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

764 XIAO-TING HE, QIANG CHEN, JUN-YI SUN, ZHOU-LIAN ZHENG AND SHAN-LIN CHEN
From (30), we may have
∂2w ∂x 2 = 4C1(3x 2 − a 2 �� �
y 2 �
) + 1 ,
4b
∂2w C1
=
∂y 2 8b2 (x2 − a 2 ) 2 ,
∂2w C1
=
∂x∂y 2b2 (x3 − a 2 x)y. (35)
Substituting (35) into (34), after integrating we obtain
�
E
U =
+ t3 1
6[1 − (µ + ) 2 ] F+ E
+
−t 3 2
6[1 − (µ − ) 2 ] F−
�
C 2 1 , (36)
where
F + = 8 a
315
9 �
16
+
b3 315 −µ+ 136
105
� a 7
The right end of (31) may be computed as
� a � b
−a
b
qwm dx dy =
−b
4003
+
75 a5b, F − = 8 a
315
9 �
16
+
b3 315 −µ− 136
105
� a � b
−a
−b
� a 7
b
+ 4003
75 a5 b. (37)
q(x 2 − a 2 ) 2�� �
y 2 �
+ 1 dx dy =
4b
98
45 qa5b. (38)
Substituting (36) and (38) into (31), we may obtain the formula for C1. After substituting it into (30),
we finally obtain the formula for w:
w =
98
15 qa5 b
E + t3 1
1 − (µ + ) 2 F+ + E−t 3 2
1 − (µ − F−
) 2
(x 2 − a 2 ) 2�� �
y 2 �
+ 1 , (39)
4b
where t1 and t2 are known and may be obtained from (14). The maximum deflection takes place at the
midpoints of the two free sides, that is, while x = 0 and y = ±b, we have
wmax =
833
120 qa9 b
E + t3 1
1 − (µ + ) 2 F+ + E−t 3 2
1 − (µ − F−
) 2
4. Comparisons and discussions
. (40)
4.1. Comparisons with FEM based on Ambartsumyan’s model. The bimodular problem attributes nonlinearity
to materials. Since existent commercial FEM programs do not include the bimodular materials
model proposed by Ambartsumyan, it is necessary to work out a numerical program based on an iterative
technique. Because the stress state of the point in question is unknown in advance, we have to begin with
a single modulus problem, thus gaining the initial stress state to form a corresponding elasticity matrix
for each element. Generally, direct iterative methods based on an incrementally evolving stiffness have
been adopted by many researchers.
Based on an incrementally evolving stiffness, Gao et al. [1998] analyzed a bimodular thin plate by
the direct iterative method. In the FEM analysis, the elements of the plate should be layered along the
direction of the plate thickness and each layer may be considered as being in the same stress state.
A rectangular thin plate fixed at its two longer sides is under the action of normal uniformly-distributed
loads, as shown in Figure 5. For convenience, all the quantities are taken as dimensionless, specifically,
the plate thickness is 0.24, the length and width of the plate are 8 and 6, respectively, and the intensity of