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Lecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
58
S. Naomis, P. C. M. Lau
Computational Tensor
Analysis of Shell Structures
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J. H. Seinfeld . P. Silvester· P. Spanos· W. Wunderlich· S. Yip
Authors
Dr. Steve Naomis
Dr. Paul C. M. Lau
Civil Engineering Department
University of Western Australia
Nedlands, WA 6009
Australia
ISBN-13:978-3-540-52852-4 e-ISBN-13:978-3-642-84243-6
001: 10.1007/978-3-642-84243-6
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PREFACE
This book presents a method which is capable of evaluating the
deformation characteristics of thin shell structures A free vibration analysis is
chosen as a convenient means of studying the displacement behaviour of the shell,
enabling it to deform naturally without imposing any particular loading conditions.
The strain-displacement equations for thin shells of arbitrary geometry are
developed. These relationships are expressed in general curvilinear coordinates and
are formulated entirely in the framework of tensor calculus. The resulting theory is
not restricted to shell structures characterized by any particular geometric form,
loading or boundary conditions.
The complete displacement and strain equations developed by Flugge are
approximated by the curvilinear finite difference method and are applied to
computing the natural frequencies and mode shapes of general thin shells. This
approach enables both the displacement components and geometric properties of the
shell to be approximated numerically and accurately.
The selection of an appropriate displacement field to approximate the
deformation of the shell within each finite difference mesh is discussed in detail. In
addition, comparisons are made between the use of second and third-order finite
difference interpolation meshes.
The method is applied to a series of flat plate, cylindrical, spherical and
conical shell structures to verify its accuracy. The computed natural frequencies
and their associated mode shapes are documented and compared with published
results.
IV
The authors wish to acknowledge the supports provided by the Civil
Engineering Department, University of Western Australia to the development of the
computational tensor analysis of shell structures.
TABLE OF CONTENTS
NOTATION IX
1 . INTRODUCTION 1
2. GENERAL THEORY 9
2.1 A Summary of the Tensorial Quantities Required in the
Formulation of a Shell Theory 11
. 2.1.1 Base Vectors 11
2.1.2 Metric Tensors 12
2.1.3 Coordinate Transfonnations 15
2.1.4 Christoffel Symbols 18
2.1.5 Covariant Derivatives 19
2.2 Surface Geometry 22
2.2.1 Curvilinear Coordinates on a Surface 22
2.2.2 Geometry of a Curved Surface 22
2.3 The Strain Tensor 30
2.4 The Stress Tensor 33
2.5 The Constitutive Equations 37
2.6 The Theory of Shells 40
2.6.1 Shell Geometry 40
2.6.2 Deformation Characteristics 43
VI
2.6.3 The Change in Curvature Tensor 47
2.6.4 The Strain-Displacement Equations 50
2.6.5 Interpretation and Discussion of dIe Strain-
Displacement Equations 56
3. NUMERICAL FUNDAMENTALS 60
3.1 The Curvilinear Finite Difference Method 62
3.2 The Numerical Computation or the Surface
Geometric Quantities 69
3.2.1 Base Vectors and Metric Tensors 70
3.2.2 The Christoffel Symbol 73
3.2.3 The Curvatufe Tensor 74
3.2.4 Covariant Derivative of the Curvature Tensor 75
3.3 The Principle or Virtual Displacements 77
3.4 Discretization and Displacement Fields 78
3.5 The Numerical Implementation or the General Surface
Stress, Strain and Displacement Components 85
3.5.1 The General Surface Displacement Components 86
3.5.2 The General Strain Tensor 87
3.5.3 The General Stress Tensor 89
3.6 Boundary Conditions 91
3.7 The Numerical Solution of the Eigenvalue Problem 98
VII
4. NUMERICAL IMPLEMENTATION 101
4.1 A Second Order Implementation 103
4.1.1 Second Order CFD Approximation 103
4.1.2 Numerical Integration Scheme 108
4.1.3 Numerical Examples 116
4.2 A Third Order Implementation 146
4.2.1 Third Order CFD Approximation 146
4.2.2 Boundary Conditions 151
5. NUMERICAL APPLICATIONS 157
5.1 Simply Supported Plate 158
5.2 Cantilever Plate 161
5.3 Spherical Cap on a Square Base 164
5.4 Cylindrical Panel 170
5.5 Curved Fan Blade 173
5.6 Conical Shell Panel 176
5.7 Cylindrical Tank 179
6. SUMMARY 182
REFERENCES 184
APPENDIX A: Displacement Transformations 196
APPENDIX B: 'Finite Difference Expressions 201
VIII
APPENDIX C: Numerical Integration of the Stiffness Matrix 233
APPENDIX D: Application of the CFD method to the Analysis
of Beam Bending Problems with Fixed Edges 240
APPENDIX E: Transformation of the Generalized Eigenvalue
Problem to Standard Form 250
APPENDIX F: Numerical Results 252
Simply supported plate 253
Cantilever plate 253
Curved fan blade 267
Spherical shell 274
Cylindrical panel 281
Cylindrical tank 288
Conoidal shell 295
SUBJECT INDEX 302
NOTATION
General
[ ] Matrix notation.
{ } Vector notation.
Partial differentiation with respect to the surface coordinates.
i
Covariant differentiation with respect to the coordinates
Oil Covariant differentiation with respect to the surface
u
coordinates xu.
Time derivative.
Transpose of a matrix.
Inverse of a matrix.
Structure's stiffness matrix
Structure's mass matrix.
Two dimensional region.
Boundary curve.
r. Integration point for a Gauss-Legendre numerical integration
I
scheme.
W. Weight factors for a Gauss-Legendre numerical integration
I
scheme.
Coordinates of a general three-dimensional curvilinear
coordinate system.
Coordinates of a general two-dimensional curvilinear
coordinate system on the shell's middle surface.
Coordinate axis normal to xu.
&i Kronecker delta.
I
Eigenvalue.
