Finite Element Analysis for Composite Structures

345 Pages·1998·9.644 MB·English

by  Lazarus Teneketzis Tenek, John Argyris (auth.)

FINITE ELEMENT ANALYSIS FOR COMPOSITE STRUCTURES
SOLID MECHANICS AND ITS APPLICATIONS
Volume 59
Series Editor: G.M.L. GLADWELL
Solid Mechanics Division, Faculty of Engineering
University of Waterloo
Waterloo, Ontario, Canada N2L 3Gl
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How much?
The aim of this series is to provide lucid accounts written by authoritative research
ers giving vision and insight in answering these questions on the subject of
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts are
monographs defining the current state of the field; others are accessible to final
year undergraduates; but essentially the emphasis is on readability and clarity.
For a list of related mechanics titles, see final pages.
Finite Element
Analysis for
Composite Structures
by
LAZARUS TENEKETZIS TENEK
University of Stuttgart,
Germany
and
JOHN ARGYRIS
University of Stuttgart,
Germany
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
Printed on acid-free paper
All Rights Reserved
ISBN 978-90-481-4975-9 ISBN 978-94-015-9044-0 (eBook)
DOI 10.1007/978-94-015-9044-0
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998.
Softcover reprint of the hardcover 1st edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface ix
Acknowledgments xi
1 Some results from continuum mechanics 1
1.1 The analysis of stress . 1
1.2 The analysis of strain .... 6
1.3 Strain energy . . . . . . . . . 9
1.4 The principle of virtual work 12
1.5 The principle of stationary energy 13
2 A brief history of FEM 17
2.1 The matrix displacement method . . . . . 17
2.2 The finite element method . . . . . . . . . 19
2.3 The natural mode finite element method. 20
2.4 The basic ideas of FEM ......... . 21
3 Natural modes for finite elements 27
3.1 The concept of natural modes ... 27
3.2 The concept of natural stiffness matrix . . 28
3.3 Natural modes for selected finite elements 31
4 Composites 37
4.1 Fundamental concepts 37
4.2 Basic material unit . . 39
4.3 Laminates....... 41
4.3.1 Special laminates . 42
4.4 Micromechanics and macromechanics . 44
vi CONTENTS
5 Composite beam element 47
5.1 Introduction.... 47
5.2 Kinematics ....... 48
5.3 The beam element . . . 51
5.4 Natural rigid body modes 52
5.5 Natural straining modes . 56
5.5.1 Mode 1: Extension 57
5.5.2 Mode 2: Symmetrical bending in the z - z plane . 60
5.5.3 Mode 3: Antisymmetrical bending and transverse
shearing in the z - z plane . . . . . . . . . . . . .. 61
5.5.4 Mode 4: Symmetrical bending in the z - y plane.. 64
5.5.5 Mode 5: Antisymmetrical bending in the z - y plane 64
5.5.6 Mode 6: Torsion about the x axis 65
5.6 Natural stiffness matrix .... 67
5.6.1 Strain operator matrix . 67
5.6.2 Constitutive relation . 68
5.6.3 Strain energy ..... . 70
5.6.4 Evaluation of integrals. 72
5.6.5 Shear correction factor. 78
5.7 Local and global stiffness matrices 81
5.8 Work of external loads ..... 86
5.9 Initial load due to temperature 89
5.9.1 Evaluation of integrals. 91
5.10 Postprocessing ........ . 95
5.10.1 Computation of forces and moments 95
5.10.2 Natural energies ....... . 96
5.10.3 Through the thickness stresses 97
5.11 Geometrical stiffness ......... . 99
5.11.1 Elastic buckling ........ . 99
5.11.2 Simplified geometrical stiffness 100
5.11.3 The natural geometrical stiffness 106
5.12 Partly simplified geometrical stiffness. 108
5.13 Computational experiments .... 114
5.13.1 Isotropic beams and frames 114
5.13.2 Composite beam structures 119
5.14 Problems ............. . 132
CONTENTS vii
6 Composite plate and shell element 135
6.1 Introduction .................. . 135
6.2 Natural kinematics of the shell element TRIC 137
6.3 Constitutive relation . . . . . . 150
6.4 Stress resultants - equilibrium . . . . . . . . . 158
6.5 Natural modes and stiffness . . . . . . . . . . 163
6.6 Total strain in the natural coordinate system 167
6.7 Axial and symmetrical bending stiffness terms. 170
6.8 Antisymmetrical bending and shearing stiffness terms 175
6.8.1 Antisymmetrical bending terms. 176
6.8.2 Antisymmetrical shearing terms. 185
6.9 Shear correction factors .... . . . 189
6.10 Simulative azimuth stiffnesses . . . . 191
6.11 Local and global cartesian stiffnesses 191
6.12 Kinematically equivalent nodal loads 196
6.13 Initial load due to temperature . . . 198
6.14 Computation of stresses and stress resultants 202
6.14.1 Computation of stress resultants . . . 202
6.14.2 Computation of through-the-thickness stresses 204
6.15 The simplified geometrical stiffness 208
6.15.1 Geometrical forces . . . . 208
6.16 Geometrically nonlinear analysis . 217
6.17 Computational experiments . . . . 218
6.17.1 Clamped isotropic plate under central load 218
6.17.2 Thick sandwich plate under uniform pressure 219
6.17.3 Pinched cylinder; Scordelis-Lo roof; pressurized shell 220
6.17.4 Pinched hemispherical shell . . . . . . . . . . . . .. 221
6.17.5 Twisted beam. . . . . . . . . . . . . . . . . . . . .. 224
6.17.6 Eight-layer (0/45/ - 45/90)5 laminate under uniform
load . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.17.7 Stresses in a sandwich plate-comparison with the elas-
ticity solution . . . . . . . . . . . . . . . . . 229
6.17.8 Deformation of a (0/90/0) square laminate ... . . . 230
6.17.9 Stresses in a (0/90/0) rectangular laminate . . . .. 233
6.17.10 Large deflections of an isotropic plate-comparison
with experimental results . . . . . . . . . . . . . 233
6.17.11 Buckling of a cross-ply (0/90/90/0) laminate . . 234
6.17.12 Elastic stability of a (0/45/ - 45/90)8 laminate. 239
CONTENTS
Vlll
6.17.13 Thermomechanical buckling of a cylindrical compos-
ite panel . . . . . . . . . . . . . . . . . . . . . . . .. 239
6.17.14 Buckling of a composite vessel under hydrostatic pres-
sure . . . . . . . . . . . . . . . . . . . . . . . . . .. 240
6.17.15 Buckling of a rocket-like composite shell under exter-
nal pressure and temperature 250
6.18 Problems ......................... . .. 258
7 Computational statistics 261
7.1 A model problem .... 261
7.2 Computational statistics report 261
8 Nonlinear analysis of anisotropic shells 267
8.1 Stable and unstable equilibrium paths 267
8.2 The incremental/iterative scheme. 268
8.3 Numerical examples ....... . 274
8.3.1 Isotropic cylindrical panels 274
8.3.2 Composite shells . . . . . . 287
9 Programming aspects 301
9.1 Memoryallocation ....... . 301
9.2 Input data ............ . 302
9.3 Program parameters and storage 309
9.4 Elemental elastic stiffness matrix 311
9.5 Assembly of the global stiffness matrices 313
9.6 Solution of the linear equations 314
9.7 Postprocessing ........... . 314
9.8 Geometrical stiffness ........ . 315
9.9 Assembly of the geometrical stiffness 319
9.10 Scholium ............... . 319
Appendices 321
A Geometry of the beam element in space 321
B Contents of floppy disk 325
Bibliography 327
Index 335
Preface
This book is an adventure into the computer analysis of three dimensional
composite structures using the finite element method (FEM). It is designed for
Universities, for advanced undergraduates, for graduates, for researchers, and
for practising engineers in industry. The text advances gradually from the
analysis of simple beams to arbitrary anisotropic and composite plates and shells;
it treats both linear and nonlinear behavior. Once the basic philosophy of the
method is understood, the reader may expand its application and modify the
computer programs to suit particular needs.
The book arose from four years research at the University of Stuttgart,
Germany. We present the theory and computer programs concisely and
systematically so that they can be used both for teaching and applications. We
have tried to make the book simple and clear, and to show the underlying
physical and mathematical ideas.
The FEM has been in existence for more than 50 years. One of the
authors, John Argyris, invented this technique in World War II in the course of
the check on the analysis of the swept back wing of the twin engined Meteor Jet
Fighter. In this work, he also consistently applied matrix calculus and introduced
triangular membrane elements in conjunction with two new definitions of
triangular stresses and strains which are now known as the component and total
measures. In fact, he was responsible for the original formulation of the matrix
force and displacement methods, the forerunners of the FEM. A distinct feature
of the present book is the consistent use of the so-called natural modes, first
proposed by Argyris in 1964, here applied to composite shells and structures.
In the present book the technique of the natural modes are described fully in
chapters 3, 5, and 6. In this way, the total displacements of an element are
defined by rigid body modes and the natural or strain modes. This specification
of the kinematics of an element leads automatically to the elimination of parasitic
phenomena like shear locking.
x Preface
An element in space has 6 possible rigid body motions: three
displacements and three rotations. An element with n degrees of freedom will
thus have 6 rigid body modes and n -6 straining modes. Only the strain modes
involve strain energy. Thus in setting up the stiffness matrix of an element we
can concentrate on the (n -6)x(n -6) matrix relating to the strain modes and then
set up the full nxn matrix by using the relations between the n degrees of
freedom and the two sets of modes, 6 rigid body and n -6 strain modes. We set
up this matrix on the computer itself.
Another distinctive feature of the method is that all the integrations
required in the calculation of the element stiffness matrices are performed in
closedform; no numerical quadratures are needed. This feature leads to more
savings in computation, and is particularly important when the FEM is being
used in each step of an incremental/iterative method, for nonlinear analysis, for
optimization or design, for example.
In the book we aim to show that this systematic use of natural modes and
closed form integrations leads to very general formulations and to considerable
savings in computer time. We show how the method applies to composite
beams, plates and shells, and equip the reader to formulate the method for other
applications.
Chapter I provides some results from continuum mechanics and forms
the Principles of Virtual Work and Potential Energy. Chapter 2 sketches the
history of the FEM, particularly in its displacement form. Chapter 3 introduces
the concepts of natural modes and natural stiffness. Chapter 4 introduces the
basic concepts of the theory of composites. Chapter 5 applies the natural mode
method to the analysis of composite beams in three dimensions. Chapter 6
introduces the composite plate and shell element. Chapter 7 show the
computational advantages of the method on a model problem. Chapter 8 deals
with nonlinear analysis of anisotropic shells. Chapter 9 discusses programming
aspects of the technique.
At this point we apologize for not including crash phenomena and the
inelastic deformations of composite structures. We realize these effects are of
the utmost importance for the behavior of composite systems. Hopefully, we
will be in a position to compose such extensions in a future edition of the present
work.