Analysis of thick composite plates using higher order Alon, Yair

Analysis of thick composite plates using higher order Alon, Yair
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1990-12
Analysis of thick composite plates using higher order
three dimensional finite elements
Alon, Yair
Monterey, California: Naval Postgraduate School
http://hdl.handle.net/10945/27545
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NAVAL POSTGRADUATE SCHOOL
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,,v STAT,,V
THESIS
STATIC ANALYSIS OF THICK COMPOSITE PLATES
USING HIGHER ORDER THREE DIMENSIONAL FINITE
F LEMENTS
BY
YAIR ALON
December 1990
Thesis Advisor:
Prof. Ramesh Kolar
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STATIC ANALYSIS OF THICK COMPOSITE PLATES USING HIGHER ORDER THREE
DIMENSIONAL FINITE ELEMENTS
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Yair Aln
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16 SUPPLEMENTARY NOTATION
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COSATI CODES
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SUB-GROUP
18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
Finite Element, nonlinear analysis, plate bending
thick plates, laminated composites, buckling,
fonstant arc length three dimensional element.
19 ABSTRACT (Continue on Ceverse if necessary &(didentify
biQk number)
A triquadratic isoparametric solielement is developed to study the behavior of thick isotropic and laminated composite plates. The element is a 27
noded Lagrangian element based on three dimensional elasticity. Material
characterisics are accounted by either using laminate plate theory or three
dimensional anisotropic theory. Element matrices for nonlinear stability
analysis are derived based on total Lagrangian formulation.
Results are presented to compare with analytical solutions to validate the
elements behavior. The effects of various integration schemes on the element performance are presented. Convergence studies for laminated composits for different fiber orientations are provided to illustrate application
An analysis for thin plated is carried out and results for thick plates are
compared with available higher order plate theories. One row of elements
in the thickness directions gives satisfactory results for thick laminates.
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Ramesh Kolar
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Analysis of Thick Composite Plates Using Higher Order Three Dimensional Finite
Elements
by
Alon Yair
Captain, Israeli Air Force
B.S.C., Israel Technion Institute of Technology, 1983
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN
AERONAUTICAL ENGINEERING
and
AERONAUTICS AND ASTRONAUTICS ENGINEERS DEGREE
from the
NAVAL POSTGRADUATE SCHOOL
December 1990
L
JUBLI ~c at 1.,ci
Author:
str b t i */
Av¢illbliA
Approved by:
Rame
G. H.
Kolar, Thesis Advisor
indse
,econ
e
E. Rol.erts Wood
Departmentof Aeroutics and Astronautics
Gordon E. Schacher
Dean of Faculty and Graduate Studies
ii
Dist
p1
Cui
ABSTRACT
A triquadratic isoparametric solid element is developed to study the behavior
of thick isotropic and laminated composite plates. The element is a 27 noded Lagrangian element based on three dimensional elasticity. Material characteristics are
accounted by either using laminate plate theory or three-dimensional anisotropic
theory. Element matrices for nonlinear stability analyses are derived based on total
Lagrangian formulation.
Results are presented to compare with analytical solutions to validate the
elements behavior. The effects of various integration schemes on the element performance are presented. Convergence studies for laminated composites for different
fiber orientations are provided to illustrate applications. An analysis of thin plates
is carried out and results for thick plates are compared with available higher order
plate theories. One row of elements in the thickness directions gives satisfactory
results for thick laminates.
iii
TABLE OF CONTENTS
INTRODUCTION
II.
.............................
1
A.
OVERVIEW ..............................
1
B.
LITERATURE REVIEW .......................
1
C.
THESIS OUTLINE
3
..........................
THEORETICAL FORMULATION ....................
4
A.
INTRODUCTION ...........................
4
B.
GENERAL DERIVATION OF FINITE ELEMENT EQUILIB-
C.
D.
E.
RIUM EQUATIONS .........................
4
INTERPOLATION SCHEME
9
....................
1.
Shape Functions (Displacement Interpolation Functions) . ..
2.
Jacobian Transformation Matrix .................
9
11
STRAIN DISPLACEMENT RELATIONS - [B]..............
12
1.
Basic Formulation .........................
12
2.
General Nonlinear Discretization
................
17
STRESS-STRAIN RELATIONS ...................
25
1.
Classical and Higher Order Laminate Theories ...........
25
2.
Three-dimensional Anisotropic Theory .............
34
F.
CONSISTENT LOADS ........................
35
G.
INTEGRATION ............................
36
1.
Gauss Quadrature
........................
39
2.
Integration Scheme ........................
39
H.
BUCKLING ANALYSIS
1.
.......................
41
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
iv
III.
IV.
2.
Implementation ..........................
42
3.
Constant Arc Length Method [Kolar and Kamel (1985)]
4.
Convergence Criterion ....
......................
45
PROGRAM IMPLEMENTATION ........................
47
A.
INTRODUCTION ................................
47
B.
LINEAR ANALYSIS ....
47
C.
NONLINEAR ANALYSIS .......................
48
D.
SOLUTION PROCEDURE ......................
48
1.
Composite Material ........................
48
2.
Linear Case ............................
48
3.
Nonlinear Case ..........................
50
.........................
54
NUMERICAL EXAMPLES ........................
...............
A. INTRODUCTION AND NOTATIONS
1.
B.
V.
. . . 44
54
54
Material Properties ........................
COLUMNS AND BARS
.......................
54
1.
B ars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.
Beam s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
C.
CLAMPED PLATES .........................
59
D.
SIMPLY SUPPORTED PLATES ...................
66
CONCLUSIONS AND SCOPE FOR FUTURE RESEARCH
A.
CONCLUSIONS ............................
B.
SCOPE FOR FUTURE RESEARCH
.....
76
76
................
77
APPENDIX A - SHAPE FUNCTIONS
.....................
78
APPENDIX B - JACOBIAN MATRIX
.....................
82
APPENDIX C - THEORIES
84
...........................
A. THEORY OF ELASTICITY SOLUTIONS
V
.............
84
B.
1.
Cantilevered bar under traction .....................
84
2.
Cantilevered Beam under end load ..................
84
CLASSICAL PLATE THEORY (CPT) ..................
84
1.
All edges clamped rectangular isotropic plate under central load 84
2.
All edges simply-supported, rectangular plate under uniformly
3.
distributed load ................................
85
Composite ...................................
85
LIST OF REFERENCES .......
.............................
INITIAL DISTRIBUTION LIST ......
.........................
vi
87
89
LIST OF TABLES
2.1
SHAPE FUNCTIONS FOR 9 NODED BIQUADRATIC ELEMENT
2.2
SAMPLING POINTS AND WEIGHTS FOR GAUSS QUADRATURE OVER THE INTERVAL-1 to 1 .................
4.1
36
41
EFFECTS OF REDUCED INTEGRATION AND REFINED MESH,
ON THE MAXIMUM DEFLECTION OF CLAMPED ISOTROPIC
CANTILEVER BAR UNDER UNIAXIAL LOAD .............
4.2
58
EFFECTS OF REDUCED INTEGRATION AND MESH CONFIGURATION ON THE MAXIMUM DEFLECTION OF CLAMPED
ISOTROPIC CANTILEVER BEAM LOADED AT THE END ....
4.3
CENTER DEFLECTION VS. ASPECT RATIO
(1)
OF AN ISOTROPIC
CANTILEVER CLAMPED BEAM LOADED AT ONE END
4.4
061
....
62
MESH COMPARISON OF AN ALL EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD (P
=
1000 lb.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
66
INTEGRATION RULES COMPARISON OF AN ALL EDGES CLAMPED
RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD
(iP = 1000)
4.6
.................................
ALL EDGES SIMPLY SUPPORTED RECTANGULAR PLATE, UNDER UNIFORMLY DISTRIBUTED LOAD ................
4.7
69
72
CENTER DEFLECTION VS. ASPECT RATIO OF SIMPLY SUPPORTED RECTANGULAR PLATE UNDER UNIFORMLY DISTRIBUTED LOAD ............................
vii
73
LIST OF FIGURES
2.1
General 3-D Body ....................................
2.2
Lagrangian Solid Element - 27 nodes .......................
10
2.3
Motion of a body in a fixed Cartesian coordinate system .........
14
2.4
Lamina coordinate system (2-D) .........................
26
2.5
Stress resultants and couples in a lamina ....................
30
2.6
Consistent loads ....................................
37
2.7
Consistent loads on a mesh .............................
38
2.8
Gauss points .......................................
40
2.9
Instability and bifurcation points .....
6
....................
43
2.10 Constant Arc Length Method ...........................
46
3.1
Thick composite plate-element arrangement ..................
49
3.2
Flow chart - linear analysis .....
51
3.3
Flow chart - Nonlinear analysis ..........................
53
4.1
Bar Sample Problems .................................
56
4.2
Clamped Bar Under Uniaxial Load .......................
57
4.3
End Loaded Beam Bending .....
60
4.4
End Loaded Beazm Deflection ......
4.5
Plate Sample Problems ................................
65
4.6
Clamped Plate Under Central Load .......................
67
4.7
Clamped Plate Integration Rules .....
68
4.8
Simply Supported Isotropic Plate .........................
74
4.9
Simply Supported Laminated Plate .......................
74
4.10 Isotropic Plate Deflections vs. Aspect Ratio ..................
75
viii
.......................
.......................
......................
....................
63
4.11 Laminated Plate Deflection vs. Aspect Ratio ..................
ix
75
I. INTRODUCTION
A.
OVERVIEW
The finite element method pro,, ides a general tool to solve problems of contin ia
such as heat conduction and fluid flow, but it is most widely used in structural
mechanics. In structural mechanics, the methodology is applicable for static and
dynamic response of structures and in predicting the elastic stability limits.
The focus of the present study is to develop tools to analyze thick laminater
composite plates and validate the model by comparing with known solutions. More
specifically, the objective of the present study is to develop a finite element for both
linear and nonlinear analysis using three dimensional elasticity relations.
By adopting such theory for thick plates, both isotropic and composite, the
solutions account for transverse shear stresses, This approach eliminates the limitations imposed by classical plate theory based on Kirchoff-Love hypothesis [Batoz,
19501 or higher order shear deformation theoies [Reddy, 1984, Lo et al., 19771.
B.
LITERATURE REVIEW
In this section, some literature pertaining to the analysis of thick composite
plates is reviewed.
The finite element method has been increasingly used as a
research tool, as well a& a design analysis tool, and the methodology is rapidly
evolving along with the development of faster and more efficient computers. Basic
concepts of the theory of finite element analysis are well documented [Cook, et ;a.,
1989]. Yang (1987) describes various two dimensional higher order elements as well
as three dimensional solid elements. Bathe (1982) discusses the general formulation
of finite elements in nonlinear analysis for one, two and three dimensional elements.
1
based on the total Lagrangian formulation and the principle of virtual displacements.
A good source for continuum formulation may be found in Malvern (1969).
Tsai and Pagano (1968) establish a notation in which composite lamina properties are invariant with respect to lamina direction. The laminate theory is well
documented by Vinson (1987), where the elasticity solution for "structures composed of composite materials" is given for various cases, such as bending of thin
plates. Based on laminate theory, Hoskin, et al. (1986) outline procedures involved
in manufacturing composite components and presents some of its applications.
A higher order shear deformation theory of laminated composite plates was
developed by Lo, et al. (1977). A higher order nonlinear theory of thick plates was
suggested by Reddy (1984a, 1984b, and 1985) and presented solutions (Reddy, 1987)
and compared numerical results to Pagano's (1969) elasticity solution for the case of
cylindrical bending. Other elasticity solutions are given by Timoshenko (1951 and
1959) and Eisley (1989) who discusses the elasticity solutions. The Heterosis finite
element was suggested by Hughes, et al. (1978) for thick and thin plate bending
problems. "Ihe effect of reduced integration in isoparametric finite elements was
presented by Zienkiewicz, et al. (1971).
In recent years, much work is concentrated on the analysis of buckling and postbuckling response of laminated plates and shells using nonlinear analysis. Ramm
(1982) applies degenerate finite elements to solve buckling of thin shells. Arnold,
et al. (1983) presents a theoretical analysis procedure for prediction of buckling
and post-buckling in laminated composite plates and compares the results to experimental results. A combined numerical and experimental study of the post-buckling
behavior of composite panel is performed by Natsiavas, et al. (1987). Gujbir et
al. (1989) use an eight noded biquadratic element to study the effects of transverse
shear on the stability of laminated plates. Some solution algorithms for nonlinear
2
analysis of structures by adapting modified Newton-Raphson and arc-length methods are given by Kolar et al (1985) and Ford et al (1987). In the literature reviewed,
there appears to be no discussion on the higher-order solid element for the analysis
of thick laminated plates.
This research addresses the problem of using a tri-quadratic displacement field
based finite element based on three-dimensional elasticity equations. A total Lagrangian formulation is used to derive relevant element nonlinear matrices, and
numerical examples are included to validate the linear portion of the development.
Analysis of typical examples include slender bars under traction and bending
loads, thin and thick plates under bending loads and effects of various integration
schemes.
C.
THESIS OUTLINE
This section provides an overview of various chapters of the thesis. The total
Lagrangian formulation for analyzing structures composed of three-dimensional elements is presented in Chapter II. Element matrices are derived for both linear and
nonlinear static analysis using the incremental load method. The material characteristics account for both linear isotropic and anisotropic behavior. Formulas are
provided to obtain work-equivalent loads for distributed body and surface forces.
Chapter III addresses aspects of computational implementation of the problem
formulated in Chapter fl. Test cases, example calculations and comparison with
classical solutions and other high order theories are given in Chapter IV. Finally,
Chapter V summarizes the results and reflects some suggestions for future work.
3
II. THEORETICAL FORMULATION
A.
INTRODUCTION
In this chapter, using the principle of virtual displacements, the stiffness matrix
will be developed for static equilibrium of triquadratic isoparametric solid elements.
In the total formulation presented, both small and large displacements are permissible for linear and nonlinear structural analysis. For both cases, small strains and
linearly elastic material will be assumed.
The element is developed for analysis of both isotropic and composite structures.
B.
GENERAL DERIVATION OF FINITE ELEMENT EQUILIBRIUM
EQUATIONS
The principle of virtual work is invoked for the general formulation of equilib-
rium [Bathe, 1982; Cook, 19891. The principle of virtual work states that a body is
in equilibrium, if and only if, the total virtual work done by the internal forces is
equal to the total virtual work done by the external forces. That is,
= 6WW.t
W.t
(2.1)
This principle is equivalent to the minimum total potential energy principle
[6fl, = 01, and holds at any given time.
Consider a three-dimensional body under arbitrary loads as shown in Figure
2.1. Using a Cartesian system, let the loads be given by
{f 8
f: f
=
4
,IT
(2.2)
{f 8 }
= [fB fVB fI]
1p, = [p
T
(2.3)
jT
24
FV,
where
{f},
(2.4)
{fB} and {F} are surface tractions, body forces, and concentrated
applied forces respectively.
The displacements of a finite element in the body due to external load is
denoted by {d}, where
{d} = [u v WIT
(2.5)
and the corresponding strains are given by,
{4
= [rz 6V ex, f. tzz f
]I
(2.6)
for which the corresponding stresses are,
{0
= [Ozz Opy 47zz O,. ffxs O,]T
The total internal virtual work for a finite element in the body is {6e}Tf{}dv
(2.7)
and
for the whole body,
6Wt = j{6c}Tfaldt
where the virtual strains, (be,
{b}
(2.8)
are
= [be.,,,btyv e,
,
.
(?,]T
(2.9)
The total external virtual work is given by:
6We, = f 6d}T{fB}dv + fj{6dTI{f°}ds + {di}T {F}
5
(2.10)
~0
LL.
CO
C3
%U
c)
a
.V4E
IL a
Figure 2.1: General 3-D Body
6
where {d
}
denotes a surface displacements and
{6d' )
represents point (nodal)
displacements corresponding to the applied loads, and the virtual displacements
{6d} are
{6d}T = [6u 6v bw]
(2.11)
On substituting equations 2.8 and 2.10 into 2.1, we get,
j6e}Tfojdv
=
j
{6d}T{fB }dv + j{&P}
T~fhl)ds
+ {6d1}jTf P}
(2.12)
It may be noted that the principle of complementary virtual work may have been
undertaken, assuming small virtual stresses with true displacements, yielding in an
analogous expression for equation 2.12.
Introducing the generalized Hooke's law for material constitutive relations,
{a = [E] {} + {a}
(2.13)
where {a.} denotes the element initial stresses and [E] denotes the elasticity matrix
of the element material.
In general, the strain-displacement relations are given by
(e) = [B]{d}
(2.14)
while the virtual strains are given by
I6e = [B] {6d}
(2.15)
Substituting equations 2.13 and 2.15 into equation 2.12 and simplifying, we have,
If{6d)T ([BI T [E] (BI) {d}dv
j{d}T~fB}dv + fj{6cP}T{P.ds
If
fj{6d}T[B] T {a.}dv+ {bd'}{fP} (2.16)
The integrations in equation 2.16 are performed over the element volume and
surface, i.e., we can evaluate every integral using the element local coordinates and
7
assemble tor the global system coordinates. Thus, we define the global displacement
vector and the global virtual displacement vector as follows:
{D}
= [uIvIwI u 2v 2w 2 ... Unvnwn ]T
(2.17)
and
{6D} = [buj1 v6vw,
...
u,,6v,,,,T
(2.18)
where n is the total number of nodal points in the body. Now we define, for m
elements,
[K] =
[k,]
[BIT [E] [B] dv
f
(2.19)
[B]T [E] [B] dv
=
(2.20)
where [K) and [kj] are the global and local stiffness matrices respectively. In addition, we define,
in
{RB}
=
{Bj=
{R.}
=
{r,}j
=
{R,)
=
V
, f [NIT {fB}d
jfN]
T
(2.21)
(2.22)
{fE}dv
.
j=1
>21[N']T{l'}da
(2.23)
j[NI T {f}d
(2.24)
(2.25)
.
j=1
rl}j = j[BIT {,}dv
where
(2.26)
{R) and {r} denote the global and local load vectors and [NJ and [N'] are
the displacement interpolation (shape functions) matrices for the volume and surface
where traction is prescribed. Using these definitions, we obtain
{6D} T [KJ(DJ
{R}
=
{6D) T ({R
=
{R 8 } + {R.}8
3
) + {R.}
{R)
-
{R,) + {fr)
+ {P}
(2.27)
(2.28)
By invoking the principle of virtual displacements and noting that {bD} is
arbitrary, we get the equilibrium equations in the following form:
[K] {D}
=
{R}
(2.29)
Equation 2.29 is the basic equation for static equilibrium, which also gives the
general form for nonlinear analysis with large displacements and strains.
C.
INTERPOLATION SCHEME
1.
Shape Functions (Displacement Interpolation Functions)
In this section, the interpolation scheme for a triquadratic isoparametric
solid element will be developed.
The one-dimensional Lagrange interpolation function based on parameters is given by
q
NP=
P
i=1
NP,+ N2P2 + ...+ NPq
(2.30)
where Ni, also called the shape functions, are given by
M
l
N,(x) =
__
(2.31)
0*j
A triquadratic solid element is a three dimensional element in which the
displacements u, v, and w are interpolated by quadratic langrangian interpolation
functions with 27 nodes. Figure 2.2 depicts an element in the local non-dimensional
coordinates (r, a, t).
For an isoparametric element, the geometry may be interpolated as,
27
Nizi
X=
i-I
27
y=
9al
II
i9I
.-
,
N
Ln
I
U.,
I.
. .. I
H
.
(fl
0
'VI0
0
0
t_,
0
-
Figure 2.2: Lagrangian Solid Element - 27 nodes
10
0
k
C
27
z =
j
(2.32)
Niz,
i=1
or, in a matrix form,
y
z
-[NJ [zjyjzj .... r27y27z2] r
(2.33)
where the shape function matrix is given by
N,
0 ...
N2 0
0 N2 0 ...
0
0 N2 ...
01
N27 0
(2.34)
0 N 27 0
0
[Ni
0 0 N
0
0 N2I
The shape functions and their derivatives in local coordinates are pre-
0
NI
0
0
sented in Appendix A.
2.
Jacobian Transformation Matrix
To obtain equilibrium equations in the global coordinate system and con-
struct stiffness matrices, we need the derivatives of the shape functions in cartesian
system. Using the chain rule for differentiation, we obtain,
Nj,,
=
Nt
z,,
y,,
z., t
z,
N,,,
t
N
(2.35)
where [J], the Jacobian matrix is given by,
[J]=
Z.r Y.r
z, y.o
X't
X~t
Z,r
z,
(2.36)
Z.1
A comma denotes differentiation, where for example, Ni,, =
etc. Using the
shape function derivatives, the elements of the Jacobian matrix may be calculated
and is given in Appendix B. The global cartesian derivatives may now be obtained
as,
rN'
N,,,
N,..
=
[J]11
rj~
NI,, 1
N,.,
N,,]
(2.37)
where the inverse of the Jacobian,
(2.38)
is given explicitly in Appendix B.
D.
STRAIN DISPLACEMENT RELATIONS
1.
-
(B]
Basic Formulation
In this section, the basic formulation for nonlinear analysis of a general
solid body is presented [Refs. Bathe (1982), and Malvern (1969)]. First, some definitions and notations will be introduced concerning the coordinate system, displacement, stress and strain measures and later on the linearized equilibrium equation
will be developed based on section II/B.
Consider the motion of a body, or an element within, in a fixed cartesian
coordinate system as shown in Figure 2.3. We have the body at time 0, t and t + At
for which the upper left superscript corresponds. The displacements at time t and
t + At are given as
tui=t Z _0 Zi
t+Aiu .F At
_0 Zi
(2.39)
(2.40)
so that the incremental displacements are
% =t+t
_ u,
(2.41)
where,
U
1
U
X 1 =r
U2 -V
U3 -W
z 2 =s
z 3 =t
12
(2.42)
we use the following notation for derivatives at time, say t + At, with respect to
coordinate at time 0 as,
+atuj = &
(2.43)
In the present approach, we use the total Lagrangian formulation, referencing all variables to the undeformed configuration at time 0, [Ref. Bathe, 1982;
Malvern, 1969]. It is assumed that at time 0 and t the equilibrium configuration is
known. Basically, equation 2.12 needs to be solved corresponding to time t + At.
Since we assume large displacements, and nonlinear constitutive relations [equation 2.14], equation 2.12 may be solved by incremental load methods [Ref. Ford &
Stiemer, 1987].
On introducing the 2nd Piola-Kirchhoff stress, it may be shown that the
2nd Piola-Kirchhoff stress tensor is energentically conjugate to the Green-Lagrange
strain tensor [Ref. Bathe, 1982; Malvern, 1969].
j
{
_+At}T
{t+&S}Odv
=
j+A,
{6e} T {t+at'}t+Atdv
(2.44)
where the 2nd Piola-Kirchhoff stress at time t is defined as,
I o,= [,X]T {tS} [,Xi det [X]
(2.45)
such that [tX], the deformation-gradient tensor is a tranformation operator from the
coordinates at time 0 to time t. Note that in the equation 2.44, the right hand side
represents internal virtual work at time t + At over the volume at that time while
the left hand side has the virtual work integrated over known configuration at the
reference volume.
Assuming linear material behaviour, we may use the linear stress-strain
relations (Generalized Hooke's Law) for the 2nd Piola-Kirchhoff stress tensor.
[ s] = [E]
13
[]
(2.46)
$44
*X
V
-H
I,
(13
'
,
YPO)
0p
'4
_-
...
..>..
.
E
4-)N
CC
1
U414
4
where the Green-Lagrange strain tensor at time t is defined as,
Si 1 (i
+' Uj, +' Uk,, 'Uk,)
(2.47)
with i, j, k = 1, 2, and 3. On subs Quting 1+u = ui + ui, we obtain at time
t + At,
t+At
+
1
+
+t Uj,i
+ u
(ui, + uji,
k,itUk)
u k., ukj + Uk,,
k,,)
(2.48)
Uks ut,,
as,
which may be written
t+atfii = f +
i
(2.49)
where, ti is defined earlier and the incremental strain e,, is given by
4, = 4 J + r/ij
(2.50)
{e} = {e} + {r/}
(2.51)
In matrix notation,
The linear incremental strain is identified as
1
ei=
+t
(,
+ U,,i +
,, Uk
+ Uki Ukj)
(2.52)
in which tukbj and 'ukj are the known displacement gradients at time t. The non-
linear incremental strains, then, are givtn by
=
1 uk.i u,
(2.53)
Rewriting the equilibrium equation as stated in equation 2.12, using the total Lagrangian approach, we have,
5(2.54)
where t+AR, the external virtual work, is assumed to be deformation independent.
Using the identity form equation 2.44, we may write the equilibrium equation in the
undeformed configuration as
j{At+dte}T{t+&ts}odv
_.=+t R
(2.55)
Noting that I{l} is displacement invariant,
{b+A'e = {&e}
(2.56)
The 2nd Piola-Kirchhoff stress at time t + At may be expressed as
{ 3 } + {1)
{t+AS} =
(2.57)
where {s} is the incremental 2nd Piola-Kirchhoff stress. On substituting Equations
2.56, 2.51, and 2.57 into 2.55 yields,
j{6 eTfs}Odv
+ I {60{'.1v+f6
Its)Odv
17 )T
(2.58)
The incremental stresses are expressed using equations 2.47 and 2.57 as
Is} = [E] I +4'e}
-
[E] {'e}
(2.59)
which in view of Equation 2.50 yields
{s} = [El {e}
(2.60)
Referring to Equation 2.51 and neglecting the nonlinear strain contribution, we get
the linearized approximation as
I1
- [E] {e}
(2.61)
{6e}T
(2.62)
and
{6C}T
-
16
On substituting these into Equation 2.58 and rearranging yields the linearized incremental equilibrium equation,
[E e 0
j{Se}T~
v+j{,}{ts}lodv = 4t R
j6eTftSOdv
-v
(2.63)
2. General Nonlinear Discretization
The general nonlinear finite element discretization for the 27-noded element is presented based on the total Lagrangian formulation discussed in the previous section.
Equation 2.52 for the linear incremental strain in cartesian form
yields
er=
evy
UI
9UI
'
tVXVX+
-= U'Y +}t U'Y uY -+-tV'W V' +O t
eVI
-
U, 1
eX
=
U,
2e
=
V,
v= + WV +t UU
2e,
=
u, + w. +t
2erv
=
u.1, + v" +9 ' u'Y + u
U
U,
1
UI
U.
+'
wW wWz
W
IV .
U~i pl
Uw,
1 WV
V13 V,+
W," W,
+ U1 'U's +t V,1 V's + V,1 tv,5 +W
u,.UV+ u= t .,+t vS v, + VIC t
. Y+tV'. v W+ v 'y
w,5 + W.V tW.X
' ?+
1
W
w) tWp
t WI. oy' + w" t'Y~
(2.64)
which in matrix form is given by
{c} = {eLa) + {eL,}
(2.65)
The first term on the right hand side is displacement independent while the second
term is displacement dependent with the engineering strains {e} represented by
{e} = le
evy e.. 2ey, 2e.
2 e,,]T
(2.66)
We define the incremental displacement gradient inthe global coordinates by
w'T w,W Wo.
{UG} = [uu.,U u, v, V,1 v,.,
1
17
T
(2.67)
Equations 2.64, 2.65, 2.66, and 2.67 result in
eLo) = [ALo] {UG}
(2.68)
{eL1} = [ALI] {UG}
(2.69)
so that [ALo] and [ALI] are given by,
1 0 0
0 0 0
0 0 0
0 0
010
0 00
000
0 0 0
001
0 0 0
0 0 1
0 1 0
0 0 1
0 0 0
1 0 0
0 1 0
100
0 0 0
[Au] =
tu,
0
0
[ALI]
0
t
(2.70)
U,.
0
0
0
t u.,
tv,
0
0
tvV
0
0
tw.
0
0
0
tv,,
0
tv,y
0
tw,
'w,
tv,
0
tw,
1wv
0
tw ,
tw,.
0
0
w
0
0
0
Sw,
(2.71)
-
0
tu,
tU,y
'U, 5
tu,
0
t v,
0
tu.,
tu.
0
v.,
9v,y
0
tv.:
5
It may be noted that the values of t uij are known at the new configuration at time
t + At. With the displacements interpolated by
27
Nkuk
U =
k-l
27
v
=
E Nkv,
k=
27
w = ENwk
(2.72)
kal
the local displacement gradients are obtained from
27
U
=r
=
N,,. uk
kni
27
U's
=
E Ni,,. uk
k-l
27
U's
N,,t uk
=
k=i
18
(2.73)
and similar expressions may be attributed to v and w. These gradients may be
represented as
U!
N,,.
Ni.,
0
0
0
0
0
0
N2 ,.
N 2 ..
N 2.#
0
0
0
0
0
0
N2 7.,
N 27'.
N27 ,
0
0
0
0
0
0
0
N1 .
0
0
N,,
0
0
N27.,
0
N,.,
0
0
N2,.
0
0
N27..
0
N,.,
U.T
U.8
1,2
V2
U's
V.,
V
W
=
0
V,,
w.,.
W,
0
Ns.,
0
0
N2, t
0
0
N27,,
0
0
N1 .,
0
0
0
0
0
0
0
0
0
0
0
N2 ,,
N2 ..
0
,,,, .
0
0
0
0
N.,
NI,,
,,
.
.
w2
0
N27 ,,
N2 7,.
N27 ,, J
•
527
P27
(2.74)
or alternatively,
{UL} = [DH] {d}
(2.75)
where the nodal displacement vector is given by,
{d}
=[u, VI W1
u2 v2 ...
(2.76)
w 2 7]T
and the local incremental displacements gradient are given by
{UL)
= [u., us u,t ,Vv,,.
v,, w,, w.
.,lT
(2.77)
As previously mentioned, in isoparametric finite elements, the same interpolation functions are used for approximating the geometry and the displacements.
Using these definitions, we may transform the displacement in global and local coordinates by similar transformations used for the geometry. Furthermore, we can
define arbitrarily the global and local coordinates to coincide at time 0 configuration,
thereby, the transformation from local coordinates at time 0 to local coordinates at
any other time, say t or t + At is identical to the transformation that relates the
global coordinates to the local coordinates at any configuration. In other words, we
19
can use the same jacobian matrix defined previously for all configurations. Writing
the relation in accordance to equations 2.37 and 2.38 we have,
{uG} = [3] {UL}
such that,
[(F] [p
[0]o
[o [r [0]
[o0 [o1 [r]
and substituting equation 2.75 into 2.78 yields,
[r 3]=
{uG}
=
(2.78)
[r3l] [DH] {d}
(2.79)
(2.80)
The incremental strains, then, may be expressed in terms of nodal displacements, and substituting equation 2.77 in 2.68 and 2.69
{e,} = [An,] [I31 [DHI] {d)
(2.81)
{eLll = [ALI]
(2.82)
[r
3] [DHI] {d}
The strain displacement operator may be identified as
[BLo] = [ALo] [13] [DH]
[BLI] = [ALI]
(r 3
[DHI]
(2.83)
(2.84)
such that,
[BL] = [BLA] + [BL1I
(2.85)
{e) = [BL {d)
(2.86)
and,
It should be underscored that using only the displacement independent strains in
equations 2.63 and 2.64 results in the linearized problem (same as linear small
20
displacement - small strain formulation) with
Ni,"
0
0
N2..
0
0
N2 7 ,
0
0
N,y
0
0
N 2,1
0
0
N2 7,.
0
0
N27 .
0
0
Nj,.
0
0
0
N27 ,1
0
N1 ,.
0
N,y
N1 ,=
0
N 2,.
N 2,1
N 2 ,y
N 2,.
0
0
N 27 ,,
N27 ,,
0
0
N2,
N2 .
0
N27 ,y
N 27 .
0
0
[BLO) =
N1 ,,
N, 1, N.
N2 r.y, N 27 ,.
(2.87)
The displacement dependent contribution to the strain-displacement operator
is given by
'U.. Nt
'u., N",.
tu.
[BL.
'U. N,
N1+1
, N,,,
III , N , 4- u,
iNI
a NW4
ta. Nil
Ni,
.'
M.
' Nf.,, ,
't
' uI Vi
, N ,, + ', .,U Ni,
M
,,
.N,+' V., N,,
N.
'u,,
U, MW..
NaT'
'U., N,
'i.
1W..N,,+W, IV ,, V NJ,,
N+, Nwo
l , N ,,
I .N ,,
'W, N,, +t W., N,'
,
in,
N
'u,' N2,,
a N
4
NJ,
U . N ,,
+, U., NU,
NW,.
**
. iNV.
Hr
, N ".
~
W. NW,,
(2.88)
To obtain the incremental nonlinear strain contribution, consider equation 2.53, which has the cartesian components,
2
S1(2
+ W )
2 (uI+*l +
2a
= 1 3 +22)
'lpv =
'Iv
(uyiU + V,,v
+
=
2
W
)
1
17X=
77,
77v
=
-(u,,u, + V.WV., + w.,w.,)
2
-(u.,u,
+ V.zv, 1 + ww,,)
1
-- i(u.xu., + Uv.v + WxW.Y)
22
21
(2.89)
With the variations at time t given by,
t v,
6?7,
= 6ul,: tu, + 6v,
br/
=
67,,
= bu,, t u,, + 6v., t v., + 6w., 'w.,
6
+ 6w,. 'v,
uLUI
'u. , + 6vb iv,V + bwV tw.,
2b%.
= 6u,,
26r7.,
=
U,,
+ uV 6 u,, + bv., iV, +
*'v.6v,, + 6 w'V
'W. +
'w11w
ut, t u., + tus6u, + 6v, Iv., + t v.,bv, + 6w,: tw. + tw6w.,
2,"v = 6u.
tu,,
+ CuX6u. + 6,v,
tv ,
+ 'v,:v,, + 6xw,: t w, + 'w bw.2.90)
(2.91)
It is worth noting that equations 2.90 are in exactly the same form as the displacement dependent strains {qIt} given in equations 2.64 and 2.65 with the incremensal
displacements derivatives replaced by their variations. Thus, we may write,
(6bq) = [ALI] {6UG}
(2.92)
(7) = [ALI] {UG}
(2.93)
such that the nonlinear incremental strain variation vector is defined as,
= [61,, 6q, 671, 2671,, 26%, 26,~v
{ b}T
6
(2.94)
and [ALI] is as given in 2.71. Observing relation 2.80 for JUG), we have,
{6uG} =
[r3] [DH] {6d)
(2.95)
and substituting for the global variations into the nonlinear strains in equation 2.90,
we get
{6,7 }T
=
{6d)T [DHIT [r3 lT [ArlIT
22
(2.96)
Using the strain-displacement relations defined thus far, we formulate the
linearized incremental equilibrium equation, given by 2.63 to take the general form
as stated in 2.29, to give,
tR}
d"+)=
[VkL] +{t+A)
-
{t+AtFI(I)
(2.97)
where [t kLl, in view of equation 2.85 is seen to be
[tkL] :
[BOn
[E] [BL] dv
(2.98)
which is the linear stiffness matrix, and i is the iteration number. This includes the
displacement dependent and independent contributions.
When small displacements are assumed, the [tkL] reduces to standard linear stiffness matrix, given by fov[BL oIT [E] [B o]0 dv. In what follows, the derivation
of [kNLI is described.
The 2nd-Piola stress vector at time t is accumulated such that,
{t s} = {+'+s} + [E] {}
(2.99)
Using equations 2.51, 2.85, 2.86, and 2.91, the incremental strains take the form
{E} =
(2 [ALI] + [ALo]) [IF3 [DHI {d}
(2.100)
or alternatively,
{c
= (2 [BLI] + [B,] Il){d}
(2.101)
Noting that equations 2.99 and 2.100 are valid at any time t, and by using equation
2.46, the 2nd Piola-Kirchhoff stress may be written as
{ s} = [E] (2 [AL)] + [ALo]) IV3] [DH] {d}
(2.102)
and the nonlinear part of equation 2.63 becomes, using the relations 2.95 and 2.101,
]T [ALIT) [E]
({6d}T [DH]T
f{b 71 r {ts} ° dv =
f
(2 [ALI] + [A,]) [173] [DH] {d}°dv
23
(2.103)
we define the nonlinear strain contribution, [kNLI, as
[kNL] =
jT[DHIT IF T [ALI]T
[E] (2 [ALI] + [ALo]) [173] [DH]° av
31
(2.104)
It may be recognized that [DH] is the strain-displacement relation given by equation
2.74 which corresponds to the linear contribution of the stiffness matrix given in
equation 2.97, and the contribution of the non-linear strains results in the 2nd
Piola-Kirchhoff matrix, [s], as
T
[S] = [r31T [ALI]
[E] (2 [ALI] + [ALo])
[I3]
(2.105)
[r3] is given in equation 2.79. It may be shown that the matrix [s] takes the form,
[ sil
1
0 101
IS [0]
[0] [S] I
[0]
[s]
[0]
(2.106)
The expression for the second term in the right hand side of equation
2.63 is evaluated in the same manner as the linear and nonlinear parts. On using
equations 2.100 and 2.101, we obtain,
[BLI]T +[BLo]T) [E] (2 [BLII+[BLo)] {d} °dv (2.107)
d}T(2
IT
j{6e}rTts}dv =
It may be seen that by defining
{ (t+AF}
=
jV(2 [BL]r +[BLo]T )[(2[BL1 + [BLo)] {d}°dv
(2.108)
the expression {d}T I+A(F} represents the work done by the external loads at time
t + At. Noting that
{ 6 i+At}
{6e}
(2.109)
We approximate for the second term in the right hand side of equation
2.62 such that
{e
T {ts }odv
/o
j{6 +AtclT{I+ "s}Odv
24
(2.110)
which represents the internal virtual work, so that the right hand side of the equilibrium equation 2.63 is the difference between the external and internal virtual
work. On substitution of relations 2.97, 2.102, 2.103, 2.107, and 2.109 into 2.63
and applying the principle of virtual displacement as shown previously, we arrive
at the incremental equilibrium equation 2.96, which may be solved by Newton-type
methods. [Ford and Stieman, 1987]
E.
STRESS-STRAIN RELATIONS
In this section, the stress-strain relations for a composite material, to be used in
the three dimensional analysis is developed. Two approaches, one based on classical
laminate theory and the other based on anisotropic material constitutive relations,
are presented.
1.
Classical and Higher Order Laminate Theories
Typical structures composed of composite materials are built using sev-
eral number of laminae, forming a laminate. Each lamina consists of, typically,
uniaxial fibers embedded in a matrix, such as epoxy-resin, forming a thin plate.
Figure 2.4 shows principal material axes, labelled 1 and 2 in directions parallel to
and normal to the fibers, respectively. It may be noted that in each lamina, there
exists a state of plane stress, as shown in Figure 2.4.
Assuming elastic orthotropic material, (i.e., the lamina possesses a plane
of elastic symmetry parallel to the x-y plane), the generalized Hooke's law may be
written as
al
a2
a3
Q1
Q 12
Q13
Q22
Q2
_
a4
symm
as
06
0
0
Q33 0
2Q4
L
0
0
0
0
2Qss
0
0
0
0
0
2Q66
25
E
f3
f4
fs
J
c
(2.111)
x
00
4-)
0
X
00
Q)
x>
0
,,
__ __ __ _ __ __ _
t
-I
"
-
x
-
tW4
Figure 2.4: Lamina coordinate system (2-D)
26
CM
where the plane-stress elastic constants are given by:
Q1I
=
QI
-
I --
E
V12
21
= 1 2 E2 '
1
-
V 12 V21
=
Q44
=
G
Q13
=
Q23 =
23
12
-
3
Q22
Qss
,
21
= G 1 3, Q6 = G12
(2.112)
0
The subscripts 1, 2, and 3 correspond to normal stresses or normal strains
while 4, 5, and 6 correspond to shear stresses or tensorial shearing strains in yz, zx,
and zy planes, respectively.
The stresses in the material coordinate axes are transformed to reference
coordinate axes (x, y, z) by the following equation:
axz
m2
n2
0
r,
n2
m2
0
o,
0
0
1
Oryz
or.
0
0
0
0
mn -mn
rY1,
0
0
0
-2mn
2mn
0
0
0
0 m
0 -n
0 0
0
a,
Or2
(2.113)
03
n
0
m
0
2
0 (m - n 2 )
0r4
as
'6J
where the direction cosines m and n are given by m = cos 0 and n = sin 0.
The straint, may be transformed in a similar manner. Introducing the
strain transformation, along with equation 2.110 into equation 2.112 results in
Q11 Q12
oQ
k
0
2Q16
C.
C
Q22
0
0
0
2Q26
0
0
Q33
0
0
0
0
0
0
2Q44
2Q45
0
0
0
0
2Q 45 2Qss
0
Q26
0
,z
Oz~y
0
Q12
az
0
0
LQi6
0
0
2Q66 J
where,
Q11
=
Q
m 4 + 2 (Q 1 2
4
+ 2Q 6 6 ) m 2 n 2 + Q 22 n
27
C2
(2.114)
fxz
-
k
012
=
(Q11 + Q22
022
=
Qun
033
Q33
Q16
Q1
-
4Q6) m 2 n2 + Q12 (m4 + n 4)
4
+ 2(Q12 + 2Q
3
n
Q22 mn'
-
6
-
4
)m 2 n 2 + Q 2 2 m
(Q12
=
Qjmn-3
Q6
=
(QI + Q22 - 2Q2) mn 2
Q44
=
Q4 4 m
(Q55 - Q4) mn
=
Q 4 4n 2 + Q 5 5 Mn2
=
+ Q6 (m
-n2)
_
-
n2)
n2)
2
2
+ Qs 5 n
Q45 =
Q5
(M2
Q 22 m 3 n + (Q12 + 2Q6) mn (m2
Q26
2
+ 2Q66) mn
(2.115)
The stress strain relations presented correspond to kth lamina.
Now,
consider a laminate composed of N laminae for which, each lamina has a different
orientation (0), with respect to the laminate x and y axes. For linear elastic plates,
the function!L, form of the displacement may be assumed to be
u(x, Y, z)
=
Uo(x,y) + zu 1 (X,y)
v(X' Y, z)
=
vo(,y) + zVI(X,y)
w(x, y)
=
WO(X,y)
(2.116)
and, the linear strains are given by,
fi=
1(uij + u,i)
so that,
Ezz
=
fyy
-= VO, 1, + ZV14 y
zz
UO,x + ZUi,x
0
28
(2.117)
1
fu
C'
+ w0,11)
1(Vi
2
=
1
=
2
I(Uo, + vo+
U1 + WO )
(u,
+ vI)
(2.118)
where uo, vo and w0 are the midplane displacements, ul and v, are related to the
rotations of the normals. It may be noted that the in-plane strains,
1
fo = Uo,X, EO = Vo, ,
(2.119)
(Uo, + Vo,)
zo =
and the curvatures are given by
= ul.,
UT¢ tc
=
vl,p,
Kz 1
=
(2.120)
(ul, + vi)
We define stress resultants for plate/shell type structures in terms of
stresses and shears (see Figure 2.5) as follows for the kth layer:
h
QX= L
2
h
~zdz
2
/M azdz
=
(2.121)
Similar expressions are applicable for Ny, Ny, Qy, Mv and M,,y, where h is the
lamina thickness.
By summing all laminae over the laminate thickness in the following manner,
{}
N..,,y
E~i
k=1
h -,
f'YO
}
dzJ~
+~
f
k_1x
zdz}
(2.122)
which, in matrix form, may be written as
{N} = [A) {o} + [B] {K}
29
(2.123)
x
~co
CH
4-)r
CI)
x
4-do
Q)
x
30)
-
where,
N
Ai 3 =
hk
(Q") k
1:
-
hk-I)
k=1
N
(h 2 - h _1 )
Bi =
(2.124)
k=1
with ij
= 1, 2, and 6 and the moment resultants are given by
{M} = [B] {co} + [D] {,}
(2.125)
where,
Dj =
E
-
I2.126)
k=1
with i,j = 1, 2, and 6.
The displacement field, as stated in equation 2.105 is linear in the thickness direction, resulting in constant shear stresses. To get better accuracy, a higher
order displacement field may be used [Reddy, 1984].
In order to account for the accurate shear distribution, shape factors are
used in computing shear energies. These factors are typically obtained by equating
the shear energies.
The procedure is outlined for linear and cubic variation of
displacement fields. The shear energy due to transverse shear stresses is given by,
Ua
=
J
(ao,, (2,Ez) + oyz (2,,)) dz dA
(2.127)
where A is the area bounded by the lamina surface dy. On using Hooke's Law, we
get
=
Uo~~~~
U.
AL
(Gxz (2cz+)2 +
=
dA
d(Gz(~~
(2fy)2) dz
(2.128)
Equating this to the linear displacement field and simplifying,
(1A
2=
[G.. (u, + Wo,.)
2
31
+ G
(v1 +Wo,,)2ldA)
h
(2.129)
Introducing a higher order displacement field yields a more realistic stress distribution, but in doing so, a shape factor is introduced to yield consistent shear energy.
Assuming a displacement field in which the displacements are expanded
as cubic functions of the thickness coordinate, while the transverse displacement is
assumed to be constant through the thickness, yields
+
U(z,i,Z)
=
UO(z 1')
+
ZUI(x,t)
V(-,%,z)
=
VO(zY)
+
ZVI(z,Y) +
W(Z,Y,Z)
=
Wo(XY)
Z U 2 (z,y,)
+ Z3 U3(x,y)
Z2 V2(z,y)
+ Z3 V3(x,y)
2
(2.130)
With u0, vo, and wo being the displacements of the midplane, the tensorial
shearing strains are evaluated as,
2fz,
=
[u,(Z,,) + 2u 2(r,y)z + 3U3 (,y)z
3
+ WO]
(2.131)
+ 2V2 (.y)z + 3U3(xy)Z 3 +
=
Using the condition that the transverse shear stresses vanish on the plate
top and bottom surfaces, we have,
OX
X(,
,+
,V Y,
= a1
0
0
(2.132)
or,
(X,
fX(V
, ±h
= Cy Y1h) 0
(2.133)
Substituting relations 2.132 into 2.130, we obtain,
U2
=
U3
=
V2 = 0
4
4 (woM + u1 )
-
4
V3
=
(W,
-
32
+ vI)
(2.134)
The displacement field then becomes
u
=
34
uo+zu 1 -z(w,+
v
=
vo + zvI -
W
=
w0
)
"Z34(wou
+ vI)
2
(2.135)
3h
(2.136)
and the shearing strains are given by
E-z
2
-(uI
=
[1
+wo,)
-4
h
(2.137)
(v+ wO.±) 1-4 (z)]
V- =
Yielding
Ui. =
(1j
1
2
G (i+w,)
+ Gy (vI + WO,Y)2] dA
~
[
2]
d
(2.138)
The shear energy ratio of the two displacement field is found to be 1, so that the
correction factor for constant shear stress using cubic displacements is
8.
It is clear from the discussion that the classical plate theory stiffens the
plate by not taking into account the higher order terms.
If we use the higher
order theory, we need to introduce a correction factor to the shearing strains of the
magnitude
Using equation 2.113, the transverse shearing stresses for the kh
layer are given by
a.xzk =
Q~IZ +
2
Q45k'EYZ
a , k = 2Q45 ,C + 2Q 44 ,f.
(2.139)
and the resultants are obtained using equation 2.122 as
QX
Q=
2(As55Ex + A4 5 fyz)
2 (A 45C.
+ A
33
44
'y)
(2.140)
Note, that equations 2.139 and 2.140 are applicable for any displacement field.
Hence, for the higher order theory presented,
-15
A,3
I-zj
k=1
(z)2] dz
[-4
1~)
k
(2.141)
or,
A
15 ~ZQJk
3
8k=1
hk-hkl4-(h 3
3h
k
2
h~3_)]
(2.142)
with i,j = 4, 5.
In the present three dimensional solid element, for which only three translational
degrees of freedom per node are defined, resultants are divided by the corresponding
thicknesses to obtain the stress-strain relations with
All
A 12 0
A 12 A 22 0
0
0
0
[1E
h
-E
A16
0
0
0
0
0
0
0
A26
0
0
0
0
2A 16
2A 26
0
0
0
0
2A44 2A 45
0
2A 45 2A 55
0
2A 66
0
(2.143)
For the special case of isotropic material, the material stiffness matrix is given by
[E]
A+2G
A
A
A+2G
A
0
0
0
A
0
0
0
A
A
0
0
0
0
0
0
A+2G 0 0 0
0
G 0 0
0 G 0
0
0 0 G
0
(2.144)
where
(2.145)
A =E
(I + v) (1 - 2v)
2.
Three-dimensional Anisotropic Theory
As an alternative to using laminate theories to obtain [E] matrix, we may
use anisotropic definition of the laminates.
ai = (Qi')
34
E
(2.146)
These relations are approximated by obtaining the Qj in the laminate x-y axes by
suitable transformations and transverse properties (thickness direction) correspond
to the matrix characteristics.
F.
CONSISTENT LOADS
In this section, we consider the element nodal loads vector, due to applied
loads. Using the virtual work principle, the distributed loads, such as surface loads
and body forces, are converted into discrete loads applied at the element nodal
points. Discretizing the distributed loads along these lines are referred to as consistent or work-equivalent loads. Consider a case where a uniform distributed load
acts on a prescribed face of the element, as seen in Figure 2.6. The consistent load
vector may be written as
{r,} -
j [N']
T
{f'}ds
(2.147)
For uniform distributed surface loads, we have
{f'} =p{l}
(2.148)
It is worth noting that the interpolation functions on a given surface, say t = 1
reduces to that of a plane biquadratic Lagrangian -mparametric element and are
presented in Table 2.1. Invoking symmetry, we observe that the forces at nodes 1,
3, 5, and 7 are equal, and similarly, forces at nodes 2, 4, 6, and 8 are equal. On
using the shape functions for the t = 1 surface, we have,
r, =
_ "l1
lr)+_ (1 -s)p dr ds -r
35
1
1
-- r2 - -r9
1
p
(2.149)
TABLE 2.1: SHAPE FUNCTIONS FOR 9 NODED BIQUADRATIC
ELEMENT
N1 N
=1 l=(1 + r) (I1- s) - 2 N8 - l N2
N3 =
1(
+ r) (I + s) - I N2 -1N4
N5 Ns=X4 (1 -r) (I +s) -2IN4,2
N2 N7=X
=14 (1 - r)(I1-
N
=
s) - 21Ns- 2 1 N6-
2 ) (Ir1 +
1 (2
1(1-r)(X-S
S)
- 1
N
4N9
- I4N9
-1N9
4
14N9
'
2
2)
2
N9 =
1N6
2
-
2
( 1 -r2) (1 _ 2)
Note: Node numbering is referred to Figure 2.2 where t = 1, upper plane.
Figure 2.6 gives consistent element nodal loads for a single element. As a check, the
total pressure loading on the surface, 2 x 2 x p = 4p, is seen to be equal to the
sum of all the discz.f'.zed nodal point forces. The procedure may be extended for
more than one element by summing loads at joint nodes, as illustrated in Figure 2.7
for four elements.
G.
INTEGRATION
In this section, we summarize the Gauss method for numerical integration,
including a discussion on some aspects of integration schemes.
36
cr.
Q
LL
4-
070
0
LLM
CC\
LL
Figure 2.6: Consistent loads
37
0)
-- -------------------
c:
0
41-)
L)
Figure 2.7: Consistent loads on a mesh
38
1.
Gauss Quadrature
The nature of finite element matrices suggests the usage of numerical
quadrature. Gauss's integration scheme is the most commonly used approach and
is adopted in the present analysis.
The method enables exact evaluation integrals, consisting of polynomials of any order, by using appropriate order of integration. In general, the Gauss
quadrature for a function O(r, s, t), has the form
I=
_ZZ w ww¢kO(r,s,t)
0(r, s, t) dr ds dt
(2.150)
The integration limit reflects the limits of non-dimensional 'master' isoparametric
elements, while O(r, s, t) represents the stiffness contribution.
Figure 2.8 demonstrates the application of the method for a two dimensional biquadratic element. Using the weighting factors as given in Table 2.3, the
element stiffness matrix is evaluated, for example, by using a 3rd order integration
scheme as follows:
55
[K] = 9 (01 + €3 + €7 +
09)
+
58
99
(02 +
04
+
06
+ 08) +
88
9 05
(2.151)
where
O, = h [B(r, S)]T [E] [B(r, s)] I J(r,s)
I
(2.152)
as is evaluated at Gauss point i as shown in the Figure.
2.
Integration Scheme
The term "full integration" refers to an integration scheme which evalu-
ates the integral exactly as shown in the previous example. In the same manner, a
lower order integration is referred to as 'reduced integration'.
In the present analysis, 'full integration' is used to evaluate the stiffness
matrices. When a crude mesh is used, a stiffer structure is obtained. In geieral, there
39
xS
0
o
I
,I
o
II
II
x
x
Iw
*.-I
4-)
CK
Figure 2.8: Gauss points
40
TABLE 2.2: SAMPLING POINTS AND WEIGHTS FOR GAUSS
QUADRATURE OVER THE INTERVAL -1 to 1
Order n
Weight Factor W
Location of Sampling Point
1
0.
2.
2
±0.57735 02691 89626 = ± 1
1.
3
±0.77459 66692 41483 = ±v 6
0.55555 55555 55555 =
9
0.
0.88888 88888 88888 =
9
4
±0.86113 63115 94053 = ±
[
L7
0.34785 48451 37454 = I2
-,
6r
±0.33998 10435 84856 = ± [3-]2
0.65214 51548 62546 = 1 +
where r = rT.
and (2r - 1) is the polinom order
are two ways to soften the structure. One way is to refine the mesh and another by
using 'reduced integration'. Thus, by using a 'reduced integration' scheme, a faster
convergence and more cost-effective, accurate solution may be obtained. However,
the method suffers such drawbacks as mesh instabilities or mechanisms, resulting in
a singular element stiffness matrix.
H.
BUCKLING ANALYSIS
1.
Introduction
It is well known that thin columns or plates under axial compression tend
to buckle. Elastic buckling occurs when the compressive stress is well below the
material stress limit. A flat plate under axial compression shortens in the direction
of the applied compressive loads. This shortening results in coupling between inplane and out-of-plane displacements.
41
As the applied compressive load increases, there is a configuration at
which the plate offers no more resistance to deform, resulting in a state of neutral
stability. The load corresponding to this configuration is referred to as the buckling
load and constitutes a limit point on the load response curve.
At this critical value, the deflection becomes very sensitive to any change
in the configuration. For some structures, beyond the limit point, the load-displacement path may take any of multiple paths. The point where the plate can take any
of the different paths is called the Bifurcation point and is illustrated in Figure 2.9.
In analyzing for nonlinear response, the incremental load method is adopted, which
may be summarized as follows: (a) the tangent stiffness matrix is formed, and solved
for displacements for an incremental load. Keeping the stiffness matrix constant,
corrections to the incremental displacements are obtained in an iterative manner
until equilibrium is achieved, (b) total displacements for this load are obtained,
(c) a new tangent stiffness matrix is formed at this new equilibrium position and
steps (a) and (b) are repeated. This procedure is continued until the desired load
is reached or the critical buckling load is reached.
2.
Implementation
In this section, the Finite-Element formulation for buckling will be pre-
sented [Bathe, (1982), Kolar, et al., (1985)].
The problem of instability can be
approached either by looking at the equilibrium of the structure in the deflected
position and transforming all quantities to the initial configuration or by solving the
system in the current configuration. The former approach, described earlier as the
total Lagrangian formulation, is adopted here. By performing an incremental load
analysis, using the nonlinear formulation described earlier, we may write
([tKLI
+
['KNL])
{d}'+')
42
_=
&t A{P
-
IF)(')
(2.153)
P
l
r0 I
rit
cn
itpointa&
10.
Patt:b
u ck
I ng
Fig. 2.9; Instability and
bifurcation points
Figure 2.9: Instability and bifurcation points
43
where {P} represents the total load applied and t+AtA is a scalar, referred to as
load parameter. The value A scales the incremental load and may be treated as a
constant or variable during iterations. Buckling load is reached when displacements
become large with no increase in the incremental load, i.e., the global stiffness of the
structure, as illustrated for a single degree of freedom system in Figure 2.9, becomes
small and [K] tend to be singular. Thus, using Newton-Raphson and modified
Newton-Raphson methods, convergence difficulties are encountered as buckling load
is approached. This is overcome by using arc length methods, described in the next
section, where the load parameter is continuously updated to reflect the state of the
structure.
3.
Constant Arc Length Method [Kolar and Kamel (1985)]
When using the Newton type iteration schemes, the stiffness matrix "be-
comes singular as limit points are approached. In order to obtain post-buckling
response, a method to overcome this singularity is needed. This is accomplished by
treating the load parameter as a variable and thus have an adaptive load incrementation. This approach differs from the conventional Newton type schemes where the
load level is held constant for all iterations at a given load step. Symbolically, at
load step m and iteration i, equation 2.153 can be rewritten as follows
[KI{d}(+ 1) = (mA + P),+ AA) {p}
-
{F}(')
(2.154)
where - [K] is the tangent stiffness matrix at load step m
A'+' = P) + AA
(2.155)
The Arc Length Method (ALM) may easily be visualized for a single degree of
freedom as shown in Figure 2.10. The displacements are updated as
{x}(i+i) =
{}
44
+ {u} ' + Au
(2.156)
such that x (' + ' ) corresponds to the displacement at the (i + 1)th iteration of load
step m.
In the constant ALM, the radius of the arc at each load step is constant.
It is clear from Figure 2.10 that the ALM is used in conjunction with the modified
Newton-Raphson method, and may also be implemented with NR iteration schemes.
The method allows one to obtain postbuckling response but bifurcation problems
require modification that will seek out multiple paths after a limit point.
4.
Convergence Criterion
For a given load step, the iterations on displacements are carried out until
a pre-set convergence is achieved. There are three convergence tests most commonly
used, (a) Displacement Convergence, (b) Residual Force Convergence, and (c) Strain
Energy Convergence. These criterion may be summarized as follows:
{mul }T{AUI}
-
{gi}T{g}
(AAI)
2
CtDISP
<
a.F.
<
aDJSPCaR.F.
{p}T{p}
f{Au}T{gi}
A'1Ai{ul}{P}
(2.157)
-
It may be noted that {Au'} is the incremental displacement at iZt iteration, {g'} is the residual force at it h iteration, AA'{P} is the incremental load at
the first iteration, {Au'} is the incremental displacement at the first iteration, and
a 's are the prescribed convergence parameters, usually in the order of 10-2 to 10- .
It is further noted that the initial load {P} used to start the analysis is
set arbitrarily and only the load parameter is modified automatically to go from
zero load to the desired load level.
45
k=k
LO
kLi +
NL
.1U)
I
I
P-
D
I
I
Figure~
2.0
~~
osatArIeghMto
46I
W
III. PROGRAM IMPLEMENTATION
A.
INTRODUCTION
This chapter presents certain aspects of computer implementation of the ele-
ments matrices developed in the previous chapter. As mentioned in Equation 2.25,
the general problem to be solved is given by
[K]{d} =
{r}
(3.1)
=
{R}
(3.2)
[K]iDI
where equations 3.1 and 3.2 are the static equilibrium equations for the element
and structural assemblage respectively. If the stiffness matrix, [K], is independent
of displacements, the analysis reduces to solving a set of linear algebraic equations.
In the case of the stiffness matrix being displacement independent, the structural
behavior is nonlinear, and an incremental load analysis together with a suitable
iterative methods has to be adopted.
B.
LINEAR ANALYSIS
In the case of linear analysis, we assume small displacements and small strains,
and the resulting force-displacement relations are solved only once. Using the displacement independent part in equation 2.61 to get the strain-displacement relations [BLo], as given by equation 2.80, equation 2.20 is used to form the element
stiffness matrix. A series of Fortran subroutines was developed incorporating the
element stiffness matrix for this element. The material characteristics may be either
isotropic, laminate theory definitions, or anisotropic description. The subroutines
are implemented in an existing computer program, FEMCOM, which is capable of
47
element assembly and subsequently calculation of the displacement solution for both
linear and incremental load methods.
C.
NONLINEAR ANALYSIS
In this research effort, geometric nonlinearities, namely, large displacements
but small strains are considered. Consequently, the element consists of a linear
displacement independent stiffness matrix, [KLo], and two other contributions. The
first contribution is due to the linear displacement dependent stiffness matrix, [KLI],
based on [BL1], as given in equation 2.81.
The other contribution comes from
nonlinear stiffness matrix, [KNL], as given by equation 2.95.
Note that the stress-strain matrix, [E], may be used both for isotropic as well
as composite materials using relations 2.131 and 2.132.
D.
SOLUTION PROCEDURE
1.
Composite Material
In order to generalize the procedure of implementing the solid element
with composite materials, the plate built of solid elements may be stacked in all three
directions. Figure 3.1 shows such a stack, where rows of elements are arranged in the
thickness direction. For each finite element, the stress-strain matrix is computed in a
subroutine separately, though it would be more efficient to compute it for the whole
row of elements, taking into account the appropriate layers. In assigning a certain
number of layers in each row, a constraint to be noted is that the total number of
layers of all rows match the number of layers of the structure being modeled.
2.
Linear Case
As mentioned earlier, the matrices corresponding to the linear displace-
ment independent part was coded into several subroutines and implemented into a
48
€4-1
~Q) Q
0))
0
U4
CUJ
-Ir
CeO
o
-.
Figure 3.1: Thick composite plate-element arrangement
49
general purpose finite element program, FEMCOM. The program does automatic
element assembly and yields solutions to prescribed loads. The flow chart shown in
Figure 3.2 shows various steps that may be summarized as follows.
1. The material properties, model geometry, applied loads, integration scheme,
boundary conditions and solution parameters are input. The material properties needed for isotropic material are Young's modulus and Poisson ratio. For
composites, data needed includes the number of layers, rows of elements, fiber
orientations, Young's moduli, shear modulus in three directions, and Poisson
ratio.
2. Using the shape function derivatives, the coordinates transformation relations,
Jacobian and the strain displacement relations are established.
3. Using the specified Gauss quadrature, the element stiffness matrix is formed
in global coordinates.
4. The element global stiffness matrix is assembled.
5. Using Gauss elimination technique, the displacement vector is computed.
6. Stresses may be computed using equations 2.14 and 2.15.
3.
Nonlinear Case
In order to obtain nonlinear response, either for studying the extension-
twist-flexure coupling or nonlinear buckling and post-buckling, the analysis procedure is termed the incremental load method, and a variation of Newton-type
iteration is used. The element formulation, assembly and equation solving proceed
as before, except that additional element stiffness contributions have to be taken
into account. The assembly and solution to get displacements needs to be done as
50
E INPUT
IR=I,,NHRON=
H
E
ko:IDTED °d'.
K
D:K-" 'k
FIG. :3.2;
Flow chartlinear analysis
Figure 3.2: Flow chart
51
-
linear analysis
frequently as the load steps increments and iterations continue, depending on the
solution strategy selected. A typical flow chart is given in Figure 3.3.
For a given load step, the incremental displacements are computed iteratively until the convergence criterion is satisfied. At that point, equilibrium is
achieved and new incremental load is applied and a new tangent stiffness matrix is
computed. The iterations continue until the new equilibrium position is obtained.
In the modified Newton-Raphson method, the tangent stiffness matrix is kept constant for all iterations for a given load step, while, for the Newton-Raphson method,
the stiffness matrix for the whole structure is formed at every iteration. By tracing
the load-displacement path, critical points, characterizing buckling, and stable and
unstable regions of post-buckling equilibrium states may be identified.
52
kLo
kil
kf
B" ERCd.
kHL
B2DL1D
LO
13'
Ff
ERdy I
F
3.3
FIG.~
hat
Nonlinear anaysi
Figre3.: Lowcat-Nnieraayi
RUL
i-l<to53e
IV. NUMERICAL EXAMPLES
A.
INTRODUCTION AND NOTATIONS
In this chapter, selected numerical examples are used to evaluate element
idiosynchrasies and demonstrate its application in solving critical structural components that use thick composites. Solutions obtained here are compared with
available elasticity solutions or other numberical solutions.
1.
Material Properties
In all the examples to be discussed, the material characteristics used are
as follows. For isotropic materials,
E = 30 x 106 psi, v
=
0.30
and for composite materials used for laminated plates, layer properties are given by
E,
= 40x106 psi
E2
=
106 psi
G 12
=
G1 3 =0.6x 106 psi
G23
=
0.5 x 106psi
v
= 0.25
An eight-layered symmetric laminate configuration using this material is selected.
All the dimensions presented in this chapter are in inches. In the discussion on
effects of numerical integration rules, L x M x N notation refers to the number of
integration points in x, y, and z directions respectively.
B.
COLUMNS AND BARS
Two simple cases have been selected as part of the element validation process.
54
1.
Bars
A bar clamped at one end and loaded at the other end with uniformly
distributed traction (Figure 4.1a) was studied and compared to the theory of elasticity. In the numerical solution, work-equivalent loads were used. The dimensions
of the bar are 10 in. x 1 in. x 1 in., and it is isotropic. The boundary conditions
are given by
U(0,h
h) =V(0,h
-h-\
= w 0, -h+)
=0
(4.1)
Figure 4.2 depicts the effects of reduced integration and mesh refinement on the
maximum deflection. It is obvious that when the mesh is refined in the thickness
direction, for instance, one element in each of x and y direction and two in z direction [1 x 1 x 2] mesh, provides a stiffer solution than for the [1 x 1 x 1] mesh.
It may be noted that the full (F) and reduced (R) integration schemes converge
to about 95% of the classical solution (See Appendix C). It may be noted that the
classical elasticity solution does not account for transverse shear stresses. A reduced
integration in the axial direction (2 x 3 x 3) gives the same results as the full integration. However, when reduced integration in the thickness directions is performed
(3 x 2 x 2), the solution converges slowly. On using (2 x 2 x 3) integration scheme
in the thickness direction for (12 x 1 x 1) mesh, spurious mode is observed. Table
4.1 summarizes the effects of various integration schemes and mesh sizes.
2.
Beams
The next example considered is a clamped, cantilever beam loaded at the
free end by a shear load. Using the clamped boundary conditions, dimensions and
material as the previous example, solutions using full and reduced (R) integration
are compared with the elasticity solution in Figure 4.3. The comparisons also include
the solution obtained using eight noded first order solid element of 'GIFTS' software.
55
(at
"Nem)
Fig. 4.1;
Bar Sample problems
Figure 4.1: Bar Sample Problems
56
Fig. 4.2; Clamped Bar Under Uniaxial Load
w
130
.
120-...
1
~9
1
... .
0
.....
.......
F Ituegraion Rule
80-S
-.-~-RR
-
-au-
70 . .....
R (2x 3 x3)
(3x2x2)
-F - Thickness Mesh
Elmsticiiy
60...............
50
0
1
100
r
2C0
300
400
500
600
#d. o. f
Figure 4.2: Clamped Bar Under Uniaxial Load
57
700
TABLE 4.1: EFFECTS OF REDUCED INTEGRATION AND REFINED MESH, ON THE MAXIMUM DEFLECTION OF CLAMPED
ISOTROPIC CANTILEVER BAR UNDER UNIAXIAL LOAD
# d.o.f.
Mesh
Integration
Rule
u-,
5.7 F(3x3x3)
R (3 x 3 x 3)
RR (2 x 2 x 3)
2-2x
x
111
F
R
3 = 3 x Ix 1
4 = 4x Ix 1
6 = 6 x 1x 1
12 = 12 x 1 x 1
99
165
219
327
651
129.56
99.43
99.84
RR
2 = 1 x 1 x2
97.49
98.17
115.98
F
R
90.77
91.20
RR
91.83
F
R
100.40
100.81
RR
112.64
F
R
101.04
101.59
RR
111.39
F
R
101.83
102.80
RR
110.74
F
R
102.74
105.00
RR
Umax
11,500.00
Uma AE
Uelasticity
pl
58
The reduced integration shows a better convergence than both the full integration
and the first order solid element. It may be noted that the eight noded solid element
converges more rapidly than the full integration scheme of the present element up
to about 300 degrees of freedom (d.o.f.).
It can be seen from Table 4.2 that mesh refinement in the thickness direction results in reduced performance and one element in the thickness direction
consistently yields good results.
In Figure 4.4, the effect of transverse shear deformation is studied for a
12 x 1 x 1 mesh using reduced integration scheme, and compared to the theory of
elasticity solution (See Appendix C). The results are summarized in Table 4.3 versus
the aspect ()
ratio.
It is clear that for thin beams where the elasticity solution is adequate,
the present element gives stiff solutions, whereas for thick beams (- < 10), better
solutions are predicted. The reason for these effects may be attributed to the transverse shear stresses. In the case of thin bars, or beams, the element aspect ratio
is very large and the parasitic shear strains appear at Gauss points, resulting in a
phenomena called 'shear locking' [Cook, 1989, and Hughes, 1978]. When the beam
is thick and the aspect ratio is of the order of 1/10, the transverse shear stresses
start to become significant, whereas in the elasticity solution, they are taken into
account only to a limited degree together with the restrictions of Saint-Venant's
principle.
C.
CLAMPED PLATES
An isotropic clamped plate of dimensions 20 in.
x 20 in. x 1 in. under
a central concentrated load is shown in Figure 4.5a. This problem is studied for
mesh sensitivity and the effects of different integration rules. The present solution
59
Fig. 4.3; End Loaded Beam Bending
110
.
.
100---
90P-( 7 0 ......
~Z
E
50 ........
40 .......
30
-P~-
3 0----
.....
0
1I
(F)
......
"GIFTS" 8solid (F)
.............
w
100
let
Prcscni Elcrncni (R)
20
0
Prcscti
200
300
_E
lasticit
400
500
#d.o.f
Figure 4.3: End Loaded Beam Bending
60
600
700
TABLE 4.2:
EFFECTS OF REDUCED INTEGRATION AND
MESH CONFIGURATION ON THE MAXIMUM DEFLECTION OF
CLAMPED ISOTROPIC CANTILEVER BEAM LOADED AT THE
END
Mesh
*
# d.o.f.
(27 solid)
f --Wma
F
p1
27 solid
R
100
8 solid t
F
I = 1x I x 1
57
5.7
26.7
9.4
3= 3xI x 1
165
37.0
54.0
47.2
4 = 4x I x 1
219
56.0
70.3
60.5
9 = 9 x 1x 1
489
91.3
95.4
85.8
12 = 12 x 1 x 1
651
95.0
97.3
89.8
3 = I x 1 x3
147
5.5
4=2xl xl
135
16.9
6 = 2 x 1x 3
273
17.0
6 =3x 1x2
285
36.0
9=3xI x3
399
39.1
w~eaotirit = 101.0
F: Full integration
e 3 x 3 x 3 for 27 solid
* 2x2x2for8solid
R: Reduced integration
* 2 x 3 x 3 for 27 solid
f 8 solid is generated in "GIFTS".
* Mesh configuration for 8 solid is twice of 27 solid in each direction, i.e., 2 x 1 x 3
for 27 solid is 4 x 2 x 6 for 8 solid.
61
TABLE 4.3: CENTER DEFLECTION VS. ASPECT RATIO
(i) OF
AN ISOTROPIC CANTILEVER CLAMPED BEAM LOADED AT ONE
END
___
We,
124.38
110.07
.5 103.90
99.49
10
100.98
97.07
50
100.04
53.43
100
100.01
17.85
2
h, lh) 3E1 2
w1I,
w
P 12
w(1, h, jh) Eh10
w
+e 3
[+(1
See Appendix C.
62
+V)
h]
102
Fig 4.4; End Loaded Beam Deflection
130j
120
....
2 100-................80...............................s...
70
S
0--
-4
Elasticity
*- Precnt Elemcmi (R)..............V
20 ............................
10
111
Lenght/Thickness [L/h]
Figure 4.4: End Loaded Beam Deflection
63
is compared with the elasticity solution (See Appendix C for details on elasticity
solutions).
Invoking symmetry conditions of the problem, a quarter of the plate is modeled
with the following boundary conditions imposed:
=
*,Y,±u (X,
, ±)2
V 1Y
)
V(X, 0 ±
W0
,±h
= (X, a,
(4.2)
=02
(4.3)
and the symmetry conditions.
u (, y, z) =v (X, a, z) =0
(4.4)
The load was taken as one quarter of the total load. Figure 4.6 shows the comparison
of a mesh composed of elements arranged in one row of elements (N x N x 1) vs.
a mesh of the type (2 x 2 x M), composed of M rows of elements arranged in the
thickness direction with 2 x 2 elements in each row. Full integration is employed
in the computations. Table 4.4 summarizes the resultant deflection and mesh sizes.
It is clea- from this and the previous examples that one row of elements in the
thickness direction is adequate to predict the response of the structures. Figure
4.7 presents the convergence characteristics of three integration schemes. It may be
noted that reduced integration (3 x 3 x 2) in the thickness direction yields very close
results to that of the full integration scheme. Reduced integration produces good
results by compensating for the estimation of finite element approximation. The
in-plane reduced integration scheme (3 x 2 x 2) shows divergence in the computed
response.
It may be mentioned that using one element to model quarter plate
resulted in much higher deflection than expected. This implies that a one element
model contains spurious modes and a one-element modeling of plate/shell problem
should be avoided. On examining the convergence plot, with less than 600 d.o.f., the
64
CC
>C'
Fig.
4. 5; Plate Sample problems
Figure 4.5: Plate Sample Problems
65
TABLE 4.4: MESH COMPARISON OF AN ALL EDGES CLAMPED
RECTANGULAR ISOTROPIC PLATE UNDER CENTRAL LOAD QP
=1000
lb.)
# d.o.f.
w
1= 1 x 1 x 1
37
13.8529
4 = 2x 2x1
145
4.2230
9 = 3 x 3x 1
325
5.0690
16 = 4 x 4 x 1
577
5.6592
8 = 2 x 2x 2
275
4.0133
12 = 2 x 2 x 3
405
3.9955
16 = 2 x 2 x 4
535
3.9597
18 = 3 x 3 x 2
591
5.0647
Mesh
h) Eh 3
w
W
2
2 2
pa
100
evaluated deflection is within 90% of the elasticity solution. Table 4.5 summarizes
the effects of various integration schemes and mesh sizes.
D.
SIMPLY SUPPORTED PLATES
Bending of a simply supported rectangular plate under uniformly distributed
force is presented herein. (See Figure 4.5b) Both isotropic and laminated plates are
investigated using one quarter of the plate, as discussed previously. The results are
66
Fig. 4.6; Clamped Plate Under Central Load
7.06.5-......................
4 .0--
- --- - - - -- - - - --- - - -
.5
35.............
30. . ...........
. .........
...
....
...
.... ...
.....
....
..-...
.. .. ..
. ..........
2 x
.
......
. . . . .
2.5-CT
2.0
0
1
i
100
200
300
400
500
#d.o.f
Figure 4.6: Clamped Plate Under Central Load
67
600
Fig. 4.7; Clamped Plate Integration Rules
9.08.5 .....
..........
8 .0 .-....
~
*
.- ....................
7.0 -...............
. . . . . . .. .
.. . . . . . . .
. . . .
. . . .
S 6.5-....................................
6.0
5
.5 -.
(F).3...3...3
......
2.5..........
2.00.5
Type
I1
100
200
30
40
#do RR 2f
00
Figure
4.7 Clamped..
.....
Plate..tegratin.Rule
3 .0
..............
c68
60
TABLE 4.5: INTEGRATION RULES COMPARISON OF AN ALL
EDGES CLAMPED RECTANGULAR ISOTROPIC PLATE UNDER
CENTRAL LOAD QP = 1000)
Mesh
#d.o.f.
w for various integration rules
(3x3x3) (3x3x2) (2x2x3)
4= 2 x2 x 1
145
4.2230
4.2575
6.9918
9= 3x 3 x 1
325
5.0690
5.0853
7.6576
16 = 4 x 4 x 1
577
5.6592
5.6763
8.1128
w
w(~
=
S,
2
h) Eh3
2
Pa
69
100
compared with Classical Plate Theory (CPT) as given in Appendix C and higher
order shear-deformation (HSDT) plate theories [Reddy, 1985, Lo et al., 1977].
The following boundary conditions are imposed,
W (0, Y, ±-.
=
W(X, 0,
)= 0
(4.5)
and the symmetry conditions are as given in equation 4.4. Reduced integration
(3 x 3 x 2) is adapted throughout all the computations presented in this section.
The convergence characteristics of the element and comparison to CPT is depicted in
Figures 4.8 and 4.9 for both isotropic and laminated plates. In the isotropic case, the
present element shows convergence within 90% of the elasticity solution for less than
200 d.o.f. In the case of laminated plates, (Figure 4.9), the classical .;olution [Vinson,
1987] gives a more flexible solution than the present element. Table 4.6 summarizes
the deflections of the isotropic and laminated plates and mesh sizes. It may be
noted that the classical solution uses laminate theory, which neglects the transverse
shear stresses, and hence the contribution of these stresses is not taken into account.
This assumes more significance for thick plates (a < 10 to 15). Furthermore, when
the plate stiffness in the thickness direction is significantly lower than its stiffness
in the in-plane direction and when the shear modulus in the thickness direction
is significant, the classical laminate theory does not predict the response of the
structure accurately. In the present example, a = 20 and -
= 40, G1 3 - G12 .
It may be concluded that using laminate theory for bending of thick plates yields
a nonconservative estimate of deflections and special attention should be given to
the stiffness ratio
[]
and shear modulus ratio
[-]
in determining the
of such plates. In Figures 4.10 and 4.11, the maximum deflection is presented for
different aspect ratios, (S), of the plate. Both isotropic and laminated plates are
analyzed and compared to CPT. In addition, the solution of the laminated plate is
70
also compared -o Higher Order Shear Deformation Theory [Reddy, 1985], as shown
in Table 4.7.
As in the beam bending c; e, shear locking is observed for thin isotropic plates.
For thick plates, (say, = 4), the computed deflections become significantly larger
than predicted by CPT, as expected.
Examining Figure 4.11, it may be deduced that even the HSDT [Reddy, 1985]
underpredicts the deflections. For thin laminated plates, the shear locking effect is
not as significant as observed for isotropic plates. This may be attributed to the
fact the laminated plate has more flexible transverse material stiffness in bending
than the coefficients than the isotrop,i': plate, so that shear locking is expected to
develop only for thin isotropic plates.
71
TABLE 4.6: ALL EDGES SIMPLY SUPPORTED RECTANGULAR
PLATE, UNDER UNIFORMLY DISTRIBUTED LOAD
# d.o.f.
Mesh
Isotropic
w
Composite
w
4 = 2 x 2x 1
186
3.2826
0.3871
9 = 3 x 3 x1
386
3.6335
0.3878
16 = 4 x 4 x 1
658
4.0541
0.4053
4.4335
*0.3634
CPT
*
Neglects G1 3 and G 23 . See Appendix C.
The uniformly distributed load is taken as the total consistent load over the area of
the quarter plate.
q
F
=
(!2
The maximum deflection is taken at upper surface.
w (z, 2, h) E2 h3 100
4
w =
q a
72
TABLE 4.7: CENTER DEFLECTION VS. ASPECT RATIO OF SIMPLY SUPPORTED RECTANGULAR PLATE UNDER UNIFORMLY
DISTRIBUTED LOAD
Isotropic
Orthotropic
Reference*
w
w
w
4
9.8275
5.1324
1.6340
10
4.9581
0.8221
0.5904
20
4.0541
0.4053
0.4336
100
1.0902
0.2406
0.3769
CPT
4.4335
0.3634
*Reference: Reddy, 1985.
73
Fig,. 4.8-, Simply SUlplpiwdt Isoltitpic P1h11C
3
41
'
... . . ... .
.. . . :..
=2
-IV
or
I(H4)
2004
4044
30(1
6004
5400
700
(1-....
o -15
. .......
Simply Supported Isotropic Plate . .
Figure 4.8:
Fig. 4.9; Sinmply Suppotrted Lanmi mlcd Iluc
o050
-- - - - --
..
------ ------- ----------------------------.......
---
LLJ 0.35
. .... . .....
u..
a .3 5 ...
:
0.25
0.20o
0
... .
..
... .. ... ....
...
.
100
200
3(o(
40)
5
600
Skitlhllickuic>s ku/i
Figure 4.9: Simply Supported Laminated Plate
74
700
Inii!. 4. 10
s();( )I
ic Pla~te Delflcclions vs. Aspect Ratio
10
9
0
.~
7
.
2
-,--
~
!r.......
.....
-- - - - - - - - - - - - - - - - -- - -
- - - - - - - - - - - - -
Side/Ilickness (;Aj
Figure 4.10: Isotropic Plate Deflections vs. Aspect Ratio
Fig. 4.11; Laminated H'awc Dclcctions vs. Aspect Ratio
4
........
i
. .......
-0.-
Row)I
i is i i
01
I o"
Io,
102
Figure 4.11: Laminated Plate Deflection vs. Aspect Ratio
75
V. CONCLUSIONS AND SCOPE FOR
FUTURE RESEARCH
A.
CONCLUSIONS
This study suggests a three-dimensional higher-order finite element to be in-
corporated in the analysis of thick plates composed of both isotropic and laminated
composites. By using a tri-quadratic Lagrangian twenty seven noded solid element,
no assumptions on transverse shear strains are introduced in the formulation. The
formulation, based on the principle of virtual work, is presented for both linear
and nonlinear analysis. The material constitutive relations for linear isotropic and
composite materials are presented. For composites, both laminate theory and three
dimensional anisotropic adaptations are described.
Several numerical examples using linear analysis are given for bars/beams and
plates using both isotropic and composite materials. Three dimensional anisotropic
relations are adapted for composites. The results show that the present element is
effective for analysis of thick beams and plates, but exhibits shear locking for thin
beam and plates.
Spurious modes are revealed for single element usage in plate modeling, as is
the case for some other finite elements.
Reduced Integration in the thickness direction for beams and plates gives satisfactory results. An interesting outcome is that one element is sufficient to capture
transverse deformation for thick laminated structures and mesh refinement in the
other two directions yields convergent solutions.
76
B.
SCOPE FOR FUTURE RESEARCH
More numerical experiments need to be performed to compare the present
soution to closed-form solutions [Pagano, 1969] to evaluate the efficacy of this element.
Implementation of buckling analysis using the nonlinear element matrices presented herein is another task that may prove useful in predicting buckling response
of thick composite cylinders subject to external pressure. By incorporating the centrifugal force in the external virtual work done by body forces, this element may be
used in modeling rotor blades.
77
APPENDIX A
Shape Functions and Derivatives for Solid Element
Shape Functions for Solid Element
Mid-edge nodes:
Mid-plane nodes:
N2 =
N4 =
N6 =-I
r(1+r)(1-s 2) t (I+t)
(1 - r 2) s (1 + a) t (1 + t)
r(1- r) (I1-s 2) t ( I + t )
N gs--
(1 - r
)
s ( 1 - s) t (
N io0= - _r(1+ r) 8(1-
I+
t)
) (1 - t
)
r (1 + r) s(1+s) (1- t2 )
N 14 = -- r (1 - r) s (1 + s) (1 - t 2 )
Nis =
r (1l- r) s(1 -s) (I - t")
N2o=--r(l+r)(1-s 2) t(I - t)
N2 2 = -1 (1 - r 2 ) 8 (1 + s) t (1- t)
N12 =
N 24 =
,- (I - r) (1 -
N 26 =
r(-
2) t
(I - t)
N 9 = .1(1 -r 2 )(
N 2 7 - (1- r 2)
Nil=
r (1 +r)
N 15 (I _r)
N13 (I- r )
1
-- s 2) t(l+t)
(1- 22)) t( 1(I-- t 2t))
(1- s
_
(1-_8
(
+
8 1
) ( 1 - t 2)
) (1 - t )
r2) s (1- s) (1- t 2 )
N17
Center node:
N 13
r 2 ) (I _ 82) (1 _ t 2 )
2
r ) s(-s)t(1-t)
Corner nodes:
N, = 1 81 (1+,-)(1 -,s)
N=
N
1 +1)(
1
(I+t) - 2(N2
+#Ns+
) 1+t
8
=
(
N7
=-=
(1 + r) (1-s)
(1 - s) (1
(I - t)t) (1-r)
N=--
+ N4
+
N12) -
1
(Nil +
N13 + N9) -
r) (
) (1 + t)-
(+
+
)+ 1+
(No
+ N6+ N 1 ) - -4(t+NTN)
(N 5 + N 1 7+
(N16+NsN
1
N)
~
I
1 tA'1
Nl
N25
=
N25
=
(1- r)(1-a)(-t)-(N24+N26++N16)-1(Ns+N17+N27)-
+ ) (1-
f-- -(N22
+ N 24
(2+2+1)
78
4
+ N 14 ) - -(N 1 3 +N
4 NsNsN
Nis
1
82
I -(I
8+
(-)l
=
-
(I2 + r)(1 + s) (1 1- t) - 1 (N2o + N22 + N12) - 1I (Ni I + N13 + N27) 1 .-
N 23
1~
111,,
1
N
N21
(N2
Nio) - 4 (Nl +.N,7 + N9) - j#1,
8
1
+ N 27 )- -NI
- 8 I
Nis
Shape Function Derivatives for Solid Element
Mid-plane nodes:
Mid-edge nodes:
N2'
N 4 ,, =
t (1+ 2r) (1 -s 2 )(1 +t)
-1rst (1 +s8) (1 + t)
Ng,,
(
Ns 61'
N2 0,, = t
N 20,,
=
-
N22,,
=
24,=
N
26,
=
Nl,,
N3,r
=
I(1-
N7r
N 2 3,,
=
a(1 2r)
+
N 25 ,,
=
2
_
2
2
r( de(:-t
Cnte
(2
_
-1
+1N_ +)
N,,)
4
,+N
2
(Ns,
+)1 7t2)
, +
-(
~( 1 ,
,
-
(N2,, + N8,, + N 1 6,,)
-
a) (1-+i)
-
(N 2 ,, + N,
+ N 1 ,)
-1(Nl,,
-
(N4,, + N,
+ N 1 ,)
(N6,, + N,3,
+ N 1 r)4 7 -
+ a) (1 + t)
S (
1+)(
-
20r+N6
~~ ~(1+8)(
81(1
'1
)-(2,
,
9
,
+ N 1 ,, + N ,)7
+ N,
N1,
(N1 3 ,,
+IO)-1
sr
-
j(N3,, + N 1 ,, + N ,,)
7
-
1,
-
Nr
-
+ N)-
N,3.
8 N18,r
N15,, +27,r)
1
(N 2 0,, + N 2 2,, + N 1 2,,)
-t
1
)
1 (Nil,, + N 1 7 ,, + N9,,)
1
N21,r=
)(1 _t2 )
S2) (I
t 2)
8) (1 + t)
1
-)
2
(1 - 2r) (1
Cntr node:+s)(1_2
_ t)
I(1+a (1 + 3) (1- t)N,
-(1
) (1 +t)
s2) (It)
(+ 2r) (I_
=
(1
s) (1 - t2)
sa (1 + ar)(1 _ t)(
t (1 + 2r) (1 - 82) (I
_I1t
((I
=
2r) (1 + 0)
-
(-(I
-Irst (1 + s) (1 - t)
=
N 1,,
-
&(I + 2r) (1
N212,, =
N
(
Ist
=
Nil::
t
2
(1 -8
=-rt
N 27 ,r =
N8 ,r = _t (- 2r) (I-s2 )(I+t)
N8,
r direction
-
24
1,)-1
2,
-
(N1
,r + N1 ,, + N 27 ,r)
N5,+N7,+N7,
,8
79
-
-Nis,
,~
Shape Function Derivatives for Solid Element - 8 direction
Mid-plane nodes:
Mid-edge nodes:
N 2 ,. - -Irst (1 + r) (1 + t)
Ng,,
rsi (1 - r) (1 + t)
Nil,
t(1-r2) (1 - 2 s ) (1 + t)
r (1 + r) (1- 2s) (1- t 2 )
Ni 5,,
N6,.
=
N,
= N l o ,, = -
lr(I+r)(1+2s)(I-t2)
N 12 ,. =
N 14 ,, = - r (1 - r) (1 + 2s) (1 - t 2 ))
N 16,.-
r (1 -r) (1-2s)
=
(1-
t
- -st (1 - r 2 ) (1 +t)
r) (1-_t2 )
-rs (I+
rs (1- r)(1-t
1(l-r
N 1 3 ,, =
N 1 7,
-2(1 -r
2
)
)
2s ) (1 - t 2 )
2
)(1-2s)(1-t 2 )
2
(1 +
t
rst (1 + r) (1 - t)
N 20 ,, =
N 22 . = -It (1 - r2) (1 + 28) (1 - t)
N 24 ', = - t rst (I - r) (I - t)
t(1-r2) (1 - 2s) (1 - t)
Ns,t=
Center node:
t2)
Nis,, = -2s(I - r2 ) (1
Corner nodes:
1 (1
N 1 ,,
=
N 3 ,.
=
(1 + r) (1 + t) -
=
N7 ,,
=
N 9,s =
-I
(1- r) ( + t) -
(N 6,. + Ns,. + N 16,.) -
=
I
N 23 ,.
N 2 5,&
=
-
I
r) (
-
(N 1 3 ,. + NI5 ,. + N 9 ,.)
-
N 1 3 ,.
N,
1
11
Nis,
+ N 17 ,. + N 9 ,)
I (Ni,, +
N 17 ,. + N 2 7 ,.)
Ns,
-
Nls
11
(I + r) (1 - t) - I (N 2 0 ,. + N 22 ,. + N 1 2 ,.)
r) (I - t) ( -)
-
I (Nis,
1
-
1
-
(N 2o,, + N 26 ,. + N 1o,.) -
+ r) (1 - t) -
+ Ng,,)
4 (Nil,. +
1
(N 4 ,, + N 6,. + N 14 ,.)
11
-I(I
-
(I - r) (1 + t) -
1
N 2 1,,
(N 2 ,, + N4,. + N 1 2 ,.)
1
1
N 5 ,.
(N 11 ,, + N 17 ,, + Ng,,) -
(N 2 ,. + N,,. + N 1o,.) -
-
1
1
_1
,(+
(1 + r) (1 + t)
-
I (N 22 ,.
+ N 24 ,. + N 14 ,.)
11r
(N 24,. + N 26 ,. + N16 .,)
80
(Nil, +
-
N83,.
+
N27,.) -
(N 1 3,. + N 1s,. + N 27 ,,)
11
I (N 5 ,. + N1 7 ,. + N 27 ,,)
Nis,
-
NIs,.
-
Nl,.
Shape Function Derivatives for Solid Element
8 (1 -r2) (I+ 8)(1 +2t)
.... r (1 - r) (I - 82) (1+ 2t)
=
=
N16 ,t
N20 ,,
=
=
N 24 ,: =
1r
N26,s,=
(I-r)
fs(l-
N1 ,
1
Irst (lI- r) (1+ 8)
-Irst (1 - r) (I - s)
-Ir (I + r) (1 - s2) (1 - 2t)
2 ) (1
(I-
r 2 ) (I1_
)(-)
-2t)
- 1(1 -r
N2 ,
,-) )
ru=I
(1
Ns
N 14.9=
t direction
Mid-plane nodes:
Mid-edge nodes:
N4,,
N6,,
-
=
2
) (1-_a
2
)(1
-2t)
-ri (I +r) (1 - 82)
2
,=s(-
(+)
2
Center node:
Nlg,t = -2t( - r 2 ) (I -
82 )
Corner nodes:
I (N 2 ,9 + Nt3,, + N1 0 ,t) - 1 (Nil,, + N 17 ,, + N 9 ,,)
N1 ,,
=
I (I1+ r) (1 -8s)-
N 3.8
=
I(1+7r) (1 +8) - I(N 2 ,t
+ N4,, + N12,,)
-
I(N 1 1 ,,
N5,,t
=
I (I-
I (N4,t
+ N6,, + N 1 4,0
-
(N 1 3,t
N7. ,
=
-
r) (I- +
+ r) (1
-
)
-
I (N2 ,, + N 2 ,, + N1 ,)
8) -
N1 =
N 25 ,9
=
)-I(~~
-~
(1-+r) (I1+,a)
-
-
( (1
a)
-
-
-
2,
81
i6,t)
N,
it
-
8
4
1)(Nlt+N7t+N70-1N
i~)
I (N 2 2 ,t + N 2 2,, + N12.0) -~
+ N 26 ,, +-
8t
-I
+ N15,, + N,) -
(N,, + N 17 ,, + N ,)7
-
2
8
- 1 +
Nigt=
+ N13 ,, + N9 ,,)
N,,
-
-
(Nil,, +~N 1 3,, + N 2 7 ,0) I (N, 5,, + N 17 ,, + N 27 ,,)
-
,
st
lt
APPENDIX B
Jacobian Matrix
Jacobian matrix elements:
27
Jl-=x,
Ni, xi
"27
y,
J
=
E
N,,, y
j=1
27
'Ni,,z i
J13--Z,r
j=1
27
x,.
J1=
=
E Ni,. x,
j=1
27
J2 = Y,.
=
1:
N,,, Y/
j=1
27
J23 =
z,s
=
1_
Nis zi
j= 1
27
J3 1
=
X,t
=
E Ni,t x,
j=1
27
J3 2 =
y,t
=
E
Ni,t yi
j=1
27
= Zt
J3
=
E Ni,t z,
j=1
Elements of the inverse Jacobian matrix:
=
(J22 J3 - J2 J32)
(J13 J32 - J1 2 J33)
r,2
=
r21 =
(J12J23 - J1 3 J22 )
(23 8 J31 -2J2 J33)
82
"722
=
r23
=
r3l
=
r'32
=
Pl IJ33 - J 3 J 3 1)
j(J
1
J
2
- J1J23)
~(J 2 1 J 3 2 -J
j(J
2
1 (Al
F3=
3
22
J3 1)
J3 -J11J32)
J22 -J 2 1 J12 )
Jacobian matrix determinant:
J det [J]
J1
J(J 22 J33 -J 23 J32 )
-J
+
12 (J21
J33 -J
23 J3 1)
J13 (J 21 J32 - J22 J131)
83
APPENDIX C
Theories
A.
THEORY OF ELASTICITY SOLUTIONS
1. Cantilevered bar under traction
PL
AE
" P = Total load
" L = Bar length
* A = Cross section area
" E = Young modulus
2.
Cantilevered Beam under end load
PL 3
-
+_____
3EI
21G
3[
+(l+V)
Reference: Timoshenko, 1951.
B.
CLASSICAL PLATE THEORY (CPT)
1. All edges clamped rectangular isotropic plate under central load
Pa
2
D
D
a
Eh=
12 (1- V2)
= 0.00560 for v = 0.3
Reference: Timoshenko, 1959.
84
2.
All edges simply-supported, rectangular plate under uniformly
distributed load
-qa
.
a
.
4
00406 for v =0.3
=0
omposite
x =W ,,
qa
8
I
H
r8.0n
m=1,3,5... n=1,3,5...
D =
4
2
D m 4 + 2 (D, 2 + 2D 6 ) (ma) + D 22 n
85
TABLE C-i
SAMPLE COMPOSITE MATERIAL DATA
Table C-1;
Sam~ple Composite Material Data
INPUT DATA;
LAMINA;
8
7
6
5
4
3
2
1
THNES
;0.12500
;0.12500
;0.12500
;0.12500
;0.12500
;0.12500
;0.12500
0.12500
; THETA ;
0.0 ;
;
;45.0
;-45.0
E2
El
0.40000E+08
0.40000E+08
;0.40000E+08
;0.40000E+08
;0.10000E+07
;0.10000E+07
;0.10000E+07
;0.10000E+07
;0.10000E+07
;90.0
;90.0 ;0.40000E+08
;-45.0 ;0.40000E+08 ;0.10000E+07
;45.0 ;0.40000E+08 ;0.10000E+07
;
0.0 ;0.40000E+08 ;0.lOOOOE+07
;Viz2
G12
;0.25
;0.25
;0.60000E+06
;0.25
;0.60000E+06
;0.60000E+06
;0.60000E+06
;0.60000E+06
;0.25 ;0.60000E+06
;0.25 ;0.60000E+06
;0.25 ;0.60000E+06
;0.25
;0.25
OUTPUT DATA;
0.12773E+08 0.55940E+07-0.81226E+06
0.55940E+07 0.17603E+08-0.30995E+07
-0.8l226E.06-0.30995E+07 0.59436E+07
0.OOOOOE+00-0.27344E-01 0.OOOOOE+00
-0.27344E-01-0.16406E+00 0.OOOOOE+00
0.OOOOOE+00 0.OOOOOE+00-0.62500E-01
D( i, j )-MATRIX
0.20743E+07 0.28699E+06 0.72461E+05
0.28699E+06 0.81544E+06 0.17998E+06
0.72461E+05 0.17998E+06 0.31612E+06
Note;
A,B and D matrices are evaluated
--- contribution i.e. G13-G23-0
,neglecting Transverse Shear
Usinq Navier's solution with n-m-200, i.e. 100 terms for each
direction, as given in the above, we have,
Wmax -
0.052328
3
Wmax*E *h
2
*100
-0.3634
q*a
Where q-90 ; a=20
86
LIST OF REFERENCES
1. Allen, D. H. and Haisler, W. E., Introduction to Aerospace Structural Analysis, John Wiley & Sons, Inc., 1985.
2. Arnold, R. R., and Mayers, J., Buckling, Postbuckling, and Crippling of Materially Nonlinear Laminated Composite Plates, Stanford University, 1983.
3. Bathe, K. J., Finite Element Procedures in Engineering Analysis, PrenticeHall, Inc., 1989.
4. CASA/GIFTS, Inc., Computer Aided Structural Analysis/GraphicalInteractive Finite Element Total System - Users Reference and Primer Manuals,
1987.
5. Cook, M. P., Concepts and Applications of Finite Element Analysis, 3rd ed.,
John Wiley & Sons, Inc., 1989.
6. Eisley, J. G., Mechanics of Elastic Structures, Prentice-Hall, Inc., 1989.
7. Ford, B. W. R. and Stiemer, S. F., "Improved Arc Length Orthogonality
Methods For Nonlinear Finite Element Analysis," Journal of Computers J
Structures, Vol. 27, No. 5, pp. 345-353, 1987.
8. Gajbir, S. and Rao, S. Y. V. M., "Buckling of Composite Plates Using Simple
Shear Flexible Finite Elements," Composite Structures, Vol. 11., #4, pp.
293-308, 1989.
9. Hoskin, B. C. and Baker, A. A., Composite Materialsfor Aircraft Structures,
AIAA, 1986.
10. Hughes, T. J. R. and Cohen, M., "The 'Hetrosis' Finite Element For Plate
Bending," Journal of Computers & Structures, Vol. 9, pp. 445-450, 1978.
11. Kolar, R. and Karmel, H. A., "On Some Efficient Solution Algorithms for
Nonlinear Finite Element Analysis," International Conference on Nonlinear
Mechanics, Trondheim, Norway, 1985.
12. Lo, K. H., Christensen, R. M. and Wu, E. M., "A High-Order Theory of Plate
Deformation, Part 1: Homogeneous Plates," Journal of Applied Mechanics,
Vol. 44, pp. 662-668, 1977.
13. Lo, K. H., Christensen, R. M. and Wu, E. M., "A High-Order Theory of Plate
Deformation, Part 2: Laminated Plates," Journalof Applied Mechanics, Vol.
44, pp. 669-676, 1977.
87
14. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium,
Prentice-Hall, Inc., 1969.
15. Natsiavas, S., Babcock, C. D., and Knauss, W. G., Postbuckling Delamination of a Stiffened Composite Panel Using Finite Element Methods, NASACR-182803, August 1987.
16. Pagano, N. J., "Exact Solutions for Composites, Laminates in Cylindrical
Bending," Journal of Composite Materials, Vol. 3, pp. 398-411, 1969.
17. Ramm, E. and Stegmuller, H., The Displacement Finite Element Method
in Nonlinear Buckling Analysis of Shells, Springer, Berlin, Heidelberg, New
York, 1982.
18. Reddy, J. N., "A Refined Nonlinear Theory of Plates With Transverse Shear
Deformation," InternationalJournalSolids and Structures, Vol. 20, pp. 293301, 1984.
19. Reddy, J. N. and Chandrashekhara, K., Nonlinear Analysis of Laminated
Shells Including Transverse Shear Strains, American Institute of Aeronautics
and Astronautics, Inc., 1984.
20. Reddy, J. N. and Phan, N. D., "Analysis of Laminated Composite Plates
Using A Higher-Order Shear Deformation Theory," International Journal
For Numerical Methods in Engineering, Vol. 21, pp. 78-91, 1985.
21. Reddy, J. N., "A Generalization of Two-Dimensional Theories of Laminated
Composite Plates," Communication in Applied Numerical Methods, Vol. 3,
pp. 212-220, 1987.
22. Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill,
Co., 1951.
23. Timoshenko, S. P. and Krieger, S. W., Theory of Plates and Shells, 2nd ed.,
McGraw-Hill, Co., 1959.
24. Tsai, S. W. and Pagano, N. J., "Invariant Properties of Composite Materiais," Composite Materials Workshop, Tsai, S. W. et al., eds., Techonomic
Publishing Co., Stanford, CT, pp. 233-253, 1968.
25. Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed
of Composite Materials, Martinus Nighoff, 1987.
26. Yang, T. Y., Finite Element Structural Analysis, Prentice-Hall, Inc., 1986.
27. Zienkiewicz, 0. C., Talor, R. L. and Too, J. M., "Reduced Integration Technique in General Analysis of Plates and Shells," International Journal for
Numerical Methods in Engineering, Vol. 3, pp. 575-586, 1971.
88
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