An Introduction to The Finite Element Method

SOLUTIONS MANUAL

for

An Introduction to The Finite Element Method

(Third Edition)

by J. N. REDDY Department of Mechanical Engineering Texas A & M University College Station, Texas 77843-3123

PROPRIETARY AND CONFIDENTIAL

This Manual is the proprietary property of The McGraw-Hill Companies, Inc. ("McGraw-Hill") and protected by copyright and other state and federal laws. By opening and using this Manual the user agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to authorized professors and instructors for use in preparing for the classes using the affiliated textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party. No part of this Manual may be reproduced, displayed or distributed in any form or by any means, electronic or otherwise, without the prior written permission of the McGraw-Hill.

McGraw-Hill, New York, 2005

ii

iii

PREFACE

This solution manual is prepared to aid the instructor in discussing the solutions to assigned problems in Chapters 1 through 14 from the book, An Introduction to the Finite Element Method, Third Edition, McGraw--Hill, New York, 2006. Computer solutions to certain problems of Chapter 8 (see Chapter 13 problems) are also included at the end of Chapter 8.

The instructor should make an effort to review the problems before assigning them. This allows the instructor to make comments and suggestions on the approach to be taken and nature of the answers expected. The instructor may wish to generate additional problems from those given in this book, especially when taught time and again from the same book. Suggestions for new problems are also included at pertinent places in this manual. Additional examples and problems can be found in the following books of the author:

1. J. N. Reddy and M. L. Rasmussen, Advanced Engineering Analysis, John Wiley, New York, 1982; reprinted and marketed currently by Krieger Publishing Company, Melbourne, Florida, 1990 (see Section 3.6).

2. J. N. Reddy, Energy and Variational Methods in Applied Mechanics, John Wiley, New York, 1984 (see Chapters 2 and 3).

3. J. N. Reddy, Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York, 1986; reprinted and marketed currently by Krieger Publishing Company, Melbourne, Florida, 1991 (see Chapters 4, 6 and 7).

4. J. N. Reddy, Theory and Analysis of Elastic Plates, Taylor and Francis, Philadelphia, 1997. 5. J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, Second Edition,

John Wiley, New York, 2002 (see Chapters 4 through 7 and Chapter 10). 6. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC

Press, Second Edition, Boca Raton, FL, 2004. 7. J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press,

Oxford, UK, 2004.

The computer problems FEM1D and FEM2D can be readily modified to solve new types of field problems. The programs can be easily extended to finite element models formulated in an advanced course and/or in research. The Fortran sources of FEM1D and FEM2D are available from the author for a price of $200.

The author appreciates receiving comments on the book and a list of errors found in the book and this solutions manual.

J. N. Reddy All that is not given is lost.

iv ? PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. All rights reserved.

1

Chapter 1 INTRODUCTION

Problem 1.1: Newton's second law can be expressed as

F = ma

(1)

where F is the net force acting on the body, m mass of the body, and a the acceleration of the body in the direction of the net force. Use Eq. (1) to determine the mathematical model, i.e., governing equation of a free-falling body. Consider only the forces due to gravity and the air resistance. Assume that the air resistance is linearly proportional to the velocity of the falling body.

Fd = cv

v Fg = mg

Solution: From the free-body-diagram it follows that

dv

m dt

=

Fg

- Fd,

Fg = mg,

Fd = cv

where v is the downward velocity (m/s) of the body, Fg is the downward force (N or kg m/s2) due to gravity, Fd is the upward drag force, m is the mass (kg) of the body, g the acceleration (m/s2) due to gravity, and c is the proportionality constant (drag

coefficient, kg/s). The equation of motion is

dv

c

+ v = g, =

dt

m

? PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. All rights reserved.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download