[Computer] Program for Solution of Beams on Bi-Moduli & Non-Linear Elastic Foundation (Translation from Bulgarian) eng. Mario Behar – Technion, Israel          Introduction          This article is a presentation of Version 1.1 of the “ELFOUND.EXE (EF.EXE)” program, made by the author of this article. The program is based on a course work instruction sheet for the course “Beams on elastic foundations” given by Prof. David Yankelewski in the Faculty of Civil Engineering in the Technion – the Technology Institute of Israel, in Haifa.          Theoretical Basis          The program is based on the theory published in [1], [2], [3] & [4]. Basic in this case is the assumption introduced by E. Winkler in 1867, that the reactions of the [elastic] support, in every single point, are proportional to the deflection of the beam at this point, namely: Eq. (1) … (see Bulgarian original) where y is the deflection, p – the force (pressure), and k – the coefficient of correlation between them – called also “support medium coefficient” or “support medium modulus”. According to [1] the elastic (deflection) line of a beam, supported on elastic foundation, can be mathematically presented by the following differential equation: Eq. (2) … (see Bulgarian original)           The general solution of (2) is as follows: Eq. (3) … (see Bulgarian original)          Considering that the first differential of (3) gives the rotation, the second differential gives the internal moment, and the third gives the shear in the beam, after certain [mathematical] operations we get the following equation: Eq. (4) … (see Bulgarian original) where: Eqs. (5), (6a) … (6d) … (see Bulgarian original) and the annotations with subscript “zero” are respectively deflection, rotation, internal moment, and shear in the left end of the beam (x = 0). These annotations are the reason the method to be called also “Method of the initial conditions”.          The current program is based on the Finite Elements Method. The finite element for a beam on elastic foundation – developed by the authors of [2], and presented there, is shown in Fig. 1, and has the following stiffness elements: Eqs. (7a) … (7f) … (see Bulgarian original) where L is the length of the element, and Eq. (8) … (see Bulgarian original)          In equations (7) the Modulus of the elastic medium k can take values corresponding to compression or tension (Fig. 2) depending on the zone in which the element is placed.          Exact solutions can be obtained when the nodes of the elements are placed at the points where concentrated vertical forces and moments are applied; where point supports are located; where the distributed loads, cross section of the beam, or the properties of the elastic foundation change their value.          For the free supported elements, the stiffness matrix has the following known format: Eq. (9) … (see Bulgarian original)          The problem is two-dimensional – with two degrees of freedom (vertical displacement and rotation) for each node.          Program Capabilities          The program is written in the TurboBASIC and ASSEMBLER computer languages. The graphic and dialog interfaces at all levels are made entirely by the author [of this article]. The basic principles of the program part that actually makes the computations are borrowed from the “Continuous Beams” section of [5]. For the matrix inversion procedure, not including the “restricted” degrees of freedom, an elegant solution from [6] has been used, in combination with the method of Gauss for matrix solution of multiple equations [with common unknowns].          As a first step in the development of the [future] complete product, this version offers the following features:          - solution of beams totally supported on elastic foundation, or totally on free supports;          - a friendly graphical interface for data input offering possibilities for fast corrections before and after the problem solution (Fig. 3);          - [possibility for] introduction of:            • fixed supports with respect to displacement and rotation, as well as local [point] sup-ports elastically restricting displacements and rotations (i.e. vertical and rotational springs);            • different compression and tension intensities of the linear elastic foundation;            • concentrated forces and moments;            • distributed vertical load;            • additional nodes (different from the above described), necessary in certain complex cases to initialize the iteration process, or simply points of special interest for the program user;          - additional tabular form of presentation of the input data;          - tabular form of presentation of the output data.          In the solution of problems related to beams on bi-moduli and non-linear elastic foundation, of great significance is to localize the points of zero deflection, i.e. the points [of the contact surface between the beam and the medium below it]  that separate a zone of compressed medium from a zone of medium in tension (Fig. 4). For the program these are point with variable position, in contrast to all [the points] described before. The search for them is being done automatically, by iterations, with a convergence criterion of L/1000 where L is the length of the beam.          Conclusion          Under development is the next version of the program, which will encompass the following features:          - solution of beams on non-linear elastic foundation, with properties of the foundation [medium] as presented on Fig. 5;          - solution of beams with portions of them on “free” supports [without elastic foundation] and other portions on elastic foundation (single-modulus, bi-moduli, or non-linear) (Fig. 3);          - possibility for operative change of the convergence criterion when one is in search of the transition points from a compressed elastic foundation medium zone to a zone under tension, and vice versa (i.e. when one is in search of the points with zero vertical displacement);          - graphical presentation of the results (in the current version they can be presented only in a tabular form);          - variable stiffness along the length of the beam;          - incorporation in the problem scheme of local vertical and rotational springs.          REFERENCE:          1. Hetenyi, M. Beams on Elastic Foundation, University of Michigan Press, Ann Arbor, Michigan, 1964.          2. Eisenberger, M., D. Yankelewski, Exact Stiffness Matrix for Beams on Elastic Foundation, Computers and Structures, Vol. 21, No. 6, 1985.          3. Adin, M., D. Yankelewski, M. Eisenberger, Analysis of Beams on Bi-Moduli Elastic Foundation, Computer Methods in Applied Mechanics and Engineering (North Holland), Vo1. 49, No. 3, 1985 (pp. 319-330).          4. Yankelewski, M., M. Eisenberger, M. Adin, Analysis of Beams on Nonlinear Winkler Foundation, Internal Publication – Dept. of Civil Engineering, Technion, Haifa, Israel, 1986.          5. Weaver, W, Jr., J. Gere, Matrix Analysis of Framed Structures, 3rd Edition, Van Nostrand Reinhold, N.Y., 1990.          6. Math Pac for HP41C, Hewlett- Packard, 1984. Back to Top home ... other publications