Optimal Design of Stiffened Plates

Ravi Shankar Bellur Ramaswamy

August 1999

A thesis siibniitted in conformity with the requirements for the degree

of Master of AppIied Science.

Graduate Department of Aerospace Science and Engineering,

University of Toronto

@Ravi Bellur Ramastvamv. 1999

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Optimal Design of Stiffened Plates Ravi S. Bellur Ramaswaniy

Degree of Master of Applied Science (August 1999)

Institute for ilerospace Studies, Unaiversitg of Toronto

4925 , Cufferin Street. Toronto, Canada hf3H 5T6

Abstract

-4 desigri nietliotloloçy for the optirnization of stiffencd plates with frequency and

biickling constrairits is prescnted. The basic idea of the rnethodology is to consider a plate

witli a fairly dense1 distribution of stiffeners. Tliickness of the plate aiid stiffeners, and the

stiffencr nidtli arc the design variables. Dcsign variable linking is accomplishcd by the use

of r a t i~na l spline surfaces. The finite element method is used for the iinülysis. The plate

is riiodelecl using iiriear SIindlin plate elemcnts and the stiffcncrs by liriear Timostienko

hcarii elemmts. Botli the plate and beani clements arc shear-locking free by forniulation:

\vit lioiit requiring ariy special techniques such as reduced integration. Results for a square

stiffcnecl plate witli three different stiffener Iayout patterns and differeiit stiffcner densityl

arc presented. The best four stiffener coiifigurations which gice the lowest mass are cliosen

and applied to ?:1 and 3:l rectangular plates. It is concluded that the preserit design

iiiethoclology giws good results. and that the stiffener pattern and stiffener density play

ari iniportant role in reducing the mass of a stiffeneci plate.

'Refers to the measure of the number of stiffeners and there spacings and NOT to the density of the

stiffener material.

Dedicated to my beloved parents and sister, whose love, support, and encouragement,

words caiinot describe.

Acknowledgement s

1 an1 clceply indebted to n- Supervisor Prof. J. S. Hansen for Iiis encouragernent:advice,

guidance. aiid financial support throughoiit the course of this work. witliout whicti n q

desirc to piirsiie gradiiatc stiidics would have remaincd just a dream.

1 would like to t h m k 11s. Nora Burnett, and 11s. Elaine Grariatstein for their assis-

taricc in gctting library material.

1 tliarik Dr. Donatiis Ogiianiariani, a good friend of mine. for Iielping nie to get started

wi t h the fini te elenient prograniming.

Sly hierici Guillaume Renaud never hesitated to answer any of ni- questions. We Iiad

many friiitful discussions on the present work. 1 enjoyed his conipany a t work. play aiid

other outdoor activities. 1 am thankful to hini.

Special thanlis to Saniir Hamdi for making my job of plotting easier by giving sorne

lidpfiil software hints. 1 sprnt some wonderfiil moments in his compaiiy. which 1 will always

rcnicmbcr.

Getting settled in a nem country and culture is not an easy task. But, this esperi-

ence mas made pleasant by niy friends Fred Wong: Patton Chan, Chris Young, and those

mentioned before. 1 thank al1 of them for their friendship.

Finally, I am grateful to al1 my parents: grandparents, uncles, and aunts for their

encouragement and initial financial support.

Contents

Abstract i i i

Acknowledgements iv

1 Introduction 1

1.1 Literaturc Siirvey and 1Iotivation . . . . . . . . . . . . . . . . . . . . . . . 3 .

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outli~ic of tiic thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Finite Element Mode1 4

2.1 Stiffener Layout Pattcrn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . 2.2 The Plate Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

2.2.1 Element Matrices for the Vibration Problern . . . . . . . . . . . . . 9

2.2.2 Eleinent SIatrices for the Buckling Problem . . . . . . . . . . . . . 12

'2.3 The Beam Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 EIement Matrices for the Vibration Probleni . . . . . . . . . . . . . 17

2.3.2 Element Matrices for the Buckling Problem . . . . . . . . . . . . . 20

2.3.3 Rotation Natris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Global Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 VibrationFroblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.3 Pre-buckling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Buckling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Validation of the Finite Element Code 25

. - 3.1 Convergence stildy for the Plate Elenient . . . . . . . . . . . . . . . . . . . 23

3.1.1 \..ibration Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Buckliiig Problern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Corivergence stiidy for the Beam EIement . . . . . . . . . . . . . . . . . . . 26

3.2.1 Vibration Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 . 2 . 2 Biickling Probleni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Gericral Coninients About the Results . . . . . . . . . . . . . . . . . . . . 27

4 The Optimization Mode1 34

-4.1 Design \.ariablcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

-4.2 Linking Design Cariables with Splirie Surfaces . . . . . . . . . . . . . . . . 35

4.3 Optimizatioii Equations of Present Work . . . . . . . . . . . . . . . . . . . 36

4.4 The Optirnization Routine Csed . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Optimal Designs 38

5 . 1 Stiffened Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.1 Frequcncy Constraint Applied on the Plate with SPDIA Stiffeners . 39

5.1.3 Frequency and Buckling Constraints Applied on the Plate with SP-

DIA Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.3 Plates wit.h Other Stiffener Patterns . . . . . . . . . . . . . . . . . 52

5.2 Rectangular Stiffened Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Concluding Remarks on the Results . . . . . . . . . . . . . . . . . . . . . . 67

6 Summary and Conclusions 86

6.1 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 87

References 88

List of Tables

3.1 Cornparison of results: plate vibration . . . . . . . . . . . . . . . . . . . . . 29

3.2 Coriiparisoii of resul ts: plate bucklirig . . . . . . . . . . . . . . . . . . . . . 99

3.3 Comparison of results: beam vibration . . . . . . . . . . . . . . . . . . . . . 29

3.4 Comparison of results: beani buckling . . . . . . . . . . . . . . . . . . . . . 99

. . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pruperties of the refererice plate 35

. . . . . . . . . . . . 5 .2 Initial design for the case of frcquency constraint alone 39

. . . . . . . . . . 5.3 Optimiini dcsign for the case of frequency constraint alone 41

. . . 5.4 Initial design For square plate mith frequericy and buckling constraints 42 - * . . . . . 3.3 Optimum design for the case of buckliiig and frequency constraints 43

5.6 >lctliocls for rcducing weight fiirtlier: Initial Design . . . . . . . . . . . . . 47 - - a . i .\ lethods for reducing weight further: Optimal Design . . . . . . . . . . . . 47

5.8 Square plate with stiffener patterns SPSQR and SPDIASQ: Initial Designs 54

5.9 Square plate with stiffener patterns SPSQR and SPDIASQ: Optimal Designs 54

5.10 Optimal masses obtained by various methods . . . . . . . . . . . . . . . . 63

5.1 l Rectangular Plate (?: 1 ratio): Initial Desigris . . . . . . . . . . . . . . . . . 68

5.12 Rectangular Plate (21 ratio): Optimal Desigris . . . . . . . . . . . . . . . 68

-- 5-13 Rectangular Plate (3: 1 ratio): Initial Designs . . . . . . . . . . . . . . . . . I (

1 ... 5.14 Rectangulûr Plate (3:l ratio): Optimal Designs . . . . . . . . . . . . . . . i i

vii

List of Figures

2.1 .A plate witli qinnietric stiffeners layed out in one of tlic possibie patterns.

1.2 A part of the plate shown with finite elcment discrctizatioii. . . . . . . . .

2.3 Thc various stiffencr patterns and there rcference narnes. . . . . . . . . . .

2.4 Dcfining the Stiffener Density Factor . . . . . . . . . . . . . . . . . . . . .

2.5 Sclieinatic of the triangiilar plate element used. . . . . . . . . . . . . . . .

2.6 Sclienlatic of the hearn elcment used. . . . . . . . . . . . . . . . . . . . . .

'2.1 Approsirtiat ion in st ifférier cross-section . . . . . . . . . . . . . . . . . . . .

3.1 Conwrgcnce plot for the plate-elernent: sirnply-supported vibrat,ing plate.

3.2 Convergence plot for the plate-element: simply-supported plate subjected

to bucklirig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Convergence plot for the beam-element : simply-supported vibrating beam.

3.4 Convergence plot for the beam-element: simply-supported beam subjected

to buckhig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The design variables at a cross-section. . . . . . . . . . . . . . . . . . . .

5.1 Envelope surface of t for the stiffeners . . . . . . . . . . . . . . . . . . . . .

5.2 Square plate mith SPDIA: Spline surface for t . . . . . . . . . . . . . . . .

5.3 Square plate with SPDIA: Spline surface for t p . . . . . . . . . . . . . . .

- 4 Square plate with SPDIA, and SPS niethod applied: Spline surface for t . . - - 3.3 Square plate with SPDIA, reduced SDF: Spline surface for t . . . . . . . .

S.6 Square plate with SPDIA7 reduced SDF and SPS method applied: Spline

surface for t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

- - . Square plate with SPDIA, reduced SDF and SPS niethod applied: Spline

surface for t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.8 Case 1 [Square plate with SPSQR (SDF = 1/2)]: Spline surfacc for t . . . . 53

.5.9 Cnse 2 [Square plate with SPSQR (SDF = 112) and SPS]: Spline surface for t . 55

5.10 Case 3 [Square plate with SPDIASQ (SDF = 1/2)]: Spliiie surface for t . . . 56

5.11 Casc 3 [Square plate with SPDIASQ (SDF = II%)]: Spline surface for t p . . 57

5.17 Case 4 [Square plate with SPDIASQ (SDF = 112) and SPS]: Spline surface

for t for the diagonal stiffeners. . . . . . . . . . . . . . . . . . . . . . . . . 58

5.13 Casc 4 [Square plate with SPDIASQ (SDF = 112) and SPS]: Spline surface

for t for tlic horizontal and vcrtical stiffeners. . . . . . . . . . . . . . . . . . 59

5 . l-l Casc 5 [Square plate with SPDIASQ (SDF = 1/4)]: Splinc siirface for t . . . 60

5.15 Case 5 [Square plate with SPDIASQ (SDF = 1/4)]: Spline siirface for t p . . 61

5-16 Cnse 6 [Sqiiarc plate witli SPDIASQ (SDF = 1 1 4 , and SPS]: Splirie surfacc

for t for the diagonal stiffeners. . . . . . . . . . . . . . . . . . . . . . . . . 62

3-17 Case 6 [Square plate with SPDIASQ (SDF = l /4) and SPS]: Splinc surface

for t for the horizontal and vertical stiffeners. . . . . . . . . . . . . . . . . . 63

5-18 Case 6 [Square plate with SPDIASQ (SDF = 114) and SPS]: Spline surface

fort,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.19 Rectangular plate (ratio 2 1 ) with SPDIA (SDF = 11-1): Spline surface for t . 67

5-20 Rectangular plate (ratio 21) with SPDIA (SDF = 114): Spline surface for t p . 69

5.31 Re~t~angular plate (ratio 21) with SPDIA (SDF = 11-1) and SPS: Spline

surface for t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.22 Rectangular plate (ratio 2 1 ) with SPDIA (SDF = 114) and SPS: Spline

surface for t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.23 Rectangular plate (ratio 2 1 ) with SPSQR (SDF = 112): Spline surface for t. 72

5.24 Rectangular plate (ratio 21) with SPSQR (SDF = 113): Spline surface for

t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.25 Rectangular plate (ratio 21) with SPSQR (SDF = 112) and SPS: Spline

surface for t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.26 Rectangular plate (ratio 21) with SPSQR (SDF = 112) and SPS: Spline

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface for t p .

5.27 Rectangular plate (ratio 2:1) with SPSQR (SDF = 112) and SPS: buckling

mode shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 . 2 s Rectaligular plate (ratio 3:l) witli SPDIA (SDF = 114): Splinc surface for t .

5.29 Rectangular plate (ratio 3:l) with SPDIA (SDF = 114): Spline surface for t p .

5.30 Rectangiilar plate (ratio 3: l ) with SPDIA (SDF = 1/4) and SPS: Splinc

siirfacc for t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.31 Rectangular plate (ratio 3 1 ) with SPDIA (SDF = l /4) and SPS: Spline

surface for t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.32 Rcctangular plate (ratio 3:l) with SPSQR (SDF = 112): Spline surface for t .

5.33 Rectangular plate (ratio 3:1) with SPSQR (SDF = 113): Spline surface for

t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.34 Rcctangular plate (ratio 3 1 ) with SPSQR (SDF = 112) and SPS: Spline

surface for t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.35 Rectangular plate (ratio 3:l) with SPSQR (SDF = 112) end SPS: Spline

surface for t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction

hlan has always bceii inspired by nature. be it art, or engineering. Perhaps one of the

dcrivat ivcs of siicli inspirat ion is stiffened engineering structures.

Sca shells. leaves. trecs. vegetables - aII of these are: in fact. stiffened structures. Ob-

serlatioiis of structures created by naturc indicate that in most cases strengt.h and rigidit~.

cIc?pciitl iiot orily or1 the niaterial but also iipon its form. This fact \vas probably noticed

long ago by soiiie slirwcl observers and resulted in the creatiori of artificial structural ele-

merits having high bearing capacity niainly due to their forni. suc11 as. girders, arches and

shells [IO. Chapter 11.

Tlie wide use of stiffened structural elernents in engineering begari in the nineteenth

century: mainly with the application of stccl plates for the hulls of ships and with dcvcl-

opment of steel bridges and aircraft structures. Stiffened plates noir find applications in

modern industry [ibid.].

Stiffeners in a stifkned plate make it possible to resist highly directional loads, and in-

troduce rnultiple load paths that may provide protection against damage and crack growth

under bot*li compressive and tensile loads. The biggest advantage of the stiffeners. tliough,

is the increased bending stiffness of the panel with a minimum of additional material, which

makes t hese structures highly desirable for out-of-plane loads and destabilizing compressive

loads [S. page 4511.

In addition to the advantages already found in using them, there should be no doubt

that stiffened plates designed with optimization techniques will be bring many benefits like

swings in material usage, cost, better performance, etc.

1.1 Literature Survey and Motivation

The riiotivation beliind the work presented here is the papcr on plate optimization by

Cheng and Ollioff [JI. They were trying to solve a plate sizing optiniizatioti. with the

plate being ~iiodeled by Kirchoff's theory. The objective was to niiniriiize the complianre.

They observed the formation of stiffeners in nuinerically cornputed 'optimal' solutions. The

nurnber of stiffeners increased when the discretization of the design ivas increased. with

the resulting decreasc in cornpliance. I t was suggested. that either the problern be relaxed.

or thc dcsign space be restrictcd.

For a relasation of the design problem. plates with irifinitcly niany, infinitell thiri

stiffcners have to be considered [3. page 2071. The bibliography in [ibid.. page 2331 @\-es

a good account of tlic research carried out in this regard. But niany of those works

considered liomogcnizatioii tecliriiques to mode1 a stiffencd plate. wliicli iriay prodiicr some

crrors whm the dcnsity of stiffeners is not so high. The present work addresses this probleni

by modeling the stiffeners as beani elements iising the finite elenient teclinique.

Ariother iriotivating factor for the present research, is the enormous improvenient in

the inanufacttiring proccsses. Almost any sliape can be rnanufacturcd using computerized

niachining. So? a plate witli unusual thickness profile can be made by direct machining! or

nioi~lds can bc made to makc high precision castings.

1.2 Objective

To develop a design rrzethodology t o op t imize the mass of stif fened plates with frequency

and buckling constraints.

To be more specific, a plate with stiffeners Iayed out in a certain pattern should be

considered. Sonie common stiffener patterns should be considered, and evaluated for their

ability to give optimal masses. The thickness of the plate and stiffeners, and the width

of the stiffeners should be considered as design variables. Using spline approximation

techniques. the design variables should be linked. Analysis should be carried out using

finite element method. Appropriate finite elements should be derived.

The optimization constraints should be in the forrn of some lower bounds on the first

nat ural frequency and the cri tical buckling Ioad.

1.3 Outline of the thesis

In Cliapter 2 the finite elenient model is described and al1 the nccessary equntions for

the plate and the l~eam elements are derived. Also, the stiffener patterns consiclcred are

introdilced.

Ihlidation of the beam and platc elements, each separately for buckling as well as

vibration, is s h o w in Chapter 3.

Cliaptcr 4 is devoted to esplaining the optimization model. Defiiiition of the design

variables of the problerii, linking them to reduce the riumber of design variables. and t h

optiniiznt ion cquations arc al1 found hcre.

Resiilts, aiid disciissions are given in Chapter 5 . Here the resiilts for a square platc

n i t li various st i ffener configurations are presented. Those configurations wliicli give the

best rcsults arc applicd to rcctangiilar plates and their results are showri.

The prcscnt work is surnmarized and concluded in the Chapter 6.

Chapter 2

The Finite Element Mode1

It was nlcntioned in Chapter 1 that the main motivation behind t h ~ o r k is the observation

of stiffcncr Formiition - Cheng and Olhoff [A]. So, the idea is to begiii the optimization

proccss II! assiiniing a fiiirly dense distribution of stiffeners with a certain pattern - for

csample Figure 2.1 shows one of several possible patterns - and discretized irito firiite

clcnicnts as in Figure 2.2.

Figure 2.1: -4 plate with symmetric stiffeners layed out in one of the possible patterns.

CHAPTER 2. THE FINTE ELEAIENT MODEL - a

Both. the plate and the beam elements, are shear-locking free by forriiulation. based

on the ideas presented in (71. -4s depicted in the inset in Figure 2.2, tlie elcmerits can

take differcnt thickness at the cnd nodes. in case of the beam elcrncnt.

i r i case of tlie plat .c element. -?

and corner nodes,

Figure 2 .2 : .A part of the plate shown with finite elenicnt discrctization.

2.1 Stiffener Layout Pattern

Ctivcri the way the plate is discretized in the present work? that is, dividing the plate into

nian- squares. dkiding eacli square into four triangular elements, as shown in Figure 2.2.

the niimber of options for stiffener layout pattern is srnall. Sonie symmetric patterns have

been chosen. and are sliomn in Figure 2.3, dong with there reference names.

It is qui te obvious that the stiffeners can be arranged wit h clifferent spacings. Hence

comes into picture the kiensity' of stiffeners. A quantity called Stifener Densitg Factor

(SDF) can be defined for the purpose of laying out the stiffeners with different spacings.

Consider the smalIest slant stiffener running between two adjacent sides of a square plate.

For esample. in Figure 2.4, BD. Let the area of the square formed !y this diagonal

be .AbTn (area ABCD in the figure). Let the area of the smallest square formed by four

triangular elements be .Apr. Again referring to Figure 2.4, it is area AEFG. Then.

Apt S B F =

C H U T E R 2. THE FINITE ELEhlENT MODEL

SPDIA SPSQR

SPDIASQ

Figure 2.3: The various stiffener patterns and there reference names.

To define SDF for SPSQR and SPDIXSQ one must consider the sniallest square formed

b the stiffeners for area Ah, and Apt remains the same as defined above.

The value of SDF cannot be any imaginable value for a given discretized plate. For

esample. the FE mode1 of SPDIA stiffened plate illustrated in Figure 2.4, the eshaustive

values of SDF are 1/4: 112 and 1. For SDF = 11.1, there are only two stiffeners along the

two diagonals of the largest square, which is the plate itseif. For SDF = 1, al1 the diagonals

of the smalIest possible squares have stiffeners.

It should be noted that the angle of the slant stiffeners SPDIA is f Go, unless oth-

erwise stated.

Figure 2.4: Defining the Stiffencr Density Fact.or

2.2 The Plate Element

The geoiiictry of the elcmcnt is given by the Cartesian coordinatcs. (.z,. y i ) . and the thick-

nesscs of the plate. t p , , (i=1,3.5).

In order to dcrive the equations. consider the plate element shown in Figure 2.5.

The plate element is basecl on Reissner-Mindlin Plate theory. Thereforc, the in-plane

displacernent fields, f i and 0 , are assurned to Vary linearly through the plate thickness and

the trariswrse displacenient. tü! is assunied constant through the plate thickness. The

kinerriatic relation is given by

whcre u ( x l y, t ) and v(x'y, t ) are mid-surface displacements, &(x, y: t ) and ~ J , ( x : y! t) are

rotation-like variables, w(z : y, t ) is the transverse displacement, and t is time. The linear

strain. is given by

5 r l r O -- Nodes for u. v. V, and y,

a -- Nodes for w 1 3

3 7

1 1 b -

X - 3 3

Global CO-ordinates Natuml CO-ordinates

Figure 2.5: Scherriatic of the triangular plate elcrncnt used.

Tlie conipoiicnts of the non-liiiear stmin vector. I r x } , wliicii are necessüry for tlie

bucklirig probleiii. are.

nliere the ü., is the partial differentiation of ïï with respect to 2: etc. Using Eqs. 2.2 in

Eqs. 2.4.

The displacements within the eleiiient are given by

{6}' = [ u o w Q~ d~~ ]

The nodal-displacemerit vector is

By rnapping froni global to natiiral coordinates,

I d } = [.VptlIqpt} (2.9)

wiirrc [:bt] is 5slS matris of interpolation functions. The interpolation fiiiictioiis for

2.2.1 Element Matrices for the Vibration Problem

The total energy of the element lias contributions from the

kinet ic cnergy. K e . Consider firs t:

ue =

strain energy, Ue. and the

If E is Young's rnodulus. G is the Shear modulus: and v is the Poisson's ratio, then,

E Qii = Q22 = (1 - v2)

CHAPTER 2. THE FIXITE ELEAfE-VT MODEL

thcn Eq. 2.12 cari be written as.

t p = Ei=1,3,5 .\y(<. q ) t p i . is the thickness of the plate at any point (c, T I ) . Now,

- " O 0 0 0 az 0 3 0 a9 0 O - " 0 0 0 a3 ZG o o o g o 0 0 0 0 $

a o o o g , o o g i o o o g 0 1

CH.4PTER 2. THE F I N T E ELEhlENT MODEL 11

The vector {e} can be approsimated by substituting for ( 6 ) from Eq. 2.9. Tlien

{el = [ B L ~ p c l { q p t }

whcrc.

[B",,j = [ L ] [ - q

Siibstituting Eq. 2.17. irito Eq. 2.14.

Tlie 18 x 18 niatris. [I<,,]. is tlie element st,iffness matris in vibration.

Considcr now the kinetic e n c r e term.

where p is the clensity of the material of the plate. Using Eq. 2.9 in Eq. 2.22,

tv here

and

The 18 x 18 matris [AIpt] is the plate element mass matris.

- p 0 0 - 2 O

O P O O - 2

O O p O O

- - O O pz2 O

O - 2 0 O p z 2 -

(2.23)

d:=

C - ptp O O O O

O ptp O O O

O O p t p O O

O O O O

L o O O O p % -

CH.4PTER 2. THE F I N T E ELEIZIELVT iZ,IODEL

2.2.2 Element Matrices for the Buckling Problem

The buckling problem is solved in two steps: the Pre-Bucklirig Probleni and the Buckling

Pro blcm.

The Pre-Buckling Problem

The loading state in the pre-buckling problem is assumed to result from the application of

a uniforni displacenient on the plate boiindary of interest. The following assurnptions are

made:

1. The applietl load results from a uniforin displacement applied t,o tlic loaded edge.

2. The application of the boundary conditions does not induce any mechanical loads.

3. Ttic problem is linear.

The eriergy witiiin an element is only due to strain. €0: because of the initial displace-

trient. and is giwn by.

where tlic prc-bucklirig strcss is given bu.

{ao) = [QI{&}

I t should be noted that the applied load results froni the specified boundary condi-

tions. and so there is no esplicit work term due to esternal forces.

By carrying out algebraic manipulations on Eq. 2.26 as in Section 2.2.1, one

ultiniately gets.

where

and

The 18 x 18 matris, [KpbPt]? is the pre-buckhg element stiffness rnatrix.

The Buckling Problem

This forniulation niakes use of the non-linear strains in Eqs. 2.5 and Eqs. 2.6. In the

dcriixtions tiiat follows it is assumed that the buckling stress at any point in the platc is

given by,

\vlirrc~ {oo} is the pre-buckling stress froin equation Eq. 2.27. Now. the strain eriergy of

t hc plat c elerntmt .

Scglecting higlier order ternis and considering the fact that {ao} rcpresrrits a state of

cqiiilibriiini. Eq. 2.32 beconics.

Sow. consider thc first terni on the R. H. S of Eq. 2.33. By followirig algebraic rnariipii-

lations sirililas to thosc shown in Section 2.2.1.

and

The 18 x 18 matris, [h'b'lLrkpt]i is the buckling plate-element stiffness inatris.

Consider nest the second term on the R. H. S. of Eq. 2.33. One can write

CHAPTER 2. THE FINITE ELEhlENT XIODEL 14

- - 0 0 O O L,, L, O O O l x l O O O O L, 0 0 - y JI, O O O O L,,

v : : O O O O O O O O O v : v O O O O O

M p / 2 -v~, = - / aod: ij = -2. -y

- t p / 2

JIiJ = - ij = xx. yy . xy

- a O O O O a~ O & O O O

o a o 0 o a,

o o g 3 0

o o $ o o 0 0 0 ~ 0

o o o g o o o o o & o o o o $ 0 0 0 1 0

0 0 0 0 1

Thc vector. {ec}. can be approsiniated by substituting for {d} from Eq. 2.9. Therefore,

{ e d = [ B ~ p t ] k p t 1 (2.41)

[ B ~ p t ] = [Lc] [-vpt]

Substituting Eq. 2.41. iiito Eq. 2.37,

The 18 x 18 matris. [kpt] is the plate-elcment gcometric stiffness niatris.

2.3 The Beam Element

Sow. t h e beani elcinent as shown in Figure 2.6. is consiclered. The geomctry of thc

eleriient is g i ~ r i i the Cartesiaii coordinates. (x,. y,). and the thicknesscs of the plate. hi :

( i = 1.3).

9 -- Nodes for u. v. %and%

a -- Nodes for w

1 2 3

Bsam Local Co-ordinates Beam Natural Co-ordinates

Figure 2.6: Schematic of the beam element used.

The beam element is based on the Tirnoshenko Beam Theory. The kinematic relation

is given by

n(x, 2, t ) = u(x, t ) - z&(x, t )

ü(x, 2, t ) = v(x: t ) - z q , (x, t )

w(x; y, ;, t ) = w(x. t ) + y ~ ~ ( x : t )

CHAPTER 2. THE FINITE ELEMEVT XIODEL 16

mlierc u ( r . t ) and ~ ( x , t ) are mid-surface displacements &(x. t ) and c?;(x. t ) are rotation-

likc \-ariahles and I L ( X . t ) is the transverse displacement. The components of tlic in-plane

iincar straiii vector. (tLbrn}. are given by

The componerits of the non-lincar strain \-ector.{~,~~,}? ah ich are uscful for the buckling

prol~lem, arc.

Ksiiig Eqs. 2.45 in Eqs. 2.47.

The displacements within the element are given by {c f } in Eq. 2.7, and the nodal-displace-

ment vector is

BJ- mapping from global to natural coordinates:

where [.V,,] is 5x1 1 matris of interpolation functions.

The interpolation functions used for w are:

- = ( - 1) .v2 = 1 - c2 LV3 = i<(< .. + 1)

arid the iriterpolatiori functions used for u, P . e, and 0, are:

= -v; 5 - 2

Approximation of the Cross-section of the Beam

Section A-A .4prox. of

Section A-A

b

Figure 2.7: (a) .A stiffener (dashed lines) superimposed along orle of tlic &es of two plate

eleinents. ( b ) -4 section of the stiffener. (c) The same section witli the approximation of

corist ant plate t hick~iess.

Iritegration aloiig tlie thickness of the stifferier becomes easier, if the cross-section of tlie

stifferier is approsimatecl as shown in Figure 2.7(c). I t is assurned that the plate thickness

is constant at a giveii stiffener cross-section, but can Vary along the stiffener length. This

approsimation should not introduce any significant errors as the optimization process tends

to make the width, and the plate thickness as small as possible and the stiffener thickness

as Iiigh as possible.

2.3.1 Element Matrices for the Vibration Problem

The total energy of the element has contributions from strain energy, libme? and kinetic

energy, Kbme . Consider first,

CH.4PTER 2. THE FINITE ELELIIENT iIIODEL

If the vector t)bfn is dcfined as.

{ e h } * = [ UV, yrqr ( w , - ) d!y,r ]

tlicn Eq. 2.53 written as.

aiid tlir elements of [Dl,,,] aftcr integrating with respect to y and z . are:

whcrc1 b is the widt 11. and h = .vi(<) h,, is the t liickness of the stiffeiier. and tp =

Z,=i,s .v:(<)tp,. is tlie tliickness of the plate at any point:(<)o d o n g thc lengtli of the beani

The vector. {ebrn}. c m be approsimated by substitutirig for 16) from Eq. 2.50:

{ebm } = [Bubm] { q b r n ) (2.58)

where

Substitiiting Eq. 2.58. into Eq. 2.55.

Thc 11 x 11 i i iatris. [I<l,bnl]. is the beain element stiffness niatris in vibration.

Considcr now the kinetic energy ternl,

wl1c.r~ p is the deiisity of the niaterial of the plate. Csing Eq. 2.50 iii Eq. 2.63.

The 11 x 11 matrix [ A L ] is the beam element mass matrk.

2.3.2 Element Matrices for the Buckiing Problem

The Pre-Buckling Problem

Here again. because of the strain energy due to the initial displacement (Section 2.2.2). a

11 x 11 prp-biickhiig stiffricss matris [IGbbm] is obtained . The pre-biickling stress is giwn

b y.

The Buckling Problem

Tliis forniiilation ninkcs use of the non-lincar strains in Eqs. 2.48. The biickling stress a t

aiiy point iri tlic stiffc~ier is giwn by.

;\s iii Section 2.2.2. neglecting higtier order ternis and considwing tlic fact that {abrno} is thc stress a t the state of cquilibrium. the strain criergy,

Siniilar to Section 2.2.1. frorn the first term of Eq. 2.68. finally oric obtains the 11 x 11

stiffness niatris. [f<buekbm]e

Corisider. now, the seco~id terni or1 the R. W. S. of Eq. 2.33.

IV here

P x 0 0 Qz O Pxy O

0 Pz O O Q, O Pz, O O P, O O O Px, & , O O R x O O O

O Q X O o s x o O

P z , O O O 0 O O 0 Pz, Px, O O O Py

CHAPTER 2. THE FINITE ELEAIENT ,IIODEL

- " 0 0 0 ax

0 ~ 0 0 0

0 0 ~ 0 0

o o o g o o o o o g O 0 0 1 0

O O O O 1

Tlic wctor. {cos, }. caii be approsimated bu substituting for {d} froni Eq. 2.50:

{ e ~ b r n ) = [ B ~ b r n ] { q b m } (2.73)

IV here

Siibstituting Eq. 2.73. into Eq. 2.69,

wliere

The 11 x 11 matris, [ I L ] ; is the beam-element geometric stiffness matrix.

CH.4PTER 2. THE FILVITE ELEhlEiVT ilIODEL 22

2.3.3 Rotation Matrix

To allow the beam elemcnts to be placed at different angles each of the element matrices

drrivcd for the beam element above should be post-multiplied by the rotation matris [RI

and prrrnultiplied by its transpose. For a beam element (Figure 2.6) with end points

(x l . ,yl ) and (x3. M) the rotation matris is giren bu.

s = sin 8 , and

Global Equations

2.4.1 Vibration Problern

Ensuring continuity between the nodes that describe bot h beam and plate eleinents! the

elenient energies are added up. The total kinetic energy! Tt and the potential energy. Ci.

of the whole plate are!

where [AI] is the global mass matrix, is the global stiffness matris, and {Q} is the

global displacement vector. By Hamilton's principle [8, pp 323-3261' a stationary value

CH.4PTER 2. THE FLVITE ELEAIE.VT ;\IODEL 23

of the Lagrangian [LI = [Tl - [L;] is sougbt. .\fter applying the pririciples of variational

calïuliis. the basic matr is eqiiat ion for the free vibration of the structure wit hout damping

is obtained:

[ J I ] { Q } + [ I i ] { Q } = O

A soliitiori for {Q} is assiimed in the forni

( 2 . SO)

{ Q } = {Q}eiUt (2.81)

tvlicre t is timc. is t h natural frequency and {Q)? the modal wctor. is a set of constant

valiirs a t the nodrs. .At the ciid of matlicniatical manipulatioris and application of boundary

coridi tioiis t htl standard eigenproblem

whcre [lit] and [.\I1] are the posi tim-defiiiitc global stiffncss and niass matrices. respectit-ely.

Subspace Iteration llethod is used to solve the eigenprobleni [ l ] .

2.4.2 Pre-buckling Problem

Tlic total putential energ! in this case is.

Taking the variation w.r. t . {Q}, aiid applying the displacement boundary conditions gives

rise to the firial equatiori,

tvherc [fit] is the positive-definite stiffness matris and {F} is the force vector obtained as

result of applying boundary conditions.

2.4.3 Buckling Problem

Total potential energy,

wliere [Iic] is the global geonietric stiffness rnatris. The final equation. after usual niath-

eniatical manipulations and application of boundary conditions. is

wtirrc X is tlic buckling factor. and [KI] and [A',!-] are, respectively. the positive-definite

stiffness and geometric stiffriess niatrices after the boundary conditions are applied. Again.

Subspacc Itcration .\let hod is used [l] to solve the eigenproblem.

Chapter 3

Validation of the Finite Element

Code

In this cliapter. tlic brani ancl the plate elcments are validated for the vibration. ancl

biirkling problcnis.

For al1 coiivcrgeiicc problcnis. the niaterial used is aluniiniiiii. witli the folloiving

properties: 1-oiing's hlodulus. E = 70.3 GPa; Poissori's ratio. u = 0.33: and Dcnsity.

p = 2712.64 kgin3. Sirripl--supporteci boundary conditions are assunicd for al1 thc studies

i r i al1 tlie cases. Sotc that for plate studies al1 the edges of the plate are considered siniply

siipported.

3.1 Convergence study for the Plate Element

The dimensions of the plate considered for both the studies in vibration and the buckling

problem are 2000 x 1000 x -O mm3.

3.1.1 Vibration Problem

The results of the present Finite Element formulation are compared (Table 3.1) ivith the

analytical solution given in [6, page 2511. The analytical angular frequency (rad/s) is,

CHAPTER 3. C:-1LID,UïOiV OF THE FINITE ELEAIEIVT CODE

where. p, is the mass per unit area, a is the length, and b is the widtli.

If h is the tliickness of the plate. then,

The convergence plots are shown in Figure 3.1.

3.1.2 Buckling Problem

.~rialyticiilly. the critical buckling load for a plate (8, pp 434-4351 of dimensions n x b x h

is given bu.

!: tlic plate is given a uniform displacement along one edge parallcl to y -a i s . wliile t h

opposite edge is held fised. the applied load is

Aa P*, = Ebh-

a

where Aa is the displacenient applied. If l a = 1 unit. then,

Hence. the buckling load factor,

Pm. k,r2h2a A,, = - - P., 12(1 - G)b2

Coin-ergence plots for the first four buckling modes can be found in Figure 3.2, and

Table 3.2 shows the comparison with the analytical solution.

3.2 Convergence study for the Beam Element

.A beam of length, 1 - 500 mm, width, b = 10 mm, and thickness, hh = 20 mm, is

considered for both cases of convergence study. The material remains the same as that

used for the plate studies above.

3.2.1 Vibration Problern

The analytical solution for the angular frequency of a beam [S. page 3311.

The convergence plot for the first 4 lowest frequcncies is sliown in Figure 3.3. and

the comparison with analytical solution is in Table 3.3.

3.2.2 Buckling Problem

Ttic analytical criticai buckling load [8, page -1131,

If tlie beani is g i w i a m i t asial displacement at one ciid whilc the othcr end is Iield fiscci,

thc app1it.d load is

Hencc. the buckling loacl factor.

The convergence plot for the first -4 buckling modes is sliown in Figure 3.4, and tlie

comparison a i t h analytical solution is in Table 3.4.

3.3 General Comments About the Results

The converged results for the rectangular plate were obtained by discretizing the plate into

16 x 16 smaller rectangles, and in turn each one divided into -4 identical triangles by the

diagonals. The beam results required 32 elements to give a sufficiently accurate result for

the first modes of vibration and buckling, although the plots for beam convergence were

drawn using up to 6-4 elements.

I t can be observed from the plots the al1 the curves converge monotonically from a

higher value for low DOF (Degrees Of Freedorn), to a lower value a t higher DOF, which

CH.4PTER 3. I:-1LIDA4TIOiV OF THE FINITE ELE:i\IENT CODE

is typical of vibration and buckling problems. This indicates that the plate and beani

clement formulation is correct.

Thc comparison tabies show ttiat for some modes of vibration and buckling, for botli

the plate and beani cases. the FE (Finite Element) results are higher than the analytical

rcsults. This is to be espected given the fact that different theories are being compared.

That is. for the plates. FE forniulation is llindlin. and compared with Classical Plate

theory. ivhereas the Tinioshenko FE forniulatiori for beanis is beirig cornpared witli Euler-

Bernoulli bcam ttieory.

Tihle 3.1: Cornparison of rcsults: plate vibration.

Table 3.2: Comparison of resui ts: plate buckling.

Table 3.3: Cornparison of results: beam vibration.

Table 3.4: Comparison of results: beam buckhg.

m

1

2

3

4

% error

-0.50

-1.97

-4.30

-7.36

Xbrn

AnaIytical

0.6580

2.6319

5.9218

10.5246

FEM

0.6547

2.5800

5.6669

9.7525

Mode (3,l) R

v -

I 1 1 L

A 1 O00 2000 3000 4000

s 1

I 1 1 I l

1 O00 2000 3000 4000 Degrees of Freedom

3000 V) al *- U c 9, 2000 J e

500

Figure 3.1 : Convergence plot for the plate-element : simpl-supported vibrating plate.

-

-

s - Mode (1 , I ) f] . Y U

2 LL

& - 1

a I I I

1 O00 2000 3000 4000

Third Mode -. \i V

Second Mode

First mode

Figure 3.2: Convergence plot for the plate-element: simply-supported plate subjected to

buckling.

- -

A

B 2 Y

VI .!! 5000 -

Second Mode L

A

lu Y A -

3 0 2 4800 1 100 200 300

1

Third Mode

200 300 Degrees of Freedom

10500

Figure 3.3: Convergence plot for the beam-element: simply-supported vibrating beam.

c. v V

-

6.3

6.2

6.1

6

5.9

5.8 Third Mode 5.7 1 7 5.6 i I L

1 O0 200 300

Second Mode A

i

I

0.665 -

First mode 0.655 - n

u

1 I 1

100 200 300 Degrees of Freedom

Figure 3.4: Convergence plot for the beam-element: sirnply-supported beam subjected to

buckling.

Chapter 4

The Optimizat ion Mode1

In the first section the design variables are dcfined. and then the issue of rediicing the

n~iriiber of design variables by design variable linking is discussed in the second section.

T h . optiiiiizatiori cquatioris of the prescnt work arc listed in the tliird section and a bricf

iiotr abolit the optiniization routines used in the computer code is given in the final section.

4.1 Design Variables

Section A-A

Figure 4.1: The design variables a t a cross-section.

Since a stiffened plate is being dealt with, the design variables may corne from the plate

andjor the stiffeners. In t,he present study: the Iength and breadth of the plate are kept

constant: howeverl the plate thickness t p at a certain point is considered as a design

variable. The stiffener width 6 is another design variable, but it is assumed that al1 the

stiffeners have the same width. As mentioned earlier, symmetric stiffeners are assurned, so

the dcpth t of the stiffener projecting out of the surface of the plate is taken as a dcsign

mriablc (sec Figure 4.1).

Diic to the F u t that Finite Element Analysis would be used, tlie plate is discretizcd.

Ideally. it ~ -ou ld be less work if t p , and t at each node are corisidered as design variables.

but the nuniber of design variables would become too large to tiandlc with the a\-ailable

resoiirces. Therefore. resort is niade to what is commonly called .design variable linking'.

4.2 Linking Design Variables with Spline Surfaces

In t h prescrit stiidy two-dinieilsional rational splines [9] havc bcen uscd for tlie purposes

of design variable linking. The follomirig provides an overview.

Let

tlenote the values of a hinction ~ ( x . y) cvaluated at the nodes of a rectangular grid

in the (.r. 9) plane. Then a srnootli surface is souglit in the x-y plane iiiterpolating the

ordinates ( t t j . In order to acliieve this end. in each of the rectangles

the function

is defined. In the above

Also. a function u = u(x , y) is defined on all R sucli that its restriction to R, coincides

with j i j . The 16(n - 1) (m - 1) coefficients ai,ki are determined then in such a way that

u ( x i , y j ) = ' U i j . The parameter p can be adjusted so that the spline fit can be any where

CHAPTER 4. THE OPTIIZIIZ.4TION MODEL 36

h m linear. for p + m! to cubic, for p + O. Interested readers shoiild rcfer [9] for details.

Knless otticrwise stated. p = 10.0 was iised.

Separate spline surfaces were iised for the plate thickness profile: and the envelope

surface fornled by the thickness profile of the stiffeners. The thickncsses tp , , and t , at t lw

intersection points (called Slaster Nodes) of the rectangular grids. Rp. and IIbTn. resper-

tiwly. dong with 6 will be called the design variables..

4.3 Optimization Equations of Present Work

The objective of the work presented is to rninirnize weight subjected to a frequency con-

straint, a bucklirig constraint. and a number of side constraints on the diniensional vari-

ables. llathernatically.

where i l . is the first natural frequency, Pcrhck is the critical buckling load, and NMN is

the numbcr of master nodes. Ci, and C2 are predefined constants.

4.4 The Optimization Routine Used

The routines in the optimization package ADS [E] were used to solve the above problem.

ADS (.\utomated Design Synthesis) Version 1.10 is a collection of FORTRAN subroutines

for the solution of non-linear constrained optimization problems. The program is segmented

into three levels: strategy. optiniizer: and one-dimensional search. k each level, several

options are al-ailable so that a total of over 100 possible combinations can be created. Of

the many possible combinations that were consistent with the problem; one was found to

give a converged feasible result with the least number of function evaluations - the strategy,

CH.4PTER -1. THE OPTIJIIZ.4TION MODEL 37

Seqiiential Linear Programming; the opt iinizer. SIodified Method of Feasible Directions:

and the one-dinierisional search option, Golden Section rnethod folloived by polynoniial

interpolation. -411 t hese are esplained in [I l] .

Chapter 5

Optimal Designs

Optimization rcsults for square and rectangulnr plates arc prcserited in this clinptcr. Tlic

platc matcrial. assiimed isotropic. is aluminum. The properties arc thc snnic as mcntionecl

in page 2 5 . Tlic plate is assunicd to be simply supported in al1 the cascs.

-4s a referencc to sct the constraints. an isotropic 2000nim x 2000rnm x 20inm plate

was corisidcreci. The propertics of t hc plate are given i r i Table 5.1. Baseci on tliis. the lower

bounds on first natural frequency and buckling load w r e set. Sorne important constraints

iniposed on the plate georiictry were: tliickncss of the plate sliould not be lcss than O.5mm:

dcpth of the stiffmers should not escced 25Onim: and the widt h of the stiffener should be

a minimum of lmrri.

Di~nensions

Naterial

11 ass

First Natural Frequency

1 Critical Buckling Load

Table 5.1: Properties of the reference plate

5.1 Stiffened Square Plate

The leiigth and breadth of the plate were maintained at 2000mm each, for al1 the square

plate cases considered.

With a view to gain a gradua1 understanding of the probleni. first a square plate with

orily the frequency constraint !vas considered. Spline interpolation was not iised. instead

liriear interpolation was used. In the nest case, the buckling coristraint was addcd. and

spline interpolation iised. Later various stiffener corifigurations were esplored.

5.1.1 F'requency Constraint Applied on the Plate with SPDIA

St iffeners

Initial Design

The first step in the optimization process is, obviously, choosing an initial design. For the

optimization strategy beirig used in the present study. that is, Sequential Linear Program-

ming, problems may arise if started with an infeasible initial design [S. pp 231:232]. Hcnce.

by trial and error a feasible design was obtained, and is s h o w in Table 5.2. The initial

design is a flüt plate with SPDIA (SDF = 112) stiffeners, al1 having the sarnc depth. and

width. SPDIA Kas chosen to try something other than the conventional SPSQR. wliich

woiilcl be considered later in some other cases. The SDF = 112 was cliosen because. it

produws a rnoderatrly dense stiffener pattern. Note that if the stifferier density is too iiigh.

then that rnay prodiicc a final optirnurn design wliich is heavier than an optimum design

produced by a lower SDF value. But a t the sanie time if the stiffener density is very low

thcri tlic stiffeners n iq- pl- a lesser role in the design.

Notice that the mass of the present stiffened platc is already 91.72% Iower than tlic

refererice plate. reinforcing the Fact that the stiffened plates have better charac teristics,

and at the same tinie have a l o w r mass.

Table 5.2: Initial design for the case of frequency constraint alone.

Dimensions

Number of Design

Vdriables ('iDV)

Mat erial

111 ass

First Yatural Frequency

2OOOmm x 2000mm; 6 = 2.0mm;

ti = 20.0mm; t p i = 0.75; i =1 to NDV

83

Aluminum

11.96kg

163.87 radis

Finite Element and Optimization Models

Because the problem is sy~nnietric. only a quarter plate with sylnrnetry boundary condi-

tions was iised for FE analysis. Hence. the plate was discretizcd into Y x S. instead of

16 x 16. squares eacli containing four siniilar triangles. (It niay be recalled that for a rect-

angiilar platc. FE convergence studies sliowed that discretizatiori irito 16 x 16 rcctanglcs.

a i th four sirnilar triangles in each. \vas siifficient for a good rcsult.)

This being the fint study. instead of choosing rational spline interpolation. linear

iritcrpolation Kas used. Tfic points where the stiffeners intersect werc choscri as the niaster

nodcs. Tlicx are 4 1 inaster nodcs for SPDI.4 witli SDF =1/? for a square platc discretizcd

into S x S scliiarcs. euch containing four sirnilar triangles. So. that becorrics S2 design

variabIes (41 for t and t p eacli) plus orle more due to the stifFericr width b. which niakcs tlie

total S3. The values of the variables at the niaster nodes w r e iised to ohtain the thickriess

at earli of the finite elemrnt nodes throiigh linear interpolation.

Optimum Design

The optiniiini design was a flat plate with thickriess, t p = O.5mni. that is t p reached its

loiver bounds. so did the wicltli b. The frequency constraint was active. The details are

sliown in Table 5.3. But. the interesting fact is that the tliickness of the stiffencrs varies

as shonn in Figure 5.1. Results sirnilar to these can also be found in [q]. It can be

obserwd that the mass of tlie stiffeners is concentrated arourid the central region. and

thc corners. Xotice the fact the stiffener width has reached its lower bound, and that the

average stiffener depth has increased from the initial design. This may be explained due

to the fact the bending stiffness of a beam varies by the cube of the depth of the stiffener.

Furthermore the mass of the stiffener lias decreased by 15.77% from the initial design,

which makes the structure verp fragile. and may buckle at even lower loads. Hence, adding

a buckling constraint becornes a necessity.

Table 5.3: Optimum design for the case of frequency constraint alone.

Dimensions

Nass

First Saturai Frequency

S q m pl& with SPDL4 (SDF = L E ) Interpolation: Linear

Consîmint (s) : Fttquency only

2000nirn x 2000mm: 6 = I.Onim:

t p i = 0.5: i =I to ND\;

9.74kg (15.77% less t han

initial design)

l53.65rad/s (active)

Figure 5.1: Envelope surface of t for the stiffeners

5.1.2 F'requency and Buckling Constraints Applied on the Plate

with SPDIA Stiffeners

Initial Design

.As inrritiorieci in the previous section. the initial design was obtaiiicd by trial and error.

and is sliowti in Table 5.4.

Niim ber of Design

Variables (YDV)

Dimensions

'\laterial

h s s

First Satura1 Frequency

Critical Buckling Load

2000mm x 2000mrn; 6 = 2.Onim;

ti = 50.0mm; t p i = 1.0: i =1 to SDC'

Table 5.4: Initial design for square plate with frequency and buckling constraints

Finite Eiement and Optimization Models

Because the buckling constraint was also being considered, half-plate mode1 was tised for

analysis. The plate )vas cut into half along the asis in the which buckling load was applied,

and appropriate symmetry boundary conditions were imposed.

If linear interpolation is used for design variable linking, the number of design vari-

ables would be roughlv double that compared to a quarter-plate model. This would take

up a lot of time for the optiniizer to converge. Hence rational spline optimization (refer

Section 4.2) j a s used. One rational spline surface was used for approsimating t p (the

plate surface)! and one for t (the envelope surface of the stiffeners). The spline surfaces

have a grid of 2 x 4 squares. The variables t 7 and t p at the intersection of the grids were the

design variables. That iso 15 design variables each for t? and t p , plus the design variable 6,

makes the total 31.

CH.4 PTER 5 . OPTIAI.41; DESIGNS

Optimum Design

The optinlum results are shown in Table 5.5, and Figures 5.2 and 5.3. It m q be rioted

that thc biickling constraint is active, meaning that it drives the design.

A coniparison wit h the result of the previous section shows that the spline surface for

t in Figure 5.2 has a shape siniilar in appcarance to that of the t-surface in Figure 5.1

n-tiere freqiiency const raint wu the lone rnajor constraint. Sornc prominent differences arc

t liat t lie nvcrage stiffcner drptli has iiicreascd. aiid the weight has approsimately doubled.

The t spliiie surface (Figure 5.3) is iiot Rat but the edges parallel to the x-ais are raised.

In ttiis regard. it woulcl he intcresting to note that the biickliiig load \vas applied parallel

to the x-mis.

1 Dimensions 1 2OOOrnm x 2000rnrn: b = 1.05nim;

Mass

Table 5 .5 : Optimum design for the case of buckling and frequency coristraints.

t and t p : sec figures 5 .2 . 3.3 resp.

19.80kg (44.07% less than

First Satiiral Frequency

Critical Biickling Load

Methods to reduce the weight further

initial design)

537.2ïrad/s

1 038 236.61X (active)

Two niethods were investigated.

First method: This may be named "Stiffener Pattern Splitting (SPS) llethod". This

rriethod essentially consists of splitting a stiffener pattern into two or more groups based

on the a t iffener orientation. and using different spline surfaces for representing different

groups. For esample. in the case of SPDIA pattern, the stiffeners can be grouped into

two. as the stifFeners are aligned at either of the t a o orientations. For a square plate

mith SPDI.4, it can be -Go, and +As0. So the group of stiffeners aligned a t - 4 5 O , have

one spline surface: and the rest tvill have another. The idea behind this approach is that

the SPS method may help stiffeners of a certain orientation which are not necessary, to

disappear, or reduce in thickness, and thus, reduce weight.

Secoiid method: Reduced SDF. For both the cases presented till now SDF was 119.

So. the idea is to reduce it to 114. By doing this the stiffener pattern imposcd lias lesser

nuinbcr of stiffeners. and thus the final mass may reduce. But. the problerri is. if the

riumbcr of stiffeners are not sufficient to niake the plate stiff enough. the plate tliickiiess

n i e incrrase. arid licnce incrcasc the m a s of the structure on the mhole. rather than

decreasing it.

Third method: Combination of both of the two mcthods discussed above. that is,

rediiring SDF. and applyirig SPS niethod for interpolation.

Tlic initial fcasiblc design for the three cases are shown in Table 5.6. Half plate

riiodcls w r e uscd for al1 ttic cases. I t should be noted that each spline surface liad 3 x 5 = 15

points. In cases 1 and 3. two spline surfaces for stiffeners. and one for the plate, were iised.

Hencc. 45 design variables. .-\dditionally, oiie for the stiffener width b. wliich rnakes the

total -16.

Thr optimal designs are shown in Table 5.7, arid Figures 5.4 to 5.7. The spline

surface for t p is shown onl'r for tlie case 3? as t p reached the lotver Gound of 0.5nirii in

d l the otlier cases. In Figures 5.4 and 5.6, DIAI refers to the negative aiiglcd diagonal

stiffencrs. and DIA2 refers to the positive angled ones. The spline surfaces for DIA1 and

DIAZ. as niight be seen. are almost niirror images of eacli other (escept for the bunip in

DIA2 of case 3) . consistent with the stiffeners' orientations. Because the stiffcncr density

is higher in case 1. the DIA1 and DIA2 stiffeners tend to zero near the right and left cdgcs.

r.espccti\-ely The t spline surface of case 2 is quite similar to that of square plate with

SPDI.4. and SDF = 112 in Figure 5.2.

-4s espected, the three cases show furtlier reductions, each being 1.9%, 26.7%. and

18.13% respectively, compared to the case of SPDIA (SDF = 1/2), and without SPS. The

best one turns out to be Square plate with SDF = 111 using spline interpolation. and

without SPS method. What is interesting is that, case 3 gave a higher rnass than case 2.

Probably. the design might have settled into one of the many local minima.

Something that is common to al1 is that the buckling constraint was active. Further-

moreo the plate tended to vanish, and in the process the plate thickness reached its lower

bound. The stiffener width was low in the majority of the cases.

Square plate wiîh SPDW (SDF =1/2)

Interpolation: Rational Spline with p = 10.0 Consmint (s ): Fi~quency and Buckling

Figure 5.2: Square plate with SPDIA: Spline surface for t

S q u m plate with SPDiA (SDF=l/Z)

Interpolation: Rational Spline with p = 10.0 v Consûaint (s): Frequency and Buckling 1 / '

Figure 5.3: Square plate with SPDIA: Spline surface for t p

CH.4 PTER 5 . OPTIM.4 L DESIGNS

Square Plate with SPDIA (SDF = 112) Constraints: Frequency and Buckling

Interpolation: Spline with Stiffener Pattern Splitting rnethod

) DIA1 Stiffei

Figure 5.4: Square plate with SPDIA, and SPS method applied: Spline surface for t

Squm plate with SPDIA (SDF =l/4) Interpolation: Rational Spline with p = 10.0

Consbnint (s): Fmquency and Buckling

Figure 5 .5 : Square plate with SPDIA, reduced SDF: Spline surface for t

CH-4 P TER 5 . OPTIJI.4 L D ESIGiVS

Square Plate with SPDlA (SDF = 114) Constraints: Frequency and Buckling

Interpolation: Spline with Stiffener Pattern Splitting method ,

Figure 5.6: Square plate with SPDI.4, reduced SDF and SPS method applied: Spline

surface for t

CH.4 PTER 5. 0PTIAI.U DESiGNS

S q m plate with SPD W (SDF =1/4) Interpolation: Spline with Stiffener Pattern Splitting method z

Constrnint (s) :Fiequency and BuckIing 1 1'

Figure 5.7: Square plate with SPDIA, reduced SDF and SPS method applied: Spline

surface for t p

5.1.3 Plates with Other Stiffener Patterns

.-\II tlie cases presented earlier weere for a plate a i t h SPDIA stiffeners. Here. tlie plates witli

stiffencr patterns SPSQR. and SPDIASQ, will be presented. For al1 tlie cases considered.

a lialf-plate mode1 \vas used wi th appropriate symrnctry boiindary conditions. .As beforc

al1 thc plates were siniply supported.

Iiiitial clesigns for the 6 cases cotisidered are shomn in Table 5.8. Sirnilar to thc

prrvious cases. a 3 x 5 point spline surface \vas used in al1 the cases. For cases 1, 3 and

5 tlicre wcrt. two spline surfaces. one each for the stiffener assembly and plate. For case 2

tlicre w r e 3 spliric surfaces - one for the plate and two for the stiffencrs. In the cases 4

ancl 6. four splinr surfarrs w r e required for the stiffeners (as SPDIASQ lias stifferiers in

four diff'wiit orientation). and one spline surface for the plate.

Tlie rcsiilts of optiniization are presented in Table 5.9, and Figures 5.8 to 5.18.

\l'hile rcfcrring to the plots where SPS niethod aas applied (Figures 5.9, 5.12, 5.13,

5.16, and 5.17). it sliould be noted t h DIA1 and D U 2 refer to diagorial stiffcncrs of

-45' arid +-Ki0. respect ive15 arid. HOR and VER refer to horizontal and vertical stiffeners.

respectively. In all tlie cases. csccpt 3. 5. and 6. the plate thickncss t p reaclicd its iower

boiinci of 0.5mm, arid hence. those are riot shown.

Thc casc of tlic plate with SPSQR ( SDF = 1/4 ) iras riot presetited. The rcason

k i n g that. in this pürticular case the number of stifferiers was not enougli to get an initial

fcasible desigii of low m a s , that is. the plate thickness had to be increased to ari estent of

about 10riirn. and the mass a a s as high as 144kg. Given the number of desigii variables it.

is reasonable enough to assume that the optimal design would settle in to a local minima

having a higher mass than al1 the cases presented.

I t can be obserl-ed that, in general, the spline surfaces for the variable t for al1 the

niethods not using SPS (cases 1, 3, and 5) look similar in shape, that is, the stiffeners

close to middle of the edges tend to vanish. So, the trend is that the mass, in the form

of stiffeners, is being concentrated in aUbulge" encompassing the whole plate, witli some

appendages around the corners.

When SPS method was used, the spline surfaces representing stiffeners of different

orientation took t here own shapes. Alt hough, spline surfaces of a particular stiffener angle

took a similar shape in al1 cases.

The optimal masses obtained by the various cases presented are cornpared in Ta-

ble 5.10. It can seeii that the stiffener configuratioii of SPDIA (SDF = 114): rational

spline interpolation arid particularly, without SPS method gives the best resiilt. The nest

bcst configuration is SPSQR (SDF = 1/2) . The stiffener pattern SPDIASQ gives the worst

results witli both SDF = 112 arid 114, eitlier with or without SPS method applied. Froni

cictors these resiilts one can conclude that while designing stiffened plates two important f.

should br coiisiderrd: a) the stiffener pattern b) the stiffener density. The density: here.

should not bc mistaken for tlie density of the material of the stiRener. I t is a measure of

thc n ~ ~ r n b e r of stiffeners and their spacing.

Squm plrite with SPSQR (SDF =1/2)

Interpolation: Rational Splina with p = 10.0 Z v

Constmint (s): Fmqucncy and Buckling

Figure 5.8: Case 1 [Square plate with SPSQR (SDF = 1/2)]: Spline surface for t.

CH.4 PTER 5. OPTLU.4 L DESIGiVS

Square Plate with SPSQR (SDF = 112) Constraints: Frequency and Buckling

Interpolation: Spline with Stiffener Pattern Splittimg method

(a) HOR Stiffeners

(b) VER Stiffeners

Figure 5.9: Case 2 [Square plate with SPSQR (SDF = 112) and SPS]: Spline surface for t .

Square plate with SPDMSQ (SDF =1/2) Interpolation: Rational Spline with p = t 0.0

Consûaint (s): Fioquency and Buckling

Figure 5.10: Case 3 [Square plate with SPDIASQ (SDF = 1/2)]: Spline surface for t .

S q u m platc with SPDLiSQ (SDF =1/2) Interpolation: Rational Spline with p = 10.0

Consimint (s) :Fmquency and Buckling

Figure 5.11: Case 3 [Square plate with SPDIASQ (SDF = 1/2)]: Spline surface for t p .

Square Plate with SPDIASQ (SDF = 112) Constraints: Frequency and BucMing

Interpolation: Spline with Stiffener Pattern Splitting method , I Y

(b) DIA2 Stil

Figure 3.12: Case 4 [Square plate with SPDIASQ (SDF = 112) and SPS]: Spline surface

for t for the diagonal stiffeners.

Square Plate with SPDIASQ (SDF = 112) Constraints: Frequency and Buckling

Interpolation: Spline with Stiffener Pattrem Splming method , I v

Figure 5.13: Case 4 [Square plate with SPDIASQ (SDF = 112) and SPS]: Spline surface

for t for the horizontal and vertical stiffeners.

Square plate with SPDLISQ (SDF =1/4) Interpolation: Rational Splirie with p = 10.0

Constnint (s): Fmquency md Buckling

Figure 5.14: Case 5 [Square plate with SPDIASQ (SDF = 1/1)]: Spline surface for t .

CHAPTER 5. OPTM4L DESIGNS

S q m plate with SPDLASQ (SDF =1/4)

Interpolation: Rational Spline with p = 10.0 Z " Constraint (s): Fmquency and Buckling kf '

Figure 5.15: Case 5 [Square plate with SPDIASQ (SDF = 1/4)]: Spline surface for t p .

Squam Plate with SPDIASQ (SDF = 114) Constraints: Frequency and Buckiing

Interpolation: Spline with Stiffener Pattern Splitting method , I Y

Figure 5.16: Case 6 [Square plate with SPDIASQ (SDF = 11.1): and SPS]: Spline surface

for t for the diagonal stiffeners.

Square Plate with SPDIASQ (SDF = 114) Constraints: Frequency and Buckling

Interpolation: S pline with Stiffenet Pattrem S plitting method

Figure 5.17: Case 6 [Square plate with SPDIASQ (SDF = 11-1) and SPS]: Spline surface

for t for the horizontal and verticaI stiffeners.

S q u m plate wiîh SPDLISQ (SDF =1/1) Interpolation: Spline with Stiffener Pattern Splitting method z

Constraint (s) :Ftpquency and Buckling 1 /'

Figure 5.18: Case 6 [Square plate with SPDIASQ (SDF = 114) and SPS]: Spline surface

for t,.

Table 5.10: Optimal masses obtainccl by various methods

5.2 Rect angular S t iffened Plat es

1 Iass

(kg)

19.80

19.49

14.53

16.21

19.03

16.91

19.73

33-87

25.43

24.58

Because of the limitations in the cornputer resoiirces. only r~ctangular plates of ratios 2 1

and 3: 1 were corisidered. The breadth of the plate was 2000mm in both the cases. The four

stiffener configuratioiis. whidi gave the lowest masses for square plates were applied to the

rectangular plates to denionstrate the present design met hodology. Half-plate models were

uscd for anal?.sis. wit h the plate being halved along the length. Finite elernent analysis on

the rectangular half-plate models of aforementioned ratios were carried out by discretizing

into 8 x 32, and 8 x -LS squares, respectively, and each square being in turn divided into

four identical triangles, by the diagonals. Doing this not only ensured that the size of the

elements remained the same as those used for square plates, but also made it possible to

get the stiffener orientations the same as those of the latter. (It is quite obvious that using

rectangles instead of squares would make the orientation of the diagonal stiffeners not

equal to i145"). The boundary condition was simply supported with symmetry conditions

included.

S1.

$0.

1 -1 - 3

1 - a

G

7

8

9

10

The initial designs can be found in Tables 5.11 and 5.13. The spline surfaces used

for the 2 1 rectangle had 3 x 9 rnaster node points, while the 3:l rectangle had 3 x 13. Spline

interpolation without SPS, as usual, required two spline surfaces, making the number of

Interpolation

Spline

SPS 9; Splinn

Spline

SPS Si Spline

Spline

SPS Si Spline

Spline

SPS 9t Spliiie

Spline

SPS k Spline

';D\,'

31

16

31

16

31

46

31

76

31

76

Stiffener

Pat tern

SPDIA

SPDIA

SPDIA

SPDIA

SPSQR

SPSQR

SPDIASQ

SPDIASQ

SPDIASQ

SPDIASQ

SDF

1/2

112

111

111

112

112

112

112

114

111

design variables 54 and 75 respectively. for 2 1 and 3:l rectangles. It required 3 spline

surfaces for interpolation with SPS! giving 81 and 117 design variables. Adding one more

design variable b. brings the totals to those shown the aforeineritioned tables.

Thc results are shown in Tables 5.12 and 5.14, as well as in Figures 5.19 to 5.35.

Soiiie of the observations made for square plate holds true for the rectangular plates. as

wcll. The buckling constraint was active in ail the cases. The plate thickness tended to

vanisli in al1 the cases. hence helping reduce the m a s . Whereas. the stiffener width for

rcctaiigiilar plates remained higher than the lower bound unlike for the square plates where

in niost of tlie cases it nttained. or remained close to. the lowr bound.

Tlic vibration modcs for al1 the 2:l plates studied were (1,l) modes. The buckling

modcs for al1 tlic 2:l platc cases, escept case 4 in Table 5.12. were (2.1) modes. For case

4. i t was (3.1). as shown in Figure 5.27. lnteresting associations car1 be made bctween

tlie splirit) siirfaces arid thc buciling modes for a given plate. For csample, compare the

splinc plots (5.25 arid 5.26) with the buckling mode shape (Figure 5.27) for case 4. In

Figure 5.25(ü). the curye at y = lOOOmm . ruiining parallel to the r - a i s . changes slope

similar to tlie comparable central ciirve of the mode shape. The rriasirna of this curw is

aroiind x = 1500112111 and 3500mm. similar to the mode shape. Siniilarly, the periodicity of

t, he plate tliickness along the long edges in Figure 5.26 is in concurrence with the lengths

of the half *'sine" surfaccs of the mode shape. This kind of an association between thc

two should corrie as no surprise given the fact that, buckling constraint is active. The

accordance with the mode sliapes and the splines is more subtle in other cases. In making

such coniparisoiis it should be borne in rriind that the vibration mode. as irientioned. earlier

is. (1.1). Both of these n i q influence the final shape aiid make the final shape of the splines

look complicated.

The buckling mode was (3J) for al1 the 3:l plates escept case 3 in Table 5.14, which

had (2.1) mode. The vibration modes were al1 (1,l). The mode sliape to spline surface

association holds true for this too.

Finally, similar to that for square plates, SPDIA (SDF = 11-1) stiffener configuration.

in general. gave the lowest masses. although for 21 rectangular plates it was with SPS,

and for 3:l plates it was without SPS. Thus! it can be concluded that SPDIA with SDF =

114 is the best stiffener configuration of al1 those studied.

5.3 Concluding Remarks on the Results

I l togctlier 3 s t iffeiier patterns were considered: SPDI-4. SPSQR. and SPDIASQ. For each

of these. Stiffener Density Factors (SDFs) of 112 and 1/1 were considered. Rational spline

interpolation was used for linking the design variables. A Stiffener Pattern Splitting (SPS)

nietliod of interpolation was introdiiced and applied on al1 the stiffener configurations. It

ivas foiind tliat SPDIA n i t h SDF = 114 with (in most cases) or without (in one case) SPS

method gave the lowest mass. Most importantly it can be concluded that the proposed

opt inial tlcsign rnct hodology gives good results.

Rcctvyulv Plate (2:l) wiîh SPDLI (SDF = 1/41 Inkrpolûtion: Rationai Splinc with p =10.0

Consmin&: Fmqucncy and Buckling

Figure 5.19: Rectangular plate (rat.io 2: l ) with SPDIA (SDF = 11-1): Spline surface for t .

CH-4 P TER 5. OP Tl'A 1.4 L DESlGiVS

H c c ~ u l e r Plate (2:l) wiîh S P D U (SDF = 1/41 Interpolation: Rationid Spline widi p 40 .0

Consmints: Frcquency anci Buckling

Figure 5.20: Rectangular plate (ratio 2: 1) with SPDIA (SDF = 114): Spline surface for t p .

HcclaiyuluPi;iIi! (21 ) with SPDLI (SDF = 1/41 Interpolation: Rationail Spline d S E Coiistninb: Fnquency uid BucWing

DLIl and D M 2 splinc surfaces çombincd z

mm)

Figure 5.21: Rectangular plate (ratio 21) with SPDIA (SDF = 114) and SPS: Spline

surface for t .

CHAPTER 5. OPTIMAL DESIGNS

RccîanguluPlate ( 2 1 ) wi îh SPDM (SDF = 11.0 inktpolation: Rationai Spline and SPS Constnints: Frrquency and Buckliq

Figure 5.22: Rectangular plate (ratio 21) with SPDIA (SDF = 114) and SPS: Spline

surface for t p .

Rcciangu1;irPi;Uc ( 2 1 ) wi th SPSQR (SDF = 1/21 Intcipol~tion: Rafional Splinc wiîh p 40 .0

Cansûainfs: Fmqucncy and Buckling

Figure 5.23: Rectangular plate (ratio 21) with SPSQR (SDF = 112): Spline surface for t .

CH--1 P TER 5. OP TI,\ 1.4 L D ESIGXS

Rccrangular Plalc (21) w i lh SPSQR (SDF = 12) Inicipol~tion: Rational Spline wi îh p 40.0

Constraints: Fmquency and Buckling

Figure 5.24: Rectangular plate (ratio 21) with SPSQR (SDF = 112): Spline surface for

t p -

CHAPTER 5. OPTIMAL DESIGiVS

Reciangular Plate (2:l) with SPSQR (SDF = 1/2) Interpolation: Rationai Spline and SPS Consbain&: Frequency and Buckling

- (b) VER spline sudace

Figure 3.25: Rectangular plate (ratio 2:l) with SPSQR (SDF = 112) and SPS: Spline

surface for t .

Rccîangulur Plate (21 1 with SPSQR (SDF = 1 / 3 inrcrp>lation: Rationai Spiine and Si% Conrtmints: Frrquency and Buckling

Figure 5.26: Rectangular plate (ratio 2:l) with SPSQR (SDF = 112) and SPS: Spline

surface for t p .

CHAPTER 5. OPTIAIAL DESIGNS

Buckling mode shape for rectangular plate (2:l) with SPSQR (SDF = 112) Interpolation: Rational Spline and SPS

Figure 5.27: Rectangular plate (ratio 21) wit-h SPSQR (SDF = 11%) and SPS: buckling

mode shape.

" 9 . a m C u 0 3

S Z k Z ao- Z M b T M cc- 06- a5 -, ce ?. 7 4 H d 4

Rccmgulw Plate (3:l) with SPDU GDF = 1/41 Intctpolation: Rationai Spline wilh p =10.0

Conîtraints: Fniqucncy and Buckling

Figure 5.28: Rectangular plate (ratio 3 3 ) with SPDIA (SDF = 114): Spline surface for t .

CH.4 P TER 5. OPTIA 1.4 L DESIGNS

Rcctuigular Plaîc (3:l) wilh SPDlA (SDF = 1/41 Interpolation: Rationai Splinc wiih p 4 0 . 0

Cansûaints: Frrqucncy d Buckiing

Figure 5.29: Rectangular plate (ratio 3:l) with SPDIA (SDF = 114): Spline surface for tp .

CH.4 PTER 5. OPTIAI.4 L DESIGNS

Rccmngular Ptatc ( k l ) w ith SPDM (SDF = 11.1) Intfipolaiion: Rational Splinc and SPS Consaaints: Frrqucncy and Buckling

DLI 1 and DLIZ splinc surCaccs combincd

Figure 5.30: Rectangular plate (ratio 3: l ) with SPDI.4 (SDF = 114) and SPS: Spliiie

surface for t .

CHA P TER 5. OP TL1 I.4 L D ESIGNS

Rccîmgul;u Marc (M) wilh SPDM (SDF = 114) Intcipolation: Rationiai Spline mi SPS Conslraints: Frcqucncy and Buckling

Figure 5.31: Rectangular plate (ratio 3:l) with SPDI.4 (SDF = 1/4) and SPS: Spline

surface for t p .

R c c ~ u l a r Plale (31) wiîh SPSQR (SDF = 1/3) Interpolation: Ratiomi Splinc with p d 0 . 0

Canstmints: hquency and Buckling

Figure 5.32: Rectangular plate (rat,io 3:l) with SPSQR (SDF = 112): Spline surface for t.

Rcctangular Plak (&Il with SPSQR (SDF = 1/21 IntErpoladon: Rational Spline with p 4 0 . 0

Consûaints: Fmqucncy and Buckliry

Figure S.33: Rectangular plate (ratio 3:1) with SPSQR (SDF = 1/2): Spline surface for

t p .

R c c ~ u l a r Plate (3:l) wiîh SPSQR (SDF = 112) inteplaiion: Rational Spline and SPS Constnints: Froquency and Buckling

Figure 5.34: Rectangular plate (ratio 3:l) with SPSQR (SDF = 112) and SPS: Spline

surface for t .

Rcct;uigdar Plate (31) with SPSQR (SDF = 112) in&rpol;icion: Rational Spline and SPS Constninb: Fmqucncy and Buckling

Figure 5.35: Rectangular plate (ratio 311) with SPSQR (SDF = 112) and SPS: Spline

surface for t p .

Chapter 6

Summary and Conclusions

This work presented a methodology for the optimal design of stiffened plates having fre-

quericy and buckling constraints. The basic idea was to assumc that a plate had a fairly

dense distribution of stiffeners in a specific pattern. The analysis of the stiffcned plate

\vas carried out by finite element analysis, with the plate being modeled by IIiridlin plate

elenients. and the stifferiers by Timoshenko beam elernents. A11 the necessary finite ele-

nient derivations wcre erilisted. The elements were validated by comparing with analytical

solutions.

The three stiffener patterns (SPDI-A, SPSQR, and SPDI-ASQ) considered, and a

quantity associated witli niimber of stiffeners and their spacings, called Stiffener Density

Factor (SDF) Kas described. A combination of one of the stiffener patterns with an SDF

gave a specific stiffener configuration.

The optimization mode1 made use of spline surfaces for linking the design variables,

tlius reducing the niimber of design variables significantly. Stiffener Pattern Splitting (SPS)

method was introduced as a way of reducing the mass further.

Results aere presented for a square plate with many stiffener configurations. The

four best configurations which gave the lowest mass were applied to rectangular plates for

ratios 2:l and 3:l. In general, it was found that SPDIA (SDF = 114) gave the best results.

Alsol the foilowing observations were made:

The deep stiffeners having smalI width were dominant.

The plate thickness in al1 the cases tended to vanish, and thus attained the Iower

bound.

There appeared to be a delicate balance between the stiffener density and the plate

thickn~ss. If the the density of the stiffeners \vas not sufficient to give enough stiffness,

then the plate thickness increased, increasing the m a s of the structure.

i 'The buckling constraint \vas active for al1 the cases considered, aiid hence drow the

design.

Interrsting associations between the buckling mode shapes and the optimal spline

surfaces werc observcd for rectangular plates.

Fiiially, it \vas coricludcd that the present methodology gives good results. and illiis-

tratcs tliat stiffenpr pattern and deiisity play a n important role in the design of stiffened

plates.

6.1 Recommendations for Future Work

The riiajor cori~traints considcred iri this work were only the frequency and biickling con-

straiiits. biit i t would be evcn more rcalistic to consider some more like the stress con-

straints.

The stiffcner patterns analyzed were of standard angles O". &-la0. a n d 90". Hetice.

there is a scopc to consider stiffeners of other angles. and differcnt patterns. The patterns

were assunied to be static in this work. It would be interesting to take a few stiffeners,

and allow them to niove? while optimizing.

Another direction in which this work can be estended, is to make use of composite

niaterials. The density of the fibers could be the design variables.

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