Performance-Based Earthquake Engineering

Introduction and Context

Performance-Based Earthquake Engineering (PBEE) is a decision-oriented framework for predicting how a structure and its occupants will perform under specified earthquake hazards. Conventional seismic design commonly relies on prescriptive rules: a structural system is selected, forces are calculated using code-defined procedures, and members are proportioned to satisfy strength and detailing requirements. Such procedures remain essential, but they do not directly state whether a building will be operational after an earthquake, how much repair will cost, or how long it will be unavailable.

PBEE addresses these questions by linking four related domains:

1. Hazard: the intensity and characteristics of earthquake ground motion.

2. Demand and response: structural and nonstructural actions caused by the hazard.

3. Damage and consequence: physical damage, casualties, repair cost, environmental effects, and downtime.

4. Decision criteria: whether predicted performance is acceptable for the owner, occupants, regulator, or society.

The essential chain is:

\[

\text{Hazard} \rightarrow \text{Demand} \rightarrow \text{Damage} \rightarrow \text{Loss and downtime} \rightarrow \text{Decision}

\]

This chapter develops that chain, explains limit states and fragility, discusses uncertainty, and demonstrates how PBEE can support design, assessment, retrofit, emergency planning, and asset management.

Learning Objectives

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

• distinguish prescriptive seismic design from performance-based assessment;
• formulate performance objectives using hazard levels and acceptance criteria;
• explain structural, nonstructural, and functional limit states;
• distinguish engineering demand parameters from intensity measures and damage measures;
• describe the hazard–demand–damage–loss framework;
• use fragility and consequence functions in a probabilistic assessment;
• separate aleatory variability from epistemic uncertainty;
• interpret downtime, repairability, and resilience requirements;
• evaluate a performance-based assessment and identify its limitations; and
• communicate results to technical and nontechnical decision-makers.

1. From Prescriptive Design to Performance-Based Assessment

A prescriptive code specifies minimum requirements considered suitable for a class of buildings. It may define design spectra, load combinations, ductility requirements, drift limits, detailing rules, and component capacities. Compliance provides evidence of regulatory acceptability, but it is not necessarily a prediction of actual loss.

A performance-based approach begins with the question: What must the building be able to do, and with what probability, under specified earthquake scenarios? The answer may include life safety, immediate occupancy, protection of critical equipment, limited repair cost, or rapid recovery of a hospital or emergency facility.

Performance-based engineering therefore involves three related activities:

• Performance specification: defining desired outcomes before analysis.
• Performance prediction: estimating response, damage, loss, and downtime.
• Performance verification: comparing predicted outcomes with acceptance criteria.

Performance objectives should be explicit. “Safe” is insufficient because it could mean avoidance of collapse, preservation of life, continued operation, or limited financial loss. Different stakeholders may select different objectives for the same facility. A warehouse, residential tower, data centre, hospital, and nuclear facility cannot be judged using identical criteria.

PBEE is not a promise that damage will not occur. It is a structured means of estimating the likelihood and consequences of damage and deciding whether those consequences are tolerable.

2. Performance Objectives and Hazard Levels

A performance objective is normally expressed as a pairing of:

• a hazard level, defined by an intensity or probability of exceedance; and
• a target performance level, defined by acceptable damage, risk, or functionality.

For example, an owner may require:

• at a frequent, relatively low-intensity earthquake: negligible structural damage and continued use;
• at a design-level earthquake: repairable damage with no threat to life; and
• at a rare or maximum-considered earthquake: very low probability of collapse and protection of life.

Hazard may be described deterministically, such as a selected earthquake scenario, or probabilistically. If an intensity measure has an annual probability of exceedance of \(p\), the corresponding return period is approximately

\[

T_R = \frac{1}{p}

\]

when \(p\) is interpreted as a constant annual exceedance rate and the approximation is suitable. More precisely, if the probability of exceedance in an interval of length \(t\) is \(P_t\), the equivalent annual rate under a Poisson model is

\[

\lambda = -\frac{\ln(1-P_t)}{t}, \qquad T_R=\frac{1}{\lambda}.

\]

Here, \(\lambda\) has units of inverse time. The Poisson model assumes independent earthquake occurrences for the purpose of occurrence probability; it does not imply that ground motions are physically independent in every respect.

Hazard descriptions may include peak ground acceleration, spectral acceleration at a specified period, peak ground velocity, duration, number of strong-motion cycles, and directionality. A single scalar intensity measure is convenient but may not capture all features relevant to a particular failure mode.

Illustrative performance-objective matrix

| Hazard level | Structural objective | Nonstructural/functional objective |

|---|---|---|

| Frequent earthquake | Essentially elastic response | No significant interruption |

| Design earthquake | Repairable damage; no life-threatening failure | Limited damage and short interruption |

| Rare earthquake | Collapse prevention; life safety | Extensive damage may occur, but critical hazards are controlled |

These descriptions are illustrative, not universal code definitions. Project-specific criteria must be established with reference to the governing regulations, owner requirements, building use, and consequences of failure.

3. Limit States and Acceptance Criteria

A limit state is a condition beyond which a specified performance requirement is no longer satisfied. Limit states may be structural, nonstructural, functional, economic, or societal.

3.1 Structural limit states

Common structural limit states include:

• Immediate occupancy or operational response: minor damage, stable gravity-load system, and limited permanent deformation.
• Damage control: damage is present but repair remains practical and residual drift is limited.
• Life safety: severe damage may occur, but gravity-load resistance and stable lateral resistance are maintained sufficiently to protect occupants.
• Collapse prevention: the structure is near its ultimate deformation capacity; collapse may be prevented, but major damage and unsafe conditions are possible.

These labels should not be treated as interchangeable with a particular drift value. Drift limits are indicators, not complete definitions of performance. A building with acceptable peak drift may still experience falling façades, damaged fire systems, residual drift, or failure of anchorage.

3.2 Nonstructural and functional limit states

Nonstructural components frequently govern economic loss and downtime. Relevant states include:

• cracking or detachment of partitions and façades;
• damage to ceilings, sprinkler pipes, elevators, and mechanical equipment;
• overturning or sliding of contents;
• loss of electrical power, water, communications, or medical gases; and
• inability to occupy or operate the building.

A functional limit state is reached when a required service is unavailable, even if the primary structural frame remains stable. For a hospital, loss of emergency power or imaging equipment may be more consequential than moderate frame damage. For a data centre, interruption of cooling or power may determine downtime.

3.3 Acceptance criteria

Acceptance criteria can be stated in terms of:

• maximum or mean interstorey drift;
• residual drift after shaking;
• component rotation, strain, acceleration, or ductility demand;
• probability of collapse;
• probability of exceeding a damage state;
• repair cost as a fraction of replacement cost;
• number of casualties;
• downtime or time to reoccupancy; and
• probability of maintaining a specified function.

Criteria must specify whether they apply to a median prediction, a percentile, or a probability. For example, “median drift less than 1%” is not equivalent to “there is no more than a 10% probability that drift exceeds 1%.” The latter explicitly addresses uncertainty.

4. The PBEE Probabilistic Framework

A useful formulation represents annual loss or risk as an integration over hazard intensity, engineering demand, damage, and consequence. A general expression for the mean annual frequency of exceeding a consequence level \(c\) is

\[

\nu(C>c)=\int \int \int P(C>c\mid DM)\,p(DM\mid EDP)\,p(EDP\mid IM)\,|d\lambda(IM)|\,dDM\,dEDP.

\]

In this expression:

• \(IM\) is an intensity measure, such as spectral acceleration;
• \(EDP\) is an engineering demand parameter, such as drift or floor acceleration;
• \(DM\) is a damage measure, such as the probability that a component is in a repair state;
• \(C\) is a consequence, such as cost or downtime;
• \(p(EDP\mid IM)\) represents demand uncertainty;
• \(p(DM\mid EDP)\) represents damage uncertainty; and
• \(|d\lambda(IM)|\) represents the seismic hazard measure over intensity.

The expression is schematic: actual implementations may use vectors of intensity measures and demands, discrete damage states, correlated component responses, and event-based simulation. It demonstrates that loss is not obtained simply by multiplying a hazard value by a damage percentage.

An alternative event-based form is

\[

\mathbb{E}[L]=\sum_{e}\lambda_e\,\mathbb{E}[L\mid e],

\]

where \(\lambda_e\) is the annual occurrence rate of event \(e\), and \(\mathbb{E}[L\mid e]\) is the expected loss conditional on that event. This form is useful when a set of earthquake ruptures, magnitudes, distances, and ground-motion simulations has been generated.

5. Hazard, Intensity Measures, and Ground-Motion Uncertainty

The hazard model describes possible earthquake sources, rupture characteristics, wave propagation, and site effects. A probabilistic seismic hazard analysis commonly considers:

• source geometry and recurrence;
• magnitude distribution;
• rupture distance and mechanism;
• attenuation or ground-motion models;
• local site classification and nonlinear site response; and
• epistemic alternatives in source and ground-motion models.

An intensity measure should be sufficiently related to the response of interest while remaining practical to predict. Spectral acceleration at the fundamental period is often effective for first-mode-dominated buildings. Peak ground acceleration may be useful for acceleration-sensitive components, while duration or spectral shape may be important for cumulative damage and near-fault response.

No intensity measure is universally optimal. A scalar measure can conceal important variation. Two records with similar spectral acceleration at one period may produce different demands because of duration, pulse effects, spectral shape, vertical motion, or higher-mode response.

Ground-motion records used in nonlinear analysis should be selected and scaled consistently with the hazard objective. Excessive scaling, inappropriate spectral matching, or neglect of record-to-record variability can produce misleadingly precise results.

6. Demand Models and Engineering Demand Parameters

An engineering demand parameter is a response quantity that can be related to component damage. Examples include:

• interstorey drift ratio;
• floor acceleration;
• floor velocity;
• plastic rotation;
• curvature ductility;
• axial force and shear demand;
• residual drift; and
• cumulative energy or cyclic deformation.

For a building subjected to a ground-motion record, nonlinear time-history analysis may produce a time history of displacement, acceleration, force, and deformation. A maximum response is commonly extracted, but the choice of maximum, residual, cumulative, or cycle-based demand must match the failure mechanism.

A demand model may take a logarithmic form such as

\[

\ln EDP = a+b\ln IM+\varepsilon,

\]

where \(a\) and \(b\) are fitted model parameters and \(\varepsilon\) is a residual term, often approximated as normally distributed with standard deviation \(\sigma_{\ln EDP}\). This relation is empirical or simulation-based; it is not a universal law. It should be calibrated for the structural system, response quantity, site, and range of intensity considered.

The median demand conditional on \(IM\) is

\[

\widehat{EDP}_{50}=e^a IM^b,

\]

if the residual has zero mean in logarithmic space. Demand dispersion is often a major contributor to risk and should not be omitted merely because the median prediction satisfies a limit.

7. Damage States and Fragility Functions

A damage state translates engineering demand into physical condition. A reinforced-concrete wall, for example, may be classified as slight, moderate, severe, or near-collapse damage. A partition may be undamaged, cracked, detached, or collapsed. Damage-state definitions must be observable or consistently inferable and should be associated with repair actions or consequences.

A fragility function gives the probability that a component reaches or exceeds a damage state at a specified demand or intensity. A commonly used lognormal fragility model is

\[

P(DS\ge ds\mid EDP)=\Phi\left[\frac{\ln(EDP)-\ln(\theta_{ds})}{\beta_{ds}}\right],

\]

where:

• \(\Phi\) is the standard normal cumulative distribution function;
• \(\theta_{ds}\) is the median demand at which the damage state is reached;
• \(\beta_{ds}\) is the logarithmic dispersion; and
• \(EDP\) must be positive and expressed in units consistent with \(\theta_{ds}\).

The function is appropriate when the logarithm of the capacity is reasonably represented by a normal distribution. It is not appropriate without modification for negative demands, strongly non-lognormal data, or mechanisms controlled by multiple correlated variables.

Fragility may be expressed against \(IM\) rather than \(EDP\), but demand-based fragility is generally more transferable across structural systems when the relationship between demand and damage is understood. Component fragilities should account for installation quality, deterioration, anchorage, orientation, and interaction with adjacent components.

If discrete damage states are defined by exceedance probabilities, the probability of being exactly in state \(ds_i\) is obtained by subtraction:

\[

P(DS=ds_i)=P(DS\ge ds_i)-P(DS\ge ds_{i+1}).

\]

The highest state requires the convention that its exceedance probability beyond the final state is zero or that a separate collapse state is defined.

8. From Damage to Loss, Downtime, and Resilience

Physical damage is not itself the final decision variable. Consequence models convert damage into:

• direct repair cost;
• replacement cost;
• contents and inventory loss;
• casualties and injuries;
• business interruption;
• temporary accommodation;
• environmental releases; and
• time to reoccupancy or functional recovery.

A simple expected repair-cost model for damage states is

\[

\mathbb{E}[L_R]=\sum_i P(DS=i)\,L_{R,i},

\]

where \(L_{R,i}\) is the repair loss associated with state \(i\). This relation assumes the state probabilities are mutually exclusive and that the loss values are conditional expected losses. In practice, repair cost may depend on quantity, access, labour availability, inflation, regional construction markets, and interaction among damaged components.

Downtime should be decomposed into activities rather than treated as a single fixed duration. A useful conceptual model is

\[

T_{recovery}=T_{inspection}+T_{engineering}+T_{procurement}+T_{repair}+T_{recommissioning},

\]

with dependencies and possible parallel activities. The terms represent inspection and safety evaluation, engineering decisions, procurement, physical repair, and testing or recommissioning. The expression is a planning model, not an assertion that all activities are sequential. Parallel work and critical-path logic may substantially reduce total time.

Functional recovery extends beyond physical repair. A building may be structurally safe but unavailable because utilities, access routes, communications, or specialized equipment are not functioning. Resilience assessment therefore considers the trajectory of service over time, including degradation, minimum service level, and recovery rate.

9. Uncertainty and Correlation

Two major uncertainty classes should be distinguished:

• Aleatory variability: inherent variation in future events, such as record-to-record differences for earthquakes with similar magnitude and distance.
• Epistemic uncertainty: lack of knowledge about models, parameters, material properties, deterioration, or component behaviour.

Aleatory variability is generally represented within probability distributions. Epistemic uncertainty can be represented using alternative models, parameter distributions, logic trees, or scenario analyses. The distinction matters because epistemic uncertainty may be reduced through research, testing, inspection, or improved data, whereas aleatory variability cannot be eliminated for a future event.

Correlation is equally important. Demands in adjacent storeys, components subjected to the same record, and repair times for related systems are not independent. Ignoring correlation can either understate or overstate system risk. For example, assuming independent failure of all sprinkler segments may underestimate the likelihood of common-cause disruption caused by a shared riser or pump.

Model uncertainty should be reported transparently. A result such as “expected loss = 2% of replacement cost” is incomplete without the analysis scope, hazard basis, modelling assumptions, dispersion, and confidence or credible interval.

10. Worked Example: Fragility-Based Damage Probability

Consider a component whose engineering demand parameter is interstorey drift ratio. Assume that under a selected earthquake hazard, the median drift ratio is 0.8%, and the logarithmic demand dispersion is 0.30. A moderate-damage state has median drift capacity \(\theta=1.0\%\) and logarithmic capacity dispersion \(\beta=0.25\). For a simplified conditional calculation, treat the demand as fixed at its median and use the fragility equation.

Using consistent percentage units,

\[

z=\frac{\ln(0.008)-\ln(0.010)}{0.25}

=\frac{\ln(0.8)}{0.25}

\approx\frac{-0.2231}{0.25}=-0.8924.

\]

Therefore,

\[

P(DS\ge \text{moderate})=\Phi(-0.8924)\approx 0.19.

\]

The estimated probability of reaching or exceeding moderate damage is approximately 19% under this simplified condition.

This is not a complete risk calculation because demand dispersion has been omitted from the conditional calculation. A fuller assessment would integrate over the distribution of demand and would consider the hazard probability, other damage states, component quantity, correlation, and consequences. The example demonstrates why comparing median drift directly with a median capacity is inadequate: even when median demand is below median capacity, a substantial probability of damage may remain.

11. Applied Case: Illustrative Regional Hospital Assessment

Background

Consider an illustrative 10-storey reinforced-concrete hospital constructed in a moderate-to-high seismic region. The hospital contains emergency, intensive-care, imaging, and surgical facilities. Its structural frame was designed to the code in force at construction, but the owner now requires improved post-earthquake functionality.

Problem

A code-compliant assessment indicates adequate life-safety strength, but hospital administrators are concerned that nonstructural damage, loss of utilities, or residual drift could prevent emergency operation. The decision is whether to accept current risk, retrofit the structure, protect equipment, or improve operational preparedness.

Analysis

The assessment team establishes three hazard levels and performance objectives. Nonlinear analyses estimate storey drift, floor acceleration, residual drift, and demands on selected components. Fragility functions are assigned to structural walls, partitions, ceilings, piping, generators, elevators, and imaging equipment. Consequence models estimate repair cost and time to restore emergency, inpatient, and diagnostic functions.

The analysis identifies the following risk concentrations:

• structural collapse probability is low but not negligible at the rare hazard level;
• upper-storey acceleration threatens suspended ceilings and sensitive equipment;
• a shared utility riser creates a common-cause vulnerability;
• residual drift in one frame configuration could make façade and elevator repair difficult; and
• generator anchorage and fuel-system protection govern emergency-power continuity.

The result illustrates a key PBEE principle: the dominant recovery constraint need not be the component with the greatest physical damage. A relatively small failure in a shared utility system can produce disproportionate downtime.

Approach and solution

The owner adopts a portfolio of measures rather than relying only on frame strengthening:

1. localized wall and connection strengthening to reduce drift and residual deformation;

2. seismic restraint and flexible connections for critical mechanical and electrical systems;

3. redundancy in emergency power and medical-gas distribution;

4. protection and anchorage upgrades for imaging and laboratory equipment;

5. inspection and procurement agreements established before an earthquake; and

6. a staged reoccupancy plan prioritizing emergency and intensive-care functions.

Outcome

The predicted collapse risk is reduced, but the greater improvement occurs in functional recovery. The hospital can restore essential services using redundant systems while noncritical areas remain under repair. The owner accepts some repair cost because the expected reduction in downtime and interruption of emergency care justifies the intervention.

Learning points

• Life safety and functional recovery are different objectives.
• Nonstructural systems can control loss and downtime.
• Shared utilities create system-level vulnerabilities.
• Retrofit selection should consider consequence reduction, not only strength increase.
• Acceptance criteria must be agreed with the owner and relevant authorities before analysis.

12. Implementation in Professional Practice

A credible PBEE study should follow a documented workflow:

1. Define decisions and stakeholders. Identify the owner, users, regulator, insurer, emergency managers, and communities affected.

2. Describe the asset. Record structural system, nonstructural components, occupancy, contents, utilities, age, deterioration, and dependencies.

3. Define hazards. Select deterministic scenarios, probabilistic hazard levels, or both; document site and record assumptions.

4. Select performance objectives. State structural, nonstructural, functional, economic, and recovery requirements.

5. Select analysis methods. Use screening, linear analysis, nonlinear static analysis, nonlinear dynamic analysis, component testing, or hybrid methods according to the decision and uncertainty.

6. Model demand and damage. Select relevant EDPs, damage states, fragilities, and correlations.

7. Estimate consequences. Include repair, casualties, downtime, business interruption, and recovery sequencing.

8. Propagate uncertainty. Report distributions or ranges, not only point estimates.

9. Compare with criteria. Identify compliance, exceedance, sensitivity, and dominant contributors.

10. Make and document decisions. Explain why a retrofit, operational measure, or acceptance of residual risk is recommended.

The method must be proportionate. A detailed probabilistic assessment is justified for high-value, high-consequence, or unusual facilities, but a simpler screening method may be more appropriate for ordinary buildings or early design decisions.

13. Limitations, Governance, and Ethics

PBEE results are model-dependent. Important limitations include:

• uncertain material and connection properties in existing buildings;
• incomplete knowledge of concealed construction;
• limited fragility data for proprietary or novel components;
• difficulties modelling soil–structure interaction and three-dimensional response;
• uncertain repair durations and labour availability;
• changing building contents and operational arrangements; and
• possible mismatch between analytical damage states and real inspection decisions.

A performance-based result should never be presented as certainty. A low calculated probability of collapse does not mean collapse is impossible, and a high predicted loss does not identify one inevitable outcome.

Ethically, engineers must disclose assumptions, omit no material adverse result, and avoid selecting models solely because they produce favourable conclusions. Decisions affecting life safety require conservative professional judgement and compliance with applicable law. Owners should not be encouraged to trade away life-safety requirements for economic savings without explicit, informed, and legally valid decision-making.

Regulators face a related governance challenge: innovative performance solutions must be transparent, reproducible, and independently reviewable. Acceptance criteria should be traceable to standards, research evidence, testing, or clearly stated stakeholder requirements.

14. Digital Tools, Data, and Artificial Intelligence

Modern PBEE increasingly uses cloud computation, building information models, sensor data, automated record selection, surrogate models, and digital twins. These tools can improve scenario generation and support rapid post-earthquake assessment. Machine-learning models may identify damage from imagery, estimate repair quantities, or emulate computationally expensive simulations.

However, AI does not remove engineering uncertainty. Training data may be biased toward particular building types, regions, or visible damage. An image-based model may miss concealed connection damage or functional failures. AI outputs should therefore be validated against physical models, inspections, and domain knowledge. Model version, training data, uncertainty, and decision thresholds should be documented.

Sensors can support condition assessment and recovery tracking, but sensor absence is not evidence of no damage. Digital tools are most valuable when integrated into a transparent engineering workflow rather than treated as autonomous decision-makers.

15. Contemporary Developments and Best Practice

Current developments include:

• explicit functional-recovery and community-resilience objectives;
• assessment of embodied carbon and repair-related environmental impacts;
• improved modelling of nonstructural systems and contents;
• regional loss and infrastructure-network analysis;
• real-time or near-real-time post-earthquake loss estimation;
• multi-hazard performance assessment involving earthquake, fire, tsunami, or landslide effects; and
• risk-informed retrofit prioritization across building portfolios.

Best practice requires that practitioners:

• define objectives before selecting an analysis method;
• use multiple relevant EDPs rather than drift alone;
• model nonstructural and utility systems explicitly where consequences justify it;
• account for residual drift and repair sequencing;
• include uncertainty and correlation;
• calibrate fragility and consequence models to local evidence;
• conduct sensitivity analysis to identify influential assumptions; and
• communicate results in terms understandable to decision-makers.

Chapter Summary

Performance-Based Earthquake Engineering connects earthquake hazard to engineering demand, physical damage, loss, downtime, and decision criteria. Its central value is that it expresses seismic safety and resilience as explicit performance objectives rather than relying only on prescriptive compliance.

A performance objective combines a hazard level with a target outcome. Limit states may concern structural stability, nonstructural damage, functionality, cost, casualties, or recovery time. Intensity measures describe earthquake excitation, engineering demand parameters describe structural response, and damage measures describe component condition. Fragility functions link demand to the probability of reaching damage states, while consequence models convert damage into cost, downtime, and functional disruption.

A complete assessment must address aleatory variability, epistemic uncertainty, correlation, modelling assumptions, and the limitations of available data. Life safety does not guarantee functional recovery, and structural damage is not always the dominant source of loss. Hospitals and other critical facilities therefore require objectives that include utilities, equipment, redundancy, and recovery sequencing.

PBEE is most effective when objectives are established with stakeholders, methods are proportionate to the decision, results are transparently reported, and engineering judgement remains central. Digital tools and AI can support analysis and response, but they must be validated and governed responsibly.

Key Terms

Acceptance criterion: A quantitative or qualitative requirement used to judge whether a performance objective is satisfied.

Aleatory variability: Inherent randomness in future earthquake events and structural response.

Collapse prevention: A severe performance level at which collapse is prevented or its probability is limited, although major damage may occur.

Consequence model: A model converting damage or functional disruption into cost, casualties, downtime, or other outcomes.

Damage measure (DM): A representation of physical damage, usually expressed through damage states.

Downtime: Time during which a building or required function is unavailable or operating below an acceptable level.

Engineering demand parameter (EDP): A response quantity, such as drift or acceleration, related to damage.

Epistemic uncertainty: Uncertainty caused by incomplete knowledge that may be reduced through better information.

Fragility function: A function giving the probability of reaching or exceeding a damage state conditional on demand or intensity.

Functional recovery: Restoration of required building services and occupancy, not merely structural safety.

Hazard level: A defined earthquake intensity or occurrence probability used in an assessment.

Intensity measure (IM): A numerical measure of ground motion, such as spectral acceleration.

Limit state: A condition beyond which a specified performance requirement is not satisfied.

Life safety: A performance objective focused on preventing unacceptable injury or death during severe shaking.

Nonstructural component: A building element that is not part of the primary gravity or lateral-force-resisting system.

Performance objective: A pairing of a hazard level and a desired response or consequence limit.

Prescriptive design: Design based primarily on specified rules, procedures, member requirements, and detailing provisions.

Residual drift: Permanent lateral deformation remaining after earthquake shaking.

Return period: The reciprocal of an event rate under the stated probability model.

Structural demand: Force, deformation, acceleration, or other response imposed on a structural component.

Review Questions

Short Answer Questions

1. What are the four principal links in the PBEE hazard-to-decision chain?

2. Distinguish an intensity measure from an engineering demand parameter.

3. Why is life safety not equivalent to functional recovery?

4. Define aleatory variability and epistemic uncertainty.

5. What information does a fragility function provide?

Long Answer Questions

1. Explain how performance objectives are formulated using hazard levels and acceptance criteria. Provide examples for an ordinary building and a hospital.

2. Describe the probabilistic relationship among hazard, demand, damage, and loss in PBEE.

3. Discuss the role of nonstructural components in repair cost and downtime assessment.

4. Explain the assumptions and limitations of a lognormal fragility function.

5. Describe a professional workflow for carrying out a performance-based seismic assessment.

Analytical / Application Questions

1. A component has a median demand capacity of 1.2% drift and logarithmic dispersion 0.30. Using the lognormal fragility equation, estimate the probability of exceeding the damage state when the drift demand is 0.9%.

2. A hospital has a low predicted probability of structural collapse but a high predicted probability of losing emergency power. Formulate a performance objective and recommend categories of intervention.

3. An assessment reports only the median repair cost and median downtime. Critically evaluate what additional uncertainty, correlation, and consequence information should be requested before accepting the result.

Multiple Choice Questions with Answers

Question 1

Which statement best describes a performance objective?

A. A material strength specified for a structural member

B. A pairing of a hazard level with a target performance outcome

C. A construction schedule for completing repairs

D. A ground-motion record selected without regard to building use

Correct Answer: B. A pairing of a hazard level with a target performance outcome

Question 2

Which quantity is an engineering demand parameter?

A. Annual probability of earthquake exceedance

B. Replacement cost of a building

C. Interstorey drift ratio

D. Probability of business interruption

Correct Answer: C. Interstorey drift ratio

Question 3

Under the stated Poisson approximation, an annual exceedance rate of 0.01 corresponds to a return period of:

A. 0.01 years

B. 0.1 years

C. 10 years

D. 100 years

Correct Answer: D. 100 years

Question 4

In a lognormal fragility function, theta represents the:

A. Standard normal cumulative distribution function

B. Median demand at which the damage state is reached

C. Annual rate of earthquake occurrence

D. Mean repair cost after damage

Correct Answer: B. Median demand at which the damage state is reached

Question 5

Which is an example of epistemic uncertainty?

A. Variation among earthquake records for similar magnitudes and distances

B. Uncertainty caused by incomplete knowledge of an existing connection detail

C. Random variation in future rupture location

D. Record-to-record variation in ground motion

Correct Answer: B. Uncertainty caused by incomplete knowledge of an existing connection detail

Question 6

Why can a building with acceptable structural drift still have unacceptable performance?

A. Drift is always unrelated to structural response

B. Nonstructural systems, utilities, or equipment may fail

C. Ground motion never affects nonstructural components

D. Acceptance criteria apply only to foundations

Correct Answer: B. Nonstructural systems, utilities, or equipment may fail

Question 7

If the probability of exceeding damage state 1 is 0.60 and that of exceeding damage state 2 is 0.20, the probability of being exactly in damage state 1 is:

A. 0.20

B. 0.40

C. 0.60

D. 0.80

Correct Answer: B. 0.40

Question 8

Which intervention most directly addresses a common-cause failure in a hospital utility system?

A. Increasing the paint thickness on interior walls

B. Removing all redundancy from the power system

C. Providing protected and redundant utility distribution

D. Reducing the number of emergency inspections

Correct Answer: C. Providing protected and redundant utility distribution

Question 9

In a recovery-time model, recommissioning refers primarily to:

A. Predicting the earthquake magnitude

B. Testing and returning repaired systems to service

C. Estimating the original construction cost

D. Selecting a fragility dispersion

Correct Answer: B. Testing and returning repaired systems to service

Question 10

What is a major limitation of using only median demand to judge performance?

A. Median demand includes every possible uncertainty exactly

B. It ignores the probability distribution and dispersion of response

C. It can only be used for nonstructural components

D. It automatically gives a conservative collapse estimate

Correct Answer: B. It ignores the probability distribution and dispersion of response