esercizi svolti costruzioni macchine

ESERCIZIO 2_1

=

o

1 0.01745rad

o

α = 1 m

σ = 250 MPa

amm

=

E 210000 MPa

υ = 0,3

=

M ?

t E 210000

= = =

G 80770 MPa

υ

2 *(1+ ) 2 *(1+ 0,3)

1*0,01745 rad

θ −5

= = 1,745*10 mm

u 1000

σ 250

τ = = = 144,34 MPa

amm

amm 3 3

Sezione chiusa

M M l

d(s) ∑

θ = =

∮

t t i

u *2 *2

dt t

4 *G * A 4 *G * A i

θ

*2

4 *G * A *

=

M u

t l

∑ i

t i

− −

l 200 12 100 10

∑ = + =

*2 * 2 52,6

i

t 10 12

i

= − − =

* 2

A (200 12) *(100 10) 16920mm

−5

2

4 *80770 *16920 *1,475*10

= = =

M 30684600Nmm 30690Nm

t 52,6

M 30690000

τ = = = 90,69 MPa

t

max * 2 *16920 *10

2 * A *t

τ 144,34

υ = = = 1,59

amm

τ 90,69

max

Sezione aperta

M

θ = t

u G * J t

θ

=

M *G * J

t u t

1 1

∑ ⎡⎣ ⎤⎦

= = + + + + =

3 3 3 3 3 3

J * l *t * 94 *10 90 *12 188*10 90 *12 94 *10 229013Nmm

t i i

3 3

−5

= =

M 1,745*10 *80770 * 229013 322800Nmm

t M *t 322800 *12

τ = = =

t max 16,9 MPa

max J 229013

t

τ 144,3

υ = = = 8,55

amm

τ 16,9

max

ESERCIZIO 2_2

σ = 235MPa

s =

G 8000009 MPa

α = o

4

= =

L 2m 2000mm

Sezione aperta

α

M

θ = =

t

G * J L

t

α

=

M *G * J

t t

L

1 ⎡⎣ ⎤⎦

= − + − − + − =

3 3 3 4

J * (50 6) *14 (120 7 5) *12 (60 6) *10 120453mm

t 3

4 *0,01745*80000 *120453

= = =

M 336304Nmm 336,4Nm

t 2000

σ 235

τ = = = 135,68MPa

s

s 3 3

M 336400 *14

τ = = =

*t 39,1MPa

t

max max

J 120453

t

Trave a sezione costante

=

L 215mm

υ = 2

=

t ?

media

=

M 336400Nmm

t 1 1

∑

= =

3 3

J * l *t * 215*t

t i i

3 3

σ 235

σ = = = 117,5MPa

s

υ

amm 2

σ 117,5

τ = = = 67,8MPa

amm

amm 3 3

1 ∑ τ

= 3

M * l *t *

t i i amm

3

336400 *3

= =

3 3

t 69,23mm

215*67,8

=

t 4,206mm

ESERCIZIO 2_7

=

P 10000N

= =

b 1m 1000mm

= =

L 2m 2000mm

σ = 240 MPa

s = = = 6

M P * b 10000 *1000 10 *10 Nmm

t = = = = 6

M M P * l 10000 * 2000 20 *10 Nmm

f xx

= =

P T 10000N

y ⎡ ⎤

1 1

= + = + − +

3 2 3 2

J * B * H B * H * y 2 * *12 *100 100 *12 *(100 6)

⎢ ⎥

xx 12 12

⎣ ⎦

⎡ ⎤

1

+2 − − = =

3 4 6 4

* *(200 12 12) *10 *(176 *0 *10) 30321493mm 30 *10 mm

⎢ ⎥

12

⎣ ⎦

È una sezione chiusa

= − − =

* 2

A (100 10) *(200 12) 16920mm

Punto A 6

M 20 *10

σ = = =

( M ) * y *100 66,67 MPa

xx

zz xx 6

J 30 *10

xx 6

M 10 *10

τ = = =

( M ) 24,63MPa

t

t * 2 *16920 *12

2 * A *t

τ =

(T ) 0

σ σ τ

= + = + =

2 2 2 2

3* 66,67 3* 24,63 79,15MPa

id σ 240

υ = = = 3,03

amm

σ

s 79,15

id

Punto B

σ =

( M ) 0

zz xx 6

M 10 *10

τ = = = −29,55MPa

( M ) t

t * 2 *16920 *10

2 * A *t

T * S

τ =

(T ) xx

J * b

xx

= − = 3

S 100 *100 *50 80 *88* 44 190240mm

xx

= =

b 2 *10 20mm

10000 *190240

τ = = −3,17

(T ) MPa

6

30 *10 * 20

τ τ τ

= + = −3,17 + = −32,72

(T ) ( M ) (−29,55) MPa

t

σ τ

= = =

3 * 3 *32,72 56,67 MPa

id σ 240

υ = = = 4,24

amm

σ

s 56,67

id

ESERCIZIO 2_8

= =

L 3m 3000mm

= 6

M 60 *10 Nmm

t

σ = 355MPa

s =

E 206000 MPa

=

G 80000 MPa

Sezione A (sezione chiusa)

= − − =

* 2

A (160 14) *(140 16) 18104mm

6

M 60 *10

τ = = = (per avere il massimo uso lo spessore minimo)

( M ) 118,36 MPa

t

max t * 2 *18104 *14

2 * A *t

σ τ

= = =

* 3 118,36 * 3 205MPa

max

σ 355

υ = = = 1,73

s

σ

s 205

M l

∑

θ = t i

u *2 t

4 *G * A i

− −

l 140 16 160 14

∑ = + =

*2 * 2 35,96

i

t 14 16

i 6

60 *10 rad

θ −5

= =

*35,96 2,057 *10 mm

u 2

4 *80000 *18104

180 180 o

θ θ −5 −3

= = =

o * 2,057 *10 * 1,18*10 mm

π π

u

θ θ −3

= = =

o o

* L 1,18*10 *3000 3,54

Sezione B (sezione aperta)

υ υ

= = 1,73

SB SA

σ

υ = s

σ

SB σ 355

σ = = = 205,20 MPa

s

υ 1,73

SB

σ 205,20

τ = = = 118,47 MPa

max 3 3

M

τ = *t

t

J t

τ * J

=

M t

t t

1 1 1

∑

= = − + − =

3 3 3 4

J * l *t *(140 8) *14 * 2 *(160 14) *16 440810mm

t i i

3 3 3

118,47 * 440810

= = =

M 3263992Nmm 3264000Nmm

t 16

M 3264000 rad

θ −5

= = = 9,25*10

t mm

u G * J 80000 * 440810

t

180 180 o

θ θ −5 −3

= = =

o * 9,25*10 * 5,3*10 mm

π π

u

θ θ −3

= = =

o o

* L 5,3*10 *3000 15,9

1 1

∑ ⎡⎣ ⎤⎦

= = − + − =

3 3 3 4

J * l *t * (140 2,5) *3,2 * 2 (80 3,2) *5 6203,73mm

t i i

3 3

M 57540

τ = = =

( M ) 0,56 MPa

t

t * 2 *10368*5

2 * A *t

ESERCIZIO 2_9

=

L 900mm

=

G 80000 MPa

σ = 190 MPa

amm 180

θ = = =

o

2 2 * 0,035rad

π

L 2

θ θ θ

= +

* * * L

tot aperta chiusa

3 3

M

θ = t

aperta G * J t

M l

∑

θ = *

t i

chiusa *2 t

4 *G * A i

= =

M F *300 300F

t

= − − =

* 2

A (140 5) *(80 3,2) 10368mm

− −

l 80 3,2 140 5

∑ = + =

*2 * 2 115,095

i

t 5 3,2

i M M

900 2

θ = + =

* *115,095* *900

t t

tot 2

80000 *6203,73 3 3

4 *80000 *10368

−7 −9 −7

= + =

6,045*10 * M 2,01*10 * M 6,0651*10 M

t t t

θ −7

= =

0,035 6,0651*10 M

tot t

0,035

= =

M 57707Nmm

−7

t 6,0651*10

M 57707

= = =

F 192N

t

300 300

Sezione chiusa

=

F 191,8N

= = =

M F *300 191,8*300 57540Nmm

t = = =

M F * l 191,8*900 172620Nmm

xx ⎛ ⎞

1 1 1

= + = + − + =

3 2 3 3 4

J * B * H B * H * y *80 *140 80 *140 *0 *73,6 *130 73,6 *130 *0 4818400mm

⎜ ⎟

⎝ ⎠

xx 12 12 12

= − − =

* 2

A (140 5) *(80 3,2) 10368mm

Punto A M 172620

σ = = =

( M ) * y *70 2,51MPa

xx

zz xx J 4818400

xx

M 57540

τ = = =

( M ) 0,56 MPa

t

t * 2 *10368*5

2 * A *t

T * S

τ =

(T ) xx

J * b

xx

= − = 3

S 73,6 *5*(70 2,5) 24840mm

xx

= =

b 2 *5 10mm

191,8* 24840

τ = =

(T ) 0,099 MPa

4818400 *10

τ τ τ

= + = + =

(T ) ( M ) 0,099 0,55 0,649 MPa

t

σ σ τ

= + = + =

2 2 2 2

3* 2,51 3*0,649 2,75MPa

id

Punto B M 172620

σ = = =

( M ) * y *65 2,33MPa

xx

zz xx J 4818400

xx

M 57540

τ = = =

( M ) 0,87 MPa

t

t * 2 *10368*3,2

2 * A *t

T * S

τ =

(T ) xx

J * b

xx

= − = 3

S 80 *5*(70 2,5) 27000mm

xx

= =

b 2 *3,2 6,4mm

191,8* 27000

τ = =

(T ) 0,168MPa

4818400 *6,4

τ τ τ

= + = + =

(T ) ( M ) 0,168 0,87 1,038MPa

t

σ σ τ

= + = + =

2 2 2 2

3* 2,51 3*1,038 2,94 MPa

id

Punto C

σ =

( M ) 0

zz xx M 57540

τ = = = −0,87

( M ) MPa

t

t * 2 *10368*3,2

2 * A *t

T * S

τ =

(T ) xx

J * b

xx

= − = 3

S 80 *70 *35 73,6 *65*32,5 40520mm

xx

= =

b 2 *3,2 6,4mm

191,8* 40520

τ = = −0,25MPa

(T ) 4818400 *6,4

τ τ τ

= + = −0,25 − = −1,12

(T ) ( M ) 0,87 MPa

t

σ τ

= = =

3 * 3 *1,12 1,94 MPa

id

Sezione aperta

= + = 3

S 2 *140 *3,2 *70 73,6 *5* 2,5 63640mm

xx = = + =

S A* y (2 *140 *3,2 73,6 *5) * y 1264 * y

xx tot tot tot

=

63640 1264 * y

tot

63640

= =

y 50,35mm

tot 1264 ⎛ ⎞

1 1 1

= + = + − + =

3 2 3 3 4

J * B * H B * H * y *80 *140 80 *140 *0 *73,6 *130 73,6 *130 *0 4818400mm

⎜ ⎟

⎝ ⎠

xx 12 12 12

⎡ ⎤

1 1

= + = + − +

3 2 3 2

J * B * H B * H * y *3,2 *140 3,2 *140 *(70 50,35) * 2

⎢ ⎥

xx 12 12

⎣ ⎦

⎡ ⎤

1

+ + − = + = =

3 2 6 4

*73,6 *5 73,6 *5*(50,35 2,5) 1809432,427 843347,7467 2652780,174 2,66 *10 mm

⎢ ⎥

12

⎣ ⎦

1 1 1

∑

= = − + + =

3 3 3 4

J * l *t * 2 *(140 2,5) *3,2 *(73,6 3,2) *5 6203,74mm

t i i

3 3 3

2 2

= = =

M F * * l 191,8* *900 115080Nmm

xx 3 3

= = =

M F *300 191,8*300 57540Nmm

t

Punto A M 115080

σ = = − =

( M ) * y *(140 50,35) 3,88MPa

xx

zz xx 6

J 2,66 *10

xx

M 57540

τ = = = −29,68MPa

( M ) *t *3,2

t

t J 6203,74

t

τ =

(T ) 0

σ σ τ

= + = + =

2 2 2 2

3* 3,88 3*(−29,68) 51,55MPa

id σ 190

υ = = = 3,68

adm

σ

s 51,55

id

Punto B M 115080

σ = = − =

( M ) * y *(70 50,35) 0,85MPa

xx

zz xx 6

J 2,66 *10

xx

M 57540

τ = = = −29,68MPa

( M ) *t *3,2

t

t J 6203,74

t

T * S

τ =

(T ) xx

J * b

xx

= + = 3

S 70 *3,2 *(35 19,65) * 2 24483,2mm

xx

= =

b 2 *3,2 6,4mm

191,8* 24483,2

τ = = −0,28MPa

(T ) 6

2,66 *10 *6,4

τ τ τ

= + = −0,28 − = −29,69

(T ) ( M ) 29,68 MPa

t

σ σ τ

= + = + =

2 2 2 2

3* 0,85 3*(−29,69) 51,43MPa

id σ 190

υ = = = 3,70

adm

σ

s 51,43

id

Punto C M 115080

σ = = = −2,18MPa

( M ) * y *50,35

xx

zz xx 6

J 2,66 *10

xx

τ =

(T ) 0 M 57540

τ = = =

( M ) *t *5 46,37 MPa

t

t J 6203,74

t

σ σ τ

= + = + =

2 2 2 2

3* 2,18 3* 46,37 80,34 MPa

id σ 190

υ = = = 2,36

adm

σ

s 80,34

id

ESERCIZIO 2_10

= =

N P 44000N

z

= =

T P 44000N

y y

= =

T P 11000N

x x

= = = 6

M P * b 27500 * 400 11*10 Nmm

xx y

= = = 6

M P * b 27500 * 400 11*10 Nmm

xx y B

= + − = + − =

M P *(c ) P * a 44000 *(200 40) 11000 *700 2860000Nmm

2

yy z x

B

= + = + = 5

M P *(c ) 27500 *(200 40) 66 *10 Nmm

2

t y

1 1 1

= + = + − + =

3 2 3 3 4

J * B * H B * H * y *80 * 200 80 * 200 *0 68*184 68*184 *0 18032810,67mm

xx 12 12 12

1 1 1

= + = + − + =

3 2 3 3 4

J * H * B B * H * x * 200 *80 80 * 200 *0 184 *68 68*184 *0 3712042,667mm

yy 12 12 12

= − − = =

* 2

A (200 8) *(80 6) 192 *74 14208mm

= − = 2

A 200 *80 68 *184 3488mm

Punto M N 44000

σ = = = −12,61MPa

(N ) A 3488 6

M 11*10 184

σ = = =

( M ) * y * 56,12 MPa

xx

xx J 18032810,67 2

xx

M 2860000

σ = = =

yy

( M ) * x * 40 30,82 MPa

yy J 3712042,6667

yy

σ σ σ σ

= + + = −12,61+ + =

(N ) ( M ) ( M ) 56,12 30,82 74,33MPa

xx yy 5

M 66 *10

τ = = =

( M ) 38,71MPa

t

t * 2 *14208*6

2 * A *t

T * S

τ = y xx

(T )

y J * b

xx ⎛ ⎞

8

= − = 3

S 80 *8* 100 61440mm

⎜ ⎟

⎝ ⎠

xx 2

= =

b 2 *6 12mm

27500 *61440

τ = = −7,81MPa

(T )

y 18032810,67 *12

T * S

τ = x yy

(T )

x J * b

yy

= − = 3

S 184 *6 *(40 3) 40848mm

yy

= =

b 2 *6 12mm

11000 * 40848

τ = = −10,09

(T ) MPa

x 3712042,667 *12

τ τ τ τ

= + + = − =

( M ) (T ) (T ) 38,71− 10,09 7,81 20,81MPa

t x y

σ σ τ

= + = + =

2 2 2 2

3* 74,33 3* 20,81 82,61MPa

id σ 355

υ = = = 4,30

s

σ

s 82,61

id

Punto N N 44000

σ = = = −12,61MPa

(N ) A 3488 6

M 11*10 184

σ = = = −56,12

( M ) * y * MPa

xx

xx J 18032810,67 2

xx

M 2860000

σ = = = −30,82

yy

( M ) * x * 40 MPa

yy J 3712042,6667

yy

σ σ σ σ

= + + = −12,61− − = −99,55MPa

(N ) ( M ) ( M ) 56,12 30,82

xx yy 5

M 66 *10

τ = = = −38,71MPa

( M ) t

t * 2 *14208*6

2 * A *t

11000 * 40848

τ = = −10,09

(T ) MPa

x 3712042,667 *12

27500 *61440

τ = = −7,81MPa

(T )

y 18032810,67 *12

τ τ τ τ

= + + = −38,71− − = −56,61MPa

( M ) (T ) (T ) 10,09 7,81

t x y

σ σ τ

= + = + =

2 2 2 2

3* (−99,55) 3*(−56,61) 139,73MPa

id σ 355

υ = = = 2,54

s

σ

s 139,73

id 5

M l 66 *10

∑ rad o

θ −6 −4

= = = =

8,3487 *10 4,784 *10

t i mm mm

u *2 2

t

4 *G * A 4 *80770 *14208 *82,5

i

E 210000

= = =

G 80770 MPa

υ

2 *(1+ ) 2 *(1+ 0,3)

⎛ ⎞

l 192 74

∑ = + =

* 2 82,5

i ⎜ ⎟

⎝ ⎠

t 6 8

i

θ θ −4

= = = o

* L 4,784 *10 * 400 0,191

tot u

ESERCIZIO 3_1

α = 0

30

σ = 180 MPa

p,o

2 N

γ = 10000 3

m

υ = 3

s σ 180

p,o

σ = = = 60 MPa

2

υ

t 3

s

= ∞

R

m 8000

= = = =

R R R 4000mm

A B t 2

R

=

R C

α

tC cos α

= = =

o

R H *tg 4500 *tg30 2598mm

c C α

W *tg 2598

= = =

R 3000mm

c α

tC o

cos cos(30 )

Punto A

= ∞

R

m σ σ

P = +

t m

t R R

t m

P * R

σ = t

t t

P * R

σ = tA

tA t A

P * R H * R 2500 * 4000

γ −9

= = = =

t * 10000 *10 * 1,6mm

tA A tA

σ σ

A 60

tA tA

Punto B (sotto i vincoli)

γ σ π

+ =

(V V ) * 2 * * R *t

1 2 m B

π 2

* R * h

=

V B cono

1 3

R 4000

= = =

h 6928mm

A

α

cono o

tg tg(30 )

π

= 2

V * R * h

2 B cilindro

=

h 5000mm

cilindro

⎛ ⎞

π 2

* R * h

γ π σ π

+ =

2

* R * h * 2 * * R *t

B cono

⎜ ⎟

B cilindro m B

3

⎝ ⎠

R * h

γ γ σ

+ =

* * R * h * 2 *t

B cono B cilindro m

3

⎛ ⎞

R * h 1

σ γ

= + R * h *

B cono

⎜ ⎟

⎝ ⎠

m B cilindro

3 2 *t

P * R h * R

σ γ

= = *

t cilindro B

t t t

σ σ σ σ σ σ

= + − =

2 2 *

IDVM m t m t amm

2 ⎡ ⎤

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

γ γ γ

2 2 2 2 2

* R h * R * h h * h * R

1

σ γ −9

= + + + − +

h * R * R * h * * *10

B cono B cil cono cil B

⎢ ⎥

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

amm cil B B cil

2 2

3 3 2 *t t

4 *t t ⎣ ⎦

⎛ ⎞ ⎛ ⎞

2

R h h * h h h

1

γ −9

= + + + − +

2 2

* * * 2 * h h h * *10

B cono cono cil cono cil

⎜ ⎟ ⎜ ⎟

⎝ ⎠

cil cil cil

t 4 9 3 3 2

⎝ ⎠

⎛ ⎞ ⎛ ⎞

2

R h 2 * h * h h h

1

γ −9

= + + + − +

2 2

t * * * h h * h *10

B cono cono cil cil cono

⎜ ⎟ ⎜ ⎟

σ ⎝ ⎠

cil cil cil

4 9 3 2 3

⎝ ⎠

amm ⎛ ⎞ ⎛ ⎞

2

10000 * 4000 1 6928 2 *6928*5000 5000 6928 −9

= + + + − + =

2 2

t * * 5000 5000 * 5000 *10 2,98mm

⎜ ⎟

⎜ ⎟ ⎝ ⎠

⎝ ⎠

60 4 9 3 2 3

ESERCIZIO 3_2

N

γ = 10000 3

m

σ = 200 MPa

p,o

2

υ = 3

s =

F 1500KN

σ 200

p,o

σ = = = 66,7 MPa

2

υ

amm 3

s

⎡ ⎤ ⎡ ⎤

D 5000