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2021 | Buch

Kapitel lesen Erstes Kapitel lesen

Statics and Influence Functions

From a Modern Perspective

verfasst von: Prof. Dr. Friedel Hartmann, Dr. Peter Jahn

Verlag: Springer International Publishing

Buchreihe : Springer Series in Solid and Structural Mechanics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This extended and revised second edition is intended for engineering students and researchers working with finite element methods in structural and mechanical analysis. Discussing numerical structural analysis from first mechanical and mathematical principles, it establishes the central role of influence functions (Green's functions) in finite element analysis, reanalysis, sensitivity analysis, parameter identification and in optimization, with a particular focus on computational aspects and questions of accuracy. It also presents a one-click reanalysis, a new technique that allows instantaneous modifications to a structure to be made by clicking on single elements. Lastly, the book features four programs that can be downloaded for the solution of the Poisson equation, 2-D elasticity, plate-bending problems and planar frames.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Foundation

Abstract

Since they form the core of modern structural analysis. As diverse as these principles and theorems are, technically they are all based on one equation: Integration by parts.

Friedel Hartmann, Peter Jahn

Chapter 2. Betti’s Theorem

Abstract

The subject of this chapter is the formulation of influence functions with Betti’s theorem. Influence functions are central to finite elements since in linear static analysis any FE-solution is an expansion in terms of the nodal influence functions $\textit{\textbf{g}}_i$ of a structure, see Fig. 2.1.

Friedel Hartmann, Peter Jahn

Chapter 3. Finite Elements

Abstract

Structures are constantly in motion, constantly changing their shape in accordance with the equilibrium conditions, and these tiny corrections are governed by influence functions. As the market woman knows, balance means equal work, $\delta W_e^{left} = \delta W_e^{right}$, means the work of the weights on both sides of the scale is the same. Install a shear hinge in a frame and spread the hinge! You will find that the work of the shear force and the load are the same—the market woman could have told you beforehand

Friedel Hartmann, Peter Jahn

Chapter 4. Betti Extended

Abstract

In the previous chapter we repeatedly made use of the fact that the FE-solution $u_h(x)$ is the superposition of the approximate influence function $G_h(y,x)$ and the load p(y)

$$\begin{aligned} u_h(x) = \int _0^{\,l} G_h(y,x)\,p(y)\,dy\,. \end{aligned}$$

This result is based on a theorem which we call Betti extended.

Friedel Hartmann, Peter Jahn

Chapter 5. Stiffness Changes and Reanalysis

Abstract

The subject of this chapter are changes in the stiffness of single structural members and we study how such shifts propagate through a structure, how they affect the stress distribution in a structure.

Friedel Hartmann, Peter Jahn

Chapter 6. Singularities

Abstract

This chapter deals with the question when and where stresses peak, and why these singularities reduce the quality of the influence functions and therewith the solution itself, not only in the vicinity of the singular points, but at all points.

Friedel Hartmann, Peter Jahn

Chapter 7. Mixed Formulations

Abstract

Variational formulations with different fields are called mixed formulations. They are based on a split of the displacement operator into a system of three equations according to the triad of strains, constitutive equations and equilibrium.

Friedel Hartmann, Peter Jahn

Chapter 8. Nonlinear Problems

Abstract

In this chapter we discuss nonlinear problems. The focus is on the questions, which parts of the algebra of finite elements can be directly transferred to the nonlinear theory and where the nonlinear theory differs from the linear theory and what are the characteristic features of nonlinear theories.

Friedel Hartmann, Peter Jahn

Chapter 9. Addenda

Abstract

Before we concentrate on details, let us briefly describe the approach to the equations of statics in this book.

Friedel Hartmann, Peter Jahn

Chapter 10. Software

Abstract

Four programs are offered for download.

Friedel Hartmann, Peter Jahn

Backmatter

Titel: Statics and Influence Functions
verfasst von: Prof. Dr. Friedel Hartmann
Dr. Peter Jahn
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-55889-5
Print ISBN: 978-3-030-55888-8
DOI: https://doi.org/10.1007/978-3-030-55889-5

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