A Finite Element Analysis of Beams on Elastic Foundation including Shear and Axial Effects

A displacement finite element method for analyzing a beam on continuous elastic foundation is presented. A three-dimensional model which accounts for the effects of both the Filonenko-Borodich and Pastemak foundation models in a consistent and complete way is used. A variational principle is introduced with the slope field due to bending only and the displacement field approximated by independent quantities subjected to variation. Numerical examples illustrate the accuracy of the element, the importance of shear, axial and shear-axial interaction effects associated with continuous elastic foundation, and finally the application of the element to a rotor supported by two hydrodynamic journal bearings. NOTATION beam cross sectional area definition in Fig. I beam diameter Young’s modulus of elasticity uniform, elastic foundation spring constants per unit length in y and z directions uniform, coupling elastic foundation spring constants per unit length in y and z directions shear modulus moments of inertia of the beam cross section with respect to y and z axes cross moment of inertia of the beam cross section shear correction factor with respect to z axis beam element length bending moment in x-y plane axial force; positive in tension critical axial force (buckling load) i = I, 2,3 shape functions uniform distributed loads in y and z directions moments of area with respect to y and z axes (Fig. 1) axial displacement of the beam axial displacement of the centroidal axis of the beam displacement of the beam in y direction virtual displacement of the beam in y direction shear forces in y and z directions dispiacement of the beam in z direction virtual displacement of the beam in z direction coordinate system shear strain in y direction on a cross section perpendicular to x direction translations of the elastic foundation in the y and z directions angle of rotation of the beam about the y axis due to bending only virtual angle of rotation of the beam about the y axis due to bending only 7 Present address: General Motors Research Laboratories, Fluid Mechanics Department, Warren, MI 48090; formerly Ph.D. Candidate. $ Professor and Chairman. Fig. 1. Definition of 6, and b,, . shear stress in y direction on a cross section perpendicular to x direction matrix inverse of a matrix column vector transpose of vector