[ ]

( )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