Dynamic analysis of laminated plates using ansys

1

CHAPTER 1

INTRODUCTION

1.1 GENERAL

In recent years, advanced composite materials are being increasingly used in

many engineering and civilian applications, ranging from fuselage of an aeroplane to

the frame of a tennis racket. The high stiffness-to-weight ratio, coupled with the

flexibility in the selection of lamination scheme that can be tailored to match the design

requirements, make the laminated plates attractive structural components for

automotive and aerospace vehicles. The increased use of the laminated plates in various

structures has created considerable interest in their analysis.

The high performance of these multilayered structures makes them ideal

candidates for use in future high-speed aircraft, spacecraft, satellite antennas and

terrestrial system reflectors.

Recent years have witnessed an increasing use of advanced composite materials

(e.g. graphite/epoxy, boron/epoxy, Kevlar/epoxy, graphite/PEEK etc.), which are

replacing metallic alloys in the fabrication of load bearing plate type structures because

of many beneficial properties, such as higher strength-to-weight ratios, longer fatigue

(including sonic fatigue) life, better stealth characteristics, enhanced corrosion

resistance and most significantly, the possibility of optimal design through the variation

of stacking pattern, fiber orientation and so forth known as composite tailoring.

A fibrous composite material generally has the fibers of glass, steel, graphite,

boron, carbon etc. that is generally bound together by embedding them using a matrix.

Few matrix materials being used are polyester, epoxy phenolics etc.

Fiber reinforced composite materials, for example contain high strength and

high modulus fibers in a matrix material. Reinforced steel bars embedded in concrete

provide an example of fiber-reinforced composites. In these composites, fibers are

principal load bearing members and matrix material keeps the fibers together, acts as a

load transfer medium between fibers and protects fibers from being exposed to the

environment.

2

1.2 LAMINATED COMPOSITE PLATES

Laminated composite plates and shells have been used in many engineering

applications in recent years because of their many beneficial properties. Composite

materials constitute a group of materials formed by putting together at least two

different materials. Fig 1.1 shows schematic representation of laminated composite

plate.

Fig 1.1 SCHEMATIC REPRESENTATION OF COMPOSITE PLATE

Composite materials are such that they inherit the superior qualities of the

combining materials. The properties which are impossible to be obtained from a single

material can be obtained from a composite due to its heterogeneous nature. All the

properties of the composites are the function of its constituent materials, their spatial

distribution and particle interaction between constituent materials.

There are two types of laminated composite plates. The first one is symmetric

laminated composite plates and the second one is anti-asymmetric laminated composite

plates.

Fig 1.2 and Fig 1.3 shows the schematic representation of symmetric and anti-

asymmetric types of laminated composite plates.

Matrix material

Fibers

3

Fig 1.2 SYMMETRIC LAMINATED COMPOSITE PLATES

Fig 1.3 ANTI SYMMETRIC LAMINATED COMPOSITE PLATES

4

1.3 NEED FOR PRESENT STUDY

Laminated composite structures have increasing applications in the aerospace,

marine, transportation, electrical and construction industries. In some of these

applications the composites are subjected to dynamic loads during their operation. The

plate and shell structures subjected to dynamic loading cause non-uniform stress field

which greatly affects the stability and dynamic behavior of structures. To avoid the

resonant behavior of the laminated composite structures, the results of the free vibration

analysis of the laminated composite structures in the structural design are very

important. The natural frequencies of the laminated composite plates have been

computed by finite element (FE) analysis software ANSYS .

1.4 ANSYS AND ITS APPLICATION

In modern world design process has been too close to precision so the use of

finite element method is extensive. It is being used as the most trustworthy tool for

designing. It helps to predict the behaviour of various products, parts, subassemblies

and assemblies. Analysing the results helps to prevent the time of prototyping and

reduces the expense due to physical test. It also increases the innovation at a faster and

more accurate way. Analysts and designers work together to find the most appropriate

answer using the most optimized tool. ANSYS is now being used in a number of

different engineering fields such as power generation, transportation, medical

components, electronic devices, and household appliances.

The first ANSYS seminar was held in 1976. The designing was improved from

2D modelling to 3D modelling. Beam models to shell and then to volumeelements were

used for modelling. Graphics were introduced for better modelling and analysis. The

substructure technique was introduced to divide the structure and analyse it element

wise. The first task was to discretize the structure into nodes and elements. ANSYS

gradually entered to a number of fields making it handy for fatigue analysis, nuclear

power plant, medical applications, to find the eigenvalues of magnet, etc. Thermal

analysis of various structures based on the thermal and mechanical loading was also

done.

For present work the analysis is done by choosing shell element from ANSYS

library. In the present work an element SHELL281 is used for the thick and thin

laminate plates.

5

1.5 SHELL281

SHELL281 is suitable for analyzing thin to moderately-thick shell structures.

The element has eight nodes with six degrees of freedom at each node: translations in

the x, y, and z axes, and rotations about the x, y, and z-axes. (When using the membrane

option, the element has translational degrees of freedom only.)

SHELL281 is well-suited for linear, large rotation, and/or large strain nonlinear

applications. Change in shell thickness is accounted for in nonlinear analyses. The

element accounts for follower (load stiffness) effects of distributed pressures.

SHELL281 may be used for layered applications for modeling composite shells

or sandwich construction. The accuracy in modeling composite shells is governed by

the first-order shear-deformation theory (usually referred to as Mindlin-Reissner shell

theory).

The element formulation is based on logarithmic strain and true stress measures.

The element kinematics allow for finite membrane strains (stretching). However, the

curvature changes within a time increment are assumed to be small.

Fig. 1.4 shows the geometry, node locations, and the element coordinate system

for this element. The element is defined by shell section information and by eight nodes

(I, J, K, L, M, N, O and P).

A triangular-shaped element may be formed by defining the same node number

for nodes K, L and O.

Fig. 1.4 SHELL281 Geometry

6

CHAPTER 2

REVIEW OF LITERATURE

2.1 GENERAL

The vibration and stability studies of composite plates are an active and

advanced field of research, because of their superior properties such as high strength,

light weight and many other attractive dynamic characteristics such as Damping and

High Stiffness. But the reliability of the materials depends on the proper assessment of

the various static and dynamic properties of the composite and their behaviour under

different loading and atmospheric conditions.

2.2 LITERATURE CONCERNING THEORETICAL ANALYSIS

OF LAMINATED COMPOSITES

Many researchers have given their contributions in this field which have been

discussed as follows:

Akbarov et al. (2010) studied the forced vibration on an initially statically

stressed rectangular orthotropic plate. Plate was simply supported on all sides. They

were studied the effect of presence of rectangular hole at edges on dynamic analysis of

laminated plates was present. They used three dimensional finite element methods for

dynamic analysis.

Ahmed et al. (2013) studied the dynamic analysis of Graphite /Epoxy

composite plates. The dynamic analysis had been done by using ANSYS 12.0 package.

The composite laminated plates were modelled by using the element SHELL99. The

boundary conditions considered in dynamic study were simply supported and clamped

boundary conditions. They concluded that the natural frequency for composite

laminated plate in clamped boundary condition was more than in simply supported

boundary condition.

Houmat (2013) studied geometrically nonlinear free vibration of laminated

composite rectangular plates with curvilinear fibers. They used finite element method

to solve the nonlinear free vibration of laminated composite rectangular plates. They

found that the fundamental linear and nonlinear frequencies and associated mode

shapes for fully clamped laminated composite square plates composed of shifted

7

curvilinear fibers. They concluded that fiber orientation angles and stacking sequence

of plies lead to changes in the fundamental mode shapes and this method can also be

applied to laminates with other shapes and other boundary conditions.

Ratnaparkhi and Sarnobat (2013) studied the free vibration of woven fiber

Glass/Epoxy composite plates in free-free boundary conditions. The specimens of

woven glass fiber and epoxy matrix composite plates were manufactured by the hand-

layup technique and elastic parameters of the plate were determined experimentally by

tensile test of specimens. An experimental investigation was carried out using modal

analysis technique, to obtain the natural frequencies. Also, this experiment was used to

validate the results obtained from the FEA using ANSYS. The effects of different

parameters including aspect ratio and fiber orientation of woven fiber composite plates

were studied in free-free boundary conditions. To model the composite plate, linear

layer shell 99 element was used. They concluded that for free-free boundary condition,

the natural frequency of plate increases with the increase of aspect ratio and natural

frequency decreases as the ply orientation increases.

8

CHAPTER 3

RESULTS AND DISCUSSION

3.1 GENERAL

Composites are being increasingly used in aerospace, marine and civil

infrastructure owing to the many advantages they offer: high strength/stiffness for lower

weight, superior fatigue response characteristics, facility to vary fiber orientation,

material and stacking pattern, resistance to electrochemical corrosion, and other

superior material properties. To avoid the resonant behaviour of the laminated

structures, the results of the free vibration analysis for the laminated composite

structures in the structural design are very important. . Also, the composite structures

whether used in civil, marine or aerospace are subjected to dynamic loads during their

operation. Therefore, there exists a need for calculating the natural frequency.

3.2 OBJECTIVE OF THE STUDY

In the present study, a four layered (0/90/90/0) symmetrical laminated

composite plate with equal thickness of layers, simply supported on the opposite sides

and clamped on the other two opposite sides has been dynamically analysed.

The material properties of graphite/epoxy composite material Ahmed et al. [1] are given

below.

E1= 175 GPa, E2 = E3 =7 GPa, G12 = G13 = 3.5 GPa, G23 = 1.4 GPa, 12 = 13 = 0.25,

23 = 0.01,

Density () = 1550 kg/m3,

Thickness of four layered laminated composite plate (h) = 0.008 m

9

Fig. 3(a) The plan of laminated plate (clamped and simply supported on opposite sides)

In the present work, the main objectives of this study are the following and following

studies have been carried out:

(i) The effect of plate side- to- thickness ratios (b/h) = 50, 100, 200, 500 and

1000 has been studied on natural frequency () of laminated plate for

modulus ratios (E1/E2) = 2, 4, 6, 8 and 10 for 1st mode, 2nd mode, 3rd mode,

4th mode and 5th mode respectively taking plate aspect ratios (b/a) = 1 to 3

in steps of 1.

(ii) The effect of change in the layer thickness (t)of only one layer at a time

has been studied on change in natural frequency () of laminated

composite plate for first mode and plate aspect ratios (b/a) = 1 to 3 in steps

of 1.

(iii) The effect of change in the fiber angles ()of only one layer at a time has

been studied on change in natural frequency () of laminated composite

plate for first mode and plate aspect ratios (b/a) = 1 to 3 in steps of 1.

10

The natural frequencies () are presented in non-dimensional frequencies () form

using the equation given below

= (

) ( /)

Where , b, h, 2 were the density, width, thickness and youngs modulus in transverse

direction of the laminated composite plate respectively.

In the present work, the following case have been studied.

3.3 Four layered (0/90/90/0) Simply Supported and Clamped on two

opposite sides.

3.3.1 The effect of plate side- to- thickness ratios (b/h) = 50, 100, 200, 500 and 1000

on non-dimensional frequencies () of laminated plate for modulus ratios (E1/E2)

= 2, 4, 6, 8 and 10 for 1st mode, 2nd mode, 3rd mode, 4th mode and 5th mode

respectively taking plate aspect ratios (b/a) = 1to 3 in steps of 1.

3.3.1.1 Variation of non-dimensional frequency () versus plate side-to-thickness

ratio (b/h) for first mode and b/a = 1 to 3 in steps of 1.

The variation of non-dimensionalized natural frequency versus plate side-to-

thickness ratio (b/h) = 50, 100, 200, 500 and 1000 having modulus ratios (E1/ E2) = 2,

4, 6, 8 and 10 for first mode and b/a = 1 to 3 in steps of 1 is shown graphically in Figs.

3.1 to 3.3.

The values of non-dimensional frequencies () with respect to plate side-to-

thickness ratio (b/h) = 50, 100, 200, 500 and 1000 are 1.69, 1.706, 1.709,1.7098 and

1.7099 for E1/ E2 = 2 ; are 2.162, 2.186, 2.193, 2.194 and 2.195 for E1/ E2 = 4 ; are

2.537, 2.577, 2.588, 2.5911 and 2.5915 for E1/ E2 = 6 ; are 2.855, 2.914, 2.930, 2.934

and 2.935 for E1/ E2 = 8 ; are 3.134, 3.214, 3.235, 3.241 and 3.242 for E1/ E2 = 10

respectively as given in Table 3.1 for b/a = 1.

The values of natural frequency () with respect to plate side-to-thickness ratio

(b/h) = 50, 100, 200, 500 and 1000 are 5.770, 5.911, 5.948, 5.959 and 5.960 for E1/ E2

= 2 ; are 7.618, 7.949, 8.038, 8.064 and 8.068 for E1/ E2 = 4 ; are 8.971, 9.524, 9.679,

9.724 and 9.730 for E1/ E2 = 6 ; are 10.0465, 10.841, 11.071, 11.1382 and 11.148 for

11

E1/ E2 = 8 ; are 10.935, 11.986, 12.299, 12.391 and 12.404 for E1/ E2 = 10 respectively

as given in Table 3.2 for b/a = 2.


(b/h) = 50, 100, 200, 500 and 1000 are 12.222, 12.864, 13.042, 13.166 and 13.173 for

E1/ E2 = 2 ; are 15.885, 17.341, 17.772, 17.899 and 17.917 for E1/ E2 = 4 ; are 18.355,

20.699, 21.438, 21.659 and 21.691 for E1/ E2 = 6 ; are 20.188, 23.434, 24.518, 24.850

and 24.899 for E1/ E2 = 8 ; are 21.618, 25.756, 27.2158, 27.669 and 27.736 for E1/ E2

= 10 respectively as given in Table 3.3 for b/a = 3.

It is observed that (i) the natural frequency increases slightly as b/h increases

from 50 to 100 (ii) there is negligible variation in the natural frequency for b/h >100

(iii) the natural frequency increases with the increase of the modulus ratios (E1/ E2). (iv)

the natural frequency increases as b/a increases.


ratio (b/h) for second mode and b/a = 1 to 3 in steps of 1.


thickness ratio (b/h) = 50, 100, 200, 500 and 1000 for different modulus ratios (E1/ E2)

= 2, 4, 6, 8 and 10 is shown graphically in Figs. 4.4 to 4.6.

The values of natural frequency () with respect to plate side-to-

thickness ratio (b/h) = 50, 100, 200, 500 and 1000 are 2.869,2.888,2.893,2.894,2.8948

for E1/ E2 = 2 ; are 3.2772,3.3088,3.3169,3.3192,3.3196 for E1/ E2 = 4 ; are

3.6360,3.6829,3.6950,3.6985 and 3.6989 for E1/ E2 = 6 ; are

3.9571,4.0212,4.0380,4.0427 and 4.0434 for E1/ E2 = 8 ; are

4.290,4.3321,4.3538,4.3601and 4.3611 for E1/ E2 = 10 respectively [Refer Table 3.4]

for b/a = 1.


(b/h) = 50, 100, 200, 500 and 1000 are 6.6153,6.7813,6.8250,6.8374and 6.8393 for E1/

E2 = 2 ; are 8.2954,8.6504,8.7473,8.7751and 8.7791 for E1/ E2 = 4 ; are

9.5731,10.1487,10.3111,10.3580 and 10.3647 for E1/ E2 = 6 ; are

10.6053,11.4223,11.6591,11.7282 and 11.7384 for E1/ E2 = 8 ; are

12

11.4723,12.5391,12.8593,12.8595 and 12.9671 for E1/ E2 = 10 respectively [Refer

Table 3.5] for b/a = 2.


(b/h) = 50, 100, 200, 500 and 1000 are 12.9263,13.6195,13.8124,14.1686 and 14.1767

for E1/ E2 = 2 ; are 16.4047,17.9118,18.3602,18.4912and 18.5104 for E1/ E2 = 4 ; are

18.7926,21.1839,21.9409,22.1672and 22.2007 for E1/ E2 = 6 ; are

20.5786,24.8818,24.9721,25.3096and 28.1637 for E1/ E2 = 8 ; are21.9818

,26.1576,27.6359,28.096 and 28.1637 for E1/ E2 = 10 respectively [Refer Table 3.6] for

b/a = 3.






ratio (b/h) for third mode and b/a = 1 to 3 in steps of 1.





thickness ratio (b/h) = 50, 100, 200, 500 and 1000 are 4.1862, 4.2444, 4.2594 4.2636and

4.2644 for E1/ E2 = 2 ; are 5.497105,5.6320,5.6675,5.6775and 5.5.6791 for E1/ E2 = 4 ;

are 6.1246, 6.1989, 6.2183, 6.2237 and 6.2245 for E1/ E2 = 6 ; are 6.5624, 6.6563,

6.6807, 6.6877and 6.6887 for E1/ E2 = 8 ; are 6.9685, 7.0832, 7.1135, 7.1221and 7.1223

for E1/ E2 = 10 respectively as noted from Table 3.7 for b/a = 1.


(b/h) = 50, 100, 200, 500 and 1000 are 8.3738, 8.5815, 8.6369, 8.6526 and 8.6549 for

E1/ E2 = 2 ; are 9.8789, 10.2703, 10.3778, 10.4088 and 10.4129 for E1/ E2 = 4 ; are

11.0892, 11.6925, 11.8635, 11.9136and 11.92028 for E1/ E2 = 6 ; are 112.1009,

12.9329, 13.1767, 13.2482 and 13.2585 for E1/ E2 = 8 ; are 12.9668, 14.0404, 14.3638,

14.4594 and 14.4773 for E1/ E2 = 10 respectively as noted from Table 3.8 for b/a = 2.

13


(b/h) = 50, 100, 200, 500 and 1000 are 14.3257, 15.0982, 15.3154, 16.0664 and 16.0736

for E1/ E2 = 2 ; are 17.5406, 19.1176, 19.59044, 19.72878 and 19.7493 for E1/ E2 = 4 ;

are 19.8181, 22.2651, 25.0462, 23.2805 and 23.3142 for E1/ E2 = 6 ;are 22.1794,

25.8325, 26.0066, 26.3514 and 26.4019 for E1/ E2 = 8 ; are 22.9333, 27.1307, 28.6267,

29.09485 and 29.1635 for E1/ E2 = 10 respectively as noted from Table 3.9 for b/a = 3.






ratio (b/h) for fourth mode and b/a = 1 to 3 in steps of 1.





thickness ratio (b/h) = 50, 100, 200, 500 and 1000 are 5.1337, 5.1750, 5.1858, 5.1887

and 5.1893 for E1/ E2 = 2 ; are 5.6488, 5.7054, 5.7202, 5.7242 and 5.7250 for E1/ E2 =

4 ; are 6.4983, 6.7264, 6.7872, 6.8046 and 68069 for E1/ E2 = 6 ; are 7.3210, 7.6532,

7.7437, 7.7696 and 7.7733 for E1/ E2 = 8 ; are 8.0230, 8.4682, 8.5914, 8.6270 and

8.6319 for E1/ E2 = 10 respectively as given in Table 3.10 for b/a = 1.75


(b/h) = 50, 100, 200, 500 and 1000 are 11.1993, 11.4790, 11.5544, 11.5758 and 11.5792


are 13.8881, 14.5445, 14.7318, 14.7860 and 14.7940 for E1/ E2 = 6 ; are 14.9582,

15.8289, 16.0849, 16.1609 and 16.1715 for E1/ E2 = 8 ; are 15.9021, 16.9963, 17.3287,

17.4267 and 17.4409 for E1/ E2 = 10 respectively as given in Table 3.11 for b/a = 2.


(b/h) = 50, 100, 200, 500 and 1000 are 16.6259, 17.5086, 17.7586, 19.0257 and 19.0374


are 21.7663, 24.2678, 25.0730, 25.3171 and 25.3523 for E1/ E2 = 6 ; are 23.4810,

14

26.8291, 27.9732, 28.3240 and 28.3756 for E1/ E2 = 8 ; are 24.8776, 29.0581, 30.5646,

31.0363 and 31.1060 for E1/ E2 = 10 respectively as given in Table 3.12 for b/a = 3.






ratio (b/h) for fifth mode and b/a = 1 to 3 in steps of 1.





thickness ratio (b/h) = 50, 100, 200, 500 and 1000 are 5.2748, 5.3512, 5.3711, 5.3767

and 5.3777 for E1/ E2 = 2 ; are 6.4059, 6.5592, 6.5994, 6.6109 and 6.6112 for E1/ E2 =

4 ; are 7.3209, 7.5662, 7.6319, 7.65077 and 7.6535 for E1/ E2 = 6 ; are 8.094, 8.4430,

8.5380, 8.5652 and 8.5691 for E1/ E2 = 8 ; are 8.7676, 9.2264, 9.3540, 9.3909 and

9.3961 for E1/ E2 = 10 respectively [Refer Table 3.13] for b/a = 1.


(b/h) = 50, 100, 200, 500 and 1000 are 15.0076, 15.4803, 15.5895, 15.6217 and 15.6260


are 18.04605, 18.8318, 19.0558, 19.1217 and 19.1311 for E1/ E2 = 6 ; are 19.2845,

20.2849, 20.5782, 20.6642 and 20.6768 for E1/ E2 = 8 ; are20.3998, 21.6232, 21.9921,

22.1004 and 22.1164 for E1/ E2 = 10 respectively[Refer Table 3.14] for b/a = 2.


(b/h) = 50, 100, 200, 500 and 1000 are 19.9414, 20.9821, 21.2814, 21.1318 and 23.1463

for E1/ E2 = 2 ; are 22.7319, 24.5162, 25.0625, 25.2240 and 25.2477 for E1/ E2 = 4 ; are

24.9021, 27.4819, 28.3210, 28.5753 and 28.6120 for E1/ E2 = 6 ; are 26.6658, 30.0459,

31.2106, 31.5700 and 31.6225 for E1/ E2 = 8 ; are 28.1425, 32.31029, 33.8232, 34.2992

and 34.3698 for E1/ E2 = 10 respectively[Refer Table 3.15] for b/a = 3.

15





3.3.2 The effect of change in the layer thickness (t) of only one layer at a time on

change in natural frequency () of laminated composite plate for first mode and

b/a = 1 to 3 in steps of 1.

The change in the natural frequency of laminated composite plate has been

studied by changing the thickness of only one layer at a time from 0.002 m to 0.0002

m in steps of 0.0002 m.

The variation of change in natural frequency versus change in the thickness of

only one layer at a time has been presented in Figs. 3.16 to 3.18 for first mode and b/a

= 1 to 3 in steps of 1.

The values of change in natural frequency are 0.97, 2.025, 3.177, 4.458, 5.91,

7.593, 9.601, 12.091 and 15.371 for change in thickness of only first or fourth layer;

are 0.69, 1.388, 2.094, 2.808, 3.53, 4.261, 5.001, 5.75 and 6.51 for change in thickness

of only second or third layer from 0.002 to 0.0002 in steps of 0.0002 as given in Table

3.16 for first mode and b/a = 1.



are 2.42, 4.89, 7.39, 9.94, 12.53, 15.17, 17.86, 20.6 and 23.39 for change in thickness

of only second or third layer from 0.002 to 0.0002 in steps of 0.0002 as given in Table

3.17 for first mode and b/a = 2.



are 4.98, 10.06, 15.24, 20.53, 25.93, 31.46, 37.1, 42.87 and 48.76 for change in

thickness of only second or third layer from 0.002 to 0.0002 in steps of 0.0002 as given

in Table 3.18 for first mode and b/a = 3.

16

It is observed that (i) the change in natural frequency increases parabolicaly for first or

fourth layer; the change in natural frequency increases linearly for second or third layer

as change in thickness of only one layer at a time increases. (ii) the change in natural

frequency increases as b/a increases.

3.3.3 The effect of change in the fiber angles () of only one layer at a time on


plate aspect ratios (b/a) = 1 to 3 in steps of 1.

The change in the natural frequency of laminated composite plate has

been studied by changing the fiber angles of only one layer at a time from 0 to 50 for

first or fourth layer and 90 to 140 for second or third layer in steps of 5.

The variation of change in natural frequency versus change in the fiber angles

of only one layer at a time has been shown graphically in Figs. 3.19 to 3.21 for first

mode and b/a = 1 to 3 in steps of 1.

The values of change in natural frequency are0, 0.209, 0.779, 1.598, 2.617,

3.825, 5.177, 6.647, 8.114, 9.374 and 10.626 for change in fiber angles of only first or

fourth layer; are 0, 0.003, 0.01, 0.023, 0.042, 0.068, 0.099, 0.134, 0.172, 0.213 and

0.256 for change in fiber angles of only second or third layer from 0 to 50 and 90 to

140 respectively in steps of 5 as noted from Table 3.19 for first mode and b/a = 1.

The values of change in natural frequency are 0, 1.76, 6.13, 11.61, 17.82, 24.89

and 32.518for change in fiber angles of only first or fourth layer; are 0, 0.01, 0.04, 0.2,

0.32, 0.45, 0.61, 0.77, 0.96 and 1.16for change in fiber angles of only second or third

layer from 0 to 50 and 90 to 140 respectively in steps of 5 as noted from Table 3.20

for first mode and b/a = 2.

The values of change in natural frequency are 0, 5.63, 18.44, 32.3, 46.52, 62.66,

80.3, 99.41, 116.17, 125.85 and 135.9 for change in fiber angles of only first or fourth

layer; are 0, 0.03, 0.13, 0.32, 0.59, 0.93, 1.3, 1.7, 2.1, 2.52 and 2.96 for change in fiber

angles of only second or third layer from 0 to 50 and 90 to 140 respectively in steps

of 5 as noted from Table 3.21 for first mode and b/a = 3.

17

It is observed that (i) the rate of change in natural frequency for change in fiber

angles of only first or fourth layer is greater than the rate of change in natural frequency

for change in fiber angles of only second or third layer of the laminated composite plate

(ii) the change in natural frequency of the laminated composite plate increases as b/a

increases.

18

3.3 Four layered (0/90/90/0) Simply Supported on

the opposite sides and Clamped on the other two

opposite sides.

Fig. 3.1 VARIATION OF NON-DIMENSIONAL FREQUENCY () VERSUS

PLATE SIDE TO THICKNESS RATIO (b/h) FOR 1ST MODE AND b/a = 1



1

1.5

2

2.5

3

3.5

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)

Plate Side- to- thickness ratio(b/h)

E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

4

5

6

7

8

9

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

19




PLATE SIDE TO THICKNESS RATIO (b/h) FOR 2ND MODE AND b/a = 1

10

15

20

25

30

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)

Side to thickness ratio(b/h)

E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

20





5

6

7

8

9

10

11

12

13

14

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)

Plate side- to- thickness ratio (b/h)

E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

10

15

20

25

30

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

21


PLATE SIDE TO THICKNESS RATIO (b/h) FOR 3RD MODE AND b/a = 1



4

5

6

7

8

9

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

7

8

9

10

11

12

13

14

15

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

22




PLATE SIDE TO THICKNESS RATIO (b/h) FOR 4TH MODE AND b/a = 1

12

15

18

21

24

27

30

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

4

5

6

7

8

9

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

23





10

12

14

16

18

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

15

18

21

24

27

30

33

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

24





4

5

6

7

8

9

10

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

12

14

16

18

20

22

24

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

25



Fig. 3.16 CHANGE OF NATURAL FREQUENCY () VERSUS CHANGE IN

THE LAYER THICKNESS (t) FOR 1ST MODE AND b/a=1

18

21

24

27

30

33

36

0 200 400 600 800 1000 1200

No

nd

ime

nsi

on

al N

atu

ral f

req

ue

ncy

(

)


E1/E2=2

E1/E2=4

E1/E2=6

E1/E2=8

E1/E2=10

0

2

4

6

8

10

12

14

16

18

0 0.0005 0.001 0.0015 0.002

CH

AN

GE

OF

FREQ

UEN

CY

(

)

CHANGE OF THE LAYER THICKNESS t (m)

1st or 4th layer 2nd or 3rd layer

26





0

10

20

30

40

50

60

70

0 0.0005 0.001 0.0015 0.002

CH

AN

GE

OF

FREQ

UEN

CY

(

)



0

20

40

60

80

100

120

140

160

0 0.0005 0.001 0.0015 0.002

CH

AN

GE

OF

FREQ

UEN

CY

(

)



27

Fig. 3.19 CHANGE OF NATURAL FREQUENCY () VERSUS CHANGE OF

FIBER ANGLES () IN DIFFERENT LAYERS FOR 1ST MODE AND b/a=1



0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60

CH

AN

GE

OF

FREQ

UEN

CY

(

)

CHANGE OF FIBER ANGLES ()


0

5

10

15

20

25

30

35

0 10 20 30 40 50 60

CH

AN

GE

OF

FREQ

UEN

CY

(

)



28



0

10

20

30

40

50

60

0 10 20 30 40 50 60

CH

AN

GE

OF

FREQ

UEN

CY

(

)



29

3.3 Four layered (0/90/90/0) simply supported on

the opposite sides and clamped on opposite sides.

TABLE 3.1 NON-DIMENSIONAL NATURAL FREQUENCY () VERSUS


b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 1.695341 2.162609 2.537176 2.855558 3.134978

100 1.706258 2.186833 2.577776 2.914774 3.214842

200 1.709081 2.193045 2.588166 2.930284 3.23581

500 1.709834 2.194795 2.591197 2.934613 3.241796

1000 1.709947 2.19504 2.591592 2.935215 3.242661



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 5.770219 7.618587 8.971829 10.0465 10.93586

100 5.911388 7.949109 9.524927 10.84175 11.98653

200 5.948506 8.038704 9.679573 11.07168 12.29951

500 5.959103 8.064303 9.724634 11.1382 12.39178

1000 5.960703 8.068067 9.730845 11.14818 12.40477



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 12.22238 15.88523 18.35568 20.18805 21.61856

100 12.86479 17.34153 20.69908 23.43436 25.75668

200 13.04217 17.77294 21.43805 24.51891 27.2158

500 13.16632 17.89924 21.65903 24.85038 27.66999

1000 13.17348 17.91712 21.6914 24.89913 27.7368

30


PLATE SIDE TO THICKNESS RATIO (b/h) FOR 2ND MODE and b/a = 1

b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 2.869769 3.277276 3.636032 3.957144 4.249081

100 2.888535 3.308879 3.682919 4.021234 4.332182

200 2.893391 3.316973 3.695078 4.038099 4.353865

500 2.894709 3.31925 3.698523 4.042786 4.360133

1000 2.894897 3.319608 3.698994 4.043445 4.361168



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 6.615348 8.295443 9.573113 10.60553 11.4723

100 6.781363 8.650435 10.1487 11.42223 12.53991

200 6.825031 8.747333 10.3111 11.6591 12.85937

500 6.837454 8.775134 10.35801 11.72829 12.95363

1000 6.839336 8.779181 10.36479 11.73845 12.96718



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 12.92634 16.40473 18.79236 20.57862 21.98183

100 13.61957 17.91185 21.18395 24.88181 26.1576

200 13.81224 18.3602 21.94099 24.97216 27.63592

500 14.16862 18.4912 22.16723 25.30965 28.09631

1000 14.17671 18.5104 22.20074 25.35877 28.1637

31



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 4.186213 5.497105 6.124646 6.562457 6.968552

100 4.244469 5.632062 6.198995 6.656381 7.083275

200 4.259452 5.667524 6.218345 6.680775 7.113542

500 4.263668 5.677518 6.223747 6.687721 7.122144

1000 4.264421 5.679118 6.224594 6.688756 7.123179



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 8.373839 9.878978 11.08926 12.10097 12.96869

100 8.581545 10.2703 11.69252 12.93293 14.04044

200 8.636958 10.37781 11.86358 13.17679 14.36381

500 8.652694 10.40883 11.91369 13.2482 14.45943

1000 8.654952 10.41297 11.92028 13.25855 14.47336



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 14.32579 17.54067 19.81819 22.17947 22.93331

100 15.09826 19.11762 22.26511 25.83235 27.13072

200 15.31547 19.59044 23.04624 26.00664 28.62673

500 16.06404 19.72878 23.28058 26.35147 29.09485

1000 16.07364 19.7493 23.31428 26.40191 29.16355

32



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 5.133737 5.648814 6.498367 7.321003 8.023082

100 5.175052 5.70547 6.726401 7.65322 8.468233

200 5.185819 5.720227 6.787235 7.743718 8.591483

500 5.188793 5.724292 6.804608 7.769637 8.627001

1000 5.189357 5.725045 6.806961 7.773307 8.631989



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 11.19937 12.65341 13.88817 14.95822 15.90217

100 11.47908 13.10948 14.54451 15.82895 16.99632

200 11.55444 13.23567 14.73183 16.08494 17.32873

500 11.57582 13.27173 14.786 16.16098 17.42679

1000 11.57921 13.27738 14.79409 16.17152 17.44091



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 16.6259 19.58008 21.76631 23.48104 24.87767

100 17.50867 21.2423 24.26782 26.82919 29.05814

200 17.75939 21.74523 25.07305 27.97322 30.56469

500 19.02576 21.89431 25.31718 28.32407 31.03639

1000 19.03743 21.91614 25.35237 28.37564 31.10603

33



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 5.274812 6.405948 7.320909 8.094795 8.767605

100 5.351231 6.559257 7.56626 8.443011 9.226403

200 5.371183 6.599462 7.631988 8.538027 9.354019

500 5.376735 6.610925 7.650773 8.565263 9.390911

1000 5.377582 6.612713 7.653596 8.569122 9.396181



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 15.00716 16.65319 18.04605 19.28457 20.3998

100 15.48036 17.23763 18.83189 20.28499 21.62326

200 15.58953 17.3995 19.05588 20.57824 21.99218

500 15.62172 17.44656 19.12176 20.66426 22.10041

1000 15.62605 17.45371 19.13117 20.67687 22.11641



b/h ( E1/E2 = 2 ) ( E1/E2 = 4 ) ( E1/E2 = 6 ) ( E1/E2 = 8 ) ( E1/E2 = 10 )

50 19.94148 22.73191 24.90214 26.66581 28.14525

100 20.98217 24.51628 27.48195 30.04595 32.31029

200 21.28145 25.06251 28.32105 31.21068 33.82324

500 23.13189 25.224 28.57535 31.57 34.29926

1000 23.14638 25.24772 28.61205 31.62252 34.36985

34

TABLE 3.16 CHANGE OF NATURAL FREQUENCY () DUE TO CHANGE

.IN THE LAYER THICKNESS (t) FOR 1ST MODE AND b/a = 1

Change of the

layer thickness

t (m)

Natural

frequency ()

due to change in

thickness of 1stor

4th layer

Natural

frequency ()

due to change in

thickness of 2nd

or 3rd layer

Change of

frequency ()

due to change in

thickness of 1stor

4th layer

Change of

frequency ()

due to change in

thickness of 2nd

or 3rd layer

0 32.876 32.876 0 0

0.0002 31.903 32.186 0.973 0.69

0.0004 30.851 31.488 2.025 1.388

0.0006 29.699 30.782 3.177 2.094

0.0008 28.418 30.068 4.458 2.808

0.001 26.966 29.346 5.91 3.53

0.0012 25.283 28.615 7.593 4.261

0.0014 23.275 27.875 9.601 5.001

0.0016 20.785 27.126 12.091 5.75

0.0018 17.505 26.366 15.371 6.51


IN THE LAYER THICKNESS (t) FOR 1ST MODE AND b/a = 2

Change of the

layer thickness

t (m)

Natural

frequency ()

due to change in

thickness of 1stor

4th layer

Natural

frequency ()

due to change in

thickness of 2nd

or 3rd layer

Change of

frequency ()

due to change in

thickness of 1stor

4th layer

Change of

frequency ()

due to change in

thickness of 2nd

or 3rd layer

0 126.09 126.09 0 0

0.0002 122.3 123.67 3.79 2.42

0.0004 118.16 121.2 7.93 4.89

0.0006 113.58 118.7 12.51 7.39

0.0008 108.43 116.15 17.66 9.94

0.001 102.51 113.56 23.58 12.53

0.0012 95.552 110.92 30.538 15.17

0.0014 87.105 108.23 38.985 17.86

0.0016 76.378 105.49 49.712 20.6

0.0018 61.7 102.7 64.39 23.39

35


IN THE LAYER THICKNESS (t) FOR 1ST MODE AND b/a = 3

Change of the

layer thickness

t (m)

Natural

frequency ()

due to change in

thickness of 1stor

4th layer

Natural

frequency ()

due to change in

thickness of 2nd

or 3rd layer

Change of

frequency ()

due to change in

thickness of 1stor

4th layer

Change of

frequency ()

due to change in

thickness of 2nd

or 3rd layer

0 275.75 275.75 0 0

0.0002 267.81 270.77 7.94 4.98

0.0004 259.1 265.69 16.65 10.06

0.0006 249.43 260.51 26.32 15.24

0.0008 238.48 255.22 37.27 20.53

0.001 225.85 249.82 49.9 25.93

0.0012 210.88 244.29 64.87 31.46

0.0014 192.57 238.65 83.18 37.1

0.0016 169.07 232.88 106.68 42.87

0.0018 136.49 226.99 139.26 48.76


IN THE FIBER ANGLES () IN DIFFERENT LAYERS FOR 1ST MODE

AND b/a=1

Change of fiber

angles ()

Natural

frequency ()

due to change in

fiber angles of

1stor 4th layer

Natural

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

Change of

frequency ()

due to change in

fiber angles of

1stor 4th layer

Change of

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

0 32.876 32.876 0 0

5 32.667 32.879 0.209 0.003

10 32.097 32.886 0.779 0.01

15 31.278 32.899 1.598 0.023

20 30.259 32.918 2.617 0.042

25 29.051 32.944 3.825 0.068

30 27.699 32.975 5.177 0.099

35 26.229 33.01 6.647 0.134

40 24.762 33.048 8.114 0.172

45 23.502 33.089 9.374 0.213

50 22.25 33.132 10.626 0.256

36



AND b/a=2

Change of fiber

angles ()

Natural

frequency ()

due to change in

fiber angles of

1stor 4th layer

Natural

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

Change of

frequency ()

due to change in

fiber angles of

1stor 4th layer

Change of

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

0 126.09 126.09 0 0

5 124.33 126.1 1.76 0.01

10 119.96 126.13 6.13 0.04

15 114.48 126.2 11.61 0.11

20 108.27 126.29 17.82 0.2

25 101.2 126.41 24.89 0.32

30 93.572 126.54 32.518 0.45

35 85.502 126.7 40.588 0.61

40 78.079 126.86 48.011 0.77

45 72.76 127.05 53.33 0.96

50 67.556 127.25 58.534 1.16



AND b/a=3

Change of fiber

angles ()

Natural

frequency ()

due to change in

fiber angles of

1stor 4th layer

Natural

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

Change of

frequency ()

due to change in

fiber angles of

1stor 4th layer

Change of

frequency ()

due to change in

fiber angles of

2nd or 3rd layer

0 275.75 275.75 0 0

5 270.12 275.78 5.63 0.03

10 257.31 275.88 18.44 0.13

15 243.45 276.07 32.3 0.32

20 229.23 276.34 46.52 0.59

25 213.09 276.68 62.66 0.93

30 195.45 277.05 80.3 1.3

35 176.34 277.45 99.41 1.7

40 159.58 277.85 116.17 2.1

45 149.9 278.27 125.85 2.52

50 139.85 278.71 135.9 2.96

37

CHAPTER 4

CONCLUSIONS AND SCOPE FOR FURTHER STUDIES

4.1 GENERAL

In the present study, a four layered (0/90/90/0) symmetrical laminated composite

plate with equal thickness of layer, simply supported on the opposite sides and clamped

on the other two opposite sides have been dynamically analyzed by using 8-noded

element (Shell 281) having six degree of freedom at each node through software Ansys.

The following studies have been carried out:

(i) The effect of plate side- to- thickness ratios (b/h) = 50, 100, 200, 500 and 1000 has

been studied on natural frequency () of laminated plate for modulus ratios (E1/E2) =

2, 4, 6, 8 and 10 for 1st mode, 2nd mode, 3rd mode, 4th mode and 5th mode respectively

taking plate aspect ratios (b/a) = 1to 3 in steps of 1.

(ii) The effect of change in the layer thickness (t)of only one layer at a time has been

studied on change in natural frequency () of laminated composite plate for first mode

andplate aspect ratios (b/a) = 1 to 3 in steps of 1.

(iii) The effect of change in the fiber angles ()of only one layer at a time has been

studied on change in natural frequency () of laminated composite plate for first mode

andplate aspect ratios (b/a) = 1 to 3 in steps of 1.

Results of the present studies bring out the following conclusions:

4.2 CONCLUSIONS:

4. 2. 1 Four layered (0/90/90/0) laminate, simply supported and clamped on two

opposite sides

4.2.1.1 The effect of plate side- to- thickness ratios (b/h) = 50, 100, 200, 500 and

1000 on natural frequency () of laminated plate for modulus ratios (E1/E2) = 2,

4, 6, 8 and 10 for 1st mode, 2nd mode, 3rd mode, 4th mode and 5th mode respectively

taking plate aspect ratios (b/a) = 1to 3 in steps of 1.

(i) The natural frequency increases slightly as b/h increases from 50 to 100.

(ii) There is negligible variation in the natural frequency for b/h >100.

(iii) The natural frequency increases with the increase of the modulus ratios (E1/ E2).

(iv) The natural frequency increases as b/a increases.

38

4.2.1.2 The effect of change in the layer thickness (t)of only one layer at a time

on change in natural frequency () of laminated composite plate for first mode

and b/a = 1 to 3 in steps of 1.

(i) The change in natural frequency increases parabolicaly for first or fourth layer; the

change in natural frequency increases linearly for second or third layer as change in

thickness of only one layer at a time increases.

(ii) The change in natural frequency increases as b/a increases.

4.2.1.3 The effect of change in the fiber angles ()of only one layer at a time on


plate aspect ratios (b/a) = 1 to 3 in steps of 1.

(i) The rate of change in natural frequency for change in fiber angles of only first or

fourth layer is greater than the rate of change in natural frequency for change in fiber

angles of only second or third layer of the laminated composite plate.

(ii) The change in natural frequency of the laminated composite plate increases as b/a

increases.

4.3 SCOPE FOR FURTHER STUDIES

The suggestions for the extension of present work are as follows:

1. The Buckling analysis of laminated plates can be included.

2. The present investigation can be extended to dynamic analysis of laminated

plates and shells subjected to hydrothermal condition.

3. Material and geometry nonlinearity may be taken into account in the

formulation for further extension of the dynamic analysis of plates.

4. The laminates with arbitrary boundary conditions can be analysed.

5. The analysis can be carried out for cyclic loading, impact loading, static loading

and sinusoidal loading.

6. The analysis may be carried out for shells with arbitrary geometry and arbitrary

boundary conditions.

7. Dynamic analysis of the laminates with holes of various shapes be carried out.

8. Dynamic analysis of the anti-symmetric laminates can be carried out.

39

REFERENCES

1. Ahmed J.K., Agarwal V.C., Pal P and Srivastav V., Static and Dynamic

Analysis of Composite Laminated Plate International Journal of Innovative

Technology and Exploring Engineering (IJITEE), Vol. 3, No. 6, pp. 56- 60, Nov

2013.

2. Akbarov S D., Yahnioglu N and Yesil U.B., Forced vibration of an initially

stressed thick rectangular plate made of an orthotropic material with a

cylindrical hole Mechanics of Composite Materials, Vol. 46, No. 3, pp. 287-

298, May 2010.

3. Houmat A., Nonlinear free vibration of laminated composite rectangular plates

with curvilinear fibers Composite Structures, Vol. 106, pp. 211- 224, Jun 2013.

4. Ratnaparkhi U.S and Sarnobat S.S., Vibration Analysis of Composite Plate

International Journal of Modern Engineering Research (IJMER), Vol. 3, No. 1,

pp. 377- 380, Jan 2013.

Documents

Dynamic analysis of laminated plates using ansys