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[ ]
( )ϕεϕεϕεε
ευευ
σ
εευευυυ
σ
⋅⋅+⋅+⋅=
⋅+⋅
−=
+⋅+⋅−⋅⋅−⋅+
=
2sinsincos
21
)()1()21()1(
22
yzxy
zzyy
yyxxnn
yyxxyy
zzyyxx
zzyyxx
zzyyxxxx
E
E
GRAÐEVINSKI FAKOTPORNOST MATERIJALA
AF=σll
llll ∆=−∆+= )(ε
AElFlt
Tl
⋅⋅+⋅∆⋅=∆ α
εσ=E
up εευ =
[ ]
=
zzyzxz
yzyyxy
xzxyxx
ij
στττστττσ
σ [ ] nijn
rr⋅= σρ [ ]
⋅=
),cos(
),cos(
),cos(
nz
ny
nx
ij
nz
ny
nx
σρρρ
222nznynxn ρρρρ ++=
22nnn
nn n
σρτ
ρσ
−=
⋅=rr
zzyzxz
yzyyxy
xzxyxx
στττστττσ
+
−−
−=
s
s
s
zzyzxz
yzsyxy
xzxysx
σσ
σ
σστττσστττσσ
00
00
00
)(31
zyxs σσσσ ++⋅=
( ) ( ) 2221
21
2,1 4 xyyyxxyyxx τσσσσσ ⋅+−⋅±+⋅=yyi
yxoi σσ
τϕ
−=tg °=+ 900201 ϕϕ ( )
yyxx
xyo σσ
τϕ
−
⋅=⋅
22tg
321 σσσσσσ ++=++ zzyyxx 321 σσσ ≥≥401 += ϕϕ
( ) ( ) ( ) ( ) ( )ϕτϕσσσσϕτϕσϕσσ ⋅⋅+⋅⋅−⋅++⋅=⋅⋅+⋅+⋅= 2sin2cos2sinsincos 21
2122
xyyyxxyyxxxyyyxxnn
( ) ( ) ( )ϕτϕσστ ⋅⋅+⋅⋅−⋅= 2cos2sin21
xyxxyynt
( ) )(5.04 2122
21
12 σστσστ −⋅±=⋅+−⋅±= xyyyxx )(5.0)()(5.0 21121 σσϕσσστ +⋅=−⋅=MAX
( ) ( )zzyyxxOKT σσσσσσσ ++⋅=++⋅= 31
32131
( ) ( ) ( )213
232
2213
1 σσσσσστ −+−+−⋅=OKT22OKTOKTOKT τσρ +=
tT
yyxxzz
xxzzyy
zzyyxxE
zzyyxx ∆⋅+
+⋅−= ασσυσε )(1
21
2xyxy
xyEGxy
γυτ τε =⋅== +⋅
zxyzxyG
zxyzxy τε ⋅= ⋅2
1
)coscoscos(2
22 γεβεαεε ⋅+⋅+⋅= zyxd )1(2 υ+⋅= EG
VVt zyxTEV ∆=++=∆⋅⋅+++⋅⋅−⋅=++= )()3()()21()( 3211
321 εεεασσσυεεεε
ZAKOVICEi
i
Mni
FnFF
nHF
VH
ρρ
⋅∑
===2
22yx FFR += DOPDOP td
Rdx
R σσττ ≤⋅
=≤Π⋅⋅
=
4
2
n = BROJ ZAKOVICA X= REZNOST ZAKOVICA
ZAVAR talax
Ftb
FDOP
⋅=≤⋅⋅
=⋅
= 7.0ττσ x=BROJ VAROVA
TORZIJA p
t
p
T
WM
IM
MAX r =⋅=τ32
4DI P⋅Π=
rI
W pp =
16
3DW p⋅Π=
lIG
M
P
t ⋅⋅
=ϕ )1(32 4
44
DdDIP −⋅Π= )1( 4
43
16 DdD
PW −⋅= ⋅Π
ρτ ⋅=p
t
IM
z
y
b/2 b/2
h/2
h/2
POSUDE TANKIH STIJENKI)
21(1
1συσε ⋅−⋅=
E hp
=+2
2
1
1ρ
σ
ρ
σ
KUGLA σσσ == 21 r== 21 ρρ
hrp
⋅⋅
=2
σ συε ⋅−=∆=Er
r 1hrp
Er
⋅⋅⋅−=∆
21
2
υ
∞== 21 ρρ rh
rp ⋅=1σ
hrp
⋅⋅
=22σ
rr
E∆=⋅⋅−= 11
5.01 συε
PRSTEN ∞== 21 ρρ r 021 == σσσ
hrp ⋅
=σhErp
rrr
⋅⋅=∆∆=∆=
2
DDε
GEOMETRIJSKE KARAKTERISTIKE PRAVOKUTNIKA
12
3hbI z⋅=
12
3bhI y⋅=
6
2bhWz =
6
2hbWy =
AI
i zz =
A
Ii y
y =
STEINEROV TEOREM AdII VLyY ⋅+= 2, ( d – UDALJENOST OD OSI y DO TEŽIŠTA A )
GLAVNE OSI PRESJEKA
2221
21
2,1 4)()( zyyzyz IIIIII ⋅+−⋅±+⋅=
MAXVMINUyz
MINVMAXUyz
IIIIII
IIIIII
=→=→<
=→=→>
( )yz
zy
II
I
−⋅−=⋅ 22tg ϕ yz IIII +=+ 21
SAVIJANJEy
MAXy
MAX
WMz
IM
xxMAX=⋅=σ
bI
ST
y
yzxz ⋅
⋅=τ DOP
yxx A
Nz
IM σσ <±⋅=
KOSO SAVIJANJE αcos⋅= MM y αsin⋅= MM z DOPIM
I
M
xx yzz
z
y
y σσ ≤⋅+⋅±= )( αϕ tgtg ⋅−=z
y
I
I
GRAFOANA PARABOLA PARABOLA 2 STUPNJA 3 STUPNJA
IEMq⋅
=
IET⋅
=ϕ
IEMw⋅
= baP ⋅= baP ⋅⋅= 5.0 baP ⋅⋅= 32
2 baP ⋅⋅= 31
1 baP ⋅⋅= 43
2 baP ⋅⋅= 41
1
axT ⋅= 5.0 axT ⋅= 31 axT ⋅= 8
52 axT ⋅= 4
31 axT ⋅= 5
32 axT ⋅= 5
41
2
2 )()()(
dxxMd
dxxdT
xq ==−dxd
EIxM ϕ=− )(
2
2)(dx
wdEI
xM =−0)( >− iax
CENTAR TORZIJE
AdzyMA
xyxzx ∫ ⋅−⋅= )( ττ ∫ ⋅−⋅==⋅A
xyxzxz dAzyMeT )( ττz
x
TM
e =
bI
ST
⋅⋅
=τ )5.0( hSttI
Ty
xy ⋅⋅⋅⋅⋅
=τ
[ ])25.0(5.05.0 22 zhthbttI
Ty
xz −⋅⋅⋅+⋅⋅⋅⋅⋅
=τ
∫ ⋅⋅⋅=⋅=b
xyxy tbdstT0
1 5.0 ττ ∫−
⋅=2
2
2
h
h
dztT xzτ
hTeTz ⋅=⋅ 1
MOŽDANICI yn
MAX
WfM
x ⋅=σ
f=PODATLJIVOST SPOJA l
I
STx
Ryb
ybMAXZ ⋅⋅
= )(
Rx= l
DOPx
bcR
oσσ ≤
⋅= DOP
x
baR
ττ ≤⋅
= DOPx
albR
ττ ≤−⋅
=)(
DOPE σσ <K
ED
σσ = Dy
EE W
M σσ <=
ALNA
a)TROOSNO DOPE σσσ ≤= 1
b)SAVIJANJE22 45.05.0 τσσσ ⋅+⋅+⋅=E
c)SAVIJANJE I TORZIJA )(5.0 22TSSE MMMM ++⋅=
2.NORMALNE DEFORMACIJE 4.POTENCIJALNA ENERGIJA DEFORMACIJA
a) DOPE σσσυσσ ≤+⋅−= )( 321 a) DOPE σσσσσσσυσσσσ ≤++⋅⋅−++= )(2 13322123
22
21
b)22 4)1(5.0)1(5.0 τσυσυσ ⋅+⋅+⋅+⋅−⋅=E b)
22 )1(2 τυσσ ⋅+⋅+=E
c)22)1(5.0)1(5.0 TSSE MMMM +⋅+⋅+⋅−⋅= υυ c)
22 )1(5.0 TSE MMM ⋅+⋅+= υ 5. POTENCIJALNA ENERGIJA PROMJENE OBLIKA
a) DOPE σσσσ ≤−= 31 a) [ ] DOPE σσσσσσσσ ≤−+−+−⋅= 213
232
221 )()()(5.0
b)22 4 τσσ ⋅+=E b)
22 3 τσσ ⋅+=E
c)22TSE MMM += c)
22 75.0 TSE MMM ⋅+=
ENERGIJA ∫ ∫ ∫ ∫ ∫ ∫ ⋅⋅+
⋅⋅+
⋅⋅+
⋅⋅⋅+
⋅⋅⋅+
⋅⋅=
l l l l l l
z
z
y
y
T
Tzz
yy IE
dxMIE
dxM
IGdxM
AGdxT
kAG
dxTk
AEdxNU
0 0 0 0 0 0
222222
222222
FU
∂∂=δ
MU
∂∂=ϕ
δ∂∂= UF
∫ ∫ ⋅++
⋅⋅
=⋅l l
y
ykykk IE
dxMM
AEdxNN
0 0
...δ
i
kiMin
ii
kiNin
ik IE
MAAENA
)(...
)( 11 ⋅⋅
++⋅⋅
= ∑∑==
δ
IZVIJANJE
2
2
i
MINKR l
EIF
⋅Π= 2
2
i
MINKRKR lA
EIA
F
⋅⋅Π
==σA
Ii MINMIN =
MIN
i
il
=λMIN
i
MIN
i
IlA
i
l 2
2
22 ⋅
==λ
pKRE σ
λσ ≤⋅Π= 2
2
pp
Eσ
λ ⋅Π=2
kKR
DOP
σσ =
EULEROVA HIPERBOLA 2
2
λσ E
KR⋅Π=
TETMAYEROV PRAVAC λσ ⋅−= baKR
VTpl SSW += TTT hAS ⋅= VVV hAS ⋅= VT AA =
0=+ UV WW ϕδ ⋅+⋅= MFWi
4
2hbpl
W ⋅=pl
Wpl
M T ⋅= σ