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CHAPTER 7
NON-LINEAR.STATIC ANALYSIS OF LAMINATED PLATES
7.1 GENERAL
In the linear analysis of structures, it is assumed that both the displacements and strains
developed in the structure are small. In other words, the geometry of the elements
remains basically unchanged during the loading process and first-order, infinitesimal,
linear strain approximation can be used. In practice, such assumptions fail frequently
eventhough actual strains may be small and elastic limits of ordinary structural materials
not exceeded. In many cases, very large displacements may occur without causing large
strains. If accurate determination of the displacements is needed, geometric non-linearity
may have to be considered in the analysis. When the displacements are large, lateral
deflections are accompanied by stretching of the middle surface, provided that the edges
or at least the comers of the plate are restrained against in-plane motion. Membrane
forces produced by such stretching can help appreciably in carrying the lateral loads.
Small deflection theory is considered to be sufficiently accurate for the analysis of
homogeneous plates if the maximum deflection is less than about half the thickness of the
plate. When the deflection is larger, membrane action also becomes significant, and the
use of a more exact plate theory, which accounts for the geometric non-linearity, has to
be used for the analysis·. It has been reported by Gorji [125] that the small deflection
theory, which is applicable to isotropic plates till about w/h = 0.4, can be valid for
composite laminates only if w/h is much lower. Hence, the analysis of composite
laminates based on large deformation theory is very important.
187
It is observed that the results available in literature are limited to some analytical
solutions and studies based on first-order shear deformation theory. The objective of the
study reported in this chapter is to investigate the effects of geometric non-linearity on
load-deflection behaviour and stress distribution in composite laminates using a finite
element model based on higher-order shear deformation theory accounting for the Von-
Karman non-linear strains. The same 4-noded element with seven degrees of freedom per
node, which was used for the linear analysis, is used in this study also.
7.2 LARGE DEFORMATION THEORY
In large deflection analysis, simultaneous bending and membrane actions resist the lateral
load .. This theory, which is based on Von-Karman's non-linear differential equation,
yields good results for small strains and moderately large deflections and rotations.
For the case of a Mindlin plate, the non-linear Green's strain vector is given as
ex
cy
Y xy ==
y xz
y yz
: + �[(:)' +(:)' +(:)'] : +�[(:)' +(:)' +(:)'] ( au + av ) + au au + av av + aw aw
oy ox ox oy ax ay ax ay
( ou aw ) ou au av av aw aw
az + ox· +oxaz +axaz + ax az
( av aw ) au au av av aw awaz + oy + oy az + ay az + ay az
188
(7.1)
Introducing the Von-Kannan assumptions, which imply that derivatives of u and v withrespect to x, y and z are small, and noting that w is independent of z, allows Green'sstrain to be rewritten as
au +tw)' ou
M:)' ax 2 ox
ax Ex av
+-1{8w
r av
M:)' -oy 2 oy oy Ey (au+ov )
+awaw ou av Yxy
= = -+- + owow (7.2) ay ox ox oy oy --Yxz
ou 8w ax ay (au+aw ) -+- 0 Yyz az ox oz ax
(: +:J av 8w -+-
oz oy 0
Toe higher-order and the product terms in the strain vector represent th� non-linearcomponents of in�plane strain and are expressed as
M:)' ax 8w
H:J' 1 ow ax
(7.3) = - 0 -
2 oy ow -
owaw ow ow oy --
-
ox ay oy ox J
Thus, the generalized Green's strain vector of Eq. (7.2) is expressed as
(7.4)
189
where [B] is the strain-displacement matrix, which gives linear strains comprising of
extensional strains, bending strains and shear strains and is same as in Eqs. (3 :7) or (3 .18)
and [BNL] represents _the non-linear strain-displacement matrix which is dependent on
nodal displacements. The strain-displacement matrix [BNL] consists of the nodal
sub-matrices as given below.
ow 0 aN·
I
[BNLi ] =
ow ox (7.5) where 0
ay aN· I
ow ow ay
ox
The equation of equilibrium between internal and external forces is expressed as
\j/(d) = fBTcrdV-R = 0 (7.6)
where \j/( d) is the sum of internal and external generalized forces and R is the external
nodal load vector.
If the displacements are large, the strains depend non-linearly on displacements and
hence the strain-displacement matrix, [B], comprises of terms which are independent of
displacements and terms which are dependent on displacements as given in Eq. (7.4).
The assembled form of the total equilibrium equation in Eq. (7.6) becomes a set of
non-linear equations in nodal displacements because of the presence of quadratic
functions in the strain-displacement matrix. The solution of the assembled non-linear
190
equilibrium equations gives the nodal displacements, using which the internal stresses are
evaluated.
7.3 FINITE ELEMENT FORMULATION
Finite element formulation for the solution of non-linear problem explained in the
previous section is given below, using a 4-noded element based on HSDT.
The development of [B] matrix in Eq. (7.4) remains the same as that in linear analysis
and is given by Eq. (3.18). [i3NLJ matrix as given in Eq. (7.5) is developed as follows.
The transverse displacement, w, is expressed using Eq. (3.16). The terms ow/ox. and
owloy within the matrix in Eq. (7.5) are obtained as
ow_ f [afi ( ) 8gi (owo) 8hi (owoJ J d--� -w +--- +--- an
OX i=l OX. O I
OX.· OX. i OX. O'j i
(7.7)
where fi, gi and hi are the interpolation functions given in Eq. (3.16) and w0, owJox. and
owJoy are the known nodal displacements.
The elements 8N/8x and 8Nj/8y within the vector in Eq. (7.5) are obtained as
191
(uo)i
(vo) i
aNi 0 0 ari 8gi Bhi 0 0 (wo)i
ax ax ax
(8:o
J = (7.8)
aNj ari agi 8hi 0 0 0 0
ay oy oy ay
(�'J (0x )i
(0y) i
= [G]{ct}
where Uo, Vo, Wo, etc. are the unknown nodal displacements. Having developed the matrix
[BNL] as given in Eq. (7.5), using Eqs. (7.7) and (7.8), the tangent stiffness matrix is
developed as given below.
From Eq. (7.6), the equilibrium equations are written as
[ JET CB dV + _!_ .fBT CBNL dV + .fBNL T CB dV + ..!.. JBNL
T CBNL dV l {d} = {R}v
2v v
2v
(7.9)
The term 1 JB1 CBNL dV is unsymmetric. Combining third and fourth terms m
Eq. (7.9), it is written as
Hence, the equilibrium equation is written as
( [K]+ [KJ+ [K0 ]){d} = {R} (7.10)
192
where ( [K ]+[Ks]+ [K cr ]) is called the secant stiffness matrix, which is unsymmetric because of [K 5]. To have a symmetric stiffness matrix, Eq. (7.10) is rewritten as [126]
where [K]= fiFCBdV, [Ksd= fiFcBNL dV + JBNLrcB dV,
V V
(7.11)
[K0d= J<FcrLG dV
The incremental equilibrium equation for the system is obtained by differentiating the equilibrium equation, f BT cr dV = R, with respect to the displacements as
V
[ fBr CB dV + f B T CBNL dV + fBNL r CB dV + fBNL r CBNLdV + f G r aG dv] {fld} = {flR} V V · V V V
[jlFQB dA+ fBTQBNL dA+ JBNLTQBdA+ JBN/QBNLdA+ JGTaG dv]{fld} = {flR} A A A A V
Hence, the incremental equilibrium equation is written as ([K]+[K NL]){fld}= {AA} where [ K] = jB T QB dA ' the linear stiffness matrix
A
and the non-linear component of stiffness matrix,
[K NJ = jB r QBNL dA + JB�L QB dA + jB�L QBNL dA + Jct a G dV (7 .12) A A A V
193
7.4 SOLUTION PROCEDURES
Basically, there are two. distinct approaches for the solution of non-linear equations of
equilibrium: (i) direct solution of the non-linear equations by iterative procedure and
(ii) piece-wise linear load incrementing method. In either of these numerical methods, it
is assumed that during each solution cycle, the stiffness analysis proceeds along a
straight-line tangent to the curve characterising the force-deflection relations of the
structure. In order to achieve such a tangent solution, the stiffness matrix of each element
should be modified to account for the accumulated stresses and the change in geometry.
7.4.1 Direct Method
For a given set of external loads, the objective of non-linear analysis is to determine the
true values of displacements and internal stresses. Since the tangent stiffness matrix is
dependent on the instantaneous strains and displacements, an iterative method of solution
is inevitable. Newton-Raphson method of successive cycles of linear analysis has been
used in the solution of a variety of non-linear structural analysis problems with extremely
satisfactory performance. The basic principles of this method are outlined below.
Using the initial geometry and the external loads, a linear stiffness analysis is performed
first. Result of this analysis is used as a trial solution for the iterative procedure. Using
this trial solution, the non-linear part of the stiffness matrix, given in Eq. (7.12), is
evaluated. At a particular node of the system, in the direction of a particular degree of
freedom, the algebraic sum of all the calculated stress resultants must be equal to the
external load applied in that particular direction. Since, the nodal displacements obtained
in the first cycle do not correspond to the true equilibrium geometry, the algebraic sum of
the internal stress resultants obtained using this first set of approximate displacements
194
will not be equal to the given external loads. The difference between the internal and
external forces constitutes the unbalanced nodal loads to be used in the next cycle of
analysis. Using the tangent stiffness matrix evaluated including the non-linear
components and the unbalanced nodal forces, a set of nodal displacements is calculated.
This gives a correction to be applied to the trial solution. When the nodal displacements
of the first and the second cycles are superimposed, the system comes closer to the actual
equilibrium configuration. As a result, the difference between the external loads and the
stress resultants, calculated on the basis of the corrected nodal displacements, is reduced.
Using the new unbalanced nodal forces and new stiffness matrix based on the corrected
nodal displacements, a new set of corrections to displacements is obtained which is
superimposed with the nodal displacements obtained in the previous cycle. A new set of
unbalanced nodal forces is calculated based on the new set of displacements. The above
iterative process is repeated until the maximum unbalanced nodal force in any direction
becomes less than a tolerable value. The tangent stiffness matrices are modified after
each cycle to include the latest strains and displacements.
The above procedure can be employed in two ways. Either the total load can be applied
in a single step or the total load can be applied in increments by dividing the total load
into a number of load steps, in which case iterations are required at each of the load steps.
The latter method improves numerical stability and gives intermediate results enabling
the understanding of load-deflection behaviour characteristics. In order to reduce the
computational effort involved in the latter method, a modified version of the above
procedure has been developed, referred to as modified Newton-Raphson method. In this
method, the tangent stiffness matrix is evaluated once only on the second iteration of
195
each load increment. In this method, relatively large load increments are generally
suitable. Behaviour predictions are independent of increment size, since solutions are
obtained to the actual non-linear model.
7.4.2 Piece-wise Linear Load Incrementing Method
Selection of linear load incrementing method exchanges the problem of solving the
multidimensional, non-linear equations for a multiplicity of linear solutions. This method
is recommended by its linearisation. The solution of sets of linear equations is readily
accomplished. The procedure involved in this method is outlined below.
The total load on the system is divided into a number of load steps and the starting
solution is obtained by performing a linear stiffness analysis for the first load step. The
displacements obtained from this step will be used to evaluate the non-linear components
of stiffness matrix. This matrix along with the linear part of stiffness matrix is used to
evaluate displacements for the next load step. These displacements are added to the
displacements for the first load step. These net displacements will be used for the
evaluation of stiffness matrix in the next load step. This process is continued until the last
load step is completed. Thus, in this method, no iteration at constant load is required,
which, in tum, means that there is no danger of a divergence of the solution. Control is
exercised over the accuracy of the incremental analysis by control of the increment size
which, in tum, depends upon load-displacement behaviour. As a consequence, the
apparent speed advantage of an incremental method can be expended in multiple
executions designed to improve or build confidence in the predicted behaviour. An
undersized increment will be wasteful, since it results in an excessive number of linear
solutions.
196
7.5 NUMERICAL RESULTS
Studies based on higher-order shear deformation theory are conducted to investigate the
effects of geometric non-linearity on load-deflection behaviour and stresses of composite
laminates.
7.5.1 Validation
The validation of the program is done by considering few examples for which results are
available in literature. Reduced integration scheme is used for the evaluation of both
bending and transverse shear terms of stiffness matrix and a mesh division of 16x 16 for
full plate is used for the analysis.
Example 7.1
Square isotropic plates of width-to-thickness ratios 10 and 100, with all the edges
clamped and subjected to a uniform load of non-dimensional value, Q = 200 and 402
respectively, for which results have been given by Shukla and Nath [63] and Pica
et aL [127], are considered first. As the method of linear piece-wise incremental
procedure necessitates smaller load steps, the load, Q, has been incremented in steps of 1.
But for the Newton-Raphson iterative procedure, the load, Q, has been incremented in
steps of 4 and a relative convergence criteria of 0.01 % at every step of loading has been
employed. The results obtained by adopting Newton-Raphson iterative technique and
piece-wise linear incremental technique for the thick plate are presented in Table 7.1 in
non-dimensional form as w = w I h and are compared with available results. Agreement
in results is found to be very good. The load-deflection variation for the thick and thin
plates is presented in Figs. 7 .1 and 7 .2 along with the corresponding linear solution.
197
·t;;Load, Q
�i \'
20
40
80
120
200
Table 7 .1 Comparison of non-dimensional central deflection for clamped isotropic plate, b/h = 10 ( v = 0.3)
w
Present analysis
Shukla and Linear load
N ewton-Raphson incrementing
Nath [63] method
method
0.3074 0.3101 0.3096
0.5441 0.5535 0.5525
0.8649 0.8849 0.8848
1.0852 1.1109 1.1128
1.3958 1.4271 1.4336
198
Turvey and Osman [128]
0.3017
0.5423
0.8655
1.0860
1.3980
3.5
... Newton-Raphson method
3.0 * Linear load incrementing method
Linear
2.5
2.0
I�
1.5
1.0
0.5
0.0
0 40 80 120 160 200
Load, Q
Figure 7 .1 Load-deflection behaviour of clamped isotropic plate (b/h = 1 O)
6.0 ,------------------,
5.0
4.0
I� 3.0
2.0
1.0
* Newton-Raphson method
Linear load incrementing metho
Linear
Exact [129)
0.0 .... ---r--�----,---.---r----j
0 80 160 240
Load, Q
320 400 480
Figure 7 .2 Load-deflection behaviour of clamped isotropic plate (b/h = 100)
199
From Table 7 .1 and Figs. 7 .1 and 7 .2, it is seen that the results predicted using a large
load step by Newton-Raphson iteration and a small load step by piece-wise incremental
procedure are almost the same. But, the latter method requires less computational effort
and also the first method exhibited some computational instability in some cases for
higher loads. Hence, the piece-wise linear incremental procedure has been employed for
further studies.
The exact central normal stress expressed in non-dimensional form, ax
= crxb2 / Eh2 for
the thin plate corresponding to the total load, Q == 402, has been reported to be 25.1 [129].
The present study gives a value of 24.66, showing good agreement.
Example 7.2
To validate the program for composite laminates, a clamped 2-layer cross-ply (0/90)
laminate given by Shukla and Nath [63] and a simply supported 2-layer angle-ply
(45/-45) laminate given by Barbero and Reddy [58] are considered. The geometry and the
material properties of the J?lates are as given below.
Cross-ply: a = b = lm, h = O.lm, E1 = 175.78 GPa , E1/E2 = 25, Gn/E2 = G13/E2 = 0.5, G23/E2 = 0.5, V12
= 0.25.
Angle-ply: a = b = lm, h = 0.002m, E1 = 250 GPa, E2 = 20 GPa, G12 = G13 = 10 GPa, G23 = 4 GPa, v12
= 0.25.
The boundary conditions used are as follows:
at x = 0 and x = a, at y = 0 and y = b,
Vo = w O = aw of oy = Sy = 0 Uo = Wo = awof8x = 9x = 0
The non-dimensional central deflections obtained are presented in Figs. 7 .3 and 7.4. In
both cases, results of the present analysis agree very well with available results.
200
' •. _·:f:
2.0 ,---------------.,Present analysis
* Linear
1.5
I� 1.0
0.5
•
+
Shukla and Nath [63]
Singh et al. [130]
0.0 --------.-----,.------'--r-----.---�
0 · 60 120 180 240 300
Load, Q
Figure 7 .3 Load-deflection behaviour of 2-layer cross-ply (0/90) laminate
4.0 -.-------------------,
3.0
I� 2.0
1.0
Present analysis
* Linear
• Barbero and Reddy [58)
0.0 �----ir----,---,-----.---;
0 15 30 45 60 75
Load, Q
Figure 7.4 Load-deflection behaviour of 2-layer angle-ply (45/-45) laminate
201
7.5.2 Parametric Study
To study the effect of various parameters on the non-linear behaviour of composite
plates, numerical studies are carried out by analyzing plates having the following
geometric and material properties.
a = b = 1.0 m, E 1 = 175.0GPa, E2 = 7.0GPa, v12 = 0.25, G12 = G13 = 3.SGPa,
G23 = l.4GPa
In all cases, a uniform load of non-dimensional value, Q = 300 is applied in steps of 1,
unless otherwise specified.
Effect of width-to-thickness ratio (b/h)
Example 7.3
To study the effect of width-to-thickness ratio on the non-linear behaviour, 4-layer
cross-ply (0/90/90/0 and 0/90/0/90) and angle-ply (45/-45/-45/45 and 45/-45/45/-45)
laminates are considered. The non-dimensional values of central deflection for symmetric
and anti-symmetric cross-ply laminates with the edges simply supported are plotted in
Figs. 7 .5 and 7 .6 respectively. From these figures, it is evident that the non-dimensional
deflection decreases with increase in b/h ratio and the behaviour of plates with width-to
thickness ratio beyond 50 is almost the same. The variation of stresses through the
thickness of thick and thin symmetric cross-ply laminates, non-dimensionalised as in
Eq. (3.14), is shown in Figs. 7.7 and 7.8, along with the linear solutions.
202
- -
2.0 bib= 10
bib =20 1.6
* bib= 50
bib== 100
1.2
I�
0.8
0.4
0.0 .----,----,-----.----.----10 60 120 180 240 300
Load,Q
Figure 7.5 Load-deflection behaviour of symmetric cross-ply (0/90/90/0) laminate
I:!=
2.0 bib= 10
bib =20
1.6 * bib= 50
bib= 100
1.2
0.8
0.4
0.0 ------..-----,r-----r----.-----i
60 120 180
Load, Q
240 300
Figure 7 .6 Load-deflection behaviour of anti-symmetric cross-ply (0/90/0/90) laminate
203
0.4746 0.0556 0.0238
I
I
0.8153 0.3919
O'x 'ixy
0-0,
0.0604
--Linear
---- Non-linear
0.4155
Tyz
V•-'YDo 0.9405
) 0)2716 0.5016
/ /,,.
Txz
Figure 7. 7 Stress variation across the thickness of symmetric cross-ply laminate (b/h = 10)
N 0 V.
I
0.8296 0.5679
crx
0.6303 0.0394 0.0232 �-��
\
\
Txy
0.0423
--Linear
---- Non-linear
4.0874 0.2544 \ \ f 0.1165 / 0.3392 I
n I
Tyz
--· . x- , 1.0280 I l
.4136 0.5483
Txz
Figure 7 .8 Stress variation across the thickness of symmetric cross-ply laminate (b/h = 100)
The non-dimensional values of central deflection for symmetric and anti-symmetric
angle-ply laminates with the edges simply supported are given in Figs. 7 .9 and 7 .10. The
behaviour is same as that of simply supported cross-ply laminates. Anti-symmetric
angle-ply laminates have lower values of deflection than for symmetric laminates, as is
seen from the figures.
1.2
1.0
0.8
I� 0.6
0.4
0.2
b/h= 10
b/h =20
b/h= 50
b/h = 100
0.0 4-----r---.----.-----.---1
0 60 120 180 240 300
Load, Q
Figure 7.9 Load-deflection behaviour of symmetric angle-ply (45/-45/-45/45) laminate
206
1.0
b/h = 10
* b/h=20
0.8 b/h = 50
b/h = 100
0.6
I�
0.4
0.2
0.0
0 60 120 180 240 300
Load, Q
Figure 7 .10 Load-deflection behaviour of anti-symmetric angle-ply ( 45/-45/45/-45) laminate
The non-dimensional values of central deflection for symmetric cross-ply (0/90/90/0)
laminates of different width-to-thickness ratios and the edges clamped are plotted in
Fig. 7 .11. It is seen that there is a decrease in deflection of around 5% from b/h = 50 to
100, whereas the load-deflection curve is almost the same for b/h = 50 and 100 in the
case of simply supported edge conditions.
207
� . .
1.0
0.8
0.6
I�
0.4
0.2
0.0
0
• b/h= 10
b/h=20
* b/h= 50
60 120 180
Load, Q
240 300
Figure 7.11 Load-deflection behaviour of clamped cross-ply (0/90/90/0) laminate
The effect of b/h ratio on the load-deflection behaviour of simply supported and clamped
symmetric cross-ply laminates corresponding to a maximum load, Q = 300 is depicted in
Fig. 7 .12. Even though the variation of linear solution is different in thick plate region for
the two edge conditions, the non-linear solution follows the same variation irrespective of
edge conditions.
208
I�
4.0
3.5
3.0
2.5
2.0
1.5
1.0 *·-
0.5
0
.t. Linear (Simply supported)
- - -.a.- - - Nonlinear (Simply supported)
* Linear (Clamped)
- - -•- - - Nonlinear (Clamped)
.... ______ _ ---"* · -------------------
20 40 60 80 100
b/h
Figure 7 .12 Comparison of linear and non-linear deflections
. .
A comparison of degree of non-linearity for simply supported symmetric cross-ply and
angle-ply laminates is given in Fig. 7 .13. From the figure, it is clear that the fibre
orientation angle does not influence the non-linear behaviour of laminates except that the
degree of non-linearity is less for angle-ply laminates. Moreover, the degree of non-
linearity increases with increase in b/h ratio.
Figure 7.14 shows the variation of degree of non-linearity for simply supported and
clamped cross-ply (0/90/90/0) laminates. It is evident that the degree of non-linearity is
less for clamped plates in thick range compared to simply supported plates. For simply
supported plates, degree of non-linearity does not vary much beyond b/h = 20, whereas in
the case of clamped plates, the degree of non-linearity goes on increasing upto b/h = 50.
209
1.0
... 0/90/90/0
i ;. 45/-45/-45/45
C:
�
is 0.8·t::� 11)
;.§ I c::
0
'-0
0.6
0
0.4 -t---.----.----.-----.----J 0 20 40 60 80 100
b/h
Figure 7.13 Variation of degree of non-linearity with b/h ratio
1.0
A Simply supported
* Clamped
t 0.8
;
b 't:: C"a 11) c::
0.6 .... I
c:: 0 c::
I...., 0
11)
11)
0.4 11)
0
0.2
0 20 40 60 80 100
b/h
Figure 7 .14 Degree of non-linearity for cross-ply laminates
210
Effect of number of layers
Example 7.4
The effect of number of layers on the non-linear behaviour is studied by considering thin
(b/h = 100) simply supported cross-ply and angle-ply laminates with anti-symmetric
arrangement. The variation of non-dimensional central deflections is shown in Figs. 7 .15
and 7 .16. It is seen that the effect of non-linearity is almost the same beyond 4 layers.
2.4 -,-------------------,
2.0
1.6
I� 1.2
0.8
0.4
0.0
0
.t. N=2
60 120 180 240 300
Load,Q
Figure 7 .15 Load-deflection behaviour of cross-ply laminates with number of layers
211
1.0
0.8
0.6
I�
0.4
0.2
• N=2
0.0 .----.----.------,----..------1
0 60 120 180 240 300
Load, Q
Figure 7 .16 Load-deflection behaviour of angle-ply laminates with number of layers
Effect of in-plane edge conditions
Example 7.5
To study the effect of in-plane edge conditions on the non-linear behaviour, two different
simply supported boundary conditions are considered for the analysis. (i) Plate with
movable edges (SS 1) and (ii) Plate with immovable edges (SS2) as given below.
SS 1: at x = 0 and x = a,
at y = 0 and y = b,
SS2: atx = Oandx= a,
at y = 0 and y = b,
Vo = Wo = EJwJay = 0y = 0
Uo = Wo = EJwJox = 0x = 0
Uo= Vo= Wo = EJwJay= 0y= 0
Uo = Vo= Wo = EJwofox = 0x = 0
212
In the case of anti-symmetric angle-ply laminates, the movable edge conditions
considered are as follows:
SSl: at x = 0 and x = a,
at y = 0 and y = b,
Uo= w0
= 8wJoy= Sy
= 0
Vo = Wo = 8wJox = Sx = 0
Cross-ply (0/90/90/0 and 0/90/0/90) and angle-ply ( 45/-45/-45/45 and 45/-45/45/-45)
laminates with SS 1 and SS2 edge conditions are analysed for both thick (b/h = 10) and
thin (b/h = 100) cases. Load-deflection curves are shown in Figs. 7.17 - 7 .24. In all cases,
deflection is more for SS 1 condition. It is seen that the difference between deflections for
SS 1 and SS2 conditions is more in thick plates, except in the case of anti-symmetric
angle-ply laminates. The difference in deflection is much more in the case of cross-ply
laminates (nearly 55% for thick plates and 50% for thin plates, the result for SSl being
taken as reference) than in the case of symmetric angle-ply laminates (20% for thick
plates and 14% for thin plates). In the case of anti-symmetric angle-ply laminates, the
difference between deflections is found to be very small for both thick and thin plates.
But there is a unique linear solution for symmetric cross-ply laminates irrespective of
edge conditions. For anti-symmetric laminates, there is a decrease in linear deflection of
around 12% with respect to that for SS 1. Figures show that as the load increases,
difference in deflections for SS 1 and SS2 conditions also increases. Membrane forces of
significant magnitude are developed in SS2 condition. Under SS 1 condition, the load is
mostly carried by bending action, and hence the plate deflects more. Under SS2
condition, membrane action contributes considerably to the load carrying capacity of the
plate and hence the deflection is less.
213
3.5 * SS1
3.0 ... SS2
2.5 Linear (SS1,SS2)
2.0
I�
1.5
1.0
0.5
0.0
0 60 120 180 240 300
Load, Q
Figure 7 .17 Effect of in-plane edge conditions on symmetric cross-ply laminate (b/h = 10)
2.0 ... SS1
* SS21.6
Linear (SS1, SS2)
1.2
I�
0.8
0.4
0.0 J£._---r----r----,----.----1
0 60 120 180 240 300
Load, Q
Figure 1.1 � Effect of in-plane edge conditions on symmetric cross-ply laminate (b/h = 100)
214
3.5
3.0
2.5
2.0
I�
1.5
1.0
0.5
0.0
0
4 SS1
* SS2
Linear, SS1
i Linear, SS2
60 120 180
Load, Q
240 300
Figure 7 .19 Effect of in-plane edge conditions on anti-symmetric cross-ply laminate (b/h =1 O)
2.5
SS1
... SS2 2.0
Linear, SS1
Linear, SS2
1.5
I�
1.0
0.5
0.0 �---..-----r---.----.----i
0 60 120 180 240 300
Load, Q
Figure 7 .20 Effect of in-plane edge conditions on anti-symmetric cross-ply laminate (b/h =l 00)
215
2.5
SS1
... SS2 2.0
Linear (SS l ,SS2)
1.5
I�
1.0
0.5
0.0 ...----,----,----.-----.-----l
0 60 120 180 240 300
Load, Q
Figure 7.21 Effect of in-plane edge conditions on symmetric angle-ply laminate (b/h =10)
I�
1.5
1.0
0.5
... SS1
* SS2
Linear (SS1, SS2)
0.0 ,M.----.-----r----r----,-----j
0 60 120 180 240 300
Load, Q
Figure 7.22 Effect of in-plane edge conditions on symmetric angle-ply laminate (b/h =100)
216
2.5 * SS1
... SS22.0
i Linear, SS1
Linear, SS2
1.5
I�
1.0
0.5
0.0
0 60 120 180 240 300
Load, Q
, re 7.23 Effect of in-plane edge conditions on anti-symmetric angle-ply laminate (b/h =10)
1.5 * SS1
... SS2
+ Linear, SS1
1.0 Linear, SS2
I�
0.5
0.0
0 60 120 180 240 300
Load, Q
Figure 7.24 Effect of in-plane edge conditions on anti-symmetric angle-ply laminate
(b/h =100)
217
Effect of aspect ratio
Example 7.6
The influence of aspect ratio on the central transverse displacement of thin (b/h = 100)
symmetric cross-ply (0/90/90/0) and angle-ply (45/-45/-45/45) laminates with simply
supported boundary conditions is shown in Figs. 7.25 and 7.26. It is seen that the non
linearity increases as alb increases in the case of both the laminates.
4.0
alb= 1.0
3.5 * alb= 1.5
3.0 alb =2.0
t alb= 2.5
2.5
I� 2.0
1.5
1.0
0.5
0.0
0 60 120 180 240 300
Load,Q
Figure 7 .25 Change in load-deflection behaviour of cross-ply laminates with aspect ratio
218
3.5
alb= 1.0
3.0 * alb= 1.5
alb= 2.0
2.5 alb= 2.5
2.0
•I�
1.5
1.0
0.5
0.0
0 60 120 180 240 300
Load, Q
Figure 7 .26 Change in load-deflection behaviour of angle-ply laminates with aspect ratio
7.6 DISCUSSION
Based on the studies conducted, the following observations are made:
1. Piece-wise linear load incrementing method and Newton-Raphson iterative
procedure give almost the same results. But, the difficulty of divergence of
solution under higher loads by the iterative procedure can be avoided by using
piece-wise linear load incrementing method.
2. Degree of non-linearity is more in cross-ply laminates than m angle-ply
laminates.
219
3. Degree of non-linearity is more in simply supported plates compared to clamped
plates in the thick range whereas the reverse is the case in thin range.
4. Angle of fibre orientation does not have any influence in the non-linear
behaviour pattern of composite plates.
5. The non-linear load deflection behaviour is the same beyond 4 layers.
6. The in-plane edge conditions play an important role in the non-linear behaviour
which is not predominant in the linear analysis.
7. The studies conducted confirm that the small deflection theory is applicable in
the case of composite laminates only if w/h is less than 0.2.
8. The simple finite element model chosen for the study is found to be sufficient for
large displacement analysis oflaminated composite plates.
Recommended