Based on the analogy of structural mechanics and optimal control, the theory of the Hamilton system can be applied in the analysis of problem solving using the theory of elasticity and in the solution of elliptic partial differential equations. With this technique, this paper derives the theoretical solution for a thick rectangular plate with four free edges supported on a Pasternak foundation by the variable separation method. In this method, the governing equation of the thick plate was first transformed into state equations in the Hamilton space. The theoretical solution of this problem was next obtained by applying the method of variable separation based on the Hamilton system. Compared with traditional theoretical solutions for rectangular plates, this method has the advantage of not having to assume the form of deflection functions in the solution process. Numerical examples are presented to verify the validity of the proposed solution method.

## References

References
1.
Westergaard
,
H. M.
,
1926
, “
Stresses in Concrete Pavements Computed by Theoretical Analysis
,”
,
7
(2), pp.
25
35
.
2.
Zhu
,
J. M.
,
1995
, “
CC Series Solution for Bending of Rectangular Plates on Elastic Foundation
,”
Appl. Math. Mech.
,
16
, pp.
593
601
.10.1007/BF02458727
3.
Wang
,
K. L.
, and
Huang
Y.
,
1986
, “
Thick Rectangular Plates With Four Free Edges on Elastic Foundation
,”
Acta Mech. Solida Sin.
,
1
, pp.
37
49
(in Chinese).
4.
Shi
,
X.
, and
Yao
,
Z.
,
1989
, “
The Solution of a Rectangular Thick Plate With Free Edges on a Pasternak Foundation
,”
J. Tongji University
,
17
(
2
), pp.
173
184
(in Chinese).
5.
Fwa
,
T. F.
,
Shi
,
X. P.
, and
Tan
,
S. A.
,
1996
, “
Analysis of Concrete Pavements by Rectangular Thick-Plate Model
,”
J. Transp. Eng.
,
122
(2), pp.
146
154
.10.1061/(ASCE)0733-947X(1996)122:2(146)
6.
Li
,
R.
,
Zhong
,
Y.
, and
Tian
,
B.
,
2011
, “
On New Symplectic Superposition Method for Exact Bending Solutions of Rectangular Cantilever Thin Plates
,”
Mech. Res. Commun.
,
38
, pp.
111
116
.10.1016/j.mechrescom.2011.01.012
7.
Li
,
R.
,
Zhong
,
Y.
, and
Li
,
M.
,
2013
, “
Analytic Bending Solutions of Free Rectangular Thin Plates Resting on Elastic Foundations by a New Symplectic Superposition Method
,”
Proc. R. Soc. A, Math. Phys. Eng. Sci.
,
469
(2153), p.
20120681
.10.1098/rspa.2012.0681
8.
Huang
,
Y. H.
,
1974
, “
Finite Element Analysis of Slabs on an Elastic Solids
,”
J. Transp. Eng. Div.
,
100
(2), pp.
403
416
.
9.
Tabatabaie
,
A. M.
, and
Barenberg
,
E. J.
,
1980
, “
Structural Analysis of Concrete Pavement Systems
,”
Transp. Eng. J.
,
106
(5), pp.
493
506
.
10.
Zhong
,
W.
,
1995
,
A New Systematic Methodology for Theory of Elasticity
,
Dalian University of Technology Press
,
Dalian, China
(in Chinese).
11.
Hu
,
H. C.
,
1981
,
Variational Principle in Elasticity and Its Applications
,
Science Press
,
Beijing
(in Chinese).
12.
Zhong
,
Y.
,
Li
,
R.
,
Liu
,
Y.
, and
Tian
,
B.
,
2009
, “
On New Symplectic Approach for Exact Bending Solutions of Moderately Thick Rectangular Plates With Two Opposite Edges Simply Supported
,”
Int. J. Solids Struct.
,
46
, pp.
2506
2513
.10.1016/j.ijsolstr.2009.02.001
13.
Yao
,
W.
, and
Zhong
,
W.
,
2002
,
Symplectic Elasticity
,
Higher Education Press
,
Beijing
(in Chinese).