Reliability Testing and Design Improvement

Introduction

Reliability engineering links design intent with performance in service. A component may satisfy strength calculations and still fail because of fatigue, wear, contamination, thermal cycling, manufacturing variation, software logic, or misuse. Reliability testing therefore has two complementary purposes: to estimate how often failures will occur and to discover weaknesses that can be removed before release.

This chapter develops methods for planning life tests, accelerated tests, and reliability demonstrations. It then connects testing with stress–strength analysis, derating, design margins, root-cause analysis, Failure Reporting, Analysis, and Corrective Action Systems (FRACAS), and reliability growth. The emphasis is on decisions: what to test, how many units are needed, how evidence should be interpreted, and how test results should change the design.

Learning Objectives

After studying this chapter, a student should be able to:

1. Distinguish life, accelerated-life, reliability-demonstration, and reliability-growth tests.
2. Select an appropriate statistical test plan and explain its assumptions.
3. Use stress–strength models, derating rules, and design margins to reduce failure probability.
4. Apply root-cause analysis and FRACAS to convert failure evidence into corrective action.
5. Interpret reliability-growth data and judge whether a design is ready for release.
6. Recognize limitations, ethical responsibilities, and governance requirements in reliability testing.

1. Reliability Testing as an Engineering Decision Process

A reliability test is not simply an endurance experiment. It is a planned process involving a requirement, a population, an operating environment, a failure definition, a statistical model, and a decision rule. A clear test plan answers six questions:

• What function must remain available?
• Under what mission profile and environmental conditions?
• What constitutes failure, degradation, or unsafe operation?
• How many units and test hours are required?
• What confidence is required in the conclusion?
• What action follows a pass, failure, or inconclusive result?

Reliability is conditional on time and conditions. The reliability function is

where T is the random failure time, F(t) is the cumulative distribution function, and R(t) is the probability of surviving beyond time t. The hazard rate is

where f(t) is the failure-time density. The hazard rate is an instantaneous conditional failure rate; it is not generally the same as the fraction of units that fail during a test.

For an exponential life model, the hazard rate is constant, λ, and

The mean time to failure (MTTF) is 1 ÷ λ. For repairable equipment, mean time between failures (MTBF) may be used, but MTBF is meaningful only with a defined population, operating regime, and repair policy. It should not be treated as a universal quality score.

A practical reliability requirement should specify the confidence level, mission duration, environmental profile, and allowable failure modes. For example, “95% reliability” is incomplete unless the mission time and confidence statement are also provided.

2. Types of Reliability Tests

2.1 Life and endurance tests

A life test operates units until failure or until a planned censoring time. It can estimate a life distribution, identify dominant failure mechanisms, and reveal wear-out behavior. Tests may be conducted under normal conditions or under a representative mission cycle.

Censoring occurs when a unit has not failed by the end of observation, is removed for another reason, or is lost to follow-up. Such observations still contain information and should not automatically be discarded. Right-censored observations are commonly analyzed using maximum likelihood, Kaplan–Meier methods, or parametric models such as Weibull analysis.

The two-parameter Weibull reliability model is

where η is the characteristic life, the time at which 63.2% of a population has failed, and β is the shape parameter. If β < 1, the hazard decreases; if β = 1, the model is exponential; and if β > 1, the hazard increases, often indicating wear-out. The model is useful only when its assumed failure mechanism and data fit are defensible.

2.2 Accelerated life testing

Accelerated life testing (ALT) applies higher-than-normal thermal, mechanical, electrical, or chemical stress to produce failures sooner. The objective is to infer life at use conditions, not merely to obtain failures quickly. Acceleration must preserve the same physical failure mechanism; otherwise extrapolation is invalid.

For temperature-driven mechanisms, the Arrhenius relationship is often used:

Here AF is the acceleration factor, E_a is activation energy in joules per mole or per particle as matched to k, k is the appropriate Boltzmann constant, and T_use and T_test are absolute temperatures in kelvin. The expression assumes a thermally activated mechanism and consistent units. Temperature must not be inserted in degrees Celsius.

For mechanical fatigue, an S–N relationship may be fitted over a defined stress range. For voltage or humidity, empirical power-law or combined models may be appropriate. Model choice must follow physics and prior evidence, not convenience.

Important ALT controls include:

• use multiple stress levels where possible;
• monitor temperature, humidity, load, vibration, and electrical variables;
• inspect failed parts to verify the mechanism;
• check for overstress, arcing, melting, or other artificial modes;
• include normal-condition confirmation tests;
• report uncertainty in the extrapolated life.

2.3 Reliability demonstration tests

A reliability demonstration test (RDT) is a decision test. It asks whether the evidence is sufficient to accept or reject a specified reliability target. It is not primarily intended to estimate the complete life distribution.

For n independent units tested for time t with zero failures under a constant exponential failure rate, the one-sided lower confidence bound on reliability at confidence level 1 − α is

This expression applies because total test exposure is n t and the zero-failure upper confidence bound on failure rate is −ln(α) ÷ (n t). More generally,

where T is total accumulated test time and t₀ is the mission time, under the exponential model and zero failures. The notation simplifies to R_L(t₀) = α^(t₀ ÷ T).

For example, suppose 20 units each operate for 100 hours with no failures. Total exposure is 2,000 hours. At 90% confidence, α = 0.10. The lower bound at 100 hours is

R_L(100) = 0.10^(100 ÷ 2000) = 0.10⁰.05 ≈ 0.891.

Thus zero failures do not prove 99% reliability. Large confidence claims require substantial exposure.

If failures occur, the test plan should specify an acceptance number, censoring rules, and the statistical model before testing. Changing the rule after observing the data creates bias.

2.4 Reliability-growth testing

Reliability-growth testing is an iterative process in which failures are analyzed, design changes are introduced, and the revised design is retested. Its purpose is to increase reliability during development rather than merely assess a fixed design.

A useful conceptual sequence is:

Figure: Reliability growth as a closed learning cycle.

The Duane model is one empirical representation of cumulative mean time between failures:

where m(t) is cumulative MTBF at test time t, K is a scale constant, and a is the growth parameter. A log–log plot of m(t) against t is approximately linear when the model is appropriate. The model describes observed improvement; it does not explain causation and should not replace physical failure analysis.

3. Planning a Statistically Defensible Test

Test planning begins with the requirement and ends with a decision rule. The engineer should define:

1. Population: production-intent units, prototypes, or a mixture.
2. Configuration: hardware, software, materials, suppliers, and assembly process.
3. Mission profile: loads, duty cycles, temperature, vibration, maintenance, and storage.
4. Failure criterion: functional failure, safety limit, performance degradation, or a specified parameter drift.
5. Observation plan: measurements, inspection intervals, censoring, and data quality controls.
6. Statistical plan: distribution, confidence level, sample size, and acceptance criteria.
7. Change control: how failures and design modifications affect test validity.

Randomization and replication help separate unit-to-unit variability from test-order effects. Calibration records, sensor uncertainty, environmental logs, and configuration identification are part of the evidence. A test that cannot be reconstructed is weak evidence regardless of the number of hours accumulated.

A useful distinction is between demonstration risk and consumer risk. Producer risk is rejecting a design that meets the requirement; consumer risk is accepting a design that does not. These risks should be agreed before testing.

4. Stress–Strength Reliability

A component succeeds when its strength exceeds the applied stress. Let S denote random stress and X denote random strength. The reliability is

If stress and strength are independent with densities f_X(x) and f_S(s), then

Equivalently, R = P(X − S > 0). The integral limits should reflect the physically possible range; the displayed form assumes nonnegative stress and strength.

When both variables are normally distributed and independent,

and therefore

where Φ is the standard normal cumulative distribution function. Independence is essential to this simple variance addition. If stress and strength are correlated, the variance becomes σ_X² + σ_S² − 2ρσ_Xσ_S, where ρ is their correlation coefficient.

Worked example. A shaft has strength normally distributed with mean 420 MPa and standard deviation 30 MPa. Its applied stress has mean 300 MPa and standard deviation 20 MPa. Assuming independence,

z = (420 − 300) ÷ √(30² + 20²) = 120 ÷ 36.06 ≈ 3.33.

The standard normal table gives Φ(3.33) ≈ 0.9996. The estimated instantaneous survival probability is therefore approximately 99.96%, provided the normal model and the specified load event are appropriate. This is not automatically the probability of surviving a year, because repeated loads, fatigue, and changing environments require a time-dependent model.

5. Derating and Design Margins

Derating means operating a component below its rated thermal, electrical, mechanical, or environmental limit. It reduces the probability that variation or transient conditions will exceed capability. Examples include reducing bearing load, capacitor voltage, motor current, junction temperature, or allowable stress.

A simple design margin is

where C is capability or allowable limit and D is design demand. A margin greater than one indicates nominal capacity exceeds nominal demand. A related factor of safety is often defined as n = C ÷ D, but terminology varies by discipline; the governing standard should be stated.

For uncertain quantities, nominal margin alone is inadequate. A conservative screening margin may compare a lower tolerance bound on capability with an upper bound on demand. For example, if capability is 100 ± 8 units and demand is 70 ± 5 units, a worst-case linear bound gives minimum margin M_min = (100 − 8) ÷ (70 + 5) = 92 ÷ 75 = 1.227. This method can be overly conservative and assumes the stated bounds are appropriate and simultaneous. Probabilistic stress–strength analysis is often more informative.

Derating must not create secondary problems. Lower voltage may reduce performance; lower speed may increase residence time and heating; reduced load may alter lubrication or control stability. Derating rules must therefore be validated under the complete mission profile.

6. Root-Cause Analysis and FRACAS

A symptom is an observed effect; a failure mode is the manner in which function is lost; a root cause is a correctable underlying condition whose removal prevents recurrence. A disciplined investigation preserves failed parts, records as-found condition, controls contamination, and separates evidence from hypotheses.

Common tools include:

• Five Whys: progressive questioning of causal chains; useful for simple systems but vulnerable to oversimplification.
• Fishbone diagrams: organize possible causes under categories such as material, method, machine, measurement, environment, and people.
• Fault-tree analysis: deductive analysis from a top event to combinations of lower-level causes using logical gates.
• Failure Mode and Effects Analysis: prospective ranking of failure modes by severity, occurrence, and detectability, while recognizing that numerical rankings are not probabilities unless explicitly defined.
• Microscopy, fractography, finite-element analysis, and laboratory replication: evidence-based confirmation of physical mechanisms.

FRACAS is a controlled closed-loop system:

1. report the failure and uniquely identify the unit and configuration;
2. quarantine or preserve evidence;
3. classify the failure mode and effect;
4. investigate and verify root cause;
5. assign corrective and preventive actions;
6. approve the design or process change;
7. verify the action by analysis, inspection, or test;
8. close the record only after objective evidence is reviewed.

A good FRACAS database links failure reports to serial numbers, suppliers, software versions, environmental records, drawings, change notices, and verification results. A corrective action that merely replaces failed parts without removing the mechanism is containment, not true correction.

7. Design Improvement Through Reliability Allocation

System reliability depends on architecture. For a series system with statistically independent component reliabilities R_i,

A series system fails when any required component fails. For a parallel active-redundant system with independent component reliabilities,

These expressions assume correct independence and complete redundancy. Common-cause failures, imperfect switching, shared power, and maintenance limitations can greatly reduce the benefit of redundancy.

Reliability allocation distributes a system target among subsystems. Allocation should consider mission criticality, physical feasibility, uncertainty, maintainability, and cost. Increasing reliability by adding redundant parts may increase weight, interfaces, complexity, and latent failure opportunities. Design improvement is therefore a multi-objective decision rather than a race toward the largest nominal margin.

8. Illustrative Case: Cooling-Pump Bearing Failures

Background. An industrial cooling skid contains four centrifugal pumps. During commissioning, two pumps experience bearing seizure after approximately 1,200 operating hours. The nominal bearing load is below the catalogue rating.

Problem. The initial conclusion—“the bearing rating is adequate”—does not explain the field-like failures. The team must determine whether the cause is load, lubrication, alignment, contamination, or a test artifact.

Analysis. FRACAS records show that both failed pumps came from the same assembly shift. Vibration data indicate a rise in axial vibration before seizure. Disassembly finds darkened grease and raceway smearing. Shaft alignment measurements show that thermal growth was not included in the cold alignment procedure. A thermal model predicts axial preload during operation. The stress–strength view is expanded: bearing strength is not a single catalogue number but a distribution affected by temperature, lubrication, contamination, and assembly variation.

Approach. The design team introduces a thermal-growth alignment allowance, specifies a controlled grease fill, adds an installation check, and derates the bearing operating temperature. A thermal-cycle endurance test is run on production-intent assemblies. Failed and surviving bearings are inspected, and vibration is monitored as a leading indicator. The revised test includes representative starts, stops, and fluid temperatures rather than constant-speed operation only.

Outcome. No seizure occurs during the planned exposure, and vibration remains below the warning threshold. The evidence supports improved robustness, but it does not prove infinite life. Continued field monitoring and a defined inspection interval remain necessary.

Learning points. Catalogue ratings are conditional; failure investigations must examine the complete system; leading indicators can shorten detection time; and corrective actions require verification under realistic stresses.

9. Limitations, Governance, and Contemporary Practice

Reliability conclusions are limited by model uncertainty, sample size, changing suppliers, nonrepresentative stresses, and unobserved failure modes. Accelerated tests can create false confidence when a new mechanism appears. Zero failures are especially easy to overinterpret. Confidence intervals, sensitivity analysis, and explicit assumptions should accompany reported estimates.

Safety-critical testing requires traceability, independent review, calibrated equipment, configuration control, and compliance with applicable standards and regulatory requirements. Engineers must not suppress failures, redefine failure criteria after results are known, or present a demonstration as a population estimate without qualification. Test data may also contain proprietary or personal information and should be governed accordingly.

Digital monitoring and machine learning can support anomaly detection, prognostics, and test prioritization. However, an AI-generated anomaly score is not a verified failure mechanism. Models require representative training data, drift monitoring, explainable thresholds where safety is involved, and human engineering review. Physics-based reasoning and physical confirmation remain essential.

Best practice is to combine complementary evidence: analytical margins, component tests, subsystem tests, system endurance, environmental qualification, FRACAS learning, and field feedback. Reliability is strongest when it is designed into architecture and process rather than inspected into a finished product.

Chapter Summary

Reliability testing is a planned decision process linking requirements, mission conditions, failure definitions, statistical models, and corrective action. Life tests estimate behavior over time; accelerated tests shorten observation only when the failure mechanism is preserved; demonstration tests provide confidence-based accept or reject decisions; and reliability-growth tests improve the design iteratively.

The exponential and Weibull models describe different hazard behaviors and must be used with justified assumptions. Stress–strength analysis estimates the probability that capability exceeds demand, while derating and design margins reduce sensitivity to variation and transients. Root-cause analysis and FRACAS convert failure observations into controlled corrective actions. Reliability growth, redundancy, and reliability allocation help improve system performance, but independence, common-cause failures, cost, and maintainability must be considered.

Sound engineering practice requires representative testing, configuration control, uncertainty reporting, ethical data handling, and verification that corrective actions remove the physical cause rather than merely treating the symptom.

Key Terms

Reliability: Probability that an item performs its required function for a stated time and conditions.

Failure time: Random time at which an item ceases to meet its defined function.

Hazard rate: Conditional instantaneous rate of failure.

Censoring: Observation ending before failure is observed.

Weibull distribution: Flexible life distribution with a shape parameter governing hazard behavior.

Accelerated life test: Test using increased stress to obtain life information sooner.

Acceleration factor: Ratio relating life or failure rate at use and test conditions.

Reliability demonstration test: Test designed to support an accept–reject reliability decision.

Confidence level: Long-run coverage associated with a statistical bound or interval.

Reliability growth: Improvement in reliability through iterative test, analysis, and redesign.

Stress–strength model: Model in which success occurs when random strength exceeds random stress.

Derating: Operating below a component’s specified maximum capability.

Design margin: Ratio or difference expressing separation between capability and demand.

Failure mode: Specific manner in which a function is lost.

Root cause: Underlying correctable cause responsible for a failure mechanism.

FRACAS: Failure Reporting, Analysis, and Corrective Action System.

Common-cause failure: Failure affecting supposedly redundant elements through a shared cause.

Mission profile: Time sequence of loads and environmental conditions experienced in service.

Configuration control: Formal management of the identity and changes of a product or test article.

Review Questions

Short Answer Questions

1. What is the difference between a life test and a reliability demonstration test?
2. State the assumptions required for the simple exponential zero-failure confidence calculation.
3. Interpret the Weibull shape parameter when it is less than, equal to, or greater than one.
4. Why can an accelerated test produce invalid conclusions?
5. What is the purpose of FRACAS closure evidence?

Long Answer Questions

1. Explain how a reliability test plan should be constructed from a product requirement.
2. Compare derating, design margin, and stress–strength reliability analysis.
3. Describe the complete reliability-growth cycle and the role of failure investigation.
4. Explain how censoring affects life-test analysis and why censored units should not automatically be discarded.
5. Discuss the benefits and limitations of redundancy in a safety-critical mechanical system.

Analytical / Application Questions

1. Twenty independent units are tested for 200 hours each with zero failures. Under an exponential model, calculate the 90% lower confidence bound for reliability at 200 hours and explain what the result does and does not demonstrate.
2. A component has strength with mean 500 MPa and standard deviation 25 MPa, while applied stress has mean 410 MPa and standard deviation 18 MPa. Assuming independent normal variables, calculate the stress–strength reliability and identify one limitation of the calculation.
3. A pump bearing fails only during hot starts, although it passes constant-temperature endurance testing. Propose a FRACAS investigation, identify likely missing elements in the test mission profile, and recommend two design or process improvements.

Multiple Choice Questions with Answers

Question 1

For a two-parameter Weibull model, what does β > 1 generally indicate?

A. Decreasing hazard rate

B. Constant hazard rate

C. Increasing hazard rate

D. Zero failure probability

Correct Answer: C. Increasing hazard rate

Question 2

Twenty units are each tested for 100 hours with zero failures. Under an exponential model, what is the 90% lower confidence bound for reliability at 100 hours?

A. 0.500

B. 0.794

C. 0.891

D. 0.990

Correct Answer: C. 0.891

Question 3

Which condition is essential for valid extrapolation in an accelerated life test?

A. The test must produce many different failure modes

B. The same physical failure mechanism must remain dominant

C. Temperature must be recorded in Celsius only

D. All test units must fail

Correct Answer: B. The same physical failure mechanism must remain dominant

Question 4

In stress–strength analysis, a component succeeds when:

A. Stress equals strength

B. Stress exceeds strength

C. Strength exceeds stress

D. Both variables have zero variance

Correct Answer: C. Strength exceeds stress

Question 5

A design has capability 120 units and demand 80 units. Using M = C ÷ D, its nominal design margin is:

A. 0.67

B. 1.50

C. 40

D. 200

Correct Answer: B. 1.50

Question 6

What is the primary purpose of FRACAS?

A. To conceal failures until production release

B. To replace all failed parts without investigation

C. To control reporting, analysis, corrective action, and verification

D. To calculate only the purchase price of components

Correct Answer: C. To control reporting, analysis, corrective action, and verification

Question 7

For independent components in a series system, system reliability is:

A. The sum of component reliabilities

B. The product of component reliabilities

C. One minus the product of component reliabilities

D. Always greater than every component reliability

Correct Answer: B. The product of component reliabilities

Question 8

Which observation is right-censored?

A. A unit fails at 500 hours

B. A unit is removed at 500 hours without failure

C. A unit is tested beyond its rated load

D. A unit has a confirmed root cause

Correct Answer: B. A unit is removed at 500 hours without failure

Question 9

Which activity best represents reliability growth?

A. Repeating an unchanged test without analyzing failures

B. Testing, identifying causes, redesigning, and retesting

C. Increasing the warranty period without design change

D. Replacing a failure report with a verbal explanation

Correct Answer: B. Testing, identifying causes, redesigning, and retesting

Question 10

Why is catalogue bearing capacity alone insufficient to establish field reliability?

A. Catalogue ratings are unrelated to any load

B. Reliability also depends on lubrication, alignment, temperature, contamination, and variation

C. Bearings never experience wear

D. Capacity ratings apply only to electrical circuits

Correct Answer: B. Reliability also depends on lubrication, alignment, temperature, contamination, and variation