Friedel Hartmann

The Somigliana identity on piecewise smooth surfaces

SummaryTheSomigliana identity is an integral representation of the displacement field of a body, i.e. the solution of the Cauchy-Navier equation in the classical theory of elasticity is represented by influence functions. This well established identity for bodies with smooth surfaces is extended in the present paper to cover two- and three-dimensional bodies with piecewise smooth surfaces.ZusammenfassungDieSomigliana Identität ist die Integraldarstellung des Verschiebungsfeldes eines Körpers, d.h. die Lösung der Cauchy-Navierschen DGL in der klassischen Elastizitätstheorie wird durch Einflußfunktionen dargestellt. Diese Identität, die für Körper mit glatter Oberfläche wohl bekannt ist, wird in diesem Anfsatz auf Körper und Scheiben mit stückweise glatten Rändern ausgedehnt.

Statik mit finiten Elementen

Finite Elemente : Programme für die Statik in Pascal und C — 2., bearb. und erw. Aufl. (Deutsch) Finite Elemente in der Statik und Dynamik. Link, Michael Finite Elemente in der Baustatik Bücher • Statik mit finiten Elementen • Structural Statik und Einflussfunktionen vom modernen Standpunkt aus, 2. Auflage Das Buch bildet die Grundlage für die Vorlesungsreihe „Finite Elemente“ und „Tragwerksdynamik“, die der Verfasser für Bauingenieurund.

Corner singularities of Kirchhoff plates and the boundary element method

Abstract The paper studies the influence of singularities of Kirchhoff plates on a boundary element solution. The solution is split into a regular and singular part. An integral representation of the stress intensity factor allows to apply an iterative so-called dual singular function method to determine the leading singular term and therewith to improve the boundary element solution.

Boundary Element Analysis of Raft Foundations on Piles

The boundary element method is used in the formulation of models for the analysis of raft foundations on piles. Two models are considered: a Kirchhoff plate on a layered elastic half-space and a Kirchhoff plate on a Winkler soil. The plates are modelled using conforming boundary elements and the piles by using linear finite elements. Mindlins solution is used as influence function within the half-space while Boussinesqs solution, a precursor although a particular case of Mindlins solution, is used to derive the deflections of the soil surface. The models are used in the analysis of some raft foundations on piles and the results and relative merits are discussed.

Integral Representations for the Deflection and the Slope of a Plate on an Elastic Foundation

With the help of two fundamental solutions, integral representations for the deflection and the slope of polygonal shaped plates on an elastic foundation are derived. The kernel functions are evaluated in terms of zero-order Kelvin- or Thomson-functions and their derivatives, respectively.

Stiffness Changes and Reanalysis

The subject of this chapter are local changes in the stiffness of single structural members which in terms of linear algebra lead to expressions such as.

The Discretization Error

The error of FE-solutions can be traced back to the error in the approximation of the Greens functions. Tools of functional analysis allow to express the energy error in powers of the mesh-width h. In goal-oriented refinement where the focus is on minimizing the error in certain functionals the adaptive refinement is steered by two errors, the error in the primal, the original, problem and the error in the dual problem, the approximation of the Greens function with finite elements. The technique of goal-oriented refinement can also be applied to nonlinear problems where the dual problem is the approximation of the Greens function at the current linearization point. In the case of nonlinear functionals the Greens function is taken as the Greens function of the linearized functionals. Because FE-solutions, in general, do not interpolate the exact solution at the nodes they exhibit a certain drift, a mismatch at the nodes. This is the prime reason why the output the approximate Greens functions produce is not exact. If the drift is uniform, or nearly so, then it is called pollution: badly resolved singularities or inconsistencies in the discretization make that the solution gets shifted in a certain direction. Often the cause of these shifts lies outside the zone where the shift is observed. Pollution is ‘silent’ it is not accompanied by oscillating stresses and it cannot be discovered solely by a local analysis. The chapter closes with a remark about the strong singularities in Greens functions. These singularities are normally much higher than the typical stress singularities at singular points on the boundary but surprisingly an FE-program manages to approximate these functions relatively well. This is a kind of a paradox: (weak) singularities on the boundary are hard to approximate while (strong) singularities in the interior can be resolved quite easily.

Finite Elements and Green’s Functions

In FE-analysis we substitute for the exact solution of the equation L u = p an approximation uh which is the exact solution of the equation L uh = ph. The special nature of the FE-solution allows to extend Betti’s Theorem (p1,u2) = (p2,u1) to the FE-solutions in the following sense (p1,uh 2,p2,u1 h) which establishes that the FE-solution is the scalar product of the approximate Green’s function Gh[x] and the original right-hand side p, namely uh(x) = (Gh[x],p). We must distinguish between weak and strong influence functions. Weak influence functions are based on the principle of virtual forces (Green’s first identity) and can only be formulated for displacement terms while strong influence functions, which can be formulated for displacement and force terms, are based on Betti’s theorem (Green’s second identity). In FE-analysis this distinction gets lost because of the approximate nature of the kernel functions. Influence functions in FE-analysis take a nodal form, that is they are evaluated by summing over the nodes (scalar product of two nodal vectors g and f or u and j). The columns of the inverse stiffness matrix are the nodal values of the Green’s functions. Infinite stresses pose a problem for influence functions because they imply that the kernel functions become unmeasurable when the dislocations, which trigger the Green’s functions, move to the singularity. The essential features of discrete FE-Green’s functions also apply in the case of mixed problems and the sensitivity of the p-method with regard to point sources also follows from the nature of the Green’s functions.

What are boundary elements

The boundary element method (BE method) is an integral equation method, or as we could say as well an influence function method. It is based on the fact that in linear problems the boundary values uniquely determine the displacements and stresses inside a structure such as the frame in Fig. 2.1, so that it suffices to discretize the edge with boundary elements only.

What are finite elements

In this introductory chapter various aspects of the FE method are studied, initially highlighting the key points.