esercizi svolti costruzioni macchine

Esercizio 2_1

0.01745 rado 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 ld(s) ∑θ = =∮ t t iu *2 *2 dt t4 *G * A 4 *G * A iθ*24 *G * A *=M ut l∑ it i− −l 200 12 100 10∑ = + =*2 * 2 52,6it 10 12i= − − =* 2A (200 12) *(100 10) 16920 mm −524 *80770 *16920 *1,475*10= = =M 30684600 Nmm 30690 Nm t 52,6 M 30690000 τ = = = 90,69 MPa tmax * 2 *16920 *102 * A *tτ 144,34 υ = = = 1,59 amm τ 90,69 max

Sezione aperta

M θ = tu G * J t θ=M *G * J t t 1 1∑ ⎡⎣ ⎤⎦= = + + + + =3 3 3 3 3 3 J * l *t * 94 *10 90 *12 188*10 90 *12 94 *10 229013 Nmm t i i3 3−5= = M 1,745*10 *80770 * 229013 322800 Nmmt 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 = 235 MPa s = G 8000009 MPa α = o4= =L 2m 2000 mm

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 120453 mm t 34 *0,01745*80000 *120453= = =M 336304 Nmm 336,4 Nm t 2000 σ 235 τ = = = 135,68 MPa ss 3 3

M 336400 *14 τ = = =*t 39,1 MPa tmax max J 120453 t

Trave a sezione costante

L 215 mm υ = 2=t ?media=M 336400 Nmm t 1 1∑= =3 3 J * l *t * 215*tt i i3 3 σ 235 σ = = = 117,5 MPa s υ amm 2 σ 117,5 τ = = = 67,8 MPa amm amm 3 3

1 ∑ τ= 3M * l *t *t i i amm 3336400 *3= =3 3 t 69,23 mm 215*67,8=t 4,206 mm

Esercizio 2_7

P 10000 N= =b 1m 1000 mm= =L 2m 2000 mm σ = 240 MPa s = = = 6M P * b 10000 *1000 10 *10 Nmm t = = = = 6M M P * l 10000 * 2000 20 *10 Nmm f xx= =P T 10000 Ny ⎡ ⎤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) 30321493 mm 30 *10 mm⎢ ⎥12⎣ ⎦È una sezione chiusa= − − =* 2A (100 10) *(200 12) 16920 mm

Punto A 6M 20 *10 σ = = =( M ) * y *100 66,67 MPa xx zz xx 6J 30 *10 xx 6M 10 *10 τ = = =( M ) 24,63 MPa tt * 2 *16920 *122 * A *tτ =(T ) 0 σ σ τ= + = + =2 2 2 23* 66,67 3* 24,63 79,15 MPa id σ 240 υ = = = 3,03 amm σ s 79,15 id

Punto B σ =( M ) 0 zz xx 6M 10 *10 τ = = = −29,55 MPa( M ) tt * 2 *16920 *102 * A *t T * S τ =(T ) xx J * b xx= − = 3 S 100 *100 *50 80 *88* 44 190240 mm xx= =b 2 *10 20 mm 10000 *190240 τ = = −3,17(T ) MPa 630 *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 3000 mm= 6M 60 *10 Nmm t σ = 355 MPa s =E 206000 MPa=G 80000 MPa

Sezione A (sezione chiusa)

= − − =* 2A (160 14) *(140 16) 18104 mm 6M 60 *10 τ = = = (per avere il massimo uso lo spessore minimo)( M ) 118,36 MPa t max t * 2 *18104 *142 * A *t σ τ= = =* 3 118,36 * 3 205 MPa max σ 355 υ = = = 1,73 s σ s 205

M l∑θ = t iu *2 t4 *G * A i− −l 140 16 160 14∑ = + =*2 * 2 35,96it 14 16i 660 *10 radθ −5= =*35,96 2,057 *10 mm u 24 *80000 *18104 180 180 oθ 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 3M 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 440810 mm t i i3 3 3118,47 * 440810= = =M 3263992 Nmm 3264000 Nm mt 16M 3264000 rad θ −5= = = 9,25 *10 t mm u G * J 80000 * 440810 t 180 180 oθ θ −5 −3= = =o * 9,25*10 * 5,3*10 mm π π u θ θ −3= = =o o* L 5,3*10 *3000 15,91

1∑ ⎡⎣ ⎤⎦= = − + − =3 3 3 4 J * l *t * (140 2,5) *3,2 * 2 (80 3,2) *5 6203,73 mm t i i3 3 M 57540 τ = = =( M ) 0,56 MPa tt * 2 *10368*52 * A *t

Esercizio 2_9

=L 900 mm=G 80000 MPa σ = 190 MPa amm 180 θ = = =o2 2 * 0,035 radπ L 2 θ θ θ= +* * * L tot aperta chiusa 3 3M θ = t aperta G * J t M l∑θ = *t i chiusa *2 t4 *G * A i= =M F *300 300F t= − − =* 2A (140 5) *(80 3,2) 10368 mm − −l 80 3,2 140 5∑ = + =*2 * 2 115,095it 5 3,2i

M M900 2 θ = + =* *115,095* *900t t tot 280000 *6203,73 3 34 *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 57707 Nmm −7 t 6,0651*10 M 57707= = =F 192 N t 300 300

Sezione chiusa

=F 191,8N= = =M F *300 191,8*300 57540 Nm mt = = =M F * l 191,8*900 172620 Nmm 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 4818400 mm⎜ ⎟⎝ ⎠xx 12 12 12= − − =* 2A (140 5) *(80 3,2) 10368 mm

Punto A M 172620 σ = = =( M ) * y *70 2,51 MPa xx zz xx J 4818400 xx M 57540 τ = = =( M ) 0,56 MPa tt * 2 *10368*52 * A *t T * S τ =(T ) xx J * b xx= − = 3 S 73,6 *5*(70 2,5) 24840 mm xx= =b 2 *5 10 mm 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 23* 2,51 3*0,649 2,75 MPa id

Punto B M 172620 σ = = =( M ) * y *65 2,33 MPa xx zz xx J 4818400 xx M 57540 τ = = =( M ) 0,87 MPa tt * 2 *10368*3,22 * A *t T * S τ =(T ) xx J * b xx= − = 3 S 80 *5*(70 2,5) 27000 mm xx= =b 2 *3,2 6,4 mm 191,8* 27000 τ = =(T ) 0,168 MPa 4818400 *6,4 τ τ τ= + = +(T ) ( M ) 0,168 0,87 1,038 MPa t σ σ τ= + = + =2 2 2 23* 2,51 3*1,038 2,94 MPa id

Punto C σ =( M ) 0 zz xx M 57540 τ = = = −0,87( M ) MPa tt * 2 *10368*3,22 * A *t T * S τ =(T ) xx J * b xx= − = 3 S 80 *70 *35 73,6 *65*32,5 40520 mm xx= =b 2 *3,2 6,4 mm 191,8* 40520 τ = = −0,25 MPa (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 63640 mm xx = = + =S A* y (2 *140 *3,2 73,6 *5) * y 1264 * y xx tot tot tot=63640 1264 * y tot 63640= =y 50,35 mm 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 4818400 mm⎜ ⎟⎝ ⎠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⎣ ⎦⎡ + + − = + = =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 6203,74 mm t i i3 3 32 2= = =M F * * l 191,8* *900 115080 Nmm xx 3 3= = =M F *300 191,8*300 57540 Nm mt

Punto A M 115080 σ = = − =( M ) * y *(140 50,35) 3,88 MPa xx zz xx 6J 2,66 *10 xx M 57540 τ = = = −29,68 MPa( M ) *t *3,2 tt J 6203,74 tτ =(T ) 0 σ σ τ= + = + =2 2 2 23* 3,88 3*(−29,68) 51,55 MPa id σ 190 υ = = = 3,68 adm σ s 51,55 id

Punto B M 115080 σ = = − =( M ) * y *(70 50,35) 0,85 MPa xx zz xx 6J 2,66 *10 xx M 57540 τ = = = −29,68 MPa( M ) *t *3,2 tt J 6203,74 t T * S τ =(T ) xx J * b xx= + = 3 S 70 *3,2 *(35 19,65) * 2 24483,2 mm xx= =b 2 *3,2 6,4 mm 191,8* 24483,2 τ = = −0,28 MPa (T ) 62,66 *10 *6,4 τ τ τ= + = −0,28 − = −29,69(T ) ( M ) 29,68 MPa t σ σ τ= + = + =2 2 2 23* 0,85 3*(−29,69) 51,43 MPa id σ 190 υ = = = 3,70 adm σ s 51,43 id

Punto C M 115080 σ = = = −2,18 MPa( M ) * y *50,35xx zz xx 6J 2,66 *10 xx τ =(T ) 0 M 57540 τ = = =( M ) *t *5 46,37 MPa tt J 6203,74 t σ σ τ= + = + =2 2 2 23* 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 44000 N z= =T P 44000 N y y= =T P 11000 N x x= = = 6M P * b 27500 * 400 11*10 Nmm xx y= = = 6M P * b 27500 * 400 11*10 Nmm xx y B= + − = + − =M P *(c ) P * a 44000 *(200 40) 11000 *700 2860000 Nmm 2 yy z x B= + = + = 5M P *(c ) 27500 *(200 40) 66 *10 Nmm 2 t y1 1 1= + = + − + =3 2 3 3 4 J * B * H B * H * y *80 * 200 80 * 200 *0 68*184 68*184 *0 18032810,67 mm xx 12 12 121 1 1= + = + − + =3 2 3 3 4 J * H * B B * H * x * 200 *80 80 * 200 *0 184 *68 68*184 *0 3712042,667 mm yy 12 12 12= − − = =* 2A (200 8) *(80 6) 192 *74 14208 mm= − = 2A 200 *80 68 *184 3488 mm

Punto M N 44000 σ = = = −12,61 MPa(N ) A 3488 6M 11*10 184 σ = = =( M ) * y * 56,12 MPa xxxx 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 74,33 MPa xx yy 5M 66 *10 τ = = =( M ) 38,71 MPa tt * 2 *14208*62 * A *t T * S τ = y xx(T )y J * b xx

⎛ ⎞8= − = 3 S 80 *8* 100 61440 mm⎜ ⎟⎝ ⎠xx 2= =b 2 *6 12 mm 27500 *61440 τ = = −7,81 MPa (T )y 18032810,67 *12 T * S τ = x yy(T )x J * b yy= − = 3 S 184 *6 *(40 3) 40848 mm yy= =b 2 *6 12 mm 11000 * 40848 τ = = −10,09(T ) MPa x 3712042,667 *12 τ τ τ τ= + + = − =( M ) (T ) (T ) 38,71− 10,09 7,81 20,81 MPa t x y

σ σ τ= + = + =2 2 2 23* 74,33 3* 20,81 82,61 MPa id σ 355 υ = = = 4,30 s σ s 82,61 id

Punto N N 44000 σ = = = −12,61 MPa(N ) A 3488 6M 11*10 184 σ = = = −56,12( M ) * y * MPa xxxx 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− − = −99,55 MPa(N ) ( M ) ( M ) 56,12 30,82 xx yy 5M 66 *10 τ = = = −38,71 MPa( M ) tt * 2 *14208*62 * A *t 11000 * 40848 τ = = −10,09(T ) MPa x 3712042,667 *1227500 *61440 τ = = −7,81 MPa (T )y 18032810,67 *12

τ τ τ τ= + + = −38,71− − = −56,61 MPa( M ) (T ) (T ) 10,09 7,81 t x y σ σ τ= + = + =2 2 2 23* (−99,55) 3*(−56,61) 139,73 MPa id σ 355 υ = = = 2,54 s σ s 139,73 id

5M l 66 *10 ∑ rad oθ −6 −4= = = =8,3487 *10 4,784 *10 t i mm mm u *2 2 t4 *G * A 4 *80770 *14208 *82,5iE 210000= = =G 80770 MPa υ2 *(1+ ) 2 *(1+ 0,3)⎛ ⎞l 192 74∑ = + =* 2 82,5i ⎜ ⎟⎝ ⎠t 6 8 iθ θ −4= = = o* L 4,784 *10 * 400 0,191 tot u

Esercizio 3_1

α = 030 σ = 180 MPa p,o 2 N γ = 10000 3m υ = 3s σ 180 p,o σ = = = 60 MPa 2 υ t 3s= ∞ Rm 8000= = = =R R R 4000 mm A B t 2 R=R Cα tC cos α= = =oR H *tg 4500 *tg30 2598 mm c C α W *tg 2598= = =R 3000 mm c α tC ocos cos(30 )

Punto A= ∞ Rm σ σ P = +t mt R Rt mP * Rσ = tt tP * Rσ = tAtA t AP * R H * R 2500 * 4000γ −9= = = =t * 10000 *10 * 1,6 mm tA A tAσ σ A 60 tA tA

Punto B (sotto i vincoli)γ σ π+ =(V V ) * 2 * * R *t1 2 m Bπ 2* R * h=V B cono1 3R 4000= = =h 6928 mm A α cono otg tg(30 ) π= 2V * R * h2 B cilindro=h 5000 mm cilindro⎛ ⎞π 2* R * hγ π σ π+ =2* R * h * 2 * * R *t B cono⎜ ⎟B cilindro m B3⎝ ⎠R * hγ γ σ+ =* * R * h * 2 *t B cono B cilindro m3⎛ ⎞R * h 1σ γ= + R * h *B cono⎜ ⎟⎝ ⎠m B cilindro3 2 *tP * R h * Rσ γ= = *t cilindro B t t tσ σ σ σ σ σ= + − =2 2 *IDVM m t m t amm2 ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞γ γ γ2 2 2 2 2* R h * R * h h * h * R1σ γ −9= + + + − +h * R * R * h * * *10B cono B cil cono cil B⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠amm cil B B cil2 23 3 2 *t t4 *t t ⎣ ⎦⎛ ⎞ ⎛ ⎞2R h h * h h h1γ −9= + + + − +2 2* * * 2 * h h h * *10B cono cono cil cono cil⎜ ⎟ ⎜ ⎟⎝ ⎠cil cil cilt 4 9 3 3 2⎝ ⎠⎛ ⎞ ⎛ ⎞2R h 2 * h * h h h1γ −9= + + + − +2 2t * * * h h * h *10B cono cono cil cil cono⎜ ⎟ ⎜ ⎟σ ⎝ ⎠cil cil cil4 9 3 2 3⎝ ⎠amm ⎛ ⎞ ⎛ ⎞210000 * 4000 1 6928 2 *6928*5000 5000 6928 −9= + + + − + =2 2t * * 5000 5000 * 5000 *10 2,98 mm⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠60 4 9 3 2 3

Esercizio 3_2

N γ = 10000 3m σ = 200 MPa p,o 2 υ = 3s =F 1500 KN σ 200 p,o σ = = = 66,7 MPa 2 υ amm 3s ⎡ ⎤ ⎡ ⎤D 5000