Reliability Evaluation of Dynamic Systems Excited in Time Domain - Redset E-Book

Achintya Haldar

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 476

Veröffentlichungsjahr: 2023

Beschreibung

RELIABILITY EVALUATION OF DYNAMIC SYSTEMS EXCITED IN TIME DOMAIN - REDSET Multi-disciplinary approach to structural reliability analysis for dynamic loadings offering a practical alternative to the random vibration theory and simulation Reliability Evaluation of Dynamic Systems Excited in Time Domain - REDSET is a multidisciplinary concept that enables readers to estimate the underlying risk that could not be solved in the past. The major hurdle was that the required limit state functions (LSFs) are implicit in nature and the lack of progress in the reliability evaluation methods for this class of problems. The most sophisticated deterministic analysis requires that the dynamic loadings must be applied in the time domain. To satisfy these requirements, REDSET is developed. Different types and forms of dynamic loadings including seismic, wind-induced wave, and thermomechanical loading in the form of heating and cooling of solder balls used in computer chips are considered to validate REDSET. Time domain representations and the uncertainty quantification procedures including the use of multiple time histories are proposed and demonstrated for all these dynamic loadings. Both onshore and offshore structures are used for validation. The potential of REDSET is demonstrated for implementing the Performance Based Seismic Design (PBSD) concept now under development in the United States. For wider multidisciplinary applications, structures are represented by finite elements to capture different types of nonlinearity more appropriately. Any computer program capable of conducting nonlinear time domain dynamic analysis can be used, and the underlying risk can be estimated with the help of several dozens or hundreds of deterministic finite element analyses, providing an alternative to the simulation approach. To aid comprehension of REDSET, numerous illustrative examples and solution strategies are presented in each chapter. Written by award-winning thought leaders from academia and professional practice, the following sample topics are included: * Fundamentals of reliability assessment including set theory, modeling of uncertainty, the risk-based engineering design concept, and the evolution of reliability assessment methods * Implicit performance or limit state functions are expressed explicitly by the extensively modified response surface method with several new experimental designs * Uncertainty quantification procedures with multiple time histories for different dynamic loadings, illustrated with examples * The underlying risk can be estimated using any computer program representing structures by finite elements with only few deterministic analyses * REDSET is demonstrated to be an alternative to the classical random vibration concept and the basic simulation procedure for risk estimation purposes * REDSET changes the current engineering design paradigm. Instead of conducting one deterministic analysis, a design can be made more dynamic load tolerant, resilient, and sustainable with the help of a few additional deterministic analyses This book describing REDSET is expected to complement two other books published by Wiley and authored by Haldar and Mahadevan: Probability, Reliability and Statistical Methods in Engineering Design and Reliability Assessment Using Stochastic Finite Element Analysis. The book is perfect to use as a supplementary resource for upper-level undergraduate and graduate level courses on reliability and risk-based design.

Leseprobe

Reliability Evaluation of Dynamic Systems Excited in Time Domain

Alternative to Random Vibration and Simulation

Achintya Haldar

University of Arizona, Tucson, Arizona, USA

Hamoon Azizsoltani

North Carolina State University, Raleigh, North Carolina, USA

J. Ramon Gaxiola‐Camacho

Autonomous University of Sinaloa, Culiacan, Mexico

Sayyed Mohsen Vazirizade

Vanderbilt University, Nashville, Tennessee, USA

Jungwon Huh

Chonnam National University, Gwangju, Korea

This edition first published 2023© 2023 John Wiley & Sons, Inc.

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The right of Achintya Haldar, Hamoon Azizsoltani, J. Ramon Gaxiola‐Camacho, Sayyed Mohsen Vazirizade, and Jungwon Huh to be identified as the authors of this work has been asserted in accordance with law.

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Library of Congress Cataloging‐in‐Publication DataNames: Haldar, Achintya, author.Title: Reliability evaluation of dynamic systems excited in time domain : alternative to random vibration and simulation / Achintya Haldar, Hamoon Azizsoltani, J. Ramon Gaxiola-Camacho, Sayyed Mohsen Vazirizade, Jungwon Huh.Description: Hoboken, NJ, USA : Wiley, [2023]Identifiers: LCCN 2022047650 (print) | LCCN 2022047651 (ebook) | ISBN 9781119901648 (hardback) | ISBN 9781119901662 (adobe pdf) | ISBN 9781119901655 (epub)Subjects: LCSH: Reliability (Engineering)—Mathematics. | Dynamics—Mathematics. | Vibration—Mathematical models.Classification: LCC TA169 .H3523 2023 (print) | LCC TA169 (ebook) | DDC 620/.004520151—dc23/eng/20221205LC record available at https://lccn.loc.gov/2022047650LC ebook record available at https://lccn.loc.gov/2022047651

Cover image: Courtesy of Achintya HaldarCover design by Wiley

1REDSET and Its Necessity

1.1 Introductory Comments

A novel reliability evaluation method, denoted hereafter as REDSET (Reliability Evaluation of Dynamic Systems Excited in Time Domain) is proposed. As the secondary heading of the book suggests, REDSET will be an alternative to the classical random vibration and simulation techniques. REDSET is expected to address a knowledge gap that will be discussed later in this chapter to help us better meet current professional needs and/or requirements.

Pierre Simon Marquis de Laplace (1749–1827) wrote a commentary on general intelligence published as “A Philosophical Essay on Probabilities.” He wrote It is seen in this essay that the theory of probabilities is at bottom only common sense reduced to calculus; it makes us appreciate with exactitude that which exact minds feel by a sort of instinct without being able ofttimes to give a reason for it. It leaves no arbitrariness in the choice of opinions and sides to be taken; and by its use can always be determined the most advantageous choice. Thereby it supplements most happily the ignorance and weakness of the human mind. (Laplace 1951). These timeless comments have a significant influence on every aspect of human activities. However, the concept of risk or reliability‐based engineering analysis, design, and planning was initiated recently, most likely by Freudenthal (1956). As a result of comprehensive efforts by different engineering disciplines, design guidelines and codes have already been modified or are in the process of being modified. The Accreditation Board of Engineering and Technology (ABET) in the United States now requires that all undergraduate students in civil engineering demonstrate the necessary knowledge to follow or interpret the requirements outlined in design guidelines and apply them in everyday practices.

1.2 Reliability Evaluation Procedures Existed Around 2000

After Freudenthal's initiative and the necessity of acquiring the required mathematical knowledge, a considerable amount of development in the reliability evaluation area was reported in the literature. The team members of the authors wrote two books earlier, published by John Wiley, titled “Probability, Reliability and Statistical Methods in Engineering Design” and “Reliability Assessment Using Stochastic Finite Element Analysis” (Haldar and Mahadevan 2000a, 2000b). The first book is now being used as a textbook all over the world. It covers basic reliability concepts for estimating the risk of structures that existed around 2000.

The reliability or risk of an engineering system is always estimated with respect to a performance requirement or limit state function (LSF). The first book presented various risk estimation methods with various levels of sophistication when LSFs are explicit and readily available that existed around 2000. However, when a structure is represented by finite elements (FEs), as in the second book, LSFs are expected to be implicit in most applications to appropriately incorporate information on different sources of nonlinearities satisfying the physics‐based formulation to accurately estimate the required structural responses. The stochastic finite element method (SFEM) concept presented in the second book is capable of estimating reliability for implicit LSFs, but the required derivatives of LFSs with respect to the design variables to estimate risk are evaluated numerically. The basic SFEM concept cannot be used to estimate the risk of complex nonlinear dynamic engineering systems (CNDES) and an alternative is urgently needed. This book will fill the vacuum.

1.3 Improvements or Alternative to Stochastic Finite Element Method (SFEM)

One of the major impacts of the SFEM concept was that it helped reliability analysis of structures represented by FEs commonly used in many engineering disciplines. The basic finite element method (FEM)‐based representation helps estimate the deterministic behavior of structures considering complicated geometric arrangements of elements made with different materials, realistic connections and support conditions, various sources of nonlinearity, and numerous sophisticated and complicated features a structure experiences from the initial to the failure state following different load paths in a very comprehensive way. However, the deterministic FEM formulation fails to incorporate the presence of uncertainty in the design variables and thus cannot estimate the underlying risk. The SFEM concept was proposed to capture the desirable features of the deterministic FEM formulation integrated with probabilistic or stochastic information on design variables. However, it was primarily developed for the static application of loadings. For ease of discussion, it can be denoted as SFEM‐static. It was a significant improvement over other available methods around the mid‐nineties.

The SFEM‐static concept or its extension cannot be used to estimate the reliability of CNDES. It can be a building block for the reliability analysis of CNDES by representing structures using FEs with improved physics‐based modeling techniques, incorporating recently introduced several advanced energy dissipation features, and exciting them in the time domain resulting in the SFEM‐dynamic algorithm. However, it should be noted that SFEM‐static was developed about three decades ago using outdated computational platforms not available at present. A considerable amount of time, money, and effort will be needed to modify it for current needs, and it may not be the best option. An improved version of SFEM‐static is needed.

The risk evaluation capability of a procedure representing structures by FEs will depend on the efficiency of the deterministic FEM formulation used. The FEM representation should also be similar to the procedures used by the deterministic community in routine applications. To estimate nonlinear responses of frame structures, the displacement‐based FEM is very commonly used. Almost all current commercially available computer programs are based on this concept. In this approach, shape functions are used to describe the displacements at the nodes of the elements. This will require a large number of elements to model members with large deformation expected just before failure, making it computationally very inefficient. It will be more efficient if the realistic structural behavior can be estimated using fewer elements and expressing the tangent stiffness matrix explicitly without updating it at every step of the nonlinear analysis. To address these issues, the assumed stress‐based FEM can be used as an alternative, especially for frame‐type structures.

The assumed stress‐based FEM formulation, although mathematically more demanding, has many advantages over the displacement‐based approach. In this approach, the tangent stiffness can be expressed in explicit form, the stresses of an element can be expressed and obtained directly, fewer elements are required in describing a large deformation configuration, and integration is not required to obtain the tangent stiffness. It is found to be accurate and very efficient in analyzing frame‐type structures just before failure.

The probability of failure of a frame needs to be estimated using two iterative schemes, one to capture the nonlinear behavior and the other for the reliability estimation to consider uncertainty in the design variables. In developing SFEM‐static, the displacement‐based FEM approach was initially used. Subsequent studies indicated that it would be practically impossible to use this approach to extract reliability information for realistic large nonlinear structural systems. Although the stress‐based FEM concept was at the early stage of development in the early eighties, it was incorporated in the subsequent developments of SFEM‐static. The book on SFEM is essentially based on the stress‐based FEM approach. The modified concept was extensively verified using available information.

Sophisticated computer programs were developed to implement the concept. They were written in the FORTRAN language available in the early eighties and no users' manual was developed. Most importantly, there is no other program using the concept readily available even today for commercial or academic research. Related subjects are not taught in most universities, and the basic concept is not fully developed to analyze complicated structural systems. The book by Haldar and Mahadevan (2000b) can be used. However, it will not satisfy the current professional needs.

In summary, it will take a considerable amount of time and effort to modify the SFEM‐static programs suitable for different computer platforms widely used at present. The most attractive option will be if users can use any computer program capable of conducting nonlinear time domain dynamic analyses using any type of FEM to estimate structural responses.

1.4 Other Alternatives Besides SFEM

Several reliability‐evaluation techniques for CNDES available at present will be briefly discussed next. Considering their deficiencies and current needs, a new concept is preferable.

1.4.1 Random Vibration

To study the stochastic behavior of dynamic systems, the classical random vibration approach (Lin 1967; Lin and Cai 2004) is expected to be the obvious choice. It is an extremely sophisticated but complicated mathematical concept. The basic random vibration concept and its derivatives include the First- or Second-Order Taylor Series Expansion, Neumann Expansion, Karhunen‐Loeve Orthogonal Expansion, Polynomial Chaos, etc. Most of them were developed for relatively linear small systems with very few numbers of dynamic degrees of freedom (DDOFs). They are unable to consider the physics‐based representation of nonlinear dynamic systems, failed to explicitly consider the statistical distribution information of system parameters, and were valid only for a small amount of randomness at the element level even when it had the potential to be significantly amplified at the system level. The dynamic loading is represented in the form of power spectral density functions and cannot be applied in the time domain as currently required for critical structures in design guidelines. Some of the novel features proposed after the Northridge earthquake of 1994 to improve dynamic response behavior of structures by making the connections more flexible cannot be incorporated (to be discussed in more detail in Chapter 8). The development of the random vibration concept was an important research topic during the latter half of the twentieth century and attracted the attention of scholars interested in applying sophisticated mathematical concepts in engineering applications. However, the overall effort had a marginal impact on the profession. A considerable knowledge gap still exists to study the stochastic behavior of CNDES. As the secondary title of the book suggests, an alternative to the classical random vibration concept is urgently needed.

1.4.2 Alternative to Basic Monte Carlo Simulation

A deterministic analysis of a realistic CNDES represented by FEs considering some of the newly developed attractive features may take several hours of computer time. To estimate the risk of low probability events using Monte Carlo simulation (MCS) is a possibility but it may require the continuous running of a computer for several years, as explained in Chapter 5. An alternative theoretical concept is necessary. As the secondary title of this book suggests, an alternative to MCS is also necessary.

1.4.3 Alternatives to Random Vibration Approach for Large Problems

Besides the random vibration approach, several uncertainty quantification methods for large computational models were proposed including the reduced order models, surrogate models, Bayesian methods, stochastic dimension reduction techniques, efficient MCS methods with numerous space reduction techniques, etc. Some of them are problem‐specific, will not satisfy current needs, and will require considerable expertise to implement.

1.4.4 Physics‐Based Deterministic FEM Formulation

The probability of failure estimation for CNDES implies that the risk needs to be estimated just before failure. The condition may be initiated by the failure of one or more structural elements in strength or if the system develops excessive vibration or deformation (deflection, rotation, etc.) making it not functional for which it was designed. Just before failure, the engineering systems are expected to go through several stochastic phases that are difficult to postulate. If the excitation is very irregular, like seismic or wave loading (discussed in Chapter 4) or thermomechanical loading (discussed in Chapter 9), it will add several additional layers of challenge. Most importantly, the deterministic community that makes the final approval decision requires that these loadings need to be applied appropriately in the time domain for the reliability estimation.

Additionally, the analytical models used to represent large engineering systems are very idealized. For example, the common assumption that supports and joints of a structure are rigidly connected or fully restrained (FR) is a major simplification; they are essentially partially restrained (PR) with different rigidities (discussed in more detail in Chapter 8). The uncertainties in the dynamic properties of structures are expected to be quite different when the connections are loading, unloading, and reloading in a very irregular fashion. Also, the failure of a member of highly indeterminate structures is expected to cause local failure without causing system failure. Current smart designs intentionally introduce alternate load transmission paths to avoid system failure. The recent trend of making connections more flexible in frame‐type structures providing more ductility to increase the energy absorption capacity needs to be appropriately incorporated into the algorithm. An intelligent risk estimation procedure should be able to incorporate these advanced features indicating a reduction in the underlying risk.

After the Northridge earthquake of 1994, the Federal Emergency Management Agency (FEMA) advocated for using the performance‐based seismic design (PBSD) concept proposed after the earthquake to reduce the economic loss by replacing the life safety criterion used in the past. It is a very sophisticated risk‐based design method, but FEMA did not specify how to estimate the underlying risk. In fact, there is no method currently available to implement PBSD.

1.4.5 Multidisciplinary Activities to Study the Presence of Uncertainty in Large Engineering Systems

Several new methods were proposed to address uncertainty‐related issues in large engineering applications in the recent past, including high‐dimensional model representation (HDMR) and explicit design space decomposition – support vector machines (EDSD – SVM). In these studies, the general objective is to develop approximate multivariate expressions for a specific response surface (discussed in detail in Chapter 3). One such method is HDMR. It is also referred to as “decomposition methods,” “univariate approximation,” “bivariate approximation,” “S‐variate approximation,” etc. HDMR is a general set of quantitative model assessment and analysis tools for capturing high‐dimensional relationships between sets of input and output model variables in such a way that the component functions are ordered starting from a constant and adding higher order terms, such as first, second, etc. The concept is reasonable if the physical model is capable of capturing the behavior using the first few lower‐order terms. However, it cannot be applied for physics‐based time domain dynamic analysis and requires MCS to extract reliability information. EDSD can be used when responses are classified into two classes, e.g. safe and unsafe. A machine learning technique known as support vector machines (SVM) was used to construct the boundaries separating different modes of failure with a single SVM boundary and refined through adaptive sampling. It suffers similar deficiencies as HDMR.

This discussion clearly indicates that both HDMR and EDSD‐SVM have numerous assumptions and limitations, and they use MCS to estimate the underlying risk. They fail to explicitly incorporate the underlying physics, sources of nonlinearities, etc., and dynamic loadings cannot be applied in the time domain.

1.4.6 Laboratory Testing

The durability or life of computer chips and solder balls was studied in the laboratory (Whitenack 2004; Sane 2007). They were subjected to thermomechanical loading caused by heating and cooling, causing significant changes in the material properties of the solder balls (discussed in more detail in Chapter 9). Conceptually, thermomechanical loading is also a time domain dynamic excitation in the presence of many sources of nonlinearity including severe material nonlinearity. Solder balls were tested in laboratories under idealistic loading conditions without explicitly addressing the uncertainty‐related issues.

In general, the results obtained from laboratory testing are limited. It may not be possible to extrapolate results with slight variations in samples used or test conditions. To reduce the duration of testing, accelerated testing conditions are used, introducing another major source of uncertainty. Test results are also proprietary in nature. The reliability of solder balls cannot be analytically estimated at present. A sophisticated reliability evaluation technique should be robust enough to consider different types or forms of dynamic loadings applied in the time domain to excite any engineering systems. The proposed REDSET approach is expected to analytically extract reliability information for CNDES providing flexibility to consider issues not considered during testing.

1.5 Justification of a Novel Risk Estimation Concept REDSET Replacing SFEM

The discussions made in the previous sections identified the knowledge gap in estimating the risk of CNDES and an urgent need to replace the existing SFEM concept. Serious attempts were made to define the desirable characteristics or features and scope of a new reliability evaluation technique. Users should be able to extract reliability information by conducting a few deterministic FEM‐based nonlinear time domain dynamic analyses using any computer program available to them. Because of numerous deficiencies of the classical random vibration and the basic MCS concepts, these methods need to be replaced. This will add a new dimension to classical engineering analysis and design. More damage‐tolerant structures can be designed using multiple deterministic analyses instead of one for any type of dynamic loading including seismic, wave, thermomechanical, etc. It will also provide an analytical approach instead of error‐prone laboratory testing, saving an enormous amount of time and money.

With an advanced conceptual understanding of the uncertainty management areas and the availability of exceptional computational power, a transformative robust reliability evaluation approach needs to be developed to evaluate the underlying risk of CNDES for multidisciplinary applications. Boundaries of the reliability estimation techniques need to extend by integrating physics‐based modeling with several advanced mathematical, computational, and statistical concepts, producing compounding beneficial effects to obtain acceptable probabilistic response characteristics/metrics/statistics. It needs to combine model reduction techniques, intelligent simulations, and innovative deterministic approaches routinely used by practicing professionals. If the fundamentals of the basic reliability evaluation procedure are sound, it should be capable to extract reliability information for different types of dynamic systems excited in the time domain.

REDSET is an innovative transformative reliability evaluation procedure that will fill the knowledge gap that currently exists in the profession.

1.6 Notes for Instructors

It will be impractical to present the reliability‐based engineering analysis and design concepts from the very early stage to the present by combining the first two books (Haldar and Mahadevan 2000a, 2000b) and a novel concept REDSET presented here in one book. The table of contents will set the road map for teaching a course. To obtain the maximum benefit from this book, the readers are requested to review some of the fundamentals of the reliability analysis concepts discussed in the first book. They are expected to be familiar with the basic risk estimation concept including the first‐order reliability method (FORM) for explicit LSFs and MCS. To make this book self‐contained, some of the essential and fundamental topics are briefly presented in Chapter 2 of this book. Advance concepts required to implement REDSET are developed gradually and in very systematic ways. If any clarification is required, the first two books can be consulted. The first book has a solutions manual. It can be obtained free of cost from John Wiley (instructors in the United States only), the publisher of this book. To improve the readability, unless it is absolutely necessary, few references are cited in developing and discussing the basic concepts. However, an extended list of publications is provided at the end of the book.

1.7 Notes to Students

Scholars with different levels of technical background are expected to be interested in the novel risk estimation procedure REDSET for CNDES. They are encouraged to be familiar with the basic FORM approach as discussed in detail in Haldar and Mahadevan (2000a). However, to master the subject, it will be beneficial if attempts are made to solve all the problems given at the end of each chapter in the aforementioned book. To obtain the maximum benefit from this book describing REDSET, interested scholars are expected to be familiar with the basic seismic analysis procedure of structures using FEM. It will be helpful if a user first selects a small example from the book and excites the structure dynamically in the time domain to check the accuracy of the results before moving forward to consider multiple time histories. Verification of the results can also be made by using a small number of simulation cycles of MCS, say about 1000. Initially, an LSF of interest can be generated using the regression analysis. After developing the necessary skills, Kriging method can be attempted. Numerous examples are given in the book. It will be very useful if attempts are made to reproduce the results for as many problems as possible. And then extend the knowledge by estimating risk for many other problems not given in the book similar to solder balls discussed in Chapter 9.

Acknowledgments

The financial support of the US National Science Foundation (NSF) was essential in developing the REDSET concept presented in this book, and it needs to be appropriately acknowledged. The title of the latest grant was CDS&E: Theoretical Foundation and Computational Tools for Complex Nonlinear Stochastic Dynamical Engineering Systems – A New Paradigm. The work was supported by a relatively new division within NSF known as Computational and Data‐Enabled Science and Engineering (CDS&E). It is an integrated division consisting of several other divisions within the NSF. The reviewers not only recommended the proposal for funding but also suggested increasing the budget considering its merit and training junior researchers to carry out the related research in the future. Some of the earlier studies on related areas were also supported by NSF including SFEM. During the study, the authors received financial assistance in the form of support of graduate students from several agencies of the government of Mexico: CONACYT, Universidad Autónoma de Sinaloa (UAS), and Dirección General de Relaciones Internacionales de la Secretaria de Educación Pública (DGRI‐SEP).

Any opinions, findings, or recommendations expressed in this book are those of the authors and do not necessarily reflect the views of the sponsors.