Keywords: variational multiscale method, composite beams, multiple-point constraints, perfect bond, interpolation error.

Composite beams that consist of two or more layers find widespread applications in a variety of engineering structures because they benefit from the best properties of their components. A practical way for the computational modelling of composite beams is to stack the layers together, while each layer is represented by a finite element, and then connect the elements at the nodes by using multiple-point constraints. However, this type of modelling does not inherit the kinematic behaviour of the continuous case and in most cases the finite element modelling results with a behaviour in which the kinematic constraint conditions imposed by the bond between the juxtaposed layers are weakened. As a consequence, finite element modelling based on multiple-point constraint application at the nodes may produce behaviour that is falsely softer than the perfectly bonded composite beam behaviour [1].

In this paper, it is shown that the source of error in multiple-point constraint applications of this type can be related to the incompatibility in the displacement field. A recent study [2] shows that the variational multiscale approach [3] can be adopted within this context and the multiple-point constraint application can be interpreted as the solution in a superfluously extended space, while the extension in the solution space arises from the weakening in the kinematic constraints. Following the approach introduced in [2], it is shown in this paper that when kinematic conditions for composite action are weakly imposed in the variational form, an interpolated finite element displacement field can be conceived as a displacement field of an extended interpolation space under a constraint condition e.g. [4]. In this study, the perfect bond kinematics is enforced in the point-wise sense by excluding the fine-scale effect from the extended solution space within the frame work of the variational multiscale approach without modifying the kinematic model or otherwise by proper selection of the interpolation field. It is shown that the cases where perfect bond between the composite beam layers are enforced as a-priori condition can be recovered. In the case of multiple-point constraint applications modifications to the element stiffness matrix and the energy equivalent load vector that are necessary to recover the perfect bond behaviour in the analysis of composite beams are identified. The selected cases show the effects of external load on the numerical error especially when shear behaviour becomes more dominant. The convergence characteristics in multiple-point constraint applications are also shown for the selected cases and compared with those based on the variational form in which the perfect bond is enforced as an a-priori condition.

1

A.K. Gupta, S.M. Paul, "Error in eccentric beam formulation", International Journal for Numerical Methods in Engineering, 11, 1473-1483, 1977. doi:10.1002/nme.1620110910

2

R.E. Erkmen, M.A. Bradford, K. Crews, "Variational multiscale approach to enforce perfect bond in multiple-point constraint applications when forming composite beams", Computational Mechanics (accepted 17th November 2011). doi:10.1007/s00466-011-0667-5

3

T.J.R. Hughes, "Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods", Computer Methods in Applied Mechanics and Engineering, 127, 387-401, 1995. doi:10.1016/0045-7825(95)00844-9

4

R. Courant, D. Hilbert, "Methods of mathematical physics", Wiley, New York, 1962. doi:10.1063/1.3057861

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