected by a hinge at the crown and supported by hinges at its two abutments.

Procedure

STEP-BY-STEP

Calculate the Global Reactions: Treat the entire arch as a single free-body. Taking the sum of moments about one support ( $\sum M_A = 0$ ) allows solving for the vertical reaction at the other support ( $V_B$ ).
Sum of Vertical Forces: Use $\sum F_y = 0$ on the entire arch to find the other vertical reaction ( $V_A$ ).
Isolate One Segment at the Crown Hinge: Because the crown hinge cannot resist bending moment ( $M_{crown} = 0$ ), pass a section through the crown hinge and consider either the left or right half of the arch as a free body.
Calculate the Horizontal Thrust ( $H$ ): Take the sum of moments about the crown hinge ( $\sum M_{crown} = 0$ ) for the isolated segment. Since the vertical reactions and external loads on that segment are known, this equation will yield the horizontal thrust ( $H$ ) at the supports.
Sum of Horizontal Forces: Use $\sum F_x = 0$ on the entire arch to find the remaining horizontal reaction.
Calculate Internal Forces at a Point ( $N, V, M$ ): Pass a section through any point on the arch. Apply equilibrium equations ( $\sum F_x = 0, \sum F_y = 0, \sum M = 0$ ) to the isolated segment to find the internal axial force ( $N$ ), shear force ( $V$ ), and bending moment ( $M$ ) acting at that cross-section. The horizontal and vertical forces must be resolved into components parallel and perpendicular to the tangent of the arch curve at the cut.

Key Takeaways

Cables are perfectly flexible elements that carry loads purely through internal axial tension. They change shape depending on the applied load configuration.
The General Cable Theorem relates cable sag to simple beam moments ( $Hy = M$ ).
Under concentrated point loads, a cable forms a funicular shape of straight segments. Under a horizontally uniform distributed load, a cable forms a parabolic curve.
The horizontal component of tension ( $H$ ) in a cable under purely vertical loading is constant throughout its entire length.
Arches primarily carry loads through axial compression. A three-hinged arch is statically determinate because the internal hinge at the crown provides a known condition where the bending moment is zero.
Thermal Stresses do not occur in three-hinged (determinate) arches but are a major design factor causing bending moments in two-hinged and fixed (indeterminate) arches.
The primary step in analyzing a three-hinged arch is to split the structure at the internal hinge and sum moments about that hinge to solve for horizontal support reactions.

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