Abstract
We present a new approach for adding Bernoulli beam reinforcements to Kirchhoff plates. The plate is discretised using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The beams are discretised by the CutFEM technique of letting the basis functions of the plate represent also the beams which are allowed to pass through the plate elements. This allows for a fast and easy way of assessing where the plate should be supported, for instance, in an optimization loop.
Highlights
Reinforcements of plates using lower–dimensional structures such as beams are often employed for the purpose of increasing buckling loads and avoiding eigenfrequencies in vibration problems
The last approach has only been used in the context of Timoshenko beams coupled to Mindlin–Reissner plates, where simple C0 approximations can be used; a similar approach was recently suggested for modeling embedded trusses by Lé, Legrain, and Moës [15]
In this paper we present a method for the coupling of Kirchhoff plates and Euler–Bernoulli beams based on this concept, together with a tangential differential approach which simplifies the implementation for arbitrarily oriented beams
Summary
Reinforcements of plates using lower–dimensional structures such as beams are often employed for the purpose of increasing buckling loads and avoiding eigenfrequencies in vibration problems. The last approach has only been used in the context of Timoshenko beams coupled to Mindlin–Reissner plates, where simple C0 approximations can be used; a similar approach was recently suggested for modeling embedded trusses by Lé, Legrain, and Moës [15]. In this paper we present a method for the coupling of Kirchhoff plates and Euler–Bernoulli beams based on this concept, together with a tangential differential approach which simplifies the implementation for arbitrarily oriented beams. This is possible thanks to the development of continuous/discontinuous Galerkin (c/dG) methods for higher order problems [6,7,9,10], avoiding the use of C1–continuity. In Ref. [12] we proposed to use the same finite element space for the beam as for the higher dimensional structure modeled by linear elasticity, using second order polynomials for elasticity and taking the restriction, or trace, of these polynomials to model the beam using c/dG
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