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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Structural Reliability Analysis and Prediction, Third Edition is a textbook which addresses the important issue of predicting the safety of structures at the design stage and also the safety of existing, perhaps deteriorating structures. Attention is focused on the development and definition of limit states such as serviceability and ultimate strength, the definition of failure and the various models which might be used to describe strength and loading. This book emphasises concepts and applications, built up from basic principles and avoids undue mathematical rigour. It presents an accessible and unified account of the theory and techniques for the analysis of the reliability of engineering structures using probability theory.
This new edition has been updated to cover new developments and applications and a new chapter is included which covers structural optimization in the context of reliability analysis. New examples and end of chapter problems are also now included.
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Veröffentlichungsjahr: 2017
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Table of Contents
Title Page
Copyright
Preface
Preface to the Second Edition
Preface to the First Edition
Acknowledgements
Chapter 1: Measures of Structural Reliability
1.1 Introduction
1.2 Deterministic Measures of Limit State Violation
1.3 A Partial Probabilistic Safety Measure of Limit State Violation—The Return Period
1.4 Probabilistic Measure of Limit State Violation
1.5 Generalized Reliability Problem
1.6 Conclusion
Chapter 2: Structural Reliability Assessment
2.1 Introduction
2.2 Uncertainties in Reliability Assessment
2.3 Integrated Risk Assessment
2.4 Criteria for Risk Acceptability
2.5 Nominal Probability of Failure
2.6 Hierarchy of Structural Reliability Measures
2.7 Conclusion
Chapter 3: Integration and Simulation Methods
3.1 Introduction
3.2 Direct and Numerical Integration
3.3 Monte Carlo Simulation
3.4 Importance Sampling
3.5 Directional Simulation*
3.6 Practical Aspects of Monte Carlo Simulation
3.7 Conclusion
Chapter 4: Second-Moment and Transformation Methods
4.1 Introduction
4.2 Second-Moment Concepts
4.3 First-Order Second-Moment (FOSM) Theory
4.4 The First-Order Reliability (FOR) Method
4.5 Second-Order Reliability (SOR) Methods
4.6 Application of FOSM/FOR/SOR Methods
4.7 Mean Value Methods
4.8 Conclusion
Chapter 5: Reliability of Structural Systems
5.1 Introduction
5.2 Systems Reliability Fundamentals
5.3 Monte Carlo Techniques for Systems
5.4 System Reliability Bounds
5.5 Implicit Limit States
5.6 Functionally Dependent Limit States
5.7 Conclusion
Chapter 6: Time-Dependent Reliability
6.1 Introduction
6.2 Time-Integrated Approach
6.3 Discretized Approach
6.4 Stochastic Process Theory
6.5 Stochastic Processes and Outcrossings
6.6 Time-Dependent Reliability
6.7 Load Combinations
6.8 Ensemble Crossing Rate and Barrier Failure Dominance
6.9 Dynamic Analysis of Structures
6.10 Fatigue Analysis
6.11 Conclusion
Chapter 7: Load and Load Effect Modelling
7.1 Introduction
7.2 Wind Loading
7.3 Wave Loading
7.4 Floor Loading
7.5 Conclusion
Chapter 8: Resistance Modelling
8.1 Introduction
8.2 Basic Properties of Hot-Rolled Steel Members
8.3 Properties of Steel Reinforcing Bars
8.4 Concrete Statistical Properties
8.5 Statistical Properties of Structural Members
8.6 Connections
8.7 Incorporation of Member Strength in Design
8.8 Conclusion
Chapter 9: Codes and Structural Reliability
9.1 Introduction
9.2 Structural Design Codes
9.3 Safety-Checking Formats
9.4 Relationship Between Level 1 and Level 2 Safety Measures
9.5 Selection of Code Safety Levels
9.6 Code Calibration Procedure
9.7 Example of Code Calibration
9.8 Observations
9.9 Performance-Based Design
9.10 Conclusion
Chapter 10: Probabilistic Evaluation of Existing Structures
10.1 Introduction
10.2 Assessment Procedures
10.3 Updating Probabilistic Information
10.4 Analytical Assessment
10.5 Acceptance Criteria for Existing Structures
10.6 Conclusion
Chapter 11: Structural Optimization and Reliability
11.1 Introduction
11.2 Types of Reliability-based Optimization Problems
11.3 Reliability Based Design Optimization (RBDO) Using First Order Reliability (FOR)
11.4 RBDO with System Reliability Constraints
11.5 Simulation-based Design Optimization
11.6 Life-cycle Cost and Risk Optimization
11.7 Discussion and Conclusion
Appendix A: Summary of Probability Theory
A.1 Probability
A.2 Mathematics of Probability
A.3 Description of Random Variables
A.4 Moments of Random Variables
A.5 Common Univariate Probability Distributions
A.6 Jointly Distributed Random Variables
A.7 Moments of Jointly Distributed Random Variables
A.8 Bivariate Normal Distribution
A.9 Transformation of Random Variables
A.10 Functions of Random Variables
A.11 Moments of Functions of Random Variables
A.12 Approximate Moments for General Functions
Appendix B: Rosenblatt and Other Transformations
B.1 Rosenblatt Transformation
B.2 Nataf Transformation
B.3 Orthogonal Transformation of Normal Random Variables
B.4 Generation of Dependent Random Vectors
Appendix C: Bivariate and Multivariate Normal Integrals
C.1 Bivariate Normal Integral
C.2 Multivariate Normal Integral
Appendix D: Complementary Standard Normal Table
D.1 Standard Normal Probability Density Function φ(
x
)
Appendix 5: Random Numbers
Appendix E: Selected Problems
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Measures of Structural Reliability
Figure 1.1 Bending moment diagrams for Example 1.1.
Figure 1.2 Example 1.2: Structure subject to overturning under lateral load
H
and with vertical load
W
and supported by two columns applying vertical forces
V
1
and
V
2
.
Figure 1.3 Reinforced concrete beam: Example 1.3.
Figure 1.4 Idealizations of actual load phenomenon for the ‘return period’ concept.
Figure 1.5 Histogram of private office live loads [after Culver, 1976].
Figure 1.6 Histogram and inferred distribution for structural steel yield strength [adapted from Alpsten, 1972 with permission of ASCE].
Figure 1.7 Schematic time-dependent reliability problem.
Figure 1.8 Space of the two random variable (
r
,
s
) and the joint density function
f
RS
(
r
,
s
), the marginal density functions
f
R
and
f
S
and the failure domain D.
Figure 1.9 Basic problem:
F
R
( )
f
S
( ) representation.
Figure 1.10 Basic problem:
f
R
( )
f
S
( ) representation.
Figure 1.11 Distribution of safety margin .
Figure 1.12 Definition of characteristic resistance.
Figure 1.13 Definition of characteristic load.
Figure 1.14 Failure probability
p
f
versus central safety factor λ
0
for lognormal (LN) and extreme value (EV-) distributions and different coefficients of variation.
Figure 1.15 Limit state violation indicators.
Chapter 2: Structural Reliability Assessment
Figure 2.1 Uncertainties in reliability assessment.
Figure 2.2 Modelling error (schematic).
Figure 2.3 Human performance function [Melchers, 1980].
Figure 2.4 Typical histogram of bending moments from analysis of loading.
Figure 2.5 Modification of resistance probability density function for human error and human intervention effects.
Figure 2.6 Schematic representation of the development of understanding of structural phenomena with time and research and the reduction of probability of failure from related technical failure, and the possible increase in the proportion of failures from human error.
Figure 2.7 Damage level for buildings in Kobe, Japan, as a function of year of construction [based on data reported by Fujino, 1996].
Figure 2.8 (a) Component costs and (b) total costs as a function of .
Figure 2.9 (a) Initial Cost curve; (b) optimal
p
fN
for : (c) optimal
p
fN
for .
Chapter 3: Integration and Simulation Methods
Figure 3.1 Inverse transform method for generation of random variates.
Figure 3.2 Use of fitted cumulative distribution function to estimate
p
f
.
Figure 3.3 Typical convergence of probability estimate with increasing sample size (schematic).
Figure 3.4 Importance sampling function ( ) in
x
space.
Figure 3.5 Highly concave limit state function in two dimensions showing obvious inefficiency of the importance sampling density.
Figure 3.6 Situations in which is not unique.
Figure 3.7 Poor choice of ( ) for case with two candidate points of maximum likelihood.
Figure 3.8 Censored region for importance sampling about .
Figure 3.9 A radial sample in direction and its intersection with the limit state at , showing also the implicit spherical limit state function.
Figure 3.10 Multi-valued solutions for in standard normal space.
Figure 3.11 Composite sampling function .
Figure 3.12 Two-dimensional load space showing a typical directional simulation and the probabilistic description of limit states and some realizations.
Figure 3.13 Schematic directional simulation in the load space showing variation of structural strength along a sample radial direction.
Chapter 4: Second-Moment and Transformation Methods
Figure 4.1 Limit state surface and its linearized version in the (2-D) space of the basic variables. Here the linearization is at point . (It is known also as the ‘checking point’ and is clearly the most probable point – MPP).
Figure 4.2 Probability density function contours and original (non-linear) and linearized limit state surfaces in the standard Normal space. The linearized version of is shown linearized at .
Figure 4.3 Marginal distribution in the space of the standardized Normal variables.
Figure 4.4 Linear limit state function in two-dimensional space.
Figure 4.5 Non-linear limit state function in two-dimensional basic variable space [note point
P
(1)
refers to Example 4.5].
Figure 4.6 One cycle of iterative solution scheme.
Figure 4.7 Breakdown of iteration through oscillation.
Figure 4.8 Inconsistency between β and
p
fN
for different forms of limit state functions.
Figure 4.9 Safety index β and
p
fN
for hyperspherical limit state surface.
Figure 4.10 Original and transformed probability density functions.
Figure 4.11 Relationships between original non-Normal
X
, standardized Normal
Y
and equivalent Normal
U
cumulative distribution functions.
Figure 4.12 Local stationary points for Example 4.7: (a) original order of dependent basic variables; (b) interchanged order.
Figure 4.13 Second order approximation to actual limit state surface at checking point in space, showing also first order (linear) approximation.
Chapter 5: Reliability of Structural Systems
Figure 5.1 (a) Simple example of load path dependence—horizontal load applied before vertical load produces a different failure mode than an attempt to apply the loads in the reverse order. (b) Typical internal action path OBC within many realistic structures.
Figure 5.2 Various strength-deformation (R-Δ) relationships.
Figure 5.3 Fault tree representation.
Figure 5.4 Event-tree representation for structure of Figure 5.3(a).
Figure 5.5 Failure-graph representation for structure of Figure 5.3(a).
Figure 5.6 Example series systems.
Figure 5.7 Basic structural system reliability problem in two dimensions showing failure domain
D
(and
D
1
).
Figure 5.8 Simple parallel systems: (a) parallel members (b) rigid-plastic frame.
Figure 5.9 Brittle material behaviour in parallel system.
Figure 5.10 Rigid frame and collapse modes: Example 5.4.
Figure 5.11 Conditional system.
Figure 5.12 Two variable system problems: (a) original space; (b) hypothetical standardized space, with failure regions shown hatched.
Figure 5.13 Resulting cdf and pdf in radial direction with multiple limit state functions each with a probabilistic representation [Melchers, 1992].
Figure 5.14 (a) Limit states for Example 5.5: (b) typical bounds for equi-correlated limit states [after Ditlevsen, 1979b].
Figure 5.15 Linearization of limit states in standard normal space.
Figure 5.16 Intersection of linear limit states.
Figure 5.17 Effect of correlation on system safety index .
Figure 5.18 Simple experimental design for a two-variable problem with as mean point.
Figure 5.19 Systematic enumeration procedure.
Chapter 6: Time-Dependent Reliability
Figure 6.1 Typical realization of random process load effect.
Figure 6.2 Realization of safety margin process
Z
(
t
) and time to failure.
Figure 6.3 Outcrossing of vector process .
Figure 6.4 Typical realizations of load effect
S
(
t
) and resistance
R
(
t
) when both are non-stationary.
Figure 6.5 Typical realizations of load effect
S
(
t
) and resistance
R
, with
R
a time-independent random variable.
Figure 6.6 Typical hazard functions
h
T
(
t
).
Figure 6.7 Typical variation of hazard function with age of structure.
Figure 6.8 Realizations
x
(
t
) of process
X
(
t
) showing clumping effect and barrier crossing.
Figure 6.9 Realization of a Borges process.
Figure 6.10 Realization of Poisson counting process.
Figure 6.11 Realization and upcrossing of a Poisson spike process.
Figure 6.12 Realization of Poisson square wave process.
Figure 6.13 Probability density function and cumulative distribution function for ‘mixed’ renewal process.
Figure 6.14 Typical realization and barrier crossing of continuous process
X
(
t
).
Figure 6.15 Segment of sample function
x
(
t
) crossing barrier
a
(
t
) in
dt
.
Figure 6.16 Integration limits in plane.
Figure 6.17 Outcrossing from safe domain
D
S
in two-dimensional space .
Figure 6.18 Vector process realization , its change in time increment and component normal to domain boundary
S
D
.
Figure 6.19 Typical realization of Poisson square wave vector process in two dimensions.
Figure 6.20 Parallel shift of hyperplane
H
l
.
Figure 6.21 Rigid-plastic frame and plastic collapse modes.
Figure 6.22 Projections of reliability indices and planes 1, 2 and 3.
Figure 6.23 Geometry to determine conditional reliability indices and correlation coefficients.
Figure 6.24 Combinations of random processes for which the ‘point-crossing’ formula is exact.
Figure 6.25 Typical realizations of mixed rectangular renewal processes with given probability density functions.
Figure 6.26 Realizations of two Borges processes, with , where
m
is an integer.
Figure 6.27 Realizations of and spectral densities for (a) wide-band, (b) white noise and (c) narrow-band random process.
Figure 6.28 Relationship between input and output spectral density functions for an example case of an offshore structure subject to hydrodynamic wave loading.
Figure 6.29 Probability density function for Rayleigh distribution.
Chapter 7: Load and Load Effect Modelling
Figure 7.1 Typical wind gust speeds for cyclones and thunderstorms plotted on a Gumbel (EV) plot [based on data from Gomes and Vickery, 1976].
Figure 7.2 Schematic representation of water particle motion, wave shape and force exerted on a tubular structural member.
Figure 7.3 Schematic time histories of typical live loads.
Figure 7.4 (a) Percentage of live load within 600 mm (2 ft) of walls [based on Culver, 1976]. (b) Observed distributions of floor loads on squares [based on Lind and Davenport, 1972]. (c) Floor load intensity for different usages, showing also the effect of area [based on data reported in Choi, 1992].
Figure 7.5 Variation of 95% level of occupancy loading with floor level number; one occupation only (i.e. 95% loads values plotted) [Mitchell and Woodgate, 1971a].
Figure 7.6 Variation of loading intensity probabilities with tributary area; floors other than lowest basements and ground floors; after one occupancy only [Mitchell and Woodgate, 1971a].
Figure 7.7 Variation of loading intensity probabilities with tributary area; floors other than lowest basements and ground floors; after 12 occupancies [Mitchell and Woodgate, 1971a].
Figure 7.8 Equivalent uniformly distributed load EUDL related to actual load realization.
Figure 7.9 Typical influence lines (in plane frames) and typical influence surfaces
Figure 7.10 Histograms of floor load intensity as modelled by distribution functions.
Figure 7.11 Single-storey beam and column fractiles as function of influence areas.
Chapter 8: Resistance Modelling
Figure 8.1 Typical histogram for yield strength of mild steel plates and shapes together with three fitted probability density functions [Adapted from Alpsten (1972) with permission from ASCE].
Figure 8.2 Typical histograms of cross-sectional dimensions for hot-rolled sections [Adapted from Alpsten (1972) with permission from ASCE].
Figure 8.3 Typical histograms for section properties for hot-rolled mild steel sections [Adapted from Alpsten (1972) with permission from ASCE].
Figure 8.4 Typical member strength simulation results for reinforced concrete beam-columns [reproduced from Ellingwood (1977) by permission of American Society of Civil Engineers].
Figure 8.5 Simulation procedure for member strength statistical properties.
Figure 8.6 Effect of discrete sizes on actual strength provided for reinforced concrete columns [based on Mirza and MacGregor (1979a) with permission from ASCE].
Figure 8.7 Typical probability density function for ratio of provided to required capacity for reinforcement concrete beams [reproduced from Mirza and MacGregor (1979a) with permission of ASCE].
Chapter 9: Codes and Structural Reliability
Figure 9.1 Variation in separation function α.
Figure 9.2 β index for steel and reinforced concrete beams in the existing code: gravity loads [Ellingwood et al., 1980].
Figure 9.3 β index for steel beams in the existing code: gravity plus wind loads [Ellingwood et al., 1980].
Figure 9.4 Variation of γ and φ factors for new safety-checking format for steel beam bending in the existing code rules [Ellingwood et al., 1980].
Figure 9.5 Flow chart for calibration of code safety-checking format.
Figure 9.6 Schematic relationship between variables in performance-based design.
Chapter 10: Probabilistic Evaluation of Existing Structures
Figure 10.1 Schematic representation of the effect of legal limits and enforcement on the upper tail of the probability density function for applied bridge loading [Melchers, 2008a].
Figure 10.2 Truncation effect on structural resistance.
Figure 10.3 Known (a priori) pdf for resistance as modified by new information (likelihood function) and modified (
posteriori
) pdf .
Figure 10.4 Predictive distributions for in-situ compressive strength of concrete (a) non-informative priors for and , (b) informative priors for both and
Figure 10.5 Typical bi-modal corrosion loss penetration as a function of exposure period showing (a) main phases in the early stages of corrosion and the effect of microbiologically influenced corrosion (MIC) and (b) the medium to longer-term linear bounding trend.
Figure 10.6 Beam in Example 10.2
Figure 10.7 Posteriori failure probability as a function of normalized proof load
q
*/
L
and the probability of failure under the application of the proof load (schematic).
Figure 10.8 Life-cycle reliability and assessment, showing effect of appropriate repairs (schematic).
Chapter 11: Structural Optimization and Reliability
Figure 11.1 Conceptual relationship between structural uncertainties in design and the increasing complexity of optimization problems.
Figure 11.2 Solutions of DDO, RBDO and LCRO problems solved using FOSM.
Figure 11.3 Probability density
f
G
(
g
) and cumulative distribution
F
G
(
g
) of the limit state function.
Figure 11.4 Interpretation of reliability constraint and generalized reliability index.
Figure 11.5 Comparison RIA x PMA: for active reliability constraint (left), points
y
*
RIA
and
y
*
PMA
are the same; for inactive constraint (right), these points are different.
Appendix A: Summary of Probability Theory
Figure A.1 Gamma probability density function.
Figure A.2 Normal probability density function.
Figure A.3 Lognormal probability density function.
Figure A.4 Beta distribution probability density function for different parameters.
Figure A.5 Extreme value distribution type I (Gumbel).
Figure A.6 Extreme value distribution type II (Frechet).
Figure A.7 Extreme value distribution type III (Weibull).
Figure A.8 Joint, marginal and conditional probability density functions.
Figure A.9 Linear dependence between two variables as a function of correlation coefficient ρ.
Appendix C: Bivariate and Multivariate Normal Integrals
Figure C.1 Bivariate normal probability density function, marginal probability density functions and regions of integration and
L
( ).
Figure C.2 Effect of correlation ρ between on the form of the bivariate Normal probability density function .
Figure C.3 (a) Integration region in original (correlated) standardized bivariate Normal space; (b) transformed integration region in transformed (independent) standardized bivariate Normal space.
Figure C.4 Bounds BPC and APD for Φ
2
( ) over region BPD.
Figure C.5 (a) Approximate and (b) reoriented equivalent limit state functions.
List of Tables
Chapter 1: Measures of Structural Reliability
Table 1.1 Typical limit states for structures
Table 1.2 Return period as function of design life
t
L
and exceedance probability
P
T
(
T
≤
t
L
).
Table 1.3 5% and 95% values for
X
k
/μ
X
Chapter 2: Structural Reliability Assessment
Table 2.1 Classification of human errors
Table 2.2 Error factors in observed failure cases
Table 2.3 Prime ‘causes’ of failure
Table 2.4 Human intervention strategies
Table 2.5 Selected risks in society (indicative)
Table 2.6 Broad indicators of tolerable risks [based on Otway et al. 1970]
Table 2.7 Typical ‘collapse’ failure rates for building structures
Table 2.8 Typical ‘collapse’ failure rates for bridges
Table 2.9 Social criteria factor
Table 2.10 Activity and warning factors
Table 2.11 Hierarchy of structural reliability measures
Chapter 3: Integration and Simulation Methods
Table 3.1
Chapter 5: Reliability of Structural Systems
Table 5.1 Basic MECE events and coefficients
c
i
for a three-component system
Chapter 6: Time-Dependent Reliability
Table 6.1 Analyses of failure probability for Example 6.2
Table 6.2 Initial conditions and state changes for upcrossings
Chapter 7: Load and Load Effect Modelling
Table 7.1 Typical average values of mean hourly wind speed data
Table 7.2 Indicative values for coefficients
C
d
and
C
m
for smooth clean cylinders
Table 7.3 Typical parameters for sustained floor loads (arbitrary-point-in-time values)
Table 7.4 Typical parameters for (multiple) extraordinary live loads
a)
Table 7.5 Example of typical basic loads for office floors, the parameters derived for the load models and the results of load combinations (see text)
Chapter 8: Resistance Modelling
Table 8.1 US yield stress
F
y
data
Table 8.2 British yield stress
F
y
data [Adapted from Baker, 1969]
Table 8.3 Swedish yield stress
F
y
data [Adapted from Alpsten, 1972]
Table 8.4 Static yield stress data [Adapted from Galambos and Ravindra, 1978]
Table 8.5 Elastic moduli of structural steel
Table 8.6 Coefficients for probability density function for yield strength of various grades of reinforcing bars [based on Mirza and McGregor, 1979b and Bournonville, et al., 2004]
Table 8.7 Variation of 'on-site' concrete compressive strength for control cylinders and cubes (between-batch)
Table 8.8 Statistical parameters for
k
cp
and
k
cr
[Stewart, 1997]
Table 8.9 Typical ratios (of test to nominal resistance) for beams in the plastic range [Yura et al., 1978]
Table 8.10 Modelling statistics (Professional factor P) [Ellingwood et al., 1980]
Table 8.11 Typical resistance statistics for reinforced concrete elements
Table 8.12 Typical statistical properties for discretization
Chapter 9: Codes and Structural Reliability
Table 9.1 Partial factors for CEB code format (CEB, 1976)
Table 9.2 Load combination factors for CEB code format
Table 9.3 Parameters for sample code calibration
Table 9.4 Overview of variables and parameters in performance-based design
Chapter 10: Probabilistic Evaluation of Existing Structures
Table 10.1 Mean and standard deviation for samples
Table 10.2 Statistical data for Example 10.2
Table 10.3 Adjustments to reliability index [Allen, 1991]
Table 10.4 Typical changes to load factors for evaluation of existing buildings [Allen, 1991]
Table 10.5 Typical resistance modification factors for bridges [after Allen, 1991]
Table 10.6 Factors influencing target failure probabilities [based on Schueremans, 2001]
Chapter 11: Structural Optimization and Reliability
Table 11.1 Interactive solution results of external optimization loop, shown at the end of each internal reliability loop
Appendix B: Rosenblatt and Other Transformations
Table B.1 Selected two-parameters distributions
Table B.2 Coefficients in formula for for selected distributions
Table B.3 Coefficients on formula for for selected Type 2 distributions
Appendix D: Complementary Standard Normal Table
Table D.1
N
(0, 1) distribution defined as Φ(−β) = 1 − Φ(β)
Structural Reliability Analysis and Prediction
Robert E. Melchers
The University of Newcastle Australia
André T. Beck
University of São Paulo Brazil
Third Edition
This edition first published 2018
© 2018 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data
Names: Melchers, R. E. (Robert E.), 1945- author. | Beck, André T., author.
Title: Structural reliability analysis and prediction / by Robert E.
Melchers, University of Newcastle, AS, André T. Beck, University of São
Paulo, BR.
Description: Third edition. | Hoboken, NJ : Wiley, 2018. | Includes
bibliographical references and index. |
Identifiers: LCCN 2017031782 (print) | LCCN 2017032813 (ebook) | ISBN
9781119266075 (pdf) | ISBN 9781119266068 (epub) | ISBN 9781119265993 (pbk.)
Subjects: LCSH: Structural stability. | Reliability (Engineering) |
Probabilities.
Classification: LCC TA656.5 (ebook) | LCC TA656.5 .M45 2018 (print) | DDC
624.1/71-dc23
LC record available at https://lccn.loc.gov/2017031782
Cover Design: Wiley
Cover Image: © maxsattana/Gettyimages
Preface
This third edition marks some 18 years since the second edition of this book appeared and what seems like half a lifetime ago—some 31 years—since the first edition was written. It has been extremely gratifying that the book has lasted this long, that it continues to be used by many and that a new edition was welcomed by Wiley.
Since the second edition the subject has consolidated and largely turned to more and more areas of application, including a renewed interest from the geotechnical engineering research community. But also in practice structural reliability increasingly is being applied, particularly for situations where quantitative, data-based risk assessment of non-elementary structural or other systems is required. Overviews of the papers contributed to conferences such as ICASP, ICOSSAR, IFIP, IALCCE and CSM shows much attention paid to applications and relatively little to sorting out some of the remaining really challenging theoretical problems such as how to deal with complex systems with a multitude of random variables or processes, and for which many potential failure modes and combination of such modes may exist. Fortunately, the availability of ever greater computational power has meant that enumeration methods, once thought to be the way forward for dealing with really complex problems, can be cast aside in favour of sheer brute force number crunching. In this sense Chapter 3 and the parts of Chapter 5 dealing with Monte Carlo methods are now more important, for practical problems, than the elegant but simpler FOSM/FOR/SOR methods that allow easier insight into ‘what was driving what’.
The present edition follows much of the second edition but updates areas such as Monte Carlo methods, systems reliability, some aspects of load and resistance modelling, code calibration, analysis of existing structures and adds, for the first time, a chapter on optimization in the context of structural reliability. The co-author for this edition, André T. Beck, has contributed much to these changes, as well as to the worked examples provided where relevant for each chapter and collected together in Appendix F. We have had the good fortune to have at hand the many comments and corrections, principally supplied by Dr. Bill Gray during his post-doctoral days at The University of Newcastle. As before, we have had to be selective in our coverage and have had to make difficult decisions about what to include and what leave out.
Now, as 18 years ago, a surf or a beach run at Newcastle's wonderful Pacific Ocean beaches, a surf or a bike ride along the south-eastern Brazilian coast, seem better ways to spend one's time than revising a book. Our spouses tell us so, our colleagues tell us so, our minds tell us so, but what do we do?
10 February 2017
Robert E. Melchers
Bar Beach, Newcastle
André T. Beck
Florianópolis, SC
Preface to the Second Edition
It is over ten years since the first edition of this book appeared and more than 12 years since the text was written. At the time structural reliability as a discipline was evolving rapidly but was also approaching a degree of maturity. Perhaps it is not surprising, then, that rather little of the first edition now seems out-dated.
This edition differs from the first mainly in matters of detail. The overall layout has been retained but all of the original text has been reviewed. Many sections have been partly rewritten to make them clearer and more complete and many, often small but annoying, errors and mistakes have been corrected. Hopefully not too many new ones have crept in. Many new references have been added and older, now less relevant, ones deleted. This is particularly the case in referring to applications, in which area there has been much progress.
The most significant changes in this edition include the up-dating of the sections dealing with Monte Carlo simulation, the addition of the Nataf transformation in the discussion of FOSM/FORM methods, some comments about asymptotic methods, additional discussion of structural systems subject to multiple loads and a new chapter devoted to the safety checking of existing structures, an area of increasing importance.
Other areas in which there have been rapid developments, such as simulation of random processes and random fields, applications in structural dynamics and fatigue and specialist refinements of theory are all of interest but beyond the scope of an introductory book. Readers might care to refer to the specialist literature, proceedings of conferences such as the ICASP, ICOSSAR and IFIP series and to journals such as Structural Safety, Probabilistic Engineering Mechanics and the Journals of Engineering Mechanics and Structural Engineering of ASCE. Overviews of various aspects of applications of structural reliability are given also in Progress in Structural Engineering and Mechanics. There are, of course, other places to look, but these should form a good starting point for keeping in touch with theoretical developments and applications.
In preparing this edition I had the good furtune to have at hand a range of comments, notes and advice. I am particularly indebted to my immediate colleagues Mark Stewart and Dimitry Val for their critical comments and their assistance with some of the new sections. Former research students have also contributed and I mention in this regard particularly H.Y. Chan, M. Moarefzadeh and X.L. Guan. Naturally, I owe a very significant debt to the international structural reliability community in general and to some key people in particular, including Ove Ditlevsen, Rudiger Rackwitz, Armen Der Kiureghian and Bruce Ellingwood—they, and many others, will know that I appreciate their forebearance and friendship.
The encouragement and generous comments from many sources is deeply appreciated. It has contributed to making the hard slog of revision a little less painful. Sometimes a beach run or a surf seemed a better alternative to spending an hour or so making more corrections to the text… As before, the forebearance of my family is deeply appreciated. Like many academic households they have learnt that academics are their own worst enemies and need occasionally to be dragged away from their Macintoshes to more socially acceptable activities.
August, 1998
Robert E. Melchers
Bar Beach, Newcastle
Preface to the First Edition
The aim of this book is to present a unified view of the techniques and theory for the analysis and prediction of the reliability of structures using probability theory. By reliability, in this context, will be understood not just reliability against extreme events such as structural collapse or facture, but against the violation of any structural engineering requirements which the structure is expected to satisfy.
In practice, two classes of problems may arise. In the first, the reliability of an existing structure at the ‘present time’ is required to be assessed. In the second, and much more difficult class, the likely reliability of some future, or as yet uncompleted, structure must be predicted. One common example of such a requirement is in structural design codes, which are essentially instruments for the prediction of structural safety and serviceability supported by previous experience and expert opinion. Another example is the reliability assessment of major structures such as large towers, offshore platforms and industrial or nuclear plants for which structural design codes are either not available or not wholly acceptable. In this situation, the prediction of safety both in absolute terms and in terms of its interrelation to project economics is becoming increasingly important. This class of assessment relies on the (usually reasonable but potentially dangerous) assumption that past experience can be extrapolated into the future.
It might be evident from these remarks that the analysis (and prediction) of structural reliability is rather different from the types of analysis normally performed in structural engineering. Concern is less with details of stress calculations, or member behaviour, but rather with the uncertainties in such behaviour and how this interacts with uncertainties in loading and in material strength. Because such uncertainties cannot be directly observed for any one particular structure, there is a much greater level of abstraction and conceptualization in reliability analysis than is conventionally the case for structural analysis or design. Modelling is not only concerned with the proper and appropriate representation of the physics of any structural engineering problem, but also with the need to obtain realistic, sufficiently simple and workable models or representations of both the loads and the material strengths, and also their respective uncertainties. How such modelling might be done and how such models can be used to analyse or predict structural reliability is the central theme of this book.
In one important sense, however, the subject matter has a distinct parallel with conventional structural engineering analysis and its continual refinement; that is, that ultimately concern is with costs. Such costs include not only those of design, construction, supervision and maintenance but also the possible cost of failure (or loss of serviceability). This theme, although not explicitly pursued throughout the book, is nevertheless a central one, as will become clear in Chapter 2. The assessment or predictions obtained using the methods outlined in this book have direct application in decision-making techniques such as cost-benefit analysis or, more precisely when probability is included, risk-benefit analysis. As will be seen in Chapter 9, one important area of application for the methods presented here is in structural design codes, which, it will be recognized, are essentially particular (if perhaps rather crude and intuitive) forms of risk-benefit methodology.
A number of other recent books have been devoted to the structural reliability theme. This book is distinct from the others in that it has evolved from a short course of lectures for undergraduate students as well as a 30-h graduate course of lectures which the author has given periodically to (mainly) practising structural engineers during the last 8 years. It is also different in that it does not attempt to deal with related topics such as spectral analysis for which excellent introductory texts are already available.
Other features of the present book are its treatment of structural system reliability (Chapter 5) and the discussion of both simulation methods (Chapter 3) and modern second-moment and transformation methods (Chapter 4). Also considered is the important topic of human error and human intervention in the relationship between calculated (or ‘nominal’) failure probabilities and those observed in populations of real structures (Chapter 2).
The book commences (Chapter 1) by reviewing traditional methods of defining structural safety such as the ‘factor of safety’, the ‘load factor’, ‘partial factor’ formats (i.e. ‘limit state design’ formats) and the ‘return period’. Some consistency aspects of these methods are then presented and their limited use of available data noted, before a simple probabilistic safety measure, the ‘safety margin’ and the associated failure probability are introduced. This simple one-load one-resistance model is sufficient to introduce the fundamental ideas of structural reliability assessment. Apart from Chapter 2, the rest of the book is concerned with elaborating and illustrating the reliability analysis and prediction theme.
While Chapters 3, 4 and 5 deal with particular calculation techniques for time-independent situations, Chapter 6 is concerned with extending the ‘return period’ concept introduced in Chapter 1 to more general formulations for time-dependent problems. The three principal methods for introducing time, the time-integrated approach, the discrete time approach and the fully time-dependent approach, are each outlined and examples given. The last approach is considerably more demanding than the other two (classical) methods since it is necessary to introduce elements of stochastic process theory. First-time readers may well decide to skip rather quickly through much of this chapter. Applications to fatigue problems and structural vibrations are briefly discussed from the point of view of probability theory, but again the physics of these problems is outside the scope of the present book.
Modelling of wind and floor loadings is described in Chapter 7 whilst Chapter 8 reviews probability models generally accepted for steel properties. Both load and strength models are then used in Chapter 9. This deals with the theory of structural design codes and code calibration, an important area of application for probabilistic reliability prediction methodology.
It will be assumed throughout that the reader is familiar with modern methods of structural analysis and that he (or she) has a basic background in statistics and probability. Statistical data analysis is well described in existing texts; a summary of probability theory used is given in Appendix A for convenience.
Further, reasonable competence in applied mathematics is assumed since no meaningful discussion of structural reliability theory can be had without it. The level of presentation, however, should not be beyond the grasp of final-year undergraduate students in engineering. Nevertheless, particularly difficult theoretical sections which might be skipped on a first reading are marked with an asterisk (*).
For teaching purposes, Chapters 1 and 2 could form the basis for a short undergraduate course in structural safety. A graduate course could take up the topics covered in all chapters, with instructors having a bias for second-moment methods skipping over some of the sections in Chapter 3 while those who might wish to concentrate on simulation could spend less time on Chapter 4. For an emphasis on code writing, Chapters 3 and 5 could be deleted and Chapters 4 and 6 cut short.
In all cases it is essential, in the author's view, that the theoretical material be supplemented by examples from experience. One way of achieving this is to discuss particular cases of structural failure in quite some detail, so that students realize that the theory is only one (and perhaps the least important) aspect of structural reliability. Structural reliability assessment is not a substitute for other methods of thinking about safety, nor is it necessarily any better; properly used, however, it has the potential to clarify and expose the issues of importance.
Acknowledgements
This book has been a long time in the making. Throughout I have had the support and encouragement of Noel Murray, who first started me thinking seriously about structural safety, and also of Paul Grundy and Alan Holgate. In more recent times, research students Michael Harrington, Tang Liing Kiong, Mark Stewart and Chan Hon Ying have played an important part.
The first (and now unrecognizable) draft of part of the present book was commenced shortly after I visited the Technical University, Munich, during 1980 as a von Humboldt Fellow. I am deeply indebted to Gerhart Schueller, now of Universität Innsbruck, for arranging this visit, for his kind hospitality and his encouragement. During this time, and later, I was also able to have fruitful discussions with Rudiger Rackwitz.
Part of the last major revision of the book was written in the period November 1984-May 1985, when I visited the Imperial College of Science and Technology, London, with the support of the Science and Engineering Research Council. Working with Michael Baker was a most stimulating experience. His own book (with Thoft-Christensen) has been a valuable source of reference.
Throughout I have been extremely fortunate in having Mrs. Joy Helm and more recently Mrs. Anna Teneketzis turn my difficult manuscript into legible typescript. Their cheerful co-operation is very much appreciated, as is the efficient manner with which Rob Alexander produced the line drawings.
Finally the forbearance of my family was important, many a writing session being abruptly concluded with a cheerful ‘How's Chapter 6 going, Dad?’
December 1985
Robert E. Melchers
Monash University
Chapter 1Measures of Structural Reliability
1.1 Introduction
The manner in which an engineering structure will respond to loading depends on the type and magnitude of the applied load and the structural strength and stiffness. Whether the response is considered satisfactory depends on the requirements that must be satisfied. These include safety of the structure against collapse, limitations on damage, or on deflections or other criteria. Each such requirement may be termed a limit state. The ‘violation’ of a limit state can then be defined as the attainment of an undesirable condition for the structure. Some typical limit states are given in Table 1.1.
Table 1.1 Typical limit states for structures
Limit state type
Description
Examples
Ultimate (safety)
Collapse of all or part of structure
Tipping or sliding, rupture, progressive collapse, plastic mechanism, instability, corrosion, fatigue, deterioration, fire.
Damage(often included in above)
Excessive or premature cracking, deformation or permanent inelastic deformation.
Serviceability
Disruption of normal use
Excessive deflections, vibrations, local damage, etc.
From observation it is known that very few structures collapse, or require major repairs, etc., so that the violation of the most serious limit states is a relatively rare occurrence. When violation of a limit state does occur, the consequences may be extreme, as exemplified by the spectacular collapses of structures such as the Tay Bridge (wind loading), Ronan Point Flats (gas explosion), Kielland Offshore Platform (local strength problems), Kobe earthquake (ductility), etc.
The study of structural reliability is concerned with the calculation and prediction of the probability of limit state violation for an engineered structural system at any stage during its life. In particular, the study of structural safety is concerned with the violation of the ultimate or safety limit states for the structure. More generally, the study of structural reliability is concerned with the violation of performance measures (of which ultimate or safety limit states are a subset). This broader definition allows the scope of application to move from structural criteria as specified in traditional design codes (Chapter 9) to broader-based performance requirements for structures, such as might be used in design optimization processes (Chapter 11).
In the simplest case, the probability of occurrence of an event such as limit state violation is a numerical measure of the chance of its occurrence. This measure either may be obtained from measurements of the long-term frequency of occurrence of the event for generally similar structures, or it may be simply a subjective estimate of the numerical value. In practice it is seldom possible to observe for a sufficiently long period of time, and a combination of subjective estimates and frequency observations for structural components and properties may be used to predict the probability of limit state violation for the structure.
In probabilistic assessments any uncertainty about a variable (expressed, as will be seen, in terms of its probability density function) is taken into account explicitly. This is not the case in traditional ways of measuring safety, such as the ‘factor of safety’ or ‘load factor’. These are ‘deterministic’ measures, since the variables describing the structure, its strength and the applied loads are assumed to take on known (if conservative) values about which there is assumed to be no uncertainty. Precisely because of their traditional and really quite central position in structural engineering, it is appropriate to review the deterministic safety measures prior to developing probabilistic safety measures.
1.2 Deterministic Measures of Limit State Violation
1.2.1 Factor of Safety
The traditional method to define structural safety is through a ‘factor of safety’, usually associated with elastic stress analysis and which requires that:
where σi(ϵ) is the i th applied stress component calculated to act at the generic point ϵ in the structure, and σpi is the permissible stress for the i th stress component.
The permissible stresses σpi are usually defined in structural design codes. They are derived from material strengths (ultimate moment, yield point moment, squash load, etc.), expressed in stress terms σui but reduced through a factor F:
where F is the ‘factor of safety’. The factor F may be selected on the basis of experimental observations, previous practical experience, economic and, perhaps, political considerations. Usually, its selection is the responsibility of a code committee.
According to (1.1), failure of the structure should occur when any stressed part of it reaches the local permissible stress. Whether failure actually does occur depends entirely on how well σi(ϵ) represents the actual stress in the real structure at ϵ and how well σpi represents actual material failure. It is well known that observed stresses do not always correspond well to the stresses calculated by linear elastic structural analysis (as commonly used in design). Stress redistribution, stress concentration and changes due to boundary effects and the physical size effect of members all contribute to the discrepancies.
Similarly, the permissible stresses that, commonly, are associated with linear elastic stress analysis are not infrequently obtained by linear scaling down, from well beyond the linear region, of the ultimate strengths obtained from tests. From the point of view of structural safety, this does not matter very much, provided that the designer recognizes that his calculations may well be quite fictitious and provided that (1.1) is a conservative safety measure.
By combining expressions (1.1) and (1.2) the condition of ‘limit state violation’ can be written as
Expressions (1.3) are ‘limit state equations’ when the inequality sign is replaced by an equality. These equations can be given also in terms of stress resultants, obtained by appropriate integration:
where Ri is the i th resistance at location ϵ and Si is the i th stress resultant (internal action). In general, the stress resultant Si are made up of the effects of one or more applied loads Qj; typically
where D is the dead load, L is the live load and W is the wind load.
The term ‘safety factor’ also has been used in another sense, namely in relation to overturning, sliding, etc., of structures as a whole, or as in geomechanics (dam failure, embankment slip, etc.). In this application, expressions (1.3) are still valid provided that the stresses σui and σi are interpreted appropriately.
1.2.2 Load Factor
The ‘load factor’ λ is a special kind of safety factor developed for use originally in the plastic theory of structures. It is the theoretical factor by which a set of loads acting on the structure must be multiplied, just enough to cause the structure to collapse. Commonly, the loads are taken as those acting on the structure during service load conditions. The strength of the structure is determined from the idealized plastic material strength properties for structural members [Heyman, 1971].
For a given collapse mode (i.e. for a given ultimate ‘limit state’), the structure is considered to have ‘failed’ or collapsed when the plastic resistances Rpi are related to the factored loads λQj by
where RP is the vector of all plastic resistances (e.g. plastic moments) and Q is the vector of all applied loads. Also, WR( ) is the internal work function and WQ( ) the external work function, both described by the plastic collapse mode being considered.
If proportional loading is assumed, as is usual, the load factor can be taken out of parentheses. Also the loads Qj usually consist of several components, such as dead, live, wind, etc. Thus (1.5) may be written in the form of a limit state equation:
with ‘failure’ denoted by the left-hand side being less than unity.
Clearly there is much similarity in formulation between the factor of safety and the load factor as measures of structural safety. What is different is the reference level at which the two measures operate: the first at the level of working loads and at the ‘member’ level; the second at the level of collapse loads and at the ‘structure level’.
1.2.3 Partial Factor (‘Limit State Design’)
A development of the above two measures of safety is the so-called ‘partial factor’ approach. For limit state i it can be expressed at the level of stress resultants (i.e. member design level) as
where R is the member resistance, φ is the partial factor on R and SD, SL are the dead and live load effects respectively with associated partial factors γD, γL. Expression (1.6) was originally developed during the 1960s for reinforced concrete codes. It enabled the live and wind loads to have greater ‘partial’ factors than the dead load, in view of the former's greater uncertainty, and it allowed a measure of workmanship variability and uncertainty about resistance modelling to be associated with the resistance R [MacGregor, 1976]. This extension of earlier safety formats had considerable appeal since it allowed better representation of the factors and uncertainties associated with loadings and resistances.
For a plastic collapse analysis at the structure level, formulation (1.6) becomes
where R and Q are vectors of resistance and loads respectively. Clearly the partial factors (φ, γ) in this expression will be different from those of expression (1.6).
Figure 1.1 Bending moment diagrams for Example 1.1.
Example 1.1
The simple portal frame of Figure 1.1(a) is subject to loads Q1 and Q2. If the relative moments of inertia of the members are known, the elastic bending moment diagram can be found as in Fig 1.1(b). The ‘limit states’ for bending capacity are then
where φ, γ1 and γ2 are partial factors described by a structural design code. The MCi are the ultimate moment capacities required at sections for the structure to be considered ‘just safe’.
If the frame is to be designed or analysed assuming rigid-plastic theory, the relative distribution of the plastic moments Mpi around the frame must be known or assumed. If they all are equal, the plastic bending moment diagram of [Figure 1.1(c) is obtained and only one limit state equation is needed for sections 1–3:]
where now Mpi is the required plastic moment capacity at sections 1, 2 and 3 and where φp, γp1 and γp2 are now code-prescribed partial factors for plastic structural systems.
1.2.4 A Deficiency in Some Safety Measures: Lack of Invariance
From Example 1.1, it will be evident that the partial factors φ and in (1.6) depend on the limit state being considered. Hence they depend on the definitions of R, SD and SL. However, even for a given limit state, these definitions are not necessarily unique, and therefore the partial factors may not be unique either. This phenomenon is termed the ‘lack of invariance’ of the safety measure. It arises because there are different ways in which the relationships between resistances and loads may be defined. Some examples of this are given below. Ideally, the safety measure should not depend on the way in which the loads and resistances are defined.
Figure 1.2 Example 1.2: Structure subject to overturning under lateral load H and with vertical load W and supported by two columns applying vertical forces V1 and V2.
Example 1.2
The structure shown in Figure 1.2 is supported on two columns. The capacity of column B is in compression. The safety of the structure can be measured in three different ways using the traditional ‘factor of safety’ F:
a.
Overturning resistance about A
b.
Capacity of column B
c.
Net capacity of column B
(resistance minus load effect of W)
All three of these factors of safety for column B apply to the same structure and the same loading, so that the difference in the values of Fi
