Force‐based derivation of exact stiffness matrix for beams onWinkler‐Pasternak foundation

Abstract

Key wordsBeam elements, Winkler-Pasternak foundation, finite element, force-based formulation, virtual force principle, soil-structure interaction, natural stiffness matrix.In this paper, the exact element stiffness matrix for a beam on Winkler-Pasternak foundation and the fixed-end force vector due to a linearly distributed load are alternatively derived based on the virtual force principle. The exact element flexibility matrix is at the core of the derivation of the exact element stiffness matrix and is formulated based on the exact force interpolation functions. The virtual force principle is employed to reveal the governing differential compatibility equations as well as the associated end-boundary compatibility conditions. The exact force interpolation functions of the beam-foundation system are derived based on the analytical solution of the governing differential compatibility equations of the problem for all combinations of foundation parameters and beam rigidity. This feature is unique to this paper. The matrix virtual force equation is employed to obtain the exact element flexibility matrix using the exact force interpolation functions. The so-called "natural" element stiffness matrix is obtained by inverting the exact element flexibility matrix.Three numerical examples confirm the accuracy and the efficiency of the natural beam element on Winkler-Pasternak foundation and show the deficiency of the widely used Winkler foundation model.

Flexural responses of nanobeams with coupled effects of nonlocality and surface energy

“…formulated a nonlocal beam element with inclusion of surface effect to investigate the effect of nonlocal parameter and surface stress on nonlinear bending and vibration characteristics of nanobeams; Sourki and Hosseini studied analytically transverse vibration of a cracked nanobeam considering the coupling effects of nonlocal and surface energy; and Ponbunyanon et al . enhanced the force‐based beam Winkler‐Pasternak foundation model of Limkatanyu et al . with the ability to account for both nonlocal and surface effects.…”

Universal analytical solution of the steady-state response of an infinite beam on a Pasternak elastic foundation under moving load

“…A fundamental limitation of the Winkler elastic foundation model is that of neglecting the interactions between adjacent foundation springs, thus overlooking for the cohesive bonds between medium particles. This may lead to unrealistic results (Limkatanyu et al (2015) [28]). To narrow down the gap between the real behaviour of continuous media and Winkler elastic foundation models, several researchers have enriched the Winkler model by introducing a coupling effect between continuous Winkler springs and different embedded structural elements.…”

“…Since A 4 = √ t≥ 0, by definition, from Eqs. (28) and Eqs. (37) it results that roots q 1 , q 2 are always placed in the lower half-plane of the complex plane, while roots q 3 , q 4 are located in the upper half-plane.…”

Exact Stiffness for Beams on Kerr-Type Foundation: The Virtual Force Approach

“…It is noted that when the upper-spring modulus 2 approaches infinite, ( 20) is reduced to a fourth-order governing differential compatibility equation of the beam on Winkler-Pasternak foundation as given by Limkatanyu et al [30] and when the shear-layer section modulus GA is equal to zero, (20) becomes a fourth-order governing differential compatibility equation of the beam on Winkler foundation as given by Limkatanyu et al [29]. Furthermore, when compared to the governing differential equation derived by Avramidis and Morfidis [19] using the principle of stationary potential energy (equivalent to the virtual displacement principle), it becomes clear that (20) and the one derived by Avramidis and Morfidis [19] are dual.…”

“…The exact element stiffness matrix can be obtained directly from the exact element flexibility matrix following the natural approach [28]. It is noted that the natural approach had been used with successes in deriving the exact element stiffness matrices for beams on Winkler foundation [29] as well as beams on Winkler-Pasternak foundation [30]. It is also imperative to emphasize that, in the proposed model, the applied distributed load does not influence the exact force interpolation functions as long as it varies uniformly along the whole length of the beam.…”