Keywords: plate bending, equilibrium element, conforming element, dual analysis.

This paper presents the application of the dual analysis concept to plate bending. In this method, a same problem is analyzed parallely by a displacement and an equilibrium model. The energetic distance between these two models is the sum of both global errors and consequently, an upper bound of each of them. After an exposition of the two models, numerical examples are presented, which illustrate the high accuracy of the method.

The best known family of finite element models is the displacement model, in which the compatibility equations are a priori verified, from the fact that the variables are displacements. The solution of the problem then leads to weak forms of the equilibrium equations. But there exists a dual family of finite elements, which are called equilibrium elements. In these elements, an equilibrated stress field is used, and the result of the computation consists in weak compatibility equations.

For plate bending, in the displacement model, continuity of deflection at the interfaces is the first requirement to satisfy. The next requirement is the continuity of normal slope between adjacent elements. Thus, - continuity requirement has to be obtained. Another reason for trying to a rigorous enforcement of normal slope continuity is to guarantee that the direct influence coefficients are actually lower bounds to the true ones in case of the homogeneous displacement boundary. From the above strict conditions, the conforming elements of assembled triangle (HCT) or assembled quadrilateral (CQ) [1] should be chosen. Moreover, in reformulating these elements, the advantage of using the area coordinates is useful when assembling and calculating the fields [5].

In the equilibrium approach, a triangular equilibrium plate element with degree zero was first introduced by Morley [2]. A regular family of equilibrium triangles and rectangles was still developed into a high level but will not be considered here. A drawback of the constant moment field is the impossibility of obtaining exact equilibrium in the presence of a constant pressure, which is a severe limitation. In order to solve this problem, an original complementary field has been added to the basic field (constant field) in order to give a general field in which the pressure is equilibrated by corner loads only. To specify that our element contains this special field, we will call it the enhanced Morley (EM) element [3,4].

The application of dual analysis to plates is shown to work as an efficient error measure. Its classical form involving strain energy comparison is limited to homogeneous boundary conditions. But its more evolved version based on the total complementary energy works for any case. The classical Morley element has been completed in order to make it able to exactly equilibrate a pressure, a fact that allows us to treat more realistic problems. This enhanced element is found to work fairly well. It has to be mentioned that, in the frame of dual analysis, there is no need to know the exact solution, as it is always between the two convergence curves. So, the Richardson's extrapolation is not necessary in the error evaluation. Our results only derive from an evolution of the energy bounds as computed by a uniform refinement of the mesh. Singular problems are more difficult to treat with uniform meshes because the convergence is slow. Thereby, a sound knowledge is necessary to generate finite element meshes based on cost-effective and accurate solutions. We thus should combine our method with an adaptive local refinement procedure in order to improve the cost effectiveness of the error bound evaluation. The result of this investigation will be shown in a further paper.

1

B.M. Fraeijs de Veubeke, "A conforming finite element for plate bending", Int.J.Solids Structures, 4, pp.95 to 108. Pergamon Press. Great Britain, 1968. doi:10.1016/0020-7683(68)90035-8

2

L.S.D. Morley, "The triangular equilibrium element in the solution of plate bending problems", Aero Quart., 19: 149-168, 1968.

3

J.F. Debongnie, "Applying pressures on plate equilibrium elements", Univ. of Liège, May 2004.

4

N.X. Hung and J.F. Debongnie, "The equilibrium finite element model and error estimation for plate bending". International Congress "Engineering Mechanics Today 2004", Ho Chi Minh City, Vietnam, August 16-20, 2004.

5

N.X. Hung, "The conforming finite element model and error estimation for plate bending", Proceedings of the 8th National Conference on Mechanics, Hai Phong City, Vietnam, August 27-28, 2004.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)