Uncertainty Modeling in Finite Element, Fatigue and Stability of Systems

by Haldar, Achintya; Guran, Ardeshir; Ayyub, Bilal M.

• ISBN13: 9789810231286
• ISBN10: 9810231288
• Format: Hardcover
• Copyright: 1997-06-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Presents an overview of the current state of uncertainty modeling, analysis, and design of large structural systems represented by finite element and boundary elements. Addresses timevarying aspect of structural reliability by considering the degradation effect.

Table of Contents

Preface	vii	(4)
Contributors	xi
STOCHASTIC FINITE AND BOUNDARY ELEMENT METHODS	1	(158)
Chapter 1. Probabilistic Finite Element Analysis of Large Structural Systems	1	(22)
Sankaran Mahadevan
1. Introduction	1	(1)
2. Probabilistic Structural Analysis	2	(2)
3. Implicit Performance Functions	4	(1)
4. Sensitivity Analysis	5	(2)
4.1 Static Loading	5	(1)
4.2 Dynamic Loading	6	(1)
4.3 Reliability Sensitivity Indices	7	(1)
5. Large Structures	7	(1)
6. Matrix Condensation and Sensitivity Analysis	7	(5)
6.1 Numerical Example - Six-story two-bay frame	9	(3)
7. Parameters with Spatial Variability	12	(4)
7.1 Selective Discretization of Random Fields	13	(1)
7.2 Numerical Example - Cantilever Beam	13	(3)
8. Structural Optimization	16	(2)
8.1 Reliability-Based Design Formulation	16	(1)
8.2 Optimization Algorithm	17	(1)
8.3 Numerical Example	17	(1)
9. Conclusion	18	(2)
10. References	20	(3)
Chapter 2. Reliability Evaluation of Structures Using Nonlinear SFEM	23	(28)
Achintya Haldar
Liwei Gao
1. Introduction	23	(1)
2. Stochastic Finite Element Methods	24	(2)
2.1 Approximation of the Limit State Function	24	(1)
2.2 Perturbation Method	25	(1)
2.3 Reliability Method	25	(1)
3. Deterministic Finite Element Method Used	26	(10)
3.1 Deterministic Nonlinear FEM Formulation	26	(7)
3.2 Material Nonlinearity	33	(2)
3.3 Flexibility of Connection	35	(1)
4. A Unified Stochastic Finite Element Method	36	(5)
4.1 Evaluation of Jacobians and Adjoint Variable Method	37	(3)
4.2 Elastic Nonlinear Case	40	(1)
4.3 Elasto-Plastic Nonlinear Case	40	(1)
5. Uncertainty Analysis	41	(2)
5.1 Uncertainty in Basic Random Variables	41	(1)
5.2 Uncertainty in Flexible Connections	42	(1)
6. Numerical Examples	43	(5)
6.1 Strength Performance Functions	44	(1)
6.2 Serviceability Performance Function	45	(1)
6.3 Results and Observations	46	(2)
7. Acknowledgment	48	(1)
8. References	48	(3)
Chapter 3. Finite Element Method for Stochastic Structures Based on Inverse of Stiffness Matrix	51	(20)
Isaac Elishakoff
Yongjian Ren
Introduction	51	(1)
1. FEM Based on Direct Exact Inverse for Bar Extension	52	(8)
1.1 Exact Inverse Based FE Solution	52	(3)
1.2 Conventional First-Order Perturbation FE Solution	55	(1)
1.3 Exact Analytic Solution	56	(2)
1.4 Bar with Stochastic Young's Modulus	58	(2)
2. FEM through Generalization of Fuch's Approach	60	(9)
2.1 New Formulation of FE Stiffness Matrix	60	(4)
2.2 Imposition of Displacement Constraints	64	(2)
2.3 Mean and Covariance of the Displacement	66	(1)
2.4 Example	66	(3)
3. Conclusion	69	(1)
4. References	69	(2)
Chapter 4. The Weighted Integral Method and the Variability Response Function as Part of an SFEM Formulation	71	(46)
George Deodatis
Lori Graham
1. Introduction	71	(1)
2. Stochastic Stiffness Matrix	72	(12)
2.1 Beam-Column Element	74	(4)
2.2 Plane Stress/Strain Element	78	(3)
2.3 Plate Bending Element	81	(3)
3. Response Variability	84	(2)
4. Variability Response Functions	86	(7)
4.1 Problems With Beam-Column Finite Elements	86	(1)
4.2 Plane Stress/Strain Problems	87	(3)
4.3 Plate Bending Problems	90	(1)
4.4 Importance of Variability Response Function	90	(1)
4.5 Spectral-Distribution-Free Bounds of Response Variability	91	(1)
4.6 Error Estimation of First-Order Taylor Expansion Approximation	92	(1)
4.7 Important Guidelines to Establish Variability Response Functions Based on Weighted Integrals	92	(1)
5. Reliability Analysis - Calculation of Safety Index	93	(1)
6. Numerical Examples	94	(15)
6.1 Problems With Beam-Column Finite Elements	94	(3)
6.2 Plane Stress/Strain Problems	97	(6)
6.3 Plate Bending Problems	103	(6)
7. Recent Developments	109	(2)
8. Some Notes on Notation	111	(1)
9. Acknowledgments	111	(1)
10. References	112	(5)
Chapter 5. Response of a Vibrating Structure to Turbulent Wall Pressure: Fluid-Loaded Structure Modes Series and Boundary Element Method	117	(42)
Paul J. T. Filippi
Daniel Mazzoni
1. Introduction	117	(4)
1.1 The Physical Systems Studied and the Hypothesis Made on the Turbulence Models	118	(1)
1.2 Short Analysis of the Existing Literature	118	(1)
1.3 Summary of the Different Sections	119	(2)
1.3.1 Response of a baffled fluid-loaded plate to a harmonic point force or to a random process excitation	119	(1)
1.3.2 Response of a baffled plate closing a cavity to a deterministic harmonic force or a random process excitation	120	(1)
1.3.3 Numerical method and results	120	(1)
2. Vibro-Acoustics Response of a Baffled Plate to a Deterministic Excitation	121	(9)
2.1 Statement of the Problem	121	(2)
2.2 Green's Representations of the Sound Pressure Fields and of the Displacement; Boundary Integral Equations	123	(2)
2.2.1 Green's representation of the pressure p(e) and p(i), integro-differential equation for the plate displacement	123	(1)
2.2.2 Boundary Integral Equations equivalent to the initial boundary value problem	123	(2)
2.3 Eigenmodes and Resonance Modes of the Physical System "Baffled Plate - External Fluid"; Modal Series Representations of the Solution	125	(5)
2.3.1 Weak form of the governing equation	126	(1)
2.3.2 Eigenvalues and eigenmodes of the system "baffled plate - external fluid"	126	(1)
2.3.3 Resonance modes and resonance angular frequencies of the system "baffled plate - external fluid"	127	(1)
2.3.4 Representation of the solution as a series of eigenmodes	128	(1)
2.3.5 Representation of the solution as a series of resonance modes	128	(2)
3. Vibro-Acoustic Response of the System Baffled Plate - Fluid to a Random Excitation	130	(3)
3.1 Relationship Between the Cross Spectral Density of the Excitation and the Cross Power Spectral Densities of the System Response	130	(2)
3.2 Representation of the Response of the System to a Random Wall Pressure by a Resonance Modes Series	132	(1)
4. Vibro-Acoustic Response of a Baffled Plate Closing a Cavity and Excited by a Deterministic Harmonic Force or Random Wall Pressure	133	(8)
4.1 Statement of the Problem	134	(1)
4.2 System of Boundary Integral Equations Equivalent to the Boundary Value Problem	135	(2)
4.3 Eigenmodes and Resonance Modes of the Physical System Baffled Plate - External Fluid - Cavity; Modal Series Representations of the Response to a Harmonic Deterministic Force	137	(3)
4.3.1 Eigenvalues and eigenmodes	137	(1)
4.3.2 Eigenmode series representation of the solution	138	(1)
4.3.3 Resonance frequencies and resonance modes of the physical system "baffled plate - external fluid - cavity"	139	(1)
4.3.4 Resonance modes series representation of the solution	140	(1)
4.4 Response of the System "Baffled Plate - External Fluid - Cavity" to a Random Wall Pressure	140	(1)
5. Numerical Solution of the Boundary Integral Equations for the Fluid Loaded Structure Problems and Examples	141	(14)
5.1 Boundary Element Method for the System "Baffled Plate - External Fluid"	142	(2)
5.2 Boundary Element Method for the System "Baffled Plate - External Fluid - Cavity"	144	(1)
5.3 Response of the Systems to Random Wall Pressure Excitation	145	(1)
5.4 Numerical Examples	145	(10)
5.4.1 Influence of the model of turbulence	146	(1)
5.4.2 Comparison between numerical predictions and experimental data	147	(5)
5.4.3 Response of a fluid-loaded plate closing a cavity	152	(3)
6. Concluding Remarks	155	(2)
7. References	157	(2)
FATIGUE RELIABILITY AND UPDATING	159	(56)
Chapter 6. Reliability-Based Structural Fatigue Damage Evaluation and Maintenance Using Non-Destructive Inspections	159	(56)
Zhengwei Zhao
Achintya Haldar
1. Introduction	159	(1)
2. Fatigue Damage Assessment	160	(4)
2.1 Fracture Mechanics-Based Fatigue Crack Growth Model	161	(1)
2.2 Geometry Functions	162	(2)
3. Fatigue Reliability Analysis	164	(7)
3.1 Uncertainty in Fatigue Damage Accumulation	165	(6)
3.1.1 Crack initiation	166	(1)
3.1.2 Material properties	167	(1)
3.1.3 Crack aspect ratio	168	(1)
3.1.4 Fatigue loading	168	(1)
3.1.5 Number of stress cycles	169	(1)
3.1.6 Critical crack size	170	(1)
3.1.7 Target reliability	170	(1)
4. Non-Destructive Inspection and Maintenance	171	(4)
4.1 Non-Destructive Inspection Techniques	171	(3)
4.1.1 Visual inspection	172	(1)
4.1.2 Magnetic particle inspection	172	(1)
4.1.3 Ultrasonic inspection	172	(1)
4.1.4 Dynamic measurement methods	173	(1)
4.1.5 Summary	174	(1)
4.2 Uncertainty in Non-Destructive Inspection	174	(1)
4.2.1 Probabilistic model of inspection capability	174	(1)
4.2.2 Human error and measurement accuracy	175	(1)
5. Fatigue Reliability Estimation Using NDI	175	(3)
5.1 Event Without Crack Detection	176	(1)
5.2 Event With Crack Detection	176	(1)
5.3 Event With Crack Detection and Size Measurement	177	(1)
5.4 Inspection Plan and Optimal Approach	177	(1)
6. Risk-Based Inspection and Maintenance Through Model Updating	178	(9)
6.1 Model Updating Without Crack Detection	178	(3)
6.1.1 Reliability updating	179	(1)
6.1.2 Updating the distribution of the random variables	180	(1)
6.2 Model Updating With Crack Detection	181	(6)
6.2.1 Crack size measurement	182	(4)
6.2.1.1 Reliability updating	182	(1)
6.2.1.2 Updating distribution of random variables	183	(3)
6.2.2 Model updating with crack detection	186	(1)
6.2.2.1 Reliability updating	186	(1)
6.2.2.2 Updating distribution of random variables	187	(1)
7. Model Updating After Repair	187	(1)
8. Risk-Based Criteria For Repair and Replacement	187	(2)
8.1 Fatigue Control Curves	187	(1)
8.2 Criteria For Repair and Replacement	188	(1)
8.3 Updating Inspection Interval	188	(1)
9. Fatigue Reliability and Maintainability for Structural Systems	189	(6)
9.1 Concepts of System Reliability	190	(1)
9.2 Reliability of Fatigue Damage for Welded Structure with Multiple Crack Sites	191	(3)
9.2.1 Distribution of weld defects	192	(1)
9.2.2 Deterministic number of crack sites	193	(1)
9.2.3 Stochastic number of crack sites	193	(1)
9.3 Inspection and Reliability Updating of Welded Structure with Multiple Crack Sites	194	(1)
9.3.1 Deterministic number of crack sites	194	(1)
9.3.2 Stochastic number of crack sites	195	(1)
10. Reliability-Based Fatigue Analysis and Maintenance for Steel Bridges	195	(12)
10.1 Validation of the Fracture Mechanics Approach	196	(1)
10.2 Sensitivity Analysis	196	(4)
10.3 Reliability Analysis of the Bridge	200	(1)
10.4 Fatigue Control Curves	200	(3)
10.5 Reliability-Based Inspection and Maintenance	203	(4)
10.5.1 Event with no crack detection	203	(1)
10.5.2 Single and multi-inspection plans	203	(2)
10.5.3 Case with crack size measurement	205	(1)
10.5.4 Updating of inspection interval	205	(2)
10.5.5 Repair decision	207	(1)
10.6 Reliability Updating After Repair	207	(1)
10.7 Low-Cycle and Deformation-Induced Fatigue	207	(1)
11. Concluding Remarks	207	(2)
12. Acknowledgment	209	(1)
13. References	209	(6)
UNCERTAINTY MODELING IN STRUCTURAL STABILITY	215	(142)
Chapter 7. Uncertainty Modeling in Structural Stability	215	(46)
B. W. Yeigh
M. Shinozuka
1. Introduction	215	(2)
2. Critical Imperfection Magnitude (CIM) Method	217	(6)
3. Simulating Structural Imperfections	223	(3)
4. Beam on Elastic Foundation	226	(8)
5. Numerical Solution	234	(23)
5.1 Computational Aspects	234	(1)
5.2 Root Mean Square Imperfection Magnitude	235	(7)
5.3 Effect of Correlation Distance I	242	(6)
5.4 Effect of Imperfection Patterns	248	(1)
5.5 Effect of Correlation Distance II	249	(6)
5.6 Effect of PSD Shapes	255	(2)
6. Conclusion	257	(1)
7. Acknowledgments	258	(1)
8. References	258	(3)
Chapter 8. Global Stability Analysis of Nonlinear Dynamical Systems	261	(37)
R. Valery Roy
1. Introduction	261	(2)
2. Noise Perturbations of A Single-Degree-of-Freedom System with Piecewise-Linear Restoring Force	263	(7)
3. Asymptotic Analysis of Randomly Perturbed Nonlinear Systems in the Limit of Weak Noise	280	(10)
4. Conclusion	290	(3)
5. Acknowledgments	293	(1)
6. References	294	(4)
Chapter 9. Dynamic Random Snap-Buckling of Composite Shallow Shells	298	(13)
R. Heuer
H. Irschik
F. Ziegler
1. Introduction	298	(1)
2. Modal Approximation of the Nonlinear Vibrations of Shallow Composite Shells	299	(6)
3. Probability of Snap-Buckling	305	(4)
4. Conclusion	309	(1)
5. References	309	(2)
Chapter 10. Buckling Analysis and Design of Imperfection-Sensitive Structures	311	(46)
G. V. Palassopoulos
1. Introduction	311	(2)
2. Overview of the Problem	313	(13)
2.1 A Simple Example of Imperfection Sensitivity in Buckling	313	(4)
2.1.1 Analysis of the conventional model	313	(2)
2.1.2 Analysis of model with structural imperfection	315	(1)
2.1.3 Case of softening supporting spring	316	(1)
2.2 Intuitive Explanation of Imperfection Sensitivity	317	(5)
2.2.1 Limit point buckling	318	(1)
2.2.2 Asymmetric bifurcation buckling	318	(1)
2.2.3 Symmetric bifurcation buckling	319	(1)
2.2.3.1 Unstable symmetric bifurcation buckling	319	(1)
2.2.3.2 Stable symmetric bifurcation buckling	320	(1)
2.2.4 Imperfection sensitivity	320	(1)
2.2.5 Limitations	321	(1)
2.3 Sources of Structural Imperfections	322	(1)
2.4 Problem Difficulties	322	(1)
2.5 Short Historical Notes	323	(3)
2.5.1 Classical stability theory	324	(1)
2.5.2 Koiter's and Koiter-based work	324	(1)
2.5.3 Critical-Imperfection-Magnitude method	325	(1)
3. General Description of the CIM Method	326	(4)
3.1 Scope	326	(1)
3.2 Terminology	326	(1)
3.3 Assumptions	327	(1)
3.4 The Four Steps of the CIM Method	328	(2)
4. Analytical Development of the CIM Method	330	(12)
4.1 Potential Energy Expressions	330	(2)
4.2 Linear Coordinate Transformation	332	(3)
4.3 Modal Coordinates and Stability Coefficients	335	(2)
4.4 Pre-Buckling Equilibrium State of Actual Structure	337	(1)
4.5 Stability Criterion of Pre-Buckling Equilibrium State	338	(2)
4.6 Symmetric Eigenvalue Problem	340	(1)
4.7 Critical Imperfection Magnitude	341	(1)
5. Significant Imperfection Sources and Components	342	(7)
5.1 Importance of the Problem for the Design Engineer	342	(1)
5.2 Limitations of Koiter's Theory	343	(1)
5.3 New Results of the CIM Method	344	(2)
5.4 Koiter's Theory Within the Context of the CIM Method	346	(1)
5.5 Reduction of the Dimension of the Problem	347	(2)
6. Advantages and Limitations of the CIM Method	349	(3)
6.1 Advantages of the CIM Method	349	(2)
6.2 Limitations of the CIM Method	351	(1)
7. Conclusions	352	(1)
8. Acknowledgment	353	(1)
9. References	353	(4)
FAULT-TOLERANT COMPUTING DESIGN	357	(38)
Chapter 11. Basic Concepts of Fault-Tolerant Computing Design	357	(38)
Chouki Aktouf
Arde Guran
Oum-El-Kheir Benkahla
1. Introduction	357	(2)
2. Basic Definitions	359	(2)
3. Principles of Fault-Tolerant Design	361	(22)
3.1 Redundancy Techniques	361	(1)
3.2 Fault Detection	362	(7)
3.2.1 Built-In Self-Test techniques	365	(2)
3.2.2 Self-checking techniques	367	(2)
3.3 Fault Diagnosis	369	(10)
3.3.1 PMC model	371	(4)
3.3.2 Generalization of test outcomes	375	(1)
3.3.3 Nature of faults	376	(1)
3.3.4 Distributed diagnosis	377	(2)
3.4 Reconfiguration Techniques	379	(1)
3.5 Fault Recovery Techniques	380	(3)
3.5.1 Recovery without checkpointing	381	(1)
3.5.2 Checkpoint-based recovery	381	(1)
3.5.3 The communication problem in recovery techniques	382	(1)
4. Examples of Fault-Tolerant Computers	383	(7)
4.1 Computers for Aerospace Applications	384	(1)
4.2 Computers for Long Life Applications	385	(1)
4.3 Computers for Commercial Applications	386	(3)
4.4 General Purpose Computers	389	(1)
5. Conclusion	390	(1)
6. References	391	(4)
Author Index	395	(8)
Subject Index	403