Earthquake Response and Seismic Analysis

Introduction and context

Earthquake analysis converts ground motion into demands on structural mass, stiffness, strength, deformation capacity, and connections. The central engineering task is not to predict one exact force, but to estimate a credible range of responses and demonstrate that the structure has adequate strength, ductility, stability, and damage-control capacity. This chapter develops equivalent-static, response-spectrum, modal, time-history, pushover, and nonlinear dynamic procedures, with particular attention to torsion, higher modes, damping, and modeling uncertainty.

The methods form a hierarchy. Equivalent-static analysis is efficient but depends on regularity and code calibration. Modal response-spectrum analysis captures elastic multimodal response without requiring a complete time record. Linear time-history analysis preserves the temporal sequence of excitation but remains dependent on the linear model. Pushover analysis estimates nonlinear capacity and approximate demand, whereas nonlinear response-history analysis provides the most detailed representation of cyclic behavior, at the cost of substantial modeling and interpretation effort.

Learning objectives

After studying this chapter, the learner should be able to:

1. Explain mass, stiffness, damping, natural frequency, mode shape, and seismic demand.

2. Perform an equivalent-static base-shear and story-force calculation.

3. Construct and apply a response spectrum, distinguishing true absolute from pseudo-spectral acceleration.

4. Formulate modal participation factors and combine modal responses.

5. Set up, scale, and interpret linear and nonlinear time-history analyses.

6. Perform a displacement-based pushover analysis and identify its assumptions.

7. Recognize torsion, higher-mode effects, damping uncertainty, and numerical/modeling uncertainty.

8. Select an analysis method appropriate to structural regularity, performance objective, and available evidence.

1. Dynamic idealization of a building

A building may be idealized as a multi-degree-of-freedom (MDOF) system:

[M]{u¨} + [C]{u˙} + [K]{u} = -[M]{r}u¨g(t) + {p(t)}

where [M], [C], and [K] are the mass, damping, and stiffness matrices; {u} is the vector of relative floor displacements; {r} is the influence vector; u¨g is ground acceleration; and {p} is any applied external-load vector. The equation assumes small deformation and linear material behavior unless [K] and the restoring-force relation are updated during analysis.

For a single-degree-of-freedom (SDOF) oscillator, the undamped natural circular frequency and period are

ωn = √(k/m), T = 2π/ωn,

where k is stiffness in force/length, m is mass, ωn is in rad/s, and T is in seconds. The critical damping coefficient is ccr = 2mωn, and the viscous damping ratio is ξ = c/ccr. For an underdamped oscillator, ωd = ωn√(1 − ξ²). The distinction between circular frequency and ordinary frequency is essential: ω is measured in rad/s, while f = ω/(2π) is measured in cycles/s.

In a modal analysis, free-vibration shapes satisfy

([K] − ωr²[M]){φr} = {0}.

The eigenvalues ωr² and eigenvectors {φr} provide periods and mode shapes. Modes are orthogonal with respect to [M] and [K] for classical, proportionally damped systems. Modal mass and participation factor are

Mr = {φr}T[M]{φr},

Γr = ({φr}T[M]{r})/({φr}T[M]{φr}).

The effective modal mass for translational excitation is

Meff,r = Γr² Mr.

The cumulative effective modal mass ratio should be checked in each principal direction; the required value is governed by the adopted standard, but an engineer commonly includes sufficient modes to capture approximately 90% or more, subject to code provisions and local response requirements.

2. Seismic response quantities and spectra

For an SDOF system subjected to ground acceleration, the relative displacement is u(t), while the absolute mass displacement is ua(t) = u(t) + ug(t). Three related spectral quantities must not be confused:

• Relative displacement spectrum Sd(T, ξ) = max |u(t)|.
• Pseudo-spectral velocity PSV = ω Sd.
• Pseudo-spectral acceleration PSA = ω² Sd.

PSA is often reported as Sa in design spectra, but mathematically it is not necessarily the maximum absolute acceleration. The true absolute spectral acceleration is

Saa(T, ξ) = max |u¨a(t)| = max |u¨(t) + u¨g(t)|.

For a lightly damped oscillator, PSA and true absolute spectral acceleration may be close over portions of the spectrum, but they are not identically equal in general. When a code labels a design ordinate “spectral acceleration,” the definition in that code must be followed.

A response spectrum is generated by applying a ground-motion record to oscillators with specified periods and damping, solving each oscillator, and recording the selected maximum response. Design spectra are usually smoothed envelopes calibrated to seismic hazard, site class, importance, damping, and code philosophy. They represent peak response by period, not a time sequence; therefore phase relationships between modes and cumulative damage effects are not retained.

3. Equivalent-static analysis

Applicability

The equivalent-static method represents earthquake action by a lateral force pattern. It is most defensible for low- to medium-rise, regular structures with limited torsional irregularity, modest higher-mode effects, and a code-permitted fundamental period. It should not be used automatically for tall, highly irregular, essential, base-isolated, or strongly nonlinear structures.

A generic code form for design base shear is

V = Cs W,

where W is seismic weight and Cs is a code-defined seismic coefficient. A typical elastic-spectrum-based form is

Cs = Sa(T1, ξ)/(R/Ie),

subject to minimum, maximum, and site-specific restrictions prescribed by the governing standard. Here Sa is the applicable design spectral ordinate, R is a response modification or behavior factor, and Ie is an importance factor. Because definitions differ among standards, coefficients must not be transferred between codes without checking their meaning.

The fundamental-period estimate may be obtained from an eigenvalue analysis or a code approximation. The code limit on an approximate period is important because an unconstrained empirical period can reduce design force unrealistically.

The vertical distribution of story forces may be represented by

Fx = V wx hx^k / Σ(wi hi^k),

where wx is the seismic weight at level x, hx is its height above the base, and k is a code-specified exponent, often related to the period. The forces must satisfy ΣFx = V. Accidental torsion is commonly represented by applying the lateral force with an additional eccentricity, often including a code-defined percentage of the plan dimension.

Worked example: base shear and force distribution

An illustrative six-storey regular building has seismic weight 12,000 kN, fundamental period T1 = 0.80 s, design spectral acceleration Sa = 0.30g, R = 5, and Ie = 1.0. Assume the code permits the simplified coefficient Cs = 0.30/5 = 0.060.

Thus,

V = 0.060(12,000) = 720 kN.

Assume equal floor weights of 2,000 kN at heights 3, 6, 9, 12, 15, and 18 m, and k = 1.0. The denominator is

Σ wi hi = 2,000(3+6+9+12+15+18) = 2,000(63) = 126,000 kN·m.

At the roof, for example,

F6 = 720[2,000(18)/126,000] = 205.7 kN.

The complete forces are approximately 34.3, 68.6, 102.9, 137.1, 171.4, and 205.7 kN from the first floor to the roof. Their sum is 720.0 kN. Story shear at a level is the sum of forces above that level; roof-level story shear is 205.7 kN, while the base shear is 720 kN. The designer must additionally apply code-required accidental torsion and check drift, strength, stability, and P-Δ effects.

4. Modal response-spectrum analysis

Procedure

Modal response-spectrum analysis is appropriate when several vibration modes contribute materially. The procedure is:

1. Define mass, stiffness, diaphragm constraints, and boundary conditions.

2. Solve the eigenvalue problem for periods and mode shapes.

3. Calculate Γr, effective modal mass, and cumulative mass participation.

4. Read the spectral ordinate for each modal period and damping ratio.

5. Calculate modal peak responses.

6. Combine modal results and then combine orthogonal directional results as required.

7. Apply accidental eccentricity, drift, stability, and code-specific scaling checks.

For a modal quantity q, the peak modal contribution is generally proportional to Γr times the spectrum ordinate and the mode-shape component. For example, modal floor displacement associated with relative displacement spectrum Sd,r is

ur,modal = Γr φr Sd,r.

Modal base shear can be calculated from the modal inertia forces or from the modal generalized response. A response-spectrum result is not a simultaneous physical state: maximum displacement in one mode and maximum shear in another need not occur at the same instant.

Modal combination

For well-separated modes, square-root-of-sum-of-squares (SRSS) combination is often acceptable:

R = √(Σ Rr²).

For closely spaced modes, complete quadratic combination (CQC) is preferred:

R = √(Σr Σs ρrs Rr Rs),

where ρrs is a correlation coefficient dependent on modal frequencies and damping. For equal damping, one commonly used form is

ρrs = [8ξ²(1+β)β3/2]/[(1−β²)² + 4ξ²β(1+β)²], β = ωs/ωr,

with consistent mode indexing and frequency ratio. Implementations should follow the adopted standard because published CQC forms and conventions can differ.

The terms “closely spaced” should be quantified, not asserted. In the worked example below, T1 = 0.85 s and T2 = 0.62 s give frequency ratio ω2/ω1 = T1/T2 = 1.371 and period separation (0.85−0.62)/0.85 = 27.1%. These modes are not extremely close in a universal mathematical sense; however, if the code or software guidance requires CQC for modal periods within a specified band, that rule governs. Moreover, CQC is conservative and generally appropriate when modal frequencies are near enough that correlation is non-negligible. A sensitivity comparison with SRSS is good practice rather than relying on the adjective “close.”

Worked modal calculation

Suppose a two-mode result for a frame gives modal story shears R1 = 420 kN and R2 = 180 kN. If SRSS is justified,

R = √(420² + 180²) = √(176,400 + 32,400) = √208,800 = 457.0 kN.

If a program calculates ρ12 = 0.20 for the actual periods and damping, CQC gives

R = √(420² + 180² + 2(0.20)(420)(180))

= √280,?

Numerically, the cross term is 30,240 kN², so the total is 239,040 kN² and R = 489.0 kN. The value is higher because the modal peaks are positively correlated. The correlation coefficient must come from the actual periods and damping, not be selected merely to increase demand.

5. Linear time-history analysis

Time-history analysis uses a digitized acceleration record u¨g(t). For each time step, the program integrates the MDOF equations and produces displacement, velocity, acceleration, member force, and support-reaction histories. A practical workflow is:

1. Select records compatible with magnitude, distance, mechanism, site, and duration.

2. Correct baseline and filter records only with defensible signal-processing procedures.

3. Select damping and define it consistently across modes.

4. Apply the record in the required direction and units.

5. Use a time step sufficiently small to resolve the highest frequency of interest; verify convergence by reducing the step.

6. Check equilibrium, energy balance where available, and peak response plausibility.

7. Scale or select records according to the governing standard.

8. Interpret peaks as record-specific values, not guaranteed simultaneous maxima.

For linear analysis, record scaling may be based on matching the mean spectrum over a prescribed period range. A common process is to calculate a scale factor α so that the scaled spectrum αSrecord lies within code limits or matches a target over the specified range. Scaling must not be used to conceal an inappropriate record set.

Small numerical application

At a selected period, a record has PSA = 0.18g and the target is 0.24g. A simple single-period scale factor is

α = 0.24/0.18 = 1.333.

The scaled ordinate is 1.333(0.18g) = 0.24g. This is only an illustrative one-period calculation; professional record selection requires a period range, multiple records, and the code’s acceptance rule.

6. Pushover analysis

Pushover analysis is a nonlinear static procedure in which lateral load is increased monotonically, or according to a prescribed adaptive pattern, while gravity loads remain applied. The analysis produces a capacity curve, usually base shear Vb versus roof displacement Δr. It is useful for estimating sequence of yielding, strength, ductility, displacement demand, and probable weak stories.

A basic procedure is:

1. Establish a gravity-load equilibrium state.

2. Define nonlinear hinges or distributed plasticity, including strength, stiffness degradation, cyclic rules, and acceptance criteria.

3. Select a lateral pattern, such as first-mode proportional, uniform, or adaptive.

4. Apply lateral load incrementally and record Vb and Δr.

5. Identify yield points, mechanism formation, softening, collapse indicators, and local demand ratios.

6. Convert the MDOF capacity curve to an equivalent SDOF representation when using a displacement-coefficient method.

7. Determine target displacement using the adopted code or performance framework.

8. Check member deformation, connection behavior, P-Δ effects, torsion, and nonstructural performance.

The first-mode pattern may be inadequate when higher modes, torsion, irregularity, or a changing mode shape is important. Adaptive pushover updates the force pattern based on the evolving structural response but remains an approximation to dynamic behavior.

Worked displacement-coefficient illustration

Assume a converted SDOF system has effective mass m* = 1,000 kN·s²/m, effective period Te = 1.0 s, and damping-adjusted spectral displacement demand Sd = 0.12 m. If the participation and transformation factor from the equivalent SDOF coordinate to roof displacement is Γφroof = 1.25, the estimated roof target is

Δtarget = Γφroof Sd = 1.25(0.12) = 0.150 m.

At Δroof = 0.150 m, the pushover curve gives Vb = 1,850 kN. The engineer then checks whether hinges have acceptable rotations, whether a soft-story mechanism has formed, and whether P-Δ instability reduces resistance. This calculation does not prove that the structure will reach the target in a real earthquake; it is a demand-capacity approximation dependent on the chosen pattern and nonlinear model.

7. Nonlinear dynamic analysis

Nonlinear response-history analysis directly integrates equations with nonlinear restoring forces:

[M]{u¨} + [C]{u˙} + {fint(u, internal variables)} = -[M]{r}u¨g(t).

The internal-force vector depends on plasticity, cracking, stiffness degradation, strength deterioration, pinching, gap/contact, impact, and other constitutive variables. The analysis is appropriate for performance-based design, irregular or important structures, retrofit verification, collapse assessment, and research, provided the model is validated.

Practical procedure

1. Define geometry, masses, diaphragm behavior, gravity loads, and foundation assumptions.

2. Choose element formulations: fiber sections, concentrated hinges, shear components, joint springs, isolator elements, or contact elements.

3. Calibrate hysteretic parameters using laboratory data, accepted empirical models, or documented standards.

4. Establish damping without double-counting energy dissipation already represented by hysteresis.

5. Apply gravity loads first and verify static equilibrium.

6. Apply each ground-motion record, checking convergence and time-step sensitivity.

7. Use multiple records and examine median, dispersion, and prescribed fractiles rather than one “representative” record.

8. Monitor drift, hinge rotation, residual displacement, shear failure, floor acceleration, collapse mechanisms, energy components, and numerical warnings.

9. Conduct sensitivity analyses for uncertain parameters.

A nonlinear result is only as credible as its failure modes. A beam-column model that permits flexural yielding but omits brittle shear failure may produce an unconservative answer. Likewise, excessive Rayleigh damping at frequencies outside the calibration range can distort response, especially after stiffness changes.

8. Torsion, higher modes, damping, and uncertainty

Torsion

Seismic torsion arises from mass–stiffness eccentricity, accidental eccentricity, diaphragm flexibility, and nonlinear redistribution. In a rigid diaphragm, lateral translation and rotation are coupled. A simple elastic estimate of torsional moment is

Mt = V e,

where V is story shear and e is the effective eccentricity. This is not a substitute for a three-dimensional analysis. Engineers should model realistic diaphragm stiffness, distribute accidental eccentricity in both signs, inspect edge-element demands, and check torsional irregularity ratios.

Higher modes

Higher modes can dominate upper-story accelerations, column shears, transfer levels, chimneys, towers, and structures with abrupt stiffness or mass changes. A first-mode equivalent-static pattern may underestimate these effects. Include enough modes for mass participation and inspect local response quantities, which may require more modes than global base shear.

Damping

Damping represents energy dissipation from material hysteresis, cracking, friction, nonstructural components, and radiation. The familiar 5% critical damping is a reference assumption, not a universal physical constant. Response-spectrum ordinates must be adjusted when damping differs from the spectrum’s reference damping. In nonlinear analysis, hysteretic energy loss and added viscous damping must be separated to avoid double counting. Rayleigh damping, C = a0M + a1K, matches selected frequencies but may generate unrealistic damping at other frequencies; stiffness-proportional damping also changes when tangent stiffness changes unless carefully formulated.

Modeling and epistemic uncertainty

Uncertainty includes record-to-record variability, uncertain stiffness and strength, cracking, soil–structure interaction, diaphragm action, connection behavior, damping, numerical integration, and construction quality. A defensible study documents assumptions, performs sensitivity studies, uses independent checks, and reports ranges or statistical summaries. False precision—such as reporting a displacement to three decimal places when material properties are highly uncertain—should be avoided.

9. Integrated illustrative case: irregular hospital wing

This fictional case is an instructional scenario. A four-storey hospital wing has a concrete moment frame, a stiff central stair core, an asymmetric plan, and heavy mechanical equipment on the roof. Immediate occupancy is required after a design-level earthquake.

Problem

A preliminary equivalent-static design gives acceptable total base shear but excessive drift at one perimeter line. The plan eccentricity also produces large diaphragm rotations. A first-mode-only model predicts low upper-story acceleration, conflicting with equipment-serviceability requirements.

Analysis approach

The design team first develops a three-dimensional mass and stiffness model with semi-rigid diaphragms and both principal directions. Eigenanalysis identifies significant second and third modes; cumulative effective mass exceeds the project criterion only after these modes are included. Modal response-spectrum analysis with accidental torsion is performed using CQC because the first two periods are sufficiently correlated under the adopted standard; SRSS is retained as a sensitivity comparison.

A suite of site-compatible records is then selected. Linear time-history analysis checks whether the spectrum-based demands are being distorted by modal combination assumptions. Because the hospital’s performance objective includes limited permanent damage, nonlinear hinges and concrete cracking are introduced. A pushover analysis identifies concentration of plastic demand at the flexible perimeter line. The lateral system is revised by adding a supplementary wall segment and improving diaphragm collectors.

Nonlinear response-history analysis is finally performed with multiple records. The model includes cyclic hinge deterioration, P-Δ effects, equipment mass, and semi-rigid diaphragm action. Record-to-record results show that median drift meets the project target, but one upper-floor acceleration limit is exceeded in several records. Equipment anchorage and isolation are therefore redesigned rather than relying on additional structural strength alone.

Outcome and learning points

The case demonstrates that acceptable base shear does not guarantee acceptable torsion, drift, acceleration, or operational performance. Equivalent-static analysis was useful for preliminary sizing; modal analysis exposed higher-mode participation; time histories tested temporal effects; pushover revealed a likely mechanism; and nonlinear dynamic analysis assessed the final performance objective. The result depended as much on model validation and uncertainty management as on software selection.

10. Selection, verification, and professional practice

Method selection should reflect the question being asked. Use equivalent-static analysis for permitted regular structures and preliminary design; modal response-spectrum analysis for routine multimodal elastic design; linear time history when record-specific elastic response matters; pushover for mechanism and deformation screening; and nonlinear dynamic analysis when nonlinear cyclic behavior, irregularity, collapse prevention, or high consequence justifies the effort.

Every analysis should include: a documented mass source; independent period and base-shear checks; mesh and time-step sensitivity where relevant; equilibrium verification; code compliance; realistic boundary conditions; review of local as well as global demands; and traceable record-selection and scaling decisions. Peer review is particularly important for nonlinear models, because a converged numerical solution can still represent the wrong physical behavior.

Digital automation and AI can assist with record classification, parameter sensitivity, anomaly detection, and surrogate modeling. They cannot replace engineering judgment, constitutive validation, code interpretation, or responsibility for life-safety decisions. Training data, extrapolation, and explainability must be documented when AI-supported tools influence design decisions.

Chapter Summary

Earthquake response analysis begins with the MDOF equation of motion and the properties of mass, stiffness, damping, and mode shapes. Equivalent-static analysis converts seismic weight into a code-calibrated base shear and vertical force pattern, but is restricted by regularity and code applicability. Response-spectrum analysis uses modal periods, participation factors, effective modal mass, and modal-combination rules; PSA, ω²Sd, must be distinguished from true absolute spectral acceleration. Time-history analysis preserves the temporal record and requires defensible selection, scaling, damping, integration, and convergence checks. Pushover analysis estimates nonlinear capacity and mechanisms but depends on the load pattern and equivalent-SDOF assumptions. Nonlinear dynamic analysis can represent cyclic degradation and collapse mechanisms, but requires validated constitutive models and multiple records.

Torsion, higher modes, damping assumptions, P-Δ effects, and model uncertainty can control design even when total base shear appears satisfactory. Professional practice therefore combines appropriate analysis hierarchy, independent verification, sensitivity studies, transparent assumptions, and performance-based interpretation.

Key Terms

Mass matrix: Matrix representing distributed structural inertia.

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

Damping ratio: Viscous damping coefficient divided by critical damping.

Natural period: Time for one cycle of free vibration.

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

Participation factor: Modal coefficient measuring participation in a specified excitation pattern.

Effective modal mass: Translational mass represented by a vibration mode.

Response spectrum: Peak oscillator response plotted against period for specified damping.

Pseudo-spectral acceleration: ω² times maximum relative displacement, ω²Sd.

Absolute spectral acceleration: Maximum absolute mass acceleration, max|u¨ + u¨g|.

Equivalent-static analysis: Lateral-force procedure replacing dynamic action with a code-based force pattern.

CQC: Complete quadratic combination of correlated modal maxima.

SRSS: Square-root-of-sum-of-squares modal combination.

Time-history analysis: Numerical calculation using an acceleration record as a function of time.

Pushover analysis: Incremental nonlinear static analysis producing a capacity curve.

Hysteresis: Loading-unloading behavior and energy dissipation under cyclic deformation.

P-Δ effect: Additional moment caused by gravity load acting through lateral displacement.

Accidental torsion: Code-prescribed torsional action representing uncertain eccentricity.

Model uncertainty: Uncertainty caused by imperfect knowledge of structural properties or behavior.

Review Questions

Short Answer Questions

1. Distinguish pseudo-spectral acceleration from true absolute spectral acceleration.

2. What information is provided by a mode shape and a natural period?

3. Why is effective modal mass checked in response-spectrum analysis?

4. State two limitations of pushover analysis.

5. Why can higher modes be important in a building with an apparently adequate first-mode response?

Long Answer Questions

1. Explain the complete workflow for modal response-spectrum analysis, including modal combination and directional effects.

2. Compare equivalent-static, response-spectrum, linear time-history, pushover, and nonlinear dynamic analysis in terms of inputs, outputs, assumptions, and appropriate applications.

3. Discuss the treatment of damping in elastic response-spectrum analysis and nonlinear response-history analysis.

4. Explain how torsion and diaphragm flexibility influence seismic demand and how they should be modeled.

5. Develop a quality-assurance plan for a nonlinear dynamic analysis of an irregular building.

Analytical / Application Questions

1. A building has seismic weight 20,000 kN and a code seismic coefficient of 0.075. Calculate the design base shear and explain three additional checks required before accepting the result.

2. Two modal story-shear contributions are 500 kN and 300 kN. Calculate their SRSS combination. Explain why this value is not necessarily a simultaneous physical story shear.

3. A pushover model predicts acceptable roof displacement but excessive column shear at a transfer level. Critically assess whether the structure passes and propose additional analyses or design changes.

Multiple Choice Questions with Answers

Question 1

For an SDOF oscillator with circular frequency ω = 4 rad/s, what is its natural period?

A. 0.25 s

B. 1.00 s

C. 1.57 s

D. 25.13 s

Correct Answer: C. 1.57 s

Question 2

Which expression defines pseudo-spectral acceleration?

A. max|u¨ + u¨g|

B. ω²Sd

C. ωSd

D. Sd/ω²

Correct Answer: B. ω²Sd

Question 3

A building has seismic weight 10,000 kN and Cs = 0.08. Its equivalent-static base shear is:

A. 80 kN

B. 800 kN

C. 1,250 kN

D. 8,000 kN

Correct Answer: B. 800 kN

Question 4

Which method explicitly preserves the time sequence of a selected ground-motion record?

A. Equivalent-static analysis

B. Response-spectrum analysis

C. Linear time-history analysis

D. SRSS combination

Correct Answer: C. Linear time-history analysis

Question 5

For modal contributions of 300 kN and 400 kN combined by SRSS, the result is:

A. 100 kN

B. 350 kN

C. 500 kN

D. 700 kN

Correct Answer: C. 500 kN

Question 6

The primary purpose of a pushover analysis is to estimate:

A. Only foundation bearing pressure

B. Nonlinear capacity and likely deformation mechanisms

C. The exact acceleration time history

D. The elastic damping ratio from field data

Correct Answer: B. Nonlinear capacity and likely deformation mechanisms

Question 7

If a mode has φᵀM r = 0, its participation factor for the specified translational excitation is:

A. Zero

B. One

C. Equal to its period

D. Undefined in all cases

Correct Answer: A. Zero

Question 8

Which issue can cause a first-mode-only analysis to underestimate upper-story demands?

A. Higher-mode participation

B. Unit consistency

C. Mass conservation

D. Static equilibrium

Correct Answer: A. Higher-mode participation

Question 9

A record has PSA 0.20g at a selected period and the target is 0.30g. A single-period scale factor is:

A. 0.667

B. 1.10

C. 1.50

D. 6.00

Correct Answer: C. 1.50

Question 10

Why must hysteretic energy dissipation and added viscous damping be checked together in nonlinear analysis?

A. To ensure the mass matrix is diagonal

B. To prevent double-counting energy dissipation

C. To eliminate all higher modes

D. To make PSA equal to absolute acceleration

Correct Answer: B. To prevent double-counting energy dissipation