This page uses CSS to present the content in the best possible manner.
If you can see this message, then CSS is not enabled in your browser, and the page will not appear as the designer intended.

Coupling Beams to 2D Solids for Efficient FE Analysis of
Laminated Composite Structures




Return to Coupling Page

Introduction



The finite element technique is becoming increasingly important as an analytical tool for composite or laminated structural design. Such structural components usually consist of many parameters including the type of material used and the material properties as well as the layer stacking sequences. Their design stages, in most cases, are further complicated by the presence of geometric discontinuities, large stress gradients, large displacements and rotations, and material nonlinearities. Various means of analysis exist for response prediction with different levels of accuracy, capability and efficiency. Some of these may affect only a small section of the structures in question, and attempting to discretize the entire domain with 3D solid elements for capturing such critical phenomena can lead to a large, prohibitively expensive analysis. Accuracy and efficiency are thus becoming two major concerns in any finite element analysis that are forcing engineers and design analysts to seek reliable and accurate yet economical methods for determining the responses of structural components.

A large percentage of finite element analyses make use of dimensionally reduced element types that are defined in terms of a reduced geometric representation with associated element properties that account for the dimensions not included. In other words, a thin sheet of material can be represented by an equivalent surface with a thickness attribute, namely plate or shell, while a long slender region can be represented by a beam, a line with cross-sectional properties. These element types produce more computationally efficient models, thus reducing analysis time and cost. Also, they require less data and are easier to construct than solid models and are thus especially suitable for conceptual design evaluation and optimisation.

Beam, plate and shell finite elements can adequately predict accurate structural responses for problems away from discontinuities. But, the simplifying assumptions of the theories make them too restrictive for capturing the critical details near any geometric or material discontinuities or near traction-free edges of a laminated model where a 3D stress field exists. One of the ideal procedures is therefore to unite the reduced or lower dimensional element types with higher dimensional elements in a single finite element model, with the former modelling long or thin regions away from boundaries or discontinuities and the latter modelling either more complex geometries or the behaviour in the boundary regions. These idealised models, however, present mathematical difficulties at the connections between the differing element types due to the incompatibility of their nodal degrees of freedom. Some scheme is therefore required for coupling the differing element types so that compatibility of displacements and stress equilibrium can be established at the dimensional transition.

A method via multi-point constraint equations for coupling two-dimensional composite structures to their equivalent one-dimensional beam elements is briefly presented below. Similar procedures are also applicable to coupling of shells and 3D solids.





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 and Armstrong, 2001; Shim et al., 2001). Consider for example a 2D-1D n-layered laminated model subjected to a terminal pure bending moment Mz as shown in Figure 1.


Figure 1. 2D-1D model subjected to bending moment Mz

Equating the work done on both sides of the interface yields
 

(1)

Since, based on the usual Mechanics of Materials,
 

(2)

where Ex(k) is the Young's modulus of ply k along the x direction and Ik is the second moment of area about the neutral z-axis,
 

(3)

Eliminating Mz from the equation and replacing the 2D continuum displacement U with the finite element nodal displacement, equation (3) becomes
 

(4)

This equation can then be expressed as a multi-point constraint equation of the form
 

(5)

and can be applied in ABAQUS using the *EQUATION command.



Analysis Results

Based on this approach, some sample results as shown below are obtained. The elements employed in the analyses were 2D eight-noded quadrilaterals and 1D quadratic elements.
 

Figure 2. Von Mises stress on a timber beam with steel reinforced plates due to axial force Fx
Figure 3. Direct stress sx on a four-layered (90/0/90/0) laminate due to bending moment Mz

 
Figure 4. Shear stress txy on a four-layered (0/90/90/0) laminate due to shear force Fy
 Figure 5. Shear stress txy on a two-layered (90/0) laminate due to shear force Fy


As can be seen from the above stress plots, the coupling procedure yielded good results for the stress contours with continuity and no disturbance at the interface. Also, it was found that the nodal values obtained from the finite element analysis agree well with those obtained using analytical solutions.



Figure 6. Application to a honeycomb sandwich flap: (a) cross-section; (b) Von Mises stress on full 2D model; (c) Von Mises stress on 2D-1D model


The current approach for 2D-1D coupling has also been applied to the analysis of a carbon fibre honeycomb sandwich flap, Figure 6(a). For comparison, a full two-dimensional model of the sandwich panel with uniformly distributed pressure on the bottom surface was analysed first and the analysis result is shown in Figure 6(b), while the corresponding 2D-1D coupling result is illustrated in Figure 6(c). For both cases, fully constrained boundary conditions were applied at the beam end node of the mixed dimensional model and the right end edge of the full 2D model. The material properties used in the analyses, for simplicity, were assumed to be isotropic homogeneous with the face sheets a thousand times stiffer than the honeycomb core. As can be seen from the Von Mises stress plots with identical contouring scales, the stress distributions on the 2D stiffened edge are virtually identical for both models. A faster analysis was achieved with the 2D-1D model, though the saving was not too significant due to the relative simplicity of the models.


Figure 7. Superelement model of a honeycomb sandwich flap

Figure 7 illustrates the use of substructuring to be integrated into mixed dimensional modelling, where the 2D reinforced region of the sandwich flap was condensed out into a stiffness superelement. The uniformly distributed load acting on the 2D elements was built into the substructure during its creation. As beam elements were coupled to the 2D edge at the dimensional interface, only one retained node was required to join the superelement with the rest of the model or beam elements. The local stresses and deformations in the 2D model of the edge, if required, could also be recovered from the substructured analysis model. Condensing out the superelement is a time-consuming operation. But once this cost has been incurred, the full 2D detail of the behaviour in the stiffened edge, including the effect of any stress concentration, is available for basically the same cost as that incurred by using a line stiffener in the global model. As substructuring can isolate possible changes outside the superelement to save time during reanalysis, rapid redesign or modification of the complete structure may be executed with ease.



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.

Shim, K. W. and Armstrong, C. G. (2001). 2D-1D coupling of composite structures. 9th ACME Conference, Birmingham, UK, 119-122.

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.