proble pentru teorie mef

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



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



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



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

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