Université d'Ottawa University of Ottawa

THEORETICAL AND EXPERIMENTAL, INVESTIGATION

OF TkiE FREE 1.IBRATIOS OF PAARA4LLE LOGRAhI PLAATES

WITH SIMPLY SUPPORTED AND CLAMPED

BOl'XDARY COSDITIOXS

Thesis presented to the School of Graduate Studies as partial fulfillment of the requirements

for the Degree o f Master o f Applied Science in Mechanical Engineering

. i ' Stefanie D. Stangier ,L '1

University of Ottawa Ottawa, Ontario, Canada

August, 1997

National Library I*I of Canada Bibliothèque nationale du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques

395 Wellington Slreel 395, rue Wellington Ottawa ON K 1 A ON4 Ottawa ON K I A ON4 Canada Canada

The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of this thesis in niicroform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de

reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette t,lièse. thesis nor substantial extracts fiotn it Ni Ia thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

A systematic approach to the free vibration analysis of thin, flat, non-rectangular

quadrilateral plates ~ v i t h combinat ions of simplc and clamprd supports 1s prrsentcid.

using the parallelogram plate as an cxampie. Modifications that Saliba made for the

right-triangular plate to the building block superposition method drvelopcd hy Gorrnan

are implerncnted. The superposition method is an analytical solution. No simplifications

arc made to take advantage of the point symmetq of the parallelogram plate. kesping the

solution general. The whole plate is divided into twvo right-triangular and one rectançular

segment. Rsctangular building blocks are supcrimposed for each of the segments to meet

the required net boundary conditions and the conditions of continuity dong thc segment

interfaces. The Lé?-type solution to the eight building blocks used art: given, dong with

thc necessary Fourier espansions and a comprehensive guide to assembling the

ei~envalue matris. Numerical results for a \vide variet' of plates are presented. including

numerous mesh and contour plots. Cornparison is made to previously published data.

The results for the fully clamprd parallelogam plate are supportcd by esperimental

results. Sis alurninum plates were tested for the fint sis resonance frequencies and the

first three mode shapes. Details of the simple, yet highly accurate esperimental method

are included.

Ac knowledgments

First of all, 1 have to thank Dr. H.T. Saliba of Lakehead University for çiving me the

opponunity to do rny research at Lakehcad, for his yidance, and for hnvinç thc

confidence that 1 would eventunlly get this thesis tïnished. 1 would also likc to thank Dr.

D.J. Gorman of the University of Ottawa for his counsel.

Thanks also go to the School of Engineering at Lakehead University for allowing me the

use of the university facilities and for providing assistance in many other ways. 1 would

like to thank Kailash Bhatia for his rxpen guidance in the machine shop. Mark Blrzzard

did a fine job of assisting me in the shop as well. Jean Eric Mesmain assisted with the

esrcution of experiments, as did Andrew Saikaley. Andrew also gave me sornr: ver).

helpful programming ideas, and rnuch needed moral support.

The Natural Science and Engineering Research Council supponed this work financial ly..

iii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables siv

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols sv

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Problem Definit~on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7 Experin~cnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . Basic Assumplions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 . Plate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Labeling Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 = Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 . The Governing Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......................................................................... 2 . Boundary Conditions

2 . 1 Simple Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . 2 Clamped Edzes ............................................................................

2.3 Slope ..........................................................................................

2.4 Bending Moment ........................................................................

2.5 Vertical Edge Rcaction .................................................................

3 . The Lévy-Type Solution ....................................................................

4 The Superposition Method ................................................................

5 . Building Block Solutions ....................................................................

5 . 1 Building Block 1 ..........................................................................

5 . 2 Building Block 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Building Block 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Building Block 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 . 5 Building Block 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 6 Building Block 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7 Building Block 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8 Buildins Block 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 . Buildhg Block Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 . Continuity Between Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 . Integral Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 .1 Building BIock I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X. 1.1 Di.vj~i(rc-errrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a . Displacement along the II = 1 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b Displacement along the 5 = 1, q = 0, < = O edges ........... c Displacement along rl = 1 . < .......................................

8 . I . 2 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . Slope along the 11 = 1 edge ............................................

b . Slope along the r; = 1 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . Slope along the tl = O edge .............................................

d . Slope along the 5 = O edge .............................................

e . Slope along the 11 = 1 - < diagonal .................................. 8.1.3 I).Iotrrent .............................................................................

a . Moment alonq the q = 1 . 6 = I . q = O and 5 = O . . . . . . . . . . . . . . . . . . . . b Monient alon- q = 1 . < .....................................................

8.1.4 I~érticcrl Edgc Rcrrction ........................................................

a . Vertical edge reaction along the q = 1 edge ...................

b . Vertical Edge Reaction along the 5 = 1 edge ................. c . Vertical edge reaction along the q = O edge ...................

d . Vertical Edge Reaction along the Z, = O edge .................

8 2 Building Block 2 ..........................................................................

8 .2 1 1)isplncerrrcnt crlorig tltc rl = 1 edge ......................................

8.7.2 i\itmcrrt cilr~rig the 7 = 1 etige . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 3 Building Block 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3.1 Displrrccrrrent .....................................................................

.... . a Displacement along the Q = 1. 11 = O and 5 = O edges b . Displacement along the < = ! edge ............................... c . Displacement along the 11 = 1 - 5 diagonal . . . . . . . . . . . . . . . . . . . . . .

8.3.2 SIopc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a . Slope along the 11 = 1 edge .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b . Slope along the 5 = 1 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c . Slope along the 11 = O edge .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d . Slope along the j = O edge .............................................

e . Slope along the q = 1 - i, diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 . .3 . 3 Jlorrrenf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . Moment along the 11 = 1. < = 1 . q = O and < = O edges . . . b . Moment along q = 1 - j ..................................................

8.3. 4 C. krticnl Edge Rcticiion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a . Vertical edge reaction along the 11 = 1 edge ... . . . . . . . . . . . . . . . .

b . Vertical edge reaction along the E, = 1 edge ...................

c . Vertical edge reaction along the q = O edge ...................

d . Vertical edge reaction along the T; = O edge ...................

8.4. Building Block 4 ..........................................................................

8.4.1 I)isplncentcnt dong the 5 = I edgc ....................................... 8.4 2 Alot?rcnr dong tltc Ç = 1 cdge ...............................................

8.5 Building Block 5 ...........................................................................

8.5. I Di.vplpkicer?icnt ......................................................................

a . Displacement along the 11 = 1, j = 1 and E, = O edges .....

b . Displacement along the 11 = O edge ................................

c . Displacement along 11 = 1 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5.2 . Vopc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Slope along the q = 1 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b Slope along < = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c Slope along the q = O edge ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d . Slope along the 5 = O edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

e . Slope along 11 = 1 - 5 ................................................... 8.5.3 i\/orricrrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . a . Moment along the q = 1. 5 = 1. 11 = O and < = O edges . . b Moment along rl = 1 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5. J I . érrictr 1 Edgc Reciction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a Vertical Edge Reaction along the 11 = 1 edge . . . . . . . . . . . . . . . .

b . Vertical Edge Reaction along the j = 1 edge . . . . . . . . . . . . . . . . .

c Vertical edge reaction along the q = O edge ... . . . . . . . . . . . . . . . .

d . Vertical Edge Reaction along the < = O edge . . . . . . . . . . . . . . . . . 8.6. Builciin- Block 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.6.1 Di.yl(iccrii~.nt d o n g the 7 = O cdgc .......................................

8.6.2 :\lontent cilong the r ] = O cdge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 7 Building Block 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X. 7.1. Di.splrrccrr~ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a . Displacement along the = 1, j = 1 and = O edges . . . .

b . Displacement along 5 = O ............................................... c . Displacement along 11 = 1 . 5 ..........................................

8 . 7.2 Skopc .................................................................................. a . Slope along the 11 = 1 edge ...........................................

b . Slope along the < = 1 edge ............................................. c . Slope along the q = O edge .............................................

d . Slope along the j = O edge ............................................

e . Slope along = 1 - E, .......................................................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b . Moment along q = 1 . 5 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.4 I érticnl GIge Rmctioti 63 a . Vertical edge reaction along the q = 1 edge . . . . . . . . . . . . . . . . . . . 63

b . Vertical Edge Reaction along the j = 1 edge . . . . . . . . . . . . . . . . . 64

c Vertical edge reaction along the q = O edge . . . . . . . . . . . . . . . . . . . 64

. . . . . . . . . . . . . . . . . . . . d Vertical edge reaction along the < = O edge 8 8 Buildins Block S .......................................................

8.8.1 Di.splclcciti~'nt (dong t/rc 5 = 0 cdge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 8.8.2 Morrtertf dorig t/ic 6 = O cdgc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 . Eigenvalue Matr~x Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Ei~envalue Matrices for SSSS Parallelogram Plates

9.2 Eigenvalue Matrices for the SSSC ParalleIogram Plate . . . . . . . . . . . . . . . . . . . . . .

9-3 Eiyenvalur Matrices for tlic SCSC ParalleIogram Plate ......................

. . . . . . . . . . . . . . . . . . . . . . 9 J Eigen\. du r Matrices for the CSSS Parallelogram Plate

...................... 9.5 Eigenvalue Matrices for the CSSC Parallelograni Plate

9.6 Eigenvalue Matrices for the CCSC ParalleIogram Plate .....................

9.7 Eigenvalue Matrices for the CSCS ParalleIogram Plate ......................

9.8 Eigenvalue Matrices for tlir CSCC Parallelosrarn Plate . . . . . . . . . . . . . . . . . . . . .

9.9 Eiçenvalue .Matrices for the CCCC ParalleIogram Plaie .....................

3 O Theoretical Results ............................................................................... 95

1 . Simply Supported Plates .................................................................... 95

2 . ClarnpedPlates .................................................................................. 98

3 . Combined Edge Supports ................................................................. 130

4 O Experirnental Analysis ......................................................................... 132

7 . ExperimentalProcedure ............................................................... 132

........................................................................ 2 . Experimental Results 132

5 O Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11

7 . Cornparison to Published Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . TheoreticalAnalysis 141

3 . Experirnental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 O Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

. 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDICES

Appendix 1 O Frequently Used lntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix 2 = Experimental Investigation . . . . . . . . . . . . . . . . . . . . . .

1 . Ob, ect~ve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 2 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 1 Test Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . 2 Longitudinal Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . 3 Cross Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Test Plates ..................................................................................

. 3 Mounting Procedure .........................................................................

4 . instrumentation ................................................................................

5 . Experimental Procedure ...................................................................

6 . Results ...........................................................................................

. Appendix 3 Plate Property Check ......................................................... A13 1 . Objective .......................................................................................... A13

2 . Theory .............................................................................................. A13

3 . Set-up ............................................................................................... A15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Procedure A16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Results .A 16

Appendix 4 . Dimensioning of Paralletograrn Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,A! 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Alternate Dimensions A17

2 . Rhornbic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . General Parallelogram Plates .A' i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Eigen value Conversions A??

Appendix 5 . Program Listings . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 SBA SE - Base And Top Are Simply Supported . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . 1 hlain Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 Subroutines For Assenibliny The Eicenvalue Matris . . . . . . . . . . . . . . . . . . . . . . . .

1 3 Subroutines For Sliape Geiieration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . CBASE - Base clamped, Top Edge Slrnply Supporîed . . . . . . . . . . . . . . . . . . . . 2 1 Main prograni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 Subroiitines for matris assetnbly with CBASE ................................

2.3 Subroutines for shape generation .................................................

3 . CTOP - 80th Base and Top Edge are Clarnped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 . l Main Pro, ( l r m , .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 2 Subrout ines for matris assen~bly ....................................................

3 3 Soubroutines for sliape generation .................................................

4 . Common Subroutines ......................................................................

List of Figures

Fiçure 1 - 1 Building block coiiibinatioiis for tlir riylit triangular plate as used b!: a Gormaii [Y j and b Saliba [-XI

Figure 1-2 Plate iayout

Figure 1 -3 Coordinate sysrem . .

Figure 2- 1 Bending tiionictits on t lie outside edges of a rectançular plate

Figure 2-2 Brndiiig iiioiiwnt aloriy an oblique line

Fiçure 2-3 Vertical edge rractioiis on a rectariylar plate

Fiçure 2-4 Building Block I

Figure 1 - 5 Building Blocl 2

Figure 2-6 Building Block -7

Fiçure 2-7 Buildiriq Bluck 4

Figure 7-8 Building Block 5

Figure 2-9 Buildi115 Block 6

Fiçure 2- 1 O Building Block 7

Fiçure 2- I 1 Biiildiiig Bluçk S

Figure 7 - 11 Buildi112 bloîk cornbinatiori O \ en.ier\

Figure 2-1 -3 Ei-envalut. iiiatrir for SSSS parallelograiii plate. 4, O 5 . . . . . . . . . . . . . . . .

Figure 7- 11 Eiyenvahie iiiatris for [lie SSSS paralleloçrani plate. O 5 < 4, < 2.0. . . . .

Fiçure 2- 15 Eiçenvalue iiiatrir for the SSSS parallelogram plate. 4, > 10. . . . . . .

Figure 7- 16 Eigenvalue niatris for the SSSC paralleloçrain plate. 4, 5 O 5 . . . . . . .

Figure 2-1 7 Eigenvalue niatris for tlie SSSC parallelogram plate. 0.5 < $, < 2.0 . . . . .

Fiçure 2- 18 Eiçenvalue matriz for rlie SSSC parallelogram plate. 4, 2.0. ......

Fiçure 2- 19 Eigenvalue niatris for tlir SCSC parallelogram plate. 4, 5 0.5. . ... . . . .......

Figure 2-20 Eigenvalue matris for [lie SCSC parallelogram plate. O 5 < $, < ?.O. ....

Figure 2-21 Eigenvalue matris for the SCSC parallelogram plate. $, 2 2.0. . . . . . . . . . . . . . .

Fiçure 2-22 Eiuenvalue + matris for the CSSS paralleloyram plate. $, < 0 . 5 . . . . . . . . . . . . .

Fiçure 2-23 Eigenvalue matris for the CSSS parallelogram plate. O. 5 < $, < 7 0 . . . . . .

Fiçure 2-24 Eiçenvalue niatris for the CSSS paralleIo-ram plate. 4, 2.0. . . . . . . . . . . . . . .

. , . . . Fiçure 2-75 Eigenvalue matris for the CSSC parallelograni plate O . 5 4, < 2.0. 80

. . Figure 2-26 Eiçenvalue matris for the CSSC parallelograrii plate $, 5 O S 81

. Figure 7-27 Eigenvalue niatris For the CSSC parallelogam plate +t 2 2 O . . . . . . . . 87 . . Figure 2-28 Eigenvalue matris for the CCSC parallelograni plate. O 5 4, 2.0. 83

. Figure 2-79 Eiçenvalue riiatrir h r the CCSC parallelogam plate 4, 5 O 5 . . . . . 84

Fiyure 2-30 Eigenvalue iiiatris for the CCSC paralleloçrani plate . 4, 2 7.0 . . . . . . . 85

Figure 2-3 1 Eigenvalur inatris for the CSCS parailelogam plate . O 5 < $, < 2.0 . . 86 Figure Eigenvalue niatrir Ior the CSCS paralleloyram plate . $, 5 O5 . . 87 Figure 2-33 Eiçenvalue niatris for the CSCS parallelograni plate . 4, ? 2 O . . . 88

Figure ?-.il Eigenvaliir niatris for the CSCC parallelograiii plate . 0.5 al --: 2.0 . . . . 89 . Figure 2-3 5 Eigenvalue niat ris for t lie CSCC parallelograrn plate $, < O . 5 . . . . . . . . . . . . . 90

Fiçure 2-36 Eiçenvalue niatris for the CSCC parallelogram plate . 4, 1 2 . 0 . . . . . . . . . . . 91 Fiçure 2-37 Eigenvalue niatris for [lie CCCC paralleloçrani plate . O . 5 < 4, ?.O . . 92

Figure 2-38 Eigenvalue niatris for the CCCC paralleloçram plate. $, 5 O 5 . . . . . . . . . . . 93

Figure 2-39 Eigenvalue inat ris for t lie CCCC parallelogram plate . 4, 7 0 . . . . . . . . . . . . . . 93 . . Figure 3- 1 CCCC paralleloçraiii plate (1, = 0.75 4, = 0.50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

. Figure 3-7 CCCC paralleloyrani plate +, = 0.75. $, = O . 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

. . Figure 3-3 CCCC parallelograni plate $, = 0 75 $, = 1 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 3-4 CCCC parallelograni plate . 4, = O . 75 . 6, = 1 2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

. . Figure -3-5 CCCC parallelograni plate $, = O75 4, = I 5 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 . Fiçure 3-6 CCCC parallelograni plate 4, = 0.75. 4, = 2.00. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 . Figure 3-7 CCCC parallelograni plate $, = 1-00. 4, = 0 . 5 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 . Fiçure 3-8 CCCC parallelogram plate 4, = 1.00. 4, = O 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 3-9 CCCC parallelogram plate. 4, = 1.00. 4, = 1 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

. . Figure 3- I O CCCC parallelograni plate 4, = 1 .OO, 4, = 1 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fiyre3-IlCCCCparallelogramplate.~,= I O O . + , = 1.50 110

. Figure 3- 12 CCCC paralleloçram plate 4, = 1 0 0 . 4, = 1.00. .................................. 1 1 1

. Figure 3- 13 CCCC parallelogram plate 4, = 1.25. 4, = 0.50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

. Figure 3- 14 CCCC parallelogram plate 9, = 1.25 . 4, = 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1;

Ficure - 3- 15 CCCC paratlelo-am plats, & = 1.25. 4, = 1 .O@ ............ .... . . . . . . . . . . . Figure 3- 16 CCCC parallelogam plate. 4, = 1.25, 4, = 1 .?Y .. . . . . .. . .. . .. . .. . . .. . .. . ... .

Fisure 3- 17 CCCC paralleloyram plate. 4, = 1.25. 0, = 1.50. . . . . . .... . . . . . . ... . . . , . . . . . .

Figure 3- 18 CCCC parallelograrn plate, ip, = 1.25, 4, = 2.00. ............ .................... . .

Figure 3- 19 CCCC parallelogam p!ate. 4, = 1.50. 4, = 0.50. . . . .. .. . . .. . ... ... . . . . . . . .... . . . .

Figure 3-20 CCCC parallrlogram plate. 4, = 1 S0. 4, = 0.75. ..................................

Figure 3-2 1 CCCC parallelogam plate. 4, - 1 S0. 4, = 1.00. ................................. Figure 3-22 CCCC parallelogram plate, 4, = 1 SO, 4, = 1 .25. ... . . .... . . . .. .. ... ... . . . . . .......

Figure 3-23 CCCC parallelogam plats. $, = 1 S0, b, = 1.50. ............. . ............. . * . . . . .

Figure 3-24 CCCC parallelogam plate. 4, = 1 S O , 6, = 1.00. ... ....... ... .. ..... . . . . . ..... . . . .

Figure 3-25 CCCC parallelogam plaie, 4, = 3.00. 4, = 0.50. . . . . . . ..................... . . . . . . .

Figure 3-26 CCCC parallelogam plate, 4, = 3.00, Q, = 0.75. ................................ ..

Figure 3-27 CCCC paral lelogam plate. 4, = 3.00. $, = 1.00. ..... . ..., . . . . . . ... . . . . . . .. ..... .

Figure 3-28 CCCC parallelogam plate. $, = 3.00, 4, = 125. .................................. Figurc 3-29 Effect of 4, on the eigrnvalue for CCCC paral lelogram plates.

#,= 0.50. ...........+..............................................................................

Figure 3-30 Effcct of #, on the eigcnvalue for CCCC parallelomam Ci plates. 4r= 1.00. .. ........................ .. ................ ........................................

Figure 3-3 1 Effcct of q$ on the eigenvalue for CCCC paral lelogram plates, @l =O.75. ........................................................................................

Figure 3-32 Effect of #[on the eigenvalue for CSSC parallelogam plates. #,= 0.50. ........................................................................................

Figure 3-33 Effect of #[ on the eigenvalue for CSSC parallelogram plates. = 1.00. ............................... u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . .................

Figure 3-34 CSSC parallelogam plates. q$ = 1.00: close-up. .................................

Figure 4- 1 CCCC parallelogram plate, a = 1 jO, 4, = 1.00. .. .. . .... .......... ...... . .. . ... ... ..

Figure 4-2 CCCC parallelograrn plate, a = 15'. Q, = 1.25. .....................................

Figure 4-3 CCCC parallelogram plate, a = 15", 4, = 1-60. ........... .......... ...............,

Figure 4-4 CCCC parallelog~am plate, a = 30". 4, = 1.00. ......................... ... ... ......

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1-5 CCCC paralIrIogram plate u = 30" 0. = 1. 7 5 .

..................................... . Figure 1-6 CCCC parallelogam plate. a = 30" 4, = 1.60. ....................................................................... Figurc A2- 1 Overview of test set-up

................................................... Figure A2-2 I'ypical plate showing the test grid

......................................................... Figure AI-3 Test platc positioned on the base

.......................................................... Figure A 2 4 Longitudinal support in position

........................................................................... Figure A2-5 Full? mounted platc

. . . . . . . . . . . . . . . Figure AI-6 Speaker microphonc and accelcrometer in place for tcstinç ............................................................. Figurc A 3 7 Close-up of the accelerometer

........................................................ Figure A2-8 The dual channel signal analyzcr

................ Figure A2-9 Animated mode shape produced by STAR Modal software

Figure A3- 1 Optimum accelerorneter positions for a 12-inch square cantilever . . . . . . . . . . platc with correspondhg eigenvalues ( from Referencc (161 )

Fisure A3-2 Squarc cantilever plate showing the location of the test points . . . . . . . . . . .

Figure A 4 1 Paral leIogram plate dimensioned in two ways .....................................

Figure A 4 2 ParalleIogram plate segmentrd into two right triançular plates ...........

.................... Figure AJ-3 Dimensioning of rhombic plates ............................... ..

Firrure A 4 4 Common lavout of the rieneral ~arallclomam date .............................

List of Tables

Table 3- 1 Eiçenvalues for the SSSS plate . QI = 2 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 . Table 3-2 Eiçenvalues for the SSSS plate 4, = 1.667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Table 3-3 Eigenvalues for the SSSS plate . $, = 1.250. . . . . . . . . . . . . . . . . . . . . . 96

Tabie 3-4 Eigenvaliies for the SSSS plate . $, = 1 000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Table 3-5 Eiçenvalues for the SSSS plate . QI = O S33 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Table 3-6 Eiyenvalues for the CCCC rlionibic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Table 3-7 Eigenvalues = (0 (7, ' JzCi/) for the CCCC parallelogam plate . . . . . . . . . 99 . Table 4- 1 CCCC paralleloçram plates a = 15" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

. Table 4-2 CCCC paralleloyram plates u = -;Oo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

List of Symbols

plate dirneiision in the s-direction.

plate dirnension in the y-direction.

plate tlexural rigidity. I ~ I I ' / I ? ( I - il:)

niodulus of elasticiti; of tlie plate material.

circular tieqiiency of plate vibration i i i Hz.

plate tliickness

lateral loading of the plate surface

subscript referring to tlie secianyular sryriierit of a parallclogam plate srnient

subscript referrino to the trianyular segnient of a pal-allelograni plate.

amplitude of-tlie plate latcral deflection.

diniensionless plate lateral deflection. II%.,-i.) L I .

spatial coordinat es

angle of ille skew side of a paralleloyraiii plate to the y- or 11-mis.

dinierisionless coordinate in tlie s-direction. .t- LI.

diniensionless coordinate in the y-direction, J h.

aspect ratio. !, a.

Poisson's ratio of the plate material. v = 0.3 is assumed in this work.

= 241,

mass density per unit area of the plate material.

circular frequency of plate vibration in radis.

eigenvalue. non-dimensional natural frequency. ma' ,/a .

1 - Introduction 1. Problem Definition

Thin plates of al1 shapcs and sizcs art: undoubtedly one 01: if not the most common . .

structural component in engineering applications. 1 hey arc Sound in evcrything. Srom

seemingly simple casings to circuit boards to solar pancls. Knowing the static rcsponsc

O S thcsc structural clemcnts is not sufiicirnt to ensure propcr. evcn salè. performance.

Thc dynamic behaviour of'platcs must bc includcd in any design projcct thal incorporatcs

them. Thc abil i t y to prcdict the vibration hehriviour without test ing saws timc and

monq spent on trial and crror approachcs. Thc resonancc flequencies have to be known

so thar thcy can bc dcal t with in one OS three ways. The' c m bc avoided in the operation

of' the structure using thc plate. The plate can be re-designed to withstand the vibration.

or thc supports or physical propcnics of thc plate can bc changtid to change its vibration

behaviour.

In thc last fkw years, the iinite element method has made the vcry problem dcpendcnt

solution methods to plate vibration obsolete in the eyes of the uninitiatcd. Mosi oihcr

approximate solutions, in particular the earliest onrs, appcar to bc limitcd to spccial

problcms. liowver. these solutions have a place as a check for the finite element

approach. It should not be Sorgotten that thry are still only approximate in nature.

Particularly in the case of skew plates, the? ail do not satisl'y cirher the boundac.

conditions or the govcrning diffèrential equation completely.

So h r , the building block superposition mrthod is the only anal'ical solution to the

problem of thin, tlar, quadrilateral plates. Solutions by this method always satislk the

govrming differential equation, and al! boundary conditions to the desirrd accuracy.

Each rectançular building block rneets this criterion for being an analyticai solution io

the rectangular plate. The building block solutions are superimposed to satisfy the net

boundary conditions, and ihey are positioned adjacent to one another to accommodate

irreyularl!. shaped plates. Continuit). of displacement. dope. bending moment and

vcrtical edçe rcaction arc accounted Ior by building blocks as well.

The spccial case of the tliin. llat parallrlograrn plates lookrd ai in this work scnrs as an

esample Ior the gencral quadri laieral plate wiih combinat ions of- simple and clampcd

suppons. With this in mind. no assumptions are made about the symmct?. or

antisyrnmrtry of the mode shapes, something that could be uscd to rcducr the

computationiil cSSm A parallelogram is point symmrtric about the centrc point (the

intersection ot' i h r diaçonals). For support combinatlons that are symrnctnc about thc

cenirc point as well, such as thc dl simply supponcd plate, thc mode shapes will have

symrnctry or antisymmetry about this ccnirc point.

In addition to dcrnonstrating the application of the building block superposition mcthod

to thc isotropic. thin, tlat parallelogram plate, exp.rimental rcsults are prcscnted tbr the

Iùlty clampcd case io support thc thcoretical rcsults The brief liicratiire review that

Ibllows shows not only the lack of an exact solution to thc problcm at hünd, but aiso to

an abscncc O( esperimental results.

2. Literature Review

A very dçtailrd review of published work dealing with the vibration of plates, including

thin. llnt parallelogram plates. has been done by Leissa [1,2,3,4] up to the year 1985. The

rcader is referred there for a more in-depth guide to the availablc literaturc up to that

timc than cm be presented here. Some of this earlirr work is highlighted hcrr. beibre

more recent research is described. Theoretical and rxperimental works are deali wirh

separately.

2.1 Theory

Two types of solution exist: approximate and analytical. Analytical solutions are the

most desirable. but not always easily atiainûble. as in the case of parallelo~~am plates.

Early solutions especially arc highly dtpcndent on the actual shapc of the plate Thc

difficulty lies in the rcquirement to satisSy both, the goveming ditfkrential equation and

dl ot'thc houndan conditions esactly.

:lpprmirtratc! Solut iorrs

The approximatc methods can bt: grouped into continuum types like thc Ray

rncthod. and discrcte types likc finitc clement and h i t e diSSerence methm

though flnitt: elcmcnt solutions have becomc the firsi choict. t'or many rescarchers with

t hc mer drcrcasing çost of computing, Rayleigh-Riiz type approaches are still popular.

Thc aim thcre is to lower ~ h c uppcr bound result of' the solution. Many authors are also

sarislied with thc Iùndamcntal Ilcquency only, and rhombic plates apprar to bc ihe test

subjeci of' choice.

It i s well known that thc Rayleigh-Ritz mcthod yiclds uppcr-bounds only. Kaul and

Cadambe [ 5 ] and Pnuelli [hl attcmpted to tind lowcr bounds. In [ 5 ] . the upper bounds

for threr plaie confiçurations are Sound by the Rayleigh-Ritz method with a single

product function approximation. Kato's principle is usrd to lind the lowcr bounds.

Durvasula [7] commrnted that iheir frequcncy expressions Sor skew plaies are raulty.

I'nuelli [6] used the membrane analog t'or simply supponcd and clamped plates. among

a variety of' other shapes. The resulting algrbraic equations depend on the plate arca

only

Durvasula is one of the more prolific researchers on the vibration of skew plaies. In

Refcrence [8], he applicd the Galerkin method, a method of weighted residuals. using a

double series of &am characteristic functions to describr the deflection of the plate. The

functions are in oblique coordinates. They satisfy the boundary conditions of the

clamped plate at the edgrs. Results for various parallrloçram plate dimensions arc çiven.

Nair and Durvasula [7] used a double series of beam characteristic functions with the

Ritz mrthod for a number of clamped and simple support combinations with various

aspect ratios and skew angles. This references includes a vcry comprehrnsive set of

nodal patterns. somcthing lacking in othcr studics The eflèct OS the sksn angle on tfic

nodal patterns and thc rcsonancc liequcncics is wcll illustrated. Durvasula and Nair 191

uscd the partition or subdomain method. I t is a modified Ibrm ol'arral collocation. The!.

dcscribc the dcflection of' the plate with a polynomial that satisfics thc boundary

conditions. Thc gnwninç di Skrential equation is thcn satistïed as bcst as pssihlc over

cach o f thc subdomains. The- Iind the convergence in the cases wherc they solvrd ihc

rcsulting set o f lincar hom«çcncous cquations to be satisVactory

Mizusawa ct al. [ I O ] uçcd R-splinc shapc lùnctions with the Rayleigh-Ritz method. Thci

R-splinc Iùnctions do not satisSy the bo1indar-y conditions on thcir own. and so the

mcthod ol' artificial springs i s applicd to deal with arbitra? boundary condittoris. For

cach cdgr. onc o f thc thrtlc spring corresponds 10 the displaccmcnt and cnch o f the othcr

two to the siope in thc s- and y-dircctions. 'I'hc cocfticicnts o f the springs arc set io zero

Sor ficc cdgcs. and to infinit! for clampcd edycs. l'hc cnerp contribution is added to the

cquation lor the masitnum potcntial mer-y or thc skcw platc Final rcsults o f thc

analysil; arc prcsented ( o r various platc dimensions and skcw ançlcs.

Kuttlèr and Siçillito [ I 1 J turned thcir attention to fïnding lowcr bounds to thc rusonancc

Srcquencies [.incar combinations o f trial lùnctions that art. not rcquircd to satisl). the

boundary conditions arc combined and optimized in a Ritz-like rnanncr. Results are

yivrn for ihc M y symmetric modes o f the clamped rhombic plate. The span betwcn

the bounds i s quite srnall, but incrcascs with increasing skew angle.

Nagaya's [II] solution is o f interest. becausc the esact diflèrential equation i s used. and

the boundary conditions arc satislied by ustnç Fourier expansions to get around the

problern created by the polar coordinates used. However, due to al~ebraic complications.

intrgrations in the solution have to be perlormed nurnerically. Rcsults yiven For simply

supporied and clamped rhornbic and trapezoidal plates are given.

Sakaia ( 1-3 1 derivrd ;in approsimation fbrinuln liir the natural frequcncics 01' simpl 1. supportrd orthotropic paralleIogram plates from the natural frcquenc ies of memtmncs or

the samc shnpc. The solution to thc govcrning diSfcrentiül equation of the membrane i s

casier than to that of the platc. The author applies his reduction mcthod to obtain

approsimations h r the sirnpl y supporicd isotropie skcw platc Srom the clamped sLcw

msrnbranc. Kcsults l'or orthotropic platcs arc includcd as wcll.

Subrahmanyain and Wah [141 looked at ycncrel quadrilateral plates. and gnvc the

l'undamcnial 1iequcncit.s 01' parallelogram plates with various support corn binations. The

plaic is divided along onc diagonal into two triançular reyions. Thc difkrcntial cquation

and the boundap conditions along al1 Sour edges are satisfkd exactly. The Srcquency is

dctcmincd by enforcing the coniinuity condition alony the diagonal.

Thc ncwcr methods, piit-tudarly thc discrets-clemsnt ones, are increas~ngly gcncral in

naturc, applving to mort. gcncral shapcs. or handling various support conditions or even

lice and Ibrcrd vihratiun. Geannakakcs [ 151 usrs thc Iinitc strip method. Arbitrarily

shnpcd plates arc. mapped into a natural coordinaic plane using Scrcndipit? hnctions.

Rcsults arc prcscntcd Iàr trapczoidat and rhombic plates, among othcrs. Katsikadclis

[16.17] applics a static hounda? clcmcnt mcthod to the gencral problcm OS fiet: and

Ibrccd vibration 01 ' plntcs No results for sksw plntcs are ofkrcd, but thc mçihod applies

to an! shopc. Matsucla c1 al.[] 81 find the fundomenial frcquencies only Tor somc

combination of clampcd and simple support, including a combination of these two

support types along the same edge.

Huang, McGce, Leissa and Kim [19] focus on hiçhly skewed, simpiy supponed rhombic

plates. They reduce the increased error found in most solutions for this type of platc by

tciking into account the stress singularities in the obtuse angles. Corner functions are

used in addition to polyomial shape functions in the Ritz rnethod. The corner Sunctions

are found to improve the convergence of the solution. The results are still upper bounds,

as is tupical of' the Ritz method. The? are however consistently lower than the results ol'

other uppcr bound solutions The authors poini to rhc imporîancc 01' considering ihc

strcss singulari tics in both. continuum and discrctc-clcmcnt büsed methods.

1.icw and Lam 1201 uscd thc Rayleigh-Riiz mcthod. Thc set o f two-dimcnsional

orthogonal shapc liinciions is çcncrated with the Gram-Schmidt orthoponalimtion

proccss Rcsults arc givcn Ibr varicius rhombic plates with combinations of simply

supponcd and clarnpcd cdycs, as wcll as caniilevcred rhomhic platcs. T h q are uppcr

hounds. and agree ~ ic l l with rcsults obtaincd by l>urvasula and Nair [9] and othcrs. [.cc

and Pon [2 1 ] also uscd a Gram-Schmidt orthogonalization proccss, but 10 cibtain a sct of'

orthogonal bcani fiirictions. On1 y the gromctric boundary conditions arc satislicd Singh

and Chakravcny (231 mappcd rcctangular and skcw platçs into a unit squarc. The

solution fi-orn then on is the samc for dl plates with the samc boundav conditions, and

ihcrrhrc has to bc done only oncc. Orthogonal polynomials that arc uscd in thc

Rayleigh-Ritz rncthod satisSy the boundary conditions. Rcsults arc includcd t o r 3 widc

variet? of plates with com binations of simply supponcd, clrimped and liec cdçcs. Thc

papcr contains a comprehcnsive list of rekrences in the I i r ld of thin platc vibration.

A difkrçnt approach to plate vibration has been taken by Wang, Siriz and Bert [3].

Thcir application of thc difirential quadrature method to the vibration and huckling of

s k w plates is similar to a miscd collocation rnethod in that the boundary conditions arc

only satisfied point-wisc. For a yivcn grid, the weighting coefficients have to bc

calculatsd only oncc. The rcsults for the simply supported rhornbic plate are in yood

agreement with those of Gorman [24].

Gornian applied the building block superposition mcthod to Mly simply supponed and

Sully clamped rhombic plûtes [74] and to the simply supponed paralleIogram plate [ 2 5 ] .

He pioneered thc method with rectangular plates [26] and applied it to riçht triangular

plates as well [27.283. With the rhombic plate [NI, Gorman makcs full use of the

symmetry of the plate about both diagonals: only one quaner of the rhombus, thüt is, one

riglit triancle. b needt; to be lookcd at, but lur al1 four possible combinations of'synmetp

and antisyrnmctry about the ascs. In Rcfkrcnce 1251, Gorman uses thc point-symmctry of'

the platc to sirnplif). thc analpis. tio~vcver. this simplification rcduces the appl icahili t),

ol' the solution to more gencral cascs, Ior csamplc thosc: fcaturing difircnt boundary

conditions on the various cdges. Only half of thc platc is actuall>~ analyzcd. This Iialf is

srymcntrd iriio a rectangular and a right triangular pan.

Sal iba applied thc supcrposiiion rncthod to sirnply supponcd and Cul 1 y clampcd

symmctrical trapczoidril plütcs [39,30,31], as an csamplc l'or the gcncral quadrilatcral

platc. Again. only half of thc platc was lookcd ai for the particulnr cases prcscntcd. and

that hall- was dividcd into rcctangular and riyht triangular scgmznts Thc solution to thc

trianyular segment is tnkcn from Goman [27.28]. with two building blocks bcing rotûtcd

to position thc Iorcing lsunctions along ihc skcw edçcs, as i l lustraicd in Fiyurc 1 - l a,

Saliba simplified the supcrposition solution (or thr right triangulnr plnic hy cnforcing thc

houndary conditions along obliquc lincs within rectanyular building blocks that arc not

Figure 1-1 Building block combinations for the right triangular plate as uscd by a. Cormrn 1271 and b. Saliba 1321. The lower triangle represents the area of intcrest.

rotatcd [32.33]. Thc sirnplilication is Srom both a derivaiion and a computation aspect.

The numbcr of'building blocks is reduced, and the rotatcd building blocks arc cl iminated.

In tàct. only two building blocks dong the longer cdgc, outsidc ol'thc arca of intcrcst. art:

rcquircd to rnliirce the boundüry conditions of simple or clarnped support almg thc

hypotenusc. as s h o w in Figure I - l b In [Ml. clainped supports arc added. T« clamp

rither caihrsus ol'thr riglit triangle. a building block with a forceci nioment dong that

edgc is addcd. I lo~rwcr. Saliba fbund that with claniping. two additional building blocks

with forcing Iùnçtions aloiig the rcmaining cdçe outside of the area of inicrcst was

nccdcd Ior good rcsults wiih aspect ratios closc to un i t - I-lc also noted in [34] thni doing

s« improvcs the modc shapes Ibr simply supponcd triangular plates In this tvork, this

simplificd triangular solution is used Ior the triangular scctions of thc parallelograrn

plate.

2.2 Expcrimcnt

N o espcrirncniül data Ibr parallclogram plates with clampcd suppns on al1 edçcs cxists

in thc litcrnturc. Thc can:ilcvercd skew plate does reccive some attention. possibl)~

hccausc of its siniilarity to the swcpt-back winys or airplanes. The prescnt work

rcprcscnts n Iirst h! prcscnting csperimental rcsults Sor thc fully clampcd parallclogram

plate.

3. Basic Assumptions

Thc platcs studied hrrc arc considcred to bc thin, that is, the plate thickncss is small

compared tu the lateral dimensions, as wcil as to the distance between the nodal lines.

Furthermore, the plate displacement is srnall rclativc to the thickness of the plate. The

effects of rotary incrtia are ncglectrd. In-plane forces art: considerrd absent in the

theorct ical analysis

4. Plate Geometry

The platès undcr consideration in this \mrk art: paral1t:lograms. Thc mèthod is ieeadiiy

npplicd to thc gcncral t h i n quadrilatcral plate where only two oppositc cdçes arc parallel.

For the purposc of gcncral ity, no sirnpt i tications are made to takc advantaçc of the point

symmeiry round \ \ i th somc support combinations. Thc plates are dividcd into thrce

segments, two triangular and one rcctangular. as shown in Figure 1-2. Many plates can

bc dividcd in this manncr in two ways. The advantage of one ovcr thc othcr is discussed

in n latcr scction. l'hc plate s i x is labrled using either ihc skew anglc a or the aspcct

ratio of thc irianyular segmcnis. 6,. and thc rcctangular aspcci ratio 9,. Conversions

betwt.cn the diinensioning conventions o f other authors and this work arc a m i labk in

Appcndis 1.

Figure 1-2 Platc layout.

5. Labeling Conventions

Suppon conditions for platcs arc labclcd eithrr 'S' for simple suppon or 'C' Tor clamped

support. The rdges arc labclcd starting with the lower edge. rnoving around the plate in a

counier-clockwise rnanner. For esample. a plaie with the base clamped and the

remaining edgcs simply supportcd would be designatrd as 'CSSS'. The SSCS plaie

would be identical to the CSSS plate, with the only diffcrence beinç the rotation by 180".

Similarly, the SSSC plate is the SCSS plate afier a 180" rotation. In Tact, the SSSC plate

can also be rotated by 90" and analyzed as a CSSS plate. The advantaçes of one over the

other are discussed in later sections. Results for nina plate configurations are provided in

this work:

with simply supported base: SSSS, SSSC, SCSC;

0 with clampsd base and simply supported upper edçe: CSSS, CSSC, CCSC:

with c tarnped base and top edge: CSCS, CSCC, CCCC.

Bouiidary conditions are abbreviatrd in fiçures as:

D - displacement, M -moment,

S - slopr=, V - vertical edge reaction.

In the software, edges are nurnbered, starting from the q = 1 cdge:

3: q = o ,

4: c = o ,

In figures of the building blocks, al

othenvise. The coordinate system used

1 edges are simply supponed, unless indicatcd

is s h o w in Figure 1-3 below.

Figure 1-3 Coordinate system.

2 - Theory

1. The Governing Differential Equation

Thc gmcrning squation !Or t h s frcc vibration OS thin. flat platcs is n partial dillkrential

cquation ol' th h u r t h ordu:

Thc last tenn on thé Icti-hand sidc ol'thc equation accounis Ibr thc incrtia. It alone i s a

lLnction of tirne. Hy scparation 01' variables using H P ( . r , y , / ) = I k r ( - r , ~ ~ ) l S ( / } . tivo

quations arc obtaincd:

and

Frorn equaiion (2-2) it is O ~ V ~ O U S that the mode shape is indcpendent of time. Equaiion

(2-3 ) is of no intcrcsi IO the prcsent analysis. For convenicnce. Equation (2 -2 ) is non-

dimensionalizcd with rcspcct io thc .*-dimension of the segment in qucstion. Tlic non-

dimensional displaccmcnt amplitude beçomrs

b . and the aspect ratio 4 = - IS intraduced.

Cl

Finally, ihe diffcrrniial equaiion follows:

where ,? = ru '1' ,/P/» is the r igenvalue. or non-di mensional resonance frequency : a)= 2 ~ / ' is the circular frrquency in radis:

p is the mass prr unit axa;

and D is the flenural ngidity of the plate

2. Boundary Conditions

Thc Sollowinç arc ttic boundary conditions for ilic classical edçe conditions in

dimcnsional and non-dimensional lbnns that are èncountcred in this work.

2.1 Simple Support

For this suppon condition. tlierc is no irioincnt or displaccment alotig ttic rdge,

Simple Support dong the s- or

2.4 Bctitiing 3foment

The posiiivc directions arc s h o w in Figurc 2-1.

Along the s- or :-asis:

Alon2 the * - or q-iisis:

M b2 I I - aD

Figure 2-1 Rcnding rnonienis on the outside d g e s of a rrctangular plate.

In an! direction:

Figure 2-2 shows the proper directions.

. Y / , = If, cos% + A I , s i n 2 a - 2 .\/,sina cosa

where O,, = c o s 2 a + v s i n 2 a

8 ,- = sin 'a + vcos' a = ( 1 - 1.) sin 7a

1 Figure 2-2 Bending niomriit dong an oblique line.

2.5 Vertical Edge Reaction

Fiçurc 3-3 shows the positive orientation of thc vcrtical cdçe rcactions.

Along thc y- or q - m i s :

Figu

4 t l 1 d h

,JJy V b' -1î-

aD

Vertical edge reactions on a rectangular plate.

sin a sin 2a where rp, = cos a + ( 1 - ip)

3 - casa sin 2a

iT, = sin n + ( 1 - 1.) - 7 -

casa sin 2u r b , = sin a - ( 1 - v)

3 - sin a sin 2a

i7, = cosa + ( 1 - 1.) 3 -

sin a sin 3a \\.herr I ; = cos a + ( 1 - if)

3

cosa sin 2a sina + ( l - Y) ---- 3 -

cosn sin2a 3 -

sin a sin 2a cosa + ( 1 - 11)

2

3. The Lévy-Type Solution

The Lkw-type solution for the free vibration or a rectangular plate requircs that the

boundary conditions on two opposite cdgrs can br satisfird by a irigonomctric funciion.

When two oppositr edgrs are simply supportrd, this condition is automatically satistird

by the single Fourier series solution of the form

This is the original I i v y t y c solution for simple support dong thc 5 -. O and 5 1 cdges. Simple support and slip shear (zero slopc and zero vertical cdgc reaction) conditions can

hc modcled \rliih cosinc. hall-sine or hülf-cosinc series. For the prescnt work. onl?. the

sine serics is uséd.

Upon substitution 01' this solution into the çovcrnins dinèrttntial equation (2.5). IWO

solutions for I;,,(qi arc found. dependent on u-hcther j: - ( r t r ~ ) ~ ' is positive or negaiive.

For both solutions. p,,, = g) JÀ: + ( r m r )'

and y,,, = #,/- --

or y , = &/=)' - A' . whichever is r d

The constants .4,,,, fi,!,. ( ',,, and I),,, arc dctemined by satisfying the boundary conditions

dong the rernaining two edçcs. Possi blt! boundary conditions include simple and

clamped suppori conditions. slip shear. and Sorced conditions such as hrced

disptaccment or slope. Solutions that have simple suppon conditions on one rdge and

either a forced hannonic displacement or a Iorcrd harmonic momeni are used in this

work. The forced moment was chosen over the forccd rotation for reasons that are

discussed in more detail in the nest section.

4. The Superposition Method

Thc 1.tivy-type solution donc cünnot bc uscd to niodel a non-rectanyular plate or cvcry

possihlc. support cornhination on ii rcctangular platc. tlowevrr. the L.iv>-type solutions to

a plate can bc a building block. Sevrral building blocks can bc supcrimposed so thrit the

nct houndüry conditions arc mci csactly

The solution can also bc applicd to simple shapes likt. right-triangular plates, as secn in

F u r 1 - 1 . The Ibrcing Iùnctions on the outsidc edgcs of rectangular building blocks arc

adjustrd to cnlime the boundary conditions along obliquc lines within the building block + +

houndan. 1 his oblique Iinc is usually convenicntly choscn as onc of thc diagonals.

t3eyond rcctangular and triangular plates, plate solutions can bc joinrd by rcquiring

continuit>, 01' displncemcnt. slope. and moment as well as a zero net vertical edge

rcaction a lmg the interlace o f segincnts. Superposition solutions to triangular and

rcctanguliir plotcs can be linked to analye gcncral quadrilateral plates.

In this work, thc parallclograin plates is uscd as an exürnple. As can bc scen in later

sections. only eight building blocks are uscd. Actually. thc solution to two building

blocks i s rotatèd by 90°, 180" and 270" to obtain thc other six. Eacti building bloçk has

ihrce sides simply supported, and a Sorced displacement or a Iorcrd btinding moment

dong the rcmaining edge. These forcing Iùnctions are both sine functions. Bending

moment i s a second derimtivc o f displacement. The second derivative of the sinc

IÙnction is a sinc lùnction also. That simplifies the solution process grcatly.

5. Building Block Solutions

5.1 Building Rlock I

13uilding Rlock I faturcs ri torccd C a Z * -

displacemcnt dong the q-1 cdgc o r the

ni- I

Tht' bending moment almg this edçe i s zcro. ' i The remaining edgcs are sirnply supporicd,

v

as shown in Figure 2 4 . Thc solution ihcn Figure 2-4 Building Block 1.

bccomcs

and

- n ) sinh p, ( '1 lm =

Y' + v#'(nrn)' siny, t m

@'(nin)' - ,O: sinhp,,, ( 'pm =

- z ) sinh y ,

l 4 ,nr = sinh p, + ,, sin y, @,Zn! =

1

s inhp, , + ( s inh 7,

5.2 Building Block 2

Building Block 2 is shotvn in Figure 2-5. I t has

no

the

displacement dong the edçt. q- 1. Instead, IJ, -L

function

dcscribcs thc forccd moment. The remaining 1

edges are simply supported. The solution for Figure 2-5 Building Rlock 2. this building block then becomes

and

- sinhp,

sin y,

- sinh P,,, sinh y

- 1

P., sinhp,, - C 1 n , 7 i siny,, - i

sinhp, + ( '21n,yb sinh 7,

5.3 Building Block 3

Building Block 3. showm in Figure 1-6. is - 7 - - b -: -

found h j rotatinç Building Block 1 b!. 90". M d d il---

The displacernent d o n g the < = I edgr is forcsd

using the function

The moment dong this edge is zero. The

rrmaining edgrs are simply supponed. Figure 2-6 Building Blocli 3.

The solution then becomes

L.

and

4'gm - iqmn)' sinhp,, CT3?,, = - #'&, - v(mrr)' sinh y,,

1 Q i i m = sinh ph + C,,, sin y,,

5.4 Building Block 4

Building Block 4 has no displacemeni d o n g 1 - , ,4 tlir edge i= 1. as indicatrd in Figure 1-7. The b , :E: = '. ,' ,- -,

describes the forced moment dong this cdge.

The remaining rdges are simply supported. Figure 2-7 Building Block 4.

The solution for this building block t h r n is

that of Building Block 7. roiated by 90':

Y,,,, = J-, whichever is real, as defined for Building Block 3, 4 and

sinh p,, C;?, = -

sinh y,,

5.5 Building Block 5

7

Building Block 5 i s obtained by rotating building b ZE: I -, ' ' -, blocks I by 180" The forcrd displacement is

applird to the q - O ttdgc. as sren in Figure (2-8):

Ttic remaining edges are simply supponed. The Figure 2-8 Building Block S.

displacerneni for Building Block 5 becomes

where fi,, = 4 II- and ,ynt = # d m or

The coefficients are identical

v = @ ,/ (nrn )' - A' , whichever is reûl. I m

to the corresponding ones for building block 1 :

sinhp, + (il, sin y, 1

sinh fl, + C,,,, sinh y rn

5.6 Building Block 6

To obtain Building Block 6, Building Block 2 1s I l

0 i

rotatrd by 180". As seen in Figure 1-9, the forced b / 8~ 5 ! i

bending moment is now applied to the q - O i i 1

edçe: l

The rrmaining edgrs are simply supported. The Figure 2-9 Building Block 6.

solution for the displacrmcnt anywhert on the

plate follows:

or y,, I ni = ( d m , whichever is real. and

sinh p,,, c i , , = - s inh y m

Q,', = 1

,ûi s inh& + ~ ~ ~ ~ ~ y i sinh y,,

These coefficients are identical to those for building block 2.

5.8 Building Block 8

Building Block 8 has no displacement along the

edge : - 0. The function

describes the forced moment alonç this edge,

while the remaining rdges are simply supponed.

This is shown in Figure (2- 1 1 ). The solution for

this building block then is that of Building Block

4, rotated by 1 80":

Figure 2-1 1 Building Block 8.

- 1 ,/T 1 4- and y,, - ()A)- (mn)- or y,, = - (mn) ( (1 ) , whichever is real, # #

as defined for Building Block 3, and

sinh p3, C , 7 8 1 m = - .

sin 7 3,

6. Building Block Seleetion

The building blocks have io br chosen with the net boundary conditions on the edges of

thc plaie to be modeled and the continuip betwern segncints in mind Figure 7-1 1 shows

the combinations chosen for this work the paralleIogram plate. There is not necessarily

one possible combination to choose from. In fact. the solution c m be checked by using a

different set of building blocks to solve the same problem. The number of distinct

building blocks that are required and computational considerations corne to ph'..

clarnped top edçe 1

t ..,. , I -- - I -. . .. .... , 1

1

I I

1 I

1 t f

Figure 2-1 2 Building block corn bina tion overview.

There is a basic set that all plate sizes require to mret the requirements for continuity

beiwren the se~ments. The conditions for simple support dong the upper and lower

rd-es are satisfisd automaticall~~. Additional building blocks are nerded to rnforce the

boundary conditions along the skew edges.

Whcn the skcw angle is very large or very small. it suffices to have the forcing functions

nredcd to obtain the boundary conditions along the skew edge dong the longer side only.

In the case of a largc value for $, (small skew angle) for example. Building Blocks 7 and

8 alone are sufflcient to control the skcw edçe of sebment 1.

For clarnping the skew edges. only the boundary conditions have to be changrd. The

buiiding block combination remains the same. If however the lower or top edçes are to

be clamped, the building blcicks with the forced bendinç moments must be added to force

the net s i o p to zero.

7. Continuity Between Segments

Four conditions have to be met dong the interfaces of the sebments of the quadrilateral

plate:

continuity of displacemcnt,

* continuity of slope,

continuity of bending moment,

and force equilibrium.

Continuity of displacement can be taken care of by the coefficients E, of each of the

building block solutions.

At the edge common to segments 1 and I I : kV1 - W,, = O, (2-25a) or at the interface between segments II and III, b, - Wl/l - O. (2-25 b)

To maintain continuity of slope. the contribution of the rectangular segment. D r in the

case of a çeneral quadnlateral plate, also the third segment, has to be adjusted bv the

ratio of the aspect ratios of the adjacent segnents, & 4,:

For thc bcnding moment. the adjustmcnt factor becomes 4)':

Vertical edge reactions of segmcnt I I are muitiplied by a factor of (4r #J ' :

The contributions of the building blocks are computed with the appropriate values of (

and 2' for that segment. Since in this work the rcference dimension for the eigenvalue is

chosen as the base lengh of the tnançular segment LI, , /.-' has to be changed for

computations involvinç the rectançular segment:

i

LI - .II, = .'; 2 7 = 4; a,-

In the analysis of a general quadrilateral plate, A' would have to be adjusted for any other

segments as well. The reference dimension for segments 1 and III for the parallelogram

plate is the same, and so this adjustment is not needed.

8. lntegral Expansions

Whrn the building blocks are superimposed, their contributions to the various boundary

conditions have to hr of a compatible f o n . This is achieved hy espandinç them in an

appropriate Fourier series, where required. The expansion is camed out for each

particular rdgs and condition. A list of integrals that are frequently used in the process is

given in Appendis 1 Once the expansions are obtained, the eigenvalue matris can be put

together. The selecrion of building blocks is rsplained in Section 6. Detûils of the

eigenvalue matris assembly are prescnted in Section 9. To understand the organization

of the expansions it should be mentioned liere that this eigenvalue matris consists of

submatrices AStr,j). where 1 indicates the row, and J, the column. Each combination of

edge. t g . q O. and condition. e.g. sloptt, has a submatrix associated with it. As the

expansions are given, their place in the appropriate submatrix is also given.

In the following. the espansions are grouped by pairs of building blocks. The expansions

for building blocks with the forcing functions located dong the same rdge differ only in

the coeflïcients and in the contribution to the displacement or moment dong the edge

along which the forcing functions are located. For example, the expansions for Building

Block 1 are alrnost idcntical to those for Building Block 2.

8.1 Building Block 1

a. Displacement along the q = 1 edge

For Building Block 1, there is a forced displacement along this edge.

which is already a sine in the desired terrns series. Hence no expansion is required.

AS0.j) = 1.0, (2-30a)

and where i #j, AS/i,j} = O. (2-30b)

b. Displacernent along the 5 = 1, q = O, F; = O edges

There is no contribuiion by Building Block I to the displacement dong these rdges:

4 . 1 . (1. (2-3 1 )

c. Displacement along q = 1 - 6 This line represents one of the diagonais of the rectançular building blocks. The

unespanded sol ut ion is

After expansion, this solution becomcs

wherc

Y,[cosï, - cos(mx + nx)] c,r4 = ri - (mn + nlr)'

The elements of the submatris then become

A S ( i , j ) = 8, ,, [ ex 1 + ex2 + C , ,, ( a 3 + er4)] for - (nin )' > O, (2-33a)

a. Slope along the q = 1 edge

AI 1 rlements that are not on the diayonal of the submatns are zero:

A S f 1 J) 0. O.

On the diagonal.

b. Slope along the 6 = 1 edge

fy (g, qq 1- = I:',_ 0, ,,(nrn)[sinh pmq + Cl ,,,, sin j',,,g] cos(mz ) ZC l ; = l m=i

In expanded fom, this becomes

,-LS(r, j ) = 8, ( m n ) cos(nin)[exl + C, ,,CI-21 for A? - ( rn l r ) '>O, (7-35a)

c. Slope along the = O edge

Elrments ofT the diagonal of the submatris are zero: r lS f i ,~ ) O.

The elcments on the diagonal are

d. Slope along the 5 = O edge

In expanded for, this then becomes

.AS( i,.j ) = 8, ,, (mn) [ex 1 + (', ,,eC] for R' - (rnn)' > O, (2-37a) and A S ( r , . j ) = 6 1 2 m ( m ~ ) [ ~ ~ + ( 7 1 2 ~ r ~ ] for 2 - (mn)' < O , (2-37b)

where ex/, er2 and cr3 are identical to al. er2 and er3 for 5 = 1 found in section

8.1.2.b:

e. Slope along the q = 1 - 6 diagonal The dope dong any oblique line is

\vliere the derivatives for this building block dong the diagonal in question are

After the expansion i s pcrformed, the elements of the submatrix can be found from

(mir - ~ t n ) sin y , ex4 = ,

y ; - (mn - nn)' (mn + nn) sinh y _

a5 = + (mIr + nn)'

y1 = P m s ' n ' P m + (mrr - nn)' - p, sinh p,

112 = pi, + (ma + nn)'

y3 = y , sin ym y:, - (mn - nn)2

y4 = Y, siny, (mn + na)? - y ;

7, sinh y , y5 =

y: + (mn - nn)' - y , sinhy,

y6 = y: + (mn + nn)'

a. Moment along the q = 1,c = 1, q = O and 5 = O

There is no contribution by Building Block 1 to the bending moment dong any ot'thc

outçide edçes: f.Sfr,j) O. (2-39)

b. Moment along q = 1 - < The moment expression that has to be expanded for Building Block 1 is

Upon expansion, the elements of the submatrix can be calculated from

Pyc

The expressions for erl to clx'.r/i are identical to those for the displacement of Building

Block 1 dong the diagonal q = I - T;, found in section 8.1.I.c. The rernaining expressions are:

(,?I,T + t ~ n ) [cosh pnJ - cos( n m + M ) ] - \ a l = -3

jji, + (mn + >ln)-

(nin + ri n ) [cos y ,, - cos( ni ir + i i ir ) ] 1.3 =

(m7r + lL7y - ( n ~ r r - nrr) [cosy, - cos(rnx - nn)]

' 4 = - (m,r - n ~ ) ~

8.1.1 I'enical Edge Reactiori

a. Vertical edge reaction along the q = 1 edge

1' 1 = 1 LIn,8 ,,,, [&, coshp,,, - C,,nlr:cosrn,] sin (mng) FI m=l

+ 2 L,#,~, [p i cosh,O, + (;,,y~cosh y,,,] sin(mni) nrk'+l

Only the rlemcnts on the diagonal of the subrnatns are non-zero:

= f S f i J j (1 for l#] .

,~ .S(+J, .J) = + ( 2 - ~ ) 4 ~ . 4 2 X , 1 whrrt. ..II:\' = 0,,,,1 [pil coshp,, - (',,,,,y;, cos;/nt 1

.A2 :\* = -8, ,,,( mir)' [& c o ~ h p , > ~ + ( cos;^,,,]

and .4 1 S = 1 9 , ~ ~ ~ [pi, COS~P,, + ( vl,,j.~tcosh y,,,]

.A2 A' = -O,,,,,(ntn)' [p, cash/?,,, + ~'i,,,y,,coshïn,] for ir' - ( m , ~ ) ' < O .

b. Vertical Edge Reaction along the 5 = 1 edge

Here the submatrix elements are defined by

.-12 .\' = QI2, i mn) cos( r n ~ )[pi, er I + ( '12,,;.: cr3] for i: - ( n m ) ? < O .

The expressions ci./. C - Y ~ and c r 3 are identical to those for the dope contribution of

Buildinç Block I dong < - 1. found in section 8.1.2.b. c. Vertical edge reaction along the q = O edge

Elements not on the diagonal of the submatrix are zero: S I O.

On the diagonal of .-K. the elements are deterrnined from

and A I N = O l 2 , [ ~ ; +

d. Vertical Edge Reaction along the 5 = O edge

a2

- L ; , , B , , , ( v I ~ ) ~ [sinh,O,q + C',?, sinh y,,,rl] m=k'+~

t h e the suhmatris elements are definrd by

Building Block 1 dong this line. found in section 8.1. l .c.

8.2 Building Block 2

The expansions for Building Block 2 are identical to those for Building Block 1 ,

where noted in the next subsections. Of course the coefficients for Building B

O,,,,,, C'21,, O??, and G,, replace the corresponding coefficients for Building B

@ln, , (7/1n1: Qlh and (h",.

8.2.1 Displacernertt along the q = 1 edge

There is no displacement along this edge: .-1Sfi,j) = 0.

8.2.2 Moment almg the q = I edge

Building Block 1 has a forced bendinç moment along this edge:

which is already a sine senes. No expansion is required:

AS,$$ = 1.0,

and where i +J, .4Sfl,j] = O.

except

lock 2 ,

lock 1 :

8.3 Building Block 3

8.3.1 Displacenierit

a. Displacement along the q = 1, q = O and r; = O edges There is no contribution by Building Block 3 io the displacerncnt alon- thrse edgcs.

4 Y , 0. ( 2 3 7 )

b. Displacement along the 6 = 1 edge

Building Block 3 Ceatures a Sorced displaccment along this edgr,

which does not require expansion, as i t is already the appropriate sine series.

c. Displacement along the q = 1 - 5 diagonal The unespanded solution is

The elements of the submatris then becomé

AS(/ , j ) = O,,, [al + er2 + C,,, (er3 + d)] for A' - (rnn)' > 0, ( 7 4 9 a )

or AS(i, j ) = O,', [er l + er2 + (lJ2,, (a5 + cr6)] for 2 - (nrx )' < O . (249b)

where cosh p,, cos(n,;.) - cosit?m)

"'1 = p,,,, pj, +(rn lr+ nn)'

COS( i i 2 T ) - COS^ Pi,, COS( \? H) ex2 = p3,,

pi, + (mn - nrr)'

a. Slope along the q = 1 edge

r

+ I:',,, O,,,,( nzx) [sinh P,,< + ( sinh ,,,,~]cos(nix) ni=k cl

In espanded fom, this becomes

and . - IS( i , , / ) = O,,, ( r n ~ ) cos(mn)[er l + (;,,cr3] for 2 - (vin )' < O . (?- job)

wherr:

2 ( n ~ ) c o s ( nn) siny,, cosh y,, cos(nir) - coshzrr) Cr 2 = i ex5 = y 3,

y;m - O m - y:,,, + (rtin + nn)" - 2(riir)cos(nn)sinhy ,,,, cos(mlr) - coshy,, cos(nj7) c r3 = ex6 = yim

y;, + (m)' + ( m n - t i r ) '

b. Slope along the = 1 edge

Ali elements that are not on the diagonal of the subrnatrix are zero:

..IS/i.// O. O for I I j. ( 2 - 5 1 a)

On the diagonal!

AS( i , j )=83,m[~ , ,cosh~,+~, , ,y ,co~ j . , , ] f o r ~ ' - ( m n ) 5 0 , (2-51bj

and AS(/, j) = O,,, [~,coshp,, + C,,, y, cosh y,,] O - ( m ) < O . (2-5 1 c )

c. Slope along the q = O edge

In espanded for, this then becomes

where er l . er2 and ex3 are identical to those of section 8.3.Z.a.:

d. Slope along the 6 = O edge

Elements off the diagonal of the subrnatrix are zero:

.-IS/i,.// - O for i F j The elernents on the diagonal are

and AS(\ , / ) = 0J2,, (&,, + ( 2 m ~ 3 , , , ) for - ( r n ~ ) ~ < 0 .

e. Slope along the q = 1 - 6 diagonal The siope along any oblique line is

n-hrrr the derkat ives for this building block are

r

+ ' : ' J n < ' ~ ~ n t [ ~ ~ n ~ COS' PJn15 + ( ;:,Y COS^ ~ 3 n , g ] ~ i n ( m a ( 1 - 5 1) nt-k * I

After the expansion i s performed, the eiements of the subrnatnx can be round frein the

( )

fol lo\ving:

k

= i : , n t 3 , n , a [ s i n h , + , , , , sin ] O -

a. Momentalong theq = 1, c = 1, q =Oand

The submatris elernrnts are definrd by

.4S( ,,>/) = -[Atr + ( 2 - if)4'.-12hr].

where -4 1 N = -O,,,,,(nin)' cos(niir)[exl + < ;,,cï2]

The expressions ex/, ex2 and e r3 are identical to those for the dope contribution of

Building Block 3 almg i = 1 . found in section 8.3.7.b.

b. Vertical edge reaction along the 5 = 1 edge

Only the eiements on thc diagonal of the submatris are non-zero:

AS'(i.j) = 0 for i # j.

On the diagonal,

c. Vertical edge reaction along the 1 = O edge

for 2 - (n t s ) ' > O ,

d. Vertical edge reaction along the 6 = O edge

Elements not on the diagonal of the submatris are zero:

On the diagonal. the elernents are detennined by

and

8.4 Building Block 1

The coefficients BJ,,. ( ',/,,. and < ;2m, are substitued Tor @,,. (';,,. &,, and rcspectively. in the contribution r ems for Building Block 3 for al1 cases escept those

specified in the subsequent sections.

8.1.1 Displacemerit alortg the 5 = 1 edge

There is no contribution to the displacement by Building Block 4 along this edge.

.-l.V~,j) = O. (2-61 )

8.42 Moment aiortg the { = I edge

The forced hamonic bendinç moment

is airrady a sine senes in the proper tems . The submatris elements on the diagonal are

.4S( j~ j = 1. (2-62a)

For the remaininç elements ahere r # j ,

.irSfi,j) = O. (2-62b)

8.5 Building Block 5

a. Displacement along the q = 1,c = 1 and 5 = O edges

Therc is no contribution by Building Block 5 to the displacement dong thesc cdges:

.-1 Sf1 , j ) O. (3-63)

b. Displacement along the q = O edge

Building Block 5 has a forcrd displacement dong this edge:

and so

nt- l

For al1 rernaining submatris elements for which r FI.

f S i O.

c. Displacement along q = 1 - 5

The unexpanded solution along this diagonal is

After espansion. this solution brcomes

I - coshp, cos(m;r + nlr) a 2 = p,

+ (ma + nx)'

Thc slcments of the submatris then becomc

or .-IS(r,,j) = 0j2m [cd + er2 + (4 + er6)] for - (nrn )' < O . (2-65b)

a. Slope along the q = 1 edge

AI1 elements that are not on the diagonal of the submatris are zero:

S i , j - 0.0 for i zj .

On the diagonal.

r l S ( r , ~ ) = - 8 , , , [ / 3 , + ~ > ~ , ~ , ] f o r A ? - ( m ~ ) ~ > O ,

and AS(i , j ) = 05,, [P,,, + ( ;2mym] for A: - ( t r z ~ ) ~ < 0 .

b. Slope along 6 = i

CM;( j. q) x.-

= E,,,,B,,,(ma) [ sinh pnl(l- p ) + C,, , sin y,( 1 - p ) ] cos(mn ) "C 1 ml

In espanded Ibrm. this bècomes

and .-IS(1,j) = O,,,, (rtrn) cos(»rn) [er l + (',,,er3] for i: - (mn)' < O , (2-67b)

c. Slope along the q = O edge

Elcments off the diagonal of the submatris are zero:

4 . 1 O forr#.j. ( 2 - 6 t h )

The clcments on the diagonal are

AS(,,.,) = - O ,ln, [p,,,coshp, + (~lmy,cosr,] for i2 - ( n m ) ' > 0 , (2-68b)

and .LS(r,j) = - 0 1 2 m [ p m ~ ~ ~ h p , + < ; 2 m ï m ~ ~ ~ h 7 , ] for >? - (r)rn) ' < 0 . ( 2 - 6 8 ~ )

d. Slope along the 6 = O edge A -

= 1 L ' , , , , ~~ ,~ ( rn i i ) [ s inh ~ ( 1 - q ) + C;,,,, sini,,,( l - 0) ]

In espanded for, this then becomes

ilS(i, j ) = Q1,, (nzrr) [ a l + c5 UZ] for A' - ( m n ) ' > 0,

and AS(,, j ) = B52m (mir) [ u l + c,,, cx3] for j? - (ntx)' < O,

e. Slope along q = 1 - 5 Thc slopc dong any obliquc line is

r

+ L n 1 & , i n w [sinh P.,-'+ ( 5 2 n l ~inh;~, ,~~]cos(nin, ' nrz A - * l

for this building block and this line. After the expansion i s pcrformcd, the elements of

the subrnairis can be found from the followinç:

- y , sinh y,,, il6 =

c (mn + n?r)2

a. Moment along the q = 1,c = 1 , q = O and 6 = O edges

There is no contribution by Building Block 5 to the bending moment dong the cdges

under considcration: rlSIi,.~) O.

b. Moment along q = 1 - 5

Z

- ~ , , 0 , , , ( m a ) ~ [sinh&,< + (;-,,,sinhYm~] sin (rnn5) m=k'+l

c%; ( g, ?7) zg:

X '

= E,B5,, [&, sinh&{ - c 5 , , y ~ sin Y sin ( m e ) 1 - m=I

L

= - E,,,B,,,,(nia)' [sinhp,J + sin sin ï,

Upon espansion, the elernents of the subrnatris can be calculated from

The cxprcssions for erl to c-1-6 are identical to those for the displacement of Building

Block 5 dong q - 1 - t. found in section 8.5.1 .c. The remaining expressions are: (mr + n x ) [coshp, - cos(rnx + nir) ]

!'I = p: + (mn + nn)"

(mn + nn) [cosï, - cos(mn + nn)] y3 =

' 7

(mn + nn)' - y ;

(m n - t m ) [cos - cos( nt n - ~ I K >1 1 4 =

y ; - (nlr - M)'

(mn + n n ) [coshym - cos(rnx + nn)] 115 =

y: + (mn + na)'

O , = cos'a + v s i n 2 a 8: = s in ' a + ifcos' a 8; = ( I - 1.) s in 2.a

a. Vertical Edge Reaction along the q = 1 edge

Only the rlemenis o n thc diagonal of the submatris are non-zero:

A.S('i,j) O for i # j.

.AIS( 1.1) = - [ A I X + ( 7 - v )# 'A~ ;v ] .

b. Vertical Edge Reaction along the 6 = 1 edge

Thc subrnatrix eicments are now defined by

The cspressions cxi, cr2 and CT.; are identical to those for the slope contribution of

Building Block 5 along the < = I edge, found in section 8.5.I.b.

c. Vertical edge reaction along the q = O edge

d. Vertical Edge Reaction along the 5 = O edge

m-k + I

The submatrix elcments are defined by

The expressions ~ p - t - 1 . L K ~ and LLTZ arc id