Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

762 XIAO-TING HE, QIANG CHEN, JUN-YI SUN, ZHOU-LIAN ZHENG AND SHAN-LIN CHEN
we obtain
τ − zx =
E− 2[1 − (µ − ) 2 ] (z2 − t 2 ∂
2 )
∂x ∇2w, τ − zy =
E− 2[1 − (µ − ) 2 ] (z2 − t 2 ∂
2 )
∂y ∇2w. − t2 ≤ z ≤ 0. (23b)
The third expression of the differential equations of equilibrium in compression is
∂σ − z
∂z = −∂τ − xz
∂x − ∂τ − yz
∂y
. (24b)
Substituting (23b) into (24b) and considering τ − xz = τ − zx and τ − yz = τ − zy as well as the stress boundary
conditions at the top of the plate,
)z=−t2 = −q, (25b)
(σ − z
we obtain
σ − z =
E− 2[1 − (µ − ) 2 �
t
]
2 z3 2
2 z − −
3 3 t3 �
2 ∇ 4 w, −t2 ≤ z ≤ 0. (26b)
Thus, all stress components have been expressed in terms of w, as shown in (9), (10), (15), (23a), (23b),
(26a), and (26b).
Using (23a) and (23b), we may compute the transverse shear forces Qx and Q y as follows:
� t1
Qx = τ
0
+ xz dz +
� 0
−t2
τ − xz dz = −D∗ ∂
∂x ∇2 w, Q y =
� t1
0
τ + yz dz +
� 0
−t2
τ − yz dz = −D∗ ∂
∂y ∇2 w. (27)
Thus, the bending moments, the torsional moment and the transverse shear forces have been expressed
in terms of w, as shown in (12), (16), and (27).
3.3. Application of the displacement variation method. For a variety of boundary conditions, the displacement
of plates along the thickness direction, w, may be taken as different expressions to satisfy the
given boundary conditions and then be determined via the differential (20). For example, for a rectangular
thin plate with four simply supported sides, the Navier solution may be adopted; for such a plate with
two opposite simply supported sides, the Levy solution may be adopted. However, in some cases, it
is convenient to use the displacement variation method to solve such a problem. For example, for a
rectangular thin plate with four sides fixed, the Galerkin approach may be adopted; for such a plate with
two opposite sides fixed, the Ritz approach may be adopted. In this paper, for the purpose of comparison
with the existent FEM results [Gao et al. 1998], we take a bimodular rectangular thin plate with two
long sides fixed as our object of study, as shown in Figure 4, where 2a and 2b are the short and long
sides, respectively, and the plate is under the action of normal uniformly distributed loads, q. Due to the
existence of free sides, we adopt the Ritz approach to solve this problem.
The displacement boundary conditions at the fixed sides is
� �
∂w
(w)x=±a = 0,
= 0, (28)
∂x x=±a
and the boundary condition at the free sides should satisfy
� �
∂w
(w)y=±b,x�=±a �= 0,
∂y
y=±b,x�=±a
�= 0. (29)