Coupling Shells to Solids for Efficient FE Analysis of
Laminated Composite Structures
Introduction
In the analysis of thin, sheet-like structures, finite elements of reduced
dimension such as plate and shell elements are usually employed for simplicity
and low cost, while near discontinuities in geometry, material or loading,
resolution of the complex stress fields requires 3D analysis. Combinations
of the reduced dimensional element types with the higher dimensional element
types require some scheme to be integrated into the analysis such that
compatibility of displacements and stress equilibrium can be achieved at
the interfaces between dimensions. An accurate and practical technique
for coupling plates/shells to 3D solids in finite element analyses of composite
laminated structures has been developed. Via multi-point constraints for
linking the incompatible nodal degrees of freedom, the proposed coupling
scheme provides an effective way of modelling transitions with little or
no stress perturbation in the vicinity of the dimensional interfaces.
Approach
In common with method adopted for
solid-beam
coupling and
solid-shell
coupling, the basic principle of the technique is that the work done
on both sides of the interface between dimensions is equated (Monaghan
et
al., 1998; McCune
et al. 2000; Shim
et al., 2001). Consider
for example a solid-plate n-layered laminated model subjected to a longitudinal
or normal force N
x as shown in Figure 1.
Figure 1. Solid-shell model subjected to normal stress resultant
Nx
It is well established that the application of an in-plane longitudinal
force to a laminate consisting of multiple orthotropic layers with arbitrary
orientations may lead to shearing, bending and twisting deformation. Coupling
of solid and plate/shell elements for the most general case must therefore
incorporate these effects. Equating the work done by N
x
acting on the edge of the plate with the work done by the longitudinal
stress
sx and the shear stress
txy
induced by the coupling stiffnesses on the 3D body at the interface gives
|
(1)
|
where u represents the longitudinal displacements of the plate edge
elements in the x direction and U and V denote the respective 3D continuum
displacements in the x and y directions at the dimensional interface. Substitution
of the relevant stress components in terms of stress resultant N
x
(determined using classical lamination theory) into equation (1) yields
|
(2)
|
where F
1(k)(z) and F
2(k)(z)
are functions relating N
x to
sx(k)
and
txy(k) respectively.
Assuming that the variation of the stress resultant along the plate edge
at the interface can be described by 1D shape functions and the nodal values,
Nx = [N
1D]{Nx}, equation (2) can be converted into a discrete
form in terms of the finite element nodal displacements:
|
(3)
|
where [N
1D] and [N
2D] denote the shape functions
of the plate elements along their edges and the 3D elements over their
faces at the solid side of the interface respectively, while n
e
in this case represents the number of 3D elements connected to the plate
edge at the transition. Equation (3) can then be simplified by eliminating
the arbitrary stress resultants {N
x} from both sides of the
equation, giving
|
(4)
|
From equation (4), a set of generalized multi-point constraints can
thus be obtained as
|
(5)
|
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. In fact, the
key requirement of this technique is to determine the assumed variation
in stress in the reduced dimensions of the structural elements, i.e. the
variation in stress through a plate or shell thickness or over a beam cross-section.
The classical lamination theory (Jones, 1999) forms the basis for the evaluation
of the in-plane stresses, while the out-of-plane shear stresses can be
determined from the three-dimensional equilibrium equations (Rolfes and
Rohwer, 1997).
Analysis Results
Based on the current procedures in deriving the
constraint equations, a number of finite element composite models with
arbitrary lamina orientations and stacking sequences has been examined.
Each ply employed in the analyses was idealized as a homogeneous, elastic
orthotropic material with the following properties:
E
1 = 138 GPa
E
2 = E
3 = 14.5 GPa
G
12 = G
13 = G
23
= 5.86 GPa
n12 =
n13
=
n23 = 0.21
The subscripts 1, 2 and 3 correspond to the longitudinal,
lateral and thickness directions respectively of a 0° ply. The models
were analysed using twenty-noded bricks representing the 3D solids, while
eight-noded reduced integration elements were used for the adjacent 2D
laminated plates. The contours plotted on the plates are the stresses predicted
on the top surfaces and should be continuous with those visible on the
top of the 3D regions.
|
|
|
Figure 2. Normal stress sx
on a two-layered (0/90) laminate due to normal force Fx
|
Figure 3. Bending stress sx
on a two-layered (15/-15) laminate due to bending moment My
|
|
|
|
Figure 4. Shear stress txf
on a four-layered (30/-30/-30/30) laminate due to bending moment Mz
|
Figure 5. Shear stress txf
on a four-layered (45/-45/-45/45) laminate due to shear force Fy
|
|
|
|
Figure 6. Shear stress txz
on a a two-layered (15/-15) laminate due to shear force Fz
|
Figure 7. Von Mises stress on a four-layered (30/-30/-30/30)
laminate due to twisting moment Mx
|
As can be seen from Figure 2 through Figure 7, proper connection between
the shell and 3D continuum elements has been achieved, with continuous
and reliable stress contours in the vicinity of the dimensional transitions.
A small but obvious stress discontinuity in stress occurred near the laminate
edges, especially in the model of Figure 5. This mainly arose from the
presence of edge effect due to the mismatch in material properties between
layers, which could not be determined using two-dimensional plate or shell
theories. However, satisfactory stress predictions are guaranteed at approximately
one laminate thickness away from the free edges, although significant stress
disturbance was developed at the 3D back region due to the application
of the fixed constraint.
|
|
|
(a)
|
(b)
|
Figure 8. Four-layered (0/90/90/0) laminate subjected to normal
strain ex:
(a) shear stress tyz; (b)
interlaminar shear stress distribution at 90-0 interface
|
|
|
(a)
|
(b)
|
Figure 8. Four-layered (0/90/90/0) laminate subjected to normal
strain ex:
(a) normal stress sz; (b)
interlaminar normal stress distribution at 90-0 interface
With only half the width and one-third of the length discretized using
3D brick elements, a finite element solid-plate model of a (0/90/90/0)
cross-ply laminate subjected to a uniform axial extension has also been
put into application. Its corresponding interlaminar stresses,
tyz
and
sz, are depicted in Figures 8(a)
and 9(a) respectively, while the variations of these out-of-plane stresses
at the 90-0 interface directly above the bottom layer and coinciding with
yz plane at the centre of the 3D region are shown in Figures 8(b) and 9(b).
As can be seen, the 3D stresses predicted at the boundary layers along
the laminate free edges show excellent agreement with Pagano’s analytical
results (Pagano, 1978). Although the shear stress
tyz
does not follow the sharp drop of the exact solution due to the relative
courseness of the mesh, these interlaminar stresses have been well captured
to a sufficient degree of accuracy.
Conclusions
-
The coupling procedures yielded continuous and reliable
stress contours at the interfaces between dimensions, where stress disturbances
can otherwise occur if conventional methods (e.g. rigid link) are used.
-
Symmetric and non-symmetric laminates with arbitrarily
oriented plies having various coupling effects can be accommodated.
-
This technique should greatly contribute to efficient
and accurate modelling of complex laminated components as far as mixed
dimensional models are supported.
References
Jones, R. M. (1999).
Mechanics of Composite
Materials. 2nd edition, Taylor & Francis.
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.
Pagano, N. J. (1978). Stress fields in composite
laminates.
International Journal of Solids and Structures,
14,
385-400.
Rolfes, R. and Rohwer, K. (1997). Improved transverse
shear stresses in composite finite element based on first order shear deformation
theory.
International Journal for Numerical Methods in Engineering,
40,
51-60.
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.
