In an article recently published in this journal, the powerful single-term extended Kantorovich method (EKM) originally proposed by Kerr in 1968 for two-dimensional (2D) elasticity problems was further extended by the authors to the three-dimensional (3D) elasticity solution for laminated plates. The single-term solution, however, failed to predict accurately the stress field near the boundaries; thus limiting its applicability. In this work, the method is generalized to the multiterm solution. The solution is developed using the Reissner-type mixed variational principle that ensures the same order of accuracy for displacements and stresses. An n-term solution generates a set of 8n algebraic-ordinary differential equations in the in-plane direction and a similar set in the thickness direction for each lamina, which are solved in close form. The problem of large eigenvalues associated with higher order terms is addressed. In addition to the composite laminates considered in the previous article, results are also presented for sandwich laminates, for which the inaccuracy in the single-term solution is even more prominent. It is shown that considering just one or two additional terms in the solution (n = 2 or 3) leads to a very accurate prediction and drastic improvement over the single-term solution (n = 1) for all entities including the stress field near the boundaries. This work will facilitate development of near-exact solutions of many important unresolved problems involving 3D elasticity, such as the free edge stresses in laminated structures under bending, tension and torsion.

References

References
1.
Mittelstedt
,
C.
, and
Becker
,
W.
, 2007, “
Free-Edge Effects in Composite Laminates
,”
Appl. Mech. Rev.
,
60
, pp.
217
244
.
2.
Fan
,
J. R.
, and
Sheng
,
H. Y.
, 1992, “
Exact Solution for Thick Laminates With Clamped Edges
,”
Acta Mech. Sinica
,
24
, pp.
574
583
.
3.
Vel
,
S. S.
, and
Batra
,
R. C.
, 2000, “
The Generalized Plane Strain Deformations of Thick Anisotropic Composite Laminated Plates
,”
Int. J. Solids Struct.
,
37
, pp.
715
773
.
4.
Kapuria
,
S.
, and
Kumari
,
P.
, 2011, “
Extended Kantorovich Method for Three Dimensional Elasticity Solution of Laminated Composite Structures in Cylindrical Bending
,”
J. Appl. Mech.
,
78
, p.
061004
.
5.
Kerr
,
A. D.
, 1968, “
An Extension of the Kantorovich Method
,”
Q. Appl. Math.
,
4
, pp.
219
229
.
6.
Dalaei
,
M.
, and
Kerr
,
A. D.
, 1996, “
Natural Vibration Analysis of Clamped Rectangular Orthotropic Plates
,”
J. Sound Vib.
,
189
, pp.
399
406
.
7.
Aghdam
,
M. M.
, and
Mohammadi
,
M.
, 2009, “
Bending Analysis of Thick Orthotropic Sector Plates With Various Loading and Boundary Conditions
,”
Compos. Struct.
,
88
, pp.
212
218
.
8.
Kantorovich
,
L. V.
, and
Krylov
,
V. I.
, 1958,
Approximate Methods of Higher Analysis
,
Interscience
,
New York
.
9.
Reddy
,
J. N.
, 2007,
Theory and Analysis of Elastic Plates and Shells
,
CRC Press
,
Boca Raton, FL
.
10.
Kerr
,
A. D.
, 1969, “
An Extended Kantorovich Method for the Solution of Eigenvalue Problems
,”
Int. J. Solids Struct.
,
5
, pp.
559
572
.
11.
Cortinez
,
V. H.
, and
Laura
,
P. A. A.
, 1990, “
Analysis of Vibrating Rectangular Plates of Discontinuously Varying Thickness by Means of the Kantorovich Extended Method
,”
J. Sound Vib.
,
137
, pp.
457
461
.
12.
Naumenko
,
K.
,
Altenbach
,
J.
,
Altenbach
,
H.
, and
Naumenko
,
V. K.
, 2001, “
Closed and Approximate Analytical Solutions for Rectangular Mindlin Plates
,”
Acta Mech.
,
147
, pp.
153
172
.
13.
Yuan
,
S.
, and
Zhang
,
Y.
, 1992, “
Further Extension of the Extended Kantorovich Method
,”
Computer Methods in Engineering and Advanced Applications
, Vol.
2
,
A. A. O.
Tay
and
K. Y.
Lam
, eds.,
World Scientific
,
Singapore
, pp.
1240
1245
.
14.
Yuan
,
S.
,
Jin
,
Y.
, and
Williams
,
F. W.
, 1998. “
Bending Analysis of Mindlin Plates by Extended Kantorovich Method
,”
J. Eng. Mech. ASCE
,
124
, pp.
1339
1345
.
15.
Shufrin
,
I.
,
Rabinovitch
,
O.
, and
Eisenberger
,
M.
, 2008, “
A Semi-Analytical Approach for the Non-Linear Large Deflection Analysis of Laminated Rectangular Plates Under General Out-of-Plane Loading
,”
Int. J. Non-Linear Mech.
,
43
, pp.
328
340
.
16.
Shufrin
,
I.
,
Rabinovitch
,
O.
, and
Eisenberger
,
M.
, 2010, “
A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Trapezoidal Plates
,”
Int. J. Mech. Sci.
,
52
, pp.
1588
1596
.
17.
Ye
,
J. Q.
, 2003,
Laminated Composite Plates and Shells: 3D Modelling
,
Springer
,
London
.
18.
Kapuria
,
S.
,
Dumir
,
P. C.
, and
Sengupta
,
S.
, 1999, “
Three-Dimensional Solution for Shape Control of a Simply Supported Rectangular Hybrid Plate
,”
J. Therm. Stresses
,
22
, pp.
159
176
.
19.
Pagano
,
N. J.
, 1970, “
Influence of Shear Coupling in Cylindrical Bending of Anisotropic Laminates
,”
J. Compos. Mater.
,
4
, pp.
330
343
.
20.
ABAQUS, 2009, ABAQUS/STANDARD User’s Manual, Version 6.9-1
21.
Kumari
,
P.
, and
Kapuria
,
S.
, 2011, “
Boundary Layer Effects in Rectangular Cross-Ply Levy-Type Plates Using Zigzag Theory
,”
Z. Angew. Math. Mech.
,
91
, pp.
565
580
.
You do not currently have access to this content.