Reliability Evaluation of Dynamic Systems Excited in Time Domain - Redset Alternative to Random Vibration and Simulation

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.

Achintya Haldar, PhD, PE, Dist. M. ASCE, Professor Emeritus, Department of Civil and Architectural Engineering, University of Arizona. Hamoon Azizsoltani, PhD, Data Scientist, Fraud and Security Intelligence, SAS Institute, North Carolina. Jungwon Huh, PhD, Professor, Department of Civil Engineering, Chonnam National University, South Korea. J. Ramon Gaxiola-Camacho, PhD, Assistant Professor, Department of Civil Engineering, Universidad Autónoma de Sinaloa, Mexico. Sayyed Mohsen Vazirizade, PhD, Post-Doctoral Research Scholar, Institute of Software Integrated Systems, Department of Electrical Engineering and Computer Science, Vanderbilt University.

Chapter 1 REDSET and Its Necessity

1.1         Introductory Comments

1.2         Reliability Evaluation Procedures Existed Around 2000

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

1.4         Other Alternatives besides SFEM

1.4.1 Random Vibration

1.4.2 Alternative to Basic Monte Carlo Simulation

1.4.3 Alternatives to Random Vibration for Large Problems

      1.4.4 Physics-Based Deterministic FEM Formulation

      1.4.5 Multi-disciplinary Activities to Study the Presence of Uncertainty in Large Problems

      1.4.6 Laboratory Testing

1.5 Justification of a Novel Risk Estimation Concept REDSET Replacing SFEM

1.6 Notes for Instructors

 1.7 Notes to Students

Acknowledgement

Chapter 2 Fundamentals of Reliability Assessment

2.1 Introductory Comments

2.2 Set Theory

2.3 Modeling of Uncertainty

               2.3.1 Continuous Random Variables

               2.3.2 Discrete Random Variables

               2.3.3 Probability Distribution of a Random Variable

               2.3.4 Modeling of Uncertainty for Multiple Random Variables

2.4 Commonly Used Probability Distributions

               2.4.1 Commonly used continuous and discrete random variables

               2.4.2 Combination of Discrete and Continuous Random Variables

2.5 Extreme Value Distributions 

2.6 Risk-Based Engineering Design Concept

2.7 Evolution of Reliability Assessment Methods

               2.7.1 – First-Order Second Moment Method (FOSM)

               2.7.2 – Advanced First-Order Reliability Method (AFOSM)

               2.7.3 – Hasofar-Lind Method

2.8 AFOSM for Non-normal Variables

               2.8.1 Two Parameters Equivalent Normal Transformation

               2.8.2 Three Parameters Equivalent Normal Transformation

2.9 Reliability Analysis with Correlated Random variables

2.10 - First-Order Reliability Method (FORM)

               2.10.1 FORM Method 1

               2.10.2 Correlated Non-normal Variables

2.11 Probabilistic Sensitivity Indices

2.12 FORM Method 2

2.13 System Reliability Evaluation

2.14 Fundamentals of Monte Carlo Simulation Technique

               2.14.1 Steps in Numerical Experimentations using Simulation

               2.14.2 Extracting Probabilistic information from N Data Points

               2.14.3 Accuracy and Efficiency of Simulation

2.15 Concluding Remarks

Chapter 3 - Implicit Performance or Limit State Functions

3.1 Introductory Comments

3.2 Implicit Limit State Functions – Alternatives

3.3 Response Surface Method (RSM)

3.4 Limitations of using the Original RSM Concept for the Structural Reliability Estimation

3.5 Generation of Response Surfaces Using the IRS Method

               3.5.1 Polynomial Representation of an Improved Response Surface

3.6 Experimental Region, Coded Variables, and Center Point

               3.6.1 Experimental Region and Coded Variables

               3.6.2 Experimental Design

               3.6.3 Saturated Design

               3.6.4 Central Composite Design

3.7 Analysis of Variance

3.8 Experimental Design for Second-Order Polynomial

               3.8.1 Experimental Design - Model 1: SD with Second-Order Polynomial without Cross Terms

               3.8.2 Experimental Design - Model 2: SD with Second Order Polynomial with Cross Terms

               3.8.3 Experimental Design - Model 3: CCD with Second Order Polynomial with Cross Terms

3.9 Comparisons of the Three Experimental Design Models

3.10 Experimental Design for Nonlinear Dynamic Problems Excited in The time domain

3.11 Selection of the most appropriate Experimental Design Model

3.12 Selection of Center Point

3.13 Generation of Limit State Functions for Routine Design

               3.13.1 Serviceability Limit State

               3.13.2. Strength Limit State Functions

               3.13.3 Interaction Equations for the Strength Limit State Functions

               3.13.4 Dynamic Effect in Interaction Equations

3.14 Concluding Remarks

Chapter 4 - Uncertainty Quantification of Dynamic Loadings Applied in the Time Domain

4.1 Introductory Comments

4.2 Uncertainty Quantification in Seismic Loadings Applied in the Time Domain

               4.2.1 Background Information

4.3 Selection a Suite of Acceleration Time Histories Using PEER Database – Alternative 1

               4.3.1 Earthquake Time History Selection Methodology

4.4 Demonstration of Selection a Suite of Ground Motion Time Histories – Alternative 1

4.5 Simulated Ground Motions using the Broadband Platform (BBP) - Alternative 2 - 

4.5.1 Broadband Platform Developed by SCEC

4.6 Demonstration of Selection and Validation of a Suite of Ground Motion Time Histories using BPP

4.7 Applications of BBP in Selecting Multiple Earthquake Acceleration Time Histories

4.8         Summary of generating multiple earthquake time histories using BPP

4.9         Uncertainty Quantification of Wind-Induced Wave Loadings Applied in the Time Domain

4.9.1 Introductory Comments

4.9.2 Fundamentals of Wave Loadings

4.9.3 Morison Equation

4.10       Modeling of Wave Loading

               4.10.1    Wave Modeling Using the New Wave Theory

               4.10.2    Wheeler Stretching Effect

               4.10.3    Three Dimensional Directionality

               4.10.4    Summary of deterministic modeling of wave loading

4.11 Uncertainty Quantifications in Wave Loadings Applied in the Time Domain

4.11.1 Uncertainty Quantification in Wave Loadings - Three Dimensional Constrained        New Wave (3D CNW) Concept

               4.11.2 Three-Dimensional Constrained New Wave (3D CNW) Concept

               4.11.3 Uncertainty in the Wave Height Estimation

               4.11.4 Uncertainty Quantification of the Wave Loading

               4.11.5 Quantification of Uncertainty in the Wave Loading

4.12 Wave and Seismic Loadings – Comparisons (Needs improvements)

4.13       Concluding Remarks

Chapter 5 - Reliability Assessment of Dynamic Systems Excited in the Time Domain – REDSET

5.1 Introductory Comments

5.2 A Novel Reliability Estimation Concept – REDSET

               5.2.1 Integration of Finite Element Method (FEM), Improved Response Surface (IRS) Method, and FORM

               5.2.2 Increase Efficiency in Generating an IRS

               5.2.3 Optimum Number of NDFEA Required for Generation of an IRS

               5.2.4 Reduction of Random variables

5.3 Advanced Sampling Design Schemes

5.4 Advanced Factorial Design Schemes

5.5 Modified Advanced Factorial Design Schemes

               5.5.1 Modified AFD Scheme 2 (MS2)

               5.5.2 Modified AFD Scheme 3 (MS3)

5.6 Optimum Number of TNDFEA Required to Implement REDSET

5.7 Improve Accuracy of Scheme MS3 further – Alternative to the Regression Analysis

               5.7.1 Moving Least Squares Method

               5.7.2 Concept of Moving Least Squares Method

               5.7.3 Improve Efficiency further of the Moving Least Squares Method

5.8 Generation of an IRS Using Kriging Method

               5.8.1      Simple Kriging

               5.8.2      Ordinary Kriging

               5.8.3 Universal Kriging

               5.8.4 Variogram Function

               5.8.5 Scheme S3 with Universal Kriging Method

               5.8.6 Scheme MS3 with Modified Universal Kriging Method

5.9 Comparisons of all Proposed Schemes

5.10 Development of Reliability Evaluation of Dynamical Engineering Systems Excited in Time Domain (REDSET)

               5.10.1 Required Steps-by-steps in Implementation of REDSET

5.11 Concluding Remarks

Chapter 6 Verification of REDET for Earthquake Loading Applied in Time Domain

6.1 Introductory Comments

6.2         Verification – Example 1 –3 Story Steel Moment Frame with W24 Columns

               6.2.1      Example 1 – Accuracy Study of all 9 Schemes

               6.2.2 Verification – Example 2 –3 Story Steel Moment Frame with W14 Columns

6.3 Case study–13-Story Steel Moment Frame

6.4         Example 4 - Site-Specific Seismic Safety Assessment of CDNES

               6.4.1 Location, Soil Condition, and Structures

6.4.2 Uncertainty Quantifications

               6.4.2.1 Uncertainty quantifications in resistance-related design variables

               6.4.2.2 Uncertainty quantifications in gravity load-related design variables

               6.4.2.3 Selection of a suite of site-specific acceleration time histories

6.5 Risk Evaluation of three Structures using REDSET

               6.5.1 Selection of Limit State functions

               6.5.2 Estimations of the Underlying Risk for the Three Buildings

6.6         Concluding Remarks

Chapter 7 Reliability Assessment of Jacket-type Offshore Platforms Using REDSET for Wave and Seismic Loadings

7.1         Introductory Comments

7.2         Reliability Estimation of a Typical Jacket-Type Offshore Platform

7.3 Uncertainty Quantifications of Jacket-Type Offshore Platform

               7.3.1      Uncertainty in Structures

               7.3.2      Uncertainty in Wave Loadings in Time Domain

7.4         Performance Functions

               7.4.1 LSF of Total Drift at the Top of the Platform

               7.4.2 Strength Performance Functions

7.5         Reliability Evaluation of JTPs

7.6         Risk Estimations of JTPs Excited by the Wave and Seismic Loadings – Comparison

7.7 Comparison of Results for the Wave and Earthquake Loadings

7.8         Concluding Remarks

Chapter 8 - Reliability Assessment of Engineering Systems using REDSET for Seismic Excitations and Implementation of PBSD

8.1 Introductory Comments

8.2 Assumed Stress-Based Finite Element Method for Nonlinear Dynamic Problems

               8.2.1 Nonlinear Deterministic Seismic Analysis of Structures

               8.2.3 Dynamic Governing Equation and Solution Strategy8.2.2 Seismic Analysis of Steel Structures

               8.2.4 Flexibility of Beam-to-Column Connection Models by Satisfying Underlying Physics - Partially Restrained (PR) Connections for Steel Structures

               8.2.5 Incorporation of Connection Rigidities in the FE Formulation Using Richard 4-Parameter Model

8.3 Pre- and Post-Northridge Steel Connections

8.4         Performance-Based Seismic Design

               8.4.1      Background information and Motivation

               8.4.2 Professional Perception of PBSD

               8.4.3 Building Codes, Recommendations, and Guidelines

               8.4.4 Performance Levels

               8.4.5 Target Reliability Requirements to Satisfy Different Performance Levels

               8.4.6 Elements of PBSD and their Sequences

               8.4.7 Explore Suitability of REDSET in Implementing PBSD

8.5 Showcasing the Implementation of PBSD

               8.5.1 Verification of REDSET- Reliability Estimation of a 2-Story Steel Frame

8.6 Implementation Potential of PBSD – 3-, 9-, and 20-Story Steel Buildings

               8.6.1 Descriptions of the Three Buildings

               8.6.2 Post-Northridge PR Connections

               8.6.3 Quantification of Uncertainties in Resistance-Related Variables

               8.6.4 Uncertainties in Gravity Loads

               8.6.5 Uncertainties in PR Beam-to-Column Connections

               8.6.6 Uncertainties in Seismic Loading

               8.6.7 Serviceability Performance Functions – Overall and Inter-Story Drifts

8.7 Structural Reliability Evaluations of the Three Buildings for the Performance Levels of CP, LS, and IO using REDSET

               8.7.1 Observations for the three performance levels

8.8 Implementation of PBSD for Different Soil Conditions

8.9 Illustrative Example of Reliability Estimation for Different Soil Condition

               8.9.1 Quantifications of uncertainties for resistance-related variables and Gravity Loads

               8.9.2 Generation of multiple design earthquake time histories for different soil conditions

               8.9.3 Implementation of PBSD for Different Soil Conditions

8.10 Concluding Remarks            

Chapter 9 - Reliability Assessment of Lead-Free Solders in Electronic Packaging Using REDSET for Thermo-Mechanical Loadings

9.1 Introductory Comments

9.2 Background Information

9.3 Deterministic Modelling of a Solder Ball

9.3.1 Solder Ball Represented by Finite Elements

9.3.2  Material Modeling of SAC Alloy

               9.3.2.1  HISS Plasticity Model

               9.3.2.2  Disturbed State Concept (DSC)

               9.3.2.3  Creep Modeling

               9.3.2.4  Rate Dependent Elasto-viscoplastic Model

9.3.3 Temperature Dependent Modeling

9.3.4 Constitutive Modeling Calibration

9.3.5 Thermo-mechanical Loading Experienced by Solder Balls

9.4         Uncertainty Quantification

               9.4.1 Uncertainty in all the parameters in a Solder Ball

               9.4.2 Uncertainty Associated with the Thermo-Mechanical Loading

9.5 The Limit State Function for the Reliability Estimation

9.6 Reliability Assessment of Lead-Free Solders in Electronic Packaging

9.7 Numerical Verification Using Monte Carlo Simulation

9.8 Verification Using Laboratory Test Results

9.9 Concluding Remarks

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