GEOMETRICALLY NONLINEAR ANALYSIS OF ...Geometrically Nonlinear Analysis of Composite Plate...

Proceedings of 3rd IRF International Conference, 18th May-2014, Hyderabad, India, ISBN: 978-93-84209-18-6

92

GEOMETRICALLY NONLINEAR ANALYSIS OF COMPOSITE PLATE

1GURANGOUDA S. PATIL, 2ANAND P.YALAWAR, 3SHRAVANKUMAR B. KERUR

1,2PG-Student, 3Associate professor, Dept. of Mechanical Engineering, Basaveshwar Engineering College, Bagalkot-587103

Abstract- Composites are being widely used in aeronautical, mechanical, civil and chemical industries due to their high strength to stiffness and low weight to stiffness ratios. This present paper discusses, the geometrically nonlinear behavior of composite plate with various boundary conditions. The finite element formulations is based on the first order shear deformation theory (FSDT). The finite element analysis has been carried out for a composite plate using an eight nodded quadrilateral element and five degree of freedom (DOF). Nonlinear equilibrium equations are linearised by using Newton-Raphson method. Numerical analysis has been carried out by developing program in analysis software ANSYS and various results are obtained. The result are compared with those available in the literature. The present paper emphasis on the effect of number of layers, stacking sequence and different boundary condition on geometrically nonlinear deflections composite plates. Keywords- FSDT, composite, geometric nonlinearity, cross-ply I. INTRODUCTION Composite materials are preferred in places where lighter materials are desired or required without sacrificing strength. They have even become essential for many applications. Composite materials are being used in a variety of applications such as the structural parts of aircrafts, automobiles, chemical equipment, transformer tubes, boats, etc. Some transmission gears make use of plastic materials in many different places such as watches, instruments, washing machines, gear pumps, etc. Composite materials are efficient in applications that required high strength to weight and stiffness to weight ratios. Brebbia and Connor [1] presented the finite element displacement formulation applicable to arbitrary plate and shell elements for geometrically nonlinear problems and also developed appropriate equation for Newton-Raphson iteration. Wood and zienkiewicz [2] studied the geometrically nonlinear analysis of beams frames and arches using total Lagrangian coordinate system by using modified isoparametric element and system nonlinear equation was solved using Newton-Raphson method. Bathe and Bolourchi [3] presented finite element method for linear, geometric, and material nonlinear analysis of plates and shell. Pica et al. [4] presented a geometrically nonlinear analysis of plates using finite element Mindilin formulation. Reddy et.al [5] developed a finite element model based on the combined theory of the yang, Norris, Stavsky and Von Karman [6] that, it accounts for the transverse shear strain, large rotations. Puctha and Reddy [7] developed a mixed shear flexible finite element with relaxed continuity, geometrically linear and nonlinear analysis of laminated anisotropic plates. Chia [8] solved the nonlinear bending of an unsymmetrically laminated angle ply rectangular plate analytically under lateral load satisfying the

von-karman type strain. Striz et al. [9] investigated the behavior of thin, circular isotropic elastic plates with the immovable edges and undergoing large deflections .they used Newton-Raphson technique to solve the nonlinear systems of equations. Z. G. Azizian and D. J. Dawe [10] studied geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method. Singh et al. [11] investigated the large deflections bending analysis of anti-symmetric rectangular cross-ply plates based on von-karman plate theory, with one–term approximation for the in-plane & transverse displacements, under sinusoidal loading. Clarke et al. [12] described various incremental- iterative technique based on the Newton-Raphson approach to analyze the geometric nonlinear behavior. II. FINITE ELEMENT FORMULATION AND METHODOLOGY The nonlinear finite element modelling and analysis is more complicated than linear one, as the stiffness matrix in nonlinear equilibrium equations is dependent on deformations. The first order shear deformation plate theory is considered in the finite element formulation. An eight- noded isoparametric plate/shell element is proposed for the geometrically nonlinear analysis of composite plates subjected to mechanical loading. The geometry and origin of the material coordinates are at the middle of the laminate as shown in Fig.1

Fig.1 Geometry of the laminated composite plate

Geometrically Nonlinear Analysis of Composite Plate


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III. DISPLACEMENT FIELD The first order shear deformation theory is used to describe the kinematics of deformation for the present analysis. The displacements components u, v, w along x, y, z respectively are expressed as

0

0

0

( , , , ) ( , , ) ( , , )( , , , ) ( , , ) ( , , )

( , , , ) ( , , )

x

y

u x y z t u x y t z x y tv x y z t v x y t z x y tw x y z t w x y t

(1) , ,u v w are total mechanical displacement at any

point in the laminate along , ,x y z axes respectively;

0 0 0, ,u v w are mid-plane (z=0) displacements in

, ,x y z axes respectively. x and y are rotations of normal to the mid plane about x and y axis respectively. Strain-displacement relation Shear deformable plate theory that include von man type geometrically nonlinearity are derived. Sander’s nonlinear strain-displacements relations associated with the displacements field are used. The strain-displacement relations are given by

0

0

0

0

0

xx xx xx

yy yy yy

xy xy xy

xz xz

yz yz

zkzk

zk

(2)

Where the strains of the middle surface are related to the displacements by

0 2

0 2

0

0

0

/ 1/ 2( / )

/ 1/ 2( / )

/ / ( / )( / )

/

/

xx

yy

xy

xz x

yz y

u x w xv y w y

u y v x w x w y

w xw y

(3) and the curvature are related as

//

/ /

xx x

xx y

xy x y

k xk yk y x

(4) The total strain ({ }) is the sum of linear ({ })L

and nonlinear ({ })NL strain components and written as

{ } { } { }L NL (5) The linear and nonlinear strain components are

/ // /

/ / ( /{ } / ( / / ))

//

Lxxx

Lyyy

Lxxy

Ly

Lxxz

Lyyz

u x z xv y z y

u y v x z yx v x u y

w xw y

(6)

2

2

1/ 2( / ){ } 1/ 2( / )

( / )( / )

NLxxNL

NL yyNLxy

w xw y

w x w y

(7) The membrane strains ( , ,xx yy xy ) are linear through the laminate thickness, while the transverse shear strains ( ,xz yz ) are constant through the thickness of the laminate. It follows that the transverse shear stress are also constant. Hence, shear correction factors is introduced to account for the non-uniform distribution of transverse shear strain across the thickness of the laminate. Constitutive relations The laminated composite plate is assumed to consist of number of thin elastic substrate laminates as shown in Fig.1 the material of each lamina is composed of an arbitrary orientation of fibers embedded in matrix material. A lamina is assumed as elastically orthotropic. The constitute equation of substrate lamina having fibre orientation ( ) with respect to material axes system (1-2-3) as shown in Fig.2 [13]may be expressed as below

Fig. 2 Lamina with material axes system.



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Stress strain relationship for a substrate lamina is given by

1 11 12 1

2 12 22 2

12 66 12

13 44 13

55 2323

0 0 00 0 0

0 0 0 00 0 0 00 0 0 0

s

s

Q QQ Q

Qk Q

k Q

(8) Where

11 12 22 2211 12 22

12 21 12 21 12 21

66 12 44 13 55 23

, , ,1 1 1

, ,

E E EQ Q Q

Q G Q G Q G

(9) The subscripts 1, 2, 3 represent the principle material axes. 11, 22 12 13 23, , ,E E G G G are elastic longitudinal, transverse and shear moduli of laminated shell respectively. sk is the correction factor. The constitutive relation is written in short as

Q (10)

Where , ,Q are stress, strain, constitutive matrix respectively. The global axis (x-y-z coordinate) lamina stiffness matrix are expressed as follows

11 12 16

12 22 26

16 26 66

44 45

54 55

0 00 00 0

0 0 00 0 0

s

s

Q Q QQ Q Q

Q Q Q Qk Q QQ k Q

(11) Where

4 4 2 211 11 22 12 66

2 2 4 412 11 22 66 12

4 2 2 422 11 22 66 22

3 316 11 12 66 12 22 66

3 326 11 12 66 12 22 66

2 266 11 12 12 6

( 2( 2 )

( 4 ) ( )( 2( 2 )

( 2 ) ( 2 )

( 2 ) ( 2 )

( 2 )

Q Q m Q m Q Q m nQ Q Q Q m n Q n mQ Q n Q Q m n Q mQ Q Q Q m n Q Q Q m nQ Q Q Q m n Q Q Q m nQ Q Q Q m n Q

4 46

2 244 44 45

45 55 45

2 255 55 45

( )

( )

m nQ Q m Q nQ Q Q mnQ Q m Q n

(12) Where cosm and sinn . The material axes (1, 2, 3) of each layer in the composite plate are oriented with respect to the laminate coordinates

(x,y,z), thus the constitutive equations of each layer is transformed to the laminate coordinates (x-y-z) using transformation matrices and integrating the constitutive relations through the composite laminate thickness , force, moment resultants in terms of the structure material stiffness matrices are given as

00

0 0

ij ij

ij ij

s sij ts

N A BkM B D

Q A

(13) Above equation is written in short as

F D (14) Where { } { }T

xx yy xyN N N N ,{ } { }Txx yy xyM M M M ,

{ } { }Txx yy xy , (15)

{ } { }Txx yy xyk k k k ,{ } { }T

ts xz yz (16) 1

2

1

[ , , ] [1, , ]k

k

zn

ij ij ij ijk z

A B D z z Q dz

, 1, 2,3i j

(17) 1

1

[ ]k

k

zn

sij s ijk z

A k Q dz

, 4,5;i j (Ks=5/6, shear

correction factor) (18) Energy equation The present analysis involves structural displacements due to external mechanical loading. The Potential energy of the system is given by

1

1 { } { }2

nk T k

pi v

T dv

(19)

Substitute the equation (10) in to (19), the above equation is rewritten as

1

1 { } [ ]{ }2

nk T k k

pi v

T Q dv

(20)

IV. VARIATIONAL PRINCIPLE The equilibrium configuration of the system can be determined from the following variational principle: an admissible force state satisfies the geometric constraints if, and only if, the variational indicator

pT vanishes for arbitrarily admissible force variations. This principle is known as Principle of Minimum Potential Energy and is given as

1

1 { } [ ]{ }2

nk T k k

pi v

T Q dv

(21)



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V. FINITE ELEMENT FORMULATION Geometrically nonlinear finite element is developed for the static analysis of laminated composite plate using first order shear deformation theory. An eight noded C0 continuous element is employed. Isoparametric element In the present analysis, an eight noded quadrilateral isoparametric element is used. The independent field variables (basic unknowns) based on the displacements field are , , , , .x yu v w The plate is discretized in to finite elements. The element configuration is shown in Fig.3. The shape function of the element are given by

21 5

2 22 6

23 7

24 8

1 1(1 )(1 )( 1); (1 )(1 )4 21 1(1 )(1 )( 1); (1 )(1 )4 21 1(1 )(1 )( 1); (1 )(1 )4 21 1(1 )(1 )( 1); (1 )(1 )4 2

N N

N N

N N

N N

(22)

Fig 3.Doubly curved isoparametric eight noded shell element.

Here and are local coordinates of the element. For the isoparametric element concept, the element geometry, displacement vectors over each element are represented by the interpolation functions.

8

1

;i ii

x N x

8

1

;i ii

y N y

1

{ } { }n

i ii

d N d

(23)

Where, 0 0 0{ } { } ,[ ]Tx y id u v w N is

the interpolation function for ith node, ,i ix y , are

geometry coordinates in x and y direction, { }id are unknown displacements vector for ith node of the

element. From the above equation, the displacement field is rewritten in the matrix form as { } [ ]{ }id N d (24) Where

Generalized strain vectors { } { } { }L NL at any point within the element are obtained by

1{ } ([ ] [ ]) { }2L NL iB B d (28)

Where, [ ]LB and [ ]NLB are linear and nonlinear strain displacements matrices respectively. Where

1 2 8[ ] [[ ] [ ]........[ ]]L L L LB B B B (29) and

0 0 0 0

0 0 0 0

0 0 0

0 0 0 0[ ] 1, 2 .. ...8

0 0 0 0

0 0 0

0 0 0

0 0 0

i

i

i i

i

L ii

i i

ii

ii

Nx

Ny

N Ny x

NxB iN

yN Ny x

N Nx

N Ny

(30) [ ] [ ] [ ]NLB A G (31)

0

[ ] 0

wx

wAy

w wy x

(32)

1 2 8[ ] ........G G G G (33)

0 0 0 0[ ] 1, 2,........8

0 0 0 0

i

ii

NxG iNy

(34)



96

The of the shape functions iN with respect to global coordinates ( x and y ) are expressed in terms of their derivate with respect to and by the following relationship

1[ ]

ii

i i

NNx JN Ny

(35)

Where [J] is Jacobian matrix given by

[ ]

x y

Jx y

(36)

Substituting the values of { } from eq. (28) in the eq. (21)

VI. RESULTS AND DISCUSSION By using the above described analysis procedure a nonlinear finite element code is developed in Ansys 11. In order to demonstrate the accuracy and applicability of the present formulation, several examples have been analyzed and the computed results are compared with published results. In all problems it is considered that all individual layers are taken to be equal thickness. In the given examples the deflection and the load are non-dimensionized as follows:

Central deflection: wWh

Load: uniformly distributed load 4 4

2Q qa E t Where a is inplane plate dimension; q is uniformly distributed load; t is thickness of plate The following boundary condition are taken in the analysis. All edges simply supported (SS):

All edges clamped (CC)

All edges hinged (HH)

VII. VALIDATION STUDY Present geometrically nonlinear formulation is validated by comparing the author results and existing literature for cross ply plates. The finite element results are compared with experimental results of Zaghloul and Kennedy [13]. The center deflection of all edges clamped symmetric bi-directional square plate under uniform loading is presented in Fig 4. The clamped boundary conditions are used for the plate. The material properties used for computation are:

The comparisons shows that the present results and experimental results are in good agreement.

Fig.4 Comparison of the present FEM results with experiments

results for a simply supported 4-ply symmetric plate



97

Numerical examples: In this section numerical analysis is carried out by considering different boundary conditions to the know the geometric nonlinear behaviour of anti-symmetric cross ply laminated plates. In the present analysis the effect of number of layers and the lamination scheme on the center deflection is examined for anti-symmetric cross-ply laminates. A square plate with clamped, simply supported, hinged boundary condition and material. The material properties of the laminates are taken as follows:

125 6

12 21.45 10 , 3.63 10 , 0.5 , 0.3EE G E From Fig. (5-7) it can be concluded that as the number of layers are increases, the center deflection decreases and number of layers reduces the nonlinearity in the cross-ply laminates.

Fig .5 Effects on number of layers on transverse displacement

of clamped anti-symmetric cross-ply plate

Fig.6 Effects on number of layers on transverse displacement

of hinged anti-symmetric cross-ply plate

Fig.7 Effects on number of layers on transverse displacement

of simply supported anti-symmetric cross-ply plate

Fig.8 Effect of boundary condition on transverse displacement

of anti-symmetric cross-ply plate In Fig.8 it clear that plate with clamped boundary condition and gives lower displacement when compared to the simply supported boundary condition. In clamped plates, all essential/natural boundary conditions are satisfied but in the simply supported laminated plate with immovable edges, moment free condition are not satisfied. CONCLUSION In this paper, the geometric nonlinear analysis of plates are carried out by using a Newton Raphson method. The validation study is carried out and validation results are good in terms of experimental results. Numerical results are obtained for nonlinear center deflections of square plates subjected to various edge conditions. It is concluded that the number of layers, stacking sequence, boundary condition have significant effect on the nonlinear deflection of the plate. REFERENCE

[1] Brebbia, C. and Connor, J. (1969). Geometrically nonlinear finite-element analysis, Journal of Engineering Mechanics Division 95: 463-483.

[2] Wood, R. D. and Zienkiewicz, O. C. (1997). Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells, Computer & Structure 7(6): 725-735.

[3] Bathe, K. J. and Bolourchi, S. (1980). A geometric and material nonlinear plate and shell element, Computer & Structure 11(1): 23-48.

[4] A.Pica R. D. Wood and E. Hinton (1979). Finite element analysis of geometrically nonlinear plate behaviour using mindlin formulation. Computer & Structure 11: 203-215

[5] J. N. Reddy and W. C. Chao (1981). Nonlinear bending of thick rectangular, laminated composite plates. Int. J. Nonlinear Mechanics 11(3/4):291-301.

[6] P. C. Yang, C. H. Norris and Y. Stavsky (1966). Elastic wave propagation in heterogeneous plates. Int. J. Solids and structure 2:665-685

[7] N. S. Puctha and J. N. Reddy (1986). A refined mixed shear felexible finite element for the nonlinear analysis of laminated plates, Computer & Structure 22(4): 529-538.



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[8] Chia CY. (1985). Large deflection of unsymmetrical laminates with mixed boundary conditions. Int J Non-Linear Mech 20(4):273-82.

[9] Striz AG, Jang SK, Bert CW. (1988). Nonlinear bending analysis of thin circular plates by differential quadrature. Thin-Walled Struct 6(1):51-62.

[10] Z. G. Azizian and D. J. Dawe (1992). Geometrically nonlinear analysis of rectangular Mindilin plates using the finite strip method. Computer & structure 21(3): 423-436,

[11] Gajbir Singh, G. Venkateswara Rao, N.G.R. Iyengar. (1991). Some observations on the large deflection bending

of rectangular anti-symmetric cross-ply plates. Compos Struct 18:77-91.

[12] Murray J. Clarke, Gregory J. Hancock. (1990) A study of incremental-iterative strategies for non-linear analyses. Int J Numer Methods Eng. 29:1365-91.

[13] Madhulit Mukhopadhya,(2012). Mechanics of Composite materials and structures, Universites press, ISBN-978 81 7371 477 1

[14] Zaghloul SA, Kennedy JB. Nonlinear behavior of symmetrically laminated plates. J Appl Mech 42: 234-6.

GEOMETRICALLY NONLINEAR ANALYSIS OF ...Geometrically Nonlinear Analysis of Composite Plate...

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