NONLINEAR RESPONSE OF THICK LAMINATED SHELLS
WITH INTERLAMINAR DEFORMATION
by
Lea Raymond Hsia
A dissertation submitted to the faculty of
The
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Department of Civil Engineering
The
April 2006
Copyright © Lea Raymond Hsia 1994
All Rights Reserved
T H E U N I V E R S I T Y O F U T A H G R A D U A T E S C H O O L
F I N A L R E A D I N G A P P R O V A L
To the Graduate Council of the
I have read the dissertation of Lea Raymond Hsia in its finalform and have found that (1) its format, citations and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the supervisory committee and is ready for submission to The Graduate School.
_____________________ _____________________________________
Date Reaz A. Chaudhuri
Chair, Supervisory Committee
Approved for the Major Department
____________________________________
Lawrence D. Reaveley
Chair/Dean
Approved for the Graduate Council
____________________________________
David S. Chapman
Dean of The
ABSTRACT
The dissertation research presents the theoretical and computational procedures that have been applied in the design of a special purpose computer program for computing static nonlinear response of laminated shell structures. A general formulation of the incremental equations of motion for laminated shell structures undergoing large displacement finite strain deformation is presented. These equations are based on the Lagrangian frame of reference, in which constitutive models of a variety of types may be introduced. The incremental equations are linearized for computational purposes, and the linearized equations are discretized using isoparametric serendipity finite elements. Computational techniques, including step-by-step incremental approach and the BFGS type iterative procedure, for the solution of nonlinear equations are also discussed in detail.
A fully nonlinear analysis for prediction of large deformation behavior of thick laminated composite shells and panels is presented. This research accounts for fully nonlinear strain-displacement relations, in contrast to the commonly used von Karman type nonlinear assumption. The formulation accounts for layerwise linear displacement distribution, in conjunction with large strain as well as large deflection/rotational behaviors. The fully nonlinear kinematic relations are employed in the present study so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted.
TABLE OF CONTENT
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives and Scope of the Present Research . . . . . . . . . . . . . . 6
2. THEORETICAL DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Kinematic Relations of a shell . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Constitutive Relations for an Orthotropic/Anisotropic Lamina . 18
2.4 Formulation of Incremental Equations of Motion . . . . . . . . . . . 22
3. FORMULATION OF 3D CUBIC THICK SHELL ELEMENT . . . . . . . 28
3.1 Total Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Updated Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Isoparametric Finite Element Discretization . . . . . . . . . . . . . . . . 32
3.4 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4. VERIFICATION PROBLEMS AND NUMERICAL RESULTS . . . . . . 47
4.1 Example 1. Large Displacement and Large Strain Analysis of a
Rubber Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Example 2. Analysis of a Homogeneous Isotropic Clamped
Thin/Shallow Cylindrical Panel under Uniform Loading . . . . . .. . 53
4.3 Example 3. Analysis of Cross-Ply [0o /90o] Clamped Cylindrical
Thin Shallow Panel under Uniform Loading. . . . . . . . . . . . . . . . . . 58
4.4 Example 4. Analysis of Homogeneous Isotropic Clamped Thin
and Thick Cylindrical Panels under Uniform Loading . . . . . . . . . . 64
4.5 Example 5. Analysis of Cross-Ply [0o /90o] Clamped Cylindrical
Thin and Thick Panel under Uniform Loading. . . . . . . . . . . . . . . . 68
5. CONCLUSIONS AND RECOMMANDATIONS . . . . . . . . . . . . . . . . . . 74
5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Recommandations for Future Research . . . . . . . . . . . . . . . . . . . . . 77
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
LIST OF FIGURES
Figure Page
2.1 Volume Element for a Curvilinear Coordinate Cystem . . . . . . . . . . . . . . 11
2.2 General Shape of a Doubly Curved Laminated Shell . . . . . . . . . . . . . . . . 14
2.3 Global and Material Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Surface Parallel Isoparametric Elements. . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Storage Arrangement of the Coefficient Matrix A . . . . . . . . . . . . . . . . . . 42
4.1 A 2D Rubber Sheet under Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Nonlinear Displacement at Each Load Step . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 The Deformation of a 2D Rubber Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 A Clamped Homogeneous Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Convergence Study of present Quadratic 16-Node Element . . . . . . . . . . 56
4.6 The Central Deflection of an Isotropic Clamped Cylindrical Shell . . . . . 56
4.7 A Cross-Ply [0o /90o] Cylindrical Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.8 Convergence Study of the Quadratic 16-Node Element for a Cross-Ply
[0o /90o] Cylindrical Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.9 Comparison with Reddy's von Karman Nonlinear Strain Solution . . . . 60
4.10 Comparison of Full and Reduced Integration Based Solutions. . . . . . . . 61
4.11 Normalized Radial Displacements of a Clamped [0o /90o] Cylindrical
Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.12 Convergence of Central deflection, Computed Using 24 Node Cubic
Elements, of a Thin Clamped Isotropic Cylindrical Panel . . . . . . . . . . . . . 66
4.13 Load Deflection (Equilibrium) Paths for a Thin Homogeneous Isotropic
Cylindrical Panel Computed Using Cubic (24 Node) Elements . . . . . . . . 66
4.14 Effect of Thickness on the Normalized Deflection of a Homogeneous
Isotropic Cylindrical Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.15 Convergence of Central deflection, Computed Using 24 Node Cubic
Elements, of a Thin Two-Layer [0o /90o] Cylindrical Panel . . . . . . . . . . . 69
4.16 Load Deflection (Equilibrium) Paths for a Thin Two-Layer [0o /90o]
Cylindrical Panel Computed Using Cubic (24 Node) Elements . . . . . . . . 70
4.17 Radial Displacements of a Clamped [0o /90o ] Cylindrical Panel . . . . . . . . 70
4.18 Thickness Study of Two Layer [0o /90o] Cylindrical Panel . . . . . . . . . . . . 71
LIST OF TABLES
Table Page
4.1 The Results for a Rubber Sheet under Tension . . . . . . . . . . . . . . . . . . . . 52
4.2 Comparison of Displacement of a Homogeneous Isotropic Cylindrical
Panel with Available Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Comparison of the von-Karman and Full Nonlinear Strain-Displacement.
Based Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Results of a Homogeneous Isotropic Panel Thickness Study . . . . . . . . . . 67
4.5 Nonlinear Thickness Effect in Homogeneous Isotropic Cylindrical Panels 68
4.6 Results of a Two Layer [0o /90o] Panel Thickness Study . . . . . . . . . . . . . . 71
4.7 Nonlinear Thickness Effect in Two Layer [0o /90o] Cylindrical Panels . . . 73
NOMENCLATURE
A Symmetrical coefficient matrix of skyline simulteneous equations (LHS)
A(i) Acceleration factors of equation solver
a Half length of a panel of square planform
BL Linear component of the matrix of strain-displacement relationship
BNN Nonlinear component of the matrix of strain-displacement relationship
b Binormal vector in Ch.2
b Skyline stored simulteneous equations RHS coefficient
b Subscript indicating bottom surface of the layer
C(i) Condition number in acceleration scheme
Cij Component of the compliance matrix
Cijrs Incremental elastic material property represented 4th order constituitive tensor
C1, C2 Constants of Mooney-Rivlin Type Material
dS Length of line segment on a deformed surface
ds Length of line segment on an undeformed surface
odV Incremental control volume of the initial configuration
E Young's modulus of the isotropic material.
ELL Young's modulus in the direction parallel to the fibers.
ETT Young's modulus in the direction perpendicular to the fibers.
Linear incremental component of the 6x1 strain vector
Linearized incremental component of the 6x1 strain vector
F Force vector
{fL} Applied load vector
t+∆t{fL} Applied load vector at the time t+∆t
{fN} Nonlinear internal force vector
t+∆t{fN}(i) Nonlinear internal force vector at the i-th iteration of the time t+∆t
GLT Longitudinal shear modulus in a plane parallel to fibers.
GTT Transverse shear modulus in a plane perpendicular to fibers.
gk Orthogonal base vector in the k-th direction
gk Coefficient of the 1st fundamental differential quadratic form of the surface in the k-th direction, k=1,2
h Total thickness of the cylinder
hi Thickness of the i-th lamina
K Stiffness matrix in Ch. 3
[KL] Linear global stiffness matrix
[KN] Nonlinear global geometric stiffness matrix
L Half length of the cylinder
Subscript indicating the longitudinal direction of a lamina
mk Half bandwidth of the total stiffness matrix
N Total number of elements
Nelm Number of elements for each layer
NL Number of layers
NT Number of elements in the circumferential direction
n Unit normal director of the surface with respect to the fixed coordinate
system
Pr, q Uniformly applied hydrostatic pressure
Qij Stiffness matrix component of the orthotropic layer
Stiffness matrix component of the k-th anisotropic layer after transformation
{oQ}(i) Incremental displacement vector due to the applied load vector
R Position vector of an arbitrary point at distance from the bottom surface of
the i-th layer
R, Rm Mean radius of the panel
Ri Inner radius of the cylinder
r Position vector of an arbitrary point on the bottom surface of the i-th layer
Incremental stress component of the k-th layer
The i-th stress component at time t, i=1 to 6
Second Piola-Kirchhoff stress tensor evaluated based on the initial configuration at t = t+∆t
The stress matrix evaluated at time t in section 2.3
The stress vector evaluated at time t in section 2.3
T Superscript indicating the transpose of the matrix
Subscript indicating the transverse direction of a lamina
TOLdisp. Tolerance of displacement convergence criterion
TOLenergy Tolerance of energy convergence criterion
TOLforce Tolerance of force convergence criterion
t Subscript indicating top surface of the layer
U Displacement vector in Ch. 3
oub(i), ovb(i), Incremental displacement component in the x, q, and z directions
owb(i) on the bottom surfaces of the i-th layer
out(i), ovt(i), Incremental displacement component in the x, q, and z directions
owt(i) on the top surface of the i-th layer
0ubk(k),0vbk(k) Incremental nodal displacement components in the x, q, and z directions,
0wbk(k) respectively, at the k-th node on the bottom surface of the i-th layer
0ubk(k),0vbk(k) Incremental nodal displacement components in the x, q, and z directions,
0wbk(k) respectively, at the k-th node on the top surface of the i-th layer
t+∆t{V}(i) Total displacement vector at the i-th iteration at time t+∆t
oV Control volume of the initial configuration
vi The i-th component of the displacement vector, i=1,2,3
W Transverse displacement component of cylindrical panel in Chapter 4
Wn Normalized Transverse displacement in Chapter 4
X Lagrangian coordinate system in Chapter 2
Deformation gradient in Chapter 4
x(i),q(i),z(i) Coordinates of a point inside an element of the i-th layer in terms of r and s
a Subscript indicating direction parallel to fiber in a lamina
b Subscript indicating direction orthogonal to fiber in a lamina
eij Physical component of Green-Lagrangian strain tensor in equation (2.9)
Incremental strain component of the k-th layer
Total Green-Lagrangian strain tensor evaluated based on the initial configuration at time t+∆t
Christoffel symbol of the 2nd kind
Nonlinear strain component vector
Variation of the nonlinear strain component vector
gij Green-Lagrangian strain tensor in section 2.2
[F] Cubic global interpolation function matrix composed of 24 node element
Yk(r,s) Cubic element interpolation function composed of 24 node element
in terms of r and s
rk(i) The k-th principal radius of curvature in the i-th layer
q Angle measured from the global x axis
t+∆t¬ External virtual work of a body
x, h, z Natural coordinates system of a parent element
ACKNOWLEDGEMENTS
The author wishes to express his gratitude and deep appreciation to Professor
Reaz A. Chaudhuri for his inspiring guidance and support throughout the course of this research. Appreciation goes to Professors E. S. Folias, K. L. DeVries, D. K. Shetty, and F. Stenger for their valuable suggestions and recommendations. A special thank is extended to author's wife Ana Ching-hua and his only son Joel Hsia for their love, patience, and assistance in achieving his academic aspirations. The author dedicates his dissertation to his parents Tien-Cheuh Hsia and Yu-Tsai Hsia for the numerous sacrifices they made throughout their lives.
T H E U N I V E R S I T Y O F U T A H G R A D U A T E S C H O O L
SUPERVISORY COMMITTEE APPROVAL
of a dissertation submitted by
Lea Raymond Hsia
This dissertation has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory.
_________________ _______________________________________
Chair: Reaz A. Chaudhuri
_________________ _______________________________________
K. Larry DeVries
__________________ _______________________________________
Danish K. Shetty
__________________ _______________________________________
Efthymios S. Folias
__________________ _______________________________________
Frank Stenger
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