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7/30/2019 Proble Pentru Teorie MEF
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'rir.r'. .',.'. .''-:.",:i{ri.',
130
llfl
7. T'lrc Rcnrrr Elenrcrrt
rmElementHomentDiagram (t 2 ,L2)
Bendrng Morncr{ CIogr
;-.1--- .:ilidiigr5r*
7.2 l''14I'LAIJ Irunctiotts Uscd l3l
3.
4.
5.
6.
thc
tlre
rcatt.ior)s at noy'Cs |,2, and 3.
rc4{s and rnonle nt.s) in cach elcntenl.
6b{iagrarn for each elctltent.
r)ronft\t diagrant for each elctltcttt.
t<-r*+t<t-+t
Fig. 7,15. lJeitrrt rvitlr'l\v<l []lentcnts ftlr Prol,rlern 7.1
probtenl .dO
CorrsitJcr thc bcarn shorvn in Figurc 1.16. Do not put a nocle ilt tlre location of
the conccntratcd loacl but usc the rnctlrorl ol'crluir,illcrttnoclitl forces for both thc
cli.stributccl loacl lrnd tlre conccntratcd lond. Givcn Ii _ 2IA CI)a, and I ._ 50 x
10-o nl'l, clctcrtttittc:
l. tlrc glolxrl .stil'frrc.ss tlt;ttrix Ior tlrc .stl'ttcttlt'c.
2. tlrc rotatiorts at ltotlcs 2, 3 and 4.
3. tlrc rcactiort.s ltt ttotlcs | , 2,3, ltrrd 4.
l. tlrc lirrcc.s (slrc;rrs lncl nl()rllcrrt.s) in citch clctttcttt.
5. tltc sltc'itt' lttrcc tliltgrnltt lilr crrclt clctttcttt.
6. tlrc bcnding nlotttcttt ditgrirlrr for clclt clctltcttt.
lOk N
23l<- ^---+i i..-
Jnt Jlll lltl llll
Itig. 7.16. [Jcitrtt rvith Distribtrtcd l.tlld frrr I'rolrlcrlr 7.2
Problent k6:@
( lorr.sitir.r'llrt: llr:rrrrr rvitlr lr sllr irr11 t'le:nlcnt_;ts sllott'rr ilr I;igttt'c:
t ,10>: l0 (inl'l ,ltl(t /,: , 50()()kNlrtt,tlelt't'tttittt':
I, lhc globrrl slil'lhcss nrlrtri.r [or tltc.stt'ttcttttc.
the forcc.s
thc shcar fi
| 1.5 2
13. Ilcnding Mclnrcrrt Diirgranr l, lcrrrcrrt 2
Bt,, r,lleeentHomentDiagrarn (f 3, L3
0ending Monrorl
Fig'
l0k N/nt
7/30/2019 Proble Pentru Teorie MEF
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132 7 . Thc fJcant Elerttcttt
4. t5c forcc.s (sllcars ancl nlonrcnts) in cach bcatn clclttcttl.
5. thc forcc itt thc spring clcntcttt.
6. tlrc shear force clingrartt fot' cach l:eattt eletttcttt.
7. the bcnclirrg nlonlcnt diirgrnrrt lor caclt bcattt clcntcttt.
E,I
nr
\*t'-o I-'Fr. "**t"rcA1
4'A
rlphtr\rl.*
ol
-!a.=...*fi
4
ffi#{,Itig. 7.17. Bcattr rvitlr n Spring for I''t'oblc'rtr 7.3
[El
4,f'
\
f:
I\_-a-v€
.-L, -l*i| L''^ I 'l-\ _V _'x--r(-_\-___--^\
c^-J
I 0kN
n"t\a*L
*rX
B The Plane Frame Elefll'"
cntcttt rvirh both local antl
trlrrs ol'clirsticiLy D, nlotncn[ '
.:rtr.i.x incl rrrli ng uxial clc[ortttirtiott
8.1 \Eiasic Equatiqns
'l-ltc1ll^nc ft'atnc .,N
global coortlittlttes. 'fhe)
oI irtcrtia /, cross-sccticltt
noclcs and is i nc I i nccl rv it
glotral X nxis ns .sltorvtt i
clctt'tcttt. sti l'lllcss tttlttrix i
(se,: rcflcrcrtccs IlJlrntl I
rt is e trvo-
{anc ft'rtln
r:ilritrcit .ll,
tl, ,\ra,rglc
in [:ig\1rc E
is gi'','.tt\Y
l til) \
clintcnsirt'
e cletttctr
anrl lcrtgti:
0 tttc:,r:,!r- .
. l. I-t:t Q =
tlrrr lilll,r'.'. '
E1..=. T
X
(" _1)",., )
, #r'
(^ -;i)"|,,rr'* 11"')\v2)
til----c"
L
/ t*r \l,r---l(:s
\ tr),ts? + Yg'
y2
6t--s
LAr;')
(^-(i'
-- ---5'II'
( ,, r'i,
- l rtc'1----\l-2
| 'r,\laI r,zl\/$I
- --5I.
r2l ,.f-
--.5
*
y2
,,, \_ lcs,t)
6T*CL
2T
6I--sL
0t- -c
4Il- *c
t.
(8' l)"'..a,
ir tlctltt-rus of'l't'e:ctlottt - tltt'cc lti
ir,' sillrr ('()rrt,'itli,tlt rtst'tl is llt:li
rrrtl t ot:ttiolt\ lll c [)()sitil'e il tlrc't'
r,,'i{ll ?t riotlt's. lltc ilotrltl stil'l'ncs"
tirrcc clcgl'ccs tl[' l'rc'ctltlttt ilt e ilcli
i . .. ! ','ll" 1,. rl'" lt't'\'ll AI:
,,/tt rllltrtc lrittttc eletltent lras two
/ntcrclockrvise l'rom the positivc
,0 itncl $ * sirr 0. In this case the
7/30/2019 Proble Pentru Teorie MEF
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168 8' The Planc l"rittttc lllctnctt' C^SU€4A
3v\:1P roblem 8.1 :
Consicier[ltcplane[ranresltorvnirrIrigtrre8.2|,GivcnDl0-2 m2, and [ -= 4 x 10-6 tn4, dtttcrtltitte:
,oI. tlrc glolxtl stil'l'lrcss f ltitlt'ix lirl'tlrc.stt'ttcttlr'(:,
7. thc displncctllctlts ltrttl t'tltittitltt ltt ltotlc 2,
3. the horizontal clisplirccnlcttI atttl rotlttiott itl. tloclc 3,
4. thc reaction.s at ttotlcs I and 3,
5. the axial force, shear forcc, ancl Llcnrling nlortlct'tt in caclt clctttcttt,
6- the axial ftlrce diltgritrtr for cltclt elcrtlcllt,
7 . the shear lorce cliagranr for eaclt eletnent,
8. the bending motl)etrt cliagrarn {'or eaq}t elcttrcnt.
l5kN.tn 3 20kN
.l x
r\ l:<
Problems:
.{*-a
ffiItig. 8.2L Plane Frutltc with J\vo Illertlcnts for ['roblcrlr 8' l
Problem 8.2:
Corrsir.lcr thc planc franrc shorvn irr l;igurc 8.22. Givcrr ri:: 210 cl'it,
10-2 fil2o and I - I x 10-5 nt4, ctctertttilte:
the globnl stiffrtcss ttt:ttrix [or tlrc stt'ttctttrc,
the displncctttcltts lttttl rotlttiolis at tloclcs 2 nntl 3, '
tltt: r'cltclitltts ltt tttttlr-s I lrrltl 'l .
tlrc lrxilrl lg;ce. sl)('i1'lir;cc, iltrrlllclrtlirrg lll()lttcttt itt cltclr clcl'llcllt'
tltc ltXilrl l'tttet:,liirlil illll lirr cltt'lt t'lcttlr-'ttl,
thc shcltt' furcc tlilrgrlttt ['t"rt' cltr-lt clctllcttt,
I lr,. lr,,r.', li,rn rr\nrllr. nl (li :t rt rit rtt t'tlf CitClt ulct t tCttt.
20kN
8.2 NIA I'Ln B [:rrttctiutts Llscd t69
5 kN/rn
K^;i-*n---,aii,=-'=t?ffl
'
l,'i1i. lf .22. l,t:rrrc li'i'rrc rvirlr I)istt'ilrtrtctl l.':ttl lirl' l't'ttlrlt'trr ti'2
Problent 8.3:
Ct,nsiclcr tlc structure corrrposetl ol'a bcanr ancl a spril]g a) sltorvrt irl Figurc 8 '23"
Givc n I! :70 Gpa, A -:- i X l0-2 nrz, / : 1 x l0-r', ol4' nncl A: : 5000 kN/m'
clctct'ttt iltc:
l. tlre global stilfncss rtuttrix lclr tlrc strttctttt'e,
2. the displacenrcnts ancJ rotation at nodc l,
3. tltc rcitt:tiotts at ttodcs 2 artt! 3,
4. thc axinl l'orce, shc{rr lrrrcc, nntl bcnding ttrotrtcttt i0 thc bcltltt' .
5. tle forcc ip tlr'. spriltg,
6. tltc axi;rl forcc clitgrapl for tlrc [cntt't,
1. tlrc slrear fcll'cc cliagrrttlt fol tlte beltttt,
8. thc lrerttlirlg lltolllcnt tli:rgrattr [ot' tltc Llclttlt"
(l Iirrt: Usc a plattc frallle clctttcttt for the belttlt so aS
cf'l.ccts. Also irsc il plunc. tr.Lrss oldnlcnt lirr'tlrc spring
inclirrrtion - in this cirsc clctcrurirrc valucs l'or -L'atttl -'l
pl' /; giycp upcl t[c lcngtI of' tlrc spri']8).
l',
xial deforntation
tuclc thc anglc of
g using the rlalue
^Vtf\/
to i rtc ltrde it
so ls to incl
lor thc sprirt
I.,,i
II
t
Lllr
I
r 0kN
1<- 4'--*;lI;ig. ti.2l. I'lltttc Ft :rttlc \vitlr it '\Prirrg
Uirr l'roltlcrrr li.3
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3. Thc Lincar Bar Elcnrcnt
0;0;0;01
-004 *
ttSIT
tB02
't77 4i c3g1,168
':it isc
iltlT n17 rn
'roth!.tfJ sl
i;6(P
Froble;,t 3.7 :
CoitsirjI ilre .stt'ucttll'c collllttlsctl ol'tlrrcrr f irrcrrr lxu's ils slrorvrr irr ltigsr.c 3.5. Ci'crrE - 7{j ' i[:'e, A = 0.00s rn2, ,I)l = lt] kN, i.ur tI lrz: t5 kN, dctcrnrine:
l. thc i: l'.:tlrrl stillne.ss rrutri.x for tlrc .stnrctur.c,2, tlrc '1is placetncnls nt noclc.s 2, 3, ilncl rl,
3. the ir';tctiort at lt,ttc I,4. lhc ..r ; csS in clch blrr"
cn<l is -0.4517x 10-a nr
tccl to tlre Icft). No callstlcc.r or I.itteorllnrlile-cqrrirccJ in this cxanrple.
.s cx:lnr;llc.
]cc
ect
rt [:t
trtlr is
rce
re(
nl,
ot
tlr
rc [r'
dir
lll c,t
Incfor
O
t
rI tlr
it is
'lile
s ilr(
red
Iet
ilrc
df
nl.
it,E
.si
nlcnI t'
that i
n rllarr-cssc.s
:r'f irrrrr
cc
tc.s
ne
lst
:.s
e(
str
)c
r 1il,
cat
Lit
\bc
rl)
ic
\lal
t.\
rrl
\.s
)1.
N cli
i,X
;T;
tgn
c(i
Ot'r
'ill
lc
Ib
ril
rir
;ig
rn(
I(
rvi
ar tlrnt the requ
r (the nrinus s
MA'I'LAB lrl:c thc clenrent
l-processing) r
;ler
nm
he
i nc,
ost
P roblc ^t.2:
, Solvc l ., rrillc 3.2 nglrirr usirrg tcrr
,* dililtlttUr ', i\f ttl lltr- I'r*rr rrrrrl rrl' llrrr
ilrstc;rtl ol' livc. I)etrr.rrrirrt: llrt.lll\lllt\fll'rr \!rltlF rrtrr'rrr.rrr rr,lrl. rl.
r)
3.2 NlAl'l"All ljurrctiorr.q Lj.sctl 4l
lrr-'l
| - lr r .- -- .--.- o --- .---. o') z I 4
1,. lln--t- 2nt.-,--. I lll--,1-" /?=---t-_*.7l
Irig.3.5.-l'lrree-Bar
Stnrclurc for Problcrrr 3. I
Problent 3.3:
Ctlttsidcr tltc .stntcture conlpo.scd of lr sprips ltrrtl 1 lirrcnr bnr as .shorvrr ip Itrigur.c 3.6.Civcn It - 200 GPit, /l :0.01 tn2, /,: : 10()0 kN/rrr, lrnd P == 2i kN, rlcrcrnrine:
l. thc glol.rirl stiffrrcss rrrrrtrix fr.'r thc strtrcturc,2, thc di.splaccnlcnt at ncld c 2,
3. thc rclrction.s at nodc.s I ancl 3,
4. thc strcss in tlrc bar,
5. the force in the spring.
v >?-h
Irig. 3.6. I-inc:tr Ilirr rvitlr ;r Sgrring ftrr. I)rrblcrrr j.l
") k*(l
v
=I
'?*+u"- 4
Ftti\ E\-->'t/
4Vi (rl