ME 6603- Finite Element Analysis 16 Marks UNIT- I 1. Determine Determine using any weighte weighte resiual resiual te!hni"ue te!hni"ue the tem#eratur tem#eraturee istri$uti%n istri$uti%n al%ng al%ng a !ir!ular !ir!ular &in %& length 6 !m an raius 1 !m. the &in is atta!he t% a $%iler wh%se wall tem#erature is 1'0(! an the &ree en is insulate. Assume !%n)e!ti%n !%e&&i!ient h*10 +,!m (!.!%nu!ti%n (!.!%nu!ti%n !%e&&i!ient !%e&&i!ient k*0 w, !m (! an T * '0(!. the g%)erning e"uati%n &%r the heat trans&er thr%ugh the &in is gi)en $y
Assume a##r%#riate $%unary !%niti%ns an !al!ulate the tem#eratures at e)ery 1 !m &r%m the le&t en. /AU MA,2UNE-014
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Deri)e Deri)e the g%)ern g%)erning ing e"uati% e"uati%n n &%r a ta#ere ta#ere r% &i5e &i5e at %ne en en an su$e!t su$e!te e t% its %wn %wn sel& weight an a &%r!e # at the %ther en as sh%wn in &ig. let the length %& the $ar $e l an let the !r%ss se!ti%n )ary linearly &r%m A1 at the &i5e en t% A at the &ree en. E an y re#resent the y%ungs m%ulus an s#e!&i! weight %& the msterial %& the $ar. 7%n)ert this e"uati%n int% its weak &%rm an hen!e etermine the matri!es &%r s%l)ing using thr rit8 te!hni"ue. /AU MA,2UNE-014
3. Using !%ll%!ati%n !%ll%!ati%n meth%9&in meth%9&in the ma5imum ma5imum is#la!ement is#la!ement %& the ta#ere ta#ere r% r% as sh%wn sh%wn in &ig E* :10 N,!m9y * 0.0 N,!m. /AU N;<,DE7-01'4
3. Using !%ll%!ati%n !%ll%!ati%n meth%9&in meth%9&in the ma5imum ma5imum is#la!ement is#la!ement %& the ta#ere ta#ere r% r% as sh%wn sh%wn in &ig E* :10 N,!m9y * 0.0 N,!m. /AU N;<,DE7-01'4
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7%nsiser 7%nsiser the $ar $ar sh%wn sh%wn in &ig etermine etermine the the n%al is#la!eme is#la!ements9 nts9 element element stresses stresses an su##%rt su##%rt rea!ti%ns. /AU N;<,DE7-01'4
. =ist an $rie&ly es!ri$e the general general ste#s ste#s %& the &inite element element meth%. meth%. /AU MA MA,2UNE,2UNE-01'4 01'4
6.
The i&&erential e"uati%n %& a #hysi!al #hen%men%n is gi)en $y
The $%unary !%niti%ns are >
%$tain %ne term a##r%5imate s%luti%n $y using galerkins meth% %& weighte resiuals. /AU MA,2UNE-01'4
. De)el%# the weak &%rm an etermine the is#la!ement &iel &%r a !antile)er $eam su$e!te t% a uni&%rmly istri$ute. =%a an a #%int l%a a!ting at the &ree en. /AU N;<,DE7-0134
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7%nsier a #alne wall with a uni&%rmly istri$ute heat s%ur!e.%$tain the &inite element &%rmulati%n &%r the a$%)e !ase $ase %n the stat%i%nary %& a &un!ti%nal. /AU N;<,DE7-0134
@. %l)e the i&&erential e"uati%n &%r a #hysi!al #r%$lem e5#resse as +ith $%unary !%niti%ns as yB0C * 0 an yB10C * 0 Using BiC %int !%ll%!ati%n meth% BiiC u$ %mine !%ll%!ati%n meth% BiiiC =east s"uares meth% an Bi)C alerkins meth% /AU MA,2UNE-0134
10. A sim#ly su##%rte $eam su$e!te t% uni&%rmly istri$ute l%a %)er entire s#an an it is su$e!te t% a #%int l%a at the !entre %& the s#an. 7al!ulate the e&el!ti%n using rayleigh rit8 meth% an !%m#are with e5a!t s%luti%ns. /AU MA,2UNE-0134
11. BiC Des!ri$e the hist%ri!al $a!kgr%un %& FEM. BiiC e5#lain the rele)an!e %& FEA &%r s%l)ing esign #r%$lems with the ai %& e5am#les. /AU N;<,DE7-0134
1. A r% &i5e at its ens is su$e!te t% a )ariying $%y &%r!e as sh%wn in &ig use rayleigh rit8 meth% with an assume is#la!ement &iel u * a0 a15 a5 t% etermine uB5C an stress gB5C. /AU N;<,DE7-0134
13. BiC is!uss the im#%ratan!e %& FEA in assisting esign #r%!ess.
BiiC s%l)e the %rinary i&&erential
u$e!t t% the $%unary !%niti%ns yB0C * yB1C*0 using galerkin meth% with trial &un!ti%ns n0B5C * 0 G N1B5C * 5B1-5C /AU MA,2UNE-0134
1'. BiC is!uss the &a!t%rs t% $e !%nsiere in e!retisati%n %& a %mine. BiiC s%l)e the &%ll%wing e"uati%ns using the gauss eliminati%n meth%.
/AU MA,2UNE-0134
1. The i&&erential e"uati%n &%r a #hen%men%n is gi)en $y The $%unary !%niti%ns are yB%C * 0 9 yBC * 0 &in the a##r%5imate s%luti%n using any !lassi!al te!hni"ue . tart with minimal #%ssi$le a##r%5imate s%luti%n. /AU MA,2UNE-014
16. BiC =ist an $rie&ly es!ri$e the general ste#s %& &inite element meth%.
BiiC Deri)e an e"uati%n t% &in the is#la!ement at n%e %& &i5e &i5e $eam su$e!te t% a5ial l%a # at n%e using Hayleigh rit8 meth%.
/AU MA,2UNE-014
1. Deri)e the !hara!teristi! e"uati%ns &%r the %ne imensi%nal $ar element $y using #ie!e wise e&ine inter#%lati%ns an weak &%rm %& the weighte resiual meth% /AU MA,2UNE-014
1?. BiC Deri)e the element le)el e"uati%n &%r %ne imensi%nal $ar element $ase %n the stati%nary %& a &un!ti%nal. BiiC =ist %ut the general #r%!eure &ir FEA #r%$lems. /AU MA,2UNE-014
1@. An all%y 1 m l%ng an 00 mm in !r%ss se!ti%n is &i5e at %ne en is su$e!te t% a !%m#ressi)e l%a %& 0 JN. I& the m%ulus %& elasti!ity &%r the all%y is 100 a9 &in the e!rease in the length %& the $ar. Als% etermine the stress e)el%#e an the e!rease in length at 0.m9 0. m an 0. m. s%l)e $y !%ll%!ati%n meth%. /AU N;<,DE7-014
0. An all%y 1 m l%ng an 00 mm in !r%ss se!ti%n is &i5e at %ne en is su$e!te t% a !%m#ressi)e l%a %& 0 JN. I& the m%ulus %& elasti!ity &%r the all%y is 100 a9 &in the e!rease in the length %& the $ar. Als% etermine the stress e)el%#e an the e!rease in length at 0.m9 0. m an 0. m. s%l)e $y Hit8 meth%. /AU N;<,DE7-014
1. E5#lain the #r%!ess %& is!retisati%n %& a stru!ture in &inite element meth% in etail9 with suita$le illustrati%ns &%r ea!h as#e!t $eing is!usse. /AU N;<,DE7-014
. A uni&%rm r% su$e!te t% a uni&%rm a5ial l%a is illustrate in &ig 9 the e&%rmati%n %& the $ar is g%)erne $y the i&&erential e"uati%n gi)en $el%w. Determine the is#la!ement using weighte resiual meth%.
/AU MA,2UNE-0114
3. Fin the n%al is#la!ement an element stresses &%r the $ar sh%wn in &ig
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'. Dis!uss the &%ll%wing meth%s t% s%l)e the gi)en i&&erential e"uati%n> +ith the $%unary !%niti%ns yB0C * 0 an yBKC * 0 BiC
AU MA,2UNE-0104
. F%r the s#ring system sh%wn in &ig 9 !al!ulate the gl%$al sti&&ness matri59 is#la!ements %& n%es an 3 9 the rea!ti%n &%r!e at n%e 1 an '. Als% !al!ulate the &%r!es in the s#ring . Assume k1*k3*100N,m9 k * 00 N,m 9 u1*u'*0 an # * 00 N.
/AU MA,2UNE-0104
6. Analy8e a sim#ly su##%rte $eam su$e!te t% a uni&%rmly istri$ute l%a th%ught using Hayleigh Hit8 meth%. A%#t %ne #arameter trig%n%metri! &un!ti%n. e)aluate the ma5imum e&le!ti%n an LM an !%m#are with the e5a!t s%luti%n. /AU N;<,DE7-0104
. %l)e the &%ll%wing set %& simultane%us e"uati%n $y aussian eliminati%n meth%
/AU N;<,DE7-0104
?. A sim#ly su##%rte $eam Bs#an = an &le5ural rigiity EIC !arries tw% e"ual !%n!entrate l%as at ea!h %& the "uarter s#an #%ints. Using Hayleigh Hit8 meth% etermining the e&le!ti%n uner the tw% l%as an tw% en sl%#es. /AU MA,2UNE-00@4
@. Use the aussian eliminati%n meth% t% s%l)e the &%ll%wing simultane%us e"uati%ns>
/AU MA,2UNE-00@4
30. BiC what is !%nstituti)e relati%nshi# E5#ress the !%nstituti)e relati%ns &%r a linear elasti! is%tr%#i! material in!luing initial stress an strain. BiiiC !%nsier the i&&erential e"uati%n &%r
su$e!te t% $%unary !%niti%ns yB0C
* 09yB1C *0 The &un!ti%nal !%rres#%ning t% this #r%$lem t% $e e5tremi8e is gi)en $y
&in the s%luti%n %& the #r%$lem using Hayleigh rit8 meth% $y !%nsiering a tw% term s%luti%n as yB5C *!5B1-5C !5B1-5C /AU N;<,DE7-00@4
31. BiC
/AU N;<,DE7-00@4
BiiC %l)e the &%ll%wing system %& e"uati%ns using gauss eliminati%n meth% .
/AU N;<,DE7-00@4
3.
E5#lain the aussian eliminati%n meth% &%r s%l)ing %& simultane%us linear alge$rai! e"uati%ns with e5am#les. 33. /AU MA,2UNE-00?4
3'. A !antile)er $eam %& length = is l%ae with a #%int l%a at the &ree en. Fin the ma5imum e&le!ti%n an ma5imum $ening m%ment using Hayleigh rit8 meth% using the &un!ti%n /AU MA,2UNE-00@4
3. 7%m#ute the )alue %& !entral e&le!ti%n &igure $y assuming 9 The $eam is uni&%rm thr%ugh an !arries a !entral #%int l%a.
36. BiC +rite sh%rt n%te %n galerkins meth%. BiiC +rite $rie&ly a$%ut aussian eliminati%n. /AU N;<,DE7-004
UNIT- II 1. Determine the ma5imum e&le!ti%n an sl%#e in the $eam9 l%ae as sh%wn in &ig . Determine als% the rea!ti%ns at the su##%rts. E* 00#a 9I* 0 10-6 m' * JN,m an =*1m
/AU MA,2UNE-014
. Deri)e using lagrangian #%lyn%mials the sha#e &un!ti%ns &%r %ne imensi%nal three n%e $ar element. #l%t the )ariati%n %& the same . Ken!e eri)e the sti&&ness matri5 an l%a )e!t%r. /AU MA,2UNE-014
3.
7%m#uter the sl%#e9 e&le!ti%n an rea!ti%n &%r!es &%r the !antile)er $eam %& length O=P !arrying uni&%rmly istri$ute l%a %& intensity &%. /AU N;<,DE7-01'4
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Determine the n%al is#la!ements9 stress an strain &%r the $ar sh%wn in &ig
/AU N;<,DE7-01'4
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A tw% n%e truss element is sh%wn in &ig 9 the n%al is#la!ements are U1* mm an u *?mm. !al!ulate the is#la!ement at 5*l,'9l,3 an l,.
/AU MA,2UNE-01'4
6.
F%r the tw% truss sh%wn in &ig 9 etermine the is#la!ements %& n%e 1 an the stress in element 13.
/AU MA,2UNE-0134
. Deri)e the sti&&ness matri5 an $%y &%r!e )e!t%r &%r a "uarati! s#ar element. /AU N;<,DE7-0134
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Analy8e the $eam sh%wn in &ig using &inite element te!hni"ue. Determine the r%tati%ns at the su##%rts. i)en E* 00a an I*'106 mm' /AU N;<,DE7-0134
@.
Determine the e5tensi%n %& the $ar sh%wn in &ig 9 ue t% sel& weight an a !%n!entrate l%a %& 600N a##lie at its en. gi)en L1*00mm 9$ *100mm an t*0mm.use tw% s#ar elements t% s%l)e the #r%$lem. take E*10 N,mm an g*0.?10-' N,mm.
/AU N;<,DE7-0134
10. A !antile)er $eam %& length 3.' m has an elati! s#ring su##%rt %& sti&&ness 30 JN,m at its &ree en9 where a #%int l%a %& 13 JN a!ts.Take y%ungs m%ulus as 00 a an area m%ment %& inertia %& the !r%ss se!ti%n as 110-' m' .etermine the is#la!ement an sl%#e at the n%e an the rea!ti%ns. /AU N;<,DE7-0134
11. Fig sh%ws the #in-%inte !%n&igurati%n. Determine the n%al is#la!ements an stresses in ea!h element.
/AU MA,2UNE-0134
1. F%r the $eam sh%wn in &ig 9 etermine BiC the sl%#es at n%e an 3 an BiiC )erti!al e&le!ti%n at the mi-#%int %& the istri$ute l%a. All the elements ha)e E*00a an I*106 mm'.
/AU MA,2UNE-0134
13. Determine the is#la!ements an sl%#es at the n%es &%r the $eam sh%wn in &ig. take k*00JN,m9 E*0a an I*10-' m'.
/AU N;<,DE7-014
1'. Determine the n%al is#la!ements an sl%#es &%r the $eam sh%wn in &ig. &in the m%ment at the mi#%int %& element 1. /AU N;<,DE7-014
1. A ta#ere $ar %& aluminimum is ha)ing a length %& 00 7m. The area %& !r%ss se!ti%n at the &i5e en is ?0 !m an the &ree en is 0!m with the )ariati%n %& the se!ti%nal area as linear. The $ar I su$e!te t% an a5ial l%a 10 JN at '0 mm &r%m the &i5e en. 7al!ulate the ma5imum is#la!ement an stress e)el%#e in the $ar. /AU N;<,DE7-014
16. A &i5e $eam AL %& m s#an !arries a #%int l%a %& 0 JN at a istan!e %& m &r%m A. Determine the sl%#e an e&e!ti%n uner the l%a. /AU N;<,DE7-014
1. Determine the sha#e &un!ti%n an element matri!es &%r "uarati! $ar element. /AU MA,2UNE-014
1?. Fin the n%al is#la!ement e)el%#e in the #laner truss sh%wn in &ig when a )erti!ally %wnwar l%a %& 1000 N is a##lie at n%e '. The re"uire ata are gi)en in the ta$le 1.
/AU MA,2UNE014
. 1@. F%r the #lane trusses su##%rte $y the s#ring at n%e 1 in &ig etermine the n%al is#la!ement an stresses in ea!h elemnt. =et E*10#a an A*.010-' m.
/AU MA,2UNE-014 .
0. A !%nsente l%a *0 JN is a##lie at the !entre %& a &i5e $eam %& length 3 m 9e#th 00 mm an with 10 mm. !al!ulate the e&le!ti%n an sl%#e at the mi#%int. Assume E*10 N,mm. /AU MA,2UNE-014
1. 7al!ulate n%al is#la!ement an elemental stresses &%r the truss sh%wn &ig E*0a !r%ss se!ti%nal area A*!m &%r all truss mem$ers.
/AU MA,2UNE-0114
. Esta$lish the sha#e &un!ti%ns %& an eight n%e "uarilateral element an re#resent them gra#hi!ally. /AU MA,2UNE-0114
3.
enerate the sti&&ness matri5 %& a n%e #rismati! $ar with %ne egree %& &ree%m at ea!h n%e. /AU N;<,DE7-0104
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Deri)e the !%nsistent l%a )e!t%r &%r a &i5e $eam su$e!te t% uni&%rmly istri$ute l%a thr%ugh. use sha#e &un!ti%ns !%rres#%ning t% tw% egrees %& &ree%m at ea!h %& the tw% n%es %& the element. A5ial e&%rmati%ns !an $e negle!te. /AU N;<,DE7-0104
. Determine the %int is#la!ements9 the %int rea!ti%ns9 element &%r!es an element stresses %& the gi)en truss elements.
/AU N;<,DE7-0104
6.
Deri)e the inter#%lati%n &un!ti%n &%r the %ne imensi%nal linear element with a length O=P an tw% n%es9 %ne at ea!h en 9 esigne as I an assume the %rigin %& the !%%rinate system is t% the le&t %& n%e i. /AU N;<,DE7-0104
. A !%lumn %& length 00 mm is l%ae a5ially as sh%wn in &ig. analy8e the !%lumn an e)aluate the stress an strains at salient #%ints. The y%ungPs m%ulus !an $e taken as E. A1*6. mm A *1 mm. /AU MA,2UNE-00@4
?. F%r the #rismati! $ar sh%wn &ig generate the sti&&ness matri5 !%rres#%ning t% the three !%%rinatePs inia!ate.use the &%ll%wing sha#e &un!ti%ns.
/AU MA,2UNE-00@4
@. The ste##e $ar sh%wn in &ig is su$e!te t% an in!rease in tem#erature T * ?0 ! . Determine the is#la!ements9 elements stresses an su##%rt rea!ti%ns.
/AU N;<,DE7-00@4
30.
7%nsier a tw% $ar truss su##%rts $y a s#ring sh%wn in 9 $%th $ars ha)e E* 10A an A*.010-' m. Lar %ne has a length %& m an tw% has a length %& 10 m. the s#ring sti&&ness is k* JN,m. etermine the h%ri8%ntal an )erti!al is#la!ements at the %int 1 an stresses in ea!h $ar. /AU N;<,DE7-00@4
31. BiC Deri)e the sha#e &un!ti%ns &%r a D $eam element. B?C BiiC Deri)e the sha#e &un!ti%ns &%r a D Truss element. B?C
/AU MA,2UNE-00?4
3.
Ea!h %& the &i)e $ars %& the #in %inte truss sh%wn in &ig9 has a !r%ss se!ti%nal area 0 s".!m an E*00a. /AU MA,2UNE-00?4
33.
+hy higher %rer elements are neee Determine the sha#e &un!ti%ns %& an eight n%e re!tangular element. /AU MA,2UNE-004
UNIT III 1.
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3. BiC a $ilinear re!tangular element has !%%rinates as sh%wn in &ig. an the n%al tem#eratures are T1 * 100 7 9 T * 60 !9 T3 * 0 7 9 T'* @0 !. !%m#ute the tem#erature at the #%int wh%se !%%rinates are B.9.C. als% etermine the ?0 7 is%therm.
BiiC Use gauss "uarature e)aluate the &%ll%wing integral
/AU MA,2UNE-014
'. F%r the &%ur n%e element sh%wn in &ig. etermine the a!%$ian an e)aluate its )alue at the #%intB1,91,3C
/AU MA,2UNE-014
. BiC eri)e the a!%$ian matri5 &%r triangular element with the B59yC !%%rinates %& the n%es are B1.9C9B93.C an B'9C at n%es I99k. /AU N;<,DE7-01'4
BiiC &in the a!%$ian trans&%rmati%n &%r &%ur n%e "uarilateral element with the B59yC !%%rinates %& the n%es are B090C9 B90C 9 B91C an B091C at n%es I99k9l als% &in the a!%$ian at #%int wh%se natural !%%rinates are B090C
6. E)aluate the integral
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Determine the sha#e &un!ti%ns N19N an N3 at the interi%r #%int # &%r the triangular element sh%wn in the &igure.
/AU MA,2UNE-01'4
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Determine the sha#e &un!ti%ns &%r a !%nstant strain triangular B7TC element in terms %& natural !%-%rinate system. /AU MA,2UNE-01'4
@.
Esta$lish the $%y &%r!e an tra!ti%n &%r!e BUD=C )e!t%r &%r a l%wer %rer "uarilateral element. /AU N;<,DE7-0134
10. Esta$lish any tw% sha#e &un!ti%ns !%rres#%ning t% %ne !%rner n%e an %ne mi n%e &%r an eight n%e "uarilateral element. /AU N;<,DE7-0134
11. 7al!ulate the )alue %& #ressure at #%int A whi!h is insie the 3 n%e triangular element as sh%wn in &ig .the n%al )ari%us are a1 * '0 Ma 9a * 3' Ma an a3 * '6 Ma9 #%int A is l%!ate at B91.C . assume #ressure is linearly )arying in the element. Als% etermine the l%!ati%n %& ' Ma !%nt%ur line. /AU MA,2UNE-0134
1. F%r the #lane stress element wh%se !%-%rinates are gi)en $y B1009100C9B'009100C an B009'00C the n%al is#la!ements are u1* .0 mm 9 )1 * 1.0 mm9 u*1.0 mm9 )* 1. mm9 )3* 0. mm9 u3* . mm . Determine the element stresses. Assume E * 00 N,man t*10 mm. all !%%rinates are in mm. /AU MA,2UNE-0134
13. Deri)e the !hara!teristi! matri5 &%r a tw% imensi%nal heat !%nu!ti%n #r%$lem using triangular element $y galerkin a##r%a!h. /AU N;<,DE7-01347LE
1'. 7%nsier a re!tangular #late %& length 300 mm an with 00 mm ha)ing a thi!kness %& 300 mm. it is su$e!te t% a uni&%rm heat s%ur!e %& 00 +,m3 a!ting %)er the wh%le $%y. The tem#erature %& the t%# sie %& the $%y is maintaine at 130 !. the $%y is insulate %n the %ther eges. Take the thermal !%nu!ti)ity %& the material ass 3 +,m!. Determine the tem#erature istri$uti%n using triangular elements. /AU N;<,DE7-01347LE
1. 7%m#ute the &inite element e"uati%n &%r the =T element sh%wn in &ig
/AU MA,2UNE-01347LE
16.
Determine the element matri!es an )e!t%rs &%r the =T element sh%wn in &ig. the n%al !%%rinates are iB191C 9 B9C an kB39C !%n)e!ti%n takes #la!e al%ng the ege k.
/AU MA,2UNE-01347LE
1.
7al!ulate n%al is#la!ement an elemental stresses &%r the truss sh%wn in &ig9 E*0 a !r%ss se!ti%nal area A* !m &%r all truss mem$ers. /AU N;<,DE7-014
1?. In a &%ur n%e re!tangular element 9 the n%al is#la!ement in mm are gi)en $y u1*0 u3*0.063 u1*0 u3* -0.063 u*0.1 u'* 0 u * 0.063 u' *0 &%r $* 0 mm9 h* mm e *510 N,mm an #%iss%ns rati% * 0.3 etermine the element strains at the !entr%i %& the element an at the !%rner n%es. /AU N;<,DE7-014
1@.
7%nsier the triangular element sh%w in &ig 9 the element is e5tra!te &r%m a thin #late %& thi!kness 0. !m . the material is h%t r%lle l%w !ar$%n steel. The n%al !%-%rinates are 51*0 yi *0 5 *09 y*-1 5k * an& yk * -1 !m. etermine the elemental sti&&ness matri5 . assuming #lane stress analysis. Take u*0.3 an E* .1510 N,!m.
/AU MA,2UNE-014
0.
/AU MA,2UNE-014
1.
/AU N;<,DE7-0147LE
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/AU N;<,DE7-014 7LE
3.
/AU MA,2UNE-0147LE
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/AU MA,2UNE-0114
6.
/AU MA,2UNE-0114
. A !ertain #r%$lem %& %ne imensi%nal steay heat trans&er with a istri$ute heat s%ur!e is g%)erne $y the e"uati%n
/AU N;<,DE7-0104
?. Analyse the truss sh%wn in &ig an e)aluate the stress resultants in mem$er9 assume area %& !r%ss se!ti%n %& all the mem$ers is same. E*50 N,mm.
/AU N;<,DE7-0104
@. Determine three #%ints %n the 0 7 !%nt%ur line &%r the re!tangular element sh%wn in &ig 9 the n%al )alues are "i * ' !9 " * ' !9 "k * 6 ! an "m * '6 !.
/AU MA,2UNE-0104
30.
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31.
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3.
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33. De)el%# sti&&ness !%e&&i!ients ue t% t%rsi%n &%r a three imensi%nal $eam element. /AU MA,2UNE-00@4
3'. 7al!ulate the tem#erature istri$uti%n in stainless steel &in sh%wn in &ig the regi%n !an $e is!reti8e int% elements an 6 n%es. /AU MA,2UNE-00@4
3.
/AU MA,2UNE-00?4
36.
/AU MA,2UNE-00?4
3.
/AU MA,2UNE-004
3?.
/AU MA,2UNE-004
UNIT I< 1.
.
3.
Determine the &irst tw% natural &re"uen!ies %& trans)erse )i$rati%n %& the !antile)er $eam sh%wn an #l%t the m%e sha#es.
/AU MA,2UNE-014
'. Determine the &irst tw% natural &re"uen!ies %& l%ngituinal )i$rati%n %& the $ar sh%wn in &ig assuming that the $ar is is!retise int% tw% elements as sh%wn in &ig E an g re#resent the y%ungs m%ulus an mass ensity %& the material %& the $ar.
/AU MA,2UNE-014
.
/AU N;<,DE7-01'4
6.
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.
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?.
/AU MA,2UNE-01'4
@.
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10.
/AU N;<,DE7-0134
11. 7%nsier a uni&%rm !r%ss se!ti%n $ar9 as sh%wn in &ig length = mae u# %& material wh%se y%ungPs m%ulus an ensity is gi)en $y E an " estimate the natural &re"uen!y %& a5ial )i$rati%n %& the $ar using $%th !%nsistent an lum#e mass matri!es. /AU MA,2UNE-0134
1. Deri)e a &inite element e"uati%n e"uati%n &%r %ne imensi%nal imensi%nal heat !%nu!ti%n !%nu!ti%n with &ree en !%n)e!ti%n. /AU MA,2UNE-0134 MA,2UNE-0134
13.
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1'.
/AU N;<,DE7-01347LE
1.
/AU MA,2UNE-01347LE MA,2UNE-01347LE
16.
/AU MA,2UNE-01347LE MA,2UNE-01347LE
1.
/AU N;<,DE7-014
1?.
/AU N;<,DE7-014
1@. Determine the natural &re"uen!ies an m%e sha#es %& trans)erse )i$rati%n &%r a $eam &i5e at $%th ens. The $eam may $e m%ele $y tw% elements9 ea!h %& length = an !r%ss se!ti%nal area A. !%nsier lum#e mass9 Matri5 a##r%a!h. /AU MA,2UNE-0114
0. Fin the res#%nse %& the system gi)en $el%w using m%al su#er#%siti%n meth%.
/AU MA,2UNE-0114
1. Deri)e the sti&&ness matri5 &%r the e"uilateral triangular element with #lane stress !%niti%ns sh%wn in the &ig. #r%)es that the resulting sti&&ness matri5 is singular. Assume u*0.
/AU N;<,DE7-0104
. The n%al !%%rinates %& the triangular element are sh%wn in &ig at the interi%r #%int the 5 !%%rinate is 3.3 an N1 *0.3 etermine the N9 N3 an y !%%rinate %& #. /AU N;<,DE7-0104
3.
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'.
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.
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6.
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27.
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28.
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29.
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30.
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31.
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32.
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UNIT < 1.
.
3.
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4.
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5.
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6.
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7.
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8.
/AU MA,2UNE-01'4 MA,2UNE-01'4
9.
/AU N;<,DE7-0134
10.
/AU N;<,DE7-0134
11. Deri)e a &inite element e"uati%n &%r %ne imensi%nal heat !%nu!ti%n with &ree en !%n)e!ti%n. /AU MA,2UNE-0134 MA,2UNE-0134
1.
In the &inite &inite element element analysis %& a tw% imensi%n imensi%nal al &l%w using using triangular triangular elements elements the )el%!ity )el%!ity !%m#%nents u an ) are assume t% )ary linearly within an element in!%m#ressi$le. /AU MA,2UNE-0134 MA,2UNE-0134
13.
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1'.
/AU N;<,DE7-0134 7LE
1.
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16.
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1.
/AU N;<,DE7-014
1?. i)e the %ne imensi%nal &%rmulati%n &%r %ne imensi%nal &l%w an ser)i!e the element sti&&ness matri5 &%r the &l%w thr%ugh a #%r%us mer!hant. /AU N;<,DE7-014
1@.
/AU MA,2UNE-014
0.
/AU MA,2UNE-014
21.
/AU N;<,DE7-0147LE
22.
/AU N;<,DE7-0147LE
3. Deri)e the element !hara!teristi!s %& a &%ur n%e "uarilateral element. /AU MA,2UNE-0147LE
24.
/AU MA,2UNE-0147LE
25.
/AU MA,2UNE-0114
26.
/AU MA,2UNE-0114
27.
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28.
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29.
/AU MA,2UNE-0104
30.
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31.
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32.
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33.
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34.
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35.
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36.
