Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method


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1 Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yungang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, China ABSTRACT Beams on elastic foundations are often encountered in the regions of civil engineering, harbour engineering and railway engineering. Analytical equations for this kind of problem are not complex to be used to analyze regular beams and plates, but they are not suitable for abnormal beams or plates. Finite element code incorporating beam elements on elastic foundation is a better choice for this problem. Generalpurpose finite element package ABAQUS provide an option of FOUNDATION to model elastic foundation, but it does not support beam elements. In this paper, the application of using plate (shell) element to represent behaviors of beams was verified firstly, and then two numerical examples of beams on elastic foundations were examined by combining plate (shell) element with FOUNDATION option. It is found that this method could be used to analyze this kind of problem without losing accuracy in comparison with analytical results. KEYWORDS: beams on elastic foundations; plate (shell) element; finite element analysis. INTRODUCTION The model of beams on elastic foundations is a simplified method to analyze forces of beam foundations or mat foundations, in which the reaction forces of subsoil are proportional to the deflection of the beam or mat at each point. That is to say, the foundations seem to lie on identical, independent, closely spaced, and linearly elastic springs. The stiffness of these springs, known as modulus of subsoil reaction k 0 [kn/m 2 /m], links intensity of reaction forces on foundation (pressure p) and deflection w, as p = k0 w. This postulate is presented first by Winkler in 1867 and still used in geoengineering designs due to its simplicity. Many simple calculation software or diagrams are formulated to analyze this kind of problems, but most of them are based on thin beam or thin plate theory and can only deal with beams or plates with simple outline, such as straight beam, regular flat plate. Some commercial finite element packages, e.g., SAP, MARC, also incorporate element type of beam on elastic foundation, then beams or plates with complex outline on elastic foundations can be analyzed
2 Vol. 17 [2012], Bund. N 2064 easily and quickly by using them. Generalpurpose finite element package ABAQUS provides an option of FOUNDATION, which acts like springs to ground and is a simple way of including the stiffness effects of a support (such as the soil under a building) without modeling the details of the support (Abaqus, 2010). This option can be used to simulate elastic foundation by prescribing the foundation stiffness per area. However, this elastic foundation option does not support beam elements, a number of spring elements are often laid under beam to accomplish this type of simulation (Charlton, 2007). This paper presents an alternative method to analysis beams on elastic foundation based on calculation framework of ABAQUS, in which the beam is modeled by plate (shell) element. Its reliableness is verified by two numerical examples comparing with the numerical solutions. BEAM AND PLATE THEORY It is well known that there are two kinds of beam theories: EulerBernoulli beam (thin beam) theory and Timoshenko beam (thick beam) theory (Hetenyi, 1950; Hughes, and Cohen, 1978). Figure 1 can illustrate the difference between them. For a beam under transverse loads, Euler Bernoulli beam assumes that the crosssectional plane being perpendicular to the neutral axis of the beam remains plane and is still perpendicular to the neutral axis after deformation. Namely, no transverse shearing deformation is taken into account, ϕ = d w/dx. For beams with relatively high crosssections compared with the beam length subjected to larger shear forces, it is inapplicable to neglect the transverse shear deformation. Timoshenko beam (thick beam) theory relaxes the perpendicular assumption but remains the plane assumption of crosssection plane to consider shear deformation γ xz, i.e. ϕ d w/dx. d w/dx γ xz Z ϕ Neutral axis dx dw X Figure 1: Bending of beam Plate theory, including thin plate theory and thick plate theory, can be viewed as extensions of onedimensional beam theory to twodimension space. For thin plate, an assumption referred to as hypothesis of straight normal, which is like what adopted in thin beam theory, is assumed. The normal to the middle plane of plate before bending remains straight and normal to the middle surface during the deformation, and the length of such elements is not altered. This also means that the vertical shear strains at cross section, γ xz andγ yz, are negligible and the normal strain may also be omitted. Thick plate theory takes the assumption which is analogous to that associated with the Timoshenko beam theory, i.e., γ xz 0 and γ 0. yz
3 Vol. 17 [2012], Bund. N 2065 From the above mentioned, one can use plate elements to substitute beam elements in finite element analysis, if the loads and supports are equivalent to that of beams. For example, line loads along cross section of the plate can substitute concentric forces acting on beams, and line loads acting on longitudinal axis of beams can be substituted by surface tractions on the plate. In the following sections, a cantilever beam subjected to concentric force was taken as an example to examine the applicable substitution of beams with plates, and examples of beam on elastic foundation were analyzed and compared with analytical solutions. NUMERICAL EXAMPLE OF BEAM SUBSTITUTED BY PLATE To illustrate that the plate elements reduces to the same behavior as beam elements, a cantilever beam subjected to concentric force, F = 10000kN, shown in Figure 2 was modeled. The elastic modulus of material of beam is 20GPa and Poisson's ratio is 0.3. The numerical model of the beam was established using beam element, plate (shell) element, and continuum element, as presented in Figure 3. F 6.0m 1.6m Figure 2: Cantilever beam 0.8m (a) Beam element (b) Shell element (c) Continuum element Figure 3: Numerical models of cantilever beam The end displacements calculated by analytical and numerical methods are presented in Table 1. It is obviously that the vertical displacements predicted by thick beam or thick plate theory are larger than that calculated by thin beam or thin plate theory, for the effect of shear deformation on displacement is considered. This example clearly indicates that plate (shell) element provided by ABAQUS can model a onedimensional beam without losing accuracy. Table 1: Vertical end displacement of cantilever beam Analytical result Beam element Plate (shell) element Continuum EulerBernoulli Timoshenko EulerBernoulli Timoshenko Thin Thick Generalelement beam Beam beam Beam shell shell Purpose shell 0.132m 0.137m 0.132m 0.139m 0.129m 0.138m 0.138m 0.137m
4 Vol. 17 [2012], Bund. N 2066 NUMERICAL EXAMPLES OF BEAMS ON ELASTIC FOUNDATIONS In this section, the option of FOUNDATION and plate (shell) element was used to model beams on elastic foundations. Two examples were analyzed and compared with analytical solutions. Consider a beam on elastic foundation with free ends. The geometrical dimensions loadings are shown in Figure 4. The modulus of elasticity of material of the beam is 20GPa, and the 4 Poisson's ratio is 0.2. The modulus of subsoil is k 0 = kn/m 2 /m. Figure 5 presents the deformed figuration of the beam modeled by plate (shell) elements, which is consistent with analytical prediction (Huang and He, 2005). The maximum vertical displacement occurred at the middle of the beam is 6.44mm, with corresponding bending moment kN m. There are only slight differences between the numerical results and the analytical ones, which are 6.46mm and kN m, separately. 4 q = kpa Example 1 F = 400kN 0.3m 5.0m 5.0m 1.0m Figure 4: Numerical example 1 Figure 5: Deflection of f example 1 Magnified by factor: 100 Example 2 Example 2 is a beam laid on elastic foundations withh 10m in length, 1.1m in width and 0.5m in depth (Parvanova, 2011). Loads that are more complexx were applied on the beam, as shown in Figure 6. The elastic modulus of material of the beam is 30GPa, and the Poisson's ratio is The modulus of subsoil is k 0 = 5 10 kn/m /m. 200kN 100kN m 200kN/ /m 0.5m 1.0m 3.0mm 1.0m 5.0m 1.1m Figure 6: Numerical example 2
5 Vol. 17 [2012], Bund. N 2067 Numerical models based on plate (shell) elements were established to analyze this problem, with thin plate element and thick plate element used separately. However, the results analyzed with these two types of elements have no remarkable difference and the results of thick plate element are presented in the following. The vertical displacements, EIv( x ) obtained by numerical method are compared with analytical results in Figure 7 and they match very well. Figure 8 shows the distribution of bending moment which is consistence with the analytical result. The maximum bending moment predicted by numerical analysis is 163.4kN m, which is 3.3percent lower that of analytical result, 169.1kN m. The bending moments at the 1m cross section and 5 m cross section away from the left end of beam are kN m and kN m, which are very close to that of analytical solutions, kN m and kN m. Length (m) EIv(x) Analytical solution Numerical solution 1600 Figure 7: Deflection along axis of the beam Figure 8: Distribution of bending moments along axis of the beam SUMMARY In this paper, reduction of thick and thin plate (shell) element to EulerBernoulli beam element and Timoshenko beam element was validated by an example of cantilever beam. Then plate (shell) element in conjugated with FOUNDATION option was used to analyze two numerical examples of beams on elastic foundations. Through comparing with numerical solutions, it can be drawn that this method could be used to analyze these kinds of problems without losing accuracy.
6 Vol. 17 [2012], Bund. N 2068 REFERENCES 1. Abaqus, (2010). User s Manual. Version Charlton,Z. I. (2007) Innovative Design Concepts for Insulated Joints, Master thesis, Virginia Polytechnic Institute and State University. 3. Hetenyi, M. (1950) A General Solution for The Bending of Beams on an Elastic Foundation of Arbitrary Continuity, Journal of Applied Physics, Vol. 21(1), Huang, Y. and He, F. S. (2005) Beams, Plates and Shells on Elastic Foundations (in chinese), Science Press of China, Beijing. 5. Hughes,T.J.R. and Cohen, M. (1978) The Heterosis Finite Element for Plate Bending, Computers and Structures, Vol. 9, Parvanova,S. (2011) Lecture notes: Structural Analysis II, University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria ejge
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