Download - Ibrahim Ekiz Yapi Statigi I
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2.4. paragrafinda da belirtildiqi gibi denge denklemleri
ve mafsal qartlari ile ~BzUmlenebilen sistemlere izostatik sis-
temler denir.lzostatik sistemler qubuklarina tesir eden iq kuvvet-
lere gBre dolu g6vdeli sistemler ve kafes sistemler olarak anilir.
Kesitlerinde kesme kuvveti.norma1 kuwet ve egilme momenti
etkiyen d4zlem qubuklardan olusan sistemlere dolu gEivdsli sistem-
ler denir.Kesit1erinde sadece eksenel kuvvet etkiyen qubuklardan
olu$an sistemlere kafes sistemler denir.Bunlarda mesnetlenme durum-
larina qare sinlflandirilir.Seki1 4.l.'de temel izostatik sistemler-
den bazilari gbsterilmigtir.
4.1- iaostad;
4.1.1- D o h Gd
Kiriqler iizerine
momenti ve kesme kuvveti
re gare boyutlandirilan
larl ve aqiklik sayilari
Bir ~Icu sabit diq
ve iizerinde gubuk ekseni
vetlerin tesiri altinda
pubuk ekseni do?rrultusun~ Y
Du durumda basit kiri~in
A ucunda normal kuvvet vc
duqu B ucunda sadece kesn
egilme momenti slflrdir.1
rumunda kirigin hiq bir 1
kig yfinii qenel olarak mor
zorlayacak biyimde segil:
senine paralel ve dik ve
olmak Uzere ilq mesnet tel
RBy9dir.Genel olarak qubr
qubuk ekseni Uzerinde ya:
netteki rubuk eksenine d.
momentlerin toplaminin s.
qubuk eksenine dik tepki
sifir olmasi gartlndan bl
Sekil 4.1. Bazi Temel tzostatik Sistemler
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Cubuk ekseni iizerindeki kuvvetlerin izdUglimlerinin toplaminin si-
fir olmasi gartida kontrol i ~ i n kullanilir.$ekil 4.2.Ide garuldugu
gibi pubuk eksenini x ekseni ve qubuk eksenine dik ekseni y ekseni
olarak sepil'irse,
zMB = 0 gartindan
ve EMA = 0 gartindan
Nx = - RAX + Pix - P. Vx = RAY + ply - P =
Mx = R ~ ~ . X - Mi - p 11 Mx = - b ( p -x)-ZMi
iy veya xB parFasi Uzerindc
bulunur.Burada px1in soldan sapa, P yukaridan agagiya ve Mi momen- Y
ti saat akrebi tersi y6nUnde pozitif alinmig yayili yUkler iqin bi-
legkeleri goz bnUne alinmigtir.Kesit tesirleride benzer gekilde tep-
kilerde bir dig yiik olarak dUgUniiliir.Norma1 kuwet kesim yapilan ke-
sit ile sagda'veya solda kalan parqanin mesnedi arasindaki kuvvetle-
rin yatay bilegenlerinin cebirsel toplamina,kesme kuweti ayni kesim-
deki kuvvetlerin pubuk eksenine dik bilegenlerinin cebirsel toplamina
ve egilme momentleride aynl kesimdeki kuvvetlerin kesim yapilan nokta-
ya gbre statik momentlerinin cebirsel toplamina egittir .
1 yazilabilir.
Daha ancede apikl, kuvvetlerin egilme momen.
dik doGrultudaki kuwetlc
dan sisteme etkiyen eeik
docrultuda bileqenlere a!
dogrultusunda etkiyen kuv
tide sadece pubuk eksenir
gdre hesaplanir.(Sekil 4. Kesit tesirleri yukarida
3.4.2. paragrafinda anlat
rida anlatilan ytintemde i
yar ipin program yapilabi
Bilgisayar ipin program v
41na veya soluna gdre den
cJidiiir.
Agagida 3.4.2. par
g6re deqigik drnekler ve
durumlarl ipin mesnet tep
verilmiatir.
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$ekil 4.4.(a)*da gbrUlen sistem,$ekil 4.4.(b) ve 4.4.(c) 'de gbriil-
dUgU gibi iki sisteme ayrllarak qaziilebilir.$ekil 4.4.(bI1deki sis-
temde sadece dU$ey mesnet tepkileri vardir ve degerleri,
py'b EMB = o dan R -
v -7 P - a
EMA = 0 dan R = BY L
olarak bulunur.AC ve CB ~ubuklarinda yilk bulunmadi~indan bu qubuklar
Uzerinde kesme kuwetinin detjeri sabittir.
dir.A ve B mesnetlerinde etJilme momenti slfir olup C kesitindeki
edilme momenti, I
P MCB. R -b = &
BY
g6rilldUtjU gibi
MCA = MCB = MC pa
dir.Diyagramlar1 $eki
$ekil 4.4. ( c ) 'I normal kuwet meydana
xpx P 0 dan RAx =
bulunur .AC ve CB ~ubul
r.inde normal kuwetin
NAC = NCA = - RAX I
NCB = NBC = 0
bulunur.Bunun diyagra
Urnek : 4.2
$ekil 4.5. 'de 1
kuvvet etkimediginden
R ~ y R ~ x r P/2 v
bulunur.Sisteme qubuk
1 kuwet yoktur.
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Urnek : 4.3.
$ekil 4.6. (a) 'da yUklemc
mesnet tepkileri,
zP, = 0 dan r RAx = (
EMB t 0 dan ;
R a8 + 50 - 70.6,00. AY ' R t (70.6,00 + 100 AY zMA = 0 dan RBy-8 - 50 - 70*2,00 R - (70.2,00 + 100 BY -
bulunur ve
ZP = - 100 + 70 + 1 Y
partindan tepkilerin do
Kesit Tesirleri :
A'nin sagindan kesilip
1 zilarak, 1
VAC 1 RAY = 100 kN I MAC = 50 kN.m.
1 AC qubuQunda kuwet o h
VCA r VAC 1 100 kN
C noktasinin solundan k
garti yazilarak,
MCA = R -2,OO + 50 AY
C dUgUm noktasinin den<
MCD = MCA = 250 kN
VCD n VCA - 70 = 10(
CD ~ubugu Uzerinde kuw
VDC = VCD = 30 kN .
D kesitinin sagindan kc
parti yazilarak,
'DB = - R P -70 kl BY
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M - - R . 3 , 0 0 + 130 = 7 0 - 3 , 0 0 t 130 r 340 kN DU - By
1) nol.ta:;lnda t c k i l illomcnt e t k i m c c l ~ ~ i l ~ d c n ,
M~)(, = = 3 4 0 k N . ~ n .
vcl I J k c $ ; i t r i n i n solunclan k u s i l i p s'14da kr1lr1n kis1111d.l L ~ L ~ I I ~ J C $ a r t 1
y,lz I Lbrrlrk,
V l j 0 = - 70 kN
M - 1.10 kN.tii. 111, -
\ 1 ~ 1 : . . i l b \ . I I I I r .
l ; ~ ~ * ; l l ~ t * : ; i i 1t:i.i ~ l i ~ ~ , t ~ ~ i ~ . r 1 1 1 1 . ~ 1 - ~ ! j~*ki 1 +I - 6 . ' ~ 1 . i {lii:iLt-~.i I i~i i$l 1 1 . .
S e k i l 4.7. ( a ) ' d a v e r i l e r d e k i yUkler in b i l e ~ k e l e ~
AC ~ u b u g u Uze r indek i yUk
- ( 1 /2 ) -40 .3 ,6 = 'AB - ve b i l e g k e n i n A n o k t a s l n
x - 3,6 . (2 /3) r 2,4 AB '
6,60 m CB gubueu Uze r indek i yUk
v e b i l e g k e n i n A nok t a s ln ,
'BC 386 + 5,40/3 :
3 , 6 m
bu lunu r . Mesnet T e p k i l e r i :
EMB = 0 dan
R AY = (50+72- 6,60+108.
ZMA = 0 dan
Kon t ro l :
z P ~ m -86+72+108-94 =
YUkUn degigim fonks iyonu ,
0 ~ ~ ~ 3 ~ 6 m i ~ i n [AC a r a
X q ( x ) = 3,6 • 40 r 1 0 - x
3 ,6<x<q m i s i n
bu lunu r . $imdi x deg igken ine bag11
0 ~ ~ ~ 3 ~ 6 m i ~ i n AC a r a s :
l a n p a r s a Uzerinde ( S e k i l
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x r 3,6 m ipin 2
VcA = 86- (5/0,9) -3,6' = 14 kN.
-
12 8
McM4(Kjm a ) ( 0 12.7 ) ( 9-X)
Mx (" l ~ l l b O M ------ Mx --4--
M \ x 4 " 1 9-X # U
(a) 03)
Sekil 4.0'.
Vx = 86-(1/2) ( 1 0 / 0 , 9 ) x - ~ Vx r 86- (5/0,9) x2 (1 )
Mx = 86-x-(1/2) ( 1 0 / 0 , 9 ) x - x . ~ -50
Mx = 86~-(5/2,7) ex3-50 (2)
ve 3,6<x<9,0 m ipin CB araslnda bir yerde kesim yaplllr ve sagda
kalan parpa (Sekil 4.8. b. ) Uzerinde denge $art1 yazllarak ,
Vx = -94+(1/2) (20/2,7) (9 ,0-X) (9 ,0-x)
Vx = -94+(10/2,7) (9 ,O-x) (3)
Mx = 94 (9,O-X) -(1/2) (20/2,7) (9,O-X) (9,O-x) (9,0-~)/3-140
Mx = 94(9,O-x)-(10/8,1) (9,0-xI3- 140 (4)
elde edilir.$imdi (l).ve (2) denklemlerinden x = 0 ipin,
'AC t 86-(5/0,9) - 0 = 86 kN.
bulunur .
M~~ = +94. (9,O-3,6)-(10/8, bulunur ve baylece denklemlex
(3 ) ve (4) denklemlerinde
x r 9-00 m konulursa,
VBC " -94 kN.
MBC = -140 kN.m. bulunur.CB qubugunda kesme ku
qubuk Uzerinde bir yerde kesm
momenti maksimum olacaktir.
Vx = -94+(10/2,7) ( 9 - ~ ) ~ = 0
(9-x)' = 94- (2 ,7/10)
(9-x)= J-25,38
9-x = 9 5,038
xo = 9-5,038 r 3,962 m
Mmax = 94(910-31962)-(10/8, bulunur.Bulunan degerlere gbre
4.7. (b) ve (c) 'de qizilmi~tir.
Urnek : 4.5
Sekil 4.9. (a) 'da gbrtilen basit
daki yUklerin bilegkesi b u l w
AC ve DB qubugundaki U ~ g e n yuk
PAC = PDB = (1/2) -30.3 - 4 5
ve bilegkenin yerinin A mesned
XAC - 2x11 xDB = (7+1) = 8m
ve B mesnedine uzakliklarl,
x1 AC = 8m , xeDB - 2m CD yubugundaki yayili yUkUn bi:
- 30.4 r 120 kN 'CD - xCD E xCD P 3+2 1 5m dir.
(3. ) ve (4.) denklemlerinde I
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Mesnet T e p k i l e r i :
R - ( 4 5 - 8 + 1 2 0 . 5 + 4 5 * 2 - 1 2 0 ) / 1 0 r 93 kN. AY -
R - (45.2+120.5+45.8+120)/10 = 117 kN. BY -
K o n t r o l :
i : p = -93+15+120+45-117 = 3 Y
d i r .
K e s i t T e s i r l e r i :
A k e s i t i n i n saGindan kcs im y a p r l r p solda k a l a n
qe ? a r t 1 y a z i l a r a k ,
> 1' = 0 dan VAC = 93 kN. Y
) M A = 0 d a n MAC I 0
B k e s i t i n i n s a o i n d a n ke s im y a p i l i p solda k a l a n
qe g a r t i y a z i l a r a k ,
EPy = 0 dan Va = 93s
rUc = 0 dan Ma = 93-
C dUgllmUnde d l $ t e k i l ma
VCD = VCA = 4 8 kN.
I McD = McA a 234 kN.
I D k e s i t i n i n solundan k e s
g e p a r t 1 y a z i l a r a k ,
VDC = -117+45 n -72 k
MDC = 117.3,O-45.1,O-
D dUghUnde d l $ t e k i l kuq
VDB = VDC - -72 kN.
= MDC = 186 kN.
B k e s i t i n i n solundan kes;
y a z l l a r a k ,
VBD = -117 kN.
MBD = -120 kN.m.
bulunur . CD ~ubugunun C ucu i l e sc
oldugundan bu ~ u b u k Uzerl
gi lme momenti maksimumduz
yiik oldugundan kesme k u w
z a k l i g l ,
xoC - VcD/30 = 48/30 = ve maksimum moment,
= MCD V ~ < / ( ~ . Z
bu lunur .Kes i t tesirleri i
m i g t i r .
Agagrda p r a t i k t e I
k u w e t i v e eg i lme mornent: tesirleri he sap l an rp d iy ,
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brnek 4 . 6 . S i m e t z i k l k i T e k i l Kuwet :
S e k i l 4 . 1 0 .
Brnek : 4 . 7 3 T e k i l Yiik
brnek : . 4 . 8
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Ornek : 4.11 -U4-.
Ornek : 4.13
brnek : 4 .15
Ornek : 4 .16
il
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S e k i l 4 . 2 1 .
S e k i l 4 . 2 2 .
brnek : 4 . 1 9 S e k i l 4 . 2 5 .
Urnek : 4 . 2 2
S e k i l 4 . 2 6 .
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A q a B
I a d l 0 R~ .I -RB = 2M.
T T V., r V-, I V
nCD = ZM.~/L-M - - M - a / ~ M
MDB = -2M.a/L nDC = - ~ M * ~ / L - M = M.a/L
Sekil 4.27.
llrnek : 4.24
llrnek : 4.25
Sekil 4.29..
llrnek : 4.26.
Sekil 4.30.
Bir ucu ankastre me
'konsol kiriqm denir.Bu dr
ankastre meenedinin bulund
ve eQilme momenti meydana
normal kuvvet,eqilme mom
cun bulundugu yerde Uq n e s
lel olan mesnet tepkisi dl
lerinin qubuk eksenine dik
ilzerindeki izdUgUmlerinin
si de dig kuvvetlerin mean
toplamina eqittir.Seki1 4.
R ~ x - 'ix - + 'nr
MA = -P ax 'PiY Xi - Pr iy 1
dir.Ancak konsol kiriglerc
tepkilerinin bulunmaslna !
Kesit tesirleri sic
ile serbest uc arasindaki
nabilir . V x = P + P + B r
1Y iy nY .
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geklinde bu1unur.B~ kiriglerde de egik yUk etkimesi durumunda bu . yUkiin qubuk.eksenine paralel bilegeni sisteme ayri etkittirilerek
buradan normal kuwetler ve qubuk eksenine dik bilegeni ve momentler
ayri etkittirilerek buradan kesme kuwetleri ve egilme momentleri he-
saplanir.
$ekil 4.31.
Sekil 4 -31. (a) Ida gdrUlen sistem $ekil 4.31. (b) ve 4.31. (c) '-
de g6riildiiqU gibi iki sisteme ayrilarak qdzUlebilir.$ekil 4.31.(b)'-
deki sistemde sadece qubuk eksenine dik mesnet tepkisi,moment mesnet
tepkisi,kesme kuweti ve eqilme momenti meydana gelir.Cubuk eksenine
paralel mesnet tepkisi ve normal kuwet meydana ge1mez.B~ sistemdeki
denge gartlarindan,
zP = O d a n R m . P Y AY Y
EMA - 0 dan MA = -P -a Y
bulunur.AC qubutju Uzerinde yUk bulunmadigindan,A'nin sagindan kesi-
lip,sag parqa Uzerinde denge garti yazilarak,
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EB pubugu Uze r inde k u w e t olmadit j indan,
VEB = VBE = 0 , MEB = MBE = 0
A k e s i t i n i n s aq indan kesim y a p i l y , s o l d a k a l a n pa rpa i l ze r inde denge
$ a r t 1 y a z i l i r s a ,
'AC ' M~~ A
C k e s i t i n i n saEiindan k e s i l i r v e s agda k a l a n p a r p a Uze r inde denge
$ a r t 1 y a z i l i r s a ,
McA = MCD - -P2 (a2-all -P3 (a3-all
VCD = VDC = P1+P2
D k e s i t i n i n sat j indan k e a i l i r , s a ? d a k a l a n pa rpa Uzerind'e denge g a r t i
y a z i l i r s a ,
MDE = MDC = -P3(a3-a21
VDE I VDC = Pg
v e E k e s i t i n i n sat j indan k e s i l i r , s a g d a k a l a n pa rpa Uzer inde denge
$ a r t i s y a z i l i r s a ,
M~~ = M~~ = 0 , vED o vEB = 0 bu lunur .
Ornek : 4.29
P
A T e p k i l e r : . B S i s t e m Uzer inde
LPx = 0 dan , RBx = 0
t py r o dan , RBy = p-L
EMB P 0 dan
MB I - p ~ ~ / 2
MCB( 1 ) p12'2 bu lunur . k&J
K e s i t ~ e s i r l e r i :
Sis temde A kesl x u z a k l i g i n d a k i b i r C
s i l i p s o l p a r p a U z e r i ~
t P m 0 dan Y
vx = - p.x
EMC = 0 dan
Mx r - p.x2/2
b u 1 u n u r . B ~ denklemlerc
x I 0 i p i n VAB = x = L i p i n
VBA = e l d e e d i l i r . (1) v e ( 2
b i t oldugundan kesme k
s i f i r v e B d e e n buyilk
t e p e n o k t a s l A da t e q e
u l agan 2. d e r e c e parabc
b r n e k : 4 . 3 0 .
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$ekil 4.34.lde gbrtilen tipgen ytik altlndaki konsol kirigte
' I Brnek : 4.31 mesnet tepkileri ,
EP =, 0 dan , P (1/2) .p-L = pL/2 I Y P~ zMA I 0 dan , MA = -(1/2)p*L.(L/3) = - pL2/6
geklinde bulunur.
A mesnedinden x uzakllglndaki C kesitindeki kesme kuweti ve et'jilme momenti sistemin bu kesitinden kesilip,sagda kalan par-
pas1 Uzerinde denge $artlarl yazllarak ,
veya L-x = x' ile gbsterilerek,
bulunur . x = 0 ipin ,
- - pL/2 'AB - MAB = - p ~ 2 / 6
VBA = 0
MBA = 0
bulunur .
C kesitinden kes
$art1 yazlllrsa (bu dtigi
kuwette olmadlgl gbz 61
ve B serbest uq oldutjunc
bulunur . Kesme kuvveti 20
A ve B u~larlnda kesme b
kuweti diyagramlarlnln
cunda kesme kuweti slfl
ml ~ubuk. eksenine teZfet
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brnek : 4.32
Sekil 4.36. YUkleme durumu verilen
Uzerindeki yUkUn bilegkesi L
ve bilegkenin yeri (A dan uzakligi
L L
Sekil 4.36.
Bilegkenin B mesnedine uzakllgl,
x i = L-(3/4)L = L/4
bulunur .
Tepkiler :
EP = 0 dan %y = Q = pL/3 Y
EMA = 0 dan MB = -Q.xi = -(pL/3).(L/4) - pL2/12 bulunur . Kesit Tesirleri :
Diferansiyel denge denklemlerinden,
elde edilir.Burada C,
Uzere A serbest ucunc
de x = 0 igin Vx = 0
lar ' yerine konularak,
I elde edilir. x = L kc
t
VBA = - pL/3
MBA = - pL2/12 bulunur .
t I brnek : 4.33
I I Sekil 4.37.
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A nin sagindan kesilir,sol parpa Uzerinde denge Barti yazilirsa,
MAB'= MA F -30 kN.m.
ve B nin solundan kesilir,satj parpa Uzerinde denge sarti yazilirsa,
MBA = -30 kN.m.
bulunur.Diyagram1ari 4.37.'de gdsterilmigtir.Seki1 4.37.Iden de gt.5-
rulebilecetji gibi,konsol kirigte sadece moment etkimesi durumunda
normal kuvvet ve kesme kuweti meydana gelmemekte,sadece etjilme mo- . . menti meydana gelmektedir.
4.1.1.1.3- Qkmalr (Konsollu) Kiriqler
Biri sabit,biri hareketli iki mesnet Uzerine oturan,bir veya
iki tarafinda qikmasi bulunan ve Uzerine pubuk ekseni duzleminde de-
gisik ybnlerde kuwetlerin tesiri altindaki sistemlere ~ikmali kirig-
ler denir.Cikmali kiriglerin egilme momentlerinin hesabinda,basit ve
konsol kirislerde oldugu gibi alt yUzUnde pekme meydana getiren mo-
mentler pozitif olacak gekilde bakig ybnU belirlenir.
$ekil 4.38. Cikmali Kirigler
Bu sistemlerde sabit mesnette 2 ve hareketli mesnette 1 olmak Uze-
re Uq mesnet tepkisi vardir ve bunlarda Uq denge denklemi yardimi
ile bu1unur.B~ kirislerin mesnetleri arasinda yUklerden qikmalarda
herhangi bir kesit tesiri meydana gelmez.Konso1 kirislerden de bi- f
lindigi Uzere qikmalarin uqlari serbest oldugundan bu uqlarda tekil
dig kuwetlerin bulunmaditji durumda kesit tesirleri sifirdir.
Yatay qikmali .kSr:
durumunda qikmalarda meyc
Cikmali kirigleri
1- TUm sistem Uzerindt
net tepkileri ve kesit tt
2- Sistem konsol ve b~
ca qbzUmlenir ve bunlarir
rak etkittirilerek basit
sol kirig mesnet tepkilez
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brnek : 4.34
S e k i l 4 . 4 0 . ( a ) Ida b o y u t l a r i v e yiikleme durumu v e r i l e n sis-
t e m S e k i l 4.40. ( b ) v e (c) ' de g6rUldUgii g i b i i k i konso l k i r i g v e ( d ) - \ de gariildU?jii g i b i b i r b a s i t k i r i g ~ e k l i n d e dUgiinU1iir.Burada tjnce kon-
s o l k i r i g l e r ~ 8 z U l U r S e k i l 4.40. ( b ) ' de
v Y = O d a n k - 5 0 k N . - AY
ZF n 0 dan RAx = 0 X
ZMA = 0 dan kA t -50.2,O = -100 kN.m.
VAC = VCA = 50 kN. -
MCA = 0 , MAC = - MA :
$ e k i l 4.40. (c) 'de
Ipy i 0 dan , %y = 21
E px = 0 dan , RBx E 0 - ZMB = 0 dan , MB = -: MDB = 0 , VDB = 0
VBD P 40 kN. , MBD = - Sulunur . Burada bulunan mt
~ l a r a k y l lk len i r ve denqe
mB = 0 dan , RAY = ( I
EMA = 0 dan , RBy = ( -
VAB = R - 100 kN. , AY -
bulunur.AB ~ubutjunda A uc
k u w e t i f a r k l i i g a r e t l i c s i f i r v e eqilme momenti 6
xiform y a y i l i yak oldu-r
~ e s n e d i n e uzak l i t j i ,
x o = VAB/p = 100/30 r
pe maksimum momentin deec
-2 Mmax = VAB /2p + MAB :
Sulunur.Bulunan bu degerl
d iyagramlar i g e k i l 4.40.1
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SDCN xPx I 0 dan , RBx = 0 EMB = 0 dan , ' d i ' R -(60-5.0-30.9-4.5-20-3,0)/9
Q ~ x AY - 4 4m , 5m + 3"' , R -161,67kN. I AY -
I ~ ~ 5 1 , 6 7 ' R ~ = ~ w . # xMA = 0 dan
ornek : 4.35
Mesnet Tepkileri : Burada AC ve BC qubukla
Sekil 4.41.Ide qbralen sistemde i$aret degigtirmedigind
tinnasi yapmaya gerek y
(b) ve (c) 'de pizilmigt
I ,P:3Offlhn
i'vI 1 1 1 1 I I Kontrol : I
Kesit Tesirleri :
Sistem A kesitinin saginflan kesilir I 8 ve sol parpa Uzerinde.denge garti ya-
(MI zilarak ,
C dtigiimtintin dengesinden, 1 VcB = VCA-60 =41,67-.60 =-18,33 kN. L Sekil 4.42.
Sekil 4.41. M~~ = MCA = 406,68 kN.m. Sistem D kesitinden kesl
Sistem B nin solundan kesilir,saQ parqa tizerinde denge garti yazila-
rak, I - -188,33+20 r -168,33 kN.
'BC - M~~ = - 2 0 ~ 3 ~ 0 = -60 kN.m.
B diiBUm noktaslnin dengesinden
MBD = MBC = -60 kN.m. VBD = VBC = -168,33+188,33 r 20 kN.
ve D ucunda,
V,, = 40 kN. , M,, = 0 I ""
bulunur .
AB araslnda kesit tesirl
0<x<6 m.
-. x = 0 ipin VAB = 66,t
x = L ipin VBA = 66,t
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x u 0 i g i n MAB = 0 2
x + L i q i n MBA = (66,667-15.6,0+0,5*6,b 1-6,O r -32 kN.
bulunur . AB qubugunun A ucu i l e B ucundaki k e s i t tesirleri f a r k l i 01-
dugundan kesme k u w e t i n i n s i f i r oldugu ye r in A k e s i t i n e u z a k l i g i ,
Vx = 66,667 - 30.X + 1,5x2 P 0
dan + xi r 17,53 m . 6 oldugundan gerqek ktlk d e g i l d i r .
x2 = 2,546 m
ve maksimum eqilme momenti, 2
max M - (66 ,667-15*2 ,546+0 ,5*2 ,516 ) -2 ,546 + 80,754 kN.m.
bulunur . Sistem B k e s i t i n i n sagindan k e s i l i p ve sagda kalan parqa Uze-
r i n d e denge g a r t i yaz i l a rak ,
'BC = 0,5* 1.2 *4 ,0 = 24 kN.
MBC = -0,5-12-4,O-(4,0/3) n -32 kN.m.
C k e s i t i s e r b e s t uq oldugundan,
VCB = 0 , MCB = 0 bulunur.
Bulunan bu dager l e re gtlre keame kuvveti ve egilrne momenti diyagram-
la r i S e k i l 4.42. (b) ve ( c ) d e g i z i l m i g t i r .
brnek : 4.37
S e k i l 4.43.(a) 'da yUkleme durumu ve b o y u t l a r i v e r i l e n sistemde
k e s i t t e s i r l e r i diyagramlarinr ~ i z i n i z ve d e g e r l e r i n i Uzerine yaz in iz .
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156
CbzUm : s
34,64+60+60+120+37,50- Sekil 4.43.(a) 'da gbrUlen gene1 yUk sistemi Sekil (b)'de
gSrUldUtJU gibi dUgey yUklere,(c)'de gbrUldUtJU gibi yatay yUklere kontrol edilir.
ayrilarak dUgey kuwetler iqin dUgey mesnet tepkileri,kesme kuwet- Sekil 4.43. (c) sis
leri ve eUilme momentleri, yatay kuwetler iqin yatay mesnet tepki- ZPx = 0 gartindan, si ve normal kuwetler hesaplanir. RBx = -20-0,5-43,30.3,
D ucundaki egik 40 kN.luk kuwetin dU$ey bilegeni, bulunur .
P - 40-sin60 = 36,64 kN. DY ' Kesit Tesirleri :
ve yatay bilegeni, AD qubugunda , PDx = 40.~0~60 = 20 kN. NDA = NAD = 20 kN. , U
bulunur.BF arasindaki egik Uqgen yUkUn F noktasindaki giddetinin MDA = 0 , MAD = -34,64 dUgey bilegeni,
AC qubuwnda , q~~
= 50.sin30 a 25 kN/m. NAC = NCA = 20 kN.
ve yatay bilegeni, 'AC = 147,94 - 34,64 n
qFx = 50.~0~360 = 43,30 kN. - 113,30 - 0,5.30- 'CA - bulunur .
MAC = MAD = -69,28 kN.
Mesnet Tepkileri : %A = 147,94-4,O-34,64 Sekil 4.43. (b) sisteminde ,
CB qubugunda , EMB = 0 gartindan,
C d U g h noktasanin R - [34,64-12,0+0,5.30.4. (10-4/3)+60.6+0,5.40-6.6/3-0,5.25. AY ' NCB = NCA = 20 kN.
3.195 7/10 VCB = 53,30-60 1 -6,70
R - 147,94 kN. AY - BFE sisteminin den
ve NX = 20 kN.
EMA = 0 gartlndan VBC = -164,ZO + 0,s-25
R ~ y = 1-34,64-2,0+0,5.30.4,0*4,0/3+60-4,0+0,5.40.6,0. (10-6/3)+ MX = -0,5.25.3,0.1,5
+0,5~25~3~0~(10+1,50)~/10 = 164,20 kN. BE qubugunda
bulunur . ve NEF = VEP = nEP L 0 CP - 0 gartindan, NFE = NFB = -0,5.43,3. Y -
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158
vFE = N~~ = 0 , 5 - 2 5 . 1 , 5 0 = 1 8 , 7 5 k ~ . CP ~ u b u q u n d a ,
M~~ = rqFB = - 0 , 5 . 2 5 . 1 , 5 - 0 , 5 = -9,38 k ~ , m . NCF = NFC = 0
N~~ = -0 ,5 ' 43 ,3 .3 ,0 = -64,95 kN. MFC ' 0 1 MCI
VnP = 0,5 .25-3 ,O = 37,50 kN.
MBF = -0 ,5 -25 .3 ,0 .1 ,50 = -56,25 kN.m. C B Cubugunda ,
NCR = NBC = 0 bulunur.Bulunan d e n e r l e r e g a r e N , V ve M d i y a g r a m l a r i s i r a y l a S e k i l
4.43. (d l , ( e l ve ( £ 1 ' d e ~ i z i l m i ~ t i r . MCn = 1 , 6 1 . 3 - 4 0
Urnek : 4.38 MBC = - 6 0 . 1 r - S e k i l d e k i s i s t e m d e k e s i t tesirleri
'Irn d i y a g r m l a r i n i q i z i n i z v e d e t j e r l e r i - BE ~ u b u ~ u n d a , n i Uze r ine y a z i n l z .
NDE = NEB = 0 Mesnet T e p k i l e r i :
zPx = 0 S a r t i n d a n MBE = - 6 0 ~ 0 kN.1
RAx = 40+22,5 r 62 ,5 kN. M1.13 = 0 E M B = 0 ~ a r t i n d a n
~ ~ u l u n u r . R =(11,25+60-60) / 7 n 1 , 6 1 kN.
AY
"UxlFIF 1.) EMA = 0
RBV =(-11,25-60+60 -8) /7 = 58,39 kN.
K e s i t T e s i r l e r i : GD qubuijunda,
NGD = NDG ' 0 , VGDE 0 V D G ~ -22,5kN.
MGD = 0 , MDG 1 -22 , s '1,5/3=
1. = -11,25 kN.m.
l y ) DAqubugunda,
NDA f NAD 1 2 2 , s kN . , VDA' VAD= 0
M~~ M~~ 1 -11,25 kN.m.
AC qubugunda,
NAC = NCA 1 - 4 0 kN. , VAc = VCA = 1 '61 kN.
(MI MAC =-11,25 kN.m.
rqCA = 1 ,61-3 -11 ,25 1 -6,42 kN.m.
S e k i l 4.44. . . ./. . . +
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4.1.1.1.4 Gerber (Birleqik) Kiri~$lcr
Birden fazla apikligi bulunan, biri sabit veya ankastre ve
digerleri hareketli mesnetler Uzerine oturan dogru eksenli pubuk-
lardan olugan dUzlem sistylere "SUrekli Xirig" denir. n apiklikli
bir mesnedi sabit (Sekil 4.45.4 sUrekli kiri~te n+2 tane bilinme-
yen mesnet tepkisi vardir.Sistem Uzerinde 3 denge denklemi yazila-
bileceginden sistem n+2-3 r n-1 inci dereceden hiperstatiktir.Bir
mesnedi ankastre olan ($ekil 4.45.b) sistemlerin hiperstatik derece-
side n+3-3 t n dir.
$ekil 4.45.
SUrekli kirigler yukarida da belirtildigi gibi hiperstatik
sistem1erdir.Hiperstatik sistemlerde,mesnetlerde pubuk eksenine dik
y6nde meydana gelen p6kmelerden etkilenirler.Ozellikle k6prii kirig-
lerinde bu mesnetler altindaki orta ayaklarin oturdugu zemin saglam
detjilse veya akarsu iperisindeki orta ayaklarin olmasi halinde bu
mesnetlerde p6kmeler meydana gelebi1ir.B~ ~6kmelerden de siirekli ki-
riglerde kesme kuwetleri ve egilme momentleri meydana gelir.Ote yan-
dan izostatik sistemlerde meanet hareketlerinden dolayi kesme kuwe-
ti ve egilme momenti meydana gelmez.Bundan dolayi hiperstatik sUrekli
kiriglerde apikliklarda uygun yerlere mafsallar konularak izostatik
duruma getirilir-Mafsal eklenerek izostatik hale getirilmig sUrekli
kiriglere 'Gerber Kirigler* denir.
Daha ancede belir iqin ilave bir denge denk
ceden hiperstatik bir sis
da mafsal eklenmesi gerek.
surekli kirigte n-1 adet,c
nafsal koymak gerekir.
Yalniz mafsallar 1
dikkat edilmeli,sistemin 1
bir kismi hiperstatik olal
a- Orta apikliklal
b- Kenar apiklikte
sinda en fazla
en az bir en fa
c- Bir orta apikll
pikllklara en f
konulmalidir.geki1 4.46.Id
Sekil 4.46. Uygun Ma
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gekil 4 . 4 6 . Uygun Mafsalla Gerber Kirig Tegkili
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Sekil 4.47.'de m; Zerilmigtir. _.----_ - C - --_ _
a) 4% n --- ----/a P .mr nm n . Gerber kiriglerir - I;r .
1- SUrekli kirig mesn
- -._ 0
b) 2; kuwetlerin doQmaslna -
fi-.-------- 4b 43s f i -- ----..#fi zeden 0lur.Halbuki mesne
=angi bir kesit tesiri m #-/-.. 2- Gerber kiriglerde
t * ---- . cl
-- 43 ----- --'fk /-a maz. @\.-*---
3- Mafsal yerleri deg
dgiaiklikler yapllablll
a~lkllklardaki egihe m a
WYa istenildigi gekilde Sekil 4.46. Uygun Olmayan Mafsal Yerlegtirilmesi
Sekil 4.46.(a)'da sisteme 3 mafsal konulmasl gerekirken 4
mafsal konulmug ve ayrlca yanyana iki a~lkliga ikiger mafsal ko-
nularak sistemin taglylcillk ozelligi bozularak labil hale gelmig-
tir.Seki1 4.46.(b)'de iki kenar a~lkllgl iki mafsal konularak o kl-
slmlar labil hale getirilmig,buna kargillk 2. ve 3. aqlkllklar hi-
perstatik olarak kalmlgtlr.Ayrlca bir mafsalda fazla konulmugtur.
Sekil 4.46.(c)'de 1. ve 2. mafsallar arasi hiperstatik kalmig,buna
kargilik 4. mesnedin bulunduqu klslm mekanizma durumuna gelmistir.
a) nm
b) -
Sekil 4.47. Mafsallarln Goruniigii ve Betonarme
Mafsal Tegkili
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Gerber Kiri~lerin Hesabi :
Gerber kirigler gene1 olarak a~agida apiklanacagi gibi iki
gekilde hesaplanabilir.Burada da bakig ybnii basit kirgilerdeki gibi
belirlenir,
1.YBntem : Tiim kirig uzerinde dogrudan dogruya denge denklem-
leri ve mafsal gartlari yazilarak mesnet tepkileri ve bunlara bag11
olarakta kesit tesirleri hesaplanlr.Mafsa1 garti uygulanirken mafsa-
lin saginda veya solunda kalan kisimdaki kuvvetlerin mafsalin bulun-
dugu yere gBre statik momentlerinin sifir oldugu gbz bniinde tutulma-
1idir.Mafsal sayisinin az olmasi durumunda bu ybntemi kullanmak daha
uygundur.Mafsa1 sayisinin artmasi durumunda bilinmeyen mesnet tepki-
side artacagindan pok bilinmeyenli denklemler ortaya pikar ve bunla-
rin ~6ziimude zaman alici o1ur.B~ ybntem uygulanirken miimkiin oldugu
kadar bir bilinmeyenli denklemler elde edecek gekilde denge ve mafsal
Sarti 962 6niine alinmalidir.
Sekil 4.49.Ida gorillen gerber kiri~te
Tiim sistemde LP = 0 gartindan, X
GC sisteminde MG = 0 gartindan
n
T h sistarde LMB = 0 gartindan
ny - (ply.bi - P b' - P . (a+aIk) -Pn(a+atn) +R .L2)/L1 R - iy i ky CY
(4.15)
Tiin sistemde EMA r 0 gartin
gartindan tepkilerin dogrul~
.Yesnet tepkileri bulundukta~
leri 3.4.2. paragrafinda be:
ramlari qizilir.
brnek : 4.39
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Sekil 4.50.(a)'da yilkleme durumu ve boyutlari verilen ger-
ber kiriate mesnet tepkileri ve kesit tesirleri agaglda hesaplan-
mlgtir.
Mesnet Tepkileri :
Turn sistemdeCPx = 0 dan , RAx = 0
GFC sisteminde M - 0 qartlndan G -
R - (50-3,0)/5 = 30 kN. CY -
Tum sistemde EMB = 0 gartindan
R - (0,s-30-5,0.2,50 - 50-5,O + 30.7,00)/5 '= 29,50 kN. AY -
Tum sistemde ZMA = 0 ~artlndan
R - (0 ,s -30 -5 ,0 .2 ,50+50 .10 ,0 -30 .12 ,0 ) /5 = 65,50 kN. BY -
bulunur ve
ZP = 0,50-30.5,OO + 50 - 29,50 - 30 - 65,50 = 0 Y
~artlndan kontrol edilir.
Kesit Tesirleri :
A mesnedinin saglndan kesilir ve solda kalan parqa iizerinde
denge ~artlarl yazllarak,
VAE = 29,50 kN , MAE = 0
E kesitinden kesilir,solda kalan parqa ilzerinde denge qartlari yazi-
larak,
'EA = 29,50 - 0,5.30.2,5 = -8,OO kN.
MEA = 29,50-2,50 - 0,5'30.2,5- (2,50/3) = 42,50 kN.m. $
bulunur . AE qubuqunun iki ucundaki kesme kuwetleri farkll iqarette ol-
dugundan bu qubuk iizerinde kesme kuvveti siflr ve egilme momenti mak-
simum olacaktir.
Sekil 4.50. (d)
Mx = 29,50.~-0,5-I
Mx = 29,50x - 2 x 3
- 29,50-2,.217 Mmax -
bulunur. E noktaslnda
VEB = VEA = -8 kN.
B mesnedinin solundan
~artlarl yazllarak,
VBE = 29,50- 0,5.30
MBE = 29,50.5,0-75
B mesnedinin sagindan
larl yazllarak,
V B G = 5 0 - 3 0 = 2
MBG = 30.7,O-50.5,
BF arasinda dl$ ytik 01
VGB = VGF = VFG = V
F noktaslnln saqlndan
ti yazllarak,
VFC = -30 kN.
MPC = 30-2,O = 60 k
ve F dilqumunun dengesi
MFG = MFC = 60 kN.m
FC qubugu uzerinde yuk
VCF = VFC =-30 kN.
bulunur . 2 .Yontern (Aylrm
ve qubuk eksenine dFk ku,
ha onceden de bilindig
kuvvetlerin kesme kuvv~
vet ve momentlerin non
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Ew sebeple sistem qubuk eksenine dik kuwet ve moment,qubuk ekse-
nine paralel kuwet iqin ayri ayri qbzhlenebilir.
Cubuk Eksenine Dik Kuwetler ve Momentlerin Etkimesi :
Bu kuwetlerin etkisi altindaki gerber kirig mafsallarinda sadece
qubuk eksenine dik mafsal kuwetleri meydana gelecek,qubuk eksenine
paralel mafsal kywetleri sifir 01acaktir.Gerber kiri~ler~herbiri
bir temel izostatik tagiyici sistem olacak gekilde,mafsallardan ay-
rilir, bu tagiyici temei izostatik sistemler,gerber kirlglerde basit
kiki$,konsol kirig ve bir veya iki tarafi qikmali kiriglerdir.Gene1
olarak basit kirigler ve tek tarafli qikmali kiriglerin qikmanin bu-
lunmadigi tarafindaki mesnedi konsol kirig veya qikrnali kirigin qik-
masinin ucuna oturacak gekilde "tagiyici sistem gemasi" belirlenir.
bnce tagiyici sistem Uzerine oturan (taginan) sistemler kendi iizeri-
ne etkiyen yUklere gore qbzUmlenir ve sonra bu taginan sistemin mes-
net tepkileri(mafsa1 kuwet1eri)tagiyan sistemde dig yiik olarak uy-
gulanir.Tagiyici sistemlerde kendi Uzerine etkiyen dig kuwetler ve
taginan sistemin mafsal kuwetleri g8z tjnUne alinarak hesaplanir.
$ekil 4.51.(a)'da yukieme durumu verilen gerber kirigte yiik-
ler $ekil 4.51.(b)'de ve 4.51.(d)'de gbriildUgU gibi bilegenlere ay-
rilir.$ekil 4.5l.(b)'deki yUklere maruz gerber kirigin tagima gemasi
$ekil 4.51.(c)'de g8sterilmigtir.Bir aqikliktaki iki mafsal arasin-
daki eleman , basit ve hareketli mesnedin bulundugu kenar aqiklik- larda mafsal olmasi durumunda dig mesnet ile mafsal arasindaki ele-
man her zaman taginan bir sistem olur.Ankastre mesnedin bulundugu
kenar aqiklikta mafsal bulunmasi durumunda ankastre mesnetle bu maf-
sal arasindaki eleman (konsol kirig) Uzerinde mafsal bulunmayan fa-
kat yanlarindaki aqiklarda mafsal bulunan eleman(iki tarafli qikmali
kiri~) her zaman tagiyici sistemdir.
$ekil 4.51.(c)'de G1 G2 basit kirigi ve G3E birer taginan sis- tem olup once bu sistemlerin mesnet tepkileri,mafsal kuwetleri ve
kesit tesirleri hesaplanir.Sonra sifayla AG1 ve G2 G3 kiriglerinde
G1,G2,G3 mafsal kuvvetleri de gbz 8nUne alinarak mesnet tepkileri ve
kesit tesirleri hesap1anir.B~ her kirige ait diyagramlar bir kiyas
dogrusu Uzerinde birlegtirilerek kesme kuweti ve egilme momenti di-
yagramlari qizilir.
Cuhuk Eksenine
Bu kuvvetlerin etkimec
kar$llayacak bir mesnc
tlndan bulunur ve ondz
bulunarak diyagram qiz
dir.
al Sistrm
4 blDii$ey yukkr
!
c l Diisry yiik tasrma Stmasr
dl Yatay yiikler
Glh
A
RA H F
8 i Ya tay yiik taslma 3tma:
s
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Cubuk eksenine paralel olarak etkiyen kuvvetlerin etkisinde
mafsallarda sadece ~ubuk eksenine paralel mafsal kuvveti vardlr.Dik
dogrultudaki mafsal kuwetleri slflrdlr.Tum kuwetler sabit veya an-
kastre mesnede akacak gekilde ta~ima gemasi ~izilerekte (Sekil 4.51.
e.) mesnet tepkileri ve normal kuwetler hesaplanabilir.Burada once
G3E sistemi hesaplanlr ve G3 deki mafsal kuweti G2G3e dlg yuk ola-
rak etkittirilerek G2G3 ve benzer gekilde slrayla GIG2 ve AGl sis-
temi icin hesap yaplllr.
Sekil 4.52.(a)'da yiikleri ve boyutlari verilen sistemde bnce
E kesitin etkiyen eqik yukun yatay ve diigey bilegeni,
ve FH ~ubutjunun kirig baglandltjl kesitindeki kesme kuweti ve egilme
moment i ,
VFH = -100 kN.
MFH = 100.1,5 = 150 kN.m.
bulunur . E kesitindeki P dugey kuweti ve F kesitindeki MFH momenti diigey
EY yiiklere gore taglma gemasl Sekil 4.52.(b) 'de goruldugii gibi ve PEx
yatay kuvvetli ve VFH = -100 kN.lu yatay yiik tagima gemasl Sekil 4.52 (e)'de gdrUldiitjU gibi Fizilir.
Dugey ytiklere gore taglma gemaslnda once GIG2 basit kirigi
hesaplanlr.Burada yiik simetrik oldu?jundan,
G2y = G3Y = 20'4,0/2 = 48 kN.
ve basit kirig oldutjundan maksimum eeilme momenti a~lkllk ortaslnda-
dlr ve degeri, 2
Mmax = 20-4,O /8 = 40 kN.m.
bulunur .
c)Kcsmc kuvveti di)a
G e h N ItVkN - RAx:200 N - 4- - 200 kN
r ) Yatay yiiklere giir
I N ) 0 D
ry
- I ) Normal kuvvct d a
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G2Y k u v v e t i G I G g t e k t a r a f l i g i k rna l l k i r i a i n G2 ucunda
tcrs ybnde d l 9 k u v v e t o l a r a k c t k i t t i r i l c r e k ,
- J
EMGl = 0 Q a r t l n d a n
b u l u n u r . EP = 173,205+40-95,923-117,282 = 0
Y
o lduqundan G1 ve R i n dofiru o ldu i ju a n l a g l l l r . BY
' G ~ E = ' E G ~ = 95 ,923 kN.
MEGl = MEB = 95,923.2,O = 1981,846 kN.
= "EB = 95,923-173,205 = -77,282 kN.
b u l u n u r . G3y m a f s a l k u w e t i G 3 1 i k i ' t a r a f l l q l k m a l i k i r i ~ i n G 3 ucun-
d a ters yBnde d l $ k u v v e t e t k i t t i r i l e r e k ,
ZM,, = 0 q a r t l n d a n
I = (40.5,0+150-0,s-30e1,5.1,50/3)/4 = 84,688 kN.
b u l u n u r . EPx = 40+22,50-84,688- ( -22,188) = 0
* . * o lduqundan y a p i l a n i g l e m l e r i n do?jru o l d u g u a n l a a l l l r .
MFC = -40-3,0+84,6€
MFD = (-22,188) -2,C
- MDI = -22,50. M~~ - V,, = 22,50 kN.
bu lunur . 1
I
Gly maf s a l k u w ~
d i g k u w e t o l a r a k e t k i ,
ZP - 0 dan , R Y - AV =I
CMA = 0 dan , MA = - M r MA = -143,88
AG1
VAGl = VGIA = 95,92'
Bulunan bu kesm
l a r i S e k i l 4.52. (c) v e
$ e k i l 4.52. (e) ' LPx = 0 dan G3, = 100 k
k i t t i r i l e r e k ,
- 100+100 = 200 G l X - 1
NX = NCF = NCG3 = N,
- N ~ 3 ~ 2 ' N ~ 2 ~ 3 = -100 N G2B = N BG2 = N~~ =
N = N ' = 7100-11 EG1 GIE
NGIA = NAGl = -200 kl
bulunmu$ v e normal kuwc
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17 4
brnek 4.41.
Sekil 4.53.(a) 'da yilkleme durumu ve boyutlarl verilen sis- ~ O O ~ N
temde bnce EF qubuqundaki kesit tesirleri,
'EF 'FE = 50 kN. , MEp = -50-0,s - -25 kN.m. , MFE = 0 ( a ) u .+..-zm-+m ; - 1 3
bulunur.GBrUldii~il gibi artlk sistemin E ucunda -25 kN.m. moment
ve PEx = 50 kN. kuvvet etkimektedir. &--SO kN
PEx kuwetinden meydana gelen mesnet tepkisi ve normal kuwetler -- $ekil 4.53. (b) den ( b ) ratay kuvvrtlrr
C
RAx = -50 kN.
ve A ile E araslnda bagka yatay yilk olmadlglndan, 1ookN 5 0 k ~ / ,
- NAD - NDA = NDB = NBC = NCB = NCE = NEC = 50 kN.
bulunur . 2
,- -3JP13kN
Diigey kuvvetler ve moment etkisi altlndaki sistemin taglma
a, T3f (c) DiiSey k ~ v v e t l r ~
gemasl Sekil 4.53.(cI1de gbsterilmigtir.
GE Sisteminde s- I d ) Normal kuwrt d;&
Mesnet Tepkileri :
ZMC = 0 gartlndan R - (50-8,0.4,0-25)/8 = 196,875 kN. GY - ZMG = 0 gartlndan 2
9. 9
R - (50-8,0.4,0+25)/8 = 203,125 kN. - CY - 8 2 4
bulunur . ( e l K a m e k~vvr t i
Kontrol : ZP = 400-196,875-203,125 = 0 dlr. Y
Kesit Tesirleri :
Uygun yerlerden kesim yaplllp denge gartlarl yazllarak, ,
\ ,
VGC = RGy =" 196,875 kN. , VCG = -203,125 kN. I vcE = - 0 'EC - MCG = MCE = -25 kN.m. , MEC -25 kN.m.
bulunur. G ile C araslnda kesme kuvveti igaret degigtirdiginden iki
kesit araslnda kesme kuweti slflr ve egilme momenti en buyuk olur.
I Bu arada yiik dUzgiin yay111 olduqundan kesme kuvvetinin slflr oldugu
I f I Egilmr momrn ti dial
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elde edilir.
AG Sistemi
*GY , AG qikmali kirisin C ucuna ters yande dl5 kuwet olarak
etkittirilip denge garti yazilarak,
CMB = 0 dan , R - (100-2,O-50.1,50.0,75-196,875-1,5)/5 AY -
= -30,313 kN.
EMA = 0 dan , RBy = (100.3,0+75-5,75+196,875.5,5)/5 = 402,188 kN.
bulunur . Kontrol : CP = 100+75-(-30,313)+4021188+1961875 = 0 dlr.
Y
bulunur.Norma1 kuwet,kesme kuweti ve eeilme momenti diyagramlari
Sekil 4.53. (d) , (el, ve (,f) 'de gbsterilmistir . - flrnek : 4.42
Sekil 4.54.(a)'daki sistemde bnce benzin deposunda tabanca
ve yanlara etkiyen hidrostatik basinq hesaplanir.
Tabandaki hidrostatik basinq,
p =.Y-h.b = 8-4,00.1,00 - 32 kN/m. Yanlardaki hidrostatik baslnq ise B ve C kesitlerinde 32 kN/m.
ve F ve I kesitlerinde sifir olan Uqgen yayili bir kuwettir.Bunlar
BF qubugunda , *
VBF = 0,5-32.4,O - 64 kN. , MBF = 64.(4,0/3) = 85,333 kN.m.
4,711 I
(C ) K c ~ m r kuvvrti diylg&r
1f 1 Egilmr momrnti diyqnml
I g1 k r m l ku wrt diyagramr
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C I gubutjunda,
vcI I 0,5*32.4,0 = 64 kN. , MCI = 64- (4 ,0 /3 ) = 85,333 kN.m.
bulunur.Burada m o m e n t l e r dtigey y t ik le re g 8 r e tagima gemasinda B ve
C k e s i t l e r i n d e d i g yUk (Seki1.4.54.c) v e kesme k u w e t l e r i y a t a y yUk-
lere g 6 r e tagima gemasinda B ve C k e s i t l e r i n d e d l $ y a t a y k u w e t
( g e k i l 4.54.d) o l a r a k gbz 8ntine a l i n a b i l i r .
$ e k i l 4 . 5 4 . ( c ) ' d e DG s i s t e m i n d e mesnet t e p k i l e r i :
EMG = 0 g a r t i n d a n
RAV P (40.3,6-8,8+0,5-50-8,0.4,0)/8 r 276,4 kN.
EMA = 0 g a r t i n d a n
R r (-144-1,8+200.4,0)/8 t 67,6 kN. GY
bulunur . Kont ro l : 144+200-276,4-67,6 = 0
K e s i t T e s i r l e r i t
'DA = 0 , %A = 0 , VAD 3 -40.3,6 = -144 kN. ,
bu1unur.E ve G a r a s i n d a kesme k u w e t i i g a r e t d e t j i g t i r d i g i n e g 6 r e
bu a r a d a kesme k u w e t i s i f x r ve egilme m m e n t i maksimum o l a c a k t i r .
9 t h ~ Vx = 6,25.(8-xI2-67,6 r 0
(8-X) - ? f 67,6/6,25
a-x 1 6 e k N 8-x = i 3,289 ~1 m 8+3,289 11,289>8
oldugundan g8zllm d e g i l d i r .
S e k i l 4.54. (h) xo o 8-3,289 n 4,711 m.
bulunur . R mafsal kuvvet i G
GY El4 a l i n a r a k ,
EMC = 0 dan ,
RBV = (67,6.13,0+85,3:
EMB = 0 dan ,
RCV = (-67,6-3,O-85,33
e l d e e d i l i r .
Kontrol : ZP = 67,6+320- Y
K e s i t T e s i r l e r i :
bulunur. C B a r a s i n d a kern
qubugu Uzerindeki k u w e t I
s l f l r olduUu y e r i n B kesi i
ve maksimum e t j i l m e momenti
Mmax = MBC + ~ & / ( 2 . 3 2 )
bulunur .
NBC = 64 kN. , NCB P 64
Keeme k u w e t i , e i j i l n e momn S e k i l 4.54. ( e l , ( f ve ( g )
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Par .... (MI
611.71
Mesnet T e p k i l e r i :
GCF s i s t e m i n d e EMC = 0 g a r t i n d a n
GCF s i s t e m i n d e E M G =' 0 g a r t l n d a n
DACB s i s t e m i n d e EMB = 0 g a r t
R - ( (1/2) -50.3.10+10C AY -
= (3722 ,5 ) /8 = 465,31
DAGB s i s t e m i n d e EMA = 0 g a r t
RBy = ( (-1/21 -50 -3 -2+100
= (4187 ,5 ) /8 e 523,44
b u l u n u r . K o n t r o l :
465,31+523,44-75-800-113,
Kesit T e s i r l e r i :
V F C = O , M F = O , 'CF =
G i l e E a r a s i n d a kesme kuvve
kesme k u w e t i O'dan gececek .
Vx = 113,75-40-X. (x/2) = 1
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A ile B arasinda kesme kuvveti i ~ a r e t deci~tirdigindcn bu arada
kesme kuvveti 0:dan gesecek ve moment en biiyiik olacaktir.
Vx = 390,31-100-x = 0 + x = 3,90 m
brnek : 4.44
u. :MZm r Parabol 6925
Kesit tesirleri :
Mc = 232,5.4-120 . 4/3) VD = -240+112,5 r -127,
Mg = 770+112,5.4-60.402
bulunur . C ile D araslnda kesme kuu
erde kesme kuweti slflr
Vx = 112,SO-60.x = 0 da
'max P 112, S2/(2.60) t7i
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. .. .
104
R - (10.10-5+0,5-20. AY - CMA = 0 sartlndan
R - (10-10-5+0,5-20- 10 kH/m k N l m BY - l l l l l l l l
E - - Kontrol : ZP = 100+100+2 -
C 10 m 1 2 m 1 3 m 1 3 m 2 m J I O ~ -I Sistemde yatay kuvvet elm I c 1 T T
RAx = 0
bulunur.
Kesit Tesirleri :
GIG? sisteminde,
VGIE = VEGl = 25 kN.
, ABGl sisteminde,
'JAB = 78,333 k ~ . , p I
VBA = -146,667+25 = -1;
MBA = MBGl = -25-2 = - 5
'gGl = VGIB = 25 kN- 1
bulunur . AB qubuqunda maksimum mome
$ekil 4.56.
p5ztim :
sistemfn. taglma gemasi Sekil 4.56.(b) 'de g6ruldu~i.i gibi ~izi-
1ir.Burada G1 G2 tasinan ABGl ve G2CD taglyan sistemdir.~istemin tii- f!,rJ6.JJ LN
mi.i simetrik oldu&jundan,sistemin yarisl iizerindeki mesnet tepkileri ve
kesit tesirleri hesanlanmigtlr.
Mesnet Tepkileri :
G1G2 sisteminde EMGz - 0 gartindan G = (50.3/6) = 25 kN. 1 Y bulunur.Bulunan, dejferlere g
A G sistemin G, ucunda G dig kuvvet olarak yiiklenerek, 4.56. (c) ve (d) 'de slrayla r B 1 Y lari qizilmiqtir.
1MB = 0 gartindan
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186
Urnek : 4.46 P noktasxndaki e g i k t e k i
Yatay b i l e g e n i : 141,
DUgey b i l e g e n i : 141,,
bulunmug ve s i s t e m i n y a t ,
(b) v e dUgey yUklere gbrc
m i g t i r . nesne t T e p k i l e r i :
DUgey yUklere gUre tagmmi
EM = 0 g a r t l n d a n R G 1 A!
EMA = 0 g a r t l n d a n G I!
bulunur . Kontrol : CPy = 60-45-15
DUgey y u k l e r e gn re t a g m i c & u p f lhr g l rr tm$tmm
EMGz = 0 gar txndan , I
EWg = 0 g a r t l n d a n , c
bulunur .
Kontrol : EP - 100-62,s- Y -
Diigey yUklere g b r e taglma
kN. v e G2 ucuna 37,50 kN.
EMC = 0 g a r t x y a z l l a r a
R = (15.12+25,981.2. BY
EMB = 0 g a r t l y a z l l a r a
R = (-15-2-51,962+20 CY
bulunur . S e k i l 4.57. 1 Kontrol : EP - 15+51,962
S e k i l 4.57.(a) ' d a b o y u t l a r l v e yUkleme durumu v e r i l e n ge rbe r k i - Y -
r i g t e GIB a r a s l n d a k i eg ik dCfegCfnpyaylll yUkUn,
I Yatay ~ U k l e r e gUre t a g l m
Yatay b i l e g e n i : 30-cos 60 = 1 5 kN/m. G z X = 100 kN. , GIG2 s
I DUgey b i l e g e n i : 3 0 - s i n 6 0 r 25,981 kN/m. A s i s t eminden RAx = 70
G 1
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Kesit Tesirleri : MI - 136,804.5-37,s.
DUqey kuwetlerden :
Vcr = -136,804-37,50 sisteminde , VCG2 = VGgC = 37850
VAH = 45,OO kN. , MA = 0 , VHA = 45-0,5-30.2 i. 15 kN.
bulunur. IC yubutjunun i M~~ = 45,00.2,0-30.(2/3) = 70 kN.m. , VEH = V = -15 kN.
EGi ducundan,
M~~ o M~~~ = M E = 15.4 = 60 kN.m. , MG1 = 0
bulunur . H ile E arasrnda kesme kuvveti igaret deqigtirdiqine gbre bu arada
bir yerde kesme kuvveti sifir,elfilrhe momenti maksimum olacaktlr.
30 *( , Vx = 15-0,5.15-~.~ = 0 t - - 4 x = J T - 1,414 m.
60 u Mx = 60t15-X-7.5.x2.x/3 3
Mmax = 60+15.1,414-2,5-1,414 = 74,14kN.m. Yatay Y(ik1ere gijre t a ~ r m
bulunur . NFD = NDF = 0
G2 D sisteminde, N ~ ~ 2 = N ~ 2 ~ = N ~ 7 ~ =
'G2F ' 'FG~ = 37,5 kN. , VFD = VDF n -62,5 kN. , MG2 = MD = 0 NBG. = -100 kN. , N G 1
MF = 37,50.5 1 187,50 kN.m. NGIE = NEG, = NEA Nj
bulunur . bulunur . G1BCG2 sisteminde, Bulunan deger lere
ma1 kuvvet diyagramlarl :
VCIB = '15 kN. , VBGl s -15-25,981.2 m -66,962 kN. mistir.
MGIB = 0 , MBGl = MBI = -15.2,O-51,962-1.0 = -81,96 kN.m.
VBI = -15-51,962+167,658 = 100,696 kN.
VIB = 100,696-0,5.5*40 = 0,696 kN. , VIC VFB = 0,696 kN.
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g e k i l 4.58.
Mesnet T e p k i l e r i :
S e k i l 4 . 5 8 . ( a I 8 d a b o y u t l a r i v e ytikleme durumu v e r i l e n ge rber
k i r i ~ t e tagima gemasi g e k i l 4 .58 . (b ) 'dek i g i b i q i z i 1 i r . B ~ taaima ge-
masinda G2ECF s i s t eminde ,
ZMC = 0 g a r t i n d a n , Gzy = (0,5.12-30.6-40.3-0,5-20.3.2)/12=75 kN.
IMG = 0 ~ a r t i n d a n , RCy = (180-6t40-15t30.14)/12 r 175 kN. 2
bulunur . Kontrol : CP = 180+40+30-75-175 -- 0 d i r .
Y
G,DBG2 s i s t e m i n d e ,
EMB = 0 e a r t ~ n d a n , G
t M - 0 e a r t l n d a n , R GI
bulunur . ~ o n t r o l : t P = 100+75-41
Y AG, s i s t e rn inde ,
Tilm s i s t e m d e LPx = 0 gar1
Kesit T e a i r l e r i :
G2F s i s t e m i n d e ,
G2 i l e E a r a s ~ n d a kesme k v e t i n i n s ~ f l r o ldugu y e r ,
Vx m 7 5 - 0 , 5 : 5 * ~ * ~ m 0
x r m 5,477 m. vl
'rnax r 75-5,477-0,50-5
VCE. = -175+40+0,5-20.3
VFC = 40 kN. , . M C = -I(
GIG2 s i s t e rn inde ,
VGID - , VDGI = 40,909 kl
A G, s i s t e m i n d e , - VACI - VClA = 40.1 909 kN
bulunur . S i s t e m l e y a t a y kuvvet olma
f ~ r d i r .
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Sekilde boyutlari ve yukleme durumu verilen gerber kirigte :
a- Mesnet tepkilerini hesaplayiniz.
b- Kritik kesitlerdeki kesit tesirlerini hesaplayiniz.
c- Gerekli yerlerde eBilme momentinin en biiyiik degerini ve yerini
hesaplayiniz.
d- Kesit tesirleri diyagramlarini qiziniz ve degerlerini uzerine ya-
$ekil 4.59. (a) ' kiriein yatay ve dUeey
de gi5sterilmigtir. C n
DUgey bilegeni : 10
Yatay bilegeni : 10
bulunur . Mesnet Tepkileri :
G2E sisteminde,
&Px = 0 gartxndan ,
EMD P 0 gartxndan , EMG2= 0 gartxndan ,
bulunur.
Kontrol : EP = 570+50- Y
GIG2 aisteminde,
Bu sistemde yukarxda
dxg yUk olarak etkittir
CPx = 0 gartxndan , EMB E 0 gartxndan , tM r 0 ~artlndan , Gl
bulunur.
Kontrol : &P = 360+256 Y
A GI sisteminde ,
Yukarxda bulunan G 1 Y
olarak etkittirilerek,
EPx = 0 gartxndan , I EP = 0,gartxndan , I Y
CMA P 0 gart~ndan , I bulunur .
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194
Kesit Tesirleri :
A sisteminde , Gl
VAGl = 182,69 kN. , VGIA = 182,69-30.3 = 92,69 kN. NAG1 =' -36.6 kN. , NGIA'= -36,6 kN.
MAGl = -413,07 kN.m. , klG1 E 0 (mafsal)
G1G2 sisteminde,
VGIB = 92,69 kN. , VBGl i 92,69-30.10 r -207,31 kN.
NGIB = -36,67 kN. , NXl= -36,67 kN.
MGIB = 0 (mafsal) , MBGl= MBG2 = MB = -256,~6.2~/2 - -573,lO kN.m. VBG2 r 256,56+30.2 = 316,56 kN. , VGZB = 256,56 kN.
NBG2 = NGZB = -36,67 kN.
GIB ~ubugunda kesme kuweti igaret degigtirdiginden,kesme kuwetinin
sif ir olduqu yer ,
va maksimum moment,
Max M r 92,692/(2.30) = 143,19 kN.m.
bulunur . G2E eisteminde,
M = 0 (mafsal) , NGzC = N = -36,67 m. G2 CGz
C dUgUm noktasinin dengainden,
NCD r 86,67-36,67 r 50 kN. , VCD = 76,56-50 r 26,56 kN.
tulunur.
C ile D araslnd, kuvvetinin slf lr olduij~
Lu lunur.
Bulunan degerlere g6re
kuvvct, k e s m tuvveti ve
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Cerqeve le r qenex s e k i l l i o l u p yapxlarda yaygln o l a r a k kul-
l an l lmaktadxr . Cerqevelerde kesme kuvve t i , eqi lme momenti ve nor-
mal kuvvet a y n l e t k i n l i k t e d i r v e bu Oq k e s i t tesiri qdzdnUnde
t u t u l a r a k b o y u t l a n d l r l l l r . Ce rqeve l e r in y a t a y veya a z e g i m l i ele- manlar lna k i r i g , dUgey veya qok e q i m l i e lemanlar lna kolon d e n i r . Genelde qerqeve e l e m a n l a r l ek sene l kuvvete k a r g l mukavemet ve ri-
j i t l i k l e r i egi lme momentinden bUyUk o l acak g e k i l d e boyut landxr l -
lrrsa s i s t emde e k s e n e l kgvve t l e r daha e t k i n egilme momenti ve kes-
m e k u w e t i daha a z e t k i n o l a b i l i r .
Cerqeve o l u g t u r a n t U m e lemanlar a y n l , d ~ z l e m i p i n d e i q e r i - '
s i n d e ise bu qe rqeve l e r e dUzlem qe rqeve l e r ve yUklerde a y n l dUz-
l e m i q i n d e ise dUzlem qerqeve sistemler denir . . Denge ve maf sa l
g a r t l a r l i l e qbz t l leb i len i z o s t a t i k qe rqeve l e r $eki1.4.6O9da gd-
rUldUgU g i b i b a s i t qe rqeve l e r , Uq maf s a l l l qe rpeve l e r , g e r g i l i
Uqmafsa l l l qe rqeve l e r v e b i r l e g i k Ug m a f s a l l x pe rqeve l e rd i r . Bu
i z o s t a t i k pe rpevs l e r uygulamada kullan1ldr4x g i b i h i p e r s t a t i k sis- t e m l e r i n qbzUmUnde b z e l l i k l e k u w e t ( e n e r j i ) ybnteminde h i p e r s t a -
. t i k s i e t emin i z o e t a t i k e s a s sistemi o l a r a k t a kargxmlza q lka r .
B i r mesnedi sab i l i qe g a r t l a r x i l e pbzUmler
Qlda bu t a r gergeveler r
Sekil.4.6l.a 'da t vede bakr$ y6nU kesik g i
eagdah so l a , E D qubu-
yukarlya dogru qeqilmi$t
ZPx - 0 gart lnda
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C D quhuqu : Sistem C D aras inda k e s i l i p sagda kalan
parqa Uzerinde denge g a r t i yaz i larak;
2Pv = 0 dan Nx == -48 I
) x * e b a g l i det j i l yulb,uk boyuna
EP - 0 dan Vx - -80 s a b i t . Y
x =-. 9 m i q i n
bulunur.
D E qubugu: S is tem D E aras inda k e s i l i p a l t t a kalan
k i s im Uzerinde denqe g a r t i yaz i larak kesim yap i lan k e s i t t e k i
k e s i t t c s i r l e r i v, deqivkenine b a q l i o larak - 0 .* Y2 a: 3 m i q i n
1 a I : O dan; V 18 - y .,
CM := 0 dan ;M - -48 y2 Y2 y2
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brnek: 4.50-
C
lbm
d ) N w m l kuwtt dipgnmt
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R ' 65 kN. AY
IllC - 0 dan r
R - ( -60~3 ,0+270~4,50+90*9~0) /9 ,0 DY
- 205 kN.
dur. I(ontro1r
IPy -- -90+270+90-65-205 - 0
,nw
d i r .
R e s i t Tee ir l er i
o c y < 3 m i p i n /-
if=+- v - 0 x ' e baQli olmadigindan Y
'AB "BA - 0 olur.
1
N - -R - 65 kN. x 'e bag11 olmadi9xn- Y AY
dan
NAB ' NBA " -65 kN.
My - 0 ve d o l a y ~ s r y l a
M A B g M B A - 0 d i r .
3 < y < 6m iqin J 1
N - -65 ; NBC - N - -65 kN. Y CB
bulunur .
MDC - - 155-9,O-1
bulunur .
M .=- 1 5 5 * 5 , 1 6 max
bulunur.
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-206-
brnek: 4.52-
"'lrN S e k i l . 4.64'de yUkleme durumu
v e b o y u t l a r i v e r i l e n s is temde:
- 6- 8,544
o s o - 8/8,544 = 0,9363
s i n a ' 3/8,544 " 0,3511
- 10/0,9363 - 10,680 m.
LBc - 2/0,9363 - 2,136 m.
d i r .
Mesnet t e p k i l e r i t
CP = 0 dan Y
R - 50*10,680 - 534 kN. Ay
M~~ 0 dan
RAx - ( -0 ,s-20-5,0(3+12/3 )5)-534,0*3,0)
R ~ X - 160,667 kN.
EMA - 0 dan
%x (0,50-20*5,0(5,0/3W34,~3,0)/8
R~~ = 210,667 kN.
Kont ro l :
cPx- 0,5*20*5,0+160,667-218,66- 0
Kesi i t tesirleri - A C 9ubuQunda
VAC = -160,667 kN.
NAC -534 kN.
MAC -; 0 (Mafsa l o.
'C A = -160,667-0,5,
NC'A - NAC -= -52 . b r
3 1 yiijkiln qubuk c k s c n i n e
uk c k : s c n i n c d i k t ~ l ~ l o g c n f
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-208-
"DC - -210,6671 v e ExC - 0 ga r t indan
M~,c 0 (M.~fsal MCB - -50=2,136*1,00 - -106,80 kN.m.
ve B serbest uq oldugundan C , O n r a s ~ n d , ~
2.31 kosmc kuvvc\ti sl VCB - NCB - MCB - 0 d r r .
C D cubu4u- : S i s t e n C n i n s a g ~ n d a n k e e i l i r ve sagda ka- l a n kAsim gOt6ntbde t u t u l a r a k
k 32 vx vcD' (P'COS~
Vx VCD-p'x 1
Mx VCD (x/'o!
Kuvvet ler in qubuk ekaeni Uzerindeki i e d U g t h l e r i n i n c e b i r -
sel toplaminin s i f rr oLneai gar t rndan Mx VCDm (X/COSO
tiCD = -50*8,544*sina-210,667-cosa - - VCD/p konul
M - - N~~
- -427,20=0,3511-210,667-0,9363 - -347,237 kN. max V ~ ~ ' V ~ ~ / '
VCD = 50*8,544*cosa-210,6674na 'ma, -' -- VCD2/ (2*p
vCD - 427,2-0,9363-210,667*0,3511 - 326,022 kt4.m.
n,, - 210,667-3,OO-50-8,54404 - -1076,80 Kt4.m. x - 326,022/50 0
D k e s i t i n i n Polundan k e s i l i p aaQdaki parqanrn dengesinda Ml"a X (326,022)
- -210,667-cosa - -210,667- 0,9363 - -197,248 kN. cdi lir,
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-210-
M e s n e t t e p k i l e r i
ZP = 0 dan; x R~
ZM = 0 d a n D
R = ( 1 5 - 2 , 0 . 8 , 0 + *Y
R = 6 1 , 3 8 9 k ~ . AY
EMA = 0 d a n
R = ( 1 5 - 2 , 0 . 1 , 0 + ] DY
K o n t r o l : ZP - 30+ Y
Kesit tesirleri
A B qubuqu:
VAR = 6 1 , 3 8 9 kN ;
VBA = 6 1 , 3 8 9 - 1 5 . 2 ,
- MAA - MBC = 6 1 , 3 8 9
B C qubugu
VBCZ 6 1 , 3 8 9 * C O S ~ - 3 1
NBC= - 6 1 , 3 8 9 S s i n a -
VCB= 6 1 , 3 8 9 . c o s a - :
VCB = - 4 3 , 6 1 1 . 0 , 8 0
NCB = - 6 1 , 3 8 9 . s i n a i
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-212-
N~~ - 43,611-0,60 - 26,167 kN.
brnek: 4.54-
M C B M M -168,611-3,O-15-5,O-2,5-50*5,0 - 6 8 , 3 3 3 kN.m. CD
B ile C a r a s i n d a kesme k u v v e t i i ~ a r e t d e g i ~ t i r d i g i n d e n
bu ~ u b u k Uzerinde b i r yerde kesme k u w e t i s i f i r ve egilme momen-
ti maksimum o l a c a k t i r .
Ornek 4.52'de oldugu q i b i kesme k u w e t i n i n s r f l r oldugu
y e r i n B nok ta s lna y a t a y u z a k l i g i
x, - vBC/15 -- 25,111/15 - 1,674 m.
ve - v / (2.15 -eoso) + M ~ ~ Mmax BC
Seki1.4.66-
Mmax - 25,1112/ (2-15-0 =8)+92,778 - 119,051 KN.m.
bu lunur . Mesnet t, p k i l
C D qubugu: B D s i s t e m i n d ,
VCD - -168,611+15-5,0*50 - -43,611 kN. Tiin1 s i s t emde
N~~ - N~~ O
VDC -- -168,611+15*2,0+50 - -88,611 kN. TUm s i s t c m d c
MDC - MDE - -15.2,O-1,O-50-2,O - -130 kN.m
'DE - 15-2,0+50 - 80 kN. Tun1 s i s t e m d c )
VDE - 50 kN.
NDE - NED " MEDg 0
Kon t ro 1 : !: 1) Y
- bulunur .
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Mesnet t e p k i l e r i
A B G s i s t e m i n d e
EMG = 0 dan
Tilm s i s t e m d e
EMF =- 0 dan ; \
2 18 R +2RAx-40.18- ( 6 0 . 1 5 Y2-100-2 a 0 AY
bulunur ve
( 1 ) dcnkleminde y e r i n e konulursa 1
1 3 H +36R -3855.4 - 3520 AY AY
RA;-9.384,9+3855=370,9 e l d e e d i l i r .
I
-18Rm+2RFHt60 I
I -18R +2Rp,, = - FV
h ve ( 4 ) dcnklemlerin
Kes i t t e s i r l e r i
v "AB RA -370,9(
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cosu =.. 3 / 5 - 0 , 6 0 ; s i n u = 4 / 5 0 , 8 0 1 1 VBG = - 6 4 , 4 5 6 . ~ ~ ~ ~
cosn . 6 / 6 , 7 0 8 2 -- 0 , 8 9 4 4 ; s i n n 2 3 / 6 , 7 0 8 2 0 , 4 4 7 2 2 NBG = - 6 4 , 4 5 6 . s i n (
K e s i t t c s i r l e r i MBG 0 ( S a b i t me:
A C q u b u r y u e G i l e B a r a s i n d a 1
F u b u k t a b i r y e r d e k e s n - V == 6 0 , 5 3 1 . s i n a + 2 , 6 2 6 - c o s n -= 5 0 , 0 kN. . Vnc cn 1 1
N NCA =- 6 0 , 5 3 1 . c o s a l - 2 , 6 2 6 . s i n a - 3 4 , 2 1 8 kN. A C 1
MAC = 0 ( B a s i t m e s n e t )
M~~ = MCG 6 0 , 5 3 1 . 1 , 5 0 + 2 , 6 2 6 . 1 , 1 2 5 - 9 2 , 7 5 1 kN.rn.
c G Cubuqunda
-100 . sin11 1
5 0 - 1 0 0 . 0 , 8 0 -30 k N . m . "cc '"GC "GA
N ~ c , . N ( ; ~ 3 4 , 2 1 8 - 1 0 0 . ~ 0 s n 2
- 2 5 , 7 0 2 kN.
M~~ = 0 ( M a f s a l o l d u q u n d a n )
Vx = 20-0150 ( 1 0 / 3 ) x
G B Cubuqunda Mx '20 ( x / c o s a k ) - 0 , 5
'G B =-- - 6 4 , 4 5 6 . c o s a . + 3 9 , 4 6 9 - s i n a + 6 7 , 0 8 2 - c o s a L 2 2
Mx = ( 2 0 * ~ - ( 5 / ~ ) ~ ~ )
V ( 6 7 , 0 8 2 - 6 4 , 4 5 6 ) . 0 , 8 9 4 4 + 3 9 , 4 6 9 . 0 , 4 4 7 2 20,O kN. GR
Mma, = ( 2 0 ~ 3 , 4 6 4 - ( 5
N - - 6 4 , 4 5 6 . s i n a -39,469-cosn2+67,082.sinfl GR 2 2
N ~ ~ 3 (67,082-64,456)~0,4472-3914699018944 - 3 4 , 1 2 7 k N .
M 0 (M‘IEs~L o l d u q u n d a n ) (: 13
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-226-
- - 2 0 ' 2 - 1 ' -40 kN.m.
brnek: 4.59-
S e k i l d e ytikleme durumu ve yii
leri v e r i l e n qerqevede :
a ) Mesnet t e p k i l e r i n i h e s a p l a y i n i z .
b) K r i t i k k e s i t l e r d e k i k e s i t tes ir- TUm s istemde g e r i n i h e s a p l a y i n i z .
L M ~ ~ - 0 , ~ a r t r n d a C ) G e r e k l i y e r l e r d e k i e n biiytik e g i l
me momenti y e r i n i v e d e g e r i n i h e s a
d ) Kesit tesirleri d iyagramln l q i z i
c o s a - 0 , 9 5 7 8 TUm s istcmde
s i n a - 0 , 2873 ):MA .-- 0 gartlnda
-46,98*0,754 DA s i s t e m i n d e
CMD - 0 Sar t indan
Titm s i s t e m d e
1% = 0 g a r t l n d a n Kontrol : ZP - 4
Y
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. a ) Mesnet t e p k i l e r i n
h) K r i t i k k c ! s i t l e r d c
C ) A q r u b u n d a GD ve
d a n a q o l e c e k e 6 i l m e
r i n i h e s a p l a y l n i z .
d ) Kesit t e s i r d i y a q
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S e k i l d e b o y u t l a r l ve yuklerne dururnu v e r i l e n qerqevede;
a ) Mesnet t e p k i l e r i n i h e s a p l a y l n l z
b ) K r i t i k k e s i t l e r d e k i k e s i t t e s i r l e r i n i h e s a p l a y l n l z .
C ) G H qubugundaki en bilyiik egi lme momentini ve y e r i n i h e s a p l a y i
n i z .
d ) K e s i t t e s i r l e r i d i y a q r a m l a r l n l q i z i n i z .
Mesnet T e p k i l e r i
G H B s i s t e m i n d e EMG== ~ a r t i n d a n
R : - (40.13.6,5-30)/11 - 304,55 kN. BY
bMB - 0 g a r t l n d a n G =- (520.4 ,5+30) /11 ?= 215,45 kN. Y
ZP . 0 dan Gx =- . 0 bulunur . X
Kontro l : 520-215,45-304,55 0 d l r -
A F G s i s t e m i n d e n ; CPx :- 0 ~ a r t i n d a n R A x = 0
EP 0 ~ a r t i n d a n RAV = 100+50+120+215,45 =.- 485,45 kN. Y
XMA - 0 q a r t i n d a n MA =:- 100.2-50.1-215,45.3-40e3.1,5 = -
; -676,35 kN.m.
K e s i t t e s i r leri;
A-$'-~I pug ~1
N nc NCA ' - 4 0 5 , 4 5 k ~ . ; VAc Vcn 0 ;
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k u v v e t i n i n s l f l r o l d u g u y e r i n G ye u z a k l l q l
max M ;. 215,45;1/(2-40) ... 5 8 0 , 2 4 kN.m. b u l u n u r .
NllI N I H 0 ; VI,,I ; ; 4 0 - 5 .-- 200 k N ;
M H I = - 4 0 - 5 * 2 , 5 + 3 0 - -470 kN.m. ; Mill -' 3 0 kN.m.
B H Cubuqu
Nrill Nlln - -304 ,55*s inrr . - 2 7 2 , 2 7 kN.
- 1 0 4 , 5 5 * c o s f s - 1 3 6 , 1 3 kN.
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-238-
B E F S i s t e m i n d e
0 , 5 ' 4 0 ' 3 - 2 = 20kN. EM = 0 mafsal ~ a r t r n d a n ; R -
E Rx 6
A C D R S i s t e m i n d c
z M - 0 m a f s a l g a r t l n d a n ; -. -30. 15' 7 , 5 + 2 7 1 , 2 5 ' 1 2 E Riu G
R ~ x = -20 kN.
b ) Kesit T e s i r l e r i
A D Cubuqu
N~~ = N~~ = - 2 7 1 , 2 5 kN ;
- 'AD - 'DA
= 2 0 k ~ . ; M~~ = 0 ; MDA ' 20- 6 z 120kN.m.
C D Cubuqu
- = 0 N~~ - N~~
; VCD = 0 ( S e r b e s t u c ) ;
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-240-
20 kN/m /- m m m Ornpk 4.63
F S i s t c m i n q e o m e t r i s i n d e n
- &924- - 7,281~1.
I c o s n 7 / 7 , 2 8 -- 0 , 9 6 1 5 I 1 1
S e k i 1 . 4 . 7 3 . a -
Mesnet T e p k i l e r i E ' d e k i m a f s a l ~ a r t i n d a n -
M - R 8+100 .1 ,2 - 0 A Ax
M* -8KAx- -120
A C D 1.: 1: S i s t c m i n d c F ' d c k i m a f s a l ~ n r t ~ n d i l r r
MA-10.R - 5 , 8 . 1 0 0 - 7 * 2 0 * 3 , 5 + 7 R Ax 0
A Y
MA-10RAx+7R = 1070 A"
A C D E F G S i s t c m i n d e G ' d e k i m a f s a l ~ a r t i n d a n
M -8Rm+14R -100*12 ,8-20 .14-7 = 0 A AY
MA-8R +14R = 3240 Ax Ay
(1) e g i t l i q i n d c k i M -8Rh A
I 2 0 ( 3 ) d c y c r i n c k o n u r s a
1 i
-120t14l l AY
3240
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Kontro l : TUm Sis temde F ' d e k i m a f s a l g a r t i n d a n
1 8 4 0 - 2 4 5 . 1 0 i 2 4 0 - 7 - 5 , 8 * 1oo-20-77 3 r5-1840+245* 1 0 - 2 4 0 . 7 + 5 , 8 * 100
t 2 0 - 7 . 3 , 5 -0 oldutjundan bulunan d e q e r l e r doqrudur.
%F - 0 (Mufsul)
b ) Y e s i t tesirleri : . S i s t e m s i m e t r i k olduqundan y a r i deqer-
leri hesaplarnak y e t e r l i d i r . F E Cubugu
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Mesnet Tepkileri
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1 J W ~ 1 3Wml LOO m 1. 6.60 m 1 S00m I I I I -
Mesnet Tepkileri
F B Sisteminde 9- 0 sartlndan
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Kontrolr ;M- 0 -
Kesit Tesirleri
C D Cubu4unda
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bulunur .
IN) B,
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-252-
VCA - 216-3- 309 s ina ' 144 kN.
ve MC = 0 ~ a r t l n d a n
MCA -120* 2,25+360= 3-30 30°1 ,5 " 675 kN/m
h c s a p l a n l r .
S c k i l .4.75. f ' d c ~ u b u k ckaczni U s c r i n d c k i izd(\.j(lm q n r t indan
258-120- cost - 186 kh.
Culwk ck~ic?ninc? clik izd(lgl\~n gnrtincli~ri
v CC.
144-120*sim1 - 48 kN.
C DUtjUmUnde t M = 0 g a r t i n d a n C
'CG 'CA - 675 kN.m.
C u h k c k s c n i Uze r indck i izdUgUm ~ n r t l n d a n
N~~ 18G-30- 9 -cosn 24 kN.
Cubuk c k s c n i n e d i k izdU~Urn g a r t i n d a n
vGc -- 48-30-9- sin0 - -168 kN.
G d e m a f s a l bulunmasindan
MGC = 0
h e b a n l a n l r . C G qubugunda kesme k u w e t i i g a r e t d e g i g t i r d i g i n d e n bu q
buk Uzer inde maksimum moment meydana g e l e c e k t i r . Burada
K e s m e k u v v e t i
Vx = 48-30 -y - s ina - 0
y a z i l a r a k
48 y =-=2"' -11.4.75- 24
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. ,
bulunur . 4.1.1.2.4- G u n l M e r
drnek: 4.66-
Mesnet T e p k i l e r i
TUm s i s temde E M ' 0 qart indan A
EMB - 0 gart indan
- - 2 0 . 1 5 ' 7 , 5 - -187 , s k ~ .
:7SkN 1 2
I1 s i s t e m i Uzerinde E M - 0 gar-
I Sisterninde EME ' 0 qartindan
Kesi t T e s i r l e r i 1- A Kes i tMe
Kontrol; EP - -187 ,5+187,5 - 0 Y
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Su lur
rf ii 1'1 i i
4 . 8 L
t 1;. I \
w.. K
I.lcsnct t e t > k l l c r i --
I!? - 0 dan Y o
CP -- I) dan X
cRHX - 50 kN.
K L D Sisteminde
R " *8+50*2 -40 -8 , DY
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VHE - 160-329,870 c o s
'HE - 160-329,87- 0,970167
VHE - 160-320,03 ' 160 kN.
!IHE = 0
E H Cubuqunda k e s n e k u w e t i i g a r e t d e b i g t i r d i o i n d e n maksi
mum moment a r a g t i n n a s i y a p i l i r . Vx - 1 6 0 - 4 0 - x - c o ~ i a ~ " 0 - x-
40 coscl 40-0,970167 38 ,81
Mmax - 160.4,12-40- 4.12 -= *cosxl - 659,2-329,36 = 329,84 H . m . 2
B H Cubugu
NBH - NHB - -312,42 kN ; VBH- VHB = 60 kP1 ;
MBH - 0 ; MHB - 60.6 -- 360 kN.m.
H I Cubuqu
NHI - NIH - -312,42+152,42 - -160 kN ;
V - VIH - 60-50 - 1 0 kN. HI
%I - MHB - 360 kN.m. ; MIH- 360+10-3 - 390 kN.a.
I G m b u g u
NIG - -90 *cosa2-160* s i n a 2
\ - -90*0,89445-160.0,44723 .
NIG = -80,50-71,56 - -152,06 kN.
- -90 . s i n a 2 t 1 6 0 - c o s a 2
160 LN - -90-0,44723+160*0,89445 .
I
I I I
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L K Cubugunda kesme k u w e t i i gare t degigtirdiginden aaksi- mum moment ara$tirmasl y a v l l i r .
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H a r e k e t l i y u k e t k i s i n e d e m a r u z s i s t e
g e n e 1 o l a r a k i k i ~ e k i l d e b o y u t l a n d i r i l i r .
1. S i s t c m i n c l c m a n l a r i i i z g r i n . d e y e t c r ' i
e l e m a n l a r i
r s i k k c s i t
qKzKniine a l i n i r v c h c r k e s i t t c s a b i t v e h a r c k c t l i y u k l e r d c n
c!qcr 1c.1
' r i n c c j i
ntcydana q c l e n k e s i t t u s i i r l c r i n i n c n bUyijk d r i t ) u l u n u r .
I-lcr k e s i t L u l u n a n bu c n bUyiik k e s i t t e s i r l c jrc boyu t l a n -
d i r i l i r . D o g a l o l a r a k b u durumda s i s t e m i n e l e r n a n l a r i d c g i q k e n
k c s i t l i o l u r . Biiyiik a q l k l i k l i s i s t e m l e r d e bu ybntem k u l l a n l l a r .
2. S i s t e m i n h c r b i r e l c m a n i n d a b o y u t l a n d i r m a y a esas o l a c a k
k e s i t t c s i r i n i n c n biiyiik d e g e r i b u l u n u r v e e l e m a n bu d e g e r
g o z o n u n e a l i n a r a k b o y u t l a n d i r i l i r . Eu durumda s i s t e m i n h e r b i r
e l e m a n i s a b i t k e s i t l i o l u r . Ktiqiik a q i k l i k l i s i s t e m l e r d e b u ycn-
t e m k u l l a n i l i r .
Burada e n biiyiik v e y a e n kiiqiik m u t l a k d e g c r c c
y a c n k i i ~ i i k a n l a m i n a d i r . Z i r a b u r a d a c c b i r s c l i g a r
lcman;l e t k i y c n s t a t i k biiyiikliiFjUn m i k t a r i o n e m l i d i - .
e k s e n e l k u v v e t du rumunda qekme v e y a 1 b a s i n c i
r m e s i d
n e n b i
urumunc
noment i
3nur a n
iyiik dea
la d ie je : H r m - X i 1
gdaoni ine a l i n i r . Veya i q a r e t d e q i ~ t i i
r e t l i tesire g 6 r e d e s i s t e m b o y u t l a n d i r i l a c a k t i r .
t o n a r m e b i r k c s i t d c e n buyUk e f l i l m e r ( + I i:
d o n a t i k e s i t i n b a k i $ y6nii t a r a f i n a c a k e n
m e moment i d e ($1 ise bu d e q e r giizSniinc a l i n m a y a c e
l i ise bu d c q e r c gHrc k e s i t i n b a k i g ynnii tcrsi t a r
yiik v e -
jer i
r i q a - n hn-
I I ~ I L I I c ~ r ~ ~ l i ~ ~ i l k L ~ r . (;oCJU x;llll;ln c n IAlyilk ;ill1 ;In11 rla I l l ; i r . ;~ 8 a g t t t l l t " I - c - 8 ,
kiiqiik an l i iminn n~inimum d a k u l l a n l l i r .
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S i s t c m l e r i n qogunda b i r q o k y i i k l e n r w r t i v a r d i r . d r n e g i n
h a r e k e t l i t r a f i k yiikii k6priiniin e l e m a n l a r i n d a yiikiin kopr i i U z e r i n -
a c k i konumuna g n r e f a r k l i l~ i iy i ik l i ik te k c s i t t e s i r i n i n o l u g ~ n a s i n a
ncdcn o l u r . Ucnzcr g e k i l d e . a g l r h a r c k e t l i yiik t a q l y a n y a p 1 sis-
t c m i p r o j e l e n d i r i l i r k e n , h u yiikiin f a r k l i z a m a n l a r d a a y n i k a t t a -
- k i f a r k l i a q i k l i k l a r a v e f a r k 1 1 k a t l a r a y i i k l e n r n e l i d i r . Coqunluk-
l a b i r e l e r n a n i n a r a n i l a n k e s i t t e s i r i n i n e n biiyiik d e q e r i n i n hem-
q i yiiklemc d u r u n ~ u n d a m y d a n a g e l c c e g i n q i k q a L e l l i c l c q i l d i r . Ik'jy-
lc L i r durumda p r o j c y i y a p a n k e s i t t cs i r lcr i i l z c r i n d c k r i t i k o l d -
L i l e c e k y a k p o z i s y o n u n u v c kombinasyonunu a r a $ t i r m n l l d i r . 4 t c s b i t i n d c q c n c l o l a r a k tcsir q i z g l l c a n i l n ~ a k l ~ ~ c l i r .
Bun un
5.1-, Tesir Cizgileri . - . M a t e r n a t i k t e G r e e n f o n k s i y o n u olilrarc c a n i m l a n a n t e s ~ r L W I ~ K -
s i y o n u , s i s ter i iGcr inde b e l i r l i g o q r u l t u d n ' h a r e k c t c d c n 1 b i r i m -
l i k h i r k u v v e t t c n d o l a y i belirli b i r k c s i t . t e l c i s t r r t i k buyiik1iic;'iin
(rnesne t t c p k i s i , k c s m kuvvc t i , e g i l r r c aloI!lc?n t i , normal Ituvve t,
dcp las rnan) d c q i g i m i n i g 3 s t e r i r . ( j r n c g i n , h a s i t k i r i q t c n c n n e t t c p -
k i s i s a d e c e yiikiin' y e r i n i n f o n k s i y o n u o l a c a k t i r . T c s i r f o n k s i y o n u
d i y a g r a m i n a " T e s i r ( ; i z g i s i l ' d e n i r . ~ c s i r q i z q i s i s i s t e r n ' i ' r c r j . n d e
b c l i r l i d o g r u l t u d a h a r c k e t e d e n k n v v e t t e n b e l i r l i k c s i t t e meyda-
n a g e l e n s t a t i k biiyiikliik b i r b i r i m l i k y a k h i 1 orcl ina t, ola-
r a k a l i n a r a k c l d c e d i l i r . S e k i l . 5.1. ' d e b i r i k u v v c t e t k i -
s i n d c I{ rncsnet t e p k i s i v e C k i s c t i n d e k i erJ i lme m m c n t i tcsir ~ i z - AY II
g i s i v c r i l m i y t i r . S c k i l . 5. l . .Bde C k e s i t i h i z t i " A ~ C
b i r i m l i k diiqey k u v v e t C k e s i t i n d e i k e n A rnes meyda ! n
m c s n e t t e p k i s i n i , S e k i l . 5 . l . c ' d e D k e s i t i h i z n s i n d a k i rg, h 4.r
b i r i r n l i k di isey k u w e t D k e s i t i n d e i k e n C k e s I
e q i lme moment in i g b s t e r m e k t e d i r .
a s i n d a ) ned inde
: L i r n a g e l e
1 b i r i m l i k di isey k u v v e t tes i r i e t k i s i n d e m s n e t t e p k i l e r i ,
k e s m k u w e t l e r i ve n o r m a l k u v v e t l e r i n tesir q i z q i l e r i b o y u t s u z
mmen t r e s n e t t e n k i s i v e c i j i l m e m o m e n t l e r i tesir q i z g i s i o r d i n a t -
l a r i n i n b i r i m i m d i r . n i r i m m r n c n t t c s i r i c t k i s i n c l c , m s n c t tcp-
k i l c r i , k c s m k u v v e t l e r i v c n o r m a l k u v v c t l e r i n tcsir q i z g l s i o r d i -
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$ e k i l . 5.1- Bi r in dUgey k u v v e t $ek i l -5 -2 - B i r i m t h n e n t E t k i s i . ~ t k i s i'.
n a t l a r m ln b i r i m l e r i l,/n, ~ o m e n t mesnet
n i n b i r i m l e r i boyu t suzdur .
5.1.1- . Tesir .Cizgilerinin Tayini
t e p k i s
g i l e r i s i r q i z . - . ..
- l z o s t a t i k s i s t e m l e r d e kuvv~ . $ i n te : s i t s t a t i k y 6 n t e m l e r l e b u l u n a b i l i r . nga t j lda v h n t e m vex
m i q t i r
ba- . -
B e l i r l i b i r k e s i t t e k i b i r s t a t i k bUyiiklUge a i t tesir q i z -
g i s i n i e l d e etmek i q i n sistem Uzer inde b e l i r l i g r u l t u y a pa-
r a l e l 1 b i r i m g i d d e t i n d e b i r k u w e t a l i n x r v e 1 e t i n y e r i
s i s t e a d e h e r h a n g i b i r nok tadan x u z a k l l g l i l e b e l i r l e n i r . Bu '
b i r i m y i ik-sabi t yUkmUg g i b i d U m l i l e r e k a r a n i l a n k e s i t t e k i s t
t i k biiyiikluk x 'e b a g l i o l a r a k y a z i l i r . 1 b i r i m l i k k u w e t kend
I
a-
isi-
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5.4.2- Konsol KMg
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l i - Kafcs Sistemler
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