Robust Engineering

EdbrtocPage 1 of 9

ROBUST ENGINEERING

DESIGN-BY-RELIABILITY

VOLUME 1

TABLE OF CONTENTS

CHAPTER 1- INTRODUCTION ...... 1
1.1- THE NEED FOR ENGINEERING DESIGN-BY-RELIABILITY ...... 1

1.2- DIFFERENCES BETWEEN MECHANICAL

AND ELECTRONIC RELIABILITY

PREDICTION METHODS ...... 2
1.3- AVAILABLE MECHANICAL RELIABILITY

PREDICTION METHODS ...... 3
1.4- COMPARISON OF THE CONVENTIONAL

DESIGN METHODOLOGY AND THE

“ENGINEERING DESIGN BY RELIABILITY”

METHODOLOGY ...... 4

1.5- THE SAFETY FACTOR AND SAFETY

MARGIN CONCEPTS IN DESIGN VERSUS THE

RELIABILITY CONCEPT ...... 25

CHAPTER 2- FIFTEEN-STEP RELIABILITY PREDICTION

AND THE “ROBUST ENGINEERING DESIGN

BY RELIABILITY” METHODOLOGY ...... 41
2.1- INTRODUCTION ...... 41

2.2- DEFINITION OF RELIABILITY ...... 42
2.3- FIFTEEN-STEP METHODOLOGY ...... 43

CHAPTER 3- THE CENTRAL LIMIT THEOREM,

AND THE MOMENTS AND THE MONTE CARLO
SIMULATION METHODS OF SYNTHESIZING

DISTRIBUTIONS ...... 69

3.1- THE SUM OF MANY INDEPENDENT

AND INDENTICALLY DISTRIBUTED (IID)

RANDOM VARIABLES ...... 69

3.2- THE CENTRAL LIMIT THEOREM ...... 75

3.3- THE METHOD OF MOMENTS ...... 80

3.4- INTERPOLATION PROCEDURE FOR

TABLES ...... 103

3.5- THE MONTE CARLO SIMULATION METHOD ...... 106

3.6- COMMENTS ON METHODS FOR SYNTHE-

SIZING DISTRIBUTIONS ...... 136

CHAPTER 4- METHODS OF DETERMINING THE FAILURE

GOVERNING STRESS DISTRIBUTION ...... 141
4.1- DETERMINATION OF THE LOAD

CHARACTERISTICS AND THE ASSOCIATED

STRESS DISTRIBUTION ...... 141
4.2- PROCEDURE FOR DETERMINING THE FAILURE

GOVERNING STRESS DISTRIBUTION ...... 150
4.3- METHODS OF SYNTHESIZING THE

FAILURE GOVERNING STRESS DISTRIBUTION ...... 159
4.4- BINARY SYNTHESIS OF DISTRIBUTIONS ...... 159

4.5- GENERATION OF SYSTEM MOMENTS ...... 166
4.6- MONTE CARLO SIMULATION ...... 170

CHAPTER 5- METHODS OF DETERMINING THE FAILURE

GOVERNING STRENGTH DISTRIBUTION ...... 179
5.1- DISTRIBUTION OF THE MATERIAL PROPERTIES

AND THE ASSOCIATED STRENGTH

DISTRIBUTION ...... 179

5.2- DATA GENERATION AND DETERMINATION

OF THE DISTRIBUTIONS OF THE MATERIAL

STRENGTH PROPERTIES ...... 181

5.3- PROCEDURE FOR DETERMINING THE FAILURE

GOVERNING STRENGTH DISTRIBUTION ...... 218
5.4- BINARY SYNTHESIS OF NORMAL DISTRIBUTIONS

METHOD ...... 229

5.5- GENERATING SYSTEM MOMENTS METHOD ...... 231

5.6- MONTE CARLO SIMULATION METHOD ...... 233

CHAPTER 6- ILLUSTRATED METHODS OF CALCULATING

THE RELIABILITY OF COMPONENTS ...... 241
6.1- INTRODUCTION ...... 241
6.2- THE GENERAL RELIABILITY EXPRESSION TO BE USED

WHEN f(S) AND f(s) ARE BOTH NEITHER NORMAL

NOR LOGNORMALLY DISTRIBUTED ...... 242
6.3- NUMERICAL INTEGRATION ...... 245
6.4- MELLIN TRANSFORMS ...... 248

6.5- MONTE CARLO SIMULATION ...... 263
6.6- NORMAL FAILURE GOVERNING STRESS

AND STRENGTH DISTRIBUTIONS ...... 263

6.7- LOGNORMAL FAILURE GOVERNING STRESS

AND STRENGTH DISTRIBUTIONS ...... 272
6.8- RELIABILITY OF COMPONENTS GIVEN THE FAILURE

GOVERNING STRESS DISTRIBUTION AND A DISCRETE,

FIXED FAILURE GOVERNING STRENGTH ...... 274

6.9- RELIABILITY OF COMPONENTS GIVEN A DISCRETE

FAILURE GOVERNING STRESS AND THE FAILURE

GOVERNING STRENGTH DISTRIBUTION ...... 279

6.10- RELIABILITY OF COMPONENTS GIVEN DISCRETE

FAILURE GOVERNING STRESS AND STRENGTH ...... 281
6.11- RELIABILITY WHEN f(s) AND f(S) ARE BOTH

NORMAL, AND WHEN ...... 283
6.12- RELIABILITY WHEN FAILURE GOVERNING

STRESS AND STRENGTH ARE BOTH DISTRIBUTED ...... 285

6.13- RELIABILITY OF COMPONENTS SUBJECTED TO

FATIGUE GIVEN A FIXED ALTERNATING
STRESS LEVEL, THE CORRESPONDING
CYCLES-TO-FAILURE DISTRIBUTION AND

A SPECIFIC LIFE REQUIREMENT ...... 286

6.14- RELIABILITY WHEN OPERATING AN ADDITIONAL

NUMBER OF CYCLES HAVING ALREADY
COMPLETED A SPECIFIC NUMBER OF CYCLES
OF OPERATION AT A SPECIFIC ALTERNATING

STRESS LEVEL AND THE ASSOCIATED f (N) ...... 296

6.15- RELIABILITY GIVEN THE DISTRIBUTION
OF THE DUTY CYCLES OF OPERATION
OF IDENTICAL COMPONENTS AND THEIR
CYCLES-TO-FAILURE DISTRIBUTION

UNDER FATIGUE LOADING ...... 301

6.16- RELIABILITY FOR A SPECIFIC LIFE GIVEN

THE FAILURE GOVERNING STRENGTH
DISTRIBUTION FOR THAT LIFE AND A
CONSTANT MAXIMUM ALTERNATING

STRESS UNDER FATIGUE LOADING ...... 311

6.17- RELIABILITY FOR A SPECIFIC LIFE GIVEN

THE FAILURE GOVERNING STRENGTH
DISTRIBUTION FOR THAT LIFE AND THE
FAILURE GOVERNING MAXIMUM ALTERNATING
STRESS DISTRIBUTION FOR THAT

LIFE UNDER FATIGUE LOADING ...... 314

6.18- RELIABILITY FOR COMPLETING AN

ADDITIONAL NUMBER OF CYCLES,
HAVING ALREADY COMPLETED A SPECIFIC
NUMBER OF CYCLES OF OPERATION
SUCCESSFULLY, GIVEN ,,
AND UNDER FATIGUE

LOADING ...... 322

6.19- RELIABILITY WITH COMBINED ALTERNATING

AND MEAN STRESS UNDER FATIGUE

LOADING ...... 325

CHAPTER 7- DETERMINATION OF THE DESIGNED-IN

RELIABILITY CONFIDENCE LIMIT

AT A SPECIFIED CONFIDENCE LEVEL ...... 341
7.1- INTRODUCTION ...... 341
7.2- DETERMINATION OF MECHANICAL

RELIABILITY ...... 342

7.3- DETERMINATION OF THE LOWER

ONE-SIDED CONFIDENCE LIMIT ON

THE RELIABILITY ...... 347

7.4- CALCULATING THE LOWER ONE-SIDED
CONFIDENCE LIMIT ON THE

RELIABILITY ...... 363

7.5- EFFECT OF CONFIDENCE LEVEL ON THE

LOWER, ONE-SIDED CONFIDENCE LIMIT

ON THE RELIABILITY ...... 367

7.6- EFFECT OF SAMPLE SIZE ON THE LOWER,
ONE-SIDED CONFIDENCE LIMIT ON

THE RELIABILITY ...... 369

7.7- HOW TO DESIGN TO A RELIABILITY GOAL AT

A SPECIFIED CONFIDENCE LEVEL ...... 370

7.8- CONCLUSIONS AND RECOMMENDATIONS ...... 374

CHAPTER 8- UNRELIABILITY AND RELIABILITY
DETERMINATION BY THE STRESS/STRENGTH

DISTRIBUTIONS' INTERFERENCE APPROACH ...... 379

8.1- INTRODUCTION ...... 379
8.2- THE FAILURE PROBABILITY AND

FAILURE FUNCTION ...... 380

8.3- FAILURE FUNCTION DETERMINATION ...... 381

8.4- THE SURVIVAL FUNCTION ...... 397

8.5- DETERMINATION OF RELIABILITY

OR UNRELIABILITY BY THE

DIFFERENCE-DISTRIBUTION METHOD ...... 402

8.6- CONCLUSIONS ...... 404

CHAPTER 9- A UNIFIED LOOK AT DESIGN SAFETY

FACTORS, SAFETY MARGINS AND

MEASURES OF RELIABILITY ...... 409
9.1- INTRODUCTION ...... 409
9.2- FAILURE GOVERNING STRESS AND STRENGTH,

AND THEIR DISTRIBUTIONS ...... 410
9.3- SAFETY FACTORS ...... 411

9.4- SAFETY MARGINS ...... 417

9.5- MEASURES OF RELIABILITY ...... 420

9.6- CONCLUSIONS ...... 439

CHAPTER 10- COMPARATIVE ACCURACY OF EVALUATING

RELIABILITY USING SIMPSON'S RULE,
THE TRAPEZOIDAL RULE AND THE

GAUSS-LEGENDRE METHOD ...... 445

10.1- INTRODUCTION ...... 445

10.2- SIMPSON'S RULE, TRAPEZOIDAL RULE,

AND GAUSS-LEGENDRE METHODS ...... 446
10.3- METHODOLOGY FOR EVALUATING

RELIABILITY ...... 448

10.4- COMPARISON OF THE ACCURACY ...... 459

10.5- CONCLUSIONS ...... 463

CHAPTER 11- EXACT AND EASY TO OBTAIN SOLUTIONS

FOR THE PREDICTION OF THE
RELIABILITY OF MECHANICAL
COMPONENTS AND STRUCTURAL

MEMBERS ...... 471

11.1- INTRODUCTION ...... 471

11.2- LOGNORMAL FAILURE GOVERNING

STRESS AND STRENGTH DISTRIBUTIONS ...... 472

11.3- GAMMA FAILURE GOVERNING STRESS

AND STRENGTH DISTRIBUTIONS ...... 475

11.4- EXPONENTIAL FAILURE GOVERNING

STRESS AND NORMAL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 481

11.5- EXPONENTIAL FAILURE GOVERNING

STRESS AND TRUNCATED NORMAL
FAILURE GOVERNING STRENGTH

DISTRIBUTIONS ...... 486

11.6- NORMAL FAILURE GOVERNING STRESS
AND EXPONENTIAL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 490

11.7- TRUNCATED NORMAL FAILURE GOVERNING

STRESS AND EXPONENTIAL FAILURE
GOVERNING STRENGTH

DISTRIBUTIONS ...... 494

11.8- EXPONENTIAL FAILURE GOVERNING

STRESS AND STRENGTH DISTRIBUTIONS ...... 498

11.9- UNIFORM FAILURE GOVERNING STRESS

AND GAMMA FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 499

11.10- GAMMA FAILURE GOVERNING STRESS

AND UNIFORM FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 502

11.11- UNIFORM FAILURE GOVERNING STRESS

AND NORMAL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 504

11.12- NORMAL FAILURE GOVERNING STRESS

AND UNIFORM FAILURE GOVERNING ...... 507

11.13- UNIFORM FAILURE GOVERNING STRESS

AND WEIBULL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 510

11.14- WEIBULL FAILURE GOVERNING STRESS

AND UNIFORM FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 513

11.15- UNIFORM FAILURE GOVERNING STRESS

AND EXTREME VALUE OF THE

MINIMA FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 517

11.16- EXTREME VALUE OF THE MAXIMA FAILURE

GOVERNING STRESS AND UNIFORM

FAILURE GOVERNING STRENGTH

DISTRIBUTIONS ...... 519

CHAPTER 12- NUMERICAL SOLUTIONS FOR THE

PREDICTION OF THE RELIABILITY
OF MECHANICAL COMPONENTS AND

STRUCTURAL MEMBERS WHEN CLOSED FORM

SOLUTIONS ARE NOT AVAILABLE ...... 527

12.1- INTRODUCTION ...... 527
12.2- WEIBULL FAILURE GOVERNING STRESS

AND WEIBULL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 528

12.3- NORMAL FAILURE GOVERNING STRESS

AND WEIBULL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 531

12.4- WEIBULL FAILURE GOVERNING STRESS

AND NORMAL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 533

12.5- LOGNORMAL FAILURE GOVERNING STRESS

AND WEIBULL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 534

12.6- WEIBULL FAILURE GOVERNING STRESS

AND LOGNORMAL FAILURE GOVERNING

STRENGTH DISTRIBUTIONS ...... 536

12.7- EXTREME VALUE OF THE MAXIMA

FAILURE GOVERNING STRESS AND
EXTREME VALUE OF THE MINIMA
FAILURE GOVERNING STRENGTH

DISTRIBUTIONS ...... 537

CHAPTER 13- MONTE CARLO SIMULATION METHOD

FOR RELIABILITY DETERMINATION ...... 543

13.1- RELIABILITY PREDICTION USING

MONTE CARLO SIMULATION ...... 543

13.2- ERROR BOUNDS AND NUMBER OF

MONTE CARLO TRIALS ...... 548

13.3- VARIANCE REDUCTION METHODS ...... 558

CHAPTER 14- FAILURE MODES, EFFECTS, AND

CRITICALITY ANALYSIS ...... 571
14.1- INTRODUCTION ...... 571
14.2- MAJOR STEPS IN A FAMECA ...... 571
14.3- METHOD 1 ...... 572

14.4- METHOD 2 ...... 618

CHAPTER 15- ADDITIONAL APPLICATIONS

OF THESE METHODOLOGIES ...... 641

CHAPTER 16- APPLICATION GUIDANCE FOR THESE

METHODOLOGIES ...... 667
16.1- FACTORS AND RESOURCES TO CONSIDER

IN ENGINEERING DESIGN BY

RELIABILITY ANALYSIS ...... 667

APPENDIX A- STANDARDIZED NORMAL

DISTRIBUTIONS’ AREA TABLES...... 671

ABOUT THE AUTHOR ...... 679
INDEX ...... 685