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2018 | Buch

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One-Dimensional Finite Elements

An Introduction to the FE Method

verfasst von: Prof. Dr. Andreas Öchsner, Prof. Dr. Markus Merkel

Verlag: Springer International Publishing

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

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Über dieses Buch

This textbook presents finite element methods using exclusively one-dimensional elements. It presents the complex methodology in an easily understandable but mathematically correct fashion. The approach of one-dimensional elements enables the reader to focus on the understanding of the principles of basic and advanced mechanical problems. The reader will easily understand the assumptions and limitations of mechanical modeling as well as the underlying physics without struggling with complex mathematics. Although the description is easy, it remains scientifically correct.

The approach using only one-dimensional elements covers not only standard problems but allows also for advanced topics such as plasticity or the mechanics of composite materials. Many examples illustrate the concepts and problems at the end of every chapter help to familiarize with the topics. Each chapter also includes a few exercise problems, with short answers provided at the end of the book.

The second edition appears with a complete revision of all figures. It also presents a complete new chapter special elements and added the thermal conduction into the analysis of rod elements. The principle of virtual work has also been introduced for the derivation of the finite-element principal equation.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this first chapter the content as well as the focus will be classified in various aspects. First, the development of the finite element method will be explained and considered from different perspectives.

Andreas Öchsner, Markus Merkel

Chapter 2. Motivation for the Finite Element Method

Abstract

The approach to the finite element method can be derived from different motivations. Essentially there are three ways:

a rather descriptive way, which has its roots in the engineering working method,
a physical or
mathematically motivated approach.

Depending on the perspective, different initial formulations result in the same principal finite element equation. The different formulations will be elaborated in detail based on the following descriptions:

matrix methods,
energy methods and
weighted residual method.

The finite element method is used to solve different physical problems. Here solely finite element formulations related to structural mechanics are considered.

Andreas Öchsner, Markus Merkel

Chapter 3. Bar Element

Abstract

The bar element describes the basic load cases tension and compression. First, the basic equations known from the strength of materials will be introduced. Subsequently the bar element will be introduced, according to the common definitions for load and deformation quantities, which are used in the handling of the FE method. The derivation of the stiffness matrix will be described in detail. Apart from the simple prismatic bar with constant cross-section and material properties also more general bars, where the size varies along the body axis will be analyzed in examples and exercises.

Andreas Öchsner, Markus Merkel

Chapter 4. Equivalences to Tension Bar

Abstract

The procedure applied to the one-dimensional tension bar can be used to describe other physical field problems. As examples, we present the one-dimensional torsion bar and the case of one-dimensional heat conduction in a bar. The first part of the chapter treats the torsion bar. First, the basic equations known from the strength of materials will be introduced. Subsequently, the torsion bar will be introduced, according to the common definitions for the torque and angle variables, which are used in the handling of the FE method. The explanations are limited to torsion bars with circular cross-section. The stiffness matrix will be derived according to the procedure for the tension bar. The second part follows a similar approach to elaborate on the hear flux bar.

Andreas Öchsner, Markus Merkel

Chapter 5. Bending Element

Abstract

This element describes the basic deformation mode of bending. First, several elementary assumptions for modeling will be introduced and the beam element used in this chapter will be differentiated from other element formulations. The basic equations from the strength of materials, meaning kinematics, equilibrium and constitutive equation will be introduced and used for the derivation of the differential equation of the bending line. Analytical solutions will conclude the section of the basic principles. Subsequently, the bending element will be introduced, according to the common definitions for load and deformation parameters, which are used in the handling of the FE method. The derivation of the stiffness matrix is carried out through various methods and will be described in detail. Besides the simple, prismatic beam with constant cross-section also variable cross-sections, generalized loads between the nodes and orientation in the plane and the space will be analyzed.

Andreas Öchsner, Markus Merkel

Chapter 6. General 1D Element

Abstract

The three basic modes of deformation, i.e. tension, torsion and bending, can occur in an arbitrary combination. This chapter serves to introduce how the stiffness relation for a general 1D element can be composed. The stiffness relations of the basic types build the foundation. For ‘simple’ loadings the three basic types can be regarded separately and can easily be superposed. A mutual dependency is nonexistent. The generality of the 1D element also relates to the arbitrary orientation within space. Transformation rules from local to global coordinates are provided. As an example, structures in the plane as well as in three-dimensional space will be discussed. Furthermore, there will be a short introduction in the subject of numerical integration.

Andreas Öchsner, Markus Merkel

Chapter 7. Plane and Spatial Frame Structures

Abstract

The procedure for the analysis of a load-bearing structure will be introduced in this chapter. Structures will be considered, which consist of multiple elements and are connected with each other on coupling points. The structure is supported properly and subjected with loads. Unknown are the deformations of the structure and the reaction forces at the supports. Furthermore, the internal reactions of the single element are of interest. The stiffness relation of the single elements are already known from the previous chapters. A global stiffness relation forms on the basis of these single stiffness relations. From a mathematical point of view the evaluation of the global stiffness relation equals the solving of a linear system of equations. As examples plane and general three-dimensional structures of bars and beams will be introduced.

Andreas Öchsner, Markus Merkel

Chapter 8. Beam with Shear Contribution

Abstract

This element describes the basic deformation mode of bending under the consideration of the shear influence. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be differentiated from other formulations. The basic equations from the strength of materials, meaning kinematics, the equilibrium as well as the constitutive equation will be introduced and used for the derivation of a system of coupled differential equations. The section about the basics is ended with analytical solutions. Subsequently, the Timoshenko bending element will be introduced, according to the common definitions for load and deformation parameters, which are used in the handling of the FE method. The derivation of the stiffness matrix is carried out through various methods and will be described in detail. Besides linear shape functions, a general concept for an arbitrary arrangement of the shape functions will be reviewed.

Andreas Öchsner, Markus Merkel

Chapter 9. Beams of Composite Materials

Abstract

The beam elements discussed so far consist of homogeneous, isotropic material. Within this chapter a finite element formulation for a special material type — composite materials — will be introduced. On the basis of plane layers the behavior for the one-dimensional situation on the beam will be developed. First, different description types for direction dependent material behavior will be introduced. Shortly a special type of composite material, the fiber reinforced materials, will be considered.

Andreas Öchsner, Markus Merkel

Chapter 10. Nonlinear Elasticity

Abstract

The case of the nonlinear elasticity, meaning strain-dependent modulus of elasticity, will be considered within this chapter. The problem will be illustrated with the example of bar elements. First, the stiffness matrix or alternatively the principal finite element equation will be derived under consideration of the strain dependency. For the solving of the nonlinear system of equations three approaches will be derived, namely the direct iteration, the complete Newton–Raphson iteration and the modified Newton–Raphson iteration. These approaches will be demonstrated with the help of multiple examples. Within the framework of the complete Newton–Raphson iteration the derivation of the tangent stiffness matrix will be discussed in detail.

Andreas Öchsner, Markus Merkel

Chapter 11. Plasticity

Abstract

The continuum mechanics basics for the one-dimensional bar will be compiled at the beginning of this chapter. The yield condition, the flow rule, the hardening law and the elasto-plastic modulus will be introduced for uniaxial, monotonic loading conditions. Within the scope of the hardening law the description is limited to isotropic hardening, which occurs for example for the uniaxial tensile test with monotonic loading. For the integration of the elasto-plastic constitutive equation, the incremental predictor-corrector method is generally introduced and derived for the fully implicit and semi-implicit backward-Euler algorithm. On crucial points the difference between one- and three-dimensional descriptions will be pointed out, to guarantee a simple transfer of the derived methods to general problems. Calculated examples and supplementary problems with short solutions serve as an introduction for the theoretical description.

Andreas Öchsner, Markus Merkel

Chapter 12. Stability-Buckling

Abstract

In common and technical parlance the term stability is used in many ways. Here it is restricted to the static stability of elastic structures. The derivations concentrate on elastic bars and beams. The initial situation is a loaded elastic structure. If the acting load remains under a critical value, the structure reacts ‘simple’ and one can describe the reaction with the models and equations of the preceding chapters. If the load reaches or exceeds the critical value, bars and beams begin to buckle. The situation becomes ambiguous, beyond the initial situation several equilibrium positions can exist. From the technical application the smallest load is critical for which buckling in either bars or beams appears.

Andreas Öchsner, Markus Merkel

Chapter 13. Dynamics

Abstract

The transient behavior of the acting loads on the structure will be introduced additionally into the analysis within this chapter on dynamics. The procedure for the analysis of dynamic problems depends essentially on the character of the time course of the loads. At deterministic loads the column matrix of the external loads is a given function of the time. The major amount of problems in engineering, plant and vehicle construction can be analyzed under this assumption. In contrast to that, randomness is relevant in the case of stochastic loads. Such cases will not be regarded here. For deterministic loads a distinction is drawn between

periodic and non-periodic,
slow and fast changing load-time functions (relatively related to the dynamic eigenbehaviour of the structure).

In the following chapter linear dynamic processes will be considered, which can be traced back to an external stimulation. The field of self-excited oscillation will not be covered.

Andreas Öchsner, Markus Merkel

Chapter 14. Special Elements

Abstract

Some elements for special applications are presented in the scope of this chapter. The first special element extends the classical Bernoulli element by consideration of an elastic foundation. In the scope of this element, the so-called Winkler foundation, which assumes that the distributed reaction forces of the foundation are proportional at every point to the deflection, is considered. The second special element considers the case of a stress singularity. A beam element with a particular mapping between the local and natural coordinates allows that the stress at one node converges to infinity. The third special element considers that the geometry can extend to infinity at a boundary. For the derivation of these elements, special shape functions for the interpolation from the local to the global positions are introduced.

Andreas Öchsner, Markus Merkel

Backmatter

Titel: One-Dimensional Finite Elements
verfasst von: Prof. Dr. Andreas Öchsner
Prof. Dr. Markus Merkel
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-75145-0
Print ISBN: 978-3-319-75144-3
DOI: https://doi.org/10.1007/978-3-319-75145-0

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