Meshless natural element method for nonlinear

Journal of Structural Engineering Vol. 42, No. 1, Apr - May 2015 pp. 57-63

No. 42-8

Meshless natural element method for nonlinear analysis of composite plates Madhukar Somireddy*, and Amirtham Rajagopal*  Email: [email protected]

*Department of Civil Engineering, Indian Institute of Technology Hyderabad, 502205, India.

In the present work, the geometrical nonlinear analysis of laminated composite plates is done using natural element method. The C1 natural neighbor interpolation function is implemented for geometric nonlinear analysis. The first order shear deformation plate theory is adopted for plate analysis. The geometric nonlinearity is based on the von Kármán’s assumptions. The nonlinear static analysis is carried out with step loading and Newton-Raphson iterative method. The formulation developed here is validated with available analytical and finite element results. The effect of plate aspect ratio on deflection has been studied. Keywords: Geometrical nonlinear; meshless method; natural neighbors; laminated composite plates.

The laminated composite panels usage in air and space industries is increased several folds in last few decades because of their high stiffness to weight ratio; for effective design of such panels the analysis should be carriedout under different load conditions. The finite element method have been used for analysis, however it faces difficulties in handling some problems such as element distortion and remeshing. To simplify such problems the meshless techniques have been developed by various researchers1,2. However, these meshless methods have restriction on imposition of essential and natural boundary conditions because of approximation made in it do not satisfy delta Kronecker property ( I ( xJ )  d IJ ). The other meshless technique to overcome this problem is natural element method (NEM), where interpolation function is based on Sibson3 natural neighbor coordinates. NEM was successfully applied to solid mechanics problems by Sukumar and Moran4. Analysis of plates by meshless methods gaining popular in recent years because of its advantages in ease handling of large deformation problems and ease refinement for very accurate results. Dinis et al.5 proposed

an improved meshless method, natural neighbor radial point interpolation method (NNRPIM). It is based on natural neighbor concept for nodal connectivity and radial point interpolators to construct interpolation functions. The NNRPIM applied to analysis of thick and laminated plates based Reissner-Mindlin plate theory. Dinis et al.6 proposed a 3D shell like approach based on NNRPIM to analysis the thin plate and shell structures. The natural neighbor Petrov-Galerkin method was developed by Li et al.7 to solve isotropic square plate bending problem and to study transient response of a rectangular plate based on Mindlin plate theory. Qian et al.8 investigated functionally graded plates based on higher order shear and normal deformable plate theory using meshless local Petrov Galerkin (MLPG) method. Ferreria et al.9 carriedout free vibration analysis of functionally graded plates based on FSDT and TSDT using collocation method and multiquadric radial basis functions. It is observed from available literature, the plate analysis using meshless methods gaining popular in recent years because of special features of meshless methods such as no element compatibility, ease insertion and deletion of nodes and accurate prediction Journal of Structural Engineering Vol. 42, No. 1, April - may 2015

57

of results. In practical cases the composite panels with small thickness often subject to large deformation and thereby geometric nonlinear analysis is needed for more accurate prediction of the deformation behavior. Zhang and Kim10 developed two simple displacement based four node quadrilateral elements for geometric nonlinear static analysis of thin to moderately thick laminated composite plates. The proposed nonlinear elements are based on first order shear deformation theory and von Kármán’s large deflection theory. Rajagopal et al.11 carried out nonlinear bending analysis of sandwich plate using first order shear deformation theory and geometric nonlinearity based on Green’s strain terms. Singha and Rupesh12 investigated the nonlinear vibration of composite plates by finite element method based on first order shear deformation plate theory. Limited literature exists on geometric nonlinear analysis of plates using meshless methods. In this work, a natural neighbor Galerkin method is employed for nonlinear analysis of composite laminated plates subjected to static bending loads. Geometrical nonlinear is based on von Kármán assumption. The C1 natural neighbor interpolants are used for analysis. The first order shear deformation plate theory is adopted for analysis of laminated composite plates. The incremental load and Newton-Raphson iterative method are used for solving nonlinear equations. The efficacy of the present method for nonlinear analysis of plates is demonstrated. The bending results of the plates using developed computer program for different plate aspect ratio are presented. The obtained results with present method are validated with finite element and analytical results available in literature.

Ι ( x ) 

AΙ ( x) A( x)

(1)

The C1 interpolant is based on Sibson C0 natural neighbor interpolant. The C1 interpolant is obtained by inserting the Sibson’s coordinates in the Bernstein-Beizer presentation of a cubic simplex. The application of C1 natural neighbor interpolant to structural analysis is presented in Sukumar and Moran4. Let a point x is surrounded by natural neighbors and the natural neighbor coordinates of x be   1 ( x ), 2 ( x ),.........n ( x )  . Φ is a barycentric coordinates of the n-gon in the plane. Then, the surface can be constructed by using barycentric coordinates with Eq.(2) given below.

(a)

(

(c)

Mathematical Formulation

(a) The natural neighbor based C0 interpolant functions 3 were introduced by Sibson . The construction of Delaunay triangles, first and second order Voronoi diagram for a set of nodes is available in Madhukar and Rajagopal13. Let a point x is introduced in to the Voronoi diagram of the set N. The shape function and its derivative of the central node in equispaced nodes in a domain is shown in Fig. 1b and Fig. 1c, respectively. The natural neighbor coordinates (shape functions, ϕ) of x with respect to a natural neighbor I are defined as the ratio of the area of overlap of their Voronoi cells to the total area of the Voronoi cell of x. 58

Journal of Structural Engineering Vol. 42, No. 1, April - may 2015

(a)

(b)

(c)

(b)

(c)

Fig. 1 (a) Nodal grid, (b) shape function (c) derivative of shape function

n

wm ( )   Bim ( ) bi

(2)

i 1

 m  m m! where Bim ( )  1i1 2i2 .....nin and   , i  i  i1 !i2 !...in ! bi is Bezier ordinates If m=3, then it is C1 natural neighbor interpolation. n

T

w ( )   Bi3 ( ) bi   ( ) w 3

(3)

i 1

where T

 ( )  1 ( ),  2 ( ),  3 ( ),.... 3n2 ( ),  3n1 ( ),  3n ( ) wT  w1 , q 1x , q 1 y ,.....wn , q nx , q ny  Strain-displacement field

The displacement filed of first order shear deformation plate theory is written as:

u ( x, y, z ) = u0 ( x, y ) + zθ x ( x, y ) v( x, y, z ) = v0 ( x, y ) + zθ y ( x, y )

(4)

w( x, y, z ) = w0 ( x, y ) where u0, v0 and w0 are displacements of a point on the plane z = 0. qx and qy are rotations of a transverse normal about y and x axis respectively. The nonlinear terms are based on Von karman assumption, the straindisplacement field is written as: 2 1 w    u         2 x    ( 0 )   ( 0 )  k (1)     x    xx  nxx xx  2       (0)      v   1 w      xx   (0)         yy   nyy   z k yy(1)  ,  yy     (1)     y   2 y    ( 0 )   ( 0 )     u v   w w     xy       xy  k xy     nxy        y x    x y  

u w  (0)     xz      xz    z x        (0)  v w     yz        yz        z y 

(5)

   

(0)

  z k , (0) n

(1)

     (0)

(6)

where

 u  0   (0)      x   xx   v    (0)   0 ,     y   yy     (0)  u v     0 0   xy      y x    

  2   1 w0     (0)   2  x        nxx    w 2   ( 0)   1 0       ,  nyy   2 y    ( 0)    w w      0 0 nxy      x y   

 q  x   (1)   x  k     xx   q   (1)   y , k    yy y       (1)    q k  q x y    xy   y  x   

(7)

 w   (0)  q  0    xz   x x      ( 0 ) w     0  yz  q y  y   

Constitutive relation

The constitutive relation for a laminated composite plate can be written in lamina coordinates (x, y, z) and the in-plane and transverse stress-strain relation for the Lth lamina is given as:    Q Q12  xx   11  yy   Q12 Q22     xy   Q16 Q26  yz   Q44     xz   Q54

Q16   xx    Q26  yy    Q66    xy 

Q45   yz    Q55   xz 

(8a)

(8b)

   Q  

(9a)

s    Q   

(9b)

where [Qij] are transformed material constants. The elements of [Qij] are given as: [Qij ]  [T1 ]1[Cij ]L [T1 ]1 (i, j = 1, 2, 6)

(10a)

1 1 and [Qij ]  [T2 ] [Cij ]L [T2 ] (i, j = 4,5)

(10b)

where [T1] and [T2] are the transformation matrices. [Cij]L is the constitutive matrix at the lamina level.  C11 C12     Cij k  C12 C22  0  0

0   0 where (i,j = 1, 2, 6), (11a) C66  


59

I1

I1

n

w0 ( X )   I ( X ) w0 I , I1

 C44 and   Cij  k   0

n

0  (i,j = 4,5). C55 

(11b)

q x ( X )   I ( X ) q x I , I1

n

q y ( X )   I ( X ) q y I (15) I1

where

Static formulation

C11  E1 / (112 21 ),

Total potential energy I = U + W

C12  12 E2 / (112 21 ),

I

C22  E2 / (112 21 ),

1      yy  yy  xy  xy  yz  yz  xz  xz  dV 2 V xx xx

  (q w)dA

C66  G12 , C55  G13 , C44  G23 material properties with respect to fibre matrix coordinate axes (1,2,3). For further details refer Madhukar and Rajagopal13. Governing equations

Strain energy part U is written as: U

1 1 T  T Q   dA dh  h V     Qs    dA dh   h A 2 2

(17)

The total potential energy I can be written as:

The equations of equilibrium for the bending analysis are obtained using principle of minimum potential energy. Let U is total strain energy due to deformation, V is the potential of external loads, and U + V = I is the total potential energy.

I

T T 1 ( ( 0 )   A ( 0 )   k (1)   B  ( 0 )   A 2 T

T

  ( 0 )   B k (1)   k (1)   D k (1)  T

T

T

T

  ( 0 )   A n( 0 )    n( 0 )   A ( 0 )   k (1)   B  n( 0 )    n( 0 )   B k (1) 

For minimum potential energy dI = 0

T

(12)

  n( 0 )   A n( 0 )  

ij

Approximation of field variables

The vector {di } is the degree of freedom at each node, T (14) di   u0 v0 w0 q x q y  where i =1,2..,N and N is the total number of nodes in plate domain Ω. u0 ( X )    I ( X ) u0 I , I1

n

v0 ( X )   I ( X ) v0 I , I1

n

w0 ( X )   I ( X ) w0 I , I1 n



n



60 Structural Engineering q x ( X )  Journal  I ( X ) qof q y(X )  xI , I ( X )q yI Vol. I1 42, No. 1, April - may I2015 1

T 1 s (0) (  ( 0 )   A    )dA  A 2

A

 A , B , D   The nodal degrees of freedom are u0, v0, w0, qx and qy the detail study about plate mathematical formulation based on FSDT available in Reddy14. Boundary conditions obtained from governing equations, the simply supported (SS) boundary conditions are u0, v0, qx = 0 for x = 0 to a, and u0, v0, qy = 0 for y = 0 to b.

(18)

  qwdA

(13)

n

(16)

A

ij

ij

hL1

hL

Qij (1, z , z 2 )dz where (i, j = 1, 2, 6) (19)

hL1

Aijs  h Qij dz where (i, j = 4, 5)

(20)

L

The strain terms in Eqn. (18 ) can expressed interms of degree of freedom vector {d} using Eqn. (15) and are given as:

 (0)  B0 d ,  n(0)  Bn0 d , k (1)  Bk1 d ,  (0)  B0 d 

(21)

For minimum total potential energy dI = 0 then Eq. (18) reduces to

 K (d )d   F  where stiffness matrix.

(22)

c

2

0

 xx   xx (0.4718a, 0.4718b, 0.5h) h 2 / a 2 q0 

A

T

and force vector F   q0 A   dA

 xy   xy (0.0282a, 0.0282a, 0.5h)h 2 / a 2 q0 Table 1

(23)

Present

Initially linear bending analysis of laminated composite plate is done and the results of the developed program for linear bending problem are validated. Then nonlinear bending analysis of plates is performed and results of the nonlinear program are compared with the available results in literature. The simply supported plates subjected to uniform distributed loading q0 (UDL) is considered for analysis and all problems are presented here. The shear correction factor for first order shear deformation plate theory is 5/6 considered and 6 point Gauss quadrature in each background triangle cell is used for integration. The plate geometry parameters are length-a, width-b and thickness-h of the plate. A tolerance of ε = 10–2 is used for convergence criteria in the Newton-Raphson iteration scheme to check for the convergence of nodal displacement for all nonlinear bending problems. The plate bending results of thick plate based on present method is validated with available results in the literature. The material properties of orthotropic material are E1=25E2, G12=G13=0.5E2, G23=0.2E2, ν12=0.25. The non-dimensional deflection and in-plane stress parameters are: w 100 wc (a / 2, b / 2, 0) E2 h3 / a 4 q0

a/h = 10 16

Pandya and Kant

Present

Pandya and Kant 16

FSDT

FSDT

HSDT

FSDT

FSDT

HSDT

w

2.6645

2.6559

2.8765

1.0217

1.0211

1.0968

σ xx

0.6516

0.6650

1.1094

0.7656

0.7851

0.8739

σ yy

0.6545

0.6625

0.7244

0.3065

0.3844

0.3945

τ xy

0.0714

0.06956 0.09463

0.0494

0.04804

0.05499

(24)

Results and discussions

Here, the second problem is nonlinear bending of a square isotropic plate with a = b = 10 in., h = 1 in., E = 1.8 x 106 psi, ν = 0.3 considered and results are obtained with simply supported boundary conditions under uniform load. The load parameter P = q0a4/E2h4 where q0 is uniform distributed load and deflection w = w0/h for all nonlinear problems. The problem considered here is thick plate a/h = 10. The results of present first order shear deformation theory are close to Reddy14 finite element method results and it can be observed from Fig. 1. 2.0 Present NEM Reddy 2004 FEM

1.5 1.0 0.5 0.0

0

50

100

150 Load P

200

250

300

Fig. 1 Center deflection versus load for a simply supported square isotropic plate

The affect of plate aspect ratio on deflection for a symmetric (00/900/900/00) square laminated composite plate subjected bending loads is investigated here.

 xx   xx (0.4718a, 0.4718b, 0.5h) h 2 / a 2 q0 yy

Non-dimensional central deflection and in-plane stresses of a simply supported composite plate (0°/90°/0°) with a/b = 1, under uniformly distributed loading a/h = 4

The final equation for plate bending analysis is given as Eqn.22, where, [K] is a function of displacements, {d} is the vector of nodal displacements and {F} is the vector of nodal forces. The nonlinear plate bending Eqn.22 is solved using Newton-Raphson technique, for more details refer Reddy15 and for linear bending cases the nonlinear terms become zero.



  yy (0.4718a, 0.4718b, 0.5h) h 2 / a 2 q0

Deflection w

K  

 B 0 T  A B 0  B1 T  B  B 0       k        0 T  1   1 T  1  Bk     B   B  Bk  Bk   D   0 T  0 0 T 0  Bn    Bn    A  B  dA  B    A   1 T  0 0 T 1  Bn    Bn    B   Bk     Bk    B     T 0 T 0 0 s  0           B A  B B A B            n  n

yy

  yy (0.4718a, 0.4718b, 0.5h) h 2 / a 2 q0

 xy   xy (0.0282a, 0.0282a, 0.5h)h 2 / a 2 q0 .


61

The material properties of lamina are E1= 25, E2= 1, G12=0.5 ν12= 0.25. From Fig. 2 it can be concluded that the nonlinear results of laminated composite plate with present method are in well agreement with the available finite element results. The affect of plate aspect ratio on deflection is shown in Fig. 3. 1.0

Deflection w

Acknowledgment

Zhang and Kim FEM Present NEM

0.8

Authors of this paper would like acknowledge greatly to the funding agency ‘Aeronautical Research and Development Board (AR&DB)’, India for funding the project on Higher order Natural Element Method for Analysis of composite plates.

0.6 0.4 0.2 0.0

References 0

50

100

150 Load P

200

250

300

Fig. 2 Center deflection versus load for a simply supported square symmetric laminated plate with a/h=10 and h=1

1.0 a/h =20 Present NEM a/h =10

Deflection w

0.8 0.6 0.4 0.2 0.0

0

50

100

150 Load P

200

250

300

Fig. 3 Center deflection versus load for a simply supported square symmetric laminated plate for a/h=10, 20 and h=1

Conclusion Initially, the C 0 natural neighbor interpolant is developed for analysis of laminated composite plates for bending load cases, then C 1 natural neighbor interpolant is implemented for geometrical nonlinear analysis of laminated plates. The present meshless method has distinct advantages for analysis of large deforming the thin structures. The efficacy of the present method for 62

nonlinear plate bending analysis is demonstrated. It is observed the predicted results with present method are in well agreement with finite element and analytical results available in literature. The limited parametric study is done on deflection of laminated plates for different plate aspect ratio.


1.

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