CHAPTER 4

BUCKLING ANALYSIS OF LAMINATED COMPOSITES

4.1 General

The mechanical behaviour of laminated composite plates are strongly dependent

on the degree of orthotropy of individual layers, the low ratio of transverse modulus to

in-plane modulus and stacking sequence of laminate. In practical situations, a plate

structure may be subjected to in-plane loads or transverse loads or both acting

simultaneously. The thickness of majority of such structural components being very

low, they are prone to buckling and hence have to be designed not just for strength but

also for stability. In order to fully exploit their strength, an accurate prediction of their

buckling load carrying capacity is essential. In this chapter the buckling equations are

derived by extending the Zeroth order Shear Defonnation Theory (ZSDT) to study the

buckling behaviour of simply supported rectangular laminated plates. Parametric

studies are perfonned by varying side to thickness ratio, plate aspect ratio, modular

ratio and number of layers. Finally a comparative study is perfonned with

Savithri[ 1991] and Reddy[ 1984a]. Review of CLPT for buckling study is also

presented.

4.2 Classical Laminated Plate Theory (CLPT)

A plate buckles when the in-plane compressive load gets so large that the

originally flat equilibrium state is no longer stable, and the plate deflects into a non flat

(wavy) configuration. The load at which the departure from the flat state takes place is

56

called the buckling load. The flat equilibrium state has only in-plane forces and

undergoes only extension, compression and shear. More comprehensively, the load at

which the plate deformed configuration suddenly changes in to a different

configuration is called the buckling load.

Analysis of plate buckling under in-plane loading involves solution of an eigen

value problem. The governing equations of cross ply laminated plates based on CLPT

are presented in Chapter 3. For buckling analysis we assume that only applied loads are

the in-plane forces and all other loads are zero. The differential equations governing the

buckling behaviour are as follows.

aN aN xy__x +--=0ax ay

aN XY aNy--+--=0ax ay

4.2.1 Solution Approach

(4.1 a)

(4.1 b)

(4.1 c)

In order to get Navier solution for cross ply laminates SS 1 type boundary

conditions are assumed. The generalised displacements can be written by assuming the

following variation:

00 00

Uo =IIUmn cosaxsin~ym=1 n=1

00 00

Vo = IIVmn sinaxcos~ym=l n=1

'" '"W o =IIWmn sin ax sin py

m=! n=J

57

(4.2a)

(4.2b)

(4.2c)

lTIn llnwhere a = ---;-;~ =b

Eqn (4.2) reduces the governing equations to the following form:

([C]-A[G]){i1} =0

where [C] refers to the flexural stiffness matrix and [0] refers to geometric stiffness

matrix and Ato the corresponding buckling parameter.

Elements of matrix [C] is 'given in Chapter 3 and elements of matrix [0] is as

follows 0 33 = (a2+k~2) and Oij = 0 for all i = j = 1 to 3 (i i- j i- 3), k is a constant given

by Ny and k = 0 for uniaxial compression.Nx

4.2.2 Numerical Results and Discussions

The non dimensionalised (dimensionless) critical buckling loads for symmetric

and anti-symmetric cross ply laminates are studied and results are presented in

Tables 4.1 and 4.2. Tables show the effect of aspect ratio and modular ratio on non

dimensionalised critical buckling loads of rectangular laminates under uniaxial and

biaxial compression. Materials properties used for calculating the numerical results are

as follows:

The non dimensionalised buckling loads used for presenting the results is as follows

Ncr = Ncr (_~_2_)1[ Dn

58

Table 4.1 Effect of plate aspect ratio and modular ratio on thedimensionless buckling loads of rectangular laminates(0/90/90/0) under uniaxial and biaxial compression.

k alb E 1/E2=5 10 15 20 25 40

0.5 13.900 18.1265 20.4288 21.8778 22.8738 24.5899

1 5.6500 6.3471 6.7242 6.9611 7.]238 7.4037

0 1.5 5.2333 5.2768 5.2975 5.3099 5.3182 5.3322

25.6500

6.2147 6.1106 6.0444 5.9988 5.9197(2,1)

0.5 11.120012.6941 13.4485 13.9222 14.2475 14.7661

(1,2) (1,2) (1,2) (1,2) (1,3)

1 2.8250 3.1735 3.3621 3.4806 3.5619 3.70191

1.5 1.6103 1.6236 1.6300 1.6338 1.6364 1.6407

2 1.2800 1.2429 1.2221 1.2089 1.1998 1.1839

Table 4.2 Effect of plate aspect ratio and modular ratio on thedimensionless buckling loads of rectangular laminates(0/90/0190) under uniaxial and biaxial compression

k alb E I /E2=5 10 15 20 25 40

0.5 4.7052 4.1565 3.9419 3.8275 3.7566 3.6465

1 2.6432 2.1893 2.0151 1.923 1.8661 1.7783

0 1.52.955 2.4868 2.3065 2.2111 2.1520 2.0608(2,1) (2,1) (2,1) (2,1) (2,1) (2,1)

22.6432 2.1893 2.0151 1.923 1.8661 1.7783(2,1) (2,1) (2,1) (2,1) (2,1) (2,1)

0.5 3.7641 3.3252 3.1535 3.0620 3.0052 2.9172

1 1.3216 1.0946 1.0075 0.9615 0.9331 0.8892

1 1.5 1.0091 0.8604 0.803 0.7726 0.7537 0.7246

2 0.9410 0.8313 0.7884 0.7655 0.7513 0.7293

59

From the results, it can be noted that the non dimensionalised buckling load

increases for symmetric laminates where as it decreases for anti-symmetric laminates as

the modular ratio increases

m;=3

m=1

I

I

\ II II III \ m=2I I I

I \ I

I \ II I\ \ II \ I

I \ /I II \ /

I \ /

\ I

\ \ \ ./ /

\ \ /

" ,,' .".'" ,m=4'" "A......'" ...... _.-.-

' __ >~: :-:-:~,,,":.-::·:::': ...._~~:-:~-:-::,:_,,_,,_,,-·m=55

25

IZ~"'0 20<II~enc

:::;e~ 15

,Q

'"'"QJ

Co.~ 10QJ

eis

o 2 3 4 5 6 7 8 9 10

Aspect ratio alb

Figure 4.1 Variation of dimensionless buckling loads with aspect ratio(a/b) for (0/90/90/0) laminate (CLPT)

25

Iz~ 20"'0

'"~en=~ 15(,l

=,Q'"'"-a 10

.S!'"=ais 5

a/b=O.5--Uniaxial.. ---.- Biaxial

a/b=1.0

a/b=1.5a/b=1.0

- -- - - - - - - - - - - ~ ------- - - - -- - - - - - . -- --.. - ---_. - -- -~ -~ -- - - - -- - - - - -- -a/b=1.5

5 10 15 20 25 30 35 40

Modular ratio E/E2

Figure 4.2 Variation of dimensionless buckling load with modular ratiofor (0/90/90/0) laminate (CLPT)

60

,Figure 4.1 shows the plot of non dimensionalised buckling loads versus plate

aspect ratio for a symmetric laminate. For aspect ratio less than 2.2 the plate buckles

into a single half wave in the x direction. Larger aspect ratios lead to higher modes of

buckling. Figure 4.2 shows the variation of non dimensionalised buckling load with

modular ratio and it is observed that the buckling load increases as the modular ratio

1l1creases.

4.3 Buckling Analysis using Higher Order Theories

The CLPT based on Kirchhoff's hypothesis is inadequate in modelling

laminated composite plates, especially dynamic aspect. The Kirchhoff's assumptions

amount to treating plates to be infinitely rigid in transverse direction neglecting the

transverse strains. This theory overestimates natural frequencies and buckling loads.

Since laminated composite materials are often very flexible in shear (and weaker in

transverse shear mode), the transverse shearing strains must be taken into account if an

accurate representation of the behaviour of the laminated plate is to be achieved.

In order to take into account the effects of low ratio of transverse shear modulus

to in-plane modulus, a number of first order shear deformation theories have been

developed. However, the assumption of displacements as linear functions of the

coordinate in the thickness direction has proved to be inadequate for predicting the

response of thick laminates. Furthennore, due to the advantage that no shear correction

factors are needed and warping of the cross section can be accounted for, to a certain

extent, higher order theories are widely preferred for analysis of composite plates.

Second and higher order shear defOlmation plate theories use higher order

polynomials in the expansion of displacement components along the plate thickness

61

and require no shear correction factors. Buckling equations are derived for composite

plates using Zeroth order Shear Deformation Theory (ZSDT) and the results are

compared with refined higher order models (Third order equivalent single layer and

layer wise).

4.3.1 Buckling Equations for Zeroth Order Model

Displacement fields, stress resultants and constitutive equations used are as

given in Chapter 3 (section 3.3.1). The differential equations governing the buckling

behaviour are derived through principle of virtual work and we have:

h/2

f H(a)5E x + a)SE y + T xy 8y xy + T yz8y yz + Txz8yxz )dxdydz-h/2

1 SS - law aw]+- 2N xy O -- dxdy = 02 ax ay

(4.3)

In buckling analysis the plate is subjected to in-plane compreSSIve loads

NxandN y only and the external load being zero. The differential equations governing

the buckling behaviour using ZSDT [Shimpi [1999]] are as follows

aN aN_x_+~=o

ax By

aN aN~+--y =0

ax By

62

(4.4a)

(4.4b)

a( 4) a( 4) ( 4 )- M --P +- M --P - --R-OOx xy 3h 2 xy By y 3h2 Y Qy h 2 Y -

4.3.2 Buckling Equations for a Layer Wise Model (LWM)

(4.4c)

(4.4d)

(4.4e)

The differential equations governing the buckling behaviour using Layer Wise

Model [Savithri, 1991] are as follows:

aN XY aNy--+-=0ax ay

&M+2&~ +&My =N &wo+N;, &wO+2~ &woax? axDy &I &2 &I &.OJ

apx + apXY _ v = 0ax ay xz

apyx apy-+--v =0ax ay yz

4.3.3 Buckling Equations for Reddy's Higher Order Model

(4.5a)

(4.5b)

(4.5c)

(4.5d)

(4.5e)

The differential equations governing the buckling behaviour using

Reddy[1984a] (HSDT) are as follows:

(4.6a)

(4.6b)

63

aMx+aMxY _ Q =0ax ay x

aM Y + aM xy _ Q = 0ay ax Y

The Stress resultants are defined as:

N Zk+J .

(Nx, Ny, NXY ) =I f (crx,cry,L XY )dzk=l Zk

N Zk+J

(Mx' My, MXY ) =I f (crx' cry, L xy )zdzk=J Zk

N Zk+1

(Px,Py,PXy ) =I f (crx,cry' Lxy )z3dzk=1 Zk

N Zk+1

(Rx, R y) =I f (Lyz, Lxz )z2dzk=1 Zk

4.3.4 Solution Approach

(4.6c)

(4.6d)

(4.6e)

(4.7a)

(4.7b)

(4.7c)

(4.7d)

Consider a simply supported rectangular composite cross ply laminate of sides a

and b with thickness h. The variables Uo, vo, wo, Qx, Qy of a general Navier solution of

the governing differential equation can be expressed as:

OCJ OCJ mnx . nnyun =,,"u cos--sm-L..JL..J mn b

m=J n=J a

64

(4.8a)

if) if) • mnx nnyVo =IIVmn sm--cos--

m~n~ a b

if) if) • mnx . nnyW o =IIWlJln sm--sm--

1Jl;) n;) a b

(4.8b)

(4.8c)

(4.8d)

(4.8e)

The variables Qx and Qy are substituted by Ul and VI in Reddy's model and layer

wise model:

([C]-A[G]){~} =0

{~}T = {UIIIII' ~11I1' W,IIII,QXIIIII,QYIIIII}

(4.9)

where [C] and [0] refers to the flexural stiffness and geometric stiffness matrices and A

to the corresponding buckling parameter. The elements of the coefficient matrix [C] is

given in chapter 3 and elements of [0] matrix are as follows:

033 = (a2+k~2) and Oij = 0 for all i= j= 1 to 5 (i i- j i- 3), k is a constant given

by Ny and k = 0 for uniaxial compression.Nx

4.4 Numerical Results and Discussions

Buckling loads of cross ply laminated composite plate with simply supported

edges are analysed using the models presented above. Both symmetric and anti-

symmetric laminations with respect to the middle plane are considered. Different

material properties have been used for parametric studies in the literature. The material

properties of the individual layers considered for the numerical examples are as

follows:

65

Symmetrical cross ply laminates such as 9 ply, 5 ply, 3 ply are considered. The

fibre orientation of the different laminae alternate between 00 and 900 with respect to x

axis and the 00 layers are at the outer surface of the laminates. The total thickness of the

zero degree and 90 degree layers in each laminate are the same. Anti-symmetric cross

ply such as 10 ply, 6 ply, 4 ply and 2 ply are considered. All laminae are assumed to be

of same thickness and made up of same orthotropic material. Detailed results for ZSDT

model are presented for the following cases.

1. Uniaxial buckling loads for isotropic, Olihotropic and laminated plate (0/90/90/0)

for different side to thickness ratios (a/h).

2. Uniaxial and biaxial buckling loads of symmetric cross ply laminates 3, 5, 7 and

9 ply for various parameters.

3. Uniaxial and biaxial buckling loads of anti-symmetric cross ply laminates for 2

ply (0/90), 4 ply (0/90h, 6ply (0/90)3 and 10 ply (0/90)5 ply for various

parameters.

A comparative study has been done for the following cases

1. Uniaxial and biaxial buckling loads of symmetric cross ply laminate for various

modular ratio and thickness ratio using higher order models.

2. Uniaxial and biaxial buckling loads of anti-symmetric cross ply laminate for

various modular ratio and thickness ratio using higher order models.

The results are presented in Tables 4.3 to 4.19. The tables contain non

dimensionalised buckling loads and the dimensionless quantities are defined as

(4.1 0)

Table 4.3 shows the non dimensionalised uniaxial buckling loads for isotropic

orthotropic and laminated square plates. Poisson's ratio for isotropic material is taken

66

as 0.3. For orthotropic and laminated plates modular ratio is taken as 40. From the

results in the table it is observed that the effect of shear deformation is quite significant

on the buckling parameter. CLPT not only results in buckling solution independent of

the side to thickness ratios but also overpredicts the buckling loads in low side to

thickness ratios. From Tables 4.3 to 4.9 it is evident that the dimensionless uniaxial and

biaxial buckling loads increase as the modular ratio or side to thickness ratio or aspect

ratio is increasing. In Tables 4.10 and 4.11 the effect produced on uniaxial buckling

load of symmetric and anti-symmetric laminates by varying number of layers and

degree of orthotropy of individual layers is presented for ZSDT, HSDT, and LWM and

the results are compared with 3D elasticity solutions. For Tables 4.10 to 4.19, results

are not available for comparison in the literature and thus are generated using the

models to compare the results. The comparison of non dimensional uniaxial buckling

loads for symmetric and anti-symmetric in tenns of % error is presented in Tables 4.12

and 4.13. From these tables it may be observed that buckling loads obtained by ESL

models are very accurate when compared with exact 3 D solutions as the errors are

within 2.5 % for symmetric laminates and 8 % for anti-symmetric laminates. Tables

4.14 to 4.19 illustrates the influence played by modular ratio, thickness ratio and aspect

ratio on uniaxial as well as biaxial buckling loads using different higher order theories.

The results of ZSDT and HSDT are exactly the same. So it can be concluded that

ZSDT can predict buckling loads very accurately like the most popular higher order

model of Reddy.

Figure 4.3 shows variation of dimensionless buckling load with thickness ratio

(a/h) for anti-symmetric laminates. Figure 4.4 shows the variation of uniaxial and

biaxial buckling loads with modular ratio for a four ply anti-symmetric square laminate.

67

Figure 4.5 illustrated the variation of buckling load with aspect ratio for anti-symmetric

laminates. Variation of buckling load with modular ratio for symmetric laminates is

presented in Figure 4.6. The effect of side to thickness ratio on buckling loads for 9 ply

square plate for various modular ratios are shown in Figure 4.7. In all the cases the

dimensionless buckling loads increases with increase in aspect ratio/modular ratio/side

to thickness ratio.

4.5 Concluding Remarks

In this chapter buckling equations are derived using a higher order ESL model

ZSDT. Buckling loads of simply supported cross ply laminated plates has been

obtained using this model. Influence played by various parameters viz, plate aspect

ratio, side to thickness ratio, modular ratio and number of layers are studied in detail.

The numerical results show that for thick composite laminates the effect of transverse

shear deformation is always to be incorporated when the material exhibits high

orthotropy ratio or when the number of layers in increased. It can be noted that global

higher order (ESL) models can predict global responses like buckling load very

accurately when compared with 3 D elasticity solutions.

68

Table 4.3 Dimensionless uniaxial buckling loads for isotropic,ortbotropic and laminated (0/90/90/0) composite plates(a/b=l, E)/Ez=40)

Isotropic

alb CLPT HSDT Present

5 4.0000 2.9512 2.9512

10 4.0000 3.4224 3.4224

20 4.0000 3.565 3.565

50 4.0000 3.6071 3.6071

100 4.0000 3.6132 3.6132

Ortbotropic

alb CLPT HSDT Present

5 36.1597 10.8138 10.8138

10 36.1597 22.1861 22.1861

20 36.1597 31.1527 31.1527

50 36.1597 35.2479 35.2479

100 36.1597 35.9272 35.9272

Laminate(0/90/90/0)

alb CLPT HSDT Present

5 36.] 597 ] 1.997] 11.9971

10 36.1597 23.3400 23.3400

20 36.1597 3] .6596 31.6596

50 36.]597 35.3467 35.3467

100 36.1597 35.9526 35.9526

69

Table 4.4 Effect of material anisotropy and thickness ratio ondimensionless uniaxial buckling loads for symmetriclaminates using ZSDT (Present Model) (a/b=l)

3 Ply(0/90/0)

E I/E2 a/h = 5 a/h = 10 a/h = 15 a/h = 20

5 5.4308 6.7700 7.0990 7.2223

10 7.1554 9.9406 10.7419 11.0566

15 8.4278 12.7643 14.1858 14.7692

20 9.4219 15.2984 17.4450 18.3631

25 10.2308 17.5894 20.5349 21.8435

30 10.9088 19.6744 23.4696 25.2162

35 11.4901 21.5832 26.2616 28.4866

40 11.9971 23.34 28.9222 31.6596

5 Ply(0/90/0/90/0)

E I/E2 a/h =5 a/h = 10 a/h = 15 a/h = 20

5 5.5227 6.8092 7.1186 7.2338

10 7.4256 10.0897 10.8212 11.1042

15 8.8777 13.0691 14.358 14.8753

20 10.0316 15.7879 17.7376 18.5474

25 10.9766 18.2812 20.9705 22.1241

30 11.7687 20.5781 24.0666 25.6092

35 12.4448 22.7028 27.0353 29.0063

40 12.9411 24.6755 29.8848 32.3191

9 Ply(0/90/0/90/0/90/0/90/0)

E I/E2 a/h =5 a/h = 10 a/h = 15 a/h = 20

5 5.5716 6.8291 7.1283 7.2394

10 7.5968 10.1772 10.8669 11.1315

15 9.1830 13.258 14.4623 14.9389

20 10.4631 16.1009 17.9195 18.6607

25 11.5213 18.7331 21.2461 22.2993

30 12.4129 21.1783 24.4495 25.8573

35 12.8600 23.4563 27.5367 29.3375

40 13.2301 25.5845 30.5141 32.7424

70

Table 4.5 Effect of material anisotropy and thickness ratio ondimensionless biaxial buckling loads for symmetric laminatesusing ZSDT(Present Model) (a/b=l)

3 Ply(0/9010)

E]/E2 a/h = 5 a/h = 10 a/h = 15 a/h = 20

5 2.7154 3.385 3.5495 3.6112

10 3.5777 4.9703 5.3709 5.5283

15 4.2139 6.3822 7.0929 7.3846

20 4.7109 7.6492 8.7225 9.1815

25 5.1154 8.7947 10.2675 10.9218

30 5.4544 9.8372 11.7348 12.6081

35 5.7450 10.7916 13.1308 14.2433

40 5.9986 11.6700 14.4611 15.8298

5 Ply(0/9010/9010)

E]/E2 a/h = 5 a/h = 10 a/h = 15 a/h = 20

5 2.7613 3.4046 3.5593 3.6169

10 3.7128 5.0448 5.4106 5.5521

15 4.4388 6.5345 7.179 7.4376

20 5.0158 7.8939 8.8688 9.2737

25 5.4883 9.1406 10.4852 11.0621

30 5.8844 10.2891 12.0333 12.8046

35 6.2224 11.3514 13.5176 14.5032

40 6.5152 12.3378 14.9424 16.1595

9 Ply(0/90/o/9010/9010/9010)

E]/E2 a/h = 5 a/h = 10 a/h = 15 a/h = 20

5 2.7858 3.4145 3.5642 3.6197

10 3.7984 5.0886 5.4335 5.5658

15 4.5915 6.6290 7.2311 7.4694

20 5.2316 8.0504 8.9597 9.3304

25 5.7606 9.3666 10.623 11.1497

30 6.2065 10.5891 12.2248 12.9287

35 6.5881 11.7281 13.7684 14.6687

40 6.9193 12.7922 15.2571 16.3712

71

Table 4.6 Effect of material anisotropy and thickness ratio ondimensionless uniaxial buckling loads for anti-symmetriclaminates using ZSDT (Present Model) (a/b=l)

2 Ply(0/90)

EI/Ez a/h = 5 a/h = 10 a/h = 15 a/h = 20

3 4.1382 4.7749 4.9153 4.9664

10 5.2621 6.2721 6.5044 6.5899

20 6.5632 8.1151 8.4896 8.6292

30 7.7205 9.8695 10.4115 10.616

40 8.7694 11.5625 12.2968 12.577

4 Ply(0/90)z

EI/Ez a/h = 5 a/h = 10 a/h = 15 a/h = 20

3 4.4794 5.2523 5.4261 5.4897

10 7.0565 9.2315 9.793 10.0063

20 9.6394 14.2543 15.6524 16.2098

30 10.6659 18.6671 21.1491 22.1837

40 11.3433 22.579 26.3201 27.9451

6 Ply(0/90)3

EI/Ez a/h = 5 a/h = 10 a/h = 15 a/h = 20

3 4.5468 5.3420 5.5213 5.587

10 7.3938 9.7762 10.3997 10.6374

20 10.0865 15.3518 16.9537 17.5975

30 11.1106 20.201 23.0718 24.2825

40 11.8198 24.4596 28.7976 30.7097

10 Ply(0/90)s

ElIE2 a/h = 5 a/h = 10 a/h = 15 a/h = 20

3 4.5819 5.3882 5.5702 5.6368

10 7.5692 10.0557 10.7106 10.9607

20 10.3119 15.9141 17.6197 18.3077

30 11.3721 20.9864 24.055 25.3559

40 12.1089 25.4225 30.0634 32.1227

72

Table 4.7 Effect of material anisotropy and thickness ratio ondimensionless biaxial buckling loads for anti-symmetriclaminates using ZSDT (Present Model) (a/b==l)

2 Ply(0/90)

E I /E2 a/h = 5 a/h == 10 a/h = 15 a/h == 20

3 2.0691 2.3874 2.4576 2.4832

10 2.631 3.1361 3.2522 3.295

20 3.2816 4.0576 4.2448 4.3146

30 3.8603 4.9347 5.2057 5.308

40 4.3847 5.7813 6.1484 6.2885

4 Ply(0/90h

E I /E2 a/h == 5 a/h == 10 a/h = 15 a/h == 20

3 2.2397 2.6261 2.713 2.7449

10 3.5283 4.6157 4.8965 5.0031

20 4.8197 7.1272 7.8262 8.1049

30 5.7358 9.3336 10.5745 11.0918

40 6.4233 11.2895 13.16 13.9726

6 Ply(0/90)3

E I /E2 a/h == 5 a/h == 10 a/h == 15 a/h == 20

3 2.2734 2.671 2.7607 2.7935

10 3.6969 4.8881 5.1999 5.3187

20 5.0964 7.6759 8.4768 8.7987

30 6.0684 10.1005 11.5359 12.1412

40 6.7873 12.2298 14.3988 15.3548

10 Ply(0/90)s

E I /E2 a/h == 5 a/h == 10 a/h == 15 a/h = 20

3 2.2909 2.6941 2.7851 2.8184

10 3.7846 5.0279 5.3553 5.4804

20 5.241 7.9571 8.8099 9.1539

30 6.2439 10.4932 12.0275 12.678

40 6.9815 12.7112 15.0317 16.0614

73

Table 4.8 Effect of aspect ratio on the dimensionless uniaxial andbiaxial buckling loads for symmetric laminates (a/h=10,E]/E2=40)

Uniaxial buckling loads (ZSDT-Present Model)

alb ply = 3 5 7 9

0.25 17.8711 17.3162 16.1968 15.4758

0.5 18.4141 18.0082 17.0293 16.3879

1 23.34 24.6755 25.3269 25.584539.162 41.758 42.2903 42.2652

1.5 (1,1) (2,1) (2,1) (2,1)47.5311 50.5711 51.6567 51.7807

2 (3,1) (3,1) (3,1) (3,1)

Biaxial buckling loads (ZSDT-Present Model)

alb ply=3 5 7 911.4932 12.3378 12.5632 12.3348

0.25 (1,5) (1,4) (1,3) (1,3)11.67 12.3378 12.6634 12.7922

0.5 (1,2) (1,2) (1,2) (1,2)

1 11.67 12.3378 12.6634 12.7922

1.5 12.0498 13.9516 15.5369 16.3603

2 14.4271 17.1613 19.6425 20.9548

74

Table 4.9 Effect of aspect ratio on the dimensionless uniaxial andbiaxial buckling loads for anti-symmetric laminates (a/h=10,E./E2=40)

Uniaxial buckling loads (ZSDT-Present Model)

alb ply = 2 4 6 10

0.25 5.0794 10.7902 11.7556 12.2488

0.5 5.7295 11.8470 12.8880 13.4202

1 11.5625 22.5790 24.4596 25.422523.7820 36.4886 38.6926 39.8566

1.5 (2,1) (2,1 ) (2,1 ) (2,1)35.0777 45.3733 47.2792 48.4357

2 (2,1) (4,1) (4,1) (4,1)

Biaxial bucklin~ loads (ZSDT-Present Model)

alb ply = 2 4 6 104.5836 9.4776 10.3104 10.7361

0.25 (l,2) (1,2) (1,2) (1,2)

0.5 4.5836 9.4776 10.3104 10.7361

1 5.7813 11.2895 12.2298 12.7112

1.5 9.4773 16.7178 17.9276 18.5526

2 14.5203 22.5975 23.9827 24.7145

75

Table 4.10 Effect of modular ratio of the individual layers on thedimensionless uniaxial buckling load for symmetric laminates(a/b=l, alh =10)

Plate No ofE]/Ez

Theories Layers 20 30 40

Exact $ 15.0191 19.3040 22.8807

Present 15.2984 19.6744 23.340

LWM@ 3 15.3287 19.7326 23.4305

HSDT* 15.2984 19.6744 23.340

CLPT 19.712 27.936 36.160

Exact 15.6527 20.4663 24.5929

Present 15.7879 20.5781 24.6755

LWM 5 15.7583 20.5271 24.6022

HSDT 15.7879 20.5781 24.6755

CLPT 19.712 27.936 36.160

Exact 15.9153 20.9614 25.3436

Present 16.1009 21.1783 25.5845

LWM 9 16.0732 21.13 25.5139

HSDT 16.1009 21.1783 25.5845

CLPT 19.712 27.936 36.160

$Noor[1975] @generated(Savithri [1991]) * generated(Reddy [1984])

76

Table 4.11 Effect of modular ratio of the individual layers on thedimensionless uniaxial buckling load for anti-symmetriclaminates (a/b=1, alh =10)

PlateEI/Ez

Theories Ply 3 10 20 30 40

Exact $ 4.6948 6.1181 7.8196 9.3746 10.8167

Present2

4.7749 6.2721 8.1151 9.8695 11.5625

LWM@ 4.7762 6.2872 8.1578 9.948 11.6842

HSDT* 4.7749 6.2721 8.1151 9.8695 11.563

Exact 5.1738 9.0164 13.7429 17.7829 21.2796

Present4

5.2523 9.2315 14.2543 18.6671 22.579

LWM 5.2493 9.2205 14.2262 18.6176 22.5059

HSDT 5.2523 9.2315 14.254 18.667 22.579

Exact 5.2673 9.6051 15.0014 19.6394 23.6689

Present6

5.342 9.7762 15.3518 20.201 24.4596

LWM 5.3391 9.7651 15.3231 20.1505 24.385

HSDT 5.342 9.7762 15.352 20.201 24.460

Exact 5.3159 9.9134 15.6685 20.6347 24.9636

Present10

5.3882 10.0557 15.9141 20.9864 25.4225

LWM 5.3854 10.0455 15.888 20.9406 25.3551

HSDT 5.3882 10.056 15.914 20.986 25.422

$Noor[1975] @ generate~ (Savithri [1991]) * generated(Reddy [1984])

77

Table 4.12 Comparison of the dimensionless uniaxial buckling load interms of percentage error from exact solutions for symmetriclaminates (a/b=l, alh =10)

Plate No ofE I /E2

Theories Layers 20 30 40

Exact - - -

Present 1.86 1.92 2.01

LWM 3 2.06 2.22 2.4

HSDT 1.86 1.92 2.01

CLPT 31.25 44.72 58.04

Exact - - -

Present 0.86 0.55 0.34

LWM 5 0.67 0.3 0.04

HSDT 0.86 0.55 0.34

CLPT 25.93 36.5 47.03

Exact - - -

Present 1.17 1.03 0.95

LWM 9 0.99 0.8 0.67

HSDT 1.17 1.03 0.95

CLPT 23.86 33.27 42.68

78

Table 4.13 Comparison of the dimensionless uniaxial buckling load interms of percentage error from exact solutions for anti­symmetric laminates (a/b~l, alh :::::10)

PlateE}/E2

Theories Ply 3 10 20 30 40

Exact - - - - -

Present2

1.71 2.52 3.78 5.28 6.89

LWM 1.73 2.76 4.33 6.12 8.02

HSDT 1.71 2.52 3.78 5.28 6.9

Exact - - - - -Present

41.52 2.39 3.72 4.97 6.11

LWM 1.46 2.26 3.52 4.69 5.76

HSDT 1.52 2.39 3.72 4.97 6.11

Exact - - - - -Present

61.42 1.78 2.34 2.86 3.34

LWM 1.36 1.67 2.14 2.6 3.03

HSDT 1.42 1.78 2.34 2.86 3.34

Exact - - - - -

Present10

1.36 1.44 1.57 1.7 1.84

LWM 1.31 1.33 1.4 1.48 1.57

HSDT 1.36 1.44 1.57 1.7 1.84

79

Table 4.14 Effect of modular ratio of the individual layers on thedimensionless biaxial buckling load for symmetric laminates(a/b=l, alh =10)

Plate No ofEI/Ez

Theories Layers 10 20 30 40

Present 4.9703 7.6492 9,8372 11.6700

LWM 3 4.9748 7.6643 9.8663 11.7152

HSDT 4.9703 7.6492 9.8372 11.6700

Present 5.0448 7.8939 10.2891 12.3378

LWM 5 5.0391 7.8791 10.2636 12.3011

HSDT 5.0448 7.8939 10.2891 12.3378

Present 5.0886 8.0504 10.5891 12.7922

LWM 9 5.0832 8.0366 10.565 12.7569

HSDT 5.0886 8.0504 10.5891 12.7922

Table 4.15 Effect of modular ratio of the individual layers on thedimensionless biaxial buckling load for anti-symmetriclaminates (afb=l, afh =10)

PlateElfEz

Theories Ply 3 10 20 30 40

Present 2.3874 3.1361 4.0576 4.9347 5.7813

LWM 2 2.3881 3.1436 4.0789 4.974 5.8421

HSDT 2.3874 3.1361 4.0576 4.9347 5.7813

Present 2.6261 4.6157 7.1272 9.3336 11.2895

LWM 4 2.6247 4.6103 7.1131 9.3088 11.253

HSDT 2.6261 4.6157 7.1272 9.3336 11.2895

Present 2.671 4.8881 7.6759 10.1005 12.2298

LWM 6 2.6696 4.8825 7.6615 10.0752 12.1925

HSDT 2.671 4.8881 7.6759 10.1005 12.2298

Present 2.6941 5.0279 7.9571 10.4932 12.7112

LWM 10 2.6927 5.0228 7.944 10.4703 12.6776

HSDT 2.6941 5.0279 7.9571 10.4932 12.7112

80

Table 4.16 Effect of thickness ratio on the dimensionless uniaxialbuckling load for symmetric laminates (a/b=l, E]/E2=40)

a/hPlate Noof

Theories Layers 5 10 15 20

Present 11.9971 23.3400 28.9222 31.6596

LWM 3 12.1412 23.4305 28.9766 31.6945

HSnT 11.9971 23.3400 28.9222 31.6596

Present 12.9411 24.6755 29.8848 32.3191

LWM 5 12.875 24.6022 29.8352 32.286

HSnT 12.9411 24.6755 29.8848 32.3191

Present 13.2301 25.5845 30.5141 32.7424

LWM 9 13.1354 25.5139 30.4689 32.7129

HSnT 13.2301 25.5845 30.5141 32.7424

Table 4.17 Effect of thickness ratio on the dimensionless biaxialbuckling load for symmetric laminates (a/b=l, E]/E2=40)

a/hPlate No of

Theories Layers 5 10 15 20

Present 5.9986 11.6700 14.4611 15.8298

LWM 3 6.0706 11.7152 14.4883 15.8473

HSnT 5.9986 11.6700 14.4611 15.8298

Present 6.5152 12.3378 14.9424 16.1595

LWM 5 6.4794 12.3011 14.9176 16.143

HSnT 6.5152 12.3378 14.9424 16.1595

Present 6.9193 12.7922 15.2571 16.3712

LWM 9 6.8800 12.7569 15.2344 16.3564

HSnT 6.9193 12.7922 15.2571 16.3712

81

Table 4.18 Effect of thickness ratio on the dimensionless uniaxialbuckling load for anti-symmetric laminates (a/b=l, E)/E2:::::40)

alhPlate

Theories Ply 5 10 15 20

Present 8.7694 11.5625 12.2968 12.577

LWM 2 9.057 11.6842 12.3576 12.6127

HSDT 8.7694 11.5625 12.2968 12.577

Present 11.3433 22.579 26.3201 27.9451

LWM 4 11.2845 22.5059 26.2748 27.9162

HSDT 11.3433 22.579 26.3201 27.9451

Present 11.8198 24.4596 28.7976 30.7097

LWM 6 11.7321 24.385 28.7511 30.6798

HSDT 11.8198 24.4596 28.7976 30.7097

Present 12.1089 25.4225 30.0634 32.1227

LWM 10 12.0238 25.3551 30.0213 32.0956

HSDT 12.1089 25.4225 30.0634 32.1227

82

Table 4.19 Effect of thickness ratio on the dimensionless biaxial bucklingload for anti-symmetric laminates (a/b=l, E)/E2=40)

alhPlate

Theories Ply 5 10 15 20

Present 4.3847 5.7813 6.1484 6.2885

LWM 2 4.5285 5.8421 6.1788 6.3064

HSDT 4.3847 5.7813 6.1484 6.2885

Present 6.4233 11.2895 13.16 13.9726

LWM 4 6.382 11.253 13.1374 13.9581

HSDT 6.4233 11.2895 13.16 13.9726

Present 6.7873 12.2298 14.3988 15.3548

LWM 6 6.7441 12.1925 14.3755 15.3399

HSDT 6.7873 12.2298 14.3988 15.3548

Present 6.9815 12.7112 15.0317 16.0614

LWM 10 6.9424 12.6776 15.0106 16.0478

HSDT 6.9815 12.7112 15.0317 16.0614

83

35 --Uniaxial------- Biaxial

(0/90)s------..

(0/90)3

(0/90)2

(0/90)1

(0/90)3 ~0/gO)5

-_ .._ :- _ - -...- .-.-.-.-.-.-.-_1.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-...-.-.-.-.- -.;.".- .. -

'~':'~------:-' .. -

'"'l''l'

~~ ..

5 (0/90)1

10 20 30 40

Side to thickness ratio(a/h)

50

Figure 4.3 Effect of side to thickness ratio on non dimensional bucklingloads for anti-symmetric square laminates (ZSDT-PresentModel)

30 ....--------- --:-.,

tl 25(Z .

'0'~

.£ 20O/J=:!2Col

.6 15

--Uniaxial··-·---Biaxial

a/h=20

a/h=15

a/h=10

a/h=20

o S 10 15 20 25 30 3S 40 45

Modular ratio E/E2

Figure 4.4 Effect of modular ratio on non dimensional buckling loads fora 4 ply (0/90/0/90) square plate (ZSDT-Present model)

84

50

--Uniaxial

1Zi> .---... Biaxial40

"Cl

'"oSOJ)

=:§ 30(,j

=..c-;=.:2 20'"=~e:a= 100 -----_.--_.-Z

O+--,---~---.--_.____r-__,_-_,____.-__,_-_,____t

0.5 1.0 1.5 2.0 2.5 3.0

Aspect Ratio alb

Figure 4.5 Effect of aspect ratio (alb) on non dimensional buckling loadsfor anti-symmetric plates alb =10, E]1E2 =40 (ZSDT-Presentmodel)

30 -,--------------------,

9 ply

5 ply

3 ply

9 ply

........"".." ,.".,,""""" ·.. ··3~~y.-­.. '

--Uniaxial------- Biaxial

5

25

10

~

co.;;;c~

.§"0CoZ

40302010O-f----.----.--.----r-~-___"T----r-___,---t

oModular ratio E/E

2

Figure 4.6 Effect of modular ratio on non dimensional buckling loads forsymmetric square plates alb = 10 (ZSDT-Present model))

85

40"("----- ....,-- ----.

--Uniaxial------- Biaxial

,-,-'-'"

E,tE2=5

E,tE2=5

5040302010o+--..-----r--,--.----r------r---.---,--.----1

o

Side to thiclmess ratio alb

Figure 4.7 Effect of side to thickness ratio on non dimensional bucklingloads for a 9 ply (0/90/0/90/0/90/0/90/0) square plate (ZSDT­Present model»

86