Handbook of Structural Life Assessment

Table of Contents

Cover

Title Page

Acknowledgements

Introduction

Acknowledgements

Part I: Fracture Mechanics Dynamics and Peridynamics

1 Fundamentals of Fracture Mechanics

1.1 Introduction and Historical Background

1.2 Classical Theory of Solid Mechanics

1.3 Stress Intensity Factor

1.4 Linear Elastic Fracture Mechanics (LEFM)

1.5 Nonlinear Fracture Mechanics

1.6 Boundary‐Layer Effect of Composites

1.7 Closing Remarks

2 Applications of Fracture Mechanics

2.1 Introduction

2.2 Fracture Mechanics of Metallic Structures

2.3 Damage of Composite Structures

2.4 Closing Remarks

3 Dynamic Fracture and Peridynamics

3.1 Introduction

3.2 Fracture Dynamics

3.3 Fracture Dynamics of Metals

3.4 Dynamic Fracture of Composites

3.5 Peridynamics

3.6 Closing Remarks

Part II: Introduction to Structural Health Monitoring

4 Structural Health Monitoring Basic Ingredients and Sensors

4.1 Introduction

4.2 Between Structural Life Assessment and Health Monitoring

4.3 Basic Ingredients of SHM

4.4 Closing Remarks

5 Statistical Pattern Recognition and Vibration‐Based Techniques

5.1 Introduction

5.2 The Statistical Pattern Recognition Paradigm

5.3 Vibration‐Based Techniques

5.4 Closing Remarks and Conclusions

Part III: Reliability and Fatigue under Extreme Loading

6 Fatigue Life and Reliability Assessment

6.1 Introduction

6.2 Fatigue Life Assessment

6.3 Design Based on Ultimate Strength of Ship Structures

6.4 Probabilistic Models of Load Effects

6.5 Climate and Environmental Effects

6.6 Closing Remarks

7 Structural Reliability and Risk Assessment Under Extreme Loading

7.1 Introduction

7.2 Historic Extreme Loading Events

7.3 Structural Life Assessment of Ocean Systems

7.4 Road Tanker Rollover

7.5 Pipes Conveying Fluids

7.6 Closing Remarks

Part IV: Environment Conditions, Joints and Crack Propagation Control

8 Corrosion and Hydrogen Embrittlement

8.1 Introduction

8.2 Corrosion of Ocean and Aerospace Structures

8.3 Fretting/Wear in Heat Exchangers

8.4 Hydrogen Embrittlement

8.5 Closing Remarks and Conclusions

9 Joints and Weldments

9.1 Introduction

9.2 Energy Dissipation and Nonlinearity of Joints

9.3 Design Considerations

9.4 Welded Joints

9.5 Closing Remarks and Conclusions

Appendix

10 Crack Control

10.1 Introduction

10.2 Basic Concept and Development of Crack Arresters

10.3 Crack Arresters of Ship Structures

10.4 Crack Control and Repair of Aerospace Structures

10.5 Pipeline Crack Arresters

10.6 Closing Remarks

References

Index

End User License Agreement

List of Tables

Chapter 8

Table 8.1 Thickness reduction of structural components by general corrosion (all thickness values are in millimeters). (Mateus and Wirz, 1998)

List of Illustrations

Chapter 1

Figure 1.1 Cartesian components of the stress vector acting on the three faces of the cubic elemental volume dV.

Figure 1.2 Forces acting along the x1‐axis on a cube of side length L. Stresses around the cube faces are developed in a Taylor series expansion about their values at x.

Figure 1.3 Large plate with a circular hole circular tunnel under remote tensile stress.

Figure 1.4 (a) Elliptic hole and (b) the limit to a flat crack of width 2a.

Figure 1.5 Crack tip coordinates for establishing the stress intensity factor.

Figure 1.6 Modes of crack loading.

Figure 1.7 Dependence of strain energy (Ue), surface energy (Us), and their sum of the crack length showing the critical crack length.

Figure 1.8 The plastic zone around a crack tip in a ductile material.

Figure 1.9 Definition of crack tip opening displacement (CTOD), crack opening stretch (COS), and clip gage displacement (CGD) or crack opening displacement (COD).

Figure 1.10 Typical relationship between the crack growth rate and the range of the stress intensity factor showing three regions of crack development for a given stress ratio.

Figure 1.11 J‐integral around a crack tip in two dimensions.

Figure 1.12 Concept of the weighting q‐function.

Figure 1.13 Contours near the crack tip showing the normal vectors for Γ0, and ; and on Γs used in the transformation from line integral to equivalent domain integral. (Song and Paulino, 2006)

Figure 1.14 Schematic diagram of composite laminate of finite width subjected to the applied force, Pz, bending moments, Mx and My, and twisting moment, Mt, all acting at the ends.

Figure 1.15 Dependence of the eigen‐value ϖ1 on the fiber orientation angle Θ of ply graphite‐epoxy composite. (Wang and Choi, 1981)

Figure 1.16 Schematic diagram of a laminate of arbitrary layup with n plies and the total thickness d showing the coordinate frame at the free‐corner edge.

Figure 1.17 Interlaminar stress σzz over the plane xy for a uniform temperature rise of for: (a) [0°/90°]‐symmetric laminate, (b) [0°/60°]‐symmetric laminate, and (c) [0°/30°]‐symmetric laminate. (Mittlestedt and Becker, 2004b)

Figure 1.18 Interlaminar tensile stress σzz over the interval , at and for a uniform temperature rise of : (a) [30°/Θ2]‐symmetric laminates and (b) [60°/Θ2]‐symmetric laminates. Mittlestedt and Becker (2004b)

Chapter 2

Figure 2.1 Crack tip opening displacement dependence on temperature for (a) ABS grade B steel plate, (b) ABS grade EH 36 steel plate: under intermediate (♢), quasi‐static (QS ♦) and 0.25 mm CTOD rate loadings (SSC‐430, 2003).

Figure 2.2 Comparison between measured and predicted crack growth rate curves for different values of stress ratio :

Figure 2.3 Dependence of fracture toughness on yield stress for three types of metals. (Wood, 1974)

Figure 2.4 Dependence of crack length parameter on yield stress for three different metals. (Wood, 1974)

Figure 2.5 Dependence of crack length parameter on structural efficiency parameter for three different metals. (Wood, 1974)

Figure 2.6 Dependence of ultimate load on temperature for carbon and glass sandwich beams. (Ayorinde et al. 2012)

Figure 2.7 Sandwich plate damaged due to transverse localized loading created by the T‐joint in a slow compression cycle. (Pilipchuk et al., 2013a)

Figure 2.8 Schematic diagram of a sandwich plate resting on a stiff base showing the model geometry and coordinate frame.

Figure 2.9 Beam model geometry and coordinate system.

Figure 2.10 Skin deflection profiles under the loads: Q₀ = 200 N (dashed line), and Q₀ = 1000 N (solid line). (Pilipchuk et al., 2013a)

Figure 2.11 Load versus strain response of the PMI beam under localized compressive loading; analytical solution and test result. (Pilipchuk et al., 2013a)

Figure 2.12 Damage formation under the load–unload cycle according to the analytical solution: (a) load, and (b) unload. (Pilipchuk et al., 2013b)

Figure 2.13 Test setup diagrams: (a) sandwich plate specimen including skin prepared for compression ‐ tension loading, and (b) a narrow piece of sandwich plate (beam) prepared for localized load‐unload cycle.

Figure 2.14 Snapshots of the crack formation during ‘localized’ load–unload test of the sandwich beam under low temperature conditions T = −60 °C, loading speed v = 0.005 in/s, and maximal effective strain , (a‐b) loading phase, and (c‐d) unloading phase. (Pilipchuk et al., 2013b)

Figure 2.15 Results of low speed ( ) ‘localized’ load–unload tests of sandwich beams under different sub‐zero temperatures and maximal effective strains: (a) , , (b) , , (c) , , (d) , . (Pilipchuk et al., 2013b)

Figure 2.16 Results of higher speed ( ) ‘localized’ load–unload tests of sandwich beams under different moderate sub‐zero temperatures and maximal effective strains: (a) , , (b) , (c) , , (d) , . (Pilipchuk et al., 2013b)

Figure 2.17 Schematic diagram of sandwich plate cross‐section showing the coordinate frame.

Figure 2.18 Temperature profiles for three different values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

Figure 2.19 Deviation of temperature profiles from the homogeneous case. (Pilipchuk et al., 2010)

Figure 2.20 Temperature gradient profiles for three different values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

Figure 2.21 Stress profiles at the cross‐section . (Pilipchuk et al., 2010)

Figure 2.22 Mean vertical displacement profile at three values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

Figure 2.23 Amplitude of the mean vertical displacement as a function of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

Figure 2.24 Dependence of the amplitude of averages vertical displacement on the relative stiffness at three different values of thickness per length ratio: hard skin region. (Pilipchuk et al., 2010)

Figure 2.25 Dependence of the averaged vertical displacement on the relative stiffness at three different values of thickness per length ratio: transition from hard to soft skin. (Pilipchuk et al., 2011)

Figure 2.26 Assembly of stress–strain curves for different samples showing variation across multiple samples at under a loading speed 1.27 mm/s. (Grace et al., 2012)

Figure 2.27 Snapshots of propagating (from the top) front of collapsing foam cells during compression at different time instances shown in µs units; see the next figure for details. (Grace et al., 2012)

Figure 2.28 Microstructure of PMI foam near the front of collapsing cells within the area 2 mm ×2mm shown for different strain values: (a) ε = 0, (b) ε = 0.1, (c) ε = 0.5, and (d) ε = 0.75. (Grace et al., 2012)

Figure 2.29 Effect of temperature on the PMI load‐displacement response at different values of loading speed: (a) , (b) , and (c) . (Grace et al., 2012)

Figure 2.30 Effect of sub‐zero temperature on compression behavior of PMI foam for a loading speed . (Grace et al., 2012)

Figure 2.31 Measured time evolution of load and acoustic emission count density for at . (Grace et al., 2012).

Figure 2.32 Time evolution of axial displacement: (a) Comparison between axial command and actual displacement, (b) Definition of displacement increment for . (Grace et al., 2012)

Figure 2.33 Acoustic emission response for at as recorded by: (a) sensor placed on the upper cap of testing machine, and (b) sensor placed on the foam side face. (Grace et al., 2012)

Figure 2.34 Compression under fluctuating speed (average speed 0.127 mm/s), (a) effect of changing value of the speed, (b) comparison with result at constant speed. (Grace et al., 2012)

Figure 2.35 Compressive cyclic stress–strain curves: (a) maximum strain 0.35 at , (b) maximum strain 0.1, 0.35, and 0.75 at , (c) maximum strain 0.1, 0.35, and 0.75 at Cycle 100. (Grace et al., 2012)

Figure 2.36 Stress versus effective strain diagrams of sandwich specimens obtained during cyclic compression–tension tests under room and low temperature conditions and loading speed 0.127 mm/s.

Figure 2.37 Energy loss in logarithmic scale during load–unload cycles at displacement speeds for maximum strain: (a) 0.1, (b) 0.35, and (c) 0.75; solid line: , and dashed line: , (Grace et al. 2012).

Figure 2.38 Geometry of the load versus displacement foam model: – solid bold line, and – thinner solid and dashed lines. (Grace et al., 2014)

Figure 2.39 Smoothing function at two different magnitudes of the parameter α.

Figure 2.40 Illustration of fitting the load versus displacement data at different speed loading: (a) , and (b) each for three different values of temperature (upper row), (middle row), and (lower row). (Grace et al., 2012).

Figure 2.41 Foam stress–train response under different conditions of loading: (a) different loading speeds at low temperature, (b) different loading speeds at high temperature, (c) different temperatures at low speed, and (d) different temperatures at high speed. (Grace et al., 2012)

Chapter 3

Figure 3.1 Computer simulations in a simple model at the atomic scale showing a transition between smoothly moving cracks and a violent branching instability that is similar to experiment. The transition is a function of the energy stored per unit length to the right of the crack. (Marder and Fineberg, 2004)

Figure 3.2 Dependence of the transitional K0,max on the initial‐notch‐to‐specimen‐width ratio a0/W as predicted (lines) and measured (symbols) in 4‐mm‐wide titanium Tiβ21s/SCS‐6 composite specimens under bending: Predicted line 1: 4.0 GPa, 2: 3.5 GPa, 3: 3.0 GPa, 4: 2.5 GPa, and 5: 2.0 GPa. ⚬: stable, ⦁: unstable. (Liu and Bown, 1999)

Figure 3.3 Crack propagation showing: (a) the dynamical mirror‐mist‐hackle transition as the crack speed increases, and (b) the crack velocity history (normalized by the Rayleigh‐wave speed). (Buehler and Gao, 2007).

Figure 3.4 Caustic formation in (a) reflection, and (b) transmission. (Liu et al., 1993)

Figure 3.5 Dependence of the normalized dynamic stress intensity factor with respect to the classical theoretical value on the radius non‐dimensional parameter (r0/υt) for Poisson’s ratio , different crack tip velocities (υ/cs). Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993).

Figure 3.6 Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for different crack tip velocities (υ/cs), for 4340 steel Poisson’s ratio , , , , . Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993)

Figure 3.7 Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for three different load levels , Poisson’s ratio , , , , and crack tip velocity ratio . Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993)

Figure 3.8 Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for different values of the distance from the screen z0; , Poisson’s ratio , , , and crack tip velocity ratio . Square points (◽) are obtained according the classical analysis, Circle points (○) according to the modified method. (Liu et al., 1993)

Figure 3.9 A sequence of snapshots of coherent grading sensing showing shear band propagation in C300 Maraging steel. (Guduru et al., 2001)

Figure 3.10 Time evolution of shear‐band advance for different for different impact speeds in C300 Maraging steel. (Guduru et al. 2001)

Figure 3.11 Time evolution of shear‐band velocity as a function of time for different impact speeds in C300 Maraging steel. (Guduru et al. (2001)

Figure 3.12 A sequence of thermal images showing the transition of crack tip plastic zone to a shear band in C300 Maraging steel. (Guduru et al., 2001)

Figure 3.13 Dependence of strength of different metals melt on temperature (Mayer and Mayer, 2015): Dependence of strength of aluminum melt on the temperature at strain rates of (‐ ‐ ‐ and •) and (_____ and ⧫) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⧫)Dependence of strength of copper melt on the temperature at strain rates of (‐ ‐ ‐ and •) and (_____ and ⧫) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⧫)Dependence of strength of nickel melt on the temperature at strain rate of (‐ ‐ ‐ and •) and (_____ and ⧫) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⧫)

Figure 3.14 Time evolutions of mixed mode dynamic stress intensity factors for different values of material gradation parameter β: (a) normalized KI(t) at the left crack tip, (b) normalized KI(t) at the right crack tip, (c) normalized KII(t) at the left crack tip, (d) normalized KII(t) at the right crack tip. (Song and Paulino, 2006)

Figure 3.15 Characteristics of non‐homogeneous materials: (a) graded material geometry along the x direction, (b) time evolutions of normalized dynamic stress intensity factors at the left and right crack tips. (Song and Paulino, 2006)

Figure 3.16 Measured time evolution of a crack tip velocity in PMMA material. After an initial jump to about 150 m/s, the crack accelerates smoothly up to a critical velocity vc shown by the dotted horizontal line. Beyond this velocity, strong oscillations in the instantaneous velocity of the crack develop and the mean acceleration of the crack slows. (Fineberg and Marder, 1999)

Figure 3.17 (a) The transition to the micro‐branching instability is a roughly a linear function of the crack acceleration prior to the transition, (b) large hysteresis observed when the micro‐branching state, shown by the shaded region, undergoes a reverse transition to the single‐crack state. (Bouchbinder et al., 2010a)

Figure 3.18 Oscillatory instability of a crack: (a) a sequence of photographs of a propagating crack, (b) photographs of XY profile (top) and (XZ) fracture surface (bottom), of a 0.2 mm thick gel sample, (c) the fracture surface is micro‐branch dominated, (d) steady‐state amplitude of oscillations versus the applied stress, (e) wavelengths of the oscillations as a function of the applied stress. (Livne et al., 2007, Bouchbinder and Procaccia, 2007, and Bouchbinder et al., 2010a)

Figure 3.19 Snapshots (increasing times to the right) of the fracture process of a square pre‐notched PMMA plate subjected to an initial uniform tensile strain rate. Color intensity indicate level of a relevant stress. (Arias et al., 2007)

Figure 3.20 Volume region (horizon) in a solid showing the bond between two points and the force density vector (pairwise) applied at both points according to Silling (2012).

Figure 3.21 Kinematics of deformation in three‐dimension.

Figure 3.22 Dependence of the normalized displacement components: (a) along the direction of x, (b) orthogonal to x.

Figure 3.23 Top monolayer of target after impact for different values of δ, and for initially unperturbed and perturbed meshes. For sufficiently large δ, crack growth is arbitrary. Perturbation of the initial mesh acts only to break symmetry of the solution. (Parks et al., 2008a)

Figure 3.24 Top and side views of time sequence of materials damage during F‐4 impact simulation.

Figure 3.25 Side and front views of time sequence of concrete damage during F‐4 impact simulation.

Figure 3.26 Micropolar constitutive model for concrete. (Gerstle et al., 2007b)

Figure 3.27 Stress–strain relations computed from plain strain micropolar peridynamic concrete model for different values of transverse stress: (a) overview, (b) close‐up of tensile region. (Gerstle et al., 2007b)

Chapter 4

Figure 4.1 Lamb waves: (a) symmetric and (b) antisymmetric.

Figure 4.2 Wavespeed dispersion curves for Lamb waves in an aluminum plate: (a) symmetric modes, and (b) antisymmetric modes. (Giurgiutiu, 2005)

Figure 4.3 EnduraTec servo‐pneumatic testing machine showing four‐point loading of composite beams.

Figure 4.4 Time evolution of (a) AE energy and load–deflection plot for carbon sandwich beam, (b) AE energy versus time magnification over 0.1 s, and (c) AE amplitude–time record of the carbon fiber sandwich beam. (Ayorinde et al., 2012)

Figure 4.5 Waveform of emission at location around point g on the carbon fiber sandwich response of Figure 4.4(a). (Ayorinde et al., 2012)

Figure 4.6 Peak acoustic emission frequency versus time for carbon fiber sandwich. (Ayorinde et al., 2012)

Figure 4.7 Three‐dimensional plot of peak acoustic emission frequency versus time and amplitude for carbon fiber sandwich beam. (Ayorinde et al., 2012)

Figure 4.8 Distribution of acoustic emission energy in (μJ) over the beam length for carbon fiber sandwich (showing location of acoustic emission sensors S1–S4). (Ayorinde et al., 2012)

Figure 4.9 Time evolution of (a) AE energy and load–deflection plot for glass fiber sandwich beam, (b) AE amplitude–time record of the glass fiber sandwich beam. (Ayorinde et al., 2012)

Figure 4.10 Waveform of emission at location around point g on the glass fiber sandwich response of Figure 4.9(b). (Ayorinde et al., 2012)

Figure 4.11 Acoustic emission peak frequency versus time for the glass fiber sandwich beam. (Ayorinde et al., 2012)

Figure 4.12 Three‐dimensional plot of peak acoustic emission frequency versus time and amplitude for glass sandwich. (Ayorinde et al., 2012)

Figure 4.13 Distribution of AE energy in (μJ) over the beam length for glass fiber sandwich (showing location of acoustic emission sensors S1–S4). (Ayorinde et al., 2012)

Figure 4.14 (a) Assembled CFRC sandwich beam failure slides (61,000 fps, Phantom video camera), (b) magnification of two slides of (a). (Ayorinde et al., 2012)

Figure 4.15 Sample CFRC sandwich beam failure slides (100,000 fps, Phantom video camera). (Ayorinde et al., 2012) Slides sequence of failure of CFRC beam, case 1Slides sequence of failure of CFRC beam, case 2

Figure 4.16 Sequence of slides of the test specimen showing the development of crack in the glass fiber Rohacell composite (GFRC) beam using the Phantom video camera. (Ayorinde et al., 2012)

Figure 4.17 Schematic diagram of the bolted joint showing the instrumented bolts (•) and the fiber Bragg gratings (▪).

Figure 4.18 Excitation plots: (a) time history of excitation signal, (b) corresponding phase space plot.

Figure 4.19 Progression of measured axial load for each of the instrumented bolts.

Figure 4.20 Attractors reconstructions for (a) sensor 1 and (b) sensor 3 for undamaged case.

Figure 4.21 Performance of the nonlinear cross‐prediction error (a) mean prediction error showing confidence intervals, (b) corresponding probability density functions. Left column of (a) and (b) belongs to sensor 2 data to forecast data taken from sensor 1, while the right column is for sensor 1 data to forecast data from sensor 2.

Figure 4.22 Schematic diagram of impact resonance method of detecting defects in a concrete specimen. (Grosse and Reinhardt, 1996)

Figure 4.23 Detected transducer signal in (a) the time domain, and (b) frequency domain. (Grosse and Reinhardt, 1996)

Chapter 5

Figure 5.1 Layout of the sensing and actuating PZT wafers for detecting delamination on a composite plate of 2 ft × 2 ft and 0.25 in thickness. The composite laminate contains 48 plies stacked according to the sequence , consisting of Toray T300 graphite fibers and a 934 epoxy matrix. (Sohn et al., 2007a)

Figure 5.2 Damage localization and quantification based on a time reversal process: (a) actual impact location, (b) actuator‐sensor paths affected by the internal delamination; and (c) the damage size and location. (Sohn et al., 2007a)

Figure 5.3 Comparison between the original input signal (solid) and the restored signal (dotted) during the time reversal acoustic process between exciting PZT wafer # 6 and sensing PZT wafer #9: (a) the time reversibility of Lamb waves for the intact composite plate and (b) the breakdown of the time reversibility due to the internal delamination. (Sohn et al., 2007a)

Figure 5.4 Threshold levels and subsequent damage identification using the baseline data: (a) threshold value using damage index values obtained from the pristine composite plate, (b) identification of damaged paths using the previously established threshold value. (Sohn et al., 2007a)

Figure 5.5 Threshold and instantaneous damage identification without using prior baseline data: (a) sorted DI values in an ascending order and the damaged paths (outliers) are identified, (b) the computed outlier probability for the first 15 largest damage index values; it reaches its maximum value at the 5th largest damage index value, (c) damage distribution function after excluding the five outliers identified in (b), the GEV is fitted to the remaining 63 DI values to estimate a new threshold value (=0.251); and (d) damage index versus wave propagation paths showing the use of the new threshold value (0.251 vs 0.243) confirms that the 5 largest DI values are outliers and the associated paths are influenced by internal delamination. (Sohn et al., 2007a)

Figure 5.6 The laser vibrometer scanning setup for composite panel: (a) The piezo‐patches are numbered while the damage is simulated with a 20 g mass added, (b) schematic diagram of the laser beam showing acquisition points. (Banerjee et al., 2009b).

Figure 5.7 Damage index distribution over the panel due excitation generated by actuator‐1 over frequency range 150–1000 Hz. The 92 small circles represent the acquisition points. The 20 g added mass is located at the black solid circle. (Banerjee et al., 2009b)

Figure 5.8 The modal assurance criterion (MAC) evaluated with excitation given by actuator‐1 for the damaged and healthy panels over frequency range150–1000 Hz. (Banerjee et al., 2009b)

Figure 5.9 The modal assurance criterion (MAC) evaluated with excitation given by actuator‐8 for the damaged and healthy panels over frequency range 150–1000 Hz. (Banerjee et al., 2009b)

Figure 5.10 Damage index distribution over the panel under excitation of actuator number 8 over frequency range 500–1000 Hz. The small circles represent the acquisition points. The 20 g added mass is located at the black solid circle. (Benerjee et al., 2009b)

Figure 5.11 (a) Schematic diagram of half symmetrical T‐joint and the total strain energy release rate distribution along the depth (y) of T‐joint according to the crack tip element analysis results for (b) thin T‐joint and (c) thick T‐joint for three different values of disbond length: ♦ 30 mm disbond, ▪ 60 mm disbond, ▴90 mm disbond. (Dharmawan et al., 2008)

Figure 5.12 Joint representation by an elastic spring system model showing the crack initiation by springs over a width d.

Figure 5.13 A small segment of the beam including localized joint.

Figure 5.14 Equivalent model for the beam with localized joint.

Figure 5.15 Dependence of modal frequencies on the joint strength parameter, ϖ0 showing the occurrence of 1:1 internal resonance between the first two modes, . Pilipchuk and Ibrahim (2011)

Figure 5.16 First three mode shapes of the beam with (a) weak joint, (b) strong joint. Pilipchuk and Ibrahim (2011)

Figure 5.17 Dependence of natural frequencies on the foundation stiffness in the weak joint case: , , . (Pilipchuk and Ibrahim, 2011)

Figure 5.18 Dependence of natural frequencies on the foundation stiffness in the strong joint case: , , . (Pilipchuk and Ibrahim, 2011)

Figure 5.19 Time evolution of the relative length of a crack based on the proposed model. (Pilipchuk and Ibrahim, 2011)

Figure 5.20 Projection of the system trajectory on the configuration planes for three different cases of the joint stiffness: (a) weak joint, (b) 1:1 resonance, (c) strong joint, under the developing fracture condition and loading: , : A‐initial crack, and B‐developed crack. (Pilipchuk and Ibrahim, 2011)

Figure 5.21 Time history records and the corresponding spectrograms for the modal coordinates in the case of a weak joint under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(a)). (Pilipchuk and Ibrahim, 2011)

Figure 5.22 Time history records and the corresponding spectrograms for the modal coordinates in the case of 1:1 resonance under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(b)). (Pilipchuk and Ibrahim, 2011)

Figure 5.23 Time history records and the corresponding spectrograms for the modal coordinates in the case of a strong joint under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(c)). (Pilipchuk and Ibrahim, 2011)

Figure 5.24 Projections of the system trajectory on configuration planes in the strong joint case with stiffness under the developing fracture condition and loading; , . (Pilipchuk and Ibrahim, 2011)

Figure 5.25 Projections of the system trajectory on the original generalized coordinate planes under the developing fracture condition and loading , . No elastic foundation assumed. (Pilipchuk and Ibrahim, 2011)

Figure 5.26 Original generalized coordinate time histories under the conditions of Figure 5.25. (Pilipchuk and Ibrahim, 2011)

Figure 5.27 Time history record of wind induced vibration of bridge hanger. (Li et al., 2015)

Figure 5.28 Time history record of the vertical acceleration due to vortex induced vibration of the bridge girder. (Li et al., 2015)

Figure 5.29 (a) Schematic diagram of the T‐joint showing the region of defect, (b) mode shape displacement profiles along the length of the stiffened panel. (Herman et al., 2013): _______ undamaged panel, – ‐ – delaminated panel, and …. porous panel

Figure 5.30 Normalized mode shape difference profiles for the first mode along the length of the stiffened panels containing (a) delamination, (b) porous region. (Herman et al., 2013): ______ profiles measured experimentally, – ‐ – and estimated using the finite element method

Figure 5.31 (a) The geometry and (b) modeling of T‐joint structure.

Figure 5.32 Amplitude–frequency response of the horizontal beam alone. (Pilipchuk et al., 2013a)

Figure 5.33 Eigen‐frequencies of the beam alone versus torsion stiffness at the left boundary. (Pilipchuk et al., 2013a)

Figure 5.34 Eigen‐frequencies of the model (Figure 5.31(b)) with no boundary springs versus stiffness of the tension‐compression spring; Point A – 1:1 internal resonance between the first two modes (Pilipchuk et al., 2013a)

Figure 5.35 Eigen‐frequencies of the model with no boundary springs versus stiffness of the central torsion spring; Point A – 1:1 internal resonance between the second and third modes. (Pilipchuk et al., 2013a)

Figure 5.36 The temporal mode shape of non‐smooth perturbation: exact (thick line) and the four‐term trigonometric expansion (thin line), the input amplitude is Y = 1. (Pilipchuk et al., 2013a)

Figure 5.37 Configuration plane trajectories under the input frequency 313.6 Hz: 1 – no damage, zero damping (analytical solution), 2 – damaged structure, zero damping (analytical solution), 3 – damaged structure, non‐zero damping (numerical steady‐state solution). (Pilipchuk et al., 2013a)

Figure 5.38 Dynamic response of the T‐joint structure on the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam; first row – strain records. (Pilipchuk et al., 2013a)

Figure 5.39 Trajectories of the T‐joint structure under the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam. (Pilipchuk et al., 2013a)

Figure 5.40 Spectral response characteristics of the T‐joint structure under the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam. (Pilipchuk et al., 2013a)

Figure 5.41 Dynamic response of the T‐joint structure on the high‐frequency load (313.6 Hz): left column – healthy structure, right column – progressive damage with bolt ‘cracking relaxation’ on one side of the T‐joint; first row – strain records. (Pilipchuk et al., 2013a)

Figure 5.42 Short‐term fragment of the high‐frequency dynamic response from the previous series of diagrams. (Pilipchuk et al., 2013a)

Figure 5.43 Trajectories of the T‐joint structure under the high‐frequency load (313.6 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam; frame ticks are preserved for comparison reason. (Pilipchuk et al., 2013a)

Figure 5.44 Short‐term fragment of the trajectories from the previous series of diagrams. (Pilipchuk et al., 2013a)

Figure 5.45 Snapshots of the horizontal beam’s centerline during 5 seconds (4.0 < t < 9.0) every 7 ms: (a) healthy structure, (b) developing damage. (Pilipchuk et al., 2013a)

Figure 5.46 Snapshots of the horizontal beam’s centerline during 0.5 seconds every 7 ms: (a) 4.0 < t < 4.5, (b) 6.0 < t < 6.5. (Pilipchuk et al., 2013a)

Figure 5.47 Stroboscopic 3D diagrams in configuration space on the interval 5.0 < t < 6.0; snapshots are taken every 1/313.6 sec. (Pilipchuk et al., 2013a)

Chapter 6

Figure 6.1 Stress spectrum.

Figure 6.2 Weibull probability density function for different values of the shape parameter k and for scale parameter .

Figure 6.3 Illustration of crack length in terms of applied stress and number of cycles.

Figure 6.4 Demonstration of random stress cycles into half stress cycles of different stress ranges.

Figure 6.5 Dependence of aluminum alloy strength on temperature under different types of loading including fatigue in reverse bending. (Grover, 1966)

Figure 6.6 Stress rupture curve showing the dependence of stress to rupture on time for two different values of temperature for nickel alloy N‐155. (Grover, 1966)

Figure 6.7 Dependence of alternating stress on the mean stress for nicker alloy N‐155 at temperature 1350 °F at 3600 cpm stress combinations for rupture. Solid curves are obtained from equation (6.13) and points from S–N curve. (Grover, 1966)

Figure 6.8 Dependence of alternating stress on the mean stress for nickel alloy N‐155 at temperature 1350 °F at 3600 cpm stress combinations for 2% creep. Solid curves are obtained from equation (6.7) and points from dynamic creep tests. (Grover, 1966)

Figure 6.9 Dependence of thermal fatigue crack length on the number of cycles under 0.5% strain range and cycle rate of 1 cpm for 0.16% carbon steel. (Taira et al., 1974, adapted from Halford, 1986)

Figure 6.10 Load–displacement curves for (a) 3 mm bolted joint: specimen 46 under pull velocity 0.02 mm/s; 42 under 5.2 mm/s; 43 under 50.5 mm/s; and 44 under 500 mm/s (b) 5 mm bolted joint: specimen 35 under 0.02 mm/s; 36 under 3.04 mm/s; 38 under 25.3 mm/s; and 39 under 250 mm/s. (Birch and Alves, 2000)

Figure 6.11 Comparison of contact stresses for load transfer ratio of 0.4 and friction coefficient μ =0.5: ____ finite element, ‐ ‐ ‐ Mindlin theory with added effect of bulk stress.

Figure 6.12 Life to failure for lap‐joint for different values of squeeze force F: ▪ F = 2500 lb; ▵ F = 3500 lb; ⚬ F = 4250 lb; and ♢ F = 5000 lb.

Figure 6.13 Schematic diagrams of sagging and hogging vessels.

Figure 6.14 Time history record of high‐pass filtered stress signal amidships for a container ship sailing in bow‐quartering sea illustrating a series of transient slamming/whipping events. Full‐scale measurements from the 9400 TEU container ship. (Andersen and Jensen, 2014)

Figure 6.15 Demonstration of the two‐node vertical bending mode of the container ship hull. (Andersen and Jensen, 2014)

Figure 6.16 Stress time history records showing unfiltered, low‐pass and high‐pass filtered time series (average of port and starboard side). The largest stresses is found at about 350 s as marked. Hogging is positive. (Andersen and Jensen, 2014)

Figure 6.17 FFT of stress time history of one hour duration from 9400 TEU containership in rough sea showing the frequency at the two‐node vertical mode close to 0.48 Hz. (Andersen and Jensen, 2014)

Figure 6.18 Detailed of short period of filtered time series record of stress average of port and starboard side for different frequency bandwidths in which hogging is positive. (Andersen and Jensen, 2014)

Figure 6.19 Preliminary evaluation of fatigue safety level. (Kühn et al., 2008)

Figure 6.20 (a): Reliability index, (b): Probability of failure plots for different values of critical crack depth. (JCSS, 2001; Kühn et al., 2008)

Figure 6.21 Concept of limit state function. (Kühn et al., 2008)

Figure 6.22 Illustration of the two‐dimensional case of a linear limit state function and standardized normally distributed variables u1 and u2. (Faber, 2012)

Figure 6.23 Proposed linearization by Hasofer and Lind (1974) in standard normal space. (Faber, 2012)

Figure 6.24 Limit state function. (Kühn et al., 2008)

Figure 6.25 Decision tree for prior and posterior decision analysis.

Figure 6.26 Load‐shortening curve demonstrating types of failure modes vs structural response. (Hess et al., 2000; Hess, 2003)

Figure 6.27 Schematic diagrams of: (a) a stiffened plate structure considered for ULS assessment by ALPS/ULSAP method, (b) ALPS/HULL model used for the progressive hull collapse analysis. (Jang et al., 2007)

Figure 6.28 Initial deflection in the form of hungry horse mode. (Carlsen and Czujko, 1978; Wang C.L.E. et al., 2006)

Figure 6.29 Variability of the reliability index with the heading angle for different cross‐sections at a given sea state and speed: (a) polar coordinates (b) Cartesian coordinates. (Decò et al. 2012; Frangopol and Soliman, 2014)

Chapter 7

Figure 7.1 Different elevation views of the north face of TWC Tower showing (a) the aircraft impacting the north face, (b) damage to the tower skin structure estimated from the computer simulation, and (c) observed as reported by ASCE/FEMA (2002). (Irfanoglu and Hoffman, 2008).

Figure 7.2 Dependence of the reduction factor on the temperature for the effective yield strength given by the ratio of σy,t ate temperature t to the yield strength σy at 20° and the ratio of the modulus of elasticity of the linear range at temperature t to its value at 20°. (Irfanoglu and Hoffman, 2008)

Figure 7.3 Dependence of axial load capacity of core columns and estimated gravity load demand at story 95 of WTC‐I. (Irfanoglu and Hoffman, 2008)

Figure 7.4 Time history records of free‐field ground motion acceleration components at Jensen Filter Plant, Generator Building. (Fenves and Ellery, 1998)

Figure 7.5 Photographs of I‐35 W Bridge: (a) before collapse and (b) after collapse (from NTSB 2008a). (Liao and Okazaki, 2009)

Figure 7.6 Panel point U10 after collapse: (a) photograph of U10W (from NTSB 2008a), and (b) reported locations of fracture. (Liao and Okazaki, 2009)

Figure 7.7 Longitudinal and vertical damage of all ships due to all causes. (SSC‐220, 1971; Wiernicki, 1986)

Figure 7.8 Experimental measurements of the dependence of crest height on the wave vertical velocity.

Figure 7.9 Probability histograms as function of free surface vertical velocity for (a) group 1, (b) group 2, and (c) group 3 of Figure 7.8.

Figure 7.10 Long‐term distribution of bending moment showing rules of minimum values.

Figure 7.11 Pressure evolution measured on the vertical wall under impacting conditions, at (a) 0.012 m and (b) 0.032 m above the deck. Two test results are given for each gauge location. The solid curve in (c) gives the numerical results before breaking occurs. (Greco et al., 2004)

Figure 7.12 Typical and equivalent profiles of impact pressure action considered by Paik and Shin (2006).

Figure 7.13 Schematic diagram of ship stiffened panel considered by Paik and Shin (2006).

Figure 7.14 Geometry of plate stiffener under impact line load considered by Paik and Shin (2006).

Figure 7.15 Dependence of permanent peak deflection ratio on the impact energy parameter as predicted by design formula without strain rate effect (‐ – ‐), design formula with strain rate effect (______), ⦁ measured for mild steel (Jones et al., 1970, and Jones, 1971), ♦ LS‐DYNA3D simulations.

Figure 7.16 Time history record of wave‐induced vertical bending moment amidships showing the minimum design wave‐induced vertical bending moments in hogging and sagging. (Andersen and Jensen, 2014)

Figure 7.17 Probability of exceedance for sagging of rigid and flexible ships over three hours (absolute sagging vertical bending moment).

Figure 7.18 Probability of exceedance for hogging of rigid and flexible ship.

Figure 7.19 Gumbel probability density function for hogging based on Weibull parameters for the individual peak, hull girder flexibility included.

Figure 7.20 Maximum bending moment for sagging and hogging from a double impact sea at aft quarter length.

Figure 7.21 Schematic diagram of a typical LNG cargo holds and No. 2 tank location considered in ABS (2006/2009)

Figure 7.22 Schematic diagram of membrane‐type LNG tank showing the major tank dimensions and filling depth considered in ABS (2006/2009).

Figure 7.23 Recent step changes in LNG delivery industry.

Figure 7.24 Concept of reliability of two structural designs (a) and (b); pdf in vertical axis = probability density function of loads, S, or capacities, R: N (mean, standard deviation) were adapted and edited (not the same) and are not the same as in the original paper.

Figure 7.25 Schematic diagram of clustered sensors at tank corner considered in ABS (2006/2009).

Figure 7.26 Time history records of local impact pressure at sensors A–P and panel pressure adapted from ABS (2006/2009): (a) sensors A–D, (b) sensors E–H, (c) sensors I–L, (d) sensors M–P, shown in Figure 7.25, (e) panel pressure.

Figure 7.27 Triangular impulse for uniform pressure considered in ABS (2006/2009).

Figure 7.28 Idealization pressure impulse by a triangular impulse considered in ABS (2006/2009).

Figure 7.29 Sloshing impact pressure time history: (a) impulse with a short rise time, (b) pressure pattern with skewness close to 1, and (c) pressure impulse with skewness greater than 1.

Figure 7.30 Example of long‐duration sloshing test.

Figure 7.31 Total losses of ship in gross tonnage during the years 1995–1998, Zhu et al. (2002): the term foundered means to sink below the surface of the water.

Figure 7.32 Damage statistics of ship grounding incidents based on: (a) damage location, (b) damage‐width ratio, and (c) damage‐length ratio. Zhu et al. (2002)

Figure 7.33 Schematic diagram of cone‐plate model for studying general grounding scenarios. (Zhu et al. 2002)

Figure 7.34 Schematic diagram showing regions of grounding damages in a cargo ship. (Zhu et al., 2002)

Figure 7.35 Comparison of probability density distributions for grounding damage length using different estimates.

Figure 7.36 Global first‐year ice loads on the Molikpaq at the Tarsiut P‐45 site during the winter of 1984/85. (Wright et al. 1986 and Timco and Frederking, 2004)

Figure 7.37 Coordinate frame and parameters used in SIMCOL code.

Figure 7.38 Probability density function of penetration ratio as estimated by SIMCOL for SH100 and DH150 compared to MARPOL and HARDER data.

Figure 7.39 Probability density function of longitudinal damage extent ratio as estimated by SIMCOL for SH100 and DH150 vessels compared to MARPOL and HARDER vessels data.

Figure 7.40 Trend of time evolution of strength, R, and load effect, S.

Figure 7.41 Time evolution of probability of failure.

Figure 7.42 Lower limit of structural life based on plate deformation (Ayyub et al., 1989): inspection 1 year, inspection 2 years.

Figure 7.43 Structural life based on fatigue failure mode (Ayyub et al., 1989).

Figure 7.44 The Bronx vehicle in a minimal rollover with frames at 0.25 s time difference. (Winkler et al., 1998)

Figure 7.45 Plan‐view sketch of the Bronx vehicle at the end of simulation. (Winkler et al., 1998)

Figure 7.46 Schematic diagram of liquid displacement in a moving tanker. (Strandberg, 1978)

Figure 7.47 Dependence of the overturning limit on the lateral acceleration for different values of harmonic oscillation frequencies and for 50% liquid volume in an elliptic tank without baffles or cross walls. (Strandberg, 1978) Harmonic oscillation frequencies: ▲: 0.3 Hz, ♦: 0.4 Hz, ■: 0.5 Hz, ●: 0.6 Hz, ▼: 0.8 Hz

Figure 7.48 Dependence of overturning limit on the frequency for different tank baffles. (Strandberg, 1978)

Figure 7.49 Side force coefficient (SC) as function of the oscillation frequency for different baffle orientation, lateral acceleration peaks: ■ 1 m/s², • 2 m/s² and ▲ 3 m/s². (Strandberg, 1978)

Figure 7.50 Time history records of the dynamic load factor of tires on different axles for liquid and frozen liquid cases showing the influence of applying the brakes at 4 s for two different values of steer angle: left column figures belong to the left side and the right column figures belong to the right side. (Kang, 2001) _____ Liquid ( ), ….. Rigid ( ), – ‐ – Liquid ( ),

Figure 7.51 Schematic diagram of a fluid volume whose free surface assumes a flat surface during lateral acceleration due to vehicle turning.

Figure 7.52 Fluid free surface configuration under steady lateral acceleration according to quasi‐static approach.

Figure 7.53 Time history records of predicted and measured of rear axle roll moment exerted by the tank with fluid of the vehicle under 40 mph lane change. (Wendel et al., 2002)

Figure 7.54 Schematic diagram of the seven‐degree of freedom pitch‐plane vehicle model as proposed by Yan and Rakheja (2010).

Figure 7.55 Schematic diagram showing free‐body diagram of the tire‐wheel assembly. (Yan and Rakheja, 2010)

Figure 7.56 Frequency spectra of slosh force components measured on the 50%‐filled unbaffled (dashed‐dotted plots) and baffled (solid plots) tanks subject to 1 m/s² lateral acceleration excitation at 0.7 Hz: (a) lateral force, (b) longitudinal force. (Yan and Rakheja, 2010)

Figure 7.57 Schematic diagram of a cantilever pipe conveying fluid and the free‐body diagram of the pipe section. (Bishop and Fawzy, 1976)

Figure 7.58 Schematic diagram of a cantilever pipes conveying fluid oscillating in three‐dimensional motion.

Figure 7.59 Schematic diagram of a cantilevered pipe (a) elevation view, (b) projection showing the restrained set of four springs. (Païdoussis et al., 2007)

Figure 7.60 Bifurcation diagram showing the dependence of pipe displacement on flow velocity, , . (Païdoussis et al., 2007)

Figure 7.61 Time history record at flow velocity showing sample of chaotic motion , (____, ζ). (Païdoussis et al., 2007)

Figure 7.62 Experimental bifurcation diagram showing average displacement ( ) and average amplitude ( ). (Païdoussis et al., 2007)

Figure 7.63 Dependence of the critical flow velocity, at which the first bifurcation occurs, on the linear spring stiffness. Hopf bifurcation, pitchfork bifurcation. (Païdoussis et al., 2007)

Figure 7.64 (a) Schematic diagram of a constrained cantilevered pipe, (b) dependence of restrained force on the displacement. (Païdoussis et al., 1989, 1991)

Figure 7.65 Bifurcation diagrams for two‐mode model showing the dependence of the free‐end displacement on the flow speed parameter for three velocity ranges: (a) velocity range 6–8.5, (b) velocity range 10–11. (Païdoussis et al., 1989, 1991)

Figure 7.66 Schematic diagram of a semicircular cantilevered pipe with constraint.

Figure 7.67 Bifurcation diagrams for the dependence of the pipe free end displacement on the fluid flow velocity under forced excitation, for mass ratio parameter , constraint stiffness parameter , constraint location : (a) no external excitation, (b) external excitation, amplitude , and frequency . (Lin et al., 2007a, and Lin and Qiao, 2008d)

Chapter 8

Figure 8.1 Time evolution of depth and rate of immersion corrosion wastage. (Soares et al., 2005b)

Figure 8.2 Evolution of cumulative corrosion loss using ballast tank corrosion model. (Gudze and Melchers, 2008).

Figure 8.3 Schematic corrosion wastage models in previous literature: 1: Southwell et al. (1979); 2: Soares and Garbatov (1998); 3: Qin and Cui (2003); 4: Paik et al. (1998a, 2003a); 5: Ivanov et al. (2004). (Compiled by Guo et al., 2008)

Figure 8.4 Schematic diagram of tanker and details of section A–A: dimensions of longitudinal plating in inches. (Akpan et al., 2002).

Figure 8.5 Instantaneous and time‐dependent probabilities of failure for a tanker with (a) corrosion, (b) cracks, and (c) different cases. (Akpan et al., 2002)

Figure 8.6 Dependence of aluminum tensile strength variation on temperature for three different types of aluminum. (SSC‐464, 2012)

Figure 8.7 Flow chart of corrosion damage algorithm for aircraft using the values of preliminary corrosion threshold values: suspended particulates = 61 µg/m³, sulfur dioxide = 43 µg/m³, ozone (O3) = 36 µg/m³, nitrogen dioxide = 64 µg/m³, absolute humidity (AH) = 7.1 g/m³, distance to sea or salt water = 4.5 km, solar radiation in July = 600 Langleys, and rainfall total =125 cm. AA: very severe rating, A: severe rating, B: moderate rating, C: mild rating.

Figure 8.8 Corrosion rates of AZ31B magnesium alloy compared with environmental ratings as obtained from different sources; A: Andrews, MD, B: Barksdale, LA, D: Davis Monthan AZ, KB: Kure Beach NC 80 ft (35‐38), N: Newark, NJ, Panama Canal, M: Marine, O: Open field, RF: Rain forest, PR: Point Reyes CA, R: Robins GA, T: Tinker OK, SC: State College PA, W‐P Wright‐Patterson OH. (Summitt and Fink, in AGARD, 1981)

Figure 8.9 Corrosion rates of 2024 T3 alclad aluminum alloy compared with environmental ratings as obtained from different sources: B: Barksdale LA, CC: Corpus Cristi TX, KB: Kure Beach 80 ft, Mc: McCook IL, N: Newark NJ, PR: Point Reyes CA, SC: State College PA, T: Tinker OK, and W‐P: Wright‐Patterson OH.

Figure 8.10 Corrosion rates of 7075 T6 aluminum alloy compared with environmental ratings as obtained from different sources: A: Andrews MD, B: Barksdale LA, D: Davis Monthan AZ, KB: Kure Beach 80 ft, N: Newark NJ, PR: Point Reyes CA, SC: State College PA, T: Tinker OK, and W: F.E. Warren WY.

Figure 8.11 Corrosion rates of 7079 T6 alclad aluminum alloy compared with environmental ratings as obtained from different sources: A: Andrews MD, KB: Kure Beach 80 ft, N: Newark NJ, R: Robins GA, SC: State College PA, and T: Tinker OK.

Figure 8.12 Typical aircraft multi‐layered coating for corrosion control. (Bierwagen and Tallman, 2001; ACW, 2014)

Figure 8.13 Dependence of crack growth rate on applied stress intensity factor for constant CMOD rate for alloy beta‐C in aqueous NaCl solution at 25 °C. Crack growth rate transitions, shown here for a CMOD rate of 0.03 µm/s, occur for CMOD rates less than 0.6 µm/s.

Figure 8.14 Dependence of threshold stress intensity factor for onset of slow crack growth from an initially static crack and onset of fast crack growth from a propagating crack plotted on the applied the crack mouth opening displacement rate. The solid curve represents predictions using the diffusion‐limited model equations. The open squares are due to Hall (2011).

Figure 8.15 Slow crack growth rate vs applied CMOD rate for applied stress intensity factors near the rate transition threshold.

Figure 8.16 Dependence of strain thresholds on austenite spacing. (Woollin and Gregori, 2004) ⚬ highest strain at which cracks do not initiate, ◾ highest strain at which cracks do not propagate.

Figure 8.17 Variation of hydrogen embrittlement stress cracking threshold strains for parent steel, cross‐weld, all‐weld metal, and simulated heat affected zone specimens.

Chapter 9

Figure 9.1 Amplitude–frequency response of the 19th mode for loose and tight bolts: – ‐ – ‐, experimental; – –, analytical. (Esteban and Rogers, 2000)

Figure 9.2 Energy dissipation of the 19th mode for loosening and tight bolts: – ‐ – ‐, tight bolts; – –, loose bolts. (Esteban and Rogers, 2000)

Figure 9.3 A schematic diagram showing a bolt subjected to prying.

Figure 9.4 Dependence of the bolt force on the external force under (a) axial tension load, (b) prying load.

Figure 9.5 Active control to vary the normal contact force in a joint by means of a piezoelectric element. (Gaul and Nitsche, 2000)

Figure 9.6 Force‐state map of an ideal linear spring dashpot joint model.

Figure 9.7 Force‐state map of an ideal dead‐band spring with Coulomb friction.

Figure 9.8 Beam element with torsional spring‐dashpot joint modeling.

Figure 9.9 Dependence of the mean square response of the top floor on (a) the fixity parameter, and (b) the damping ratio.

Figure 9.10 Sensitivity curves showing the derivative of the second‐order moment of the top story displacement with respect to (a) the damping coefficient of the connections of the ith story versus fixity factor v for , (b) the fixity factor of each story versus damping coefficient for .

Figure 9.11 Stochastic panel response amplitude in the presence of spatial uncertainty of Young’s modulus, grid, correlation length CL = 0.10 L. (Lindsley et al., 2002a)

Figure 9.12 Response surface to variability in boundary condition parameter β: : pinned condition, : clamped condition. (Lindsley et al., 2002a,b).

Figure 9.13 Three‐dimensional pdf plot and its projection on the amplitude‐B.C. uncertainty plane showing the contours of equal probability density function. (Lindsley et al., 2002a)

Figure 9.14 Dependence of preload relaxation on average vibration life in cycles for two different values of initial pre‐load. (Finkelston, 1972)

Figure 9.15 Time history records of excitation, responses for two different values of the joint bending rigidity to the torsional stiffness parameter z = 0.0 and z = 0.5057. (Qiao et al., 2001)

Figure 9.16 Power spectra of excitation and response for two different values of the joint bending rigidity to the torsional stiffness parameter z = 0.0 and z = 0.5057. (Qiao et al., 2001)

Figure 9.17 Conditional mean square response of the system for two different values of the joint bending rigidity to the torsional stiffness parameter z = 0.0 and z = 0.5057. (Qiao et al., 2001)

Figure 9.18 Dependence of the response central frequency on the joint bending rigidity to the torsional stiffness parameter, z. (Qiao et al., 2001)

Figure 9.19 Dependence of the response mean squares on the joint bending rigidity to the torsional stiffness parameter, z. (Qiao et al., 2001)

Figure 9.20 Modified experimental residual preload versus the number of cycles.

Figure 9.21 Time evolution of the linear natural frequency and nonlinear coefficient: (a) natural frequency (b) nonlinear coefficient. (Qiao et al., 2001)

Figure 9.22 Time history records under sinusoidal excitation for excitation amplitude : (a) direct numerical integration, (b) one‐step averaging. (Qiao et al., 2001)

Figure 9.23 Time history records under sinusoidal excitation for excitation amplitude : (a) direct numerical integration, (b) one‐step averaging. (Qiao et al., 2001)

Figure 9.24 Frequency and time domains of the response: (a) response spectrum, and (b) response amplitude time history for excitation amplitude . (Qiao et al., 2001)

Figure 9.25 Frequency and time domains of the response: (a) response spectrum, and (b) response amplitude time history for excitation amplitude . (Qiao et al., 2001)

Figure 9.26 Relaxation of the joint stiffness α(t) and response statistics for slope parameter under excitation spectral density . (Qiao et al., 2001)

Figure 9.27 Relaxation of the joint stiffness α(t) and response statistics for slope parameter under excitation spectral density . (Qiao et al., 2001)

Figure 9.28 Relaxation of the joint stiffness α(t) and response statistics for slope parameter under excitation spectral density . (Qiao et al., 2001)

Figure 9.29 Response spectra estimated over short interval of time history record of duration 30 s, each revealing the shift of response frequency due to joint relaxation. (Qiao et al., 2001)

Figure 9.30 Time history response and response statistics for slope parameter under excitation spectral density . (Qiao et al., 2001)

Figure 9.31 Time history response and response statistics for slope parameter under excitation spectral density . (Qiao et al., 2001)

Figure 9.32 Schematic diagram of a two‐dimensional panel with boundary conditions relaxation.

Figure 9.33 Dependence of real and imaginary parts of the panel natural frequency on dynamic pressure for : (a) real parts, (b) imaginary parts ___z = 0.001,……z = 0.1, ……..z = 1. (Beloiu et al., 2005)

Figure 9.34 Dependence of real and imaginary parts of the first and second natural frequencies on relaxation parameter z for : (a) real parts, (b) imaginary parts, , , . (Beloiu et al., 2005)

Figure 9.35 Boundaries of panel flutter on the plane for different values of in‐plane load and for , . (Beloiu et al., 2005)

Figure 9.36 Boundaries of panel flutter for different values of damping factor showing the reversal effect of damping for , . (Beloiu et al., 2005)

Figure 9.37 Boundaries of panel flutter on the plane for different values of relaxation parameter, z, and for . Dashed curve indicates the critical damping ratio that separate between stabilizing and destabilizing damping effects. (Beloiu et al., 2005)

Figure 9.38 Bifurcation diagram showing the regions of different panel regimes for damping parameter and different values of relaxation parameter. (I) statically stable, (II) static buckling (divergence), (III) limit cycle oscillation, and (IV) multi‐period oscillations and chaos.

Figure 9.39 Bifurcation diagram for different values of relaxation parameter for , , and . (Beloiu et al., 2005)

Figure 9.40 Bifurcation diagram for different values of relaxation parameter for , , and . (Beloiu et al., 2005)

Figure 9.41 Three‐dimensional plots of amplitudes time evolutions and their dependence on dynamic pressure for . (Beloiu et al., 2005)

Figure 9.42 (a) Relaxation of boundary conditions and (b) time history record of panel deflection at , for , , , , and . (Beloiu et al., 2005)

Figure 9.43 Relaxation and mode convergence time history records: (a) relaxation of boundary conditions and time history records of the panel deflection showing convergence of the numerical results as the number of modes increases; (b) two‐mode interaction, (c) four‐mode interaction, and (d) six‐mode interaction. , , , , and . (Beloiu et al., 2005)

Figure 9.44 Bifurcation diagram and the corresponding largest Lyapunov exponent for , , , , and . (Beloiu et al., 2005)

Figure 9.45 Phase plots for , , , , and corresponding to Figure 9.44 for: (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) ; (h) ; (e) . (Beloiu et al., 2005)

Figure 9.46 FFT plots for , , , , and corresponding to Figure 9.44 for: (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) ; (h) ; (e) . (Beloiu et al., 2005)

Figure 9.47 Bifurcation diagram of the first return and largest Lyapunov exponent for , , , , and . (Beloiu et al., 2005)

Figure 9.48 Chaos boundaries for , , : (a) ; (b) ; (c) ; (d) . (Beloiu et al., 2005)

Figure 9.49 Bifurcation diagram of the first return and largest Lyapunov exponent along the path line A of Figure 9.38, for , , , , and . (Beloiu et al., 2005)

Figure 9.50 Bifurcation diagram of the first return and largest Lyapunov exponent along the path line A of Figure 9.38 for , , , , and . (Beloiu et al., 2005)

Figure 9.51 Bifurcation diagram of the first return and Lyapunov exponent along the path line B of Figure 9.38 for , , , , and . (Beloiu et al., 2005)

Figure 9.52 Phase diagrams for , , , for two sets of initial conditions: (1)

. (2)

. (a) ; (b) ; (c) ; (d) .

Figure 9.53 Section of bifurcation diagram of the first return for , , , , and for initial conditions (1)

, and initial conditions (2)

. (Beloiu et al., 2005)

Figure 9.54 FFT plots and spectrograms for : (a) and (b) . (Beloiu et al., 2005)

Figure 9.55 Wavelet transform results (a) time history, (b) modulus of WT, (c) phase of WT, (d) three‐dimensional plot of modulus of WT for , , , , , and . (Beloiu et al., 2005)

Figure 9.56 Wavelet transform results (a) time history, (b) modulus of WT, (c) phase of WT, (d) three‐dimensional plot of modulus of WT for , , , , , and . (Beloiu et al., 2005)

Figure 9.57 Types of cracks in welded joints.

Figure 9.58 Schematic diagram showing stress categories used in finite element model of hot spots used in DNV (2003).

Figure 9.59 Dependence of fatigue crack growth rates on the range of stress intensity factor for different stress ratios, . (Alam, 2005)

Figure 9.60 Dependence of the stress intensity factor range on fatigue life (in cycles) for different stress ratios,

. (Alam, 2005)

Figure 9.61 Qualitative residual stress distribution caused by spot heating as described by Booth (1991).

Chapter 10

Figure 10.1 Crack arresters: (a) riveted seam type, (b) inserted type, (c) welded patch type, (d) riveted patch type, (e) stiffener type, and (f) ditch type. (SSC 265, 1977).

Figure 10.2 Effect of test temperature on the crack arrest toughness of 50.8 mm thick A36 steel tested in the L‐T direction. (Ripling and Crosley, 1982)

Figure 10.3 Dependence of crack initiation KJ(IT) and crack arrest KIA master curves for the HSLA‐100 (with To = −166 °C and TKIA = −135 °C and HY‐100 (with To = −124 °C and TKIA = −64 °C) steel plates. (Link et al., 2009)

Figure 10.4 Schematic diagrams of two arresters: (a) splice type arrester (Hirose et al., 2009b), (b) semi‐cylindrical crack arrester. (Takeda et al., 2009)

Figure 10.5 Energy release rate distribution over the length in the absence and presence of splice‐type crack arrester under constant load of 100 N. (Hirose et al., 2009b)

Figure 10.6 Schematic of crack detection technique. (Minakuchi et al., 2011a)

Figure 10.7 Schematics of sensor deformation demonstrated in finite element analysis. (a) double cantilever beam specimen, (b) end‐notch flexure specimen. (Minakuchi et al., 2011a)

Figure 10.8 Principal strains in the optical fiber core depending on L. Double cantilever beam specimen: (a) sensor A, (b) sensor B; end‐notch flexure specimen: (c) sensor A, and (d) sensor B. (Minakuchi et al., 2011a)

Figure 10.9 Frame sections (a) with crack arrester, (b) without crack arrester. (Kotari et al., 2014)

Figure 10.10 Fail‐safety scenarios for conventional and integral structure: (a) multiple‐piece fail‐safe, (b) integral fail‐safe, (c) fuselage section. (Pettit et al., 2000)

Figure 10.11 Boeing C‐17 tail damage. (Dorworth, 2007)

Figure 10.12 Schematic diagrams of two different designs of crack stoppers in a fuselage, (a) continuous crack stopper bands around the fuselage, stopping fatigue cracks in the lap joints and other cracks outside the lap joint, and (b) local crack stoppers at the lap joints only. (Schijve, 2009)

Figure 10.13 A schematic diagram of the sample under four point bending: B = 7.8 mm, H1 = H2 = 0.75 mm, L = 4 mm, and 2D = 27 mm (Huang et al., 2005)