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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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dlsf.001
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dlsf.001
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brti.002REAZIONI 797733 Beretta Igor
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brti.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brti.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnrd.003REAZIONI 811839 Bonora Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ
LXoCD = ∫
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
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( 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
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
xc = 24. mmyc = 10. mmvc = -20.69 mmσc = N/A-Mv/Ju = -155.4 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnrd.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnrd.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chsg.004REAZIONI 813529 Chiesa Giancarlo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ
LXoBC = ∫
o
b(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ
LXoCB = ∫
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
LXoEA = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoAE = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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
xc = 24. mmyc = 11. mmvc = -22.38 mmσc = N/A-Mv/Ju = 161.2 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chsg.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chsg.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dmns.005REAZIONI 817109 Damian Sebastiano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoCB = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dmns.005
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.elds.006REAZIONI 819711 El Dosoky Yasmin
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
yg = 33.72 mmN = 1015. NTy = -2900. NMx = -1870500. Nmmxm = 12. mmum = -9. mmvm = -33.72 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 21. mmyc = 13. mmvc = -20.72 mmσc = N/A-Mv/Ju = -128.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.elds.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.elds.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmm.007REAZIONI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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
yg = 29.14 mmN = -3544. NTy = -3730. NMx = -2545730. Nmmxm = 12. mmum = -12. mmvm = -29.14 mmσm = N/A-Mv/Ju = -219.7 N/mm2
xc = 24. mmyc = 13. mmvc = -16.14 mmσc = N/A-Mv/Ju = -123.3 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmm.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmm.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnst.008REAZIONI 843782 Benassai Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 1224. mm2
Ju = 388508. mm4
Jv = 128736. mm4
yg = 31.94 mmN = -1339. NTy = -3825. NMx = -2782690. Nmmxm = 12. mmum = -12. mmvm = -31.94 mmσm = N/A-Mv/Ju = -229.9 N/mm2
xc = 24. mmyc = 13. mmvc = -18.94 mmσc = N/A-Mv/Ju = -136.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnst.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnst.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnsg.009REAZIONI 845411 Danesi Gabriele
@ 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.dnsg.009AZIONI INTERNE 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]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 = 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnsg.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnsg.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bvra.010REAZIONI 845742 Beverina Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
stat
ica
X=
1
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bvra.010
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bvra.010
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.awda.011REAZIONI 847541 Awad Alaa Mohamed Hassan
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
ed
@ 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
.aw
da.0
11P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4754
1 A
wad
Ala
a M
oham
ed
@ 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.awda.011PROCEDIMENTO E RISULTATI 847541 Awad Alaa Mohamed
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ
LXoCD = ∫
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
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( 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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.awda.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.awda.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grnd.012REAZIONI 849412 Gerna Diego
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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Die
go
@ A
dolfo
Zav
elan
i Ros
si, P
olite
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o di
Mila
no, v
ers.
27.0
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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
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4941
2 G
erna
Die
go
@ 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.grnd.012PROCEDIMENTO E RISULTATI 849412 Gerna Diego
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ
LXoBC = ∫
o
b(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ
LXoCB = ∫
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
LXoEA = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoAE = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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
xc = 21. mmyc = 41. mmvc = 20.72 mmσc = N/A-Mv/Ju = -134.5 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grnd.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grnd.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnmc.013REAZIONI 850426 Dioni Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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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
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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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
yg = 18.13 mmN = 3750. NTy = -3000. NMx = -1575000. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 35.87 mmσm = N/A-Mv/Ju = 229.6 N/mm2
xc = 21. mmyc = 44. mmvc = 25.87 mmσc = N/A-Mv/Ju = 166.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnmc.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnmc.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.espa.014REAZIONI 850941 Esposto Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
yg = 24.86 mmN = 8945. NTy = -4835. NMx = -2719690. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 29.14 mmσm = N/A-Mv/Ju = 239.9 N/mm2
xc = 24. mmyc = 41. mmvc = 16.14 mmσc = N/A-Mv/Ju = 136.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.espa.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.espa.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmt.015REAZIONI 868168 Boara Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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Mila
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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
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si, P
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Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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.brmt.015PROCEDIMENTO E RISULTATI 868168 Boara Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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 = 1152. mm2
Ju = 348030. mm4
Jv = 122688. mm4
yg = 20.63 mmN = -4369. NTy = -3495. NMx = -2123210. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 33.38 mmσm = N/A-Mv/Ju = 199.8 N/mm2
xc = 24. mmyc = 43. mmvc = 22.38 mmσc = N/A-Mv/Ju = 132.7 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmt.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brmt.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnr.016REAZIONI 868395 Bernasconi Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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ISU
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TI 8
6839
5 B
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scon
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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
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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.brnr.016PROCEDIMENTO E RISULTATI 868395 Bernasconi Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
yg = 21.35 mmN = 4301. NTy = 2325. NMx = 1499630. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 32.65 mmσm = N/A-Mv/Ju = -209.6 N/mm2
xc = 21. mmyc = 46. mmvc = 24.65 mmσc = N/A-Mv/Ju = -156.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnr.016
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnr.016
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dmcm.017REAZIONI 870485 D’Amico Michele Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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CE
DIM
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TO
E R
ISU
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7048
5 D
’Am
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Mic
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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
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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
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.dmcm.017PROCEDIMENTO E RISULTATI 870485 D’Amico Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoCB = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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.dmcm.017PROCEDIMENTO E RISULTATI 870485 D’Amico Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
xc = 21. mmyc = 8. mmvc = -23.11 mmσc = N/A-Mv/Ju = 162.3 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dmcm.017
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dflm.018REAZIONI 871570 De Flammineis Margherita
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 786. mm2
Ju = 237222. mm4
Jv = 71478. mm4
yg = 34.93 mmN = 770. NTy = -2200. NMx = -1567500. Nmmxm = 12. mmum = -9. mmvm = -34.93 mmσm = N/A-Mv/Ju = -229.8 N/mm2
xc = 21. mmyc = 9. mmvc = -25.93 mmσc = N/A-Mv/Ju = -170.3 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dflm.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dflm.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cltf.019REAZIONI 871912 Calati Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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 = 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
xc = 24. mmyc = 10. mmvc = -19.46 mmσc = N/A-Mv/Ju = -159.6 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cltf.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cltf.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dgtn.020REAZIONI 876780 De Gaetano Salvatore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 1104. mm2
Ju = 327740. mm4
Jv = 113472. mm4
yg = 32.5 mmN = -1761. NTy = -5030. NMx = -1999430. Nmmxm = 12. mmum = -12. mmvm = -32.5 mmσm = N/A-Mv/Ju = -199.9 N/mm2
xc = 24. mmyc = 11. mmvc = -21.5 mmσc = N/A-Mv/Ju = -132.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dgtn.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dgtn.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cste.021REAZIONI 877793 Castiglione Ettore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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.cste.021PROCEDIMENTO E RISULTATI 877793 Castiglione Ettore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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 = 642. mm2
Ju = 237823. mm4
Jv = 37350. mm4
yg = 28.63 mmN = 5200. NTy = -4160. NMx = -1809600. Nmmxm = 12. mmum = -9. mmvm = -28.63 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 21. mmyc = 47. mmvc = 18.37 mmσc = N/A-Mv/Ju = 147.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cste.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cste.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.agsr.022REAZIONI 877947 Agosti Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 858. mm2
Ju = 281777. mm4
Jv = 74286. mm4
yg = 32.75 mmN = 7705. NTy = -4165. NMx = -1967960. Nmmxm = 12. mmum = -9. mmvm = -32.75 mmσm = N/A-Mv/Ju = -219.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.agsr.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.agsr.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brgf.023REAZIONI 877968 Bergui Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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.brgf.023PROCEDIMENTO E RISULTATI 877968 Bergui Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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 = 960. mm2
Ju = 313227. mm4
Jv = 65088. mm4
yg = 27.92 mmN = -6144. NTy = -4915. NMx = -2506650. Nmmxm = 12. mmum = -12. mmvm = -27.92 mmσm = N/A-Mv/Ju = -229.9 N/mm2
xc = 24. mmyc = 47. mmvc = 19.08 mmσc = N/A-Mv/Ju = 146.3 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brgf.023
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brgf.023
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dcml.024REAZIONI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
.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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 1176. mm2
Ju = 365284. mm4
Jv = 119520. mm4
yg = 31.06 mmN = 9102. NTy = 4920. NMx = 2730600. Nmmxm = 12. mmum = -12. mmvm = -31.06 mmσm = N/A-Mv/Ju = 239.9 N/mm2
xc = 24. mmyc = 13. mmvc = -18.06 mmσc = N/A-Mv/Ju = 142.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dcml.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dcml.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grcg.025REAZIONI 878061 Greco Giovanni
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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-
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- 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(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
LXoED = ∫
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
A = 1200. mm2
Ju = 364306. mm4
Jv = 127584. mm4
yg = 21.43 mmN = -6993. NTy = 3780. NMx = 2239650. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 31.57 mmσm = N/A-Mv/Ju = -199.9 N/mm2
xc = 24. mmyc = 41. mmvc = 19.57 mmσc = N/A-Mv/Ju = -126.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grcg.025
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grcg.025
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldg.026REAZIONI 878411 Baldon Giacomo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- 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( 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
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
yg = 23.04 mmN = -7869. NTy = 2580. NMx = 1625400. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 29.96 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 21. mmyc = 41. mmvc = 17.96 mmσc = N/A-Mv/Ju = -130.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldg.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldg.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.csrf.027REAZIONI 879105 Caserta Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 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( 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
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
xc = 24. mmyc = 7. mmvc = -15.51 mmσc = N/A-Mv/Ju = -121.7 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.csrf.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.csrf.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brga.028REAZIONI 881254 Broggi Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- 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( 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
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
yg = 19.6 mmN = 8555. NTy = 2805. NMx = 1998560. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 33.4 mmσm = N/A-Mv/Ju = -229.8 N/mm2
xc = 21. mmyc = 41. mmvc = 21.4 mmσc = N/A-Mv/Ju = -143.7 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brga.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brga.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.allf.029REAZIONI 886516 Allegra Filippo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- b +1/4 b +1/6 b ) Fb 1/EJ = -7/12 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
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
yg = 17.34 mmN = -4300. NTy = 2000. NMx = 1500000. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 35.66 mmσm = N/A-Mv/Ju = -239.9 N/mm2
xc = 21. mmyc = 42. mmvc = 24.66 mmσc = N/A-Mv/Ju = -167.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.allf.029
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.allf.029
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpdv.030REAZIONI 886876 De Padova Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (1/2 b -1/2 b +1/8 b ) Fb 1/EJ = 1/8 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(-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
A = 984. mm2
Ju = 322959. mm4
Jv = 73152. mm4
yg = 24.16 mmN = 5206. NTy = -5480. NMx = -2178300. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 28.84 mmσm = N/A-Mv/Ju = 199.8 N/mm2
xc = 24. mmyc = 41. mmvc = 16.84 mmσc = N/A-Mv/Ju = 118.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpdv.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpdv.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bssm.031REAZIONI 886998 Bassi Marianna
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- b +1/4 b +1/6 b ) Fb 1/EJ = -7/12 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
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
yg = 19.93 mmN = 10535. NTy = 4900. NMx = 2131500. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 33.07 mmσm = N/A-Mv/Ju = -209.9 N/mm2
xc = 24. mmyc = 42. mmvc = 22.07 mmσc = N/A-Mv/Ju = -137. N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bssm.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bssm.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brss.032REAZIONI 887236 Berselli Samuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
totali79/24Fb2/EJ5/3Xb/EJ
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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(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
LXoCB = ∫
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
LXoEA = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoAE = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 594. mm2
Ju = 206801. mm4
Jv = 40230. mm4
yg = 20.38 mmN = 2864. NTy = 3015. NMx = 1424590. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 32.62 mmσm = N/A-Mv/Ju = -219.9 N/mm2
xc = 21. mmyc = 44. mmvc = 23.62 mmσc = N/A-Mv/Ju = -157.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brss.032
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.czzg.033REAZIONI 887287 Cazzagon Giovanni
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-3/2 b +3/4 b ) Fb 1/EJ = -3/4 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( 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
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
yg = 34.05 mmN = -5466. NTy = 3215. NMx = 1711990. Nmmxm = 12. mmum = -9. mmvm = -34.05 mmσm = N/A-Mv/Ju = 229.8 N/mm2
xc = 21. mmyc = 8. mmvc = -26.05 mmσc = N/A-Mv/Ju = 173.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.czzg.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.czzg.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnda.034REAZIONI 887657 Donadio Aurora
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- 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
xc = 21. mmyc = 10. mmvc = -26.77 mmσc = N/A-Mv/Ju = -174.6 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnda.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnda.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmc.035REAZIONI 887860 Colombo Cristina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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.clmc.035PROCEDIMENTO E RISULTATI 887860 Colombo Cristina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-3/2 b +3/4 b ) Fb 1/EJ = -3/4 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( 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
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 = 984. mm2
Ju = 337339. mm4
Jv = 77472. mm4
yg = 31.85 mmN = 5678. NTy = 3340. NMx = 2054100. Nmmxm = 12. mmum = -12. mmvm = -31.85 mmσm = N/A-Mv/Ju = 199.7 N/mm2
xc = 24. mmyc = 11. mmvc = -20.85 mmσc = N/A-Mv/Ju = 132.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmc.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmc.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cnns.036REAZIONI 887904 Cannarsa Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (- 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
yg = 34.22 mmN = 343. NTy = -3430. NMx = -2263800. Nmmxm = 12. mmum = -12. mmvm = -34.22 mmσm = N/A-Mv/Ju = -209.9 N/mm2
xc = 24. mmyc = 12. mmvc = -22.22 mmσc = N/A-Mv/Ju = -136.2 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cnns.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cnns.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmg.037REAZIONI 888192 Colombo Giacomo Paolo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
bo 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
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
ISU
LTA
TI 8
8819
2 C
olom
bo 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=
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 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(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
yg = 31.57 mmN = 4163. NTy = 2775. NMx = 1935560. Nmmxm = 12. mmum = -9. mmvm = -31.57 mmσm = N/A-Mv/Ju = 219.6 N/mm2
xc = 21. mmyc = 13. mmvc = -18.57 mmσc = N/A-Mv/Ju = 131.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmg.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clmg.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnda.038REAZIONI 888197 Bendo Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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.bnda.038PROCEDIMENTO E RISULTATI 888197 Bendo Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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 = 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
yg = 34.65 mmN = 6101. NTy = -2905. NMx = -2156960. Nmmxm = 12. mmum = -9. mmvm = -34.65 mmσm = N/A-Mv/Ju = -229.8 N/mm2
xc = 21. mmyc = 13. mmvc = -21.65 mmσc = N/A-Mv/Ju = -141.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnda.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnda.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frsa.039REAZIONI 888228 Farisè Alberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ
LXoCD = ∫
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
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 - 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
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
yg = 30.3 mmN = -7697. NTy = -3665. NMx = -2858700. Nmmxm = 12. mmum = -12. mmvm = -30.3 mmσm = N/A-Mv/Ju = -239.7 N/mm2
xc = 24. mmyc = 13. mmvc = -17.3 mmσc = N/A-Mv/Ju = -140. N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frsa.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frsa.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bsns.040REAZIONI 888270 Biasinutto Sara
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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.bsns.040PROCEDIMENTO E RISULTATI 888270 Biasinutto Sara
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ
LXoBC = ∫
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
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(-3/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ
LXoAE = ∫
o
b(1 -7/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [ x -7/4 x2/b +5/6 x3/b2 ]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
yg = 32.79 mmN = -9473. NTy = 6315. NMx = 2604940. Nmmxm = 12. mmum = -12. mmvm = -32.79 mmσm = N/A-Mv/Ju = 199.9 N/mm2
xc = 24. mmyc = 13. mmvc = -19.79 mmσc = N/A-Mv/Ju = 117.7 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bsns.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bsns.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dvta.041REAZIONI 888341 De Vita Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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-5/2Fx-3/2Fx+5/2Fx2/bx
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
DC b14Fb-4Fx4Fb-4Fx1
DE 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
ED bx/b9/2Fx-1/2qx2
9/2Fx2/b-1/2qx
3/bx
2/b
2
totali83/24Fb2/EJ5/3Xb/EJ
iperstatica X=WCD-83/40Fb
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.dvta.041PROCEDIMENTO E RISULTATI 888341 De Vita Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(-3/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ
LXoCB = ∫
o
b(1 -7/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [ x -7/4 x2/b +5/6 x3/b2 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
xc = 24. mmyc = 41. mmvc = 18.32 mmσc = N/A-Mv/Ju = -117.2 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dvta.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clla.042REAZIONI 888424 Collini Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-1/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -5/6 x3/b2 ]o
b Fb 1/EJ
= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoCD = ∫
o
b(-3 +11/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-3 x +11/4 x2/b -5/6 x3/b2 ]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
yg = 24.63 mmN = 8288. NTy = -3855. NMx = -1908230. Nmmxm = 30. mmym = 55. mmum = 9. mmvm = 30.37 mmσm = N/A-Mv/Ju = 219.8 N/mm2
xc = 21. mmyc = 41. mmvc = 16.37 mmσc = N/A-Mv/Ju = 123.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clla.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clla.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bstf.043REAZIONI 888472 Busetto Ferruccio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
etto
Fer
rucc
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
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
IME
NT
O E
RIS
ULT
AT
I 888
472
Bus
etto
Fer
rucc
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
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.bstf.043PROCEDIMENTO E RISULTATI 888472 Busetto Ferruccio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ
LXoDE = ∫
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
LXoED = ∫
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 = 960. mm2
Ju = 334225. mm4
Jv = 69408. mm4
yg = 24.09 mmN = -13626. NTy = -4955. NMx = -2638540. Nmmxm = 36. mmym = 55. mmum = 12. mmvm = 30.91 mmσm = N/A-Mv/Ju = 229.8 N/mm2
xc = 24. mmyc = 7. mmvc = -17.09 mmσc = N/A-Mv/Ju = -149.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bstf.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bstf.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frru.044REAZIONI 888759 Ferretti Umberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-1/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -5/6 x3/b2 ]o
b Fb 1/EJ
= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoCD = ∫
o
b(-3 +11/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-3 x +11/4 x2/b -5/6 x3/b2 ]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
yg = 20.95 mmN = -8718. NTy = -4055. NMx = -2341760. Nmmxm = 30. mmym = 55. mmum = 9. mmvm = 34.05 mmσm = N/A-Mv/Ju = 239.9 N/mm2
xc = 21. mmyc = 41. mmvc = 20.05 mmσc = N/A-Mv/Ju = 137.4 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frru.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.frru.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.fngl.045REAZIONI 888972 Fenoglio Lucia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -7/4 b +5/6 b ) Fb 1/EJ = 1/12 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(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 = 846. mm2
Ju = 274282. mm4
Jv = 78138. mm4
yg = 18.9 mmN = 3788. NTy = 2525. NMx = 1552880. Nmmxm = 30. mmym = 55. mmum = 9. mmvm = 36.1 mmσm = N/A-Mv/Ju = -199.9 N/mm2
xc = 21. mmyc = 46. mmvc = 27.1 mmσc = N/A-Mv/Ju = -148.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.fngl.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.fngl.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnzs.046REAZIONI 889282 Donzelli Sara
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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 = 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 = 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
xc = 24. mmyc = 41. mmvc = 15.45 mmσc = N/A-Mv/Ju = 113.8 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnzs.046
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dnzs.046
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blll.047REAZIONI 889389 Bello Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ
LXoCD = ∫
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
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 - 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
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
yg = 21.32 mmN = -7581. NTy = -3610. NMx = -2517980. Nmmxm = 36. mmym = 55. mmum = 12. mmvm = 33.68 mmσm = N/A-Mv/Ju = 219.7 N/mm2
xc = 24. mmyc = 44. mmvc = 22.68 mmσc = N/A-Mv/Ju = 145.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blll.047
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blll.047
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnl.048REAZIONI 889394 Brandizi Leonardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ
LXoBC = ∫
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
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(-3/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ
LXoAE = ∫
o
b(1 -7/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [ x -7/4 x2/b +5/6 x3/b2 ]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
yg = 22.3 mmN = -3398. NTy = 2265. NMx = 1681760. Nmmxm = 30. mmym = 55. mmum = 9. mmvm = 32.7 mmσm = N/A-Mv/Ju = -229.7 N/mm2
xc = 21. mmyc = 47. mmvc = 24.7 mmσc = N/A-Mv/Ju = -174.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnl.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brnl.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crmp.049REAZIONI 889561 Cremona Paolo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
CA b-13Fb-3Fx-3Fb+3Fx1-3/2Fb
2/EJXb/EJ
AC b1-3Fx-3Fx1
DC b-x/b1/2Fb+3Fx-1/2qx2
-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
totali-59/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAB59/40Fb
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
LXoAB = ∫
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
LXoBA = ∫
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
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(-3 +5 x/b -3/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-3 x +5/2 x2/b -1/2 x3/b2 -1/8 x4/b3 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 648. mm2
Ju = 233472. mm4
Jv = 46872. mm4
yg = 33.47 mmN = 6969. NTy = -2285. NMx = -1748030. Nmmxm = 12. mmum = -9. mmvm = -33.47 mmσm = N/A-Mv/Ju = -239.9 N/mm2
xc = 21. mmyc = 8. mmvc = -25.47 mmσc = N/A-Mv/Ju = -180. N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crmp.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crcr.050REAZIONI 889619 Cura Curà Pietro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ
A = 864. mm2
Ju = 252075. mm4
Jv = 83808. mm4
yg = 36.1 mmN = 13560. NTy = -3715. NMx = -1504580. Nmmxm = 12. mmum = -9. mmvm = -36.1 mmσm = N/A-Mv/Ju = -199.8 N/mm2
xc = 21. mmyc = 9. mmvc = -27.1 mmσc = N/A-Mv/Ju = -146.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crcr.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crcr.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grnl.051REAZIONI 889847 Guarneri Ludovico
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
CA b-13Fb-3Fx-3Fb+3Fx1-3/2Fb
2/EJXb/EJ
AC b1-3Fx-3Fx1
DC b-x/b1/2Fb+3Fx-1/2qx2
-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
totali-59/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAB59/40Fb
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
LXoAB = ∫
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
LXoBA = ∫
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
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(-3 +5 x/b -3/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-3 x +5/2 x2/b -1/2 x3/b2 -1/8 x4/b3 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 972. mm2
Ju = 321252. mm4
Jv = 77328. mm4
yg = 31.31 mmN = -13832. NTy = -4535. NMx = -2006740. Nmmxm = 12. mmum = -12. mmvm = -31.31 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 24. mmyc = 11. mmvc = -20.31 mmσc = N/A-Mv/Ju = -141.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grnl.051
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnga.052REAZIONI 890108 Bongiovanni Antonio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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ISU
LTA
TI 8
9010
8 B
ongi
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nton
io
@ A
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Zav
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i Ros
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olite
cnic
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Mila
no, v
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27.0
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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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnga.052
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnga.052
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cmzk.053REAZIONI 890185 Camozzi Kevin
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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CE
DIM
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TO
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LTA
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9018
5 C
amoz
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evin
@ A
dolfo
Zav
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i Ros
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olite
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o di
Mila
no, v
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27.0
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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
E R
ISU
LTA
TI 8
9018
5 C
amoz
zi K
evin
@ 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-5/2Fx-3/2Fx+5/2Fx2/bx
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
DC b14Fb-4Fx4Fb-4Fx1
DE 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
ED bx/b9/2Fx-1/2qx2
9/2Fx2/b-1/2qx
3/bx
2/b
2
totali83/24Fb2/EJ5/3Xb/EJ
iperstatica X=WCD-83/40Fb
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.cmzk.053PROCEDIMENTO E RISULTATI 890185 Camozzi Kevin
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(-3/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ
LXoCB = ∫
o
b(1 -7/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [ x -7/4 x2/b +5/6 x3/b2 ]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.cmzk.053PROCEDIMENTO E RISULTATI 890185 Camozzi Kevin
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 720. mm2
Ju = 272496. mm4
Jv = 49680. mm4
yg = 31.02 mmN = 3205. NTy = 3770. NMx = 1979250. Nmmxm = 12. mmum = -9. mmvm = -31.02 mmσm = N/A-Mv/Ju = 229.8 N/mm2
xc = 21. mmyc = 13. mmvc = -18.02 mmσc = N/A-Mv/Ju = 135.4 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cmzk.053
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bslf.054REAZIONI 890213 Basilico Filippo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-1/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -5/6 x3/b2 ]o
b Fb 1/EJ
= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoCD = ∫
o
b(-3 +11/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-3 x +11/4 x2/b -5/6 x3/b2 ]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
yg = 34.02 mmN = 8417. NTy = -3915. NMx = -2202190. Nmmxm = 12. mmum = -9. mmvm = -34.02 mmσm = N/A-Mv/Ju = -239.8 N/mm2
xc = 21. mmyc = 13. mmvc = -21.02 mmσc = N/A-Mv/Ju = -144.7 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bslf.054
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bslf.054
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bcga.055REAZIONI 890313 Bacigalupo Ada
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b -5/4 b +1/2 b ) Fb 1/EJ = 1/4 Fb2/EJ
LXoDE = ∫
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
LXoED = ∫
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 = 1044. mm2
Ju = 354848. mm4
Jv = 83376. mm4
yg = 29.78 mmN = -10258. NTy = -3730. NMx = -2265980. Nmmxm = 12. mmum = -12. mmvm = -29.78 mmσm = N/A-Mv/Ju = -200. N/mm2
xc = 24. mmyc = 13. mmvc = -16.78 mmσc = N/A-Mv/Ju = -117. N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bcga.055
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bcga.055
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gnnl.056REAZIONI 890623 Giannini Lucrezia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-1/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-1/4 x2/b -5/6 x3/b2 ]o
b Fb 1/EJ
= (-1/4 b -5/6 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoCD = ∫
o
b(-3 +11/2 x/b -5/2 x2/b2 ) Fb 1/EJ dx = [-3 x +11/4 x2/b -5/6 x3/b2 ]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
yg = 32.21 mmN = -8245. NTy = -3835. NMx = -2473580. Nmmxm = 12. mmum = -12. mmvm = -32.21 mmσm = N/A-Mv/Ju = -210. N/mm2
xc = 24. mmyc = 13. mmvc = -19.21 mmσc = N/A-Mv/Ju = -127.9 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gnnl.056
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gnnl.056
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.glzt.057REAZIONI 890711 Galeazzi Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
.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
Quadro contributi PLV per iperstatica X=WAB
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
CA b-13Fb-3Fx-3Fb+3Fx1-3/2Fb
2/EJXb/EJ
AC b1-3Fx-3Fx1
DC b-x/b1/2Fb+3Fx-1/2qx2
-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
totali-59/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAB59/40Fb
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
LXoAB = ∫
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
LXoBA = ∫
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
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(-3 +5 x/b -3/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-3 x +5/2 x2/b -1/2 x3/b2 -1/8 x4/b3 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 1236. mm2
Ju = 393000. mm4
Jv = 129744. mm4
yg = 22.25 mmN = 11636. NTy = -3815. NMx = -2603740. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 31.75 mmσm = N/A-Mv/Ju = 219.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.glzt.057
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.ambf.058REAZIONI 890818 Ambiveri Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ
A = 696. mm2
Ju = 265885. mm4
Jv = 43992. mm4
yg = 24.15 mmN = 9654. NTy = -2645. NMx = -1924240. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 29.85 mmσm = N/A-Mv/Ju = 229.9 N/mm2
xc = 21. mmyc = 40. mmvc = 15.85 mmσc = N/A-Mv/Ju = 128.6 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.ambf.058
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.ambf.058
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.abbg.059REAZIONI 890974 Abbenda Giovanni
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
CA b-13Fb-3Fx-3Fb+3Fx1-3/2Fb
2/EJXb/EJ
AC b1-3Fx-3Fx1
DC b-x/b1/2Fb+3Fx-1/2qx2
-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
totali-59/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAB59/40Fb
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
LXoAB = ∫
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
LXoBA = ∫
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
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(-3 +5 x/b -3/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-3 x +5/2 x2/b -1/2 x3/b2 -1/8 x4/b3 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 948. mm2
Ju = 318391. mm4
Jv = 69264. mm4
yg = 23.61 mmN = -10477. NTy = -3435. NMx = -2627780. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 30.39 mmσm = N/A-Mv/Ju = 239.7 N/mm2
xc = 24. mmyc = 7. mmvc = -16.61 mmσc = N/A-Mv/Ju = -148.2 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.abbg.059
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crsn.060REAZIONI 891208 Crispiatico Nicol
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 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
xc = 21. mmyc = 40. mmvc = 19.44 mmσc = N/A-Mv/Ju = -116. 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crsn.060
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crsn.060
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brsa.061REAZIONI 891317 Borsani Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(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
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
xc = 21. mmyc = 45. mmvc = 26.45 mmσc = N/A-Mv/Ju = -156.6 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brsa.061
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.brsa.061
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.btta.062REAZIONI 891364 Battelino Ada Sonia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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
yg = 25.05 mmN = -23765. NTy = 4900. NMx = 2352000. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 28.95 mmσm = N/A-Mv/Ju = -219.8 N/mm2
xc = 24. mmyc = 40. mmvc = 14.95 mmσc = N/A-Mv/Ju = -124.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.btta.062
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.btta.062
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chtc.063REAZIONI 891692 Chitanu Cristian
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ
A = 1164. mm2
Ju = 356609. mm4
Jv = 123696. mm4
yg = 20.9 mmN = -18396. NTy = -5040. NMx = -2646000. Nmmxm = 36. mmym = 54. mmum = 12. mmvm = 33.1 mmσm = N/A-Mv/Ju = 229.8 N/mm2
xc = 24. mmyc = 44. mmvc = 23.1 mmσc = N/A-Mv/Ju = 155.6 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chtc.063
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.chtc.063
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clml.064REAZIONI 892410 Colombo Leonardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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
yg = 21.86 mmN = 16733. NTy = 3450. NMx = 1940630. Nmmxm = 30. mmym = 54. mmum = 9. mmvm = 32.14 mmσm = N/A-Mv/Ju = -239.7 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clml.064
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.clml.064
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dlcj.065REAZIONI 892644 Dolci Jacopo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoDE = ∫
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
LXoED = ∫
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
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
yg = 32.08 mmN = -9658. NTy = 2445. NMx = 1448660. Nmmxm = 12. mmum = -9. mmvm = -32.08 mmσm = N/A-Mv/Ju = 199.8 N/mm2
xc = 21. mmyc = 46. mmvc = 13.92 mmσc = N/A-Mv/Ju = -109.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dlcj.065
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dlcj.065
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blla.066REAZIONI 892814 Biella Astrid
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/6 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ
LXoEA = ∫
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 = 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 = 822. mm2
Ju = 238712. mm4
Jv = 77634. mm4
yg = 35.22 mmN = 6435. NTy = -2340. NMx = -1474200. Nmmxm = 12. mmum = -9. mmvm = -35.22 mmσm = N/A-Mv/Ju = -209.6 N/mm2
xc = 21. mmyc = 9. mmvc = -26.22 mmσc = N/A-Mv/Ju = -154.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blla.066
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.blla.066
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cppn.067REAZIONI 892907 Capparelli Nicola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoDE = ∫
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
LXoED = ∫
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
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
xc = 24. mmyc = 10. mmvc = -20.16 mmσc = N/A-Mv/Ju = 151.3 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cppn.067
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.cppn.067
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.emdr.068REAZIONI 892994 Emad Ragab Abdel Hamid
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
.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
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
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
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
totali83/24Fb2/EJ5/3Xb/EJ
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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 -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
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(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
LXoCB = ∫
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
LXoEA = ∫
o
b(-3/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +5/6 b ) Fb 1/EJ = 1/12 Fb2/EJ
LXoAE = ∫
o
b(1 -7/2 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [ x -7/4 x2/b +5/6 x3/b2 ]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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 1140. mm2
Ju = 330567. mm4
Jv = 122544. mm4
yg = 32.78 mmN = -2792. NTy = 3285. NMx = 2340560. Nmmxm = 12. mmum = -12. mmvm = -32.78 mmσm = N/A-Mv/Ju = 229.7 N/mm2
xc = 24. mmyc = 11. mmvc = -21.78 mmσc = N/A-Mv/Ju = 151.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.emdr.068
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bltt.069REAZIONI 893013 Beltran Toledo Italo Jose
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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.bltt.069PROCEDIMENTO E RISULTATI 893013 Beltran Toledo Italo Jose
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/2 b -1/4 b +1/3 b ) Fb 1/EJ = -5/12 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 = 678. mm2
Ju = 249963. mm4
Jv = 43506. mm4
yg = 29.63 mmN = -10028. NTy = 2865. NMx = 2148750. Nmmxm = 12. mmum = -9. mmvm = -29.63 mmσm = N/A-Mv/Ju = 239.9 N/mm2
xc = 21. mmyc = 46. mmvc = 16.37 mmσc = N/A-Mv/Ju = -155.5 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bltt.069
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bltt.069
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bzzs.070REAZIONI 893038 Bozzini Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
ozzi
ni S
ilvia
@ 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
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
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
9303
8 B
ozzi
ni S
ilvia
@ 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.bzzs.070PROCEDIMENTO E RISULTATI 893038 Bozzini Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
yg = 33.1 mmN = 7659. NTy = -4505. NMx = -1790740. Nmmxm = 12. mmum = -9. mmvm = -33.1 mmσm = N/A-Mv/Ju = -199.9 N/mm2
xc = 21. mmyc = 13. mmvc = -20.1 mmσc = N/A-Mv/Ju = -118. N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bzzs.070
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bzzs.070
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crlm.071REAZIONI 893348 Carlino Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/2 b -1/4 b +1/3 b ) Fb 1/EJ = -5/12 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 = 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
xc = 24. mmyc = 13. mmvc = -15.63 mmσc = N/A-Mv/Ju = 122.7 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crlm.071
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crlm.071
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnny.072REAZIONI 893640 Banani Yassine
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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
yg = 31.37 mmN = -8993. NTy = -5290. NMx = -2499530. Nmmxm = 12. mmum = -12. mmvm = -31.37 mmσm = N/A-Mv/Ju = -219.9 N/mm2
xc = 24. mmyc = 13. mmvc = -18.37 mmσc = N/A-Mv/Ju = -131.8 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnny.072
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnny.072
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnmh.073REAZIONI 894745 Ben M’Hamed Ines
@ 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.bnmh.073AZIONI INTERNE 894745 Ben M’Hamed Ines
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]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 = 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
xc = 24. mmyc = 40. mmvc = 18.37 mmσc = N/A-Mv/Ju = -135.2 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnmh.073
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bnmh.073
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gstc.074REAZIONI 895222 Giusto Carola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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
A = 678. mm2
Ju = 249963. mm4
Jv = 43506. mm4
yg = 23.37 mmN = 1086. NTy = -3620. NMx = -2009100. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 29.63 mmσm = N/A-Mv/Ju = 239.8 N/mm2
xc = 21. mmyc = 40. mmvc = 16.63 mmσc = N/A-Mv/Ju = 135.3 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gstc.074
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.gstc.074
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crpl.075REAZIONI 896189 Carpani Lorenzo Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-1/4 b +1/6 b ) Fb 1/EJ = -1/12 Fb2/EJ
LXoCD = ∫
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
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( 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
A = 924. mm2
Ju = 296396. mm4
Jv = 68112. mm4
yg = 22.84 mmN = -999. NTy = -3330. NMx = -1973030. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 30.16 mmσm = N/A-Mv/Ju = 199.7 N/mm2
xc = 24. mmyc = 43. mmvc = 20.16 mmσc = N/A-Mv/Ju = 133.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crpl.075
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crpl.075
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldd.076REAZIONI 912028 Baldin Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (7/2 b -2 b +1/6 b ) Fb 1/EJ = 5/3 Fb2/EJ
LXoBC = ∫
o
b(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (7/2 b -3 b +5/6 b ) Fb 1/EJ = 4/3 Fb2/EJ
LXoCB = ∫
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
LXoEA = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoAE = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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
xc = 21. mmyc = 40. mmvc = 20.1 mmσc = N/A-Mv/Ju = -127.1 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldd.076
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.bldd.076
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crra.077REAZIONI 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
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
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
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.crra.077PROCEDIMENTO E RISULTATI 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= ( 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(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/2 b ) Fb 1/EJ = -1/4 Fb2/EJ
LXoCB = ∫
o
b(-3/2 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/2 x3/b2 ]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.crra.077PROCEDIMENTO E RISULTATI 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
A = 822. mm2
Ju = 238712. mm4
Jv = 77634. mm4
yg = 17.78 mmN = -2071. NTy = 2180. NMx = 1471500. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 35.22 mmσm = N/A-Mv/Ju = -219.6 N/mm2
xc = 21. mmyc = 44. mmvc = 26.22 mmσc = N/A-Mv/Ju = -164.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.crra.077
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.arta.078REAZIONI 914905 Artene Alexandru Ionut
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 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
xc = 24. mmyc = 7. mmvc = -17.37 mmσc = N/A-Mv/Ju = -137.4 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.arta.078
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.arta.078
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.drfd.079REAZIONI 916289 D’Auria Federica
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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 = 1140. mm2
Ju = 330567. mm4
Jv = 122544. mm4
yg = 20.22 mmN = -3097. NTy = -3260. NMx = -2445000. Nmmxm = 36. mmym = 53. mmum = 12. mmvm = 32.78 mmσm = N/A-Mv/Ju = 239.8 N/mm2
xc = 24. mmyc = 42. mmvc = 21.78 mmσc = N/A-Mv/Ju = 158.4 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.drfd.079
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.drfd.079
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpnd.080REAZIONI 916623 Deponti Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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 = 606. mm2
Ju = 215454. mm4
Jv = 40698. mm4
yg = 20.92 mmN = -1194. NTy = -3410. NMx = -1355480. Nmmxm = 30. mmym = 53. mmum = 9. mmvm = 32.08 mmσm = N/A-Mv/Ju = 199.9 N/mm2
xc = 21. mmyc = 45. mmvc = 24.08 mmσc = N/A-Mv/Ju = 149.6 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpnd.080
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.dpnd.080
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grts.081REAZIONI 917200 Gritcul Serghei
@ 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.grts.081AZIONI INTERNE 917200 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
-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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (-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(7/2 -6 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [7/2 x -3 x2/b +5/6 x3/b2 ]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 = 690. mm2
Ju = 250916. mm4
Jv = 53046. mm4
yg = 34.75 mmN = -1017. NTy = 3390. NMx = 1525500. Nmmxm = 12. mmum = -9. mmvm = -34.75 mmσm = N/A-Mv/Ju = 209.8 N/mm2
xc = 21. mmyc = 9. mmvc = -25.75 mmσc = N/A-Mv/Ju = 155.1 N/mm2
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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grts.081
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.grts.081
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.xxxx.082REAZIONI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
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:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
= (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(-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
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
xc = 21. mmyc = 10. mmvc = -26.96 mmσc = N/A-Mv/Ju = -159.9 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
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.xxxx.082
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19
Ing Civ.xxxx.082
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 31.05.19