Computers & Srruc~es Vol. 38, No. 516, pp. 669-671. 1991 0045-7949/91 $3.00 + 0.00

Printed in Great Britain. 0 1991 Pergamon Press plc

TECHNICAL NOTE

AN IMPORTANT NOTE ON SYMMETRY LINE BOUNDARY CONDITIONS IN FIBRE-REINFORCED

LAMINATED ANISOTROPIC COMPOSITES

MALLIKARJUNAT

Structural Engineering Research Centre, Madras 600 113, India

(Received 6 February 1990)

Abstract-Symmetry line boundary conditions of symmetrical and unsymmetrical laminated fibre- reinforced anisotropic composite structures are discussed. A simple Co isoparametric finite element formulation based on the Reissner-Mindlin first-order shear deformation theory is used. The effects of boundary conditions on the laminated scheme and the orientation of fibres are studied. Numerical results for deflections presented herein should be of interest to composite-structure designers, experimentalists, and numerical analysts in verifying their results.

INTRODUCTION

During the last two decades, many attempts have been made to investigate the exact behaviour of fibre-reinforced orthotropic/anisotropic composite laminates. This is evident from the number of publications which have appeared in the literature. While analysing composite laminates, most of the researchers consider the half or quarter plate/shell due to the uniaxial or biaxial symmetry of the structures. The present author[l] has found that many mistakes have been made by others in specifying the symmetry line boundary con- ditions, particularly in orthotropic and/or anisotropic structures. The present paper presents an important note on symmetry line boundary conditions in fibre-reinforced orthotropic/anisotropic composite structures.

ANALYSIS AND DISCUSSION

A first-order shear deformation (Reissner-Mindlin) theory (FOST) in conjunction with the simple Co isopara- metric finite element is used with five degrees of freedom (u,,, or,, w,, 0:, 0!) per node, where u, and v,, are the inplane displacements m the x- and y-directions, and w0 is the transverse displacement in the z-direction, of a generic point (x, y) on the mid-plane; 0, and 0 are the rotations of the normals to the mid-plane about t h e y- and x-axis, respect- ively (see Fig. 1). The detailed theoretical formulation of FOST is oresented in 121. A four-noded linear element is employed’and the reduced integration technique, i.e. 2 x 2 Gauss quadrature rule, is adopted to integrate the mem- brane, flexure, coupling between inplane and bending and shear terms in the energy expression for evaluation of the element stiffness property.

A clamped, square plate with 4 x 4 mesh discretization (see Fig. 1) is analysed using the following data (all units are non-dimensional):

a/b = 1 a/h = 10 a =50 h =5 q =O.l

E, /Ez = 25 Ez = 1 G,, = G,, = OS&

G,, = 0.2Ez vu = 0.25.

7 Present address: Department of Civil Engineering, Lava1 University, Quebec, Canada GlK 7P4.

The results are tabulated in Tables 1, 2 and 3 for different orientations of fibres in a laminate. From these tables, it is observed that, if a quarter/half plate is to be analysed, then certain boundary conditions should be specified corresponding to orientation of fibres. However, to date, researchers have used only one type of symmet- ric line boundary conditions for all types of laminates. From the present study, it is considered that there are three types of fibre-oriented composite laminates. For the laminates with different orientations of fibres as listed in Table 1, the following symmetric line boundary conditions can be used in the case of uniaxial/biaxial symmetry:

u0 = ex = 0 at x = a/2

v,=e,=O aty=b/2. (1)

The laminates, which are given in Table 2, can be analysed with the following conditions:

v0 = e, = 0 at x = a/2

uo=tJ,,=o aty=b/2. (2)

The fibre-reinforced composite laminates with differ- ent orientations presented in Table 3 should be analysed with full plate/shell structures irrespective of the geometry.

It must be noted that the inplane displacements u, and v0 can be dropped in the symmetrically laminated plate (but not in shells), as the mid-plane represents the neutral surface of the deformed laminate.

CONCLUSION

In analysing the fibre-reinforced orthotropic/anisotropic composite laminates which are listed in Table 3, a full plate/shell structure should invariably be used. In the case of uniaxial or biaxial symmetry, the quarter or half plate/shell can be used, depending upon the fibre orientations, but symmetric line boundary conditions

669

670 Technical Note

Table 1

Node No.

D.O.F. Orientation of fibres (degrees) 0 o/90 0/90/0/90/0

uo 0.00 00 0.00

8 M'O 14.23 k 0.00 *Y -9.67

uo 0.00 t’ 0 0.00

12 wo 14.47 0, -3.51 0, 0.00

% 0.00 00 0.00

13 w 8’

20.86

i 0.00 0.00

0.00 0.00 -1.08 0.00 17.09 13.55 0.00 0.00

-0.92 -0.61

1.08 0.00 0.00 0.00

17.09 14.06 -0.92 -0.33

0.00 0.00

0.00 0.00 0.00 0.00

26.03 19.80 0.00 0.00 0.00 0.00

Fig. I. Plan of 4 x 4 mesh laminated plate (orientation of fibres is from x-axis).

Node No.

8

12

13

D.O.F.

MO 00

WO

0, 0,

UO

00

W A0 8:

uo t’ 0 M’O e.

Case I

0.75 0.00

17.79 0.00

- 1.08

0.00 1.66

17.74 -0.86

0.00

0.00 0.00

26.99 0.00

Table 2

O~entation of fibres Case II Case III Case IV Case V Case VI Case VII Case VIII

1.16 0.22 0.12 0.21 1.01 0.93 0.64 0.00 0.00 0.00 0.00 0.00 0.00 0.00

18.22 14.17 13.47 15.61 15.49 17.12 15.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.99 -0.81 -0.57 -0.33 -0.88 -1.13 -1.08

0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.16 0.15 0.08 0.42 0.76 I.11 0.44

18.22 14.98 14.08 14.35 16.53 17.83 16.41 -0.99 -0.33 -0.33 -0.96 -0.58 -0.74 -0.41

0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

21.18 19.70 22.06 23.08 26.54 23.43 0.00 0.00 0.00 0.00 0.00 0.00

6; 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Case I: 30/- 30”; Case Ilk O/45; -45/45/-45/O”; Case V: 90/30/ - 30/90”; Case II: 45/- 45”; Case IV: 0/~/4Sl-45/90/O”; Case VI: 30/~190/ - 30”; Case VII: 30/45/ -45J - 30”; Case VIII: 0/3Oi45/60~ - 60/-45!-- 30/o”.

Node No.

8

12

13

D.O.F.

UO

00

w e? 4

UO

00 II/ e” e:

% I‘0

‘I’0 8..

30

0.00

0.00

14.99 0.32

-0.94

0.00

0.00 15.24

-0.59 0.41

0.00 0.00

22.52 0.00

Table 3

Orientation of fibres (degrees) 45 o/90/45 30/60/90/45

0.00 -0.83 0.20 0.00 -0.17 -0.36

15.39 16.13 15.08 0.44 0.25 0.39

-0.8 1 -0.01 -0.85

0.00 0.78 0.35 0.00 -0.81 -0.08

15.39 16.95 15.83 -0.81 -0.69 -0.64

0.44 0.22 0.33

0.00 0.00 0.00 0.00 0.00 0.00

23.12 24.97 23.04 0.00 0,oo 0.00

o/45/-45

0.00 0.00

14.15 0.11

-0.81

0.00 0.00

14.95 -0.33

0.14

0.00 0.00

21.16 0.00

e;, 0.00 0.00 0.00 0.00 0.00

Technical Note 671

should be employed accordingly, which are given in eqns (1) and (2). Irrespective of the boundary conditions (simply- supported/clamped) along the boundary edges, the above mentioned conditions must be taken into consideration.

REFERENCES

1. Mallikarjuna, Refined theories with Co finite elements for free vibration and transient dynamics of anisotropic comnosite-sandwich nlates. Ph.D. thesis, Indian Insti- tute bf Technology, Rombay (1988).

Acknowledgemenr-The author is grateful to the Director, Structural Engineering Research Centre, Madras, for his continuous support and encouragement, and also for permission to publish this paper.

2. T. Kant and Mallikarjuna, Transient dynamics of com- posite plate using 4-, 8- and 9-noded isoparametric elements. Int. J. appl. Finite Elements Comput. Aided Engng-Finite Elements Anal. Des. 5, 307-318 (1989).