Structural Analysis and Modelling of Bridges

Introduction and Context

Bridge analysis is the process of converting a physical bridge, its actions, supports, materials, construction sequence and environment into an idealised structural model from which forces, deformations, reactions, vibration characteristics and failure modes can be predicted. The quality of an analysis depends not only on the mathematical method but also on whether the model represents the real structural behaviour with appropriate fidelity.

Bridge structures are particularly demanding because their behaviour is three-dimensional and highly interactive. Loads are distributed through decks, girders, diaphragms, bearings, piers, foundations and surrounding soil. Traffic is moving and may be dynamic; temperature produces restraint forces; construction stages alter the load path; and cracking, creep, yielding, contact and geometric changes may introduce nonlinearity. Consequently, a suitable analysis may range from a simple beam model to a nonlinear, staged, soil–structure-interaction finite-element model.

This chapter develops the principal analytical models used in bridge engineering. It emphasises the relationship between structural idealisation, assumptions, numerical implementation, verification and engineering judgement.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain the principal load paths and structural idealisations used for bridges.

2. Apply classical beam and grillage concepts to preliminary and detailed analysis.

3. Select appropriate finite-element types, material models and boundary conditions.

4. Formulate staged-construction, geometric and material nonlinear analyses.

5. Explain soil–structure interaction and dynamic bridge analysis.

6. Distinguish verification, validation, sensitivity analysis and calibration.

7. Critically assess modelling assumptions and communicate analytical results professionally.

1. Structural Behaviour and Load Paths

A bridge should first be understood as a load-resisting system rather than as a collection of members. A typical highway bridge may contain a deck slab, longitudinal girders, transverse diaphragms, bearings, piers, pile caps, foundations and approach structures. A vertical wheel load is transferred through the wearing surface and deck slab to longitudinal and transverse members, then through bearings to piers or abutments, and finally into the ground.

The load path changes with the action considered. For example:

• Vertical gravity loads are primarily resisted by flexure and shear in the deck and girders.
• Braking and traction forces are transferred longitudinally through the deck, bearings and substructure.
• Wind may produce transverse deck forces, torsion and overturning reactions.
• Temperature changes may be accommodated by movement bearings or converted into restraint forces.
• Earthquake actions may involve inertial forces, bearing restraint, pier ductility, foundation flexibility and soil response.

The analyst must identify whether the structure behaves as a beam, grillage, plate, shell, spatial frame or continuum. The idealisation should be no more complex than necessary, but sufficiently detailed to answer the engineering question.

2. Classical Analysis of Bridge Components

2.1 Beam idealisation

A simply supported prismatic beam under a uniformly distributed load w over span L has maximum sagging moment and support shear:

M_max = wL²/8

V_support = wL/2

where w is force per unit length and L is the span. These expressions assume a statically determinate beam, small displacements, constant stiffness, a uniform load over the complete span and ideal simple supports. They are useful for preliminary sizing and checking computer output, but are not generally adequate for continuous, skew, curved or highly irregular bridges.

For an Euler–Bernoulli beam, plane cross-sections remain plane and shear deformation is neglected. The curvature–moment relationship is:

κ = M/(EI)

where κ is curvature, M is bending moment, E is Young’s modulus and I is the second moment of area. The assumption is appropriate for slender members in which span-to-depth ratio is large and shear deformation is small.

For deep girders, short spans, prestressed concrete members with substantial web depth or members subjected to high shear, Timoshenko beam theory is more appropriate because it includes shear deformation. The total rotation is then composed of bending and shear contributions. The shear strain is approximately:

γ = V/(κ_s GA)

where V is shear force, G is shear modulus, A is cross-sectional area and κ_s is the shear-correction factor. The value of κ_s depends on the cross-sectional shape and the adopted theory.

2.2 Continuous beams and compatibility

In a continuous bridge, support moments arise because rotations and displacements must be compatible at internal supports. Classical methods include the force method, slope-deflection method, moment distribution and matrix stiffness analysis. Modern software generally uses the matrix stiffness method, but classical solutions remain essential for checking the sign, magnitude and distribution of computed actions.

For a linearly elastic structure, the global relation is commonly expressed as:

[K]{u} = {F}

where [K] is the assembled stiffness matrix, {u} is the vector of unknown nodal displacements and {F} is the vector of applied nodal loads. Once displacements are obtained, member forces follow from element force–deformation relationships. This formulation assumes small displacements, known connectivity, elastic material behaviour and correctly imposed boundary conditions.

2.3 Influence lines and moving loads

Bridge live loads move across the structure. An influence line gives the value of a response quantity—such as reaction, shear or moment—at a fixed location as a unit load moves along the bridge. For a determinate structure, influence lines can be obtained by statics; for indeterminate structures, they are commonly obtained using analysis software or virtual-work principles.

For a moving load system, the response is found by placing the load train in positions that maximise the selected response. The critical position for a support reaction will generally differ from that for midspan moment or torsion. Dynamic amplification, multiple lanes, braking, lane distribution and load models specified by the governing code must be treated consistently.

3. Grillage Analysis

3.1 Concept

A grillage model represents a bridge deck as an interconnected network of longitudinal and transverse beam elements. Longitudinal members represent girders or equivalent strips; transverse members represent deck slab action, diaphragms and load distribution. Torsional stiffness may be assigned to members to reproduce Saint-Venant torsion and, where appropriate, warping-related effects.

A grillage is particularly useful for reinforced-concrete and steel-girder decks subjected to eccentric wheel loads. It provides a practical balance between a simple line-girder model and a detailed shell or solid model.

3.2 Modelling decisions

Important decisions include:

• spacing and orientation of grillage members;
• representation of the deck slab and effective flange widths;
• torsional rigidity of longitudinal and transverse members;
• location of supports and bearings;
• modelling of diaphragms and cross-girders;
• treatment of skew and curvature;
• application and distribution of wheel loads.

The spacing should be fine enough to capture changes in response near supports, concentrated loads and discontinuities, but excessive refinement can create misleading numerical precision. Equivalent stiffness properties must not double-count slab and girder material. For example, if a girder element already includes the effective slab flange, the same slab area must not also be assigned independently to an overlapping longitudinal member.

3.3 Interpretation and limitations

Grillage results include member moments, shears, torsions, reactions and deflections. These are model-dependent quantities. A grillage does not directly provide the local two-dimensional stress field around openings, concentrated bearing plates, anchorage zones or severe geometric discontinuities. Such regions require shell, solid or local submodels.

A grillage is also sensitive to torsional stiffness assumptions. Assigning unrealistically high torsional rigidity may suppress twist and redistribute load in a nonphysical manner. Conversely, omitting torsional resistance may exaggerate differential girder effects. Calibration against analytical solutions, refined models or measured data is therefore good practice.

4. Finite-Element Modelling

4.1 Element types

Finite-element models divide the structure into elements connected at nodes. Common element types are:

• Truss elements: axial force only; useful for bracing or cables under suitable tension assumptions.
• Beam elements: axial force, bending, shear and possibly torsion; efficient for girders, piers and piles.
• Plate or shell elements: membrane and bending action; suitable for slabs, webs, diaphragms and walls.
• Solid elements: three-dimensional stress and strain representation; useful for bearings, anchorage zones, pile caps, joints and complex details.
• Contact elements: compression-only, frictional or separation interfaces between components.

The element type should correspond to the intended response. A shell model is not automatically more accurate than a beam model: poor mesh quality, incorrect thickness, unsuitable constraints or erroneous material properties can make a detailed model less reliable.

4.2 Mesh selection

A mesh should resolve the stress and deformation gradients relevant to the decision being made. Mesh refinement is usually required near supports, load introduction points, holes, connections, changes in section, cracks and contact interfaces. A mesh-convergence study compares the selected response as element size is reduced. Convergence of one response, such as global displacement, does not guarantee convergence of local peak stress.

Point loads applied to a single node may cause artificial stress singularities in continuum models. Loads should normally be distributed over a realistic contact area or applied through rigid loading patches. Similarly, sharp re-entrant corners may produce mathematically unbounded elastic stresses; engineering interpretation should focus on averaged, integrated or fracture-mechanics-relevant quantities rather than an isolated peak.

4.3 Boundary conditions and connections

Boundary conditions represent restraints against translation and rotation. They must reflect the actual bearing arrangement. A bridge may have one fixed bearing line, guided bearings in the transverse direction and free movement longitudinally at other supports. Restraining every degree of freedom at every support can create artificial forces and prevent thermal movement.

Connections may be modelled as rigid, pinned, semi-rigid, frictional or contact interfaces. Bearing stiffness can be represented explicitly where it affects load distribution or dynamic behaviour. The analyst should check that the model is stable without being over-restrained. Reaction sums must balance applied loads, and rigid-body modes should be absent unless intentionally retained for a dynamic or substructure analysis.

5. Staged Construction and Time-Dependent Behaviour

5.1 Why construction stages matter

The final bridge is not necessarily loaded as a completed structure. Members may be erected sequentially, supports may be temporary, concrete may gain stiffness with age, prestressing may be applied at particular stages, and formwork or construction loads may be removed. The final stress state depends on the sequence of activation, loading, prestressing, creep, shrinkage and relaxation.

A staged analysis activates or deactivates elements and loads in a specified sequence. At each stage, the solver determines equilibrium using the current stiffness, material state and accumulated history. The state variables—such as plastic strains, creep strains and damage—must be transferred from one stage to the next.

5.2 Prestressed and composite bridges

For a prestressed concrete bridge, tendon stressing, anchorage, grouting, transfer and subsequent losses should be represented in their actual sequence when these effects are significant. For a composite steel–concrete bridge, the concrete slab may be activated after steel erection. The steel carries construction-stage loads alone, while later loads are shared by the composite section according to its connection and stiffness.

Time-dependent effects include concrete creep, shrinkage and ageing, steel relaxation where relevant, and changing modulus. The exact constitutive laws should follow the adopted design standard or a defensible research model. Mixing parameters from incompatible standards can produce internally inconsistent results.

6. Geometric and Material Nonlinearity

6.1 Geometric nonlinearity

Geometric nonlinearity occurs when changes in geometry significantly affect equilibrium or stiffness. Examples include cable-supported bridges, slender piers, large deck displacements, buckling and P–Δ effects. In a second-order analysis, an axial force P acting through a laterally displaced member produces an additional moment approximately equal to PΔ, where Δ is the lateral displacement relative to the relevant reference configuration. This approximation is valid only when displacements remain sufficiently small for the chosen second-order formulation and the member remains within the assumed stability regime.

Large-displacement formulations update geometry and equilibrium throughout the analysis. Initial imperfections, residual stresses and load eccentricities may be essential when studying buckling or instability. A mathematically converged solution is not necessarily physically stable; load–displacement paths and alternative equilibrium branches must be examined.

6.2 Material nonlinearity

Material nonlinearity arises when stress is not proportional to strain or when stiffness changes due to cracking, yielding, crushing, fracture, creep or damage. Typical models include:

• elastic–perfectly plastic or bilinear steel models;
• elastoplastic models with isotropic and kinematic hardening;
• concrete cracking and compression-crushing models;
• damage-plasticity models;
• nonlinear tendon and cable constitutive laws;
• hyperelastic or viscoelastic models for bearings.

The model must reflect the purpose of analysis. A linear elastic model may be suitable for service-load distribution, whereas ultimate capacity or seismic ductility requires nonlinear constitutive behaviour. Parameters such as tensile strength, fracture energy, hardening ratio and confinement should be supported by testing, standards or documented calibration.

6.3 Solution and convergence

Nonlinear equilibrium is often solved incrementally using Newton–Raphson or modified Newton methods. At load increment i, the residual is:

{r_i} = {F_ext,i} - {F_int,i}

where {F_ext,i} is the externally applied load vector and {F_int,i} is the internal resisting-force vector. Convergence is assessed using residual forces, displacement increments, energy norms or combinations of these criteria. Tightening a tolerance does not correct an incorrect model; convergence checks must be accompanied by examination of load paths, deformation patterns and physical plausibility.

Common causes of nonconvergence include excessive load increments, unstable equilibrium, inappropriate contact stiffness, poor mesh quality, abrupt material degradation and inadequate constraints. Remedies should be physically justified, not limited to manipulating solver settings.

7. Soil–Structure Interaction

7.1 Significance

The assumption of perfectly fixed pier bases or rigid abutments can substantially distort bridge response. Soil flexibility changes natural periods, foundation forces, displacement demands and load distribution. Pile groups may experience axial, lateral and rocking stiffness, while abutment backfill may contribute passive resistance and damping.

Soil–structure interaction can be represented using:

1. Winkler springs, in which soil reaction is related locally to displacement;

2. p–y, t–z and q–z curves for nonlinear pile response;

3. elastic continuum or impedance models;

4. three-dimensional finite-element soil domains with appropriate constitutive laws and absorbing boundaries.

A simple translational spring follows F = ku, where F is force, k is stiffness and u is displacement. This relation assumes linear behaviour about the selected operating point. Real soil response may be nonlinear, pressure-dependent, hysteretic and affected by drainage, cyclic loading and gapping.

7.2 Modelling considerations

Foundation stiffness should be compatible with the load level and frequency range. Soil properties must account for stratigraphy, groundwater, density, stiffness degradation and potential liquefaction where relevant. Boundaries of a continuum soil model must be sufficiently remote or treated with appropriate transmitting boundaries to avoid artificial wave reflection.

Interaction models should be validated against geotechnical calculations, field data or accepted correlations. Uncertainty in soil properties is often greater than numerical uncertainty, so sensitivity studies are essential.

8. Dynamic Analysis of Bridges

8.1 Modal properties

A linear undamped free-vibration model is expressed as:

[M]{ü} + [K]{u} = {0}

where [M] is the mass matrix, [K] is the stiffness matrix and {u} is the displacement vector. Assuming harmonic motion, the eigenvalue problem is:

([K] − ω²[M]){φ} = {0}

where ω is circular natural frequency and {φ} is a mode shape. The natural frequency is f = ω/(2π). The mass matrix may be lumped or consistent; the choice should be appropriate to the element formulation and the dynamic response of interest.

Modes should be examined for physical meaning: vertical bending, lateral bending, torsion, local deck modes and substructure modes. Frequencies that are unexpectedly high or low may indicate incorrect mass, stiffness, constraints or units.

8.2 Forced vibration and damping

For a linear damped system:

[M]{ü} + [C]{u̇} + [K]{u} = {P(t)}

where [C] is the damping matrix and {P(t)} is the time-dependent load. Damping may be represented by modal damping, Rayleigh damping or more advanced frequency-dependent models. Rayleigh damping takes the form [C] = α[M] + β[K], but its damping ratio varies with frequency; therefore α and β should not be selected without checking the frequency range of interest.

Dynamic bridge actions include vehicle–bridge interaction, pedestrian loading, wind, vortex shedding, seismic excitation, construction vibration and impact. Moving-load analysis may be adequate when vehicle inertia and suspension interaction are negligible. Vehicle–bridge interaction is required when bridge and vehicle dynamics are strongly coupled, particularly for high-speed rail or irregular decks.

8.3 Seismic analysis

Seismic methods include equivalent-static analysis, response-spectrum analysis, modal time-history analysis and nonlinear time-history analysis. The chosen method depends on structural regularity, ductility, importance, code requirements and the need to capture nonlinear response. Ground motions should be selected and scaled consistently with the hazard definition and site conditions. Bearings, expansion joints, pounding, abutment participation and foundation flexibility may govern the response.

9. Verification, Validation and Quality Assurance

Verification asks whether the equations have been solved correctly; validation asks whether the model represents the real bridge adequately. These are distinct activities.

A rigorous verification process should include:

• hand calculations for simple reactions, moments and frequencies;
• equilibrium checks of forces and moments;
• dimensional and unit checks;
• comparison with closed-form beam or plate solutions;
• mesh-convergence studies;
• alternative modelling approaches;
• review of deformed shapes and mode shapes;
• sensitivity to material, support and load assumptions.

Validation may use laboratory tests, load testing, monitoring data, previous bridges or calibrated component tests. Calibration should not merely force agreement with one measurement; it should produce parameters that remain credible under other load cases.

Model uncertainty should be documented. A professional analysis report should state geometry, material properties, load models, combinations, boundary conditions, element types, mesh density, nonlinear settings, convergence criteria, software version, assumptions, limitations and quality-control checks.

10. Illustrative Application: Grillage and Finite-Element Assessment of a Skew Bridge

Background

Consider an illustrative two-span reinforced-concrete highway bridge with a 30° skew, four longitudinal girders, a cast-in-place deck slab and elastomeric bearings. The engineering objective is to estimate girder live-load distribution, support reactions and service deflection, and to determine whether skew-induced torsion requires a refined model.

Problem

A line-girder analysis cannot represent transverse load distribution or skew effects reliably. A full solid model would require substantial effort and could obscure the global load path. The analyst therefore needs a staged modelling strategy.

Analysis approach

First, a beam model is prepared to establish approximate global reactions and span moments. Second, a grillage is created with four longitudinal beam lines and transverse members at regular intervals, with additional refinement near supports and the skew abutment. The deck contribution is assigned once through equivalent section properties. Bearings are represented by their measured or specified translational stiffnesses, with longitudinal movement permitted where intended.

Wheel loads are applied over realistic contact patches and moved across the deck. The most adverse positions are identified separately for positive moment, negative moment, torsion and bearing reaction. Equilibrium is checked for every critical position.

A shell finite-element model is then developed for the deck and diaphragms. It includes the actual skew geometry, bearing locations and local support regions. The grillage and shell results are compared for global moments, reactions and deflections. Local shell stresses adjacent to bearings are not compared directly with grillage member stresses; instead, integrated forces over corresponding sections are used.

Outcome

The preliminary beam model predicts total reactions accurately but underestimates the difference between inner and outer girder reactions. The grillage captures transverse distribution and indicates increased torsional demand near the skew support. The shell model confirms the global trend but reveals local stress concentrations around bearing zones that are not represented by the grillage. A local solid submodel is recommended for bearing-seat reinforcement, while the grillage remains adequate for routine load distribution and design-envelope generation.

Learning points

1. Model complexity should follow the engineering question.

2. Global equilibrium is necessary but does not prove local accuracy.

3. Skew and support regions require special attention.

4. Different models should be compared using compatible response quantities.

5. Local refinement is often more efficient than making the entire model unnecessarily detailed.

11. Professional Practice, Governance and Emerging Technology

Bridge analysis is safety-critical. Engineers must maintain traceability between design requirements, analysis assumptions and decisions. Software output must never be accepted without independent checks. Proprietary software should be used in accordance with licensing, documented theory and recognised verification procedures.

Artificial intelligence and machine learning can assist with anomaly detection in monitoring data, surrogate modelling, mesh adaptation, damage identification and rapid parametric studies. However, data-driven models may fail outside their training range and may not preserve equilibrium, compatibility or physical constraints. AI-generated predictions should therefore be treated as supplementary evidence, subject to engineering validation and clear accountability.

Digital twins combine geometric information, analytical models, sensor data and inspection records. Their usefulness depends on data quality, sensor reliability, model updating procedures and cybersecurity. Automated model updating should not conceal uncertainty; confidence intervals and alternative hypotheses remain important for asset management.

Best practice includes:

• defining the question before selecting the model;
• using the simplest defensible model first;
• checking units, signs, mass and stiffness;
• documenting all assumptions;
• performing sensitivity and convergence studies;
• separating model verification from physical validation;
• retaining an auditable analysis record;
• obtaining independent review for critical decisions.

Challenges and Limitations

No structural model represents every aspect of a real bridge. Important limitations include uncertain material properties, construction tolerances, hidden damage, imperfect bearing behaviour, unknown soil conditions, simplified traffic models and incomplete monitoring data. Linear analysis may conceal redistribution and failure mechanisms; nonlinear analysis may introduce uncertain parameters and numerical instability. Dynamic predictions are sensitive to mass, stiffness, damping and boundary conditions, which are often poorly known.

The correct response to uncertainty is not automatically to build the largest possible model. It is to identify influential assumptions, quantify their effects where possible, adopt conservative but realistic bounds, and communicate residual uncertainty to decision-makers.

Chapter Summary

Bridge analysis begins with understanding load paths and selecting an idealisation suited to the engineering objective. Classical beam methods provide essential preliminary estimates and independent checks. Grillage models efficiently represent deck distribution, eccentric loading and torsion, while shell and solid finite-element models are needed for local stress fields and complex details.

Staged construction is essential when erection sequence, prestressing, composite action or time-dependent behaviour affects the final state. Geometric nonlinearity is relevant to second-order effects, cables and instability; material nonlinearity is required for cracking, yielding, crushing and ductility assessment. Soil–structure interaction can alter both static and dynamic response, and dynamic analysis requires credible mass, stiffness, damping and excitation models.

Reliable results depend on verification, validation, convergence studies, sensitivity analysis and careful documentation. Computational sophistication cannot compensate for incorrect boundary conditions, double-counted stiffness, unrealistic load application or unexamined assumptions. Professional bridge analysis is therefore an iterative process combining mechanics, numerical modelling, field knowledge and engineering judgement.

Key Terms

Structural idealisation: Representation of a real bridge by an analytical model.

Load path: Route by which actions are transferred through structural components to the ground.

Grillage model: Network of interconnected longitudinal and transverse beam elements representing a deck.

Finite element: Discrete computational region used to approximate structural behaviour.

Shell element: Element representing membrane and bending behaviour of a thin surface.

Solid element: Three-dimensional element used to represent continuum stress and strain.

Stiffness matrix: Matrix relating nodal displacements to corresponding forces in a linear model.

Influence line: Variation of a response quantity as a unit load moves through a structure.

Staged construction: Analysis in which elements, loads and material states are activated sequentially.

Geometric nonlinearity: Change in response caused by deformation-dependent geometry.

Material nonlinearity: Nonlinear response caused by constitutive behaviour such as yielding or cracking.

P–Δ effect: Additional moment caused by axial load acting through lateral displacement.

Soil–structure interaction: Mutual influence of foundation movement and surrounding soil response.

Modal analysis: Evaluation of natural frequencies and mode shapes.

Damping: Mechanism by which vibration energy is dissipated.

Verification: Demonstration that the mathematical and numerical solution has been implemented correctly.

Validation: Assessment of whether a model represents the physical bridge adequately.

Mesh convergence: Reduction of response changes as the finite-element mesh is refined.

Sensitivity analysis: Study of response changes caused by variations in assumptions or parameters.

Digital twin: Integrated digital representation updated using structural, inspection and monitoring information.

Review Questions

Short Answer Questions

1. What is meant by a bridge load path?

2. State two assumptions of Euler–Bernoulli beam theory.

3. Why are grillage models useful for bridge decks?

4. Distinguish verification from validation.

5. What is the purpose of a mesh-convergence study?

Long Answer Questions

1. Explain the selection, formulation and limitations of beam, grillage, shell and solid finite-element models for bridge analysis.

2. Discuss how construction sequence and time-dependent material behaviour influence the final response of a prestressed or composite bridge.

3. Explain geometric and material nonlinearity, including situations in which each should be considered.

4. Describe the principal methods of representing soil–structure interaction and discuss their limitations.

5. Develop a quality-assurance procedure for a nonlinear finite-element analysis of a bridge.

Analytical / Application Questions

1. A continuous bridge model gives support reactions that do not sum to the applied vertical load. Identify possible modelling causes and propose a systematic diagnostic procedure.

2. A skew bridge exhibits substantially different inner- and outer-girder reactions under an eccentric wheel load. Explain the structural mechanisms involved and recommend an appropriate modelling strategy.

3. A bridge monitoring programme reports natural frequencies significantly lower than those predicted by the analytical model. Propose a structured investigation considering mass, stiffness, boundary conditions, damage and soil–structure interaction.

Multiple Choice Questions with Answers

Question 1

Which model is generally most suitable for efficient global analysis of a bridge deck while representing transverse load distribution?

A. A grillage model

B. A single truss element

C. A one-dimensional cable model

D. A purely rigid-body model

Correct Answer: A. A grillage model

Question 2

For a simply supported beam carrying a uniform load w over span L, the maximum sagging moment is:

A. wL²/8

B. wL²/2

C. wL/2

D. wL²/12

Correct Answer: A. wL²/8

Question 3

Which statement best distinguishes validation from verification?

A. Verification checks numerical implementation, whereas validation assesses physical representativeness

B. Verification concerns field testing, whereas validation concerns mesh density only

C. Verification is used only for nonlinear models, whereas validation is used only for linear models

D. They are identical terms for checking software output

Correct Answer: A. Verification checks numerical implementation, whereas validation assesses physical representativeness

Question 4

The principal purpose of staged-construction analysis is to:

A. Represent sequential activation, loading and changing material states

B. Eliminate the need to define boundary conditions

C. Guarantee elastic behaviour at all load levels

D. Replace all dynamic analyses

Correct Answer: A. Represent sequential activation, loading and changing material states

Question 5

Which equation represents the undamped linear free-vibration problem?

A. [M]{ü} + [K]{u} = {0}

B. [K]{u} = {F} only

C. {F} = [C]{u̇} only

D. {r} = {F_ext} + {F_int}

Correct Answer: A. [M]{ü} + [K]{u} = {0}

Question 6

A simple linear soil spring is commonly represented by:

A. F = ku

B. F = k/u

C. F = u/k²

D. F = k + u

Correct Answer: A. F = ku

Question 7

Which phenomenon is directly associated with geometric nonlinearity?

A. P–Δ effects

B. Concrete shrinkage only

C. Steel hardening only

D. Unit conversion error

Correct Answer: A. P–Δ effects

Question 8

Why should a concentrated load in a solid finite-element model usually be distributed over a realistic area?

A. To avoid artificial local stress singularities

B. To remove the need for a stiffness matrix

C. To make the structure statically determinate

D. To ensure that all elements remain elastic

Correct Answer: A. To avoid artificial local stress singularities

Question 9

In a nonlinear analysis, the residual force vector is generally the difference between:

A. External and internal force vectors

B. Mass and damping matrices

C. Moment and curvature only

D. Mesh size and element thickness

Correct Answer: A. External and internal force vectors

Question 10

Which is a sound professional practice in bridge modelling?

A. Compare computer results with independent checks and document assumptions

B. Accept software output without examining reactions

C. Use the finest possible mesh regardless of the objective

D. Treat convergence as proof that the model is physically correct

Correct Answer: A. Compare computer results with independent checks and document assumptions