main-content

## Über dieses Buch

This textbook presents finite element methods using exclusively one-dimensional elements. The aim is to present 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 easily understands the assumptions and limitations of mechanical modeling as well as the underlying physics without struggling with complex mathematics. But 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 like 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.

## Inhaltsverzeichnis

### 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 formulations result, which however all result in a common principal equation of the finite element method. The different formulations will be elaborated in detail based on the following descriptions:
• matrix methods,
• physically based working and 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 [1, 57, 912].
Andreas Öchsner, Markus Merkel

### Chapter 3. Bar Element

Abstract
On the basis of the bar element, tension and compression as types of basic load cases will be described. 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 [19] and exercises.
Andreas Öchsner, Markus Merkel

### Chapter 4. Torsion Bar

Abstract
The basic load type torsion for a prismatic bar is described with the help of a 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 [16].
Andreas Öchsner, Markus Merkel

### Chapter 5. Bending Element

Abstract
By this element the basic deformation bending will be described. First, several elementary assumptions for modeling will be introduced and the element used in this chapter will be outlined towards other 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 bar with constant cross-section and load on the nodes 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
Within the application the three basic types 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 gained. The stiffness relation 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
Within this chapter the procedure for the analysis of a load-bearing structure will be introduced. 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 on 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 total stiffness relation forms on the basis of these single stiffness relations. From a mathematical point of view the evaluation of the total 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
By this element the basic deformation bending under consideration of the shear influence will be described. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be distinguished 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 with the definitions for load and deformation parameters which are commonly accepted at the handling via the FE method. The derivation of the stiffness matrix at this point also takes place via 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 introduced.
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, M Merkel

### Chapter 10. Nonlinear Elasticity

Abstract
Within this chapter, the case of the nonlinear elasticity, meaning strain-dependent modulus of elasticity, will be considered. 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 NewtonRaphson iteration and the modified NewtonRaphson iteration, and will be demonstrated with the help of multiple examples. Within the framework of the complete NewtonRaphson 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
Within the chapter on dynamics the transient behavior of the acting loads on the structure will be introduced additionally into the analysis. The procedure for the analysis of dynamic problems depends essentially on the character of the time course of the loads. At deterministic loads the vector 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, the coincidence is relevant in the case of stochastic loads. Such cases will not be regarded here. For deterministic loads a distinction is drawn between $$bullet$$ periodic and non-periodic, $$bullet$$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

### Backmatter

Weitere Informationen