CHAPTER - 1 INTRODUCTION
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1 .1 INTRODUCTION
Man took a le a f out of the book of nature to apprise him self of the p o te n tia lity of st iffe n e d structures. One comes across a wide variety o f these natural s t iffe n e d structures in day to day l i f e - leaves# sea sh e lls / mushrooms/ vegetables# f is h e s are some o f the g la rin g examples. In the past the development o f the st iffe n e d structural form was slow due to m an's
lim ite d knowledge and lim ited a v a i l a b i l i t y of known m a teria ls. Innovation and introductio n of structural m aterials lik e ste e l/ reinforced concrete/ aluminium a llo y s and composites have brought about a re v o lu tio n in structural desig n . It is only in the last century that a new era of s t if f e n e d structures has been started by engineers and techno lo gists through continuous research and a zeal for e x p lo it in g the p ro p erties of these
m aterials to t h e ir f u ll e s t exten t. New m aterials alone are not responsible for the advancement in stru ctura l d e s ig n s . Continuous e ffo r ts by engineers and re searchers have paved the way fo r the development o f new s t if f e n e d
structural forms and t h e ir design procedures.
Since the beg inn in g o f the nineteenth century/
s t if f e n e d stru ctural elements have found wide a p p lic a tio n s in en g in e e rin g . These s t i f f e n e d p late elements through th e ir lig h t w eights p ro v id e an e f f i c i e n t and
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e ff e c t iv e means to cater to the requirements o f the d e s ig n s. Their response to the a p p lie d external fo rces and bending strength make them id e a lly suited fo r
ap p lica tio n in v ario u s engineering structures. Ty p ical examples where t h in p lates strengthened by means o f s t iffe n e r s in the form of lo n g itu d in a ls and beams or frames are ocean structures - ships/ offshore d r i l l i n g and production platform s and other sim ilar stru ctu re s.
Much of the knowledge of s t iffe n e d p lates has advanced w ith the development of bridge en g in eering . Various types of bridge configurations# v i z . bridge slabs# box g irders and c e l l u l a r structures h av ing s t r a i ^ i t , skew or h o rizo n ta lly curved plan forms# have posed great challenges to d esig n ers and a n a ly s t s . Apart from
bridges# c i v i l engineers have also been confronted with the problem o f d e s ig n in g conposite concrete-steel beams which fin d wide a p p lic a tio n in flo o r co nstructio n.
Aeronautical engineers have been moving in p a r a l l e l with other engineers i n contributing to the knowledge o f the
structural mechanics associated w ith s t iffe n e d p l a t e s . The development o f lig h t weight metal construction for a ir c r a f t follow ed the discovery th at a structure b u i l t o f two sets of m utually p erp en d ic u la r beams covered w ith a th in metal sheet could s a fe ly carry loads even
i f the skin b u c k le d. Further/ th ere are various other structures where s t if f e n e d p l a t e s have found w ide a p p lic a tio n s .
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looking into the s t iffe n e d p la t e stru ctural problems of to-day# one can say th at there are two
b a sic types of p la t e structures - (i) a p late s t if f e n e d by 'open* s t iffe n e r s or 'c lo s e d ' s t iffe n e r s on one side or both sides o f the plate# ( i i ) two skin p la t e s
(u n stiffe n e d or s t iff e n e d as in (i) above) separated by webs. Depending on the requirements and the design#
the plan form o f the plate may be square, re cta ng u la r, skewed, h o rizo n ta lly curved, polygon or of any o th er a rb itra ry shape. The stiffe n e r s can occupy p o s itio n s which are p a r a lle l to the edges or in c lin e d in an a rb itra ry manner to the edges. A ll the s t iffe n e r s may have an id e n tic a l cross-section or the geometrical p ro p erties may vary from s t iffe n e r to s t if f e n e r . The
s t iffe n e r s may be p lac ed at equal spacings or they may have unequal spacings between them. In the case o f a
double p late structure, commonly known as c e l l u l a r structure, the spacing between the two skin p l a t e s can be uniform or ta p erin g or can have any other type of v a r i a t i o n .s t i f f e n e d plate panels are used in the v ario u s components o f the h u ll o f . a ship (namely - deck# bulkhead# sid e shell# bottom sh ell e t c .)# as w ell as i n bridge decks# lock gates# helidecks# flo o r s o f b u ild in g s etc. The c e l lu l a r type o f structure i s used
in bridge# a ir c r a f t wing# dockgates# as w ell as in the double bottom# a w ing tank or a rudder of a ship and
so o n . Depending on the use# a s t if f e n e d p la t e p an el
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has a set of boundary and support conditions and can be subjected to a wide variety o f external a p p lie d
lo ads. Such structures pose d i f f i c u l t i e s for stru ctural a n aly sis owing to t h e ir complicated structural arrange
ments. No exact a n a ly tic a l procedure is available# but v ario us approximate methods for a n a ly s is can be u sed . A
s ig n ific a n t amount o f an aly tical and experimental inv estigatio n s have been conducted on the problem.
Various approaches have been suggested fo r the a n a ly s is of s t iff e n e d plate problem s. Amongst the
d iff e r e n t id e a l is a t io n s proposed are the orthotropic p la t e concept and the plated g r il l a g e concept fo r ships
stru ctures. In the orthotropic p la t e model the s t i f f e n ed p la t e is replaced by a bare p la t e having ortho tro pic p la t e p ro p e rties. In the g r illa g e an aly sis the p la t e is considered as the fla n g e o f the beams. In many actu al en gin eering structures the rib s are placed only on one sid e o f the p l a t e . This makes the s t iffe n e d p l a t e
problem even more d i f f i c u l t and i t becomes necessary to include the ecc en tric ity o f the s t iffe n e r in the form ulation o f the d i ff e r e n t ia l equations fo r the plate-beam system.
The advent o f the corrputer has changed the
outlook in many f i e l d s of engineering# and sh ip b u ild in g i s no exception. The impact o f the computer can be seen
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c le a r ly in the f i e l d o f structural en gineering. Ocean structural engineers are becoming more dependant on computational methods of structural analysis u sin g various numerical methods to solve structural problems which were not attempted e a rlie r owing to the tremendous computational labour involved. For the an aly sis o f
complex structures such as s t iffe n e d plates# v arious numerical techniques have been a p p lie d for o b ta in in g the solution. The more important methods which have been a p p lie d for the so lu tion of s t iffe n e d plates are the f i n i t e d iffe re n ce method# the dynamic relaxatio n method#
the f i n i t e strip method and the f i n i t e element method.
These methods along w ith th eir advantages and d is advantages have been discussed in d e t a il in the next chapter.
Following the advent o f the high speed computer#
the f i n i t e element method has proved to be undoubtedly the most v e r s a t ile o f these approaches. The e x i s t i n g app lica tio n s o f the f i n i t e element method to s t if f e n e d p la t e problems have certa in disadvantages# such as - correct representation o f the s t if f e n e r in the p la t e element# correct evaluation of the beam and the p la t e stresses# f i t t i n g i n o f ir r e g u la r boundaries o f the
plate# consideratio n o f the tran sv erse shear deformations o f the p late and the s t if fe n e r i n th e form ulation and o rie n ta tio n o f the s t if f e n e r in the p l a t e . The work
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described in th is th e sis is an attempt to overcome these lim ita tio n s.
1.2
OBJECT AND SCOPE OF THE WORKThe main purpose of the present work i s to obtain solutions of s t if f e n e d p late structures in bending by an e f f i c i e n t and accurate f i n i t e element method which i s r e a d ily a p p licab le to a wide v a r ie ty of ocean / marine p la t e d stru c tu r es' such as bulkheads, decks, s h e ll
p la t in g , double bottoms, wing tanks, h e lid e c k s , d o c k gates, rarrps and o ther engineering structures. Isopara
m etric quadratic s t iffe n e d p late bending elements w ith three degrees freedom per node and f iv e degrees o f
freedom per node have been developed here to achieve t h is purpose. The main feature o f th e elements i s th at
*
the form ulation i s b ased in a general manner w ith o ut d istu rb in g the p o s it io n or the p ro p erties o f the s t i f f e n
e r s. Even a s t if f e n e r having a rb itra r y o r ie n t a t io n w ith in an element remains undisturbed i n the form ulation. A
s im ila r approach was made e a r lie r in the case o f the
s t if f e n e d p la t e p lan e stress problem (Mukhopadhyay# .1 9 8 1 ).
F urther, as the form ulation is b ased on the isoparam etric quadratic element# irr e g u la r b o un d aries o f the structure as also the tran sverse shear deform ations o f th e p l a t e and the s t iffe n e r s can be very conveniently taken care o f .
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The method i s more th eo retically accurate than any of the f i n i t e element approaches made so fa r . A comparison
is made between the present method and the lunped model to study the extent o f inaccuracy inherent in the l a t t e r approach . A new scheme for o b ta in in g s t iffe n e r stresses for the lumped model is also proposed. A p p licatio n has been made to a h e lid e c k on the b a s is of the element
developed. At the in c lin e d boundaries of the octagonal helideck# the elements are t r ia n g u la r . With the help of the isoparametric c h a ra cteristic s o f the element by c o lla p sin g a couple o f nodes# rectangular elements have been transformed into the tr ia n g u la r boundary elem ents.
A v arie ty o f problems such as rectangular
s t if f e n e d plates# composite bridges# box girders# c e ll
u la r bridges# curved stiffen ed bridges# skew s t if f e n e d plates# dock gates and h elidecks are analysed by t h is method. loadings considered are p o in t load# uniform ly d is t r ib u t e d load# patch load and h ydro static load#
depending on the problem. D iffe r e n t support co n d itio n s for d iffe r e n t problems are c o n sid ered . Results o b ta in e d by the proposed method are compared w ith those i n
p u b lish e d l i t e r a t u r e . Also the r e s u lt s ob tain ed by conducting model experiments w ith models made from p erspex have been analysed by the proposed f i n i t e
element approach*
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In order to obtain the so lu tio n of the
problems# two conputer programs have been developed i n FORTRAN-IV. One program i s b ased on the element w ith three degrees o f freedom p er node whereas the other program i s b ased on the element with fiv e
degrees of freedom p er node. The s a lie n t featu res of the programs are l i s t e d below I
i) Automatic mesh generation
i i ) Element s t if f n e s s matrix generation i i i ) Assembly o f overall s t if f n e s s matrix in
half-band width
iv) Element lo ad in g matrix generation by a s s ig n in g any type o f load in c lu d in g patch loading
v) Generation o f overall lo a d in g matrix
v i) F it t in g in the boundary conditions of the structure
v i i ) C a lc u la tio n o f the nodal displacements v i i i ) C a lc u la tio n of plate stress resultants
ix ) C a lcu la tio n o f s t iffe n e r stress r e s u lt a n t s .
The inp ut data required fo r the a n a ly s is are as follow s
1
a) Linear p la t e dimensions together w ith the in t e n s ity o f loading
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b) Mesh d iv is io n details# number o f s t if f e n e r s i n eith er direction# type o f loading and mode of in te g ra tio n
c) Boundary conditions of the p late
d) M aterial properties of the p late and the s t iffe n e r s
e) S t iffe n e r p o sitio n s and geometrical p ro p e rties .
The program was developed and run on a B6700 computer of the Regional Computer Centre# C a lc u tta .
1 .3 NOMENCLATURE
a p late dimension in lo n g itu d in a l d ire c tio n A b cross- sectional area of the s t iffe n e r b p late dimension in transverse direction
strain m atrix
D G r strain m atrix associated w ith the rth node D G r ig id it y m atrix
d ] m odified r i g i d it y matrix
r i g i d it y m atrix for the stiffen er/cross- w eb
[ * J m odified r i g i d i t y m atrix for the s t i f f e n e r / cross-web
E Y o u n g's modulus
displacem ent f i e l d vector G modulus of r i g i d it y
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distance between the two skin p lates
matrix connecting stra in and m odified s t r a in vectors for the plate
matrix connecting stra in and m odified stra in vectors for the stiffener/cross- w eb
id entity m atrix
second moment of the s t if f e n e r area about the reference plane of the structure along x-axis second moment of the s t if f e n e r area about the reference plane of the structure along y-axis p o lar moment of in e r t ia o f the s t if fe n e r
Jacobian m atrix
Jacobian determinant
element s t if fn e s s matrix
stress - resultants
m atrix o f shape functions
matrix o f shape functions o f the rth node uniform load inten sity per u n it area
shear forces
a rb itrary node number vector o f reactions
subscript to denote stiffen er/c ro ss- w eb shear area o f the stiffen er/c ro ss- w eb shear r i g i d i t i e s
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I d
t plate th ickn ess
T transpose
transform ation matrix
U /V displacements in the p lane o f the p late
U ,V ,W displacements of any p o in t in the co-ordinate directio ns
w la t e r a l d e fle c t io n o f the plate x * y * z co-ordinates
x » x co-ordinate of the cen tro id of the s t if f e n e r cross-section from web mid-line
y « y co-ordinate of the cen tro id of the s t if f e n e r cross-section from web mid-line
a in c lin a t io n of st iffe n e r to x-axis
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v a r ia tio n a l operator{ * } vector o f nodal displacements
{ 6 > r vector o f nodal displacements at the rth node strain vector
{t } m odified s t r a in vector {*} stress vector
{ a } stress- resultant vector
rotations o f the middle surface normals
^ x '^ y shear ro tatio n s o f the m iddle surface normals
