Keywords—Elastic foundation, Finite Element Method, beam element, structures, contact element, theory, applications, biomechanics, traumatology, external fixators, mining, mining supports, rack-railway, drop-in test Abstract—This paper contains numerical methods and approaches used in the solution of plane beams and frames and 3D structures on an elastic foundation. In the first case, the solution uses beam element BEAM54 in the program ANSYS and the derivation of the stiffness matrix for this element is presented. The second approach uses a beam element in a combination with a contact element with the description of the derivative of the stiffness matrix applied for the frame on elastic foundation. Both solutions are compared with theoretical solution. The influence of the number of divisions for the beam element on the accuracy of the solution is shown. There are also presented some other application of structures on elastic foundation (biomechanics & traumatology – external fixators for treatment of complicated bone fractures, mining industry - pressure distributions in the contact between mining supports and foot-wall, rack-railway and drop-in test as a problem of 3D body on elastic foundation). I. INTRODUCTION OLUTION of frames and beams on elastic foundation often occur in many practical cases for example, solution of building frames and constructions, buried gas pipeline systems and in design of railway tracks for railway transport, etc., see Fig.1. Fig. 1 Example of a beam resting on an elastic foundation. This beam is loaded by force F, couple M and distributed loading q. Assoc. Prof. M.Sc. Karel FRYDRÝŠEK, Ph.D., ING-PAED IGIP is with the Department of Mechanics of Materials, Faculty of Mechanical Engineering, VŠB – Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic, (corresponding author, phone: +420 597323495, e-mail: karel.frydrysek@vsb.cz). Assoc. Prof. M.Sc. Roland JANČO, Ph.D., ING-PAED IGIP is with the Institute of Applied Mechanics and Mechatronics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, Nam. Slobody 17, 812 31 Bratislava, Slovak Republic (e-mail: roland.janco@stuba.sk). Prof. M.Sc. Horst GONDEK, DrSc. is with Department of Production Machines and Design, Faculty of Mechanical Engineering, VŠB – Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic, (e-mail: horst.gondek@vsb.cz). Solution of beam on elastic foundation is a statically indeterminate problem of mechanics. In this case, we have the beam with elastic foundation along the whole length and width or only over some part of the length or width. Detailed explanation of theoretical solution can be found in [2], [3] [4] and [10]. Not all problems can be solved by theoretical approach (i.e. sometimes, the theoretical solution is very complicated). In solution of these complicated problems, the Finite Elements Method (FEM) can be applied. In this paper, FEM is applied for the solution of 2D and 3D beams, frames structures on elastic Winkler's foundation including theory and practice. II. THEORETICAL BACKGROUND FOR 2D BEAM ON ELASTIC FOUNDATION The Winkler's foundation model is easy to formulate using energy concepts. The analysis of bending of beams on an elastic foundation (Winkler's model) is developed on the assumption that: • The strains are small. • The resisting pressure 2 R p / Nm / Kv − = in the foundation are proportional at every point to the deflection v = v(x) /m/ normal to its surface at that point. Displacement and resting pressure etc. can be expressed as functions of variable / /m x . The parameter () / /Nm 3 − = x K K is the modulus of the foundation. • The surrounding foundation is utterly unaffected, see Fig.2a. Fig. 2 Deflection of structure on elastic foundation under pressure p or distributed loading q, (a) Winkler foundation, (b) elastic solid foundation • The general problem of the beam on elastic foundation (Winkler's theory) is described by ordinary differential equation. In the most situations, the influences of normal Solutions of Beams, Frames and 3D Structures on Elastic Foundation Using FEM Karel FRYDRÝŠEK, Roland JANČO, Horst GONDEK S INTERNATIONAL JOURNAL OF MECHANICS Issue 4, Volume 7, 2013 362