Structural Dynamics, Wind and Earthquake Response

Introduction and Context

Structural analysis is often introduced through static equilibrium, but buildings are rarely subjected only to slowly applied, unchanging loads. Wind gusts, machinery, footfall, traffic, blast, and earthquakes produce forces that vary with time. When the loading frequency approaches a structure’s natural frequency, dynamic amplification may become substantial even when the applied force is modest.

Structural dynamics provides the concepts and tools needed to predict these effects. For construction technologists, the subject connects structural theory with practical decisions about stiffness, mass distribution, damping, connection detailing, serviceability, wind-induced motion, and seismic safety. This chapter develops idealized single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) models, then applies them to simplified wind and earthquake response of buildings.

Learning Objectives

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

1. Idealize a structural system as an SDOF or MDOF model.
2. Derive and use the equation of motion for free and forced vibration.
3. Calculate natural frequency, period, damping ratio, and resonance response.
4. Explain modal analysis and its use for building structures.
5. Interpret response spectra and estimate seismic design forces.
6. Distinguish static, dynamic, gust, and resonant components of wind response.
7. Identify assumptions, limitations, and professional responsibilities in simplified dynamic analysis.

1. Dynamic Idealization of Structures

A real building has continuously distributed mass and stiffness. Exact analysis therefore involves differential equations in space and time. In engineering practice, the structure is discretized into floors, beams, columns, walls, and finite elements. A useful idealization retains the dominant physical properties:

• Mass resists acceleration.
• Stiffness provides restoring force when the structure is displaced.
• Damping dissipates energy.
• Excitation is the externally applied or support motion that produces response.

An SDOF model has one independent displacement, usually the horizontal displacement of a floor mass. An MDOF model has several independent coordinates, such as the lateral displacement of each floor. The choice of model must reflect the purpose of analysis. An SDOF model may be adequate for preliminary estimates or a regular low-rise structure, whereas an MDOF model is normally required to evaluate higher-mode effects, torsion, irregularity, and distribution of forces.

1.1 The SDOF Equation of Motion

For a mass m connected to a linear spring of stiffness k and viscous damper c, subjected to force p(t), Newton’s second law gives:

Here x(t) is displacement relative to the ground, ẋ(t) is velocity, ẍ(t) is acceleration, m has units kg, c has units N·s/m, k has units N/m, and p(t) has units N. The equation assumes small deformation, linear elastic stiffness, lumped mass, and viscous damping.

For free vibration, p(t) = 0. If damping is initially neglected, the equation becomes:

The undamped circular natural frequency is:

where ωₙ is in rad/s. The natural frequency in cycles per second and the period are:

A larger stiffness increases frequency; a larger mass decreases it. Dimensional consistency follows because k/m has units (N/m)/kg = 1/s².

1.2 Damping

The critical damping coefficient is the value separating oscillatory and non-oscillatory free response:

The damping ratio is:

For most building structures, damping is commonly represented by a small equivalent viscous damping ratio, often approximately 2–5% for ordinary elastic response, although the appropriate value depends on structural system, nonstructural components, amplitude, and code provisions. It must not be assumed indiscriminately.

For an underdamped system, 0 < ζ < 1, the damped frequency is:

The damped free-vibration response is:

A and B are determined from initial displacement and velocity. The exponential term demonstrates that damping reduces amplitude with time.

2. Forced Vibration and Resonance

Consider harmonic loading p(t) = P₀ sin(ωt), where P₀ is force amplitude and ω is forcing frequency. The steady-state displacement amplitude is:

where:

is the frequency ratio. The dynamic amplification factor is:

The static displacement P₀/k is amplified by D. At low frequency, r → 0 and D → 1. Near r = 1, response is large. For an undamped model, the theoretical resonant amplitude is unbounded; real structures always have damping, nonlinear behavior, or changing frequency that limits the response.

For small damping, the maximum displacement occurs close to, but not exactly at, r = 1. At r = 1, the amplification is approximately 1/(2ζ). Thus a system with ζ = 0.05 can have approximately ten times its static harmonic response at the resonant frequency. This is a central reason why vibration control and frequency separation matter.

Worked Example 1: Natural Frequency and Harmonic Amplification

A floor system is idealized as an SDOF oscillator with m = 180,000 kg and k = 72,000,000 N/m. Its damping ratio is ζ = 0.05. Determine the natural period and the amplification factor for a harmonic force whose frequency is 90% of the natural frequency.

First,

Therefore,

For r = 0.90,

The two terms are 0.19² = 0.0361 and 0.09² = 0.0081. Hence:

The harmonic displacement is therefore approximately 4.76 times the corresponding static displacement P₀/k. This result applies to steady harmonic forcing and a linear SDOF model; it is not a universal factor for arbitrary wind or earthquake loading.

3. Multi-Degree-of-Freedom Systems

For an MDOF structure with n coordinates, the linear equation of motion is:

[M], [C], and [K] are the mass, damping, and stiffness matrices; {u} is the displacement vector; and {p(t)} is the applied-force vector. The matrices must be assembled consistently from the structural model. Units must be compatible, for example kg, N·s/m, and N/m in an SI formulation.

For undamped free vibration, assume {u} = {ϕ} sin(ωt). Substitution gives:

A nonzero mode shape {ϕ} exists only when:

This eigenvalue equation produces n natural frequencies and corresponding mode shapes. The lowest frequency is the fundamental frequency. Higher modes generally contain more changes in displacement direction and are important for short-period response, local force distribution, and irregular buildings.

For classical damping, modal transformation expresses displacement as:

where [ϕ] contains mode shapes and {q(t)} contains modal coordinates. With suitable normalization, the equations become approximately uncoupled:

The subscript r identifies mode r. Modal analysis is efficient because a limited number of modes often captures most of the effective mass in a particular direction. Code requirements determine acceptable modal mass participation and combination rules.

Shear-Building Idealization

A simplified n-storey shear building assumes each floor diaphragm translates laterally without in-plane deformation, masses are concentrated at floors, columns and walls provide storey stiffness, and vertical deformation and torsion are neglected. This model is useful for understanding mode shapes and preliminary seismic calculations, but it may be inadequate where diaphragm flexibility, torsion, plan irregularity, or significant axial-flexural interaction exists.

4. Base Excitation and Earthquake Response

Earthquake motion is usually represented by ground acceleration ẍ_g(t). If u(t) is the mass displacement relative to the ground, the SDOF equation is:

The effective earthquake force is therefore related to mass times ground acceleration, but the actual response depends on period, damping, and the complete acceleration time history. A static force equal to mass times peak ground acceleration does not generally predict peak structural force because maximum displacement, velocity, and acceleration need not occur at the same time.

For an MDOF building under uniform horizontal ground motion:

where {r} is the influence vector, commonly a vector of ones for a translational direction. Modal participation for mode r is characterized by:

The effective modal mass is:

These expressions depend on the selected normalization of the mode shape, although the physical modal contribution is invariant when used consistently.

5. Response Spectra

A response spectrum summarizes the maximum response of a family of SDOF oscillators subjected to one ground-motion record or a prescribed design spectrum. For each period T and damping ratio ζ, the spectral acceleration Sₐ(T, ζ), spectral velocity Sᵥ(T, ζ), or spectral displacement S_d(T, ζ) is recorded.

For linear SDOF response, pseudo-spectral quantities are related by:

These are pseudo-spectral relationships when Sₐ is based on ωₙ² times maximum relative displacement; they should not be confused automatically with maximum absolute acceleration. For design, the applicable building code defines the spectrum, damping modification, site class, importance category, and combination rules.

A response-spectrum procedure generally follows these steps:

1. Establish mass, stiffness, and damping assumptions.
2. Calculate periods and mode shapes.
3. Obtain spectral ordinates at each modal period.
4. Calculate modal participation and modal responses.
5. Combine modal results using an approved rule such as SRSS or CQC.
6. Check total base shear, drift, torsion, force distribution, and code minimums.

SRSS is commonly written for well-separated modes as:

where Rᵣ is the response from mode r. Closely spaced modes require a correlation-based method such as CQC, because their modal maxima are not statistically independent. The exact combination coefficients are code- and spectrum-dependent.

Equivalent Lateral Force Method

For regular, low- to medium-rise buildings within the limits of a governing seismic code, a static lateral-force procedure may be permitted. In generic form, the design base shear is:

where V is design base shear, C_s is a code-defined seismic response coefficient, and W is effective seismic weight. The coefficient incorporates hazard, soil, period, ductility or response modification, and importance factors according to the applicable code. It must not be substituted with a universal value.

A typical vertical distribution is expressed as:

where Fₓ is lateral force at level x, wₓ is seismic weight at level x, hₓ is height above the base, and k is a code-specified exponent related to period. The formula is a simplified distribution, not a replacement for dynamic analysis where irregularity or higher-mode effects are significant.

Drift and Ductility

Interstorey drift is the relative horizontal displacement between adjacent floors. A drift ratio may be written:

where uₓ and uₓ₋₁ are floor displacements and hₓ is storey height. Excessive drift can damage partitions, façades, elevators, utilities, and structural members. Seismic design therefore aims not only to resist force but also to provide ductility, stable hysteretic behavior, continuity, redundancy, and a reliable load path.

6. Wind Response of Buildings

Wind loading includes a mean component and fluctuating turbulence. For a building with tributary area A, a simplified pressure estimate is:

where q is velocity pressure in N/m², ρ is air density in kg/m³, and V is wind speed in m/s. This expression is dimensionally consistent because kg/m³ × m²/s² = kg/(m·s²) = N/m². Actual design pressure includes exposure, directionality, topographic, importance, gust, and pressure coefficients specified by the governing wind standard.

A simplified force estimate is:

where C_f is an appropriate force coefficient. This is useful for preliminary assessment, but it does not by itself capture dynamic amplification, local cladding pressure, vortex shedding, or across-wind effects.

Wind response may be classified as:

• Along-wind response: primarily caused by turbulence in the direction of mean wind.
• Across-wind response: caused by fluctuating lift, vortex shedding, and aerodynamic instability.
• Torsional response: caused by eccentricity between the wind resultant and the centre of rigidity or mass.
• Background response: quasi-static response from large-scale turbulence.
• Resonant response: dynamic response near a structural frequency.

For a preliminary SDOF model, fluctuating wind force can be represented as p(t), and the same dynamic equation applies. However, wind excitation is stochastic rather than a single sinusoid. Professional wind engineering commonly uses power spectral density functions, aerodynamic admittance, mode shape integration, and gust-response factors. Wind-tunnel testing or computational fluid dynamics may be needed for tall, flexible, unusual, or closely surrounded buildings.

7. Worked Seismic Application: Illustrative Three-Storey Model

Consider an illustrative three-storey regular building with equal floor masses. Suppose a preliminary model gives the fundamental period T₁ = 0.60 s and the design response spectrum at 5% damping gives Sₐ(T₁) = 0.80g, where g = 9.81 m/s². Let the effective modal mass in the first mode be 75% of the total seismic mass, M = 900,000 kg.

The effective modal mass is:

The approximate first-mode base shear contribution is:

This is a modal elastic-response estimate, not automatically the final design base shear. A complete design must include all required modes, modal combination, code scaling or minimum base shear provisions, accidental torsion, diaphragm behavior, P–Δ effects where relevant, and strength and drift checks. If the structure were highly irregular, the simplified first-mode estimate could be misleading even though its arithmetic is correct.

8. Applied Case: Wind and Seismic Screening of an Illustrative Office Building

Background

An illustrative eight-storey reinforced-concrete office building has a rectangular plan, regular floor-to-floor heights, reinforced-concrete moment frames, and a central stair-and-elevator core. The owner requests a preliminary study before selecting the lateral system.

Problem

The architect proposes a relatively flexible open perimeter. The project team must determine whether preliminary wind motion and seismic drift are likely to govern the selection of moment frames, shear walls, or a dual system.

Analysis

The engineer develops an MDOF shear-building model with floor masses at each level. An eigenvalue analysis identifies the first three periods and mode shapes. The first mode contributes most of the translational mass, but the second and third modes contribute materially to upper-floor forces. A response-spectrum analysis is therefore preferred over a single equivalent static force.

For wind screening, the team calculates mean velocity pressure using the applicable wind standard, then applies exposure and force coefficients. The fundamental period is compared with the dominant range of turbulent and vortex-shedding excitation. Because the building is flexible, a gust-response procedure is used rather than a purely static pressure estimate.

The analysis shows acceptable member strength but excessive predicted top-floor acceleration under a preliminary wind case and excessive interstorey drift under the seismic case. The team evaluates alternatives: increasing core wall thickness, adding perimeter moment frames, improving diaphragm continuity, and redistributing stiffness symmetrically to reduce torsion.

Approach and Solution

A dual system is selected: a stiff central core controls drift and acceleration, while perimeter frames provide strength, redundancy, and resistance to overturning. The model is updated with realistic cracked stiffness assumptions, diaphragm constraints, and mass from permanent and applicable imposed loads. Modal participation is checked in each principal direction, and torsional response is included.

Outcome

The revised scheme reduces fundamental period, interstorey drift, and wind-induced acceleration. The final design still requires code-compliant detailed analysis, connection design, ductile detailing, construction inspection, and review of nonstructural components.

Learning Points

1. Strength alone does not demonstrate satisfactory dynamic performance.
2. Wind comfort and seismic drift may govern different design decisions.
3. Stiffness distribution and torsional balance are as important as total strength.
4. Preliminary models should be refined as architectural and structural information develops.
5. The engineer must distinguish screening calculations from code-level design.

9. Limitations, Safety, and Professional Practice

Dynamic analysis is sensitive to modelling assumptions. Important sources of uncertainty include:

• cracked versus uncracked stiffness;
• actual mass, including partitions and services;
• damping and energy dissipation;
• soil–structure interaction;
• diaphragm flexibility;
• accidental eccentricity and torsion;
• nonstructural component participation;
• nonlinear behavior and strength degradation;
• uncertainty in future earthquake and wind records.

A computer-generated modal result is not automatically credible. Analysts should check units, mass participation, mode shapes, boundary conditions, mesh or discretization quality, convergence, reactions, equilibrium, and sensitivity to reasonable changes in assumptions. Results should be compared with hand estimates and code limits.

Safety-critical structural decisions must follow the legally applicable building, wind, and seismic codes and be reviewed by appropriately licensed professionals. Ethical practice requires transparent assumptions, conservative treatment of uncertainty where justified, documentation of limitations, and refusal to present preliminary estimates as guaranteed performance. AI-assisted modelling can help automate parameter studies, anomaly detection, and report checking, but it cannot replace engineering judgment, code interpretation, independent verification, or responsibility for public safety.

10. Contemporary Developments and Best Practices

Current practice increasingly combines conventional modal analysis with nonlinear time-history analysis, performance-based seismic design, digital twins, structural health monitoring, and operational wind measurements. Sensors can identify changes in modal frequency or damping that may indicate damage, although environmental effects must be separated from structural deterioration.

Best practice includes:

1. Define the performance objective before selecting the analysis method.
2. Use the simplest model that captures the governing behavior, then refine it systematically.
3. Maintain consistent units and mass conventions.
4. Verify natural periods against approximate empirical expectations and physical intuition.
5. Include relevant higher modes and torsion.
6. Treat damping as an assumption requiring justification.
7. Check both strength and deformation-related performance.
8. Coordinate structural response with façades, partitions, equipment, and services.
9. Use code-approved spectra, coefficients, and modal-combination procedures.
10. Record model version, assumptions, loads, software settings, and quality-control checks.

Chapter Summary

Structural dynamics models the interaction of mass, stiffness, damping, and time-varying excitation. The SDOF equation provides the foundation for understanding natural frequency, period, damping, forced vibration, dynamic amplification, and resonance. MDOF models represent floor-by-floor building behavior and are solved through eigenvalue and modal analysis. Earthquake response is commonly evaluated using ground acceleration and response spectra, while wind response includes mean, turbulent, resonant, across-wind, and torsional effects. Simplified equations are valuable for screening, but building-code procedures and professional judgment are essential for design. Reliable analysis requires realistic modelling, verification, attention to drift and acceleration, and recognition of uncertainty.

Key Terms

Natural frequency: Frequency at which a structure freely vibrates after disturbance.

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

Damping: Energy dissipation that reduces vibration amplitude.

Damping ratio: Actual damping divided by critical damping.

Resonance: Large response caused by excitation near a natural frequency.

Dynamic amplification factor: Ratio of dynamic response to corresponding static response.

SDOF system: Idealized system with one independent displacement coordinate.

MDOF system: System with multiple independent displacement coordinates.

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

Modal participation factor: Measure of how strongly a ground motion excites a mode.

Effective modal mass: Mass associated with the contribution of a mode to a specified excitation.

Response spectrum: Maximum response of SDOF oscillators plotted against period.

Base shear: Total horizontal design force transferred to a structure’s foundation.

Interstorey drift: Relative lateral displacement between adjacent floors.

Gust response: Structural response caused by fluctuating wind turbulence.

Vortex shedding: Alternating aerodynamic loading produced by flow separation around a body.

Ductility: Capacity to undergo inelastic deformation while sustaining resistance.

Torsional response: Rotation caused by eccentric or uneven lateral loading.

Influence vector: Vector describing support-motion participation of structural coordinates.

Performance-based design: Design approach linking hazards to explicit structural performance objectives.

Review Questions

Short Answer Questions

1. What physical properties are represented by mass, stiffness, and damping in a structural model?
2. Define natural frequency and natural period.
3. What does a damping ratio of 0.05 signify?
4. Why are higher modes relevant in an MDOF building?
5. Distinguish along-wind response from across-wind response.

Long Answer Questions

1. Derive the undamped SDOF natural frequency and explain the influence of mass and stiffness.
2. Explain resonance and dynamic amplification using the frequency-ratio formulation.
3. Describe the eigenvalue problem for an MDOF structure and explain the meaning of mode shapes.
4. Explain how a response-spectrum analysis is performed for a building.
5. Discuss the limitations of equivalent lateral-force procedures for irregular structures.

Analytical / Application Questions

1. An SDOF oscillator has m = 100,000 kg and k = 25,000,000 N/m. Calculate its circular natural frequency and period. If ζ = 0.05 and r = 1.0, estimate the harmonic amplification factor.
2. A four-storey building has significant torsional irregularity and flexible diaphragms. Recommend an analysis strategy and justify why a simple one-dimensional shear-building model may be inadequate.
3. A preliminary wind calculation predicts acceptable member strength but uncomfortable top-floor acceleration. Identify at least three structural or nonstructural measures that could reduce the dynamic response, and explain possible trade-offs.

Multiple Choice Questions with Answers

Question 1

Which expression gives the undamped circular natural frequency of an SDOF system?

A. ω_n = √(k/m)

B. ω_n = √(m/k)

C. ω_n = c/k

D. ω_n = 2πm/k

Correct Answer: A. ω_n = √(k/m)

Question 2

For an SDOF system with damping ratio ζ = 0.05 under harmonic loading at r = 1, the approximate dynamic amplification factor is:

A. 0.05

B. 2

C. 10

D. 20

Correct Answer: C. 10

Question 3

In the MDOF equation [M]{ü} + [C]{u̇} + [K]{u} = {p(t)}, [K] represents the:

A. Mass matrix

B. Damping matrix

C. Stiffness matrix

D. Influence vector

Correct Answer: C. Stiffness matrix

Question 4

A response spectrum primarily presents maximum response as a function of:

A. Building occupancy only

B. Natural period and damping

C. Concrete strength only

D. Foundation depth only

Correct Answer: B. Natural period and damping

Question 5

The simplified velocity-pressure relation for wind is:

A. q = ρV

B. q = ρV²

C. q = 1/2 ρV²

D. q = 2ρ/V²

Correct Answer: C. q = 1/2 ρV²

Question 6

In a base-excited SDOF earthquake model, the effective excitation term is:

A. +m ẍ_g

B. −m ẍ_g

C. +kx_g

D. −c x_g

Correct Answer: B. −m ẍ_g

Question 7

Which condition most strongly indicates that higher-mode effects should be considered?

A. A perfectly rigid single-storey shed

B. A regular structure with only static vertical loading

C. A tall building with significant upper-storey force response

D. A massless structure

Correct Answer: C. A tall building with significant upper-storey force response

Question 8

The interstorey drift ratio is calculated from:

A. The sum of adjacent floor displacements divided by storey height

B. The difference in adjacent floor displacements divided by storey height

C. The base shear divided by total mass

D. The period divided by damping ratio

Correct Answer: B. The difference in adjacent floor displacements divided by storey height

Question 9

For wind loading, vortex shedding primarily produces:

A. Alternating aerodynamic excitation

B. Constant vertical gravity load

C. Uniform material damping

D. Foundation settlement only

Correct Answer: A. Alternating aerodynamic excitation

Question 10

A major limitation of treating a complex irregular building as a one-dimensional shear model is that it may omit:

A. The unit of force

B. Torsion and diaphragm flexibility

C. The definition of mass

D. The concept of equilibrium

Correct Answer: B. Torsion and diaphragm flexibility