2.5 Dimensions of container, Duct Seal and soil samples. . . . . . . . . . . . ...... 6.1 Vertical stress distribution under rigid circular punch of radius a on ho- mogeneous ...... 9.5 VC/VE accelerance [m/s2/N] at 33g for solid pile in inclusion R5 with ...... (1991) have also shown that (i) rectangular containers are more effective in min-.
Iowa State University From the SelectedWorks of Jeramy C. Ashlock
2006
Computational and Experimental Modeling of Dynamic Foundation Interactions With Sand, PhD Thesis Jeramy C Ashlock, University of Colorado Boulder
Available at: http://works.bepress.com/jeramy_ashlock/11/
Computational and Experimental Modeling of Dynamic Foundation Interactions With Sand by Jeramy Curtis Ashlock B.S., Civil Engineering, University of Colorado at Boulder, 1997 M.S., Civil Engineering, University of Colorado at Boulder, 2000
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Civil, Environmental and Architectural Engineering 2006
This thesis entitled: Computational and Experimental Modeling of Dynamic Foundation Interactions With Sand written by Jeramy Curtis Ashlock has been approved for the Department of Civil, Environmental and Architectural Engineering
Ronald Y. S. Pak
Prof. Martin Dunn
Prof. Hon-Yim Ko
Prof. Richard Regueiro
Prof. Stein Sture
Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.
iii Ashlock, Jeramy Curtis (Ph.D., Civil Engineering) Computational and Experimental Modeling of Dynamic Foundation Interactions With Sand Thesis directed by Professor Ronald Y. S. Pak
The subject of dynamic soil-structure interaction (SSI) is fundamental to the design of structures and foundations to withstand dynamic events such as earthquakes, explosions and vibrations from traffic, wind, waves and machinery. With the tools of modern experimental and computational methods, a critical evaluation of current concepts and capabilities for the dynamic characterization of both surface foundations and embedded piles in granular soils is the first target. Well suited for this kind of fundamental study, a variety of scaled-model surface and pile foundations are designed and tested in a large geotechnical centrifuge using dynamic forced excitations. Used as a rigorous theoretical setting for synthesis of the measurements, an advanced boundary element code is extended to address some complex aspects of three-dimensional boundary value problems. Through the integrated study, significant analytical, numerical and physical insights are gained, and a number of fundamental resolutions and milestones are realized. On the analytical side, they include the identification and solution of a problem related to slender elements in the boundary element method, and a new class of elements for discretization methods in general. On the physical side, they encompass the development and generation of a significant experimental database on soil-structure interaction, as well as the implementation and validation of the hybrid-mode dynamic test method for surface and deep foundations. As a new framework of understanding for theoretical research and engineering practice, these developments are used to formulate (i) an advanced inclusion model for surface foundations under multi-directional loading, and (ii) a comparable theory for the corresponding pile problem, both of which can rationally explain many key experimental observations.
Dedication
To the memories of Robert J. Driscoll (1922-2002) and Jean Ashlock (1922-2005).
v
Acknowledgements
I would like to thank my advisor, Professor Ronald Y. S. Pak for his wisdom, inspiration, patience, friendship and dedication. He has certainly gone above and beyond the call of duty of an advisor. Particularly, I cherish the many philosophical and analytical discussions we shared on topics ranging from science and engineering to effective presentation and writing style. I would especially like to thank Farzad Abedzadeh for his friendship, guidance, thoughtful discussion and many important contributions to this research, and Bojan Guzina for his inspiration and for laying the foundation for much of this work. Thanks to Mahdi Soudkhah, Satoshi Kurahashi, Kari Leach and Ned Turner for their assistance in the centrifuge experiments, and to Robb Wallen for his assistance in programming the network analyzer in Labview. Without the unconditional love and support of my wife Shelina and the encouragement of my family, this work would not have been possible. I am grateful to Professors Martin Dunn, Hon-Yim Ko, Richard Regueiro and Stein Sture for serving on my thesis committee. This study was supported by the National Science Foundation through Grants CMS 9712835 and CMS 0201353, and a GAANN Graduate Fellowship, for which I am appreciative.
Contents
Chapter 1 Introduction
1
1.1
Overview of Dynamic Soil-Foundation Interaction . . . . . . . . . . . . .
1
1.2
General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Scope of Research
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PART I: EXPERIMENTAL & COMPUTATIONAL DEVELOPMENTS
5
Chapter 2 Experimental Modeling
6
2.1
Centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3
Soil Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.1
Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.2
Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.4
Container and Boundary Effects . . . . . . . . . . . . . . . . . . . . . .
13
2.5
Scaled Model Footings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.6
Scaled Model Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.7
Excitation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.8
Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.9
Data Acquisition Systems and Measurement Approach . . . . . . . . . .
24
vii 2.9.1
Spectral Dynamics SigLab System . . . . . . . . . . . . . . . . .
24
2.9.2
Wireless National Instruments LabView System . . . . . . . . . .
25
2.9.3
Data Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.9.4
Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.9.5
Frequency Response Function . . . . . . . . . . . . . . . . . . . .
34
2.10 New Experimental Methodologies . . . . . . . . . . . . . . . . . . . . . .
40
2.10.1 Method of Hybrid-Mode Testing . . . . . . . . . . . . . . . . . .
40
2.10.2 Ambient Vibration Tests . . . . . . . . . . . . . . . . . . . . . . .
42
2.10.3 Friction Reducing Dumpling
. . . . . . . . . . . . . . . . . . . .
42
2.10.4 Method of FRF Measurement by Chaotic Impact Loading . . . .
44
2.10.5 Method of Free-Field Modulus Determination . . . . . . . . . . .
47
2.10.6 Modeling of Instrumentation Effects . . . . . . . . . . . . . . . .
47
2.11 Theoretical Accelerance Calculation at an Arbitrary Point on a General Superstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.12 Theoretical Accelerance Calculation for Footing . . . . . . . . . . . . . .
57
2.13 Instrumentation Size Effects: Accelerance Correction for Load Cell Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.14 Calculation of Inertial Properties of Model Foundations . . . . . . . . .
65
Chapter 3 Computational Modeling 3.1
3.2
67
Regularized Direct Boundary Integral Equation Method . . . . . . . . .
68
3.1.1
BIE Discretization and Interpolation . . . . . . . . . . . . . . . .
69
Fundamental Numerical Aspects . . . . . . . . . . . . . . . . . . . . . .
71
3.2.1
Effects of Element Configuration on Integration Accuracy . . . .
77
3.2.2
Adaptive Elemental Integration Scheme for Uniform Accuracy .
84
3.2.3
Proximity-Based Mapping of Nearby Elements . . . . . . . . . .
92
3.2.4
Integrand Discontinuities of Multi-Layered Green’s functions . .
93
viii 3.3
3.4
Adaptive-Gradient Elements for Mixed Boundary Value Problems . . .
98
3.3.1
1D AG Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.3.2
2D AG Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Numerical Quadrature of Weakly-Singular and Nearly-Singular Integrals in Boundary Element Methods . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.1
Singular Transformations for Quadrilateral Elements . . . . . . . 114
3.4.2
Numerical Examples of Nearly-Singular and Weakly-Singular Integral Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5
3.4.3
Singular Transformations for Triangular Elements . . . . . . . . 126
3.4.4
Gradient-Reduction Mapping for Integration over AG Elements . 138
3.4.5
Integration over AG Elements Far From the Collocation Node . . 145
3.4.6
Integration over AG Elements Near the Collocation Node . . . . 147
AG Element Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3.5.1
Example 1: Frictionless versus Bonded Contact of a Rigid Punch 157
3.5.2
Example 2: Load Transfer to a Medium with Material Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
PART II:
SURFACE FOUNDATIONS
Chapter 4 Experimental Footing Test Results
165 166
4.1
Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2
General Footing Accelerance Characteristics . . . . . . . . . . . . . . . . 167 4.2.1
VC Excitation Tests . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.2.2
VE Excitation Tests . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.2.3
HC Excitation Tests . . . . . . . . . . . . . . . . . . . . . . . . . 170
ix Chapter 5 Capability of Current Engineering Analytical Frameworks for Dynamic Footing Problems
182
5.1
Performance of Existing Analytical Frameworks for Dynamic Footings . 182
5.2
Numerical Evaluation of Dynamic Impedances . . . . . . . . . . . . . . 183
5.3
Accelerance Error Measure for Evaluation of Continuum Models . . . . 186
5.4
The Homogeneous Half-Space Model . . . . . . . . . . . . . . . . . . . . 187
5.5
The Pure Square-Root Half-Space Model
5.6
The I.M.F.-enhanced Homogeneous Half-Space Model . . . . . . . . . . 194
. . . . . . . . . . . . . . . . . 193
Chapter 6 Advanced Continuum Models for Surface Footings
208
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.2
Infinitesimal Strain Shear Modulus . . . . . . . . . . . . . . . . . . . . . 209
6.3
A Next-Level Continuum Model for Dynamic Response of Surface Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.3.1
A Fundamental Model to Account for Both Near and Far-Field Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.3.2
Impedance and Accelerance Characteristics of Ellipsoidal Inclusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.4
Synthesis of Experimental Data via Advanced Continuum Model . . . . 231 6.4.1
Synthesis Level 1: Validation and Capability of Inclusion Model
231
6.4.2
Synthesis Level 2: Parameter Space of the Inclusion Model . . . 236
6.4.3
Synthesis Level 3: Relations to Physical Foundation-Soil Parameters237
6.4.4
Synthesis Level 4: Further Model Refinement for Theoretical Mechanics: Contact Stress Distribution by the Ellipsoidal Inclusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.4.5
Synthesis Level 5: Impedance Comparison and Implications for I.M.F.s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
x
PART III:
PILE FOUNDATIONS
318
Chapter 7 Experimental Pile Test Results
319
7.1
Identification and Usefulness of Ambient Vibrations . . . . . . . . . . . 320
7.2
General Dynamic Characteristics of Measured Pile Accelerance Functions 331
7.3
7.4
7.2.1
Solid Pile Versus Hollow Pile Accelerance Characteristics . . . . 335
7.2.2
Dependence of Accelerance on g-level
. . . . . . . . . . . . . . . 336
VC, HC and VE Forcing Configurations . . . . . . . . . . . . . . . . . . 346 7.3.1
Measurement of Lateral Rocking Response . . . . . . . . . . . . 347
7.3.2
Measurement of Vertical Response . . . . . . . . . . . . . . . . . 351
Effect of Static Prestress Force in VE Tests . . . . . . . . . . . . . . . . 351
Chapter 8 Existing Analytical Frameworks for Piles
365
8.1
Computational Modeling of the Experimental Pile Tests . . . . . . . . . 366
8.2
The Homogeneous Half-Space Model . . . . . . . . . . . . . . . . . . . . 369
8.3
The Pure Square-Root Half-Space-Pile Model . . . . . . . . . . . . . . . 369
8.4
The I.M.F.-enhanced Homogeneous Half-Space Model . . . . . . . . . . 370
8.5
The I.M.F.-enhanced Square-Root Half-Space-Pile Model . . . . . . . . 371
8.6
Modeling of Weakened Pile-Soil Contact at Soil Surface . . . . . . . . . 372
8.7
Effect of Hysteretic Damping in Soil Medium . . . . . . . . . . . . . . . 373
Chapter 9 Advanced Continuum Models for Piles 9.1
9.2
391
A New Model for Dynamic Pile-Soil Interaction . . . . . . . . . . . . . . 392 9.1.1
Inclusion Mesh and Layering Considerations . . . . . . . . . . . . 394
9.1.2
Computational Implementation of Pile Inclusion Model . . . . . 396
Application of 2-Soil Zone Inclusion Model to 3D Pile-Sand Interaction
400
9.2.1
Development and Selection of Inclusion Models . . . . . . . . . . 400
9.2.2
Performance of Proposed 2-Zone Soil Model for Pile in Sand . . 415
xi 9.3
9.4
General Modeling Capabilities of 2-Soil Zone Inclusion Model . . . . . . 445 9.3.1
Effect of Inclusion Parameter α . . . . . . . . . . . . . . . . . . . 445
9.3.2
Effect of Inclusion Parameter β . . . . . . . . . . . . . . . . . . . 449
9.3.3
Effect of Inclusion Parameter z d
9.3.4
Effect of Inclusion Parameter n . . . . . . . . . . . . . . . . . . . 456
9.3.5
Effect of Inclusion Parameter ξ 0
. . . . . . . . . . . . . . . . . . 453
. . . . . . . . . . . . . . . . . . 459
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Chapter 10 Conclusions
Bibliography
463
466
Appendix A Footing Tests
476
B Pile Tests
510
Tables
Table 2.1
Scaling relations for centrifuge tests. . . . . . . . . . . . . . . . . . . . .
11
2.2
Typical physical properties of F-75 silica sand. . . . . . . . . . . . . . .
11
2.3
Grain size analysis results for F-75 silica sand.
11
2.4
Soil sample properties for centrifuge tests in this study.
. . . . . . . . .
12
2.5
Dimensions of container, Duct Seal and soil samples. . . . . . . . . . . .
13
2.6
Properties of aluminum alloy 6061-T6 used for hollow piles and high-
. . . . . . . . . . . . . .
strength alloy 7075-T6 used for footings, pile caps and monolithic pile-cap assemblies.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.7
Properties of Model Footings A, B and C. . . . . . . . . . . . . . . . . .
16
2.8
Properties of Model Footings D, E and F. . . . . . . . . . . . . . . . . .
16
2.9
Properties of Model Foundation G (B2/3). . . . . . . . . . . . . . . . . .
19
2.10 Properties of Model Foundation H (B1/3). . . . . . . . . . . . . . . . . .
19
2.11 Properties of Model Foundation I (B0). . . . . . . . . . . . . . . . . . .
19
2.12 Typical surface footing contact pressures [kPa], assuming instrumentation arrangement of two load cells at 12.06gr each and three accelerometers at 2.50gr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Properties of scale model piles. † measured, ? calculated.
20
. . . . . . . . .
21
2.14 Properties of B&K electromagnetic exciters. . . . . . . . . . . . . . . . .
22
2.15 Mass of components of load cell assembly. . . . . . . . . . . . . . . . . .
66
xiii 3.1
Nodal correspondence between triangular domain A of Fig. 3.24(a) and its quadrilateral image in Fig. 3.24(b) for the degenerate mapping. . . . 115
3.2
Nodal correspondence between triangular domain B of Fig. 3.24(a) and its quadrilateral image in Fig. 3.24(c) for the degenerate mapping. . . . 117
3.3
Gauss points required for integration of BIE integrals shown in Figs. 3.26 to 3.28 with integration tolerance of 0.1%. Unit point load in plane of element (tangential approach). w = element half-width. . . . . . . . . . . 122
3.4
Gauss points required for integration of BIE integrals shown in Figs. 3.26 to 3.28 with integration tolerance of 0.1%. Unit point load located on element normal vector at node 7 (normal approach). w = element halfwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5
Nodal correspondence between triangular domain of Fig. 3.30(a) and its quadrilateral image in Fig. 3.30(b) for the degenerate mapping. . . . . . 130
6.1
Masses and normalized avg. contact pressures for instrumented footings used in this chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.1
Section properties of BEM meshes normalized by those of a circular crosssection of radius a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
8.2
Comparison of boundary element (BEM) and mechanics of materials (MOM) static stiffnesses for fixed base beam/column with continuum Green’s functions. Actual mesh geometry used for section properties. . . 370
9.1
9.2
Inclusion zone shear modulus and damping profiles for cases L1–D5 of p Fig. 9.4. Outer zone square-root modulus profile G O (z) = GO 0.5 z/a. . . 407 Inclusion zone shear modulus and damping profiles for cases E–E19 of p Fig. 9.4. Outer zone square-root modulus profile G O (z) = GO 0.5 z/a. . . 408
xiv 9.3
Effect of α on coordinates of theoretical accelerance peaks (f pk , |A|pk ) in [Hz,m/s2 /N]. Hollow Pile B at 33g in inclusion R5 with β = 0.1, z d = 5, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
9.4
Effect of β on coordinates of theoretical accelerance peaks (f pk , |A|pk ) in [Hz,m/s2 /N]. Hollow Pile B at 33g in inclusion R5 with α = 1.1, z d = 5, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
9.5
Effect of zd on coordinates of theoretical accelerance peaks (f pk , |A|pk ) in [Hz,m/s2 /N]. Hollow Pile B at 33g in inclusion R5 with α = 1.1, β = 0.1, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
9.6
Effect of n on coordinates of theoretical accelerance peaks (f pk , |A|pk ) in [Hz,m/s2 /N]. Hollow Pile B at 33g in inclusion R5 with α = 1.1, β = 0.1, zd = 5 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
9.7
Effect of ξ0 on coordinates of theoretical accelerance peaks (f pk , |A|pk ) in [Hz,m/s2 /N]. Hollow Pile B at 33g in inclusion R5 with α = 1.1, β = 0.1, zd = 5 and n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
A.1 Inertial configurations for footing tests used in this study. . . . . . . . . 476 A.2 VC excitation tests of Footing E at nominal g-level 33. . . . . . . . . . . 477 A.3 VC excitation tests of Footing E at nominal g-level 44. . . . . . . . . . . 477 A.4 VC excitation tests of Footing E at nominal g-level 66. . . . . . . . . . . 477 A.5 VE excitation tests of Footing E at nominal g-level 33. . . . . . . . . . . 478 A.6 VE excitation tests of Footing E at nominal g-level 44. . . . . . . . . . . 478 A.7 VE excitation tests of Footing E at nominal g-level 55. . . . . . . . . . . 478 A.8 VE excitation tests of Footing E at nominal g-level 66. . . . . . . . . . . 478 A.9 VE excitation tests of Footing B at nominal g-level 33. . . . . . . . . . . 479 A.10 VE excitation tests of Footing B at nominal g-level 44. . . . . . . . . . . 480 A.11 VE excitation tests of Footing B at nominal g-level 55. . . . . . . . . . . 481
xv A.12 VE excitation tests of Footing B at nominal g-level 66. . . . . . . . . . . 482 A.13 HC excitation tests of Footing B at nominal g-level 33. . . . . . . . . . . 483 A.14 HC excitation tests of Footing B at nominal g-level 44. . . . . . . . . . . 483 A.15 HC excitation tests of Footing B at nominal g-level 55. . . . . . . . . . . 484 A.16 HC excitation tests of Footing B at nominal g-level 66. . . . . . . . . . . 484 A.17 VC excitation tests of Footing G (B2/3) at nominal g-level 33. . . . . . 485 A.18 VC excitation tests of Footing G (B2/3) at nominal g-level 44. . . . . . 486 A.19 VC excitation tests of Footing G (B2/3) at nominal g-level 55. . . . . . 486 A.20 VC excitation tests of Footing G (B2/3) at nominal g-level 66. . . . . . 487 A.21 VE excitation tests of Footing G (B2/3) at nominal g-level 33. . . . . . 488 A.22 VE excitation tests of Footing G (B2/3) at nominal g-level 44. . . . . . 490 A.23 VE excitation tests of Footing G (B2/3) at nominal g-level 55. . . . . . 491 A.24 VE excitation tests of Footing G (B2/3) at nominal g-level 66. . . . . . 492 A.25 HC excitation tests of Footing G (B2/3) at nominal g-level 33. . . . . . 494 A.26 HC excitation tests of Footing G (B2/3) at nominal g-level 55. . . . . . 494 A.27 HC excitation tests of Footing G (B2/3) at nominal g-level 66. . . . . . 494 A.28 VC excitation tests of Footing H (B1/3) at nominal g-level 33. . . . . . 495 A.29 VC excitation tests of Footing H (B1/3) at nominal g-level 44. . . . . . 495 A.30 VC excitation tests of Footing H (B1/3) at nominal g-level 55. . . . . . 496 A.31 VC excitation tests of Footing H (B1/3) at nominal g-level 66. . . . . . 496 A.32 VE excitation tests of Footing H (B1/3) at nominal g-level 33. . . . . . 497 A.33 VE excitation tests of Footing H (B1/3) at nominal g-level 44. . . . . . 499 A.34 VE excitation tests of Footing H (B1/3) at nominal g-level 55. . . . . . 500 A.35 VE excitation tests of Footing H (B1/3) at nominal g-level 66. . . . . . 501 A.36 HC excitation tests of Footing H (B1/3) at nominal g-level 33. . . . . . 502 A.37 HC excitation tests of Footing H (B1/3) at nominal g-level 44. . . . . . 503 A.38 HC excitation tests of Footing H (B1/3) at nominal g-level 55. . . . . . 503
xvi A.39 HC excitation tests of Footing H (B1/3) at nominal g-level 66. . . . . . 504 A.40 VC excitation tests of Footing I (B0) at nominal g-level 33. . . . . . . . 504 A.41 VC excitation tests of Footing I (B0) at nominal g-level 44. . . . . . . . 505 A.42 VC excitation tests of Footing I (B0) at nominal g-level 55. . . . . . . . 505 A.43 VC excitation tests of Footing I (B0) at nominal g-level 66. . . . . . . . 506 A.44 VE excitation tests of Footing I (B0) at nominal g-level 33. . . . . . . . 507 A.45 VE excitation tests of Footing I (B0) at nominal g-level 44. . . . . . . . 508 A.46 VE excitation tests of Footing I (B0) at nominal g-level 55. . . . . . . . 509 A.47 VE excitation tests of Footing I (B0) at nominal g-level 66. . . . . . . . 509 B.1 Inertial configurations for pile tests used in this study. . . . . . . . . . . 510 B.2 VE excitation tests of Pile A at nominal g-level 33. . . . . . . . . . . . . 511 B.3 VE excitation tests of Pile A at nominal g-level 44. . . . . . . . . . . . . 512 B.4 VE excitation tests of Pile A at nominal g-level 55. . . . . . . . . . . . . 513 B.5 VE excitation tests of Pile A at nominal g-level 66. . . . . . . . . . . . . 513 B.6 HC excitation tests of Pile A at nominal g-level 33. . . . . . . . . . . . . 514 B.7 HC excitation tests of Pile A at nominal g-level 44. . . . . . . . . . . . . 514 B.8 HC excitation tests of Pile A at nominal g-level 55. . . . . . . . . . . . . 514 B.9 HC excitation tests of Pile A at nominal g-level 66. . . . . . . . . . . . . 514 B.10 VC excitation tests of Pile B at nominal g-level 33. . . . . . . . . . . . . 515 B.11 VC excitation tests of Pile B at nominal g-level 44. . . . . . . . . . . . . 516 B.12 VC excitation tests of Pile B at nominal g-level 55. . . . . . . . . . . . . 517 B.13 VC excitation tests of Pile B at nominal g-level 66. . . . . . . . . . . . . 517 B.14 VE excitation tests of Pile B at nominal g-level 33. . . . . . . . . . . . . 518 B.15 VE excitation tests of Pile B at nominal g-level 44. . . . . . . . . . . . . 520 B.16 VE excitation tests of Pile B at nominal g-level 55. . . . . . . . . . . . . 522 B.17 VE excitation tests of Pile B at nominal g-level 66. . . . . . . . . . . . . 524
xvii B.18 HC excitation tests of Pile B at nominal g-level 33. . . . . . . . . . . . . 525 B.19 HC excitation tests of Pile B at nominal g-level 44. . . . . . . . . . . . . 526 B.20 HC excitation tests of Pile B at nominal g-level 55. . . . . . . . . . . . . 527 B.21 HC excitation tests of Pile B at nominal g-level 66. . . . . . . . . . . . . 528
Figures
Figure 2.1
Schematic of 400 g-ton Centrifuge at the University of Colorado at Boulder (from Ko (1988a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
7
Photograph of 400 g-ton Centrifuge at the University of Colorado at Boulder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Grain size distribution of F-75 silica sand. . . . . . . . . . . . . . . . . .
10
2.4
Preparation of a soil sample via pluviation . . . . . . . . . . . . . . . . .
12
2.5
Soil Container and Duct Seal Walls. . . . . . . . . . . . . . . . . . . . .
14
2.6
Location of instrumentation holes for model Footings A through F. . . .
17
2.7
Scaled-model Footings D, E and F. . . . . . . . . . . . . . . . . . . . . .
17
2.8
Location of instrumentation holes for model Footings G, H and I. . . . .
20
2.9
Location of instrumentation holes for scaled-model pile caps. . . . . . .
20
2.10 Scaled-model pile and pile-cap assemblies A: solid monolithic pile and cap, B: threaded hollow pile and cap, C: filleted solid-monolithic pile and cap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.11 Large and small exciters used in vertical and horizontal excitation tests.
23
2.12 Schematic of B&K 4809 Exciter (from Bruel & Kjaer Company). . . . .
23
2.13 Frequency responses of passive single-pole low-pass RC filter and higherorder Butterworth filtering from K-H multichannel filter. f Break = 1/2πRC = break frequency, fc =cutoff frequency. . . . . . . . . . . . . . . . . . . . .
27
xix 2.14 Oscilloscope measurements of random analog signals from SigLab hardware versus unfiltered (a,c) and filtered (b,d) output from NI (Labview) hardware. Butterworth filtering of the Labview output is achieved with the K-H multichannel filter. (a,b): shorter time scale, (c,d): longer time scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.15 Rectangular window. (a) Time Window. (b) Spectral Window. . . . . .
32
2.16 Hanning window. (a) Time Window. (b) Spectral Window. . . . . . . .
33
2.17 Auto-spectral density of acceleration (a) and force (b) during ambient and forced horizontal vibration tests of Footing E at 66g. Ambient test ilebbg66.am2, forced vibration test ilebbg66.dat. . . . . . . . . . . . . .
38
2.18 Accelerance and coherence functions for forced horizontal vibration test ilebbg66.dat of Footing E at 66g. . . . . . . . . . . . . . . . . . . . . . .
38
2.19 Typical measured vertical accelerance and coherence functions for eccentric vertical loading of footing B at 66g. Test VE-B72. . . . . . . . . . .
39
2.20 Typical measured horizontal accelerance and coherence functions for eccentric vertical loading of footing B at 66g. Test VE-B72. . . . . . . . .
39
2.21 Notation for measurement locations for surface, embedded and pile foundation tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.22 33g tests of Pile B. (a): HC accelerometer ASD peak frequency from ambient-vibration test p3032617.vna, (b): HC/VE accelerance peak from forced-vibration test p3032607.vna. . . . . . . . . . . . . . . . . . . . . .
43
2.23 Oil filled friction reducing dumpling resting on load cell button in VE test configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.24 VE tests on Pile C at 55g using random load with dumpling versus chaotic impact loading. (a): VC/VE and HC/VE accelerance magnitudes. (b): time-histories of VC acceleration. (c): time-histories of VE load. Random excitation test p5062002.vna, impact test p5070905.vna. . . . . . . . . .
46
xx 2.25 A substructure formulation of the cap-pile-soil interaction problem. P: general point of interest. C: centroid of rigid sub-structure. V/H: points of application of vertical/horizontal load. T/B: top/bottom points of deformable pile-stem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.26 Monolithic scale model pile and cap A with attached accelerometer and load cell assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2.27 Disassembled components of load cell assembly. . . . . . . . . . . . . . .
59
2.28 Free body diagram of upper and lower portions of load cell assembly obtained by making a cut at mid-height of load cell. Shear force and moment resultants of internal stresses are omitted as they are not required. 61 2.29 Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of pile model A in square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.30 Effect of accounting for inertial properties of load cell assembly components on HC/VE accelerance [m/s2 /N] of pile model A in square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.31 Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of Footing I (B0) on square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.32 Effect of accounting for inertial properties of load cell assembly components on VE/VE accelerance [m/s2 /N] of Footing I (B0) on square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.33 Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of Footing E on square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
xxi 2.34 Effect of accounting for inertial properties of load cell assembly components on HC/VE accelerance [m/s2 /N] of Footing E on square-root half-space at 33g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
2.35 Cross-section of idealized hollow cylinder used to calculate inertial properties of accelerometers. . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.1
Three dimensional solid with finite interior region Ω i internal to Γ = Γu ∪Γt . 70
3.2
Three dimensional solid with infinite region Ω e external to Γ. . . . . . .
3.3
Two non-adjacent triangular elements. Numbers in parentheses are local node numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
76
BEM displacement solution for 3D cylinder with inadequate gauss integration rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
71
81
Magnitude of displacement vector U along the line (x, y) = (0, 1) of cylinder for applied RBM of top surface with inadequate Gauss integration rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Magnitude of traction vector T on cylinder top surface for applied RBM rotations of top surface with inadequate gauss integration rules. . . . . .
3.7
81
82
Error in displacement vector U at (x, y, z) = (0, 1, 20) versus Gauss points for quadrilateral integration regions for applied rotation Θ z of top surface. 82
3.8
Resultant moment Mz of all tractions versus Gauss points for quadrilateral integration regions for applied rotation Θ z of top surface. . . . . . .
3.9
83
Histogram of Gauss rules at convergence for quadrilateral regions. Tolerance = 10%. 88
3.10 Gauss rules at convergence for quadrilateral regions vs. distance from collocation point to nearest node of integration element. Tolerance = 10%. 88 3.11 Magnitude of displacement vector U along the line (x, y) = (0, 1) of cylinder for applied RBM of top surface. Adaptive integration with tolerance = 10%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
xxii 3.12 (a): Error in displacement vector at (x, y, z) = (0, 1, 20) for applied rotation Θz of top surface, (b): corresponding maximum required Gauss rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.13 (a): 4x8x4 cylindrical surface mesh and (b): half-space shear modulus distribution for 8 layers with height h/a = 0.25. . . . . . . . . . . . . . .
94
3.14 Close-up of real component of multi-layered displacement Green’s functions along the line shown in Fig. 3.13. . . . . . . . . . . . . . . . . . . .
95
3.15 Real component of regular part of multi-layered stress Green’s functions along the line shown in Fig. 3.13. . . . . . . . . . . . . . . . . . . . . . .
96
3.16 Imaginary component of regular part of multi-layered stress Green’s functions along the line shown in Fig. 3.13. . . . . . . . . . . . . . . . . . . . √ 3.17 Quadratic interpolation of the function 1/ 1 − x2 using (a) 4 and (b) 10
97
elements. Rightmost node is at x = 0.999. . . . . . . . . . . . . . . . . .
99
3.18 The AG kernel γ(η) for various values of m and n. . . . . . . . . . . . . 102 3.19 AG shape functions with m = 1. . . . . . . . . . . . . . . . . . . . . . . . 103 3.20 AG shape functions with n = 1. . . . . . . . . . . . . . . . . . . . . . . . 103 3.21 Node numbering convention for 9-node and 8-node quadrilateral AGelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.22 8-node AG-Edge element shape functions for (m, n) = (1, 10). . . . . . . 110 3.23 8-node AG-Corner element shape functions for (m, n) = (1, 10). . . . . . 110 3.24 Degenerate triangular subdivision and mapping for integration of a weak singularity having a pole at corner node k = 4 of a bi-unit square domain. 116 3.25 Degenerate triangular subdivisions for collocation at nodes 1–9 of quadrilateral element. Collocation at (a): corner nodes, (b): mid-side nodes, (c): center node. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
xxiii (3)
3.26 Product of standard serendipity shape function N T
and displacement
Green’s function component U3Kelvin for unit point load in plane of element acting in ξ3 -direction near node 7. Distance |y (n) −ξ (7) | = 0.25. . . 123 (3)
3.27 Product of standard serendipity shape function N U and traction Green’s function component T2Kelvin for unit point load in plane of element acting in ξ3 -direction near node 7. Distance |y (n) −ξ (7) | = 0.25. . . . . . . . . . 124 3.28 Traction Green’s function component T 2Kelvin for unit point load in plane of element acting in ξ3 -direction near node 7. Distance |y (n) −ξ (7) | = 0.25. 125 3.29 Transformation of triangular surface elements to standard triangular parent domain in areal coordinates. (a): Physical element surface in global Cartesian domain (ξ1 , ξ2 , ξ3 ), (b): Parametrized element surface in areal coordinate domain (ζ1 , ζ2 , ζ3 ). . . . . . . . . . . . . . . . . . . . . . . . . 127 3.30 Two possible transformations for a triangular region defined in the Cartesian (η1 , η2 ) frame. Left: Singular transformation of the triangular region (a) to a quadrilateral domain (b). Right: Nonsingular transformation of the triangular region (c) to intrinsic triangular areal coordinate domain (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.31 Direct degenerate transformation having a pole at corner node 3 from areal coordinate triangular domain to Cartesian coordinate quadrilateral domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 (P )
(P )
(P )
3.32 Triangular subdivision about a point P (ζ 1 , ζ2 , ζ3 ) followed by singular transformations of Eqn.s (3.70) to (3.72). (a): physical element in global Cartesian domain ξi . (b): parent element in areal coordinate domain ζi with sub-domains A, B and C. (c): auxiliary standard triangular domains ζiA , ζiB and ζiC for sub-domains A, B and C. (d): standard bi-unit quadrilateral domains for the singular transformations.
. . . . . 135
3.33 (a): Mapping of Eqn. (3.92) and (b): resulting Jacobian of Eqn. (3.93).
141
xxiv 3.34 Effect of mapping of Eqn. (3.92) for the Jacobian exponents α J = (1, 2, 3) on the 1-D AG shape functions with (m, n) = (0.05, 2). . . . . . . . . . . 143 3.35 Sequential mappings for AG-edge element not having a node within distance r1 of collocation node. . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.36 Sequential mappings for AG-edge element with Node 7 coincident or within distance r1 of collocation node. . . . . . . . . . . . . . . . . . . . 149 (3)
3.37 Product of AG-edge element traction shape function N T
(3)
= NAG−8E
with displacement Green’s function component U 3Kelvin for unit point load in ξ3 -direction at Node 7 of a planar 8-node AG-edge element. AGsmoothing mapping of Eqn. (3.95) is not used. . . . . . . . . . . . . . . 152 (3)
3.38 Product of standard serendipity displacement shape function N U with traction Green’s function component T 2Kelvin for unit point load in ξ3 direction at Node 7 of a planar 8-node AG-edge element. AG-smoothing mapping of Eqn. (3.95) is not used. . . . . . . . . . . . . . . . . . . . . . 153 (3)
3.39 Product of AG-edge element traction shape function N T
(3)
= NAG−8E
with displacement Green’s function component U 3Kelvin for unit point load in ξ3 -direction at Node 7 of a planar 8-node AG-edge element. AGsmoothing mapping of Eqn. (3.95) is used for Quadrilateral domains A II and BII with αJ = 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 (3)
3.40 Product of standard serendipity displacement shape function N U with traction Green’s function component T 2Kelvin for unit point load in ξ3 direction at Node 7 of a planar 8-node AG-edge element. AG-smoothing mapping of Eqn. (3.95) is used for Quadrilateral domains A II and BII with αJ = 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 R1 R1 (3) 3.41 Convergence of integral I = −1 −1 NAG−E (η1 , η2 )dη1 dη2 of AG-edge shape function for parameters (m, n) = (0.05, 2), with and without AG-
smoothing mapping of Eqn. (3.95). . . . . . . . . . . . . . . . . . . . . . 157
xxv 3.42 Basic 10x16 mesh of standard 8-node quads and 6-node triangles with 8-node AG-Edge elements for BEASSI. . . . . . . . . . . . . . . . . . . . 159 3.43 Normalized vertical traction for unit vertical displacement of rigid frictionless punch for 10x16 mesh of 8-node elements, (a) full range, (b) close-up of last two elements. . . . . . . . . . . . . . . . . . . . . . . . . 159 3.44 Vertical traction for unit vertical displacement of rigid bonded punch for 10x16 mesh of 8-node elements, (a) full range, (b) close-up of last element.160 3.45 Normalized vertical traction for unit vertical displacement of rigid bonded punch for the uniform 10x16 mesh of 8-node elements, (a) full range, (b) close-up of last two elements. . . . . . . . . . . . . . . . . . . . . . . . . 161 3.46 An axially-loaded cylinder embedded in a bi-material elastic half space: G1 /G2 /G0 = 1/10/1000, l/a = 2, h1 /a = 1, ν = 0.25. . . . . . . . . . . . 161 3.47 Meshes for embedded cylinder:(A) coarse grid with regular elements, (B) coarse grid with AG elements , (C) fine grid with AG elements. . . . . . 163 3.48 Computed vertical shear traction on the cylindrical surface of embedment.163 4.1
Wide scatter in ratio of predicted and measured vertical vibration amplitude of foundations on sand (from Richart et al., 1970). . . . . . . . . 167
4.2
Discrepancy between theoretical and measured lateral-rocking foundation responses (from Richart et al., 1970). . . . . . . . . . . . . . . . . . . . . 168
4.3
Representative VC/VC accelerance measurements [m/s 2 /N] for Footing E at 33, 44 and 66g. Tests VC-E2/11/18. . . . . . . . . . . . . . . . . . 172
4.4
Representative VC/VC accelerance measurements [m/s 2 /N] for Footing G at 33, 44, 55 and 66g. Tests VC-G6/27/40/53. . . . . . . . . . . . . . 172
4.5
Representative VC/VC accelerance measurements [m/s 2 /N] for Footing H (B1/3) at 33, 44, 55 and 66g. Tests VC-H6/17/35/49. . . . . . . . . . 173
xxvi 4.6
Representative VC/VC accelerance measurements [m/s 2 /N] for Footing I (B0) at 33, 44, 55 and 66g. Tests VC-I6/32/50/67. . . . . . . . . . . . 174
4.7
Representative VE/VC accelerance measurements [m/s 2 /N] for Footing I (B0) at 33, 44, 55 and 66g. Tests VC-I6/32/50/67. . . . . . . . . . . . 174
4.8
Representative VC/VE accelerance measurements [m/s 2 /N] for Footing B at 33, 44, 55 and 66g. Tests VE-B12/35/56/72. . . . . . . . . . . . . . 175
4.9
Representative HC/VE accelerance measurements [m/s 2 /N] for Footing B at 33, 44, 55 and 66g Tests VE-B12/35/56/72. HC accelerometer at hole 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.10 Representative VC/VE accelerance measurements [m/s 2 /N] for Footing G (B2/3) at 33, 44, 55 and 66g. Tests VE-G50/87/117/167. . . . . . . . 176 4.11 Representative HC/VE accelerance measurements [m/s 2 /N] for Footing G (B2/3) at 33, 44, 55 and 66g. Tests VE-G50/87/117/167. HC accelerometer at hole 3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.12 Representative VC/VE accelerance measurements [m/s 2 /N] for Footing H (B1/3) at 33, 44, 55 and 66g. Tests VE-H36/67/93/135. . . . . . . . . 177 4.13 Representative HC/VE accelerance measurements [m/s 2 /N] for Footing H (B1/3) at 33, 44, 55 and 66g. Tests VE-H36/67/93/135. HC accelerometer at hole 3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.14 Representative VC/VE accelerance measurements [m/s 2 /N] for Footing I (B0) at 33, 44, 55 and 66g. Tests VE-I11/33/44/56.
. . . . . . . . . . 178
4.15 Representative VE/VE accelerance measurements [m/s 2 /N] for Footing I (B0) at 33, 44, 55 and 66g. Tests VE-I11/33/44/56. VE accelerometer at hole 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.16 Representative prototype-scale VC/VE accelerances [m/s 2 /N] for Footing B at 33, 44, 55 and 66g. Tests VE-B12/35/56/72. . . . . . . . . . . . . . 179
xxvii 4.17 Representative prototype-scale HC/VE accelerance [m/s 2 /N] for Footing B at 33, 44, 55 and 66g Tests VE-B12/35/56/72. HC accelerometer at hole 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.18 Representative HC/HC accelerance measurements [m/s 2 /N] for Footing B at 33, 44, 55 and 66g. Tests HC-B13/20/25/38. HC accelerometer at hole 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.19 Close-up of representative HC/HC accelerance measurements [m/s 2 /N] for Footing B at 33, 44, 55 and 66g Tests HC-B13/20/25/38. HC accelerometer at hole 3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.20 Representative HC/HC accelerance measurements [m/s 2 /N] for Footing H at 33, 44, 55 and 66g. Tests HC-H6/24/39/47. HC accelerometer at hole 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.21 Close-up of representative HC/HC accelerance measurements [m/s 2 /N] for Footing H at 33, 44, 55 and 66g Tests HC-H6/24/39/47. HC accelerometer at hole 3. 5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Coordinate system for a fundamental soil-foundation interaction problem with an interface mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2
VC/VE accelerance for homogeneous half-space fit of vertical mode for Footing E at 33g. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . 189
5.3
HC/VE accelerance for homogeneous half-space fit of vertical mode for Footing E at 33g. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . 189
5.4
HC/VE accelerance for homogeneous half-space fit of lateral mode for Footing E at 33g. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . 190
5.5
VC/VE accelerance for homogeneous half-space fit of lateral mode for Footing E at 33g. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . 190
xxviii 5.6
VC/VE accelerance for homogeneous half-space fit of vertical mode for Footing H(B1/3) at 33g. Tests VE-H33 to H38. . . . . . . . . . . . . . . 191
5.7
HC/VE accelerance for homogeneous half-space fit of vertical mode for Footing H(B1/3) at 33g. Tests VE-H33 to H38. . . . . . . . . . . . . . . 191
5.8
HC/VE accelerance for homogeneous half-space fit of lateral mode for Footing H(B1/3) at 33g. Tests VE-H33 to H38. . . . . . . . . . . . . . . 192
5.9
VC/VE accelerance for homogeneous half-space fit of lateral mode for Footing H(B1/3) at 33g. Tests VE-H33 to H38. . . . . . . . . . . . . . . 192
5.10 VC/VE accelerance of Footing E at 33g for pure square-root half-space model. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.11 HC/VE accelerance of Footing E at 33g for pure square-root half-space model. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.12 VC/VE accelerance of Footing H (B1/3) at 33g for pure square-root halfspace model. Tests VE-H33 to H38. . . . . . . . . . . . . . . . . . . . . 196 5.13 HC/VE accelerance of Footing H (B1/3) at 33g for pure square-root halfspace model. Tests VE-H33 to H38. . . . . . . . . . . . . . . . . . . . . 196 5.14 Effect of I.M.F. αhh on HC accelerance for VE excitation of footings (VE accelerance for Footing I) on homogeneous half-space. α mm = αmh = 1. p ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.15 Effect of I.M.F. αmm on HC accelerance for VE excitation of footings (VE
accelerance for Footing I) on homogeneous half-space. α hh = αmh = 1. p ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.16 Effect of I.M.F. αmh on HC accelerance for VE excitation of footings (VE
accelerance for Footing I) on homogeneous half-space. α hh = αmm = 1. p ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
xxix 5.17 Effect of I.M.F. αhh on HC accelerance for HC excitation of footings (VC excitation, VE accelerance for Footing I) on homogeneous half-space. p αmm = αmh = 1. ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . 201
5.18 Effect of I.M.F. αmm on HC accelerance for HC excitation of footings
(VC excitation, VE accelerance for Footing I) on homogeneous half-space. p αhh = αmh = 1. ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . . 202
5.19 Effect of I.M.F. αmh on HC accelerance for HC excitation of footings
(VC excitation, VE accelerance for Footing I) on homogeneous half-space. p αhh = αmm = 1. ω ¯ = ωb/ Geq.hom. /ρ. . . . . . . . . . . . . . . . . . . . 203
5.20 Error of Eqn. (5.13) for HC/VE accelerance vs. α hh and αmm with weights (wV , wpk , wH ) = (0, 0.5, 0, 5). Minimum error at (α hh , αmm ) = (0.75, 0.7).
HC/VE accelerance averaged from tests VE-E1, VE-E2, VE-E3. . . . . 205 5.21 VC/VE accelerance of Footing E at 33g for I.M.F.-augmented homogeneous half-space model. Best-fit Geq.hom. = 74M P a. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.22 HC/VE accelerance of Footing E at 33g for I.M.F.-augmented homogeneous half-space model. Geq.hom. = 74M P a, αhh = 0.75, αmm = 0.7. Tests VE-E1 to E3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.23 VC/VE accelerance of Footing H (B1/3) at 33g for I.M.F.-augmented homogeneous half-space model. Best-fit G eq.hom. = 58M P a. Tests VEH33 to H38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.24 HC/VE accelerance of Footing H (B1/3) at 33g for I.M.F.-augmented homogeneous half-space model. Geq.hom. = 58M P a, αhh = 0.85, αmm = 0.65. Tests VE-H33 to H38. . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.1
Vertical stress distribution under rigid circular punch of radius a on homogeneous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
xxx 6.2
Confining stress and resulting shear Modulus underneath the rigid foundation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3
Schematic of static normal contact stress distribution beneath rigid footing on (a) perfectly elastic medium, (b) granular medium (large strain) (c) granular medium (small-strain). . . . . . . . . . . . . . . . . . . . . . 211
6.4
Small-strain shear modulus Gmax . . . . . . . . . . . . . . . . . . . . . . 212
6.5
Measured shear moduli from resonant column and torsional shear tests with shear strains in the range 10−6 < γ < 10−2 (Iwasaki et al., 1978). . 213
6.6
Ellipse cross-section in x-z plane as defined by aspect ratio l z /lx and the three points (x, y, z) = (±b, 0, 0) and (0, 0, h f ).
. . . . . . . . . . . . . . 217
6.7
Lam´e Curves of Eqn. (6.18) with b = d. . . . . . . . . . . . . . . . . . . . 218
6.8
Rigid surface footing on homogeneous elastic inclusion in semi-infinite half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.9
Ellipsoidal inclusion mesh with l z /lx = 1.8, hf /b = 3, and s(z) = 2 + 0.1(z/b)−1.5 .219
6.10 Images in x–z plane of family of ellipsoidal inclusions with l z /lx = 1.8. . 222 6.11 Cross-section through a family of 3D ellipsoidal inclusion BEM meshes with lz /lx = 1.8 and s(z) = 2 + 0.1(z/b)−1.5 . . . . . . . . . . . . . . . . . 222 ¯ on impedances K ¯ vv , K ¯ hh , K ¯ mm , K ¯ mh and K ¯ tt with modulus 6.12 Effect of h ¯ = 1.0. Solid lines: Real, dashed lines: Imaginary. . . . . . . . . . 224 ratio G ¯ on impedances K ¯ vv , K ¯ hh , K ¯ mm , K ¯ mh and K ¯ tt with modulus 6.13 Effect of h ¯ = 1.0. Solid lines: Magnitude, dashed lines: Phase angle. ratio G
. . . . 225
¯ on impedances K ¯ vv , K ¯ hh , K ¯ mm , K ¯ mh and K ¯ tt with inclusion 6.14 Effect of G ¯ = 2.5. Solid lines: Real, dashed lines: Imaginary. . . . . . . . . . 226 depth h ¯ on impedances K ¯ vv , K ¯ hh , K ¯ mm , K ¯ mh and K ¯ tt with inclusion 6.15 Effect of G ¯ = 2.5. Solid lines: Magnitude, dashed lines: Phase angle. . . . . 227 depth h ¯ on impedances K ¯ ij /K ¯ ij (0) with modulus ratio G ¯ = 1.0. Solid 6.16 Effect of h lines: Real, dashed lines: Imaginary. . . . . . . . . . . . . . . . . . . . . 228
xxxi ¯ = 2.5. Solid ¯ on impedances K ¯ ij /K ¯ ij (0) with inclusion depth h 6.17 Effect of G lines: Real, dashed lines: Imaginary. . . . . . . . . . . . . . . . . . . . . 229 6.18 Self-Similar impedances for ellipsoidal inclusions with l z /lx = 1.8. Solid lines: Real, dashed lines: Imaginary. . . . . . . . . . . . . . . . . . . . . 232 6.19 Self-Similar impedances for ellipsoidal inclusions with l z /lx = 1.8. Solid lines: Magnitude, dashed lines: Phase angle. . . . . . . . . . . . . . . . . 233 6.20 Inclusion depth vs. modulus ratio for one possible set of self-similar impedances of the ellipsoidal inclusion defined by Eqns. (6.19) and (6.20). 234 6.21 Normalized VC/VE accelerance A i ρO b3 of Footing (B2/3) for self-similar impedances of Fig. 6.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.22 Normalized HC/VE accelerance A i ρO b3 of Footing (B2/3) for self-similar impedances of Fig. 6.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 ¯ = 3.5. ¯ = 1.4 for ellipsoidal inclusion with h 6.23 Determination of best-fit G VC/VE accelerance of footing B at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). . . . . . . . . . . . . . . . . . . . . . . . . 240 ¯ = 3.5. ¯ = 1.4 for ellipsoidal inclusion with h 6.24 Determination of best-fit G HC/VE accelerance of footing B at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). . . . . . . . . . . . . . . . . . . . . . . . . 240 ¯ = 1.5. ¯ = 1.6 for ellipsoidal inclusion with h 6.25 Determination of best-fit G VC/VE accelerance of footing H (B1/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . 241 ¯ = 1.5. ¯ = 1.6 for ellipsoidal inclusion with h 6.26 Determination of best-fit G HC/VE accelerance of footing H (B1/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . 241 6.27 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
xxxii 6.28 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 6.29 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b4011446.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.30 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b4011446.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.31 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b5011458.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.32 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b5011458.vna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.33 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b6011465.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.34 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing B at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b6011465.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.35 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g3012208.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.36 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g3012208.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
xxxiii 6.37 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 44g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g4012216.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.38 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 44g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g4012216.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.39 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 55g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g5012221.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.40 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 55g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g5012221.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.41 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g6012228.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.42 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing G (B2/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test g6012228.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.43 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h3011704.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.44 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 33g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h3011704.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.45 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 44g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h4011712.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
xxxiv 6.46 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 44g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h4011712.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.47 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 55g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h5011719.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.48 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 55g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h5011719.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.49 Best-fit VC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.50 Best-fit HC/VE accelerances for ellipsoidal inclusions of varying depth. Footing H (B1/3) at 66g, with weights w V = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.51 Best-fit VC/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 33g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i3091306.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.52 Best-fit VE/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 33g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i3091306.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.53 Best-fit VC/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 44g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i4091318.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.54 Best-fit VE/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 44g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i4091318.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
xxxv 6.55 Best-fit VC/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 55g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i5091331.vna,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 6.56 Best-fit VE/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 55g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i5091331.vna,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 6.57 Best-fit VC/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 66g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i6091342.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.58 Best-fit VE/VC accelerances for ellipsoidal inclusions of varying depth. Footing I (B0) at 66g, with weights w V = 0.5, wpk = 0, wH = 0.5 (test i6091342.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.59 Best-fit ellipsoidal inclusion modulus ratios for 33g tests. . . . . . . . . . 258 6.60 Best-fit ellipsoidal inclusion modulus ratios for 44g tests. . . . . . . . . . 258 6.61 Best-fit ellipsoidal inclusion modulus ratios for 55g tests. . . . . . . . . . 259 6.62 Best-fit ellipsoidal inclusion modulus ratios for 66g tests. . . . . . . . . . 259 6.63 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 33g tests.260 6.64 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 33g tests.260 6.65 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 44g tests.261 6.66 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 44g tests.261 6.67 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 55g tests.262 6.68 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 55g tests.262 6.69 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 66g tests.263 6.70 Normalized error for best-fit ellipsoidal inclusion modulus ratios, 66g tests.263 6.71 Best-fit ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 33g tests. . . . . . . . . . . . . . . . . . . . . . . . . . . 264
xxxvi 6.72 Best-fit ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 44g tests. . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.73 Best-fit ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 55g tests. . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.74 Best-fit ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 66g tests. . . . . . . . . . . . . . . . . . . . . . . . . . . 265 ¯ p relation of Eqn. (6.29) based on average of all four g-levels for ellip6.75 G–¯ soidal inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 ¯ G ¯ model of Eqn. (6.29) based on average of all four g-levels for 6.76 p¯–h– ellipsoidal inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 ¯ vv at frequency ω 6.77 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 1.8, (b): Homogeneous half(a): Ellipsoidal inclusion with h space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 ¯ hh at frequency ω 6.78 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 1.8, (b): Homogeneous half(a): Ellipsoidal inclusion with h space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 ¯ mm at frequency ω 6.79 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 1.8, (b): Homogeneous half(a): Ellipsoidal inclusion with h space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 ¯ mh at frequency ω 6.80 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 1.8, (b): Homogeneous half(a): Ellipsoidal inclusion with h space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 ¯ hm at frequency ω 6.81 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 1.8, (b): Homogeneous half(a): Ellipsoidal inclusion with h space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6.82 Original and modified families of ellipses. . . . . . . . . . . . . . . . . . 278 6.83 Family of modified ellipsoidal inclusion meshes. . . . . . . . . . . . . . . 278
xxxvii ¯ vv at frequency ω 6.84 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 2.4, (b): Homoge(a): Modified ellipsoidal inclusion with h neous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 ¯ hh at frequency ω 6.85 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 2.4, (b): Homoge(a): Modified ellipsoidal inclusion with h neous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 ¯ mm at frequency ω 6.86 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 2.4, (b): Homoge(a): Modified ellipsoidal inclusion with h neous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 ¯ mh at frequency ω 6.87 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 2.4, (b): Homoge(a): Modified ellipsoidal inclusion with h neous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 ¯ hm at frequency ω 6.88 Traction on interface z = 0 for impedance K ¯ = 0.05. ¯ = 3 and G ¯ = 2.4, (b): Homoge(a): Modified ellipsoidal inclusion with h neous half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.89 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing B at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). 284 6.90 Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing B at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b3011437.vna). 284 6.91 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing B at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b4011446.vna). 285 6.92 Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing B at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b4011446.vna). 285 6.93 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing B at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b5011458.vna). 286 6.94 Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing B at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b5011458.vna). 286
xxxviii 6.95 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing B at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b6011465.vna). 287 6.96 Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing B at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test b6011465.vna). 287 6.97 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g3012208.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.98 Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g3012208.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.99 Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g4012216.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.100Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g4012216.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.101Best-fit VC/VE accelerances for modified ellipsoidal inclusions Footing G (B2/3) at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g5012221.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 6.102Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g5012221.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 6.103Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g6012228.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
xxxix 6.104Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing G (B2/3) at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test g6012228.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.105Best-fit VC/VE accelerances for modified ellipsoidal inclusions Footing H (B1/3) at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h3011704.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6.106Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing H (B1/3) at 33g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h3011704.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6.107Best-fit VC/VE accelerances for modified ellipsoidal inclusions. Footing H (B1/3) at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h4011712.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.108Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing H (B1/3) at 44g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h4011712.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.109Best-fit VC/VE accelerances for modified ellipsoidal inclusions Footing H (B1/3) at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h5011719.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.110Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing H (B1/3) at 55g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h5011719.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.111Best-fit VC/VE accelerances for modified ellipsoidal inclusions Footing H (B1/3) at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.112Best-fit HC/VE accelerances for modified ellipsoidal inclusions. Footing H (B1/3) at 66g, with weights wV = 0.3, wpk = 0.4, wH = 0.3 (test h6010778.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
xl 6.113Best-fit VC/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 33g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i3091306.vna).296 6.114Best-fit VE/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 33g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i3091306.vna).296 6.115Best-fit VC/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 44g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i4091318.vna).297 6.116Best-fit VE/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 44g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i4091318.vna).297 6.117Best-fit VC/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 55g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i5091331.vna).298 6.118Best-fit VE/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 55g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i5091331.vna).298 6.119Best-fit VC/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 66g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i6091342.vna).299 6.120Best-fit VE/VC accelerances for modified ellipsoidal inclusions. Footing I (B0) at 66g, with weights wV = 0.5, wpk = 0, wH = 0.5 (test i6091342.vna).299 6.121Best-fit ellipsoidal inclusion modulus ratios for 33g tests. . . . . . . . . . 300 6.122Best-fit ellipsoidal inclusion modulus ratios for 44g tests. . . . . . . . . . 300 6.123Best-fit ellipsoidal inclusion modulus ratios for 55g tests. . . . . . . . . . 301 6.124Best-fit ellipsoidal inclusion modulus ratios for 66g tests. . . . . . . . . . 301 6.125Normalized error for best-fit ellipsoidal inclusion modulus ratios, 33g tests.302 6.126Normalized error for best-fit ellipsoidal inclusion modulus ratios, 33g tests.302 6.127Normalized error for best-fit ellipsoidal inclusion modulus ratios, 44g tests.303 6.128Normalized error for best-fit ellipsoidal inclusion modulus ratios, 44g tests.303 6.129Normalized error for best-fit ellipsoidal inclusion modulus ratios, 55g tests.304 6.130Normalized error for best-fit ellipsoidal inclusion modulus ratios, 55g tests.304 6.131Normalized error for best-fit ellipsoidal inclusion modulus ratios, 66g tests.305
xli 6.132Normalized error for best-fit ellipsoidal inclusion modulus ratios, 66g tests.305 6.133Best-fit modified ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 33g tests. . . . . . . . . . . . . . . . . . . . . . . 306 6.134Best-fit modified ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 44g tests. . . . . . . . . . . . . . . . . . . . . . . 306 6.135Best-fit modified ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 55g tests. . . . . . . . . . . . . . . . . . . . . . . 307 6.136Best-fit modified ellipsoidal inclusion modulus ratios as a function of footing contact pressure for 66g tests. . . . . . . . . . . . . . . . . . . . . . . 307 ¯ p relation of Eqn. (6.36) based on average of all four g-levels for mod6.137 G–¯ ified ellipsoidal inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 ¯ G ¯ model of Eqn. (6.36) based on average of all four g-levels for 6.138 p¯–h– modified ellipsoidal inclusion. . . . . . . . . . . . . . . . . . . . . . . . . 308 6.139Four prescribed inclusion models (large black dots) based on average κ = 0.8 for k = 0.7 from all four g-levels. . . . . . . . . . . . . . . . . . . . 309 6.140VC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing B at 55g. . . . . . . . . . 310 6.141HC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing B at 55g. . . . . . . . . . 310 6.142VC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing G (B2/3) at 55g.
. . . . 311
6.143HC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing G (B2/3) at 55g.
. . . . 311
6.144VC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing H (B1/3) at 55g. . . . . . 312 6.145HC/VE accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing H (B1/3) at 55g. . . . . . 312
xlii 6.146VC/VC accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing I (B0) at 55g. . . . . . . 313 6.147VE/VC accelerance prediction of modified ellipsoidal inclusion model ¯ 2 = (1 + 0.8¯ ¯ = 1 + 0.1h with G p0.7 )0.5 for footing I (B0) at 55g. . . . . . . 313 6.148Best-fit VC/VE accelerances for homog. half-space, homog. half-space with I.M.F.s, original and modified inclusion models for footing B at 55g (test b5011458.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6.149Best-fit HC/VE accelerances for homog. half-space, homog. half-space with I.M.F.s, original and modified inclusion models for footing B at 55g (test b5011458.vna). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6.150Close up of fundamental HC/VE accelerance peak for homog. half-space, homog. half-space with I.M.F.s, original and modified inclusion models for footing B at 55g (test b5011458.vna). . . . . . . . . . . . . . . . . . . 315 6.151Normalized impedances for homogeneous half-space, homogeneous halfspace with I.M.F.s (αhh , αmm ) = (0.85, 0.85), original and modified ellipsoidal inclusions. Solid lines: Real, dashed lines: Imaginary. ω ¯ = q O ωb/ GO 0.5 /ρ. Geq.hom. /G0.5 =1.59. . . . . . . . . . . . . . . . . . . . . . . 316
6.152Normalized impedances for homogeneous half-space, homogeneous half-
space with I.M.F.s (αhh , αmm ) = (0.85, 0.85), original and modified ellipsoidal inclusions. Solid lines: Magnitude, dashed lines: Phase angle. q O ω ¯ = ωb/ GO 0.5 /ρ. Geq.hom. /G0.5 =1.59. . . . . . . . . . . . . . . . . . . . . 317 7.1
Measured VC response of Pile A under ambient and impact vibrations at 33g. Tests VE-PA1 (forced) and VE-PA2 (ambient). (a): VC/VE accelerance, (b): VC/VE coherence, (c): time history of VC accelerometer (hole 7) (d): time history of load cell, (e): ASD of VC accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
xliii 7.2
Measured HC response of Pile A under ambient and impact vibrations at 33g. Tests VE-PA1 (forced) and VE-PA2 (ambient). (a): HC/VE accelerance, (b): HC/VE coherence, (c): time history of HC accelerometer (hole 2) (d): time history of load cell, (e): ASD of HC accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
7.3
Measured VE response of Pile A under ambient and impact vibrations at 33g. Tests VE-PA1 (forced) and VE-PA2 (ambient). (a): VE/VE accelerance, (b): VE/VE coherence, (c): time history of VE accelerometer (hole 6) (d): time history of load cell, (e): ASD of VE accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
7.4
Measured VC response of Pile A under ambient and impact vibrations at 66g. Tests VE-PA45 (forced) and VE-PA44 (ambient). (a): VC/VE accelerance, (b): VC/VE coherence, (c): time history of VC accelerometer (hole 7) (d): time history of load cell, (e): ASD of VC accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7.5
Measured HC response of Pile A under ambient and impact vibrations at 66g. Tests VE-PA45 (forced) and VE-PA44 (ambient). (a): HC/VE accelerance, (b): HC/VE coherence, (c): time history of HC accelerometer (hole 2) (d): time history of load cell, (e): ASD of HC accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.6
Measured VE response of Pile A under ambient and impact vibrations at 66g. Tests VE-PA45 (forced) and VE-PA44 (ambient). (a): VE/VE accelerance, (b): VE/VE coherence, (c): time history of VE accelerometer (hole 6) (d): time history of load cell, (e): ASD of VE accelerometer, (f): ASD of load cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
xliv 7.7
Representative VC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PA6, VE-PA20, VE-PA34, VE-PA49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
7.8
Representative VC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12, VE-PA27, VE-PA41, VE-PA51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
7.9
Representative HC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PA6, VE-PA20, VE-PA34, VE-PA49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
7.10 Representative HC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12, VE-PA27, VE-PA41, VE-PA51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 7.11 Representative VE/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PA6, VE-PA20, VE-PA34, VE-PA49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 7.12 Representative VE/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12, VE-PA27, VE-PA41, VE-PA51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 7.13 Representative VC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PB47/127/153. . . . 337 7.14 Representative VC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 337 7.15 Representative HC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PB47/127/153. . . . 338 7.16 Representative HC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 338
xlv 7.17 Representative VE/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at VC location. Tests VE-PB47/127/153. . . . 339 7.18 Representative VE/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 339 7.19 VC/VE accelerance [m/s2 /N] for Piles A and B at 33g with unused load cell at HC location. Pile A test VE-PA12, Pile B test VE-PB38. . . . . 340 7.20 VC/VE accelerance [m/s2 /N] for Piles A and B at 66g with unused load cell at HC location. Pile A test VE-PA51, Pile B test VE-PB150. . . . . 340 7.21 HC/VE accelerance [m/s2 /N] for Piles A and B at 33g with unused load cell at HC location. Pile A test VE-PA12, Pile B test VE-PB38. . . . . 341 7.22 HC/VE accelerance [m/s2 /N] for Piles A and B at 66g with unused load cell at HC location. Pile A test VE-PA51, Pile B test VE-PB150. . . . . 341 7.23 VE/VE accelerance [m/s2 /N] for Piles A and B at 33g with unused load cell at HC location. Pile A test VE-PA12, Pile B test VE-PB38. . . . . 342 7.24 VE/VE accelerance [m/s2 /N] for Piles A and B at 66g with unused load cell at HC location. Pile A test VE-PA51, Pile B test VE-PB150. . . . . 342 7.25 Prototype-scale VC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12/27/41/51. . . 343 7.26 Prototype-scale VC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 343 7.27 Prototype-scale HC/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12/27/41/51. . . 344 7.28 Prototype-scale HC/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 344 7.29 Prototype-scale VE/VE accelerance [m/s 2 /N] for Pile A at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PA12/27/41/51. . . 345
xlvi 7.30 Prototype-scale VE/VE accelerance [m/s 2 /N] for Pile B at 33, 44, 55 and 66g with unused load cell at HC location. Tests VE-PB38/83/122/150. . 345 7.31 VC acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PA16, HC test HC-PA3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 7.32 VC acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PA51, HC test HC-PA12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 7.33 HC acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PA16, HC test HC-PA3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 7.34 HC acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PA51, HC test HC-PA12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 7.35 VE acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PA16, HC test HC-PA3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.36 VE acceleration response [m/s 2 /N] of Pile A to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PA51, HC test HC-PA12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.37 VC acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PB24, HC test HC-PB19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.38 VC acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PB150, HC test HC-PB75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
xlvii 7.39 HC acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PB24, HC test HC-PB19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 7.40 HC acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PB150, HC test HC-PB75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 7.41 VE acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 33g (experiment and theoretical reference). VE test VE-PB24, HC test HC-PB19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.42 VE acceleration response [m/s 2 /N] of Pile B to VE versus HC loading at 66g (experiment and theoretical reference). VE test VE-PB150, HC test HC-PB75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.43 VC acceleration response [m/s 2 /N] of Pile B to VE versus VC loading at 33g. VE tests VE-PB1 to PB7. VC tests VC-PB1 to PB10. . . . . . . . 358 7.44 VC acceleration response [m/s 2 /N] of Pile B to VE versus VC loading at 66g. VE tests VE-PB130 to PB132. VC tests VC-PB38 to PB40. . . . . 358 7.45 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (VC response). Tests VE-PB46 and VE-PB47. . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.46 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (HC response). Tests VE-PB46 and VE-PB47. . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.47 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (VE response). Tests VE-PB46 and VE-PB47. . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
xlviii 7.48 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (VC response). Tests VE-PB46 and VE-PB47. (a): VC/VE accelerance, (b): VC/VE coherence, (c): time history of VC accelerometer (hole 7) (d): time history of load cell, (e): ASD of VC accelerometer, (f): ASD of load cell. . . . . . . 362 7.49 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (HC response). Tests VE-PB46 and VE-PB47. (a): VC/VE accelerance, (b): VC/VE coherence, (c): time history of VC accelerometer (hole 2) (d): time history of load cell, (e): ASD of VC accelerometer, (f): ASD of load cell. . . . . . . 363 7.50 Effect of negative DC prestress in relieving friction in a 33g hybrid-mode VE test of hollow Pile B with oil-filled dumpling (VE response). Tests VE-PB46 and VE-PB47. (a): VC/VE accelerance, (b): VC/VE coherence, (c): time history of VC accelerometer (hole 6) (d): time history of load cell, (e): ASD of VC accelerometer, (f): ASD of load cell. . . . . . . 364 8.1
BEM meshes for embedded pile segment with A,B: 4-node quads and 3-node triangles, C,D,E: 8-node quads and 6-node triangles. . . . . . . . 375
8.2
Transverse displacements of pile fixed at z/a = 26.586 per unit infinitesimal rotation at z = 0. Continuum Green’s functions for pile. . . . . . . 376
8.3
Impedance comparison for coarse vs. fine meshes of linear elements. Continuum Green’s functions for pile at 66g with G pile /GExp = 890. . . . 377
8.4
Impedance comparison for coarse vs. fine meshes of quadratic elements. Continuum Green’s functions for pile at 66g with G pile /GExp = 890. . . . 378
8.5
Impedance comparison for coarse vs. fine meshes of linear elements. Structural Green’s functions for pile at 66g with G pile /GExp = 890. . . . 379
xlix 8.6
Impedance comparison for coarse vs. fine meshes of quadratic elements. Structural Green’s functions for pile at 66g with G pile /GExp = 890. . . . 380
8.7
¯ mh at frequency ω Traction on interface z = 0 for impedance K ¯ = 0.05. Coarse quadratic Mesh C with standard vs. AG-edge elements at edges.
8.8
381
Impedance comparison for meshes of quadratic elements having AG-edge elements with (m, n, η1s ) = (0.5, 2, −1) at top and bottom edges. Continuum Green’s functions for pile at 66g with G pile /GExp = 890. . . . . . . 382
8.9
Impedance comparison for meshes of quadratic elements having AG-edge elements with (m, n, η1s ) = (0.5, 2, −1) at top and bottom edges. Structural Green’s functions for pile at 66g with G pile /GExp = 890. . . . . . . 383
8.10 Impedance comparison for Mesh E with quadratic or AG-edge elements having (m, n, η1s ) = (0.5, 2, −1), for continuum vs. structural Green’s functions. Gpile /GExp = 890. . . . . . . . . . . . . . . . . . . . . . . . . . 384 8.11 Effect of pile Green’s function on HC/VE pile accelerance [m/s 2 /N] for pile in pure square-root half-space with G pile /GExp = 890. . . . . . . . . 385 8.12 VC/VE accelerance for pile in homogeneous half-space (G = 60MPa) versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 8.13 HC/VE accelerance for pile in homogeneous half-space (G = 60MPa) versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 8.14 VE/VE accelerance for pile in homogeneous half-space (G = 60MPa) versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 8.15 VC/VE accelerance for pile in square-root half-space versus measured accelerances at 33g. Inactive load cell at top center position.
. . . . . . 387
l 8.16 HC/VE accelerance for pile in square-root half-space versus measured accelerances at 33g. Inactive load cell at top center position.
. . . . . . 388
8.17 VE/VE accelerance for pile in square-root half-space versus measured accelerances at 33g. Inactive load cell at top center position.
. . . . . . 388
8.18 Exploded view of upper portion of mesh for no pile-soil contact over first two rows of embedded elements. Red circles denote nodes 2-6-3 of AG-edge elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 8.19 VC/VE accelerance for pile in square-root half-space with loss of contact versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 8.20 HC/VE accelerance for pile in square-root half-space with loss of contact versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 8.21 VE/VE accelerance for pile in square-root half-space with loss of contact versus measured accelerances at 33g. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 9.1
Schematic of pile segment ΩP embedded in modified inner zone of disturbance ΩI surrounded by outer far-field zone Ω O . . . . . . . . . . . . . 394
9.2
Representative 2 soil-zone 3-domain continuum pile-soil interaction model (Case R5/E2).
9.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Examples of inclusion geometry. (a) geometry R10: cylindrical inclusion of radius r/a = 10, (b) geometry R10T5: Tapered inclusion, 5 ≤ r/a ≤ 10, (c) geometry R5: cylindrical inclusion of radius r/a = 5. . . . . . . . . . 402
9.4
Example pile-inclusion shear modulus profiles studied in this investigation.406
li 9.5
VC/VE accelerance [m/s2 /N] at 33g for solid pile in inclusion R5 with linear shear modulus profiles L1, L2 and L3. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
9.6
HC/VE accelerance [m/s2 /N] at 33g for solid pile in inclusion R5 with linear shear modulus profiles L1, L2 and L3. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
9.7
VE/VE accelerance [m/s2 /N] at 33g for solid pile in inclusion R5 with linear shear modulus profiles L1, L2 and L3. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
9.8
VC/VE accelerance [m/s2 /N] at 33g for pile in inclusion R5 with nonlinear shear modulus profiles A, B and C. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
9.9
HC/VE accelerance [m/s2 /N] at 33g for pile in inclusion R5 with nonlinear shear modulus profiles A, B and C. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
9.10 VE/VE accelerance [m/s2 /N] at 33g for pile in inclusion R5 with nonlinear shear modulus profiles A, B and C. Inactive load cell at top center position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 9.11 Inclusion modulus and damping profiles for Case R5/E2. . . . . . . . . . 415 9.12 VC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. . . . . . 417 9.13 VC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. . . . . . . . . . 417 9.14 HC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. . . . . . 418 9.15 HC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. . . . . . . . . . 418
lii 9.16 VE/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. . . . . . 419 9.17 VE/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. . . . . . . . . . 419 9.18 VC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top center position. . . . . . 420 9.19 VC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at side position. . . . . . . . . . 420 9.20 HC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top center position. . . . . . 421 9.21 HC/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at side position. . . . . . . . . . 421 9.22 VE/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top center position. . . . . . 422 9.23 VE/VE accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at side position. . . . . . . . . . 422 9.24 VC/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. . . . 423 9.25 VC/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top off-center position. . . . 423 9.26 HC/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. . . . 424 9.27 HC/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top off-center position. . . . 424 9.28 VE/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. . . . 425
liii 9.29 VE/HC accelerance for solid Pile A in square-root half-space versus inclusion R5/E2 at 66g. Inactive load cell at top off-center position. . . . 425 9.30 VC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 9.31 VC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
9.32 HC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 9.33 HC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
9.34 VE/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 9.35 VE/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
9.36 VC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 9.37 VC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
liv 9.38 HC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 9.39 HC/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
9.40 VE/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 9.41 VE/VE accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at side position. Stem length l0 =1.75cm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
9.42 VC/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 9.43 VC/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 9.44 HC/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 9.45 HC/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 9.46 VE/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 33g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
lv 9.47 VE/HC accelerance for hollow Pile B in square-root half-space versus inclusion R5/E2 at 55g. Inactive load cell at top off-center position. Stem length l0 =1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 9.48 VC/VE accelerance for solid Pile A and hollow Pile B in inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 = 1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 9.49 HC/VE accelerance for solid Pile A and hollow Pile B in inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 = 1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 9.50 VE/VE accelerance for solid Pile A and hollow Pile B in inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 = 1.75cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.51 Horizontal displacement of hollow pile and solid pile at 1 st accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . . . . . . . . . 441 9.52 Horizontal displacement of hollow pile and solid pile at 2 nd HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . 441 9.53 Axial displacement of hollow pile and solid pile at 1 st HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . . . . . . . . . 442 9.54 Axial displacement of hollow pile and solid pile at 2 nd HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . . . . . . . . . 442 9.55 Normalized horizontal displacement of hollow and solid piles at 1 st HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . 443
lvi 9.56 Normalized horizontal displacement of hollow and solid piles at 2 nd HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . 443 9.57 Normalized axial displacement of hollow and solid piles at 1 st HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . 444 9.58 Normalized axial displacement of hollow and solid piles at 2 nd HC/VE accelerance peak frequency. (a): Re, (b): Im. Inclusion R5/E2 at 55g. Inactive load cell at side position. Hollow pile stem length l 0 =1.75cm. . 444 9.59 Inclusion and half-space shear modulus profiles for α = (0.9, 1.0, 1.1) with β = 0.1, zd = 5, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . 447 9.60 Effect of inclusion parameter α on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 447
9.61 Effect of inclusion parameter α on HC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 448
9.62 Effect of inclusion parameter α on VE/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 448
9.63 Inclusion and half-space shear modulus profiles for β = (0.1, 0.3, 0.5) with α = 1.1, zd = 5, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . 451 9.64 Effect of inclusion parameter β on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 451
9.65 Effect of inclusion parameter β on HC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 452
9.66 Effect of inclusion parameter β on VE/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 452
9.67 Inclusion and half-space shear modulus profiles for z d = (5, 7, 9) with α = 1.1, β = 0.1, n = 3 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . . 454
lvii 9.68 Effect of inclusion parameter z d on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 454
9.69 Effect of inclusion parameter z d on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 455
9.70 Effect of inclusion parameter z d on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 455
9.71 Inclusion and half-space shear modulus profiles for n = (3, 5, 7) with α = 1.1, β = 0.1, zd = 5 and ξ0 = 0.3. . . . . . . . . . . . . . . . . . . . . 457 9.72 Effect of inclusion parameter n on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 457
9.73 Effect of inclusion parameter n on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 458
9.74 Effect of inclusion parameter n on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 458
9.75 Inclusion and half-space shear modulus profiles for ξ 0 = 0, 0.2, 0.3, 0.4 with α = 1.1, β = 0.1, zd = 5 and n = 3. . . . . . . . . . . . . . . . . . . . 460 9.76 Effect of inclusion parameter ξ 0 on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 460
9.77 Effect of inclusion parameter ξ 0 on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 461
9.78 Effect of inclusion parameter ξ 0 on VC/VE pile-cap accelerance in [m/s 2 /N] for hollow Pile B at 33g, inactive load cell at side position.
. . . . . . . 461
Chapter 1
Introduction
1.1
Overview of Dynamic Soil-Foundation Interaction The subject of dynamic soil-structure interaction (SSI) is fundamental to the de-
sign of buildings, bridges, highways and dams to withstand dynamic events such as earthquakes, explosions and vibrations from traffic, wind, waves or machinery. The fundamental notion behind SSI is simple and logical: Under general loading and material configurations, a superstructure should be analyzed as an interactive sub-region of the structure-foundation-soil system, and not as an isolated entity (Wolf, 1985; Clough and Penzien, 1993; Kramer, 1996). In such a broader framework, issues such as the deformability and inertial effects of each subsystem, classical and non-classical structural damping, and energy radiation in the soil domain can naturally be accounted for without excessive assumptions. Undoubtedly, a key technical challenge in this field is to find a practical representation of the soil-foundation domain that can capture the essence of various geotechnical aspects without undue complexity nor over-simplicity. The lack of progress in this task despite years of research stems not only from the unboundedness of the ground which is always three-dimensional in nature, but also from the myriad of nonlinear geo-material behavior whose characterization is difficult even for small deformation. With past observations of inconsistent performance even for the most basic soil-foundation interaction problems as in Richart et al. (1970) and Novak (1985), legitimate questions have persisted in the engineering community of the
2 merits of advanced analytical methods for dynamic soil-foundation interaction and the uncertainties associated with more empirical treatments.
1.2
General Background Analytical research in dynamic soil-foundation interaction dates back to Reissner
(1936). Since his seminal work on the dynamics of a circular surface footing, there are the studies by Reissner and Sagoci (1944), Robertson (1966), Gladwell (1968), Veletso and Wei (1971), Luco and Westmann (1971), Kausel and Roesset (1975), Pak and Gobert (1991) and Guzina and Pak (1998) on various surface footing problems associated with both homogeneous and inhomogeneous soil media. On dynamic pile-soil interaction, there are also analytical attempts such as those in Kuhlemeyer (1979a,b), Novak (1974, 1977), Novak and Sheta (1980), Pak and Jennings (1987), Trochanis et al. (1991b) and Pak and Ji (1993). In terms of solution techniques, they range from generalized eigenfunction expansions, integral transforms and integral equations for mathematically exact treatments, to numerical methods such as finite element and boundary element methods for their adaptability and flexibility. While they vary in theoretical rigor, these treatments are all significant analytical accomplishments in theoretical and computational mechanics. As to their direct relevance and adequacy in geotechnical and structural engineering, however, the issue has been a subject of continuing debate among both researchers and practitioners because of their lack of predictive capability and consistency even for some very basic soil configurations. For the footing vibration problem for instance, Fry (1963), Richart and Whitman (1967) and Novak (1970) found that the classical continuum solution for vertical vibration of footings can barely predict the resonant curve of such foundations on sand, with error up to 200%. For lateral vibration, the predictions are even worse, with the theoretical horizontal-rocking response being much stiffer than what can be observed from field testing. More recent field experiments have been conducted and analyzed as in Novak (1985) and Dobry et al. (1986)
3 and Crouse et al. (1990), and have found similar fundamental discrepancies between theory and observations. With interpretation obliging largely to conventional wisdom, the analytical-experimental disagreements have spurred fair questions on the adequacy of the specific site characterization as well as questionable concepts such as the need to allow non-unique soil moduli for differing modes of vibration (e.g. Gazetas and Stokoe, 1991). With the ability to eliminate some of the major uncertainties associated with field testing, dynamic centrifuge scaled modeling studies have confirmed the usefulness of the experimental technique as well as the need of much more fundamental research in this class of problems (e.g. Lenke et al., 1991; Gobert and Pak, 1994; Pak and Guzina, 1995; Pak and Ashlock, 2000a; Kurahashi, 2002).
1.3
Scope of Research To advance the field of dynamic soil-structure interaction, this study is focused
on understanding and solving the fundamental problem of dynamic characterization of both surface and deep foundations in granular soils. Aimed to generate a comprehensive experimental database for the intended investigation, a variety of scaled model surface and pile foundations were designed and tested dynamically in a large geotechnical centrifuge with the latest technology. Computationally, a boundary element platform was enhanced at several levels to enable a rigorous treatment of the corresponding elastodynamic boundary value problems. Through the integrated study, general as well as specific analytical, numerical and physical challenges have been confronted and dealt with. They include the validation of a new hybrid-mode dynamic test method, the identification and resolution of a problem related to slender elements in boundary element methods, a new class of elements for singular or high-gradient solutions, the development of the concept of Impedance Modification Factors for experimental and practical engineering designs, and two next-level continuum mechanics models for both footing and pile dynamics problems.
4 Because of the extent and scope of the research conducted, this thesis is divided into Parts I, II and III. Part I contains a delineation of the new experimental techniques and computational methods developed that are useful for advanced studies of a large class of dynamic soil-foundation-structure interaction problems. Part II is a description of an in-depth experimental-analytical investigation of the dynamic behavior of a surface footing under multi-directional excitations while Part III contains the same for the single-pile problem.
5
PART I:
EXPERIMENTAL & COMPUTATIONAL DEVELOPMENTS
Chapter 2
Experimental Modeling
2.1
Centrifuge For the physical foundation vibration experiments in this investigation, the 400 g-
ton centrifuge at the University of Colorado at Boulder was used (Figs. 2.1 and 2.2). This centrifuge consists of an asymmetric rotor arm with a hanging platform at one end, and counterbalance compartments on the other. The platform is designed to accommodate sample containers up to 4ft by 4ft by 3 ft in height and weighing up to 2 metric tons (2000kg or 4400lbf), which may be subjected to a maximum centrifugal acceleration of 200g. The radius from the center of rotation to the platform is 18ft in the fully extended position. Two high strength steel tension straps are designed to bear the centrifugal load during spinning, while an external box-girder structure supports the 330kN weight of the arm. The power train of the centrifuge consists of a General Electric 684 kW DC electrical motor connected through shear pins to a horizontal drive shaft, which enters a right angle gear box having a 6.4 to 1 speed reduction ratio. During spin-down of the centrifuge, the DC motor acts as a generator, returning electrical power to the campus electrical grid. During N-g tests, the high velocity of the platform causes a stirring of the cylindrical air mass enclosed in the centrifuge chamber, leading to heat generation. To minimize thermal effects on the soil specimens, pipes carrying chilled water line the walls and are used to stabilize the chamber temperature. More detailed information about this centrifuge may be found in Ko (1988a).
7
Figure 2.1: Schematic of 400 g-ton Centrifuge at the University of Colorado at Boulder (from Ko (1988a)).
Figure 2.2: Photograph of 400 g-ton Centrifuge at the University of Colorado at Boulder.
8
2.2
Scaling Relations In addition to scaling the physical dimensions in scaled-model centrifuge tests,
other relevant prototype quantities should be scaled according to the relations given in Table 2.1. These scale factors can be derived using either dimensional analysis or the governing differential equations. A common concern in geotechnical centrifuge tests which use the prototype soil is the effect of the increased ratio of grain size to length of the scale model structure or foundation. To minimize the grain-size effects in this study, a very fine silica sand was used so that the mean particle diameter was small relative to the smallest model foundation dimension. To verify that these scale effects have been adequately addressed, “modeling of models” tests were performed as in Gillmor (1999) and Ashlock (2000). Modeling of models involves comparing scaled test results from different sized models of the same prototype at g-levels inversely proportional to their length scales. If the measurements are in close agreement when scaled to the prototype scale, then modeling of models is said to be achieved and grain size effects are not significant. Note that three different time scale factors may be obtained depending on the form of the differential equation which governs the experimental phenomenon. For the dry dynamic tests in this investigation, the dynamic time scale factor of 1/n is appropriate.
2.3 2.3.1
Soil Sample Physical Properties Physical experiments in this investigation were performed on samples of F-75
Ottawa sand; a uniform, fine silica sand having the typical physical properties given in Table 2.2. Results of grain size analysis tests provided by the U.S. Silica company are shown in Table 2.3 and Figure 2.3. From the grain size analysis, one can obtain the
9 coefficient of uniformity Cu and coefficient of concavity Cc as D10 = 0.116mm, Cu =
D30 = 0.152mm,
D60 = 1.71, Cc = D10
2 D30 D10 D60
D60 = 0.198mm,
= 1.01,
(2.1)
where Dn is the width in millimeters of the sieve opening through which n% of the sample passes. The coefficient of uniformity is a measure of the slope of the gradation curve. A large Cu indicates a poorly graded (well sorted) particle size distribution and a small Cu indicates a well graded (poorly sorted) distribution. The coefficient of curvature is a measure of the shape of the curve between D 10 and D60 , and indicates a gap in grading when significantly different from 1. According to tests performed by the U.S. Bureau of Reclamation, the minimum and maximum achievable dry densities are ρd,min = 1469
kg , m3
ρd,max = 1781
kg , m3
(2.2)
giving the minimum and maximum void ratios emin = emax =
ρs ρd,max ρs ρd,min
−1=
2.65 · 1000kg/m 3 − 1 = 0.4879 1781kg/m 3
−1=
2.65 · 1000kg/m 3 − 1 = 0.8039, 1469kg/m3
(2.3)
where ρs = G s ρw ,
(2.4)
in which ρs and Gs are the mass density and specific gravity of the soil solids, respectively, and ρw is the density of water. 2.3.2
Sample Preparation To obtain a soil sample with a uniform density throughout, the sand was rained
from a calibrated height through a slotted plate attached to the bottom of a suspended container known as a hopper (Figure 2.4). This commonly used method, known as pluviation through air, allows one to control the soil density for a given traversal speed
10
100.0
% Finer
80.0
60.0
40.0
20.0
0.0 1.00
0.10 Grain Size [mm]
0.01
Figure 2.3: Grain size distribution of F-75 silica sand.
of the hopper by changing the flux of the sand via the slot dimensions or raining height. For the density used in this study and given in Table 2.4, the slotted plate had an opening of approximately 17.5in by 3/16in, and the density was calibrated by raining the sand into a smaller container with dimensions 40.7cm by 40.7cm by 39.4cm deep. During the raining process, the hopper traversed the soil sample with the aid of an overhead crane, and the raining height was reset to the desired value after every eighth pass of the hopper, corresponding to approximately 1cm of soil deposition. The dry density and void ratios reported in Table 2.4 were obtained by raining the sand from a height of 68in. All samples were dry, with water contents less than 0.1%. The void ratios and relative densities in this table are calculated as e=
Gs · ρ w − 1, ρb
and
Dr =
emax − e . emax − emin
(2.5)
11 Table 2.1: Scaling relations for centrifuge tests. Quantity
Prototype
Length Area Volume Mass Density Mass Strain Displacement Velocity Acceleration Energy Density Energy Stress Force Time (viscous flow) Time (dynamic) Time (seepage)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Model at n · g 1/n 1/n2 1/n3 1 1/n3 1 1/n 1 n 1 1/n3 1 1/n2 1 1/n 1/n2
Table 2.2: Typical physical properties of F-75 silica sand. Property Mineral Grain Shape Specific Gravity, Gs Surface Area
Description Quartz, 99.8% SiO2 Rounded 2.65 162 cm2 /gram
Table 2.3: Grain size analysis results for F-75 silica sand. US Standard Sieve Size 40 50 70 100 140 200 270 Pan
Diameter [mm] 0.425 0.300 0.212 0.150 0.106 0.075 0.053
% Retained on Sieve 0.3 6.3 25.4 39.9 22.0 5.6 0.8 Trace
% Passing 99.7 93.7 68.3 28.4 6.4 0.8 0.0
12
Table 2.4: Soil sample properties for centrifuge tests in this study. Property Raining Height Water content Dry density Void ratio Relative Density Qualitative Description Friction angle* Coeff. of earth pressure at rest *From Shertmann’s chart
Value 68 in w≈0 ρd = 1735.21kg/m 3 e = 0.5272 Dr = 87% Very Dense φ = 40◦ K0 = 1 − sinφ = 0.3572 of φ vs. Dr (1978)
Figure 2.4: Preparation of a soil sample via pluviation .
13
2.4
Container and Boundary Effects One of the goals of the experimental portion of this study is to provide a high-
quality experimental database for modeling the dynamic response of full-scale foundations. To simulate the half-space condition in reality, it is essential that the threedimensional radiation condition is satisfied as much as possible in the experiments. Left untreated in the model experiments, the finite boundary of the relatively rigid soil container can produce unwanted reflection of waves back towards the foundation, thereby contaminating the measured response with multiple complex modes of vibration. Lenke et al. (1991) have also shown that (i) rectangular containers are more effective in minimizing unwanted reflections, as a circular container tends to focus the reflected wave energy back to the center of the sample, and (ii) lining the vertical container walls with an oil-based putty such as the commercially available product Duct Seal can help to mitigate the reflected wave energy (see also Hushmand, 1983; Guzina, 1992). Proven in Pak and Guzina (1995) as an effective mitigation measure, a large rectangular container of the size detailed in Table 2.5 was used for the centrifuge tests in this study, with Duct Seal putty lining the vertical walls. Table 2.5: Dimensions of container, Duct Seal and soil samples. Item Container Soil Sample Duct Seal
2.5
Width [m] 1.00 0.93
Length [m] Depth [m] 1.20 0.61 1.13 0.46 ≈ 0.035 thick
Scaled Model Footings In a long series of centrifuge studies of this class of problems (Guzina, 1992; Pak
and Guzina, 1995; Brown, 1995; Gillmor, 1999; Ashlock, 2000; Sloan, 2000; Kurahashi, 2002), high strength aluminum with the properties given in Table 2.6 was used for the scaled-model footings. In the surface foundation phase of this study, selected scaled-
14
Figure 2.5: Soil Container and Duct Seal Walls.
model footings from Gillmor (1999), Ashlock (2000) and Kurahashi (2002) were utilized based on the range of contact pressure they provided, and for correlation to the database of test results from the aforementioned studies. The first group of foundations used is from Gillmor (1999), consisting of footings named A, B and C, with B and C scaled from Footing A for the purpose of modeling-of-models verification. These foundations have five evenly distributed instrumentation holes on two opposing vertical sides, and three symmetrically placed instrumentation holes on top, as detailed in Table 2.7 and Fig. 2.6. The second group of footings named D, E and F are from Ashlock (2000) and were designed such that the mid-sized Footing E had the same overall dimension as Footing B, but was equipped with additional asymmetrically located instrumentation holes on its top for investigating the effects of eccentric vertical loads. For modeling-ofmodels tests, Footings D and F were then created as scale models of Footing E, with slightly different scale factors than those used for Footings A and C (see Table 2.8).
15 Table 2.6: Properties of aluminum alloy 6061-T6 used for hollow piles and high-strength alloy 7075-T6 used for footings, pile caps and monolithic pile-cap assemblies. Property Density Young’s modulus Poisson’s Ratio Shear Modulus
Alloy 6061-T6 ρ = 2800kg/m3 E = 68.95GP a (10, 00ksi) ν = 1/3 G = 25.86GP a (3750ksi)
Alloy 7075-T6 ρ = 2800kg/m3 E = 71.71GP a (10, 400ksi) ν = 1/3 G = 26.89GP a (3900ksi)
Due to the fixed size of the threaded holes used for instrumentation, the actual mass and moment of inertia of model Footings B, C, D and F do not precisely scale according to the scaling relations of Table 2.1. However, as indicated in Tables 2.7 and 2.8, the maximum of these variations is only 0.3% for the footings alone. The use of the same set of instrumentations, which typically weigh 20–30 grams, for load and acceleration measurements for all footings will increase these variations by a few percent. To investigate the effects of foundation contact pressure on the dynamic response, three model foundations were designed in Kurahashi (2002) to have the same width as Footing B, but with heights that are approximately 2/3, 1/3 and 1/18 that of Footing B. These are referred to as Footing G (or B2/3), H (or B1/3) and I (or B0), respectively. Their actual dimensions and inertial properties are listed in Tables 2.9 to 2.11. They all have five instrumentation holes on their tops which are 1.8cm apart and aligned in two orthogonal directions. Additionally, Footings G (B2/3) and H (B1/3) have four instrumentation holes on two opposing sides, the locations and notational conventions of which are detailed in Tables 2.9, 2.10, 2.11 and Fig. 2.8. In the database of past footing tests from Gillmor (1999), the soil surface glevels 45g, 60g, 75g and 90g were used for Footing A in order to model four prototype foundations having widths 1.8m, 2.4m, 3.0m and 3.6m, respectively. Based on its scale factor of 1.375 (see Table 2.7), the corresponding g-levels for modeling the same four prototype footing widths using the larger Footing B in Gillmor (1999) as well as the identically sized Footing E in Ashlock (2000) were therefore 32.73g, 43.64g, 54.55g, 65.45g. These four g-levels are used as the nominal g-levels 33g, 44g, 55g and 66g in
16 this and the aforementioned studies. To examine the effects of contact pressure for the same four prototype footing widths, Footings B, G (B2/3), H (B1/3) and I (B0) were also tested at the nominal g-levels 33g, 44g, 55g and 66g in this study and in Kurahashi (2002). The resulting combinations of footing mass and g-level provided 16 prototype bearing pressures typically ranging from 10kPa to 256kPa (see Table 2.12), with slight variations from these values depending on the number of instrumentations used.
Table 2.7: Properties of Model Footings A, B and C.
A B C
x/b y/b
Width [cm] 4.00 5.50 7.00
Height [cm] 10.00 13.75 17.50
1/9 ±1.000 4.650
Scaled Scaled Mom. of Mom. Centroid Mass Mass Inertia of Inertia Height 2 2 [kg] [kg] [kg cm ] [kg cm ] [cm] 0.4510 0.4510 4.354 4.354 4.991 1.173 1.172 21.47 21.40 6.870 2.421 2.417 71.77 71.46 8.747 Hole # 3/11 4/12 5/13 6 7 8 ±1.000 ±1.000 ±1.000 -0.650 0.000 0.650 3.150 2.400 1.650 5.000 5.000 5.000
Length Scale Factor 1.000 1.375 1.750
2/10 ±1.000 3.900
Table 2.8: Properties of Model Footings D, E and F.
D E F
Width 2b [cm] 4.40 5.50 6.60
Height [cm] 11.00 13.75 16.50
Length Scale Factor 0.800 1.000 1.200
x/b y/b
1/13 ±1.000 4.650
2/14 ±1.000 3.900
x/b y/b
7 -0.800 5.000
8 -0.533 5.000
Scaled Mom. of Mass Mass Inertia [kg] [kg] [kg cm2 ] 0.5966 0.5966 6.932 1.164 1.164 21.22 2.010 2.012 52.86 Hole # 3/15 4/16 ±1.000 ±1.000 3.150 2.400 Hole # 9 10 -0.267 0.000 5.000 5.000
Scaled Mom. of Inertia [kg cm2 ] 6.952 21.22 52.79
Centroid Height [cm] 5.491 6.869 8.246
5/17 ±1.000 1.650
6/18 ±1.000 0.900
11 0.400 5.000
12 0.800 5.000
CL 6
x8
7
xC
11 12 13
7
8
9 10
17
CL
C
yC
1
13
2
14
3
15
4
16
5
y4
17
x 12
8 9 10
11
12 1 2
xC
3 4
C
yC
5
18
b
b
Footings A, B & C
6
b
b
Footings D, E & F
Figure 2.6: Location of instrumentation holes for model Footings A through F.
Figure 2.7: Scaled-model Footings D, E and F.
y4
18
2.6
Scaled Model Piles For the experimental study on piles in Part III of the thesis, the three scaled-
model pile and cap systems shown in Fig. 2.10 were fabricated. These models include two monolithic high-strength aluminum rod and pile-cap assemblies, one of which has a right-angle pile/cap junction (Pile A), while the other has a filleted pile/cap junction (Pile C). A hollow standard-strength aluminum tube threaded 1cm into a separate pile cap made of high-strength aluminum was also fabricated (Pile B). The properties of these pile-cap systems are given in Table 2.13, and the labeling of their instrumentation holes is shown in Fig. 2.9. Each pile cap has five instrumentation holes on the top, arranged in two orthogonal lines crossing at the center, and a row of three instrumentation holes on each of two opposing sides, as shown in Fig. 2.10. For Pile Caps A and C, the distance between adjacent holes is 1.4cm, and the holes on the side are located a distance of 0.6cm from the bottom of the pile cap. For the larger Pile Cap B, the distance between adjacent holes is 1.8cm, and the holes on the side are centered a distance of 1cm from the top and bottom of the pile cap. The main focus of the pile tests in this study involved the monolithic Pile A shown instrumented in Fig. 2.10 and the hollow threaded Pile B. Both piles were inserted into the soil at 1g until an un-embedded pile segment of length l0 =1.5cm remained above the soil surface, as indicated in Table 2.13. For convenience and correlation with the same soil conditions used for the surface footings described in the previous section, the nominal g-levels of 33, 44, 55 and 66g used in footing tests were also used for vibration tests of the piles.
19
Table 2.9: Properties of Model Foundation G (B2/3). Width 2b [cm] 5.50
x[cm] y[cm]
x[cm] y[cm]
Mom. of Centroid Mass Inertia Height [kg] [kg cm2 ] [cm] 0.7644 7.042 4.495 Hole # 1/10 2/11 3/12 4/13 ±2.750 ±2.750 ±2.750 ±2.750 8.000 6.000 4.000 2.000 Hole # 5 6 7 8 9 -1.800 0.000 1.800 0.000 0.000 9.000 9.000 9.000 9.000 9.000 Height [cm] 9.00
Table 2.10: Properties of Model Foundation H (B1/3). Width 2b [cm] 5.50
x[cm] y[cm]
x[cm] y[cm]
Mom. of Centroid Mass Inertia Height 2 [kg] [kg cm ] [cm] 0.3807 1.593 2.247 Hole # 1/10 2/11 3/12 4/13 ±2.750 ±2.750 ±2.750 ±2.750 3.500 2.500 1.500 0.500 Hole # 5 6 7 8 9 -1.800 0.000 1.800 0.000 0.000 4.500 4.500 4.500 4.500 4.500 Height [cm] 4.50
Table 2.11: Properties of Model Foundation I (B0). Width 2b [cm] 5.50
Height [cm] 0.75
x[cm] y[cm]
1 -1.800 0.750
Mom. of Centroid Mass Inertia Height 2 [kg] [kg cm ] [cm] 0.06323 0.1623 0.3745 Hole # 2 3 4 5 0.000 1.800 0.000 0.000 0.750 0.750 0.750 0.750
20 Table 2.12: Typical surface footing contact pressures [kPa], assuming instrumentation arrangement of two load cells at 12.06gr each and three accelerometers at 2.50gr. Footing B G (B2/3) H (B1/3) I (B0)
33g 127.8 84.5 43.7 10.1
Nominal g-level 44g 55g 66g 170.4 213.0 255.6 112.6 140.8 168.9 58.3 72.9 87.5 13.4 16.8 20.1
CL 8 (front) 9 (back) 5
x7 6
7
10
1
xC 11
2
CL 12 13
3
yC
4 (front) 5 (back) 1
y2
C 4
b
b
b
Footings G (B2/3) & H (B1/3)
x3 2
3
b
Footing I (plate/B0)
Figure 2.8: Location of instrumentation holes for model Footings G, H and I.
9 (on back) 10 11
7
4 5
6
8
1 2 3
Figure 2.9: Location of instrumentation holes for scaled-model pile caps.
21 Table 2.13: Properties of scale model piles. † measured, ? calculated. Pile model Type Pile Aluminum alloy Outer radius a [cm] Inner radius ain [cm] Pile length lp [cm] Un-embedded length l0 [cm] Embedded length le [cm] Embedment ratio le /a Young’s Modulus, E [GPa] EA [MN] EI [N-m2 ] Mass of pile and cap Mass of pile cap [gr]? Pile cap width [cm] Pile cap height [cm]
[gr]†
Pile cap mom. of inertia [g-cm2 ] Pile cap centroid height [mm]
A rod 7075-T6 0.4572 – 13.655 1.50 12.155 26.6 71.71 4.71
B tube 6061-T6 0.4699 0.3493 14.166 1.50 12.667 27.0 68.95 2.14
C rod 7075-T6 0.4521 – 14.166 1.50 12.667 28.0 71.71 4.61
24.61
18.34
23.53
114.92 87.42 4.00 2.00
149.44 136.35 5.00 2.00
116.50 87.42 4.00 2.00
145.25 9.998
331.33 10.018
145.25 9.998
A B
C
Figure 2.10: Scaled-model pile and pile-cap assemblies A: solid monolithic pile and cap, B: threaded hollow pile and cap, C: filleted solid-monolithic pile and cap.
22
2.7
Excitation System For applying vertical loads, a Br¨ uel & Kjær type 4809 electromagnetic exciter
was used. It was mounted to an aluminum beam and bolted onto two channels of a stiff reaction frame, which was in turn bolted to the container as shown in Fig. 2.11. For rectilinear motion, the exciter’s moving element is supported internally by a guidance system of grouped radial and transverse flexures, and is attached to a drive coil that moves over a permanent magnet which is rigidly attached to the exciter body (Fig. 2.12). For horizontal excitation forces, a smaller B&K type 4810 mini-shaker was used. This mini-exciter was suspended via hose clamps from a rectangular steel tube bolted to two 3/8 in steel bars which were, in turn, suspended from the large exciter’s support frame (Fig. 2.11). The force ratings and other pertinent properties of these two exciters are detailed in Table 2.14. Both exciters were driven by a two-channel, 750Watt Techron Model 5515 power amplifier with variable gain. Table 2.14: Properties of B&K electromagnetic exciters. Property Peak force rating [N] Diameter [cm] Height [cm] Moving element mass [gr] Total mass [kg] Max. peak to peak displacement [mm] Max. input current [A rms] Axial stiffness of flexures [N/mm] Frequency range [kHz] First axial resonance [kHz]
2.8
Model 4809 44.5 14.9 14.3 60 8.3 8 5 12 0.01-20 20
Model 4810 10 7.6 7.5 18 1.1 6 1.8 2 0-18 18
Instrumentation To monitor the intended planar motion of the models, the foundations were in-
strumented with three or more stud-mounted PCB Model 303-B67 and 352-C67 piezoelectric quartz accelerometers, each having a mass of approximately 2 grams, a mea-
23
Figure 2.11: Large and small exciters used in vertical and horizontal excitation tests.
Figure 2.12: Schematic of B&K 4809 Exciter (from Bruel & Kjaer Company).
24 surement range of ±50g, and a resolution of 0.0003g rms. These accelerometers were hand-selected by PCB Piezotronics to have exceptionally low cross-sensitivity (≤ 1.5%). Accelerometer power and signal conditioning were provided by a PCB Model 483A signal conditioner mounted on the centrifuge arm. For force measurements, two Kistler washer shaped quartz piezoelectric load cells (models 9001 and 9001A) were directly mounted on the foundation so that the point of force application was precisely known. These miniature high rigidity load cells have a mass of 3 grams and a measurement range of 7.5kN. The charge signals from these load cells were conditioned by two Kistler charge amplifiers (types 5010A and 5004), which amplify and convert the low-level charge to a proportional voltage. Due to the low-levels of piezoelectric charge output by the load cell, Kistler Model 1631C high impedance, low capacitance cables are used between the load cell and charge amplifier to minimize static discharge and triboelectric noise. For uniform load transfer and accurate control of the point of force application, each load cell was integrated into a stinger assembly with a hemispherical cap or “button” and a load-distribution cap (see Ashlock (2000) and Fig. 2.27).
2.9 2.9.1
Data Acquisition Systems and Measurement Approach Spectral Dynamics SigLab System The majority of tests in this study were performed using a pair of Spectral Dynam-
ics SigLab Model 20-42 dynamic signal and system analyzers having 20 kHz bandwidth capabilities. They are controlled via a virtual instrument software measurement suite implemented in Matlab. These analyzers were located in the control room outside the centrifuge chamber, and connected to the instrumentation through the centrifuge sliprings. Each of the SigLab spectrum analyzers allow the measurement and digitization of four differential input signals and the generation of two independent output signals. A
25 16-bit D/A converter with a 9th order elliptic filter is used for smoothing of the output signals, which were amplified by a Techron Model 5515 power amplifier and then sent over the centrifuge slip rings to the exciter. A drawback to the use of slip-rings for all electrical signals passing between the data acquisition system and the excitation/instrumentation hardware is that unwanted electro-mechanical noise is injected into the signals. This is a result of the metal to metal contact of each rotating slip-ring with its corresponding stationary point, which inevitably introduces some noise during the transmission process. To alleviate this effect, following conditioning of the load cell and accelerometer signals on the arm, a gain of 10:1 was applied using either a Kron-Hite (K-H) Model 3905A multichannel filter or an in-house designed voltage amplifier. The amplification of the voltage signals prior to routing over the slip-rings improves the final signal to noise ratio of the contaminated signals, which are then routed to the SigLab analyzer in the control room.
2.9.2
Wireless National Instruments LabView System During the course of this investigation, the centrifuge data acquisition and con-
trol systems were overhauled, and new wireless data acquisition and real-time control systems were implemented (Wallen, 2005). The SigLab analyzers in the control room were replaced by an 8-channel National Instruments (NI) PXI-4472 dynamic signal acquisition module residing on the centrifuge arm. The NI data acquisition system is implemented using Labview software, which is controlled using a wireless remote desktop connection. This eliminates the need for signals to pass outside the centrifuge chamber via the slip-rings in most situations. For example, in the new foundation instrumentation setup, the accelerometers are connected directly to the NI PXI-4472 card, as are the outputs from the load cell charge amplifiers. The PCB signal conditioner is no longer needed, as the PXI-4472 provides the necessary accelerometer excitation current and signal conditioning. Because the slip-rings are no longer used for the load cell
26 and accelerometers, the issue of electro-mechanical slip-ring noise is eliminated, and the amplifier for these signals is no longer needed. This reduction in both the number of electronic components and the length of signal paths offers a simpler and significantly more efficient experimental setup. With the Labview system, the input signals for the exciter are now provided by a National Instruments PXI-6052E 16-bit analog multifunction DAQ. However, in contrast to the SigLab hardware, this function generator does not have smoothing filters on the analog output. Analysis of the raw analog voltage output of this card on an oscilloscope revealed piecewise constant output which also suffered from the Gibbs phenomenon, evidenced by overshoots of the digitally stepped output. Both problems resulted in additional high-frequency components, which were audible in the exciter’s output. To alleviate this problem, an external passive single-pole low-pass RC filter was found to be effective when applied to the PXI-6052E output. However, due to the gentle attenuation of single-pole filters, the gain was not unity over the 0-5kHz frequency range of interest for the centrifuge tests (see Fig. 2.13). To provide a sharper attenuation with frequency, the K-H filter was employed in low pass mode with unity gain and Butterworth filtering with a cutoff frequency of 10kHz (Fig. 2.13). The result is an output signal that is nearly identical in smoothness to that of the SigLab system. The original unfiltered output voltage from the NI PXI-6052E is compared to the smoother output of SigLab for two different time scales in Figs. 2.14(a) and (c). Following application of the external hardware filter to the Labview output signal, it is seen to be comparable in smoothness to that of the SigLab system (Figs. 2.14(b), (d)). Because the heavy internal components of the Techron 5515 power amplifier are mounted to relatively flexible circuit boards, it is unlikely that it would survive the increased g-levels were it mounted on the centrifuge arm. Therefore, the power amplifier was placed outside the centrifuge chamber, next to the slip-ring tower. The exciter input voltage is the only signal still routed over the slip-rings in the new NI hardware/Labview
27
Figure 2.13: Frequency responses of passive single-pole low-pass RC filter and higherorder Butterworth filtering from K-H multichannel filter. f Break = 1/2πRC =break frequency, fc =cutoff frequency.
(a)
Labview (unfiltered)
SigLab
(b)
Labview (filtered)
SigLab
(c)
Labview (unfiltered)
SigLab
(d)
Labview (filtered)
SigLab
Figure 2.14: Oscilloscope measurements of random analog signals from SigLab hardware versus unfiltered (a,c) and filtered (b,d) output from NI (Labview) hardware. Butterworth filtering of the Labview output is achieved with the K-H multichannel filter. (a,b): shorter time scale, (c,d): longer time scale.
28 software data acquisition configuration. Exciter input signals are now routed from the NI output card through the slip-rings, into the K-H filter followed by the Techron power amplifier, and finally back down the slip-rings to the exciters on the centrifuge platform. Contamination of the exciter input signals by electrical slip-ring noise is usually not as serious an issue as was the case for the lower intensity load cell and accelerometer signals. This is because the actual force applied to the foundation by the exciter is the measured stimulus of the system. The slip-ring noise only serves to alter the force applied to the foundation, although thereby affecting the intended spectral density of excitation. When using larger exciters that require more than 10Amps of current, the output of the power amplifier must be distributed over multiple pairs of slip rings and re-combined at the exciter. This is a consequence of the electrical current limit of 10Amps per slip-ring pair as a result of gauge of wire connected to them. However, preliminary seismic experiments using a large (100lbf) exciter requiring 20-30Amps of current have revealed some non-linearity and cross-talk over adjacent pairs of slip-rings in such cases. This problem may be solved in the future by using a more rugged highpower amplifier that can withstand increased g-levels and be placed on the centrifuge arm. Alternatively, one may seek to improve the isolation and power handling capacity of the slip-rings. While the wireless data acquisition system has numerous advantages over the previous hard-wired slip-ring system, the National Instruments Labview software did not contain a complete implementation of a network analyzer. With most needed software modules available in the Sound and Vibration Software Toolkit, however, a full network analyzer virtual instrument program was written using the SigLab network analyzer as a target, while incorporating additional features deemed useful. A benefit of this approach is that complete control over the network analyzer is now available to the experimentalist who wishes to add necessary features to the spectrum analyzer’s program code. For both data acquisition systems, random excitation signals which have uni-
29 form spectral content were used to excite the foundations over an entire bandwidth at once. The resulting foundation acceleration and force measurements were sampled simultaneously over a number of time windows, and converted into the frequency domain through the use of the Fast Fourier Transform (FFT). For these vibration tests, the primary network analyzer functions of interest are the time histories of acceleration and force, the transfer- or frequency response functions (FRF), the coherence functions (COH) and the auto-spectral density (ASD), also known as the power spectral density (PSD) functions. The frequency response functions used in this study are defined as the ratio of directional acceleration at various points on the foundation to the force applied at a fixed point. This FRF will be referred to as the “accelerance function”, or simply “accelerance”. The accelerance, coherence and auto-spectral density functions are described in detail in the following sections.
2.9.3
Data Digitization To convert the analog data signals into digital format, they are first sampled at
a selected time interval ∆t, the inverse of which is known as the sampling rate s in samples per second (Hz). As a consequence of the sampling theory in the time domain (see Bendat and Piersol, 1986), the critical highest frequency that can be obtained for a given sampling rate is fc =
1 , 2∆t
(2.6)
which is known as the Nyquist, or folding frequency. Frequencies in the original data above fc will appear below fc after application of the Fourier transform, and will be confused with the original data from this lower frequency range. This problem is referred to as aliasing, and the practical way to remove the aliasing error is to employ analog low pass filters (“anti-aliasing filters”) to remove all information above f c prior to the analog-to-digital conversion. Since it is impossible to construct a low pass filter with an infinitely sharp roll off, is is customary to set the cutoff frequency of the anti-aliasing
30 filter to 70 − 80% of the Nyquist frequency to ensure that all data above f c is sufficiently attenuated. The SigLab analyzer uses anti-aliasing filters which set the cutoff frequency (fcutoff ) equal to the selected bandwidth (B) of measurement. This analyzer was designed with adjustable filters such that B is always 78% of the Nyquist frequency. Using Equation (2.6), one may relate the sampling rate to the Nyquist frequency through s fc = , 2
(2.7)
from which one can obtain a relation between the desired bandwidth of measurement B and the required sampling rate s: B = fcutoff = 0.78fc =
0.78s → s = 2.56B. 2
(2.8)
The SigLab analyzer always uses a sampling rate that is 2.56 times the selected bandwidth of measurement in order to minimize the aliasing error. This commonly used sampling principle (Eqn. (2.8)) was also implemented in the programming of the new network analyzer in Labview.
2.9.4
Fourier Transforms Fourier series are commonly used in dynamics problems to express periodic or
nearly periodic forcing functions as a sum of sinusoidal components, after which the principle of superposition for linear problems is used to obtain the overall system response as the sum of the responses to the individual sinusoidal components. For transient non-periodic data, however, such discrete spectral representations are not possible. Instead, a continuous spectral representation X(f ) of the transient data x(t) may be obtained through the use of a Fourier transform, one possible definition of which is Z ∞ X(f ) = x(t) e−i2πf t dt, (2.9) −∞
where i =
√
−1. The inverse of this transform is given by Z ∞ X(f ) ei2πf t df. x(t) = −∞
(2.10)
31 For a real valued function x(t), X(f ) is generally complex. In digital data analysis, the discrete version of the Fourier transform is evaluated for a function sampled over a time interval (0, T ) at N equally spaced points which are a distance ∆t apart. The sampling times are tn = n∆t, n = 0, 1, 2, . . . N − 1.
(2.11)
Usually, the discrete frequency values are chosen such that fk =
k k = , T N ∆t
k = 0, 1, 2, . . . , N − 1.
(2.12)
Note that k = N/2 corresponds to the Nyquist frequency. Denoting x(t n ) by xn , the discrete forms of Equations (2.9) and (2.10) may be written as X(fk ) = ∆t
N −1 X
xn e−i2πkn/N ,
n=0
k = 0, 1, 2, . . . , N − 1.
(2.13)
n = 0, 1, 2, . . . , N − 1,
(2.14)
and N −1 1 X Xk ei2πkn/N , x(tn ) = N ∆t k=0
where Xk = X(fk ) and ∆f has been replaced with 1/N ∆t in Equation (2.14). In the SigLab network analyzer, the Discrete Fourier Transform (DFT) given by Equation (2.13) is implemented via the Fast Fourier Transform (FFT) algorithm (see Bendat and Piersol, 1986). This algorithm is most efficient if the number of sampling points is a power of 2. Consequently, the analyzer allows a choice of N in the range of 64 to 8192 by powers of 2. For all experiments in this investigation, N = 4096 was chosen for its resolution and speedy data acquisition. Because the finite Fourier transform in (2.13) is evaluated for only a portion 0 ≤ t ≤ T of the actual data signal x(t), one may view X(f k ) as the Fourier transform of an unlimited time history record v(t) multiplied by a rectangular time window u(t), where u(t) =
1
0
0≤t≤T otherwise
(2.15)
32 1 u(t)
0
0
t
T
(a) | U(f) |
−3/T
−2/T
−1/T
0 (b)
1/T
2/T
3/T
f
Figure 2.15: Rectangular window. (a) Time Window. (b) Spectral Window.
so that x(t) = u(t) v(t).
(2.16)
As a result, the Fourier transform of x(t) becomes a convolution of the Fourier transforms of u(t) and v(t); X(f ) =
Z
∞ −∞
U (α) V (f − α) dα.
(2.17)
The Fourier transform of the rectangular time window (Fig. 2.15(a)) given by (2.15) is U (f ) = T
sin(πf T ) πf T
e−iπf T .
(2.18)
The magnitude of this complex-valued function is shown in Figure 2.15(b). The large side lobes of |U (f )| allow leakage of power at frequencies well away from the main lobe of this spectral window and may cause significant distortions of X(f ). If the data v(t) is periodic with period Tp and T is chosen to be an integer multiple of T p , then this leakage problem will not occur. For such a case, the Fourier components at f = kfp = (k/Tp ), k = 1, 2, . . . will not leak into the main lobe as U (f ) is zero at these frequencies. It should be noted, however, that if T 6= kT p then leakage will occur even if the data is periodic.
33 1 u(t)
0
0
t
T
(a) | U(f) |
−3/T
−2/T
−1/T
0 (b)
1/T
2/T
3/T
f
Figure 2.16: Hanning window. (a) Time Window. (b) Spectral Window.
To mitigate the effects of the leakage problem, one may use a time window that tapers the time history of the data in order to eliminate discontinuities at the beginning and end of the data record v(t), 0 ≤ t ≤ T . One of the original and most commonly used windows for this purpose is called the cosine squared or Hanning window, defined by
u(t) =
1 − cos2 0
πt T
0≤t≤T
,
(2.19)
otherwise
which is shown along with the magnitude of its Fourier transform in Figure 2.16. Use of a Hanning window greatly reduces side lobe leakage, but reduces the magnitude of p the Fourier transform by a factor of 3/8. Therefore, it is necessary to multiply the p p right hand sides of (2.13) and (2.14) by 8/3 and 3/8, respectively, when using this window. Due to its generality, the Hanning window was used in this investigation.
As mentioned previously, when using the FFT the Nyquist frequency occurs at k = N/2 (see Equation (2.12)), and consequently the first N/2 + 1 spectral values at k = 0, 1, . . . N/2 define the discrete Fourier transform in the range 0 ≤ f ≤ f c . The last N/2 − 1 values may be interpreted as the discrete Fourier transform for −f c ≤ f ≤ 0.
34 However, the one-sided spectrum from 0 ≤ k ≤ N/2 is usually presented in network analyzers.
2.9.5
Frequency Response Function For a constant parameter linear system characterized by a single input and sin-
gle output, the dynamic characteristics may be described by a unit impulse response function h(τ ), which is defined as the output of the system at any time due to a unit impulse input applied at time τ = 0. The output y(t) of the system for an arbitrary input x(t) is given by the convolution integral y(t) =
Z
∞
−∞
h(τ ) x(t − τ ) dτ.
(2.20)
The corresponding characterization in the frequency domain is given by the Fourier transform of the unit impulse response function, and is referred to as the frequency response function H(f ); H(f ) =
Z
∞
h(τ ) e−i2πf τ dτ,
(2.21)
−∞
which is typically complex valued. For non-linear systems H(f ) also depends on the excitation, and for systems in which the parameters are not constant, H(f ) will be a function of time. Taking the Fourier transform of both sides of (2.20) gives Y (f ) = H(f )X(f ),
(2.22)
and therefore reduces the convolution integral to a simple linear algebraic expression in the frequency domain in terms of the frequency response function and the Fourier transforms of the input and output. In physical tests, the DFT of the input x(t) and output y(t) are computed, in which case the frequency response function of the system can be computed as H(fk ) =
Y (fk ) , X(fk )
k = 0, 1, 2, . . . N − 1
(2.23)
35 for a system with zero measurement noise. However, since measurements of most physical systems contain some degree of measurement noise, a statistical definition of (2.23) is desired. Bendat and Piersol (1986) demonstrate that the FRF of a single-input/singleoutput system with output noise may be calculated such that the effect of noise is minimized in a least-squares sense using H(fk ) =
Gxy (fk ) , Gxx (fk )
(2.24)
in which Gxx is the one-sided auto-spectral density function; Gxx (fk ) =
2 X ∗ (fk )X(fk ), N ∆t
k = 0, 1, 2, . . . N − 1,
(2.25)
and Gxy is the one-sided cross-spectral density function; Gxy (fk ) =
2 X ∗ (fk )Y (fk ), N ∆t
k = 0, 1, 2, . . . N − 1,
(2.26)
where X ∗ (fk ) is the complex conjugate of X(fk ). To further reduce the effect of random error on the measured FRF, an ensemble of input and output records is used to calculate the average auto-spectral and cross spectral density functions, defined as; n
¯ xx (fk ) = G
d X 2 Xi∗ (fk )Xi (fk ), nd N ∆t
i=1
k = 0, 1, 2, . . . N − 1,
(2.27)
k = 0, 1, 2, . . . N − 1,
(2.28)
n
¯ xy (fk ) = G
d X 2 Xi∗ (fk )Yi (fk ), nd N ∆t
i=1
where nd is the number of averages. The improved estimate of the FRF is then ¯ xy (fk ) G H(fk ) = ¯ . Gxx (fk )
(2.29)
It can be shown that when using this averaging technique, the error spectrum is proporp tional to 1/nd . For most tests in this investigation, 30 averages were used to reduce
the effects of random noise in the data.
In the study, both random and chaotically periodic impact loadings were used to excite the foundations. The force measured by the load cell was used as the input f (t)
36 and the directional accelerations measured at n various points on the foundation were used as the outputs x ¨n (t). This single-input multiple-output system was modeled as n separate single-input single-output systems. Recall that the FRF used in this study is the directional accelerance function, defined as the ratio of an acceleration component of a point on the foundation to the to applied force applied at another point of the foundation, in the frequency domain. The n accelerance functions are then given by ¯ f x¨ (fk ) G Ai (fk ) = ¯ i , Gf f (fk )
i = 1, 2, . . . n,
(2.30)
where n is typically 3 or 4. In addition to the FRF, the real coherence function (or coherency squared function) was also calculated from the measured data. This function has the definition 2 γxy (fk )
G ¯ xy (fk ) 2 = ¯ ¯ yy (fk ) , Gxx (fk )G
(2.31)
and is used to measure the linearity and amount of noise in the system. Physically, the coherence function may be thought of as the fraction of output power that is directly correlated to the input power. For a linear single-input/single-output system with constant properties and zero noise, the coherence function will assume a value of unity for all frequencies. A deviation of the coherence function from unity could indicate; (1) A change in the properties of a linear system with time. (2) Non-linearities in the system. (3) Additional random error input to the system, such as electrical noise in the data signals or acceleration of the footing in addition to that caused by the measured force. Typical plots of measured auto-spectral density functions for a horizontally oriented accelerometer and load cell on Footing E at 66g are shown in Fig. 2.17. This plot contains the auto-spectral densities for both an ambient and an immediately preceding
37 forced vibration test. The auto-spectral density of the accelerometer in in Fig. 2.17(a) indicates that a that a fair amount of horizontal vibration is taking place in the ambient test, even with the exciter quiescent and retracted from the footing. Note that the ambient vibration primarily occurs below a frequency of 500Hz. When random excitation is applied to the footing, the auto-spectral density of the accelerometer increases slightly, while that of the load cell increases significantly. The resulting accelerance and coherence functions for the forced vibration test of Fig. 2.17 are shown in Fig. 2.18. The measured acceleration in the forced vibration test will be caused not only by the force applied by the exciter, but also by the sources of the ambient acceleration below 500Hz evidenced in Fig. 2.17(a). As discussed above, this will contribute to the poor coherence below 500Hz shown in Fig. 2.18(c). Although the ambient vibration adversely affects the spectral measurements of forced vibration tests, it will be shown in later sections to be useful for gauging the quality of forced-vibration tests. Typical plots of measured accelerance and coherence functions for eccentric vertical excitation of Footing B at 66g are illustrated in Figs. 2.19 and 2.20. From the plots of coherence, one can see that the system is relatively linear and time invariant with little noise, with the exception of the range 0 ≤ f ≤ 500 Hz, where coherence is degraded due to a low signal-to-noise ratio and ambient accelerations. Furthermore, above 500Hz, the coherence of the vertical mode (Fig. 2.19(c)) is typically better than that of the horizontal mode (Fig. 2.20(c)). A comparison of Figs. 2.18(a) and 2.20(a) illustrates that the horizontal accelerance due to horizontal load is approximately 20 times that due to vertical load. Also, the quality of coherence for horizontal accelerometers is higher for vertical forcing (Fig. 2.20(c)) than for horizontal forcing (Fig. 2.18(c)).
Acceleration ASD [m/s2rms]
38
0.2 Forced vibration, fpk=172.5Hz
0 (a) Force ASD [Nrms]
Ambient vibration, fpk=170Hz
0.1
0
500
1000 f [Hz]
1500
2000
1500
2000
0.15 0.1 Forced vibration, fpk=96.25Hz Ambient vibration
0.05 0 0
500
1000 f [Hz]
(b)
Accelerance [m/s2/N]
Figure 2.17: Auto-spectral density of acceleration (a) and force (b) during ambient and forced horizontal vibration tests of Footing E at 66g. Ambient test ilebbg66.am2, forced vibration test ilebbg66.dat.
Coherence
(b)
Accelerance [m/s2/N]
(a)
(c)
20
fpk=167.5Hz
10 0 0 10
Mag 500
1000
1500
2000
500
1000
1500
2000
500
1000 f [Hz]
1500
2000
Re 0 Im −10 0 1 0.5 0
0
Figure 2.18: Accelerance and coherence functions for forced horizontal vibration test ilebbg66.dat of Footing E at 66g.
39
Accelerance [m/s2/N]
3
1
Mag
Im
0 −1
Re
−2
(b) Coherence
2
0 1
Accelerance [m/s2/N]
(a)
Vertical eccentric force Vertical centric acceleration
1 0.5 0
1000 1500 2000 f [Hz] Figure 2.19: Typical measured vertical accelerance and coherence functions for eccentric vertical loading of footing B at 66g. Test VE-B72. Accelerance [m/s2/N]
(c)
0
500
1 Vertical eccentric force Horizontal acceleration 0.5 Mag
(a) Accelerance [m/s2/N]
0 1
Re
0 Im
−1
Coherence
(b) 1 0.5 0
1000 1500 2000 f [Hz] Figure 2.20: Typical measured horizontal accelerance and coherence functions for eccentric vertical loading of footing B at 66g. Test VE-B72. (c)
0
500
40
2.10
New Experimental Methodologies
2.10.1
Method of Hybrid-Mode Testing
To excite the fundamental vertical, horizontal and rocking modes of the foundation, three basic dynamic configurations were used in the surface foundation study and extended to pile tests. These consist of the vertical-centric (VC), horizontal-centric (HC), and vertical-eccentric (VE) loading configurations shown in Fig. 2.21. For the VC configurations, the dynamic load is applied vertically at the foundation centerline, aiming to invoke only the vertical mode of vibration for a symmetrically instrumented foundation. In the HC test, a horizontal load is applied to the foundation so as to excite the coupled horizontal-rocking mode without participation of the vertical mode. To enable a more efficient characterization of all three modes, the idea of hybrid-mode VE tests in which a vertical force is applied to the foundation top at a horizontallyeccentric location was developed in Ashlock (2000) and successfully employed in Sloan VC VE VC VE VC VE
HE
HC
HC HE
HE HC
Piles Surface footings
Embedded footings
Figure 2.21: Notation for measurement locations for surface, embedded and pile foundation tests.
41 (2000) and Kurahashi (2002). The VE load is equivalent to the combination of a vertical force and moment at the foundation centroid, and therefore invokes both the vertical and coupled horizontal-rocking modes simultaneously. This allows characterization of vertical, horizontal and rocking modes in a single test, as opposed to separate vertical and lateral tests as commonly used in the past. In any new testing configuration, it is also prudent to verify the consistency of measured responses from separate VC, HC and VE tests in order to assure that sufficient care has been taken to account for any physical factors which may affect the experimental outcome. Examples of such factors are the presence of friction (Ashlock, 2000) or accurately accounting for the inertial effect of the instrumentation (see Sec. 2.11). For both surface and pile foundations, it can be shown from a comparison of measurements from VC, VE and HC excitation tests that the fundamental characteristics of the dynamic response of the vertical and coupled horizontal-rocking modes of vibration may be obtained from a single carefully executed VE test. Considering that most real-life foundations are subject to multi-directional loadings and their response is multi-modal in nature, the proposed VE testing approach offers a practical, efficient and relevant alternative to single mode tests. To capture the motion of the foundation in a vertical plane, the applied force and resulting acceleration were measured in at least 3 non-collinear VC, VE or HC directions. Because of its effectiveness in fully characterizing the time-domain behavior of a linearized dynamic system, the transfer function of the acceleration relative to the dynamic force stimulus, defined as the accelerance function in Section 2.9.4, is the primary measurement of interest. For conciseness, the points of acceleration measurement will be denoted in the same manner as force measurements, so that accelerance functions may henceforth be referred to as VC/VE, HC/VE and VE/VE, for the example of a test with VE loading. In this notation, the prefix denotes the acceleration measurement location and direction, and the suffix denotes the load location and direction.
42 2.10.2
Ambient Vibration Tests
To obtain a measure of the lateral-rocking peak frequency, ambient vibration measurements without exciter-foundation contact were made prior to execution of forcedvibration tests. In the ambient test, a minimal dynamic excitation is experienced as a result of mechanical and air-flow induced vibrations of the centrifuge arm, soil model and container during spinning. Although accelerance functions are not meaningful in this condition due to the supposedly zero force measurement, the auto-spectral density of the HC and VE accelerometers can carry useful information, such as the frequency of the fundamental lateral-rocking peak under minimal excitation conditions. This resonance peak can be compared to the lateral-rocking peak in forced-vibration tests for collaboration and detection of the effects of any unwanted friction, and to track any changes in the site conditions and the overall vibration level. For example, in the footing data of Fig. 2.17(a), the peak frequency of the HC accelerometer’s auto-spectral density (ASD) under ambient vibration is 170Hz. In the subsequent forced-vibration test, the peak frequencies of the HC acceleration and HC force ASD are quite different (172.5Hz and 96.25Hz in Fig. 2.17). However, their HC/HC accelerance ratio in the properly executed forced vibration test gives a peak frequency of 167.5Hz (Fig. 2.18(a)), very close to the accelerometer’s ambient-test ASD peak frequency. The same technique is useful for pile tests under favorable conditions. For example, a peak ASD frequency of 184.375Hz was obtained for an HC accelerometer on Pile B at 33g, as shown in Fig. 2.22(a). On the preceding spin of the centrifuge, the same frequency was obtained for the HC/VE accelerance peak in a forced-vibration test (Fig. 2.22(b)).
2.10.3
Friction Reducing Dumpling
As discussed in Ashlock (2000) and Kurahashi (2002), un-lubricated contact of the exciter with the load stinger introduces a transverse frictional resistance or excitation
acceleration ASD [m/ssrms]2
43
0.03 0.02 0.01 0
Accelerance [m/s2/N]
(a)
0
20
500
1000
1500
2000
HC/VE Accelerance from forced−vibration test fpk=184.375Hz
10
0 0 (b)
HC Auto−spectral density from ambient test fpk=184.375Hz
500
1000 f [Hz]
1500
2000
Figure 2.22: 33g tests of Pile B. (a): HC accelerometer ASD peak frequency from ambient-vibration test p3032617.vna, (b): HC/VE accelerance peak from forcedvibration test p3032607.vna.
in tests involving horizontal or rocking motion. This frictional force can increase the natural frequency of the measured fundamental HC/VE accelerance peak. As shown in Ashlock (2000), an effective approach to eliminate this friction in such tests was through the use of an oil-filled dumpling or pillow. Constructed of two rectangular pieces of latex membrane glued together at the edges and injected with lubricating oil using a syringe, its placement on top of the load cell button in VE tests (see Fig. 2.23) can greatly decrease of the fundamental HC/VE accelerance peak frequencies to values close to those of the ambient tests. One minor drawback of the dumpling approach, however, is that the compliance of the latex membrane affects the load transfer at the high end of the frequency spectrum, resulting in a decrease of the measured coherence and limiting its usefulness in tests with HC load, which require low contact forces. Care
44
Figure 2.23: Oil filled friction reducing dumpling resting on load cell button in VE test configuration.
must also be exercised to ensure that the dumpling does not fall off the load cell.
2.10.4
Method of FRF Measurement by Chaotic Impact Loading
For collaboration of data as well as convenience, an alternative method of minimizing frictional effects in forced vibration tests by means of “chaotic impact loading” was also investigated. In this technique, the dumpling is not used, but a small amount of Tri-Flow or equivalent Teflon based lubricant is applied to the rounded surface of the button. A DC current is then applied to the exciter in addition to a small random AC signal to locate a position where the exciter does not have continuous contact with the button. Under such conditions, a randomly-timed series of impacts with the model foundation can be observed at a desirable level (e.g. see Fig. 2.24). By viewing the time histories of force and acceleration, the user can adjust the DC offset until the delays be-
45 tween impacts are timed such that the subsequent inertial free-vibration responses do not die out completely. During the intermittent periods of free-vibration response, the mass of the instrumentation above the load cell (see Fig. 2.27) may create an inertial loading which ensures a non-zero force measurement and meaningful accelerance measurements. The resulting accelerance and coherence measurements are slightly better than those obtained with a dumpling and random-loading, with clean fundamental peaks for the lateral-rocking modes (e.g. see Fig. 2.24(a)), which are also in agreement with those of ambient tests. Owing to the smaller foundation mass and larger horizontal motion for pile tests compared to typical footing tests, it was found that the friction problem was more pronounced in pile tests. This was evidenced by the fact that random-loading pile tests with dumplings required that the exciter contact force be reduced by applying a negative DC current in order to remove a spurious third peak from the HC/VE accelerance and to reduce the frequency of the fundamental peak to agree with that obtained in an ambient test. An comparison of such a reduced-force dumpling test against a chaotic impact test is shown in Fig. 2.24. Compared to continuous-contact random loading, the chaotic impact loading method produces higher intensities of acceleration and force over shorter time intervals, as shown in the time-histories of acceleration and force in Fig. 2.24(b) and (c).
46
Mag(Accelerance) [m/s2/N]
15
chaotic impact
VC/VE accelerance
random load with dumpling
10
5 HC/VE accelerance 0 0
VC Acceleration [m/s2]
(a)
40
1000
2000 3000 Frequency [Hz]
4000
5000
chaotic impact
20 0 −20 0
(b)
random load with dumpling 0.05
0.1
0.15
0.1
0.15
time [s] 8
chaotic impact
Force [N]
6 4 2 0 random load with dumpling (c)
−2 0
0.05 time [s]
Figure 2.24: VE tests on Pile C at 55g using random load with dumpling versus chaotic impact loading. (a): VC/VE and HC/VE accelerance magnitudes. (b): time-histories of VC acceleration. (c): time-histories of VE load. Random excitation test p5062002.vna, impact test p5070905.vna.
47 2.10.5
Method of Free-Field Modulus Determination
As will be discussed later, a key prerequisite to a comprehensive analysis of foundations on sand is the knowledge of the value of the shear modulus as a result of the gravity-induced and foundation contact pressure-induced effects on the in-situ stress field. A novel method of determining the free-field modulus variation due to the soil’s self-weight while minimizing the effect of the foundation contact pressure was described in Pak et al. (2003b) and Kurahashi (2002) for centrifuge testing. The approach involves testing of a light-weight foundation embedded to a depth such that the change to the vertical free-field stress profile under the foundation is minimal. The parametric identification approach to determine the in-situ modulus profiles will be further discussed in Parts II and III of this thesis.
2.10.6
Modeling of Instrumentation Effects
As the size of the scaled-model foundation decreases, the fixed size of the instrumentation detailed in previous sections can have an increasingly significant effect on the overall inertial properties of the foundation-instrumentation system. To achieve a level of precision appropriate to the measurements’ resolution, extra consideration is given to ensure a proper account of the possible inertial effect of the instrumentation on the foundation accelerance functions. As will be shown in detail later, such an effect is quite significant for the accelerance functions of the relatively small pile caps, even though it is negligible for heavier footings, such as footing B.
2.11
Theoretical Accelerance Calculation at an Arbitrary Point on a General Superstructure To validate and calibrate various continuum models against the experimentally
measured dynamic responses of scaled-model foundations, one might attempt to use the experimental measurements of force and acceleration to calculate a set of “experimental”
48 or “measured” impedances at the level of the soil surface. As shown in Guzina and Pak (1998) and Ashlock (2000), however, such calculations are numerically ill-conditioned, as small levels of experimental noise in the force and acceleration measurements will result in large errors in the calculated impedances. To circumvent the difficulty, a system identification approach via accelerance matching is adopted in this study. It entails calculating theoretical accelerances from a chosen impedance theory and matching them with the experimental accelerances through a variation of the impedance theory’s parameters. To calculate such theoretical accelerances, an efficient approach is to use the method of sub-structuring. In this approach, the super-structure above ground can be treated separately, while the below-ground system such as the soil can be dealt with by means of the boundary element method. The superstructure system can be further sub-structured into individual systems whose treatment can be separately chosen for efficiency and accuracy. The application of such an approach in this investigation is illustrated in Fig. 2.25 for the treatment of the pile/soil system, with the footing problem as a degenerate case. For the proposed substructure analysis of the cap-pile-soil interaction problem shown, it is important that the sign conventions and notations for all interactive quantities are defined explicitly, as shown in Fig. 2.25. Because only planar motion is considered for the superstructure, a 2-D Cartesian (x, y) coordinate system is sufficient. For consistency with the experimental measurements in which the load cells generate a positive signal for compressive loads, the external forces qV and qH are taken to be directed towards the foundation as shown in the schematic. In the following, capital letters denote the Fourier transform of the corresponding time-dependent quantities. For the displacement u x (x, y, t), e.g. Ux (x, y, ω) = F[ux (x, y, t)] =
Z
∞
−∞
ux (x, y, t)eiωt dt,
(2.32)
49
Superstructure substructure 1: rigid pile cap qH (t)
qV (t) V
H T
substructure 2: deformable pile-stem
Point of interest P
C
fy (T, t)
m(T, t)
fy + dfy
fx (T, t)
m + dm fx + dfx fx m
T l0
dy
B
fy
fy (B, t) m(B, t)
fx (B, t) B y, uy (t)
Soil-foundation substructure
θ(t)
x, ux(t)
Origin at point B
Figure 2.25: A substructure formulation of the cap-pile-soil interaction problem. P: general point of interest. C: centroid of rigid sub-structure. V/H: points of application of vertical/horizontal load. T/B: top/bottom points of deformable pile-stem.
50 and the dependence on frequency ω will henceforth be suppressed in the notation for brevity. Accordingly, one may write the displacement and force vectors at any point P = (xP , yP ) as Uy (xP , yP ) U(P ) = Ux (xP , yP ) Θ(xP , yP )
Fy (xP , yP ) F(P ) = Fx (xP , yP ) M (xP , yP )
,
.
(2.33)
The distance from the centroid located at point C = (x C , yC ) to any other point P = (xP , yP ) on the pile-cap is defined by eP = x P − x C ,
hP = yP − yC .
(2.34)
Using the foregoing definitions, a general rigid-body kinematic transformation matrix TP C which relates the centroidal displacements U(C) at point C to the displacements U(P ) at any point P on the rigid pile-cap for small rotations may be defined by Uy (P ) = Uy (C) + eP Θ(C) or
Ux (P ) = Ux (C) − hP Θ(C)
U(P ) = TP C U(C),
(2.35)
Θ(P ) = Θ(C) where
TP C
1 0 eP = 0 1 −hP 0 0 1
.
(2.36)
Note that the inverse of this kinematic transformation may be easily obtained by inspection after rearranging Eqns. (2.35) to the form U(C) = TCP U(P ),
(2.37)
giving
TCP ≡
T−1 PC
1 0 −eP = 0 1 hP 0 0 1
.
(2.38)
51 The frequency domain rigid-body equations of motion for the pile cap having a mass m and centroidal polar moment of inertia J, and subjected to vertical force q V (t) applied at point V and horizontal load 0 0 −QV 1 − 0 1 0 QH −QV eV − QH hH eT −hT 1
q H (t) applied at point H are Fy (T ) m 0 0 Fx (T ) = 0 m 0 M (T ) 0 0 J
F[¨ uy (C)] , F[¨ u (C)] x ¨ F[θ(C)] (2.39)
where point T = (xT , yT ) denotes the point at the connection of the cap and pile, and point C = (xC , yC ) denotes the centroid of the cap and instrumentation (see Fig. 2.25). Defining the pile cap mass matrix as
the forcing vector Q as
m 0 0 M = 0 m 0 , 0 0 J
Q=
−QV
QH −QV eV − QH hH
and the Fourier transforms of acceleration as 2 F[¨ uy (C)] −ω Uy (C) F[¨ ux (C)] = −ω 2 Ux (C) ¨ F[θ(C)] −ω 2 Θ(C) Equation (2.39) may then be written compactly as
(2.40)
,
(2.41)
= −ω 2 U(C),
Q − TTT C F(T ) = −ω 2 MU(C).
(2.42)
(2.43)
Using the kinematic transformations defined for an arbitrary point P in (2.35) and (2.37), Eqn. (2.43) may be written in terms of the forces and displacements at point P = T as F(T ) = TTCT Q + ω 2 TTCT MTCT U(T ).
(2.44)
52 For the above-ground segment of the pile subjected only to end loadings, it is practically sufficient to model it as a beam-column. For axial deformation, the equation of motion is 1 ∂ 2 uy (y, t) ∂ 2 uy (y, t) − = 0, ∂y 2 Cp2 ∂t2 where Cp =
(2.45)
p Ep /ρp is the 1-D compressional wave speed of the pile which has a mass
density ρp , cross-sectional area Ap , and Young’s modulus Ep . The axial force at any point in the pile segment is fy (y, t) = Ep Ap
∂uy (y, t) . ∂y
(2.46)
In the frequency domain, Eqns. (2.45) and (2.46) translate to the ordinary differential equations d2 Uy (y, ω) + dy 2
ω Cp
2
Uy (y, ω) = 0,
(2.47)
and Fy (y, ω) = Ep Ap
dUy (y, ω) . dy
(2.48)
The general solution to Eqn. (2.47) is Uy (y, ω) = C1 sin(αy) + C2 cos(αy),
(2.49)
and the corresponding axial force is Fy (y, ω) = Ep Ap α [C1 cos(αy) − C2 sin(αy)] ,
(2.50)
where α ≡ ω/Cp . For the flexural motion of the un-embedded portion of the pile, with the sign conventions given in Fig. 2.25, the cross-sectional rotation, curvature, moment and
53 shear force, respectively can be related to the kinematics by ∂ux (y, t) , ∂y ∂θ(y, t) ∂ 2 ux (y, t) κ(y, t) = =− , ∂y ∂y 2 ∂ 2 ux (y, t) , m(y, t) = Ep Ip κ(y, t) = −Ep Ip ∂y 2 ∂m(y, t) ∂ 3 ux (y, t) fx (y, t) = = −Ep Ip . ∂y ∂y 3 θ(y, t) = −
(2.51)
Accordingly, the equation of motion is ∂ 4 ux (y, t) ρp Ap ∂ 2 ux (y, t) + = 0. ∂y 4 Ep Ip ∂t2
(2.52)
Taking the Fourier transforms of Eqns. (2.51) and (2.52) gives dUx (y, ω) , dy d2 Ux (y, ω) dΘ(y, ω) , =− K(y, ω) = dy dy 2 d2 Ux (y, ω) M (y, ω) = Ep Ip K(y, ω) = −Ep Ip , dy 2 d3 Ux (y, ω) dM (y, ω) = −Ep Ip , Fx (y, ω) = dy dy 3 Θ(y, ω) = −
(2.53)
and the Fourier transformed governing differential equation is d4 Ux (y, ω) − β 4 Ux (y, ω) = 0, dy 4
(2.54)
where β 4 ≡ ω2
ρp Ap . Ep Ip
(2.55)
The general solution to (2.54) may be written Ux (y, ω) = C3 sin(βy) + C4 cos(βy) + C5 e−βy + C6 eβ(y−l0 ) ,
(2.56)
where l0 is the length of the beam section between the soil surface and pile cap. The
54 corresponding cross-sectional rotation, moment and shear force are Θ(y, ω) = −βC3 cos(βy) + βC4 sin(βy) + βC5 e−βy − βC6 eβ(y−l0 ) M (y, ω) = Ep Ip β 2 C3 sin(βy) + Ep Ip β 2 C4 cos(βy) − Ep Ip β 2 C5 e−βy − Ep Ip β 2 C6 eβ(y−l0 ) Fx (y, ω) = Ep Ip β 3 C3 cos(βy) − Ep Ip β 3 C4 sin(βy) + Ep Ip β 3 C5 e−βy − Ep Ip β 3 C6 eβ(y−l0 ) . (2.57) The beam-column Equations (2.45) through (2.57) are valid for points P = (xP , yP ) having coordinates (xP = 0, yP ∈ [0, l0 ]). At any such cross section defined by the yP -coordinate, the Fourier-transformed beam-column structural element Equations (2.49), (2.50), (2.56) and (2.57) may be written in matrix form as U(P ) S(P )C = , F(P )
in which the 6 × 6 structural beam-column coefficient matrix S(P ) is cos(αyP ) 0 sin(αyP ) 0 0 sin(βyP ) 0 0 −βcos(βyP ) S(P ) = Ep Ap αcos(αyP ) −Ep Ap αsin(αyP ) 0 0 0 Ep Ip β 3 cos(βyP ) 0 0 Ep Ip β 2 sin(βyP ) 0
0
cos(βyP )
e−βyP
βsin(βyP )
βe−βyP
0
0
−Ep Ip β 3 sin(βyP )
Ep Ip β 3 e−βyP
Ep Ip β 2 cos(βyP )
−Ep Ip β 2 e−βyP
0
(2.58)
β(y −l ) 0 P e −βeβ(yP −l0 ) , 0 3 β(y −l ) 0 P −Ep Ip β e −Ep Ip β 2 eβ(yP −l0 )
(2.59)
55 and the 6 × 1 unknown-constant and right-hand-side vectors are C1 Uy (yP , ω) C U (y 2 x P , ω) C3 U(P ) Θ(yP , ω) C= , and = . C4 F(P ) Fy (yP , ω) C F (y 5 x P , ω) C6 M (yP , ω)
(2.60)
The constants in C may now be eliminated from the accelerance calculations by en-
forcing (2.58) at the end points of the beam-column, i.e. at top point T = (0, l 0 ) and bottom point B = (0, 0), giving U(T ) −1 S (T ) F(T )
= S−1 (B)
U(B)
.
(2.61)
F(B)
Using the soil impedance relations F(B) = K(ω)U(B) at bottom point B, and the
pile-cap equations of motion referred to top point T (i.e. Eqn. 2.44) in (2.61) gives U(B) U(T ) −1 . (2.62) = S(T )S (B) KU(B) TT Q + ω 2 TT MTCT U(T ) CT CT This equation may be rearranged as I3×3 I3×3 03×1 −1 U(B), = S(T )S (B) U(T ) + K TTCT Q ω 2 TTCT MTCT
(2.63)
Eqn. (2.63) can be rearranged I3×3 S(T )S−1 (B) K
(2.64)
where I3×3 is the 3 × 3 identity matrix and 03×1 is a 3 × 1 vector of zeros. Finally, to
03×1 U(B) −I3×3 . = T 2 T TCT Q U(T ) −ω TCT MTCT
The 6 × 6 system of Equations (2.64) may be programmed and solved for U(B)
and U(T ) under an arbitrary vertical load Q V (with QH = 0), horizontal load QH
56 (with QV = 0), or any combination thereof. To facilitate the numerical solution of this equation in the case of a single applied force, both vectors in (2.64) can be divided by the non-zero forcing term QV or QH , resulting in the following numerical forms of the forcing vector: −1 Q = 0 QV −eV
0 Q for vertical load, and = 1 QH −hH
for horizontal load
(2.65)
for which the corresponding solution vectors will be U(B) U(B) QV QH for vertical load, and for horizontal load. U(T ) U(T ) QV QH
(2.66)
At any point P = (xP , yP ) of interest on the pile cap, the accelerance vector, defined as the ratio of acceleration to force in the frequency domain, may then be obtained by extracting the Fourier-transformed displacement vector at point T from the above solution vector, i.e. A(P ) ≡
−ω 2 U(P ) U(T ) = −ω 2 TP T , Q Q
(2.67)
where Q is taken as QV or QH and
TP T
1 0 (xP − xT ) = 0 1 −(yP − yT ) 0 0 1
(2.68)
is the kinematic transformation matrix from point T to point P . While the 6×6 system of equations in (2.64) is efficient for programming purposes, it is also useful to formulate the equations without performing any matrix inversions or eliminating the vector of unknowns C in the process. Doing so results in the 18 × 18
57 system of equations
−I6×6 S(T ) 06×6 S(B) 2 T −ω TCT MTCT 06×6 03×3
C 06×1 06×6 U(T ) 06×1 −I6×6 F(T ) = , (2.69) T TCT Q I3×3 03×3 03×3 U(B) 0 −K I3×3 03×3 3×1 F(B)
which, although less succinct than Eqn. (2.64), may also be programmed for calculation of the theoretical accelerances and all related quantities.
2.12
Theoretical Accelerance Calculation for Footing To obtain the theoretical accelerance functions for a rigid surface footing resting
directly on the soil, one may consider the free pile-stem in the formulation of the previous section to have zero length by taking point T to coincide with point B. The foundation impedance relation F(T ) = K(ω)U(T ) can then be used in the rigid-body equations of motion (2.44), giving KU(T ) = TTCT Q + ω 2 TTCT MTCT U(T ).
(2.70)
By an inversion of (2.70), the interfacial displacement vector can be found as −1 T TCT Q, U(T ) = K − ω 2 TTCT MTCT
(2.71)
which may be used in (2.67) to obtain the accelerance at any point P . This degenerate formulation is similar to the direct one given in Ashlock (2000).
2.13
Instrumentation Size Effects: Accelerance Correction for Load Cell Assembly As the size or mass of the scale model footing or pile cap decreases, the inertial
properties of the instrumentation can comprise an increasingly significant portion of the
58 total mass and moment of intertia of the foundation-instrumentation assembly. For pile cap A instrumented as shown in Fig 2.10, for example, the mass of the instrumentations constitutes about 11% of the combined mass of the cap and attachments. As shown in Figs. 2.26 and 2.27, the upper half of the load cell, the load distribution cap, button, one-half of the sleeve and approximately one-fourth of the screw all represent masses which are located between the external force applied to the button by the exciter and the measured force at the mid-section of the load cell. Owing to the inertia of these masses, the difference between the externally applied and internally measured forces is proportional to frequency, and may therefore have a more pronounced effect on the accelerance in the higher frequency range for a given foundation. For research rigor, it is prudent to incorporate such details in the analytical synthesis of the experiments. To account for the fact that the experimentally measured force corresponds to that at the mid-height of the load cell, one should calculate the theoretical accelerance relative to this force internal to the load cell. For this purpose, one may draw a freebody diagram of the components above the center of the load cell, accounting for the resultant axial force, shear force and moment of the internal stresses. The accelerance would then be calculated using these three internal stress resultants in the forcing vector Q, as opposed to the single external force Q V or QH , which are not directly known in reality. To be consistent, it is also necessary that the foundation-attachment inertial properties be reduced to account for the separation of the upper and lower halves of the load cell assembly. To provide three additional equations for the three unknown internal resultants introduced in this approach, the equations of motion of the upper portion of the load cell assembly can be written in terms of its mass and mass moment of inertia. Analytically, however, a simpler alternative is to use the solution to Equation (2.67), which gives the theoretical accelerance in terms of the external forces Q V or QH , with the entire load cell assembly considered to be rigidly attached to the foundation. Then
59
Load cell assembly
Pile cap Pile
Accelerometer Figure 2.26: Monolithic scale model pile and cap A with attached accelerometer and load cell assembly.
Load cell cable
Screw Sleeve
Load distribution cap Button
Load cell: Measurement plane Figure 2.27: Disassembled components of load cell assembly.
60 only the equation of motion in the direction of the applied force need be considered for the free body diagram of the upper portion of the load cell assembly (Fig. 2.28). Calculations of the shear and moment resultants of the internal stresses at the mid-plane of the load cell are then not required, and one needs only to estimate the mass of the upper portion, which may be calculated with more certainty than its moment of inertia. Denoting the force measured internal to the load cell as either Q V i or QHi respectively, the frequency-domain translational equation of motion for the upper portion of the load cell assembly is −QV + QV i = −ω 2 muLC Uy (V )
(2.72)
QH − QHi = −ω 2 muLC Ux (H)
(2.73)
for vertical load, and
for horizontal load, where mLC is the total mass of the load cell assembly, and m uLC is the mass of the upper portion above the center of the load cell. As defined previously, points V = (xV , yV ) and H = (xH , yH ) are the points of application of the vertical or horizontal loads, respectively. For the case of vertical load, the accelerance A i (P ) in terms of the internally measured force is obtained from the existing numerical solution (2.67) and Equation (2.72) as
QV Uy (V ) −1 = A(P ) 1 − ω 2 muLC QV i QV Uy (T ) Θ(T ) −1 = A(P ) 1 − ω 2 muLC , − ω 2 muLC (xV − xT ) QV QV
Ai (P ) ≡ A(P )
where
Uy (T ) QV
and
Θ(T ) QV
(2.74)
are extracted from the solution vector U(T ) in Equation (2.66),
and use is made of the small-rotation kinematic relation Uy (V ) = Uy (T ) + (xV − xT )Θ(T ).
(2.75)
Similarly, the accelerance in terms of the force internal to the load cell for the case of
“Upper” portion, mass = muLC
“Lower” portion
61
Point H
QH , Externally applied force
QHi , Force internal to load cell
Pile cap or foundation
Figure 2.28: Free body diagram of upper and lower portions of load cell assembly obtained by making a cut at mid-height of load cell. Shear force and moment resultants of internal stresses are omitted as they are not required.
horizontal load is −1 QH 2 u Ux (H) = A(P ) 1 + ω mLC Ai (P ) ≡ A(P ) QHi QH Θ(T ) −1 2 u Ux (T ) 2 u = A(P ) 1 + ω mLC − ω mLC (yH − yT ) , QH QH where
Ux (T ) QH
and
Θ(T ) QH
(2.76)
are extracted from the solution vector U(T ) in Eqn. (2.66), and
use is made of Ux (H) = Ux (T ) − (yH − yT )Θ(T ).
(2.77)
As shown in Figs. 2.29 and 2.30, there is a clear difference in the theoretical accelerance at 33g for Pile A embedded in a square-root half-space when the correction for the load cell assembly is neglected as in (2.67), or incorporated as in (2.74) and (2.76). A similar effect is shown for the smallest Footing I (B0) in Figs. 2.31 and 2.32. In contrast, because of the larger mass and moment of inertia of Footing E, the difference in accelerance is relatively small whether or not the load-cell effect is considered, as shown in Figs. 2.33 and 2.34. The above effects of neglecting the inertial distribution of the load cell assembly should warrant serious attention by other researchers.
62
Mag
10
With correction for load cell assy.
Without correction for load cell assy.
Re
0 10 0
Im
−10
0 −10 −20 0
1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 2.29: Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of pile model A in square-root half-space at 33g.
Mag
500
1000
20 10
Re
0 20
0
Without correction for load cell assy. With correction for load cell assy.
Im
−20 20
0
−20 0
500
1000
1500 2000 2500 3000 3500 4000 Frequency [Hz] Figure 2.30: Effect of accounting for inertial properties of load cell assembly components on HC/VE accelerance [m/s2 /N] of pile model A in square-root half-space at 33g.
63
Mag
20
10
0 10 With correction for load cell assy.
Re
0 −10
Without correction for load cell assy.
Im
0 −10 −20 0
500
1000
1500 2000 2500 3000 Frequency [Hz] Figure 2.31: Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of Footing I (B0) on square-root half-space at 33g.
Mag
20 10 0 Without correction for load cell assy.
Re
10 0
With correction for load cell assy.
−10 −20
Im
20 10 0 −10 0
500
1000
1500 2000 2500 3000 Frequency [Hz] Figure 2.32: Effect of accounting for inertial properties of load cell assembly components on VE/VE accelerance [m/s2 /N] of Footing I (B0) on square-root half-space at 33g.
64
Mag
2
1 Correction for load cell assembly:
included neglected
Re
0 2
0
Im
−2 2
0
−2 0
500
1000 1500 2000 Frequency [Hz] Figure 2.33: Effect of accounting for inertial properties of load cell assembly components on VC/VE accelerance [m/s2 /N] of Footing E on square-root half-space at 33g. 1 Mag
Correction for load cell assembly:
included neglected
0.5
Re
0 1
0
Im
−1 1
0
−1 0
500
1000 1500 2000 Frequency [Hz] Figure 2.34: Effect of accounting for inertial properties of load cell assembly components on HC/VE accelerance [m/s2 /N] of Footing E on square-root half-space at 33g.
65
Figure 2.35: Cross-section of idealized hollow cylinder used to calculate inertial properties of accelerometers.
2.14
Calculation of Inertial Properties of Model Foundations For the solution of Equations (2.67), the centroid location (x C , yC ), mass m and
moment of inertia J of the pile cap or foundation and its instrumentation are required. For higher precision in modeling, the instrumentation holes are considered and idealized as cylindrical bore 3mm in diameter, ranging from 6.5mm for Footing I (B0) to 10mm deep for the pile cap. The accelerometers are each modeled as a hollow cylinder having a mass of 2.5gr, outer length of 11.4mm, outer diameter of 7.1mm, and a wall thickness of 1mm (Fig. 2.35), resulting in a centroid that is 5.7mm from the surface of the foundation and a polar moment of inertia of 4.419·10 −8 kg m2 . Table 2.14 lists the measured masses of the components of the load cell assembly, which is idealized as a cylinder having mass mLC = 12.06gr, length L = 16mm and radius a = 5mm, giving a centroid located 8mm away from the surface of the foundation. The corresponding polar centroidal moment of inertia of the load cell assembly is J0 =
mLC (3a2 + L2 ) = 3.327 · 10−7 kg m2 , 12
(2.78)
which is used in the parallel axis theorem along with the properties of each accelerometer and any remaining load cells to calculate the overall centroid location, mass and moment of inertia of the entire foundation-instrumentation assembly. The above mass (m LC ) and moment of inertia (J0 ) of the entire load cell assembly is used in Equations (2.40) and (2.67) to calculate the theoretical accelerance relative to the externally applied force. Then the theoretical accelerance relative to the internal force measured at the load cell
Table 2.15: Mass of components of load cell assembly. object button load dist. cap load cell & sleeve screw cable total
66
mass [gr] 2.68 4.70 3.55 0.93 0.20 12.06
center is calculated using Equation (2.74) or (2.76), where the components above the mid-height of the load cell include the button, load distribution cap, one half of the load cell and sleeve, one fourth of the screw, and one half of the cable, constituting a mass of muLC = 9.4875 gr.
Chapter 3
Computational Modeling
As mentioned in Chapter 1, one of the main difficulties in analyzing dynamic soil-structure interaction problems is the choice of an appropriate representation for the soil domain. Numerical methods of solving the governing partial differential equations include finite difference, finite element and boundary element methods, all of which can be considered as some form of the method of weighted residuals (Brebbia et al., 1984). For seismic or wave propagation problems, the model of a semi-infinite halfspace is generally taken to be the most reasonable representation of the soil domain, as it allows the infinite propagation of elastic wave energy without the unintended reflection or refraction that would be encountered at boundaries of a truncated finite domain. In the computational phase of this study, the soil domain will be treated as a semi-infinite half-space. While methods of simulating infinite boundaries have been developed for use in the finite element method, they are ultimately approximate in nature. The boundary element method (BEM), however, affords an exact rigorous treatment of infinite boundaries through the use of fundamental solutions which satisfy the radiation conditions. Because of its inherent ability to account for unbounded domains rigorously, the boundary element method will be used in this investigation to analyze the dynamic response of the semi-infinite half-space. For general, large dynamic loading, there exist complex material models in continuum mechanics that can incorporate non-linear effects such as friction, nonlinear
68 elasticity, plasticity and periodic contact phenomena. For low magnitude forced accelerations such as those pursued in this study, a linear visco-elastic material model for the soil should be sufficient. Consistent with the experimental measurement approach as well, an advanced frequency-domain boundary element method formulation is therefore employed for the analytical phase of the investigation. The fundamental solutions available in the Green’s function library of the BEM program include those for an isotropic homogeneous linear elastic full-space and half-space, as well as a piecewise isotropic homogeneous multi-layered elastic and visco-elastic medium. Employed suitably, the latter can also be used to model discrete heterogeneous profiles having continuous variation of material parameters in the vertical direction (Pak and Guzina, 2002). To provide a brief summary of the analytical framework for later reference, the regularized format of the conventional direct boundary integral equation for threedimensional elastostatics (Rizzo and Shippy, 1977; Pak and Guzina, 1999) will be first presented, followed by a discussion of some important numerical issues relevant to general boundary element methods.
3.1
Regularized Direct Boundary Integral Equation Method Within a Cartesian coordinate frame ξ = (0, ξ 1 , ξ2 , ξ3 ), consider a three-dimensional
elastic body occupying a regular region Ω with a closed boundary Γ having the unit outward normal vector n (Fig 3.1). The closed boundary Γ is the union of Γ u over which displacement boundary conditions are specified and tractions are unknown, and Γ t , on which traction boundary conditions are specified and displacements unknown. Fouriertransforms of the components of boundary displacement and traction are denoted U i (ξ) and Ti (ξ; n), respectively, for ξ ∈ Γ. The fundamental integral statement from which the BIE is derived is an integral form of a reciprocity relation in terms of the boundary tractions and internal displacement and body-force fields of two separate permissible elastic states of the body. One of the elastic states is considered to correspond to a
69 known fundamental solution (or set of Green’s functions) for a three-dimensional unit point load acting at a source point x ∈ Ω. Upon careful consideration of the limiting form of the fundamental integral statement as the source point approaches the boundary, i.e. x → y ∈ Γ, a regularized form of the boundary integral equation (BIE) for zero body-force fields can be derived as Z Z k ˆ [Ui (ξ) − Ui (y)] [Tˆik (ξ, y; n)]1 dΓξ Ti (ξ; n) Ui (ξ, y) dΓξ − Γ
Γ
−
Z
Γ
Ui (ξ) [Tˆik (ξ, y; n)]2 dΓξ =
0, Internal domain
1, External domain
(3.1) Uk (y),
y∈Γ
ˆ k (ξ, y) is the displacement for k = 1, 2, 3 (Pak and Guzina, 1999). In this equation, U i fundamental solution for a unit point load acting in the k th -direction and applied at point y ∈ Γ. The functions [Tˆik (ξ, y; n)]1 and [Tˆik (ξ, y; n)]2 represent the singular and regular components, respectively, of a decomposition of the traction fundamental solution Tˆik (ξ, y; n) of the form Tˆik (ξ, y; n) = [Tˆik (ξ, y; n)]1 + [Tˆik (ξ, y; n)]2 (see Pak and Guzina, 1999). For an internal domain Ω i , Γ is the closed boundary of the finite region having an outward unit normal vector as depicted in Fig. 3.1. For an external domain Ωe , Γ is the closed internal boundary surrounded by an infinite domain having a unit normal as depicted in Fig. 3.2.
3.1.1
BIE Discretization and Interpolation As only boundary values of the unknown displacement and traction fields are
involved in the integrals in Eqn. (3.1), their numerical solution in the BEM is facilitated by a spatial discretization of the boundary into a mesh of elements over which geometry, displacement and traction are interpolated based on nodal values via sets of interpolation functions. In Pak and Guzina (1999), the displacement and traction fields so approximated are reduced to a finite number of nodal unknowns which are solved for by applying the method of collocation, in which Eqn. (3.1) is enforced at N unique
70
Γt
n
Γu
Ωi ξ2
x y ξ1
ξ3 Figure 3.1: Three dimensional solid with finite interior region Ω i internal to Γ = Γu ∪Γt .
source locations y taken to be the nodal coordinates y (n) , n = 1, . . . N of the boundary mesh. Collocating for each of the source directions (k = 1, 2, 3) at each of these N nodes gives 3N integral equations, each containing the 3N nodal displacements and 3N nodal tractions. Factoring the constant nodal values outside the integrals then allows the formulation of a linear system of equations HU = GT, where U and T are the global vectors of nodal displacements and tractions, respectively. The full matrices H and G are matrices of influence coefficients. The matrix H consists of integrals of the product of displacement shape functions and traction Green’s functions, with an identity matrix added to the main diagonal for external problems (coming from the right-hand-side of Eqn. (3.1)), while coefficients in G consist of integrals of the product of traction shape functions and displacement Green’s functions. Specification of either a traction or displacement boundary condition for all three directions at each node allows for the solution of the remaining 3N nodal unknowns.
71
Γρ
Γ
ρ
n
x
ξ2
y Ωe ξ1
ξ3 Figure 3.2: Three dimensional solid with infinite region Ω e external to Γ.
3.2
Fundamental Numerical Aspects For a more specific description of the details of the BIE discretization, the dis(m)
placement and traction over an element based on its nodal values U i
(m)
and Ti
may
be approximated by
Ui
(ξ) =
me X
Ui
me X
Ti
(m)
(m)
NiU (ξ)
m=1 Ti
(ξ; n) =
(3.2) (m)
(m)
NiT (ξ),
m=1 (m)
(m)
where “me ” is the number of nodes for element number e, and N iT (ξ) and NiU (ξ) are the chosen interpolation functions for the i th component of traction and displacement at the node with local number m. By virtue of the interpolated fields in Eqn. (3.2), the boundary integral equation (3.1) can be converted to a linear algebraic system by collocation at all nodes with global node numbers n and nodal coordinates y (n) , leading to
72
nel X e=1
"Z
Ti
ˆik (ξ, y(n) ) dΓξ − (ξ; n) U
Z
Ui (ξ) [Tˆik (ξ, y(n) ; n)]2 dΓξ
#
Γe
−
Z
Γe
Γe
[Ui (ξ)− Ui (y(n) )] [Tˆik (ξ, y(n) ; n)]1 dΓξ
=
k = 1, 2, 3,
0, Internal domain
1, External domain
Uk (y(n) ),
(3.3)
n = 1, 2, . . . N,
in which the boundary Γ is discretized into a mesh of n el elements, each having a ge
ometrically interpolated elemental domain Γe . For a given collocation node y (n) , the left-hand side of Eqn. (3.3) will contain contributions from integrals over a number “n c ” of surface elements which contain the collocation node, and contributions from the remaining nel − nc elements which do not contain the collocation node. In general, the ratio of nc to nel − nc will depend upon the element order, local element size and specific location of the collocation point. The regularization afforded by the displacement subtraction in the second integral of Eqn. (3.3) will only be realized for integration over the elements which contain the collocation node. This point can be clarified by considering a conceptual reassignment of the arbitrary global element numbers so that the n c
(n)
elements appear first in the above summation, and noting that Ui (y(n) ) = Ui
where
n = 1, . . . N is the global node number. Accordingly, Eqns. (3.2) and (3.3) combine to
73 give a discretized and interpolated form of the BIE, i.e. "Z me nc 3 X X X (m) (m) ˆ k (ξ, y(n) ) dΓξ U T N (ξ) i i iT Γe i=1 m=1
e=1
+
nel X
e=nc +1
−
Z
−
Z
me 3 X X
Γe i=1 m=1
me 3 X X
Γe i=1 m=1
"Z
−
Ti
3 X me X
Ui
+
Z
Γe i=1 m=1
(n)
Γe
−
Z
=
Ui
h i (m) (m) NiU (ξ) − NiU (y(n) ) [Tˆik (ξ, y(n) ; n)]1 dΓξ
(m) (m) Ui NiU (ξ)
3 X me X
Γe i=1 m=1
Z
(m)
Ui
[Tˆik (ξ, y(n) ; n)]2
dΓξ
(m)
(m) ˆik (ξ, y(n) ) dΓξ NiT (ξ) U
(m)
(m) NiU (ξ) [Tˆik (ξ, y(n) ; n)]1 dΓξ
#
(3.4)
[Tˆik (ξ, y(n) ; n)]1 dΓξ
3 X me X
Γe i=1 m=1
(m)
Ui
(m)
NiU (ξ) [Tˆik (ξ, y(n) ; n)]2 dΓξ
#
0, Internal domain (n) U , 1, External domain k
k = 1, 2, 3,
n = 1, 2, . . . N.
(m)
In the second integral of the above equation, the interpolation function N iU (y(n) ) will take the value 1 at the local node m that coincides with the global collocation node n, and 0 at the remaining nodes. For each node with local number m ∈ [1, 2, . . . m e ], let its corresponding global node number be denoted by m gl ∈ [1, 2, . . . N ]. Such notation (m)
allows the displacement interpolation function N iU (y(n) ) in Eqn. (3.4) to be expressed using the Kronecker delta symbol as (m)
NiU (y(n) ) = δnmgl .
(3.5)
To obtain a linear equation for the unknown nodal values in (3.4), the order of integra-
74 tion and summation may be interchanged, which is then followed by a factoring of the constant nodal values from the integrals, leading to " Z nc X 3 X me X (m) (m) ˆik (ξ, y(n) ) dΓξ Ti NiT (ξ) U Γe
e=1 i=1 m=1
+
nel X me 3 X X
e=nc +1 i=1 m=1
(m) −Ui
Z
(m) −Ui
Z
(m) Ti
Z
(m) −Ui
Z
(n) +Ui
Z
(m) −Ui
Z
"
=
Γe
Γe
Γe
Γe
Γe
Γe
h
(m)
NiU (ξ) − δnmgl
(m) NiU (ξ)
i
[Tˆik (ξ, y(n) ; n)]1 dΓξ
[Tˆik (ξ, y(n) ; n)]2 dΓξ
#
(m) ˆ k (ξ, y(n) ) dΓξ NiT (ξ) U i
(3.6)
(m) NiU (ξ) [Tˆik (ξ, y(n) ; n)]1 dΓξ
[Tˆik (ξ, y(n) ; n)]1 dΓξ
(m) NiU (ξ)
[Tˆik (ξ, y(n) ; n)]2
dΓξ
#
0, Internal domain (n) U , 1, External domain k
k = 1, 2, 3,
n = 1, 2, . . . N.
In this format, it should be clear that the analytical regularization which converts the strongly-singular integrand involving [ Tˆik (ξ, y(n) ; n)]1 into a weakly-singular integrand via the displacement field subtraction shown in Eqn. (3.1) benefits only those elements that contain the collocation node, which are represented by the first set of summations above. While the integrals over elements which do not contain the collocation node (i.e. the second set of summations in Eqn. (3.6)) are theoretically regular, one may still have to contend with nearly-singular integrands such as those involving only [ Tˆik (ξ, y(n) ; n)]1
for Γe in the neighborhood of the collocation point y (n) , and their severity can worsen in
75 the case of progressive mesh refinement or thin structures (see, e.g. Huang and Cruse, 1993; Hayami, 2005). To help illustrate the issues involved, consider a pair of nonadjacent three-node P el triangular elements of the discretized surface mesh Γ= ne=1 Γe , which have elemental
surfaces Γ1 and Γ2 as shown in Figure 3.3. When the collocation point y (n) is coincident
with local node m = 1 of element 1 (i.e. global node m gl = n = 10), the second integral in Eqn. (3.3) will have the form Z (10) [ U (ξ)− U )] [Tˆik (ξ, y(10) ; n)]1 dΓξ i i (y Γ1
=
3 Z X i=1
=
3 X i=1
"
+
Γ1
h
(10)
Ui
(10) Ui
Z
(11) Ui
Z
Γ1
Γ1
(1)
(11)
NiU (ξ) + Ui
(2)
(12)
NiU (ξ) + Ui
(3)
(10)
NiU (ξ) − Ui
i
[Tˆik (ξ, y(10) ; n)]1 dΓξ
(1) NiU (ξ) − 1 [Tˆik (ξ, y(10) ; n)]1 dΓξ
(2) NiU (ξ)[Tˆik (ξ, y(10) ; n)]1 dΓξ
+
(12) Ui
Z
Γ1
(3) NiU (ξ)[Tˆik (ξ, y(10) ; n)]1 dΓξ
#
(3.7) for integration over the first element. For integration over the second element, the above integral will have the form Z (10) [ U (ξ)− U )] [Tˆik (ξ, y(10) ; n)]1 dΓξ i i (y Γ2
=
3 Z X i=1
=
3 X i=1
"
+
Γ2
h
(3)
(1)
(4)
(2)
(5)
(3)
(10)
Ui NiU (ξ) + Ui NiU (ξ) + Ui NiU (ξ) − Ui
(3) Ui
Z
(5) Ui
Z
Γ2
Γ2
(1) NiU (ξ)[Tˆik (ξ, y(10) ; n)]1 dΓξ
(3) NiU (ξ)[Tˆik (ξ, y(10) ; n)]1 dΓξ
+
−
(4) Ui
Z
(10) Ui
Z
i
[Tˆik (ξ, y(10) ; n)]1 dΓξ
(2)
Γ2
Γ2
NiU (ξ)[Tˆik (ξ, y(10) ; n)]1 dΓξ
[Tˆik (ξ, y(10) ; n)]1 dΓξ
#
. (3.8)
For each of the collocation nodes y (n) , integration over all elements containing the
76
10(1)
Collocation point y
(10)
3(1)
Γ2
12(3)
Γ1
5(3)
4(2) 11(2)
Global(local) node numbers
Figure 3.3: Two non-adjacent triangular elements. Numbers in parentheses are local node numbers
collocation node will give rise to integrals of the form of Eqn. (3.7), which appear in the first summation in Eqn. (3.6), while integration over elements not containing the collocation node will give rise to integrals of the form of Eqn. (3.8), which appear in the second summation in Eqn. (3.6). i h (m) For interpolation functions of linear or higher order, the term NiU (ξ) − 1 in
the second integral of Eqn. (3.6) will be H¨older continuous as required, i.e. α (m) NiU (ξ) − 1 = O ξ − y(n)
as
ξ → y(n) ,
0 < α ≤ 1,
(3.9)
where it can be shown that α = 1 for all Lagrangian polynomial interpolation functions, and 0 < α ≤ 1 for the new adaptive-gradient or AG elements to be developed later. All of the remaining terms for which mgl 6= n in the summation over m in the second integral of h i (m) Eqn. (3.6) have the form NiU (ξ) − 0 , and are also H¨older continuous. Therefore, the
integrands corresponding to all nodes m = 1, 2, . . . , m e of the nc elements which contain the collocation point will indeed be regularized despite the higher-order singularity of the traction Green’s function in this discretized version of the regularized BIE. For the elements which do not contain the current collocation point, however, (n)
the coefficient of Ui
in the second summation of Eqn. (3.6) is the integral of the
77 strongly-singular part of the traction Green’s function alone. This integral, Z
Γe
[Tˆik (ξ, y(n) ; n)]1 dΓξ
(3.10)
will become nearly-strongly-singular as the domain of integration over ξ (the element
surface Γe ) approaches the collocation node y (n) , due to the strong singularity of [Tˆik (ξ, y(n) ; n)]1 at the point y(n) , which behaves as ! 1 [Tˆik (ξ, y(n) ; n)]1 = O ξ − y(n) 2
as
ξ → y(n) .
(3.11)
Consequently, when collocating on small or slender elements, accurate integration of (3.10) over their neighboring elements not containing the collocation node may require the use of a significantly higher-order Gauss integration rule compared to other terms in Eqn. (3.6). If a fixed Gauss rule were used for all integrals, the accuracy of the components in the H and G matrices could actually decrease as the mesh is refined, which runs counter to the common goal of higher accuracy from mesh refinement. For the solution to retain the property of convergence as element size is decreased, as is commonly relied upon in numerical methods, it should be evident that that an accurate evaluation of all such nearly-singular integrals is a prerequisite. The distribution of the n el − nc elemental areas not containing the collocation node n and thus requiring the non-regularized integration of Eqn. (3.10) may be different for each of the N collocation points, and depends on the design of the mesh. For robustness and consistent integration accuracy, a versatile strategy that can handle arbitrary topological configurations of the mesh is clearly desirable. The need and development of such an algorithm will be discussed with numerical examples in the next two sections.
3.2.1
Effects of Element Configuration on Integration Accuracy In the analysis of three-dimensional mixed boundary value problems in linear elas-
ticity using the boundary element method, discretization of a slender physical domain
78 may often be more efficiently accomplished using elements which are slender, skewed, or non-planar. However, the use of slender elements may induce folding and skewing of the already complicated integrands (Pak et al., 2004). For finite element methods, such mesh configurations are well known to lead to inaccuracy and numerical problems due to the skewing of the mapping and the Jacobian going to zero. For boundary element methods, similar difficulties are known to occur although some of the issues are of a different nature (e.g. Ainsworth et al., 1999; Rodin and Steinbach, 2003). The use of slender elements in the regularized BEM can lead to a decrease in integration accuracy due to the prospect of nearly-singular integrands as discussed above, while the quality of BEM solutions is known to be dependent upon the accuracy of integration of weakly-singular and nearly-weakly-singular integrals (Lachat and Watson, 1976; Rizzo and Shippy, 1977; Hayami, 1990; Huang and Cruse, 1993; Liu et al., 1993; Sladek and Sladek, 1998; Hayami, 2005; Cruse and Aithal, 1993). Because of these factors, the degree of slenderness or skewness of an element can directly affect the accuracy of the elemental matrices, resulting in the common recommendation to avoid slender elements altogether, with a rule of thumb being to limit aspect ratios to 3 : 1 or less (Felippa, 2005; Cook et al., 1989). As a result of such concerns, use is often made of gradated and transitional meshes with progressively many smaller elements in regions of high stress gradients or discontinuities in geometry. Especially for cases in which the stresses or boundaries have discontinuities or high gradients in one direction only, e.g. at the edge of a circular punch on a half-space, the combined requirements of near-unity aspect ratios and refined meshing near the edge can result in an excessive number of elements to the detriment of the practicality of the overall computational effort. The nearly-singular integral problem discussed in the previous section will only be compounded by excessive mesh refinement, even for unity aspect ratios. Without careful attention to identify and resolve the fundamental underlying issues, such conditions can lead to serious errors in the final numerical solution of general boundary value problems even in a regularized
79 boundary integral equation setting. As an example consider the rigid-body displacement problem of a traction free, slender, solid cylinder with length-to-radius ratio l/a = 20 and prescribed planar motions of the top surface (see Fig. 3.4(a)). For its treatment by BEM, a surface mesh of 8-node quads and 6-node triangles and Kelvin’s fundamental solution were used. Owing to the slenderness of the cylindrical domain and the simple kinematics of the solution, it is a logical attempt to use a mesh as shown in the figure where four elements of uniform length in the radial and vertical (z) directions and eight elements of constant angle in the angular direction are used. Displacements U = (U x , Uy , Uz ) corresponding to an infinitesimal rigid-body rotation are prescribed on the top end, while the surface traction is prescribed to be zero on the remainder of the boundary. To accommodate properly the mixed boundary conditions at the upper corner, double nodes were used at the upper perimeter while single nodes were used everywhere else. To further improve the integration accuracy of the weakly singular integrals contained in the regularized formulation when an element contains the collocation node, elemental areas were subdivided into triangular regions which were subsequently mapped to degenerate collapsed quadrilaterals (Luchi and Rizzuti, 1987; Guzina, 1996). For numerical quadrature, a 7 × 7 Gauss rule was used for all quadrilateral regions, including the quadrilateral parent domains of all degenerately mapped triangular regions with the collocation point at their vertex, whereas a 25-point triangular Gauss rule was adopted for integration over triangles not containing the collocation point. For collocation on nodes of triangular elements, the triangular domains were also degenerately mapped to collapsed quadrilateral domains as will be described in Section 3.4.3. Despite the extra attention, the results obtained are less than satisfactory. Shown in Fig. 3.4(b) is the result for the case of an infinitesimal rocking rigid-body motion rotation Θy (x, y, 0) of the top surface about the y-axis. As can be seen from the figure, there is an artificial parasitic curvature which should not exist in the displacement solution.
80 The simulation result of all four fundamental rigid-body motion cases imposed at the top end of the cylindrical domain are summarized in Fig. 3.5 where the magnitudes of the displacement vectors along the line (x, y, z) = (0, 1, z), 0 ≤ z ≤ 20a are plotted. While the pure vertical and horizontal translation cases fare quite well, the significant errors in the rotational Θy and Θz cases are obvious. Contradictory to physical expectations as well, the corresponding surface traction vectors T(x, y, 0) are also typically non-zero on the top surface (see Fig. 3.6(a) and Figure 3.6(b) for two of the cases), with the error in the traction solution being the largest under an imposed rigid-body torsional rotation Θz (x, y, 0).
81
(a) Original Configuration r/a = (x/a,y/a,z/a)
(b) Exaggerated Displaced Configuration r/a + (U/a)/Θy
0 0 5 Exact RBM
10 z/a
z/a
5
10
15 15 20 20 −1 0 y/a 1
10
20
1
0 −1 x/a
y/a −101
0
x/a
Figure 3.4: BEM displacement solution for 3D cylinder with inadequate gauss integration rules.
0
0
(d) RBM Θz
(c) RBM Θy
(b) RBM Ux
(a) RBM Uz
0
0
z/a
Exact RBM 5
5
5
5
10
10
10
10
15
15
15
15 BEM
20
20
20
20
0 10 20 30 10 20 1 2 0 1 2 0 | U(0,1,z) | | U(0,1,z) | | U(0,1,z) | | U(0,1,z) | Figure 3.5: Magnitude of displacement vector U along the line (x, y) = (0, 1) of cylinder for applied RBM of top surface with inadequate Gauss integration rules. 0
( |T(x,y,0)|/G ) / Θy
82
0.1
0.05 0 −1
−0.5 y/a
( |T(x,y,0)|/G ) / Θz
(a)
−1
0
0.5
1
−1
0.5
0
−0.5
1
x/a
2 1 0 −0.5
0
(b)
0.5
1
−1
0.5
0
−0.5
1
y/a x/a Figure 3.6: Magnitude of traction vector T on cylinder top surface for applied RBM rotations of top surface with inadequate gauss integration rules.
% Error in |U(0,1,20)/a| / Θz(x,y,0)
25 20 15 10 5 0 (−2594% Error for Gauss rule = 7) −5 0
50
100
150
200
250
300
350
400
450
2
Gauss points for quadrilateral regions = (Gauss rule)
Figure 3.7: Error in displacement vector U at (x, y, z) = (0, 1, 20) versus Gauss points for quadrilateral integration regions for applied rotation Θ z of top surface.
83
Resultant moment (Mz/G) / Θz(x,y,0)
0.04
0.03
0.02
0.01
0 (Mz/G/Θz(x,y,0) = −3.27 for Gauss rule = 7)
−0.01 0
50
100
150
200
250
300
350
400
450
Gauss points for quadrilateral regions = (Gauss rule)2 Figure 3.8: Resultant moment Mz of all tractions versus Gauss points for quadrilateral integration regions for applied rotation Θ z of top surface.
3.2.1.1
Main Source of Error: Integration
Shown in Figs. 3.7 and 3.8 are, respectively, the error in the magnitude of the displacement vector at the bottom corner of the cylinder for an applied RBM rotation Θz (x, y, 0), and the corresponding resultant moment of the top-end traction which should be zero in such circumstances, plotted against the integration effort for convergence. For these plots, a fixed 73-point Gauss rule was used for the triangular elements at the terminal ends when not containing the collocation point, while the order of the Gauss rule for all quadrilateral integration regions was increased systematically. From these plots, it should be evident that the problem is largely related to integration accuracy. To address the problem, it is useful to note that the integration rule necessary to suppress the solution errors of the type presented in Figs. 3.4 through 3.8 inherently depends on many factors such as the nature of the Green’s functions and shape
84 functions, the degree of regularization in the analytical formulation, the distance between integration element and collocation point, and the distribution of element sizes as well as their aspect ratios. Because of these multiple sources, previous approaches to increasing the integration accuracy by pre-determined coordinate transformations (Cruse, 1974; Hayami, 2005), element subdivision (h-strategy) (Lachat and Watson, 1976), and increasing the number of Gauss points (p-strategy) (Sladek et al., 1997; Huang and Cruse, 1993) may yield only limited performance by themselves for general cases.
3.2.2
Adaptive Elemental Integration Scheme for Uniform Accuracy As discussed in Section 3.2 and illustrated in the previous section, integration
over elements which are near but do not contain the collocation node may require the numerical evaluation of nearly-singular integrals, whose specific nature can be mesh- and problem-dependent. While the BEM literature contains many different approaches for addressing the integration accuracy of the weakly-singular and nearly-weakly-singular BIE integrals (see, e.g. Huang and Cruse, 1993; Sladek and Sladek, 1998; Hayami, 2005), many are based on a-priori error estimates and pre-determined Gauss rules, and while they may be efficient, their actual accuracy has often not been addressed. It should be clear that the integration strategy developed herein, with its adaptivity and a-posteriori error calculation, emphasizes accuracy over efficiency. To achieve a uniform level of integration accuracy for all integrals in Eqn. (3.6) in the face of such mesh-dependency, an adaptive integration scheme may be the only viable approach to the problem. To this end, the structure of the boundary element program BEASSI was modified to include an iterative numerical quadrature scheme which allows for user-defined convergence control. In this approach, the complex modulus of each (m)
coefficient of Ui
(n)
, Ui
(m)
and Ti
in Eqn. (3.6) was evaluated for a chosen initial Gauss
rule and compared to the value obtained using the available next-order Gauss rule. For triangles, the available Gauss integration rules were those having 3, 7, 13, 19, 25, 37
85 and 73 points. These rules all possess positive weights and have coordinates inside the standard triangle (Dunavant, 1985). For quadrilateral integration domains, including all parent domains of degenerately mapped triangular integration regions (see Guzina, 1996), the starting n × n rule was incremented to the next (n + 1) × (n + 1) Gauss rule. If the percent change in modulus of any of the coefficients for a given element
surface Γe and collocation point y (n) was above a specified tolerance (typically 0.1%), the Gauss rule was incremented to the next available rule and the integrals were reevaluated. When the percent change of all coefficients with a modulus greater than a prescribed threshold (typically 10 −14 for this study) was below the specified tolerance, the numerical quadrature of the integral was deemed to have converged for the current collocation point. To provide further insight, the following statistical information was also generated in the execution of the enhanced integration scheme: (1) Coordinates y(n) of the current collocation point. (2) The final Gauss rule used for each element or element sub-domain. (3) The distance from the source point to the nearest node of the current element. (4) The maximum percent difference of all elemental integrals in Eqn. (3.6) when convergence was achieved, for each collocation point. The availability of the foregoing information will, for example, allow quantification as well as the design of the integration effort necessary to achieve uniform accuracy for a given mesh, while elucidating other numerical issues that may require attention. As an illustration, the rigid-body motion tests presented in Figs. 3.4 through 3.6 were reanalyzed using the adaptive scheme with a tolerance of 10%, which was found to be sufficient to reduce the “artificial curvature” problem of Fig. 3.4(b). As part of the result, the histogram of Gauss rules needed for convergence is shown in Fig. 3.9. This figure indicates that for the majority of combinations of collocation points and quadri-
86 lateral integration regions (including the parent domains of quadrilaterals degenerately mapped onto triangular sub-regions over the elements containing the collocation point), convergence is achieved at Gauss rules of order 3×3, 4×4, and 5×5, with some integration regions requiring 6×6 through 17×17 Gauss rules. The triangular regions not containing the collocation point and therefore evaluated using triangular integration rules all converge at 13, 19 or 25 points. To examine the dependence of the Gauss rules needed on the proximity of the collocation point to the integration region, each instance of convergence for a quadrilateral region is also plotted versus the distance from the collocation point to the nearest node of each integration element in Fig. 3.10. For elements containing the collocation point, this distance is zero, and integration is achieved at Gauss rules below 10 × 10, showing the effectiveness of the analytically-regularized integrands and the singular coordinate transformations employed. An even more important observation from this figure, though, is that the elements requiring the highest Gauss rules are the ones with non-regularized and non-mapped integrands, whose element domains are close to but not including the collocation point. As discussed previously, these integrands can become quasi-singular as their domain of definition draws closer to the collocation point. If fixed but insufficient Gauss rules were used, the solution error would increase, for instance, as a mesh is refined, defying the latter’s usual purpose. By virtue of the adaptive elemental integration scheme, on the other hand, one can ensure that all coefficients in the linear system of equations have a uniform level of accuracy, avoiding the difficulty in predicting the required distribution of Gauss rules for a given mesh, or the usage of overly high-order Gauss rules for all integrals. To illustrate more fully the performance of the adaptive integration scheme, the magnitudes of the displacement vectors along the cylinder for the adaptive scheme with a prescribed tolerance of 10% are shown in Fig. 3.11. This figure illustrates that the numerical displacement solutions are now much closer to the exact rigid-body motion.
87 Tightening the tolerance to a level of 1% or 0.1% will further improve the solution for this and other general boundary value problems. The required tolerance, however, will depend upon the desired level of accuracy in the numerical displacement and traction solutions, as well as the particular mesh configuration. For example, Fig. 3.12(a) demonstrates that to achieve an error of less than 1% in the magnitude of the displacement vector at bottom edge of the cylinder for the RBM rotation Θ z , a tolerance of 1% is required, for which the maximum required Gauss rule for quadrilateral regions is 22×22 (Fig. 3.12(b)). It should be noted, however, that the triangle’s Gauss rule reaches the upper limit of 73 points at a tolerance of 1% and below for this problem.
88
800 700
# of converged integration regions
# of converged integration regions
15
600 500 400 300
10
5
0 100
200
150 200 250 Gauss points
300
100 0
0
50
100
150 Gauss points
200
250
300
Figure 3.9: Histogram of Gauss rules at convergence for quadrilateral regions. Tolerance = 10%.
300
300
200
150 100
200
100
0
0
1
2
3
dmin/a
50
0
Gauss points
Gauss points
250
0
5 10 15 dmin/a, distance to nearest node of element
20
Figure 3.10: Gauss rules at convergence for quadrilateral regions vs. distance from collocation point to nearest node of integration element. Tolerance = 10%.
89
(a) RBM Uz
0
(b) RBM Ux
0
0
(c) RBM Θy
0
5
5
5
10
10
10
10
15
15
15
15
z/a
5
20 0
0.5
1
| U(0,1,z) |
20 0
0.5
1
| U(0,1,z) |
20 0
10
20
| U(0,1,z) |
(d) RBM Θz
20 0
0.5
1
| U(0,1,z) |
% Error in |U(0,1,20)/a|
Figure 3.11: Magnitude of displacement vector U along the line (x, y) = (0, 1) of cylinder for applied RBM of top surface. Adaptive integration with tolerance = 10%.
4
2
0
Max. Gauss rule (1D)
(a)
10 40
1
0.1
0.01
0.001
1
0.1 Tolerance [%]
0.01
0.001
30
20 (b)
10
Figure 3.12: (a): Error in displacement vector at (x, y, z) = (0, 1, 20) for applied rotation Θz of top surface, (b): corresponding maximum required Gauss rule.
90 3.2.2.1
Numerical Implementation of Adaptive Integration Scheme in BEASSI
Going beyond general description of the adaptive integration scheme that was incorporated into the boundary element code BEASSI given in the previous section, the details of its programming are provided in this section for user’s reference. In the numerical implementation of Eqn. (3.6) for a given domain and collocation node y(n) , the first summation is achieved by a loop over all elements bounding a domain. In the loop, contributions from each element are assembled into the global H and G matrices according to the collocation node number (n) and the global node numbers mgl ∈ [1, 2, . . . N ] of the integration element with number e. The second and third loops over the component directions (i) and element nodes (m) are not performed as such during assembly. Instead, they are carried out by matrix multiplication only for the specified boundary conditions in the global displacement and traction vectors
T2
T1
T3
T2
(2)
(1)
(1)
(1)
(2)
U2
U1
U3
U2
(2)
(1)
(1)
(1)
U = [U1
(N )
(2)
U3 · · · U 1
(N )
U2
(N ) T
]
(3.12)
(N ) T
(3.13)
U3
and T = [T1
(2)
(2)
(N )
T3 · · · T 1
(N )
T2
T3
] .
This is done after the fully-formed system of equations HU = GT is re-partitioned and arranged into AY = F in order to solve for the mixed nodal unknowns which constitute the vector Y (see Guzina, 1996). For the nodal unknowns represented in Y, summation over i and m is only implied through the Gaussian elimination process. Within the element loop, numerical integration in triangular or quadrilateral natural coordinate domains is performed using one or two loops, respectively, over Gauss integration points. For a given integration element e, the coordinates of its l th receiver (Gauss) point will be denoted as ξ l , and the product of the corresponding Gauss weights as wl . The product of the Jacobians of all applied mappings will be denoted J l . With the above notations, the implementation of the first integral from either group of
91 summations in Eqn. (3.6) as programmed in BEASSI may be written as (1) (m ) 1 1 1 ˆ ˆ ˆ U 1 U 2 U 3 N 1T 0 0 N 1T e 0 0 nG X ˆ2 ˆ2 ˆ2 (1) (m ) cofu ≡ wl Jl U ... 0 N 2T e 0 1 U 2 U 3 0 N 2T 0 l=1 (1) (m ) ˆ3 U ˆ3 U ˆ3 0 0 N 3T U 0 0 N 3T e 1 2 3
=
ˆ 1 N (1) U ˆ 1 N (1) U ˆ 1 N (1) ˆ 1 N (me ) U ˆ 1 N (me ) U ˆ 1 N (me ) U U 1 1T 2 2T 3 3T 1 1T 2 2T 3 3T nG X ˆ 2 (1) ˆ 2 (1) ˆ 2 (1) ˆ 2 (me ) U ˆ 2 N (me ) U ˆ 2 N (me ) wl Jl U 2 2T 3 3T 1 N 1T U 2 N 2T U 3 N 3T . . . U 1 N 1T l=1 ˆ 3 N (1) U ˆ 3 N (1) U ˆ 3 N (1) ˆ 3 N (me ) U ˆ 3 N (me ) U ˆ 3 N (me ) U U 1 1T 2 2T 3 3T 1 1T 2 2T 3 3T
,
(3.14)
where nG is the number of Gauss points for the element. In Eqn. (3.14), the displacement Green’s functions and traction shape functions are evaluated at each of the Gauss points ξ l , l = (1, 2, . . . nG ) for the current collocation point y (n) . For the integrals involving the traction Green’s functions over elements containing the collocation point, the second and third integrals in the summation over the first n c elements of Eqn. (3.6) are added to obtain the matrix coft. For example, when local node 1 is the collocation point, ˆ1 ˆ 1 ˆ 1 T1 T2 T3 n G X wl Jl Tˆ12 Tˆ22 Tˆ32 coft ≡ l=1 Tˆ 3 Tˆ 3 Tˆ 3 1
2
3
ˆ1 ˆ 1 ˆ1 T1 T2 T3 + Tˆ12 Tˆ22 Tˆ32 Tˆ13 Tˆ23 Tˆ33
1
2
(1) N1U −1
0
0
(1)
0
0
N2U −1
0
0
(1) N 1U
0 (1)
0
(1)
(m ) N 1U e
(1)
0 N 3U
...
0
(m )
0 (m )
0 N 3U e
0 0
0
(m )
0
0 N 2U e 0
0
0 N 2U e
...
N3U −1
0
0 N 2U 0
(m ) N 1U e
(m )
0 N 3U e
,
(3.15)
while cofs ≡ 0. For integration over the elements which do not contain the collocation point (i.e. the second set of summations in Eqn. (3.6)), the regularization by displacement subtraction is not performed, and the coft and cofs matrices are instead evaluated
92 as
ˆ1 ˆ1 ˆ 1 T1 T2 T3 n G X coft ≡ wl Jl Tˆ12 Tˆ22 Tˆ32 l=1 Tˆ13 Tˆ23 Tˆ33
Tˆ11
Tˆ21
Tˆ31
+ Tˆ12 Tˆ22 Tˆ32 Tˆ13 Tˆ23 Tˆ33 and
1
2
(1) N 1U
0
0
(1)
0 N 2U 0 0 (1) N 1U
(1)
0
0 (m ) N 1U e
0
(1)
0
0
(m )
0
0 N 2U e
...
0 N 3U
0 N 2U 0 0
(m ) N 1U e
(m )
0 N 3U e 0
0
(m )
0
0 N 2U e
...
(1)
0 N 3U
0
(m )
0 N 3U e
(3.16)
ˆ1 ˆ 1 ˆ 1 T1 T2 T3 n G X wl Jl Tˆ12 Tˆ22 Tˆ32 cofs ≡ − l=1 Tˆ13 Tˆ23 Tˆ33
.
(3.17)
1
After integration over the current element via completion of the summation over the n G Gauss points in Eqns. (3.14)–(3.17), the coefficients of the matrices cofu, coft and cofs are stored, and the matrices are re-computed using the available next-order Gauss rule as part of the adaptive algorithm. The magnitude of the complex-valued components of these matrices are then compared for the two Gauss rules. If the magnitude of each component is below the threshold (usually 10 −14 ) or its change is below the chosen tolerance, convergence is considered to have been obtained for the current element or element sub-domain. Following such convergence, the cofu matrix for each element is then assimilated into the G matrix in the global system of equations HU = GT. Similarly, the coft and cofs matrices of Eqns. (3.15)–(3.17) are assimilated into the H matrix, along with the remaining term on the right-hand side of Eqn. (3.6).
3.2.3
Proximity-Based Mapping of Nearby Elements As a way to further improve the convergence of the adaptive Gauss integration
scheme for the nearly-singular integrands belonging to both quadrilateral and triangular
93 elements near the collocation point, it was found in this study that an extension of a singular coordinate transformation technique (see Sec. 3.4) to all elements having a node within a prescribed distance from the collocation point, with the pole of the singular mapping placed at the node closest to the collocation point, can be effective. Such a mapping has the effect of stretching the non-regularized and nearly-singular integrands in the general direction of the collocation point, or equivalently, concentrating the Gauss points towards the element node closest to the collocation point. For meshes with small or slender elements, the transformation can accelerate the convergence by allowing the nearly-singular integrals of elements very close to the collocation point to converge at lower Gauss rules. It can also remove the restriction of the maximum Gauss rule for triangles, as triangular domains are mapped to a degenerate quadrilateral parent domain when a node is within the sphere centered at the collocation point and having a user specified radius r1 .
3.2.4
Integrand Discontinuities of Multi-Layered Green’s functions As discussed in the first part of this chapter, one of the fundamental solutions
available in BEASSI is that of a piecewise isotropic homogeneous multi-layered elastic and visco-elastic medium Pak and Guzina (2002). For a rigorous treatment, a critical numerical issue that arises from such material domains is the discontinuous behavior of the multi-layered Green’s functions. To illustrate the problem, a small cylindrical domain of radius a, depth 2a, shear modulus 1000G ref , density ρref and Poisson’s ratio ν = 0.25 is embedded in a half-space for which multi-layered Green’s functions are used to model a linear increase in shear modulus with depth from G s /Gref = 1 to Gs /Gref = 9 for a half-space with uniform material density ρ s = ρref and Poisson’s ratio νs = 0.25. Standard eight-node quads and six-node triangles are used for the cylindrical surface mesh which is composed of 4x8x4 elements in the radial, circumferential and vertical directions. For the external half-space domain, 8 layers of uniform height h = a/4 but
94
0
(a)
(b)
0
1
1 z/a
z/a
0.5
1.5
2
2 −1 0 x/a1 1 Node 195
0 y/a
−1 Element #41
3 0
5 G(z)/Gref
10
Figure 3.13: (a): 4x8x4 cylindrical surface mesh and (b): half-space shear modulus distribution for 8 layers with height h/a = 0.25.
variable modulus overlaying a homogeneous half-space are used. To illustrate the issue, the collocation at a node on the bottom edge of the cylinder (node 195 of Fig. 3.13(a)) is considered and the behavior of the Green’s functions are examined along a vertical line defined in cylindrical coordinates by (r, θ, z) = (1.1a, π/8, z), 0 ≤ z ≤ 2a, which is shown near the surface of the cylinder in Fig. 3.13(a). A normalized frequency of p ω ¯ = ωa/ Gref /ρref = 0.1 was used to analyze the case of a uniform vertical displacement imposed on the cylinder surface at z = 0.
Close-up plots of all 9 components of the displacement Green’s functions in Fig. 3.14 show that they have sharp cusps at the layer interfaces. Similar plots of the regular part of the 6-component tensor of the stress Green’s functions for each direction of point load are shown in Figs. 3.15 and 3.16. In the formulation of the multi-layered fundamental solutions, enforcement of traction continuity was only required for the k , k = 1, 2, 3 normal to the layer interfaces. Nearly components of the stress tensor σ ˆ zz
all the secondary components of stress in Fig. 3.15 contain discontinuities at the layer interfaces. From these displays, it should be evident that use of conventional polynomial
ag replacements
95 z/a
0
0
0.25
0.25
ˆX U x
0.5
8
9
0 0.25
0.25
ˆX U y
0.5
0.75
1 11 −3 −1.6 x10 0
10
1 −1.4
−1.2
−1
0.25
−1.2
−1
0
0.25
ˆX U z
0.5
10
2
4
6
1 8 −5 −2 x10
−2
−1.8
0
0.5
0.75
1
11 −3 x10
0.25
ˆY U z
0.5
0.75
8.5
x10−5
1 9
0
0.25
8
0.75 8
x10−6
7.5
ˆZ U y
0.5
1 −1.4
7
0
0.75
1 −1.6
6.5
x10−6
ˆY U y
0.5
0.75
ˆZ U x
0.5
0.75
1
z/a
0.25
ˆY U x
0.5
0.75
z/a
0
−1.6
−1.4
x10−4
ˆZ U z
0.75 −1.5
−1
1 −0.5 0.012 0.014 0.016 0.018 0.02 −4 x10
Figure 3.14: Close-up of real component of multi-layered displacement Green’s functions along the line shown in Fig. 3.13.
interpolation functions can lead to integration difficulties when an element contains a material interface within its domain. To deal with such C 1 and C 0 discontinuities of the Green’s functions, it is useful to divide the elements into integration sub-domains which are bounded by the layer interfaces. As another numerical enhancement, such a sub-element Green’s function integration scheme was implemented in the BEASSI code. To use the scheme most effectively, the user should create a BEM mesh having horizontal element boundaries parallel to the horizontal layer interfaces, and specify a desired number of uniform subregions of integration in parent coordinates. For finite internal or semi-infinite external domains characterized by the layered Green’s functions, a set of possible layer coordinates coincident with the boundaries of such elemental sub-regions are then obtained in a pre-processing step via a uniform partition of the element’s geometry interpolation
96 0
0
0
1
2 −6
1
2 −4
−2
0
x10
5
10
z/a
z/a
z/a
1
1
2 −6
−4
−2
0
0
x10−3
10
2 −0.2 −0.15 −0.1 −0.05
−3
x10
0
2
[ˆ σ Zzz]2 z/a
1
1
2 −6
−4
−2
x10
10
15
0
2 6
8
0
1
2 −0.06
x10−4
−0.02
−5
z/a
z/a
1
2
4
6
8
−4
−4
x10
0
1
2 2
4 −3
x10
0
2
−2
−1
4 x10−3
[ˆ σ Zxy]2 z/a
z/a
[ˆ σ Yxy]2
0
−2
0
[ˆ σ Xxy]2
−2
1
2 0
2
10 x10−3
[ˆ σ Zzx]2
2
1
5
0
0
−0.02
0
−0.05
1
[ˆ σ ]
−0.04
−0.1
0
Y zx 2
1
−0.15
2 −0.04
0
[ˆ σ Xzx]2
2 −0.06
−0.2
[ˆ σ Zyz]2 z/a
1
4
x10−3
[ˆ σ Yyz]2 z/a
[ˆ σ Xyz]2
2
1
2 5
−3
0
0
1
[ˆ σ Yzz]2 z/a
z/a
5
0
[ˆ σ Xzz]2
z/a
[ˆ σ Zyy]2
[ˆ σ Yyy]2
0
z/a
−0.2 −0.15 −0.1 −0.05 x10−3 0
[ˆ σ Xyy]2
z/a
15 x10−3
0
2 −8
1
2 0
−3
0
PSfrag replacements
[ˆ σ Zxx]2 z/a
[ˆ σ Yxx]2 z/a
z/a
[ˆ σ Xxx]2
0
1 x10−3
1
2 −20
−10
0 x10−4
Figure 3.15: Real component of regular part of multi-layered stress Green’s functions along the line shown in Fig. 3.13.
97 0
0
0
1
2 −8
1
2 −6
−4
−2
0
1
1
0
0.5
z/a
z/a
z/a 2 x10
2
6 x10
x10−6
0
−6
−4
−2
0
6
8 −8
x10
1
2 −4
1 x10−9
[ˆ σ Zxy]2 z/a
z/a
2
0
0
[ˆ σ Yxy]2
1
4 x10−9
1
2 −1
0 x10−13
[ˆ σ Xxy]2
2
[ˆ σ Zzx]2
1
2 −8
8
0
0
z/a
z/a 6
1.5 x10−5
1
2 −2
8 −6
0
2
4
4
1
[ˆ σ Zyz]2
[ˆ σ Yzx]2
1
2
0.5
0
z/a
z/a
z/a
0
[ˆ σ Xzx]2 z/a
0
−9
1
5
0
z/a
4
0
x10−13
4
1
2 0
2 0
2 x10−4
[ˆ σ Zzz]2
[ˆ σ Yyz]2
1
2
1
0
2
[ˆ σ Xyz]2
0
0
x10−7
1
1
0
PSfrag replacements
1.5
0
x10−9
0
1
[ˆ σ Yzz]2
0
−5
1
2 0
[ˆ σ Xzz]2 1
2 x10−4
[ˆ σ Zyy]2 z/a
z/a
z/a −2
0
−1
1
0
2 −4
x10−9
2 −10
0
x10−8
[ˆ σ Yyy]2
1
2 −2
2
0
[ˆ σ Xyy]2
−6
1
2 0
x10−8
0
2 −8
[ˆ σ Zxx]2 z/a
[ˆ σ Yxx]2 z/a
z/a
[ˆ σ Xxx]2
1
2 −3
−2
−1
0 −8
x10
0
1
2 −8
x10
Figure 3.16: Imaginary component of regular part of multi-layered stress Green’s functions along the line shown in Fig. 3.13.
98 functions in the parent coordinate domain. Within the program, each element’s natural coordinate axis having a vertical-orientation is determined from the numbering of the nodes, and the parent element is then subdivided into the desired number of rectangular strips along that coordinate direction. Each strip is then mapped to a standard bi-unit square for Gauss integration. With this layer-bounded integration scheme implemented in BEASSI, solution of the radiation problem for the mesh of Fig. 3.13 using adaptive integration with a tolerance of 0.1% reduces the computation time by 99% as well as the error in nodal traction solutions.
3.3
Adaptive-Gradient Elements for Mixed Boundary Value Problems In mixed boundary value problems in the theory of elasticity such as those defined
in this thesis, the solution often possesses a very high to singular gradient of variation, or exhibits very localized behavior. To capture these characteristics effectively by numerical methods, mathematical approximations that are generically capable of following sharp variations and permitting their development in the solution would clearly be desirable. In computational mechanics, the polynomial basis in the form of Lagrange elements has been widely adopted as the fundamental interpolation platform in discretization methods because of its simplicity and versatility for general applications. While it has undisputed success in many applications, the use of the polynomial basis in tackling problems where the solution and/or its gradient can become singular is a different issue. Beyond a certain gradation, for example, mesh refinement via such elements typically leads to deviations in the singular zone where accurate details of the solution can be of critical interest. To achieve a fundamental advance in the computational treatment of singular problems, an expansion of current element paradigms may be necessary in order to realize the next level of performance. During the course of this study, a new concept and class of elements which may help to bring the field closer
99
20
√ 1/ 1 − x2
20
Quadratic interpolation f(x)
f(x)
Quadratic interpolation
10
10
0
PSfrag replacements (a)
0
√ 1/ 1 − x2
0 0.5 x
1
0 (b)
0.5 x
1
√ Figure 3.17: Quadratic interpolation of the function 1/ 1 − x2 using (a) 4 and (b) 10 elements. Rightmost node is at x = 0.999.
to such end was realized. Termed the “Adaptive-Gradient” or “AG” element method (Pak and Ashlock, 2007 in press), it represents a generalized analytical platform of interpolation which can be parametrically localized and adapted to handle high-gradient, localized, or singular problems, while encompassing regular and singular elements as special cases. Because of its analytical richness and computational flexibility, the AG element method can bring considerable benefits to discretization methods, as will be illustrated later, such as an improvement in the convergence rate, a reduction in the size and complexity of the mesh, and an increase in robustness and efficiency in the solution of difficult linear as well as nonlinear boundary value problems. To see the basic interpolation difficulties of polynomial-based elements in high-gradient problems, consider the √ function f (x) = 1/ 1 − x2 in 0 ≤ x ≤ 1 and its approximation by quadratic Lagrange elements in a coarse mesh and a fine mesh wherein exact nodal values are enforced to minimize the discrepancy. Despite such favorable input treatment, one can easily see from Fig. 3.17(a) a pronounced limitation of the interpolative capability of the quadratic element in the vicinity of a singularity: there is a significant “dip” followed by an “overshoot” near the singular point x = 1. With a finer mesh, the convergence difficulty only
100 gets squeezed into a smaller region, but it will not go away (see Fig. 3.17(b)). With higher-order polynomial-based interpolations, such ills will not only persist but become more erratic, although their appearance will be delayed in terms of the threshold scale of the elements.
3.3.1
1D AG Elements To understand the core concept behind the AG element (Pak and Ashlock, 2007
in press), consider a 1-dimensional interpolation function of the form: f (η) = A + Bη + Cγ(η),
η ∈ [−1, 1],
(3.18)
for a 3-node element where f is the function of interest, A, B and C are constants, η is the natural coordinate of the standard parent domain [−1, 1], and γ(η) is the AG kernel. For a large class of problems, the following choice can be recommended: !m n η+1 2 (3.19) γ(η) = 1 − 1 − 2 where m and n are any constants. The gradient of the AG-kernel; 2 !m ! 2 !m n−1 η + 1 η + 1 γ 0 (η) = sgn 1− 1− mn 1− 1− 2 2 !m−1 η+1 2 η+1 1− 2 2
(3.20)
is adaptive through the parameters m and n, and is singular if m < 1 or n < 1, whereas the AG-kernel itself is finite for m > 0 and n > 0. To ensure the existence of a solution through three arbitrary points, the restrictions m 6= 0 and (m, n) 6= (1, 0.5) are enforced. The AG kernel function in (3.19) is C ∞ continuous, and by varying m and n judiciously, the interpolation function in Eqn. (3.18) can deliver a great variety of analytical behavior. As can be seen from the plots of γ for different combinations of m and n in Fig. 3.18, the function can be localized and sharpened toward the AG-end of the parent domain via the two parameters. For non-negative choices of m and n, one
101 can impose the usual nodal interpolation requirements at η = 1, 0 and −1, leading to the equations f (−1) = A − B ≡ f (1) , f (0) = A + Cγ0 ≡ f (2) ,
(3.21)
f (1) = A + B + C ≡ f (3) , where γ0 = γ(0) = 1 −
Inverting (3.21), one finds
2 !m n 1 1− . 2
A=
−f (1) γ0 + f (2) − f (3) γ0 , (1 − 2γ0 )
B=
f (1) (γ0 − 1) + f (2) − f (3) γ0 , (1 − 2γ0 )
C=
f (1) − 2f (2) + f (3) . (1 − 2γ0 )
(3.22)
(3.23)
By virtue of (3.23), the AG format can be written as f (η) =
3 X
(i)
f (i) NAG−3 (η)
(3.24)
i=1
where (1)
NAG−3 (η) = (2) NAG−3 (η)
(3) NAG−3 (η)
−γ0 + (γ0 − 1)η + γ(η) , (1 − 2γ0 )
1 + η − 2γ(η) , = (1 − 2γ0 )
−γ0 − γ0 η + γ(η) = (1 − 2γ0 )
(3.25)
are the fundamental AG-shape functions for the 3-node AG element. As expected, (i)
they satisfy the interpolation requirement of N AG−3 (η k ) = δik , where η k is the parent coordinate of node k. Because the interpolation depends on only the nodal values, while the shape functions defined above sum to unity, one can easily show that the element
102
γ(η)
1
(a) m=1
0.5
n=1
0 −1
−0.5
0
n=2
n=10 n=100 0.5 1
1 γ(η)
(b) n=1 0.5 0 −1
γ(η)
1
−0.5
0
m=1 m=0.5 m=0.1 m=0.01 0.5 1
(c) n=2 m=1 m=0.5 m=0.05
0.5 0 −1
−0.5
0 η
0.5
1
Figure 3.18: The AG kernel γ(η) for various values of m and n.
is capable of representing arbitrary constant and linear variations, i.e. it is at least C 0 complete in all cases. Examples of the shape functions are shown in Figs. 3.19 and 3.20 for a wide range of positive m and n. With m = n = 1, one should note that the foregoing AG element’s shape functions will degenerate to those of the regular quadratic 3-node element, a property that can, for example, be utilized to facilitate smoother element class transitions and enhance solution convergence even in a uniform mesh setting. As n or 1/m tends to ∞, on the other hand, the AG interpolation would gradually approach a Heaviside step function, allowing discontinuities to be regularized as a smooth limit sequence if desired. Allowing m to be negative with n = 1, one can see that the AG kernel in Eqn. (3.19) will become singular at η = 1. This enables the AG element to turn into a true singular element with a singularity exponent of m, by replacing the interpolation condition at η = 1 in Eqn. (3.21) by one at a slightly offset node at η = c < 1, as in Guzina et al. (2006). It is noteworthy that the resulting AG-singular
103
N(1) (η) AG−3
2 m=1 0 n=1, 2, 10, 100 −2
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
N(2) (η) AG−3
2 0
−2 N(3) (η) AG−3
2 0
−2
0 0.5 η Figure 3.19: AG shape functions with m = 1.
1
N(1) (η) AG−3
2 n=1
1 0 −1 −2
−1
−0.5
0
−1
−0.5
0
−1
−0.5
m=1, 0.5, 0.1, 0.01 0.5 1
N(2) (η) AG−3
2 1 0 −1 −2
0.5
1
N(3) (η) AG−3
2 1 0 −1 −2
0 0.5 η Figure 3.20: AG shape functions with n = 1.
1
104 element retains the ability to handle arbitrary linear variation, a potent feature that is not commonly present in past singular element formulations. As one can infer, there are additional AG-element candidates in the parametric space of m and n that can be useful, but their exploration is beyond the purpose of this section. To examine the practical range of negative m which leads to AG-singular elements, one may substitute r=
1−η , 2
−1 ≤ η ≤ 1,
1≥r≥0
(3.26)
into the AG kernel expression (3.19) to obtain m n 2 . γ(r) = 1 − 2r − r
(3.27)
Near the AG-edge r ≈ 0, the above gives to the 1 st order
which, for m < 0, gives
n m γ(r) ≈ 1 − (2r) ,
r → 0.
n n −|m| = (2r)−|m| − 1 , γ(r) ≈ 1 − (2r)
(3.28)
r → 0, m < 0.
(3.29)
Rearranging the above equation gives the leading order of the singularity at r = 0 for m < 0 and n > 0 as 1 − (2r)|m| γ(r) ≈ (2r)|m|n
n
,
r → 0, m < 0, n > 0,
(3.30)
whose integrability in 1D requires |m|n < 1. 3.3.2
(3.31)
2D AG Elements By virtue of the one-dimensional results described earlier, the AG element ap-
proach can be extended to 2D. For instance, with reference to the parent domain and node numbering convention depicted in Fig. 3.21 where the AG edge is assumed to be
105 located at η1 = 1, one may obtain a two-dimensional, 9-node quadrilateral AG-Edge element by multiplying the one-dimensional 3-node AG element’s shape functions of Eqn. (3.25) in the η1 -direction with the one-dimensional 3-node Lagrange interpolation functions in the η2 -direction, which are (1)
NL (η2 ) =
η2 (η2 − 1) 2
(2)
NL (η2 ) = (1 − η2 )(1 + η2 ) (3)
NL (η2 ) =
(3.32)
η2 (η2 + 1) . 2
The result is (1) NAG−9E (η1 , η2 )
(2) NAG−9E (η1 , η2 )
=
(1) (1) NAG−3 (η1 )NL (η2 )
=
(3) (1) NAG−3 (η1 )NL (η2 )
(3)
(3)
−γ0 + (γ0 − 1)η1 + γ(η1 ) η2 (η2 − 1) = (1 − 2γ0 ) 2
−γ0 − γ0 η1 + γ(η1 ) η2 (η2 − 1) = (1 − 2γ0 ) 2
(3)
NAG−9E (η1 , η2 ) = NAG−3 (η1 )NL (η2 ) = (4) NAG−9E (η1 , η2 )
(5) NAG−9E (η1 , η2 )
(6) NAG−9E (η1 , η2 )
(7) NAG−9E (η1 , η2 )
=
(1) (3) NAG−3 (η1 )NL (η2 )
=
(2) (1) NAG−3 (η1 )NL (η2 )
=
(3) (2) NAG−3 (η1 )NL (η2 )
=
(2) (3) NAG−3 (η1 )NL (η2 )
(8)
(1)
(2)
=
(2) (2) NAG−3 (η1 )NL (η2 )
−γ0 − γ0 η1 + γ(η1 ) η2 (η2 + 1) (1 − 2γ0 ) 2
−γ0 + (γ0 − 1)η1 + γ(η1 ) η2 (η2 + 1) = (1 − 2γ0 ) 2
1 + η1 − 2γ(η1 ) η2 (η2 − 1) = (1 − 2γ0 ) 2
−γ0 − γ0 η1 + γ(η1 ) = (1 − η22 ) (1 − 2γ0 )
1 + η1 − 2γ(η1 ) η2 (η2 + 1) = (1 − 2γ0 ) 2
NAG−9E (η1 , η2 ) = NAG−3 (η1 )NL (η2 ) = (9) NAG−9E (η1 , η2 )
−γ0 + (γ0 − 1)η1 + γ(η1 ) (1 − η22 ) (1 − 2γ0 )
1 + η1 − 2γ(η1 ) = (1 − η22 ). (1 − 2γ0 )
(3.33)
Similarly, the shape functions for a 9-node quadrilateral AG-corner element with nearsingular behavior at the edges η1 = 1 and η2 = 1 may be constructed by forming products
106 of the one dimensional AG interpolation functions. They are (1)
(1)
(1)
(2)
(3)
(1)
(3)
(3)
(3)
(4)
(1)
(3)
(5)
(2)
(1)
NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = −γ01 + (γ01 − 1)η1 + γ1 (η1 ) −γ02 + (γ02 − 1)η2 + γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = −γ01 − γ01 η1 + γ1 (η1 ) −γ02 + (γ02 − 1)η2 + γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = −γ01 − γ01 η1 + γ1 (η1 ) −γ02 − γ02 η2 + γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = −γ01 + (γ01 − 1)η1 + γ1 (η1 ) −γ02 − γ02 η2 + γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = 1 + η1 − 2γ1 (η1 ) −γ02 + (γ02 − 1)η2 + γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) (6) NAG−9C (η1 , η2 )
(7) NAG−9C (η1 , η2 )
=
(3) (2) NAG−3 (η1 )NAG−3 (η2 )
=
(2) (3) NAG−3 (η1 )NAG−3 (η2 )
(8)
(1)
1 + η2 − 2γ2 (η2 ) (1 − 2γ02 )
−γ02 − γ02 η2 + γ2 (η2 ) (1 − 2γ02 )
−γ01 − γ01 η1 + γ1 (η1 ) = (1 − 2γ01 )
1 + η1 − 2γ1 (η1 ) = (1 − 2γ01 )
(2)
NAG−9C (η1 , η2 ) = NAG−3 (η1 )NAG−3 (η2 ) = −γ01 + (γ01 − 1)η1 + γ1 (η1 ) 1 + η2 − 2γ2 (η2 ) (1 − 2γ01 ) (1 − 2γ02 ) (9) NAG−9C (η1 , η2 )
=
(2) (2) NAG−3 (η1 )NAG−3 (η2 )
1 + η1 − 2γ1 (η1 ) = (1 − 2γ01 )
1 + η2 − 2γ2 (η2 ) , (1 − 2γ02 ) (3.34)
where the AG kernel is, for simplicity, taken to be Eqn. (3.19) for both directions so that γ1 = γ2 = γ and γ01 = γ02 = γ0 .
4
7
8
9
1
3
6
5
107
η2
η2
2
4
η1
3
7
6
8
1
5
η1
2
Figure 3.21: Node numbering convention for 9-node and 8-node quadrilateral AGelements.
To obtain an 8-node serendipity-type quadrilateral AG element which can be computationally attractive, one may start from the 9-node element and employ a node− ejection technique. As an example, one may subtract 41 N (9) from shape functions N (1) through N (4) , and add 12 N (9) to shape functions N (5) through N (8) above. Since the 9-node shape functions of the previous section sum to unity, the 8-node shape functions so constructed will also sum to unity, as this approach is equivalent to distributing N (9)
108 to the other eight nodes. The 8-node AG-edge element shape functions so derived are (1) NAG−8E (η1 , η2 )
(2)
−1 − η1 − (1 − 2γ0 )(1 − η1 )η2 + 2γ(η1 ) (1 − η2 ) = 2(1 − 2γ0 ) 2
NAG−8E (η1 , η2 ) = (3) NAG−8E (η1 , η2 )
(4) NAG−8E (η1 , η2 )
(5) NAG−8E (η1 , η2 )
−1 − η1 − (1 − 2γ0 )(1 + η1 )η2 + 2γ(η1 ) (1 − η2 ) 2(1 − 2γ0 ) 2
−1 − η1 + (1 − 2γ0 )(1 + η1 )η2 + 2γ(η1 ) (1 + η2 ) = 2(1 − 2γ0 ) 2
−1 − η1 + (1 − 2γ0 )(1 − η1 )η2 + 2γ(η1 ) (1 + η2 ) = 2(1 − 2γ0 ) 2 1 + η1 − 2γ(η1 ) (1 − η2 ) = (1 − 2γ0 ) 2
(6)
NAG−8E (η1 , η2 ) = (1 − η22 ) (7) NAG−8E (η1 , η2 )
(8)
(1 + η1 ) 2
1 + η1 − 2γ(η1 ) (1 + η2 ) = (1 − 2γ0 ) 2
NAG−8E (η1 , η2 ) = (1 − η22 )
(1 − η1 ) , 2
(3.35)
109 and the 8-node AG-corner element shape functions are (1)
NAG−8C (η1 , η2 ) = 4[γ01 (η1 − 1) − η1 + γ(η1 )][γ02 (η2 − 1) − η2 + γ(η2 )] − [1 + η1 − 2γ(η1 )][1 + η2 − 2γ(η2 )] 4(1 − 2γ01 )(1 − 2γ02 ) (2)
NAG−8C (η1 , η2 ) = 4[−γ01 (η1 + 1)+γ(η1 )][γ02 (η2 − 1) − η2 + γ(η2 )] − [1 + η1 − 2γ(η1 )][1 + η2 − 2γ(η2 )] 4(1 − 2γ01 )(1 − 2γ02 ) (3)
NAG−8C (η1 , η2 ) = 4[γ(η1 ) − γ01 (η1 + 1)][γ02 (η2 − 1) − η2 + γ(η2 )] − [1 + η1 − 2γ(η1 )][1 + η2 − 2γ(η2 )] 4(1 − 2γ01 )(1 − 2γ02 ) (4)
NAG−8C (η1 , η2 ) = 4[γ01 (η1 − 1) − η1 + γ(η1 )][γ(η2 ) − γ02 (η2 + 1)] − [1 + η1 − 2γ(η1 )][1 + η2 − 2γ(η2 )] 4(1 − 2γ01 )(1 − 2γ02 ) (5) NAG−8C (η1 , η2 )
1 + η1 − 2γ(η1 ) (1 − η2 ) = (1 − 2γ01 ) 2 1 + η2 − 2γ(η2 ) (1 + η1 ) (6) NAG−8C (η1 , η2 ) = (1 − 2γ02 ) 2 (7) NAG−8C (η1 , η2 )
(8) NAG−8C (η1 , η2 )
1 + η1 − 2γ(η1 ) (1 + η2 ) = (1 − 2γ01 ) 2 1 + η2 − 2γ(η2 ) (1 − η1 ) = (1 − 2γ02 ) 2
(3.36)
Examples of the foregoing set of 8-node edge and corner AG element shape functions with sharp edge gradients are shown in Figures 3.22 and 3.23, respectively. Alternative 8-node and other reduced AG elements can also be derived via different mathematical philosophies.
110
N5
N2
N1 1
1
1
0
0
0
1 −1
0
η1
1
0 −1 η2
−1
0
η1
1
0 −1 η2
1
0 η −1 2
1 0
η1
1
0 −1 η2
N8
N6 1
1
0
0
1
1 −1
1 −1
0
η1
1
0 η −1 2
−1
0
η1
Figure 3.22: 8-node AG-Edge element shape functions for (m, n) = (1, 10). N3
N2
N1 1 0
1
1
0
0 1 −1
1 −1
0
η1
1
0 −1 η2
−1
0
η1
1
0 −1 η2
0
η1
1
0 −1 η2
N6
N5 1
1
0
0
1
1 −1
1
0
η1
1
0 −1 η2
−1
0
η1
1
0 −1 η2
Figure 3.23: 8-node AG-Corner element shape functions for (m, n) = (1, 10).
111
3.4
Numerical Quadrature of Weakly-Singular and Nearly-Singular Integrals in Boundary Element Methods While significant effort has been devoted to the topic of singular integrals in
BIE/BEM, it has been shown that the mathematical singularities introduced into the BIE by the use of singular fundamental solutions of the non-singular governing equations of elasticity (as opposed to physical singularities introduced by boundary conditions of mixed BVPs) can be completely removed. The approach is to further regularize the nearly-weakly singular integrals of the regularized format by using either analytical integration (Cruse and Aithal, 1993) or properties of the fundamental solutions (Liu and Rudolphi, 1999) to generate a completely non-singular BIE/BEM formulation. However, the former approach precludes the use of curved elements as it is not possible for quadratic and higher order elements (Cruse and Aithal, 1993), and the latter raises some questions regarding the theoretical requirement of a higher order (C 1 ) continuity of the displacement field (see Liu and Rudolphi, 1999), and may not have the merit suitable for the BEM procedure, according to Liu (2000). Cauchy-, weakly-, hyper- and nearly-singular integrals in the BEM have been studied extensively, and a wide range of analytical and numerical approaches have been used to improve their rate of convergence (Huang and Cruse, 1993; Sladek and Sladek, 1998). Some examples include polar coordinate transformation followed by analytical integration for the case of Kelvin’s solution used with flat elements with linear interpolation (Cruse, 1974), and projection onto auxiliary surfaces of integration (Liu et al., 1993). Other examples include polar transformations on a local tangent plane going through the node nearest the source point followed by either Taylor expansions with semi-analytical integration for flat surfaces (Cruse and Aithal, 1993) or radial variable transforms for curved surfaces (Hayami, 2005). Additionally, numerous h-adaptive and p-adaptive Gaussian integration strategies have been utilized, as well as the use of a
112 degenerate mapping of a triangle to a square for which the Jacobian approaches zero at the pole, used with quadratic elements by Lachat and Watson (1976). A novel lineintegral approach for homogeneous media was proposed by Liu et al. (1993) in which Stokes’ theorem and properties of the solid angle integral are used to transform the nearly-singular terms into line integrals over the contour boundary of the surface element, although its extension to the more complex multi-layered Green’s functions of interest in this study (see Pak and Guzina, 2002) is unclear, as is also the case for the numerous semi-analytical integration approaches presented in the literature. Following an analytical regularization of the singular integrals in the BIE by the one-term Taylor series displacement field subtraction as in Pak and Guzina (1999) (see also Rizzo and Shippy, 1977; Liu and Rudolphi, 1991; Liu, 2000), the resulting integrals comprising the regularized boundary integral equation (3.1) are at most weakly singular at the collocation point, and are therefore integrable by ordinary Gauss integration. As demonstrated earlier in this chapter, however, substantial numerical effort can still be required for uniform integration accuracy in establishing the H and G matrices, depending on the particular topology of the boundary element mesh. The integrals constituting the regularized BIE will be weakly-singular over elements whose integration domains contain the collocation point, but can be nearly-singular over adjacent elements which do not contain the collocation point. If the element’s aspect ratio deviates significantly from unity, the surfaces of such weakly-singular integrands can also become folded and skewed. This is because the Taylor expansion of the displacement field will be multiplying the singular traction-Green’s function, which can create a sharply varying integrand which is forced to zero at the collocation point (Pak et al., 2004). Furthermore, the strength of the nearly-singular integrands will generally increase with element slenderness ratio for thin-body problems, or proximity to the boundary for calculation of stresses at internal points (Huang and Cruse, 1993), making the numerical integration of the nearly-singular integrals sometimes even more computationally de-
113 manding than the weakly-singular integrals after regularization (Hayami, 1990; Cruse and Aithal, 1993). As discussed above, many of the previously used techniques for addressing weaklysingular integrals in BEM are either not valid for, or can not easily be extended to, the multi-layered Green’s functions and general isoparametric and sub-parametric element unknown interpolations for higher-order and curved elements. However, to accelerate convergence and improve the accuracy of the numerical evaluation of both weaklysingular and nearly-singular integrals over higher-order elemental surfaces featured in the discretized version of the regularized BIE used in this study (i.e. Eqn. (3.6)), a powerful technique is the use of analytical mappings or changes of variables. To deal with such issues in this study, parent elements are therefore subdivided, if necessary, into triangular regions which are then mapped to quadrilaterals, similar to the approaches of Lachat and Watson (1976), Luchi and Rizzuti (1987) and Guzina et al. (2006). Specifically, a parent element’s bi-unit square domain is first subdivided, depending on the location of the collocation node, into a suitable number of triangular sub-regions with common vertexes located at the weakly singular collocation point. For each of the subdivisions, use is made of singular transformations from the triangular sub-regions to quadrilateral regions which, by the collapsing of an edge to a point coinciding with the collocation node, generates a Jacobian which aids the integration. A similar strategy was also implemented for triangular elements in this investigation, as will be explained in the following sections. To speed the convergence of the nearly-singular integrals over the “nearby–element” domains described in Section 3.2.3, the above subdivision-singular transformation approach, originally applied in the program BEASSI only to the weakly-singular integrals of elements which actually contain the collocation node, was also extended to the nearlysingular integrals over elements in close proximity to the collocation node, with the pole of the transformation placed at the element node closest to the collocation node. As will
114 be demonstrated in the following sections, the proposed treatment can aptly accelerate the convergence of the nearly-singular integrals over such nearby-element domains as well.
3.4.1
Singular Transformations for Quadrilateral Elements To illustrate how the approach is implemented for quadrilateral elements, the
case of having the collocation (i.e. a weak singularity) at the corner node k = 4 of a bi-unit square in the parent η1 –η2 domain is considered (see Fig. 3.24(a)). The element is subdivided into two triangular regions, A and B, having a common vertex at the collocation point. To describe Triangle A in the η 1 –η2 domain in Fig. 3.24(a), one may consider the triangle as the limiting case of a quadrilateral shown in Fig. 3.24(b) with node 20 collapsed to node 30 (coincident with node k = 4), during a mapping from the η 10 – η20 to the η1 –η2 domain. To arrive at a generalized expression valid for collocation at all four corners, the η10 -axis will always be oriented in the η 1 –η2 domain such that it points towards the collocation point after collapsing the edge η 10 = 1. Furthermore, the η10 -axis before collapsing will be oriented parallel to the η 2 axis for Triangle A, and parallel to the η1 axis for Triangle B. In Fig. 3.24, nodes k = 1, 2, 3, 4 in the η 1 –η2 domain correspond to the nodes of the physical element, and are associated with the element interpolation functions N (1) (η1 , η2 ) through N (4) (η1 , η2 ) of a 4 to 9 node quadrilateral element. The auxiliary nodes 10 , 20 , 30 , 40 of Fig. 3.24(b) and Fig. 3.24(c) can be used to formulate the mapping of the square η10 –η20 domains onto the triangular η1 –η2 sub-domains A and B in terms of the four standard bi-linear interpolation functions (1 − η10 ) (1 − η20 ) , 2 2 (1 + η10 ) (1 − η20 ) 0 N (2 ) (η10 , η20 ) = , 2 2 (1 + η10 ) (1 + η20 ) 0 N (3 ) (η10 , η20 ) = , 2 2 (1 − η10 ) (1 + η20 ) 0 . N (4 ) (η10 , η20 ) = 2 2 0
N (1 ) (η10 , η20 ) =
(3.37)
115 Table 3.1: Nodal correspondence between triangular domain A of Fig. 3.24(a) and its quadrilateral image in Fig. 3.24(b) for the degenerate mapping. Triangle A nodal coords. orig. node, k (η1 , η2 ) 1 2 4
Quadrilateral A nodal coords. image node (η10 , η20 ) 40 10 2 0 , 30
(−1, −1) (1, −1) (−1, 1)
(−1, 1) (−1, −1) (1, −1), (1, 1)
For the orientation of quadrilateral domain A shown in Fig. 3.24(b) for example, the nodal correspondence in the degenerate mapping between the triangular domain A and its quadrilateral image in the η10 –η20 domain is given in Table 3.1. With the aid of (3.37), the analytical expression for the degenerate mapping between the two domains may be written in closed form as (1)
0
(2)
0
(4)
ηi (η10 , η20 ) = ηi N (4 ) + ηi N (1 ) + ηi (k)
in which ηi
0 0 N (2 ) + N (3 ) ,
i = 1, 2,
(3.38)
are the coordinates of node k in the original η 1 –η2 domain. Written out
explicitly, Eqns. (3.38) with the nodal data of Table 3.1 yield
which simplify to
0 0 0 0 η1 (η10 , η20 ) = −N (4 ) + N (1 ) − N (2 ) + N (3 ) 0 0 0 0 η2 (η10 , η20 ) = −N (4 ) − N (1 ) + N (2 ) + N (3 ) ,
η1 (η10 , η20 ) = −η20 η2 (η10 , η20 )
=
(3.39)
(1 − η10 ) (1 + η10 ) 1 − = −1 1 − (1 − η10 )(1 − η20 ) 2 2 2
(3.40)
η10 .
Similarly, for triangle B with coordinates oriented as shown in Fig. 3.24(c) which results in the nodal correspondence given in Table 3.2, the degenerate mapping from the η 10 –η20 to the η1 –η2 domain can be written as (2)
0
(3)
0
(4)
ηi (η10 , η20 ) = ηi N (4 ) + ηi N (1 ) + ηi
0 0 N (2 ) + N (3 ) ,
i = 1, 2,
(3.41)
116
Original bi-unit square domain η2
Collocation point= pole of transformation
20 (1, −1)
k = 4(−1, 1)
B k = 3(1, 1)
B
η10 B
(a)
k=1 A (η1 = −1, η2 = −1)
η10 B
η1
(c)
k = 2(1, −1)
Collapse
η10 A
η20 B 40 (−1, 1)
30 (1, 1)
30 (1, 1)
Parent of Triangle B
20 (1, −1) Collapse
η10 A
η10 A = const. η20 A = const.
η20 A
(b)
10 (η10 = −1, η20 = −1)
A 0
4 (−1, 1)
10 (η10 = −1, η20 = −1)
Parent of Triangle A
Figure 3.24: Degenerate triangular subdivision and mapping for integration of a weak singularity having a pole at corner node k = 4 of a bi-unit square domain. η2
η2 4
7
4
3
η2
7
3
(a)
8
η10
A
9
6
η1
9
8
2
5
7
3
η10
η1
8
η1
8
BI BII
2
1
3
2
1
η1
8
6
8
5
2
1
A 1
4
3
7
3
9
BI 6 BII
AI 6
9
η1
8
AII 5
2
5 η2
BII
AI
η1
A
2
7 BI
B
6
η10 9
A
5
4
AII
BII 5
6
η2
9
3 η10
B
9 η10
A
AI 6
BI 1
6
7
4
AII 9
8
η10
A
5
7 B
η2
AI
4
A
1
η2 4
3
B
B 1
η10
7 B
η10 B
η10 B
(b)
4
A
A
η2
η1
AII 2
1
5
2
η2 4
7 AI
(c)
8
BI BIV
9
AIV 1
3
=Collocation node / pole of transformation
AII BII 6 BIII
η1
AIII 5
2
Figure 3.25: Degenerate triangular subdivisions for collocation at nodes 1–9 of quadrilateral element. Collocation at (a): corner nodes, (b): mid-side nodes, (c): center node.
117 Table 3.2: Nodal correspondence between triangular domain B of Fig. 3.24(a) and its quadrilateral image in Fig. 3.24(c) for the degenerate mapping. Triangle B nodal coords. orig. node, k η1 –η2 2 3 4
Quadrilateral B nodal coords. image node, m0 (η10 , η20 ) 40 10 2 0 , 30
(1, −1) (1, 1) (−1, 1)
(−1, 1) (−1, −1) (1, −1), (1, 1)
or explicitly as 0 0 0 0 N (4 ) + N (1 ) − N (2 ) + N (3 ) 0 0 0 0 η2 (η10 , η20 ) = −N (4 ) + N (1 ) + N (2 ) + N (3 ) .
η1 (η10 , η20 ) =
(3.42)
In turn, Eqs. (3.42) simplify to η1 (η10 , η20 ) = −η10 η2 (η10 , η20 ) = −η20
1 (1 − η10 ) (1 + η10 ) + = 1 1 − (1 − η10 )(1 + η20 ) . 2 2 2
(3.43)
Utilizing the same sub-division philosophy and repeating the degenerate mapping for all other corner collocation nodes (i.e. k = 1, 2, 3 of Fig. 3.24(a)) under the condition that the η10 -axis of all bi-unit square domains points toward the chosen pole after collapsing, one can summarize the results in the form of 1 (k) (k) (k) 0 0 1 − (1 − η1 )(1 + η1 η2 η2 ) , η1 = η 1 2 1 (k) (k) (k) 0 0 η2 = η 2 1 − (1 − η1 )(1 − η1 η2 η2 ) , 2
(k)
η2 = η2 η10
for Triangle A, (3.44)
η1 =
(k) η1 η10
for Triangle B,
with the convention that the η10 –axis of the auxiliary square domain before collapsing is parallel to the η2 –axis for Triangle A, and parallel to the η 1 –axis for Triangle B. The determinant of the Jacobian of the degenerate triangular mapping (3.44) for both triangles A and B is |JT r | ≡
∂η1 ∂η10
∂η2 ∂η10
∂η1 ∂η20
∂η2 ∂η20
1 = (1 − η10 ), 2
(3.45)
118 which is linear in η10 and goes to zero at the pole η10 = 1. For collocation at a quadrilateral element’s mid-side or internal node, the element is first subdivided about the collocation node into 2 or 4 rectangular sub-regions, respectively (see Fig. 3.25(b) and (c) for a summary of the subdivision strategy). Each rectangular sub-region is then mapped to the appropriate bi-unit square region of Fig. 3.25(a) followed by the degenerate mapping of Eqn. (3.44), such that the pole of the transformation is placed at the collocation node. To understand the usefulness of the proposed analytical mapping, recall from calculus that the double integral of a function f (η 1 , η2 ) under a change of variables or mapping such as that given by (3.44) is transformed according to Z
1 −1
Z
1
f (η1 , η2 )dη1 dη2 = −1
+
Z
Z
1
Z
1
−1 −1 1 Z 1
−1
−1
f (η1 (η10 A , η20 A ), η2 (η10 A , η20 A ))|JT rA |dη10 A dη20 A
(3.46)
f (η1 (η10 B , η20 B ), η2 (η10 B , η20 B ))|JT rB |dη10 B dη20 B .
Properly chosen, the Jacobian of transformation (3.44) can be utilized to mitigate the strength of the weakly-singular integrands in the same manner as a true polar-coordinate transformation. To see this, a true polar-coordinate transformation of the η 1 –η2 domain of Fig. 3.24(a) with the pole or origin at node k = 4 can be written as p −1 η2 − 1 2 2 θ(η1 , η2 ) = tan , r(η1 , η2 ) = (η1 + 1) + (η2 − 1) , η1 + 1 −1 ≤ η1 ≤ 1,
(3.47)
− 1 ≤ η2 ≤ 1,
the inverse of which may be described by η1 (r, θ) = r cos(θ) − 1, −2 , sin(θA ) 2 , 0≤r≤ cos(θB ) 0≤r≤
η2 (r, θ) = r sin(θ) + 1, −π −π ≤ θA ≤ , 2 4 −π ≤ θB ≤ 0, 4
(3.48)
in which θA and θB are the angular coordinates of Triangular sub-regions A and B, respectively. The surface integral of a function f (η 1 , η2 ) under this true polar-coordinate
119 mapping is therefore transformed according to Z
1
−1
Z
1
f (η1 , η2 )dη1 dη2 =
−1
+
Z Z
−π 4 −π 2
0 −π 4
Z
−2 sin(θA )
f (η1 (r, θA ), η2 (r, θA )) r drdθA
0
Z
2 cos(θB )
(3.49) f (η1 (r, θB ), η2 (r, θB )) r drdθB ,
0
in which the Jacobian determinant of transformation, ∂η1 ∂η2 ∂r ∂r |J| ≡ =r ∂η1 ∂η2 ∂θ ∂θ
(3.50)
can serve to mitigate integrable singularities in 2D, defined as those having order
O(1/r α ) with α < 2. The relation between the polar-coordinates and the degenerate quadrilateral coordinates may be obtained by first solving for the inverse of mapping (3.40) for Triangle A, i.e. η10 (η1 , η2 ) = η2 ,
η20 (η1 , η2 ) =
1 + η2 + 2η1 , η2 − 1
(3.51)
and the inverse of mapping (3.43) for Triangle B, i.e. η10 (η1 , η2 ) = −η1 ,
η20 (η1 , η2 ) =
1 − η1 − 2η2 . η1 + 1
(3.52)
Combining the above two expressions with (3.48), one obtains η10 (r, θA ) = 1 + r sin(θA ),
η20 (r, θA ) = 1 + 2 cot(θA )
(3.53)
η20 (r, θB ) = −(1 + 2 tan(θB ))
(3.54)
for Triangle A, and η10 (r, θB ) = 1 − r cos(θB ), for Triangle B, from which −r sin(θA ) 1 (1 − η10 (r, θA )) = , 2 2 1 r cos(θB ) |JT rB | = (1 − η10 (r, θB )) = , 2 2 |JT rA | =
(3.55)
so that the Jacobian of the degenerate mapping, while dependent on θ, is linearly proportional to r along lines of constant θ (i.e. lines of constant η 20 ). In the same manner
120 as the translated polar-coordinate transformation (η 1 + 1, η2 − 1) = (r cos θ, r sin θ) of Eqn. (3.48) is singular because it collapses the line (r, θ) = (0, θ) in the r–θ domain to the point (η1 , η2 ) = (−1, 1) in the η1 –η2 domain, the degenerate transformation of Eqn. (3.44) is singular in that it collapses the line (η 10 , η20 ) = (1, η20 ) to the point (k)
(k)
(η1 , η2 ). The degenerate transformation is therefore an alternative to the use of a polar-coordinate transformation, the latter requiring a variable upper limit of integration r(θ) in the radial direction, as employed in Cruse (1974), Rizzo and Shippy (1977) and Hayami (2005), among others. Another advantage of the degenerate mapping is that the resulting integrals in Eqn. (3.46) are defined over bi-unit square regions, and ordinary Gauss quadrature can thus be performed immediately without any further transformation.
3.4.2
Numerical Examples of Nearly-Singular and Weakly-Singular Integral Treatment As discussed in Section 3.2.3, integrals over elements that are near the collocation
node can be even more difficult to integrate numerically than those containing the collocation node (Hayami, 1990; Cruse and Aithal, 1993), a fundamental concern for mesh refinement. As discussed earlier, the degenerate triangular mappings given in Eqn. (3.44) for weakly-singular integrals over elements containing the collocation node can also be used to improve the convergence of the nearly-singular integrals over elements that are close to the collocation node. Such elements are represented by the second group of summations in the discretized and regularized BIE of Eqn. (3.6). These integrands, having the forms (m) ˆ k (ξ, y(n) ), NiT (ξ) U i
(3.56)
(m) NiU (ξ) [Tˆik (ξ, y(n) ; n)]1 ,
(3.57)
121 and [Tˆik (ξ, y(n) ; n)]1
(3.58)
for static problems, have generally strong variations, with the first integrand becoming nearly-singular, and the last two integrands becoming nearly-strongly-singular as the
collocation point approaches the elemental domain of integration Γe . As an illustration of this nearly-singular behavior, selected components of the above three integrands for a planar, square 8-node serendipity element of half-width w are plotted in Figs. 3.26 to 3.28 for collocation at a point y (n) in the plane of the element, located a distance |y(n) − ξ (7) | = 0.25w from node 7. These plots illustrate the complexity and localization of the integrand surfaces, which will slow the integration convergence in the original η 1 –η2 domains. As shown in these figures, applying the triangular subdivision and degenerate transformation about a pole corresponding to the element node nearest the source point can greatly alleviate the problem by smoothing the integrands in the η 10 –η20 domains. The Gauss rules required for integration of the surfaces shown in these three figures using a tolerance of 0.1% are tabulated for the case of the source point approaching node 7 in the plane of the element in Table 3.3 (tangential approach), and for the more difficult case of the source point approaching the element along its normal at node 7 in Table 3.4 (normal approach), for which the term in Kelvin’s traction solution involving the normal derivative is activated (Cruse and Aithal, 1993). As the distance between node 7 and the collocation point decreases, integration in the degenerately transformed η 10 –η20 domains is seen to be increasingly more efficient than integration in the original η 1 –η2 domain. Furthermore, the order of the Gauss rules increases only gradually with decreasing distance, demonstrating the robustness of the performance.
122
Table 3.3: Gauss points required for integration of BIE integrals shown in Figs. 3.26 to 3.28 with integration tolerance of 0.1%. Unit point load in plane of element (tangential approach). w = element half-width.
Pole distance |y(n) − ξ (7) | 0.25w 0.2w 0.1w
Gauss points required η1 –η2 domain η10 –η20 domain (No mapping) (AI + BI + AII + BII ) 152 = 225 182 = 324 342 = 1156
82 + 72 + 82 + 62 = 213 82 + 72 + 82 + 72 = 226 112 + 92 + 112 + 92 = 404
Table 3.4: Gauss points required for integration of BIE integrals shown in Figs. 3.26 to 3.28 with integration tolerance of 0.1%. Unit point load located on element normal vector at node 7 (normal approach). w = element half-width.
Pole distance |y(n) − ξ (7) | 0.25w 0.2w 0.1w
Gauss points required η1 –η2 domain η10 –η20 domain (No mapping) (AI + BI + AII + BII ) 192 = 361 232 = 529 452 = 2025
112 + 102 + 112 + 102 = 442 132 + 112 + 132 + 112 = 580 172 + 132 + 172 + 152 = 972
123
Source point y(n)
1
8 AI AII
5
ξ3
0 2
−1 −1
6
4
BI
N(3) UKelvin |JG| T 3
1
7
BII
−1
3 0
0 ξ 2
1
1
0.04 0.02 0
ξ1
0
−1
|JG||JS||JTr|
0
−1
−1 0 0
η′2
1
η′1
1
−3 −1
0 1
′ 1 η1
(e) Quadrilateral domain AII
N(3) UKelvin |JG||JS||JTr| T 3
N(3) UKelvin |JG||JS||JTr| T 3
−1
η′2
0 0
η′2
1
η′1
1
(d) Quadrilateral domain BI
x10−3
0
−1
−2
(c) Quadrilateral domain AI
PSfrag replacements
η1
x10−3
N(3) UKelvin T 3
N(3) UKelvin |JG||JS||JTr| T 3
x10−3
0 −0.5 −1 −1.5 −2 −2.5 −1
0 η 2
1 1 (b) bi−unit square subdivision in natural coords
(a) original square element in global coords
0 −0.5 −1 −1.5 −2 −2.5 −1
−1
0.02 −1
0.01 0
0 −1
0
η′2
1
1
η′1
(f) Quadrilateral domain BII (3)
Figure 3.26: Product of standard serendipity shape function N T and displacement Green’s function component U3Kelvin for unit point load in plane of element acting in ξ3 -direction near node 7. Distance |y (n) −ξ (7) | = 0.25.
124
Source point y(n)
1
8 AI AII
5
ξ3
0 2
−1 −1
6
4
BI
N(3) TKelvin |JG| U 2
1
0.02 0.01 0 −0.01
7
BII
−1
3 0
0 ξ 2
1
1
ξ1
−1
η′2
1
η′1
1
−5
0 η′2
1
0
−1
1
(e) Quadrilateral domain AII
η′1
N(3) TKelvin |JG||JS||JTr| U 2
N(3) TKelvin |JG||JS||JTr| U 2 PSfrag replacements
−1
0
−1
−10
0
η′2
1
1
η′1
(d) Quadrilateral domain BI
x10−4
−1
η1
x10−4 0
(c) Quadrilateral domain AI
0 −2 −4 −6 −8 −10
η2
−15
0 0
0
1 1 (b) bi−unit square subdivision in natural coords
N(3) TKelvin |JG||JS||JTr| U 2
N(3) TKelvin |JG||JS||JTr| U 2
−1
0
−1
(a) original square element in global coords
x10−4 0 −2 −4 −6 −8 −10
−1
x10−3 3 2 1 0
−1
−1 0 0
η′2
1
1
η′1
(f) Quadrilateral domain BII (3)
Figure 3.27: Product of standard serendipity shape function N U and traction Green’s function component T2Kelvin for unit point load in plane of element acting in ξ 3 -direction near node 7. Distance |y (n) −ξ (7) | = 0.25.
125
Source point y(n)
AI AII
5
ξ3
0 2
−1 −1
6
4
BI 7
BII
−1
3
0
ξ2
TKelvin |JG||JS||JTr| 2
1 1 (a) original square element in global coords
ξ1
−1
−1 0 0
η′′′ 2
1
1
η′′′ 1
PSfrag replacements
8 x10−3 −1
4 0 1
1 η′′′ 2 (e) Quadrilateral domain AII
η2
η1
20 x10−3 −1
10 0 −1
0 0
η′′′ 2
1
1
η′′′ 1
(d) Quadrilateral domain BI
η′′′ 1
TKelvin |JG||JS||JTr| 2
TKelvin |JG||JS||JTr| 2
(c) Quadrilateral domain AI
0
0
1 1 (b) bi−unit square subdivision in natural coords
4
0 −1
−1
0.1 0
8 x10−3
0 −1
0.2
0
TKelvin |JG||JS||JTr| 2
1
8
TKelvin |JG| 2
1
20 −3 x10 −1
10 0 −1
0 0
η′′′ 2
1
1
η′′′ 1
(f) Quadrilateral domain BII
Figure 3.28: Traction Green’s function component T 2Kelvin for unit point load in plane of element acting in ξ3 -direction near node 7. Distance |y (n) −ξ(7) | = 0.25.
126 3.4.3
Singular Transformations for Triangular Elements For intrinsic triangular element domains, the conventional areal coordinates (ζ 1 , ζ2 , ζ3 )
of straight sided triangular domains are relevant and related to the 2D Cartesian coordinates (x, y) through (1) x(2) x(3) x x (1) y = y (2) y (3) y 1 1 1 1
ζ1 ζ2 , ζ3
0 ≤ ζi ≤ 1,
(3.59)
where (x(i) , y (i) ), i = 1, 2, 3 are the Cartesian coordinates of the triangle’s vertexes. The
areal coordinates may be expressed in terms of the Cartesian coordinates as (2) (3) x x(3) −x(2) x(2) y (3) −x(3) y (2) ζ1 y −y 1 (3) (1) (1) (3) (1) (3) (1) (3) = y , x −x y x −x y ζ2 2A y −y (1) (2) (2) (1) (1) (2) (1) (2) ζ3 1 y −y x −x x y −y x
(3.60)
where the signed area A of the triangle (positive for counterclockwise vertex numbering) is given by
x(1)
2A = det y (1) 1
x(2)
x(3)
y (2)
y (3)
1
1
.
(3.61)
For triangular surface elements with sufficiently regular integrands in the 3D Cartesian coordinates (ξ1 , ξ2 , ξ3 ), integration in the standard triangular domain (Fig. 3.29) via the areal coordinates (ζ1 , ζ2 , ζ3 ) can be achieved by first using the global-to-parent element mapping ξi
(ζ1 , ζ2 , ζ3 ) =
me X
(m)
ξi
(m)
NG (ζ1 , ζ2 , ζ3 ),
i = 1, 2, 3
(3.62)
m=1 (m)
in which the element’s nodal Cartesian coordinates ξ i (m)
tion functions NG
and the geometry interpola-
for me = 3, 6, or 7–noded triangles together define the Cartesian
coordinates ξi of the interpolated elemental geometry. The Jacobian determinant of the
(a)
ζ2
Global to parent element
∂ξ/∂ζ2
ˆ2 ξ2 , e
ζ3 = const.
6(1/2,0,1/2)
∂ξ/∂ζ1
3
3(0,0,1)
ζ2 = const.
4
7
ζ3
ζ3 = 1 − ζ 1 − ζ 2
Mapping 1:
2 5
127
(b)
6
5(0,1/2,1/2) 7(1/3,1/3,1/3)
1
ξ
2(0,1,0)
1(1,0,0)
ζ1
ζ1
ˆ1 ξ1 , e ˆ3 ξ3 , e
4(1,2/1,2,0)
ζ2
ζ1 = const.
Figure 3.29: Transformation of triangular surface elements to standard triangular parent domain in areal coordinates. (a): Physical element surface in global Cartesian domain (ξ1 , ξ2 , ξ3 ), (b): Parametrized element surface in areal coordinate domain (ζ 1 , ζ2 , ζ3 ).
surface parametrization, or mapping, from three independent global Cartesian coordinates to two independent areal coordinates in Eqn. (3.62) can be determined by (see Brebbia et al., 1984) e ˆ1 ∂ ξ ∂ ξ = ∂ ξ1 × |JG | ≡ ∂ζ1 ∂ζ ∂ζ 2 1 ∂ ξ1 ∂ζ2
ˆ2 e
ˆ3 e
∂ ξ2 ∂ζ1
∂ ξ3 ∂ζ1
∂ ξ2 ∂ζ2
∂ ξ3 ∂ζ2
=
v 2 2 2 u u u ∂ξ2 ∂ξ3 ∂ξ3 ∂ξ2 + ∂ξ3 ∂ξ1 − ∂ξ1 ∂ξ3 + ∂ξ1 ∂ξ2 − ∂ξ2 ∂ξ1 , t − ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ2
(3.63)
ˆi are unit-vectors for the ξi –axes. Because the areal coordinates (ζ 1 , ζ2 , ζ3 ) where e represent a parametrization of a 2D surface, any two may be considered independent, with the remaining coordinate defined implicitly by the last of equations (3.59). One may therefore use the chain rule in (3.63) to obtain alternate expressions for the Jacobian determinant in terms of any two of the areal coordinates as well, i.e. ∂ ξ ∂ ξ ∂ ξ ∂ ξ ∂ ξ ∂ ξ = = × × × |JG | = . ∂ζ1 ∂ζ2 ∂ζ2 ∂ζ3 ∂ζ3 ∂ζ1
(3.64)
128 The interpolated geometry of Eqn. (3.62) is then used as an argument to the interpolation and Green’s functions of the BIE (3.6) for performing numerical quadrature using Gauss rules defined in terms of areal coordinates. As proven in previous sections, weakly-singular and nearly-singular integrands over quadrilateral elements can be effectively treated by subdividing the element into triangular sub-domains which are mapped to square domains in a Cartesian coordinate system. A similar subdivision and degenerate mapping technique can likewise be applied to 3, 6 and 7–node triangular elements having interpolation functions defined in terms of the areal coordinates (ζ1 , ζ2 , ζ3 ) ∈ [0, 1]. For triangular elements, the parent element domain in areal coordinates is divided into 1, 2, or 3 triangular sub-regions for corner, mid-side and center collocation nodes, and each such sub-region is degenerately mapped to a square quadrilateral domain of which two corner nodes are collapsed to their midpoint. In addition to the regularization of the resultant integrand afforded by such transformations, the mapping of a triangle to a quadrilateral allows a circumvention of the upper limit of known areal-coordinate Gauss rules of 73 points for a triangular domain, as mentioned previously (Dunavant, 1985). To illustrate the approach more fully, consider the case of a weak singularity having a pole at corner node 3 of a triangular element (see Fig. 3.30(a)). Analogous to the treatment of quadrilateral elements in the previous section, one may parametrize the triangular region defined in the Cartesian η 1 –η2 frame through a degenerate transformation from a square η10 –η20 domain during which nodes 20 and 30 are collapsed to their mid-point, to coincide with the vertex node 3 in the original triangular element domain as illustrated in Figs. 3.30(a) and (b). Stated precisely in the format of nodal correspondence in Table 3.5, this transformation may be written using the bilinear interpolation
η2
(a)
129
(c)
Pole 3, 20 , 30 (0, 1)
η2 ζ3 3(0, 1)
η10 η1
η1
η20
1, 40 (−1, −1)
ζ1
(b)
(d) 0
2(1, −1)
1(−1, −1)
2, 10 (1, −1)
ζ2
ζ3
0
3 (1, 1)
2 (1, −1)
3(0,0,1)
ζ2 = const.
ζ3 = const.
η10 η20
η10 = const. η20 = const. 1(1,0,0)
40 (−1, 1)
10 (−1, −1)
ζ1
2(0,1,0)
ζ2
ζ1 = const.
Figure 3.30: Two possible transformations for a triangular region defined in the Cartesian (η1 , η2 ) frame. Left: Singular transformation of the triangular region (a) to a quadrilateral domain (b). Right: Nonsingular transformation of the triangular region (c) to intrinsic triangular areal coordinate domain (d).
functions of Eqn. (3.37) as (1)
0
(2)
0
(3)
ηi (η10 , η20 ) = ηi N (4 ) + ηi N (1 ) + ηi − 1 ≤ ηi0 ≤ 1,
−1 ≤ ηi ≤ 1,
0 0 N (2 ) + N (3 ) ,
(3.65)
i = 1, 2.
To apply this transformation to standard triangular elements which have interpolation functions defined in terms of the areal coordinates, a relation for the transformation from areal coordinates (ζ1 , ζ2 , ζ3 ) to degenerate quadrilateral coordinates η 10 –η20 is necessary. For this purpose, one may make use of Eqn. (3.59) to relate the areal coordinates to the
130 Table 3.5: Nodal correspondence between triangular domain of Fig. 3.30(a) and its quadrilateral image in Fig. 3.30(b) for the degenerate mapping. Triangle nodal coords. orig. node, m η1 –η2 1 2 3
Corresponding quad. nodal coords. image node, m0 (η10 , η20 ) 40 10 2 0 , 30
(−1, −1) (1, −1) (0, 1)
(−1, 1) (−1, −1) (1, −1), (1, 1)
Cartesian ones via (1)
(2)
(3)
ηi (ζ1 , ζ2 , ζ3 ) = ηi ζ1 + ηi ζ2 + ηi ζ3 , 0 ≤ ζi ≤ 1,
−1 ≤ ηi ≤ 1,
(3.66)
i = 1, 2,
as shown schematically in Fig. 3.30(c) and (d). Denoting this relation in vector notation as η(ζ), and that of Eqn. (3.65) as η(η0), one may invert the former (as in Eqn. (3.60)) to obtain the desired relation through a series of successive mappings as ζ = ζ(η(η 0 )),
(3.67)
which corresponds physically to transformations from the ζ to η to η 0 domains of Fig. 3.30(d), (a) and (b), respectively. However, rather than performing the inversion to get ζ(η), the same relation can be obtained more expediently if η is eliminated by requiring that the two physical descriptions of the triangle in the η domains of Figs. 3.30(a) and (c) in terms of η 0 and ζ correspond, by setting η(η 0 ) = η(ζ).
(3.68)
For the current case with the pole of transformation placed at node 3, this is achieved by equating (3.65) and (3.66), giving (1)
0
(2)
0
(3)
ηi N (4 ) + ηi N (1 ) + ηi
0 0 (1) (2) (3) N (2 ) + N (3 ) = ηi ζ1 + ηi ζ2 + ηi ζ3 ,
i = 1, 2, (3.69)
which more directly gives the desired transformation ζ = ζ(η 0 ) from the triangular to
131
ζ3 30 (1, 1)
3, 20 , 30 (0, 0, 1) Degenerate triangular mapping
η10
20 (1, −1) η10
η20
η10 = const. η20 = const.
η20 2, 10 (0, 1, 0)
1, 40 (1, 0, 0) ζ1
40 (−1, 1)
ζ2
10 (−1, −1)
Figure 3.31: Direct degenerate transformation having a pole at corner node 3 from areal coordinate triangular domain to Cartesian coordinate quadrilateral domain.
the Cartesian square domain as (1 − η10 ) (1 + η20 ) , 2 2 (1 − η10 ) (1 − η20 ) 0 ζ2 (η10 , η20 ) = N (1 ) = , 2 2 0 0 (1 + η10 ) , ζ3 (η10 , η20 ) = N (2 ) + N (3 ) = 2 0
ζ1 (η10 , η20 ) = N (4 ) =
|JT3 | ≡
∂ζ1 ∂η10
∂ζ2 ∂η10
∂ζ1 ∂η20
∂ζ2 ∂η20
1 = (1 − η10 ), 8
(3.70)
where |JT3 | is the determinant of the Jacobian of this transformation (see the schematic in Fig. 3.31). Repeating this procedure with the η 10 –axis directed towards a weak singularity at corner number 2 in the triangular domain gives (1 − η10 ) (1 − η20 ) , 2 2 (1 + η10 ) 0 0 , ζ2 (η10 , η20 ) = N (2 ) + N (3 ) = 2 (1 − η10 ) (1 + η20 ) 0 . ζ3 (η10 , η20 ) = N (4 ) = 2 2 0
ζ1 (η10 , η20 ) = N (1 ) =
|JT2 | =
1 (1 − η10 ), 8
(3.71)
Similarly, if the pole of the transformation is placed at the original triangle’s corner number 1, 0
(1 + η10 ) , 2 (1 − η10 ) (1 + η20 ) = , 2 2 (1 − η10 ) (1 − η20 ) . = 2 2 0
ζ1 (η10 , η20 ) = N (2 ) + N (3 ) = 0
ζ2 (η10 , η20 ) = N (4 ) 0
ζ3 (η10 , η20 ) = N (1 )
1 |JT1 | = (1 − η10 ), 8
(3.72)
The singular transformations (3.70), (3.71) and (3.72) represent in essence a stretching of the integrand away from edge 20 –30 in the η10 –η20 domain, or equivalently, a concen-
132 tration of mapped quadrilateral Gauss points near corners 1, 2 and 3 respectively, in the (ζ1 , ζ2 , ζ3 ) domain. Similar to the case of quadrilaterals in the previous sections, the linear Jacobian determinants of transformation, |J Ti | = (1 − η10 )/8, will weaken any integrable singularities having a pole at corner node i by one order. As noted earlier, because integration is performed in the quadrilateral η 10 –η20 domain, the restriction of the currently known maximum 73–point triangular Gauss rule is overcome, and quadrilateral Gauss rules of any order may be used in the adaptive integration scheme for all triangles containing the collocation point and thus having weakly-singular integrands, as well as for nearby triangles having a node within a sphere of user-specified radius r1 centered at the collocation point, which have nearlysingular integrands. The areal to Cartesian mappings given by Eqns. (3.70) to (3.72) are analogous to the formulation of the triangular element’s linear interpolation functions through degeneration of the bilinear quadrilateral interpolation functions as given in Hughes (2000), but they are also relevant as coordinate transformations in formulating higher-order triangular elements as well.
Under transformations (3.70) to (3.72), the surface integral of a function f ( ξ)
over an interpolated triangular element domain Γ is transformed according to Z 1 Z 1−ζ2 ZZ f ( ξ) dΓ = f (ξ (ζ)) |JG | dζ1 dζ2 ξ 0
0
Γe
=
Z
1
−1
Z
1
−1
0
f (ξ (ζ(η ))) |JG | |JTi |
(3.73)
dη10 dη20 ,
where ξ= (ξ 1 , ξ 2 , ξ 3 ) is the interpolated Geometry of Eqn. (3.62) in terms of the areal coordinates ζ = (ζ1 , ζ2 , ζ3 ), which are mapped to the quadrilateral coordinates η 0 = (η10 , η20 ) as ζ(η 0 ) = (ζ1 (η10 , η20 ), ζ2 (η10 , η20 ), ζ3 (η10 , η20 )), etc. While the above singular transformations are developed for collocation at a triangular element’s corner nodes, they are also useful as a foundation for treating collocation at mid-side or interior nodes. In such cases, one should first subdivide the standard triangle into two or three triangular sub-regions having corners at the collocation point.
133 Each of these sub-regions are then mapped to another standard triangle 0 ≤ ζ i ≤ 1, so that the appropriate of transformations (3.70) to (3.72) can be easily incorporated. All these scenarios can be summarized by the general case where the pole of the transforma(P )
(P )
(P )
tion is allowed to be located an arbitrary point P defined by ζ (P ) = (ζ1 , ζ2 , ζ3 ) in the conventional triangular element domain. Following the transformation of the phys
ical element geometry from the global Cartesian coordinates ξ to the plane triangular parent domain in areal coordinates ζ through Eqn. (3.62), the triangle is subdivided about the pole into three triangular sub-regions denoted by A, B and C for which the pole is at corner nodes 1A , 2B and 3C , respectively (Fig. 3.32(b)). These triangular sub-domains are then mapped to the standard triangular domains with 0 ≤ ζ iA ≤ 1, 0 ≤ ζiB ≤ 1 and 0 ≤ ζiC ≤ 1, i = 1, 2, 3 (Fig. 3.32(c)). The resulting standard triangular domains are then mapped to their corresponding mid-point collapsed quadrilateral regions −1 ≤ ηi0A ≤ 1, − 1 ≤ ηi0B ≤ 1 and −1 ≤ ηi0C ≤ 1, i = 1, 2 using the singular transformations of Eqns. (3.72), (3.71) and (3.70), respectively (Fig. 3.32(d)). For conceptual clarity, the steps for this process are summarized in Fig. 3.32. The element geometry interpolation first transforms the element from the global Cartesian
ξ coordinate domain into a planar triangular element parent domain defined in areal coordinates ζ = (ζ1 , ζ2 , ζ3 ) through Eqn. (3.62) with the Jacobian determinant |J G | of Eqn. (3.64). The linear transformation of each of the triangular sub-domains A, B and C of the domain 0 ≤ ζi ≤ 1 in Fig. 3.32(b) to the standard triangular element domains 0 ≤ ζiA ≤ 1, 0 ≤ ζiB ≤ 1 and 0 ≤ ζiC ≤ 1 of Fig. 3.32(c) is then obtained using Eqn. (3.59). Such a linear transformation between two areal coordinate domains may be obtained by noting that replacing the orthogonal Cartesian coordinates on the right-hand side Eqn. (3.59) with non-orthogonal areal coordinates results in a linear transformation between two areal coordinate domains. For example, the transformation from sub-domain A in the (ζ1 , ζ2 , ζ3 ) domain of Fig. 3.32(b) to the standard triangular
134 domain (ζ1A , ζ2A , ζ3A ) of Fig. 3.32(c) is obtained from Eqn. (3.59) as (1) (2) (3) ζ1 ζ1 ζ1 ζ 1A ζ1 0 ≤ ζiA ≤ 1, (1) (2) (3) = , ζ ζ ζ ζ ζ 2 2A 2 2 2 ζi ∈ [0, 1], 1 ζ3 1 1 1 A (i)
(3.74)
(i)
where (ζ1 , ζ2 ), i = 1, 2, 3 are the areal coordinates triangular sub-domain A’s vertices.
Because ζi are themselves areal coordinates, one can replace the last component of the vector on the left hand side of the above with (ζ 1 + ζ2 + ζ3 ), which, when combined with the first two equations represented in (3.74) can be shown to give (2)
(1)
(3)
ζ 3 = ζ 3 ζ 1A + ζ 3 ζ 2A + ζ 3 ζ 3A .
(3.75)
Combining Eqns. (3.75) and (3.74) gives the general linear transformation between the areal coordinate domain ζi and (1) ζ1 ζ1 = ζ2(1) ζ 2 (1) ζ3 ζ3
its sub-domain (3) (2) ζ1 ζ1 (3) (2) ζ2 ζ2 (3) (2) ζ ζ 3
3
ζiA as ζ 1A ζ 2A , ζ3
0 ≤ ζiA ≤ 1,
(3.76)
ζi ∈ [0, 1].
A
Making use of Eqn. (3.76), the sub-regions A, B and C of the standard triangle in the (ζ1 , ζ2 , ζ3 ) domain may each be parametrized and thus transformed to standard triangular regions 0 ≤ ζiA , ζiB , ζiC ≤ 1 via (P )
ζi (ζ1A , ζ2A , ζ3A ) = ζi
(2)
(3)
for Triangle A,
(3)
for Triangle B,
ζ 1A + ζ i ζ 2A + ζ i ζ 3A
(1)
(P )
(1)
(2)
ζi (ζ1B , ζ2B , ζ3B ) = ζi ζ1B + ζi
ζ 2B + ζ i ζ 3B (P )
ζi (ζ1C , ζ2C , ζ3C ) = ζi ζ1C + ζi ζ2C + ζi
ζ 3C
(3.77)
for Triangle C,
(j)
for i = 1, 2, 3. Substituting the known nodal values ζ i = δij , j 6= P in the above gives (P )
ζ 1 = ζ 1 ζ 1A ,
(P )
for Triangle B,
(P )
for Triangle C, (3.78)
(P )
ζ 3 = ζ 3 ζ 2B + ζ 3B
ζ 2 = ζ 2 ζ 2B ,
(P )
ζ 2 = ζ 2C + ζ 2 ζ 3C ,
ζ 1 = ζ 1C + ζ 1 ζ 3C ,
for Triangle A,
ζ 3 = ζ 3 ζ 1A + ζ 3A
(P )
ζ 1 = ζ 1B + ζ 1 ζ 2B ,
(P )
(P )
ζ 2 = ζ 2 ζ 1A + ζ 2A ,
(P )
ζ 3 = ζ 3 ζ 3C
135
ζ2
ζ3
Mapping 1:
2
Global to
5
(a)
∂ξ/∂ζ2
(b)
∂ξ/∂ζ1
3
6
ζ3
6
(P )
, ζ3 )
7 2(0,1,0)
1(1,0,0)
ζ1
4
ζ1
ξ1
ξ3
(P )
P (ζ1 , ζ2 B A C 5
1
ξ
ξ2
(P )
4
7
P
3(0,0,1)
parent element
ζ2
Mapping 2: subdivision
3A (0, 0, 1)
(c)
η10 C
η20 B
η20 C
B
η10 A η20 A 1A (1, 0, 0)
3C (0, 0, 1)
3B (0, 0, 1)
A
ζ1 A
ζ3 C
ζ3 B
ζ3 A
ζ2 A
2A (0, 1, 0)
η10 B
ζ1 B 1B (1, 0, 0)
ζ2 B
C
ζ1 C
2B (0, 1, 0)
1C (1, 0, 0)
ζ2 C
2C (0, 1, 0)
Mapping 3: degenerate triangular mapping
30 (1, 1)
20 (1, −1)
30 (1, 1)
η10 A
30 (1, 1)
20 (1, −1) η10 B
20 (1, −1) η10 C
(d) η20 A
40 (−1, 1)
η20 B
10 (−1, −1)
40 (−1, 1)
η20 C
40 (−1, 1)
10 (−1, −1) (P )
(P )
(P )
10 (−1, −1)
Figure 3.32: Triangular subdivision about a point P (ζ 1 , ζ2 , ζ3 ) followed by singular transformations of Eqn.s (3.70) to (3.72). (a): physical element in global Cartesian domain ξi . (b): parent element in areal coordinate domain ζ i with sub-domains A, B and C. (c): auxiliary standard triangular domains ζ iA , ζiB and ζiC for sub-domains A, B and C. (d): standard bi-unit quadrilateral domains for the singular transformations.
136 where the pole P corresponds to corners 1, 2 and 3 of regions A, B and C, respectively. The Jacobians of the subdivision and linear transformation between the areal-coordinate systems represented by (3.78) may be obtained formally as ∂ζ1 ∂ζ2 ∂ζ1A ∂ζ (P ) |JSA | = 1A for Triangle A, = ζ1 ∂ζ1 ∂ζ2 ∂ζ2 ∂ζ2 A
A
|JSB | =
(P ) ζ2
for Triangle B,
(P )
for Triangle C.
|JSC | = ζ3
(3.79)
One should note that the result in Eqn. (3.79) can also be deduced as a consequence of the definition of areal coordinates of straight-sided triangles, in the sense that the (P )
coordinate ζi
(P )
(P )
(P )
represents the ratio of the area subtended by the point (ζ 1 , ζ2 , ζ3 ) (P )
and corners j and k to the total area ∆, i.e. ζ i
= ∆jk /∆, where i, j, k are cyclic
permutations. For linear one-to-one mappings of each of the three triangular sub-regions having areas ∆jk to their corresponding conventional triangular element domains having areas ∆, the Jacobians of the transformations are exactly equal to these area ratios. Upon application of the singular transformations (3.72), (3.71) and (3.70) having poles at nodes 1, 2 and 3 to triangles A, B and C, respectively (see Fig. 3.32), one finds 0
(1 + η10 A ) , 2 (1 − η10 A ) (1 + η20 A ) , = 2 2 (1 − η10 A ) (1 − η20 A ) . = 2 2 0
ζ1A (η10 A , η20 A ) = N (2 ) + N (3 ) = 0
ζ2A (η10 A , η20 A ) = N (4 ) 0
ζ3A (η10 A , η20 A ) = N (1 )
1 |JT1 | = (1 − η10 A ), 8
(3.80)
for Triangle A, (1 − η10 B ) (1 − η20 B ) , 2 2 (1 + η10 B ) 0 0 ζ2B (η10 B , η20 B ) = N (2 ) + N (3 ) = , 2 (1 − η10 B ) (1 + η20 B ) 0 ζ3B (η10 B , η20 B ) = N (4 ) = . 2 2 0
ζ1B (η10 B , η20 B ) = N (1 ) =
|JT2 | =
1 (1 − η10 B ), 8
(3.81)
137 for Triangle B, and (1 − η10 C ) (1 + η20 C ) , 2 2 (1 − η10 C ) (1 − η20 C ) 0 ζ2C (η10 C , η20 C ) = N (1 ) = , 2 2 (1 + η10 C ) 0 0 ζ3C (η10 C , η20 C ) = N (2 ) + N (3 ) = 2 0
ζ1C (η10 C , η20 C ) = N (4 ) =
1 |JT3 | = (1 − η10 C ), 8
(3.82)
for Triangle C. Combining the foregoing general formulas for subdivision and sequential mappings given in Eqns. (3.78) through (3.82) for a pole placed at an arbitrary point ζ (P ) , the special cases of collocation at nodes 1–7 of a standard triangular element may then be obtained by taking point ζ (P ) to coincide with the nodal locations. For collocation at the center node 7, the Jacobians for triangles A, B and C in Eqn. (3.79) all become 1/3. For the degenerate cases of collocation at the three mid-side nodes 4, 5 and 6, one (P )
of the three coordinates ζi
will be zero and the area and Jacobian for that sub-region
are therefore zero according to (3.79), while the Jacobians of the remaining two regions are equal to 1/2. For collocation at the corner nodes 1, 2 and 3, two Jacobians become zero, corresponding to zero area of integration, while the remaining Jacobian becomes unity as the subdivision mapping (3.74) becomes an identity for the remaining domain, and the mappings of Eqs. (3.70) to (3.72) are recovered. Under the general subdivision and singular transformations about an arbitrary
point ζ (P ) as given by Eqns. (3.78) to (3.82), the surface integral of a function f ( ξ) over
138
an interpolated triangular element domain Γ is transformed according to Z 1 Z 1−ζ2 ZZ f (ξ) dΓξ = f (ξ (ζ)) |JG | dζ1 dζ2 0
Γe
=
Z
Z
Z =
Z
Z
Z
0
1 0
1−ζ2A
Z
0
1
1−ζ2B
Z
0
0 1
Z
1−ζ2C
Z
1
0
0 1
−1 −1 1 Z 1
−1 −1 1 Z 1
−1
−1
f (ξ (ζ(ζ A ))) |JG | |JSA | dζ1A dζ2A +
f (ξ (ζ(ζ B ))) |JG | |JSB | dζ1B dζ2B +
f (ξ (ζ(ζ C ))) |JG | |JSC | dζ1C dζ2C
(3.83)
f (ξ (ζ(ζ A (η 0A )))) |JG | |JSA | |JT1 | dη10 A dη20 A +
f (ξ (ζ(ζ B (η 0B )))) |JG | |JSB | |JT2 | dη10 B dη20 B +
f (ξ (ζ(ζ C (η 0C )))) |JG | |JSC | |JT3 | dη10 C dη20 C ,
where ξ= (ξ 1 , ξ 2 , ξ 3 ), ζ = (ζ1 , ζ2 , ζ3 ), ζ A = (ζ1A , ζ2A , ζ3A ), η 0A = (η10 A , η20 A ), ζ A (η 0A ) = (ζ1A (η10 A , η20 A ), ζ2A (η10 A , η20 A ), ζ3A (η10 A , η20 A )), etc. As will be illustrated numerically in subsequent sections, the proposed strategy of subdivision and degenerate transformations used in this investigation can greatly accelerate the convergence and improve the accuracy of numerical integration of the weakly-singular and nearly-singular integrals encountered in the regularized BIE of Eqn. (3.6) as a consequence of the use of singular fundamental solutions. In the next section, a next-level analytical transformation will be discussed which can be used to accelerate numerical convergence of the BEM integrals over elements with singular or quasi-singular shape functions, which are pertinent to a rigorous treatment of many types of mixed boundary value problems.
3.4.4
Gradient-Reduction Mapping for Integration over AG Elements In the previous sections, the method of domain subdivision and analytical trans-
formation is proposed for treating Green’s function-induced singularities in elemental
139 integration. Such ideas can be extended to cases where localized or singular shape functions are involved, such as those associated with AG elements. Specifically, one may seek a transformation so that the localized or singular nature of the shape functions can be reduced or eliminated for the purpose of numerical quadrature prior to the treatment of the Green’s function singularities detailed in Sections 3.4.1 and 3.4.3. As an illustration, consider the one-dimensional case of a singular shape function, as would be obtained using a negative value for the AG element parameters m or n. As demonstrated in Luchi and Rizzuti (1987), a one-dimensional integral having an integrand in the form of a product of a regular factor f (η 1 ) and an integrable (weak) singularity of order 0.5 at η1 = −1 as in Z
1 −1
f (η1 ) dη1 (1 + η1 )0.5
(3.84)
can be mapped onto the domain −1 ≤ η10 ≤ 1 via the change of variable 1 (1 + η1 )0.5 = √ (1 + η10 ) 2
(3.85)
so that a bounded integral is obtained in the transformed domain of integration via the aid of the Jacobian of transformation. Such a change of variable can be generalized to the case of one-dimensional improper integrals with a singularity of order α at η 1 = 1 of the form Z
1
−1
f (η1 ) dη1 , (1 − η1 )α
(3.86)
which is integrable for α < 1. This is the form encountered for 1D singular or AGsingular elements. For such a case, the singularity in the integrand above can be nullified by the Jacobian of a one-to-one mapping from the η 1 to the η10 domain defined by (1 − η1 )α = B(1 − η10 )C ,
0
