References

Finite Element Analysis Books

1. Logan, D., A First Course in the Finite ElementMethod, Second Edition, PWS PublishingCompany, ITP, 1993.

2. Logan, D., A First Course in the Finite Element Method Using Algor, Second Edition,Brooks/Cole Publishing Company, ITP, 2001.

3. Cook, R., Malkus, D. and Plesha, M., Concepts and Applications of Finite Element Anal­ysis, Third Edition, John Wiley & Sons, 1989.

4. Cook, R., Finite Element Modeling for Stress Analysis, John Wiley & Sons, 1995.5. Zienkiewicz, O. and Taylor,R., The Finite Element Method: Volumes I, 11, and lIl , Fifth

Edition, Butterworth-Heinemann, 2000.6. Reddy, 1., An Introduetion to the Finite Element Method, Second Edition, McGraw-HiII,

1993.7. Bathe, K-J., Finite Element Procedures, Prentice Hall, 1996.8. Chandrupatla, T.and Belegundu,A., Introduetion to Finite Elements in Engineering,Sec­

ond Edition, Prentice Hall, 1997.9. Lewis, P. and Ward,1.,The Finite Element Method: Principles andApplications, Addison­

Wesley, 1991.10. Henwood, D. and Bonet, J., Finite Elements: AGentJe Introduction, Macmillan Press,

1996.11. Fagan, M., Finite Element Analysis: Theory and Practice, Longman, 1992.12. Knight, C.,The Finite Element Method in MechanicalDesign, PWS PublishingCompany,

ITP, 1993.13. Burnett, D., Finite Element Analysis: From Concepts to Applications, Addison-Wesley,

1988.14. Bickford, W , A First Course in the Finite Element Method, Second Edition, Irwin Pub­

lishing, 1994.15. Pintur, D., Finite Element Beginnings: Mathcad Electronic Book, MathSoft, 1994.16. Buchanan, G., Schaum's Outlines: Finite Element Analysis, Second Edition, McGraw­

Hili, 1995.17. Kwon, Y. and Bang, H., The Finite Element Method Using MATLAB, Second Edition,

CRC Press, 2000.18. Sennett, R., Matrix Analysis of Structures, Prentice Hall, 1994.

MATLAB Books

19. Etter, D., Engineering Problem Solving with MATLAB, Second Edition, Prentice Hall,1996.

20. Part-Enander,E. and Sjoberg, A., Tbe MATLAB 5 Handbook,Addison-Wesley, 1999.

360 References

21. Biran , A. and Breiner, M., MATLAB 5 for Engineers, Addison-Wesley, 1999.22. Higham, D. and Higham, N., MATLAB Guide, SIAM, 2000.23. Hanselman, D. and Littlefield, B., Mastering MATLAB 6, Prentice Hall, 2000.24. Palm, W., Introduetion to MATLAB 6 for Engineers, McGraw-HiII, 2000.25. Chen, K., Giblin, P. and Irving, A., Mathematical Explorations with MATLAB, Cambridge

University Press, 1999.26. Polking, 1. and Arnold, D., Ordinary Differential Equations Using MATLAB, Second

Edition, Prentice Hall, 1999.27. Fausett, L., Applied Numerical Analysis Using MATLAB, Prentice Hall, 1999.

MATLAB Tutorials on the Internet

28. Cavers, 1., An Introductory Guide to MATLAB , Second Edition, 1998,http ://www.cs.ubc .ca/spider/cavers/MatlabGuide/guide.html

29. Gockenback, M., A Practical Introduetion to MATLAB: Updated for MATLAB 5,http ://www.math .mtu .edu/~msgocken/intro/intro.html

30. Huber, T., MATLAB Tutorials: Envision-It! Workshop 1997,http ://physics .gac .edu/~huber/envision/tutor2/

31. Maneval, J., Helpful Information for Using MATLAB,http ://www.facstaff.bucknell.edu/maneval/help211/helpmain.html

32. Recktenwald, G., MATLAB Hypertext Reference, 1996,http ://www.me.pdx .edu/~gerry/MATLAB/

33. University of Texas, MATLAB Online Reference Documentation,http ://www.utexas .edu/cc/math/Matlab/Manual/ReferenceTOC.html

34. Southem Illinois University at Carbondale, MATLAB Tutorials,http ://www.math .siu .edu/matlab/tutorials .html

35. MATLAB Educational Sites,http://www.eece.maine .edu/mm/matweb.html

Answers to Problems

Answersto all the problemsare givenbelow. Foreach problemtwo resultsare given:the displacements and the reactions.

Problem 2.1:

u =

0 .0222

F =

-4 .444410 .0000-5 .5556

Problem 2.2:

u =

0.14710 .22060.3676

F =

-25.00000 .00000 .0000

25.0000

Problem 3.1:

u =

1 .0e-003 *

362 Answers to Problems

0.01430.10000.1429

F

-5.0000-10.0000-0 .000015.0000

Problem 3.2:

u =

1.0e-004 *-0.4582-0.3554-0.2819-0.2248-0.1780-0.1385-0.1042-0.0739-0.0469-0 .0224

It is clear from the results above that the displacement at the free end is-0.04582 mm which is very close to that obtained in Example 3.2. However, theresult obtained in this problem is more accurate because we have used more elementsin the discretization.

Problem 3.3:

u =

2 .4975e-005

F

-24 .975025 .0000-0 .0250

Answers 10 Problems 363

Problem 4.1:

u =

1.0e-004 *

-0 .4211-0.2782-0 .1353-0.0677

Thus it is clear that the displacement at the free end is -0.4211 x 10-4 mor -0.04211 mm which is very close to that obtained in Example 3.2 whichwas -0.04517 mm and that obtained in the solution of Problem 3.2 which was-0.04582 mmo

Problem 4.2:

u =

0.00750.00790 .0081

F

-15.00000 .0000

10 .00005 .0000

Problem 5.1:

u =

1.0e-003 *

0 .2083-0 .03330.0106

-0.03330.17660.01070.0212

-0.0516

364 Answers to Problems

F =

-8.8889-9.333320.0000

0.00000 .0000

o0.0000

oo

-0 .0000-11.1111

9.3333

Problem 5.2:

u =

0 .0000- 0. 00000.0034

F =

-2.6489- 1.9866-5 .3645

o-1 .9866

1 .9866-0 .0000-0 .000010.0000

Problem 6.1:

u =

1.0e-003 *0 .23510.0003

- 0. 3671

F =

0 .000016 .64719 .9883

-6 .4151-10 .6919-0 .0000-0 .0000

-16 .686210 .0117-8.584910 .73110 .0000

15 . 0000-0.0000

-20.0000

Problem 7.1:

ti =

1.0e-004 *

0.2273- 0 .4545

0.2273

F =

- 1.5584o

-3 .2143-15 .0000

4 .7727o

Problem 7.2:

ti =

0.0006- 0 .0012

0 .0020

Answers 10 Problems 365

366 Answers to Problems

F =

3.99463.9946

-8.47837 .50007.7242

-15 .0000-3 .240515.0000

Problem 7.3:

u =

-0 .0016-0 .0002

0.0009

F =

1 .52252 .4913

-10.00000.00000 .6920

o7 .7855

Problem 8.1:

u =

0.18650 .0000

-0 .02980.18650 .0149

F =

-20 .0000-4 .687546.2501

0.0000o

15.000020.00004.6875

- 0. 0000

Problem 8.2:

u =

0.0013-0.0009-0 .0005

0 .00120.00080.0003

F =

2.02838.40588.0296

20.0000-12.5000-10.4170

- 0 . 0000-12.5000

10.4170-22.0283

16 .594215.1226

Problem 8.3:

u =

0.0001-0.0056

0.0021

F =

-0.0000-10.0000-0 .0000

-13 .0903

Answers 10 Problems 367

368 Answers 10 Problems

0.1822- 0 . 729013.09039 .8178

Problem 9.1:

u =

-0 .00480 .0000

-0 .0018

F =

-10 .00000.0000

o5.0000

- 13 .890520 .0000

5 .000013 . 890520.0000

Problem 10.1:

u =

-0 .00040.0000

-0 .00060 .0000

- 0 . 00040.0000

-0 .00210 .0000

- 0 .00060 .0000

-0 .00040 .0002

-0.00210.00000 .00060 .0000

-0.00040.0002

-0.00040 .00000.00060.0000

-0.00040.0000

F =

1.15992.50541. 00912.67190.3008

-3.27376.33245.74841.00912.67190.3008

-17.69376.3481

-5.7484-1.0091-2.6719

0.3019-17.7439

1.1596-2.5054- 1. 0091-2.6719

0.3019-3 .2733

o0.0000

o0.00000 .00000.0000

o0 .0000

o0.0000

Answers to Problems 369

370 Answers to Problems

0 .00000.0000

-15.00000 .00000 .00000 .00000.00000.0000

oo

0.00000 .00000 .00000 .0000

Problem 11.1:

u =

1.0e-005 *0 .69280.07140 .6928

-0 .07140 .32710 .0000

F =

-9 .3750-3.7540

9 .37500 .00009 .37500 .0000

-9 .37503.7540

o0.0000

Problem 11.2:

u =

1.0e-003 *

- 0. 0200-0.0225-0 .0291-0 .0581-0 .0305-0 .0854-0.0008-0 .0173-0 .0072-0 .0585-0.0077-0 .0867

0 .0001-0.01760.0010

-0.06390 .0064

-0 .09600.0207

-0.01990.0346

-0.06350.0356

-0.1167

F =

18.80541.0788

o0 .00000.00000.00000.00000.00001.33669.25380 .00000.00000 .00000 .00000 .00000.00000 .91050 .2247

Answers to Problems 371

372 Answers to Problems

o0.00000.00000.00000.00000 .0000

-21.05259.44270 .0000

oo

0.00000 .0000

-20.0000

Problem 11.3:

u =

0.00020.00440.00020.00440 .00020.00440 .00020 .0044

F =

o17.5000

o17.50000.00000.00000.00000.0000

-17.5000-17.5000

Problem 12.1:

u =

1.0a-005 *

0.35000 .05900.70060.04150.16530.01720.52860.02880 .34540 .00000 .70800 .00000 .1653

-0.01720 .5286

-0 .02880 .3500

-0 .05900 .7006

-0 .0415

F =

-3 .4469-1.5335

o0 .00003.12500 .0000

o0 .00000 .00000.0000

-11.85620 .00000 .00000.0000

12 .50000.0000

Answers to Problems 373

374 Answers ta Problems

0 .00000.00000.0000

o-3 .4469

1.53350.00000 .00003.1250

o

Problem 13.1:

u =

1.0e-005 *0.17680.05520.35000.05480.52840.05360.70710.05350.16480.00000 .34960.00000 .52870.00000.70710 .00000.1768

-0.05520.3500

-0 .05480.5284

-0 .05360 .7071

-0.0535

F =

-4 .9836

-1. 25800 .00000.00000 .00000.00000.00000.00004 .68750.0000

-8.78290 .00000 .00000 .0000

o0.00000 .00000.00009 .37500.0000

-4.98361 .25800 .0000

o0 .00000 .00000 .00000.00004 .68750.0000

Problem 13.2:

u =

1.0e-003 *-0.0299-0.0284-0.0402-0.0753-0 .0386-0.1102

0 .0015-0.0203-0.0068

Answers 10 Problems 375

376 Answers 10 Problems

-0 .0800-0 .0123-0.1088-0 .0021-0.0185-0 .0023-0 .0824

0 .0047-0.12240.0307

-0.02600.0489

-0 .07580 .0565

-0.1589

F =

17 .65703 .44500.00000 .00000 .00000 .00000.00000 .00007 .48067.03140.00000 .00000 .00000.00000 .00000.0000

-7.93216 .74160.00000 .00000.00000 .00000 .00000.0000

-17 .20542.7819

o

o0.00000.00000.0000

-20.0000

Problem 13.3:

u =

-0.00030 .0029

-0.00030.0029

-0.00030.0029

-0.00030.0029

-0 .00030.0029

-0 .00030 .0029

F =

0 .00008.75000 .0000

17 .50000.00008 .75000 .00000.00000 .00000.0000

o0.0000

- 11. 6553-11. 6894-11 .6553

Answers to Problems 377

378 Answers to Problems

Problem 14.1:

u =

-0 .00020.0029

-0 .00020 .0029

-0.00020 .0029

-0 .00020.0029

-0.00020.0029

-0 .00020 .0029

-0 .00020 .0029

- 0 . 00020 .0029

F =

o5 .83330.0000

23 .33330.00005 .83330 .00000 .0000

oo

0.00000.00000 .00000.00000 .00000 .0000

- 11. 6419-1 1. 7 162- 11. 6419

Problem 15.1:

u =

1.0e-005 *

0.01850.67100.14850.00910.66990.14890 .01830.58090.03190.00740.57950.0317

F =

-51.1925-3 .0565-4 . 484251.2090-6 .3185-3.03680.00003 .12500 .00000 .00006 .25000 .0000

-29 .2553-6 .31854.6493

29 .2388-3 .0565

2.87170.00006 .2500

o0.00003 .1250

o

Answers 10 Problems 379

Contents of the Accompanying CD-ROM

The accompanying CD-ROM includes two folders as follows:

M-Files: This folder includes the 75 MATLAB functions written specifically to beused with this book. In order to use them they should be copied to the workingdirectory in your MATLAB folder on the hard disk or you can set the MATLAB pathto the correct folder that includes these files.

Solutions Manual : This folder includes the Solutions Manual for the book. Twoversions of the Solutions Manual are provided. The file "Solutions to Problems"includes detailed solutions to all the problems in the book and is over 300 pages inlength. The other file "Solutions to Problems2" is a reduced version of the solutionsin about 50 pages only. The two Solutions Manuals are provided in both DOC andRTF formats.

Index

AAuea 211,242 ,267,303Assembly 1,14,19,31,39,48,54,67,76,

97, 111, 120, 142, 158, 177,201,218,231, 255,278,291 ,320,344

Axial Force 136, 189Axial Force Diagram 136,189

BBar- Linear 27-43- Quadratic 44-59Beam 106-132BeamA ssemble 3, 107, 108BeamElementForces 3, 107, 108BeamElementMomentDiagram 3,107 ,

108BeamElementShearDiagram 3,107 ,108BeamElementStijfness 3, 107Bending Moment 136,138 ,189 ,190,196Bending Moment Diagram 136,138 ,189 ,

190, 196Bilinear Quadrilateral 267-302BilinearQuadAssemble 4, 270, 272BilinearQuadElementArea 4,270,271BilinearQuadElementPStresses 4, 271,

274BilinearQuadElementStijfness 4, 270, 271BilinearQuadEIementStijfness2 4, 270BilinearQuadElementStresses 4, 270, 273Boundary Conditions 1,15,21,32,40,49,

55,68,82,100,111 ,123,145,160,180,204,220,234,259,280 ,293 ,322,347

CCD-ROM 2, 381Continuous System

DDiagram- Axia1 Force 136, 189- Bending Moment 136,138,189,190,

196- ShearForce 136,138 ,189,195,196Discrete System 1Discretization 1,2,217 ,226 ,253,275,

288,318,339Displacement 1Domain 1,217 ,226 ,253 ,275 ,288,318,

339

EElasticElement 1Element Stiffness Matrix 1,14,18,30,

37,47,53,65,73,94,110 ,119,140,157 ,175,198,217,226,253,276,289,318,339

FFinite ElementForce 2Frame- Plane 133-169- Space 187-210

GGaussian Elimination 2G1oba1 Stiffness Matrix 1,14,19,31 ,39,

48,54,67,76,97, lIl , 120, 142, 158,177,201,218 ,231,255,278,291,320,344

Grid 170-186GridAssemble 3, 172, 173GridElementForces 3, 172, 173GridElementLength 3, 171 , 172GridElementStijfness 3,171 ,172

384 Index

IInclined Support 61,62,72,135,136

LLength 61,62,135 ,136,171,172,189,

190Linear 1Linear Bar 27-43LinearBarAssemble 2, 28, 29LinearBarElementForces 2, 28, 29LinearBarElementStiffness 2,28LinearBarElementStresses 2, 28, 29Linear Tetrahedron 329-357Linear Triangle 211-241LinearTriangleAssemble 3,213,214Linear TriangleElementArea 3,213LinearTriangleElementPStresses 3,213,

216LinearTriangleElementStiffness 3,213,

214LinearTriangleElementStresses 3,213,

215

MMATLAB 1,2,4Matrix 1,6,8,14,18,19,30,31 ,37,39,

47,48,53,54,65,67,73 ,76,94,97,110,111,119,120,140,142,157,158 ,175,177,198,201,217,218,226,231,253,255,276,278,289,291 ,318,320,339,344

NNode 1,11,27 ,44,60,89,106,133 ,170,

187,211,242,267 ,303 ,329

pPartitioning 2Plane Frame 133-169PlaneFrameAssemble 3, 135, 137PlaneFrameElementAxiaLDiagram 3, 136,

138PlaneFrameElementForces 3, 135, 137PlaneFrameElementLength 3, 135, 136PlaneFrameElementMomentDiagram 3,

136, 138PlaneFrameElementShearDiagram 3,

136, 138PlaneFrameElementStiffness 3, 135, 136PlaneFramelnclinedSupport 3, 136, 139Plane Truss 60-88PlaneTrussAssemble 2, 62, 63PlaneTrussElementForce 2, 62, 63PlaneTrussElementLength 2, 61, 62

PlaneTrussElementStiffness 2, 61, 62PlaneTrussElementStress 2, 62, 63PlaneTrusslnclinedSupport 2,62, 64Plane Strain 211,242,267,303Plane Stress 211, 217, 226, 242, 253, 267,

276,288,303,318Plot 9, 10, 107, 108, 136, 138Post-processing 2,16,23,33,50,57,70,

84,102 ,113 ,125 ,148,162,183,206,222,237 ,262,284,295,325,350

QQuadratic Bar 44-59QuadraticBarAssemble 2,45,46QuadraticBarElementForces 2,45,46QuadraticBarElementStiffness 2, 45QuadraticBarElementStresses 2,45 ,46Quadratic Quadrllateral 303-328QuadraticQuadAssemble 4, 306, 309QuadraticQuadElementArea 4, 306, 307QuadraticQuadElementPStresses 4, 307,

317QuadraticQuadElementStiffness 4, 306,

307QuadraticQuadElementStresses 4, 306,

315Quadratic Triangle 242-266QuadTriangleAssemble 4,244,247QuadTriangleElementArea 4, 244, 245QuadTriangleElementPStresses 4, 245,

252QuadTriangleElementStiffness 4, 244, 245QuadTriangleElementStresses 4, 244, 250Quadrilateral- Bilinear 267-302- Quadratic 303-328

RReaction 2

SShearForce 136,138,189,195,196Shear Force Diagram 136,138 ,189 ,195,

196Space Frame 187-210SpaceFrameAssemble 3,189,191SpaceFrameElementAxiaLDiagram 3, 189,

195SpaceFrameElementForces 3,189 ,194SpaceFrameElementLength 3,189,190SpaceFrameElementMomentYDiagram 3,

190,196

SpaceFrameElementMomentZDiagram 3,189,196

SpaceFrameElementShearYDiagram 3,189, 195

SpaceFrameElementShearZDiagram 3,189,196

SpaceFrameElementStiffness 3,189 ,190SpaceFrameElementTorsionDiagram 3,

189, 196Space Truss 89-105SpaceTrussAssemble 3,90,91SpaceTrussElementForce 3, 90, 92SpaceTrussElementLength 3,90,91SpaceTrussElementStiffness 3,90,91SpaceTrussElementStress 3,91 ,93Spring 11-26SpringAssembIe 2, 12SpringElementForces 2,12,13SpringElementStiffness 2, 12Stiffness Matrix- Element 1,14,18,30,37,47 ,53,65,

73,94 , IlO, 119, 140, 157, 175, 198,217,226,253 ,276,289,318,339

- Global 1,14,19,31,39,48,54,67,76,97, lil , 120, 142, 158, 177,201,218,231,255,278,291,320,344

Strain 211,242,267,303

Index 385

Stress 211, 217,226, 242, 253, 267, 276,288,303 ,318

Structure- Continuous- Discrete 1Support- Inclined 61,62 ,72,135 ,136SymbolicMath Tooibox 242,267,303

TTetrahedron- Linear 329-357TetrahedronAssemble 4, 332, 334TetrahedronElementPStresse s 4, 332, 338TetrahedronElementStiffness 4, 332, 333TetrahedronElementStresses 4, 332, 337TetrahedronElementVolume 4,332Triangle- Linear 211-241- Quadratic 242-266Truss- Plane 60-88- Space 89-105Tutorial 4, 360

VVolume 329