30-10-2012, 01:37 PM
An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates
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ABSTRACT
An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses
of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of
the system stiffness matrix is performed by using the strain smoothing technique over the smoothing
domains associated with edges of elements. In order to avoid the transverse shear locking and to improve
the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG)
method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete
shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG
method is free of shear locking and achieves the high accuracy compared to the exact solutions and others
existing elements in the literature.
Introduction
Static, free vibration and buckling analyses of plate structures
play an important role in engineering practices. Such a large
amount of research work on plates can be found in the literature
reviews [1,2], and especially major contributions in free vibration
and buckling areas by Leissa [3–6], and Liew et al. [7,8].
Owing to limitations of the analytical methods, the finite element
method (FEM) becomes one of the most popular numerical
approaches of analyzing plate structures. In the practical applications,
lower-order Reissner–Mindlin plate elements are preferred
due to its simplicity and efficiency. However, these low-order plate
elements in the limit of thin plates often suffer from the shear locking
phenomenon which has the root of incorrect transverse forces
under bending. In order to eliminate shear locking, the selective reduced
integration scheme was first proposed [9–12].
Formulation of ES-DSG3
In the ES-FEM, we do not use the compatible strain fields as in
(13) but ‘‘smoothed” strains over local smoothing domains associated
with the edges of elements. Naturally the integration for the
stiffness matrix and the geometrical stiffness matrix is no longer
based on elements, but on these smoothing domains. These local
smoothing domains are constructed based on edges of the
elements such that X ¼
SNed
k¼1XðkÞ and X(i) \ X(j) = ; for i–j, in which
Ned is the total number of edges of all elements in the entire
problem domain. For triangular elements, the smoothing domain
X(k) associated with the edge k is created by connecting two
end-nodes of the edge to centroids of adjacent elements as shown
in Fig. 3.
Numerical results
The present element formulation has been coded using Matlab
program. For practical applications, we define rotations hx, hy about
the corresponding axes. Hence, the relations hx = by and hy = bx
have been used to establish the stiffness formulations, see Fig. 1.
For comparison, several other elements such as DSG3, MIN3 [30]
and MITC4 have also been implemented in our package.
Free vibration of plates
In this section, we investigate the accuracy and efficiency of the
ES-DSG3 element for analyzing natural frequencies of plates. The
plate may have free (F), simply (S) supported or clamped © edges.
The symbol, CFSF, for instance, represents clamped, free, supported
and free boundary conditions along the edges of rectangular plate.
A non-dimensional frequency parameter - is often used to stand
for the frequencies and the obtained results use the regular
meshes. The results of the present method are then compared to
analytical solutions and other numerical results which are available
in the literature.
Circle plates
In this example, a circular plate with the clamped boundary is
studied as shown in Fig. 16. The material parameters are Young’s
modulus E = 2.0 1011, Poisson’s ratio m = 0.3, the radius R = 5
and the density mass q = 8000. The plate is discretized into 848 triangular
elements with 460 nodes. Two thickness-span ratios h/
(2R) = 0.01 and 0.1 are considered. As shown in Table 6, the frequencies
obtained from the ES-DSG3 element are closer to analytical
solutions in Refs. [3,71] than that of the DSG3 element and is a
good competitor to quadrilateral plate elements such as the Assumed
Natural Strain solutions (ANS4) [72] and the higher order
Assumed Natural Strain solutions (ANS9) [73]. In case of the thickness-
span ratio h/(2R) = 0.1, the ES-DSG3 results also are very good
in comparison to the ANS4 element that used 432 quadrilateral
elements (or 864 triangular elements), cf. Table 7.
Conclusions
An edge-based smoothed finite element method with the stabilized
Discrete Shear Gap technique using triangular elements is
formulated for static, free vibration and buckling analyses of Reissner–
Mindlin plates. Through the formulations and numerical
examples, some concluding remarks can be drawn as follows:
The ES-DSG3 uses only three DOFs at each vertex node without
additional degrees of freedom and no more requirement of high
computational cost.
The ES-DSG3 element is more accurate than the DSG3, MIN3
triangular elements, and often found more accurate than the
well-known MITC4 element when the same sets of nodes are
used for all cases studied. The results of the ES-DSG3 element
are also in a good agreement with analytical solution and compared
well with results of several other published elements in
the literature.
For free vibration and buckling analyses, no spurious non-zero
energy modes are observed and hence the ES-DSG3 element is
stable temporally. The ES-DSG3 element gives more accurate
results than the DSG3 element and shows also a strong competitor
to existing complicated models such as the Rayleigh–Ritz
method, the pb-2 Ritz method, the spline finite strip and the
meshfree approaches.