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