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Vrijednosti 'd' u smjeru X- osi , a očitanja širina (ordinata) u smjeru Y -osiVrijednosti integrala od X-osi do krivulje:

Trapezno pravilo

Koeficijenti: 0.5-0.5 tj. 0.5, 1, 1, 1,…, 1, 0.5

1

2

1

2

1 1

3

1

2 3 2

1 1

2

3

n

VL i ii

n

X i ii

n n

Y i i i i ii i

n

X i ii

n n

Y i i i i ii i

A d k y

d M k y

M d k y x d k y i

d I k y

I d k y x d k y i

=

=

= =

=

= =

= ⋅ ⋅

= ⋅ ⋅

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅

= ⋅ ⋅

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅

∑

∑

∑ ∑

∑

∑ ∑

Simpsonovo I pravilo

Koeficijenti: 0.5-2-0.5 tj. 0.5,2,1,2,1,…,1,2,0.5

1

2

1

2

1 1

3

1

32 2

1 1

2

3

1 2

2 3

2 2

3 3

1 2

3 3

2 2

3 3

n

VL i ii

n

X i ii

n n

Y i i i i i

i in

X i ii

n n

Y i i i i ii i

d A k y

d M k y

d d M k y x k y i

d I k y

d d I k y x k y i

=

=

= =

=

= =

= ⋅ ⋅

= ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅

∑

∑

∑ ∑

∑

∑ ∑

Y X VL VL

VL VLi

M M A

x d i X A

Y = == ⋅

Korekcijski koeficijenti a

a

d k d = 0.5 ili 2 ili 1a a ak k k k = ⋅ ⋅ ⋅

Steinerovi dodaci

Dodatak je:- pozitivan ako od neutralne osi tijela preračunavamo moment na neku drugu os - negativan ako od neke druge osi tijela preračunavamo moment na neutralnu os - nula ako je os X ili Y jednaka neutralnoj osi površine

2

2

B X VL VL

L Y VL VL

I I A Y

I I A X

= ± ⋅

= ± ⋅

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Centracija, početni stabilitet i pomaci tereta

( )

0 0

0

X

1

1

2

0 0

M je moment volumena oko osi X ili u smjeru osi Z

L L

B L L

X

n

i ii

n

ii

M G M F KF KG M G M F KF KG

I I M F M F

V V M KF

V

m Z KG

m

m d f B M G M G V

tg T ρ

ϕ

=

=

= + − = + −

= =

=

⋅

=

⋅ ⋅ = = ∆ = ⋅ ⋅∆

∑

∑

( ) 11

0 0

0.01

t L Lt B G

PP PP

VLVL

A F A F

PP PP

A VL A F VL F

A A A F F F

M I M Gt M m d X X M

M L L

mT p A

A

L Lt t t t

L L

L X X L X X

T T T t T T T t

ρ ρ

⋅∆= = ⋅ = ∆ ⋅ − = =

∆ = = ⋅ ⋅⋅

= ⋅ = ⋅

= − = −

= + ∆ + = + ∆ +