Abstract
This paper alternatively derives the exact element stiffness equation for a beam on Kerr-type foundation. The shear coupling between the individual Winkler-spring components and the peripheral discontinuity at the boundaries between the loaded and the unloaded soil surfaces are taken into account in this proposed model. The element flexibility matrix is derived based on the virtual force principle and forms the core of the exact element stiffness matrix. The sixth-order governing differential compatibility of the problem is revealed using the virtual force principle and solved analytically to obtain the exact force interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix based on the exact force interpolation functions. The so-called “natural” element stiffness matrix is obtained by inverting the exact element flexibility matrix. One numerical example is utilized to confirm the accuracy and the efficiency of the proposed beam element on Kerr-type foundation and to show a more realistic distribution of interactive foundation force.
Highlights
As a numerical counterpart of the continuous medium model, the continuum finite element model has been widely used by geotechnical researchers in studying several complex soil-structure interaction (SSI) problems due to drastic advances in computer technology
The interactive foundation force distribution obtained with the Kerr-type foundation model corresponds well to the observation made by Foppl [12] that there exists a certain degree of discontinuities in system responses between the loaded and the unloaded regions of the actual beamfoundation system. This feature is unique to the Kerr-type foundation model and clearly indicated by incompatibilities between the upper and the lower foundation spring forces at
The “natural” element stiffness matrix and the fixed-end force vector for a beam on elastic foundation subjected to a uniformly distributed load are derived in this paper
Summary
As a numerical counterpart of the continuous medium model, the continuum finite element model has been widely used by geotechnical researchers in studying several complex soil-structure interaction (SSI) problems due to drastic advances in computer technology. Morfidis [20, 21] derived the exact beamfoundation stiffness matrix based on the exact solution of the governing differential equilibrium equations for static and dynamic analyses, respectively, and calibrated the foundation parameters with the analysis results obtained with high fidelity 2D finite element models. The modified Kerr-Reissner hybrid foundation model is attractive to practicing geotechnical engineers since it combines the advantages of both mechanical subgrade model and simplified continuum model as comprehensively discussed in Colasanti and Horvath [25]. It is 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 This finding renders the proposed flexibility-based model attractive since the analytical solution to the governing differential compatibility equation requires only the homogeneous part. A 2D finite element package VisualFEA [33] is used to analyze this numerical example for comparison purpose
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