´ GEOMETRICHE PROPRIETA DI ALCUNE AREE PIANE a
Autore: Fabrizio Barpi, Gennaio 2006 (http://ulisse http://ulisse.polito .polito.it/ma .it/matdid/1i tdid/1ing ng aer B4600 TO 0/).
a
1
SEZI SEZION ONII SOTT SOTTIL ILII (δ ) yG xG
A = δ,
G
I x
G
xG
=
1 3 δ sin2 α, 12
G
1 3 δ cos2 α, 12
=
yG
yG = 0; I x
G
Sx = (− cos ϑ2 + cos ϑ1 )δR 2 ,
A = (ϑ2 − ϑ1 )δ R , y
I y
xG = 0,
1 3 δ sin α cos α. 12
=
yG
S y = (sin ϑ2 − sin ϑ1 )δR2 ;
1 (ϑ2 − ϑ1 − sin ϑ2 cos ϑ2 + sin ϑ1 cos ϑ1 )δR 3 , 2 1 (ϑ2 − ϑ1 + sin ϑ2 cos ϑ2 − sin ϑ1 cos ϑ1 )δR 3 , I yy yy = 2 1 (− cos(2ϑ2 ) + cos(2 ϑ1 ))δR3 . I xy xy = 4
I xx xx = R
2
1
1.1
x
Casi Casi pa part rtic icol olar arii y, yG R
A = 2πδR ,
x, xG
I x
G
G
xG
3 = I xx xx = πδ R ,
I y
G
yG
A = πδR ,
y, yG xG
I x
G R
G
xG
=
π
2
−
4
δR3 ,
A =
yG
y
xG
I x
G
G
R
xG
=
π
4
−
2 π
3
δR ,
x
I xx xx =
2
I y
G
π
4
xG =
δR , yG
δR3 ,
xG
I x
G
xG
=
1 sin2 α α + sin(2α) − 2 2 α
x
I xx xx =
=
π
4
I yy yy =
A = 2αδ R ,
y, yG
G R
π
I y
G
π
π
4
π
δR ,
1 α + sin(2α) δR3 , 2
I y
I yy yy =
G
π
δR ,
=
α−
yG
= I xy xy = 0 .
R; I x
G
yG
= 0;
I xy xy = 0.
2 π
R;
I x
G
I xy xy =
yG = yG
G
δR3 ,
3
δR3 ,
xG = 0, 3
2
I x
2
yG =
R,
2
−
π
=
δR3 ,
2
2
π
yG =
yG
I yy yy =
2
yG = 0;
3 = I yy yy = πδ R ,
xG = 0,
π π 3 I xx δR , xx =
x
xG = 0,
α−
=−
π
−
2
δR3 ;
1 3 δR . 2
sin α α
yG
2 1
R;
1 sin(2α) δR3 , 2
1 sin(2α) δR3 , 2
I x
I xy xy = 0.
G
yG
= 0;
2
SEZIONI COMPATTE yG
y
A = bh, xG
h
I x
G
b
yG
y
=
xG
G
1 bh, 2
xG
G
G
x
b
2 x
y, yG
G
G
xG =
G
G
G
G
G
2
y
2
1
b2
a
A = πab,
x, xG
2b
G
1 1 b, yG = h; 3 3 1 3 1 3 1 I x x = bh , I y y = b h, I x y = − b2 h2 ; 36 36 72 1 3 1 3 1 2 2 I xx = bh , I yy = b h, I xy = b h . 12 12 24 A =
h
1 h; 2
yG =
1 3 1 3 bh , I y y = b h, I x y = 0; 12 12 1 1 1 I xx = bh3 , I yy = b3 h, I xy = b2 h2 . 3 3 4
G
x
1 b, 2
xG =
G
I x
G
xG
π
= I xx =
4
ab3 ,
I y
G
xG = 0, yG
= I yy
yG = 0; π = a3 b, I x
G
4
yG
= I xy = 0 .
2a
2 x
y, yG
2
y
2
1
b2
a
A =
xG
b G
I x
x
G
xG
=
2a
yG
y
b G
π
8
−
I xx =
2 x a
2
y
2
b2
xG
I x
x
G
xG
=
π
16
−
4
9π
3
ab ,
a
I xx =
π
8
I y
ab3 ,
ab3 ,
G
16
yG
I y
G
yG
π
I yy =
8
=
ab3 ,
=
π
16
π
8
a3 b,
4 a, 3π 4 π − 16 9π
I yy =
4 b; 3π
yG =
xG =
ab,
4
π
xG = 0,
ab,
2 8 9π
π
A =
1
π
a3 b,
3
a b,
a3 b,
G
yG
= 0;
I xy = 0.
yG =
I x
4 b; 3π I x
G
yG
=−
4 1 9π
−
8
1 2 2 a b . 8
I xy =
y, yG
A =
xG
b
G b y
2
2 x
a
x
2a
I x
G
4 ab, 3
16 3 ab , 175 4 I xx = ab3 , 7
xG
=
2
R2
R1
1
x
I y
G
yG
I yy =
=
yG =
4 3 a b, 15
4 3 a b, 15
3 b; 5 I x
G
yG
= 0;
I xy = 0 .
1 2 (R − R21 )(ϑ2 − ϑ1 ); 2 2 1 1 S x = (R32 − R31 )(− cos ϑ2 + cos ϑ1 ), S y = (R32 − R31 )(sin ϑ2 − sin ϑ1 ); 3 3 1 I xx = (R42 − R41 )(ϑ2 − ϑ1 − sin ϑ2 cos ϑ2 + sin ϑ1 cos ϑ1 ), 8 1 I yy = (R42 − R41 )(ϑ2 − ϑ1 + sin ϑ2 cos ϑ2 − sin ϑ1 cos ϑ1 ), 8 1 (R42 − R41 )(− cos(2ϑ2 ) + cos(2 ϑ1 )). I xy = 16 A =
y
xG = 0,
a2 b2 ;
2.1
Casi particolari: y, yG R
A = πR 2 , xG = 0, yG = 0; π π = I xx = R4 , I y y = I yy = R4 , I x
x, xG
I x
G
G
xG
G
4
A =
y, yG R
xG
G
I x
G
=
xG
x
π
8
xG
I x
G
xG
=
G
π
16
R2 ,
8 9π π
8
xG = 0, R4 ,
R4 ,
G
4
I y
G
I yy =
π
8
π
=
8
R4 ,
= I xy = 0.
4 R; 3π
yG =
yG
yG
R4 ,
I x
G
yG
= 0;
I xy = 0 .
4 4 R, yG = R; 4 3π 3π 4 4 π − R4 , I y y = R4 , I x y = − 9π 16 9π 1 π 4 π 4 I xx = R , I yy = R , I xy = R4 . 16 16 8 A =
yG
R
2
−
I xx =
y
π
G
π
R2 ,
−
x
G
xG =
G
G
G
4 1 9π
−
8
R4 ;
y, yG R2
R1
A = π (R22 − R21 ),
x, xG
I x
G
G
R2
G
4
π
A =
y, yG R1
π
= I xx =
xG
xG
I x
G
xG
=
x
π
8
2
4 2
(R
(R42 − R41 ),
I y
(R22 − R21 ),
xG = 0,
−
4 1
R )−
I xx =
π
8
xG
G x
I x
G
xG
1 = 4
1 α + sin(2α) R4 , 2
y, yG
G
I x
G
xG x
xG
I y
G
π
8
R4 , I y
I y
G
yG
yG
=
π
8
= I xy = 0.
I x
G
yG
= 0;
I xy = 0.
2 sin α R; 3 α 1 1 = α − sin(2α) R4 , I x 4 2
G
yG
yG =
α−
1 sin(2α) R4 , 2
G
yG
= 0.
I xy = 0.
2 sin α R21 + R1 R2 + R22 ; 3 α R1 + R2
−
4 (R32 − R31 )2 sin2 α , 9 R22 − R21 α
1 1 = (R42 − R41 ) α − sin(2α) , 4 2
1 1 I xx = (R42 − R41 ) α + sin(2α) , 4 2
(R42 − R41 ),
(R42 − R41 ),
1 1 = (R42 − R41 ) α + sin(2α) 4 2
yG
yG =
1 I yy = 4
xG = 0,
G
4 R21 + R1 R2 + R22 ; 3π R1 + R2
yG =
xG = 0,
I x
4
I yy =
1 16 sin2 α α + sin(2α) − 2 9 α
1 I xx = 4
yG
(R42 − R41 ),
A = (R22 − R21 )α,
R
G
8 (R32 − R31 )2 , 9π R22 − R21
A = αR 2 ,
y, yG R
xG = 0, yG = 0; π = I yy = (R42 − R41 ),
I x
G
yG
= 0;
1 1 I yy = (R42 − R41 ) α − sin(2α) , 4 2
I xy = 0.
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