Fundamentals of Structural Dynamics

Introduction and Context

Structural dynamics concerns the response of structures to loads that vary with time. Earthquakes, machinery, traffic, wind gusts, impacts, explosions, and human activity can produce forces whose magnitude, direction, or point of application changes rapidly. Unlike static analysis, dynamic analysis must account for inertia and energy dissipation in addition to stiffness and applied loading.

For earthquake engineering, structural dynamics provides the conceptual and computational basis for understanding why two structures subjected to the same ground motion may experience very different responses. The principal controlling properties are mass, stiffness, damping, natural frequency, mode shape, and the characteristics of the excitation. These properties govern displacement, velocity, acceleration, internal force, ductility demand, and energy absorption.

The simplest useful idealization is the single-degree-of-freedom (SDOF) oscillator. Although real buildings possess many degrees of freedom, the SDOF model explains the essential physics and remains central to response-spectrum analysis. Multi-degree-of-freedom (MDOF) systems are treated using matrix equations and modal decomposition, which often transforms a coupled problem into several approximately independent modal SDOF problems.

Learning Objectives

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

1. Formulate the equation of motion for free and forced vibration of SDOF and MDOF systems.

2. Explain natural frequency, period, damping, resonance, mode shapes, and modal participation.

3. Derive and use the principal solutions for undamped and damped SDOF vibration.

4. Apply modal analysis to linear MDOF systems.

5. Explain direct numerical integration, including the Newmark method.

6. Interpret response spectra and identify their use and limitations in earthquake-resistant design.

7. Select appropriate dynamic-analysis methods while recognizing modelling and numerical limitations.

1. Idealization of a Structural System

A continuous structure has infinitely many possible deformation patterns. For analysis, it is represented by discrete degrees of freedom (DOFs), such as lateral floor translations and rotations. A DOF is an independent displacement coordinate required to describe the configuration of the idealized system.

An ideal linear SDOF system consists of:

• mass, m, representing inertia;
• stiffness, k, representing resistance to deformation; and
• viscous damping, c, representing energy dissipation.

The displacement relative to a fixed reference is u(t), and the applied force is p(t). Under linear assumptions, the equation of motion is

m u¨(t) + c u˙(t) + k u(t) = p(t).

Here u˙ is velocity and u¨ is acceleration. The equation is valid when mass, damping, and stiffness can be treated as constant and the force–deformation relation is linear. SI units are kg, N·s/m, and N/m, respectively, so every term has units of force.

For an MDOF model, the corresponding matrix equation is

[M]{u¨} + [C]{u˙} + [K]{u} = {p(t)},

where [M], [C], and [K] are the mass, damping, and stiffness matrices; {u} is the displacement vector; and {p(t)} is the force vector. For a stable unconstrained structure, [M] and [K] are normally symmetric and positive definite after rigid-body freedoms are appropriately handled.

External and Base Excitation

Forces may be applied directly to structural coordinates. Earthquake excitation is more naturally described as prescribed ground acceleration, ü_g(t). If all support points undergo the same horizontal ground motion and u is the relative displacement of the mass with respect to the ground, the SDOF equation becomes

m u¨ + c u˙ + k u = -m ü_g(t).

The absolute acceleration of the mass is u¨ + ü_g. The negative sign on the right-hand side reflects the inertial effect of ground acceleration in relative coordinates. This distinction is fundamental: earthquake response is usually calculated in relative displacement and velocity, but absolute acceleration is often needed for equipment and nonstructural components.

2. Free Vibration of an Undamped SDOF System

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

m u¨ + k u = 0.

Assume a solution u = A exp(st). Substitution gives

m s² + k = 0,

so

s = ±iω_n,

where

ω_n = √(k/m)

is the circular natural frequency in radians per second. The natural period and frequency are

T_n = 2π/ω_n, f_n = 1/T_n = ω_n/(2π).

The general displacement response is

u(t) = u_0 cos(ω_n t) + (v_0/ω_n) sin(ω_n t),

where u_0 and v_0 are the initial displacement and velocity. Equivalently,

u(t) = U cos(ω_n t − φ),

where U and φ are determined by initial conditions.

The system oscillates indefinitely because no energy is dissipated. Its total mechanical energy is

E = 1/2 m u˙² + 1/2 k u²,

which remains constant. At maximum displacement, velocity is zero and energy is entirely strain energy; at the equilibrium position, displacement is zero and energy is entirely kinetic.

Worked Example 1: Undamped Free Vibration

A linear oscillator has m = 1000 kg and k = 400,000 N/m. Determine its natural circular frequency, period, and frequency.

ω_n = √(400,000/1000) = √400 = 20 rad/s.

Therefore,

T_n = 2π/20 = 0.314 s,

and

f_n = 1/0.314 = 3.18 Hz.

The oscillator completes approximately 3.18 cycles per second. Increasing stiffness reduces the period, whereas increasing mass increases it.

3. Damped Free Vibration

Real structures dissipate energy through material hysteresis, friction, cracking, connections, foundations, and nonstructural components. Viscous damping is a mathematical idealization that produces a force proportional to velocity, c u˙.

The damped free-vibration equation is

m u¨ + c u˙ + k u = 0.

Define the critical damping coefficient and damping ratio as

c_c = 2mω_n = 2√(km),

ζ = c/c_c.

The damping ratio is dimensionless. Critical damping, ζ = 1, is the boundary between oscillatory and non-oscillatory response. Most ordinary building structures have small damping, commonly approximated as about 2–7% of critical damping depending on structural and nonstructural characteristics, although the appropriate value depends on the design situation and standard being used.

For underdamping, 0 < ζ < 1, the damped natural frequency is

ω_d = ω_n√(1 − ζ²),

and the response is

u(t) = e^(−ζω_n t)[u_0 cos(ω_d t) + ((v_0 + ζω_nu_0)/ω_d) sin(ω_d t)].

The exponential term is the decay envelope. The logarithmic decrement, useful for estimating damping from successive peaks, is

δ = ln(u_i/u_{i+1}) = 2πζ/√(1 − ζ²),

for successive peaks of an underdamped free response. For small damping, δ is approximately 2πζ.

For ζ = 1, the response does not oscillate and returns to equilibrium as rapidly as possible without overshooting. For ζ > 1, the response is overdamped and returns without oscillation more slowly than the critically damped case.

4. Harmonic Forced Vibration and Resonance

Consider a harmonic force

p(t) = p_0 sin(ωt),

where p_0 is the force amplitude and ω is the forcing circular frequency. After transient effects decay, the steady-state response is

u(t) = U sin(ωt − θ),

where

U = (p_0/k) / √[(1 − r²)² + (2ζr)²],

r = ω/ω_n,

and

tan θ = (2ζr)/(1 − r²),

with the phase angle interpreted using the correct quadrant.

The dimensionless ratio

D(r,ζ) = U/(p_0/k)

is the dynamic amplification factor. At low forcing frequency, r approaches zero and D approaches one, corresponding to quasi-static response. Near r = 1, resonance may occur. Damping limits the amplitude and changes the phase. At high frequency, displacement amplitude decreases, although acceleration response may remain important.

For an undamped system, exact resonance produces an unbounded mathematical response because energy is added at the natural frequency without dissipation. Real structures do not generally exhibit unbounded response because damping, nonlinear behaviour, yielding, strength limits, and changing properties intervene. Nevertheless, resonance can generate severe demands.

The steady-state solution does not represent the complete response immediately after loading begins. The total response consists of a forced component plus a transient free-vibration component. Numerical simulations and earthquake analysis must account for initial conditions and transient effects.

5. Energy and Dynamic Characteristics

A useful interpretation of vibration is through energy transfer. External work is converted into kinetic energy and strain energy, while damping removes energy. Under earthquake excitation, input energy is distributed among kinetic energy, elastic strain energy, damping energy, and, in nonlinear structures, hysteretic energy.

The natural period is not simply a property of the material; it is a property of the complete mass–stiffness system. Changes in infill walls, cracking, foundations, partitions, or heavy equipment can alter both mass and stiffness. A structure with an unexpectedly long period may attract lower spectral acceleration in some portions of a design spectrum but may experience larger displacement demands.

6. Multi-Degree-of-Freedom Systems

For an undamped free MDOF system,

[M]{u¨} + [K]{u} = {0}.

Assume {u} = {φ}e^(iωt). Substitution gives

([K] − ω²[M]){φ} = {0}.

A nontrivial mode shape exists only if

det([K] − ω²[M]) = 0.

This is the generalized eigenvalue problem. Its solutions are the natural frequencies ω_r and corresponding mode shapes {φ_r}. For an n-DOF system there are generally n modes, provided the system is properly constrained and the matrices satisfy the usual linear assumptions.

Orthogonality

For distinct modes r and s of a classically undamped linear system,

{φ_r}^T[M]{φ_s} = 0,

and

{φ_r}^T[K]{φ_s} = 0, r ≠ s.

The modal mass and modal stiffness are

M_r = {φ_r}^T[M]{φ_r},

K_r = {φ_r}^T[K]{φ_r} = ω_r²M_r.

Mode shapes can be scaled arbitrarily. Consequently, their absolute amplitudes have no physical significance until a normalization is selected. Common normalizations include maximum component equal to one, mass normalization, or a specified roof displacement.

Modal Transformation

The displacement vector is expanded as

{u(t)} = [Φ]{q(t)} = Σ {φ_r}q_r(t),

where [Φ] contains the mode shapes and q_r are modal coordinates. Premultiplying the equation by [Φ]^T and using orthogonality gives uncoupled modal equations when damping is classically proportional:

M_r q¨_r + C_r q˙_r + K_r q_r = p_r(t),

where

p_r(t) = {φ_r}^T{p(t)}.

For base acceleration, the modal equation is commonly written

q¨_r + 2ζ_rω_r q˙_r + ω_r²q_r = −Γ_r ü_g(t),

where the participation factor is

Γ_r = ({φ_r}^T[M]{ι})/({φ_r}^T[M]{φ_r}),

and {ι} is the influence vector defining the direction of ground motion. The effective modal mass is

M^*_r = Γ_r² M_r.

The sum of effective modal masses should approach the total participating mass as sufficient modes are included. A low modal-mass ratio indicates that important modes have been omitted or that the excitation direction has been defined incorrectly.

If damping is nonclassical, the classical real-mode uncoupling procedure may not apply. State-space methods, complex modes, or direct integration may then be required.

Worked Example 2: Two-DOF Modal Interpretation

Consider a two-storey shear building with equal floor masses and identical storey stiffnesses. The first mode has both floors moving in the same direction, with the upper floor generally moving more than the lower floor. The second mode has a node between floors: one floor moves in one direction while the other moves oppositely.

The first mode usually has the longer period and dominates overall lateral translation. The second mode contributes significantly to interstorey deformation and higher-frequency response, particularly in taller buildings or when the response quantity varies rapidly over height. It is therefore unsafe to assume that the first mode alone always represents maximum storey force or drift.

7. Numerical Integration of Dynamic Equations

Closed-form solutions are available for simple linear systems and idealized loads. Earthquake records, nonlinear material behaviour, changing contact conditions, and complicated damping usually require numerical integration.

A time-stepping method estimates displacement, velocity, and acceleration at discrete times t_i. Important requirements are:

• a sufficiently small time step to resolve the highest relevant frequency and the input record;
• consistent initial conditions;
• accurate treatment of stiffness and damping;
• monitoring of numerical stability and energy behaviour; and
• appropriate handling of nonlinear iterations when stiffness depends on displacement or history.

Newmark Method

The Newmark family uses parameters β and γ. Over one time step Δt, the standard relations are

u_{i+1} = u_i + Δt v_i + Δt²[(1/2 − β)a_i + βa_{i+1}],

v_{i+1} = v_i + Δt[(1 − γ)a_i + γa_{i+1}].

Here u, v, and a denote displacement, velocity, and acceleration. For the average-acceleration method, β = 1/4 and γ = 1/2. For a linear undamped system, this method is unconditionally stable, meaning that stability does not impose a finite upper limit on Δt; however, accuracy can still be poor if Δt is too large. Stability is not the same as accuracy.

At each step, equilibrium is enforced using

m a_{i+1} + c v_{i+1} + k u_{i+1} = p_{i+1}.

For a linear system, substitution of the Newmark relations produces an effective algebraic equation for u_{i+1}. For a nonlinear system, the step generally requires Newton–Raphson or another iterative procedure, with the tangent stiffness updated until displacement and force residuals satisfy prescribed tolerances.

Other important methods include the central-difference explicit method, Wilson-type methods, Hilber–Hughes–Taylor methods, and state-space integration. Explicit methods may be efficient for very large systems but often have conditional stability. Algorithms with numerical damping can suppress high-frequency response, which may be beneficial for computational robustness but can distort quantities of engineering interest.

8. Response Spectra

A response spectrum summarizes the maximum response of a family of SDOF oscillators subjected to a specified ground-motion record, usually as a function of natural period and damping ratio. Common spectral ordinates are:

• spectral displacement, S_d: maximum relative displacement;
• spectral velocity, S_v: maximum relative velocity, often pseudo-velocity in design spectra; and
• spectral acceleration, S_a: maximum absolute acceleration or pseudo-acceleration, depending on the definition.

For an oscillator with circular frequency ω,

S_{pa} = ω² S_d,

is the pseudo-spectral acceleration associated with maximum relative displacement. This is not necessarily equal to the maximum absolute acceleration, although the values may be close under some conditions. Definitions must therefore be checked carefully.

A spectrum is not a time history. It gives peak values but generally does not preserve the time at which different modal maxima occur or their signs. Consequently, modal peak responses cannot usually be added algebraically. Common combination rules include the square-root-of-sum-of-squares (SRSS) method for well-separated modes and the complete quadratic combination (CQC) method for closely spaced modes. CQC accounts for modal correlation and damping; the precise correlation coefficients depend on the formulation and damping assumptions.

For a design spectrum, code provisions define spectral shape, hazard level, soil or site class, damping modification, and combinations of horizontal and vertical components. A design spectrum is not a prediction of one specific earthquake; it is a prescribed envelope or target representation for design.

Earthquake Response Using Modal Spectra

For each mode r, the modal SDOF oscillator is assigned period T_r and damping ratio ζ_r. The spectrum supplies a peak modal response. The physical response is reconstructed using the mode shape and participation factor. For example, a modal displacement contribution is proportional to

{u_r,max} ≈ {φ_r} Γ_r S_{d,r},

subject to the adopted response-spectrum convention. Because peaks occur at different times, modal contributions are combined statistically or by code-specified rules rather than by direct summation.

Response-spectrum analysis is efficient and widely used for linear seismic design. It is less suitable when detailed time sequencing, residual drift, cumulative damage, pulse effects, soil–structure interaction, or strongly nonlinear behaviour is important. Nonlinear time-history analysis may then be necessary.

9. Illustrative Application: Seismic Assessment of a Four-Storey Office Building

Background

Consider an existing four-storey reinforced-concrete office building in a moderate-to-high seismic region. A preliminary assessment indicates irregular infill distribution and possible soft-storey behaviour at the ground floor. The objective is to estimate whether a first-mode response approximation is adequate and to identify the next analytical step.

Problem

The building has significant mass at each floor, lateral stiffness concentrated in columns and frames, and uncertain contribution from masonry infill. A simple model using only the first mode predicts acceptable roof displacement, but the ground-storey drift appears high. The question is whether a first-mode response-spectrum analysis can be trusted.

Analysis

A four-DOF shear-building model is formed with floor translations as coordinates. Two models are developed:

1. a bare-frame model, excluding infill stiffness; and

2. an infilled-frame model, including idealized diagonal strut stiffness with documented uncertainty.

The mass and stiffness matrices are assembled, and the generalized eigenvalue problem is solved. The first mode is predominantly translational, but higher modes produce curvature changes over the height. Effective modal masses are calculated in the horizontal direction. The first mode captures most, but not all, participating mass. The second and third modes contribute materially to storey shear and drift near the stiffness discontinuity.

A response-spectrum analysis is performed using the applicable code spectrum and damping assumption. SRSS is used only after checking modal separation; CQC is selected where modal frequencies are close. The model is then subjected to a suite of ground-motion records in nonlinear time-history analysis, with modelling uncertainty represented by alternative infill and hinge assumptions.

Approach and Outcome

The analysis shows that roof displacement is dominated by the first mode, but maximum ground-storey drift is not. Higher-mode effects and the soft-storey mechanism concentrate deformation at the first level. The nonlinear analyses also show that stiffness degradation lengthens the effective period and increases residual drift in some records.

The resulting engineering decision is not to rely on first-mode response alone. Strengthening measures are considered at the ground storey, and the infill model is treated as an uncertainty rather than a fixed beneficial assumption. A refined assessment is documented with sensitivity studies and record-selection justification.

Learning Points

• A modal mass ratio is a useful completeness check but does not guarantee accurate local responses.
• First-mode dominance for roof displacement does not imply first-mode dominance for storey drift or member force.
• Infill walls can alter period, mode shapes, torsional characteristics, and failure mechanisms.
• Linear response spectra are valuable screening tools, but nonlinear time histories may be required for irregular or vulnerable structures.
• Engineering judgement and transparent uncertainty documentation are essential.

10. Modelling, Computational, and Professional Limitations

Dynamic analysis results are only as reliable as the model and input data. Important sources of uncertainty include:

• inaccurate mass estimates, especially for partitions, façade, equipment, and stored contents;
• uncertain cracked stiffness in reinforced-concrete and masonry members;
• inappropriate damping assumptions;
• neglected soil–structure interaction;
• torsional and three-dimensional effects;
• inadequate representation of connections and diaphragm flexibility;
• insufficient modes or unsuitable modal-combination rules;
• time steps that fail to resolve important frequencies; and
• ground-motion records that do not represent the hazard, site, duration, or pulse characteristics.

A model should be calibrated where observations are available and subjected to sensitivity analyses. A highly detailed model is not automatically more accurate: additional parameters may introduce poorly known assumptions and obscure governing behaviour.

Ethically, engineers must not select modelling assumptions merely to obtain favourable results. Analysis files, input records, damping values, convergence tolerances, and design decisions should be traceable and reviewable. Regulatory requirements concerning seismic hazard, load combinations, record selection, nonlinear acceptance criteria, and quality assurance must be followed. Software verification is also important: benchmark problems, limiting-case checks, unit checks, and independent calculations can reveal implementation errors.

11. Contemporary Developments and Best Practice

Modern structural-dynamics practice increasingly combines finite-element modelling, automated parameter studies, cloud computation, structural health monitoring, and probabilistic seismic assessment. Digital twins and sensor data can assist in identifying changes in natural frequencies or damping, but measured changes require careful interpretation because temperature, occupancy, damage, and sensor uncertainty may produce similar effects.

Machine learning can classify vibration signatures, estimate modal properties, or accelerate surrogate modelling. It should not replace mechanics-based verification, physical reasoning, or code compliance. Training-data bias and extrapolation outside the observed range are major risks.

Best practice includes:

1. Define the analysis objective before selecting the model.

2. Establish coordinates, units, mass sources, boundary conditions, and excitation direction explicitly.

3. Check natural periods and mode shapes against engineering expectations.

4. Confirm modal mass participation and investigate omitted modes.

5. Perform limiting-case checks, such as zero damping, rigid-body motion, and static loading.

6. Use a time step appropriate to both the record and the highest frequency of interest.

7. Distinguish absolute from relative response quantities.

8. Report assumptions, uncertainties, convergence information, and sensitivity results.

9. Use nonlinear analysis when the expected response exceeds the reliable range of linear methods.

10. Apply independent review for safety-critical decisions.

Chapter Summary

Structural dynamics represents structures as systems in which mass, stiffness, damping, and time-dependent excitation interact. The SDOF equation m u¨ + c u˙ + k u = p(t) is the fundamental model for free, forced, and damped vibration. Natural frequency and period follow from the mass–stiffness ratio, while damping controls decay, resonance amplitude, and phase.

MDOF systems are formulated using mass, damping, and stiffness matrices. Solving the generalized eigenvalue problem gives natural frequencies and mode shapes. Orthogonality permits modal transformation, and participation factors quantify the contribution of each mode to ground-motion response. Effective modal mass and modal-completeness checks are essential in practical analysis.

Numerical integration, particularly the Newmark method, enables analysis of realistic records and nonlinear systems. Stability, accuracy, time-step selection, convergence, and numerical damping must be distinguished. Response spectra efficiently provide peak SDOF responses for seismic design, but they omit time sequencing and may be inadequate for strongly nonlinear, irregular, or cumulative-damage problems.

Reliable dynamic analysis depends on appropriate modelling, transparent assumptions, code compliance, validation, and engineering judgement. The central professional lesson is that a mathematically sophisticated analysis cannot compensate for an inappropriate physical model.

Key Terms

Degree of freedom: An independent coordinate needed to describe structural motion.

Mass matrix: Matrix representing inertial properties in an MDOF system.

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

Damping: Mechanism by which vibration energy is dissipated.

Natural frequency: Frequency of free vibration of a system under specified boundary conditions.

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

Damping ratio: Damping coefficient divided by the critical damping coefficient.

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

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

Modal mass: Mass associated with a particular mode and its normalization.

Participation factor: Quantity measuring how strongly a mode is excited by a specified loading direction.

Effective modal mass: Participating mass associated with a mode under a specified excitation.

Orthogonality: Mathematical property allowing distinct modes to decouple under mass and stiffness products.

Base excitation: Structural excitation caused by prescribed support or ground motion.

Response spectrum: Peak response of SDOF oscillators plotted against period or frequency.

Spectral displacement: Maximum relative displacement of an oscillator under a specified excitation.

Newmark method: Time-stepping family for integrating structural dynamic equations.

SRSS: Modal combination rule based on the square root of the sum of squared modal responses.

CQC: Modal combination rule accounting for correlation between closely spaced modes.

Numerical stability: Property indicating whether computational errors remain bounded during integration.

Review Questions

Short Answer Questions

1. Define natural frequency and natural period for an undamped SDOF system.

2. What is the physical meaning of the damping ratio?

3. Why are mode shapes important in MDOF analysis?

4. Distinguish spectral displacement from absolute acceleration.

5. State one difference between numerical stability and numerical accuracy.

Long Answer Questions

1. Derive the free-vibration solution of an undamped SDOF system and explain the associated energy exchange.

2. Explain the response of a damped SDOF system under harmonic loading, including resonance and phase angle.

3. Derive the generalized eigenvalue problem for an undamped MDOF system and discuss modal orthogonality.

4. Explain modal transformation and the role of participation factors in earthquake response analysis.

5. Describe the Newmark average-acceleration method, its assumptions, and its practical limitations.

Analytical / Application Questions

1. An oscillator has m = 1500 kg, k = 600,000 N/m, and damping ratio ζ = 0.05. Calculate its natural circular frequency, natural period, critical damping coefficient, and actual damping coefficient.

2. A four-storey building has most of its effective modal mass in the first mode, but a stiffness discontinuity at the ground floor. Explain why first-mode response may be adequate for roof displacement but inadequate for ground-storey drift, and recommend an analysis strategy.

3. A numerical simulation produces stable displacements but unexpectedly low high-frequency acceleration. Identify possible causes and propose checks involving time step, numerical damping, input record, and output definitions.

Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system, the natural circular frequency is

A. √(m/k)

B. √(k/m)

C. k/m

D. m/k

Correct Answer: B. √(k/m)

Question 2

The critical damping coefficient of an SDOF system is

A. mω_n

B. k/ω_n

C. 2mω_n

D. ω_n/(2m)

Correct Answer: C. 2mω_n

Question 3

For an underdamped oscillator, the damped natural frequency is

A. ω_n√(1−ζ²)

B. ω_n(1+ζ²)

C. ω_n/ζ

D. ω_n√(1+ζ²)

Correct Answer: A. ω_n√(1−ζ²)

Question 4

At very low forcing frequency, the displacement amplification factor of a damped SDOF system approaches

A. zero

B. one

C. the damping ratio

D. infinity

Correct Answer: B. one

Question 5

The MDOF free-vibration problem requires solving

A. [M]{u}={0}

B. ([K]−ω²[M]){φ}={0}

C. [C]{u˙}={p}

D. [K]{u˙}={0}

Correct Answer: B. ([K]−ω²[M]){φ}={0}

Question 6

The effective modal mass depends on

A. only the absolute scale of the mode shape

B. the excitation direction and modal participation

C. only the damping coefficient

D. only the time-step size

Correct Answer: B. the excitation direction and modal participation

Question 7

For the Newmark average-acceleration method, the usual parameters are

A. β=0 and γ=0

B. β=1/4 and γ=1/2

C. β=1/2 and γ=1/4

D. β=1 and γ=1

Correct Answer: B. β=1/4 and γ=1/2

Question 8

A response spectrum primarily provides

A. the complete time history of a structure

B. peak SDOF response as a function of period and damping

C. only the static base shear

D. the exact damage sequence in a building

Correct Answer: B. peak SDOF response as a function of period and damping

Question 9

The CQC modal combination method is especially useful when modes are

A. identical in mass only

B. closely spaced in frequency

C. undamped and rigid

D. excluded from the model

Correct Answer: B. closely spaced in frequency

Question 10

For an oscillator with m=1000 kg and k=400,000 N/m, the natural period is approximately

A. 0.0314 s

B. 0.314 s

C. 3.14 s

D. 31.4 s

Correct Answer: B. 0.314 s