m.tech structural engg fea important questions

8/10/2019 M.TECH STRUCTURAL ENGG FEA IMPORTANT QUESTIONS

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MTech (Struct. Engg) FINITE ELEMENT ANALYSIS 2013

IMPORTANT QUESTIONS FINITE

ELEMENT ANALYSIS

Q1. WRITE DOWN THE VARIOUS STEPS IN FINITE ELEMENANALYSIS.

The following !e" #$e "e$fo$%e& fo$ fini!e ele%en! #n#l'i.1. Discretisation of the continuum:The (on!in))% i &i*i&e& in!o n)%+e$ of ele%en! +' i%#gin#$' line o$ )$f#(e. The in!e$(onne(!eele%en! %#' h#*e &iffe$en! i,e #n& h#"e.

-. Identification of variables: The ele%en! #$e #)%e& !o +(onne(!e& #! !hei$ in!e$e(!ing "oin! $efe$$e& !o # nol "oin!. Ae#(h no&e )n/nown &i"l#(e%en! #$e !o +e "$e($i+e&.0. Choice of approximating functions:Di"l#(e%en! f)n(!ion i !h!#$!ing "oin! of !he %#!he%#!i(#l #n#l'i. Thi $e"$een! !he *#$i#!ioof !he &i"l#(e%en! wi!hin !he ele%en!. The &i"l#(e%en! f)n(!ion %#+e #""$oi%#!e& in !he fo$% # line#$ f)n(!ion o$ # highe$2o$&e$ f)n(!ion

A (on*enien! w#' !o e"$e i! i +' "ol'no%i#l e"$eion. The h#"o$ geo%e!$' of !he ele%en! %#' #lo +e #""$oi%#!e&.3. Formation of the element stiffness matrix: Af!e$ (on!in))% &i($e!ie& wi!h &ei$e& ele%en! h#"e !he in&i*i&)#l ele%en! !iffne%#!$i i fo$%)l#!e&. 4#i(#ll' i! i # %ini%i,#!ion "$o(e&)$e wh#!e*e%#' +e !he #""$o#(h #&o"!e&. Fo$ (e$!#in ele%en! !he fo$% in*ol*e

g$e#! &e#l of o"hi!i(#!ion. The geo%e!$' of !he ele%en! i &efine& i$efe$en(e !o !he glo+#l f$#%e. 5oo$&in#!e !$#nfo$%#!ion %)! +e &onfo$ ele%en! whe$e i! i ne(e#$'.6. Formation of overall stiffness matrix:Af!e$ !he ele%en! !iffne%#!$i(e in glo+#l (oo$&in#!e #$e fo$%e& !he' #$e #e%+le& !o fo$%!he o*e$#ll !iffne %#!$i. The #e%+l' i &one !h$o)gh !he no&e

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whi(h #$e (o%%on !o #&7#(en! ele%en!. The o*e$#ll !iffne %#!$i '%%e!$i( #n& +#n&e&.8. Formation of the element loading matrix:The lo#&ing fo$% #

een!i#l "#$#%e!e$ in #n' !$)(!)$#l enginee$ing "$o+le%. The lo#&inini&e #n ele%en! i !$#nfe$$e& #! !he nol "oin! #n& (oni!enele%en! %#!$i i fo$%e&.9. Formation of the overall loading matrix:Li/e !he o*e$#ll !iffne%#!$i !he ele%en! lo#&ing %#!$i(e #$e #e%+le& !o fo$% !he o*e$#lo#&ing %#!$i. Thi %#!$i h# one (ol)%n "e$ lo#&ing (#e #n& i! ei!he$ # (ol)%n *e(!o$ o$ # $e(!#ng)l#$ %#!$i &e"en&ing on !he n)%+eof lo#&ing (#e.:. Incorporation of boundary conditions: The +o)n$' $e!$#i(on&i!ion #$e !o +e i%"oe& in !he !iffne %#!$i. The$e #$e *#$io)!e(hni;)e #*#il#+le !o #!if' !he +o)n$' (on&i!ion. One i !he i,of !he !iffne %#!$i %#' +e $e&)(e& o$ (on&ene& in i! fin#l fo$%. Te#e (o%")!e$ "$og$#%%ing #"e(! #n& !o eleg#n!l' in(o$"o$#!e !h

+o)n$' (on&i!ion !he i,e of o*e$#ll %#!$i i /e"! !he #%e.


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Q-. WHAT ARE NATURAL 5OORDINATES>

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N#!)$#l (oo$&in#!e '!e% i +#i(#ll' # lo(#l (oo$&in#!e '!e% whi(#llow !he "e(ifi(#!ion of # "oin! wi!hin !he ele%en! +' # e! o

&i%enionle n)%+e$ whoe %#gni!)&e ne*e$ e(ee& )ni!'. Th(oo$&in#!e '!e% i fo)n& !o +e *e$' effe(!i*e in fo$%)l#!ing !hele%en! "$o"e$!ie in fini!e ele%en! fo$%)l#!ion. Thi '!e% i &efinein )(h !h#! !he %#gni!)&e #! nol "oin! will h#*e )ni!' o$ ,e$o o$ (on*enien! e! of f$#(!ion. I! #lo f#(ili!#!e !he in!eg$#!ion !o (#l()l#!ele%en! !iffne.E#%"le?

1 !ne Dimensional "ine #lementsThe line ele%en! #$e )e& !o $e"$een! "$ing !$) +e#% li/%e%+e$ fo$ !he fini!e ele%en! #n#l'i ")$"oe. S)(h ele%en! #$;)i!e )ef)l in #n#l',ing !$) (#+le #n& f$#%e !$)(!)$e.

He$e !he n#!)$#l (oo$&in#!e of #n' "oin! P (#n +e &efine& # follow.

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Whe$e i $e"$een!e& in 5#$!ei#n (oo$&in#!e '!e%. Si%il#$l' (#n +e $e"$een!e& # in n#!)$#l (oo$&in#!e '!e%. Th) !he #+o*

e"$eion (#n +e $ew$i!!en in !he fo$% of n#!)$#l (oo$&in#!e '!e% #gi*en +elow.

Now !he $el#!ionhi" +e!ween n#!)$#l #n& 5#$!ei#n (oo$&in#!e (#+e e"$ee& #?

Si%il#$l' fo$ !h$ee no&e line ele%en! !he h#"e f)n(!ion (#n +e &e$i*e&wi!h !he hel" of n#!)$#l (oo$&in#!e '!e% whi(h %#' +e e"$ee& #

follow?

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$ %&o Dimensional %riangular #lements

The n#!)$#l (oo$&in#!e '!e% fo$ # !$i#ng)l#$ ele%en! i gene$#l

(#lle& # !$i#ng)l#$ (oo$&in#!e '!e%. The (oo$&in#!e of #n' "oin!

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ini&e !he !$i#ngle i x, y in 5#$!ei#n (oo$&in#!e '!e%. He$e !h$e

(oo$&in#!e L1 L2#n& L3(#n +e )e& !o &efine !he lo(#!ion of !he "oin

in !e$% of n#!)$#l (oo$&in#!e '!e%. The "oin! P (#n +e &efine& +' !h

following e! of #$e# (oo$&in#!e?

Whe$e1B A$e# of !he !$i#ngle P-0

-B A$e# of !he !$i#ngle P100B A$e# of !he !$i#ngle P1-

ABA$e# of !he !$i#ngle 1-0

Th)B1C-C0

#n&1C-C0B1

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The$efo$e !he n#!)$#l (oo$&in#!e of !h$ee no&e will +e? no&e 1 1==no&e - =1= #n& no&e 0 ==1.

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The #$e# of !he !$i#ngle (#n +e w$i!!en )ing 5#$!ei#n(oo$&in#!e (oni&e$ing ' # (oo$&in#!e of #n #$+i!$#$' "oin!P ini&e o$ on !he +o)n$ie of !he ele%en!?

The $el#!ion +e!ween !wo (oo$&in#!e '!e% !o &efine "oin! P(#n +e e!#+lihe& +' !hei$ nol (oo$&in#!e #?

Whe$e

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The in*e$e +e!ween n#!)$#l #n& 5#$!ei#n (oo$&in#!e %#' +ee"$ee& #?

Q0. HOW TO DERIVE SHAPE FUN5TIONS FROM AREA5OORDINATES>

5oni&e$ing # line#$ &i"l#(e%en! *#$i#!ion of # !$i#ng)l#$

ele%en! # hown in Fig)$e !he &i"l#(e%en! #! #n' "oin! (#n+e w$i!!en in !e$% of i! #$e# (oo$&in#!e.

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He$e A i !he !o!#l #$e# of !he !$i#ngle. I! i i%"o$!#n! !o no!e!h#! !he #$e# (oo$&in#!e #$e &e"en&en! # G1C G-C G0B1 . I!%#' +e een f$o% fig)$e !h#! #! no&e 1 L1B 1 while L-B L0B =.

Si%il#$l' fo$ o!he$ !wo no&e? #! no&e - L-B 1 while L1B L0B =#n& L0 B 1 while L- B L1 B =. Now )+!i!)!ing !he #$e#(oo$&in#!e fo$ no&e 1 - #n& 0 !he &i"l#(e%en! (o%"onen!#! no&e (#n +e w$i!!en #

Th) f$o% !he #+o*e e"$eion one (#n o+!#in !he )n/nown(oeffi(ien! G?

OR

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The #+o*e e"$eion (#n +e w$i!!en in !e$% of in!e$"ol#!ionf)n(!ion #

Si%il#$l' !he &i"l#(e%en! *#$i#!ion * in Y &i$e(!ion (#n +e

e"$ee& # follow.

Th) fo$ !wo &i"l#(e%en! (o%"onen! ) #n& * of #n' "oin!ini&e !he ele%en! (#n +e w$i!!en #?

Th) !he h#"e f)n(!ion of !he ele%en! will +e(o%e

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Q3. SHAPE FUN5TIONS OF TRIANULAR ELEMENT INTERMS OF 5ARTESIAN 5OORDINATES.

In!e$"ol#!ion f)n(!ion (#n +e w$i!!en # Refe$ No!e

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Q3. SHAPE FUN5TIONS OF RE5TANULAR ELEMENT IN TERMS OF 5ARTESIAN 5OORDINATES.A)%e !he &i"l#(e%en! fiel& #?

A""l'ing nol (on&i!ion !he #+o*e e"$eion %#' +e w$i!!enin %#!$i fo$% #

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The )n/nown "ol'no%i#l (oeffi(ien! %#' +e o+!#ine& f$o% !he#+o*e e;)#!ion wi!h !he )e of nol fiel& *#$i#+le.

Th) !he fiel& *#$i#+le #! #n' "oin! ini&e !he ele%en! (#n +e&e($i+e& in !e$% of nol *#l)e #

F$o% !he #+o*e e"$eion !he h#"e f)n(!ion Ni (#n +e&e$i*e& #n& will +e # follow.

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Now )+!i!)!ing !he nol (oo$&in#!e in !e$% of 1 '1 ## + #! no&e 1 - '- # # + #! no&e - 0 '0 # # +#! no&e 0 #n& 3 '3 # # + #! no&e 3 !he #+o*e e"$eion

(#n +e $e2w$i!!en #?

Q6. WHAT ARE LARANEJS AND SERENDIPITYELEMENTS>

"agrange's element? In!e$n#l No&e P$een!. Two &i%enion#lL#g$#nge ele%en! (on!#in in!e$n#l no&e whi(h #$e no!(onne(!e& !o o!he$ no&e.E#%"le? Nine No&e& Re(!#ng)l#$ Ele%en! #n& 18 No&e&

Re(!#ng)l#$ Ele%en!

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Serendipity #lement:Highe$ o$&e$ L#g$#nge ele%en! (on!#in in!e$n#l no&e whi(h&o no! (on!$i+)!e !o !he in!e$2ele%en! (onne(!i*i!'. Howe*e$

!hee (#n +e eli%in#!e& +' (on&en#!ion "$o(e&)$e whi(hnee& e!$# (o%")!#!ion. The eli%in#!ion of !hee in!e$n#lno&e $e)l! in $e&)(!ion in i,e of !he ele%en! %#!$i(e.Al!e$n#!i*el' one (#n &e*elo" h#"e f)n(!ion of !wo&i%enion#l ele%en! whi(h (on!#in no&e onl' on !he+o)n$ie. Thee ele%en! #$e (#lle& e$en&i"i!' ele%en!.E#%"le? 32 No&e& :2 No&e& #n& 1- No&e& Re(!#ng)l#$ele%en!.

Tho)gh +' )ing e$en&i"i!' ele%en! %#!$i i,e (#n +e$e&)(e& +)! i! i o+e$*e& !h#! !he L#g$#nge ele%en! h#*e #+e!!e$ &eg$ee of (o%"le!ene in "ol'no%i#l f)n(!ion (o%"#$e!o e$en&i"i!' ele%en! Lee$ n)%+e$ of "#$#i!i( !e$%.The$efo$e L#g$#nge ele%en! "$o&)(e (o%"#$#!i*el' f#!e$ #n&+e!!e$ #(()$#('.

Q8. WHAT IS EOMETRI5 INVARIAN5E>Di"l#(e%en! h#"e ho)l& no! (h#nge wi!h # (h#nge in lo(#l(oo$&in#!e '!e%. Thi (#n +e #(hie*e& if "ol'no%i#l i+#l#n(e& in (#e #ll !e$% (#nno! +e (o%"le!e&. Thi K+#l#n(e&J$e"$een!#!ion (#n +e #(hie*e& wi!h !he hel" of P#(#l !$i#ngle

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in (#e of !wo2&i%enion#l "ol'no%i#l. Fo$ e#%"le fo$ #"ol'no%i#l h#*ing fo)$ !e$% !he in*#$i#n(e (#n +e o+!#ine& if!he following e"$eion i ele(!e& f$o% !he P#(#l !$i#ngle.

The geo%e!$i( in*#$i#n(e (#n +e en)$e& +' !he ele(!ion of !he(o$$e"on&ing o$&e$ of !e$% on ei!he$ i&e of !he #i of'%%e!$' of P#(#lJ T$i#ngle hown in fig)$e.

Q8. WHAT IS ISOPARAMETRI5 FORMULATION ANDMENTION ITS ADVANTAES IN FEA>

The !wo o$ !h$ee &i%enion#l ele%en! #$e of $eg)l#$ geo%e!$'e.g. !$i#ng)l#$ #n& $e(!#ng)l#$ ele%en! h#*ing !$#igh! e&ge.Hen(e fo$ !he #n#l'i of #n' i$$eg)l#$ geo%e!$' i! i &iffi()l! !o

)e )(h ele%en! &i$e(!l'. If we )e !$i#ng)l#$ ele%en! fo$%ehing # ()$*e& )$f#(e !he $e)l! i hown in fig)$e #.

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Tho)gh !$i#ng)l#$ ele%en! (#n )i!#+le #""$oi%#!e !he(i$()l#$ +o)n$' of !he (on!in))% +)! !he ele%en! (loe !o!he (en!e$ +e(o%e len&e$ #n& hen(e #ffe(! !he #(()$#(' offini!e ele%en! ol)!ion. One "oi+le ol)!ion !o !he "$o+le% i!o $e&)(e !he heigh! of e#(h $ow of ele%en! # we #""$o#(h !o!he (en!e$. 4)! )nne(e#$' $efining of !he (on!in))%gene$#!e $el#!i*el' l#$ge n)%+e$ of ele%en! #n& !h)

in($e#e (o%")!#!ion !i%e. Al!e$n#!i*el' when %ehing i&one )ing $e(!#ng)l#$ ele%en! # hown in Fig)$e +. !he #$e#of (on!in))% e(l)&e& f$o% !he fini!e ele%en! %o&el iignifi(#n!l' #&e;)#!e !o "$o*i&e in(o$$e(! $e)l!.

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In o$&e$ !o i%"$o*e !he #(()$#(' of !he $e)l! one (#n gene$#!e%eh )ing *e$' %#ll ele%en!. 4)! !hi will ignifi(#n!l'in($e#e !he (o%")!#!ion !i%e. Ano!he$ "oi+le w#' i !o )e #

(o%+in#!ion of +o!h $e(!#ng)l#$ #n& !$i#ng)l#$ ele%en!.4)! )(h !'"e of (o%+in#!ion %#' no! "$o*i&e !he +e! ol)!ionin !e$% of #(()$#(' in(e &iffe$en! o$&e$ "ol'no%i#l #$e )e&!o $e"$een! !he fiel& *#$i#+le fo$ &iffe$en! !'"e of ele%en!.Alo !he !$i#ng)l#$ ele%en! %#' +e len&e$ #n& !h) (#n #ffe(!!he #(()$#('.In Fig)$e ( !he #%e (on!in))% i &i($e!i,e& wi!h$e(!#ng)l#$ ele%en! ne#$ (en!e$ #n& wi!h fo)$2no&e;)#&$il#!e$#l ele%en! ne#$ +o)n$'. Thi fo)$2no&e;)#&$il#!e$#l ele%en! (#n +e &e$i*e& f$o% $e(!#ng)l#$ ele%en!)ing !he (on(e"! of %#""ing. Uing !he (on(e"! of %#""ing$eg)l#$ !$i#ng)l#$ $e(!#ng)l#$ o$ oli& ele%en! in n#!)$#l(oo$&in#!e '!e% /nown # "#$en! ele%en! (#n +e!$#nfo$%e& in!o glo+#l 5#$!ei#n (oo$&in#!e '!e% h#*ing

#$+i!$#$' h#"e wi!h ()$*e& e&ge o$ )$f#(e.

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Fig)$e hown +elow how !he "#$en! ele%en! in n#!)$#l(oo$&in#!e '!e% #n& !he %#""e& ele%en! in glo+#l 5#$!ei#n'!e%.

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The geo%e!$' of #n ele%en! %#' +e e"$ee& in !e$% of !hein!e$"ol#!ion f)n(!ion # follow.

Whe$enBNo.of No&eNiBIn!e$"ol#!ion F)n(!ioni'i,iB5oo$&in#!e of Nol Poin! of !he Ele%en!

A !he #%e h#"e f)n(!ion #$e )e& fo$ +o!h !he fiel& *#$i#+le#n& &e($i"!ion of ele%en! geo%e!$' !he %e!ho& i /nown #

io"#$#%e!$i( %#""ing. The ele%en! &efine& +' )(h # %e!ho&i /nown # #n io"#$#%e!$i( ele%en!. Thi %e!ho& (#n +e )e&!o !$#nfo$% !he n#!)$#l (oo$&in#!e of # "oin! !o !he 5#$!ei#n(oo$&in#!e '!e% #n& *i(e *e$#.

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(umerical )roblem:

Determine the Cartesian coordinate of the point P (= 0.8,

= 0.9) as shown in Figure.

The relaton !et"een t"o coor#nate $%$te&$can !e re're$ente# through ther nter'olatonfuncton$. Therefore the alue$ of thenter'olaton functon at 'ont P "ll !e

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Thu$ the coor#nate of 'ont P n *arte$ancoor#nate $%$te& can !e calculate# a$

h!s the coordinate of point P (= 0.8, = 0.9) in

Cartesian coordinate s"stem wi## $e %.&', %.'%.

Import!t" +egar#le$$ of the nu&!er ofno#e$ or 'o$$!le curature of e#ge$ the$ol# ele&ent $ ,u$t l-e a 'lane ele&ent"hch $ &a''e# nto the $'ace of natural coor#nate$ .e

/. *once't of aco!an &atr an# t$ utlt%.A aret% of #erate$ of the nter'olatonfuncton$ "th re$'ect to the glo!alcoor#nate$ are nece$$ar% to for&ulate theele&ent $tne$$ &atrce$. A$ the !othele&ent geo&etr% an# araton of the $ha'e

functon$ are re're$ente# n ter&$ of thenatural coor#nate$ of the 'arent ele&ent$o&e a##tonal &athe&atcal o!$tacle ar$e$.For ea&'le n ca$e of ealuaton of the$tran ector the o'erator &atr $ "th

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re$'ect to an# % !ut the nter'olatonfuncton $ "th and . Therefore the o'erator&atr $ to !e tran$for&e# for ta-ng

#erate "th and . The relaton$h'!et"een t"o coor#nate $%$te&$ &a% !eco&'ute# !% u$ng the chan rule of 'artal#erentaton a$

The a!oe e4uaton$ can !e e're$$e# n&atr for& a$ $ho"n !elo"5

The &atr 67 $ #enote# a$ aco!an &atr"hch $5

Snce for $o'ara&etrc for&ulaton$

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8ence

"here n $ the nu&!er of no#e$ n anele&ent.S&larl% one can calculate the other ter&$ 12

21an# 22of the aco!an &atr. 8ence

Snce

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8ence

Su''o$e

$ the ele&ent &atr re're$entngthe ner$e of 67

Therefore

S&larl% for t#r$$ %&m$!'&o!( )'$ thefollo"ng relaton e$t$ !et"een the

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#erate o'erator$ n the glo!al an# thenatural coor#nate $%$te&.

9here

or

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/:. E;AL+IS>PA+AMET+I* ELEMENT.

9e hae $een that

For t"o #&en$onal 'lane $tre$$?$tranfor&ulaton the $tran ector can !ere're$ente# a$

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>r u$ng 67 &atr "e can "rte5

The a!oe e're$$on can !e re"rtten n&atr for& a$5

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For an n no#e ele&ent the #$'lace&ent ucan !e re're$ente# a$

an# $&larl% for @ ".Therefore

Therefore the $tran &atr can !e "rtten a$5

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*o&'arng the a!oe e4uaton "th $tran#$'lace&ent e4uaton5

9e get the 67 &atr a$5

It $ nece$$ar% to tran$for& ntegral$ fro&*arte$an to the natural coor#nate$ a$ "ellfor calculaton of the ele&ental $tne$$

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&atr n $o'ara&etrc for&ulaton. The#erental area relaton$h' can !ee$ta!l$he# fro& a#ance# calculu$ an# the

ele&ental area n *arte$an coor#nate can !ere're$ente# n ter&$ of area n naturalcoor#nate$ a$5

8ere

$ the #eter&nant of the aco!an &atr. The$tne$$ &atr for a t"o #&en$onal ele&ent&a% !e e're$$e# a$ 5

The a!oe e're$$on n *arte$an coor#nate$%$te& can !e change# to the naturalcoor#nate $%$te& a$ follo"$ to o!tan theele&ental $tne$$ &atr

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N*m$r&)( Pro+($m"*alculate the aco!an &atr an# the $tran#$'lace&ent &atr for four no#e t"o

#&en$onal 4ua#rlateral ele&ent$corre$'on#ng to the gau$$ 'ont (0.B3B0.B3B) a$ $ho"n n Fgure

The aco!an &atr for a four no#e ele&ent $gen !%

For the four no#e ele&ent one can Cn# thefollo"ng relaton$.

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No" for a four no#e 4ua#rlateral ele&entthe aco!an &atr "ll !eco&e5

P!tting the a#!es of * a$ 0.B3B an# 0.B3Bre$'ectel% one "ll o!tan the follo"ng.

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/D. 98AT A+E IS>PA+AMET+I*S


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the ele&ent are #eCne# !% the $ha'efuncton$ (NG) a$ !elo".

If N H NG then the ele&ent $ calle#$o'ara&etrc.

If the geo&etr% of ele&ent $ #eCne# !%$ha'e functon$ of or#er hgher than that forre're$entng the araton of #$'lace&ent$then the ele&ent$ are calle#'*p$rprm$tr&),

If the geo&etr% of ele&ent $ #eCne# !%

$ha'e functon$ of or#er lo"er than that forre're$entng the araton of #$'lace&ent$then the ele&ent$ are calle#'*+prm$tr&)

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/10. 98AT A+E T8E IFFE+ENT TYPES >FELEMENTS


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!. +ectangular ele&entc./ua#rlateral ele&ent#. /ua#rlateral for&e# !% t"o

trangle$e. /ua#rlateral for&e# !% four

trangle$

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3) Three #&en$onal ele&ent$ (for 3'ro!le&$ l-e te&'erature $tre$$ o"eloct%)

a. Tetrahe#ron!. +ectangular !rc- ele&ent

/11. NN IN FEA.The ntegraton$ "e generall% encounter nCnte ele&ent &etho#$ are 4ute co&'lcate#

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an# t $ not 'o$$!le to Cn# a clo$e# for&$oluton$ to tho$e 'ro!le&$. Eact an#e'lct ealuaton of the ntegral a$$ocate#

to the ele&ent &atrce$ an# the loa#ngector $ not al"a%$ 'o$$!le !ecau$e of thealge!rac co&'let% of the coeKcent of the#erent e4uaton (.e. the $tne$$ nuencecoeKcent$ ela$tct% &atr loa#ngfuncton$ etc.). In the Cnte ele&ent anal%$$"e face the 'ro!le& of ealuatng thefollo"ng t%'e$ of ntegraton$ n one t"o an#three #&en$onal ca$e$ re$'ectel%. The$eare nece$$ar% to co&'ute ele&ent $tne$$an# ele&ent loa# ector.A''ro&ate $oluton$ to $uch 'ro!le&$ are'o$$!le u$ng certan nu&ercal techn4ue$.

Seeral nu&ercal techn4ue$ are aala!le n&athe&atc$ for $olng #eCnte ntegraton'ro!le&$ nclu#ng 'ont ruletra'eo#alrule S&'$on$ 1?3r# ruleS&'$on$ 3?:th rule an# Jau$$ /ua#raturefor&ula. A&ong the$e Jau$$ /ua#raturetechn4ue $ &o$t u$eful one for $olng'ro!le&$ n Cnte ele&ent &etho#.

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The conce't of Jau$$ /ua#rature $ Cr$tllu$trate# n one #&en$on n the contet ofan ntegral n the for& of

To tran$for& fro& an ar!trar% nteral of 1 2to an nteral of 1 1 "e nee# tochange the ntegraton functon fro& f() to() according#". h!s, for a #inear ariation in one

dimension, one can write the fo##owing re#ations.

Nu&ercal ntegraton !a$e# on Jau$$/ua#rature a$$u&e$ that the functon () wi##$e ea#!ated oer an intera# + 1. *on$#erngan one#&en$onal ntegral Jau$$

/ua#rature re're$ent$ the ntegral () in theform of

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-here, the 1, 2, 3, ..., nre're$ent$ n nu&!er$ of'ont$ -no"n a$ Jau$$ Pont$ an# thecorre$'on#ng coeKcent$ "1 "2 "3 O "n

are -no"n a$ "eght$. The locaton an#"eght coeKcent$ of Jau$$ 'ont$ arecalculate# !% Legen#re 'ol%no&al$. 8enceth$ &etho# $ al$o $o&et&e$ referre# a$Jau$$Legen#re /ua#rature &etho#. The$u&&aton of the$e alue$ at n $a&'lng'ont$ ge$ the eact $oluton of a 'ol%no&alntegran# of an or#er u' to 2n1. For ea&'lecon$#erng $a&'lng at t"o Jau$$ 'ont$ "ecan get eact $oluton for a 'ol%no&al of anor#er (221) or 3. The u$e of &ore nu&!erof Jau$$ 'ont$ ha$ no eect on accurac% ofre$ult$ !ut ta-e$ &ore co&'utaton t&e.

O!$- Po&!t Form*(*on$#erng n H 1 then

T.o-Po&!t Form*(If "e con$#er n H 2 then

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Jau$$ 'ont$ an# corre$'on#ng "eght$

N*m$r&)( Pro+($m"Ealuate

u$ng one t"o an# three 'ont gau$$/ua#rature.efore a''l%ng the Jau$$ /ua#raturefor&ula the e$tng l&t$ of ntegraton

$houl# !e change# fro& 60 17 to 61 Q17.A$$u&ng

the u''er an# lower #imit can $e changed. i.e., at =0, = + and at = , = /. h!s, p!tting these

conditions and so#ing for a * $, we get a = + and $ = &.

he re#ation $etween two coor#nate $%$te&$ "ll!eco&e

Therefore the ntal e4uaton can !e "rttena$

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U'&!/ o!$ po&!t /*'' Q*%rt*r$"

U'&!/ T.o po&!t /*'' Q*%rt*r$"

U'&!/ T#r$$ po&!t /*'' Q*%rt*r$"

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/12. EFINE STATI* *>NENSATI>N.The hgher or#er Lagrangan ele&ent$ (.e.nne no#e $teen no#e rectangular ele&ent)

contan nu&!er of nternal no#e$. Th$ $nece$$ar% $o&et&e$ for the co&'letene$$ ofthe #e$re# 'ol%no&al u$e# n #$'lace&entfuncton for #eraton of nter'olatonfuncton. The$e nternal no#e$ are notconnecte# to the a#,onng ele&ent$ n the

a$$e&!lage.

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Thu$ the #$'lace&ent$ of the$e no#e$ arenot re4ure# to for&ulate oerall e4ul!ru&

e4uaton$ of the $tructure. Th$ l&t$ theu$efulne$$ of the$e ele&ent$. A techn4ue-no"n a$ R$tatc con#en$aton can !e u$e#to $u''re$$ the #egree$ of free#o&a$$ocate# "th the nternal no#e$ n the Cnalco&'utaton. The techn4ue of $tatccon#en$aton $ e'lane# !elo". The

e4ul!ru& e4uaton for a $%$te& aree're$$e# n the Cnte ele&ent for& a$

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9here FU 6V7 an# #U are the loa# ector$tne$$ &atr an# #$'lace&ent ector forthe entre $tructure. The a!oe e4uaton can

!e rearrange# !% $e'aratng the releantter&$ corre$'on#ng to nternal an# eternalno#e$ of the ele&ent$.

8ere #U an# #eU are the #$'lace&entector$ corre$'on#ng to nternal an# eternalno#e$ re$'ectel%. S&larl% FU an# FeUare force ector$ corre$'on#ng to nternalan# eternal no#e$. No" the a!oee're$$on can !e "rtten n the follo"ng

for& $e'aratel%.

The $tne$$ &atr an# no#al loa# ector

corre$'on#$ to the nternal no#e$ can !e$e'arate# out an# can !e re"rtten a$5

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Su!$ttutng the alue of #U o!tane# fro&the a!oe e4uatonW the follo"ng e're$$on"ll !e o!tane#.

8ere the e4uaton$ are re#uce# to a for&

nolng onl% the eternal no#e$ of theele&ent$. The a!oe re#uce# $u!$tructuree4uaton$ are a$$e&!le# to achee theoerall e4uaton$ nolng onl% the !oun#ar%un-no"n$. Thu$ the a!oe e4uaton can !ere"rtten a$

>r

8ere 6Vc7 $ calle# con#en$e# or re#uce#$tne$$ &atr an# FcU $ the con#en$e# or

eecte no#al loa# ector corre$'on#ng toeternal no#e$ of the ele&ent$. In th$'roce$$ the $e of the &atr for ner$on"ll !e co&'aratel% $&all. The un-no"n#$'lace&ent$ of the eteror no#e$ #eU can

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!e o!tane# !% nertng the &atr 6Vc7.>nce the alue$ of #eU are o!tane# the#$'lace&ent$ of nternal no#e$ #U can !e

foun# fro& e4uaton5

/13. S8>+T N>TE >N A=ISYMMET+I*ELEMENTS.Man% three#&en$onal 'ro!le&$ $ho"$%&&etr% a!out an a$ of rotaton. If the'ro!le& geo&etr% $ $%&&etrc a!out an a$an# the loa#ng an# !oun#ar% con#ton$ are$%&&etrc a!out the $a&e a$ the 'ro!le&

$ $a# to !e a$%&&etrc. Such three#&en$onal 'ro!le&$ can !e $ole# u$ngt"o#&en$onal Cnte ele&ent$. Thea$%&&etrc 'ro!le&$ are most conenient#"defined $" po#ar coordinate s"stem with coordinates (r, ,

1) as shown in Figure.

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Thu$ for a$%&&etrc anal%$$ follo"ngcon#ton$ are to !e $at$Ce#.1. The #o&an $houl# hae an a$ of$%&&etr% an# $ con$#ere# a$ a$.&. he #oadings on the domain has to $e s"mmetric a$o!t

the ais of reo#!tion, th!s the" are independent of

circ!mferentia# coordinate .3. The !oun#ar% con#ton an# &ateral'ro'erte$ are $%&&etrc a!out the $a&e a$

an# "ll !e n#e'en#ent of crcu&ferentalcoor#nate.2is"mmetric so#ids are of tota# s"mmetr" a$o!t the ais

of reo#!tion (i.e., 1+ais), the fie#d aria$#es, s!ch as the

stress and deformation is independent of rotationa# ang#e

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.Therefore the Cel# ara!le$ can !e #eCne#a$ a functon of (r ) an# hence the 'ro!le&!eco&e$ a t"o #&en$onal 'ro!le& $&lar

to tho$e of 'lane $tre$$?$tran 'ro!le&$.A$%&&etrc 'ro!le&$ nclu#e$ crcularc%ln#er loa#e# "th unfor& eternal ornternal 're$$ure crcular "ater tan-'re$$ure e$$el$ ch&ne% !oler crcularfootng re$tng on $ol &a$$ etc.

R$(t&o! +$t.$$! Str&! !%&'p()$m$!tAn a$%&&etrc 'ro!le& $ rea#l% #e$cr!e#n c%ln#rcal 'olar coor#nate $%$te&5 r and. 3ere, meas!res the ang#e $etween the p#ane

containing the point and the ais of the coordinates"stem.

2t = 0, the radia# and aia# coordinates coincide withthe g#o$a# Cartesian 4 and 5 coordinates.

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6f the #oading consists of radia# and aia# components that

are independent of and the materia# is either isotropic or

orthotropic and the materia# properties are independent of

, the disp#acement at an" point wi## on#" hae radia#uran# u co&'onent$. The onl% $tre$$

co&'onent$ that "ll !e nonero are7rr, 7, 7and r.A #erental ele&ent of the !o#% n the r'lane $ $ho"n n Fgure. The ele&entun#ergoe$ #efor&aton n the ra#al #recton.

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Therefore t ntate$ ncrea$e ncrcu&ference an# a$$ocate# crcu&ferental$tran.

Let #enote the ra#al #$'lace&ent a$ u thecrcu&ferental #$'lace&ent a$ an# the

aal #$'lace&ent a$ ". a$he# lnere're$ent$ the #efor&e# 'o$ton$ of the !o#%n Fgure. The ra#al $tran can !e calculate#fro& the a!oe #agra& a$

r

uudr

r

uu

drr

=

+=

,

z

wwdz

z

ww

dzz

=

+= ,

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*on$#erngthe origina# arc #ength ers!s the deformedarc #ength, the differentia# e#ement !ndergoes an

epansion in the circ!mferentia# direction. efore

deformation, #et the arc #ength is ass!med as ds = rd.2fter deformation, the arc #ength wi## $ecome ds = (r/!)

d. h!s, the tangentia# strain wi## $e

S&larl% the $hear $tran "ll !e

Thu$ there are four $tran co&'onent$'re$ent n th$ ca$e an# $ gen !%

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/1X. ere the $ha'e functon$ n ter&$ ofnatural coor#nate $%$te&.S#p$ *!)t&o! or TO !o%$ +r$($m$!tLet u$ con$#er the natural coor#nate of thecenter of the ele&ent a$ 0 an# the naturalcoor#nate of the no#e

s and & are + and /respectie#". herefore, the nat!ra# coordinate at an"

point can $e represented $"


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rp#&)( r$pr$'$!tt&o! o S#p$ *!)t&o!' or t.o !o%$ +r $($m$!t

S#p$ *!)t&o! or TREE !o%$% +r$($m$!t

For a three no#e !ar ele&ent a$ $ho"n nFgure the $ha'e functon "ll !e 4ua#ratc nnature.

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rp#&)( r$pr$'$!tt&o! o S#p$ *!)t&o!' or t#r$$ !o%$ +r$($m$!t

S#p$ *!)t&o! or t.o %&m$!'&o!($($m$!t'

Four node rectangular element

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Nine node rectangular element

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Two dimensional eight node rectangular

element

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Q15, r&t$ '#ort !ot$ o! S#$(( $($m$!t,

A $hell $ a cure# $urface "hch !% rtue ofther $ha'e can "th$tan# !oth &e&!ranean# !en#ng force$. A $hell $tructure can ta-ehgher loa#$ f &e&!rane $tre$$e$ are're#o&nant "hch $ 'r&arl% cau$e# #ue ton'lane force$ ('lane $tre$$ con#ton).

8o"eer locale# !en#ng $tre$$e$ "lla''ear near loa# concentraton$ or geo&etrc#$contnute$. The $hell$ are analogou$ toca!le or arch $tructure #e'en#ng on "hetherthe $hell re$$t$ ten$le or co&'re$$e$tre$$e$ re$'ectel%.Fe" a#antage$ u$ng $hell ele&ent$ are 5

1. 8gher loa# carr%ng ca'act%2. Le$$er thc-ne$$ an# hence le$$er #ea#loa#3. Le$$er $u''ort re4ure&entX. Larger u$eful $'ace

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B. 8gher ae$thetc alue.

The ea&'le of $hell $tructure$ nclu#e$

large$'an roof coolng to"er$ ''ng$%$te& 're$$ure e$$el arcraft fu$elageroc-et$ "ater tan- arch #a&$ an# &an%&ore. Een n the Cel# of !o&echanc$ $hellele&ent$ are u$e# for anal%$$ of $-ull*ru$tacean$ $ha'e re# !loo# cell$ etc.Shell &a% !e cla$$Ce# "th $eeralalternate$ "hch are 're$ente# n Fgure.

e'en#ng u'on #eecton n tran$er$e#recton #ue to tran$er$e $hear force 'erunt length the $hell can !e cla$$Ce# nto$tructurall% thn or thc- $hell. Further

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#e'en#ng u'on the thc-ne$$ of the $hell nco&'ar$on to the ra# of curature of the $urface the $hell $ referre# to a$

geo&etrcall% thn or thc- $hell. T%'call% fthc-ne$$ to ra# of curature $ le$$ than0.0B then the $hell can !e a$$u&e# a$ a thn$hell. For &o$t of the engneerng a''lcatonthe thc-ne$$ of $hell re&an$ "thn 0.001 to0.0B an# treate# a$ thn $hell.

A''*mpt&o!' or T#&! S#$(( T#$orThn $hell theore$ are !a$call% !a$e# onLoeVrcho a$$u&'ton$ a$ follo"$.1. A$ the $hell #efor&$ the nor&al to the un#efor&e# #le $urface re&an $traght an#nor&al to the #efor&e# #le $urface

un#ergo no eten$on. .e. all $tranco&'onent$ n the #recton of the nor&al tothe #le $urface $ ero.2. The tran$er$e nor&al $tre$$ $ neglecte#.

Thu$ a!oe a$$u&'ton$ re#uce the three#&en$onal 'ro!le&$ nto t"o #&en$onal.

O$r&$. o S#$(( F&!&t$ E($m$!t' F(t '#$(( $($m$!t

The geo&etr% of the$e t%'e$ of ele&ent$ $a$$u&e# a$ at. The cure# geo&etr% of $hell

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$ o!tane# !% a$$e&!lng nu&!er of atele&ent$. The$e ele&ent$ are !a$e# onco&!naton of &e&!rane ele&ent an#

!en#ng ele&ent that enforce# Vrcho$h%'othe$$. It $ &'ortant to note that thecou'lng of &e&!rane an# !en#ng eect$#ue to curature of the $hell $ a!$ent n thenteror of the n##ual ele&ent$.+ *r$% '#$(( $($m$!t*ure# $hell ele&ent$ are $%&&etrcal a!outan a$ of rotaton. A$ n ca$e of a$%&&etrc'late ele&ent$ &e&!rane force$ for the$eele&ent$ are re're$ente# "th re$'ecte# to&er#an #recton. A$ (" N M) an# ncrcu&ferental #recton$ a$(" N M).8o"eer the #Kculte$ a$$ocate# "th the$e

ele&ent$ nclu#e$ #Kcult% n #e$cr!nggeo&etr% an# acheng nterele&entalco&'at!lt%. Al$o the $at$facton of rg#!o#% &o#e$ of !ehaour $ acute n cure#$hell ele&ent$.) So(&% '#$(( $($m$!t

Though u$e of 3 $ol# ele&ent $ anothero'ton for anal%$$ of $hell $tructure #ealng"th too &an% #egree$ of free#o& &a-e$ tunecono&c n ter&$ of co&'utaton t&e.Further #ue to $&all thc-ne$$ of $hell

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ele&ent the $tran nor&al to the $urface$ a$$ocate# "th er% large $tne$$coeKcent$ an# thu$ &a-e$ the e4uaton$ ll

con#tone#.% $/$!$rt$% '#$(( $($m$!t'8ere ele&ent$ are #ere# !% #egeneratng a3 $ol# ele&ent nto a $hell $urface ele&ent!% #eletng the nter&e#ate no#e$ n thethc-ne$$ #recton an# then !% 'ro,ectng theno#e$ on each $urface to the $urface a$$ho"n n Fgure.

Th$ a''roach ha$ the a#antage of !engn#e'en#ent of an% 'artcular $hell theor%.

Th$ a''roach can !e u$e# to for&ulate ageneral $hell ele&ent for geo&etrc an#&ateral nonlnear anal%$$. Such ele&ent ha$!een e&'lo%e# er% $ucce$$full% "hen u$e#"th D or n 'artcular 1Z no#e$. 8o"eer

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the 1Zno#e ele&ent $ 4ute e'en$e nco&'utaton. In a #egenerate# $hell &o#elthe nu&!er$ of un-no"n$ 're$ent are Ce 'er

no#e (three $urface #$'lace&ent$ an#t"o #rector rotaton$). Mo#eratel% thc- $hell$can !e anal%$e# u$ng $uch ele&ent$.8o"eer $electe an# re#uce# ntegratontechn4ue$ are nece$$ar% to u$e #ue to $hearloc-ng eect$ n ca$e of thn $hell$. Thea$$u&'ton$ for #egenerate# $hell are $&larto the +e$$nerMn#ln a$$u&'ton$./1Z. 9rte Short Note on Prnc'le of ;rtual9or-.

he princip#e of irt!a# wor: is a er" !sef!# approach

for so#ing arieties of str!ct!ra# mechanics pro$#em.

-hen the force and disp#acement are !nre#ated to theca!se and effect re#ation, the wor: is ca##ed irt!a# wor:.

herefore, the irt!a# wor: ma" $e ca!sed $" tr!e force

moing thro!gh imaginar" disp#acements or ice ersa.

h!s, the princip#e of irt!a# wor: can $e diided into

two categories; (a) princip#e of irt!a# forces and ($)

princip#e of irt!a# disp#acements. he princip#e of irt!a#

forces esta$#ishes the compati$i#it" conditions. heprincip#e of irt!a# disp#acements esta$#ishes the

conditions of e

mode# of the finite e#ement techni

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he eterna# irt!a# wor: is the wor: done $" rea# #oad

moing thro!gh imaginar" disp#acements in a str!ct!re.

hese #oads inc#!de $oth the #oad distri$!ted oer the

entire s!rface and o#!me. h!s, the irt!a# wor: done $"the eterna# force is;

-here, !, and w are the components of the irt!a#

disp#acements in , " and 1 direction respectie#". F>, F>"and F>1are the s!rface forces and F?, F?"and F?1are the

$od" forces in , " and 1 direction respectie#". 6n the

a$oe e

entire s!rface in the first term and oer the entire o#!me

in the second term. he a$oe epression can $e rewrittenas;

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2ccording to princip#e of irt!a# wor:, the wor: done $"eterna# forces d!e to the irt!a# disp#acement of a

str!ct!re in e


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nter'olatng functon $ generall% a 'ol%no&al"th n #egree "hch auto&atcall% 'ro#e$ a$nglealue# an# contnuou$ Cel#. In Cnte

ele&ent lterature th$ nter'olaton functon (N ) $referre# to RSha'e functon a$ "ell. For lnearnter'olaton n "ll !e 1 an# for 4ua#ratcnter'olaton n "ll !eco&e 2 an# $o on. There aret"o t%'e$ of nter'olaton functon$ na&el% ()Lagrange nter'olaton an# () 8er&tannter'olaton. Lagrange nter'olaton functon $"#el% u$e# n 'ractce. 8ere the a$$u&e# functonta-e$ on the $a&e alue$ a$ the gen functon at$'ecCe# 'ont$. In ca$e of 8er&tan nter'olatonfuncton the $lo'e$ of the functon al$o ta-e the$a&e alue$ a$ the gen functon at $'ecCe#'ont$.

m.tech structural engg fea important questions

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