Rigorous amendment of Vlasov's theory for thin elastic plates on elastic Winkler foundations, based on the Principle of Virtual Power

• Published in: European journal of mechanics, A, Solids A, Solids, 2019-01, Vol.73, p.449-482
• Main Authors: Höller, R., Aminbaghai, M., Eberhardsteiner, L., Eberhardsteiner, J., Blab, R., Pichler, B., Hellmich, C.
• Format: Article
• Language: English
• Subjects: Bending problem; Boundary conditions; Deformation; Differential equations; Elastic deformation; Elastic foundations; Elastic plates; Engineering; Finite element method; Fourier series; Kirchhoff plate theory; Materials elasticity; Plate on elastic foundation; Principle of Virtual Power; Shear forces; Shear stress; Swimming; Systematic errors; Thin plates; Winkler foundation model
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2018.07.013

Deflection modes relevant for plates with rigidly supported edges are commonly used as kind of “approximation” for the deformation behavior of plates which are freely swimming on an elastic foundation. However, this approach entails systematic errors at the boundaries. As a remedy to this problem, we here rigorously derive a theory for elastically supported thin plates with arbitrary boundary conditions, based on the Principle of Virtual Power. Somewhat surprisingly, it appears that the well-known Laplace-type differential equation for the deflections needs to be extended by additional boundary integrals entailing moments and shear forces, so as to actually “release” the boundaries from “spuriously” acting external moments and shear forces. When approximating the deflections through 2D Fourier series, the Principle of Virtual Power yields an algebraic system of equations, the solution of which provides the Fourier coefficients of the aforementioned series representation. The latter converges, with increasing number of series members, to the true solution for the plate deflections. The new method is applied to relevant problems in pavement engineering, and it is validated through comparison of the numerical results it provides, with predictions obtained from Finite Element analysis. With respect to the latter, the new series-based method reduces the required computer time by a factor ranging from one and a half to almost forty. •Vlasov's governing equation entails systematic errors at the plate's boundaries.•The Principle of Virtual Power identifies missing internal power terms.•The Fourier-series based solution procedure is more efficient than FE-modeling.•The new method significantly speeds up pavement engineering simulations.