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

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.

Semi analytical solution of a rigid pavement under a moving load on a Kerr foundation model

“…where , are the flexural rigidities of the plate in the direction and the direction respectively, is the torsional rigidity of a plate and is foundation response. Because the Kerr model [14] consists of two axial springs ( and ) and shear spring layer ( ), the deflection of the plate can be given as [13]:…”

“…Based on the modified Bolotin method, the two unknowns real numbers and in Eqs. 13, (14) can be solved from the two auxiliary Levy's type problems [15].…”