Coupling Shells to Solids for Efficient FE Analysis
Introduction
A new technique for coupling three dimensional
solid elements to curved shell elements in finite element stress analysis
is presented here. The method adopted follows an approach developed by
McCune
et al. (2000), who showed that proper connections between
plate and beam elements, and plate and solid elements can be achieved by
equating the work done on either side of the mixed dimensional interface.
This can then generate multi-point constraint equations and provide compatible
relationships of nodal degrees of freedom between the differing element
types. Monaghan
et al. (1998) successfully extended this work on
coupling
beams to 3D solids. Coupling of shells and solids is thus the logical
next step.
Approach
The basic principle of the technique is that the
work done on both sides of the interface between dimensions is equated.
Consider the case when a solid-shell interface is subjected to a normal
force N
x. Equating the work done by the normal stress resultant
acting on the shell edge with the work done by the surface stresses of
the 3D body at the interface, the following equation results:
|
(1)
|
where u and U are the longitudinal displacements
of the shell edge elements and the 3D continuum at the interface respectively.
In a finite element model, it is assumed here that the stress resultant
N
x along an element edge can be approximated from the nodal
stress resultant {N
x} using N
x = [N
1D]{N
x}
= {N
x}
T[N
1D]
T, where [N
1D]
represents the shape functions of the nodes on the shell edge along that
edge. Equation (1) can then be converted into a discrete form in terms
of nodal stress resultants {N
x}, the finite element nodal displacements
{u, U} and shape functions as
|
(2)
|
where [N
2D] denotes the shape function
of the 3D elements over their faces at the solid side of the interface,
and F(z) is the function that relates N
x to
sx.
Eliminating the arbitrary stress resultants {N
x} from both sides
of equation (2), the resulting equation has the form
From this, a set of generalised constraints can
be obtained as
which can then be implemented in the ABAQUS commercial finite element
package using the *EQUATION command. Similar derivations can be applied
for other load cases, while the full technical details are available
here
(Shim
et al., 2001).
Analysis Results
Figure 1 illustrates a solid-shell model of a
square plate with a circular hole subjected to uniaxial tensile load. The
model has a width of 80 mm and a thickness of 5 mm, while the diameter
of the hole is 20 mm. The material employed in constructing the model was
characterized of isotropic properties with E = 209 GPa. Due to the symmetric
geometry and boundary condition of the problem, only one quarter of the
specimen was modelled. Although it is not necessary, a non-planar or curved
3D-2D interface was employed to illustrate the generality of the present
technique for coupling. As can be seen from the Von Mises stress plot,
the approach yielded perfect continuity in stress at and around the dimensional
interface. Also, the stress concentration factor (K
t =
smax/
snom)
determined based on the analysis result K
t = 2.49 compares reasonably
well with that obtained from theory, i.e. K
t = 2.42 (Roark and
Young, 1975).
Figure 1. Von Mises stress on a square plate with circular hole
Figure 2 illustrates a model of a stiffened panel in an aircraft structure,
where one side of the shell edges was fully constrained while the other
side was subjected to a pressure loading. For brevity, only the solid-shell
model is demonstrated. But, a close-up view of the detailed regions is
given for comparison with a full 3D and a substructure representation.
As identical contouring scales were used, it can be seen that the stress
distribution of the solid-shell model correlates closely with that of the
3D model, and since substructuring does not introduce any additional approximation,
the stresses match exactly with those recovered from the superelement analysis
model. However, the time taken for the solid-shell analysis, though not
too significant due to the relative simplicity of the models, was 191 seconds,
while a full 3D analysis required 333 seconds. The superelement model,
on the other hand, was solved in 98 seconds. It is important to note that
269 seconds are required to condense out the superelement. But, once this
cost has been incurred, the full 3D detail of the behaviour in the stiffener,
including the effect of any stress concentration, lightening holes etc.,
is available for basically the same cost as that incurred by using a line
stiffener in a large global model.
Figure 2. Von Mises stress on a stiffened panel: (a) Solid-shell
model; (b) Full 3D model; (c) Substructure recovered from superelement
model
The present approach for solid-shell coupling has also been applied
to a finite element model of an automotive wheel as shown in Figure 3(a).
The wheel model, being homogeneous throughout, was assumed to have material
properties with E = 70 GPa. The wheel centre consisted of four stud holes
and 16 vent holes and was connected to the rim by a tee-section joint.
3D modelling with 20-noded bricks was used for the tee joint as a means
for appropriate load transfer between the wheel centre and the rim, while
eight-noded quadratic shell elements were coupled to this connection at
a distance of two thickness away from the blended corners. A full constraint
was applied along the edges of the four stud holes and a point load was
applied to the edge of the rim. The constraint equations, formed using
the procedure outlined above, led to the analysis result as shown in Figure
3(b). A full 3D analysis of the wheel model with similar boundary and loading
conditions has also been carried out, the result of which is illustrated
in Figure 3(c). As identical contouring scales were used, it can be seen
that the stress distribution of the solid-shell model correlates closely
with that of the 3D model. However, a saving in cost over the full 3D model
of a factor of approximately 2.5 was achieved, with the solid-shell model
being solved in 327 seconds and the full 3D model in 795 seconds.
(a)
|
|
|
(b)
|
(c)
|
Figure 3. Application to an automotive wheel: (a) Solid-shell model;
(b) Von Mises stress on solid shell model; (c) Von Mises stress on full
3D model
References
McCune, R. W., Armstrong, C. G. and Robinson,
D. J. (2000). Mixed dimensional coupling in finite element models.
International
Journal for Numerical Methods in Engineering,
49, 725-750.
Monaghan, D. J., Doherty, I. W., McCourt, D. and
Armstrong, C. G. (1998).
Coupling
1D beams to 3D bodies.
7th Inthernational Meshing Roundtable,
Sandia National Laboratories, Dearborn, Michigan, 285-293.
Roark, R. J. and Young, W. C. (1975). Formulas
for Stress and Strain. 5th edition, McGraw-Hill.
Shim, K. W., Monaghan, D. J. and Armstrong, C.
G. (2001),
Mixed
dimensional coupling in finite element stress analysis,
10th
International Meshing Roundtable, Sandia National Laboratories, Newport
Beach, California, 269-277.
