82

dimensional stresses xx and xz

, respectively of a square sandwich plate under point load at center (TYPE-3). It is seen that

xxis almost same for clamped and simply

supported plates whereas transverse shear

stress xzis more for simply supported

sandwich plate.

Table-1 Nonlinear transverse central deflection ( )w

of a simply supported [0/90/90/0] cross-ply square plate

subjected to uniformly distributed load (Load TYPE-5) , (Mat-1, TSF-2)

P Present References

RBF4 RBF4 RBF4 RBF1 RBF2 RBF3

Kant [22] Kant

[22]

Zhang

[2]

Zhang [2]

THE-1 THE-2 THE-3 THE-3 THE-3 THE-3 Analytical HOST

RDKQ-

NL24

RDKQ-

NL20

50 0.3387 0.3379 0.3393 0.3434 0.3258 0.3434 0.3560 0.3704 0.3704 0.3631

100 0.4954 0.4947 0.4960 0.4988 0.4665 0.4965 0.5100 0.5249 0.5249 0.5141

150 0.6033 0.6026 0.6039 0.6059 0.5610 0.6021 0.6100 0.6290 0.6290 0.6158

200 0.6874 0.6866 0.6881 0.6881 0.6456 0.6817 0.6890 0.7099 0.7099 0.6949

250 0.7421 0.7416 0.7423 0.7458 0.7120 0.7457 0.7470 0.7770 0.7770 0.7605

Table-2 Linear and non-linear transverse central deflection ( )w

of a simply supported [0/90/90/0] cross-ply square

Plate subjected to various types of loadings,( Mat-1, TSF-2,RBF-4)

P Linear Nonlinear

TYPE-5 TYPE-4 TYPE-3 TYPE-2 TYPE-1 TYPE-5 TYPE-4 TYPE-3 TYPE-2 TYPE-1

10 0.10375 0.06853 0.31400 0.17475 0.14364 0.09891 0.06716 0.23716 0.15917 0.13450

15 0.15563 0.10280 0.47100 0.26213 0.21547 0.14133 0.09829 0.28637 0.21987 0.18903

20 0.20750 0.13707 0.62800 0.34950 0.28729 0.17885 0.12736 0.30777 0.26824 0.23456

25 0.25938 0.17133 0.78500 0.43688 0.35911 0.21213 0.15431 0.31539 0.30630 0.27202

30 0.31125 0.20560 0.94200 0.52425 0.43093 0.24189 0.17917 0.31688 0.33624 0.30287

40 0.41500 0.27414 1.25600 0.69900 0.57458 0.29353 0.22342 0.36958 0.34574

50 0.51875 0.34267 1.57000 0.87375 0.71822 0.33791 0.26191 0.36664

60 0.62250 0.41120 1.88400 1.04850 0.86187 0.37207 0.29594

83

70 0.72625 0.47974 2.19800 1.22325 1.00551 0.40719 0.32637

Fig. 5

Fig. 6

Fig. 7

Fig.8.

Fig. 9.

Fig.10:

5*5 7*7 9*9 11*11 13*13 15*15

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.015

w

Number of Nodes

Present

Timoshenko [21]

0 5000 10000 15000 20000 25000 300000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

q0

w

TYPE-5

TYPE-4

TYPE-3

TYPE-2

TYPE-1

5x5 7x7 9x9 11x11 13x13 15x15 17x170.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

1000P

w

Number of Nodes

TYPE-1

TYPE-2

TYPE-3

TYPE-4

TYPE-5

0 50 100 150 200 250 300 350 400 450 500 5500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

P

w

TYPE-5

TYPE-4

TYPE-3

TYPE-2

TYPE-1

0 2000 4000 6000 8000 10000 12000 14000 160000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

q0

w TYPE-5

TYPE-4

TYPE-3

TYPE-2

TYPE-1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

R= 5

q0 = 1000

( )w

Length of the plate (a)

TYPE-1

TYPE-2

TYPE-3

TYPE-4

TYPE-5

84

Fig. 11.

Fig. 12.

Fig. 13

Fig. 14.

Fig. 15.

Fig. 16. Conclusions The nonlinear displacement response of laminated and sandwich composite plates subjected to different type of loadings are obtained using various RBFs and quadratic extrapolation technique. Effects of different loadings, core thickness and radial basis functions on nonlinear flexural responses are presented. It is observed that Gaussian function used as RBF underestimates the transverse deflection. The results presented herein show the applicability of present solution methodology. It is seen that selection of shape parameters plays important role in convergence of the solution. However, choice of shape parameter for better and faster convergence of deflection for the plate under point loads is still matter of further investigations.

0 5000 10000 15000 20000 25000 300000.0

0.2

0.4

0.6

0.8

1.0

1.2

q0

w

RBF-1

RBF-2

RBF-3

RBF-4

0 5000 10000 15000 20000 25000 300000.0

0.2

0.4

0.6

0.8

1.0

1.2

q0

w h

c = 0.8h

hc = 0.6h

hc = 0.333h

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-50 -40 -30 -20 -10 0 10 20 30 40 50

q0 = 10000

xx

z/h

TYPE-1

TYPE-2

TYPE-4

TYPE-5

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-4 -3 -2 -1 0 1 2 3 4

q0 = 10000

xz

z/h

TYPE-1

TYPE-2

TYPE-4

TYPE-5

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-200 -150 -100 -50 0 50 100 150 200

q0 = 5000

xx

z/h

TYPE-3 SSSS

TYPE-3 CCCC

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-15 -10 -5 0 5 10 15

q0 = 5000

xz

z/h

TYPE-3 SSSS

TYPE-3 CCCC

85

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