Nonlinear Stability Characteristics of Composite ...insa.nic.in/writereaddata/UpLoadedFiles/PINSA/2016_Art19.pdfNonlinear Stability Characteristics of Composite Cylindrical Panel

Research Paper

Nonlinear Stability Characteristics of Composite Cylindrical PanelSubjected to Non-uniform In-plane Mechanical and Localized ThermalLoadingsRAJESH KUMAR, L S RAMACHANDRA* and BISWANATH BANERJEEDepartment of Civil Engineering, IIT Kharagpur 721 302, India

(Received on 20 April 2016; Accepted on 27 April 2016)

The non-linear stability analysis of composite cylindrical panel subjected to concentrated in-plane mechanical and localizedthermal loadings are reported here. The buckling of composite panel subjected to concentrated in-plane mechanical loading/localized thermal loading is solved in two steps as the prebuckling stress distribution within the panel is not known a priori.In the first step, the semi-analytical expressions for the pre-buckling stresses within the composite cylindrical panel underin-plane mechanical/ thermal loadings are developed by solving in-plane elasticity/thermoelasticity problem. Subsequently,using these in-plane stresses within the cylindrical panel, the governing equations for nonlinear stability of layeredcomposite panel are formulated using variational principle. The cylindrical panel is modeled based on Donnell’s shallowshell theory considering higher order shear deformation theory and incorporating von-Kármán geometric nonlinearity. TheGalerkin’s method is used to solve the non-linear governing partial differential equations. The influence of different typesof mechanical and thermal loadings, initial geometric imperfections, and radius of curvature on the postbuckling equilibriumpaths is investigated.

Keywords: Composite Cylindrical Panel; Non-Uniform Mechanical Loading; Localized Thermal Loading;Galerkin Method; Nonlinear Stability

*Author for Correspondence: E-mail: [email protected]

Proc Indian Natn Sci Acad 82 No. 2 June Spl Issue 2016 pp. 271-288Printed in India. DOI: 10.16943/ptinsa/2016/48419

Introdunction

The classical buckling problem of composite cylindricalpanel subjected to uniform in-plane mechanical loadsor uniform thermal loading over plate area is welldocumented in literature. However, studies on bucklingof composite panels subjected to concentrated in-plane mechanical loading/localized thermal loading aremeagre. Due to localized heating source at the centreof the panel or due to non-uniform exposure of thepanel to thermal loading T(x, y), the panel is subjectedto localized thermal heating with maximumtemperature at the centre and minimum temperatureat the support. This has been idealized as (i) localizedheating over central rectangular or circular region withconstant temperature within the region or (ii) domeshaped heating T(x, y). In the present article, the non-uniform mechanical loadings are modelled asconcentrated load and partial edge load. Recent

developments have shown that under the action ofnon-uniform mechanical in-plane loads, all three

components of stresses ( , ,xx yy xy ) are developedwithin the plate/panel. Thus, the buckling and postbuckling problems of panel subjected to localizedthermal loading or non-uniform mechanical loadinginvolves two steps. In the first step, the prebucklingstress distributions within the panels are evaluated.Subsequently, using these stress distributions, thegoverning stability equations for panel are derivedwhich are solved by Galerkin’s method in the presentarticle.

The numerous studies on the buckling andpostbuckling response of composite flat panel underuniform temperature distribution over plan andthrough thickness are reported in the literature(Tauchert and Huang, 1987; Chen and Chen, 1989;Meyers and Hyer, 1991; Singh et al., 1993; Singha et

Published Online on 29 June 2016

272 Rajesh Kumar et al.

al., 2001; Jones, 2005; Libtrescu et al. 2000, Ounis etal., 2014). Birman and Bert (1993) and Girish andRamachandra (2006) traced the postbucklingequilibrium path of composite cylindrical shell panelssubjected to combined mechanical and thermal loads(uniform temperature rise through the thickness) usingone term and multi term Galerkin’s methodrespectively. Yang et al. (2006) reported thepostbuckling equilibrium path of functionally gradedcylindrical panel with temperature dependentproperties under thermomechanical loads within theframework of differential quadrature method. In theabove papers, first, the panel was subjected to uni-axial mechanical load and then uniform temperatureloading was applied. Dongyun et al. (2016) studiedthe postbuckling behaviour of stiffened curved panelexperimentally and numerically (finite element method)under shear and in-plane mechanical loadings. Thepostbuckling response of composite cylindrical panel(Panda and Ramachandra, 2010) and sandwich panel(Dey and Ramachandra, 2014) under parabolicallydistributed edge loading was studied using multi termGalerkin’s method. Authors used Ritz method forevaluating the in-plane stresses within curved paneldue to non-uniform loading.

Chen et al. (1991) used finite element methodto analyze the thermal buckling of composite flat panelunder tent-like non-uniform temperature distribution.Mead (2003) used Rayleigh-Ritz method to analyzethe buckling and free vibration of free-free isotropicflat panel under non-uniform in-plane thermal stresses.Author expressed the stress functions as orthonormalbeam function and assumed that the flat panel is stress-free at all the edges to calculate in-plane stresses.Morimoto et al. (2006) evaluated the prebucklingthermal stress distribution within functionally gradedrectangular plates due to localized thermals loadingsby solving the thermoelastic problem. Using these in-plane stresses the governing equations for the platestability problem are formulated which are solved byGalerkin method to compute critical bucklingtemperature.

From the literature survey, it is observed thatanalytical expression is not available for prebucklingstress distribution within the layered compositecylindrical panel subjected to in-plane concentratedloading/localized thermal loading. In the presentinvestigation, the semi-analytical expressions for the

in-plane stresses within the layered compositecylindrical panel due to non-uniform in-planemechanical / thermal loadings are developed by solvingthe in-plane elasticity problem. Subsequently, usingthese in-plane stresses within the panel, the governingnonlinear stability equations of layered compositecylindrical panel are formulated in terms of

displacement (u-v-w) and rotation ( ,x y ) variables.

Using Galerkin’s method, the governing partialdifferential equations satisfying the boundaryconditions are reduced into a system of nonlinearalgebraic equations. These equations are solved usingNewton-Raphson method in conjunction with Riksapproach to trace the postbuckling equilibrium paths.The effects of radius-to-width ratio, geometricimperfections and different types of non-uniformmechanical/localized thermal loadings on the nonlinearstability analysis of the layered composite cylindricalpanel are examined.

Formulation

Kinematic EquationsConsider a composite cylindrical panel of length a,breadth b, thickness h and composed of N orthotropiclayers of uniform thickness and subjected to non-uniform in-plane mechanical and thermal loading asshown in Fig. 1. Let the coordinate system of themiddle surface of the panel coincide with x-y axes ofthe cartesian co-ordinate (x-y-z) system and the linesof principal curvature (1/R) of the middle surface,and z coordinate is normal to the middle surface. Inthe present study the governing equation of thecylindrical panels is derived based on Donnell’s shelltheory considering higher order shear deformationtheory. The strains at a distance z from the middlesurface can be expressed as (Soldatos, 1991),

( )

o axx xxxx xxo a

yy yy yy yy

o axy xyxy xy

z f z

(1)

Where , ,o o oxx yy xy , , ,xx yy xy ,

, ,a a axx yy xy and ,yz xz are membrane strains,

bending strains, additional bending strains due to

Nonlinear Stability Characteristics of Composite Cylindrical Panel 273

additional changes of curvature and shear strainsrespectively and can be defined as,

2

, , , ,

2

, , , ,

, , , , , , , ,

1

21

/2

o o oo x x x xxx

o o o oyy y y y y

o o o o o o oxy y x x y x y y x

u w w w

v w R w w w

u v w w w w w w

(2)

,

,

,2

oxxxxo

yy yy

oxy yy

w

w

w

(3)

,

,

, ,

axx x x

ay yyy

a x y y xxy

(4)

'( )yz y

xz x

f z

(5)

Here, , andu v w are the displacements at ageneric point that is ‘z’ distance away from the middlesurface of the panels along orthogonal co-ordinatesx, y and z, respectively. u0, v0 and w0 thecorresponding displacements of a generic point onthe middle surface, and x and y are the rotation ofcross sections initially perpendicular to the x and yaxes, respectively. (.),x and (.),y denote the partialdifferentiation with respect to x and y respectively. Inthe equation (2), the von Kármán strain-displacement(nonlinear strains) relations are used to account forthe deformed configuration of the shallow cylindricalpanels with moderate displacements and smallrotations. The initial geometric imperfection w* (i.edeviation of the middle surface from the perfect panel)is assumed as,

cos cosx y

w ea b

(6)

Where, e is the amplitude of imperfection.

Governing Differential Equations

The partial differential equations governing the non-linear stability of composite shallow cylindrical panelsubjected to non-uniform in-plane mechanical andthermal loading in the cartesian co-ordinate systemare expressed in terms of in-plane force, moment,additional moment and shear resultants respectivelyusing variation principle as,

, ,

, ,

, , ,

, ,

, ,

ˆ ˆ 0

ˆ ˆ 0

ˆ( / ) 2

ˆ ˆ ˆ ˆ( , , ), ( , , ), 0

- Q 0

- Q 0

xx x xy y

xy x yy y

yy y xx xx xy xy yy yy

xx x xy y x yy y xy x y

axx x xy y xx

axy x yy y yy

N N

N N

N R M M M

N w N w N w N w

P P

P P

(7a-e)

Here,

Here, ˆ ( ) ( , , )ij ij ij T ijN N n n i j x y (8)

In equation (8), nij and (nT)ij are the panelinternal stress resultants due to non-uniform in-planemechanical/localized thermal loadings respectively,and Nij are the secondary stress resultants due to the

large deformation. Therefore, ˆijN are the resultant

cylindrical panel internal stress resultants. Mij and Pij

Fig. 1: Geometry and non-uniform mechanical and localizedthermal loading for cylindrical panel


are the moment and additional moment resultants

respectively, and axxQ and a

yyQ are additional

transverse shear force resultants.

The force, moment, additional moment and shearresultants are related to membrane, bending, additionalbending and shear strains respectively, through theconstitutive relations:

ˆ 0A B C aTN N (9)

T 0B D E aTM M (10)

T 0C E Fa a aTM M (11)

HQ (12)

Where, panel stiffness A, B, C, D, E, F and Hare defined in terms of the transformed lamina

stiffness Q as,

/ 22

/ 2

2

1 1

/ 2

/ 2

1 1

/ 2' '

/ 2

1' '

( , , ) (1, , )

(1, , ) , 1, 2,6

( , , ) (1, , ( ))

(1, , ( )) ( ) , 1, 2,6

( ) ( )

( ) ( )

h

ij ij ij ijh

zN k

ijk zk

h

ij ij ij ijh

zN k

ijk zk

h

ij ijh

zk

ijzk

A B D Q z z dz

Q z z dz i j

C E F Q z f z dz

Q z f z f z dz i j

H Q f z f z dz

Q f z f z dz

1

, 4,5N

ki j

(13)

Where, N is the number of orthotropic layers. In theabove equations (9)-(11), thermal force

( ) ( ) ( )TT T xx T yy T xyN N NN , thermal

moment ( ) ( ) ( )TT T xx T yy T xyM M MM , and

additional thermal moment TaTM =

( ) ( ) ( )a a aT xx T yy T xyM M M resultants are related

to transformed thermal expansion coefficients

Txx yy xy by the relation,

1 1

, ,

(1, , ( )) ( , , )zN k

k zk

z f z T x y z dz Q

aT T TN M M

(14)

Here, xx, yy and xy are defined as,

2 21 2

2 21 2

1 2

= cos sin

= sin cos

= 2( )sin cos

xx

yy

xy

(15)

where, is the angle between x-axis and fiberdirection. 1 and 2 are respectively the coefficientsof thermal expansion in the principal material axes 1and 2. In the present case, the thermal moment andadditional thermal moment resultants

( 0 aT T

M M ) are zero due to uniform

temperature rise though the panel thickness. It is alsoassumed that there is no heat exchange between thepanel and the surroundings. Using equations (8)-(12),the non-linear governing partial differential equationsof the layered composite cylindrical panel areexpressed in terms of displacement (uo, vo, wo) androtation (x, y) variables, the details of which aregiven in Appendix A.

In-Plane Elasticity Problem

In this section, semi-analytical expression for the in-plane pre-buckling stresses (xx, yy and xy) withinthe layered composite shallow cylindrical panel underconcentrated in-plane mechanical and localizedthermal loadings are developed. For localized thermalloadings, the temperature distribution is assumed tobe symmetrical about the co-ordinate axes anduniform across the thickness for ease of analyticaldevelopment for the in-plane prebuckling stressexpression within the cylindrical panel. Since the


applied in-plane loading is not uniform, all threecomponents of stresses exist within the compositecylindrical panel. These stresses have been computedby solving the plane elasticity problem and satisfyingthe in-plane stress boundary conditions for mechanicalloads and in-plane displacement boundary conditionsfor localized thermal loadings.

The different distributions of in-plane mechanicalloadings adopted in the present case along an edge ofthe panel are,

2

2

1( ) exp

yy

c c

for concentrated load;

1

2 2sin cos

n

b d n d n y

d b n b b

for partial load;

= 1 for uniform load (16)

Here, c is the intensity of concentrated loadwhose value depends on numerical study and d is thewidth of partial edge load.

The temperature distributions over the panel fordifferent types of nonuniform heating are,

Full heating: ( , ) FT x y T

/2 and / 2x a y b (17)

Dome profile heating:

( , ) cos / cos /DT x y T x a y b

/2 and / 2x a y b (18)

Central circular heating:

2 2

2 2

at

and( , )

0 at /2

and / 2

CT y r x

x rT x y

r x y b

r x a

(19)

Central rectangular heating:

0

0

0

0

at /2

and / 2( , )

0 at /2 /2

and / 2 /2

RT x a

y bT x y

a x a

b y b

(20)

where, TF, TD, TR and TC are respectively the uniformtemperature rise through its thickness for full heating,dome profile heating, central rectangular heating andcentral circular heating. r is the radius of circularheating and, a0 and b0 are respectively the sides ofthe central rectangular heating area.

(a) Stress distribution within the cylindrical panelunder in-plane mechanical loading:

Inplane stress equilibrium equation (in the absence ofthermal loading) in terms of stress function () forcomposite cylindrical panel are obtained using strain-compatibility conditions and are,

4 4 4

22 12 66 114 2 2 4(2 ) 0a a a a

x x y y

(21)

Here, the compatibility equation is written interms of xy coordinates instead of R coordinates asthe shell panel is shallow. Airy’s stress function (isdefined by

2 2 2

2 2, ,xx yy xyn n n

x yy x

(22)

where, aij is the flexibility coefficient of the laminatedpanel, and can be obtained using the followingequations,

111 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

a a a A A A

a a a A A A

a a a A A A

(23)

Airy’s stress solutions is assumed in terms ofseries as,


1

20

1

( , ) ( ) cos( )

( ) cos( )

i ii

j jj

x y r y x

s x y R y

(24)

where,2 2

,i ji j

a b

Substituting the above solutions in the inplane

stress equilibrium equation (21) and equating the

coefficients of cos(ix) and cos(j y) results in two

ordinary differential equations in ri(y) and sj(x)

respectively,

4 22 4

11 12 66 224 2

4 22 4

22 12 66 114 2

( ) ( )(2 ) ( ) 0

( ) ( )(2 ) ( ) 0

i ii i i

j jj j j

d r y d r ya a a a r y

dy dy

d s x d s xa a a a s x

dx dx

(25)

Substituting ri(y) = exp(x) and sj(x) = exp(x)

in the above equations, the functions ri(y) and sj(x)are computed by considering the symmetry conditionabout x and y axes (i.e ri(y) = ri(–y) and sj(x) = sj(–x). Thus,

1 1 2 2

1 1 2 2

( ) cosh( ) cosh( )

( ) cosh( ) cosh( )i i i i i

j j j j j

r y R y R y

s x S x S x

(26)

Substituting the expressions for ( )ir y and ( )js x

in equation (24), the expression for Airy’s stresssolution is written as,

1 1

2 21

1 1 20

2 21

cosh( )( , ) cos( )

cosh( )

cosh( ) + cos( )

cosh( )

i ii

i ii

j jj

j jj

R yx y x

R y

S xy R y

S x

(27)

The inplane stress resultants are obtained bysubstituting the stress function (equation (24)) inequation (22). Thus,

21 1 1

21 2 2 2

1 1 20

2 21

cosh( )cos( )

cosh( )

cosh( ) cos( ) 2

cosh( )

i i ixx i

i i i i

i jj j

i jj

R yn x

R y

S xy R

S x

(28)

1 1 2

2 21

21 1 1

21 2 2 2

cosh( )cos( )

cosh( )

cosh( ) + cos( )

cosh( )

i iyy i i

i ii

i j jj

j i j j

R yn x

R y

S xy

S x

(29)

1 1 1

2 2 21

1 1 1

2 2 21

sinh( )sin( )

sinh( )

sinh( ) + sin( )

sinh( )

i i ixy i i

i i ii

i j jj j

i j jj

R yn x

R y

S xy

S x

(30)

The coefficients, Ri1, Ri2, Sj1, Sj2 in expressions

nxx(x, y), nyy(x, y) and nxy(x, y) are determined using

stress boundary conditions, which is defined as,

, ( ), , 0,2 2

, 0, , 02 2

xx xy

xy yy

a bn y y n x

a bn y n x

(31)

Satisfying the boundary conditions results insimultaneous equations in terms of the unknowncoefficients as,

2

21 2

11

sinh2

sinh2

i

ii i

ii

b

R Rb

(32)

2

21 2

11

sinh2

sinh2

j

jj j

jj

a

S Sa

(33)


2 2 2 1 22

1

21 12

21 11

2 2

cosh coth sinh2 2 2

sinhcos2 22

sinh sinh2 2

i i i ii i

i

jjjj

jj jj

j

b b bR

abI

Sa aa

I

(34)

2 2 1 222

1

21 32

0 21 11

2 4

cosh coth sinh2 2 2

sinhcos2 2 22

sinh sinh2 2

j j j jj j

j

iiii

ii ii

i

a a aS

baI

I Rb bb b

I

(35)

/ 20 0

1( )

bR y dy

b (36)

Here, / 20 0

2 ( )cos( ) ;b

jI y y dy

/ 21 10

2 cosh( ) cos( ) ;a

j iI x x dx

/ 22 20

2 cosh( ) cos( ) ;a

j iI x x dx

/ 23 10

2 cosh( ) cos( ) ;b

i iI y y dy

/ 24 20

2 cosh( ) cos( ) .b

i iI y y dy

Solving the above simultaneous equations, the

unknown coefficients, Ri1, Ri2, Sj1, Sj2 are determined.

Now, the explicit expression for stress distribution oflayered composite panel under non-uniformmechanical loading are computed using equations(28)-(30).

(b) Stress distribution within the cylindrical panelunder localized thermal loading:

The temperature distribution over partial rectangularand circular regions are expressed over the completedomain of the panel by using Fourier series as,

1 1( , ) cos( ) cos( )ij i j

i jT x y f x y

(37)

where,

2 2

0

16 cos( )sin

rC

ij i jj

Tf x r x dx

ab

for rectangular heating at the centre.

20 0

16sin( / 2)sin( / 2)R

ij i ji j

Tf a b

ab

for rectangular heating at the centre.

and,(2 1) (2 1)

, .i ji j

a b

Now, the thermal force resultants TTN =

( ) ( ) ( )T xx T yy T xyN N N in equation (14) can be

expressed in terms of in-plane temperature distributionas,

1 1cos( ) cos( )C ij i j

i jf x y

TN N (38)

where, T11 21 31N N NCN is termed as

coefficient of thermal force resultants and is,

1 1

zN k

Ck zk

dz QN (39)

In-plane stress equilibrium equation (presenceof thermal loading) in terms of Airy’s stress function() is derived using strain-compatibility conditions andare defined as,

4 4 4

22 12 66 114 2 2 4

22 2

11 122 2 2

2 2

22 662

(2 )

( )( ) ( )

( ) ( )

T yyT xx T xx

T yy T xy

a a a ax x y y

NN Na a

y x y

N Na a

x yx

(40)


In the above equation, (NT)xy is zero due to

symmetric laminated composite cylindrical panel. The

general solution (x, y) of equation (40) has two

components: complementary solution CS(x, y) and

particular integralPI(x, y),

CS PI( , ) ( , ) ( , )x y x y x y (41)

Following Morimoto et al. (2006), thecomplementary solution is assumed in terms of seriesas,

CS1

1

( , ) ( ) cos( )

( ) cos( )

i ii

j jj

x y r y x

s x y

(42)

Substituting the above complementary solutionsin the in-plane stress equilibrium equation (40) andequating the coefficients of and results in two ordinarydifferential equations in and respectively,

4 22

11 12 664 2

422

4 22

22 12 664 2

411

( ) ( )(2 )

( ) 0

( ) ( )(2 )

( ) 0

i ii

i i

j jj

j j

d r y d r ya a a

dy dy

a r y

d s x d s xa a a

dx dx

a s x

(43)

Substituting ri(y) = exp(y) and sj(x) = exp(x)

in the above equations, the functions and are

computed by considering the symmetry about x and y

axes (i.e., ri(y) = ri(–y) and sj(x) = sj(–x). Thus,

1 1 2 2

1 1 2 2

( ) cosh( ) cosh( )

( ) cosh( ) cosh( )i i i i i

j j j j j

r y R y R y

s x S x S x

(44)

Substituting the expressions for ri(y) and sj(x)

in equation (42), the expression for complementary

solution is written as,

1 1CS

1 2 2

1 1

1 2 2

cosh( )( , ) cos( )

cosh( )

cosh( ) + cos( )

cosh( )

i ii

i i i

j j

jj j j

R yx y x

R y

S xy

S x

(45)

The particular integral of equation (40) is writtenas,

PI1 1

2 212 11 22 21 11 11 12 21

4 2 2 422 12 66 11

( , ) cos( ) cos( )

( ) ( )

(2 )

ij i ji j

i j

i i j j

x y f x y

a N a N a N a N

a a a a

(46)

The in-plane thermal stress resultant due tolocalized thermal loadings is obtained by substitutingthe stress function (equation (41)) in equation (22).Thus,

21 1 1

212 2 2

1 1 2

1 2 2

2

1 1

2 212 11 22 21 11 11 12 21

22

cosh( )( ) cos( )

cosh( )

cosh( ) cos( )

cosh( )

+ cos( )cos( )

i i iT xx i

ii i i

i j

j jj i j

ij i j ji j

i j

R yn x

R y

S xy

S x

f x y

a N a N a N a N

a

4 2 2 4

12 66 11(2 )i i j ja a a

(47)

1 1 2

1 2 2

21 1 1

212 2 2

cosh( )( ) cos( )

cosh( )

cosh( ) + cos( )

cosh( )

i iT yy i i

i i i

i j j

jj

i j j

R yn x

R y

S xy

S x


2

1 1

2 212 11 22 21 11 11 12 21

4 2 2 422 12 66 11

+ cos( ) cos( )

(2 )

ij i j ii j

i j

i i j j

f x y

a N a N a N a N

a a a a

(48)

1 1 1

1 2 2 2

1 1 1

1 2 2 2

1 1

212 11 22 21 11 1

sinh( )( ) sin( )

sinh( )

sinh( ) + sin( )

sinh( )

+ sin( )sin( )

i i iT xy i i

i i i i

i j j

j jj i j j

ij i j i ji j

i

R yn x

R y

S xy

S x

f x y

a N a N a N

21 12 21

4 2 2 422 12 66 11(2 )

j

i i j j

a N

a a a a

(49)

For the case of thermal loading, the in-planedisplacements are zero at the boundary. Therefore,in-plane displacements are obtained by substitutingthe in-plane thermal stress resultants

( ) , ( ) , ( )T xx T yy T xyn n n and thermal forces

( ) , ( ) , ( )T xx T yy T xyN N N in strains expression

(equation (9)). Integrating strains expression, the in-plane displacement are expressed as,

( , )ou x y

21

1 1 11 12

212

2 2 11 12

sin( ) cosh( )

sin( ) cosh( )

ii i i i

i

ii

i i i ii

R x y a a

R x y a a

2

1 1 11 12 11

21

2 2 11 12 22

sinh( )cos( )

+

sinh( )cos( )

jj j j j

j

jj

j j j jj

S x y a a

S x y a a

1 1

2 2 211 22 12 11 11 66 11

2 212 11 22 12 66 21

4 2 2 422 12 66 11

sin( ) cos( )

( )

( )

(2 )

ij i i ji j

i j

j

i i j j

f x y

a a a N a a N

a a a a a N

a a a a

(50)

( , )ov x y

2

1 1 22 12 11

21

2 2 22 12 22

cos( )sinh( )

cos( )sinh( )

ii i i i

i

ii

i i i ii

R x y a a

R x y a a

21

1 1 22 12

212

2 2 22 12

cosh( )sin( )

cosh( )sin( )

jj j j j

j

jj

j j j jj

S x y a a

S x y a a

1 1

2 2 211 22 12 21 22 66 21

2 212 11 22 12 66 11

4 2 2 422 12 66 11

cos( )sin( )

( )

( )

(2 )

ij j i ji j

j i

i

i i j j

f x y

a a a N a a N

a a a a a N

a a a a

(51)

The coefficients, Ri1, Ri2, Sj1, Sj2 in expression

u(x, y) and v(x, y) are determined using immovable


edge boundary conditions, u0 = 0, v0 = 0. Satisfyingthe immovable edge boundary conditions results insimultaneous equations in terms of the unknowncoefficients as,

21 1

11 12

2 122 2

11 12

cosh2

cosh2

i ii

ii i

i ii

i

ba a

R Rb

a a

(52)

21 1

22 12

2 122 2

22 12

cosh2

cosh2

j jj

j

j jj j

jj

aa a

S Sa

a a

(53)

21 1

11 12

2 22 1 1

2 2 2 21 2

1 12 2

1

4sin

2

cosh sin2 2

sinh cos2 2

i ii

i

j i i ji

i j i j

ji i

i j

aa a

b

bb

bb

1

2 212 2

22

2 21

11 12 1 11 12 21 2

cosh2 sinh cos

2 2cosh2

sinh2

i

ji ii

ii j

j j jj j

j j

bbb

Rb

aa a a a

21

122 122

1222

22 12

cosh2 sinh

2cosh

2

jjj

j jj

jjj

j

aa aa

Sa

a a

1

2 2 211 22 12 11 11 66 11

2 212 11 22 12 66 21

4 2 2 422 12 66 11

sin2

( )

( )0

(2 )

iij i

i

i j

j

i i j j

af

a a a N a a N

a a a a a N

a a a a

(54)

21

22 12 11

21

11 122

22 12 2 22 2

11 12

sinh2

i ii

i

ii

iii

i ii

i

ba a

a a

a a

a a

1

21

2

21

22 12

cosh2 sinh

2cosh2

4sin

2

i

ii

i

j jj

j

bb

Rb

ba a

a

2 22 1 1

2 2 2 21 2

1 1

2 21

cosh sin2 2

sinh cos2 2

i j j j i

j i j i

j j i

j i

a a

a a

1

2 212 2

22

1

cosh2 sinh cos

2 2cosh

2

sin2

j

j j ij

jj i

jij j

j

aa a

Sa

bf


2 2 211 22 12 21 22 66 21

2 212 11 22 12 66 11

4 2 2 422 12 66 11

( )

( )0

(2 )

j i

i

i i j j

a a a N a a N

a a a a a N

a a a a

(55)

Solving the above simultaneous equations, the

unknown coefficients, Ri1, Ri2, Sj1, Sj2 are determined.

Now, the explicit expression for thermal stressdistribution of layered composite panel under non-uniform thermal loading are computed using equations(47)-(49). Subsequently, suing these stressdistributions, the governing equations for shallow shellpanel stability are formulated in displacement androtation variables which are given in Appendix A.

Solution Technique

Boundary Conditions of the Problem

In the present investigation, simply supported boundaryconditions are considered in the out of plane directionof the cylindrical panel. The movable edge in-planeboundary conditions are assumed for mechanical loadsand immovable edge in-lane boundary conditions areassumed for thermal loading. The following two setsof simply supported boundary conditions are assumedfor cylindrical panel,

Movable edge boundary conditions:

0

/ 2, / 2

0

/ 2, / 2

o o oxx xx y xx

o o oyy x yy yy

N v w P M

at x a a

u N w P M

at y b b

(56)

Immovable edge boundary conditions:

0

/ 2, / 2

0

/ 2, / 2

o o o oxx y xx

o o o ox yy yy

u v w P M

at x a a

u v w P M

at y b b

(57)

To satisfy the above simply supported boundaryconditions, the following sets of displacement fieldsare assumed in the Galerkin’s method.

Displacement fields for movable edge boundaryconditions are,

1 1

1 1

1 1

1 1

1

sin cos

cos sin

cos cos

sin cos

cos sin

jio

mnm n

jio

mnm n

jio

mnm n

ji

x mnm n

j

y mnn

m x n yu U

a b

m x n yv V

a b

m x n yw W

a b

m x n yK

a b

m xK

a

1

i

m

n y

b

(58)

Displacement fields for immovable edgeboundary conditions are,

1 1

1 1

1 1

1 1

1

2sin cos

2cos sin

cos cos

2sin cos

cos

jio

mnm n

jio

mnm n

jio

mnm n

ji

x mnm n

j

y mnn

m x n yu U

a b

m x n yv V

a b

m x n yw W

a b

m x n yK

a b

m xK

a

1

2sin

i

m

n y

b

(59)

Where, m and n denote the number of halfwaves in x and y directions. The resulting algebraicequations after the Galerkin projection of the nonlineargoverning equations are solved using Newton-Raphson method in conjunction with Riks approachto trace the post-buckling equilibrium path.

Results and Discussions

To validate the present formulation, the postbucklingequilibrium paths of layered composite cylindrical panelunder uniform uni-axial compressive loading and fullthermal loading over the panel are compared withliterature and ABAQUS results in Fig. 2a and Fig.2b. It is observed from these figures that thepostbuckling equilibrium path obtained for in-planemechanical loading compares well with that of Girishand Ramachandra (2006) and the equilibrium path


for thermal loading compares fairly well with that ofABAQUS results. Here, Nxcr and Tcr are the criticalbuckling load and of a shallow panel and criticaltemperature of a flat panel respectively.

Following material properties are used forvalidating the postbuckling equilibrium path ofcylindrical panel under in-plane mechanical / thermalloading.

After validating the formulation, some newresults for postbuckling equilibrium paths of compositecylindrical panel subjected to different types of non-uniform in-plane mechanical and thermal loadings arepresented in this section.

Following material properties are used in thepresent study:

(i)

11 22 12 13 22

23 22 12

5 51 2

/ 25, 0.5 ,

0.2 , 0.25,

0.1 10 , 0.2 10

E E G G E

G E v

for

tracing the postbuckling equilibrium paths of cylindricalpanel under in-plane mechanical loading.

(ii)

11 22 12 13 2

23 2 12

5 51 2

/ 20, 0.5 ,

0.2 , 0.25,

0.1 10 , 0.2 10

E E G G E

G E v

for

tracing the postbuckling equilibrium paths of cylindricalpanel under thermal loading.

The postbuckling equilibrium paths of thecylindrical panel are traced for three types of in-planemechanical loadings: concentrated, partial and uniformload. To trace the thermal postbuckling equilibriumpaths, two sets of thermal loadings have beenconsidered. In the first set, four cases of the partialcentral rectangular heating have been considered.The length and width of heating region for four casesare,

Case 1: 0 0/ / 0.25,a b b b

Case 2: 0 0/ / 0.50,a b b b

Case 3: 0 0/ / 0.75,a b b b

Fig. 2A: Postbuckled equilibrium paths of a three layeredsymmetric cross-ply [0/90//0] simply supportedcomposite cylindrical panel (a/b = 1, a/h = 50, R/a =5) under uniform uni-axial in-plane loading

Fig. 2B: Postbuckled equilibrium path of a four layeredsymmetric cross-ply [0/90/90/0] simply supportedc o m p o s i t e c y l i n d r i c a l p a n e l ( a/b = 1, b/h = 100, R/b =10) under full thermal loading

Fig. 3A: The dimensionless stress distribution ( xx ) for a

composite panel along line x = 0 for differentnonuniform in-plane mechanical loadings


Case 4: 0 0/ / 1.00.a b b b

In the above four cases, width of the panel (b)is kept constant and the area of heating region

0 0 0A a b is varied.

In the second set, four different types of thermalheating have been considered over square plate. Theyare full heating, dome profile heating, central circularheating and central rectangular heating. The radius

of circular heating (r) is taken as2 2

band side for

rectangular heating is a0/b = 0.50 and b0/b = 0.50.

Fig. 3A and Fig. 3B, show the dimensionless in-plane normal stress distribution for a four layeredcross-ply [0/90/90/0] shallow cylindrical compositepanel (a/b = 1, a/h = 100) under different types ofnonuniform mechanical in-plane loadings and fourcases partial central rectangular heating respectively.The normal stresses due to nonuniform mechanicalloading/localized heating are made dimensionless bydividing it by stresses corresponding to uniform in-plane mechanical loading/full heating. The distributionsof loadings are only different but the total load appliedis kept the same for all types of mechanical loadingswhile calculating the prebuckling stress distribution.The total heat supplied for each case of centralrectangular heating is the same. Therefore, theuniform temperature rise through the panel thicknessfor four cases can be related by,

31 2 49

16 4 16 1RR R RTT T T

where, TR1, TR2, TR3 and TR4 are temperature risethrough the panel thickness for case 1, case 2, case 3and case 4 respectively.

As per St. Venant’s principle, the applied loadingmust diffuse into a state of uniform stress away fromthe boundary. It is well known that the stressdistribution becomes uniform for the case of uniformmechanical loading and full thermal heating for squarecomposite panel. However, the prebuckling stressdistribution become uniform only for panel with higheraspect ratio (a/b > 9) under nonuniform mechanicalin-plane loading (Fig. 3C).

The postbuckling equilibrium paths arerepresented as a plot of dimensionless in-plane load / or /xx xcr crN N T T against dimensionless out-of-plane displacement (w/h) at the centre. In Fig. 4, thepostbuckling equilibrium paths for simply supportedfour layered symmetric cross-ply [0/90/90/0]composite cylindrical panel (a/b = 1, b/h = 100, R/b =10) subjected to concentrated in-plane loading aretraced considering different number of terms in thedisplacement fields. The converged result for

Fig. 3B: The dimensionless in-plane thermal normal stress

( ( ) ( ) /( )FT xx T xx T xx ) distribution for composite

panel along line x = 0 for four cases of partial centralrectangular heating

Fig. 3C: The dimensionless in-plane stress ( xx )

distribution for a composite panel along line x = 0under concentrated in-plane mechanical loadingfor different aspect ratios


equilibrium path is obtained for displacement field withf o u r t e r m s i n e a c h o f u, v, w, x and y. Thesubsequent results are reported with four terms inthe displacement fields. In Fig. 5, the postbucklingequilibrium paths of simply supported four layeredsymmetric cross-ply [0/90/90/0] composite cylindricalpanel (a/b = 1, b/h = 100, R/b =10) is given for varioustypes of inplane loading distribution. Here, Nxcr is thecritical load of panel under concentrated loading.Bifurcation buckling is observed for panels underuniform loading only. Bifurcation point is not observedfor other loading cases as non-uniform loading inducescompressive stress as well tensile stress within thepanel. The tensile stress in the unloading directioncauses the panel to deflect out-of-plane as soon asthe load is applied due to curvature of the panel. Inall cases snap through buckling is observed.

In Fig. 6, the postbuckling equilibrium paths ofsimply supported four layered symmetric cross-ply[0/90/90/0] composite cylindrical panel (a/b = 1, b/h= 100) subjected to concentrated inplane loading isgiven for different radius to width (R/b) ratios. It isobserved from the figure that the limit point and snapthrough behaviour exist for panels with R/b < 10. Forhigher radius to width ratios the postbuckledequilibrium path shows hardening type of behaviour.The influence of initial geometric imperfections onthe postbuckling equilibrium path of a simply supportedfour layered symmetric cross-ply [0/90/90/0]composite cylindrical panel (a/b = 1, b/h = 100, R/b =10) under concentrated in-plane loading is presentedin Fig. 7.

It is observed from the figure that the bifurcationbuckling does not occur in case of perfect panels underconcentrated in-plane mechanical loading. Snapthrough behaviour is observed due to initial inward(+ve) imperfections (0.01h, 0.05h, 0.1h) and for smallmagnitude of outward (–ve) imperfection (–0.01h).The panel deflects outward for higher amplitude ofoutward imperfections (–0.05h, –0.1h) and showshardening behaviour.

The influence of load ratio

0 /yy xxN N N on the postbuckling equilibrium

path of a simply supported four layered symmetriccross-ply [0/90/90/0] composite cylindrical panel (a/

b = 1, b/h = 100, R/b = 10) under concentrated in-plane loading is presented in Fig. 8. The load ratio isdefined as the ratio of compressive/tensile edge loadin the y-direction to compressive edge load in the x-direction. Here, compressive loading is taken aspositive and tensile in-plane loading is taken asnegative. It is observed from the curves that the paneldeflects in the inward direction for all negative loadratios. Also, for small positive value of load ratio (N0= 0.01, 0.015) the panel deflect in the inward directionand exhibit snap though behaviour, and for higher valueof load ratio (N0 = 0.025) the panel deflects in theoutward direction and shows hardening behaviour.After limit point the load carrying capacity of shellpanels (Figs. 6, 7 and 8) initially decrease due todecrease in curvature of the deformed panel. After

Fig. 5: The effect of different types of nonuniform in-planeloadings on the equilibrium path of a simply supportedcomposite cylindrical panel

Fig. 4: Postbuckled equilibrium paths of a simply supportedcomposite cylindrical panel under concentrated in-plane loading


flattening of shell panel, it develops curvature in theopposite direction which results in increased loadcarrying capacity.

Fig. 9 shows the postbuckling equilibrium pathof a four layered cross-ply [0/90/90/0] simplysupported composite cylindrical panel (a/b = 1, b/h =100, R/b = 10) under four cases of partial centralrectangular heating. In the figure, Tcr is the criticalload of the flat panel under full heating. Due to thermalloadings the biaxial compressive stresses aredeveloped within the heating region which causes thepanel to deflect outwards and hence curvature of thepanel is increased which results in hardeningbehaviour. The temperature deflection curve is almost

linear for the case of full heating and it becomesnonlinear for localized heating.

Fig. 10 shows the postbuckling equilibrium pathof a four layered cross-ply [0/90/90/0] simplysupported composite cylindrical panel (a/b = 1, b/h =100, R/b = 10) under different types of thermalheating. Here, Tcr is the critical load of the flat panelunder full heating. It is observed from the figure thatthe panel deflects in the outward direction due tocompressive stress within heating region in both thedirections. This results in hardening type of behaviour.It is observed from the figure that the panel deflectsin the outward direction for all the cases of localizedthermal loadings.

Fig. 6: The influence of radius-to-width (R/b) ratio on theequilibrium path of a simply supported compositecylindrical panel under concentrated in-planeloading

Fig. 7: The influence of geometric imperfections on thepostbuckling equilibrium path of a simply supportedcomposite cylindrical panel under concentrated in-plane loading

Fig. 8: The influence of load ratio (N0) on the postbucklingequilibrium path of a simply supported compositecylindrical panel under concentrated in-planeloading

Fig. 9: Influence of four cases of the localized thermalloading on the equilibrium path of a simply supportedcomposite cylindrical panel


The influence of radius to width ratios on thepostbuckling equilibrium path of a simply supportedfour layered symmetric cross-ply [0/90/90/0]composite cylindrical panel (a/b = 1, b/h = 100) underpartial central rectangular heating (a0/b = b0/b = 0.50)is presented in Fig. 11. Here, Tcr is the critical load ofthe flat. It is observed from the figure that the paneldeflects in the outward direction due to biaxialcompressive stresses within heating region. Theequilibrium path of the flat panel is symmetrical asshown in the figure under localized thermal loading.Hardening behaviour of composite panel increaseswith decrease of radius to width ratios.

Conclusions

In this article for the first time, semi-analyticalexpressions for the in-plane stresses within the layeredcomposite cylindrical panel due to non-uniform in-plane mechanical/localized thermal loadings aredeveloped by solving the in-plane elasticity/thermoelasticity problem. Subsequently, using theseprebuckling stress distributions, the governingnonlinear stability equations of panel are derived.Panels are modelled based on Donnell’s shallow shelltheory considering higher order shear deformationtheory and incorporating von-Kármán geometricnonlinearity. These are solved using Galerkin method.

It is observed that for nonuniform in-planemechanical loading the prebuckling stresses becomeuniform only for higher aspect ratio. The localizedthermal heating induces all three componentsprebuckling stresses within the panel. In case oflocalized heating, both and are compressive withinthe heating region and stresses suddenly drop andbecome tensile outside the heating region and theygradually become zero at the boundary. The magnitudeof tensile stress decreases with the increase oflocalized heating area. Whereas in case of non-uniform mechanical loading is tensile at some locationsof the panel and there is no sharp change in themagnitude of stress as in the case of localized thermalloading. Bifurcation buckling is not observed in casepanels subjected to non-uniform mechanical loadingsdue to tensile stresses developed in the unloadeddirection. The tensile stress in the unloaded directioncauses the panel to deflect out-of-plane as soon asthe load is applied due to curvature of the panel. Underthermal loadings the panel exhibit hardening type ofequilibrium paths due to in-plane stress distributionwithin the panel. Due to biaxial compression underthermal loading, the panel initially deflects outwards(-ve direction) and hence curvature of the panel isincreased which results in hardening behaviour. Underconcentrated loading, snap through behaviour isobserved due to initial inward imperfections and forsmall magnitude of outward imperfection. The paneldeflects outward for higher amplitude of outwardimperfections and shows hardening type of behaviour.The deformation of the panel is sensitive to the typeof mechanical load applied and magnitude and directionof initial imperfections.

Fig. 10: The influence of different types of nonuniformheating on the postbuckling equilibrium path of asimply supported composite cylindrical panel

Fig. 11: The influence of radius-to-width ratio on theequilibrium path of a simply supported compositecylindrical panel under localized (a0/b = b0/b = 0.50)central rectangular heating


Appendix A

Non-linear governing partial differential equations ofsymmetric cross-ply composite cylindrical panel interms of displacement (u0, v0, w0) and rotation

( ,x y ) variables are given below:

11 , 66 , 12 66 ,

* *11 , , 66 , , ,

*11 , 66 , , 12 66

* *, , , , , ,

, , , ,

( )

{ ( ) ( )}

( ) ( )

( )

( ) ( ) 0

o o oxx yy xy

o o oxx xx yy yy x

o oxx yy x

o o o oy xy y xy y xy

xx x T xx x xy y T xy y

A u A u A A v

A w w A w w w

A w A w w A A

w w w w w w

n n n n

(A-1)

12 66 , 66 , 22 ,

* *66 , , 22 , , ,

*66 , 22 , , 12 66

* *, , , , , ,

, , , ,

( )

{ ( ) ( )}

( ) ( )

( )

( ) ( ) 0

o o oxy xx yy

o o oxx xx yy yy y

o oxx yy y

o o o ox xy x xy x xy

xy x T xy x yy y T yy y

A A u A v A v

A w w A w w w

A w A w w A A

w w w w w w

n n n n

(A-2)

2 *12 , , , , 22 ,

2 *, , ,

11 , 12 66 , 22 ,

11 , 12 66 , 22 ,

12 66 , 11 , 12 66

1/ [ { 0.5( ) } { /

0.5( ) } ( ) ]

{ 2( 2 ) }

( 2 )

( 2 ) ( 2 )

(

o o o o ox x x x y

o oy y y yy T yy

o o oxxxx xxyy yyyy

x xxx x xyy y yyy

y xxy x xxx

R A u w w w A v w R

w w w n n

D w D D w D w

E E E E

E E E E E

2, , 22 , 11 , ,

* 2 *, , 12 , , , ,

*, , 66 , ,

* * *, , , , , , , ,

12 ,

) [ ( 0.5( )

) ( / 0.5( ) )

( ) ]( ) 2[ (

) ( ) }]( )

[ ( 0.5(

o ox xyy y xxy y yyy x x

o o o o ox x y y y y

o o oxx xx xx y xT xx

o o o o ox y x y x y xy xy xyT xy

ox

E A u w

w w A v w R w w w

n n w w A u v

w w w w w w n n w w

A u 2 *, , , 22 ,

2 * *, , , , ,

) ) ( /

0.5( ) ) ( ) ]( ) 0

o o o ox x x y

o o oy y y yy yy yyT yy

w w w A v w R

w w w n n w w

(A-3)

11 , 12 66 ,

11 , 66 , 12 66 ,

55

( 2 )

( )

0

o oxxx xyy

x xx x yy y xy

x

E w E E w

F F F F

H(A-4)

22 , 12 66 ,

66 , 22 , 12 66 ,

44

( 2 )

( )

0

o oyyy xxy

y xx y yy x xy

y

E w E E w

F F F F

H(A-5)

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