Ing Civ.dlsf.001 AZIONI INTERNE ... - intranet dica:...

Ing Civ.dlsf.001REAZIONI 772937 D’Alessio Francesca

@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19

F

3/20F

3/20F1/2Fb

A

B

3/20F3/2Fb

3/20F33/20Fb

B C3F

3/20F33/20Fb

4F

3/20F37/20Fb

C

D4F

17/20F37/20Fb

4F

17/20FFb

DE

Ing Civ.dlsf.001AZIONI INTERNE 772937 D’Alessio Francesca


-3/20 -3/20

0

-3/20 -3/20

4

F

1 0

3/20

-3 -4

17/2

0

F

0 1/2

3/2

33/2

0

33/20-37/20

-37/

20-1

Fb

Ing

Civ

.dls

f.001

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 772

937

D’A

less

io F

ranc

esca

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2 0 0-7

/2-7/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dls

f.001

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 772

937

D’A

less

io F

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dolfo

Zav

elan

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si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

CB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.dlsf.001PROCEDIMENTO E RISULTATI 772937 D’Alessio Francesca


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o

b(3 x/b +1/2 x2/b2 ) Fb 1/EJ dx = [3/2 x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o

b(7/2 -4 x/b +1/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -2 x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoED = ∫

o

b( x/b +5/2 x2/b2 ) Fb 1/EJ dx = [1/2 x2/b +5/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

A = 612. mm2

Ju = 225968. mm4

Jv = 40716. mm4

yg = 32.65 mmN = -993. NTy = 3310. NMx = 1464680. Nmmxm = 12. mmum = -9. mmvm = -32.65 mmσm = N/A-Mv/Ju = 210. N/mm2

xc = 21. mmyc = 8. mmvc = -24.65 mmσc = N/A-Mv/Ju = 158.1 N/mm2

τc = 8.568 N/mm2

σo = √σ2+3τ2 = 158.8 N/mm2

S* = 3510. mm3mm 0 12 18 24 30 42x

0

6

48

54

y

8σc,τc

σm

u

v

Ing Civ.dlsf.001


Ing Civ.dlsf.001


Ing Civ.dlsf.001


Ing Civ.brti.002REAZIONI 797733 Beretta Igor


F

3/20F7/20Fb

3/20F3/20Fb

A

B

17/20F3/20Fb

17/20FFb

B C

F

3/20F

3/20F1/2Fb

D

E

3/20F1/2Fb

3/20F7/20Fb

EA

Ing Civ.brti.002AZIONI INTERNE 797733 Beretta Igor


3/203/20

0

3/203/20

0

F

-10

-17/

20

-10

-3/2

0

F

7/20-3/20

-3/2

0-1

0-1/21/

27/

20

Fb

Ing

Civ

.brt

i.002

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 797

733

Ber

etta

Igor

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/2 -1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brt

i.002

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 797

733

Ber

etta

Igor

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.brti.002PROCEDIMENTO E RISULTATI 797733 Beretta Igor


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoCB = ∫

o

b( x/b -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x2/b -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoEA = ∫

o

b(-1/2 x/b +1/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

A = 828. mm2

Ju = 250978. mm4

Jv = 77652. mm4

yg = 35.87 mmN = 966. NTy = -3220. NMx = -1545600. Nmmxm = 12. mmum = -9. mmvm = -35.87 mmσm = N/A-Mv/Ju = -219.7 N/mm2

xc = 21. mmyc = 10. mmvc = -25.87 mmσc = N/A-Mv/Ju = -158.1 N/mm2

τc = 9.021 N/mm2

σo = √σ2+3τ2 = 158.9 N/mm2

S* = 4219. mm3mm 0 12 18 24 30 42x

0

6

42

54

y

10σc,τc

σm

u

v

Ing Civ.brti.002


Ing Civ.brti.002


Ing Civ.brti.002


Ing Civ.bnrd.003REAZIONI 811839 Bonora Davide


F

3/20F

3/20F1/2Fb

A

B

3/20F1/2Fb

3/20F7/20Fb

BCF

3/20F7/20Fb

3/20F3/20Fb

C

D

17/20F3/20Fb

17/20FFb

D E

Ing Civ.bnrd.003AZIONI INTERNE 811839 Bonora Davide


-3/2

0-3

/200

-3/2

0-3

/20

0

F

-10

-3/20

-10 -17/20

F

0-1

/21/27/20

7/20

-3/2

0-3/20 -1

Fb

Ing

Civ

.bnr

d.00

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1183

9 B

onor

a D

avid

e

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

0-1/2-1

/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bnr

d.00

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1183

9 B

onor

a D

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@ A

dolfo

Zav

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i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JD

C b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

DE

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bnrd.003PROCEDIMENTO E RISULTATI 811839 Bonora Davide


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDE = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

A = 936. mm2

Ju = 311455. mm4

Jv = 68256. mm4

yg = 30.69 mmN = -1325. NTy = -4415. NMx = -2317880. Nmmxm = 12. mmum = -12. mmvm = -30.69 mmσm = N/A-Mv/Ju = -229.8 N/mm2


τc = 5.997 N/mm2

σo = √σ2+3τ2 = 155.8 N/mm2

S* = 5077. mm3mm 0 12 18 30 36 48x

0

6

48

54

y

10σc,τc

σm

u

v

Ing Civ.bnrd.003


Ing Civ.bnrd.003


Ing Civ.bnrd.003


Ing Civ.chsg.004REAZIONI 813529 Chiesa Giancarlo


3F

3/20F33/20Fb

4F

3/20F37/20Fb

A

B4F

17/20F37/20Fb

4F

17/20FFb

BC

F

3/20F

3/20F1/2Fb

D

E3/20F

3/2Fb3/20F

33/20Fb

E A

Ing Civ.chsg.004AZIONI INTERNE 813529 Chiesa Giancarlo


3/20

3/20

-4

3/20

3/20

0

F

-3-4

17/20

10

3/20

F

33/2

0-3

7/20

-37/20-1

01/

2

3/2 33/20

Fb

Ing

Civ

.chs

g.00

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1352

9 C

hies

a G

ianc

arlo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-7/2

-7/2

-10 1/2 3/

20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.chs

g.00

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1352

9 C

hies

a G

ianc

arlo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

BA

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

BC

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3 x

/b +

1/2

x2 /b2 )

Fb

1/E

J dx

= [3

/2 x

2 /b +

1/6

x3 /b2 ] ob F

b 1/

EJ

= (3

/2 b

+1/

6 b

) Fb

1/E

J =

5/3

Fb2 /E

J

LXo

BA =

∫ ob (7/2

-4

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[7/2

x -

2 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.chsg.004PROCEDIMENTO E RISULTATI 813529 Chiesa Giancarlo


= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

A = 1152. mm2

Ju = 348030. mm4

Jv = 122688. mm4

yg = 33.38 mmN = 1328. NTy = 4425. NMx = 2489060. Nmmxm = 12. mmum = -12. mmvm = -33.38 mmσm = N/A-Mv/Ju = 239.8 N/mm2


τc = 6.216 N/mm2

σo = √σ2+3τ2 = 161.5 N/mm2

S* = 5867. mm3mm 0 12 18 30 36 48x

0

6

42

54

y

11σc,τc

σm

u

v

Ing Civ.chsg.004


Ing Civ.chsg.004


Ing Civ.chsg.004


Ing Civ.dmns.005REAZIONI 817109 Damian Sebastiano


F

19/40F

19/40F1/2Fb

A

B

19/40F3/2Fb

19/40F79/40Fb

B C4F

19/40F79/40Fb

4F

19/40F81/40Fb

C

D4F

21/40F81/40Fb

4F

61/40FFb

DE

Ing Civ.dmns.005AZIONI INTERNE 817109 Damian Sebastiano


-19/40 -19/40

0

-19/40

44

F

1 0

19/4

0

-4

21/4

061

/40

F

0 1/2

3/2

79/4

0

79/40-81/40

-81/

40-1

Fb

Ing

Civ

.dm

ns.0

05P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1710

9 D

amia

n S

ebas

tiano

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2 0 0-4

-4-1M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dm

ns.0

05P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1710

9 D

amia

n S

ebas

tiano

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Quadro contributi PLV per iperstatica X=WCD

→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx

AB b0Fx-1/2qx2

0000

BA b0-1/2Fb+1/2qx2

00

BC b-x/b3/2Fb-3/2Fx-3/2Fx+3/2Fx2/bx

2/b

2

-1/4Fb2/EJ1/3Xb/EJ

CB b1-x/b-3/2Fx-3/2Fx+3/2Fx2/b1-2x/b+x

2/b

2

CD b-1-4Fx4Fx12Fb

2/EJXb/EJ

DC b14Fb-4Fx4Fb-4Fx1

DE b-1+x/b-4Fb+5/2Fx+1/2qx2

4Fb-13/2Fx+2Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

37/24Fb2/EJ1/3Xb/EJ

ED bx/bFb+7/2Fx-1/2qx2

Fx+7/2Fx2/b-1/2qx

3/bx

2/b

2

totali79/24Fb2/EJ5/3Xb/EJ

iperstatica X=WCD-79/40Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

Ing Civ.dmns.005PROCEDIMENTO E RISULTATI 817109 Damian Sebastiano


= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDE = ∫

o

b(1 -2 x/b + x2/b2 ) 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o

b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXXED = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o

b(4 -13/2 x/b +2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [4 x -13/4 x2/b +2/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (4 b -13/4 b +2/3 b +1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

LXoED = ∫

o

b( x/b +7/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [1/2 x2/b +7/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (1/2 b +7/6 b -1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

Ing Civ.dmns.005PROCEDIMENTO E RISULTATI 817109 Damian Sebastiano


A = 684. mm2

Ju = 262207. mm4

Jv = 43524. mm4

yg = 30.16 mmN = -2774. NTy = 2920. NMx = 1773900. Nmmxm = 12. mmum = -9. mmvm = -30.16 mmσm = N/A-Mv/Ju = 200. N/mm2

xc = 21. mmyc = 47. mmvc = 16.84 mmσc = N/A-Mv/Ju = -118. N/mm2

τc = 9.941 N/mm2

σo = √σ2+3τ2 = 119.2 N/mm2

S* = 5356. mm3mm 0 12 18 24 30 42x

0

12

48

54

y

47σc,τc

σm

u

v

Ing Civ.dmns.005


Ing Civ.dmns.005


Ing Civ.elds.006REAZIONI 819711 El Dosoky Yasmin


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.elds.006AZIONI INTERNE 819711 El Dosoky Yasmin


-3-3

7/40

0

7/407/40

F

33/4

073

/40

-3

7/40

-10

F

-53/

400

67/40-53/40

1/2

27/4

0

0-1/2

Fb

Ing

Civ

.eld

s.00

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1971

1 E

l Dos

oky

Yas

min

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2 2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.eld

s.00

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

1971

1 E

l Dos

oky

Yas

min

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.elds.006PROCEDIMENTO E RISULTATI 819711 El Dosoky Yasmin


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -1/2 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o

b(-2 +7/2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [-2 x +7/4 x2/b -1/2 x3/b2 ]o

b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 900. mm2

Ju = 299017. mm4

Jv = 80460. mm4



τc = 9.884 N/mm2

σo = √σ2+3τ2 = 129.6 N/mm2

S* = 6115. mm3mm 0 12 18 24 30 42x

0

12

42

54

y

13σc,τc

σm

u

v

Ing Civ.elds.006


Ing Civ.elds.006


Ing Civ.elds.006


Ing Civ.brmm.007REAZIONI 829837 Bormolini Matteo


F

19/40F

19/40F1/2Fb

A

B

19/40F1/2Fb

19/40F1/40Fb

BC

19/40F1/40Fb

19/40F1/40Fb

C

D

21/40F1/40Fb

61/40FFb

D E

Ing Civ.brmm.007AZIONI INTERNE 829837 Bormolini Matteo


-19/

40-1

9/400

-19/

40

0 0

F

-10

-19/40

0

-21/40 -61/40

F

0-1

/21/21/40

1/40

1/40

1/40-1

Fb

Ing

Civ

.brm

m.0

07P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

2983

7 B

orm

olin

i Mat

teo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

00

0-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brm

m.0

07P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

2983

7 B

orm

olin

i Mat

teo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

tota

li1/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-1

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.brmm.007PROCEDIMENTO E RISULTATI 829837 Bormolini Matteo


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoDE = ∫

o

b(1/2 x/b -1/2 x3/b3 ) Fb 1/EJ dx = [1/4 x2/b -1/8 x4/b3 ]o

b Fb 1/EJ

= (1/4 b -1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoED = ∫

o

b( x/b -3/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [1/2 x2/b -1/2 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

A = 1008. mm2

Ju = 343131. mm4

Jv = 74304. mm4



τc = 6.219 N/mm2

σo = √σ2+3τ2 = 123.8 N/mm2

S* = 6865. mm3mm 0 12 18 30 36 48x

0

12

48

54

y

13σc,τc

σm

u

v

Ing Civ.brmm.007


Ing Civ.brmm.007


Ing Civ.brmm.007


Ing Civ.bnst.008REAZIONI 843782 Benassai Tommaso


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.bnst.008AZIONI INTERNE 843782 Benassai Tommaso


33

-7/4

0

0

-7/4

0-7

/40

F

33/4073/40

-3

7/40

-10

F

-53/400

67/4

0-5

3/40

1/2 27/40

0-1

/2

Fb

Ing

Civ

.bns

t.008

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 843

782

Ben

assa

i Tom

mas

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

qq

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2

2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bns

t.008

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 843

782

Ben

assa

i Tom

mas

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.bnst.008PROCEDIMENTO E RISULTATI 843782 Benassai Tommaso


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 1224. mm2

Ju = 388508. mm4

Jv = 128736. mm4



τc = 6.321 N/mm2

σo = √σ2+3τ2 = 137.2 N/mm2

S* = 7704. mm3mm 0 12 18 30 36 48x

0

12

42

54

y

13σc,τc

σm

u

v

Ing Civ.bnst.008


Ing Civ.bnst.008


Ing Civ.bnst.008


Ing Civ.dnsg.009REAZIONI 845411 Danesi Gabriele


F

3/20F

3/20F1/2Fb

A

B

3/20F3/2Fb

3/20F33/20Fb

B C3F

3/20F33/20Fb

4F

3/20F37/20Fb

C

D4F

17/20F37/20Fb

4F

17/20FFb

DE

Ing Civ.dnsg.009AZIONI INTERNE 845411 Danesi Gabriele


-3/20 -3/20

0

-3/20 -3/20

4

F

1 0

3/20

-3 -4

17/2

0

F

0 1/2

3/2

33/2

0

33/20-37/20

-37/

20-1

Fb

Ing

Civ

.dns

g.00

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

4541

1 D

anes

i Gab

riele

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2 0 0-7

/2-7/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dns

g.00

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

4541

1 D

anes

i Gab

riele

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

CB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.dnsg.009PROCEDIMENTO E RISULTATI 845411 Danesi Gabriele


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

A = 1224. mm2

Ju = 388508. mm4

Jv = 128736. mm4

yg = 22.06 mmN = -1140. NTy = 3800. NMx = 2907000. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 31.94 mmσm = N/A-Mv/Ju = -239.9 N/mm2

xc = 24. mmyc = 41. mmvc = 18.94 mmσc = N/A-Mv/Ju = -142.7 N/mm2

τc = 6.28 N/mm2

σo = √σ2+3τ2 = 143.1 N/mm2

S* = 7704. mm3mm 0 12 18 30 36 48x

0

12

42

54

y

41σc,τc

σm

u

v

Ing Civ.dnsg.009


Ing Civ.dnsg.009


Ing Civ.dnsg.009


Ing Civ.bvra.010REAZIONI 845742 Beverina Andrea


F

3/20F7/20Fb

3/20F3/20Fb

A

B

17/20F3/20Fb

17/20FFb

B C

F

3/20F

3/20F1/2Fb

D

E

3/20F1/2Fb

3/20F7/20Fb

EA

Ing Civ.bvra.010AZIONI INTERNE 845742 Beverina Andrea


3/203/20

0

3/203/20

0

F

-10

-17/

20

-10

-3/2

0

F

7/20-3/20

-3/2

0-1

0-1/21/

27/

20

Fb

Ing

Civ

.bvr

a.01

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

4574

2 B

ever

ina

And

rea

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/2 -1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

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ica

X=

1

Ing

Civ

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a.01

0P

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TI 8

4574

2 B

ever

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And

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27.0

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31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.bvra.010PROCEDIMENTO E RISULTATI 845742 Beverina Andrea


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

A = 684. mm2

Ju = 262207. mm4

Jv = 43524. mm4

yg = 23.84 mmN = 1275. NTy = -4250. NMx = -1721250. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 30.16 mmσm = N/A-Mv/Ju = 199.8 N/mm2

xc = 21. mmyc = 41. mmvc = 17.16 mmσc = N/A-Mv/Ju = 114.5 N/mm2

τc = 14.38 N/mm2

σo = √σ2+3τ2 = 117.2 N/mm2

S* = 5324. mm3mm 0 12 18 24 30 42x

0

6

42

54

y

41σc,τc

σm

u

v

Ing Civ.bvra.010


Ing Civ.bvra.010


Ing Civ.bvra.010


Ing Civ.awda.011REAZIONI 847541 Awad Alaa Mohamed Hassan


F

3/20F

3/20F1/2Fb

A

B

3/20F1/2Fb

3/20F7/20Fb

BCF

3/20F7/20Fb

3/20F3/20Fb

C

D

17/20F3/20Fb

17/20FFb

D E

Ing Civ.awda.011AZIONI INTERNE 847541 Awad Alaa Mohamed Hassan


-3/2

0-3

/200

-3/2

0-3

/20

0

F

-10

-3/20

-10 -17/20

F

0-1

/21/27/20

7/20

-3/2

0-3/20 -1

Fb

Ing

Civ

.aw

da.0

11P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

4754

1 A

wad

Ala

a M

oham

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@ A

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i Ros

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olite

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no, v

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27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

0-1/2-1

/2-1

Mo

fless

ione

da

caric

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sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

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Ing

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.aw

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no, v

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27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JD

C b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

DE

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.awda.011PROCEDIMENTO E RISULTATI 847541 Awad Alaa Mohamed


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDE = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

A = 936. mm2

Ju = 311455. mm4

Jv = 68256. mm4

yg = 23.31 mmN = -1455. NTy = -4850. NMx = -2146130. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 30.69 mmσm = N/A-Mv/Ju = 209.9 N/mm2

xc = 24. mmyc = 44. mmvc = 20.69 mmσc = N/A-Mv/Ju = 141. N/mm2

τc = 6.588 N/mm2

σo = √σ2+3τ2 = 141.5 N/mm2

S* = 5077. mm3mm 0 12 18 30 36 48x

0

6

48

54

y

44σc,τc

σm

u

v

Ing Civ.awda.011


Ing Civ.awda.011


Ing Civ.awda.011


Ing Civ.grnd.012REAZIONI 849412 Gerna Diego


3F

3/20F33/20Fb

4F

3/20F37/20Fb

A

B4F

17/20F37/20Fb

4F

17/20FFb

BC

F

3/20F

3/20F1/2Fb

D

E3/20F

3/2Fb3/20F

33/20Fb

E A

Ing Civ.grnd.012AZIONI INTERNE 849412 Gerna Diego


3/20

3/20

-4

3/20

3/20

0

F

-3-4

17/20

10

3/20

F

33/2

0-3

7/20

-37/20-1

01/

2

3/2 33/20

Fb

Ing

Civ

.grn

d.01

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

4941

2 G

erna

Die

go

@ A

dolfo

Zav

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i Ros

si, P

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cnic

o di

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no, v

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27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-7/2

-7/2

-10 1/2 3/

20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.grn

d.01

2P

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DIM

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TI 8

4941

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dolfo

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no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

BA

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

BC

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3 x

/b +

1/2

x2 /b2 )

Fb

1/E

J dx

= [3

/2 x

2 /b +

1/6

x3 /b2 ] ob F

b 1/

EJ

= (3

/2 b

+1/

6 b

) Fb

1/E

J =

5/3

Fb2 /E

J

LXo

BA =

∫ ob (7/2

-4

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[7/2

x -

2 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.grnd.012PROCEDIMENTO E RISULTATI 849412 Gerna Diego


= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

A = 900. mm2

Ju = 299017. mm4

Jv = 80460. mm4

yg = 20.28 mmN = 1226. NTy = 4085. NMx = 1960800. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 33.72 mmσm = N/A-Mv/Ju = -219.8 N/mm2


τc = 13.92 N/mm2

σo = √σ2+3τ2 = 136.7 N/mm2

S* = 6115. mm3mm 0 12 18 24 30 42x

0

12

42

54

y

41σc,τc

σm

u

v

Ing Civ.grnd.012


Ing Civ.grnd.012


Ing Civ.grnd.012


Ing Civ.dnmc.013REAZIONI 850426 Dioni Michele


3F

11/8F11/8Fb

3F

11/8F

AB

3F

3/8F13/8Fb

3F

3/8F11/8Fb

C

A

5/8F1/2Fb

3/8F5/8Fb

D C

F

5/8F

5/8F1/2Fb

E

D

Ing Civ.dnmc.013AZIONI INTERNE 850426 Dioni Michele


-3

-3/8

00

5/85/8

F

11/8

-3

5/8

-3/8

-10

F

-11/

80

13/8-11/8

1/2

5/8

0-1/2

Fb

Ing

Civ

.dnm

c.01

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

5042

6 D

ioni

Mic

hele

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2 2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dnm

c.01

3P

RO

CE

DIM

EN

TO

E R

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TI 8

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6 D

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Mic

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dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Quadro contributi PLV per iperstatica X=WAB


AB b-1+x/b001-2x/b+x2/b

2

01/3Xb/EJBA bx/b00x

2/b

2

CA b-13Fb-3Fx-3Fb+3Fx1-3/2Fb

2/EJXb/EJ

AC b1-3Fx-3Fx1

DC b-x/b1/2Fb+2Fx-1/2qx2

-1/2Fx-2Fx2/b+1/2qx

3/bx

2/b

2

-19/24Fb2/EJ1/3Xb/EJ

CD b1-x/b-2Fb+Fx+1/2qx2

-2Fb+3Fx-1/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00

totali-55/24Fb2/EJ5/3Xb/EJ

iperstatica X=WAB11/8Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.dnmc.013PROCEDIMENTO E RISULTATI 850426 Dioni Michele


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-1/4 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-1/4 b -2/3 b +1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

LXoCD = ∫

o

b(-2 +3 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-2 x +3/2 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (-2 b +3/2 b -1/6 b -1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

A = 828. mm2

Ju = 250978. mm4

Jv = 77652. mm4



τc = 8.405 N/mm2

σo = √σ2+3τ2 = 167.5 N/mm2

S* = 4219. mm3mm 0 12 18 24 30 42x

0

12

48

54

y

44σc,τc

σm

u

v

Ing Civ.dnmc.013


Ing Civ.dnmc.013


Ing Civ.dnmc.013


Ing Civ.espa.014REAZIONI 850941 Esposto Andrea


3/40F3/40Fb

3/40F3/40Fb

A

B

43/40F3/40Fb

43/40FFb

B C

F

37/40F

37/40F1/2Fb

D

E

37/40F1/2Fb

3/40F3/40Fb

EA

Ing Civ.espa.014AZIONI INTERNE 850941 Esposto Andrea


-3/40

0

37/4037/40

00

F

0

-43/

40

-10

-37/

403/

40

F

3/403/40

3/40

-1

0-1/21/

23/

40

Fb

Ing

Civ

.esp

a.01

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

5094

1 E

spos

to A

ndre

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00 0

-1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.esp

a.01

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

5094

1 E

spos

to A

ndre

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

Fx+

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /bx2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

qx2

-1/2

Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

tota

li1/

8Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B-3

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

BC =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

= (1

/2 b

-1/

3 b

) Fb

1/E

J =

1/6

Fb2 /E

J

LXo

CB =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.espa.014PROCEDIMENTO E RISULTATI 850941 Esposto Andrea


= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoEA = ∫

o

b(-1/2 x/b + x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-1/4 x2/b +1/3 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoAE = ∫

o

b(-1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-1/6 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

A = 1008. mm2

Ju = 343131. mm4

Jv = 74304. mm4



τc = 8.061 N/mm2

σo = √σ2+3τ2 = 137.5 N/mm2

S* = 6865. mm3mm 0 12 18 30 36 48x

0

6

42

54

y

41σc,τc

σm

u

v

Ing Civ.espa.014


Ing Civ.espa.014


Ing Civ.espa.014


Ing Civ.brmt.015REAZIONI 868168 Boara Matteo


3F

11/8F11/8Fb

3F

11/8F

AB

3F

3/8F13/8Fb

3F

3/8F11/8Fb

C

A

5/8F1/2Fb

3/8F5/8Fb

D C

F

5/8F

5/8F1/2Fb

E

D

Ing Civ.brmt.015AZIONI INTERNE 868168 Boara Matteo


3

3/8

0 0

-5/8

-5/8

F

11/8

-3

5/8-3/8

-10

F

-11/80

13/8

-11/

8

1/2 5/8

0-1

/2

Fb

Ing

Civ

.brm

t.015

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 868

168

Boa

ra M

atte

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2

2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brm

t.015

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 868

168

Boa

ra M

atte

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1+x/b001-2x/b+x2/b

2

01/3Xb/EJBA bx/b00x

2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-2Fx2/b+1/2qx

3/bx

2/b

2



-2Fb+3Fx-1/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.brmt.015PROCEDIMENTO E RISULTATI 868168 Boara Matteo


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -2/3 b +1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +3/2 b -1/6 b -1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

A = 1152. mm2

Ju = 348030. mm4

Jv = 122688. mm4



τc = 4.909 N/mm2

σo = √σ2+3τ2 = 133. N/mm2

S* = 5867. mm3mm 0 12 18 30 36 48x

0

12

48

54

y

43σc,τc

σm

u

v

Ing Civ.brmt.015


Ing Civ.brmt.015


Ing Civ.brmt.015


Ing Civ.brnr.016REAZIONI 868395 Bernasconi Riccardo


4F

3/40F77/40Fb

4F

3/40F83/40Fb

A

B4F

43/40F83/40Fb

4F

43/40FFb

BC

F

37/40F

37/40F1/2Fb

D

E37/40F

3/2Fb3/40F

77/40Fb

E A

Ing Civ.brnr.016AZIONI INTERNE 868395 Bernasconi Riccardo


-3/4

0

-4

37/4

037

/40

0 0

F

-4

43/40

10

37/40-3/40

F

77/4

0-8

3/40

-83/40-1

01/

2

3/2 77/40

Fb

Ing

Civ

.brn

r.01

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

6839

5 B

erna

scon

i Ric

card

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-4

-4-1

0 1/2 3/2

0

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brn

r.01

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

6839

5 B

erna

scon

i Ric

card

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JB

A b

14F

b-4F

x4F

b-4F

x1

BC

b-1

+x/

b-4

Fb+

3Fx

4Fb-

7Fx+

3Fx2 /b

1-2x

/b+

x2 /b2

3/2F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb+

3Fx

Fx+

3Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

Fx-

1/2q

x2-3

/2F

x+F

x2 /b+

1/2q

x3 /bx2 /b

2

-7/2

4Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-2F

x+1/

2qx2

-2F

x+5/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

tota

li77

/24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

7/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (4 x

/b )

Fb

1/E

J dx

= [2

x2 /b

] ob Fb

1/E

J

= (2

b )

Fb

1/E

J =

2 F

b2 /EJ

LXo

BA =

∫ ob (4 -

4 x/

b ) F

b 1/

EJ

dx =

[4 x

-2

x2 /b ] ob F

b 1/

EJ

Ing Civ.brnr.016PROCEDIMENTO E RISULTATI 868395 Bernasconi Riccardo


= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o

b(4 -7 x/b +3 x2/b2 ) Fb 1/EJ dx = [4 x -7/2 x2/b + x3/b2 ]o

b Fb 1/EJ

= (4 b -7/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

LXoCB = ∫

o

b( x/b +3 x2/b2 ) Fb 1/EJ dx = [1/2 x2/b + x3/b2 ]o

b Fb 1/EJ

= (1/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

LXoEA = ∫

o

b(-3/2 x/b + x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-3/4 x2/b +1/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-3/4 b +1/3 b +1/8 b ) Fb 1/EJ = -7/24 Fb2/EJ

LXoAE = ∫

o

b(-2 x/b +5/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [- x2/b +5/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (- b +5/6 b -1/8 b ) Fb 1/EJ = -7/24 Fb2/EJ

A = 612. mm2

Ju = 225968. mm4

Jv = 40716. mm4



τc = 6.018 N/mm2

σo = √σ2+3τ2 = 156.9 N/mm2

S* = 3510. mm3mm 0 12 18 24 30 42x

0

6

48

54

y

46σc,τc

σm

u

v

Ing Civ.brnr.016


Ing Civ.brnr.016


Ing Civ.brnr.016


Ing Civ.dmcm.017REAZIONI 870485 D’Amico Michele Francesco


F

19/40F

19/40F1/2Fb

A

B

19/40F3/2Fb

19/40F79/40Fb

B C4F

19/40F79/40Fb

4F

19/40F81/40Fb

C

D4F

21/40F81/40Fb

4F

61/40FFb

DE

Ing Civ.dmcm.017AZIONI INTERNE 870485 D’Amico Michele Francesco


-19/40 -19/40

0

-19/40

44

F

1 0

19/4

0

-4

21/4

061

/40

F

0 1/2

3/2

79/4

0

79/40-81/40

-81/

40-1

Fb

Ing

Civ

.dm

cm.0

17P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7048

5 D

’Am

ico

Mic

hele

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2 0 0-4

-4-1M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dm

cm.0

17P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7048

5 D

’Am

ico

Mic

hele

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b0Fx-1/2qx2

0000

BA b0-1/2Fb+1/2qx2

00


2/b

2

-1/4Fb2/EJ1/3Xb/EJ


2/b

2

CD b-1-4Fx4Fx12Fb

2/EJXb/EJ



4Fb-13/2Fx+2Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

37/24Fb2/EJ1/3Xb/EJ


Fx+7/2Fx2/b-1/2qx

3/bx

2/b

2



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

Ing Civ.dmcm.017PROCEDIMENTO E RISULTATI 870485 D’Amico Michele


= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXXED = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (4 b -13/4 b +2/3 b +1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b +7/6 b -1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

Ing Civ.dmcm.017PROCEDIMENTO E RISULTATI 870485 D’Amico Michele


A = 570. mm2

Ju = 206355. mm4

Jv = 34542. mm4

yg = 31.11 mmN = -2085. NTy = 2195. NMx = 1481630. Nmmxm = 12. mmum = -9. mmvm = -31.11 mmσm = N/A-Mv/Ju = 219.7 N/mm2


τc = 5.895 N/mm2

σo = √σ2+3τ2 = 162.6 N/mm2

S* = 3325. mm3mm 0 12 18 24 30 42x

0

6

48

53

y

8σc,τc

σm

u

v

Ing Civ.dmcm.017


Ing Civ.dmcm.017


Ing Civ.dflm.018REAZIONI 871570 De Flammineis Margherita


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.dflm.018AZIONI INTERNE 871570 De Flammineis Margherita


-3-3

7/40

0

7/407/40

F

33/4

073

/40

-3

7/40

-10

F

-53/

400

67/40-53/40

1/2

27/4

0

0-1/2

Fb

Ing

Civ

.dflm

.018

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 871

570

De

Fla

mm

inei

s

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2 2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dflm

.018

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 871

570

De

Fla

mm

inei

s

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.dflm.018PROCEDIMENTO E RISULTATI 871570 De Flammineis


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 786. mm2

Ju = 237222. mm4

Jv = 71478. mm4



τc = 6.093 N/mm2

σo = √σ2+3τ2 = 170.7 N/mm2

S* = 3942. mm3mm 0 12 18 24 30 42x

0

6

42

53

y

9σc,τc

σm

u

v

Ing Civ.dflm.018


Ing Civ.dflm.018


Ing Civ.dflm.018


Ing Civ.cltf.019REAZIONI 871912 Calati Francesca


F

19/40F

19/40F1/2Fb

A

B

19/40F1/2Fb

19/40F1/40Fb

BC

19/40F1/40Fb

19/40F1/40Fb

C

D

21/40F1/40Fb

61/40FFb

D E

Ing Civ.cltf.019AZIONI INTERNE 871912 Calati Francesca


-19/

40-1

9/400

-19/

40

0 0

F

-10

-19/40

0

-21/40 -61/40

F

0-1

/21/21/40

1/40

1/40

1/40-1

Fb

Ing

Civ

.cltf

.019

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 871

912

Cal

ati F

ranc

esca

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

00

0-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cltf

.019

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 871

912

Cal

ati F

ranc

esca

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

tota

li1/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-1

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.cltf.019PROCEDIMENTO E RISULTATI 871912 Calati Francesca


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/4 b -1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

A = 888. mm2

Ju = 285133. mm4

Jv = 59040. mm4

yg = 29.46 mmN = -2902. NTy = -3055. NMx = -2291250. Nmmxm = 12. mmum = -12. mmvm = -29.46 mmσm = N/A-Mv/Ju = -240. N/mm2


τc = 4.322 N/mm2

σo = √σ2+3τ2 = 159.8 N/mm2

S* = 4840. mm3mm 0 12 18 30 36 48x

0

6

48

53

y

10σc,τc

σm

u

v

Ing Civ.cltf.019


Ing Civ.cltf.019


Ing Civ.cltf.019


Ing Civ.dgtn.020REAZIONI 876780 De Gaetano Salvatore


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.dgtn.020AZIONI INTERNE 876780 De Gaetano Salvatore


33

-7/4

0

0

-7/4

0-7

/40

F

33/4073/40

-3

7/40

-10

F

-53/400

67/4

0-5

3/40

1/2 27/40

0-1

/2

Fb

Ing

Civ

.dgt

n.02

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7678

0 D

e G

aeta

no S

alva

tore

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

qq

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2

2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dgt

n.02

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7678

0 D

e G

aeta

no S

alva

tore

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.dgtn.020PROCEDIMENTO E RISULTATI 876780 De Gaetano Salvatore


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 1104. mm2

Ju = 327740. mm4

Jv = 113472. mm4



τc = 7.275 N/mm2

σo = √σ2+3τ2 = 133.4 N/mm2

S* = 5688. mm3mm 0 12 18 30 36 48x

0

6

42

53

y

11σc,τc

σm

u

v

Ing Civ.dgtn.020


Ing Civ.dgtn.020


Ing Civ.dgtn.020


Ing Civ.cste.021REAZIONI 877793 Castiglione Ettore


3F

11/8F11/8Fb

3F

11/8F

AB

3F

3/8F13/8Fb

3F

3/8F11/8Fb

C

A

5/8F1/2Fb

3/8F5/8Fb

D C

F

5/8F

5/8F1/2Fb

E

D

Ing Civ.cste.021AZIONI INTERNE 877793 Castiglione Ettore


-3

-3/8

00

5/85/8

F

11/8

-3

5/8

-3/8

-10

F

-11/

80

13/8-11/8

1/2

5/8

0-1/2

Fb

Ing

Civ

.cst

e.02

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7779

3 C

astig

lione

Etto

re

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2 2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cst

e.02

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7779

3 C

astig

lione

Etto

re

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1+x/b001-2x/b+x2/b

2

01/3Xb/EJBA bx/b00x

2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-2Fx2/b+1/2qx

3/bx

2/b

2



-2Fb+3Fx-1/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.cste.021PROCEDIMENTO E RISULTATI 877793 Castiglione Ettore


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -2/3 b +1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +3/2 b -1/6 b -1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

A = 642. mm2

Ju = 237823. mm4

Jv = 37350. mm4



τc = 13.72 N/mm2

σo = √σ2+3τ2 = 149.8 N/mm2

S* = 4706. mm3mm 0 12 18 24 30 42x

0

12

48

53

y

47σc,τc

σm

u

v

Ing Civ.cste.021


Ing Civ.cste.021


Ing Civ.cste.021


Ing Civ.agsr.022REAZIONI 877947 Agosti Riccardo


3/40F3/40Fb

3/40F3/40Fb

A

B

43/40F3/40Fb

43/40FFb

B C

F

37/40F

37/40F1/2Fb

D

E

37/40F1/2Fb

3/40F3/40Fb

EA

Ing Civ.agsr.022AZIONI INTERNE 877947 Agosti Riccardo


-3/40

0

37/4037/40

00

F

0

-43/

40

-10

-37/

403/

40

F

3/403/40

3/40

-1

0-1/21/

23/

40

Fb

Ing

Civ

.ags

r.02

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7794

7 A

gost

i Ric

card

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00 0

-1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.ags

r.02

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7794

7 A

gost

i Ric

card

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

Fx+

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /bx2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

qx2

-1/2

Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

tota

li1/

8Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B-3

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

BC =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

= (1

/2 b

-1/

3 b

) Fb

1/E

J =

1/6

Fb2 /E

J

LXo

CB =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.agsr.022PROCEDIMENTO E RISULTATI 877947 Agosti Riccardo


= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

A = 858. mm2

Ju = 281777. mm4

Jv = 74286. mm4


xc = 21. mmyc = 13. mmvc = -19.75 mmσc = N/A-Mv/Ju = -129. N/mm2

τc = 14.53 N/mm2

σo = √σ2+3τ2 = 131.4 N/mm2

S* = 5900. mm3mm 0 12 18 24 30 42x

0

12

42

53

y

13σc,τc

σm

u

v

Ing Civ.agsr.022


Ing Civ.agsr.022


Ing Civ.agsr.022


Ing Civ.brgf.023REAZIONI 877968 Bergui Francesco


3F

11/8F11/8Fb

3F

11/8F

AB

3F

3/8F13/8Fb

3F

3/8F11/8Fb

C

A

5/8F1/2Fb

3/8F5/8Fb

D C

F

5/8F

5/8F1/2Fb

E

D

Ing Civ.brgf.023AZIONI INTERNE 877968 Bergui Francesco


3

3/8

0 0

-5/8

-5/8

F

11/8

-3

5/8-3/8

-10

F

-11/80

13/8

-11/

8

1/2 5/8

0-1

/2

Fb

Ing

Civ

.brg

f.023

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 877

968

Ber

gui F

ranc

esco

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2

2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brg

f.023

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 877

968

Ber

gui F

ranc

esco

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1+x/b001-2x/b+x2/b

2

01/3Xb/EJBA bx/b00x

2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-2Fx2/b+1/2qx

3/bx

2/b

2



-2Fb+3Fx-1/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.brgf.023PROCEDIMENTO E RISULTATI 877968 Bergui Francesco


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -2/3 b +1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +3/2 b -1/6 b -1/8 b ) Fb 1/EJ = -19/24 Fb2/EJ

A = 960. mm2

Ju = 313227. mm4

Jv = 65088. mm4



τc = 7.392 N/mm2

σo = √σ2+3τ2 = 146.8 N/mm2

S* = 5653. mm3mm 0 12 18 30 36 48x

0

12

48

53

y

47σc,τc

σm

u

v

Ing Civ.brgf.023


Ing Civ.brgf.023


Ing Civ.brgf.023


Ing Civ.dcml.024REAZIONI 877976 Di Camillo Lorenzo


4F

3/40F77/40Fb

4F

3/40F83/40Fb

A

B4F

43/40F83/40Fb

4F

43/40FFb

BC

F

37/40F

37/40F1/2Fb

D

E37/40F

3/2Fb3/40F

77/40Fb

E A

Ing Civ.dcml.024AZIONI INTERNE 877976 Di Camillo Lorenzo


-3/4

0

-4

37/4

037

/40

0 0

F

-4

43/40

10

37/40-3/40

F

77/4

0-8

3/40

-83/40-1

01/

2

3/2 77/40

Fb

Ing

Civ

.dcm

l.024

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 877

976

Di C

amill

o Lo

renz

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

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0-4

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0 1/2 3/2

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sseg

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-1 -1

-10

0 0 0-1

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X=

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Ing

Civ

.dcm

l.024

PR

OC

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976

Di C

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27.0

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31.0

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Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JB

A b

14F

b-4F

x4F

b-4F

x1

BC

b-1

+x/

b-4

Fb+

3Fx

4Fb-

7Fx+

3Fx2 /b

1-2x

/b+

x2 /b2

3/2F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb+

3Fx

Fx+

3Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

Fx-

1/2q

x2-3

/2F

x+F

x2 /b+

1/2q

x3 /bx2 /b

2

-7/2

4Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-2F

x+1/

2qx2

-2F

x+5/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

tota

li77

/24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

7/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (4 x

/b )

Fb

1/E

J dx

= [2

x2 /b

] ob Fb

1/E

J

= (2

b )

Fb

1/E

J =

2 F

b2 /EJ

LXo

BA =

∫ ob (4 -

4 x/

b ) F

b 1/

EJ

dx =

[4 x

-2

x2 /b ] ob F

b 1/

EJ

Ing Civ.dcml.024PROCEDIMENTO E RISULTATI 877976 Di Camillo Lorenzo


= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (4 b -7/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

LXoEA = ∫

o

b(-3/2 x/b + x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-3/4 x2/b +1/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-3/4 b +1/3 b +1/8 b ) Fb 1/EJ = -7/24 Fb2/EJ

LXoAE = ∫

o

b(-2 x/b +5/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [- x2/b +5/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (- b +5/6 b -1/8 b ) Fb 1/EJ = -7/24 Fb2/EJ

A = 1176. mm2

Ju = 365284. mm4

Jv = 119520. mm4



τc = 8.351 N/mm2

σo = √σ2+3τ2 = 143.5 N/mm2

S* = 7440. mm3mm 0 12 18 30 36 48x

0

12

42

53

y

13σc,τc

σm

u

v

Ing Civ.dcml.024


Ing Civ.dcml.024


Ing Civ.dcml.024


Ing Civ.grcg.025REAZIONI 878061 Greco Giovanni


F

37/40F

37/40F1/2Fb

A

B

37/40F3/2Fb

3/40F77/40Fb

B C4F

3/40F77/40Fb

4F

3/40F83/40Fb

C

D4F

43/40F83/40Fb

4F

43/40FFb

DE

Ing Civ.grcg.025AZIONI INTERNE 878061 Greco Giovanni


-37/40 -37/40 00

3/40

4

F

1 0

37/4

0-3

/40

-4

43/4

0

F

0 1/2

3/2

77/4

0

77/40-83/40

-83/

40-1

Fb

Ing

Civ

.grc

g.02

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7806

1 G

reco

Gio

vann

i

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2 0 0-4

-4-1M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.grc

g.02

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7806

1 G

reco

Gio

vann

i

@ A

dolfo

Zav

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i Ros

si, P

olite

cnic

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

Fx-

1/2q

x2-3

/2F

x+F

x2 /b+

1/2q

x3 /bx2 /b

2

-7/2

4Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-2F

x+1/

2qx2

-2F

x+5/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

CD

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JD

C b

14F

b-4F

x4F

b-4F

x1

DE

b-1

+x/

b-4

Fb+

3Fx

4Fb-

7Fx+

3Fx2 /b

1-2x

/b+

x2 /b2

3/2F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb+

3Fx

Fx+

3Fx2 /b

x2 /b2

tota

li77

/24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

7/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+ x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

1/3

b +

1/8

b ) F

b 1/

EJ

= -

7/24

Fb2 /E

J

LXo

CB =

∫ ob (-2 x

/b +

5/2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

x2 /b

+5/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.grcg.025PROCEDIMENTO E RISULTATI 878061 Greco Giovanni


= (- b +5/6 b -1/8 b ) Fb 1/EJ = -7/24 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (4 b -7/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b + b ) Fb 1/EJ = 3/2 Fb2/EJ

A = 1200. mm2

Ju = 364306. mm4

Jv = 127584. mm4



τc = 6.159 N/mm2

σo = √σ2+3τ2 = 126.6 N/mm2

S* = 7123. mm3mm 0 12 18 30 36 48x

0

12

42

53

y

41σc,τc

σm

u

v

Ing Civ.grcg.025


Ing Civ.grcg.025


Ing Civ.grcg.025


Ing Civ.bldg.026REAZIONI 878411 Baldon Giacomo


61/40F3/2Fb

21/40F19/40Fb

AB

F

61/40F

61/40F1/2Fb

C

A

F

19/40F21/40Fb

F

19/40FFb

D E

F

21/40F19/40Fb

F

21/40F21/40Fb

B

D

Ing Civ.bldg.026AZIONI INTERNE 878411 Baldon Giacomo


00

-61/40 -61/40

-1

-21/40

F

-61/

40-2

1/40

1 0

-19/

40

-1

F

3/2

19/4

0

0 1/2

-21/

40-1

19/40-21/40

Fb

Ing

Civ

.bld

g.02

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7841

1 B

aldo

n G

iaco

mo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

3/21

01/

2

0-1

10

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bld

g.02

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

7841

1 B

aldo

n G

iaco

mo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

Fx+

1/2q

x2-3

/2F

x+F

x2 /b-1

/2qx

3 /bx2 /b

2

-13/

24F

b2 /EJ

1/3X

b/E

JB

A b

1-x/

b-F

b-1/

2qx2

-Fb+

Fx-

1/2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-7

/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WD

E21

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+ x

2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

1/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

1/3

b -1

/8 b

) F

b 1/

EJ

= -

13/2

4 F

b2 /EJ

LXo

BA =

∫ ob (-1 +

x/b

-1/

2 x2 /b

2 +1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[- x

+1/

2 x2 /b

-1/

6 x3 /b

2 +1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.bldg.026PROCEDIMENTO E RISULTATI 878411 Baldon Giacomo


= (- b +1/2 b -1/6 b +1/8 b ) Fb 1/EJ = -13/24 Fb2/EJ

LXoDE = ∫

o

b( x/b - x2/b2 ) Fb 1/EJ dx = [1/2 x2/b -1/3 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 666. mm2

Ju = 245945. mm4

Jv = 43038. mm4



τc = 8.661 N/mm2

σo = √σ2+3τ2 = 131.4 N/mm2

S* = 4954. mm3mm 0 12 18 24 30 42x

0

6

42

53

y

41σc,τc

σm

u

v

Ing Civ.bldg.026


Ing Civ.bldg.026


Ing Civ.bldg.026


Ing Civ.csrf.027REAZIONI 879105 Caserta Francesco


F

37/40F

37/40F1/2Fb

A

B

37/40F1/2Fb

3/40F3/40Fb

BC

3/40F3/40Fb

3/40F3/40Fb

C

D

43/40F3/40Fb

43/40FFb

D E

Ing Civ.csrf.027AZIONI INTERNE 879105 Caserta Francesco

@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19-3

7/40

-37/

4000

3/40

0

F

-10

-37/403/40

0

-43/40

F

0-1

/21/23/40

3/40

3/40

3/40-1

Fb

Ing

Civ

.csr

f.027

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 879

105

Cas

erta

Fra

nces

co

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

00

0-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.csr

f.027

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 879

105

Cas

erta

Fra

nces

co

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

Fx+

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /bx2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

qx2

-1/2

Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

tota

li1/

8Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WC

D-3

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+ x

2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/4

x2 /b +

1/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

1/4

b +

1/3

b -1

/8 b

) F

b 1/

EJ

= -

1/24

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x2 /b

2 +1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[-1/

6 x3 /b

2 +1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.csrf.027PROCEDIMENTO E RISULTATI 879105 Caserta Francesco


= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

A = 912. mm2

Ju = 289000. mm4

Jv = 67104. mm4

yg = 22.51 mmN = -5883. NTy = -3180. NMx = -2146500. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 30.49 mmσm = N/A-Mv/Ju = 220. N/mm2


τc = 5.329 N/mm2

σo = √σ2+3τ2 = 122. N/mm2

S* = 5812. mm3mm 0 12 18 30 36 48x

0

6

48

53

y

7σc,τc

σm

u

v

Ing Civ.csrf.027


Ing Civ.csrf.027


Ing Civ.csrf.027


Ing Civ.brga.028REAZIONI 881254 Broggi Andrea


61/40F3/2Fb

21/40F19/40Fb

AB

F

61/40F

61/40F1/2Fb

C

A

F

19/40F21/40Fb

F

19/40FFb

D E

F

21/40F19/40Fb

F

21/40F21/40Fb

B

D

Ing Civ.brga.028AZIONI INTERNE 881254 Broggi Andrea


00

61/4

061

/40

1

21/4

0

F

-61/40-21/40

10

-19/40

-1

F

3/219/40

01/

2

-21/40 -1

19/4

0-2

1/40

Fb

Ing

Civ

.brg

a.02

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8125

4 B

rogg

i And

rea

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

1

0 1/2

0-1

1 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brg

a.02

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8125

4 B

rogg

i And

rea

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

Fx+

1/2q

x2-3

/2F

x+F

x2 /b-1

/2qx

3 /bx2 /b

2

-13/

24F

b2 /EJ

1/3X

b/E

JB

A b

1-x/

b-F

b-1/

2qx2

-Fb+

Fx-

1/2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-7

/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WD

E21

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+ x

2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

1/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

1/3

b -1

/8 b

) F

b 1/

EJ

= -

13/2

4 F

b2 /EJ

LXo

BA =

∫ ob (-1 +

x/b

-1/

2 x2 /b

2 +1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[- x

+1/

2 x2 /b

-1/

6 x3 /b

2 +1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.brga.028PROCEDIMENTO E RISULTATI 881254 Broggi Andrea


= (- b +1/2 b -1/6 b +1/8 b ) Fb 1/EJ = -13/24 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 882. mm2

Ju = 278746. mm4

Jv = 79974. mm4



τc = 9.485 N/mm2

σo = √σ2+3τ2 = 144.7 N/mm2

S* = 5655. mm3mm 0 12 18 24 30 42x

0

12

42

53

y

41σc,τc

σm

u

v

Ing Civ.brga.028


Ing Civ.brga.028


Ing Civ.brga.028


Ing Civ.allf.029REAZIONI 886516 Allegra Filippo


43/40F3/2Fb

43/40F17/40Fb

AB

F

43/40F

43/40F1/2Fb

C

A

F

3/40F23/40Fb

F

37/40FFb

D E

F

43/40F17/40Fb

F

43/40F23/40Fb

B

D

Ing Civ.allf.029AZIONI INTERNE 886516 Allegra Filippo


0

-43/40 -43/40

-1-1

-43/40

F

-43/

40

1 0

3/40

-37/

40

-1

F

3/2

17/4

0

0 1/2

-23/

40-1

17/40-23/40

Fb

Ing

Civ

.allf

.029

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 886

516

Alle

gra

Fili

ppo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o3/21

01/

2

0-1

10

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.allf

.029

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 886

516

Alle

gra

Fili

ppo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

1/2F

x-3

/2F

x+1/

2Fx2 /b

x2 /b2

-7/1

2Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-Fb-

1/2F

x-F

b+1/

2Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-2

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E23

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/6

b ) F

b 1/

EJ

= -

7/12

Fb2 /E

J

LXo

BA =

∫ ob (-1 +

1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[- x

+1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.allf.029PROCEDIMENTO E RISULTATI 886516 Allegra Filippo


= (- b +1/4 b +1/6 b ) Fb 1/EJ = -7/12 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/4 b -1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 810. mm2

Ju = 227958. mm4

Jv = 77166. mm4



τc = 5.819 N/mm2

σo = √σ2+3τ2 = 167.8 N/mm2

S* = 3980. mm3mm 0 12 18 24 30 42x

0

12

48

53

y

42σc,τc

σm

u

v

Ing Civ.allf.029


Ing Civ.allf.029


Ing Civ.allf.029


Ing Civ.dpdv.030REAZIONI 886876 De Padova Matteo


19/40F1/40Fb

19/40F1/40Fb

A

B

21/40F1/40Fb

61/40FFb

B C

F

19/40F

19/40F1/2Fb

D

E

19/40F1/2Fb

19/40F1/40Fb

EA

Ing Civ.dpdv.030AZIONI INTERNE 886876 De Padova Matteo


19/4000

19/4019/40

0

F

0

-21/

40-6

1/40

-10

-19/

40

F

1/401/40

1/40

-1

0-1/21/

21/

40

Fb

Ing

Civ

.dpd

v.03

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8687

6 D

e P

adov

a M

atte

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

00 0

-1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dpd

v.03

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8687

6 D

e P

adov

a M

atte

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li1/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-1

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

BC =

∫ ob (1/2

x/b

-1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/4

x2 /b

-1/

8 x4 /b

3 ] ob Fb

1/E

J

= (1

/4 b

-1/

8 b

) Fb

1/E

J =

1/8

Fb2 /E

J

LXo

CB =

∫ ob ( x/b

-3/

2 x2 /b

2 +1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

2 x3 /b

2 +1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.dpdv.030PROCEDIMENTO E RISULTATI 886876 De Padova Matteo


= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

A = 984. mm2

Ju = 322959. mm4

Jv = 73152. mm4



τc = 9.008 N/mm2

σo = √σ2+3τ2 = 119.9 N/mm2

S* = 6370. mm3mm 0 12 18 30 36 48x

0

6

42

53

y

41σc,τc

σm

u

v

Ing Civ.dpdv.030


Ing Civ.dpdv.030


Ing Civ.dpdv.030


Ing Civ.bssm.031REAZIONI 886998 Bassi Marianna


43/40F3/2Fb

43/40F17/40Fb

AB

F

43/40F

43/40F1/2Fb

C

A

F

3/40F23/40Fb

F

37/40FFb

D E

F

43/40F17/40Fb

F

43/40F23/40Fb

B

D

Ing Civ.bssm.031AZIONI INTERNE 886998 Bassi Marianna


0

43/4

043

/40

1 1

43/4

0

F

-43/40

10

3/40-37/40

-1

F

3/217/40

01/

2

-23/40 -1

17/4

0-2

3/40

Fb

Ing

Civ

.bss

m.0

31P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8699

8 B

assi

Mar

iann

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

1

0 1/2

0-1

1 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bss

m.0

31P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8699

8 B

assi

Mar

iann

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

1/2F

x-3

/2F

x+1/

2Fx2 /b

x2 /b2

-7/1

2Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-Fb-

1/2F

x-F

b+1/

2Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-2

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E23

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/6

b ) F

b 1/

EJ

= -

7/12

Fb2 /E

J

LXo

BA =

∫ ob (-1 +

1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[- x

+1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bssm.031PROCEDIMENTO E RISULTATI 886998 Bassi Marianna


= (- b +1/4 b +1/6 b ) Fb 1/EJ = -7/12 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/4 b -1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 1128. mm2

Ju = 321538. mm4

Jv = 121536. mm4



τc = 6.952 N/mm2

σo = √σ2+3τ2 = 137.5 N/mm2

S* = 5474. mm3mm 0 12 18 30 36 48x

0

12

48

53

y

42σc,τc

σm

u

v

Ing Civ.bssm.031


Ing Civ.bssm.031


Ing Civ.bssm.031


Ing Civ.brss.032REAZIONI 887236 Berselli Samuele


4F

19/40F79/40Fb

4F

19/40F81/40Fb

A

B4F

21/40F81/40Fb

4F

61/40FFb

BC

F

19/40F

19/40F1/2Fb

D

E19/40F

3/2Fb19/40F

79/40Fb

E A

Ing Civ.brss.032AZIONI INTERNE 887236 Berselli Samuele


19/4

0

-4-4

19/4

019

/40

0

F

-4

21/4061/40

10

19/40

F

79/4

0-8

1/40

-81/40-1

01/

2

3/2 79/40

Fb

Ing

Civ

.brs

s.03

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8723

6 B

erse

lli S

amue

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-4

-4-1

0 1/2 3/2

0

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brs

s.03

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8723

6 B

erse

lli S

amue

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1-4Fx4Fx12Fb

2/EJXb/EJ

BA b14Fb-4Fx4Fb-4Fx1

BC b-1+x/b-4Fb+5/2Fx+1/2qx2

4Fb-13/2Fx+2Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

37/24Fb2/EJ1/3Xb/EJ

CB bx/bFb+7/2Fx-1/2qx2

Fx+7/2Fx2/b-1/2qx

3/bx

2/b

2

DE b0Fx-1/2qx2

0000

ED b0-1/2Fb+1/2qx2

00

EA b-x/b3/2Fb-3/2Fx-3/2Fx+3/2Fx2/bx

2/b

2

-1/4Fb2/EJ1/3Xb/EJ

AE b1-x/b-3/2Fx-3/2Fx+3/2Fx2/b1-2x/b+x

2/b

2


iperstatica X=WAB-79/40Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

Ing Civ.brss.032PROCEDIMENTO E RISULTATI 887236 Berselli Samuele


LXXCB = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXEA = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXAE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBA = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (4 b -13/4 b +2/3 b +1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b +7/6 b -1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

Ing Civ.brss.032PROCEDIMENTO E RISULTATI 887236 Berselli Samuele


A = 594. mm2

Ju = 206801. mm4

Jv = 40230. mm4



τc = 8.081 N/mm2

σo = √σ2+3τ2 = 158.5 N/mm2

S* = 3326. mm3mm 0 12 18 24 30 42x

0

6

48

53

y

44σc,τc

σm

u

v

Ing Civ.brss.032


Ing Civ.brss.032


Ing Civ.czzg.033REAZIONI 887287 Cazzagon Giovanni


17/20F3/2Fb

17/20F13/20Fb

AB

F

17/20F

17/20F1/2Fb

C

A

2F

3/20F17/20Fb

2F

3/20FFb

D E

F

17/20F13/20Fb

2F

17/20F17/20Fb

B

D

Ing Civ.czzg.033AZIONI INTERNE 887287 Cazzagon Giovanni


0

-17/20 -17/20

-2

-17/20 -17/20

F

-17/

20

1 0

-3/2

0

-1 -2

F

3/2

13/2

0

0 1/2

-17/

20-1

13/20-17/20

Fb

Ing

Civ

.czz

g.03

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8728

7 C

azza

gon

Gio

vann

i

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o3/23/2

01/

2

0-1

3/2

0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.czz

g.03

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8728

7 C

azza

gon

Gio

vann

i

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb

-3/2

Fx

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-3/2

Fb

-3/2

Fb+

3/2F

x1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

BD

b-1

3/2F

b-F

x-1/

2qx2

-3/2

Fb+

Fx+

1/2F

x2 /b1

-5/6

Fb2 /E

JX

b/E

JD

B b

1-2

Fx+

1/2q

x2-2

Fx+

1/2F

x2 /b1

tota

li-1

7/12

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E17

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

) F

b 1/

EJ

dx =

[-3/

4 x2 /b

] ob Fb

1/E

J

= (-

3/4

b ) F

b 1/

EJ

= -

3/4

Fb2 /E

J

LXo

BA =

∫ ob (-3/2

+3/

2 x/

b ) F

b 1/

EJ

dx =

[-3/

2 x

+3/

4 x2 /b

] ob Fb

1/E

J

Ing Civ.czzg.033PROCEDIMENTO E RISULTATI 887287 Cazzagon Giovanni


= (-3/2 b +3/4 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBD = ∫

o

b(-3/2 + x/b +1/2 x2/b2 ) Fb 1/EJ dx = [-3/2 x +1/2 x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (-3/2 b +1/2 b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

LXoDB = ∫

o

b(-2 x/b +1/2 x2/b2 ) Fb 1/EJ dx = [- x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (- b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

A = 654. mm2

Ju = 244740. mm4

Jv = 46890. mm4



τc = 8.053 N/mm2

σo = √σ2+3τ2 = 174.4 N/mm2

S* = 3678. mm3mm 0 12 18 24 30 42x

0

6

48

55

y

8σc,τc

σm

u

v

Ing Civ.czzg.033


Ing Civ.czzg.033


Ing Civ.czzg.033


Ing Civ.dnda.034REAZIONI 887657 Donadio Aurora


2F

21/20F21/20Fb

2F

21/20F

AB

3F

1/20F29/20Fb

2F

1/20F21/20Fb

C

A

1/20F1/2Fb

1/20F9/20Fb

D C

F

1/20F

1/20F1/2Fb

E

D

Ing Civ.dnda.034AZIONI INTERNE 887657 Donadio Aurora


-2

-1/20-1/20

0

-1/20-1/20

F

21/2

0

-3-2

-1/2

0

-10

F

-21/

200

29/20-21/20

1/2

9/20

0-1/2

Fb

Ing

Civ

.dnd

a.03

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8765

7 D

onad

io A

uror

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

5/2

0

1/2 3/2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dnd

a.03

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8765

7 D

onad

io A

uror

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

BA

bx/

b0

0x2 /b

2

CA

b-1

5/2F

b-3F

x+1/

2qx2

-5/2

Fb+

3Fx-

1/2F

x2 /b1

-7/6

Fb2 /E

JX

b/E

JA

C b

1-2

Fx-

1/2q

x2-2

Fx-

1/2F

x2 /b1

DC

b-x

/b1/

2Fb+

Fx

-1/2

Fx-

Fx2 /b

x2 /b2

-7/1

2Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-3/2

Fb+

Fx

-3/2

Fb+

5/2F

x-F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-7

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B21

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

CA =

∫ ob (-5/2

+3

x/b

-1/2

x2 /b

2 ) F

b 1/

EJ

dx =

[-5/

2 x

+3/

2 x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

5/2

b +

3/2

b -1

/6 b

) F

b 1/

EJ

= -

7/6

Fb2 /E

J

LXo

AC =

∫ ob (-2 x

/b -

1/2

x2 /b2 )

Fb

1/E

J dx

= [-

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.dnda.034PROCEDIMENTO E RISULTATI 887657 Donadio Aurora


= (- b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b - x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -1/3 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b -1/3 b ) Fb 1/EJ = -7/12 Fb2/EJ

LXoCD = ∫

o

b(-3/2 +5/2 x/b - x2/b2 ) Fb 1/EJ dx = [-3/2 x +5/4 x2/b -1/3 x3/b2 ]o

b Fb 1/EJ

= (-3/2 b +5/4 b -1/3 b ) Fb 1/EJ = -7/12 Fb2/EJ

A = 870. mm2

Ju = 264856. mm4

Jv = 83826. mm4

yg = 36.77 mmN = -298.5 NTy = -2985. NMx = -1723840. Nmmxm = 12. mmum = -9. mmvm = -36.77 mmσm = N/A-Mv/Ju = -239.7 N/mm2


τc = 8.147 N/mm2

σo = √σ2+3τ2 = 175.1 N/mm2

S* = 4338. mm3mm 0 12 18 24 30 42x

0

6

42

55

y

10σc,τc

σm

u

v

Ing Civ.dnda.034


Ing Civ.dnda.034


Ing Civ.dnda.034


Ing Civ.clmc.035REAZIONI 887860 Colombo Cristina


17/20F3/2Fb

17/20F13/20Fb

AB

F

17/20F

17/20F1/2Fb

C

A

2F

3/20F17/20Fb

2F

3/20FFb

D E

F

17/20F13/20Fb

2F

17/20F17/20Fb

B

D

Ing Civ.clmc.035AZIONI INTERNE 887860 Colombo Cristina


0

17/2

017

/20

2

17/2

017

/20

F

-17/20

10

-3/20

-1-2

F

3/213/20

01/

2

-17/20 -1

13/2

0-1

7/20

Fb

Ing

Civ

.clm

c.03

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8786

0 C

olom

bo C

ristin

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

3/2

0 1/2

0-1

3/2 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.clm

c.03

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8786

0 C

olom

bo C

ristin

a

@ A

dolfo

Zav

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i Ros

si, P

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cnic

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb

-3/2

Fx

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-3/2

Fb

-3/2

Fb+

3/2F

x1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

BD

b-1

3/2F

b-F

x-1/

2qx2

-3/2

Fb+

Fx+

1/2F

x2 /b1

-5/6

Fb2 /E

JX

b/E

JD

B b

1-2

Fx+

1/2q

x2-2

Fx+

1/2F

x2 /b1

tota

li-1

7/12

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E17

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

) F

b 1/

EJ

dx =

[-3/

4 x2 /b

] ob Fb

1/E

J

= (-

3/4

b ) F

b 1/

EJ

= -

3/4

Fb2 /E

J

LXo

BA =

∫ ob (-3/2

+3/

2 x/

b ) F

b 1/

EJ

dx =

[-3/

2 x

+3/

4 x2 /b

] ob Fb

1/E

J

Ing Civ.clmc.035PROCEDIMENTO E RISULTATI 887860 Colombo Cristina


= (-3/2 b +3/4 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBD = ∫

o


b Fb 1/EJ

= (-3/2 b +1/2 b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

LXoDB = ∫

o


b Fb 1/EJ

= (- b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

A = 984. mm2

Ju = 337339. mm4

Jv = 77472. mm4



τc = 4.584 N/mm2

σo = √σ2+3τ2 = 133. N/mm2

S* = 5556. mm3mm 0 12 18 30 36 48x

0

6

48

55

y

11σc,τc

σm

u

v

Ing Civ.clmc.035


Ing Civ.clmc.035


Ing Civ.clmc.035


Ing Civ.cnns.036REAZIONI 887904 Cannarsa Simone


2F

21/20F21/20Fb

2F

21/20F

AB

3F

1/20F29/20Fb

2F

1/20F21/20Fb

C

A

1/20F1/2Fb

1/20F9/20Fb

D C

F

1/20F

1/20F1/2Fb

E

D

Ing Civ.cnns.036AZIONI INTERNE 887904 Cannarsa Simone


2

1/20

1/20

0

1/20

1/20

F

21/20

-3-2

-1/20

-10

F

-21/200

29/2

0-2

1/20

1/2 9/20

0-1

/2

Fb

Ing

Civ

.cnn

s.03

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8790

4 C

anna

rsa

Sim

one

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

5/20

1/2

3/2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cnn

s.03

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8790

4 C

anna

rsa

Sim

one

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

BA

bx/

b0

0x2 /b

2

CA

b-1

5/2F

b-3F

x+1/

2qx2

-5/2

Fb+

3Fx-

1/2F

x2 /b1

-7/6

Fb2 /E

JX

b/E

JA

C b

1-2

Fx-

1/2q

x2-2

Fx-

1/2F

x2 /b1

DC

b-x

/b1/

2Fb+

Fx

-1/2

Fx-

Fx2 /b

x2 /b2

-7/1

2Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-3/2

Fb+

Fx

-3/2

Fb+

5/2F

x-F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-7

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B21

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

CA =

∫ ob (-5/2

+3

x/b

-1/2

x2 /b

2 ) F

b 1/

EJ

dx =

[-5/

2 x

+3/

2 x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

5/2

b +

3/2

b -1

/6 b

) F

b 1/

EJ

= -

7/6

Fb2 /E

J

LXo

AC =

∫ ob (-2 x

/b -

1/2

x2 /b2 )

Fb

1/E

J dx

= [-

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.cnns.036PROCEDIMENTO E RISULTATI 887904 Cannarsa Simone


= (- b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b - x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -1/3 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b -1/3 b ) Fb 1/EJ = -7/12 Fb2/EJ

LXoCD = ∫

o

b(-3/2 +5/2 x/b - x2/b2 ) Fb 1/EJ dx = [-3/2 x +5/4 x2/b -1/3 x3/b2 ]o

b Fb 1/EJ

= (-3/2 b +5/4 b -1/3 b ) Fb 1/EJ = -7/12 Fb2/EJ

A = 1200. mm2

Ju = 368598. mm4

Jv = 131904. mm4



τc = 4.894 N/mm2

σo = √σ2+3τ2 = 136.4 N/mm2

S* = 6312. mm3mm 0 12 18 30 36 48x

0

6

42

55

y

12σc,τc

σm

u

v

Ing Civ.cnns.036


Ing Civ.cnns.036


Ing Civ.cnns.036


Ing Civ.clmg.037REAZIONI 888192 Colombo Giacomo Paolo


F

3/4F

3/4F1/2Fb

A

B

3/4F3/2Fb

3/4F3/4Fb

B C 3F

3/4F7/4Fb

4F

3/4F7/4Fb

C

D4F

7/4F7/4Fb

4F

7/4F

DE

Ing Civ.clmg.037AZIONI INTERNE 888192 Colombo Giacomo Paolo


3/4 3/4

0

3/4 3/4

4

F1 0

-3/4

-3 -4

7/4

F0 1/2

3/2

3/4

7/4-7/4

-7/4

0

Fb

Ing

Civ

.clm

g.03

7P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8819

2 C

olom

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olite

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no, v

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27.0

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31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2-1

0-7

/2-7/2

0M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.clm

g.03

7P

RO

CE

DIM

EN

TO

E R

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27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

5/2F

x-3

/2F

x+5/

2Fx2 /b

x2 /b2

1/12

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

Fb-

5/2F

xF

b-7/

2Fx+

5/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+7/

2Fx

7/2F

b-7F

x+7/

2Fx2 /b

1-2x

/b+

x2 /b2

7/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

7/2F

x7/

2Fx2 /b

x2 /b2

tota

li35

/12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/4F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+5/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+5/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

5/6

b ) F

b 1/

EJ

= 1

/12

Fb2 /E

J

LXo

CB =

∫ ob (1 -

7/2

x/b

+5/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-7/4

x2 /b

+5/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.clmg.037PROCEDIMENTO E RISULTATI 888192 Colombo Giacomo


= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o

b(7/2 -7 x/b +7/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -7/2 x2/b +7/6 x3/b2 ]o

b Fb 1/EJ

= (7/2 b -7/2 b +7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoED = ∫

o

b(7/2 x2/b2 ) Fb 1/EJ dx = [7/6 x3/b2 ]o

b Fb 1/EJ

= (7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

A = 726. mm2

Ju = 285657. mm4

Jv = 49698. mm4



τc = 9.126 N/mm2

σo = √σ2+3τ2 = 132.5 N/mm2

S* = 5637. mm3mm 0 12 18 24 30 42x

0

12

48

55

y

13σc,τc

σm

u

v

Ing Civ.clmg.037


Ing Civ.clmg.037


Ing Civ.clmg.037


Ing Civ.bnda.038REAZIONI 888197 Bendo Alessandro


F

21/20F9/20Fb

21/20F1/20Fb

A

B1/20F

1/20Fb1/20F

B C

F

21/20F

21/20F1/2Fb

D

E21/20F1/2Fb

21/20F11/20Fb

EA

Ing Civ.bnda.038AZIONI INTERNE 888197 Bendo Alessandro


21/2021/20

0

21/2021/20

0

F

-10

1/20

-10

-21/

20

F

9/20-1/20

-1/2

00

0-1/21/

2-1

1/20

Fb

Ing

Civ

.bnd

a.03

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8819

7 B

endo

Ale

ssan

dro

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/20

0-1

/2

1/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bnd

a.03

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8819

7 B

endo

Ale

ssan

dro

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b+1/

2Fx

1/2F

b-F

x+1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

1/2F

x1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JA

E b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

tota

li3/

4Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B-9

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.bnda.038PROCEDIMENTO E RISULTATI 888197 Bendo Alessandro


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o

b(1/2 - x/b +1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCB = ∫

o

b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o

b Fb 1/EJ

= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoAE = ∫

o

b(1 -5/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [ x -5/4 x2/b +1/2 x3/b2 ]o

b Fb 1/EJ

= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

A = 942. mm2

Ju = 316348. mm4

Jv = 86634. mm4



τc = 9.673 N/mm2

σo = √σ2+3τ2 = 142.1 N/mm2

S* = 6321. mm3mm 0 12 18 24 30 42x

0

12

42

55

y

13σc,τc

σm

u

v

Ing Civ.bnda.038


Ing Civ.bnda.038


Ing Civ.bnda.038


Ing Civ.frsa.039REAZIONI 888228 Farisè Alberto


F

21/20F

21/20F1/2Fb

A

B

21/20F1/2Fb

21/20F11/20Fb

BCF

21/20F9/20Fb

21/20F1/20Fb

C

D

1/20F1/20Fb

1/20F

D E

Ing Civ.frsa.039AZIONI INTERNE 888228 Farisè Alberto


-21/

20-2

1/200

-21/

20-2

1/20

0

F

-10

-21/20

-10

1/20

F

0-1

/21/2-11/209/

20-1

/20-1/20

0

Fb

Ing

Civ

.frsa

.039

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

228

Far

isè

Alb

erto

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

-1

0-1/2-1

/20

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.frsa

.039

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

228

Far

isè

Alb

erto

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JC

B b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

CD

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JD

C b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

DE

b-1

+x/

b-1

/2F

b+1/

2Fx

1/2F

b-F

x+1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

1/2F

x1/

2Fx2 /b

x2 /b2

tota

li3/

4Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WC

D-9

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/2

b ) F

b 1/

EJ

= 1

/4 F

b2 /EJ

LXo

CB =

∫ ob (1 -

5/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-5/4

x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.frsa.039PROCEDIMENTO E RISULTATI 888228 Farisè Alberto


= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o

b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o

b Fb 1/EJ

= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

A = 1056. mm2

Ju = 372596. mm4

Jv = 83520. mm4



τc = 5.911 N/mm2

σo = √σ2+3τ2 = 140.4 N/mm2

S* = 7211. mm3mm 0 12 18 30 36 48x

0

12

48

55

y

13σc,τc

σm

u

v

Ing Civ.frsa.039


Ing Civ.frsa.039


Ing Civ.frsa.039


Ing Civ.bsns.040REAZIONI 888270 Biasinutto Sara


3F

3/4F7/4Fb

4F

3/4F7/4Fb

A

B4F

7/4F7/4Fb

4F

7/4F

BC

F

3/4F

3/4F1/2Fb

D

E

3/4F3/2Fb

3/4F3/4Fb

E A

Ing Civ.bsns.040AZIONI INTERNE 888270 Biasinutto Sara


-3/4

-3/4

-4

-3/4

-3/4

0

F

-3-4

7/4

10

-3/4

F

7/4

-7/4

-7/40

01/

2

3/2 3/4

Fb

Ing

Civ

.bsn

s.04

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8827

0 B

iasi

nutto

Sar

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-7/2

-7/2

00 1/2 3/

2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bsn

s.04

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8827

0 B

iasi

nutto

Sar

a

@ A

dolfo

Zav

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olite

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

BA

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

BC

b-1

+x/

b-7

/2F

b+7/

2Fx

7/2F

b-7F

x+7/

2Fx2 /b

1-2x

/b+

x2 /b2

7/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

7/2F

x7/

2Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

5/2F

x-3

/2F

x+5/

2Fx2 /b

x2 /b2

1/12

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

5/2F

xF

b-7/

2Fx+

5/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li35

/12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/4F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3 x

/b +

1/2

x2 /b2 )

Fb

1/E

J dx

= [3

/2 x

2 /b +

1/6

x3 /b2 ] ob F

b 1/

EJ

= (3

/2 b

+1/

6 b

) Fb

1/E

J =

5/3

Fb2 /E

J

LXo

BA =

∫ ob (7/2

-4

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[7/2

x -

2 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bsns.040PROCEDIMENTO E RISULTATI 888270 Biasinutto Sara


= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (7/2 b -7/2 b +7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoCB = ∫

o

b(7/2 x2/b2 ) Fb 1/EJ dx = [7/6 x3/b2 ]o

b Fb 1/EJ

= (7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

A = 1272. mm2

Ju = 412017. mm4

Jv = 137952. mm4



τc = 10.17 N/mm2

σo = √σ2+3τ2 = 119. N/mm2

S* = 7960. mm3mm 0 12 18 30 36 48x

0

12

42

55

y

13σc,τc

σm

u

v

Ing Civ.bsns.040


Ing Civ.bsns.040


Ing Civ.bsns.040


Ing Civ.dvta.041REAZIONI 888341 De Vita Alessandro


F

17/40F

17/40F1/2Fb

A

B

17/40F3/2Fb

17/40F43/40Fb

B C 4F

17/40F83/40Fb

4F

17/40F77/40Fb

C

D4F

57/40F77/40Fb

4F

97/40F

DE

Ing Civ.dvta.041AZIONI INTERNE 888341 De Vita Alessandro


17/40 17/40

0

17/40

44

F1 0

-17/

40

-4

57/4

097

/40

F0 1/2

3/2

43/4

0

83/40-77/40

-77/

400

Fb

Ing

Civ

.dvt

a.04

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8834

1 D

e V

ita A

less

andr

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2-1

0-4

-40

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dvt

a.04

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8834

1 D

e V

ita A

less

andr

o

@ A

dolfo

Zav

elan

i Ros

si, P

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Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b0Fx-1/2qx2

0000

BA b0-1/2Fb+1/2qx2

00


2/b

2

1/12Fb2/EJ1/3Xb/EJ

CB b1-x/bFb-5/2FxFb-7/2Fx+5/2Fx2/b1-2x/b+x

2/b

2

CD b-1-4Fx4Fx12Fb

2/EJXb/EJ



4Fb-15/2Fx+3Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

11/8Fb2/EJ1/3Xb/EJ

ED bx/b9/2Fx-1/2qx2

9/2Fx2/b-1/2qx

3/bx

2/b

2



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

Ing Civ.dvta.041PROCEDIMENTO E RISULTATI 888341 De Vita Alessandro


= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXXED = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o

b(4 -15/2 x/b +3 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [4 x -15/4 x2/b + x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (4 b -15/4 b + b +1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

LXoED = ∫

o

b(9/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= (3/2 b -1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

Ing Civ.dvta.041PROCEDIMENTO E RISULTATI 888341 De Vita Alessandro


A = 1248. mm2

Ju = 413282. mm4

Jv = 129888. mm4

yg = 22.68 mmN = 5172. NTy = 6085. NMx = 2738250. Nmmxm = 36. mmym = 55. mmum = 12. mmvm = 32.32 mmσm = N/A-Mv/Ju = -210. N/mm2


τc = 10.16 N/mm2

σo = √σ2+3τ2 = 118.5 N/mm2

S* = 8281. mm3mm 0 12 18 30 36 48x

0

12

42

55

y

41σc,τc

σm

u

v

Ing Civ.dvta.041


Ing Civ.dvta.041


Ing Civ.clla.042REAZIONI 888424 Collini Andrea


3F

3/40F57/40Fb

3F

37/40FFb

AB

3F

43/40F63/40Fb

3F

43/40F57/40Fb

C

A

43/40F1/2Fb

43/40F63/40Fb

D C

F

43/40F

43/40F1/2Fb

E

D

Ing Civ.clla.042AZIONI INTERNE 888424 Collini Andrea


-3-3

43/40

0

43/4043/40

F

-3/4

037

/40

-3

43/4

0

-10

F

-57/

40-1

63/40-57/40

1/2

63/4

0

0-1/2

Fb

Ing

Civ

.clla

.042

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

424

Col

lini A

ndre

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CDE

W

F

W X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2 3

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.clla

.042

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

424

Col

lini A

ndre

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-3

/2F

x+1/

2qx2

3/2F

x-2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

5/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

bF

b-1/

2Fx-

1/2q

x2F

x-1/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

5/2F

x-1

/2F

x-5/

2Fx2 /b

x2 /b2

-13/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-3

Fb+

5/2F

x-3

Fb+

11/2

Fx-

5/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-1

9/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B57

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3/2

x/b

-2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [3

/4 x

2 /b -

2/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (3

/4 b

-2/

3 b

+1/

8 b

) Fb

1/E

J =

5/2

4 F

b2 /EJ

LXo

BA =

∫ ob ( x/b

-1/

2 x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.clla.042PROCEDIMENTO E RISULTATI 888424 Collini Andrea


= (1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 5/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +11/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

A = 702. mm2

Ju = 278693. mm4

Jv = 44010. mm4



τc = 13.11 N/mm2

σo = √σ2+3τ2 = 126. N/mm2

S* = 5687. mm3mm 0 12 18 24 30 42x

0

6

42

55

y

41σc,τc

σm

u

v

Ing Civ.clla.042


Ing Civ.clla.042


Ing Civ.clla.042


Ing Civ.bstf.043REAZIONI 888472 Busetto Ferruccio


F

11/8F

11/8F1/2Fb

A

B

11/8F1/2Fb

11/8F7/8Fb

BC

11/8F1/8Fb

11/8F1/8Fb

C

D

3/8F1/8Fb

5/8F

D E

Ing Civ.bstf.043AZIONI INTERNE 888472 Busetto Ferruccio


-11/

8-1

1/80

-11/

8

0 0

F

-10

-11/8

0

3/8-5/8

F

0-1

/21/2-7/81/

81/

8

1/80

Fb

Ing

Civ

.bst

f.043

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

472

Bus

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Fer

rucc

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@ A

dolfo

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i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

-1

00

00

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bst

f.043

PR

OC

ED

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RIS

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AT

I 888

472

Bus

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Fer

rucc

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dolfo

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cnic

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Mila

no, v

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27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JC

B b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b1/

2Fx-

1/2q

x2-1

/2F

x+F

x2 /b-1

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3 /b1-

2x/b

+x2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

ED

bx/

b-1

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x+1/

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-1/2

Fx2 /b

+1/

2qx3 /b

x2 /b2

tota

li5/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-1

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b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/2

b ) F

b 1/

EJ

= 1

/4 F

b2 /EJ

LXo

CB =

∫ ob (1 -

5/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-5/4

x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bstf.043PROCEDIMENTO E RISULTATI 888472 Busetto Ferruccio


= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

A = 960. mm2

Ju = 334225. mm4

Jv = 69408. mm4



τc = 7.764 N/mm2

σo = √σ2+3τ2 = 149.7 N/mm2

S* = 6284. mm3mm 0 12 18 30 36 48x

0

6

48

55

y

7σc,τc

σm

u

v

Ing Civ.bstf.043


Ing Civ.bstf.043


Ing Civ.bstf.043


Ing Civ.frru.044REAZIONI 888759 Ferretti Umberto


3F

3/40F57/40Fb

3F

37/40FFb

AB

3F

43/40F63/40Fb

3F

43/40F57/40Fb

C

A

43/40F1/2Fb

43/40F63/40Fb

D C

F

43/40F

43/40F1/2Fb

E

D

Ing Civ.frru.044AZIONI INTERNE 888759 Ferretti Umberto


33

-43/

40

0

-43/

40-4

3/40

F

-3/4037/40

-3

43/40

-10

F

-57/40-1

63/4

0-5

7/40

1/2 63/40

0-1

/2

Fb

Ing

Civ

.frru

.044

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

759

Fer

retti

Um

bert

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

W

X

X

qq

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2

3

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.frru

.044

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

759

Fer

retti

Um

bert

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-3

/2F

x+1/

2qx2

3/2F

x-2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

5/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

bF

b-1/

2Fx-

1/2q

x2F

x-1/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

5/2F

x-1

/2F

x-5/

2Fx2 /b

x2 /b2

-13/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-3

Fb+

5/2F

x-3

Fb+

11/2

Fx-

5/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-1

9/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B57

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3/2

x/b

-2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [3

/4 x

2 /b -

2/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (3

/4 b

-2/

3 b

+1/

8 b

) Fb

1/E

J =

5/2

4 F

b2 /EJ

LXo

BA =

∫ ob ( x/b

-1/

2 x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.frru.044PROCEDIMENTO E RISULTATI 888759 Ferretti Umberto


= (1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 5/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +11/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

A = 918. mm2

Ju = 319684. mm4

Jv = 80946. mm4



τc = 13.89 N/mm2

σo = √σ2+3τ2 = 139.5 N/mm2

S* = 6570. mm3mm 0 12 18 24 30 42x

0

12

42

55

y

41σc,τc

σm

u

v

Ing Civ.frru.044


Ing Civ.frru.044


Ing Civ.frru.044


Ing Civ.fngl.045REAZIONI 888972 Fenoglio Lucia


F

3/4F

3/4F1/2Fb

A

B

3/4F3/2Fb

3/4F3/4Fb

B C 3F

3/4F7/4Fb

4F

3/4F7/4Fb

C

D4F

7/4F7/4Fb

4F

7/4F

DE

Ing Civ.fngl.045AZIONI INTERNE 888972 Fenoglio Lucia


3/4 3/4

0

3/4 3/4

4

F1 0

-3/4

-3 -4

7/4

F0 1/2

3/2

3/4

7/4-7/4

-7/4

0

Fb

Ing

Civ

.fngl

.045

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

972

Fen

oglio

Luc

ia

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2-1

0-7

/2-7/2

0M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.fngl

.045

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 888

972

Fen

oglio

Luc

ia

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

5/2F

x-3

/2F

x+5/

2Fx2 /b

x2 /b2

1/12

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

Fb-

5/2F

xF

b-7/

2Fx+

5/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+7/

2Fx

7/2F

b-7F

x+7/

2Fx2 /b

1-2x

/b+

x2 /b2

7/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

7/2F

x7/

2Fx2 /b

x2 /b2

tota

li35

/12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/4F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+5/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+5/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

5/6

b ) F

b 1/

EJ

= 1

/12

Fb2 /E

J

LXo

CB =

∫ ob (1 -

7/2

x/b

+5/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-7/4

x2 /b

+5/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.fngl.045PROCEDIMENTO E RISULTATI 888972 Fenoglio Lucia


= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (7/2 b -7/2 b +7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoED = ∫

o

b(7/2 x2/b2 ) Fb 1/EJ dx = [7/6 x3/b2 ]o

b Fb 1/EJ

= (7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

A = 846. mm2

Ju = 274282. mm4

Jv = 78138. mm4



τc = 6.819 N/mm2

σo = √σ2+3τ2 = 149.4 N/mm2

S* = 4444. mm3mm 0 12 18 24 30 42x

0

12

48

55

y

46σc,τc

σm

u

v

Ing Civ.fngl.045


Ing Civ.fngl.045


Ing Civ.fngl.045


Ing Civ.dnzs.046REAZIONI 889282 Donzelli Sara


F

21/20F9/20Fb

21/20F1/20Fb

A

B1/20F

1/20Fb1/20F

B C

F

21/20F

21/20F1/2Fb

D

E21/20F1/2Fb

21/20F11/20Fb

EA

Ing Civ.dnzs.046AZIONI INTERNE 889282 Donzelli Sara


21/2021/20

0

21/2021/20

0

F

-10

1/20

-10

-21/

20

F

9/20-1/20

-1/2

00

0-1/21/

2-1

1/20

Fb

Ing

Civ

.dnz

s.04

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8928

2 D

onze

lli S

ara

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/20

0-1

/2

1/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dnz

s.04

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8928

2 D

onze

lli S

ara

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b+1/

2Fx

1/2F

b-F

x+1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

1/2F

x1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JA

E b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

tota

li3/

4Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B-9

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.dnzs.046PROCEDIMENTO E RISULTATI 889282 Donzelli Sara


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCB = ∫

o

b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o

b Fb 1/EJ

= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

A = 1032. mm2

Ju = 363732. mm4

Jv = 75456. mm4

yg = 25.55 mmN = 7949. NTy = -3785. NMx = -2498100. Nmmxm = 36. mmym = 55. mmum = 12. mmvm = 29.45 mmσm = N/A-Mv/Ju = 210. N/mm2


τc = 6.376 N/mm2

σo = √σ2+3τ2 = 114.4 N/mm2

S* = 7353. mm3mm 0 12 18 30 36 48x

0

6

42

55

y

41σc,τc

σm

u

v

Ing Civ.dnzs.046


Ing Civ.dnzs.046


Ing Civ.dnzs.046


Ing Civ.blll.047REAZIONI 889389 Bello Lorenzo


F

21/20F

21/20F1/2Fb

A

B

21/20F1/2Fb

21/20F11/20Fb

BCF

21/20F9/20Fb

21/20F1/20Fb

C

D

1/20F1/20Fb

1/20F

D E

Ing Civ.blll.047AZIONI INTERNE 889389 Bello Lorenzo


-21/

20-2

1/200

-21/

20-2

1/20

0

F

-10

-21/20

-10

1/20

F

0-1

/21/2-11/209/

20-1

/20-1/20

0

Fb

Ing

Civ

.blll

.047

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

389

Bel

lo L

oren

zo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

-1

0-1/2-1

/20

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.blll

.047

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

389

Bel

lo L

oren

zo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JC

B b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

CD

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JD

C b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

DE

b-1

+x/

b-1

/2F

b+1/

2Fx

1/2F

b-F

x+1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/6F

b2 /EJ

1/3X

b/E

JE

D b

x/b

1/2F

x1/

2Fx2 /b

x2 /b2

tota

li3/

4Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WC

D-9

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/2

b ) F

b 1/

EJ

= 1

/4 F

b2 /EJ

LXo

CB =

∫ ob (1 -

5/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-5/4

x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.blll.047PROCEDIMENTO E RISULTATI 889389 Bello Lorenzo


= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoED = ∫

o

b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o

b Fb 1/EJ

= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ

A = 1176. mm2

Ju = 375010. mm4

Jv = 123840. mm4



τc = 5.018 N/mm2

σo = √σ2+3τ2 = 146.1 N/mm2

S* = 6256. mm3mm 0 12 18 30 36 48x

0

12

48

55

y

44σc,τc

σm

u

v

Ing Civ.blll.047


Ing Civ.blll.047


Ing Civ.blll.047


Ing Civ.brnl.048REAZIONI 889394 Brandizi Leonardo


3F

3/4F7/4Fb

4F

3/4F7/4Fb

A

B4F

7/4F7/4Fb

4F

7/4F

BC

F

3/4F

3/4F1/2Fb

D

E

3/4F3/2Fb

3/4F3/4Fb

E A

Ing Civ.brnl.048AZIONI INTERNE 889394 Brandizi Leonardo


-3/4

-3/4

-4

-3/4

-3/4

0

F

-3-4

7/4

10

-3/4

F

7/4

-7/4

-7/40

01/

2

3/2 3/4

Fb

Ing

Civ

.brn

l.048

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

394

Bra

ndiz

i Leo

nard

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-7/2

-7/2

00 1/2 3/

2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brn

l.048

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

394

Bra

ndiz

i Leo

nard

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

BA

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

BC

b-1

+x/

b-7

/2F

b+7/

2Fx

7/2F

b-7F

x+7/

2Fx2 /b

1-2x

/b+

x2 /b2

7/6F

b2 /EJ

1/3X

b/E

JC

B b

x/b

7/2F

x7/

2Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

5/2F

x-3

/2F

x+5/

2Fx2 /b

x2 /b2

1/12

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

5/2F

xF

b-7/

2Fx+

5/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li35

/12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/4F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3 x

/b +

1/2

x2 /b2 )

Fb

1/E

J dx

= [3

/2 x

2 /b +

1/6

x3 /b2 ] ob F

b 1/

EJ

= (3

/2 b

+1/

6 b

) Fb

1/E

J =

5/3

Fb2 /E

J

LXo

BA =

∫ ob (7/2

-4

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[7/2

x -

2 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.brnl.048PROCEDIMENTO E RISULTATI 889394 Brandizi Leonardo


= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (7/2 b -7/2 b +7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoCB = ∫

o

b(7/2 x2/b2 ) Fb 1/EJ dx = [7/6 x3/b2 ]o

b Fb 1/EJ

= (7/6 b ) Fb 1/EJ = 7/6 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

A = 630. mm2

Ju = 245181. mm4

Jv = 41202. mm4



τc = 5.898 N/mm2

σo = √σ2+3τ2 = 175.1 N/mm2

S* = 3830. mm3mm 0 12 18 24 30 42x

0

6

48

55

y

47σc,τc

σm

u

v

Ing Civ.brnl.048


Ing Civ.brnl.048


Ing Civ.brnl.048


Ing Civ.crmp.049REAZIONI 889561 Cremona Paolo


3F

19/40F59/40Fb

3F

19/40FFb

AB

3F

21/40F61/40Fb

3F

21/40F59/40Fb

C

A

61/40F1/2Fb

21/40F61/40Fb

D C

F

61/40F

61/40F1/2Fb

E

D

Ing Civ.crmp.049AZIONI INTERNE 889561 Cremona Paolo


-3

21/40

00

61/4061/40

F

19/4

0

-3

61/4

021

/40

-10

F

-59/

40-1

61/40-59/40

1/2

61/4

0

0-1/2

Fb

Ing

Civ

.crm

p.04

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8956

1 C

rem

ona

Pao

lo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CDE

W

F

W X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2 3

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crm

p.04

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8956

1 C

rem

ona

Pao

lo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1+x/b-FxFx-Fx2/b1-2x/b+x

2/b

2

1/6Fb2/EJ1/3Xb/EJ

BA bx/bFb-FxFx-Fx2/bx

2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-3Fx2/b+1/2qx

3/bx

2/b

2

-9/8Fb2/EJ1/3Xb/EJ

CD b1-x/b-3Fb+2Fx+1/2qx2

-3Fb+5Fx-3/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.crmp.049PROCEDIMENTO E RISULTATI 889561 Cremona Paolo


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBA = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b -3 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-1/4 x2/b - x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-1/4 b - b +1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +5/2 b -1/2 b -1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

Ing Civ.crmp.049PROCEDIMENTO E RISULTATI 889561 Cremona Paolo


A = 648. mm2

Ju = 233472. mm4

Jv = 46872. mm4



τc = 5.886 N/mm2

σo = √σ2+3τ2 = 180.2 N/mm2

S* = 3609. mm3mm 0 12 18 24 30 42x

0

6

47

54

y

8σc,τc

σm

u

v

Ing Civ.crmp.049


Ing Civ.crmp.049


Ing Civ.crcr.050REAZIONI 889619 Cura Curà Pietro


33/40F7/40Fb

33/40F7/40Fb

A

B7/40F

7/40Fb7/40F

B C

F

73/40F

73/40F1/2Fb

D

E73/40F1/2Fb

33/40F33/40Fb

EA

Ing Civ.crcr.050AZIONI INTERNE 889619 Cura Curà Pietro


33/40

0

73/4073/40

00

F

0

-7/4

0

-10

-73/

40-3

3/40

F

7/407/40

7/40

0

0-1/21/

2-3

3/40

Fb

Ing

Civ

.crc

r.05

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8961

9 C

ura

Cur

à P

ietr

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00 0 0

0-1

/2

1/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crc

r.05

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

8961

9 C

ura

Cur

à P

ietr

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

CB

bx/

b0

0x2 /b

2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

2Fx+

1/2q

x2-1

/2F

x+2F

x2 /b-1

/2qx

3 /bx2 /b

2

7/24

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

Fx-

1/2q

x2F

b-2F

x+1/

2Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

tota

li7/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

EA =

∫ ob (-1/2

x/b

+2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/4

x2 /b +

2/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

1/4

b +

2/3

b -1

/8 b

) F

b 1/

EJ

= 7

/24

Fb2 /E

J

LXo

AE =

∫ ob (1 -

2 x/

b +

1/2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [

x -

x2 /b +

1/6

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

Ing Civ.crcr.050PROCEDIMENTO E RISULTATI 889619 Cura Curà Pietro


= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ

A = 864. mm2

Ju = 252075. mm4

Jv = 83808. mm4



τc = 10.05 N/mm2

σo = √σ2+3τ2 = 147.1 N/mm2

S* = 4090. mm3mm 0 12 18 24 30 42x

0

6

41

54

y

9σc,τc

σm

uv

Ing Civ.crcr.050


Ing Civ.crcr.050


Ing Civ.crcr.050


Ing Civ.grnl.051REAZIONI 889847 Guarneri Ludovico


3F

19/40F59/40Fb

3F

19/40FFb

AB

3F

21/40F61/40Fb

3F

21/40F59/40Fb

C

A

61/40F1/2Fb

21/40F61/40Fb

D C

F

61/40F

61/40F1/2Fb

E

D

Ing Civ.grnl.051AZIONI INTERNE 889847 Guarneri Ludovico


3

-21/

40

0 0

-61/

40-6

1/40

F

19/40

-3

61/40 21/40

-10

F

-59/40-1

61/4

0-5

9/40

1/2 61/40

0-1

/2

Fb

Ing

Civ

.grn

l.051

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

847

Gua

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i Lud

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@ A

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olite

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2

3

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.grn

l.051

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 889

847

Gua

rner

i Lud

ovic

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19




2/b

2

1/6Fb2/EJ1/3Xb/EJ


2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-3Fx2/b+1/2qx

3/bx

2/b

2

-9/8Fb2/EJ1/3Xb/EJ


-3Fb+5Fx-3/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.grnl.051PROCEDIMENTO E RISULTATI 889847 Guarneri Ludovico


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBA = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b - b +1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +5/2 b -1/2 b -1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

Ing Civ.grnl.051PROCEDIMENTO E RISULTATI 889847 Guarneri Ludovico


A = 972. mm2

Ju = 321252. mm4

Jv = 77328. mm4



τc = 6.407 N/mm2

σo = √σ2+3τ2 = 141.6 N/mm2

S* = 5446. mm3mm 0 12 18 30 36 48x

0

6

47

54

y

11σc,τc

σm

u

v

Ing Civ.grnl.051


Ing Civ.grnl.051


Ing Civ.bnga.052REAZIONI 890108 Bongiovanni Antonio


4F

39/40F81/40Fb

4F

39/40F79/40Fb

A

B4F

79/40F79/40Fb

4F

79/40F

BC

F

1/40F

1/40F1/2Fb

D

E

1/40F3/2Fb

39/40F41/40Fb

E A

Ing Civ.bnga.052AZIONI INTERNE 890108 Bongiovanni Antonio


-39/

40

-4

1/40

1/40

0 0

F

-4

79/40

10

1/40-39/40

F

81/4

0-7

9/40

-79/400

01/

2

3/2 41/40

Fb

Ing

Civ

.bng

a.05

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9010

8 B

ongi

ovan

ni A

nton

io

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-4

-40

0 1/2 3/2

-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bng

a.05

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9010

8 B

ongi

ovan

ni A

nton

io

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JB

A b

14F

b-4F

x4F

b-4F

x1

BC

b-1

+x/

b-4

Fb+

4Fx

4Fb-

8Fx+

4Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

4Fx

4Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

2Fx-

1/2q

x2-3

/2F

x+2F

x2 /b+

1/2q

x3 /bx2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

3Fx+

1/2q

x2F

b-4F

x+7/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

tota

li27

/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-8

1/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (4 x

/b )

Fb

1/E

J dx

= [2

x2 /b

] ob Fb

1/E

J

= (2

b )

Fb

1/E

J =

2 F

b2 /EJ

LXo

BA =

∫ ob (4 -

4 x/

b ) F

b 1/

EJ

dx =

[4 x

-2

x2 /b ] ob F

b 1/

EJ

Ing Civ.bnga.052PROCEDIMENTO E RISULTATI 890108 Bongiovanni Antonio


= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o

b(4 -8 x/b +4 x2/b2 ) Fb 1/EJ dx = [4 x -4 x2/b +4/3 x3/b2 ]o

b Fb 1/EJ

= (4 b -4 b +4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoCB = ∫

o

b(4 x2/b2 ) Fb 1/EJ dx = [4/3 x3/b2 ]o

b Fb 1/EJ

= (4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoEA = ∫

o

b(-3/2 x/b +2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-3/4 x2/b +2/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-3/4 b +2/3 b +1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoAE = ∫

o

b(1 -4 x/b +7/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [ x -2 x2/b +7/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= ( b -2 b +7/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

A = 1188. mm2

Ju = 350338. mm4

Jv = 131760. mm4

yg = 33.62 mmN = 238.5 NTy = 4770. NMx = 2289600. Nmmxm = 12. mmum = -12. mmvm = -33.62 mmσm = N/A-Mv/Ju = 219.9 N/mm2

xc = 24. mmyc = 11. mmvc = -22.62 mmσc = N/A-Mv/Ju = 148. N/mm2

τc = 6.713 N/mm2

σo = √σ2+3τ2 = 148.5 N/mm2

S* = 5917. mm3mm 0 12 18 30 36 48x

0

6

41

54

y

11σc,τc

σm

u

v

Ing Civ.bnga.052


Ing Civ.bnga.052


Ing Civ.bnga.052


Ing Civ.cmzk.053REAZIONI 890185 Camozzi Kevin


F

17/40F

17/40F1/2Fb

A

B

17/40F3/2Fb

17/40F43/40Fb

B C 4F

17/40F83/40Fb

4F

17/40F77/40Fb

C

D4F

57/40F77/40Fb

4F

97/40F

DE

Ing Civ.cmzk.053AZIONI INTERNE 890185 Camozzi Kevin


17/40 17/40

0

17/40

44

F1 0

-17/

40

-4

57/4

097

/40

F0 1/2

3/2

43/4

0

83/40-77/40

-77/

400

Fb

Ing

Civ

.cm

zk.0

53P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9018

5 C

amoz

zi K

evin

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dolfo

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27.0

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31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2-1

0-4

-40

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cm

zk.0

53P

RO

CE

DIM

EN

TO

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9018

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Mila

no, v

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27.0

3.13

31.0

5.19



AB b0Fx-1/2qx2

0000

BA b0-1/2Fb+1/2qx2

00


2/b

2

1/12Fb2/EJ1/3Xb/EJ

CB b1-x/bFb-5/2FxFb-7/2Fx+5/2Fx2/b1-2x/b+x

2/b

2

CD b-1-4Fx4Fx12Fb

2/EJXb/EJ



4Fb-15/2Fx+3Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

11/8Fb2/EJ1/3Xb/EJ

ED bx/b9/2Fx-1/2qx2

9/2Fx2/b-1/2qx

3/bx

2/b

2



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

Ing Civ.cmzk.053PROCEDIMENTO E RISULTATI 890185 Camozzi Kevin


= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXXED = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (4 b -15/4 b + b +1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (3/2 b -1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

Ing Civ.cmzk.053PROCEDIMENTO E RISULTATI 890185 Camozzi Kevin


A = 720. mm2

Ju = 272496. mm4

Jv = 49680. mm4



τc = 12.72 N/mm2

σo = √σ2+3τ2 = 137.2 N/mm2

S* = 5517. mm3mm 0 12 18 24 30 42x

0

12

47

54

y

13σc,τc

σm

u

v

Ing Civ.cmzk.053


Ing Civ.cmzk.053


Ing Civ.bslf.054REAZIONI 890213 Basilico Filippo


3F

3/40F57/40Fb

3F

37/40FFb

AB

3F

43/40F63/40Fb

3F

43/40F57/40Fb

C

A

43/40F1/2Fb

43/40F63/40Fb

D C

F

43/40F

43/40F1/2Fb

E

D

Ing Civ.bslf.054AZIONI INTERNE 890213 Basilico Filippo


-3-3

43/40

0

43/4043/40

F

-3/4

037

/40

-3

43/4

0

-10

F

-57/

40-1

63/40-57/40

1/2

63/4

0

0-1/2

Fb

Ing

Civ

.bsl

f.054

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

213

Bas

ilico

Fili

ppo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CDE

W

F

W X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2 3

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bsl

f.054

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

213

Bas

ilico

Fili

ppo

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-3

/2F

x+1/

2qx2

3/2F

x-2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

5/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

bF

b-1/

2Fx-

1/2q

x2F

x-1/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

5/2F

x-1

/2F

x-5/

2Fx2 /b

x2 /b2

-13/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-3

Fb+

5/2F

x-3

Fb+

11/2

Fx-

5/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-1

9/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B57

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3/2

x/b

-2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [3

/4 x

2 /b -

2/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (3

/4 b

-2/

3 b

+1/

8 b

) Fb

1/E

J =

5/2

4 F

b2 /EJ

LXo

BA =

∫ ob ( x/b

-1/

2 x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.bslf.054PROCEDIMENTO E RISULTATI 890213 Basilico Filippo


= (1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 5/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +11/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

A = 936. mm2

Ju = 301116. mm4

Jv = 86616. mm4



τc = 13.39 N/mm2

σo = √σ2+3τ2 = 146.6 N/mm2

S* = 6181. mm3mm 0 12 18 24 30 42x

0

12

41

54

y

13σc,τc

σm

u

v

Ing Civ.bslf.054


Ing Civ.bslf.054


Ing Civ.bslf.054


Ing Civ.bcga.055REAZIONI 890313 Bacigalupo Ada


F

11/8F

11/8F1/2Fb

A

B

11/8F1/2Fb

11/8F7/8Fb

BC

11/8F1/8Fb

11/8F1/8Fb

C

D

3/8F1/8Fb

5/8F

D E

Ing Civ.bcga.055AZIONI INTERNE 890313 Bacigalupo Ada


-11/

8-1

1/80

-11/

8

0 0

F

-10

-11/8

0

3/8-5/8

F

0-1

/21/2-7/81/

81/

8

1/80

Fb

Ing

Civ

.bcg

a.05

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9031

3 B

acig

alup

o A

da

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

-1

00

00

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bcg

a.05

5P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9031

3 B

acig

alup

o A

da

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JC

B b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b1/

2Fx-

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /b1-

2x/b

+x2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

ED

bx/

b-1

/2F

x+1/

2qx2

-1/2

Fx2 /b

+1/

2qx3 /b

x2 /b2

tota

li5/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-1

/8F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/2

b ) F

b 1/

EJ

= 1

/4 F

b2 /EJ

LXo

CB =

∫ ob (1 -

5/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[ x

-5/4

x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bcga.055PROCEDIMENTO E RISULTATI 890313 Bacigalupo Ada


= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

A = 1044. mm2

Ju = 354848. mm4

Jv = 83376. mm4



τc = 6.18 N/mm2

σo = √σ2+3τ2 = 117.4 N/mm2

S* = 7055. mm3mm 0 12 18 30 36 48x

0

12

47

54

y

13σc,τc

σm

u

v

Ing Civ.bcga.055


Ing Civ.bcga.055


Ing Civ.bcga.055


Ing Civ.gnnl.056REAZIONI 890623 Giannini Lucrezia


3F

3/40F57/40Fb

3F

37/40FFb

AB

3F

43/40F63/40Fb

3F

43/40F57/40Fb

C

A

43/40F1/2Fb

43/40F63/40Fb

D C

F

43/40F

43/40F1/2Fb

E

D

Ing Civ.gnnl.056AZIONI INTERNE 890623 Giannini Lucrezia


33

-43/

40

0

-43/

40-4

3/40

F

-3/4037/40

-3

43/40

-10

F

-57/40-1

63/4

0-5

7/40

1/2 63/40

0-1

/2

Fb

Ing

Civ

.gnn

l.056

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

623

Gia

nnin

i Luc

rezi

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

W

X

X

qq

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2

3

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.gnn

l.056

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

623

Gia

nnin

i Luc

rezi

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-3

/2F

x+1/

2qx2

3/2F

x-2F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

5/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

bF

b-1/

2Fx-

1/2q

x2F

x-1/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

5/2F

x-1

/2F

x-5/

2Fx2 /b

x2 /b2

-13/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-3

Fb+

5/2F

x-3

Fb+

11/2

Fx-

5/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-1

9/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B57

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3/2

x/b

-2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [3

/4 x

2 /b -

2/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (3

/4 b

-2/

3 b

+1/

8 b

) Fb

1/E

J =

5/2

4 F

b2 /EJ

LXo

BA =

∫ ob ( x/b

-1/

2 x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.gnnl.056PROCEDIMENTO E RISULTATI 890623 Giannini Lucrezia


= (1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 5/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +11/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ

A = 1260. mm2

Ju = 391706. mm4

Jv = 137808. mm4



τc = 6.353 N/mm2

σo = √σ2+3τ2 = 128.4 N/mm2

S* = 7786. mm3mm 0 12 18 30 36 48x

0

12

41

54

y

13σc,τc

σm

u

v

Ing Civ.gnnl.056


Ing Civ.gnnl.056


Ing Civ.gnnl.056


Ing Civ.glzt.057REAZIONI 890711 Galeazzi Tommaso


3F

19/40F59/40Fb

3F

19/40FFb

AB

3F

21/40F61/40Fb

3F

21/40F59/40Fb

C

A

61/40F1/2Fb

21/40F61/40Fb

D C

F

61/40F

61/40F1/2Fb

E

D

Ing Civ.glzt.057AZIONI INTERNE 890711 Galeazzi Tommaso


-3

21/40

00

61/4061/40

F

19/4

0

-3

61/4

021

/40

-10

F

-59/

40-1

61/40-59/40

1/2

61/4

0

0-1/2

Fb

Ing

Civ

.glz

t.057

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

711

Gal

eazz

i Tom

mas

o

@ A

dolfo

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elan

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si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CDE

W

F

W X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2 3

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.glz

t.057

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 890

711

Gal

eazz

i Tom

mas

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19




2/b

2

1/6Fb2/EJ1/3Xb/EJ


2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-3Fx2/b+1/2qx

3/bx

2/b

2

-9/8Fb2/EJ1/3Xb/EJ


-3Fb+5Fx-3/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.glzt.057PROCEDIMENTO E RISULTATI 890711 Galeazzi Tommaso


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBA = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b - b +1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +5/2 b -1/2 b -1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

Ing Civ.glzt.057PROCEDIMENTO E RISULTATI 890711 Galeazzi Tommaso


A = 1236. mm2

Ju = 393000. mm4

Jv = 129744. mm4


xc = 24. mmyc = 40. mmvc = 17.75 mmσc = N/A-Mv/Ju = 127. N/mm2

τc = 6.551 N/mm2

σo = √σ2+3τ2 = 127.5 N/mm2

S* = 8098. mm3mm 0 12 18 30 36 48x

0

12

41

54

y

40σc,τc

σm

u

v

Ing Civ.glzt.057


Ing Civ.glzt.057


Ing Civ.ambf.058REAZIONI 890818 Ambiveri Francesco


33/40F7/40Fb

33/40F7/40Fb

A

B7/40F

7/40Fb7/40F

B C

F

73/40F

73/40F1/2Fb

D

E73/40F1/2Fb

33/40F33/40Fb

EA

Ing Civ.ambf.058AZIONI INTERNE 890818 Ambiveri Francesco


33/40

0

73/4073/40

00

F

0

-7/4

0

-10

-73/

40-3

3/40

F

7/407/40

7/40

0

0-1/21/

2-3

3/40

Fb

Ing

Civ

.am

bf.0

58P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9081

8 A

mbi

veri

Fra

nces

co

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00 0 0

0-1

/2

1/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.am

bf.0

58P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

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8 A

mbi

veri

Fra

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co

@ A

dolfo

Zav

elan

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si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

CB

bx/

b0

0x2 /b

2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

2Fx+

1/2q

x2-1

/2F

x+2F

x2 /b-1

/2qx

3 /bx2 /b

2

7/24

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

Fx-

1/2q

x2F

b-2F

x+1/

2Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

tota

li7/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

EA =

∫ ob (-1/2

x/b

+2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/4

x2 /b +

2/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

1/4

b +

2/3

b -1

/8 b

) F

b 1/

EJ

= 7

/24

Fb2 /E

J

LXo

AE =

∫ ob (1 -

2 x/

b +

1/2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [

x -

x2 /b +

1/6

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

Ing Civ.ambf.058PROCEDIMENTO E RISULTATI 890818 Ambiveri Francesco


= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ

A = 696. mm2

Ju = 265885. mm4

Jv = 43992. mm4



τc = 9.223 N/mm2

σo = √σ2+3τ2 = 129.6 N/mm2

S* = 5563. mm3mm 0 12 18 24 30 42x

0

6

41

54

y

40σc,τc

σm

u

v

Ing Civ.ambf.058


Ing Civ.ambf.058


Ing Civ.ambf.058


Ing Civ.abbg.059REAZIONI 890974 Abbenda Giovanni


3F

19/40F59/40Fb

3F

19/40FFb

AB

3F

21/40F61/40Fb

3F

21/40F59/40Fb

C

A

61/40F1/2Fb

21/40F61/40Fb

D C

F

61/40F

61/40F1/2Fb

E

D

Ing Civ.abbg.059AZIONI INTERNE 890974 Abbenda Giovanni


3

-21/

40

0 0

-61/

40-6

1/40

F

19/40

-3

61/40 21/40

-10

F

-59/40-1

61/4

0-5

9/40

1/2 61/40

0-1

/2

Fb

Ing

Civ

.abb

g.05

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9097

4 A

bben

da G

iova

nni

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

30

1/2

3

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.abb

g.05

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9097

4 A

bben

da G

iova

nni

@ A

dolfo

Zav

elan

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si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19




2/b

2

1/6Fb2/EJ1/3Xb/EJ


2/b

2


2/EJXb/EJ

AC b1-3Fx-3Fx1


-1/2Fx-3Fx2/b+1/2qx

3/bx

2/b

2

-9/8Fb2/EJ1/3Xb/EJ


-3Fb+5Fx-3/2Fx2/b-1/2qx

3/b1-2x/b+x

2/b

2

ED b0-Fx+1/2qx2

0000

DE b01/2Fb-1/2qx2

00



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

Ing Civ.abbg.059PROCEDIMENTO E RISULTATI 890974 Abbenda Giovanni


LXXAC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXCD = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoBA = ∫

o


b Fb 1/EJ

= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b - b +1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-3 b +5/2 b -1/2 b -1/8 b ) Fb 1/EJ = -9/8 Fb2/EJ

Ing Civ.abbg.059PROCEDIMENTO E RISULTATI 890974 Abbenda Giovanni


A = 948. mm2

Ju = 318391. mm4

Jv = 69264. mm4



τc = 5.522 N/mm2

σo = √σ2+3τ2 = 148.5 N/mm2

S* = 6142. mm3mm 0 12 18 30 36 48x

0

6

47

54

y

7σc,τc

σm

u

v

Ing Civ.abbg.059


Ing Civ.abbg.059


Ing Civ.crsn.060REAZIONI 891208 Crispiatico Nicol


4F

39/40F81/40Fb

4F

39/40F79/40Fb

A

B4F

79/40F79/40Fb

4F

79/40F

BC

F

1/40F

1/40F1/2Fb

D

E

1/40F3/2Fb

39/40F41/40Fb

E A

Ing Civ.crsn.060AZIONI INTERNE 891208 Crispiatico Nicol


-39/

40

-4

1/40

1/40

0 0

F

-4

79/40

10

1/40-39/40

F

81/4

0-7

9/40

-79/400

01/

2

3/2 41/40

Fb

Ing

Civ

.crs

n.06

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9120

8 C

rispi

atic

o N

icol

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-4

-40

0 1/2 3/2

-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crs

n.06

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9120

8 C

rispi

atic

o N

icol

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JB

A b

14F

b-4F

x4F

b-4F

x1

BC

b-1

+x/

b-4

Fb+

4Fx

4Fb-

8Fx+

4Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

4Fx

4Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

2Fx-

1/2q

x2-3

/2F

x+2F

x2 /b+

1/2q

x3 /bx2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

Fb-

3Fx+

1/2q

x2F

b-4F

x+7/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

tota

li27

/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-8

1/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (4 x

/b )

Fb

1/E

J dx

= [2

x2 /b

] ob Fb

1/E

J

= (2

b )

Fb

1/E

J =

2 F

b2 /EJ

LXo

BA =

∫ ob (4 -

4 x/

b ) F

b 1/

EJ

dx =

[4 x

-2

x2 /b ] ob F

b 1/

EJ

Ing Civ.crsn.060PROCEDIMENTO E RISULTATI 891208 Crispiatico Nicol


= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (4 b -4 b +4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoCB = ∫

o

b(4 x2/b2 ) Fb 1/EJ dx = [4/3 x3/b2 ]o

b Fb 1/EJ

= (4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoEA = ∫

o

b(-3/2 x/b +2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [-3/4 x2/b +2/3 x3/b2 +1/8 x4/b3 ]o

b Fb 1/EJ

= (-3/4 b +2/3 b +1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoAE = ∫

o

b(1 -4 x/b +7/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [ x -2 x2/b +7/6 x3/b2 -1/8 x4/b3 ]o

b Fb 1/EJ

= ( b -2 b +7/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

A = 912. mm2

Ju = 304351. mm4

Jv = 80928. mm4

yg = 20.56 mmN = 224.8 NTy = 4495. NMx = 1820480. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 33.44 mmσm = N/A-Mv/Ju = -199.8 N/mm2


τc = 15.81 N/mm2

σo = √σ2+3τ2 = 119.2 N/mm2

S* = 6424. mm3mm 0 12 18 24 30 42x

0

12

41

54

y

40σc,τc

σm

u

v

Ing Civ.crsn.060


Ing Civ.crsn.060


Ing Civ.crsn.060


Ing Civ.brsa.061REAZIONI 891317 Borsani Alessio


F

1/40F

1/40F1/2Fb

A

B

1/40F3/2Fb

39/40F41/40Fb

B C 4F

39/40F81/40Fb

4F

39/40F79/40Fb

C

D4F

79/40F79/40Fb

4F

79/40F

DE

Ing Civ.brsa.061AZIONI INTERNE 891317 Borsani Alessio


-1/40 -1/40 00

39/40

4

F1 0

1/40

-39/

40

-4

79/4

0

F0 1/2

3/2

41/4

0

81/40-79/40

-79/

400

Fb

Ing

Civ

.brs

a.06

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9131

7 B

orsa

ni A

less

io

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2-1

0-4

-40

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.brs

a.06

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9131

7 B

orsa

ni A

less

io

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

2Fx-

1/2q

x2-3

/2F

x+2F

x2 /b+

1/2q

x3 /bx2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

Fb-

3Fx+

1/2q

x2F

b-4F

x+7/

2Fx2 /b

-1/2

qx3 /b

1-2x

/b+

x2 /b2

CD

b-1

-4F

x4F

x1

2Fb2 /E

JX

b/E

JD

C b

14F

b-4F

x4F

b-4F

x1

DE

b-1

+x/

b-4

Fb+

4Fx

4Fb-

8Fx+

4Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

4Fx

4Fx2 /b

x2 /b2

tota

li27

/8F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-8

1/40

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

2/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

2/3

b +

1/8

b ) F

b 1/

EJ

= 1

/24

Fb2 /E

J

LXo

CB =

∫ ob (1 -

4 x/

b +

7/2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [

x -2

x2 /b

+7/

6 x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.brsa.061PROCEDIMENTO E RISULTATI 891317 Borsani Alessio


= ( b -2 b +7/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (4 b -4 b +4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoED = ∫

o

b(4 x2/b2 ) Fb 1/EJ dx = [4/3 x3/b2 ]o

b Fb 1/EJ

= (4/3 b ) Fb 1/EJ = 4/3 Fb2/EJ

A = 840. mm2

Ju = 261182. mm4

Jv = 78120. mm4

yg = 18.55 mmN = -174.5 NTy = 3490. NMx = 1544330. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 35.45 mmσm = N/A-Mv/Ju = -209.8 N/mm2


τc = 9.699 N/mm2

σo = √σ2+3τ2 = 157.5 N/mm2

S* = 4355. mm3mm 0 12 18 24 30 42x

0

12

47

54

y

45σc,τc

σm

u

v

Ing Civ.brsa.061


Ing Civ.brsa.061


Ing Civ.brsa.061


Ing Civ.btta.062REAZIONI 891364 Battelino Ada Sonia


97/40F3/2Fb

57/40F17/40Fb

AB

F

97/40F

97/40F1/2Fb

C

A

F

17/40F17/40Fb

F

17/40F

D E

F

57/40F23/40Fb

F

57/40F17/40Fb

B

D

Ing Civ.btta.062AZIONI INTERNE 891364 Battelino Ada Sonia


00

-97/40 -97/40

-1

-57/40

F

-97/

40-5

7/40

1 0

17/4

0

-1

F

3/2

-17/

40

0 1/2

-17/

400

23/40-17/40

Fb

Ing

Civ

.btta

.062

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 891

364

Bat

telin

o A

da S

onia

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

3/20

01/

2

0 0

10

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.btta

.062

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 891

364

Bat

telin

o A

da S

onia

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

2Fx+

1/2q

x2-3

/2F

x+2F

x2 /b-1

/2qx

3 /bx2 /b

2

-5/2

4Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-Fx-

1/2q

x2-F

x+1/

2Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

ED

bx/

b0

0x2 /b

2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-1

7/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E17

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

2/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

2/3

b -1

/8 b

) F

b 1/

EJ

= -

5/24

Fb2 /E

J

LXo

BA =

∫ ob (- x/

b +

1/2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/2

x2 /b +

1/6

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

Ing Civ.btta.062PROCEDIMENTO E RISULTATI 891364 Battelino Ada Sonia


= (-1/2 b +1/6 b +1/8 b ) Fb 1/EJ = -5/24 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 1020. mm2

Ju = 346417. mm4

Jv = 75312. mm4



τc = 8.474 N/mm2

σo = √σ2+3τ2 = 125.6 N/mm2

S* = 7189. mm3mm 0 12 18 30 36 48x

0

6

41

54

y

40σc,τc

σm

u

v

Ing Civ.btta.062


Ing Civ.btta.062


Ing Civ.btta.062


Ing Civ.chtc.063REAZIONI 891692 Chitanu Cristian


F

73/40F

73/40F1/2Fb

A

B

73/40F1/2Fb

33/40F33/40Fb

BC33/40F

7/40Fb

33/40F7/40Fb

C

D

7/40F7/40Fb

7/40F

D E

Ing Civ.chtc.063AZIONI INTERNE 891692 Chitanu Cristian


-73/

40-7

3/4000

-33/

40

0

F

-10

-73/40-33/40

0

-7/40

F

0-1

/21/2-33/407/

407/

40

7/40 0

Fb

Ing

Civ

.cht

c.06

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9169

2 C

hita

nu C

ristia

n

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

WX

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

-1

00

00

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cht

c.06

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9169

2 C

hita

nu C

ristia

n

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

2Fx+

1/2q

x2-1

/2F

x+2F

x2 /b-1

/2qx

3 /bx2 /b

2

7/24

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

Fb-

Fx-

1/2q

x2F

b-2F

x+1/

2Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

ED

bx/

b0

0x2 /b

2

tota

li7/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/4

x2 /b +

2/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

1/4

b +

2/3

b -1

/8 b

) F

b 1/

EJ

= 7

/24

Fb2 /E

J

LXo

CB =

∫ ob (1 -

2 x/

b +

1/2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [

x -

x2 /b +

1/6

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

Ing Civ.chtc.063PROCEDIMENTO E RISULTATI 891692 Chitanu Cristian


= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ

A = 1164. mm2

Ju = 356609. mm4

Jv = 123696. mm4



τc = 6.899 N/mm2

σo = √σ2+3τ2 = 156. N/mm2

S* = 5858. mm3mm 0 12 18 30 36 48x

0

12

47

54

y

44σc,τc

σm

u

v

Ing Civ.chtc.063


Ing Civ.chtc.063


Ing Civ.chtc.063


Ing Civ.clml.064REAZIONI 892410 Colombo Leonardo


97/40F3/2Fb

57/40F17/40Fb

AB

F

97/40F

97/40F1/2Fb

C

A

F

17/40F17/40Fb

F

17/40F

D E

F

57/40F23/40Fb

F

57/40F17/40Fb

B

D

Ing Civ.clml.064AZIONI INTERNE 892410 Colombo Leonardo


00

97/4

097

/40

1

57/4

0

F

-97/40-57/40

10

17/40

-1

F

3/2-17/40

01/

2

-17/400

23/4

0-1

7/40

Fb

Ing

Civ

.clm

l.064

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

410

Col

ombo

Leo

nard

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

0

0 1/2

00

1 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.clm

l.064

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

410

Col

ombo

Leo

nard

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

2Fx+

1/2q

x2-3

/2F

x+2F

x2 /b-1

/2qx

3 /bx2 /b

2

-5/2

4Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-Fx-

1/2q

x2-F

x+1/

2Fx2 /b

+1/

2qx3 /b

1-2x

/b+

x2 /b2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

ED

bx/

b0

0x2 /b

2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-1

7/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E17

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+2

x2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

2/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

3/4

b +

2/3

b -1

/8 b

) F

b 1/

EJ

= -

5/24

Fb2 /E

J

LXo

BA =

∫ ob (- x/

b +

1/2

x2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/2

x2 /b +

1/6

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

Ing Civ.clml.064PROCEDIMENTO E RISULTATI 892410 Colombo Leonardo


= (-1/2 b +1/6 b +1/8 b ) Fb 1/EJ = -5/24 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 624. mm2

Ju = 234015. mm4

Jv = 41184. mm4


xc = 21. mmyc = 7. mmvc = -14.86 mmσc = N/A-Mv/Ju = 150. N/mm2

τc = 11.9 N/mm2

σo = √σ2+3τ2 = 151.4 N/mm2

S* = 4844. mm3mm 0 12 18 24 30 42x

0

6

47

54

y

7σc,τc

σm

u

v

Ing Civ.clml.064


Ing Civ.clml.064


Ing Civ.clml.064


Ing Civ.dlcj.065REAZIONI 892644 Dolci Jacopo


79/40F3/2Fb

79/40F19/40Fb

AB

F

79/40F

79/40F1/2Fb

C

A

F

39/40F19/40Fb

F

1/40F

D E

F

79/40F21/40Fb

F

79/40F19/40Fb

B

D

Ing Civ.dlcj.065AZIONI INTERNE 892644 Dolci Jacopo


0

-79/40 -79/40

-1-1

-79/40

F

-79/

40

1 0

39/4

0-1

/40

-1

F

3/2

-19/

40

0 1/2

-19/

400

21/40-19/40

Fb

Ing

Civ

.dlc

j.065

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

644

Dol

ci J

acop

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o3/20

01/

2

0 0

10

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dlc

j.065

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

644

Dol

ci J

acop

o

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b1/

2Fx-

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /b1-

2x/b

+x2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

ED

bx/

b-1

/2F

x+1/

2qx2

-1/2

Fx2 /b

+1/

2qx3 /b

x2 /b2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-1

9/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E19

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

BA =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.dlcj.065PROCEDIMENTO E RISULTATI 892644 Dolci Jacopo


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 606. mm2

Ju = 215454. mm4

Jv = 40698. mm4



τc = 8.703 N/mm2

σo = √σ2+3τ2 = 110.5 N/mm2

S* = 4601. mm3mm 0 12 18 24 30 42x

0

6

47

53

y

46σc,τc

σm

u

v

Ing Civ.dlcj.065


Ing Civ.dlcj.065


Ing Civ.dlcj.065


Ing Civ.blla.066REAZIONI 892814 Biella Astrid


11/8F1/8Fb

11/8F1/8Fb

A

B

3/8F1/8Fb

5/8F

B C

F

11/8F

11/8F1/2Fb

D

E

11/8F1/2Fb

11/8F7/8Fb

EA

Ing Civ.blla.066AZIONI INTERNE 892814 Biella Astrid


11/800

11/811/8

0

F

0

3/8

-5/8

-10

-11/

8

F

1/81/8

1/8

0

0-1/21/

2-7

/8

Fb

Ing

Civ

.blla

.066

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

814

Bie

lla A

strid

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

00 0 0

0-1

/2

1/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.blla

.066

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 892

814

Bie

lla A

strid

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

00

10

Xb/

EJ

BA

b1

00

1

BC

b-1

+x/

b1/

2Fx-

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /b1-

2x/b

+x2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

CB

bx/

b-1

/2F

x+1/

2qx2

-1/2

Fx2 /b

+1/

2qx3 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

3/2F

x-1

/2F

x+3/

2Fx2 /b

x2 /b2

1/4F

b2 /EJ

1/3X

b/E

JA

E b

1-x/

bF

b-3/

2Fx

Fb-

5/2F

x+3/

2Fx2 /b

1-2x

/b+

x2 /b2

tota

li5/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-1

/8F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

BC =

∫ ob (-1/2

x/b

+ x

2 /b2 -

1/2

x3 /b3 )

Fb

1/E

J dx

= [-

1/4

x2 /b +

1/3

x3 /b2 -

1/8

x4 /b3 ] ob F

b 1/

EJ

= (-

1/4

b +

1/3

b -1

/8 b

) F

b 1/

EJ

= -

1/24

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x2 /b

2 +1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[-1/

6 x3 /b

2 +1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.blla.066PROCEDIMENTO E RISULTATI 892814 Biella Astrid


= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ

A = 822. mm2

Ju = 238712. mm4

Jv = 77634. mm4



τc = 6.499 N/mm2

σo = √σ2+3τ2 = 154.5 N/mm2

S* = 3978. mm3mm 0 12 18 24 30 42x

0

6

41

53

y

9σc,τc

σm

u

v

Ing Civ.blla.066


Ing Civ.blla.066


Ing Civ.blla.066


Ing Civ.cppn.067REAZIONI 892907 Capparelli Nicola


79/40F3/2Fb

79/40F19/40Fb

AB

F

79/40F

79/40F1/2Fb

C

A

F

39/40F19/40Fb

F

1/40F

D E

F

79/40F21/40Fb

F

79/40F19/40Fb

B

D

Ing Civ.cppn.067AZIONI INTERNE 892907 Capparelli Nicola


0

79/4

079

/40

1 1

79/4

0

F

-79/40

10

39/40-1/40

-1

F

3/2-19/40

01/

2

-19/400

21/4

0-1

9/40

Fb

Ing

Civ

.cpp

n.06

7P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9290

7 C

appa

relli

Nic

ola

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

0

0 1/2

00

1 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.cpp

n.06

7P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9290

7 C

appa

relli

Nic

ola

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b1/

2Fx-

1/2q

x2-1

/2F

x+F

x2 /b-1

/2qx

3 /b1-

2x/b

+x2 /b

2

-1/2

4Fb2 /E

J1/

3Xb/

EJ

ED

bx/

b-1

/2F

x+1/

2qx2

-1/2

Fx2 /b

+1/

2qx3 /b

x2 /b2

BD

b-1

Fb-

Fx

-Fb+

Fx

1-1

/2F

b2 /EJ

Xb/

EJ

DB

b1

-Fx

-Fx

1

tota

li-1

9/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WD

E19

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

BA =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.cppn.067PROCEDIMENTO E RISULTATI 892907 Capparelli Nicola


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ

LXoBD = ∫

o

b(-1 + x/b ) Fb 1/EJ dx = [- x +1/2 x2/b ]o

b Fb 1/EJ

= (- b +1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

LXoDB = ∫

o

b(- x/b ) Fb 1/EJ dx = [-1/2 x2/b ]o

b Fb 1/EJ

= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ

A = 924. mm2

Ju = 296396. mm4

Jv = 68112. mm4

yg = 30.16 mmN = 11909. NTy = 3015. NMx = 2035130. Nmmxm = 12. mmum = -12. mmvm = -30.16 mmσm = N/A-Mv/Ju = 220. N/mm2


τc = 4.217 N/mm2

σo = √σ2+3τ2 = 151.5 N/mm2

S* = 4975. mm3mm 0 12 18 30 36 48x

0

6

47

53

y

10σc,τc

σm

u

v

Ing Civ.cppn.067


Ing Civ.cppn.067


Ing Civ.cppn.067


Ing Civ.emdr.068REAZIONI 892994 Emad Ragab Abdel Hamid


4F

17/40F83/40Fb

4F

17/40F77/40Fb

A

B4F

57/40F77/40Fb

4F

97/40F

BC

F

17/40F

17/40F1/2Fb

D

E

17/40F3/2Fb

17/40F43/40Fb

E A

Ing Civ.emdr.068AZIONI INTERNE 892994 Emad Ragab Abdel Hamid


-17/

40

-4-4

-17/

40-1

7/40

0

F

-4

57/4097/40

10

-17/40

F

83/4

0-7

7/40

-77/400

01/

2

3/2 43/40

Fb

Ing

Civ

.em

dr.0

68P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9299

4 E

mad

Rag

ab A

bdel

@ A

dolfo

Zav

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i Ros

si, P

olite

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o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-4

-40

0 1/2 3/2

-1

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.em

dr.0

68P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9299

4 E

mad

Rag

ab A

bdel

@ A

dolfo

Zav

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i Ros

si, P

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cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b-1-4Fx4Fx12Fb

2/EJXb/EJ

BA b14Fb-4Fx4Fb-4Fx1

BC b-1+x/b-4Fb+7/2Fx+1/2qx2

4Fb-15/2Fx+3Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

11/8Fb2/EJ1/3Xb/EJ

CB bx/b9/2Fx-1/2qx2

9/2Fx2/b-1/2qx

3/bx

2/b

2

DE b0Fx-1/2qx2

0000

ED b0-1/2Fb+1/2qx2

00

EA b-x/b3/2Fb-5/2Fx-3/2Fx+5/2Fx2/bx

2/b

2

1/12Fb2/EJ1/3Xb/EJ

AE b1-x/bFb-5/2FxFb-7/2Fx+5/2Fx2/b1-2x/b+x

2/b

2


iperstatica X=WAB-83/40Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

Ing Civ.emdr.068PROCEDIMENTO E RISULTATI 892994 Emad Ragab Abdel


LXXCB = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXEA = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXXAE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXoAB = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBA = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (4 b -15/4 b + b +1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (3/2 b -1/8 b ) Fb 1/EJ = 11/8 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ

Ing Civ.emdr.068PROCEDIMENTO E RISULTATI 892994 Emad Ragab Abdel


A = 1140. mm2

Ju = 330567. mm4

Jv = 122544. mm4



τc = 4.758 N/mm2

σo = √σ2+3τ2 = 152. N/mm2

S* = 5746. mm3mm 0 12 18 30 36 48x

0

6

41

53

y

11σc,τc

σm

u

v

Ing Civ.emdr.068


Ing Civ.emdr.068


Ing Civ.bltt.069REAZIONI 893013 Beltran Toledo Italo Jose


7/4F3/2Fb

7/4F1/4Fb

AB

F

7/4F

7/4F1/2Fb

C

A

2F

3/4F3/4Fb

2F

3/4F

D E

F

7/4F3/4Fb

2F

7/4F3/4Fb

B

D

Ing Civ.bltt.069AZIONI INTERNE 893013 Beltran Toledo Italo Jose


0

-7/4 -7/4

-2

-7/4 -7/4

F

-7/4

1 0

3/4

-1 -2

F

3/2

-1/4

0 1/2

-3/4

0

3/4-3/4

Fb

Ing

Civ

.bltt

.069

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 893

013

Bel

tran

Tol

edo

Italo

Jos

e

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o3/21/2

01/

2

0 0

3/2

0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

00

-10

-1-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bltt

.069

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 893

013

Bel

tran

Tol

edo

Italo

Jos

e

@ A

dolfo

Zav

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i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

Fx

-3/2

Fx+

Fx2 /b

x2 /b2

-5/1

2Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-1/2

Fb-

Fx

-1/2

Fb-

1/2F

x+F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

ED

bx/

b0

0x2 /b

2

BD

b-1

3/2F

b-F

x-1/

2qx2

-3/2

Fb+

Fx+

1/2F

x2 /b1

-5/6

Fb2 /E

JX

b/E

JD

B b

1-2

Fx+

1/2q

x2-2

Fx+

1/2F

x2 /b1

tota

li-5

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WD

E3/

4Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+ x

2 /b2 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

1/3

x3 /b2 ] ob F

b 1/

EJ

= (-

3/4

b +

1/3

b ) F

b 1/

EJ

= -

5/12

Fb2 /E

J

LXo

BA =

∫ ob (-1/2

-1/

2 x/

b +

x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

2 x

-1/4

x2 /b

+1/

3 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bltt.069PROCEDIMENTO E RISULTATI 893013 Beltran Toledo Italo Jose


= (-1/2 b -1/4 b +1/3 b ) Fb 1/EJ = -5/12 Fb2/EJ

LXoBD = ∫

o


b Fb 1/EJ

= (-3/2 b +1/2 b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

LXoDB = ∫

o


b Fb 1/EJ

= (- b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

A = 678. mm2

Ju = 249963. mm4

Jv = 43506. mm4



τc = 9.998 N/mm2

σo = √σ2+3τ2 = 156.4 N/mm2

S* = 5234. mm3mm 0 12 18 24 30 42x

0

12

47

53

y

46σc,τc

σm

u

v

Ing Civ.bltt.069


Ing Civ.bltt.069


Ing Civ.bltt.069


Ing Civ.bzzs.070REAZIONI 893038 Bozzini Silvia


2F

3/20F23/20Fb

2F

3/20FFb

AB

3F

17/20F27/20Fb

2F

17/20F23/20Fb

C

A

17/20F1/2Fb

17/20F27/20Fb

D C

F

17/20F

17/20F1/2Fb

E

D

Ing Civ.bzzs.070AZIONI INTERNE 893038 Bozzini Silvia


-2

17/2017/20

0

17/2017/20

F

3/20

-3-2

17/2

0

-10

F

-23/

20-1

27/20-23/20

1/2

27/2

0

0-1/2

Fb

Ing

Civ

.bzz

s.07

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9303

8 B

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ni S

ilvia

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dolfo

Zav

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i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CDE

W

F

W X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

5/2

0

1/2 5/2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bzz

s.07

0P

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DIM

EN

TO

E R

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LTA

TI 8

9303

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ni S

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@ A

dolfo

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Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JB

A b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

CA

b-1

5/2F

b-3F

x+1/

2qx2

-5/2

Fb+

3Fx-

1/2F

x2 /b1

-7/6

Fb2 /E

JX

b/E

JA

C b

1-2

Fx-

1/2q

x2-2

Fx-

1/2F

x2 /b1

DC

b-x

/b1/

2Fb+

2Fx

-1/2

Fx-

2Fx2 /b

x2 /b2

-11/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-5

/2F

b+2F

x-5

/2F

b+9/

2Fx-

2Fx2 /b

1-2x

/b+

x2 /b2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-2

3/12

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B23

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

= (1

/2 b

-1/

3 b

) Fb

1/E

J =

1/6

Fb2 /E

J

LXo

BA =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.bzzs.070PROCEDIMENTO E RISULTATI 893038 Bozzini Silvia


= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-5/2 +3 x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-5/2 x +3/2 x2/b -1/6 x3/b2 ]o

b Fb 1/EJ

= (-5/2 b +3/2 b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoAC = ∫

o

b(-2 x/b -1/2 x2/b2 ) Fb 1/EJ dx = [- x2/b -1/6 x3/b2 ]o

b Fb 1/EJ

= (- b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -2/3 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b -2/3 b ) Fb 1/EJ = -11/12 Fb2/EJ

LXoCD = ∫

o

b(-5/2 +9/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [-5/2 x +9/4 x2/b -2/3 x3/b2 ]o

b Fb 1/EJ

= (-5/2 b +9/4 b -2/3 b ) Fb 1/EJ = -11/12 Fb2/EJ

A = 894. mm2

Ju = 284424. mm4

Jv = 80442. mm4



τc = 15.78 N/mm2

σo = √σ2+3τ2 = 121.1 N/mm2

S* = 5978. mm3mm 0 12 18 24 30 42x

0

12

41

53

y

13σc,τc

σm

u

v

Ing Civ.bzzs.070


Ing Civ.bzzs.070


Ing Civ.bzzs.070


Ing Civ.crlm.071REAZIONI 893348 Carlino Mauro


7/4F3/2Fb

7/4F1/4Fb

AB

F

7/4F

7/4F1/2Fb

C

A

2F

3/4F3/4Fb

2F

3/4F

D E

F

7/4F3/4Fb

2F

7/4F3/4Fb

B

D

Ing Civ.crlm.071AZIONI INTERNE 893348 Carlino Mauro


0

7/4

7/4

2

7/4

7/4

F

-7/4

10

3/4

-1-2

F

3/2-1/4

01/

2

-3/40

3/4

-3/4

Fb

Ing

Civ

.crlm

.071

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 893

348

Car

lino

Mau

ro

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

3/2

1/2

0 1/2

00

3/2 0

Mo

fless

ione

da

caric

hi a

sseg

nati

0-1

0 0

-10

-1 -1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crlm

.071

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 893

348

Car

lino

Mau

ro

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WD

E

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-x

/b3/

2Fb-

Fx

-3/2

Fx+

Fx2 /b

x2 /b2

-5/1

2Fb2 /E

J1/

3Xb/

EJ

BA

b1-

x/b

-1/2

Fb-

Fx

-1/2

Fb-

1/2F

x+F

x2 /b1-

2x/b

+x2 /b

2

CA

b0

Fx-

1/2q

x20

00

0A

C b

0-1

/2F

b+1/

2qx2

00

DE

b-1

+x/

b0

01-

2x/b

+x2 /b

2

01/

3Xb/

EJ

ED

bx/

b0

0x2 /b

2

BD

b-1

3/2F

b-F

x-1/

2qx2

-3/2

Fb+

Fx+

1/2F

x2 /b1

-5/6

Fb2 /E

JX

b/E

JD

B b

1-2

Fx+

1/2q

x2-2

Fx+

1/2F

x2 /b1

tota

li-5

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WD

E3/

4Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BA =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

BD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXo

AB =

∫ ob (-3/2

x/b

+ x

2 /b2 )

Fb

1/E

J dx

= [-

3/4

x2 /b +

1/3

x3 /b2 ] ob F

b 1/

EJ

= (-

3/4

b +

1/3

b ) F

b 1/

EJ

= -

5/12

Fb2 /E

J

LXo

BA =

∫ ob (-1/2

-1/

2 x/

b +

x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

2 x

-1/4

x2 /b

+1/

3 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.crlm.071PROCEDIMENTO E RISULTATI 893348 Carlino Mauro


= (-1/2 b -1/4 b +1/3 b ) Fb 1/EJ = -5/12 Fb2/EJ

LXoBD = ∫

o


b Fb 1/EJ

= (-3/2 b +1/2 b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

LXoDB = ∫

o


b Fb 1/EJ

= (- b +1/6 b ) Fb 1/EJ = -5/6 Fb2/EJ

A = 996. mm2

Ju = 326526. mm4

Jv = 74160. mm4

yg = 28.63 mmN = 17640. NTy = 5040. NMx = 2192400. Nmmxm = 12. mmum = -12. mmvm = -28.63 mmσm = N/A-Mv/Ju = 210. N/mm2


τc = 8.633 N/mm2

σo = √σ2+3τ2 = 123.6 N/mm2

S* = 6712. mm3mm 0 12 18 30 36 48x

0

12

47

53

y

13σc,τc

σm

u

v

Ing Civ.crlm.071


Ing Civ.crlm.071


Ing Civ.crlm.071


Ing Civ.bnny.072REAZIONI 893640 Banani Yassine


2F

3/20F23/20Fb

2F

3/20FFb

AB

3F

17/20F27/20Fb

2F

17/20F23/20Fb

C

A

17/20F1/2Fb

17/20F27/20Fb

D C

F

17/20F

17/20F1/2Fb

E

D

Ing Civ.bnny.072AZIONI INTERNE 893640 Banani Yassine


2

-17/

20-1

7/20

0

-17/

20-1

7/20

F

3/20

-3-2

17/20

-10

F

-23/20-1

27/2

0-2

3/20

1/2 27/20

0-1

/2

Fb

Ing

Civ

.bnn

y.07

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9364

0 B

anan

i Yas

sine

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1

5/20

1/2

5/2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bnn

y.07

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9364

0 B

anan

i Yas

sine

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-F

xF

x-F

x2 /b1-

2x/b

+x2 /b

2

1/6F

b2 /EJ

1/3X

b/E

JB

A b

x/b

Fb-

Fx

Fx-

Fx2 /b

x2 /b2

CA

b-1

5/2F

b-3F

x+1/

2qx2

-5/2

Fb+

3Fx-

1/2F

x2 /b1

-7/6

Fb2 /E

JX

b/E

JA

C b

1-2

Fx-

1/2q

x2-2

Fx-

1/2F

x2 /b1

DC

b-x

/b1/

2Fb+

2Fx

-1/2

Fx-

2Fx2 /b

x2 /b2

-11/

12F

b2 /EJ

1/3X

b/E

JC

D b

1-x/

b-5

/2F

b+2F

x-5

/2F

b+9/

2Fx-

2Fx2 /b

1-2x

/b+

x2 /b2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-2

3/12

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B23

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

= (1

/2 b

-1/

3 b

) Fb

1/E

J =

1/6

Fb2 /E

J

LXo

BA =

∫ ob ( x/b

- x

2 /b2 )

Fb

1/E

J dx

= [1

/2 x

2 /b -

1/3

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.bnny.072PROCEDIMENTO E RISULTATI 893640 Banani Yassine


= (1/2 b -1/3 b ) Fb 1/EJ = 1/6 Fb2/EJ

LXoCA = ∫

o

b(-5/2 +3 x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-5/2 x +3/2 x2/b -1/6 x3/b2 ]o

b Fb 1/EJ

= (-5/2 b +3/2 b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoAC = ∫

o

b(-2 x/b -1/2 x2/b2 ) Fb 1/EJ dx = [- x2/b -1/6 x3/b2 ]o

b Fb 1/EJ

= (- b -1/6 b ) Fb 1/EJ = -7/6 Fb2/EJ

LXoDC = ∫

o

b(-1/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -2/3 x3/b2 ]o

b Fb 1/EJ

= (-1/4 b -2/3 b ) Fb 1/EJ = -11/12 Fb2/EJ

LXoCD = ∫

o

b(-5/2 +9/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [-5/2 x +9/4 x2/b -2/3 x3/b2 ]o

b Fb 1/EJ

= (-5/2 b +9/4 b -2/3 b ) Fb 1/EJ = -11/12 Fb2/EJ

A = 1212. mm2

Ju = 369093. mm4

Jv = 128592. mm4



τc = 8.998 N/mm2

σo = √σ2+3τ2 = 132.8 N/mm2

S* = 7533. mm3mm 0 12 18 30 36 48x

0

12

41

53

y

13σc,τc

σm

u

v

Ing Civ.bnny.072


Ing Civ.bnny.072


Ing Civ.bnny.072


Ing Civ.bnmh.073REAZIONI 894745 Ben M’Hamed Ines


F

3/20F

3/20F1/2Fb

A

B

3/20F3/2Fb

3/20F33/20Fb

B C3F

3/20F33/20Fb

4F

3/20F37/20Fb

C

D4F

17/20F37/20Fb

4F

17/20FFb

DE

Ing Civ.bnmh.073AZIONI INTERNE 894745 Ben M’Hamed Ines


-3/20 -3/20

0

-3/20 -3/20

4

F

1 0

3/20

-3 -4

17/2

0

F

0 1/2

3/2

33/2

0

33/20-37/20

-37/

20-1

Fb

Ing

Civ

.bnm

h.07

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9474

5 B

en M

’Ham

ed In

es

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2 0 0-7

/2-7/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bnm

h.07

3P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9474

5 B

en M

’Ham

ed In

es

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

CB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bnmh.073PROCEDIMENTO E RISULTATI 894745 Ben M’Hamed Ines


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

A = 1212. mm2

Ju = 369093. mm4

Jv = 128592. mm4

yg = 21.63 mmN = -1583. NTy = 5275. NMx = 2690250. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 31.37 mmσm = N/A-Mv/Ju = -230. N/mm2


τc = 8.972 N/mm2

σo = √σ2+3τ2 = 136.1 N/mm2

S* = 7533. mm3mm 0 12 18 30 36 48x

0

12

41

53

y

40σc,τc

σm

u

v

Ing Civ.bnmh.073


Ing Civ.bnmh.073


Ing Civ.bnmh.073


Ing Civ.gstc.074REAZIONI 895222 Giusto Carola


F

3/20F7/20Fb

3/20F3/20Fb

A

B

17/20F3/20Fb

17/20FFb

B C

F

3/20F

3/20F1/2Fb

D

E

3/20F1/2Fb

3/20F7/20Fb

EA

Ing Civ.gstc.074AZIONI INTERNE 895222 Giusto Carola


3/203/20

0

3/203/20

0

F

-10

-17/

20

-10

-3/2

0

F

7/20-3/20

-3/2

0-1

0-1/21/

27/

20

Fb

Ing

Civ

.gst

c.07

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9522

2 G

iust

o C

arol

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/2 -1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.gst

c.07

4P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 8

9522

2 G

iust

o C

arol

a

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.gstc.074PROCEDIMENTO E RISULTATI 895222 Giusto Carola


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

A = 678. mm2

Ju = 249963. mm4

Jv = 43506. mm4



τc = 12.57 N/mm2

σo = √σ2+3τ2 = 137. N/mm2

S* = 5207. mm3mm 0 12 18 24 30 42x

0

6

41

53

y

40σc,τc

σm

u

v

Ing Civ.gstc.074


Ing Civ.gstc.074


Ing Civ.gstc.074


Ing Civ.crpl.075REAZIONI 896189 Carpani Lorenzo Marco


F

3/20F

3/20F1/2Fb

A

B

3/20F1/2Fb

3/20F7/20Fb

BCF

3/20F7/20Fb

3/20F3/20Fb

C

D

17/20F3/20Fb

17/20FFb

D E

Ing Civ.crpl.075AZIONI INTERNE 896189 Carpani Lorenzo Marco


-3/2

0-3

/200

-3/2

0-3

/20

0

F

-10

-3/20

-10 -17/20

F

0-1

/21/27/20

7/20

-3/2

0-3/20 -1

Fb

Ing

Civ

.crp

l.075

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 896

189

Car

pani

Lor

enzo

Mar

co

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

0-1/2-1

/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crp

l.075

PR

OC

ED

IME

NT

O E

RIS

ULT

AT

I 896

189

Car

pani

Lor

enzo

Mar

co

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JD

C b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

DE

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.crpl.075PROCEDIMENTO E RISULTATI 896189 Carpani Lorenzo Marco


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoDE = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

A = 924. mm2

Ju = 296396. mm4

Jv = 68112. mm4



τc = 4.658 N/mm2

σo = √σ2+3τ2 = 133.4 N/mm2

S* = 4975. mm3mm 0 12 18 30 36 48x

0

6

47

53

y

43σc,τc

σm

u

v

Ing Civ.crpl.075


Ing Civ.crpl.075


Ing Civ.crpl.075


Ing Civ.bldd.076REAZIONI 912028 Baldin Daniele


3F

3/20F33/20Fb

4F

3/20F37/20Fb

A

B4F

17/20F37/20Fb

4F

17/20FFb

BC

F

3/20F

3/20F1/2Fb

D

E3/20F

3/2Fb3/20F

33/20Fb

E A

Ing Civ.bldd.076AZIONI INTERNE 912028 Baldin Daniele


3/20

3/20

-4

3/20

3/20

0

F

-3-4

17/20

10

3/20

F

33/2

0-3

7/20

-37/20-1

01/

2

3/2 33/20

Fb

Ing

Civ

.bld

d.07

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1202

8 B

aldi

n D

anie

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

BC

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-7/2

-7/2

-10 1/2 3/

20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1 -1

-10

0 0 0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.bld

d.07

6P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1202

8 B

aldi

n D

anie

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

BA

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

BC

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

DE

b0

Fx-

1/2q

x20

00

0E

D b

0-1

/2F

b+1/

2qx2

00

EA

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (3 x

/b +

1/2

x2 /b2 )

Fb

1/E

J dx

= [3

/2 x

2 /b +

1/6

x3 /b2 ] ob F

b 1/

EJ

= (3

/2 b

+1/

6 b

) Fb

1/E

J =

5/3

Fb2 /E

J

LXo

BA =

∫ ob (7/2

-4

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[7/2

x -

2 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.bldd.076PROCEDIMENTO E RISULTATI 912028 Baldin Daniele


= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

A = 894. mm2

Ju = 284424. mm4

Jv = 80442. mm4

yg = 19.9 mmN = 862.5 NTy = 2875. NMx = 1811250. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 33.1 mmσm = N/A-Mv/Ju = -209.8 N/mm2


τc = 10.07 N/mm2

σo = √σ2+3τ2 = 128.3 N/mm2

S* = 5978. mm3mm 0 12 18 24 30 42x

0

12

41

53

y

40σc,τc

σm

u

v

Ing Civ.bldd.076


Ing Civ.bldd.076


Ing Civ.bldd.076


Ing Civ.crra.077REAZIONI 914406 Corradino Andrea


F

19/40F

19/40F1/2Fb

A

B

19/40F3/2Fb

19/40F79/40Fb

B C4F

19/40F79/40Fb

4F

19/40F81/40Fb

C

D4F

21/40F81/40Fb

4F

61/40FFb

DE

Ing Civ.crra.077AZIONI INTERNE 914406 Corradino Andrea


-19/40 -19/40

0

-19/40

44

F

1 0

19/4

0

-4

21/4

061

/40

F

0 1/2

3/2

79/4

0

79/40-81/40

-81/

40-1

Fb

Ing

Civ

.crr

a.07

7P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1440

6 C

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dino

And

rea

@ A

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Zav

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no, v

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27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o0

1/2

3/2 0 0-4

-4-1M

o fle

ssio

ne d

a ca

richi

ass

egna

ti0

0

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.crr

a.07

7P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1440

6 C

orra

dino

And

rea

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19



AB b0Fx-1/2qx2

0000

BA b0-1/2Fb+1/2qx2

00


2/b

2

-1/4Fb2/EJ1/3Xb/EJ


2/b

2

CD b-1-4Fx4Fx12Fb

2/EJXb/EJ



4Fb-13/2Fx+2Fx2/b+1/2qx

3/b1-2x/b+x

2/b

2

37/24Fb2/EJ1/3Xb/EJ


Fx+7/2Fx2/b-1/2qx

3/bx

2/b

2



Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

Ing Civ.crra.077PROCEDIMENTO E RISULTATI 914406 Corradino Andrea


= ( b ) 1/EJ = b/EJ

LXXDC = ∫

o

b(1 ) 1/EJ dx = [ x ]o

b 1/EJ

= ( b ) 1/EJ = b/EJ

LXXDE = ∫

o


b 1/EJ

= ( b - b +1/3 b ) 1/EJ = 1/3 b/EJ

LXXED = ∫

o

b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o

b 1/EJ

= (1/3 b ) 1/EJ = 1/3 b/EJ

LXoBC = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o

b(4 x/b ) Fb 1/EJ dx = [2 x2/b ]o

b Fb 1/EJ

= (2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDC = ∫

o

b(4 -4 x/b ) Fb 1/EJ dx = [4 x -2 x2/b ]o

b Fb 1/EJ

= (4 b -2 b ) Fb 1/EJ = 2 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (4 b -13/4 b +2/3 b +1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b +7/6 b -1/8 b ) Fb 1/EJ = 37/24 Fb2/EJ

Ing Civ.crra.077PROCEDIMENTO E RISULTATI 914406 Corradino Andrea


A = 822. mm2

Ju = 238712. mm4

Jv = 77634. mm4



τc = 6.055 N/mm2

σo = √σ2+3τ2 = 164.5 N/mm2

S* = 3978. mm3mm 0 12 18 24 30 42x

0

12

47

53

y

44σc,τc

σm

u

v

Ing Civ.crra.077


Ing Civ.crra.077


Ing Civ.arta.078REAZIONI 914905 Artene Alexandru Ionut


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.arta.078AZIONI INTERNE 914905 Artene Alexandru Ionut


-3-3

7/40

0

7/407/40

F

33/4

073

/40

-3

7/40

-10

F

-53/

400

67/40-53/40

1/2

27/4

0

0-1/2

Fb

Ing

Civ

.art

a.07

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1490

5 A

rten

e A

lexa

ndru

Ionu

t

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2 2

0-1

/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.art

a.07

8P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1490

5 A

rten

e A

lexa

ndru

Ionu

t

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.arta.078PROCEDIMENTO E RISULTATI 914905 Artene Alexandru Ionut


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 996. mm2

Ju = 326526. mm4

Jv = 74160. mm4

yg = 24.37 mmN = 1281. NTy = -3660. NMx = -2607750. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 28.63 mmσm = N/A-Mv/Ju = 230. N/mm2


τc = 5.948 N/mm2

σo = √σ2+3τ2 = 137.8 N/mm2

S* = 6368. mm3mm 0 12 18 30 36 48x

0

6

41

53

y

7σc,τc

σm

u

v

Ing Civ.arta.078


Ing Civ.arta.078


Ing Civ.arta.078


Ing Civ.drfd.079REAZIONI 916289 D’Auria Federica


F

19/40F

19/40F1/2Fb

A

B

19/40F1/2Fb

19/40F1/40Fb

BC

19/40F1/40Fb

19/40F1/40Fb

C

D

21/40F1/40Fb

61/40FFb

D E

Ing Civ.drfd.079AZIONI INTERNE 916289 D’Auria Federica


-19/

40-1

9/400

-19/

40

0 0

F

-10

-19/40

0

-21/40 -61/40

F

0-1

/21/21/40

1/40

1/40

1/40-1

Fb

Ing

Civ

.drf

d.07

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1628

9 D

’Aur

ia F

eder

ica

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

C

DE

W

F

W

X

X

q

q

Sch

ema

di c

alco

lo ip

erst

atic

o

0-1/2

1/2

0

00

0-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1-1-1-1

0

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.drf

d.07

9P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1628

9 D

’Aur

ia F

eder

ica

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

-Fx+

1/2q

x20

00

0B

A b

01/

2Fb-

1/2q

x20

0

BC

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

00

10

Xb/

EJ

DC

b1

00

1

DE

b-1

+x/

b-1

/2F

x-1/

2qx2

1/2F

x-1/

2qx3 /b

1-2x

/b+

x2 /b2

1/8F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb-

3/2F

x+1/

2qx2

Fx-

3/2F

x2 /b+

1/2q

x3 /bx2 /b

2

tota

li1/

24F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-1

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

= (-

1/4

b +

1/6

b ) F

b 1/

EJ

= -

1/12

Fb2 /E

J

LXo

CB =

∫ ob (-1/2

x/b

+1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-1/

4 x2 /b

+1/

6 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.drfd.079PROCEDIMENTO E RISULTATI 916289 D’Auria Federica


= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (1/4 b -1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 Fb2/EJ

A = 1140. mm2

Ju = 330567. mm4

Jv = 122544. mm4



τc = 4.722 N/mm2

σo = √σ2+3τ2 = 158.6 N/mm2

S* = 5746. mm3mm 0 12 18 30 36 48x

0

12

47

53

y

42σc,τc

σm

u

v

Ing Civ.drfd.079


Ing Civ.drfd.079


Ing Civ.drfd.079


Ing Civ.dpnd.080REAZIONI 916623 Deponti Daniele


3F

33/40F53/40Fb

3F

73/40F

AB

3F

7/40F67/40Fb

3F

7/40F53/40Fb

C

A

7/40F1/2Fb

7/40F27/40Fb

D C

F

7/40F

7/40F1/2Fb

E

D

Ing Civ.dpnd.080AZIONI INTERNE 916623 Deponti Daniele


33

-7/4

0

0

-7/4

0-7

/40

F

33/4073/40

-3

7/40

-10

F

-53/400

67/4

0-5

3/40

1/2 27/40

0-1

/2

Fb

Ing

Civ

.dpn

d.08

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1662

3 D

epon

ti D

anie

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

CD

EW

F

WX

X

qq

Sch

ema

di c

alco

lo ip

erst

atic

o

00

30

1/2

2

0-1/2

Mo

fless

ione

da

caric

hi a

sseg

nati

-10

-1-1

0-1

00

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.dpn

d.08

0P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1662

3 D

epon

ti D

anie

le

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

+x/

b-1

/2F

x+1/

2qx2

1/2F

x-F

x2 /b+

1/2q

x3 /b1-

2x/b

+x2 /b

2

1/24

Fb2 /E

J1/

3Xb/

EJ

BA

bx/

b1/

2Fx-

1/2q

x21/

2Fx2 /b

-1/2

qx3 /b

x2 /b2

CA

b-1

3Fb-

3Fx

-3F

b+3F

x1

-3/2

Fb2 /E

JX

b/E

JA

C b

1-3

Fx

-3F

x1

DC

b-x

/b1/

2Fb+

3/2F

x-1

/2F

x-3/

2Fx2 /b

x2 /b2

-3/4

Fb2 /E

J1/

3Xb/

EJ

CD

b1-

x/b

-2F

b+3/

2Fx

-2F

b+7/

2Fx-

3/2F

x2 /b1-

2x/b

+x2 /b

2

ED

b0

-Fx+

1/2q

x20

00

0D

E b

01/

2Fb-

1/2q

x20

0

tota

li-5

3/24

Fb2 /E

J5/

3Xb/

EJ

iper

stat

ica

X=

WA

B53

/40F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

BA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

AC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CD =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob (1/2

x/b

- x

2 /b2 +

1/2

x3 /b3 )

Fb

1/E

J dx

= [1

/4 x

2 /b -

1/3

x3 /b2 +

1/8

x4 /b3 ] ob F

b 1/

EJ

= (1

/4 b

-1/

3 b

+1/

8 b

) Fb

1/E

J =

1/2

4 F

b2 /EJ

LXo

BA =

∫ ob (1/2

x2 /b

2 -1/

2 x3 /b

3 ) F

b 1/

EJ

dx =

[1/6

x3 /b

2 -1/

8 x4 /b

3 ] ob Fb

1/E

J

Ing Civ.dpnd.080PROCEDIMENTO E RISULTATI 916623 Deponti Daniele


= (1/6 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ

LXoCA = ∫

o

b(-3 +3 x/b ) Fb 1/EJ dx = [-3 x +3/2 x2/b ]o

b Fb 1/EJ

= (-3 b +3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoAC = ∫

o

b(-3 x/b ) Fb 1/EJ dx = [-3/2 x2/b ]o

b Fb 1/EJ

= (-3/2 b ) Fb 1/EJ = -3/2 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (-1/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (-2 b +7/4 b -1/2 b ) Fb 1/EJ = -3/4 Fb2/EJ

A = 606. mm2

Ju = 215454. mm4

Jv = 40698. mm4



τc = 9.08 N/mm2

σo = √σ2+3τ2 = 150.4 N/mm2

S* = 3442. mm3mm 0 12 18 24 30 42x

0

6

47

53

y

45σc,τc

σm

u

v

Ing Civ.dpnd.080


Ing Civ.dpnd.080


Ing Civ.dpnd.080


Ing Civ.grts.081REAZIONI 917200 Gritcul Serghei


F

3/20F

3/20F1/2Fb

A

B

3/20F3/2Fb

3/20F33/20Fb

B C3F

3/20F33/20Fb

4F

3/20F37/20Fb

C

D4F

17/20F37/20Fb

4F

17/20FFb

DE

Ing Civ.grts.081AZIONI INTERNE 917200 Gritcul Serghei


-3/20 -3/20

0

-3/20 -3/20

4

F

1 0

3/20

-3 -4

17/2

0

F

0 1/2

3/2

33/2

0

33/20-37/20

-37/

20-1

Fb

Ing

Civ

.grt

s.08

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1720

0 G

ritcu

l Ser

ghei

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

AB

C

D

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

01/

2

3/2 0 0-7

/2-7/2-1

Mo

fless

ione

da

caric

hi a

sseg

nati

00

0-1

-1-1

-10

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.grt

s.08

1P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI 9

1720

0 G

ritcu

l Ser

ghei

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WC

D

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b0

Fx-

1/2q

x20

00

0B

A b

0-1

/2F

b+1/

2qx2

00

BC

b-x

/b3/

2Fb-

3/2F

x-3

/2F

x+3/

2Fx2 /b

x2 /b2

-1/4

Fb2 /E

J1/

3Xb/

EJ

CB

b1-

x/b

-3/2

Fx

-3/2

Fx+

3/2F

x2 /b1-

2x/b

+x2 /b

2

CD

b-1

-3F

x-1/

2qx2

3Fx+

1/2F

x2 /b1

5/3F

b2 /EJ

Xb/

EJ

DC

b1

7/2F

b-4F

x+1/

2qx2

7/2F

b-4F

x+1/

2Fx2 /b

1

DE

b-1

+x/

b-7

/2F

b+5/

2Fx

7/2F

b-6F

x+5/

2Fx2 /b

1-2x

/b+

x2 /b2

4/3F

b2 /EJ

1/3X

b/E

JE

D b

x/b

Fb+

5/2F

xF

x+5/

2Fx2 /b

x2 /b2

tota

li11

/4F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WC

D-3

3/20

Fb

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

BC =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

CB =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CD =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DC =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

DE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

ED =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXo

BC =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

= (-

3/4

b +

1/2

b ) F

b 1/

EJ

= -

1/4

Fb2 /E

J

LXo

CB =

∫ ob (-3/2

x/b

+3/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[-3/

4 x2 /b

+1/

2 x3 /b

2 ] ob Fb

1/E

J

Ing Civ.grts.081PROCEDIMENTO E RISULTATI 917200 Gritcul Serghei


= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ

LXoCD = ∫

o


b Fb 1/EJ

= (3/2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDC = ∫

o


b Fb 1/EJ

= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ

LXoDE = ∫

o


b Fb 1/EJ

= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

LXoED = ∫

o


b Fb 1/EJ

= (1/2 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ

A = 690. mm2

Ju = 250916. mm4

Jv = 53046. mm4



τc = 8.826 N/mm2

σo = √σ2+3τ2 = 155.8 N/mm2

S* = 3920. mm3mm 0 12 18 24 30 42x

0

6

47

55

y

9σc,τc

σm

u

v

Ing Civ.grts.081


Ing Civ.grts.081


Ing Civ.grts.081


Ing Civ.xxxx.082REAZIONI Nome:


F

3/20F7/20Fb

3/20F3/20Fb

A

B

17/20F3/20Fb

17/20FFb

B C

F

3/20F

3/20F1/2Fb

D

E

3/20F1/2Fb

3/20F7/20Fb

EA

Ing Civ.xxxx.082AZIONI INTERNE Nome:


3/203/20

0

3/203/20

0

F

-10

-17/

20

-10

-3/2

0

F

7/20-3/20

-3/2

0-1

0-1/21/

27/

20

Fb

Ing

Civ

.xxx

x.08

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI N

ome:

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

A

B

CD

E

W

F

W

X

X

q

qS

chem

a di

cal

colo

iper

stat

ico

0-1

/2

-1/2 -1

0-1

/2

1/20

Mo

fless

ione

da

caric

hi a

sseg

nati

-1-1

-10

00

0-1

Mx

fless

ione

da

iper

stat

ica

X=

1

Ing

Civ

.xxx

x.08

2P

RO

CE

DIM

EN

TO

E R

ISU

LTA

TI N

ome:

@ A

dolfo

Zav

elan

i Ros

si, P

olite

cnic

o di

Mila

no, v

ers.

27.0

3.13

31.0

5.19

Qua

dro

cont

ribut

i PLV

per

iper

stat

ica

X=

WA

B

→M

x(x)

Mo(

x)M

xMo

MxM

x∫M

xMo/

EJd

x∫X

MxM

x/E

Jdx

AB

b-1

-Fx+

1/2q

x2F

x-1/

2Fx2 /b

11/

3Fb2 /E

JX

b/E

JB

A b

11/

2Fb-

1/2q

x21/

2Fb-

1/2F

x2 /b1

BC

b-1

+x/

b-1

/2F

b-1/

2Fx

1/2F

b-1/

2Fx2 /b

1-2x

/b+

x2 /b2

1/3F

b2 /EJ

1/3X

b/E

JC

B b

x/b

Fb-

1/2F

xF

x-1/

2Fx2 /b

x2 /b2

DE

b0

-Fx+

1/2q

x20

00

0E

D b

01/

2Fb-

1/2q

x20

0

EA

b-x

/b1/

2Fb-

1/2F

x-1

/2F

x+1/

2Fx2 /b

x2 /b2

-1/1

2Fb2 /E

J1/

3Xb/

EJ

AE

b1-

x/b

-1/2

Fx

-1/2

Fx+

1/2F

x2 /b1-

2x/b

+x2 /b

2

tota

li7/

12F

b2 /EJ

5/3X

b/E

J

iper

stat

ica

X=

WA

B-7

/20F

b

Svi

lupp

i di c

alco

lo ip

erst

atic

a

LXX

AB =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BA =

∫ ob (1 )

1/E

J dx

= [

x ] ob 1

/EJ

= (

b )

1/E

J =

b/

EJ

LXX

BC =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXX

CB =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

EA =

∫ ob ( x2 /b

2 ) 1

/EJ

dx =

[1/3

x3 /b

2 ] ob 1/E

J

= (1

/3 b

) 1

/EJ

= 1

/3 b

/EJ

LXX

AE =

∫ ob (1 -

2 x/

b +

x2 /b

2 ) 1

/EJ

dx =

[ x

- x2 /b

+1/

3 x3 /b

2 ] ob 1/E

J

= (

b -

b +

1/3

b )

1/E

J =

1/3

b/E

J

LXo

AB =

∫ ob ( x/b

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x2 /b

-1/

6 x3 /b

2 ] ob Fb

1/E

J

= (1

/2 b

-1/

6 b

) Fb

1/E

J =

1/3

Fb2 /E

J

LXo

BA =

∫ ob (1/2

-1/

2 x2 /b

2 ) F

b 1/

EJ

dx =

[1/2

x -

1/6

x3 /b2 ] ob F

b 1/

EJ

Ing Civ.xxxx.082PROCEDIMENTO E RISULTATI Nome:


= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoBC = ∫

o

b(1/2 -1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/6 x3/b2 ]o

b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoCB = ∫

o


b Fb 1/EJ

= (1/2 b -1/6 b ) Fb 1/EJ = 1/3 Fb2/EJ

LXoEA = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

LXoAE = ∫

o


b Fb 1/EJ

= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ

A = 906. mm2

Ju = 265632. mm4

Jv = 89982. mm4

yg = 36.96 mmN = 961.5 NTy = -3205. NMx = -1586480. Nmmxm = 12. mmum = -9. mmvm = -36.96 mmσm = N/A-Mv/Ju = -219.7 N/mm2


τc = 8.772 N/mm2

σo = √σ2+3τ2 = 160.7 N/mm2

S* = 4362. mm3mm 0 12 18 24 30 42x

0

6

41

55

y

10σc,τc

σm

u

v

Ing Civ.xxxx.082


Ing Civ.xxxx.082


Ing Civ.xxxx.082


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