Uploaded by omarsanybappy

FEM MSc Class note

advertisement
M E -6 1 8 3
Ot/tc/oS
l1 r{\6fks DtskibuJlOn
frn6l Exam
AsdEttwrls
,,n"1
da
u
r^thaf is
-
fl,rita E,evnay,fMefll"d I rn r.nt?
uelhd
A t1u$e\ical alal3sts
elSinearing
for solvhg a
ytida varr'ety of
Probl4ths.
industrr'es.
MaJsr atar*lon rh hgirreerirrg SChoo\s and
Pesaarcta rir
- So\id li€oh8nic5
- Aarcs?^caer$neesl,tg
€vgg.
- AulowroHt€
- bio- Mech€nics
.
- 4rvil 4rucf uYaEr$g'
'Ilerual
enttg.
' - Ftuid {low '4nabEt3
- Eleciromagrafic
s+udies
Availablz sq"435
x (ovrrr'rarciallg
- ANsys- sui.fabk{ot ganoralpurporc solufion.
- A6Ao,uJ
- t{onjirrodv ln alvsis
- Ffl4UA/cor"UoL
- Sr.u4aHe
$ot mulll-physicsTrobhrns
- loY elzctnca\analgsis
l^sc MAR.K
&r sohiCs.
l--s DYilA- for impact ard damagea''AINs.
x flislor5
"l
FE/v\'.
ioans., 8vP th con{irruunr mqhanlcs.
Applted rn6lhern61
Phgsicisl: ?iscq-,^lisa{unctrons lor
Engireers" skossos ri.
Ln 1q43, an
Mahl'ng of
ctnh'nr,luwruechanr)cs
frrc{rawta skurc{up,
(ourant
app\ed l,\al+renaheiavr
torsional
ilsa this for
problarnswil4^ -ftr'angr,rlarr€VNns,
&rsf inifialcd s'liFlnqssmafrin
t,t1416, flt)tner avrd C.lough,At
&r {russ , baar ad other 4rr.rctural e\euanB''usin1
elaslcr\
-: ln
lq6o,
khwram.
f'lough:
Soluhbn of
of kiangutav elpty1.;".yk
ayd
plarv probl4h4iof ekstr'ci{y
gavl'+ha ytar1.e'FEM,.
t( Why Fe,vr7
€nhanred
Be-nat'Js
4'tlic FF/\ rnclr'tdo thcraasd accuY6c9,
d€3i3n ?a{a/'efors,
design and bc.ffor insignt rnlo 6ritical
ltsltr
, {av{ar hardwava ptot"{gpas, a
vixttaa( proFolSPing
groduetr)vitY'
znA loss €xpcnsiv€ dosign cycla , fncraased
znJ fncraaSsdYevQhUe.
g
lvrveslrqa+ibn
I
Theoye+rcal
6xpevrnrental
NuNerical
t
,t
r-_'-__--__--l
v.tv
I rrm I
I-F[l
I
lBElrl
o Finding Lod- baxir,E 6?aL\\
having a no. ol
6d
noh- 11n
ifsrtu
o R6l€ of
c ophbnl,
vniftrm
61q'n-
<lrflneg drd hole< of
p.lalz
avbilary shaPas
orr'etrfahon.
o De+orminafrbnd
rI
ofa
po\lutanLsrh
airnospheri6air tritl"
Condifions.
4'a'bitaryg1qa.
4luH {low deferminafion fn a chanvrot
Neqbe+ alt Ne,enon-uni{ovrnity arA rqducp)ha
poblarn {6
arnfla
ole.
- Sowe cA38S, Apyoxivnations
41a accejfaHc.
- lhosl\ho casas, -1h'rs sir"rpli{rtah
ovr C uses €(r'ous
"f,
thSccuracv o\ r{Yong answQ(,
, OdrhI :
Retairr 6U \he
ron- r-rni{orr'nr-fl'Bs
, br,rt vafhor ?'rodtlce
f--
corrrPu+ins(na(htnet'
- approximak solu,h'ovl by
x Fini-la Dil-lerancerterhd CtrD^!
-kovrdssagoirrt-wr';capproxnrahon16'lltegoverningqn'
- I
is a diroci n€l&d of
- Tho bd3 or
hwv
soluhbrr.
1fi4 ragion ramainsa
spocific poin{s (catled node$
1lr3\QbcJy' htt wa Sdei'l
on +h€ doynainby € -
rvleghnefWork .
* fini{c F],ovwd /l€{4od I
- SrovtJos a
eql"
wieo aPproxifia+lahto ihe govatn\ng
Pl€cQ-
solu}iovr'SgstrnnresPoYls€
an indirect rvretl,rl "f
'
t'r advance
Wi{1rin trrQ dew\ant t! assurueA
'rof crnoln
- Tne. solufrbn donrain "f Ltta body does
- I{ i:
2- sin4\0,b.dv , rathav rt
4 aplacivrg if v.tili
ragions oy vlewer{s.
ra
'rnabted
,/ approrimalod
inaY'lg s^^l\ iwlarconnecfoisLtb-
- elouuer*to elaruvnl scnn€cd^o"rorrly lirougl" nde
Cenenls,
5 6 <pnrqrrr .r pVfe .
- nol
[, Tna
boundary.
5iryle ehrtent ; ?rcparfy romar'ns con4enb.
o \4ccurrcres
Fini{a elevnan,t 6ppyoxrmafi'sn
I'r Fp,(, ongina(Pds {apk(ed by a lgebric eql .
o hhY
FEM
Over fDM:
1 v(fl'/,alLftan+
c-omPhxboundaYysh€rJ.
"{
be evwQn'
-(tn rrrvt amv KinA gelo*ek! ol elerhe\a{
,aan
buf
th FDM
o\4lg Y1@. 6Yr- accessibV)
-r Non-unisr,nitf o{ Eyshrrr qaramaiors,
(wifr rh lhe olement
proprtr'esrarnain const albatt tt cYntgcslrou eleuenl
Io edna*) , ean^'l bz <olved W FD^A'
==:
-
lzl t2lo)
of r en'r.
B Alvav*oges
1.
-lVw
2. verll
.
nrefhod wovks evon whsl aU &e rnathds|ail
goo'1 in
rnal\agi/t3
shaPzs(arbilarg bo'.rrdarv
con'Plieaicd
\.\
s. Qood rn dea\ing v{ith nan-\rh)lQrmmaleriatfreldProPrh?5.
a Drsdvarrllgas.f, rr,lt
f" ba no* rir a <rnglc hdy, bu&
reatizad
is
bodyfegiorr
aw
\.
vathar an assamblagod' elemen\s csnnecloAonly at fhz vrade
fhen $aF f*tr-|t'o"s
2' fo*ox' {.sPonso fs pra-cla{ermined
g*lorpolai\ {uncio'rs).
aLrass eleme'rl
s. Field {ariablee av usual(J di<contrvtuot"ls
s.
boundarr'e
*frlon
tr. xurvrbad' .rariablasat eaeh 'rde ts ovnetr"nresVliEher
'
frrrzases)
those "t FDr^. (r$ Vari6bla 'hc(aaS€Jequalron
1l can not
hardla
il4ora'lhan f
(has condrlr\trs
sing'.rlarrtg.
tcinfof si'lubrilg
slpulJ haveJ
BoundarJ ElcYrranflrtethd CBED
- A nunrcrical analysri vuethd.
* Boxr-rrdargt'fl+agYalequa{ion tllpfhd.
.. Not-wrte\ \isd rh schools6s !'{alt6s i'dLlskr'€s.
PopulavrnZlnlg rh lhr
acouslrbs.
"f
Uso (veen3 Theorcm16 feduce the. dr'rnensionahly
of the proDle
fibld
rlolurua grob ------> Surface prob
S*tfa*
grob ----> l-r?a ProP.
4he rnelfid
usa n o
connected al6'43 fl,p
Wr"nal rndes, orrly r+edes(aeshesl
ex{effel boundarlof Fhs region.
rzu
F Basic Sa'pn S'[ops:
"f
i.. Dr'scratzahono{ subdivtgionof
! --r2 3
at
r
+
-a- g ---z
a
2 3 + 5 6-
tha donain/regiot, I vrldy.
Ele'tnelt N 0 .
ilodado.
g,
4 P" 4 4 1, --?Fr'ed"ariabb
xt x2 x3 x4 '5 xC
irrlernal .r?des
boundarl nodes
af 'rdes .
lalsral
4eneraldtladxi
a\awouvYs
hr e\eywant
Reclangu
1;7 -L)-)
\
w
l.t
& nrdad
t ^t^-^r$,
llQAAh'(]'Iat e<trtz'
A
4-nodrd Mta nQA{61
eleY4eftI.
2. jplec{ron oF (ntargo\ahonFrJ|rcfrors(shaye funchovis)
go[nomia\{unctrbrsof ltr'lcr . ovders'
- usu6lt5 fn tsvms
"t
-1n6 llnchbns aYa uqd )h describz lho possible
-!ie\d 'JariaHeo{ar a sidr-4<n
'l
?rte
bolrdviorlvariahbn
3. >elermine the elevnenl-propartgqua+tbn.
*he
-'A malrfx€quahon G derived dhfch erPtass
sal+lad 566p2
chorac{qnstic o\ lne deman{ vrsing fve
[uncdbn.
*
3 @foa<lres-
opplicationo{ conf}rr
diree*
,{yproc-h
199
Dirccl
equatioD
-R*z rvletrd)
Griaftovu\ t(e\'hodlRayteigh
-: riros'l
wr&! ''tsed'
,{pproaehes
ReSflL{a\
- Qalerkns l(e&"d'
(-1
tKl iV\ 1r l4
L-
*'"
"...tu w.da| paramafurs'
unknownrtdal tariable
I co-etgttent flaktx CKnownJ
lt'\atr'x
5M-----' Kr s+lS{Yless
P, loJo\ fttea nta\rix
t 6t the nodeg'
di:ptaeenen
Asserrrblingof +p
asserrrblingtha matrice .9f
demetls. (j-+
eaeh elenen'll
- Asarnding oF the lrrdivid'raat elevnq4
- propetb mafrrtes .l.o
o}ain 6
sing\e global matrix equahblr, \^{hth hlfll
descrfuaItta bahovior oF fta er+ira lomain/region.
lrT { u} = {rl
5. lmgosifion .f
,[d+rix€quah'o4.
-.' QlobalStrffness
ttte giventhglcal <ondr'hbrr<
al thu hundery
noAes.
6. Nrrvnerical
soft,rt'on .f
ftr
global tr1s'tervr
€qvaliD4.
silabr_ &v -too ef )
C(ausselfmf'nahbn
-:f
- &rrfon-RaThsovruelhd. &r
tha sgshm rron-hi€aY
hriihar neshing ttorahve rnefhodt? used)
- frr
l.
FE/A, "fvon-lal solu-tr'on
,tle.ihd" spciall5 pto&rable.
catcqla'hbn4
of infertst. other
dr'Fferqni para@ic13
fhan {reU 'tariabbs.
19/t2/o)
T'I
O@
ffi
I
k2K,3
ur
wz
F1
LJz
r-
F2
Elayy'eri !,.
hhear{xing
o
_____o
d,o_trT___
o
Ut
e)
uz
fi
\2
{r= r,a
["^_
(pnry
Non-\-vraar
t2
F=ka
. k, ( \.rr-ui) = \ur-Kirra
7
1l = -(ur +Kuz
I
In,
I
1rtt1
I
\
|
'lt?')-<,\
-t?
|\
L-h
nl
\,
|'
[ -2r
f nl l - )
\
|f
I
r '')
l
tl\
sr{fnJss
I't ui.r,{"""
v€ch(
makix.
NodalUnMot'trt
Veclov
tr €or elbmerrh 2 .
lu- -n-l)u"1= I fi"I
L<" ",)l ut) \+fJ
+ Sfi$rress rnalrlt< Rr ehmpti !
tkl
,
,ftt
l*' -Kr olI
l-(r
Lo
n, 0
o
al
.l
--1
l-"
tv
I
Lo
B
KfKz
-v2-
ol
I
-wl
-l
* l{.r-:l I
6r the global S5slunr
l.
l-\
L"
-K
\vrKL
'R7
n1{\ \tH\
I^IJ"J=111
--- BorLndarJ ^t'r6;+bns
tr,
= \,lnh1o!\n
'I
C--
n
.1
O
I
I
Lkl-=lovz-krl
-tl
-vl
[-] =l'.]'* lnl".
dernertt 2
-k2 u'l
-t"l
K2_
t
In'"
| -K"
vz=
-'t
'
?
( U:-?
I
il
L us i
g€dqc€d
\oi
=
(
lDl
)
,_>
€tobal 5'6q1;"rt
Eqo
it,\=f:I
?
iui)
tD J
-h tl" - P,
1*-1
L1=L2=l
Ur=Ur.O
zA,
)l
E
L
I
r
l->?
t--
' - oI
i
L"
ul
Ur
q
l2
6
J
u3
-
4eV7
E.
')
,ZA' E
r---6-_-l
-z
t+-x
u,
\.,l.,
E
t,L
EA
t1 = E',94 *A=
!)i?
(u,-rrz;
Ll
eEA (u,_J2)
o - zEL lu.w)
.A
-'L
-EA
L''
,,
=!'
fu,-fu,
- n#u,+f
.u =l,l
-tf rur )
r Lr
=
T 1.,,11;itJ"tt
2EAft
'frr
e{errenl z
FA
T
L_ _CA
L-
'J3 =
EA.,
\t, +
T.u?r
rr
t.2
rz
+ L lltll =1$1
;i ;lli:l ={?!}
71.^
Lo
v-
-r
t lLtl:l
-z
r.tfz
ol(oJ
1r=l
(nr I
Tl',2+t -, llwl=1pl
L€ -r 'l I
rs
t" J
I
'3ut
f
J
!-
az =- ?L
@
6,
Ee'= t.(i4i9)/
\r
'
Fe
3eh
6.=.E€L' " F. (Ll?dr)
L
g,T
T-
/'/f
( ll---Tn
\ffi
le- L' ------:l
o@
d2
0r
-r
tta& ! .
.qJ
L
'(41-ezl
=1
1-
gJ
-t-
T
(-Q+o! --Tz
.&
L
T
T=
aO
r: -t
^r
>r=9lna
02
F-
t-----1
,2
'1L
TI
dl
8z
FnergJ rGngnvalrbnq\
.ldti
-.
r^. .-i ^r'-
Law
ot
&=
dX
_kA
Ndel:
KA
L
Nd€ 2:
KA
-f
'
\
t'r+'2./
?
O1
0z
f, :lfll "ls'i.
_x4
I
lr- l2-,/ =
Sr?
2tA2/09
vL
u3
@x:
FL
Fe
tl,.
F,
Fihd tnrtt =
l lrJ
aerl'ar* q ,
etobat nodes"-,1
Kr+ |(.-+ 19 -f
\=
"l
kl
It'l
=
"l
ul
-kl
:l
o
[r"] =
K2
o
- Yt-
-KL
VL
"J
;l
"l
o
lu'J
E1\.
ti
(,
o
K?
-3
'K3
k9
o
;l
O
fxul =
o
K4
-Kll
il
---->
4X4
rnaf rix
-h
-' f Ke'l
111+K2_+Kj - R 2
H
I,tJ
.:
tt'z+\+\
- Y9-K3
- I,q
Er.o,nlb'.
r,l
' Pq
t 2 In t'
'
ll
Eletnart.4
Qldalndes=r
IOJ
't 1
t"
s ?a'
L,
ci@
rj .l
5
V = Elash'cnrodu(qr
/.
t
T,,
: ts6dy torc€s/Volul44€
Tr. Ty.'Tq -
?oir$
{cvces
rr
[r']
I
Tr
-v
-Ai
-1|
xt'
d i - diavrre{er'
?t , ?z
i"' -'I I
f k']
s',rrfac forca /'larqlx
ttr1"il1
(3O
oo
o
L't
o
lI
to
o
15
o
oo
o
6o
a
cO
6
lr,l =
m
o
t.
-Artz
E o
o
-ry,|.2
0
o
Ur-t
o
o
o
_f
-o
o
[r'I =
o
o
o
o
o
o
AA
t4
-43
-/,ltt
ht/vt
t-
o
o
o
-70
,X)
U7-
o
Tr,4
I LNI-
o
o
4!
o
i-4
L4
=
Kr+ K' + Kj+
l,{.r
- A'llu
u5
Lt
L9
lrql
Vt
o
o
0
kq
tlrlr
, I!
-T1- 2
lF l =
(L*'"8)'h|
g
t(+tl. ('t!E
i!L')
+
'2\-<
/
r s. AaLr
.r.!- \ )
2
1-J
\
I
je+ a@+I SaqLi. tql-q1|
\
( sArrk . r'.La
\2L
z
rl
nl
2-)
|
o
1l
t
@t
['
6.
['
6ti
\:
6/
Ll
+ slefllenr
I
L4
-rf
- 3'nptrtfrahbn
lo
Wluce
prevrov<,poHarir.
ShaPc Funclon
- lnforpolafing
f approximahns{uncfion
of
- They a.r4 useA to ex precs thc variadon / bchauio'r
the eAw* 'rvr tor''tts "t
lhz {bld ',iatiable witxn
inaepnawttvoriables (PoItrorn(aq
- Polgnonial funch'ons ava wr&lJ useA' iocautz s\mple and
be apptrzd lt I*tpm.
eosi\
can
kechn\que
rh{egration
Spoc)al
4-D probkrts
RG) = a" -t a'x
h'nzar
shaPa {'incfttrr
hQq - ?6+ dtx I BzyL
2-D &oblens
brt(y)' ?.+ arx+ azY
h U,9) .
'4"1 atx* Gz\ -l 69t-! I
a4*L+ 2rgL
fl
l
PYobl2m
3-D
Prlx,u,z) = €o * 6rx+ azlJ+ a3z
b"V,!,2)
#
= €.161x{?2Y *asz+
aq*qaryaaa4zL + arxJ+agV?ra9zx
Urear shapa{un<*rons in l-D
gtobkrtr
U> a"+ar x
Ltt= Zlo+ arxr
I
Vt " A.'+ 6tx,
. U,- tlr
.al
\AL
Xt
'<2_
4-rLf
X1-x,
?o
\lt
^-
--1r
lr{lX2-U2Xl
=
)e- xt
U'
UrXz- \'{zxr
Xz- Xt
- rX"-X
l-
L
-lXL- ll
'r
X2-Xpl
/v\
x
/X-Xr
. r' _ .\' _
lL-ltr
\.,
) ( 4 r-
--+ infsrms o{ Yrodal.ld\iabl€
= Ltt()ur + LzG) tlz
LrE) ard LaVl
la tx)
LzB)
AYQ
/-2-Y-
X- xr
Yz- *t
NEAT
shapc-{unciions
tr 6enera\ Prqcttrbs
Lrt(D= I
6l
rro& t
2. LzEr)= o
sf
noda aL
t.
Lr(xz). a
l.{x,l-
I
t-rlx)+t-z(x)=t
U=
LrUr+ Lzaz
.X = LrXl -r LzX2
dl-r(t . dr-rtx)
-dl
JK
..!.\x
X2, U7
Xr,ut
i=hUr + LzVt
$t 'l
rzl
=
1
1uI tL'
I 1''l:.l
/
tBu v6(.
f-
tlodalv ar .
lil I u1
I
L *,ape {unc}r'tn mafrix
04ot/to
It
\L
U : L\(x) tlr t L{*) Vz
2- X.
*l
4--
LrG)
l.inear shapn{unc{rbn.
X2-x t
h(x) = x-xl
(1-Xt
e
al coordi42iP
Nalu.r
l
- A n ovmali zed
dene*
inalp
<-ooxd
syslarvr is u,rsed widelY ftr
.l,.ri vina l
s+i{Inesse{.
- &(in
lhe. coordinatP s5stem is 6f 16a micHle "P tire
of
LVwt+ @lth,
- 'lhis coordinde
- fiiis
-l
4shtn vaYier Erowr
pr{icular coordinato
applying an elfictenf
!ahi& s
<eguird to
1l
)
-t <
at
J
X= Xr
X= X:_
IJ
tL
< +a
1.-1
.w
I
very
sui*abk for
simple Melkcl oS infog<ahbn'
desive lhe z/s/n e4s'
6Yd
x-rr
" /\ - ( e - x r)lt - L ,r
fa
sJ<}an s
ro a-'
- - -,rA-
3="
slat:&rd l-D elenrent.
tir'neat
r---
t6\
,
J,d
L________J_,y__:_.
.}.E,
i
5ol
I=o
Xt
'
)='
X2
x"^xr- )e
= r-T
N,t3t
z
Xr-- X
h (rl
Xz- Xr
V6)
N,(J) =
X- Xr
xt- xl
''
I
2
U - Lr(x) Vr -r Lr (x)ur
= N,(T)u, + N'(T;ti,
. t'r,xJ li',]
= [r.t]| ul
J =n(i1^)_,
X-*r =
T
Cr-r;
x : xr.t T 0.--t
* =tJ)*, n f$;*.
X=NrG)*'tN,$;xz
= trr, N^,]{, x,
I
xt I
= Url {r}.
q =t^rl{uJ
x = trl{xl
-Ve
ftfmulaio'n
in
vrhich bolh the
coordinah patar4s+qy
and thz
drsplacerrent S'€ld v{i{-hin ftte dqeft+ cb\} tu
ex$es€d rh ]arns o!
Ihp Sarne <haY {.,rrrcfionr_
r!
Kv10\{nas
Tsoyara
uei<i <
S<rvruIa+rbn tYr FEM
t'--
j= -l
4-D
J
-€- |
'''
rluad^rahkelewrent
U: 8"+6$ t Azd
t3
---1
F-1"
'
= Xz-Xt
tJ = Lr(() Ust LzE) \,t1 a 136) %
Lp@Lzg,L3(x)
aR
quedrahl
*aPz
{unc}r'o' th x.
\'l
Ft.
-_-_--_-'6
.7
j=o
3=-r
,w' l
t,l-,
z,
r=l
nt
X2
I
LrG)
'qtj )
loG)
'..t-(!
4ut
nls(.3)
X')
^ / X-xr r
J
\X2-Xrl
r,rr(1)
- i (,-1)
N'L3) il, (t+\1
$(T)
(r+1) Lr-j)
a,5+drJz
l=
'"n
Tkree qtladtahc <haf frl^nclron 1
6"
/\-4
:7 N
t a13-'na"$
+ Ortr) + 6zLtl
+
O
t0.
6".0
'A^. - ol
2L= 11 a\
-- \- Qz-
-2^ - V^
6t = -Y"
'l' t:) =
''].
_l
*.6;= j/{,'r;
u(;t=e3;
t-},
3z
J=o
J. r
glrr;r io$}
v
'r{
L
Ug
32_
o
l'-r
\"t
re _i
L-I
- Sl-avrlard 1-o quadraht 4uttenl
tL
U= Nrtlr + N z_V:_
= f,.r, r.rl
{| I,
1
u7)
N,/z'
-;,\r/ -^ \-E
-(
I
\
N' Gl -lT
)
ment fi€ld
dt"Ela<e
I
-
( r.r,r\
A /r..r
= ai
*l
gu'
1 ,,]/
-- 1-dr..rr dr.tz-l( ut )
L-a^ r"l ).1,I
-- TBtlytl
d\'1
(}X
'-
fdl.t| --Ti
dt
rbl =Ld&=
dr.r2
dr I
--E -ro\
d)(
Iu
-L-
2
xL- xl
Ibl =.B
9--
I
).e
dx l
^
L'
' JI
il
ftr
url
it
- J
I, ,€t t-'
L
Linear L.'o derrnel;
6"- E[B] \Ltt
tar
Jur 1.
L"r
\UzI
E r '
-1-.L
{*y
rl l
J
u=ttt {u}
r = tBl{u}
lUz\
--> 6=
"I
\'1
!
,
f-r
dx.= i;4
.
*z--At )
-l
clla
I
\21
|
Xz- Xr
(I, .Yrramdrlplacenenirnakix)
lq2\
;r5u'?.
'l
gDu{<eW
t{ = Nrr-lr+N2U2t
= [N, N2 Nt
\r(\)= QJ? A-9
\u. I
,
qu€drah( <hEB S..tnchorr
N3t{_i
iqr 1
luzl
t ,rfu t
"
"i-
-12
'c _du
- --{x
=
f ar,
4N4
d>-
LZ;
d9l
fii1
(ur
= lsl 1u"\I
lu3 )
IF.-'l
L-J
f ,,
1 dNl
I
L
Ti
A
- .?,
*I z
lt^
dNz
ql
dT
tl't+I)
.r
4e
lrr-l
l\
\
z,
@$k
Qi1)
d'lr
-+t
dxj
^"t
'/an
\-zs) ]
C"H)
dir
-r;
tr:Jlu]
o\-
f" Lt1-,,
G*h)
crl)l 1'ri
o\/ot/ ro
n
FEM Wj
rroa+rlr
B4r1$t gropcr*g
by
(a) girBet mefhd
ol
melhd
Variottonat
Lc) Residual l'!t@{had
L-D WareWALrt
h
tt (i-\
/
p:T:"
trll
l,L-
xl
lZ
r- '
7't
i<-_-- t-->l
rk"'l=1rf'
Le L-l
I
[r1{u}={ ri
II
rnei?rd
Variatrbnal
Variatiorral
Aethd
gslenlia\ eneql
t. lv\ivrirrrr,trtr
z. ?agleigh-Rilz
rdnod.
aVptoae\ ->
5o1idrnechanic<'
t: l,lrhlnu,.r,r pui€nlal energy $e.Nhd
-l-ctal
?
2ulen{ial €'4er{} = ri,g1 in&rnalerrcrgy
6lorrd bj lhe bdy due to dfirrnahon
=
TI
Slrain enzrg.l U
+
Tr
Workdona 0n tte
bad! by he exhrnal
toadr.
WorK Pui"rfihl
(wp)
u- v'{P
6rr
t
=0
5 tu- blpJ
Vafr'atronalfaft YdelLr
iP
ir rpri'q =
Fnerfl <1orad
T-T__
I llr
=| *@rcain S7riq)xdisT\acenervl
+
Av
4K
=\_ ^ K,\.txu
I
-7-^-7-T
=f, ru2
\{P= Pu
Tt = !w" -?u
-:)
*
- f,.zuu-r
)
(ut-P
,-.------------1
=
+ ff=tcr,ti +
L--
ku=P
U=+[<redv
,^r
vtP=
t'-
Jl; udv = | ur -q"av
,tl
tt= \ra =trr lE
Y
Wp-=
|^ r
''L
u ras
J
T- {rac{ion
"5a.ft.tn
J
= lcev.
wp^= Lqvt
*,=
+ [{ultrar* I urp,
'fir1'Rdv
'
|
lr
t
is
-r.T^
'tl
| ^- )' t. .tel ,
z-
,' =+' 2 - .t{ct' { rTdvl{uJ f"au_I juJ'n,,_
Jl
..vi
F Tr - f. J-,--lf. =JT .
-J
u'.f"eocx
?
Ju rdxl<' € Aedt
Z , t.,D,
e2e
? n, = ?ljo'.
n
'l
u' d\ - + ai P;
nnax-?f,rt.l"e.a^-l-J
12
,,
wQ 'i
I
f
-
) ste Aadx
Xp
4,4t - NodaldrrPlace*rent
Nr4r+ Nz9z
ShaF fr;ncHort
Nr Na-
tNJ{q}
I .l - rbl\q\
{(1 -- E rBt{q}
3v
.
Tu1= E-u.a:
d^
dT
LdY
JTI|
--r
Lt )-
^::- o
r,
)e JY,
rl
h)
l>it
l-l
4eL
ll
)
N t =- *
:-,
.'
N z ' - Tl , j ' I
tL
tr
, ^J,
1r (rrai{aU
( trj {q}J +" dx
ue=
r.
tl
t'
)<l
=11
:>aT'#*,.ar=fra
Xr-
Aaax
Cv7
r
\al
rc
{qi'
YI
Ve
.d*-$J\
-z {ct'(!n AeL6rrr6t
4 a\) Iq\
-l
ur
i
r;
{11
\^f(ry'f
H";'
-r
\
1.*_--/
-z
_l
:J
o
V\2
ylPr
dI) 1ct
1,1'Q
+[llillir){ql
-ll)
(T'
\qr' I-l r'i
-^'T
i
rl
^ /X-7t r
''5;;,J-l
-z 101' rnt q1.
f
----=
40,sing\efur e\er'enl
_I
J
J(t^l{qfr"o. + "J
1'qi'{"Ae dx
lT
-l
,i 1 \
/
1f
T
{q
I
Aq ls
fo
---;-
tNl' d]
I
I
( tlr1r\lzi-r
nele {
/ {o-t-
\--
_l
I
neln ( )
$"
'--'7I
I
.l!
o\
\t
-t
)
rllIE
Wn--
1"1)
(
{,}'
WorK donc
)la+!"ih tlo A
wp"=
on the bdg
lon.{
W kn
bodY 10rc€.
I t't'T dx,
tt
I
= [rqf Ln{]tT
J
4! < r !
I
,T
= '{qt.,
.Jolrdj
le
-l
J{z d3
\)
. {qlt !
t7-
t
I
I
rKtlqt
1cI
,ra
T''e=i
rl
---> dis+ribv+?d
qltally at {'oro''rdes'
I
c ^or / ft Aele
1YI \ -=-
{t}
\-
{.neln,11n
+h
-
- {cf
aTe = j{e1'rr:{q}
i--'-T
\
{r,tule
--L
=I lo\'c"t
J
-
.lr1
*.-
+\Lre+ b"
I
bar eluna'rt
(^a
LKl lqf =
.
L
- \qi lr|
Spring elanant
Ir = +ulklu
a,
)-
\q{
ITt? t r l l
\21'r,/
fAts ft
-:ll^;,\
"tll
{{}
1,1 Ndanunbar.
_'
o91o
tno
B,
t14a1 -nducfion .quation L3-il
-(4a--4*+F)*e=
(( -?r
.d1\T**-xj,az,/,i-..'in\
/wlutx
mfatnal heal greralionf avsorbfto"
qr' - K+
=-rE d n
4v'-k*
qz . _K
+'
A /,,Trl
6 r^7;/
(q^l
+
3Z
+Q = (c /?t
+$ t^F,)*& C-tl
Ir
o
o-l !)%" I
* "i\>2" I
J"
-1.
=
10,f
o (J latn'\
\sotro?ic^^lprial
Vr)
o'I
! q., =-1
l-o*
o Ky
{ f
lr") 1",
t-tl
(,'1
lo' (
j
l{z
L'
l- K,,
I
I
o,
L *r'
.1J arAx\
I
11u{qria
[xlhot1e7\c
u(r,
(_o \\
*rlf o'rt.J
Yrz
<u
lTAr
(
I
\Itry
.
\(
*'\\
Kzz
Kt1
Kte'Kz\
xzt)
,n,
.t41: Y-ti'
-.,
,Anigo\xofl'e
1-
K1y. Vzg
l
1
Ytoat Cnndugtiorlqqa
\erP< al
tr l-o
?ro{)+&,=o
-
i(
ox/
a-1-=-.
-
g--,
,--
*
| ---------+)
|
X.b
CasaT: T rs 9iven,
(asafi .
Ta,
-Tt
tred {lur ls qiven 1^'9,
cass.11:
(9c;
Convecdsnhurdar5 eordi{ion.
d
dT\
6xJ
a
)=
(-
2?
..i-
!l
dx
^ l.
,L
righi
tov
tov letF end
-ILHA (
a
ll r- Id-E
L'u , )
evtl
,.lf
A tlo = HAltq- Hi
= t tr-t*)
#l
-,.
\g
6rall
kins nrethd
la{(ki
Dt&s.
- An apptoximaio ^u11*t6t'lral nre'fhdSr <olving
- a resrdual nethol ' ltr.retgnka resfdual vnetlid)
fi
6nides a
'
d'Y
---_-i' t
d*-
O.o's--)
ganoral P€\ '
jI + {(^l 'l = F
or*t
I \'/
s]\
SFapU.,
la+t-v1
n
hvtat+ or g'(a) =L, ?
,t
L h.,etb)
t qz c'(b, =rL I
{
(egdue,
€ G<)= LHs ol
DFa -
RHs of
DEd
I
KesidLla\e\ror
iuncfton
= +**hk) Adx+qt*)Y-F
g,rrbslitu&a
€(x) .o
\r.rhaneract
e(s /"
'
.{ov approrirnato Eol*Hon
1bpz: €Jtuiliry an
soludovri5
n .,rxorir\att
-rl
so\qtto"r
vfu) =Q AQ) + cz{x(t)- g{tlx)'''
-
= 2 q" #it*)
Q - arb\{arg cons{an}g
OiG) - baea {unc{ions'
- vGi
shouH eaHe{.J E.c a
4iCx) 'dsz dnear\y ;rde2ondonl fqnctioas'
di{"cHJ '.rith
9
.
boundarj. g6Yd+onr
gorqning €q4 and
^rN16y1 andvanlage
.
eq[
Salarkrn's
J
z (4 wior)dx = o
Nr'(x) - r^lerlhtadttrnctr'on, ivrthts <ace,
t_
'^iiU, .
$ (x;
'la-
ri
X2-
?(I
eawat €{uatron,
e"u
*C-+)n
e(^)--&cKs) o q
vtx) = c,Q,6) + QVzlb+ GQ"tlt--= cr qcx) + 94-tx)
z noJer'
as ltp'z V ontT
= Nr(I)Tr+ Nzk)fz
=[rtx) .nr,t-rl l],]
= rNllrl
YL
(ahrkn! pql
J
X1
-
6(xrutiGl 'Jr = o
4qnchb
N, r{, -unaarldcrptiahng
(shapdrrcrbnsJ
le.
n
(
)
NiG)
leG)
J
dx -o
*1
XL
+
l{*
=)
cX9+o'} NiGrdr'o
o
t\ Ni(x)dx.=
[*l I l.r ^ 4,
)*
-,
J*, 1 F't'' a*
xL
Y,
& Jx
*
dx
riLx,
,<]Lt
["u,,..1
f
[
'''*''
r 6K'
+
dr
J
o
)
XL
{"t^"
r#' (
-t,
"-l
ro
+ rui(x)
\
ft
I
'xt
r
L
*,
(
f
>
d.
r
\\,\
I
i
r.a _ 0\,
9(x('T-:r) d
t
,*
J)tX \
x
r
J
)6-dr.
+lNi 8)
xL
Xr-
l,r,rn*S.1-- f $! r-?r
lv .rr+ [ r.riGl
-' dx
o
J
L''*'''--&-Jx,' { 1 J &
*t
Xr
X2_
f"
J
= klu,o,*]*'*
? * [*$ *.0*
u,.
JO*
(x{.) adx
I
/l
*1
t1
-{r
{\
K
*)( J ,,
LNitx)
=
r/l
K Llr4 j
+"
Grl\ t'=|
Lx"- Ny("'l
$^
/-
=-k*lx,
- u,l;*^*ol
- [-.r4,*; (-")
' k +0,
du
)t
-&
=
T-n
\rNtlriJ
..,
= ldr!-l{T}
I I'
LdxJ
; dNr dlrl t x 1
=
Lzx
dxJl
tLt.
/y
-7 [ a'.ri
J "R'
[$l {r1.a"
Xr
Xl
I a*,
lr]
-U I-4r t-rqla*
dj( l
lar'
J
Yg
tt
d*
tJ ._f"
t+
{j}l
?'}
dxr
:; lot
l1
f(dur I
:
\?Tif+
rl
I r
Jla*.
1xr I Ex- /,re,
dx
, sr't
flax] o*
lr'l
Rowr eqn A)
ii++
j--^
xL
ax
l{,ar,t"r I
r,.r'r ,.r1 (-**'l\-dr
{.1,-si
1r- ]r" \ =o{ a,,,
="-l
!
i"
*t'r
I
lJl'o*t\
L*+, * +Jo.
*\
u,(1)-_ -2-r-t
/ x- xl L
nr,(1)=
T
)
' u\lfx,fa
2-
&
xL-xt
2
\2,
ff4lr
\1"
Yla,q
,LT5
:J
I
x
rt
d.s dMdx
Ill I'o
4Nr dT
A"9
dil
dJ
Lc1-
Vu"l
Jx I
Ar"
d3
dxl
(X?
at1,;l'
l5
t
d^: I
b dJ t(l
IT")
-vr.-1t?" dJ
( l'"'
44-
dx
'
I r.
t
4l<
1i
dX
J l-Yu'4
-l
), l-ro-
&Nz
.-1-
-1n"1
l+
js.4 I
dt
dt' d t
dT dx
Vu- )
_,J
I t'1
ITz\.
[-"
=K
l1; l
l-| *r-u1*
( (lte/^
'' l
)
7
lq! l
*txt J
I
I
J
& d3
o r.1(1;.
-l
]=r
I
r sn,tt)B.d 5
J
- Elotutrt\
*,n,
\
a.t4-
)
slffnnrs ?rop€r-|3€qa
v..li$ intcrnal
apptrd
heal
+or
{nr
!.0
4itgle ebnul
genardion'
gloFa\
shffrry
"41
"noltr ringle&
au
K2
LL
K.Y"
-+:Ll 44
-K2
l2
o
0rc
- rK z -
K z . k-iz
-r\1
- !
--
o -
o
A
*1'
l\
I
LL
-k,
Tt
K3.ulq.-tr -., o
L{l
L3
Lq
I
Xa
-
kt
(rlobal stffness me*rix €q4
\r
kt 2-
Kt3
(zz
Kr?
Klq
Kzl
.
--
Rtn
n
Kzyl
l
7
K9t
K32
K?)
Kjq
"t-. (Al I
dr'/
{
I
F.nr
Khi
krt
vn2'
hr
Vzt
K3t
KyL
knq
Vn$
q3
Y.:29
Y13
---
ktq
- - - Ktn
h4 -
- '
k34
---K?n
Tr
knn
' Kzt\
I
o+
&la
T? =K
o+
A\0
:
Knr
\nL
Kn3
lGq - -'
Krq^
ql.o
-KdJ6D.F
d{
lTr-
:
-t
fd J <t*"t
'/
11
qF
Ktr
+
4z
1+tt
?w-
Ylt
Y4^
l<2-9 - -
v33 -
i
Knr
K4,
t-
hs--.--
Vnz
- ki: + O'tal.
atx
-
kzn
d.l.€.
K3^
,ek
HTa+0Q,
ta/o(ro
U3
t{r
--t--------V--
tr
d2
----->
L
x.
I
x3
X1
r.-t
I--
J
1,1.
inea<'&metl
t
EA I' -rTtvrl 5Fr'l
-z]rr rl1u.J=1r"I
l
ils9r
N1Q,
+ N2t, 1
= tfliqf
=-u-3u\)
r,l1
z
t{3
J " '(**)-,
= | ] (r+i;
dx=4lt
-t'
= Q'? t-''i)
| 4l
tBl1ql
-
J<1 - F tBl{q}
\ tt P .)t
,-v
-
-
d[-]
J-dNr qa
'
4r
dx \
lex,
Lf| -
\-2Y
------:
t.r-2T
122
^'!-.LT
'I
{ {<r'12}Adx J t j l-J
-2
I
TF
t.f
_J
t\\r?
''r u/'
2 l\u\ I,e dx - f {uJrrdx- 2uifr,
J'
T.
(r tal{q})'frsr.{q}JA aQ/"d1
\-L-ll|'1,/\L-JE''/).
'l
.t-
.-!c
2-71
q{(ry
\lut'tst.rtI{qJ.
''I L'IJ'
J
-l
. a+ \e{ tnl{io?
Eovrre# stif?ness. f Let = E*
.J
tfsl're)
ai
-t
I r
", Fi{€ ) \-r-ri
2
--\s/tll'* 2_ I <'
\( I FzT '+' . -2! ldr
.
I lt+z]
tr
lLa
_l\ . / l z l,rr\
ft l r
-tre ^€
jr
--:-
JU
t
7
-8 -l '
-el^
tt I 3
l-e. -"3
Fz=og,t\
Jl"f*
^t
ir d l _
F"Al{
\'{
?l
I
{ul'
r0fiu_
({
,)l
Itt"
NrdI I
t
tt'd5 |
|
I ,fs{3
rJ
J
-,-'
ce'
f,; .---- SodJ &t
F"AIs
=
\^r1
tk 1
to
=
?Y.
I
lg"tl
\q1sJ
Vt
I
C
"?
t]
--> rnaffr\
)Xb
_______-___+\
F__ L
Sdrctfle structure cqnsidering2 {urad Fts.
A1 Vz \trs t1q th
-T
'-l
i- .
.
r
o
.11
qNB
|
'2
i
LNB
I
e)
"z
2
+
5
bar
LND
L local r'toJe 'nrrnrber
i't3
'
tr
l*')=Ed I,
lz
r r.a-t
Lrr=
-sl
-tllz
'8
d-
-8
I
/l
e
2-
teI l-=
r.-r-slt6
l1+
l,
-8
lt
EA
7
=
lVr-l
L iJ
z
!-J
I
_s\2
"lL) s
I
141
o
EA
(J\
-8
1{
L
-$'
o
o
-s
o
o
L
It
5Lr
*3L
-c
l6
Ut
I
I
Uh
tu5
.
:9
{+l}= rnr.(7' JI;
)23
1n
+/,I
|
{n'}
t7.
= FAlt
\vol
1vr)
I
2
z
q
lll6l
\/
3
q
tt
Err
4l
ol (uz)
, rb
-8
-8
1+
1"1=
-
l--
r
t*l
-*l(u, l=1b I
t
ll
f ^1
I
lD
L
re-Jlut'l
-8
0
(or
q\obel
q^1
t' u - l { q 5
r'l'
, Y/"o
|
-
t 1:ll
-'
@
(t
,^'nN
@
..1
tl
h-
d
'7
I
\Az,ru,r\.?.
)
I=-t
1.o
1 -+4
I
E
t;l=lil
-ll (
Er{ [r
tr L' 'J1
Al =
u1Ca9E
r- qr"q
r.l/ - U24osts+,zii'€
"!= eOSv
y1. gin€
Xz-Xr.
= --
G-
I
4Q
v.-q,
pIJ''*rr'-r)"
------1-
;
lo
(urr?
fi
r
o
1u;l= l_" o r
t u')1
['i
\u/-J
LD
.\\:l
";l
ul
Y'4
{u't = tRr{q}.
o
4z
:'l
.4 ,\ )'
t
llr
t nliql
D
,r, = i 1,litr"dr{i}
= |' (re:{oi}.trlt I*l { 1
= l.z { q1T | .,'rr T#l
ltt
\LFr
IR]
1q1
= .1
ioit tkt{qt.
-'l
'
['
f
rl Lo
L-l
L
I l.^
I
=
tr11
-T:
lll
o
d
I
t
L€
- lrh
€^l
-;
tr-
tn
L
lr^
n
I|
-[n
,,1-
L
L-*
-*
.L
rrl
')-
zx\
(\n )
trrt
,a
M,
-lnt
-lm
_*
"{
-r'f
- Itrl
tL-
(n
lfil
n\
rL
1'1-
!m
ttt\
f|l
,,
t'x1
-tJ\
')D1
-lvtt
1rl
Fyl
62'
Fxt
R.
o
lSlor lto
( ,i 1
1',I
- ftlr 1
ra o
"i
o ,l ,J 2qi
lrl
=[:
lvz I
=ffltu'L
[€ = le1rgq6R]
=#t 4 [,,'lJ
[+Kq
L
u("'
)-:!--,
F,<
,
(xr- Yz) "
.,-_+ Fx,
Cx,9r)
L=
xz-xt
le
m=
ve"
f,
L
La = I tx,- {,t t-
6= E€
{.L-*r.f
-
t?
-
I
( t!'1
v[-r'tt
E
= r-l
l
_le
-,tt
,tl
)_
f' 'l \ui I.
'l "l r[ t
Ti l-r
lz
E
i
.1.
t?
[-r
U IRt{u1
Iur ')
" "1J",\
nl- \u, {
'l L,,J
r
-' Iu'
"J1"' )
l''{'l
teL
=:-
E
-m
l-L
,,r]
It ull .
J
I
U2.
Z-D
(xr
Y,?D
(" v"zt
ul r {ur + mut FnNl
Vl = [W+ qu;-t nWl
L. @s(xlx).n, cc6[/,y)
Hz-Vr
Le
Xl- K
n. cosfx,z;
=
.Za_Zt
--I;-
Le = f cx,-r,r'
+ (s-ct- + (2,+l'f
J
1l'1' t:
rFl4O
oot
= [R_]{u}
o ol
u
(ur l
)'rr I
, 'l 11;)
lu, I
l.t{J
g
(ul = [p]rrr;l
Ltn
LN
1l-
-(m
)J1
tlg
L4
LA
m
mn
tnn
t2:
-Lwr
-ln
- {lvl
4
_m_
_nn
n
- ln
-/rn
-nL
-l'4
-1n
IL
tm
l4
-n(
-w\n
Ul
tt,-
t|n
-nL
'l.n
fin
nt'
4-
-mn
lxl
\rl
I kel
LR']
rv,
\'lr
uL
Vr
ll
ryl
F91
Fzt
dt$ness rqatrix €q\4ahtn far
Eleraaent
in s-D +ace.
a *r vtss elevrrg*
If
rlabrial gropq(ly. F
Area *4
I
,t. {DCos46"
ra= ,$h95'=
tm
-Lx'
fi2-
'Ln
LL
qh
-ItA
-rnl
-I$
J
t,.
F,
.,L
r,-
Lkl
21
-1,
-y2_
[+
-,.
l-?
v
l1
l-
l-y" +
l.
-Yz
UY"
.-lrr^I
-q,l
rrh
Lltrrr
rh"
-tzl /
-bt ,
_l,z_
-L
,z
Y2-
t.
,2
v-
-4
,l/
,t)
- v2-
I E/l
\)"
I
J
I
. l:0oslg5=-h
14=.t4!|35
=
+
I
J.
€A
-f)-
:-
t22,
,2,
I
T,-
Trrl
_t2-
Il+
fkcl =
I
.i
>,'I
-v.
)
I-L
" i-n'
-/z-
L
-\
o
lo
I
o
4
lo
t7,
o
!
1.
-,L
1t
.!
z
I
' 2l )
L
-/Lt .
IJ
,l--"
axi9
Cteukat
\t
&oss^Secfion o\ beavYt
\r-
--T-
,.4
E
,Q-
dvM
-i-"
:
-=:EJ
U . tefbelion garane{er.
r: Fxgression{or
bar ot
inlarn al 6l-orad €nergY due fo
bending of c
lengthdx.
rf
du = b lc-edeax
= J- [s"4pdx
'tF
= * l) ( r"s'r dnax
J E \
I"/
tfnL
=+\L.-_vianax
=J'
r r.-, dtu ? tdA dx
' { -ttz
J
l" ?x"J
, ){vlL
e
^
= + ELT') Jc'rr&
A
.
'L
t1-
= !' t - E1 ( o,r, )dx
\
dXLt
Total po{ential€nergJ +r
fia fuam
F. disl-ribqfed lod.
n = U_v,Jp
-
L
^'
: +- 1., (*-)"d*- \ bua"J
J"'td*"''
Oo
lood al x: x,n,
Fn-^Concentrated
lt< = Monerd
--2
t
kr,{z-
_l
)-
"l
fte cougleaf 1=ao
Fr,l
Urku -f:v.
I P"ur - IMI'*
m
tr
tl
,.
I
.d
o------:-;--
t
/ 2=1)_
^
\q [=--fFF-1
^
) ___*rv2_
I
\
ur
p,
''
c.r
-;.
4xq
v2ft- 'xz
M2
x,
'"1 'e,
)-/l
y-
de!(ecltovr af nde
I
6 = jY
rotatrbn 4+ nde
dF
r.
| | ,2ffI,.
-
-
Nochl {c*ce
Nodal nrotnent.
Hertni+ lfilaryo\a+i$t/ Shape
{unciirn
H;(f] = A+h'T +cti'+di]'
llr '.
6r-tbr3 + Q 1o + 4i1
ft2
= 2'
tb
=As+b?]
t{q
+b"\
= 6q+bq\
-L
+o\
' -3
rozj
, +st. -L + arJ,
+qr1
a-
,J
+q4!
I
-L
A
t"l
g1oPe.4
'd5
T!
II
s"l
- __J*
2
.,{\
hpetttes d'
Hermi{ $op {unctior
coordina{Ps3s}am'
f, Four <frpe Suncftws
&( fJ
?raTuraI
baavne\ewn+
r+r$)=f ('-r)"tzt9
Hs(T)=f, Lr+1)(+9
HrG)= | l*t)(j*9
tt+(I) -; (r+91'z(1-!
"9=
cuht vaxiatrbn oF the Pata'aeloc
(tlermit {uncfionsJ
. .lro-
rr,", r rrr(${)"i HEuz+ H,r(* )".
,= 6rr{r}
tz/ot lto
|
r
/e'7
2-
\Yez
Vva
q
tc rlL
+"
\
\,^
1lr9r
\'l'
r.z Mr-
F- l.-l
's noda\ deRbchonz' gign convenhbn
beam derv€r\+
.
tigl l-D
- 6hay 'funchbns.
| (r-])'(r* r)
Hs= I trttf (e-:)
u
' | l-
-
lL
{ tr 11r1i+r;
Hq
+ Crtf 1"1-9
L
,
-Ue=
(, ,.v .2
d"
; tr'JQs,.)
+ 1r{ ik"l{c}
o
vG) = ttrv,* *{j1nn), *
r
J= ^/'_ x-X
x:--xl
,2( --
o5
,Jv I
\d3/l
dv
-\
-
f vL
d^
HaVr t
I -l
ar
d,j
=d'
?x' 11
/dv.
Hq["'g)z
1/- = 9r g'
rdv\
-
\ot-/2
-" -'O't
\r(i) =+\ul+FL .lel^e, + H3YA+ HLI
f
=lHr Ie/"A"
L'
'-1
Hz
azn,,
) Lu,
' lrJ
-.T
8al
i
l,t
-
l
Lo"J
= fHl{q}
-
Av _l
-dx
l
dt
tHt1{l = d n
a
d"{/
--
J- It J X !
\2
att
Lq
_{!* {r1
l,
ivrt '-aZ,
dj
--rt tq\
t ,
2
'qdH 1cl.
ld\
l€
),--4
, ,t- -L
{
-];
,
dH
,r.
tI - -u. . : - .dH
.'...-
-
(^r\_-2
.41t
JI"
\l""
4l-
f,i-J - J lt,
tv,r,
L- 2'
4)
t-t:
i ^ -'
r
fzE I
I
( ,,
\ jl-
J Q r4
-L
\t
9,,11)
nj
,^drduHrT
ltll
ll
L'r
\ ) r- zt J
t,r.(*,"it*rtt+
rr'
--
-7,j ,ry ,rt
la./ .-l+9!
z<
dY f| ----a
-:'(-
t'\/
t-
^q,tt-\tl
f*I
rql
-1
'
T
,^T
+ Er lqt
33
)BE:l
| -;
--,a
'
*t
\
La -lrJt
-;-,
='
-
ru
JL
\aL[3T
l-;
L-
,141
'.ail
z-J
4
Iq
tL .
-\+g"t - GT
7
lu12,
'
r+}t lJ?'f
lqf
";) ) I l t.
:
'l3sl^ I 'r?H''-\'t?'
=+{qrr?r,,
Lg
[l e-'ry
i
'nr.
\ Lp-ll
I
(
^,
3
lt ,i
L t +ry)
1r
\-
*xt
--
+
11.{f lK€/
[r"-] =
{q}
f' (
\ JI
gEL
Jr
LQ
ldt
r
]"-'t
lx t
d+) .
- sEr t [-rr"rr.'1
Jrix4
tt.l
E!
.'_^.
1p2
-
\
|
I\',,/
f''
-t2
61.<
+6 ""-*
l'o
^ ate
-',
|
L*
C[€
-l
":
zte"
\
-ak I
"t
u*"J
1L
- 6lz
!"l,eL
L
a'
rr4
\r]
\{ lql
a-r
U,,
J-
o
,t
Lrnlr l
-/'
c,f
ul
''-;-
q}
. , 1 ' = I'
1a
I
6IA
1
t.
6\a
'{&
I
nr] (( u,
*'
) -1*' \
\ u' \-
- br?
:\
L
(,fu'l
\r,' \
lts'J
)
L*,
,{ lo
&sl(ihrtc
tod r!
gresen
E
wr,'
.L
r1v)rdx
J
ot
*,"or
= r (rutlol)'
J
-t
l_
=\cf (+ f [n]'a'l)
\-".
I
L€Hz,
fr
= lqtr ( Ple
Lnr' z
*
I
r''
-t
,
fl3'
.T
SHrJ
\
dT
- {r?' , | I't(',-rl'tr?)
I
Pl4
.L
'o*
i' O*i't'-')'tr*'lar
!
\
, \'u*rr"(r-T)'dl
I
\
Prl
DtL
-t\
'('+ill(')-r)dl
t Bi
Jqx,
- heo
^l
= loitl
itl = FI
{d
dv
dx"
lr
Ie"
=
t
11
Lt"==
-r rI
td-
7d"H\
Vt'l lx4{qi4xl
'Yr-'Y
I d'!
l')
l7
L'
l
fr' I
qt -
- 72t
r+3.!
t"t -, l
tal
l
nil
\tl- \
g," J
-
dM
-'1---
dX
tr
cr
*"'T-
0
T- vq'a
)
d
I
d?
le-
L-
I ut, ,t\*
'r
=bE
3l
t
=
-1, , ?tn1
-{)
L
l€e
le 1+e1
-;-'-*
f9
q
le r+ejl/
(', I
1u'I
lu'l
L('.j
_11
Er,/:
Na{at de{lecHon
Sheat ftrce
- 6tih ends lixed
|earn
Reactton
116rxPn'l= ?
+x+ - t-ou\ s\i\t"ess mafxix
6Kb-
Q\oh\
rIl
I u"l
s\i{hess vna}<i<
\- tr-
Te \u'
I-'o
-5L
a^\
€T I'o
-l-,
-tbler-
L otde[befion
Sbpz JdeQkohbn
IL
qr--.l
-6L
L6L
Ik"1"
o',I
"'- II
-eu
6L
6L
-12-
rrLry
-bL
- '6L
IL
gY
-61
=9
^rl
r-r: I
_nt
'.ir l\
Jl lra
Lr"
a-)
\-k4l=F.*1,+ [K.J'6L
IL
2(
o
,tl
o
-lz
btl
o
EL
-6L'
.t 2-
6L
l-
lz
-6L
2L-
-€,L
o
6L
t24
l
LO
--'
aav
-b
4f
;f th1={;i1
rlf:' C:.;cci,t:t, . -- , cc.Vi
Pc-,il
y(d
Fl
.9r
/'rr\
V1
-P
D2
ll
W"
F2
bl"
My
--
2 5 / O tl t o
t^
('tnb"+*n
'e9oF
n
.
Paravrrelers)
(Load'lrg pararoreh$
I{p
Equivabntfid6t {trces and rnortanft dqe 6
dislriLL{l€dlmding
I rr {u}= {o}
Ff
l2
6le
-10
Ele
Gl€
{tA
- 6le
2\t
-12
-6lz
tz
^'"I
lf-\
-{le
\
-G\e
di:\tit'iC
\"ril"t1,
Arf l
( ?te1-
Fur
lll
1\
E"
l{z
r,
\V"
\+vtE"
l_P,1
Llz
tr
$Ue lite
z-eleraPntrnsdel)
&.art, Prtbbnr b7 Pg^l ('faxe a
1
s tarrent
r
lt2
[K"'] =
trrl
t3
-l
-ilz
6L
stl \
I
-!q
I 6L
4L
I -t2
-61
la
-"1
ot$
-6L
qL- I
I
& = lkll
')
+ [k[l
-to
a'
,1,
xl
6L
,l^l
|'t
;I-l
L
L-
?4
IL
o
6L
-5L
a9
2L
il I"J
\*
-(L
-6L
|
Fi, I
1-pte
/Ar I
i,pti
dl \-
y'l+<""
't#) l'.i,
-'" i
l
'
lu"o\
)
L&,,
n
$t,,a +he bednt sugporfed b/
Et ing.Teke lt1" be'am as
6irrgle eleMent'
t-.
KVz
(as
l.
l5L
l-r'
I
L6L
I l,
a
"pv,rarJ;
CL
ET f''
ft
tl- -n
Systen'r
Cttsu.
1
'"
-t,
4*
-bL
- bL
l2
2g
2xz
fi
- 6L
- aLI ( r,z1
1-r_F'
)
o
{
lrr
\3q"
-
ldAt
(-*"'-B
'
,'/
o [0,- ur]1
lg
^- \-q ?,
f'rz1K -6LlrvL\
L'
qr'l\e'l
l.gl
L-,,
rl
Jl
&nErder c deY'e"\\.
e\evnenfy' anothersPrtng'
6L
IL
Lk'l =
. -bl-
4g
LL.
_6L
-la
6L
r. 11
LK l =
qf
-".]
[R
L-K
{*}
kJ \q )
-6L
r t2tK
D
+X,L- , - tzl- -
o
o
-6L
e{
-K
6L
o
2f
o
-il
-k
LrJ2!
E-l
K
;1
1
i;i=
+l
[]:,.
j
tl
fv Xrrle elenenf
par rnodeJ
; 6x6 nakir
(e-ote
!@2
6"---^
'1
_,2U2
111
r,
{\ t [__-,------rJ?l
4
\-
/l
\1,/
\://
tr
0;
;l
/
/ A = J ' !-
-
h'I,A
3 -r
4
\v E------Tl-r--x \l_/
J-
!,lr
tJr
€q
tl2
Vz
cz
Rafie
€l€Menf.
@upe<iosiibn)
_ \,{r
IEA
rJ1 _
EA I
-T
Lk*.] = i'I -CA
I
\1 11r
q1 qq
EA I
l-i-i-l
LLI-
2 356- &aht
tl1
81 {r,
\rl
v1
J+
oJ+
o o
-
L
OL
4o
o "l
"l[u'[u'
vl
I lt'
'+ '+
'+
-:+
f, :+ ry
+'-t(#\
.#l\q
o
6F-L 14
L
CEA o
T
-!Lt rnl \
o
V
-
l-Lrl
E-I
L
o
Jq
y.
O
I
ol
I lu'
I
10"
-'
-YE"g,o '4 -FJ I
o
Ir
L-
6E
{1
I
Ir
o -6tf
fltItr1
o
Xz
fY,lL
w I
6xA
5C1l
VLLYTJ
\oad
Aitfnbt^W
lr- |ransversn
1tad.
disl{ibuk'l
q- axial
*
$i.Snes
wralnV &r
sirnglr {ratr'tp
nh^efr'
3dor/to
tr
g,1 ,A'L
16-Z----#
-__-___-+V2
-----+Ut
A
Irarrp ebuent
lvz
lq
(e
\ \.?
u2
.
dof
- beam dewre.*
--
l)----
matrorr4
rlilh nodal axial 4nfov
--------
r6d
t-I l1n
(
,142_
rir
iY'
|
\
2
^
_-____?
t-v^
,x\
axialrca
-
,,"t*,
/lrV
lding garanrehrr'
lr;
*
oo#.
o
o
vyo
v
veo
V
o
)e
*
-
t-l'
Dl
d7
o
60*
o
vvo
d
w90
{kom bar ebr,er*
{ro rln beavnelener*
v
V
v
v,
VL
rrt
tr
v,
r-lr'= rlr Cos0 + rl1S1h
0
u/ -- - r.llsirrO+ urCos$
I
I,
x
4eonre*rg ol
{va '" e
eler*ev*
-dr-
lv"
(u
lri
Lqs
lc
4 c,,
s-
----
'"'
th:
)
' ; lof lret \r't - 1oit+t
€
vz w2
fr-
-
ulCcs0
q5in0
"
r.!$in0 + vr CosE
-0"
ulihx-axis
r'l
a,'sg- t
U 1 / = l r . - 1 11 r 1 r . 9 1
Srh€ = Yrt
rll = -mrlr + lur
$l =y62a mv,
Vi ' -rn\iz+ l\tz
ut
Lll
iql -
v,'
!a'\ ll-
d,'
Vt
0r
lJ2-
w2
VL
u
0z
ui
t
D,,
-m
I
0r'
o
o
o
w1
\t;
g,'
tl
t
o
o
o
0
o
a
o
o
o
,\
0
o
o
o
o
(l
o
- Yl1
o
o
o
c)
L
- l*lex,
1qi,^,
lo'1r_,
rnakix
ft66ftrmado'1
coo<rlineta
ti
i
' t ,,7
=
Us { lq'f 1r"I { r'}
'f
.
r
^\
{q})' [#l uer117
= + 1q1'[el ' [rJ] LRI'fqt
-I(nt\ -
-2
U1
U1
0{
rl2
vL
02
Ia
\r-}'[re:'rkirrc {cl
=* {qi' fu4\ql
6Kc
+
io'l'{r.}
= (lnt {t})' { t"
,}
= {qJ'(tt# frl),-,
- \qlr {te\6K,
fe=
fl
0
W
1Fa?=
II
'
IPJ
r*'v
pil"
o
.PV"
-?(r',
d.'
[r"] = [R.-]rlkd] LRI
5oo{1?r
t]
-f
Jocc lb
E = :o xro6p;i
I
I = Al inq
s
4 . 6,8 \'nz
,n'-)
Fo< e)evqerl' L.
|/t'
n1=0.
t-
--R.z
a
o
F+"
FJr
\\\
o
I
Inl =
o
o
to{ €bn€rh
t-
2.
J
o
o
o
o
o
o
o
-l= 0 ,
1
o
o
1_
oo
o
to
o
6L
o
o
OO
I4t.'1.
tr^
oz
[R]-'- =
Ol]
H
t 11il
oo
DO
oD
tr
= - 300a
,
r*t'{o;1
{*"t
-?2000
0
- JOo0
7200
Boundaqcorrdiio
n.
Fy,= 3oo o
l3
="\
:OJ
rJr: u
-$
AAI
t{q. o
R2=o
u4S\'
Fgr= o
*
lNz = o
I'
0r/ o zlto
L.l
p=arrd nl
€ = 2 o xlb3 lt/ilr\L
(a)
ftb1d
Find the
1dr:Placemer*
\
sttesst'tu
\
-
I
l.i"reaselerrcnts
uc\ng (D quad<ahLeleqeflfs
\eactloY15
J
I s"PPoot
cons'(de( ba{ denrent.
(,a
-1-1
EA fl
LKr=T
L-,
rl
t42 '+i'
" F'
EA ft
I ksl
ll'l
-l
o
u.
I
L
2
l-t
I
Lu
1,r3=fg
-l
-r
I
htm
,{'o
-rl Ju'l \)<1
l<
7 -
- l-'
e* f z
'l\ul\
-/
, t12 ttttlt
\.42
= 1,5 nrn
x.
'c
=
-
!O XN
{)
Isc i
V3
\o'I
<)41
\rrl
( (,?
_L
<t
|LlzI
zoo ttlrirk
a
EX fl - l ,
i-l
\ |
- \o rt/mnnL
5upporlr€6c+ion.
d
=
=
wt- L
--i-
_5O KN
\Ur
lvs
06.02.to
If
I
L
t
992 yA
F--L
lll FN
uL- L54 A
z
€\ = 6,'€ xrdoN/m
Er - ro'? 11Jo * l"t
L3 = 3'59 m
rt/nf
e, = ro 'a xtJ!
\= 2'54n
A1:> $2'!
Az- 3s'A em\
43 = 25. fi, crnL
itt€ \hras na^ba( iY\lss-
FrnJ sr
- Nodat d€tlecnurI
- MernbQr
-
SftqsJ
(€a(h'ors '
va'- yr^oor
l - F
a.
,J
I:
t:
L;
fk"l -
nI
915 lo
l.
l-r
I
a,
-l
o
IK'] = 1;.7.7
r+
I
-l
o
o
o
I
t
4: 9
qu
I
-l
,^
r
l
CtttL
9s
9q
4t
Qz-
4
I
,
5
I
l K'l
_-
-I:
n".r
-t
{
l
L;
-l
1
I
il
1t
I
-,1q"
-;lI
2
q-
t
-t
-l
6
ut
zzz'<N
q
IIt KN
ur-
o
as trxed.
Sr,rpporfre6clron
Vj
tlj
7\5+74.c
- f +,t
-b\5
o
o
-7't.U
h.+
o
*715
o
A'
L<'J =
o
o
-74'Lt
?4.\
l%1\
?ri.+
L-
-74.q
o
-z+.+l \Q'l
1+.+
r+u
I
-74.4
|
0
a7.7
e
1\.q
-?q.q
- d 671
l
t
?+'tt
:+h.V
"l
-37.?
I
-7'f'ttl
81,71
+"r{.!J
(2l1.
=
,,\
'+r-i
1 1-','I
{r.
4L'
tr -rheot!
oq <bsh'citJ
U
4c
l
*
I sv"
s**
r t'-
n,ffi-
------2x
Qt
{.}
'
=
\u)
scy
caz
zyJ
tzz
zxy
uxy
L.ta
6Ja
9-.
6zx
€q\
equiltibrir'tttt 4
S
b (x*
b(xy -r-
bfzx -:"p":o
bx'dg
?6yy
-Dy- n
Ddzz
__:_
A7
Z
'dx
bY{
[*-
a;
I
bz
.*- P1=' o
- )Se
b?
d(z:/ * ?, -- O
. -PSz -. -T7
-
A_t
- r-'<-w
f6u, - v 6xi
'c.
-6
tdxg
o bodvin
L-"
LQ* - tcn^
1------:fouard whal
Lt axi. porp.'l'ot,l.
6xr 'z t r lsr.,- cyu- 6al
- crv\
er. =-[
[6rz- e7+
.V,.=
- oYiJ
Lsex_ 6'q
i
z.a{ = -Ti
-rx
cry =
-T
2r* = -]a.
, = -5!
b\r l\,
zxy
{- Ti
6q,
,- _-
XU
i\,1
r-E
5t
a.*=t* *#
( .i?
luJ' 1"1
lwl
aPcoach {or
tsasic
r-0.
as
,assqllad
bodJ
3-D
E
- Plan€ s'|r?ss
(zo) *:9 groblern
g^}
6*.€v,s;y
- ?tah€ st\ 6tn Vo*atl. (zD)
- urfur y'ane dro,s alSurn$ott I
{nr-1""'
tl:'\t
l<"v t
vJhen one dinrcrrsiont! for <tort.
bff* S.*r.,.'""
?(yy + )(ry I gy: o
Fbx
('"1 l* "l
{ul
) €,vrl ^
*,
t\ri
\t
1
l"
l6'l
X--l
r?
{"}
L&
t'
\-v
t+
C e<**-
elr) --- Strain pery.{o the plane.
Plane strain problern
- lrlhen a
' -lhal
bd) {s 16l lorrg
sbiect. 9Y.
can be assqmeAat 2-n
PrassurfTedPiPe'
'€x:,, €yv
s-*, (lr
>tr_.
'-:}l )- #"xK.., + F =o
d'l
dx
Dsry
?)
'
bf
tx
-+ Fv = o
v
-'1
\,<-
oez?
E
i_ F3
:O
+-6rr)'
62, = if<xx
F
exy
, siv. <>z
:lbtu
tf ?la.rp F{agh'c Roblens
a-
-R
[rea< io6rithrdernenk
d'a
diecsetizahon
u,l
rl-
11t
"t)
L4o
\-
lqu
l4c
r7l
[:'
R,.
\
) :E' . .
lr'
txL
4,
lt,Fx7
\4c
lFx"
\u
[.t' 1 3
'^^)tc-:"/u fi clocKwts€ r
iflq
trutnrbo<
\,{ = N,dr ; N2ll2+N3\i13
Ll -
-t N1v3
: N'u, + Nz-Va
s\4e
+ Ikl tul = {r}
\:,
\qu
\-r
FE:
usin€ 4<iarrgular
bdY
Ir
ul
url_
U,
I
\,{q I
lq,
q,
frl .n,
2o
fi'trtc'tions'
preterabte.
Nr4r+ NrQl+ NgQs
'NrQz-F Nl9 t{ + }'la9t
q
It aq,
I
" I
(ul
f"l,
-l
-[o
1"1
OtlzON3
Nr
dZ
o
o
"l
\
Nel lo,
q,{
\ I
\qr i
\
O.
a {uJ=trl{
J
Tteee Lnea.c shapa {un6{ionsh naturalcoordinat< syslen(Nc9
Can ba taken at
\tr,3
Nr,
Nl.
N1t,t'L+N)= I
P<operi'.es:
Nt,o
nod'. t
dl-
node 2l't
"tnoda 2
Nr" I
*l-
o
r.l1=
af node 1t'g
Ne"{
af
N3:o
I f a '1-]
rl
nr " l-1-"1
Nr' t
-
el
qod'c 3
vPle 2'l
!'t4<l
),
l -
-TL
_-_)-
NrtNz+ tl: . o
5
lineac 4"ria'ngrrlatebrneni
[r-rr)
bnstavt stcatn tr i angle
N1Q1
1Nz43t
.^
-4-\
'b)1 .t
\'1r
1\ , 4. , - Q. r.J\ - in -,r ar-g
Q'-ua) ] t (,ar-ur)? +,t:
Nr9r t N2qqr
5qlid mechanics)
N:9s
T4, + nq.
r- (r- !-n),
.l t)n5
I t, , \"
U=
t\n
N3gb
r(r_T-,,1),.rS
J ur+ t Ltz
= 342+'l(++(r1-1yu =
J Lq,_qr)*rt
ll+-1e+tt
:Qe-ur)j t 1f,r,-u3;.'v,
U --.Qtr-uf +Lr.{z-ul)l+ \.t3= urr} + rtul _r_
t(t
J
u - Lur-tt,\ + Lu,.-rf)
1+ g -- vral _ruzs!+ vc
fo\
'
iso-paranretric{ormulahon
x = Lxr-x3)T + (xr- x3)'l t *3 - xtal *xz3l1y3
Y = Y,r\ -t lzr'l + Yj
ts61din6l-a
gtotzl c 'nafr,l<al
sys{awr<elahbrtshi?'
du
dY
2u -r:
)x
-fr
irl
,I _-i:-:
#.,,"
-r,
\q
tx
dYI
I
?'j
=-
o.1 oJ
)
iw
.
-!L,t
I-
.,
.lr3
' ix -.br
-tT' )v
;i
?1
" S*'n, n-$'$.,
t )u I
r
(bu',)
Ll J sr y
i\
{
[*rc
1T{} = I
jn
l!-\
szai I br^ I
.J t\
|
"1
I bt't
>
l .\
|
j"r]
I I!.r
fr
'l
l.
4ransitrnafr
L **r,un vnalrix
s{ coordinafp
-
+{
br.l
aul
OX
oE(
-
0u
I
ITI
al
b.t
dvl
:
5rl
I
:-
I
bu I
ur)
1..
@'I lTl
def[r]
aLl I
I
-$l
U-^
I
I -x.|
'trrother relation {ar
ldei [J] | = lr
Aarea __tL
xr3_l
6eomeky "l
-r',1{#'l
e",
f
L:"r,
-''l1fiI
--tr)
qY'l
a1 |'
tu
I
-l
!alrila<)Y]
AI J
-Friangle
Au*a.
der lT I
Ar+ /q-+ 43 = A
o'. ** * ** ='t
)
N r-'l- Nz +
Nl3-' 1
.,
\:
,$r
i-,
-r=F , ^a,*
t1la2/t"
f-]
U : \lrUl + Nztlz
+ Nsv:
N,91+ Nzua+ Nov.3
U: Nq{,f Nz43+r{S9r
E.,X)
rJ = Nffr -t Nu4rt + Ns9{
l-inear{"rian6,,lar elenrent
=l:
--l-
o
N2_
{.,1
ur-\
Nz
o
t{1
43 lqqe
| [0,
- e'
.\o]ro
=
[Nlr*.
1vt
'
2_Xr
tlr.
Nz. vl
J ,
-aq
I
I
*l =
[r]-'JLu
)
[
I bu
I
b€tlrl
6l
I
t
:
'f o"
I| -7o
L-'
bd- LJ1 = xls y23 -
t+ [^'
L xz3
1
IT
LHI I
''l
tl3 . l-]-
z
9rs^4
ill
bu
E
iu
q
I
-_^
s,sl
I
r..r"".
|'--
Del{Tl L:}L?
_!,"-l
xrg _l
-
1n'*
l- &, 6-u3
uettrl
rJ.- -Irl
5u
" t,
**,a
*' l
5irr i Iarly,
lzv
b!
6X
-;r
LTI
bv
^-
0J
-
\iT
[235]
l
1l0v
detkl
1"1
\"\ r*,
=
(€"1
{tvr l -
l.*,)
l-ro\
\F
1bv
\aJ
I
| 0u
0)
'n
'l
-
I
Fr-a
* -,rfiJ
I bt\
U29 -18 -l
l-*" -19
,,r+l
dv
5i
6U
5T
-u* -r-n:
#
$
-*. + +xnfi *v.rS-v,rfi
u: Qr-rdl +Lql-qtl t c5
(ez-eu)5 +Q+-9e)lt lr
- crslag-l4
'
-F-61 IYBLq''qt
1.1=
+ x1e(4t-4u)
{-rry.-r,l
q,l
Yzr(%-qoJ-vptt+
!1,-cr) 1 xrI (le-4sJ
\-xrr
gs
{, - urais- Qzr-!ra)
t9z:
g,
d"tFf l "urtxrlqq + Q<"r-*rr)
II >tzrlt
[-
I
I
| 9zr4zt xrzg:- yrg9v+(Xz:-*'4r5- (ve3-vr3)
9t
J
(9zrgr- cragr* lrr'Qo
,l t l
I
e6.
l+ ttt 1-xugr-Fxr3{q+ x621)
ll
I
vzsq"+ Xtlgr -Jr3tq txzr{,s+ v119c
J
f"234rt
I
9rc1L | 921171 Ysls
xtt4z * xralq + xzlQ6
)d trt
(
+YrrYe
I
xSzlr + Y2j{1 }x13i1+ 9j1 {q +Xztls
o
,n3
_1
\?z o
U=,tSf
vt92
-
tl
-
\4"
lA\
'')
L
lar
U4
*r)
-u
osizo
x'5
o
xzr
'14
!a
AL., 9rt
q-
{'1
7xL L j Axt
l;
o
Yrt
*7L
o
rlrt
*r3
YIL
O
o
I
xr,
I
v?r xlr
Xa
9r
d6r \J_J
q rolal pobrhal sng1g1 of
,a
t' = t
the bodv
fr
n
rT
nlv
ieldv_ itut
liel
-\
J'
T
z
lul'TdA\l,
_,/
{ui} r,.
lLt
'
PoinN
diskibu+ed
body
'
-J{u}r
rraeI {u1 r+aL- i,{nrfli \*=t'r
=+ (nt'Tt'tt'^
J
)'
t
\
" 1I
15
= =
i""\
I s,ntJ
['*l
t1t
[u*I
;J
1"\
\€-'l
v!
\- va
plarFsh€ssawtu<.
oq
- Lol\n\
t\ q"Iq
f'-"
E
lrs
:
(u- [ r"J6o
I
Ld
I"')
= lo]lnt
t'r,,,
\t
L-ls
rl
6
Plarp Slca\n
;
'Yy
l
a?rox.
6^y
F
- --z :- ^
l- rJ"
U'>.-:-*
..)
T
r-
+ liet Lrl lef
IT
e
T dtr -
e
tut {ol
rtr'
u" =*- J lgrltal)t
-*
r
/'t
I
/
1
--?
'
\ Lt1 1-tr1 tDlT
rsl lq\
{u} rta* Jl*l"t'L'
+an
{dr
-- l2 -
qaY-1
.
-4.
2 1ufI rt
=i lol' ( Jtro'u'trsl da) I'J
= -l- JqlT
2 \'l
- rt \(
L2
\K"l*e
=
I t,ir'r"toJotl't
q1. (tn"I*1trn'rer)h!
,
\ql
tr"l tql
tAe
["tt to-ttttr
"T
{ Ndl€ numb6ing 3nould bo
ecw.
2olo2l1p
rl
[re] = +A" tBlr tDlrtpl
?r_ Di
l=l
els+xets
P/ skaln
ne=.| lants1
# t5dal share al el<msi bodv ftrce (ft)
J{u}r'1 av
tf {li
ru
J
e levrrsrt bcd:
= {Q'o'.
J
flJ t an
+ (r'qr* *tqu1xts9e)
r,r-o^ .r No.9cl {x
1,(t'r' {nan)+ $1{
ae
Jx, .
+dA=J(,.4&rurv)+dn
o'f*9
1r(J+"t'a9*
n u\I+^rt-dt
}+rt1iy do)
+ q,"(tNefldt
r/olumeof tte tek hedron
:! x ba€aroaxheight. = -lAexr = + Ae
r lx
fo, 1t
4y
iq
gs
Nr',A
Jr
CJ
' -__t
i
N 1d
lfx,ll Nt7_, 4t
+frI
l,N,_dIA
t{ i
\f' d l
N2i
s'
r'rraA
I fr \
fn ^''11 ^
:'LY\
\t
^
11
= 5q1r 1c"'1
.
'
I.)
^- a.1 4k
lS -ta]i
c'9 ndet
eorrngonents
1-,n
bodYtorce'
[\
r{{""}=ry lr," rr"
*
'Tx
'
n'l
{r'
fu--, f"o t' Pr'
huHon ftrce'.
Ndal share sf
Elemenf +rdc'tion{orce
_f
-j
fr
t.{
_f
-J
r)( t
{
,tt r
Jr
e -t..,:-l
,l
dI
-'(t l
.I
Lr-
I
J
I
$, YD
TY
x
Totd
as al 1 t'
force dqljd4 \nto 4 patls or cofiPemenls
U^ r..lr9r1'
r,h9r
lx = Nl lxr +NZTXL
l, = tlr Qr t t'lr9q
-!
= {1Ty,+ Na\,
tocdon tzrc€ = \IuJl
5o. 'e.terrrent
I
zl9,,t',%
t:
l*
/-
-
l:l
1
L'YJ
td(
l_L
- l(ur*+
-J-
uT:v)tdl
= [1 nr,r,tu.t)(or'\o*,r*i+Q\qu+r2lf
$,1,r
JLtr-z
=
f
'1,
u,nro
t, + N,N2 q,r
{
xl1N2Tx,9:t llrr,.1,
^,lrrlzTt, 9! + Ni TvLl q) t dL
t
; N3rn 9z t N,Nb1'z'9r
Jwar
r-rn
J
? n
'$,*. d,= *
ar = -=-l-rr
[,'r/
J-
Lro
91
.1L
Nr tx
1r, I
,1
+
,-l
N r N t. t >aZ
x
N rN z
ll
- Ta
.1
'1L
c4
I
'-F
ry"
,"1
+NL
1T^ 1 1 Tx2
x2-
f Lrr
L'l
I
Nr' i ! xXl-
N 1ll1T1t t(1 NxTy,
a,
Vz I
(
2 Ty, +rTTYt
\Yz
J*2
Txr 1 - 1J
1-T*:
Tv^
Tyz
+ 22-'l
\r
n.T
tql
r -o
1
lrl
L'l4, + Txz
wherB, 1rnl- tli!
2Tv,4.
\ll t
/:-\
LDltBjll+ApLbl
-6X6
"
9r
4z
'Tv1
z rxa
f{
r-x
Fxl
a,
"t
q.,)
q/
s- Elbl{at
-t.
fDl
Fyr
tAe
t
2-fy,+ T*.
1Y
+
+L,
-t-
'16
11
?*g
11"
2'ftr+ Tx2
{1
1Y
.
T4 +2\z
fyi t zTyz
o
o
-_ elactl'
Progar\
l--
lEr^rTq*rl9t.*,
&orn coordinzte
Lrl
Y1
I
I
Yrt
I
q-
I
x
U=
+ Nsus + N6u6
N r U l + N2Uu+ N9u?+ AtqLlq
\J r
l{r lrt + Nr-vr*
N3\rl + r'l1vq -l- NE\'< t
N6vt '
'.l.- r.l1{1* xl2! 3 * Nsqs + Nq471d54c + Ne4U
V= Nr4r + tb4rr tNggg -f Nqgg+ r'ls9roa Nc91z
('..{1
1
1 ,I
.Nt
-
I nt, o\tzoNsor{qor'lso
I
Lo
Nr
Ne o
n
r'lc o
N4 o
Ns
uool
o*l
2](tL
1ur= [nt1r*,"
lalrr,
U.)"
U't* +bs rY i
bg!
r
t
+bz-x
h
bt tbSr+
6** = *
€vv' y
€t]y
=
{
j
l2t
b, Y-
b', Y'
brrxY*
I
btoxl
bg'i +
= ht%qx+bs9
= $+ bslx-r?brrY
b5 + ['x
\
] lb1qx 4 b r r y
+bg
rb6Y
1
1
8.ud€)rc shaPn
*.rnchbn.
.,
S{rains ?rs lr'n€ar.
Stress 6'
6frain va{r)e< [.r"neor\y.
S Shap {uvrcftonsin
Nr =
r.lstuxal cocrdinalzsgslen:
5 (r1- r;
Aiz
.l(E-,)
N?
e Qvr)
N4
triargle)
l tt,ear strui"
, ut:1
Ny ' \ 1\P
tl6
" 4 Y\
\o
^ -.fv )
it'
1l
6,\)
Lkl .
JJ
+ tb-l-rtDl
t{ &
t Cel+rol tsl drdY
JJ
vi
"lJ
tD1Le.) d€ttlldldS
dxdy= &t ld Jt d1 .
-
q,'l,:'ol'^
-_lv-rIv
tl
gr\0
linea<ty
.ratgrng
?regglt(€
t=40 4h.
M?A
7-
$m,eo)
Fi'd th"
0
41,)_
co01P0nen1idf
Yt
-T
-L
L*,-*r)' + (q- Hif]
c
a-=
^12'
-Lr
1*t
-kacttbn force
?t flode ?,3, 9.
- o-bl t
=
-_
LfL
=2f
Xr- Xz
= 0,8
lrt
ll1
_T-.
lYz
-
=-5f1
= -
z E
>l-z
^'
IJ
t_-
t
e (--=,
( \/
ti
4r.bnNs
2r- I
L*.1
!i
\,
-T_
lgx
= - 0.8
Tlv = - o'6
= -1.6
Ttv= -t.z
| [te-
la-e!
"-7-
lTxr t- Tx:_
2Ty,+ TYe
Txt t 2tr:_
\ + zt-r"
,l
{000lb
tr
' r,t\
.f ^ n,q
A
E = 3oxto" psi lileet)
\ ;
a.rd ehrrent
t rnd ihe nodal dirPlaceTnenr:
-l _
Q,2T
SkeSSes.
,*
(aye 4 lriangulat elev(en\.
Vsa
ill
plane stf€ss
ximahbn,
aPP{o
,,i,+
El
FE sDluirhn o{
z-D elaslic Pr0bl€l?u USing l-rnear
quadrilaleral Q@meil(.
rh
(iz vz]
qs
Li
Ur
' (xr
Y)
(tq Yq)
N1U11 (2Ur
+
N1r31{ N2V1
+ N1u, -r
Msvlg+ tlqUq
N r ,D , ,
0\
U=N,o'rtl
V:
Nr9t t
NrQr -r
Nr94 t tt39p + Nq9s
r-
\ut
<l t - l l-
lul
Nz9s+ \9r
lN,
I
o
Ld
N1
N3
o
Nz
o
41
Nuo
xlq
N3
o.
:J
Qy
YLI
\
u]r
L-r
3
IN]
x\
:l ;t)
IT
1
2- (l
,)
l
tt
9o
Iq1
2Y\g l '$xl
,(l
-t
;
|
z-\
_
J
Lt
_)
rl -4 I
- t r zl'
\Mas
-
L
i.l I
t,h-
0
o
o
I
\
0
cl
N3
Nqlo
r'r,"efr_j1
qu-
Nr
.lt
N)
Nl1
q
0
o
L
o
t
r(r,-r;
2 (\rt)
(r1.)
t -- cxq
Nt=
o l
,r)
+ c= \74
1)
t'-9
r+1)(\-1)
=i( r+]) [tt !
.+( t-1)(t+11
x = NrXrt tlr- Xr + NgX: + N,rxq
7 - NrYrt Nrh
+ N3Y3 + *q )rr
lil =F'Jl:
I;
- tlo-pataruetvic
&rrvrtl\ation
shapr{r-,rnction5.
-
ri =-1,
(rn ? ir) (r+
,t
1 li)
whqe
5i,
U -- Nigr t
}f
d3
E
N.t1+
N:9? * N g 9 S t
N g g ( + Nq98
oF
-a-y
Zl
u
)v
dT
I
coor&hotpa{ nodcs.
=* &.s +
^)f
-
bf
?4
are the
1;
NrqL + N:-1,r +
\t -
t; |,2, 3,t+
}L
frFl
\tri.
t-)
\df I
lSnl
LI-
)X
\l
b3
E
. -!l
t'{
r b+]
l"
I
<(
lbt
D4
I
I
L" t
s-l
dxl
gr
'tx
"l
tf
o't
)
?t\
OF
_-l
I = LJ]
;.1 .)
EE
)
l+
b'l
1.
-(\,.l);]
_0r =
(r-1)t+(r-1)xz
r.(rr1Jx3
+ LbT
)
- i'*'i;':lr',1"\1t1)$
-tL-$!r
*lr-r)Y1
z8 ^ 1 f
J2X:9
ix
q
tr t Ir!]
'i')q
1
*
*
t
-- L- t, g){r (r*3)v, (-,'
T-.
ag
--!
i
5-1 = .L- (,-a)'Jt- (r15)g_+!r+1)
0..,u-)\l
-
Tr,
JSl _fr,, ",,rJ
5;
l
lgI-1", '"1\E
01-
I
€r ttTl
wJ
r {LJl
f* u""
"11"5)
It -
JrsJ
L--*r2l
( 1q
16I
hK
=
iz? lto,{
1 lv
t€o J
\ "drl . 1 \ 9
+-\
I 58 t f , *
f bu')
\5x t
<r
lbu
.-
I
J
[57
aul
5iI
nrf
ryJ
:--
I
II ',z.
dsLf,JL-q,
I
ftL
de+
It t"r
-'l'i'
u")
dl\
Trr
ql
Ivl
dal
I
Trr I
=
.*1
IDu
)*
=
1tu
{o}
l5Yl0q
,T
tul
-)
[vl
EI
b\r
t''
xzzzT
zT
:\\,{
+zrfr+4,
$
og - L )
-Jzr\u + .-rtu
,
"'za
t'I J.,tt t Tz'\
'\r7=
-"-l
- lit
rzt
-521
o
{zr
dr
Tzt
"u \
.r'
-"" I
J
4rI
VN\
5;
0u,
5;
-l
7\
' [p,] fa
aq
hJ
b,1
4xt
{,1
8(l
(r
r,lL
' '
\r1- tRl
qe
?xs
Lal . [n] [q]
'1rt
t7
A!
q.
IT
1e
ft\)
-- lol
}
{u
\,1 , f>-lLAItcl {
KflONN
4)
'l
covd
.t
I
'
r t
ve--2 +lj6f
^T (.",
1rl+e d A
}\
A
(t,
.r.,
, ^rT..-(^l
dxdl
jqj
-_ r+-l(l-br
Lq
_
L _ , f, b-llq])'
I r/
L_. 1|\
Jl_
:
_.-
=\
Car-T
=l+,1[qlt
-2iJJl
fBlrtdfr,,l !q1 dxJr
tL"r
h
'(
E_
r ))
'(lu
ral [Bl defTrl d?
-- iqJ
[s-]r
JJ
,,
I
\ql
=det-Fl* d?
dxdy
qt
e1r
tr..l
{
t
r(
=>1
'1,
rl
ro
ti
=6J
x"1
i
i
-{ -l
lJ dl
ant
fsl
[B]' LDILB-I
r
-^
8r8
of rhtet'Po\ahon'
nrethd
$ Garrss \ira\radure
r-[qutdx
-J
-l
1-: o1{(x')+ cof txrl+crfu4 t
--
+Cltr9-Trapezoida
- - = vlrt8i 1*1f.t*z) + w?+k3)+
n
+wnfSd
-. ,
= Lu.li ttxi)
|
1i=
\Nr = \laightinq €eeFot
Yi -- qawo PointsGP)
{kr
Ut
\l^lr'\
t\-J
€rx
0 = ?"t
unknOV.]n .
I
Error '
I
\fl.odx
fa dx -Wr
6:v):o
J
-t
Ira
1 i 'x) Ax
[ (a"*t
J)
']''
lr-
'
v.',(a;1a1t) O
--l
3{' I
> lax+
L
zJ
-l
2a" -W1 !"
+
r
- wt?o-ulrdrxr
:o
- w 1 3 1 X 1- - o
a.[2-wr,ts
(- a,w,x4_ru
3" ( z-wJ= o
d qW , x 1:
?w--z
> Xl :o
+ 2- point
r.f=
'
I
f(xt dx
"l
J
-l
= wrf(xr)fw2fLx2)
--+ l+ unKnown cubic
-,
po\ynomioJ,
f(xl = a"+ rx -f 4zx2+ 2y.3
I
-(
Lrro( = I
t(x)dr
J
- 1^11-l
(Xr)- Wzt (xrl = O
-l
., e I
- ,wrT(x,
-\^/2t U, _-o
dx
alx
?zxl
I
)
lf G+orx+
t
J
-l
:)
fr-* * tl:o *
Ir
?"=
'
u |
rvr
I
+"*
|
t
f /. .
. ",
_ u'.r
/ +,.'./
( x , \ _ h l 2 T t x z - ,=) 0
--l
) 2a.n tu" - w, (20-f 31x11 arxf+ ajxl)-uA-(a"+ 2sr+a,t{+a3f)
t-*r,
=2 2" (2er'-wrJ- a (w1
I ) - 2t (w,x1+wrt]) -o
r
rr rrnrzxd
+ 6'( 3 -r^r'N
V,l1it{1' Q
\ N 1 x11W 2 X z = o
r^lrxr!- wuxI=
2lt
W,121h12X31 -n
= 1-
W1=W2
.n. t :
-
-
I
: -0 5?7
\t
Xr - +!,t3
' O,Ei4
t" f(-tr)*f(+,)
#
-t -l
J3
* Tor higher accl4(acyS-poin{
.'l
I= J{Cxl{x
-l
= \^,tf[x) +NLfkL]+ N3ff,
i
'
l\
\o'
'
-\-or7
-
\
0'tt5
o.8t
I
\
\
u'??
o.r,
rt
r! i'bw,r= [ f {G,0d]1
JJ
EoM t
o r
t3
used'
r
l,
Il l Z/s
-
J \.=r
w, t(1'.
) d,1
\ ' r - 4l
r/./
|
I
I
I ui- (Fr"rif (q,
-"' .'r;l)
57
,:'r
:
\,
crj
22_
f
;l-
t
t-
(r-9
\^Jiv,{;
' ., €(r..
"-"'];)
)-r
vtl
= |',lr"f (j,,1,) + v,l'Nr.{(11,'1,)rwrvlz'|
tl-,
-.
\rt., /r/ tWzte,1z)
r ={(}r,-g;* I G,-St.-r bfi,h,)rr[},,#)
06,03.10
a ptisvnaficbar under torslon
E fiess. analysu
a r-tnitorrvrba 4 any
.hntslinS6or,rra,t u r,rsuat\ lakenas
Ittp disglaceuenlf'eu rf
Sechonunder
tJ= - BZ$
\) = 07x
u1 = 0VG,1)
r,rnlt lengtt f+e bar
+ilisl
W{
argle
0"f
{^'Filg
'\nl
'{orsion
on'.
f rrncl-t
funcliorr'o\
Y-
€xx
)rl
= -:=
-/\
"xg
AY
€us =
9Uz
-/\
-::
o\l
-.DEt
-
'Jxx
)bl
_-,
= U
-&
* S =t#-no
= s(* -9
6vz
sZZ
=v
SVt
= (J
b ( X e ="
!6N
^
-'v"'--6,
Ax
d"gl
't.
?szz
--i=-
qoLH -9
, :\J
"xv =,
usxx
-
lxz
-^
-\J
. 014 = n/)
qL_/+ 6^ )\l
tEE
irg
-6U
" z-N
fi
--E
oJ
€zz
il,
----rt\
<e(fi+x)
\'r
b6x! -[o6{z _ ,..,_ - -C')
r
5;f
6(xz
+ 'D*
+-0-6ia-o-
-Q'J
b{
't
DCxz
DdVz=
-trr 't -;J"
--
.to sdve
4over"rlng Eqr
-torsionalprobienr.
(
b(
'bx
n
/ )\u
tEb\ii
.?
-Vlf
+
tGo-a"J
, c ) -E!
-1-
ff' * *"
=t
& 1e"($j "xl|=o
(4e )Bur]= o
-------->
hpla"ee{-
Boundari qondition
':\r)
?
/
v' -V)
r-a(
-"\ bx
r
drr€eHonoo$ne.
n--
Z=gr.4x+Cg2.ns=o
**)nv =o
r\y+'40 ($
Yl1= CDS0.
ng = 6in0
u(*-r)+-ffi*l
This has to be sals$ied
of t'he x- 6ec*ton.
dx
ds
bu
A
ds.
--d-rds
_N
r.
rl
tte bounatng Sufrcce
,E=-5e
-'
DY
?fi
6;, =#
1
ur='-# I
skegs fqrr+'on
ft6 lppsienal problems'
66'J) v
$xs= Qdt#-U
>ct
6lz =eS +x)e.
=N
-
/
,,n
6!
H *&rn-= - se #r-qq+ eeffi-ae
= - 2-q6
{fi
b(r-
>v
-+ -:__t_ r_ zaa,a
\-,t v
t
-rr
-v
€overning eq! ip terrxs
Skess funch ot' d6,tt
6iz.n1 +\2'rty'g
+
dxr ='
bd '.-d6
d,l +
., -6;J'\:
i9r/ -ds
\J
Ty
+ff t+
dF
E3
oP
?x
<IX
ds -"
- consfonf along the boulndary.
f=o
tr
No|^/,
- q' ' axae
' dxdvx
vl
Lq"
J .J
=JJL-g
- u-!gl
" bsJ
-= -
"JJ
dxdg
flp,{axtt
6 (xzvz)
'2-
g=Nrdr+M/r+Nl/,
-
(s'l
l-"
N1
*l 1t"l
\Qg)
-- I-r'11
u'l
?g
d\
{r}'
oF lx .,__-Q!.::
0* 11 ry bj.
zF -fiMtx
b1
E-
bJ
-,bFT-u'E.
+
\
!r
q
)rl I osl
lbx
l5--
bF
-rT
l"
I
l
ttx
x= NrX+ Nr(2 +Ngxj
v^
TrJ 1w1
-l td n
\rx
t'tgC-t-'?
f-{l ^- t
,,-,
-"\tr
\
=lrj
I''ax
59
Lu
d€llfl
0v
5? 1
[t
J
I=
lr"
IIJ
fu
L-"tg
Ttu
ddD't"r,
( af \
\5TJ
dp
\ar
v,r-l
Lxzs !r4
t d61
Iar
@
rlcfr{,
ttz"tl
t
lrl =f*',
- [r)] ap
[
tlg
IAJ
Itr J
L-
-[
-
(dp
\
dF
xl vr + N2U2.+N3V3
N1
b6 1
q
EIJ*I
tv l\tK I
-r--l
xtc)
fFrl
. -'l lr.l
-r I
- r'ul f'
I
01
lr, l\
L4
,zr)
')-i2
xsJ
-,-.[4)
d31
-r'o
=Lo 1 -'Jlid,\
p,l
---/
)
larl
(ql
( p/'l
coo -\3 Q+zs*YIf
.T
) 0,, [ _
-]_
- dui6l
llrr\
l-xzt
xle
(
-*t)
1as
lxze
_l \ fu )
l-rs'
. :
-lfvzr +$st
defJ l ru, xra
= rst {sl
fbl .
s,rl \ql
."\\k\
I
&J
fua
L
Nsr xta
,nl
lzr J
"2 ure
ttEfhod.
€alarhn
Qowtnivrg E+
tF
--tE n,oo
-'"
Dgx
bxx
e(x,\) = Db*tg
'
bJx +288
bx2
Ief ug aonsl"derkiat
sduh'on: vlx, vJ
Qalarkinr equdtlDn
'o
fr €tx,uJvltx,9 dxdS
ll
JJ
=o
-'S, -,-'onl
dxdu
==)
J! "L*"
A
-!- f ,'nlFr
bx \-
bx/
," fls
"6
c
No\r.t,
u!?
bg!
= 't*$o^ #
=Eb to gu*)
=i;
t f\ rr lEt
?g J
\F
),,, -ix
i,''
bv
od
bU
AC
Y'J"*
rf [* t,+rt- H # +&tY,)r'e'
JJ
A
-o
drdv
.. A+ zJJvqo
d*d
*
rzJJne
#
e#).
{i!,^u,
t
fr
*
Jar
"JJl$
1jE
t
A
fk
^-\
.
--. 1
l^frr," rr,
1,r " --.r
lc
lo l-r"nefrrfegrat'
Sutr8acB
o Qreens -tt€otzn( I
L"rre s ' tr
ft'r z a@a A' blundd bv bourdrhg
r1ll0
5rtrtobl5srnoofh
{uncibns, 48'v; ad Fg, Y;
(rr
arG,$
6(x,sds.
a^ar= h rs,9d^1
t \f 591)_
I
I
J
ax
bY
'o
'
t4+egr6,
(.5urtbce
\\L
JU
;
( Lhe rht€gral)
\
-bx
r\
-,0
- -DE
>\r
-_l
ll
I
lt
vJ
|
li
r
\cJ\
I
u
I i" \g \.1dxdU=
-lZ- lrO vPl
I
I! - br \"t;e ^ t t-f -ni
'J
\" lcJj
|
A
dx
.gH_l.E
J
tr-
vI
E
Jt-
i6
nv
l
&
=Q
'
eoflditbn
boun)tdar3'
lat aclrrHo
n <hpubl saHsNg
"JJ
fau
L5r'
?F
'
tl
#.H#l,o
"
ln
Ibu -:- +
-!a
TE
0x
by
L0x
\6
\U
li--.
=,JIt*
'
_(
f'0"
aF
2l\ u Qe dxdY
dJ
dA. "JJ" ana*av
|u l{ D X
.-oqll
-l
rn trQ€ Axdy--e
nJl
bp
OJ
:' JJct{*1J'
( rsl\v
.
t)dA
'J
=z\
cn
I
J J
q e fur{v} J"JY
{v}-qt"uat v'irtual skegsfurcHon
'JI
=ncteJJt-fr-f"t
u=g'rii v]
ta:tpi'ao
tarr
lv1'
=r {nJ'
{'pittnt
,t
=
{rr
(ee
lpt
(rur.ru:
iliii}
il,)
.i^
fr NidA= +4o.r
=A
VJ
ll
+LkeJ
1e1'
U1=lvi' i a1
t
=AeIsl L0l
tK€l
,qeo, \t,\
{
-'r
rel=
{
vJ' (r*l ls t - lftij =o
----;-
<'l
l')
^'?
.
^
"
\
Elanenf sfrtfness equ atrbn'
r
1
_
=
1";rl
!cy.l
'
f rol
\il{
=rnr{r}
i;;|
ut
Iiv)
L,I
%-
probbng
" # t#).{- 2q€ ,o (rorsio-n.
#(*l
I
= skess €unchbnfin torsional analgsi:.
*(^41)*
(l^ \
0tr
*(*]rt
0X\ tf
_/
T. rewr.
+ e =o
e(&.*d!)--e
\ ?xbs'J
Eleckicat Problenn
4 - etedric Aolanfral
ivit-g of l\e
6 - Per,rr.l+f
contluctor
'
e - Volurire
ohargedensify
'F[+(])*L/lg)l=o
au \0y/J
L0K\dx./
u-
rnqnehL Potential
4 t{te nal'arial
P - perrrreaH.L,lg
rt
46
I
at2,
z- cz
ft, vd
k',D
Ql .o)
8t"D
C",-D
(o,-,
Staidard qudtatrc quadrilafaral e\evnenf
{qil6xl
tul=r':
r-xl6
2xI
I o
'l^\ Nr oNro
LONr o
rl1 . O
N,t
N9
o
q,l
't
qrl
oNloN6
,lT ' o
ika
Al4oNbo
oNi
o$l
:i
46
U:
Nr4r tNr4st
'10 -
tJr4r t r.lr-4+
lsopcarndrtc
Srmulaf(ovt
r.rlgg + Nq gr t
\4q
+,t'r'5(n t r(71rs1 Nr 4rS
'
-l
2 r.tiX,' = N1X1* Nr,{l+
L tt,Ui = Nrgr + Nz.Jl+
-
_^ \, l.^; or ,! ,b
Ngxt
+ NrUB
Nr' I = - -\'
N.^
-4L
|
-t
LrnSY)tf*$i)
(t-3:;-4tr)
Nq=-L
Nat-
-h (r-l)("1)(,,
i-0
Ns=)- \ - / (r-1') ('-1)
Nt" t
r,l''r
Ng
* tr+j) (r-1)
= t(t-3') (r+1)
" t t'-9t'-9
IN
tr Thr€e djnrensionolprob\ems
r
1
t
3
9o\E
q-
4.
(q-ndu'd; Sotrt elen€nf.
rtslrahedral
'
n11"sh5nle5
t
/r\
'I
t
//(Qo,
J
i,lasfer elerrrer'rlfiY lhear 4uetrilah2ftl elem€nt
{'}rr, = [u]r","1'4t,
ql
qL
=
[u'
lo
I
o
o
Nr o
o
Lo
Nr
Nz
o
o
r.lr
o
o
Nl
-l
o
o.ilqo'l
o
0
o'ilr
NL
oor.l3ooN!
9V
o N1 ol
I
alllr+
X=
>
Nixi
-1
Lr{iY.'
t:_ t
= N1X1
i l!-Xrt
NgYa-t n\rtx4
4
. r..l,Y,+ }.laVr+Nsq3i- r.l1St, -', 7= ? NiZi
tat
Shap {uncfiuns
ll. - ?
.J
r.t".
I
N3rV
Na --t-T.-n-!,
tl
iFCB]
+ {4')
lD5 I
<l
Et\lt
I
a; l()T I
)Y Ki \# t
"l
t"
l.
\?ct:
\q
l0x
l'l
II
lnu
}I
lax
?9
lq i
-
f--J
|
,ltl
\r;
-l
LV
2v1
av\
I
wl
I
lbv I
IDTJ
=[r] {$\
\;.^\
\?-'J
[H
.J.,Q
t $.t' l I
-1
l/t l L:J-
lTl
LJJ
xrj -- xi- x'l
\
-4\
I
7e+)
a't I
al-tI
I r,r1
lq \
=lAl )!s 7
0'^l )
irl
\"1 I
lrl
\ul
IbY
J
]="
t.l
'rl
as
lx
A.\
05
*t\
:"\
z2l' )
0v
bu. b$
5qr
br
bvl
+T
?g . ) 4
-D<
lb,^tI
|.6T,J
o
,#
*
C.
l5i I
\* t
J
rt
I
Iau I
{+}=t^r1H
lH
ll
\^
I
lDt 1
Lv
l\
tet
''l
I bu
\i"- [
IdZ
tl
t5i
I
t ;.,1 J
.
azl fut l
a9
fal
llr
il
tf,
u= Nt9r+ Nz94
tlrQ-
t
N3gB 1 N49rr
Nr4r- + Nrqi t
ilr ,ta+
N2q6
+ Nqlro
-r N3qq 1 r.t4?r-
- 1il6*,,' lol
t 'r lzxl
{ti*
= fp-l
- '
i(l , |
6xb
ext
.|il
- 'L*tc-
\ oq
'1
)
{ 'rlzx I
tDl =
OO
*
o
o
o
o
q-q
o
^E
'
q?
--
I | { .ls}r 1e1dv
)'
V
, .J,
\ { t(r't tsl lqll ( rDl{q]}v
JJJ\u-^n
=L
L
^1
'iqi
tslJq1dv.
[D]'tgl
lf
l
JJJ
T
T ,
.
z(r+lt
a
))
-t
--T
0
b
* = {(tr,}")
lr
.O
*q
o
U
r
O.
,L
*
It
o
O
+
=t^.
llr
llll ' {ql' L[;ir,t'
f '!{r}
b)L
J ) )
= t{q}'
'LL
'
Ju
(^
r
)
( ta'l'tul'IBlv.J 11 |
\
't {o\-tri {e}
ftl
1
, r(€J
vlq
r--1
: ve . Lhl
-'f
lr^-'rlf
v"=t I d€+
trl I
.--l
LDI LDl
=JJJ{ufr'a"
dV
' llf(t-'lq])'n
-jffiof
r^.,1,,
al
d\
to
= lql'\
Ur
-r,\
(f(
-/
t6llltNl ov/
\
'.rJ
=101',F
1:l
'I
ry
I
I
{ul'{r"
= I 11,]'r an
Np
an
= [f(r*r{q"f)rr.
tl\--r,,f
JJ
( ffl^riIr.d,
. \q].
\ .,
\.r,,,
=1q\'t! T.*Tl, .Tz, r*,T., .T7, Tr,!
\t
\
l-
1 =?l{'f
[r
lK t2q
It
q
r lrlt
r)
, Tz , o. o,
, "1
\Pl I
I
'(
J+
I
\tr
It
,i'
a^
q't
,{o;oO'
L1r
t)
(1,v'z';
-t 13 a+l
-t 4 Yl_1rt
- | z 9 4 +l
17 {re + n'tr
Ns116
1zz
1
u=rlrgr*Nr9rr
+
N,11,.
rNgg+t
1N5q13
1J: f{1qltNz95a N3{9+ r|\11
9rr t nlsqh + Ns 1r+ uH^ t ^ls129
,.1+9a + lr 12,t
1^1
. N11]
+ r.19161
ilgqq+ N4lrr1ilg{1y + ila9rr +
!t
/\fr
=
lul
| Nl
l:..
[nJ.
lxn\
o
Il .o'|
'r 2l{xl
Nr"'l
slr
8
x -- : Nixi
t:1
s
Y .. 'l
|tr'vi
t.t
8
.,.-.
7 -- a N'4
tir
elenPd
"hexahedtal
sotld
Shapfuvrehorts
Ni=f(rt3j,)tr+l '1i,)'t I+ v vc)
Q=
't n v"
l( -'
?(e
'' t't' - '
*\
of
tlt<- coo<dinates
eac^ vrde ln
116tur6l
Jrv
'
co-o<dinato9$shei\
r tl
trrdv =I t ld€fLrJd1d1d{
l\ul'frt tgl
tq =r"I
JJ J"
. JS'
JJJ
It
Il{a', JJJaxrsr
1..
fi4
1,
dl/ dr4' dY
l'+ t-r'l
u<,
LJJ
Download