Long-Span and Special Bridge Systems

Introduction

Long-span bridges are governed not only by material strength but also by structural form, geometric stiffness, aerodynamic stability, construction sequence and whole-life performance. As span increases, self-weight becomes a dominant action, deflections become geometrically significant, and the completed bridge may behave very differently from its partially erected states. Structural analysis must therefore represent the load path, cable or arch geometry, material behaviour, boundary conditions and erection history with appropriate fidelity.

This chapter examines arch, cable-stayed, suspension, movable and integral bridges. It emphasises system selection, cable behaviour, nonlinear response, wind effects, erection engineering and system-specific design risks.

Learning Objectives

After studying this chapter, readers should be able to:

• distinguish the load paths and suitable applications of major special bridge systems;
• analyse elementary arch and cable behaviour using equilibrium and compatibility;
• explain geometric and material nonlinearities in long-span bridges;
• assess flutter, vortex shedding, buffeting and related aerodynamic phenomena;
• evaluate erection strategies and temporary structural states;
• identify critical design issues in movable and integral bridges; and
• formulate a rational system-selection and risk-management strategy.

1. Structural Form and System Selection

Long-span systems use geometry to transfer loads efficiently. Arches carry loads mainly through compression, whereas cables carry them through tension. Cable-stayed and suspension bridges combine tension systems with bending-resistant decks and compression towers. Movable bridges provide navigational clearance mechanically, while integral bridges eliminate deck joints by connecting the deck continuously to the substructure.

| System | Principal load path | Typical controlling issues |

|---|---|---|

| Arch | Deck → arch rib → abutments | Horizontal thrust, buckling, erection |

| Cable-stayed | Deck → inclined stays → towers → foundations | Stay fatigue, tower-deck interaction, erection balance |

| Suspension | Deck → suspenders → main cables → towers and anchorages | Aerodynamics, cable corrosion, deck stiffness |

| Movable | Deck → primary structure, machinery and supports | Machinery reliability, fatigue, alignment |

| Integral | Deck and abutments act continuously | Thermal movement, soil interaction, cyclic displacement |

System selection depends on span, foundation conditions, navigation or clearance requirements, erection access, wind climate, seismicity, maintainability, durability, aesthetics and life-cycle cost. Published span ranges are only indicative: local geology, construction capability and risk may outweigh span efficiency.

2. Arch Bridges

2.1 Structural behaviour

An arch develops horizontal thrust because its supports restrain outward spreading. If its shape coincides with the funicular line for a particular loading, bending moment under that loading is theoretically zero. Real bridges depart from perfect funicular action because of moving loads, temperature changes, support movement, asymmetric loading and construction imperfections.

Common forms include:

• three-hinged arches, which are statically determinate;
• two-hinged arches, which are externally indeterminate to the first degree;
• fixed arches, which have greater indeterminacy and sensitivity to support movement;
• tied arches, in which a tension tie resists horizontal thrust; and
• deck, through and half-through arches, classified by deck position.

For a symmetrical three-hinged parabolic arch of span L and rise f carrying a uniform load w per unit horizontal length, equilibrium gives

H = wL²/(8f),

where H is the horizontal reaction. Each vertical reaction is

V = wL/2.

These expressions apply to a symmetrical arch, equal-level supports and full-span uniform loading measured on the horizontal projection.

2.2 Compatibility of a two-hinged arch

Consider a two-hinged arch with H chosen as the redundant horizontal reaction. Let M⁰(s) be the bending moment in the released primary structure due to external actions, and let y(s) be the arch ordinate measured from the support chord. With the sign convention that the redundant reduces positive beam moment,

M(s) = M⁰(s) − Hy(s).

Neglecting axial and shear deformation, zero relative horizontal displacement between the hinges requires

0 = ∫ [M(s)/(EI)](∂M/∂H) ds.

Because ∂M/∂H = −y,

0 = ∫ [(M⁰ − Hy)(−y)/(EI)] ds.

Consequently,

H = [∫ M⁰y/(EI) ds] / [∫ y²/(EI) ds].

The integrals extend along the arch axis. E is Young’s modulus, I is the second moment of area and ds is an element of arch length. This expression is a bending-only compatibility result. If axial deformation, temperature, support movement or fabrication error is significant, its compatible displacement contribution must be included in the virtual-work equation. It must not be represented by adding a second bending term that duplicates the effect of H.

2.3 Worked example: arch reactions

A symmetrical three-hinged parabolic arch has span 120 m and rise 24 m. It carries a uniform load of 60 kN/m over the horizontal span.

The horizontal reaction is

H = 60(120²)/(8 × 24) = 4,500 kN.

Each vertical reaction is

V = 60(120)/2 = 3,600 kN.

The magnitude of the resultant reaction at each support is

R = √(4,500² + 3,600²) = 5,763 kN approximately.

Increasing the rise would reduce horizontal thrust under the stated loading, although clearance, aesthetics, rib stability and construction requirements constrain the rise.

2.4 Design challenges

Arch ribs must be checked for in-plane and out-of-plane buckling, second-order effects, local instability and imperfections. Concrete arches may be influenced by creep, shrinkage and staged loading. Steel arches require fatigue and corrosion assessment, particularly at hanger and bracing connections. Since thrust is transferred into the ground, weak foundations may favour a tied arch rather than a thrust-resisting arch.

3. Cable Mechanics

3.1 Ideal cable behaviour

An ideal cable has negligible flexural rigidity and carries tension only. Its shape changes until equilibrium is achieved. Cable-supported bridges therefore exhibit strong interaction among geometry, tension and loading.

For a cable spanning horizontally between equal-level supports, subjected to a uniform vertical load q per unit horizontal length, the cable is parabolic. If y is positive downward,

−H(d²y/dx²) = q,

where H is the constant horizontal component of cable tension. With zero ordinate at the supports and sag f at midspan,

H = qL²/(8f).

The vertical reaction at each support is

V = qL/2,

and the maximum cable force at a support is

Tmax = √(H² + V²).

Reducing sag increases H and generally increases the total support cable force. It does not reduce the vertical reaction for the same symmetrical vertical loading: V remains fixed by global equilibrium.

When loading is uniform per unit actual cable length, as for cable self-weight, the exact shape is a catenary. Taking the lowest point as the origin,

y = a cosh(x/a) − a,

where a = H/wc and wc is weight per unit actual cable length. Practical bridge analysis commonly uses elastic catenary or geometrically nonlinear cable elements rather than assuming a fixed parabola.

3.2 Worked example: sag and cable force

A suspension cable has a horizontal span of 800 m, sag of 80 m and uniform load of 120 kN/m of horizontal span.

H = 120(800²)/(8 × 80) = 120,000 kN = 120 MN.

V = 120(800)/2 = 48,000 kN = 48 MN.

Therefore,

Tmax = √(120² + 48²) = 129.24 MN.

If the sag is reduced to 64 m under the same loading,

H = 120(800²)/(8 × 64) = 150 MN,

while V remains 48 MN. The support tension becomes

Tmax = √(150² + 48²) = 157.49 MN.

The example demonstrates the force penalty associated with a smaller sag-to-span ratio.

3.3 Geometric stiffness

Prestress gives a cable transverse stiffness even though its bending stiffness is negligible. Consider two straight, symmetric cable segments meeting at midspan. Let L be the full horizontal span and H the initial horizontal tension component. For a small transverse midpoint displacement z, each half has transverse slope approximately z/(L/2). The total restoring force is therefore

F ≈ 2H[z/(L/2)] = (4H/L)z.

Thus, the midpoint geometric stiffness is

kg ≈ 4H/L.

This is a special result for a symmetric two-segment cable, small transverse movement and approximately constant initial H. Elastic stretching, sag, distributed mass, support flexibility and continuous cable curvature require a more complete tangent-stiffness formulation.

For example, with H = 20 MN and L = 1,000 m,

kg = 4(20,000)/1,000 = 80 kN/m.

4. Cable-Stayed Bridges

In a cable-stayed bridge, inclined stays transfer deck loads directly to one or more towers. The towers carry predominantly compression and bending, while the deck resists axial compression, local bending and global bending between stay anchorages.

Stay arrangements include fan, semi-fan and harp systems. Fan systems are structurally efficient but may create congested tower anchorages. Harp arrangements are visually ordered but generally impose larger tower bending because the stays are nearly parallel.

Deck systems may be steel, concrete or composite. Multiple stay planes improve torsional resistance, whereas a single central plane requires a deck with substantial torsional stiffness.

Critical design issues include:

• fatigue from traffic, wind and rain-wind vibration;
• anchorage stress concentrations;
• cable replacement provisions;
• corrosion protection and dehumidification;
• tower stability during erection;
• construction-stage force control;
• stay-force optimisation; and
• nonlinear interaction among stays, deck and towers.

Stay cables exhibit sag, although less dramatically than suspension cables. Their axial response is therefore softer than that of a perfectly straight bar. Detailed analysis uses catenary or cable elements incorporating initial strain, self-weight and geometric stiffness.

5. Suspension Bridges

A suspension bridge carries deck loading through suspenders to the main cables. Main-cable forces pass over tower saddles and are resisted by anchorages in an earth-anchored bridge. In a self-anchored suspension bridge, horizontal cable forces are closed through the deck, producing a highly sequence-dependent structural system.

The principal components are:

• main cables;
• suspenders or hangers;
• stiffening girder or aerodynamic box deck;
• towers and saddles;
• anchorages; and
• bearings, expansion joints and damping devices.

The main cable is efficient in tension, but the flexibility of the cable-deck system creates susceptibility to wind-induced motion. The deck distributes concentrated traffic loads, limits local distortion and contributes torsional and aerodynamic stability.

Maintenance is especially important. Cable wires may corrode internally despite external wrapping. Modern strategies include cable dehumidification, acoustic wire-break monitoring and replaceable suspenders. Hanger replacement itself changes the force distribution and requires a controlled temporary works procedure.

6. Nonlinear Structural Response

Long-span bridges commonly require nonlinear analysis because equilibrium depends on the deformed configuration. Important nonlinearities include:

1. Geometric nonlinearity: cable sag, large displacement, P–Δ effects and changing force directions.

2. Material nonlinearity: yielding, cracking, creep, shrinkage and nonlinear soil response.

3. Boundary nonlinearity: bearing uplift, friction, joint gaps, cable slackening and contact.

4. Construction-stage nonlinearity: activation or removal of elements, temporary supports and changing dead-load geometry.

The tangent equilibrium equation may be expressed incrementally as

KT Δu = ΔP,

where KT is the tangent stiffness matrix, Δu is the displacement increment and ΔP is the load increment. KT includes material and geometric stiffness. Iterative procedures such as Newton–Raphson update equilibrium until residual forces satisfy specified tolerances.

A credible analysis should distinguish between the fabricated geometry, unstressed geometry and target completed geometry. Initial cable strains or pretensions must be calibrated so that the desired deck profile and forces are obtained after dead load, creep and staged erection. Eigenvalue buckling alone is insufficient for imperfection-sensitive systems; geometrically and materially nonlinear analysis with representative imperfections may be required.

7. Aerodynamic Stability

7.1 Wind phenomena

Long-span decks, towers and cables interact dynamically with wind. The main phenomena are:

• buffeting: forced response to turbulent wind;
• vortex shedding: oscillation associated with alternating wake vortices;
• flutter: self-excited coupled motion, commonly involving vertical and torsional modes;
• galloping: low-frequency instability associated with aerodynamic force variation; and
• divergence: static aerodynamic instability, often torsional.

The vortex-shedding frequency may be estimated by

fs = St U/D,

where St is the Strouhal number, U is wind speed and D is a characteristic cross-sectional dimension. This relationship estimates shedding frequency; it does not by itself predict response amplitude or lock-in.

Flutter cannot generally be assessed through a simple static pressure coefficient. It requires sectional-model testing, validated computational fluid dynamics, flutter-derivative analysis or full aeroelastic modelling. The critical flutter speed must exceed the specified design threshold with the required safety margin.

7.2 Aerodynamic design measures

Mitigation measures include streamlined box girders, edge fairings, vents, guide vanes, increased torsional stiffness and mass dampers. Cable vibration may be reduced using surface treatments, cross-ties, external dampers and improved drainage. Aerodynamic performance must be checked during erection because incomplete decks may have lower stiffness and less favourable cross-sections than the completed bridge.

8. Erection Strategy and Temporary States

Construction method is part of the structural design. Typical strategies include:

• arch bridges: temporary falsework, cable-stayed cantilever erection, rotation or lifting of arch halves;
• cable-stayed bridges: balanced cantilever erection from towers with sequential stay installation and stressing;
• suspension bridges: tower and anchorage construction, cable spinning or strand erection, hanger installation and deck-segment lifting;
• movable bridges: off-site assembly of mechanical and structural units followed by controlled installation;
• integral bridges: staged deck-abutment construction with careful control of backfill sequence.

Each construction stage must be checked for strength, stability, deformation, wind response and accidental imbalance. Survey data and stay-force measurements are used to compare actual and predicted behaviour. Geometry-control models may require updating for temperature, measured material properties and time-dependent effects.

Balanced cable-stayed erection reduces tower bending, but perfect symmetry is rarely possible. The design must allow for segment-weight variation, temporary equipment, wind and delayed installation. Temporary conditions can govern permanent member sizes.

9. Movable Bridges

Movable bridges provide intermittent navigation clearance where a high fixed bridge would be impractical. Principal forms are:

• bascule bridges, rotating about a horizontal axis;
• swing bridges, rotating about a vertical axis;
• vertical-lift bridges, raising a span between towers; and
• retractile bridges, translating the movable span.

Counterweights reduce operating energy but introduce large concentrated forces and inertia. Design must integrate structural engineering, machinery, electrical power, control systems and navigation requirements.

Key issues include fatigue from repeated operating cycles, gear and bearing wear, rail or roadway alignment, wind limits for operation, locking devices, emergency power and fail-safe control logic. Structural deflections must remain compatible with machinery tolerances. Redundant sensors and interlocks should prevent opening when traffic is present or closing when the navigation channel is obstructed.

Maintenance planning is central because failure may disrupt both road and water transport. Whole-life assessment should include spare components, inspection access, cybersecurity of control systems and manual recovery procedures.

10. Integral Bridges

An integral bridge has no conventional expansion joint between the deck and abutment. The deck, abutments and often the piers interact continuously. Eliminating joints reduces leakage, corrosion and maintenance, but thermal and time-dependent deck movements are transferred into the abutments, foundations and surrounding soil.

For an unconstrained length L subjected to a uniform temperature change ΔT, free thermal movement is

Δ = αLΔT,

where α is the coefficient of thermal expansion. This is a free-movement estimate, not the actual movement of a restrained integral system.

For example, a 150 m deck with α = 12 × 10⁻⁶/°C and ΔT = 40°C has free movement

Δ = 12 × 10⁻⁶ × 150 × 40 = 0.072 m = 72 mm.

In an integral bridge, soil resistance and structural restraint modify this movement and generate forces. Repeated expansion and contraction can compact backfill, increase earth pressure and cause settlement or pavement distress. Skew introduces three-dimensional response and may increase rotation and lateral soil interaction.

Design requires soil–structure interaction models, suitable approach slabs, flexible foundations where appropriate, drainage and careful backfill specification. Maximum practical lengths and skews vary by jurisdiction, climate, foundation type and institutional experience; they should not be treated as universal constants.

11. Illustrative Applied Case: Estuary Crossing

Background

A fictional estuary crossing requires a 680 m navigation span, a 55 m clearance envelope and minimal construction within the channel. The banks can support deep foundations, but large gravity anchorages would be costly. Strong seasonal winds occur, and the bridge must carry highway and light-rail traffic.

Problem

The project team compares an arch, suspension bridge and cable-stayed bridge. The evaluation considers structural efficiency, navigation, foundations, erection, aerodynamic risk, inspection access and life-cycle cost.

Analysis

A through arch would require complex erection over the navigation channel and would create large thrust or tie forces. A suspension bridge is feasible but requires major anchorages and has a relatively high cable-maintenance burden. A cable-stayed system falls within contemporary feasible practice for the proposed span and avoids large external anchorages.

For a representative stay inclined at 25° to the horizontal and carrying a factored vertical component of 10.5 MN, the required stay force from equilibrium is

T = 10.5/sin 25° = 24.85 MN.

If the preliminary allowable design stress for sizing is 796.5 MPa, corresponding to 0.45 of a specified 1,770 MPa tensile strength, the indicative metallic area is

A = 24.85 × 10⁶/(796.5 × 10⁶) = 0.0312 m².

This is only preliminary sizing. Final design must account for resistance factors, fatigue, anchorage efficiency, bending, corrosion protection, construction tolerances and cable replacement.

Approach and Solution

A twin-plane cable-stayed bridge with streamlined composite deck is selected. Balanced cantilever construction limits activity in the navigation channel. Wind-tunnel testing is specified for completed and erection-stage configurations. Replaceable stays, external dampers, dehumidified anchorage zones and permanent structural-health monitoring are incorporated.

A staged nonlinear model is used to determine stay stressing, tower movements and target deck geometry. Closure is scheduled within a controlled temperature range, and contingency analyses address loss of lifting equipment, temporary stay failure and extreme erection-stage wind.

Outcome

The selected system provides the required clearance without large gravity anchorages. The decision is based not solely on span economy but on constructability, aerodynamic performance, maintenance access and risk allocation.

Learning Points

• Structural form must be evaluated with foundations and erection strategy.
• Preliminary equilibrium calculations remain valuable for checking numerical models.
• Construction-stage wind and imbalance may govern design.
• Replaceability and inspection access should be designed from project inception.

12. Governance, Ethics and Contemporary Practice

Long-span bridge projects have low tolerance for fragmented responsibility. Independent design checking, aerodynamic peer review, verified software, traceable assumptions and formal management of interfaces are essential. Designers have an ethical duty to report unfavourable findings, uncertain data and construction deviations rather than adjusting models merely to reproduce expected outcomes.

Climate change may alter wind, temperature, flood, scour and corrosion exposure. Robustness studies should examine hazards beyond traditional historical averages. Navigation authorities, transport operators, environmental regulators and emergency services should participate in governance.

Structural-health monitoring increasingly combines accelerometers, strain sensors, displacement measurement, cable-force estimation and weather data. Machine-learning methods can assist anomaly detection, but outputs depend on representative training data and stable sensor performance. Artificial intelligence should support—not replace—mechanics-based interpretation, engineering judgement and accountable decisions.

Best practice includes designing for inspection, cable or bearing replacement, access to anchorages, safe maintenance and controlled decommissioning. Life-cycle carbon, durability and disruption costs should supplement initial-cost comparisons.

Chapter Summary

Long-span and special bridges achieve efficiency through structural form. Arches principally carry compression and develop support thrust; cables carry tension and acquire stiffness through geometry and prestress. Cable-stayed bridges transfer deck loads directly to towers, while suspension-bridge suspenders transfer deck loads to main cables. Movable bridges integrate structures with machinery and controls, whereas integral bridges exchange joint maintenance for thermal soil–structure interaction.

Long-span analysis must address geometric nonlinearity, construction sequence, cable pretension, instability and aerodynamic response. Equilibrium-based calculations provide essential checks on advanced numerical models. Safe and economical design also depends on erection engineering, durability, monitoring, replaceability, independent review and life-cycle risk management.

Key Terms

Funicular shape: A geometry that carries a specified loading through axial force without bending.

Horizontal thrust: Horizontal support reaction developed by an arch or cable system.

Two-hinged arch: An arch hinged at both supports and externally indeterminate to the first degree.

Tied arch: An arch in which a tension tie resists horizontal thrust.

Cable sag: Vertical distance between a cable chord and the cable at a specified location.

Catenary: The equilibrium shape of a cable under load uniform per unit actual cable length.

Geometric stiffness: Stiffness arising from existing axial force and structural geometry.

Stay cable: Inclined tension member connecting a bridge deck to a tower.

Suspender: Vertical or inclined member transferring suspension-bridge deck load to the main cable.

Flutter: Self-excited aerodynamic instability involving structural motion and aerodynamic forces.

Buffeting: Forced vibration caused by turbulent wind fluctuations.

Vortex shedding: Alternating wake-vortex formation that can excite structural vibration.

Tangent stiffness: Incremental stiffness used in nonlinear equilibrium calculations.

Construction-stage analysis: Analysis accounting for the sequence of element installation, loading and support changes.

Bascule bridge: Movable bridge rotating about a horizontal axis.

Integral bridge: Bridge constructed without conventional deck joints at its abutments.

Soil–structure interaction: Mutual influence between structural deformation and soil response.

Structural-health monitoring: Continuous or periodic measurement used to evaluate structural condition and performance.

Review Questions

Short Answer Questions

1. What is meant by the funicular shape of an arch?

2. Why does reducing cable sag increase horizontal cable force under unchanged loading?

3. What load path do suspenders provide in a suspension bridge?

4. Distinguish buffeting from flutter.

5. Why are integral bridges sensitive to cyclic soil interaction?

Long Answer Questions

1. Derive the bending-only compatibility expression for horizontal thrust in a two-hinged arch.

2. Compare cable-stayed and suspension bridges with respect to load path, foundations, erection and maintenance.

3. Explain the sources of nonlinear response in long-span bridges and their treatment in analysis.

4. Discuss the structural, mechanical and operational design requirements of movable bridges.

5. Evaluate aerodynamic risks during both construction and service of a long-span bridge.

Analytical / Application Questions

1. A parabolic cable spans 500 m with a sag of 50 m and carries 80 kN/m horizontally. Calculate H, the vertical support reaction and maximum cable tension.

2. Develop a staged-analysis and geometry-control plan for balanced cantilever erection of a cable-stayed bridge.

3. Compare an arch, cable-stayed bridge and suspension bridge for a 600 m navigation crossing using structural, geotechnical, aerodynamic and life-cycle criteria.

Multiple Choice Questions with Answers

Question 1

A symmetrical three-hinged parabolic arch has a span of 100 m, rise of 20 m and uniform load of 40 kN/m. What is its horizontal reaction?

A. 1,250 kN

B. 2,500 kN

C. 5,000 kN

D. 10,000 kN

Correct Answer: B. 2,500 kN

Question 2

Neglecting axial and shear deformation, which expression gives the horizontal reaction H of a two-hinged arch when M = M⁰ − Hy?

A. H = (∫M⁰y/EI ds)/(∫y²/EI ds)

B. H = (∫M⁰/EI ds)/(∫y/EI ds)

C. H = ∫M⁰y/EI ds

D. H = (∫y²/EI ds)/(∫M⁰y/EI ds)

Correct Answer: A. H = (∫M⁰y/EI ds)/(∫y²/EI ds)

Question 3

A parabolic cable has span 600 m, sag 60 m and uniform load 100 kN/m of horizontal span. What is its horizontal tension component?

A. 37.5 MN

B. 60 MN

C. 75 MN

D. 150 MN

Correct Answer: C. 75 MN

Question 4

For a symmetrical cable carrying an unchanged full-span vertical load, what is the effect of reducing sag?

A. Both horizontal and vertical reactions decrease

B. Horizontal force increases while the vertical reaction remains fixed by equilibrium

C. Horizontal force remains constant while the vertical reaction decreases

D. Both horizontal and vertical reactions become zero

Correct Answer: B. Horizontal force increases while the vertical reaction remains fixed by equilibrium

Question 5

A symmetric two-segment cable has full horizontal span 600 m and initial horizontal tension component 15 MN. Under the stated small-displacement approximation, what is its midpoint transverse geometric stiffness?

A. 25 kN/m

B. 50 kN/m

C. 100 kN/m

D. 400 kN/m

Correct Answer: C. 100 kN/m

Question 6

What is the correct gravity load path through the suspenders of a conventional suspension bridge?

A. Main cable to deck

B. Tower to anchorage

C. Deck to main cable

D. Anchorage to foundation through the deck

Correct Answer: C. Deck to main cable

Question 7

Which wind phenomenon is a self-excited instability commonly coupling vertical and torsional bridge motion?

A. Buffeting

B. Flutter

C. Static drag

D. Thermal expansion

Correct Answer: B. Flutter

Question 8

A 150 m bridge has a coefficient of thermal expansion of 12 × 10⁻⁶/°C and a uniform temperature change of 40°C. What is its unconstrained thermal movement?

A. 7.2 mm

B. 48 mm

C. 72 mm

D. 720 mm

Correct Answer: C. 72 mm

Question 9

Which movable bridge type rotates primarily about a vertical axis?

A. Bascule bridge

B. Vertical-lift bridge

C. Swing bridge

D. Retractile bridge

Correct Answer: C. Swing bridge

Question 10

Why is balanced cantilever erection commonly used for cable-stayed bridges?

A. It eliminates all cable tension

B. It reduces unbalanced tower and foundation actions during erection

C. It makes aerodynamic assessment unnecessary

D. It removes the need for geometry control

Correct Answer: B. It reduces unbalanced tower and foundation actions during erection