Structural Dynamics and Seismic Response

Introduction

Structural dynamics examines how structures respond to loads that vary with time. Unlike static analysis, in which loads are assumed to act sufficiently slowly for inertia to be neglected, dynamic analysis recognizes that mass, stiffness, damping, and the characteristics of excitation jointly determine structural response. Earthquakes are among the most important dynamic actions because they impose rapidly changing ground motion, may excite several vibration modes, and can produce nonlinear deformation, strength degradation, and instability.

For postgraduate civil engineering practice, structural dynamics is not merely the study of vibration periods. It provides the analytical basis for estimating displacements, accelerations, internal forces, energy dissipation, damage potential, and collapse risk. It also supports the interpretation of seismic design codes, numerical modelling, performance-based design, structural health monitoring, and earthquake-resistant detailing.

Learning Objectives

On completing this chapter, the reader should be able to:

1. formulate equations of motion for single- and multi-degree-of-freedom systems;
2. explain natural frequency, mode shape, damping, resonance, and dynamic amplification;
3. derive and apply modal superposition for linear structures;
4. distinguish force excitation from base excitation and formulate seismic response equations;
5. use response spectra and modal combination procedures appropriately;
6. interpret nonlinear seismic response, ductility, hysteresis, and energy dissipation;
7. evaluate modelling assumptions, numerical methods, and limitations;
8. connect analytical results with seismic design, detailing, and performance objectives.

1. Fundamental Concepts of Structural Dynamics

A structural system possesses three fundamental dynamic properties:

• mass, which resists acceleration;
• stiffness, which resists deformation; and
• damping, which dissipates mechanical energy.

A dynamic action is described by its magnitude, direction, duration, frequency content, phase, and spatial distribution. Two earthquakes with similar peak ground acceleration may cause very different structural responses because their frequency contents and durations differ.

A degree of freedom (DOF) is an independent displacement coordinate required to describe the configuration of a structural model. A single-storey idealization may have one lateral DOF, whereas a multistorey building may be represented by one lateral displacement per floor. A continuous structure technically has infinitely many DOFs, but finite-element idealization converts it into a finite set.

For a linear system, the principal equation is

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u(t) is the displacement vector, p(t) is the applied load vector, and overdots denote derivatives with respect to time. In consistent units, M ü, C u̇, K u, and p all have units of force.

The equation assumes small displacements, linear elastic material behaviour, constant system properties, and adequately defined mass, damping, and stiffness matrices. If stiffness or damping changes with response history, the equation must be modified or interpreted incrementally.

Free vibration of an undamped single-degree-of-freedom system

For free vibration, p(t) = 0 and damping is neglected:

Assuming u(t) = U sin(ωt + φ), the circular natural frequency is

where m is mass, k is stiffness, and ω_n is in radians per second. The natural period is

and the cyclic frequency is f_n = ω_n/(2π), measured in hertz. The expression is valid for a linear undamped SDOF system. Increasing mass lengthens the period; increasing stiffness shortens it.

For a damped system, the damping ratio is

where c is viscous damping coefficient. For underdamped systems, 0 < ξ < 1, the damped frequency is

The damping ratio is dimensionless. It is not itself a force or a percentage of stiffness; it compares actual viscous damping with critical damping.

Forced vibration and dynamic amplification

For harmonic force p(t) = p₀ sin(ωt), the steady-state displacement amplitude is

where r = ω/ω_n is the frequency ratio. The dynamic amplification factor is

At low excitation frequency, the response approaches the static displacement p₀/k. Near resonance, r is close to 1 and response can become large, although damping limits the amplitude. At high frequency, displacement response decreases, but acceleration response may remain important.

The formula applies to steady-state harmonic loading and does not include transient response caused by initial conditions or sudden application of the load. Real earthquakes are non-periodic, so time-history or response-spectrum methods are generally more appropriate.

2. Modal Analysis of Multi-Degree-of-Freedom Systems

For undamped free vibration of a linear MDOF system,

Let u(t) = φ sin(ωt), where φ is a mode shape. Substitution gives

A non-trivial mode shape exists only when

This eigenvalue equation produces n natural frequencies and corresponding mode shapes for an n-DOF system. The mode shapes are relative patterns of deformation; their absolute scaling is arbitrary unless a normalization is specified.

For classical systems, distinct modes satisfy orthogonality relations:

The superscript T denotes transpose. Modal mass and modal stiffness are

If the displacement is expanded as u(t) = Σ φ_i q_i(t), q_i(t) is the generalized modal coordinate. With proportional damping, the equations decouple into independent modal equations:

where p_i(t) = φ_iᵀ p(t) is the modal load and ξ_i is modal damping ratio.

Modal participation in base excitation

For uniform horizontal ground acceleration ü_g(t), the relative displacement vector u satisfies

where r is the influence vector, usually containing ones for DOFs moving with the ground in the considered direction. The modal equation becomes

with participation factor

The effective modal mass is

and the effective mass ratio is M_i^*/M_total. A sufficient number of modes should be included so that the participating mass in each important direction meets the applicable design-code requirement. The exact threshold varies by code and structural system; it should not be assumed universally.

3. Seismic Response and Response Spectra

An earthquake is usually represented by an acceleration time history of the ground. The total absolute displacement of a mass is the sum of ground displacement and relative displacement. For a linear SDOF oscillator, the relative-motion equation is

where u_r is relative displacement and ü_g is ground acceleration. Absolute acceleration is

This distinction is essential. Structural deformation is generally related to relative displacement, whereas inertial force is related to absolute acceleration.

A response spectrum gives the maximum response of a family of SDOF oscillators with specified damping subjected to the same ground-motion record. Common spectra include spectral displacement S_d, spectral velocity S_v, and spectral acceleration S_a. For a linear oscillator,

when S_a is expressed as pseudo-spectral acceleration. Similarly,

for pseudo-spectral velocity. These relationships are exact definitions for pseudo-spectral quantities, not necessarily equal to the maximum absolute acceleration or velocity.

The design lateral force associated with a mode is often expressed as

where V_i is modal base-shear contribution, M_i^ is effective modal mass, and S_{a,i} is spectral acceleration at the modal period and damping. Unit consistency requires S_a to be acceleration and M_i^ to be mass, producing force.

Modal combination

Peak modal responses do not generally occur at the same time. Therefore, direct algebraic addition is usually overly conservative, while simple absolute addition may be inappropriate. Two common combination methods are:

Square-root-of-sum-of-squares (SRSS):

This is suitable when modal frequencies are well separated and modal correlation is small.

Complete quadratic combination (CQC):

where ρ_ij is the modal correlation coefficient. CQC is preferable when modes are closely spaced or when modal correlation is significant. The precise expression for ρ_ij depends on frequencies and damping assumptions and should follow the adopted standard or validated software implementation.

Modal combination is a statistical or approximate procedure. It does not reconstruct the exact time at which all peak modal responses occur. Spatial components of earthquake excitation may also require a separate orthogonal-component combination rule.

4. Worked Example: Fundamental Properties of an SDOF System

Consider a lumped mass m = 200,000 kg and lateral stiffness k = 80,000,000 N/m. Assume viscous damping ratio ξ = 0.05.

The circular natural frequency is

The period is

The damping coefficient is

Thus,

Suppose the harmonic excitation frequency is ω = 18 rad/s. Then r = 18/20 = 0.90. The dynamic amplification factor is

Therefore, the steady-state displacement is approximately 4.76 times the static displacement produced by the same force amplitude. This large factor illustrates why response near the natural frequency is critical, even when the applied load is not large in static terms.

5. Damping, Energy, and Hysteresis

Damping represents mechanisms that remove energy from a vibrating system. Sources include material microcracking, friction, joint slip, radiation into the ground, aerodynamic effects, and interaction with nonstructural components. In structural models, damping is often idealized as viscous damping because it is mathematically convenient.

Rayleigh damping approximates the damping matrix as

where α and β are coefficients selected to produce target damping ratios at selected frequencies. For mode i,

Rayleigh damping should be calibrated carefully. Excessive damping at high frequencies can suppress physically important response, while poorly selected coefficients may produce unrealistic damping in the frequency range governing the structure.

In nonlinear analysis, energy dissipation is commonly represented by hysteresis. The area enclosed by a force–displacement cycle is energy dissipated during that cycle:

where F is restoring force and u is displacement. Hysteretic behaviour may include yielding, pinching, stiffness degradation, strength deterioration, and cyclic accumulation of damage. A model that matches the initial stiffness but has unrealistic hysteretic energy may produce misleading seismic demands.

6. Nonlinear Seismic Response

Linear analysis is useful for elastic response estimation and for many ordinary design procedures, but strong earthquakes can cause yielding. The nonlinear equation may be written as

where f_s is the nonlinear restoring-force vector and depends on current deformation and loading history.

Important nonlinear concepts include:

• yield displacement: deformation at first significant yielding;
• ductility: ratio of maximum displacement to yield displacement;
• overstrength: strength above the nominal design strength;
• post-yield stiffness: tangent stiffness after yielding;
• cyclic degradation: reduction in stiffness or strength under repeated cycles;
• P–Δ effect: additional moment caused by axial load acting through lateral displacement;
• residual drift: permanent lateral deformation after shaking.

A displacement ductility measure is

where Δ_max is maximum displacement and Δ_y is yield displacement. Ductility is not synonymous with safety. A structure may possess high ductility but still suffer unacceptable damage, excessive residual drift, connection fracture, or instability.

Nonlinear static pushover analysis estimates capacity through a gradually increasing lateral load pattern. It can clarify likely plastic mechanisms and approximate displacement capacity, but it does not reproduce the changing inertia forces and higher-mode effects of an earthquake. Nonlinear response-history analysis directly integrates the equations of motion using ground-motion records and is more informative, but it requires reliable material, connection, damping, and degradation models.

7. Numerical Time-History Analysis

For a time-discretized linear system, common integration methods include Newmark methods, Wilson-type methods, and state-space algorithms. In the Newmark family,

where β and γ are integration parameters. The average-acceleration choice β = 1/4 and γ = 1/2 is unconditionally stable for linear systems, subject to implementation assumptions, but numerical stability does not guarantee accuracy. The time step must be sufficiently small to resolve the highest important frequency and the sharpest features of the input record.

Good practice includes checking equilibrium residuals, energy balance, time-step sensitivity, convergence tolerances, modal participation, and the effect of record scaling. Numerical results should be reviewed physically: unreasonable drifts, abrupt force jumps, excessive damping energy, or nonphysical collapse may indicate modelling or integration errors rather than unusual structural behaviour.

8. Seismic Design and Performance-Based Assessment

Traditional force-based design reduces elastic earthquake demand using response modification factors and then checks strength and detailing. Performance-based earthquake engineering instead relates hazard intensity to engineering demand parameters and damage states. Typical performance objectives may include:

These labels are not universal substitutes for code provisions; their precise definitions depend on the adopted standard and project requirements.

Important engineering demand parameters include peak interstorey drift, floor acceleration, residual drift, plastic rotation, connection strain, foundation demand, and nonstructural component response. A robust assessment considers uncertainty in hazard, structural properties, record selection, modelling, and damage-state thresholds.

Illustrative applied case: seismic assessment of a reinforced-concrete frame

Background. An existing eight-storey reinforced-concrete moment frame was designed under an older code. A preliminary review identified irregular infill walls, uncertain reinforcement detailing, and a soft first storey.

Problem. A code-spectrum analysis indicated acceptable nominal base shear but large first-storey drift. A force-only conclusion could therefore be misleading because the weak storey might form a concentration of plastic demand.

Analysis. Engineers developed a three-dimensional model including cracked-section stiffness, diaphragm action, gravity-load P–Δ effects, and nonlinear beam-column hinges. Seven pairs of spectrum-compatible ground motions were selected. The model was checked through modal periods, mass participation, gravity equilibrium, and sensitivity to damping assumptions.

Approach. Nonlinear response-history analyses were performed. Results were evaluated using maximum interstorey drift, hinge rotations, column axial-force interaction, residual drift, and foundation demand. A supplemental pushover analysis was used to interpret the likely mechanism, not as a replacement for the dynamic analyses.

Outcome. The study showed that global strength was adequate, but first-storey drift and column rotation demand exceeded the selected life-safety limits. The retrofit scheme introduced reinforced-concrete jacketing of critical columns, strengthened selected beam-column joints, and added a carefully located structural wall to redistribute lateral demand. A second analysis showed reduced drift concentration and improved mechanism formation, although foundation demands increased and had to be checked.

Learning points. Seismic safety depends on deformation capacity, hierarchy of strength, regularity, ductile detailing, and credible modelling—not on base shear alone. Retrofit measures can shift demand to foundations, diaphragms, collectors, and nonstructural components; the complete load path must therefore be reviewed.

9. Modelling, Uncertainty, and Professional Responsibility

Dynamic analysis is sensitive to assumptions that are often uncertain. Major sources include:

1. mass modelling: omitted partitions, façade, services, and live-load participation;
2. stiffness modelling: cracked concrete, connection flexibility, shear deformation, and soil–structure interaction;
3. damping: uncertain physical mechanisms and potentially excessive numerical damping;
4. ground motions: record-to-record variability, duration, near-fault pulses, and vertical excitation;
5. nonlinear properties: cyclic degradation, fracture, buckling, and low-cycle fatigue;
6. boundary conditions: foundation flexibility, diaphragm constraints, and adjacent-building interaction.

Model sophistication should be proportional to the decision being made. A complex model with poorly supported assumptions is not automatically more reliable than a simpler validated model. Engineers should document assumptions, version-controlled input data, selected records, scaling procedures, convergence results, and acceptance criteria.

Seismic design also has an ethical dimension. Underestimating demand, concealing uncertainty, or presenting a single deterministic result as certainty can endanger occupants and obstruct informed decisions. Regulatory compliance is a minimum obligation; professional judgement must address unusual geometry, vulnerable details, construction quality, existing damage, and changes in use.

10. Contemporary Developments

Current practice increasingly combines high-fidelity finite-element modelling, automated record selection, cloud-based computation, structural health monitoring, and probabilistic risk assessment. Machine learning can assist with damage classification, surrogate modelling, anomaly detection, and rapid generation of approximate response surfaces. However, AI outputs require engineering validation because training data may not represent rare earthquakes, unusual structural systems, or changing material behaviour.

Digital twins and sensor networks can compare measured modal properties with analytical predictions and identify stiffness loss after an earthquake. Base isolation and supplemental damping devices modify the dynamic system by lengthening periods or increasing energy dissipation. Their design must consider displacement capacity, temperature and ageing effects, device variability, maintenance, and failure modes.

Research directions include physics-informed machine learning, real-time hybrid simulation, resilience-based design, embodied-carbon assessment of retrofit alternatives, and integrated modelling of structural and nonstructural damage. These developments do not remove the need for fundamental dynamic understanding; they increase the importance of it.

Best-Practice Checklist

Before accepting a seismic dynamic analysis, confirm that:

• the mass and stiffness idealizations represent the intended load path;
• units and coordinate conventions are consistent;
• periods and mode shapes are physically reasonable;
• sufficient modal mass is included;
• damping assumptions are documented and justified;
• P–Δ effects and torsion are considered where relevant;
• ground-motion records represent the hazard and site conditions;
• nonlinear material and connection models are calibrated or justified;
• convergence, energy balance, and sensitivity checks are completed;
• demand measures are compared with appropriate performance criteria;
• retrofit or design changes are traced through the entire structural system.

Chapter Summary

Structural dynamics describes the response of structures subjected to time-varying actions through the interaction of mass, stiffness, damping, and excitation. The SDOF equation establishes natural frequency, period, damping ratio, resonance, and dynamic amplification. MDOF systems are analysed through eigenvalues, mode shapes, modal participation, and modal superposition. For earthquakes, base excitation produces a relative-motion equation driven by ground acceleration, and response spectra provide an efficient representation of maximum oscillator response.

SRSS and CQC methods combine modal maxima approximately, while nonlinear response-history analysis captures yielding, hysteresis, degradation, P–Δ effects, and residual drift when credible models and records are available. Seismic assessment must consider uncertainty, record variability, modelling assumptions, performance objectives, and the complete load path. Modern computational and AI tools can improve efficiency, but they cannot replace engineering verification, transparent assumptions, and professional responsibility.

Key Terms

Mass matrix: Matrix representing distribution of inertial mass in a structural model.

Stiffness matrix: Matrix relating structural displacement to restoring force in a linear model.

Damping ratio: Ratio of actual viscous damping to critical damping.

Natural frequency: Frequency of free vibration of a system.

Natural period: Time required for one cycle of free vibration.

Resonance: Large response occurring when excitation frequency approaches a natural frequency.

Degree of freedom: Independent coordinate required to describe structural configuration.

Mode shape: Relative deformation pattern associated with a natural frequency.

Modal participation factor: Coefficient describing the contribution of a mode to a specified excitation.

Effective modal mass: Equivalent mass participating in a particular vibration mode.

Base excitation: Dynamic excitation imposed through support or ground motion.

Response spectrum: Maximum response of SDOF oscillators over a range of periods and damping ratios.

Hysteresis: Cyclic force–deformation behaviour involving energy dissipation.

Ductility: Capacity to undergo inelastic deformation while maintaining resistance.

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

Pushover analysis: Nonlinear static analysis using incrementally increasing lateral load.

Response-history analysis: Analysis using a time-varying excitation record.

Interstorey drift: Relative lateral displacement between adjacent floors divided, commonly, by storey height.

Performance-based design: Design framework linking hazard, response, damage, and functional objectives.

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

Review Questions

Short Answer Questions

1. What physical roles are represented by mass, stiffness, and damping in a dynamic structural model?
2. Define natural frequency and natural period, and state how each changes with stiffness.
3. Why are mode shapes important in the analysis of multistorey buildings?
4. Distinguish relative displacement from absolute displacement during earthquake excitation.
5. What is the purpose of a response spectrum?

Long Answer Questions

1. Derive the natural frequency and period of an undamped single-degree-of-freedom system and discuss the assumptions involved.
2. Explain modal superposition for a linear MDOF system, including modal mass and participation factor.
3. Compare SRSS and CQC modal-combination methods and identify situations in which each is appropriate.
4. Discuss the physical meaning and modelling limitations of viscous and hysteretic damping.
5. Explain the principal differences between linear response-spectrum analysis, pushover analysis, and nonlinear response-history analysis.

Analytical / Application Questions

1. A building floor is idealized as a 150,000 kg mass supported by a lateral stiffness of 60,000,000 N/m. Calculate its circular natural frequency and natural period. Comment on the effect of doubling the stiffness.
2. A structural model shows adequate total base shear but excessive drift concentrated in the first storey. Identify likely mechanisms, propose additional analyses, and recommend possible retrofit strategies.
3. Design an analysis plan for an irregular existing building in a high-seismic region. Justify the mass model, damping model, ground-motion selection, nonlinear components, performance measures, and validation checks.

Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system, the circular natural frequency is given by ω_n = √(k/m). If stiffness increases while mass remains constant, what happens to the natural period?

A. It increases

B. It decreases

C. It remains unchanged

D. It becomes zero for all stiffness values

Correct Answer: B. It decreases

Question 2

Which equation represents relative-motion response of an SDOF structure under horizontal ground acceleration?

A. mü + ku = 0

B. mü + cū + ku = p(t)

C. mü_r + cū_r + ku_r = −mü_g(t)

D. mü_g + cū_g + ku_g = mr

Correct Answer: C. mü_r + cū_r + ku_r = −mü_g(t)

Question 3

The damping ratio is defined as:

A. c/(2√(km))

B. 2c√(km)

C. k/(2√(cm))

D. m/(2√(kc))

Correct Answer: A. c/(2√(km))

Question 4

The eigenvalue equation for undamped free vibration of an MDOF system is:

A. det(K + ω²M) = 0

B. det(K − ω²M) = 0

C. det(C − ωM) = 0

D. det(M − K) = 1

Correct Answer: B. det(K − ω²M) = 0

Question 5

Which modal combination method is generally more suitable when modal frequencies are closely spaced?

A. Direct addition only

B. CQC

C. Static equilibrium only

D. Ignoring higher modes

Correct Answer: B. CQC

Question 6

For a harmonic force, resonance is associated most closely with which condition?

A. The excitation frequency is much lower than the natural frequency

B. The excitation frequency is approximately equal to the natural frequency

C. The damping ratio is exactly zero only in every practical structure

D. The mass is equal to the stiffness numerically

Correct Answer: B. The excitation frequency is approximately equal to the natural frequency

Question 7

The area enclosed by a force–displacement hysteresis loop represents:

A. Elastic stiffness

B. Natural period

C. Energy dissipated during the cycle

D. Modal participation factor

Correct Answer: C. Energy dissipated during the cycle

Question 8

Which analysis directly uses time-varying earthquake records and can represent nonlinear cyclic response?

A. Nonlinear response-history analysis

B. Elastic static analysis only

C. Eigenvalue analysis only

D. Uniform dead-load analysis

Correct Answer: A. Nonlinear response-history analysis

Question 9

The displacement ductility ratio is commonly defined as:

A. Δ_y/Δ_max

B. Δ_max/Δ_y

C. Δ_max + Δ_y

D. Δ_max − Δ_y

Correct Answer: B. Δ_max/Δ_y

Question 10

A major limitation of a simple pushover analysis is that it:

A. Cannot estimate any deformation

B. Cannot represent static gravity loads

C. Does not fully reproduce changing inertia forces and higher-mode earthquake effects

D. Always requires no material model

Correct Answer: C. Does not fully reproduce changing inertia forces and higher-mode earthquake effects