non-linear dynamic analysis of laminated...
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CHAPTERS
NON-LINEAR DYNAMIC ANALYSIS OF LAMINATED PLATES
8.1 GENERAL
Dynamic behaviour of isotropic and homogeneous plates has long been a subject of
interest. However, the linear and non-linear transient analysis of laminated composite
plates has not received much attention. Large amplitude dynamic analysis of laminated
composite plates using higher-order shear deformation theory has not been studied so far,
except for the work of Kant and co-workers [72, 73]. Effect of damping has not been
studied at all.
The large amplitude free and forced vibration analysis of laminated composite plates
using the 4-noded finite element based on higher-order shear deformation theory is
presented in this chapter. Effect of damping is also studied.
8.2 FREE VIBRATION ANALYSIS
It is well known that the linear frequencies of vibration of a system remain the same
irrespective of the amplitude of vibration. But, it has been experienced that the non-linear
frequencies of vibration depend on the amplitude of vibration and, hence, vary with the
change in amplitude. This has led researchers to study the effect of amplitude of vibration
on the non-linear frequencies by varying the amplitude. The few results available in
literature are the values of frequency ratio (coNdcoL) for various values of amplitude ratio
(w/h), where w represents the maximum amplitude of vibration. To check the accuracy
and efficiency of the present model, a similar analysis is carried out and the results are
compared with results available in literature.
221
8.2.1 Analysis Procedure
Evaluation of non-linear frequencies of vibration of a system involves the iterative
solution of equation of motion of the form ([K]+[KNd){$}-o} [Ml{$}= 0, where
[KNL] from Eq. (7.12) denotes the non-linear component of stiffness matrix. The
sequence of steps in the iterative procedure to evaluate the non-linear frequencies are
summarised as follows:
1. Use the subspace iteration technique to find the fundamental frequency and
the associated mode shape vector by performing a linear analysis.
2. The eigen vector obtained is scaled appropriately so that the maximum
displacement is equal to the desired amplitude.
3. Based on this scaled eigen vector, the new stiffness matrix 1s evaluated
including the non-linear components, as explained in section 7.3.
4. Find the non-linear frequency using this new stiffness matrix by subspace
iteration technique.
5. If the difference between frequencies obtained during two consecutive
iterations is less than a small prescribed value, the solution is converged;
otherwise return to step 2.
8.2.2 Numerical Results
Numerical results are obtained using the 4-noded element, based on higher-order shear
defom1ation theory, with seven degrees of freedom per node, as explained in Chapter 3.
Reduced integration and exact integration schemes are used for the evaluation of stiffness
and mass matrices respectively. A mesh division of 16xl6 for full plate is used for the
analysis.
222
Example 8.1
Square simply supported isotropic plates of width-to-thickness ratio 1 O and 1000
presented by Ganapathi et al. [70] are analysed, varying the amplitude ratio from 0.2 to
1.0. The in-plane conditions for the edges are considered as immovable. i.e., u0 = v0 = O
on edges (SS2). The variation of frequency ratio with amplitude ratio for these plates is
presented in Figs. 8.1 and 8.2 and are compared with the values given by Ganapathi
et al. [70]. They have used 8-noded serendipity element based on first-order shear
deformation theory, with 2x2 mesh for quarter plate and exact integration for evaluating
the stiffness matrix. The present results agree very well in both cases.
2.00
1.75
� 1.50 z 8
1.25
1.00
0.0 0.2 0.4
* Present analysis
• Ganapathi et al. [70]
0.6 0.8
w/h
1.0
Figure 8.1 Variation of frequency ratio with amplitude ratio for an isotropic plate (b/h = 10)
223
2.00 .---------------
1.75
':::i 1.50 z
1.25
0.0 0.2 0.4
Present analysis
... Ganapathi et al. [70]
0.6 0.8 1.0
w/h
Figure 8.2 Variation of frequency ratio with amplitude ratio for an isotropic plate (b/h = 1000)
Similar results for an orthotropic plate with SS2 edge conditions and having the
,, following material properties as given by Ganapathi et al. [70] are also presented for two
different thickness ratios (b/h = 10 and b/h = 1000) in Figs. 8.3 and 8.4. Agreement in
results is found to be good.
Material - Argonite crystal
Ex = 1.0, Ey
= 0.543103, Gxy
= 0.262931, Gyz = 0.26681, Gxz = 0.159914, Yxy
= 0.23319
224
2.00 i----------------.*
1.75
Present analysis
Ganapathi et al. [70]
11.50
1.25
1.00 +-======�----.----,.... _____J 0.0 0.2 0.4 0.6 0.8 1.0
w/h
Figure 8.3 Variation of frequency ratio with amplitude ratio for an orthotropic plate (b/h = 10)
2.00 -,-.-----------------.
1.75
* Present analysis
• Ganapathi et al.[70]
1 1.SO
1.25
1.00 ... =:::::::.:;:_ __ -r---.----.-------1
0.0 0.2 0.4 0.6 0.8 1.0
w/h
Figure 8.4 Variation of frequency ratio with amplitude ratio for an orthotropic plate (b/h = 1000)
225
In the studies conducted on non-linear free vibration analysis of composite plates by
researchers [70, 72], the non-linear components of stiffness matrix were evaluated using
the scaled eigen vectors obtained from the linear analysis. The scaling was done such that
the maximum amplitude of vibration became h, 2h, 3h, etc. That is, the frequency ratios
corresponding to amplitude ratios of 1, 2, 3, etc. were reported. In the present study, the
procedure of scaling the eigen vectors obtained from the linear analysis is avoided. That
is, the non-linear components of stiffness matrix are evaluated directly based on the eigen
vectors obtained from linear analysis, which are M-orthonormalised.
The effect of various parameters on the non-linear fundamental frequency is investigated
by analysing laminates of different types. The geometry and the properties of the
material of the plate considered are a = b = 25cm, E2 = 1.0, E 1/E2= 25, G12 = G13 = 0.5E2,
G23 = 0.2E2, v12 = 0.25, p = 1.0. The in-plane edge conditions for the simply supported
plates are considered as immovable (SS2) in all cases, unless otherwise specified.
Effect of width-to-thickness ratio (b/h)
Effect of width-to-thickness ratio on non-linear fundamental frequency is studied by
considering simply supported 4-layer cross-ply and angle-ply laminates with symmetric
and anti-symmetric arrangements of layers. The variation of frequency ratio (roNdroL)
with b/h ratio is presented in Fig. 8.5.
226
,, i.rf ... -
1.4 ,.
0/90/90/0
0/90/0/90
1.3 ... 45/-45/-45/45
+ 45/-45/45/-45
8 ':::i 1.2
8
1.1
1.0
0 20 40 60 80 100
b/h
Figure 8.5 Variation of frequency ratio with b/h ratio
It is seen that the frequency ratio increases with increase in b/h ratio and remains
practically the same for both symmetric and anti-symmetric arrangements in the case of
cross-ply laminates. But, in the case of angle-ply laminates, the frequency ratio is less for
anti-symmetric arrangement than symmetric arrangement, the difference being small. It is
seen that the effect of non-linearity is predominant only in the case of thin (b/h � 40)
plates.
The variation of non-linear fundamental frequency, non-dimensionalised as
�L = coNLb2
�, with b/h ratio for the same plates is presented in Fig. 8.6. It is seen h VEz
that there is a sudden increase in the value of coNL in the thick plate range (b/h � 20),
beyond which the increase is at a slow rate.
227
..J
z 19
25.0
20.0
15.0
10.0
0 20 40
b/h
* 0/90/90/0
0/90/0/90
45/-45/-45/45
45/-45/45/-45
60 80 100
Figure 8.6 Variation of non-dimensional fundamental frequency with b/h ratio
Behaviour similar to that of 4-layer laminates has been observed for 2-layer cross-ply and
angle-ply laminates with b/h ratio and the results are presented in Fig. 8. 7. Non-linearity
. is more in cross-ply laminates because of the smaller stiffness.
228
1.5
1.4
1.3
1.2
1.1
1.0
0
* 0/90
"" 45/-45
20 40 60 80 100
b/h
Figure 8. 7 Variation of frequency ratio with b/h ratio for 2-layer laminates
Effect of fibre orientation angle
Example 8.3
The variation of frequency ratio with increase in the fibre orientation angle, for simply
supported 4-layer symmetric and anti-symmetric laminates is presented in Fig. 8.8 for
b/h = 10 and 100. It is seen that the fibre orientation angle doesn't have any non-linear
influence on the frequency in the case of thick plates, irrespective of lamination
sequence. But in the case of thin plates, the frequency ratio decreases with increase in
fibre orientation angle. Moreover, there is a reduction of around 4% in the frequency ratio
for anti-symmetric arrangement with respect to symmetric arrangement for 45°
fibre
orientation angle.
229
1.4
1.3
1.2 al-a/al-a {b/h = 100)
. ..J ... al-al-ala {b/h = 100)
. ·� 1.1 . z
al-a/al-a (b/h = 10) '· 8
* al-al-ala (b/h = 10)1.0
0.9
0.8
0 15 30 45
Angle of fibre orientation, a.
Figure 8. 8 Variation of frequency ratio with fibre orientation angle
Effect of number oflayers
Example 8.4
Effect of number of layers on non-linear fundamental frequency is studied by considering
both thin and thick cross-ply and angle-ply laminates with anti-symmetric lay up. The
variation of frequency ratio with number of layers is presented in Fig. 8.9. Even though
there is no change in the frequency ratio with increase in number of layers in the case of
thick plates, there is a small change from 2 to 4 layers in the case of thin plates and
afterwards the value remains practically constant. This is in agreement with the earlier
observation that 2-layer plates behave differently from multi-layered plates.
230
1.5
1.4
1.3
..J 1.2 8
8 1.1
1.0
0.9
0.8
0 2
* *
4 6
Number of layers
*
8 10
-I• angle-ply (b/h = 100)
"' cross-ply (b/h = 100)
angle-ply (b/h = 10)
* cross-ply (b/h = 10)
Figure 8.9 Variation of frequency ratio with number oflayers
Effect of material anisotropy
Example 8.5
To study the effect of material anisotropy on the non-linear fundamental frequency,
4-layer symmetric cross-ply and angle-ply laminates with b/h = 10 and 100 are
considered and the variation of frequency ratio is presented in Fig. 8.10. The increase in
material anisotropy (E1/E2) from 5 to 40, keeping E2 constant, causes an increase in non-
· linear frequency in the case of thin cross-ply plates, but the percentage increase is very
small.
231
,, .
!f rr,, f/
�'
1.5
1.4
1.3
...J 1.2 8
8 1.1
1.0
0.9
0.8
0 10 20 30 40
angle-ply (b/h = 100)
... cross-ply (b/h = I 00)
angle-ply (b/h = 10)
* cross-ply (b/h = I 0)
Figure 8.10 Variation of frequency ratio with material anisotropy
Effect of in-plane edge conditions
Example 8.6
The effect of movable (SS 1) and immovable (SS2) in-plane edge conditions of the simply
supported plate on the non-linear fundamental frequency is studied by analyzing 4-layer
/; cross-ply and angle-ply laminates. The variation of frequency ratio for various b/h values
is presented in Figs. 8.11 and 8.12. It is seen that there is an increase in frequency ratio of
around 33% for thin (b/h = 100) plates with SS2 edge condition (values for SSI being
· taken as reference) irrespective of stacking sequence in the case of cross-ply laminates. In
other words, the frequency of vibration in non-linear domain remains almost the same as
232
that in linear range, if the in-plane movement of supports is permitted. In the case of
symmetric angle-ply laminates there is an increase of only 9% in the frequency ratio with
SS2 edge condition. But, in the case of anti-symmetric angle-ply laminates, the variation
is the same for both the edge conditions.
1.5
1.4
1.3
0/90/90/0, ss l
1.2 0/90/90/0, SS2 8 --::i 0/90/0/90, ss l 8 1.1
0/90/0/90, SS2
1.0
0.9
0.8
0 20 40 60 80 100
b/h
Figure 8.11 Variation of frequency ratio with in-plane edge conditions (cross-ply)
233
,'
.
�, . ... ?;'.; �·'
1.5
1.4
1.3
...J 1.2
I. I
1.0
0.9
0.8
0 20 40
b/h
60 80 100
* 45/-45/-45/45, ss 1
45/-45/-45/45, SS2
45/-45/45/-45, ss 1
45/-45/45/-45, SS2
Figure 8.12 Variation of frequency ratio with in-plane edge conditions (angle-ply)
8.3 TRANSIENT ANALYSIS
Though the linear transient analysis of structures can be simplified by adopting mode
superposition method enabling uncoupling of equations of motion, it is not possible in the
r case of a system where the physical properties (stiffness, mass or damping coefficients)
r
f change during the response period. In the case of a large deflection problem, the stiffness
1,,
influence coefficients are altered by the change in geometry and, hence, it becomes
necessary to do the numerical integration of the coupled equations of motion in a step-by-
· step manner. Accordingly, the computational effort involved will be enormous.
234
One potential difficulty in the step-by-step response integration of multi-degree of
freedom systems is that the damping matrix, [C], must be defined explicitly rather than in
terms of modal damping ratios. It is very difficult to estimate the magnitudes of the
damping influence coefficients of a complete damping matrix. In general, the most
effective means for deriving a suitable damping matrix is to assume appropriate values of
modal damping ratios for all the modes which are considered to be important and then to
compute an orthogonal damping matrix which has the same properties as those of the
original system [ 120]. In this procedure, the complete diagonal matrix, [ c] of generalised
damping coefficients is obtained by pre- and post-multiplying the damping matrix, [C] by
the mode shape matrix, [<l>].
s1co1M1 0 0 0
0 s2co2M2 0 0 [c] = [<1>f[c][<1>] = 2 (8.1)
0 0 SnCOnMn
From Eq. (8.1 ), it is evident that the damping matrix, [C], can be obtained by
post-multiplying [c] by the inverse of [<l>] and pre-multiplying by the inverse of [<l>]T .
(8.2)
As the inversion of mode shape matrix involves a large computational effort, a simplified
procedure has been adopted by using the orthogonality properties of the mode shapes
related to the mass matrix. The diagonal generalized mass matrix of the system is
obtained by pre- and post-multiplying the mass matrix by the complete mode shape
matrix.
235
·
,· ."!
Pre-multiplying Eq. (8.3) by the inverse of the generalized mass matrix, we get
That is, the inverse of the mode shape matrix is given by,
Substituting Eq. (8.4) in Eq. (8.2), we get,
Eq. (8.5) is rewritten as
2ro.):. wheret:; i =� M.
I
(8.3)
(8.4)
(8.5)
(8.6)
In practice, each modal damping ratio provides an independent contribution to the
damping matrix as Ci
= M$it:;
i$
iTM. Thus the total damping matrix is obtained as the
sum of the modal contributions as
(8.7)
Equation (8. 7) is rewritten as
(8.8)
236
where p is the total number of modes of vibration under consideration. Equation (8.8)
gives the contribution to the damping matrix from each mode proportional to the
damping ratio.
Eventhough the non-linear analysis of structures requires explicit evaluation of damping
matrix as given in Eq. (8.8) by considering all modes of vibration, the evaluation of all
mode shape vectors for a plate with a fine finite element mesh is computationally
expensive. The non-linear transient analysis of laminated composite plates is carried out
by employing linear load incrementing method, explained in Chapter 7. As the response
history is divided into short, equal time increments and the response is calculated during
each increment, it becomes necessary to divide the total load also into a number of
suitable load steps for each time increment. A linear analysis will be performed for the
first load step to get the dynamic characteristics such as displacement, velocity and
acceleration at a particular time. Based on this displacement, the stiffness matrix will be
modified incorporating the non-linear components as given in Eq. (7.12) and the analysis
is carried out for the next load step at the same time and this process will be continued till
the completion of all load steps. It must be noted that linear analysis has to be performed
for the first load step of all time increments, and for each load step of successive time
increments the response obtained at the previous time step corresponding to that load step
will have to be used. The dynamic displacement at each step is obtained by solving the
coupled equations of motion using Wilson-8 method [119].
237
8.3.1 Numerical Results
Example 8.7
The correctness of the program is ensured by analyzing an isotropic plate having the
geometry and material properties as given by Reddy [69], which are presented below.
a = b = 243.8 cm, h = 0.635 cm, E = 7.03lxl05 N/cm2
, p =2.547x10-6 Nsec2/cm
4,
VJ2 = 0.25.
The boundary conditions for the plate are as given below.
at X = 0 and a, U0 = W0 = aw Jax= 8x = 0 and at y = 0 and b, Vo = W0
= aw Joy= 8y = 0
The plate is subjected to a suddenly applied pressure load, q = 4.882x10-4
N/cm2. The
load, q, is divided into ten equal load steps for the analysis. The time step taken is same
· as that used by Reddy [69]. The central deflection obtained for various loads are shown
in Fig. 8.13. The results agree fairly well with those presented in Figure 7 of Reddy [69],
using a first-order shear deformation theory.
1.4
1.2
1.0
--
0.8
0.6
0.4
0.2
o.o
O.OE+O 4.0E+4 8.0E+4
Time (µs)
* L,q
l.2E+5
NL,q
NL,2q
NL,Sq
l.6E+5
Figure 8.13 Displacement response of an isotropic plate
238
To study the effect of various parameters on the non-linear transient response of
laminated plates, simply supported plates are analysed. The following geometry and
material properties are used in all the examples, unless otherwise specified.
6 2 I a = b = 25 cm, E2 = 2.lxlO N/cm, E1 E2 = 25, Y12 = 0.25, G12 = G13 = 0.5E2,
G23 = 0.2E2, p = 8x10-6
Nsec2/cm
4. In all cases a suddenly applied uniformly distributed
load of non-dimensional value, Q = 200, is considered.
Example 8.8
To decide upon the number ofload steps to be used for getting accurate results, numerical
experiments are conducted by analysing 4-layer simply supported cross-ply (0/90/90/0)
laminates of width-to-thickness ratios 10 and 100 and the results are presented in
Table 8.1.
Table 8.1 Convergence study for load steps
Time step Number of Time of occurrence Peak b/h (µs) load steps of peak displacement
displacement (µs) (cm)
50 90 5.6128
10 10 100 90 5.6549
50 850 0.4809
100 50 100 850 0.4859
It is observed that the difference in the values of peak displacements obtained by using 50
load steps and 100 load steps is only of the order of 1 %, irrespective of the thickness of
239
the plate. Hence, to reduce the computational effort, a load step of 50 is used for further
studies.
The size of one time step is also critical in obtaining accurate results. To recommend a
suitable time step, studies are conducted on the same plate used for load step study.
Results of this study are presented in Table 8.2.
Table 8.2 Convergence study for time step (load step= 50)
Time step Time of occurrence Peak b/h (µs)
of peak displacement displacement (µs) (cm)
5 80 5.7467
10 10 90 5.6128
25 100 4.9969
10 810 0.4946
100 25 825 0.4910
50 850 0.4809
It is found that a time step of 5µs for thick plates (b/h = 10) and 25µs for thin plates
(b/h = 100) gives good results and, hence, the same is adopted for further studies.
Example 8.9
To have an idea about the difference in the linear and non-linear dynamic behaviour,
4-layer symmetric cross-ply (0/90/90/0) laminates simply supported on all edges
(at X::::: 0 and a, Vo = Wo = ow of ox= 0x = 0 and at y = 0 and b, Uo = Wo = ow Joy= 0y = 0),
240
with b/h = 10 and 100, are analysed and the results are shown in Figs. 8.14 and 8.15
respectively along with the linear solutions.
14.0
10.0
- 6.0
�
2.0
-2.0
0
, I
, ' ,
, , , ... ..
- ..,'
100
' ' ' ' '
' '
Time (µs)
' '
200
Non-linear
Linear
300
Figure 8.14 Displacement response of a 4-layer cross-ply laminate (b/h = 10)
1.0
,' 0.5
0.0
-0.5
0 500
Non-linear
... ..
- - ... ------- Linear ' ' ' ' ' ' ' ' ' ' ' '
1000 1500
Time (µs)
' '
2000 2500
Figure 8.15 Displacement response of a 4-layer cross-ply laminate (b/h = 100)
241
From these figures, it is seen that the peak amplitude as well as the time at which it
occurs changes very much in the case of large deflection analysis. This change is more
pronounced in thick plates.
Effect of width-to-thickness ratio (b/h)
Example 8.10
To study the effect of width-to-thickness ratio on the non-linear dynamic behaviour,
4-layer symmetric cross-ply (0/90/90/0) and angle-ply (45/-45/-45/45) laminates simply
supported on all edges are analysed and the displacement response for different width-to-
thickness ratios are given in Figs. 8.16 and 8.17 respectively. It is observed that the peak
amplitude increases manifold as the thickness of the plate increases.
6.0
4.0
* b/h = 10
2.0 ... b/h = 20
b/h = 100
0.0
-2.0
0 so 100 150 200 250 300
Time (µs)
Figure 8.16 Displacement response of 4-layer cross-ply laminates with b/h ratio
242
6.0
4.0
2.0
0.0
-2.0
0 50 100 150
Time (µs)
200
b/h = 10
b/h = 20
b/h = 100
250 300
Figure 8.17 Displacement response of 4-layer angle-ply laminates with b/h ratio
To have a comparison of non-linear behaviour of cross-ply and angle-ply .laminates, the
displacement responses of 4-layer symmetric cross-ply and angle-ply laminates with
different thicknesses (b/h = 10 and 100) are presented in Figs. 8.18 and 8.19. It is seen
that the amplitude of vibration is more and frequency is less for cross-ply laminates, both
for thin and thick plates. This fact is also evident from Figs. 8.16 and 8 .17.
243
..-..
6.0
5.0
4.0
3.0
2.0
1.0
0.0
-1.0
0
45/-45/-45/45
, - - - - - - - 0/90/90/0 :'
100 200
Time (µs)
300
Figure 8.18 Displacement response of 4-layer laminates (b/h = 10)
0.75
0.50
0/90/90/0
- - - - - - - 45/-45/-45/45
�
! 0.25:s
0.00
-0.25 +----....-----.-------.-----,------.
0 500 1000 1500 2000 2500
Time (µs)
Figure 8.19 Displacement response of 4-layer laminates (b/h = 100)
244
Effect of number of layers
Example 8.11
To study the effect of number of layers on the displacement response, 2-layer and 4-layer
anti-symmetric cross-ply and angle-ply laminates with b/h = 10 are considered. Only
2-layer and 4-layer laminates have been considered because the difference in amplitude
of vibration was found to be high for these two cases in linear dynamic analysis. The
displacement response is shown in Figs. 8.20 and 8.21 respectively. It is seen that the
effect of number of layers on the non-linear dynamic displacement is small. It is also
found that the peak displacements occur almost at the same time.
8.0
N=2
6.0
..-.. 4.0
2.0
0.0
-2.0
0 100 200 300 400
Time (µs)
Figure 8.20 Displacement response of cross-ply laminates with different number of layers
245
4.0
3.0
� 2.0 �
1.0
0.0
0 100 200
Time (µs)
300 400
Figure 8.21 Displacement response of angle-ply laminates with different number of layers
Effect of in-plane edge conditions
Example 8.12
From the non-linear static and free vibration analyses, it has been observed that the
in-plane edge conditions play an important role in the non-linear analysis. Hence, 4-layer
symmetric cross-ply laminates with movable (SS1) and immovable (SS2) in-plane edge
conditions are analysed and the displacement response is shown in Figs. 8.22 and 8.23 for
b/h = 10 and 100 respectively. It is seen that the amplitude of vibration is more and the
frequency of vibration is less in the case of plates with movable edge conditions. This is
true for both thin and thick plates. This is because the load is mostly carried by bending
action under SS 1 condition, whereas membrane action contributes considerably to the
load carrying capacity of the plate under SS2 condition. Membrane action tends to reduce
the transverse displacement.
246
6.0
4.0
� 2.0 �
0.0
0 100 200 300
Time (µs)
Figure 8.22 Displacement response of cross-ply laminates with different in-plane edge conditions (b/h= 10)
0.75 SS1
- - - - - - - SS2
0.50
0.25 �
0.00
�0.25
0 500 1000 1500 2000 2500
Time (µs)
Figure 8.23 Displacement response of cross-ply laminates with different in-plane edge conditions (b/h= 100)
247
Effect of fibre orientation
Example 8.13
Studies conducted so far indicate that the effect of non-linearity is significant in thin
plates. Hence, the effect of fibre orientation on dynamic displacement is studied by
analyzing a 4-layer thin (b/h = 100) symmetric laminate and the displacement history is
presented in Fig. 8.24. The behaviour is found to be the same as that in linear
analysis (Fig. 4.18).
0.75
0.50
:[ 0.25�
0.00
-0.250 500 1000
Time (µs)
----- a = 0°
,. a = 15°
* a = 30°
1500 2000
Figure 8.24 Displacement response of 4-layer laminates with different fibre orientation angle
8.3.2 Analysis of damped systems
The method of evaluation of damping matrix as explained in section 8.3 leads to a fully
populated matrix. But the stiffness and mass matrices are in banded form. To take
advantage of the banded nature of stiffness and mass matrices, the effect of damping is
considered in the form of mass-proportional damping, wherein the damping matrix is
248
evaluated as [C] = ao[M], a0 being the coefficient selected to obtain a specified value of
damping ratio in any one mode. In the present study, the fundamental mode is considered
to evaluate ao as a0 = 2ro1�1- The mass matrix being diagonal, the resulting damping
matrix will also be diagonal. The time-domain analysis is carried out using Wilson-9
method as in the case of undamped systems.
Example 8.14
To study the effect of damping on the non-linear transient response, 4-layer symmetric
cross-ply and angle-ply laminates of width-to-thickness ratios 10 and 100 are analysed.
Both symmetric and anti-symmetric arrangements are considered. The displacement
responses of these plates with a damping ratio of 5% are presented in Figs. 8.25 - 8.32. A
fine mesh of 16x16 for full plate and a reasonably small time step (5µs for b//h = 10 and
25 µs for b/h = 100) are used so as to obtain good converged results.
6.0
4.0
2.0
0.0
0 250 500 750
Time (µs)
Figure 8.25 Non-linear displacement response of a 4-layer symmetric cross-ply laminate (b/h = 10)
249
,' .
"'
g,
6.0
4.0
2.0
0.0
0 250 500 750
Time (µs)
Figure 8.26 Non-linear displacement response of a 4-layer anti-symmetric cross-ply laminate (b/h = 10)
0 250 500 750
Time (µs)
Figure 8.27 Non-linear displacement response of a 4-layer symmetric angle-ply laminate (b/h= 10)
250
e u
..._,
0 250 500 750
Time (µs)
Figure 8.28 Non-linear displacement response of a 4-layer anti-symmetric angle-ply laminate (b/h = 10)
0 1000 2000
it Time (µs) ·;;,!'; ··
3000 4000
:igure 8.29 Non-linear displacement response of a 4-layer symmetric cross-ply laminate �l;, (b/h = 100) -,� i:,
w·
251
0 1000 2000
Time (µs)
3000 4000
gure 8.30 Non-linear displacement response of a 4-layer anti-symmetric cross-ply laminate (b/h = 100)
0.3
1 2, 0.2
0.1
0 1000 2000
Time (µs)
3000 4000
Figure 8.31 Non-linear displacement response of a 4-layer symmetric angle-ply laminate (b/h = 100)
252
',:;
·,
· .
.
.
.
.
� S·l·.� 0.3
0.2
0.1
0.0
0 1000 2000
Time (µs)
3000 4000
Figure 8.32 Non-linear displacement response of a 4-layer anti-symmetric angle-ply laminate (b/h = 100)
8.4 DISCUSSION
1. The frequency ratio increases with increase in width-to-thickness ratio in the case of
cross-ply and angle-ply laminates, with higher values for cross-ply laminates. But, the
variation remains practically the same for symmetric and anti-symmetric arrangement
of cross-ply laminates. The frequency ratios are slightly less in the case of anti-
symmetric angle-ply laminates than symmetric angle-ply laminates.
2. Non-linear fundamental frequency of vibration decreases with mcrease in fibre
orientation angle for thin plates, the maximum decrease being for a. = 45°.
3. Number of layers does not have any significant influence on non-linear frequency of
vibration beyond 4-layers.
253
4. The percentage increase in the frequency ratio with increase in E1/E2 ratio is very
small.
5. The in-plane edge conditions play an important role in the non-linear frequency of
composite plates, particularly in cross-ply laminates.
6. Effect of non-linearity is seen significant for plates of width-to-thickness ratio greater
than 40.
7. The non-linear amplitude of vibration is small compared to linear amplitude with a
change in the periodicity of vibration.
8. The peak amplitude of vibration increases manifold as the thickness of the plate
increases.
9. In the case of both thin and thick cross-ply laminates, the amplitude of vibration is
more and the frequency is less compared to angle-ply laminates.
10. Effect of number of layers on the non-linear dynamic displacement is found to be
small.
11. The amplitude of vibration is more and the frequency of vibration is less in the case
of plates with movable edge conditions.
12. Plates with 45° fibre orientation have less amplitude of vibration with a change in the
periodicity for different values of orientation angle.
13. The simple finite element model chosen for the study is found to be sufficient for the
non-linear dynamic analysis of laminated composite plates. Results presented can
serve as bench mark solutions where such results are not available in open literature.
254