Probabilistic Finite Element Analysis of Structures using ... Probabilistic Finite Element Analysis

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  • Probabilistic Finite Element Analysis of

    Structures using the Multiplicative

    Dimensional Reduction Method

    by

    Georgios Balomenos

    A thesis

    presented to the University of Waterloo

    in fulfillment of the

    thesis requirement for the degree of

    Doctor of Philosophy

    in

    Civil Engineering

    Waterloo, Ontario, Canada, 2015

    ©Georgios Balomenos 2015

  • ii

    AUTHOR'S DECLARATION

    I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,

    including any required final revisions, as accepted by my examiners.

    I understand that my thesis may be made electronically available to the public.

  • iii

    Abstract

    It is widely accepted that uncertainty may be present for many engineering problems, such as

    in input variables (loading, material properties, etc.), in response variables (displacements,

    stresses, etc.) and in the relationships between them. Reliability analysis is capable of dealing

    with all these uncertainties providing the engineers with accurate predictions of the

    probability of a structure performing adequately during its lifetime.

    In probabilistic finite element analysis (FEA), approximate methods such as Taylor series

    methods are used in order to compute the mean and the variance of the response, while the

    distribution of the response is usually approximated based on the Monte Carlo simulation

    (MCS) method. This study advances probabilistic FEA by combining it with the

    multiplicative form of dimensional reduction method (M-DRM). This combination allows

    fairly accurate estimations of both the statistical moments and the probability distribution of

    the response of interest. The response probability distribution is obtained using the fractional

    moments, which are calculated from M-DRM, together with the maximum entropy (MaxEnt)

    principle. In addition, the global variance-based sensitivity coefficients are also obtained as a

    by-product of the previous analysis. Therefore, no extra analytical work is required for

    sensitivity analysis.

    The proposed approach is integrated with the OpenSees FEA software using Tcl programing

    and with the ABAQUS FEA software using Python programing. OpenSees is used to analyze

    structures under seismic loading, where both pushover analysis and dynamic analysis is

    performed. ABAQUS is used to analyze structures under static loading, where the concrete

    damage plasticity model is used for the modeling of concrete. Thus, the efficient applicability

    of the proposed method is illustrated and its numerical accuracy is examined, through several

  • iv

    examples of nonlinear FEA of structures. This research shows that the proposed method,

    which is based on a small number of finite element analyses, is robust, computational

    effective and easily applicable, providing a feasible alternative for finite element reliability

    and sensitivity analysis of practical and real life problems. The results of such work have

    significance in future studies for the estimation of the probability of the response exceeding a

    safety limit and for establishing safety factors related to acceptable probabilities of structural

    failures.

  • v

    Acknowledgements

    At this point, I would like to express my sincere gratitude to my supervisor Professor Mahesh

    D. Pandey for his encouragement, guidance and support from the beginning till the end of my

    Ph.D. research. His valuable advices, knowledge, prompt responses to my queries and

    extreme help were decisive for the accomplishment of this research.

    I would like to thank my committee members, Professor Andrzei S. Nowak, Professor Maria

    Anna Polak, Professor Susan L. Tighe and Professor Sagar Naik for their time, valuable

    comments and constructive feedback that enhanced my work. Especially, I would like to

    thank Professor Maria Anna Polak for her discussions and advices in the area of reinforced

    concrete.

    I wholeheartedly would like to express my gratitude to Professor Stavroula J. Pantazopoulou

    for inspiring me to continue my studies towards the doctorate level and for her

    encouragement and guidance to elaborate my Ph.D. in Canada.

    I would like to extend my deepest sincere gratitude to Professor Jeffrey S. West for serving

    as his Teaching Assistant for several courses, which gave me a lot of confidence and

    knowledge on how to deliver effectively any course material, making me to feel more than

    lucky to have served as one of his TAs.

    From the bottom of my heart, I would like to express my thanks and appreciation to

    Aikaterini Genikosmou for her patience, continuous support, encouragement, unremitting

    help and longtime discussions which gave me confidence and strength to reach my goals.

    I thankfully acknowledge my Greek friends Georgios Drakopoulos and Anastasios

    Livogiannis for inspiring and persuading me that programing is actually fun and Dr. Nikolaos

    Papadopoulos for his valuable advices and longtime discussions.

  • vi

    Many thanks go to my colleagues and friends; Kevin Goorts, Dr. Xufang Zhang, Olivier

    Daigle, Shayan Sepiani, Wei Jiang, Paulina Arczewska, Martin Krall, María José Rodríguez

    Roblero, Joe Stoner, Jeremie Raimbault, Dainy Manzana, Joe Simonji, Chao Wu and other

    graduate students with whom the days in Waterloo where wonderful.

    The consistent financial support in form of Research Assistantship by the Natural Science

    and Engineering Research Council (NSERC) of Canada and the University Network of

    Excellence in Nuclear Engineering (UNENE) is gratefully appreciated and acknowledged.

    The financial support in form of Teaching Assistantship by the Department of Civil and

    Environmental Engineering, University of Waterloo, is also gratefully appreciated.

    Last but not least, words are not enough to express my thankfulness and gratitude to my

    adorable parents Kalliopi and Panagiotis and my loving sister Vasiliki for their incessant

    support, encouragement and love throughout my life and for standing by me to all my

    choices.

    Georgios Balomenos,

    Waterloo, Fall 2015

  • vii

    To

    My Family

  • viii

    Table of Contents

    AUTHOR’S DECLARATION ................................................................................................. ii

    Abstract .................................................................................................................................... iii

    Aknowledgenemts ..................................................................................................................... v

    Dedication ............................................................................................................................... vii

    Table of Contents ................................................................................................................... viii

    List of Figures ......................................................................................................................... xv

    List of Tables ........................................................................................................................ xxii

    Chapter 1 – Introduction........................................................................................................ 1

    1.1 Motivation ....................................................................................................................... 1

    1.2 Objective and Research Significance .............................................................................. 3

    1.3 Outline of the Dissertation .............................................................................................. 4

    Chapter 2 – Literature Review .............................................................................................. 6

    2.1 Reliability Analysis ......................................................................................................... 6

    2.1.1 Monte Carlo Simulation ........................................................................................ 7

    2.1.2 First Order Reiability Method ............................................................................... 9

    2.2 Finite element Analysis ................................................................................................. 10

    2.3 Probabilistic Finite Element Analysis ........................................................................... 12

    2.4 Sensitivity Analysis ....................................................................................................... 14

    Chapter 3 – Multiplicative Dimensional Reduction Method ............................................ 17

    3.1 Introduction ................................................................................................................... 17

    3.1.1 Background .......................................................................................................... 17

    3.1.2 Objective ..........................................................................