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2018 | Book

Representation of Curves on a Plane Read chapter Read first chapter

An Isogeometric Approach to Beam Structures

Bridging the Classical to Modern Technique

Author: Buntara S. Gan

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book proposes a novel, original condensation method to beam formulation based on the isogeometric approach to reducing the degrees of freedom to conventional two-node beam elements. In this volume, the author defines the Buntara Condensation Formulation: a unique formulation in condensing the dynamic equilibrium equation for beam structures, suitable for reducing the number of unlimited dynamic equations necessary to yield a classic two-node beam element. Professor Buntara’s method overcomes the problem of the isogeometric approach where the number of degrees of freedom is increased along with the complexity of the geometrical beam element and facilitates implementation of the codes into the existing beam structures programs, and CAD geometrical data into the conventional FE beam element codes. The book proposes a new reduction method where the beam element can be treated as under the conventional beam element theory that has only two nodes at both ends.

Table of Contents

Frontmatter
Chapter 1. Representation of Curves on a Plane
Abstract
In this chapter, we will discuss in detail several methods of representing curves in geometric modeling. Unlike the other books which usually start from the standard basic theories behind the equations, this chapter will lead us directly on how to find equations or functions from a given set of data points. This following and connected subsection will reveal the techniques of representing the curves in real applications of beam geometric modeling. The objective of this chapter is to bring the reader to understand the concept of the nonuniform rational B-spline (NURBS) which is the basis foundation for the construction of beam element formulations in the Isogeometric approach.
Buntara S. Gan
Chapter 2. Numerical Integration
Abstract
In this chapter, we will learn about the procedure, algorithm, and technique in the numerical analysis to obtain solutions of the definite integral of a line. In the finite element formulation of general beam element, we will deal with curvilinear coordinate, Jacobian operator, and curvature of a general curved beam element where the integration must be done numerically. To stick with the most basic concepts of beam element formulation using numerical integration, we will focus our description on a one-dimensional integration using the Gauss-Legendre quadrature method.
Buntara S. Gan
Chapter 3. Finite Element Formulation of Beam Elements
Abstract
In this chapter, various types of beams on a plane are formulated in the context of finite element method. The formulation of the beam elements is based on the Euler-Bernoulli and Timoshenko theories. The kinematic assumptions, governing equations via Hamilton’s principle and matrix formulations by using shape functions, are described in detail. In constructing the beam element formulations, the shape functions which are derived from the homogeneous governing equations lead to high-accuracy beam analyses. The theories discussed and derived herewith will be used in the subsequent chapters when we deal with the Isogeometric approach to beams.
Buntara S. Gan
Chapter 4. Isogeometric Approach to Beam Element
Abstract
This chapter presents the formulation of beam elements in the finite element method by using the isogeometric approach. The implementation of the NURBS functions as either the geometry or the shape functions to various types of beam formulations on a plane is highlighted in detail and accompanied by MATLAB program lists. This chapter will use the program lists and concept of the NURBS from Chap. 1. The numerical integration introduced in Chap. 2 will be adopted in constructing the beam element matrices from the NURBS functions. Based on the beam theories developed in Chap. 3, the implementation of the isogeometric approach to beam element will be discussed.
Buntara S. Gan
Chapter 5. Condensation Method
Abstract
This chapter is the core of this book. There are rapid demands in using NURBS functions for any kind of free curved beam element for design practices. Inevitably, the computing efforts are increasing exponentially with the higher degree of real geometry to be taken into design analyses. The idea of condensing the increasing number of degrees of freedom using Isogeometric modeling is the utmost important for beam analysis practitioners. The formulation of beam elements in the finite element method by using the Isogeometric approach is rather a new subject for the designer. Provided with present powerful condensation method where the number of degrees of freedom in the formulation can be condensed precisely into a two-node six degree of freedom common beam element could ease practitioners to adopt. This new condensation method is discussed in detail and provided by MATLAB function list. The condensation method is applied to the same examples of the beam by using NURBS Chap. 3 to show its effectiveness.
Buntara S. Gan
Chapter 6. Straight Beam Element Examples
Abstract
In this chapter, the isogeometric approach is applied to straight beam element. The numerical solutions for some examples for static analysis and free vibration analysis are presented and compared with the exact solutions.
Buntara S. Gan
Chapter 7. Circular Curved Beam Element Examples
Abstract
In this chapter, the Isogeometric approach is applied to circular curved beam element. The numerical solutions for some examples for static analysis and free vibration analysis are presented and compared with the results of published studies.
Buntara S. Gan
Chapter 8. General Curved Beam Element Examples
Abstract
In this chapter, the isogeometric approach is applied to general curved beam element. The general curved beams considered in this section include parabolic, elliptical, and sinusoidal shapes. The numerical solutions for some examples in free vibration problems are presented and compared with the results of published studies.
Buntara S. Gan
Chapter 9. Free Curved Beam Element Examples
Abstract
In this chapter, the Isogeometric approach is applied to free curved beam element. The free curved beams considered in this section were modeled by the least element number which is necessary. The numerical solutions for the examples in static and free vibration problems are presented to show the effectiveness of the NURBS functions in modeling free curved beams.
Buntara S. Gan
Backmatter
Metadata
Title
An Isogeometric Approach to Beam Structures
Author
Buntara S. Gan
Copyright Year
2018
Electronic ISBN
978-3-319-56493-7
Print ISBN
978-3-319-56492-0
DOI
https://doi.org/10.1007/978-3-319-56493-7

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