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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

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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, the Finite Element Method,

An Introduction to

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.

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PROPRIETARY MATERIAL.

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1

Chapter 1 INTRODUCTION

Problem 1.1:

Newton’s second law can be expressed as F

=

(1)

ma

where F is the net force acting on the body,

mass of the body, and a the

m

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 m

where

dv dt

=

Fg

− Fd ,

=

Fg

mg,

Fd

=

cv

is the downward velocity (m/s) of the body, Fg is the downward force (N or 2 kg m/s ) due to gravity, Fd is the upward drag force, m is the mass (kg) of the body, 2 g the acceleration (m/s ) due to gravity, and c is the proportionality constant (drag v

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

PROPRIETARY MATERIAL.

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

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2

AN INTRODUCTION TO THE FINITE ELEMENT METHOD

Problem 1.2:

(or head)

A cylindrical storage tank of diameter D contains a liquid at depth Liquid is supplied to the tank at a rate of qi (m3 /day) and drained

h(x, t).

at a rate of q0 (m3 /day). Use the principle of conservation of mass to arrive at the governing equation of the flow problem. Solution:

The conservation of mass requires time rate of change in mass = mass inflow - mass outflow

The above equation for the problem at hand becomes d dt

(ρAh) =

or

ρqi − ρq0

d(Ah) dt

where A is the area of cross section of the tank (A = of the liquid. Problem 1.3:

= qi

− q0

π D2 /4) and ρ is the mass density

Consider the simple pendulum of Example 1.3.1. Write a computer

program to numerically solve the nonlinear equation (1.2.3) using the Euler method. Tabulate the numerical results for two different time steps

∆t = 0.05 and ∆t = 0.025

along with the exact linear solution. Solution: In order to use the finite difference scheme of Eq. (1.2.3) as a pair of first-order equations d

θ

dt

=

dv

v,

dt

=

(1.3.3), we rewrite

−λ2 sin θ

Applying the scheme of Eq. (1.3.3) to the two equations at hand, we obtain

θi+1

= θi + ∆t vi ;

vi+1

= vi

− ∆t λ2 sin θi

The above equations can be programmed to solve for (θi , vi ). Table P1.3 contains representative numerical results. Problem 1.4:

An improvement of Euler’s method is provided by Heun’s method,

which uses the average of the derivatives at the two ends of the interval to estimate the slope. Applied to the equation du dt

=

f (t, u)

(1)

Heun’s scheme has the form ui+1

=

ui

+

∆t h 2

f (ti , ui )

PROPRIETARY MATERIAL.

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+ f (ti+1, u 0i+1 )

,

0

ui+1

=

ui

+ ∆t

f (ti , u i )

(2)

°

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

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SOLUTIONS MANUAL

Table P1.3:

Comparison of various approximate solutions of the (d2 θ /dt2 ) + λ2 sin θ = 0 with its exact linear solution.

Exact θ

t 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.80 1.00

0.78540 0.76965 0.72302 0.64739 0.54578 0.42229 0.28185 0.13011 -0.02685 -0.18274 -0.33129 -0.58310 -0.78356 -0.50591

Approx. solution θ

Exact

∆t = .05

∆t = .025

0.78540 0.78540 0.75694 0.70002 0.58980 0.50496 0.37123 0.21803 0.05023 -0.12628 -0.30481 -0.63965 -1.05068 -0.94062

0.78540 0.77828 0.74276 0.67944 0.56482 0.47627 0.34225 0.19218 0.03148 -0.13374 -0.29690 -0.59131 -0.91171 -0.74672

3

equation

Approx. solution v ∆t = .05

v -0.00000 -0.62801 -1.23083 -1.78428 -2.26615 -2.65711 -2.94148 -3.10785 -3.14955 -3.06491 -2.85732 -2.11119 0.21536 2.41051

-0.00000 -0.56922 -1.13844 -1.69123 -2.20984 -2.67459 -3.06403 -3.35605 -3.53018 -3.57060 -3.46921 -2.85712 -0.50399 2.29398

∆t = .025 -0.00000 -0.56922 -1.13027 -1.66622 -2.15879 -2.58816 -2.93371 -3.17573 -3.29791 -3.29007 -3.15014 -2.50787 -0.28356 2.19765

In books on numerical analysis, the second equation in (2) is called the predictor equation and the first equation is called the corrector equation. Apply Heun’s method to Eqs. (1.3.4) and obtain the numerical solution for Solution:

∆t = 0.05.

Heun’s method applied to the pair d

θ

dt

=

dv

v,

dt

=

−λ2 sin θ

yields the following discrete equations:

θ 0i+1

= θi + ∆t vi

vi+1

= vi

θi+1

− λ2

= θi +

∆t ³

∆t 2

2

0

sin θi + sin θ i+1

´

(vi + vi+1)

The numereical results obtained with the Heun’s method and Euler’s method are presented in Table P1.4.

PROPRIETARY MATERIAL.

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AN INTRODUCTION TO THE FINITE ELEMENT METHOD

Table P1.4:

Numerical solutions of the nonlinear equation d2θ/dt2 + λ2 sin θ = 0 along with the exact solution of the linear equation d2θ/dt2 + λ2θ = 0.

Exact

Approx. solution θ

t

θ

Euler’s

Heun’s

0.00 0.05 0.10 0.20 0.40 0.60 0.80 1.00

0.785398 0.769645 0.723017 0.545784 -0.026852 -0.583104 -0.783562 -0.505912

0.785398 0.785398 0.756937 0.615453 0.050228 -0.639652 -1.050679 -0.940622

0.785398 0.771168 0.728680 0.564818 0.015246 -0.544352 -0.787095 -0.587339

Exact v -0.000000 -0.628013 -1.230833 -2.266146 -3.149552 -2.111190 0.215362 2.410506

Approx. solution v Euler’s

Heun’s

-0.000000 -0.569221 -1.138442 -2.209838 -3.530178 -2.857121 -0.503993 2.293983

-0.000000 -0.569221 -1.121957 -1.121957 -3.073095 -2.194398 -0.114453 2.023807

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.

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

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