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2009 | Buch

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Structural Analysis with the Finite Element Method

Linear Statics

verfasst von: Eugenio Oñate

Verlag: Springer Netherlands

Buchreihe : Lecture Notes on Numerical Methods in Engineering and Sciences

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

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Über dieses Buch

STRUCTURAL ANALYSIS WITH THE FINITE ELEMENT METHOD

Linear Statics

Volume 1 : The Basis and Solids

Eugenio Oñate

The two volumes of this book cover most of the theoretical and computational aspects of the linear static analysis of structures with the Finite Element Method (FEM). The content of the book is based on the lecture notes of a basic course on Structural Analysis with the FEM taught by the author at the Technical University of Catalonia (UPC) in Barcelona, Spain for the last 30 years.

Volume1 presents the basis of the FEM for structural analysis and a detailed description of the finite element formulation for axially loaded bars, plane elasticity problems, axisymmetric solids and general three dimensional solids. Each chapter describes the background theory for each structural model considered, details of the finite element formulation and guidelines for the application to structural engineering problems. The book includes a chapter on miscellaneous topics such as treatment of inclined supports, elastic foundations, stress smoothing, error estimation and adaptive mesh refinement techniques, among others. The text concludes with a chapter on the mesh generation and visualization of FEM results.

The book will be useful for students approaching the finite element analysis of structures for the first time, as well as for practising engineers interested in the details of the formulation and performance of the different finite elements for practical structural analysis.

STRUCTURAL ANALYSIS WITH THE FINITE ELEMENT METHOD

Linear Statics

Volume 2: Beams, Plates and Shells

Eugenio Oñate

The two volumes of this book cover most of the theoretical and computational aspects of the linear static analysis of structures with the Finite Element Method (FEM).The content of the book is based on the lecture notes of a basic course on Structural Analysis with the FEM taught by the author at the Technical University of Catalonia (UPC) in Barcelona, Spain for the last 30 years.

Volume 2 presents a detailed description of the finite element formulation for analysis of slender and thick beams, thin and thick plates, folded plate structures, axisymmetric shells, general curved shells, prismatic structures and three dimensional beams. Each chapter describes the background theory for each structural model considered, details of the finite element formulation and guidelines for the application to structural engineering problems Emphasis is put on the treatment of structures with layered composite materials.

The book will be useful for students approaching the finite element analysis of beam, plate and shell structures for the first time, as well as for practising engineers interested in the details of the formulation and performance of the different finite elements for practical structural analysis.

Inhaltsverzeichnis

Frontmatter

1. Introduction to the Finite Element Method for Structural Analysis

The Finite Element Method (FEM) is a procedure for the numerical solution of the equations that govern the problems found in nature. Usually the behaviour of nature can be described by equations expressed in differential or integral form. For this reason the FEM is understood in mathematical circles as a numerical technique for solving partial differential or integral equations. Generally, the FEM allows users to obtain the evolution in space and/or time of one or more variables representing the behaviour of a physical system.

When referred to the analysis of structures the FEM is a powerful method for computing the displacements, stresses and strains in a structure under a set of loads. This is precisely what we aim to study in this book.

2. 1D Finite Elements for Axially Loaded Rods

The objective of this chapter is to introduce the basic concepts of the FEM in its application to the analysis of simple one-dimensional (1D) axially loaded rods.

The organization of the chapter is as follows. In the first section the analysis of axially loaded rods using 2-noded rod elements is presented. Particular emphasis is put in the analogies with the solution of the same problem using the standard matrix analysis techniques studied in the previous chapter for bar structures. Here some examples of application are given. In the last part of the chapter the matrix finite element formulation adopted throughout this book is presented.

3. Advanced Rod Elements and Requirements for the Numerical Solution

The analysis of the simple axially loaded rod problem using the 2-noded rod element studied in the previous chapter is of big interest as it summarises the basic steps for the analysis of a structure by the FEM. However, a number of important questions still remain unanswered, such as: Can higher order rod elements be effectively used? What are their advantages versus the simpler 2-noded rod element? Can it be guaranteed that the numerical solution converges to the exact one as the mesh is refined? What are the conditions influencing the error in the numerical solution? The reader who faces the application of the FEM for the first time will certainly come across these and similar questions. In this chapter we will see that there are not definitive answers for many of the questions, and in some cases only some practical hints are possible. For simplicity we will mostly refer to the axially loaded rod problem as it allows a simple explanation of topics which are of general applicability to more complex problems.

The chapter is organized as follows. In the next section the derivation of the one-dimensional (1D) shape functions is presented. Such functions are very useful for obtaining the shape functions for two- (2D) and three- (3D) dimensional elements in the next chapters. An example of the derivation of the relevant matrix expressions for a quadratic 3-noded rod element is given. The concepts of isoparametric element and numerical integration are presented next. These concepts are essential for the derivation of highorder 2D and 3D elements. Finally, the requirements for the convergence of the numerical solution are discussed, together with a description of the more usual solution errors.

4. 2D Solids. Linear Triangular and Rectangular Elements

This chapter initiates the application of the FEM to structures which satisfy the assumptions of two-dimensional (2D) elasticity (i.e. plane stress or plane strain). Many of the concepts here studied will be useful when dealing with other structural problems in the subsequent chapters. Therefore, this chapter is introductory to the application of the FEM to continuous 2D and 3D structures.

There are a wide number of structures of practical interest which can be analyzed following the assumptions of 2D elasticity. All these structures have a sort of prismatic geometry. Depending on the relative dimensions of the prism and the loading type, the following two categories can be distinguished: Plane stress problems. A prismatic structure is under plane stress if one of its dimensions (thickness) is much smaller than the other two and all the loads are contained in the middle plane of the structure. The analysis domain is the middle section (Figure 4.1). Amongst the structural problems that can be included in the plane stress category we find the analysis of deep beams, plates and walls under in-plane loading, buttress dams, etc.

5. Higher Order 2D Solid Elements. Shape Functions and Analytical Computation of Integrals

This chapter extends the concepts studied in the previous one for the analysis of solids under the assumptions of 2D elasticity using higher order triangular and quadrilateral elements.

The chapter is organized as follows. In the first sections we detail the systematic derivation of the shape functions for rectangular and triangular elements of different order of approximation. Next, some rules for the analytical computation of the element integrals over rectangles and straight side triangles are given. Finally the performance of linear and quadratic triangular and rectangular elements is compared in some academic examples.

6. Isoparametric 2D Solid Elements. Numerical Integration and Applications

In the previous chapter we have described how to obtain the shape functions for 2D solid elements of triangular and rectangular shape and how to compute analytically the stiffness matrix and the equivalent nodal force vector for straight-sided triangular elements and rectangular elements.

This chapter explains how to derive 2D solid elements of arbitrary shape (i.e. irregular quadrilaterals and curve-sided triangles) using an isoparametric formulation and numerical integration. The basis of the isoparametric formulation for 2D solid elements is described in the next section. Then, the quadrature rules for the numerical integration of the stiffness matrix and the equivalent nodal force vector for triangular and quadrilateral elements are explained. The patch test for 2D solid elements is presented. Some hints on the organization of a computer program for FEM analysis of 2D solids are given. The chapter concludes with examples of the application of some of the 2D solid elements studied to the analysis of real structures.

7. Axisymmetric Solids

This chapter treats the analysis of solids with axial symmetry. Thus, only solids with geometrical and material properties independent of the circumferential coordinate θ are considered (Figure 7.1). This property allows the inherent 3D behaviour of a solid to be expressed with a much simpler 2D model.

If the loading is also axisymmetric, the displacement vector has only two components in the radial and axial directions. The analysis of axisymmetric solid structures by the FEM is not difficult and follows very similar steps to those explained in the previous chapters for plane elasticity problems. For arbitrary non-axisymmetric loading a full 3D analysis is needed. However, even in these cases the axial symmetry of the structure allows important simplifications to be introduced. For instance, the loading can be expanded in Fourier series and the effect of each harmonic term can be evaluated by a simpler 2D analysis. The final result is obtained by adding the contributions from the different 2D solutions. This avoids costly 3D computations. This chapter focuses only on the analysis of axisymmetric solids under axisymmetric loading. Axisymmetric solids under arbitrary loading will be studied in Volume 2 [On].

8. Three Dimensional Solids

Many structures have geometrical, mechanical or loading features which make it impossible to use the simple plane stress/plane strain and axisymmetric models studied in previous chapters; or even the plate and shell models to be described in the second volume of the book [On]. The only alternative is to perform a full three dimensional (3D) analysis based on general 3D elasticity theory [TG].

Examples of these situations are found in solids with irregular shapes and in the study of prismatic solids with heterogeneous material properties or arbitrary loading. Figure 8.1 shows some examples of typical structures requiring a full 3D analysis.

9. Miscellaneous: Inclined Supports, Displacement Constrains, Error Estimation, Mesh Adaptivity Etc.

This chapter deals with topics of general interest in finite element structural analysis not covered in previous chapters. Boundary conditions in inclined supports are presented first. Then, methods to link different element types and for prescribing general constraints in the nodal displacements are studied. The three following sections deal with condensation and recovery algorithms, mesh symmetries and elastic supports. The last part of the chapter is devoted to the computation of nodal stresses, the estimation of the solution error and its application to adaptive mesh refinement.

10. Generation of Analysis Data and Visualization of Numerical Results

A significant task in the finite element analysis of a structure is the generation and the specification of all the data required for the computations. This work, usually called “preprocessing”, includes the definition of the geometry of the structure in parametric form, either by hand or, what is more usual, by means of advanced computed-aided design (CAD) tools, the generation of a mesh and the assignment of the material properties, the boundary conditions and the loading. These tasks which are trivial for simple academic structural shapes, can be extremely complex for real structures. Here the use of advanced preprocessing tools is mandatory in practice.

A similar diffculty arises for the visualization of the results from the finite element computations. The so called \postprocessing" of the numerical outputs in the form of vector isolines or contours of displacements, strains and stresses is usually required. For practical problems these displays can only be performed with the help of modern graphical tools, specially for 3D problems.

11. Learning to Program the Fem with Matlab and Gid

As for any other numerical method, the application of the FEM is linked to the programming language and software tools chosen. Historically the first programming language for practical use of the FEM was FORTRAN. Since then many routines, algorithms and programs associated to the method have been programmed in this language. With the development of computers new languages have appeared, each one with capabilities and specific tools for diverse fields of application. The common objective is to simplify the coding of the algorithms and to optimize the computer resources.

Although FORTRAN continues being a language of reference for the FEM, the new languages and programming tools allow simplifications in the coding work. At the same time specific libraries can be used that optimize the memory and computer resources. This is a key feature of MATLAB that besides being a research tool, it allows us to write codes that it can be interpreted at the time of execution. From an optimal programming point of view, interpretive languages are quite slow. However, MATLAB allows us to make use of all the implemented matrix routines for optimizing the calculations up to the point to compete effciently with other compiled languages.

Francisco Zárate

Backmatter

Titel: Structural Analysis with the Finite Element Method
verfasst von: Eugenio Oñate
Verlag: Springer Netherlands
Electronic ISBN: 978-1-4020-8733-2
Print ISBN: 978-1-4020-8732-5
DOI: https://doi.org/10.1007/978-1-4020-8733-2