Fourth Edition: Engineering Finite Element Methods

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"工程中的有限元方法第四版英文版，由Tirupathi R. Chandrupatla和Ashok D. Belegundu合著，是有限元分析领域的一本经典教材，现提供pdf格式的原版下载。" 《工程中的有限元方法》第四版是一部深入探讨有限元理论及其在工程应用中的著作。该书由Rowan University的Tirupathi R. Chandrupatla和The Pennsylvania State University的Ashok D. Belegundu共同撰写，涵盖了广泛的有限元（FEM）主题，旨在帮助读者理解和掌握这一强大的数值计算技术。有限元方法是一种广泛应用于解决各种工程和科学问题的数值分析方法。它通过将复杂区域分解成许多简单的“元素”，然后用这些元素的组合来近似整个区域的解。这种方法特别适用于非线性、多物理场和高维问题，如结构力学、流体力学、热传导、电磁学等领域。本书的内容可能包括但不限于以下几个方面： 1. **有限元的基本概念**：介绍有限元的基本思想，包括变分原理、弱形式和Galerkin方法。 2. **一维有限元分析**：对杆件和梁进行分析，讲解如何构建一维有限元模型并求解。 3. **二维和三维有限元分析**：涵盖平面应力、平面应变以及三维问题的处理，包括三角形和四边形单元的应用。 4. **几何和材料非线性**：讨论大变形、塑性、蠕变等问题，以及如何处理材料的非线性行为。 5. **时间相关问题**：涉及瞬态热传导、结构动力学等动态问题的有限元方法。 6. **边界条件和约束**：详述如何正确施加边界条件，包括固定边界、铰接、滑移边界等。 7. **求解器和后处理**：介绍如何使用线性代数求解器求解大型稀疏系统，并讲解结果的可视化和解释。 8. **实际工程案例**：通过真实的工程问题，演示有限元分析的完整过程，帮助读者理解并应用所学知识。 9. **软件应用**：可能会涉及商业有限元软件如ANSYS、ABAQUS等的使用，介绍如何在这些软件中设置和求解问题。 10. **高级主题**：可能涵盖接触问题、流固耦合、复合材料分析等进阶内容。此书适合工程专业的学生、研究人员和工程师学习使用，不仅可以作为教学教材，也是从事有限元分析工作的重要参考书。对于想要深入了解有限元方法的人来说，这是一份宝贵的学习资料。由于是英文版，阅读过程中需要一定的英语基础，但书中的数学表述和实例应该可以帮助读者跨越语言障碍，深入理解这一关键的工程计算工具。

Preface xv

minimize. Galerkin’s formulation for steady-state heat transfer, torsion, potential flow,

seepage flow, electric and magnetic fields, fluid flow in ducts, and acoustics are

presented.

Chapter 11 introduces dynamic considerations. Element mass matrices are given.

Techniques for evaluation of eigenvalues (natural frequencies) and eigenvectors (mode

shapes) of the generalized eigenvalue problem are discussed. Methods of inverse

iteration, Jacobi, tridiagonalization, and implicit shift approaches are presented.

Preprocessing and postprocessing concepts are developed in Chapter 12 . Theory

and implementation aspects of two-dimensional mesh generation, least-squares

approach to obtain nodal stresses from element values for triangles and quadrilaterals,

and contour plotting are presented.

At the undergraduate level some topics may be dropped or delivered in a

different order without breaking the continuity of presentation. We encourage the

introduction of the Chapter 12 programs at the end of Chapter 6 . This helps the students

to prepare the data in an efficient manner.

We thank Professor Hongbing Fang, Mechanical Engineering and Engineering

Science, UNC Charlotte; Professor Kishore Pochiraju, Department of Mechanical

Engineering, Stevens Institute of Technology, Hoboken, New Jersey; Professor

Subramaniam Rajan, Ira A. Fulton School of Engineering, Arizona State University;

Professor Chris H. Reidel, A. Leon Linton Department of Mechanical Engineering,

Lawrence Technological University, Michigan; and Professor Nicholas J. Zabaras,

Sibley School of Mechanical and Aerospace Engineering, Cornell University, who

reviewed our third edition and gave many constructive suggestions that helped us

improve the book.

Complete self-contained computer programs with source codes in Visual Basic,

Excel-based Visual Basic, MATLAB, FORTRAN, JAVASCRIPT and C to accompany

the text are available at www.pearsonhighered.com/chandrupatla.

Tirupathi Chandrupatla expresses his gratitude to J. Tinsley Oden, whose teaching

and encouragement have been a source of inspiration to him throughout his career. He

expresses his thanks to many students at Rowan University and Kettering University

who took his courses. He expresses his thanks to his colleague Paris von Lockette who

gave valuable feedback after teaching the course using the second and third editions.

Ashok D. Belegundu thanks his students at Penn State for their feedback on the

course material and programs.

We thank Marcia Horton who provided us guidance with the present and previous

editions of the book. We thank our editor Norrin Dias, Tacy Quinn, Debbie Yarnell,

and Clare Romeo all from Prentice Hall who turned this project into a pleasant chore.

We thank Maheswari PonSaravanan, project manager and her team at TexTech

International, Chennai, India for efficient handling of copyediting and page proofs.

T  R. C 

A  D. B 

About the Authors

Tirupathi R. Chandrupatla is Professor and Founding Chair of Mechanical Engineering

at Rowan University, Glassboro, New Jersey. He received the B.S. degree from National

Institute of Technology (formerly REC), Warangal. He received the M.S. degree in

design and manufacturing from the Indian Institute of Technology (IIT), Bombay. He

started his career as a design engineer with Hindustan Machine Tools, Bangalore. He

then taught in the Department of Mechanical Engineering at IIT, Bombay. He pursued

his graduate studies in the Department of Aerospace Engineering and Engineering

Mechanics at the University of Texas at Austin and received his Ph.D. in 1977. He

subsequently taught at the University of Kentucky. Prior to joining Rowan, he was a

Professor of Mechanical Engineering and Manufacturing Systems Engineering at

Kettering University (formerly GMI Engineering and Management Institute/General

Motors Institute), where he taught for 16 years.

Dr. Chandrupatla has broad research interests, which include finite element anal-

ysis, design, optimization, quality and reliability, and manufacturing engineering. He

has published widely in these areas and serves as a consultant to industry.

Dr. Chandrupatla is a registered Professional Engineer and also a Certified Manufac-

turing Engineer. He is a member of ASEE, ASME, SAE, and SME.

Ashok D. Belegundu has been a Professor of Mechanical Engineering at The

Pennsylvania State University, University Park, since 1986. Prior to this, he taught at

GMI, now Kettering University, in Michigan. He received his B. Tech. degree from IIT,

Madras and his Ph.D. from the University of Iowa. He has been a principal investigator

on research projects involving finite element based optimization for several agencies,

including the National Science Foundation, Army Research Office, NASA, SERC

(UK), MacNeal-Schwendler Corporation, Gentex Corporation, and Ingersoll-Rand.

Adetailed list of his publications and projects can be found at http://www.mne.psu.edu/

Directories/Faculty/Belegundu-A.html . He has given short courses on finite elements

to the Forging Industry Association, Hazleton Pumps, Infosys (India).

xvi

CHAPTER 1

Fundamental Concepts

1.1 INTRODUCTION

The finite element method has become a powerful tool for the numerical solution of a

wide range of engineering problems. Applications range from deformation and stress

analysis of automotive, aircraft, building, and bridge structures to field analysis of heat

flux, fluid flow, magnetic flux, seepage, and other flow problems. With the advances in

computer technology and CAD systems, complex problems can be modeled with rela-

tive ease. Several alternative configurations can be tested on a computer before the first

prototype is built. All these suggest that we need to keep pace with these developments

by understanding the basic theory, modeling techniques, and computational aspects

of the finite element method. In this method of analysis, a complex region defining a

continuum is discretized into simple geometric shapes called finite elements. The mate-

rial properties and the governing relationships are considered over these elements and

expressed in terms of unknown values at element corners. An assembly process, duly

considering the loading and constraints, results in a set of equations. Solution of these

equations gives us the approximate behavior of the continuum.

1.2 HISTORICAL BACKGROUND

Basic ideas of the finite element method originated from advances in aircraft structural

analysis. In 1941, Hrenikoff presented a solution of elasticity problems using the “frame

work method.” Courant’s paper, which used piecewise polynomial interpolation over

triangular subregions to model torsion problems, was published in 1943. Turner et al.

derived stiffness matrices for truss, beam, and other elements and presented their find-

ings in 1956. The term finite element was first coined and used by Clough in 1960.

In the early 1960s, engineers used the method for approximate solution of prob-

lems in stress analysis, fluid flow, heat transfer, and other areas. A book by Argyris in

1955 on energy theorems and matrix methods laid a foundation for further develop-

ments in finite element studies. The first book on finite elements by Zienkiewicz and

2 Chapter 1 Fundamental Concepts

Cheung was published in 1967. In the late 1960s and early 1970s, finite element analysis

was applied to nonlinear problems and large deformations. Oden’s book on nonlinear

continua appeared in 1972.

Mathematical foundations were laid in the 1970s. New element development, con-

vergence studies, and other related areas fall in this category.

Today, the developments in mainframe computers and availability of powerful

microcomputers have brought this method within the reach of students and engineers

working in small industries.

1.3 OUTLINE OF PRESENTATION

In this book, we adopt the potential energy and the Galerkin approaches for the pre-

sentation of the finite element method. This method originated from the field of solids

and structures, and we start our study with these ideas to solidify understanding. For this

reason, several early chapters deal with rods, beams, and elastic deformation problems.

The same steps are used in the development of material throughout the book, so that

the similarity of approach is retained in every chapter. The finite element ideas are then

extended to field problems in Chapter 10 . Every chapter includes a set of problems and

computer programs for interaction.

We now recall some fundamental concepts needed in the development of the finite

element method.

1.4 STRESSES AND EQUILIBRIUM

A three-dimensional body occupying a volume V and having a surface S is shown in

Fig. 1.1 . Points in the body are located by x , y , z coordinates. The boundary is constrained

on some region, where displacement is specified. On part of the boundary, distributed

force per unit area T , also called traction, is applied. Under the force, the body de-

forms. The deformation of a point x

x, y, z

is given by the three components of

its displacement:

u =

u, v, w

(1.1)

The distributed force per unit volume, for example, the weight per unit volume, is the

vector f given by

f = 3f

, f

(1.2)

The body force acting on the elemental volume dV is shown in Fig. 1.1 . The surface trac-

tion T may be given by its component values at points on the surface:

T = 3T

, T

(1.3)

Examples of traction are distributed contact force and action of pressure. A load P act-

ing at a point i is represented by its three components:

= 3P

, P

(1.4)

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