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IPERΣ01.bhng.002REAZIONI 781715 Bao Hong Da
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
39/40F
1/2F19/40Fb
1/40F
1/2F
A
B
1/2F
1/2FFb
1/2F
1/2F1/2Fb
C
D
79/40F
1/2F5/2Fb
79/40F
1/2F21/40Fb
E
C
1/40F
5/2F5/2Fb
1/40F
5/2F
EB
59/40F19/40Fb
59/40F19/40Fb
A C
1/2F
1/2F
Fb
1/2F
1/2F
F
A
1/2F
1/2F
1/2F
1/2F
GF
1/2F
1/2F1/2Fb
1/2F
1/2F
DG
IPERΣ01.bhng.002AZIONI INTERNE 781715 Bao Hong Da
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2
-1/2
-1/2
-1/2
1/40
59/40
0
-1/2-1/2 -1/2
F
39/4
0-1
/40
-1/2
-79/
40
5/2
0
-√2/2
-1/21/2
-1/2
F
-19/
400
11/
25/
221
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-5/20
19/40 19/40
1
0
00 1/20
Fb
IPE
RΣ0
1.bh
ng.0
02P
RO
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DIM
EN
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ISU
LTA
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ao H
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Da
@ A
dolfo
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olite
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Mila
no, v
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27.0
3.13
03.0
9.18
A
B
C
D E
FG
F
W
XX
q
q
0 0
11/2 5/21
-5/2
000
1
000
1/2
0
Mo
fless
ione
da
caric
hi a
sseg
nati
1 0
00 01 00
-1-1
0
000
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.bh
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DIM
EN
TO
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TI 7
8171
5 B
ao H
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Da
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WA
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b1-
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
b-x
/b-1
/2F
x+1/
2qx2
1/2F
x2 /b-1
/2qx
3 /bx2 /b
2
CD
b0
Fb-
1/2F
x0
00
0D
C b
0-1
/2F
b-1/
2Fx
00
EC
bx/
b5/
2Fb-
3/2F
x5/
2Fx-
3/2F
x2 /bx2 /b
2
3/4F
b2 /EJ
1/3X
b/E
JC
E b
-1+
x/b
-Fb-
3/2F
xF
b+1/
2Fx-
3/2F
x2 /b1-
2x/b
+x2 /b
2
EB
b0
-5/2
Fb+
5/2F
x0
00
0B
E b
05/
2Fx
00
AC
b-1
00
10
Xb/
EJ
CA
b1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li19
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
C-1
9/40
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.bhng.002PROCEDIMENTO E RISULTATI 781715 Bao Hong Da
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
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
LXoBA = ∫
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
LXoEC = ∫
o
b(5/2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [5/4 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (5/4 b -1/2 b ) Fb 1/EJ = 3/4 Fb2/EJ
LXoCE = ∫
o
b(1 +1/2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [ x +1/4 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= ( b +1/4 b -1/2 b ) Fb 1/EJ = 3/4 Fb2/EJ
IPERΣ01.bhng.002PROCEDIMENTO E RISULTATI 781715 Bao Hong Da
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 756. mm2
Ju = 165564. mm4
Jv = 74844. mm4
yg = 39. mmN = 15.25 NTy = 1525. NMx = -930250. Nmmxm = 18. mmum = -3. mmvm = -39. mmσm = N/A-Mv/Ju = -219.1 N/mm2
xc = 21. mmyc = 16. mmvc = -23. mmσc = N/A-Mv/Ju = -129.2 N/mm2
τc = 4.569 N/mm2
σo = √σ2+3τ2 = 129.5 N/mm2
S* = 2976. mm3mm 0 18 24 42x
0
42
54
y
16σc,τc
σm
u
v
IPERΣ01.bhng.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.bhng.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.bnst.003REAZIONI 843782 Benassai Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
3F
F2Fb
2F
F1/2Fb
A
B
4/5F4/5Fb
4/5F
C
D
4/5F
4F4Fb
4/5F
4F
ED
14/5F
F4Fb
14/5F
F6/5Fb
E
A
FFb
F
F
C
1/5F4/5Fb
1/5F4/5Fb
C A
F
F1/2Fb
F
BG
FGF
IPERΣ01.bnst.003AZIONI INTERNE 843782 Benassai Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
11
0
-4/5
1
-√2/2
-1/5
111
F
32
-4/5
-4
14/5
-√2/2
0
-100
F
-21/
2
4/5
0 40
-4-6
/5
1
0
-4/5 -4/5
1/2000
Fb
IPE
RΣ0
1.bn
st.0
03P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4378
2 B
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sai T
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aso
@ A
dolfo
Zav
elan
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olite
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Mila
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27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q
q
-21/2
00
40
-4 -2
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
10
00
0 1
0
0
-1-1
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.bn
st.0
03P
RO
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ISU
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-2F
b+3F
x-1/
2qx2
00
00
BA
b0
-1/2
Fb+
2Fx+
1/2q
x20
0
CD
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
DC
b-x
/b0
0x2 /b
2
ED
b0
4Fb-
4Fx
00
00
DE
b0
-4F
x0
0
EA
bx/
b-4
Fb+
2Fx
-4F
x+2F
x2 /bx2 /b
2
-4/3
Fb2 /E
J1/
3Xb/
EJ
AE
b-1
+x/
b2F
b+2F
x-2
Fb+
2Fx2 /b
1-2x
/b+
x2 /b2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b-1
00
10
Xb/
EJ
AC
b1
00
1
BG
b0
1/2F
b-F
x+1/
2qx2
00
00
GB
b0
-1/2
qx2
00
GF
b0
00
00
0F
G b
00
00
tota
li-4
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
A4/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.bnst.003PROCEDIMENTO E RISULTATI 843782 Benassai Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(-4 x/b +2 x2/b2 ) Fb 1/EJ dx = [-2 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-2 b +2/3 b ) Fb 1/EJ = -4/3 Fb2/EJ
LXoAE = ∫
o
b(-2 +2 x2/b2 ) Fb 1/EJ dx = [-2 x +2/3 x3/b2 ]o
b Fb 1/EJ
= (-2 b +2/3 b ) Fb 1/EJ = -4/3 Fb2/EJ
IPERΣ01.bnst.003PROCEDIMENTO E RISULTATI 843782 Benassai Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 864. mm2
Ju = 251424. mm4
Jv = 62208. mm4
yg = 33. mmN = -528. NTy = -2640. NMx = 1742400. Nmmxm = 18. mmum = -6. mmvm = -33. mmσm = N/A-Mv/Ju = 228.1 N/mm2
xc = 24. mmyc = 14. mmvc = -19. mmσc = N/A-Mv/Ju = 131.1 N/mm2
τc = 3.822 N/mm2
σo = √σ2+3τ2 = 131.2 N/mm2
S* = 4368. mm3mm 0 18 30 48x
0
48
54
y
14σc,τc
σm
u
v
IPERΣ01.bnst.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.bnst.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.brts.004REAZIONI 811986 Beretta Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
3F
11/20F5/2Fb
2F
11/20F
A
B
F
31/20F5/2Fb
F
31/20F19/20Fb
A C
F
2F3/2Fb
F
2F1/2Fb
C D
11/20F11/20Fb
11/20F
EB
F
F1/2Fb
F
D
F
F
F
G
F
Fb
F
G
E
9/20F11/20Fb
9/20F11/20Fb
E
C
IPERΣ01.brts.004AZIONI INTERNE 811986 Beretta Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
11/2
011
/20
-1 -1
0
-1-1
-1
√2/2
9/20
F
-3-2
31/20 2
-11/20
-10
0-√2/2
0
F
5/2
0
-5/2 -19/20 -3/21/2
11/200
1/2
00
01
0
-11/
20-1
1/20
Fb
IPE
RΣ0
1.br
ts.0
04P
RO
CE
DIM
EN
TO
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ISU
LTA
TI 8
1198
6 B
eret
ta S
tefa
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Zav
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olite
cnic
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Mila
no, v
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27.0
3.13
03.0
9.18
A
B
CD
EFG F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
5/20
-5/2
-3/2
-3/2
1/2
00
1/2000
1
0 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
0000
0
0 1 1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.br
ts.0
04P
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DIM
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TO
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ISU
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TI 8
1198
6 B
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ta S
tefa
no
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WE
B
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
5/2F
b-3F
x+1/
2qx2
00
00
BA
b0
-2F
x-1/
2qx2
00
AC
b-x
/b-5
/2F
b+F
x5/
2Fx-
Fx2 /b
x2 /b2
11/1
2Fb2 /E
J1/
3Xb/
EJ
CA
b1-
x/b
3/2F
b+F
x3/
2Fb-
1/2F
x-F
x2 /b1-
2x/b
+x2 /b
2
CD
b0
-3/2
Fb+
2Fx
00
00
DC
b0
-1/2
Fb+
2Fx
00
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
1/2F
b-F
x+1/
2qx2
00
00
FD
b0
-1/2
qx2
00
FG
b0
00
00
0G
F b
00
00
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li11
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WE
B-1
1/20
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.brts.004PROCEDIMENTO E RISULTATI 811986 Beretta Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(5/2 x/b - x2/b2 ) Fb 1/EJ dx = [5/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (5/4 b -1/3 b ) Fb 1/EJ = 11/12 Fb2/EJ
LXoCA = ∫
o
b(3/2 -1/2 x/b - x2/b2 ) Fb 1/EJ dx = [3/2 x -1/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/4 b -1/3 b ) Fb 1/EJ = 11/12 Fb2/EJ
IPERΣ01.brts.004PROCEDIMENTO E RISULTATI 811986 Beretta Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 1080. mm2
Ju = 276955. mm4
Jv = 116640. mm4
yg = 35.4 mmN = 577.5 NTy = -3150. NMx = 1863750. Nmmxm = 18. mmum = -6. mmvm = -35.4 mmσm = N/A-Mv/Ju = 238.8 N/mm2
xc = 24. mmyc = 15. mmvc = -20.4 mmσc = N/A-Mv/Ju = 137.8 N/mm2
τc = 4.76 N/mm2
σo = √σ2+3τ2 = 138.1 N/mm2
S* = 5022. mm3mm 0 18 30 48x
0
42
54
y
15σc,τc
σm
u
v
IPERΣ01.brts.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.brts.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.brmm.006REAZIONI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
F
F
F1/2Fb
AB
1/5F1/5Fb
1/5F
C D
2F
1/5F2Fb
2F
1/5F
E
D
F
11/5F2Fb
F
11/5F1/5Fb
EA
F
Fb
F
F
C
6/5F1/5Fb
6/5F1/5Fb
C
A
F
F1/2Fb
F
B
G
F
G
F
IPERΣ01.brmm.006AZIONI INTERNE 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
11
0
-1/5
1
√2/2
6/5
-1-1
-1
F
-10
1/5
2
-11/5
-√2/2
0
10
0
F
0-1/2
-1/50
-20
2-1/5
1
0
1/5
1/5-1/2
00
0
Fb
IPE
RΣ0
1.br
mm
.006
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 829
837
Bor
mol
ini M
atte
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
0-1
/2
00
-20
20
1
0
00
-1/20 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
-10
0 0
0-1
0
0
11
0 0 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.br
mm
.006
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 829
837
Bor
mol
ini M
atte
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
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
CD
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
DC
bx/
b0
0x2 /b
2
ED
b0
-2F
b+2F
x0
00
0D
E b
02F
x0
0
EA
b-x
/b2F
b-2F
x-2
Fx+
2Fx2 /b
x2 /b2
-1/3
Fb2 /E
J1/
3Xb/
EJ
AE
b1-
x/b
-2F
x-2
Fx+
2Fx2 /b
1-2x
/b+
x2 /b2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b1
00
10
Xb/
EJ
AC
b-1
00
1
BG
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0G
B b
01/
2qx2
00
GF
b0
00
00
0F
G b
00
00
tota
li-1
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
D1/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.brmm.006PROCEDIMENTO E RISULTATI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(-2 x/b +2 x2/b2 ) Fb 1/EJ dx = [- x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (- b +2/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
LXoAE = ∫
o
b(-2 x/b +2 x2/b2 ) Fb 1/EJ dx = [- x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (- b +2/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
IPERΣ01.brmm.006PROCEDIMENTO E RISULTATI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 612. mm2
Ju = 149427. mm4
Jv = 27756. mm4
yg = 36.88 mmN = -104. NTy = 1040. NMx = -842400. Nmmxm = 12. mmum = -3. mmvm = -36.88 mmσm = N/A-Mv/Ju = -208.1 N/mm2
xc = 15. mmyc = 16. mmvc = -20.88 mmσc = N/A-Mv/Ju = -117.9 N/mm2
τc = 3.216 N/mm2
σo = √σ2+3τ2 = 118. N/mm2
S* = 2773. mm3mm 0 12 18 30x
0
42
54
y
16σc,τc
σm
u
v
IPERΣ01.brmm.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.brmm.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.btts.007REAZIONI 808392 Bottini Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
109/40F
1/2F7/2Fb
69/40F
1/2F51/40Fb
A
B
29/40F
7/2F7/2Fb
29/40F
7/2F
A C
29/40F
1/2F29/40Fb
29/40F
1/2F
D
C
3/2F
1/2F2Fb
3/2F
1/2F1/2Fb
B
E
3/2F
1/2F
3/2F
1/2F
F G
3/2F
1/2F1/2Fb
3/2F
1/2F
E F
9/40F29/40Fb
9/40F29/40Fb
DB
3/2F
1/2F
Fb
3/2F
1/2F
G
D
IPERΣ01.btts.007AZIONI INTERNE 808392 Bottini Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
29/40
1/2
1/2
-3/2 -3/2-3/2
-9/40
√2
F
109/
4069
/40
-7/2
-29/
40
3/2
1/2-1/2
1/2
0
-√2/2
F
-7/2
-51/
40
7/2 0
29/4
00
-2-1
/2
0 0-1/2
0
-29/40-29/40
1
0
Fb
IPE
RΣ0
1.bt
ts.0
07P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0839
2 B
ottin
i Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A B
C
D
EF
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-7/2 -2
7/2
0 00
-2 -1/2
00
-1/2
000
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
00
-10
0 0
00
001
1
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.bt
ts.0
07P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0839
2 B
ottin
i Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WBD
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b-x/b-7/2Fb+2Fx-1/2qx2
7/2Fx-2Fx2/b+1/2qx
3/bx
2/b
2
29/24Fb2/EJ1/3Xb/EJ
BA b1-x/b2Fb+Fx+1/2qx2
2Fb-Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
AC b07/2Fb-7/2Fx0000
CA b0-7/2Fx00
DC b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJCD bx/b00x
2/b
2
BE b0-2Fb+3/2Fx0000
EB b01/2Fb+3/2Fx00
FG b01/2Fx-1/2qx2
0000
GF b0-1/2Fx+1/2qx2
00
EF b0-1/2Fb+1/2Fx0000
FE b01/2Fx00
DB b10010Xb/EJ
BD b-1001
GD √2b0Fb-√2/2Fx0000
totali29/24Fb2/EJ5/3Xb/EJ
iperstatica X=WBD-29/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.btts.007PROCEDIMENTO E RISULTATI 808392 Bottini Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(7/2 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [7/4 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (7/4 b -2/3 b +1/8 b ) Fb 1/EJ = 29/24 Fb2/EJ
LXoBA = ∫
o
b(2 - x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [2 x -1/2 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (2 b -1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 29/24 Fb2/EJ
IPERΣ01.btts.007PROCEDIMENTO E RISULTATI 808392 Bottini Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 792. mm2
Ju = 225759. mm4
Jv = 30240. mm4
yg = 31.36 mmN = 377. NTy = -1820. NMx = 1565200. Nmmxm = 12. mmum = -6. mmvm = -31.36 mmσm = N/A-Mv/Ju = 217.9 N/mm2
xc = 18. mmyc = 13. mmvc = -18.36 mmσc = N/A-Mv/Ju = 127.8 N/mm2
τc = 2.606 N/mm2
σo = √σ2+3τ2 = 127.9 N/mm2
S* = 3879. mm3mm 0 12 24 36x
0
48
54
y
13σc,τc
σm
u
v
IPERΣ01.btts.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.btts.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.clnl.008REAZIONI 832133 Calonico Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
5/2F
1/2F3/2Fb
3/2F
1/2F1/2Fb
A
B
13/20F
1/2F13/20Fb
13/20F
1/2F
C
D
13/20F
7/2F7/2Fb
13/20F
7/2F
ED
53/20F
1/2F7/2Fb
53/20F
1/2F17/20Fb
E
A
1/2F
1/2F
Fb
1/2F
1/2F
F
C
3/20F13/20Fb
3/20F13/20Fb
C A
1/2F
1/2F1/2Fb
1/2F
1/2F
BG
1/2F
1/2F
1/2F
1/2F
GF
IPERΣ01.clnl.008AZIONI INTERNE 832133 Calonico Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
1/2
-13/20
1/2
0
3/20
1/21/21/2
F
5/2
3/2
-13/
20
-7/2
53/2
0
-√2/2
0
-1/2-1/21/2
F
-3/2
1/2
13/2
00 7/20
-7/2
-17/
20
1
0
-13/20 -13/20
1/2000
Fb
IPE
RΣ0
1.cl
nl.0
08P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3213
3 C
alon
ico
Lore
nzo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q
q
-3/21/2
00
7/2
0
-7/2 -3/2
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
10
00
0 1
0
0
-1-1
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.cl
nl.0
08P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3213
3 C
alon
ico
Lore
nzo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-3/2
Fb+
5/2F
x-1/
2qx2
00
00
BA
b0
-1/2
Fb+
3/2F
x+1/
2qx2
00
CD
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
DC
b-x
/b0
0x2 /b
2
ED
b0
7/2F
b-7/
2Fx
00
00
DE
b0
-7/2
Fx
00
EA
bx/
b-7
/2F
b+2F
x-7
/2F
x+2F
x2 /bx2 /b
2
-13/
12F
b2 /EJ
1/3X
b/E
JA
E b
-1+
x/b
3/2F
b+2F
x-3
/2F
b-1/
2Fx+
2Fx2 /b
1-2x
/b+
x2 /b2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b-1
00
10
Xb/
EJ
AC
b1
00
1
BG
b0
1/2F
b-1/
2Fx
00
00
GB
b0
-1/2
Fx
00
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
tota
li-1
3/12
Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WC
A13
/20F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.clnl.008PROCEDIMENTO E RISULTATI 832133 Calonico Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(-7/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [-7/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-7/4 b +2/3 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoAE = ∫
o
b(-3/2 -1/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [-3/2 x -1/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-3/2 b -1/4 b +2/3 b ) Fb 1/EJ = -13/12 Fb2/EJ
IPERΣ01.clnl.008PROCEDIMENTO E RISULTATI 832133 Calonico Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 936. mm2
Ju = 248849. mm4
Jv = 52704. mm4
yg = 33.46 mmN = -344.5 NTy = -1855. NMx = 1688050. Nmmxm = 12. mmum = -6. mmvm = -33.46 mmσm = N/A-Mv/Ju = 226.6 N/mm2
xc = 18. mmyc = 14. mmvc = -19.46 mmσc = N/A-Mv/Ju = 131.6 N/mm2
τc = 2.762 N/mm2
σo = √σ2+3τ2 = 131.7 N/mm2
S* = 4446. mm3mm 0 12 24 36x
0
42
54
y
14σc,τc
σm
u
v
IPERΣ01.clnl.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.clnl.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.cmna.009REAZIONI 867174 Comana Alberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
109/40F7/2Fb
1/2F
69/40F51/40Fb
A B
7/2F
29/40F7/2Fb
7/2F
29/40F
A
C
1/2F
29/40F29/40Fb
1/2F
29/40F
DC
1/2F
3/2F2Fb
1/2F
3/2F1/2Fb
B E
1/2F
3/2F
1/2F
3/2F
F
G
1/2F
3/2F1/2Fb
1/2F
3/2F
E
F
9/40F29/40Fb
9/40F29/40Fb
D
B
1/2F
3/2F
Fb
1/2F
3/2F
G
D
IPERΣ01.cmna.009AZIONI INTERNE 867174 Comana Alberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2 -1/2
-29/
40
-1/2
-1/2
3/2
3/2
3/2
9/40
-√2
F
109/40 69/40
-7/2
-29/40
3/2
1/2
-1/2
1/20
-√2/2
F
-7/2 -51/40
7/2
0
29/400
-2 -1/2
00
-1/2
0-29/
40-2
9/40
1
0
Fb
IPE
RΣ0
1.cm
na.0
09P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6717
4 C
oman
a A
lber
to
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-7/2
-2
7/2 0
00
-2-1
/2
0 0-1/2
0
00
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
0 0
-10
00
0 00 0
11
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.cm
na.0
09P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6717
4 C
oman
a A
lber
to
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WBD
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b-x/b-7/2Fb+2Fx-1/2qx2
7/2Fx-2Fx2/b+1/2qx
3/bx
2/b
2
29/24Fb2/EJ1/3Xb/EJ
BA b1-x/b2Fb+Fx+1/2qx2
2Fb-Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
AC b07/2Fb-7/2Fx0000
CA b0-7/2Fx00
DC b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJCD bx/b00x
2/b
2
BE b0-2Fb+3/2Fx0000
EB b01/2Fb+3/2Fx00
FG b01/2Fx-1/2qx2
0000
GF b0-1/2Fx+1/2qx2
00
EF b0-1/2Fb+1/2Fx0000
FE b01/2Fx00
DB b10010Xb/EJ
BD b-1001
GD √2b0Fb-√2/2Fx0000
totali29/24Fb2/EJ5/3Xb/EJ
iperstatica X=WBD-29/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.cmna.009PROCEDIMENTO E RISULTATI 867174 Comana Alberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(7/2 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [7/4 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (7/4 b -2/3 b +1/8 b ) Fb 1/EJ = 29/24 Fb2/EJ
LXoBA = ∫
o
b(2 - x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [2 x -1/2 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (2 b -1/2 b -1/6 b -1/8 b ) Fb 1/EJ = 29/24 Fb2/EJ
IPERΣ01.cmna.009PROCEDIMENTO E RISULTATI 867174 Comana Alberto
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 936. mm2
Ju = 248849. mm4
Jv = 52704. mm4
yg = 20.54 mmN = -384.3 NTy = -1855. NMx = 1780800. Nmmxm = 24. mmym = 54. mmum = 6. mmvm = 33.46 mmσm = N/A-Mv/Ju = -239.9 N/mm2
xc = 18. mmyc = 40. mmvc = 19.46 mmσc = N/A-Mv/Ju = -139.7 N/mm2
τc = 2.762 N/mm2
σo = √σ2+3τ2 = 139.8 N/mm2
S* = 4446. mm3mm 0 12 24 36x
0
12
54
y
40σc,τc
σm
u
v
IPERΣ01.cmna.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.cmna.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dmns.011REAZIONI 817109 Damian Sebastiano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/5F
1/2F
1/5F
1/2F
A B
4/5F
1/2F
4/5F
1/2F4/5Fb
A
C
1/2F
1/2FFb
1/2F
1/2F1/2Fb
C
D
1/5F
1/2F1/5Fb
1/5F
1/2F
E
B
3/2F
1/2F1/2Fb
3/2F
1/2F
D F
3/2F
1/2F
3/2F
1/2F
F G
3/2F
1/2F
Fb
3/2F
1/2F
G
E
13/10F1/5Fb
13/10F1/5Fb
EC
IPERΣ01.dmns.011AZIONI INTERNE 817109 Damian Sebastiano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/5 -1/5
-1/2
-1/2
-1/2
3/2 3/2 3/2
-√2
-13/10
F
1/2-1/2
-4/5
1/2
-1/5
1/2 1/2-1/2
-√2/2
0
F
0 00-4
/5-1
-1/2
1/5
0
-1/20 0 0
1
0
-1/5-1/5
Fb
IPE
RΣ0
1.dm
ns.0
11P
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
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0-1-1-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
00
00
0
0
11
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.dm
ns.0
11P
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
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
E
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
1/2F
x-1/
2qx2
00
00
BA
b0
-1/2
Fx+
1/2q
x20
0
AC
b-x
/b-F
xF
x2 /bx2 /b
2
1/3F
b2 /EJ
1/3X
b/E
JC
A b
1-x/
bF
b-F
xF
b-2F
x+F
x2 /b1-
2x/b
+x2 /b
2
CD
b0
-Fb+
1/2F
x0
00
0D
C b
01/
2Fb+
1/2F
x0
0
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
-1/2
Fb+
1/2F
x0
00
0F
D b
01/
2Fx
00
FG
b0
1/2F
x-1/
2qx2
00
00
GF
b0
-1/2
Fx+
1/2q
x20
0
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li1/
3Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WC
E-1
/5F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.dmns.011PROCEDIMENTO E RISULTATI 817109 Damian Sebastiano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b( x2/b2 ) Fb 1/EJ dx = [1/3 x3/b2 ]o
b Fb 1/EJ
= (1/3 b ) Fb 1/EJ = 1/3 Fb2/EJ
LXoCA = ∫
o
b(1 -2 x/b + x2/b2 ) Fb 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= ( b - b +1/3 b ) Fb 1/EJ = 1/3 Fb2/EJ
IPERΣ01.dmns.011PROCEDIMENTO E RISULTATI 817109 Damian Sebastiano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 864. mm2
Ju = 251424. mm4
Jv = 62208. mm4
yg = 21. mmN = -3946. NTy = -1973. NMx = 1562400. Nmmxm = 30. mmym = 54. mmum = 6. mmvm = 33. mmσm = N/A-Mv/Ju = -209.6 N/mm2
xc = 24. mmyc = 40. mmvc = 19. mmσc = N/A-Mv/Ju = -122.6 N/mm2
τc = 2.856 N/mm2
σo = √σ2+3τ2 = 122.7 N/mm2
S* = 4368. mm3mm 0 18 30 48x
0
6
54
y
40σc,τc
σm
u
v
IPERΣ01.dmns.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dmns.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dnsg.012REAZIONI 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
103/40F
F3Fb
63/40F
F37/40Fb
A
B
23/40F
3F3Fb
23/40F
3F
A C
23/40F23/40Fb
23/40F
D
C
F
F3/2Fb
F
F1/2Fb
B
E
FF G
F
F1/2Fb
F
E F
23/40F23/40Fb
23/40F23/40Fb
DB
FFb
F
G
D
IPERΣ01.dnsg.012AZIONI INTERNE 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
11
23/40
0
1
-1-1 -1
-23/40
√2/2
F
103/
4063
/40
-3
-23/
40
1
01 0
0
-√2/2
F
-3-3
7/40
3 0
23/4
00
-3/2
-1/2
0 0-1/2
0
-23/40-23/40
1
0
Fb
IPE
RΣ0
1.dn
sg.0
12P
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
03.0
9.18
A B
C
D
EF
G
F
W
X
X
q
q
-3 -3/2
30 00
-3/2 -1/2
00
-1/2
000
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
00
-10
0 0
00
001
1
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.dn
sg.0
12P
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
03.0
9.18
Quadro contributi PLV per iperstatica X=WBD
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b-x/b-3Fb+2Fx-1/2qx2
3Fx-2Fx2/b+1/2qx
3/bx
2/b
2
23/24Fb2/EJ1/3Xb/EJ
BA b1-x/b3/2Fb+Fx+1/2qx2
3/2Fb-1/2Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
AC b03Fb-3Fx0000
CA b0-3Fx00
DC b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJCD bx/b00x
2/b
2
BE b0-3/2Fb+Fx0000
EB b01/2Fb+Fx00
FG b000000
GF b0000
EF b0-1/2Fb+Fx-1/2qx2
0000
FE b01/2qx2
00
DB b10010Xb/EJ
BD b-1001
GD √2b0Fb-√2/2Fx0000
totali23/24Fb2/EJ5/3Xb/EJ
iperstatica X=WBD-23/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.dnsg.012PROCEDIMENTO E RISULTATI 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(3 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -2/3 b +1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
LXoBA = ∫
o
b(3/2 -1/2 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x -1/4 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -1/4 b -1/6 b -1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
IPERΣ01.dnsg.012PROCEDIMENTO E RISULTATI 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 612. mm2
Ju = 149428. mm4
Jv = 27756. mm4
yg = 17.12 mmN = 276. NTy = -1440. NMx = 878400. Nmmxm = 18. mmym = 54. mmum = 3. mmvm = 36.88 mmσm = N/A-Mv/Ju = -216.4 N/mm2
xc = 15. mmyc = 38. mmvc = 20.88 mmσc = N/A-Mv/Ju = -122.3 N/mm2
τc = 4.453 N/mm2
σo = √σ2+3τ2 = 122.5 N/mm2
S* = 2773. mm3mm 0 12 18 30x
0
12
54
y
38σc,τc
σm
u
v
IPERΣ01.dnsg.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dnsg.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dvng.013REAZIONI 774590 D’Avino Guido
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
33/40F13/40Fb
7/40F
A
B
F1/2Fb
F1/2Fb
C
D
73/40F
F2Fb
73/40F
F7/40Fb
E
C
7/40F
2F2Fb
7/40F
2F
EB
73/40F13/40Fb
73/40F13/40Fb
A C
FFb
F
F
A
FGF F
F1/2Fb
F
DG
IPERΣ01.dvng.013AZIONI INTERNE 774590 D’Avino Guido
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
00
-1-1
7/40
73/40
√2/2
-1 -1-1
F
33/4
0-7
/40
0-7
3/40
2
0
-√2/2
0-1
0
F
-13/
400
1/2
1/2
27/
40
-20
13/40 13/40
1
0
00 1/20
Fb
IPE
RΣ0
1.dv
ng.0
13P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
7459
0 D
’Avi
no G
uido
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
C
D E
FG
F
W
XX
q
q
0 0
1/21/2 21/2
-200
0
1
000
1/2
0
Mo
fless
ione
da
caric
hi a
sseg
nati
1 0
00 01 00
-1-1
0
000
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.dv
ng.0
13P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
7459
0 D
’Avi
no G
uido
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WA
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b1-
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
b-x
/b-1
/2F
x+1/
2qx2
1/2F
x2 /b-1
/2qx
3 /bx2 /b
2
CD
b0
1/2F
b0
00
0D
C b
0-1
/2F
b0
0
EC
bx/
b2F
b-3/
2Fx
2Fx-
3/2F
x2 /bx2 /b
2
1/2F
b2 /EJ
1/3X
b/E
JC
E b
-1+
x/b
-1/2
Fb-
3/2F
x1/
2Fb+
Fx-
3/2F
x2 /b1-
2x/b
+x2 /b
2
EB
b0
-2F
b+2F
x0
00
0B
E b
02F
x0
0
AC
b-1
00
10
Xb/
EJ
CA
b1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
00
00
0F
G b
00
00
DG
b0
1/2F
b-F
x+1/
2qx2
00
00
GD
b0
-1/2
qx2
00
tota
li13
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
C-1
3/40
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.dvng.013PROCEDIMENTO E RISULTATI 774590 D’Avino Guido
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
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
LXoBA = ∫
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
LXoEC = ∫
o
b(2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [ x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= ( b -1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
LXoCE = ∫
o
b(1/2 + x/b -3/2 x2/b2 ) Fb 1/EJ dx = [1/2 x +1/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (1/2 b +1/2 b -1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
IPERΣ01.dvng.013PROCEDIMENTO E RISULTATI 774590 D’Avino Guido
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 756. mm2
Ju = 165564. mm4
Jv = 74844. mm4
yg = 15. mmN = 127.8 NTy = 1460. NMx = -963600. Nmmxm = 24. mmym = 54. mmum = 3. mmvm = 39. mmσm = N/A-Mv/Ju = 227.2 N/mm2
xc = 21. mmyc = 38. mmvc = 23. mmσc = N/A-Mv/Ju = 134. N/mm2
τc = 4.374 N/mm2
σo = √σ2+3τ2 = 134.2 N/mm2
S* = 2976. mm3mm 0 18 24 42x
0
12
54
y
38σc,τc
σm
u
v
IPERΣ01.dvng.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dvng.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dvnl.014REAZIONI 867268 D’Avino Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
5/2F
2/5F2Fb
3/2F
2/5F
A
B
1/2F
7/5F2Fb
1/2F
7/5F3/5Fb
A C
1/2F
3/2FFb
1/2F
3/2F1/2Fb
C D
1/2F
2/5F2/5Fb
1/2F
2/5F
EB
1/2F
1/2F1/2Fb
1/2F
1/2F
D
F
1/2F
1/2F
1/2F
1/2F
F
G
1/2F
1/2F
Fb
1/2F
1/2F
G
E
1/10F2/5Fb
1/10F2/5Fb
E
C
IPERΣ01.dvnl.014AZIONI INTERNE 867268 D’Avino Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
2/5
2/5
-1/2 -1/2
-1/2
-1/2
-1/2
-1/2
0
1/10
F
-5/2
-3/2
7/5 3/2
-2/5
-1/2
-1/2
1/2
-√2/2
0
F
20
-2 -3/5 -11/2
2/50
1/2
00
01
0
-2/5
-2/5
Fb
IPE
RΣ0
1.dv
nl.0
14P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6726
8 D
’Avi
no L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CD
EFG F
W
X X
q
q
20
-2-1
-11/
2
00
1/2000
1
0 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
0000
0
0 1 1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.dv
nl.0
14P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6726
8 D
’Avi
no L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
E
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
2Fb-
5/2F
x+1/
2qx2
00
00
BA
b0
-3/2
Fx-
1/2q
x20
0
AC
b-x
/b-2
Fb+
Fx
2Fx-
Fx2 /b
x2 /b2
2/3F
b2 /EJ
1/3X
b/E
JC
A b
1-x/
bF
b+F
xF
b-F
x2 /b1-
2x/b
+x2 /b
2
CD
b0
-Fb+
3/2F
x0
00
0D
C b
0-1
/2F
b+3/
2Fx
00
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
1/2F
b-1/
2Fx
00
00
FD
b0
-1/2
Fx
00
FG
b0
-1/2
Fx+
1/2q
x20
00
0G
F b
01/
2Fx-
1/2q
x20
0
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li2/
3Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WC
E-2
/5F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.dvnl.014PROCEDIMENTO E RISULTATI 867268 D’Avino Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(2 x/b - x2/b2 ) Fb 1/EJ dx = [ x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= ( b -1/3 b ) Fb 1/EJ = 2/3 Fb2/EJ
LXoCA = ∫
o
b(1 - x2/b2 ) Fb 1/EJ dx = [ x -1/3 x3/b2 ]o
b Fb 1/EJ
= ( b -1/3 b ) Fb 1/EJ = 2/3 Fb2/EJ
IPERΣ01.dvnl.014PROCEDIMENTO E RISULTATI 867268 D’Avino Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 792. mm2
Ju = 225759. mm4
Jv = 30240. mm4
yg = 22.64 mmN = 484. NTy = -3025. NMx = 1718200. Nmmxm = 24. mmym = 54. mmum = 6. mmvm = 31.36 mmσm = N/A-Mv/Ju = -238.1 N/mm2
xc = 18. mmyc = 41. mmvc = 18.36 mmσc = N/A-Mv/Ju = -139.1 N/mm2
τc = 4.331 N/mm2
σo = √σ2+3τ2 = 139.4 N/mm2
S* = 3879. mm3mm 0 12 24 36x
0
6
54
y
41σc,τc
σm
u
v
IPERΣ01.dvnl.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dvnl.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dcml.015REAZIONI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
F
103/40F3Fb
F
63/40F37/40Fb
A B
3F
23/40F3Fb
3F
23/40F
A
C
23/40F23/40Fb
23/40F
DC
F
F3/2Fb
F
F1/2Fb
B E
F
F
G
F
F1/2Fb
F
E
F
23/40F23/40Fb
23/40F23/40Fb
D
B
F
Fb
F
G
D
IPERΣ01.dcml.015AZIONI INTERNE 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1 -1
-23/
40
0
-1
11
1
23/4
0
-√2/2
F
103/40 63/40
-3
-23/40
1
01
0
0
-√2/2
F
-3 -37/4030
23/400
-3/2 -1/2
00
-1/2
0-23/
40-2
3/40
1
0
Fb
IPE
RΣ0
1.dc
ml.0
15P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7797
6 D
i Cam
illo
Lore
nzo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-3-3
/2
3 0
00
-3/2
-1/2
0 0-1/2
0
00
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
0 0
-10
00
0 00 0
11
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.dc
ml.0
15P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7797
6 D
i Cam
illo
Lore
nzo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WBD
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b-x/b-3Fb+2Fx-1/2qx2
3Fx-2Fx2/b+1/2qx
3/bx
2/b
2
23/24Fb2/EJ1/3Xb/EJ
BA b1-x/b3/2Fb+Fx+1/2qx2
3/2Fb-1/2Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
AC b03Fb-3Fx0000
CA b0-3Fx00
DC b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJCD bx/b00x
2/b
2
BE b0-3/2Fb+Fx0000
EB b01/2Fb+Fx00
FG b000000
GF b0000
EF b0-1/2Fb+Fx-1/2qx2
0000
FE b01/2qx2
00
DB b10010Xb/EJ
BD b-1001
GD √2b0Fb-√2/2Fx0000
totali23/24Fb2/EJ5/3Xb/EJ
iperstatica X=WBD-23/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.dcml.015PROCEDIMENTO E RISULTATI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(3 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -2/3 b +1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
LXoBA = ∫
o
b(3/2 -1/2 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x -1/4 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -1/4 b -1/6 b -1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
IPERΣ01.dcml.015PROCEDIMENTO E RISULTATI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 1080. mm2
Ju = 276955. mm4
Jv = 116640. mm4
yg = 18.6 mmN = -391. NTy = -2040. NMx = 1550400. Nmmxm = 30. mmym = 54. mmum = 6. mmvm = 35.4 mmσm = N/A-Mv/Ju = -198.5 N/mm2
xc = 24. mmyc = 39. mmvc = 20.4 mmσc = N/A-Mv/Ju = -114.6 N/mm2
τc = 3.083 N/mm2
σo = √σ2+3τ2 = 114.7 N/mm2
S* = 5022. mm3mm 0 18 30 48x
0
12
54
y
39σc,τc
σm
u
v
IPERΣ01.dcml.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.dcml.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.frtg.017REAZIONI 789093 Frattini Giulietta
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/5F
2F2Fb
1/5F
2F
A B
11/5F
F2Fb
11/5F
F1/5Fb
A
C
F
F
F1/2Fb
C
D
1/5F1/5Fb
1/5F
E
B
F
F1/2Fb
F
D FF
F G
FFb
F
G
E
6/5F1/5Fb
6/5F1/5Fb
EC
IPERΣ01.frtg.017AZIONI INTERNE 789093 Frattini Giulietta
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/5
-1-1
-1
0
1 1 1
-√2/2
-6/5
F
2
-11/
5-1
0
1/5
1 0 0-√2
/2
0
F
-202
-1/5
0-1
/2
-1/5
0
-1/20 0 0
1
0
1/51/5
Fb
IPE
RΣ0
1.fr
tg.0
17P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
8909
3 F
ratti
ni G
iulie
tta
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
Sch
ema
di c
alco
lo ip
erst
atic
o
-20
200-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
00
00
0
0
11
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.fr
tg.0
17P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
8909
3 F
ratti
ni G
iulie
tta
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
E
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-2F
b+2F
x0
00
0B
A b
02F
x0
0
AC
b-x
/b2F
b-2F
x-2
Fx+
2Fx2 /b
x2 /b2
-1/3
Fb2 /E
J1/
3Xb/
EJ
CA
b1-
x/b
-2F
x-2
Fx+
2Fx2 /b
1-2x
/b+
x2 /b2
CD
b0
-Fx+
1/2q
x20
00
0D
C b
01/
2Fb-
1/2q
x20
0
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0F
D b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li-1
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
E1/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.frtg.017PROCEDIMENTO E RISULTATI 789093 Frattini Giulietta
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(-2 x/b +2 x2/b2 ) Fb 1/EJ dx = [- x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (- b +2/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
LXoCA = ∫
o
b(-2 x/b +2 x2/b2 ) Fb 1/EJ dx = [- x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (- b +2/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
IPERΣ01.frtg.017PROCEDIMENTO E RISULTATI 789093 Frattini Giulietta
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 498. mm2
Ju = 141019. mm4
Jv = 31734. mm4
yg = 35.17 mmN = 102. NTy = 1020. NMx = -867000. Nmmxm = 18. mmum = -3. mmvm = -35.17 mmσm = N/A-Mv/Ju = -216.1 N/mm2
xc = 21. mmyc = 15. mmvc = -20.17 mmσc = N/A-Mv/Ju = -123.8 N/mm2
τc = 3.003 N/mm2
σo = √σ2+3τ2 = 123.9 N/mm2
S* = 2491. mm3mm 0 18 24 42x
0
48
53
y
15σc,τc
σm
u
v
IPERΣ01.frtg.017
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.frtg.017
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.grts.018REAZIONI 792732 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
17/40F
1/2F17/40Fb
17/40F
1/2F
A
B
1/2F
1/2FFb
1/2F
1/2F1/2Fb
C
D
97/40F
1/2F5/2Fb
57/40F
1/2F23/40Fb
E
C
17/40F
5/2F5/2Fb
17/40F
5/2F
EB
37/40F17/40Fb
37/40F17/40Fb
A C
1/2F
1/2F
Fb
1/2F
1/2F
F
A
1/2F
1/2F
1/2F
1/2F
GF
1/2F
1/2F1/2Fb
1/2F
1/2F
DG
IPERΣ01.grts.018AZIONI INTERNE 792732 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2
-1/2
-1/2
-1/2
-17/40
37/40
0
-1/2-1/2 -1/2
F
17/4
0
-1/2
-97/
40-5
7/40
5/2
0
-√2/2
-1/21/2
-1/2
F
-17/
400
11/
25/
223
/40
-5/20
17/40 17/40
1
0
00 1/20
Fb
IPE
RΣ0
1.gr
ts.0
18P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
9273
2 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
03.0
9.18
A
B
C
D E
FG
F
W
XX
q
q
0 0
11/2 5/21
-5/2
000
1
000
1/2
0
Mo
fless
ione
da
caric
hi a
sseg
nati
1 0
00 01 00
-1-1
0
000
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.gr
ts.0
18P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
9273
2 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
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WA
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
BA
b-x
/b0
0x2 /b
2
CD
b0
Fb-
1/2F
x0
00
0D
C b
0-1
/2F
b-1/
2Fx
00
EC
bx/
b5/
2Fb-
2Fx+
1/2q
x25/
2Fx-
2Fx2 /b
+1/
2qx3 /b
x2 /b2
17/2
4Fb2 /E
J1/
3Xb/
EJ
CE
b-1
+x/
b-F
b-F
x-1/
2qx2
Fb-
1/2F
x2 /b-1
/2qx
3 /b1-
2x/b
+x2 /b
2
EB
b0
-5/2
Fb+
5/2F
x0
00
0B
E b
05/
2Fx
00
AC
b-1
00
10
Xb/
EJ
CA
b1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li17
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
C-1
7/40
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.grts.018PROCEDIMENTO E RISULTATI 792732 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(5/2 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [5/4 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (5/4 b -2/3 b +1/8 b ) Fb 1/EJ = 17/24 Fb2/EJ
LXoCE = ∫
o
b(1 -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [ x -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= ( b -1/6 b -1/8 b ) Fb 1/EJ = 17/24 Fb2/EJ
IPERΣ01.grts.018PROCEDIMENTO E RISULTATI 792732 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 714. mm2
Ju = 156210. mm4
Jv = 68670. mm4
yg = 38.15 mmN = -174.3 NTy = 1025. NMx = -922500. Nmmxm = 18. mmum = -3. mmvm = -38.15 mmσm = N/A-Mv/Ju = -225.5 N/mm2
xc = 21. mmyc = 16. mmvc = -22.15 mmσc = N/A-Mv/Ju = -131. N/mm2
τc = 3.165 N/mm2
σo = √σ2+3τ2 = 131.1 N/mm2
S* = 2894. mm3mm 0 18 24 42x
0
42
53
y
16σc,τc
σm
u
v
IPERΣ01.grts.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.grts.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.lbtt.019REAZIONI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
2F
F3/2Fb
2F
F1/2Fb
A
B
11/20F11/20Fb
11/20F
C
D
11/20F
3F5/2Fb
11/20F
2F
ED
31/20F
F5/2Fb
31/20F
F19/20Fb
E
A
FFb
F
F
C
9/20F11/20Fb
9/20F11/20Fb
C A
F
F1/2Fb
F
BG
FGF
IPERΣ01.lbtt.019AZIONI INTERNE 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
0
-11/20-11/20
1
-√2/2
-9/20
111
F
2
-11/
20
-3-2
31/2
0
-√2/2
0
-100
F
-3/2
1/2
11/2
00 5/20
-5/2
-19/
20
1
0
-11/20 -11/20
1/2000
Fb
IPE
RΣ0
1.lb
tt.01
9P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
5029
6 La
bate
Tom
mas
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q q
-3/21/2
00
5/2
0
-5/2 -3/2
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
-10
00
0-1
0
0
11
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.lb
tt.01
9P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
5029
6 La
bate
Tom
mas
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
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
-3/2
Fb+
2Fx
00
00
BA
b0
-1/2
Fb+
2Fx
00
CD
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
DC
bx/
b0
0x2 /b
2
ED
b0
5/2F
b-3F
x+1/
2qx2
00
00
DE
b0
-2F
x-1/
2qx2
00
EA
b-x
/b-5
/2F
b+F
x5/
2Fx-
Fx2 /b
x2 /b2
11/1
2Fb2 /E
J1/
3Xb/
EJ
AE
b1-
x/b
3/2F
b+F
x3/
2Fb-
1/2F
x-F
x2 /b1-
2x/b
+x2 /b
2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b1
00
10
Xb/
EJ
AC
b-1
00
1
BG
b0
1/2F
b-F
x+1/
2qx2
00
00
GB
b0
-1/2
qx2
00
GF
b0
00
00
0F
G b
00
00
tota
li11
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
D-1
1/20
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.lbtt.019PROCEDIMENTO E RISULTATI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(5/2 x/b - x2/b2 ) Fb 1/EJ dx = [5/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (5/4 b -1/3 b ) Fb 1/EJ = 11/12 Fb2/EJ
LXoAE = ∫
o
b(3/2 -1/2 x/b - x2/b2 ) Fb 1/EJ dx = [3/2 x -1/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/4 b -1/3 b ) Fb 1/EJ = 11/12 Fb2/EJ
IPERΣ01.lbtt.019PROCEDIMENTO E RISULTATI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 816. mm2
Ju = 230061. mm4
Jv = 52992. mm4
yg = 31.79 mmN = -401.5 NTy = -2190. NMx = 1733750. Nmmxm = 18. mmum = -6. mmvm = -31.79 mmσm = N/A-Mv/Ju = 239.1 N/mm2
xc = 24. mmyc = 13. mmvc = -18.79 mmσc = N/A-Mv/Ju = 141.1 N/mm2
τc = 3.13 N/mm2
σo = √σ2+3τ2 = 141.2 N/mm2
S* = 3946. mm3mm 0 18 30 48x
0
48
53
y
13σc,τc
σm
u
v
IPERΣ01.lbtt.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.lbtt.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.lnge.020REAZIONI 806366 Longoni Emanuele Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
4F
4/5F4Fb
4F
4/5F
A
B
F
14/5F4Fb
F
14/5F6/5Fb
A C
F
3F2Fb
F
2F1/2Fb
C D
4/5F4/5Fb
4/5F
EB
F
F1/2Fb
F
D
F
F
F
G
F
Fb
F
G
E
1/5F4/5Fb
1/5F4/5Fb
E
C
IPERΣ01.lnge.020AZIONI INTERNE 806366 Longoni Emanuele Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
4/5
-1 -1 -1
0
-1-1
-1
√2/2
1/5
F
-4
14/5 3 2
-4/5
-10
0-√2/2
0
F
40
-4 -6/5 -21/2
4/50
1/2
00
01
0
-4/5
-4/5
Fb
IPE
RΣ0
1.ln
ge.0
20P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0636
6 Lo
ngon
i Em
anue
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CD
EFG F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
40
-4-2
-21/
2
00
1/2000
1
0 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
01
00
10
0000
0
0
-1 -1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.ln
ge.0
20P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0636
6 Lo
ngon
i Em
anue
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WE
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
4Fb-
4Fx
00
00
BA
b0
-4F
x0
0
AC
bx/
b-4
Fb+
2Fx
-4F
x+2F
x2 /bx2 /b
2
-4/3
Fb2 /E
J1/
3Xb/
EJ
CA
b-1
+x/
b2F
b+2F
x-2
Fb+
2Fx2 /b
1-2x
/b+
x2 /b2
CD
b0
-2F
b+3F
x-1/
2qx2
00
00
DC
b0
-1/2
Fb+
2Fx+
1/2q
x20
0
EB
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
BE
b-x
/b0
0x2 /b
2
DF
b0
1/2F
b-F
x+1/
2qx2
00
00
FD
b0
-1/2
qx2
00
FG
b0
00
00
0G
F b
00
00
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b-1
00
10
Xb/
EJ
CE
b1
00
1
tota
li-4
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WE
C4/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.lnge.020PROCEDIMENTO E RISULTATI 806366 Longoni Emanuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(-4 x/b +2 x2/b2 ) Fb 1/EJ dx = [-2 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-2 b +2/3 b ) Fb 1/EJ = -4/3 Fb2/EJ
LXoCA = ∫
o
b(-2 +2 x2/b2 ) Fb 1/EJ dx = [-2 x +2/3 x3/b2 ]o
b Fb 1/EJ
= (-2 b +2/3 b ) Fb 1/EJ = -4/3 Fb2/EJ
IPERΣ01.lnge.020PROCEDIMENTO E RISULTATI 806366 Longoni Emanuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 1032. mm2
Ju = 260495. mm4
Jv = 107424. mm4
yg = 34.56 mmN = 600. NTy = -3000. NMx = 1500000. Nmmxm = 18. mmum = -6. mmvm = -34.56 mmσm = N/A-Mv/Ju = 199.6 N/mm2
xc = 24. mmyc = 15. mmvc = -19.56 mmσc = N/A-Mv/Ju = 113.2 N/mm2
τc = 4.674 N/mm2
σo = √σ2+3τ2 = 113.5 N/mm2
S* = 4870. mm3mm 0 18 30 48x
0
42
53
y
15σc,τc
σm
u
v
IPERΣ01.lnge.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.lnge.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mnfa.021REAZIONI 853708 Manfrin Alves Fernando
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
17/40F17/40Fb
1/2F
17/40F
A B
1/2F
1/2FFb
1/2F
1/2F1/2Fb
CD
1/2F
97/40F5/2Fb
1/2F
57/40F23/40Fb
EC
5/2F
17/40F5/2Fb
5/2F
17/40F
E
B
37/40F17/40Fb
37/40F17/40Fb
A
C
1/2F
1/2F
Fb
1/2F
1/2F
F
A1/2F
1/2F
1/2F
1/2F
G
F
1/2F
1/2F1/2Fb
1/2F
1/2F
D
G
IPERΣ01.mnfa.021AZIONI INTERNE 853708 Manfrin Alves Fernando
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2 1/21/2
17/4
0
-37/
40
0
1/2
1/2
1/2
F
17/40
-1/2 -97/40-57/40
5/20
-√2/2
-1/2
1/2
-1/2
F
-17/400
11/2 5/223/40
-5/2
0
17/4
017
/40
1
0
00
1/2
0
Fb
IPE
RΣ0
1.m
nfa.
021
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
708
Man
frin
Alv
es
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
CD
E
F
G
F
W
X X
q
q Sch
ema
di c
alco
lo ip
erst
atic
o
00
11/
25/
21
-5/20
0 0
1
0
00 1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
-10
00
0-1
00
1 1
0
0
00 00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.m
nfa.
021
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
708
Man
frin
Alv
es
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→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
CD
b0
Fb-
1/2F
x0
00
0D
C b
0-1
/2F
b-1/
2Fx
00
EC
b-x
/b5/
2Fb-
2Fx+
1/2q
x2-5
/2F
x+2F
x2 /b-1
/2qx
3 /bx2 /b
2
-17/
24F
b2 /EJ
1/3X
b/E
JC
E b
1-x/
b-F
b-F
x-1/
2qx2
-Fb+
1/2F
x2 /b+
1/2q
x3 /b1-
2x/b
+x2 /b
2
EB
b0
-5/2
Fb+
5/2F
x0
00
0B
E b
05/
2Fx
00
AC
b1
00
10
Xb/
EJ
CA
b-1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li-1
7/24
Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WC
A17
/40F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.mnfa.021PROCEDIMENTO E RISULTATI 853708 Manfrin Alves
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-5/2 x/b +2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-5/4 x2/b +2/3 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (-5/4 b +2/3 b -1/8 b ) Fb 1/EJ = -17/24 Fb2/EJ
LXoCE = ∫
o
b(-1 +1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [- x +1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (- b +1/6 b +1/8 b ) Fb 1/EJ = -17/24 Fb2/EJ
IPERΣ01.mnfa.021PROCEDIMENTO E RISULTATI 853708 Manfrin Alves
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 438. mm2
Ju = 124871. mm4
Jv = 12114. mm4
yg = 33.08 mmN = 242.3 NTy = 1425. NMx = -783750. Nmmxm = 12. mmum = -3. mmvm = -33.08 mmσm = N/A-Mv/Ju = -207. N/mm2
xc = 15. mmyc = 14. mmvc = -19.08 mmσc = N/A-Mv/Ju = -119.2 N/mm2
τc = 4.166 N/mm2
σo = √σ2+3τ2 = 119.4 N/mm2
S* = 2190. mm3mm 0 12 18 30x
0
48
53
y
14σc,τc
σm
u
v
IPERΣ01.mnfa.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mnfa.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mrna.022REAZIONI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
F1/2Fb
F1/2Fb
AB
1/20F1/20Fb
1/20F
C D
F
1/20F1/2Fb
1/20F
E
D
F
19/20F1/2Fb
F
19/20F9/20Fb
EA
F
Fb
F
F
C
19/20F1/20Fb
19/20F1/20Fb
C
A
F
F1/2Fb
F
B
G
F
G
F
IPERΣ01.mrna.022AZIONI INTERNE 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
0
1/20
1/20
1
√2/2
19/2
0
-1-1
-1
F
0
-1/20
10
-19/20
-√2/2
0
10
0
F
-1/2-1/2
1/20 0
-1/2
0
1/2-9/20
1
0
-1/2
0-1
/20-1/2
00
0
Fb
IPE
RΣ0
1.m
rna.
022
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 835
477
Mar
anga
And
rea
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
-1/2
-1/2
00
-1/20
1/2
-1/2
1
0
00
-1/20 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
10
0 0
01
0
0
-1-1
0 0 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.m
rna.
022
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 835
477
Mar
anga
And
rea
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-1/2
Fb
00
00
BA
b0
1/2F
b0
0
CD
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
DC
b-x
/b0
0x2 /b
2
ED
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0D
E b
01/
2qx2
00
EA
bx/
b1/
2Fb-
Fx
1/2F
x-F
x2 /bx2 /b
2
-1/1
2Fb2 /E
J1/
3Xb/
EJ
AE
b-1
+x/
b1/
2Fb-
Fx
-1/2
Fb+
3/2F
x-F
x2 /b1-
2x/b
+x2 /b
2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b-1
00
10
Xb/
EJ
AC
b1
00
1
BG
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0G
B b
01/
2qx2
00
GF
b0
00
00
0F
G b
00
00
tota
li-1
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
A1/
20F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.mrna.022PROCEDIMENTO E RISULTATI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
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 = -1/12 Fb2/EJ
LXoAE = ∫
o
b(-1/2 +3/2 x/b - x2/b2 ) Fb 1/EJ dx = [-1/2 x +3/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +3/4 b -1/3 b ) Fb 1/EJ = -1/12 Fb2/EJ
IPERΣ01.mrna.022PROCEDIMENTO E RISULTATI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 582. mm2
Ju = 140714. mm4
Jv = 25506. mm4
yg = 36.03 mmN = 1004. NTy = -1004. NMx = 852000. Nmmxm = 12. mmum = -3. mmvm = -36.03 mmσm = N/A-Mv/Ju = 219.9 N/mm2
xc = 15. mmyc = 15. mmvc = -21.03 mmσc = N/A-Mv/Ju = 129. N/mm2
τc = 3.053 N/mm2
σo = √σ2+3τ2 = 129.1 N/mm2
S* = 2567. mm3mm 0 12 18 30x
0
42
53
y
15σc,τc
σm
u
v
IPERΣ01.mrna.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mrna.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mrtm.024REAZIONI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
3/2F
1/2FFb
3/2F
1/2F1/2Fb
A
B
2/5F
1/2F2/5Fb
2/5F
1/2F
C
D
2/5F
5/2F2Fb
2/5F
3/2F
ED
7/5F
1/2F2Fb
7/5F
1/2F3/5Fb
E
A
1/2F
1/2F
Fb
1/2F
1/2F
F
C
1/10F2/5Fb
1/10F2/5Fb
C A
1/2F
1/2F1/2Fb
1/2F
1/2F
BG
1/2F
1/2F
1/2F
1/2F
GF
IPERΣ01.mrtm.024AZIONI INTERNE 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
-2/5-2/5
1/2
0
-1/10
1/21/21/2
F
3/2
-2/5
-5/2-3/2
7/5
-√2/2
0
-1/2-1/21/2
F
-11/
2
2/5
0 20
-2-3
/5
1
0
-2/5 -2/5
1/2000
Fb
IPE
RΣ0
1.m
rtm
.024
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 846
219
Mar
tigno
ni M
atte
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q
q
-11/2
00
20
-2 -1
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
10
00
0 1
0
0
-1-1
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.m
rtm
.024
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 846
219
Mar
tigno
ni M
atte
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-Fb+
3/2F
x0
00
0B
A b
0-1
/2F
b+3/
2Fx
00
CD
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
DC
b-x
/b0
0x2 /b
2
ED
b0
2Fb-
5/2F
x+1/
2qx2
00
00
DE
b0
-3/2
Fx-
1/2q
x20
0
EA
bx/
b-2
Fb+
Fx
-2F
x+F
x2 /bx2 /b
2
-2/3
Fb2 /E
J1/
3Xb/
EJ
AE
b-1
+x/
bF
b+F
x-F
b+F
x2 /b1-
2x/b
+x2 /b
2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b-1
00
10
Xb/
EJ
AC
b1
00
1
BG
b0
1/2F
b-1/
2Fx
00
00
GB
b0
-1/2
Fx
00
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
tota
li-2
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
A2/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.mrtm.024PROCEDIMENTO E RISULTATI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(-2 x/b + x2/b2 ) Fb 1/EJ dx = [- x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
LXoAE = ∫
o
b(-1 + x2/b2 ) Fb 1/EJ dx = [- x +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
IPERΣ01.mrtm.024PROCEDIMENTO E RISULTATI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 900. mm2
Ju = 233812. mm4
Jv = 48816. mm4
yg = 32.66 mmN = -492. NTy = -3075. NMx = 1722000. Nmmxm = 12. mmum = -6. mmvm = -32.66 mmσm = N/A-Mv/Ju = 240. N/mm2
xc = 18. mmyc = 14. mmvc = -18.66 mmσc = N/A-Mv/Ju = 136.9 N/mm2
τc = 4.725 N/mm2
σo = √σ2+3τ2 = 137.1 N/mm2
S* = 4311. mm3mm 0 12 24 36x
0
42
53
y
14σc,τc
σm
u
v
IPERΣ01.mrtm.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.mrtm.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.msts.026REAZIONI 846325 Musto Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
1/2FFb
1/2F
1/2F1/2Fb
AB
1/2F
1/5F1/5Fb
1/2F
1/5F
C D
1/2F
1/5F
1/2F
1/5F
E
D
1/2F
4/5F
1/2F
4/5F4/5Fb
EA
1/2F
3/2F
Fb
1/2F
3/2F
F
C
13/10F1/5Fb
13/10F1/5Fb
C
A
1/2F
3/2F1/2Fb
1/2F
3/2F
B
G
1/2F
3/2F
1/2F
3/2F
G
F
IPERΣ01.msts.026AZIONI INTERNE 846325 Musto Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
1/5
1/5
1/2
√2
13/1
0
-3/2
-3/2
-3/2
F
1/2
-1/5
1/2
-1/2
-4/5
-√2/2
01/2
1/2
-1/2
F
-1-1/2
1/5 0
00
0-4/5
1
0
-1/5
-1/5-1
/20
00
Fb
IPE
RΣ0
1.m
sts.
026
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 846
325
Mus
to S
imon
e
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q Sch
ema
di c
alco
lo ip
erst
atic
o
-1-1
/2
00
0 0
0-1
1
0
00
-1/20 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
10
0 0
01
0
0
-1-1
0 0 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.m
sts.
026
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 846
325
Mus
to S
imon
e
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-Fb+
1/2F
x0
00
0B
A b
01/
2Fb+
1/2F
x0
0
CD
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
DC
b-x
/b0
0x2 /b
2
ED
b0
1/2F
x-1/
2qx2
00
00
DE
b0
-1/2
Fx+
1/2q
x20
0
EA
bx/
b-F
x-F
x2 /bx2 /b
2
-1/3
Fb2 /E
J1/
3Xb/
EJ
AE
b-1
+x/
bF
b-F
x-F
b+2F
x-F
x2 /b1-
2x/b
+x2 /b
2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b-1
00
10
Xb/
EJ
AC
b1
00
1
BG
b0
-1/2
Fb+
1/2F
x0
00
0G
B b
01/
2Fx
00
GF
b0
1/2F
x-1/
2qx2
00
00
FG
b0
-1/2
Fx+
1/2q
x20
0
tota
li-1
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
A1/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.msts.026PROCEDIMENTO E RISULTATI 846325 Musto Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(- x2/b2 ) Fb 1/EJ dx = [-1/3 x3/b2 ]o
b Fb 1/EJ
= (-1/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
LXoAE = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb 1/EJ dx = [- x + x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (- b + b -1/3 b ) Fb 1/EJ = -1/3 Fb2/EJ
IPERΣ01.msts.026PROCEDIMENTO E RISULTATI 846325 Musto Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 462. mm2
Ju = 129608. mm4
Jv = 14346. mm4
yg = 19.18 mmN = 1443. NTy = -721.2 NMx = 816000. Nmmxm = 18. mmym = 53. mmum = 3. mmvm = 33.82 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 15. mmyc = 39. mmvc = 19.82 mmσc = N/A-Mv/Ju = -121.7 N/mm2
τc = 2.09 N/mm2
σo = √σ2+3τ2 = 121.7 N/mm2
S* = 2253. mm3mm 0 12 18 30x
0
6
53
y
39σc,τc
σm
u
v
IPERΣ01.msts.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.msts.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.nrde.027REAZIONI 835674 Nardi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/20F
3/2F3/2Fb
1/20F
3/2F
A B
41/20F
1/2F3/2Fb
41/20F
1/2F11/20Fb
A
C
1/2F
1/2F1/2Fb
1/2F
1/2F1/2Fb
C
D
1/20F
1/2F1/20Fb
1/20F
1/2F
E
B
3/2F
1/2F1/2Fb
3/2F
1/2F
D F
3/2F
1/2F
3/2F
1/2F
F G
3/2F
1/2F
Fb
3/2F
1/2F
G
E
31/20F1/20Fb
31/20F1/20Fb
EC
IPERΣ01.nrde.027AZIONI INTERNE 835674 Nardi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/20
-1/2
-1/2
-1/2
-1/2
3/2 3/2 3/2
-√2
-31/20
F
3/2
-41/
20-1
/21/
2
1/20
1/2 1/2-1/2
-√2/2
0
F
-3/203/
2-1
1/20
-1/2
-1/2
-1/2
00
-1/20 0 0
1
0
1/201/20
Fb
IPE
RΣ0
1.nr
de.0
27P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3567
4 N
ardi
Edo
ardo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
Sch
ema
di c
alco
lo ip
erst
atic
o
-3/2
03/2
-1/2-1/2-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
00
00
0
0
11
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.nr
de.0
27P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3567
4 N
ardi
Edo
ardo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
E
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-3/2
Fb+
3/2F
x0
00
0B
A b
03/
2Fx
00
AC
b-x
/b3/
2Fb-
2Fx
-3/2
Fx+
2Fx2 /b
x2 /b2
-1/1
2Fb2 /E
J1/
3Xb/
EJ
CA
b1-
x/b
1/2F
b-2F
x1/
2Fb-
5/2F
x+2F
x2 /b1-
2x/b
+x2 /b
2
CD
b0
-1/2
Fb-
1/2F
x+1/
2qx2
00
00
DC
b0
1/2F
b+1/
2Fx-
1/2q
x20
0
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
-1/2
Fb+
1/2F
x0
00
0F
D b
01/
2Fx
00
FG
b0
1/2F
x-1/
2qx2
00
00
GF
b0
-1/2
Fx+
1/2q
x20
0
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li-1
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
E1/
20F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.nrde.027PROCEDIMENTO E RISULTATI 835674 Nardi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(-3/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +2/3 b ) Fb 1/EJ = -1/12 Fb2/EJ
LXoCA = ∫
o
b(1/2 -5/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [1/2 x -5/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (1/2 b -5/4 b +2/3 b ) Fb 1/EJ = -1/12 Fb2/EJ
IPERΣ01.nrde.027PROCEDIMENTO E RISULTATI 835674 Nardi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 852. mm2
Ju = 238569. mm4
Jv = 62064. mm4
yg = 20.54 mmN = 63. NTy = 1890. NMx = -1606500. Nmmxm = 30. mmym = 53. mmum = 6. mmvm = 32.46 mmσm = N/A-Mv/Ju = 218.6 N/mm2
xc = 24. mmyc = 39. mmvc = 18.46 mmσc = N/A-Mv/Ju = 124.4 N/mm2
τc = 2.824 N/mm2
σo = √σ2+3τ2 = 124.5 N/mm2
S* = 4277. mm3mm 0 18 30 48x
0
6
53
y
39σc,τc
σm
u
v
IPERΣ01.nrde.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.nrde.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.omrm.028REAZIONI 743445 Omara Mohamed
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
17/8F
F3Fb
17/8F
F7/8Fb
A
B
1/8F
3F3Fb
1/8F
3F
A C
9/8F5/8Fb
1/8F
D
C
F
F3/2Fb
F
F1/2Fb
B
E
FF G
F
F1/2Fb
F
E F
9/8F5/8Fb
9/8F5/8Fb
DB
FFb
F
G
D
IPERΣ01.omrm.028AZIONI INTERNE 743445 Omara Mohamed
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
1/8
00
1
-1-1 -1
-9/8
√2/2
F
17/8
-3
-9/8
-1/8
1
01 0
0
-√2/2
F
-3-7
/8
3 0
5/8
0
-3/2
-1/2
0 0-1/2
0
-5/8-5/8
1
0
Fb
IPE
RΣ0
1.om
rm.0
28P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
4344
5 O
mar
a M
oham
ed
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A B
C
D
EF
G
F
W
X
X
q
q
-3 -3/2
30 00
-3/2 -1/2
00
-1/2
000
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
00
-10
0 0
00
001
1
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.om
rm.0
28P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 7
4344
5 O
mar
a M
oham
ed
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
D
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
Fb+
3/2F
x3F
x-3/
2Fx2 /b
x2 /b2
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
3/2F
b+3/
2Fx
3/2F
b-3/
2Fx2 /b
1-2x
/b+
x2 /b2
AC
b0
3Fb-
3Fx
00
00
CA
b0
-3F
x0
0
DC
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
CD
bx/
b1/
2Fx-
1/2q
x21/
2Fx2 /b
-1/2
qx3 /b
x2 /b2
BE
b0
-3/2
Fb+
Fx
00
00
EB
b0
1/2F
b+F
x0
0
FG
b0
00
00
0G
F b
00
00
EF
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0F
E b
01/
2qx2
00
DB
b1
00
10
Xb/
EJ
BD
b-1
00
1
GD
√2b
0F
b-√2
/2F
x0
00
0
tota
li25
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WB
D-5
/8F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.omrm.028PROCEDIMENTO E RISULTATI 743445 Omara Mohamed
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(3 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoBA = ∫
o
b(3/2 -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoDC = ∫
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
LXoCD = ∫
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
IPERΣ01.omrm.028PROCEDIMENTO E RISULTATI 743445 Omara Mohamed
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 606. mm2
Ju = 141406. mm4
Jv = 27738. mm4
yg = 16.76 mmN = 41.25 NTy = -990. NMx = 891000. Nmmxm = 18. mmym = 53. mmum = 3. mmvm = 36.24 mmσm = N/A-Mv/Ju = -228.3 N/mm2
xc = 15. mmyc = 38. mmvc = 21.24 mmσc = N/A-Mv/Ju = -133.8 N/mm2
τc = 3.018 N/mm2
σo = √σ2+3τ2 = 133.9 N/mm2
S* = 2587. mm3mm 0 12 18 30x
0
12
53
y
38σc,τc
σm
u
v
IPERΣ01.omrm.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.omrm.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.pdzl.030REAZIONI 871692 Peduzzi Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
7/2F
13/20F7/2Fb
7/2F
13/20F
A
B
1/2F
53/20F7/2Fb
1/2F
53/20F17/20Fb
A C
1/2F
5/2F3/2Fb
1/2F
3/2F1/2Fb
C D
1/2F
13/20F13/20Fb
1/2F
13/20F
EB
1/2F
1/2F1/2Fb
1/2F
1/2F
D
F
1/2F
1/2F
1/2F
1/2F
F
G
1/2F
1/2F
Fb
1/2F
1/2F
G
E
3/20F13/20Fb
3/20F13/20Fb
E
C
IPERΣ01.pdzl.030AZIONI INTERNE 871692 Peduzzi Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
13/2
0
-1/2 -1/2 -1/2
-1/2
-1/2
-1/2
-1/2
0
-3/2
0
F
-7/2
53/20 5/2 3/2
-13/20
-1/2
-1/2
1/2
-√2/2
0
F
7/2
0
-7/2 -17/20 -3/21/2
13/200
1/2
00
01
0
-13/
20-1
3/20
Fb
IPE
RΣ0
1.pd
zl.0
30P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7169
2 P
eduz
zi L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CD
EFG F
W
X
X
Sch
ema
di c
alco
lo ip
erst
atic
o
7/20
-7/2
-3/2
-3/2
1/2
00
1/2000
1
0 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
0000
0
0 1 1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.pd
zl.0
30P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7169
2 P
eduz
zi L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WE
B
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
7/2F
b-7/
2Fx
00
00
BA
b0
-7/2
Fx
00
AC
b-x
/b-7
/2F
b+2F
x7/
2Fx-
2Fx2 /b
x2 /b2
13/1
2Fb2 /E
J1/
3Xb/
EJ
CA
b1-
x/b
3/2F
b+2F
x3/
2Fb+
1/2F
x-2F
x2 /b1-
2x/b
+x2 /b
2
CD
b0
-3/2
Fb+
5/2F
x-1/
2qx2
00
00
DC
b0
-1/2
Fb+
3/2F
x+1/
2qx2
00
EB
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
BE
bx/
b0
0x2 /b
2
DF
b0
1/2F
b-1/
2Fx
00
00
FD
b0
-1/2
Fx
00
FG
b0
-1/2
Fx+
1/2q
x20
00
0G
F b
01/
2Fx-
1/2q
x20
0
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li13
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WE
B-1
3/20
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.pdzl.030PROCEDIMENTO E RISULTATI 871692 Peduzzi Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(7/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [7/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (7/4 b -2/3 b ) Fb 1/EJ = 13/12 Fb2/EJ
LXoCA = ∫
o
b(3/2 +1/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [3/2 x +1/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (3/2 b +1/4 b -2/3 b ) Fb 1/EJ = 13/12 Fb2/EJ
IPERΣ01.pdzl.030PROCEDIMENTO E RISULTATI 871692 Peduzzi Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 780. mm2
Ju = 214152. mm4
Jv = 30096. mm4
yg = 22.16 mmN = 513.5 NTy = -2765. NMx = 1382500. Nmmxm = 24. mmym = 53. mmum = 6. mmvm = 30.84 mmσm = N/A-Mv/Ju = -198.4 N/mm2
xc = 18. mmyc = 40. mmvc = 17.84 mmσc = N/A-Mv/Ju = -114.5 N/mm2
τc = 4.085 N/mm2
σo = √σ2+3τ2 = 114.7 N/mm2
S* = 3797. mm3mm 0 12 24 36x
0
6
53
y
40σc,τc
σm
u
v
IPERΣ01.pdzl.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.pdzl.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.ptrm.031REAZIONI 868523 Petronio Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
F
17/8F3Fb
F
17/8F7/8Fb
A B
3F
1/8F3Fb
3F
1/8F
A
C
9/8F5/8Fb
1/8F
DC
F
F3/2Fb
F
F1/2Fb
B E
F
F
G
F
F1/2Fb
F
E
F
9/8F5/8Fb
9/8F5/8Fb
D
B
F
Fb
F
G
D
IPERΣ01.ptrm.031AZIONI INTERNE 868523 Petronio Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1
-1/8
00
-1
11
1
9/8
-√2/2
F
17/8
-3
-9/8-1/8
1
01
0
0
-√2/2
F
-3 -7/830
5/80
-3/2 -1/2
00
-1/2
0-5/8
-5/8
1
0
Fb
IPE
RΣ0
1.pt
rm.0
31P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6852
3 P
etro
nio
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-3-3
/2
3 0
00
-3/2
-1/2
0 0-1/2
0
00
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
0 0
-10
00
0 00 0
11
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.pt
rm.0
31P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6852
3 P
etro
nio
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
D
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
Fb+
3/2F
x3F
x-3/
2Fx2 /b
x2 /b2
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
3/2F
b+3/
2Fx
3/2F
b-3/
2Fx2 /b
1-2x
/b+
x2 /b2
AC
b0
3Fb-
3Fx
00
00
CA
b0
-3F
x0
0
DC
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
CD
bx/
b1/
2Fx-
1/2q
x21/
2Fx2 /b
-1/2
qx3 /b
x2 /b2
BE
b0
-3/2
Fb+
Fx
00
00
EB
b0
1/2F
b+F
x0
0
FG
b0
00
00
0G
F b
00
00
EF
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0F
E b
01/
2qx2
00
DB
b1
00
10
Xb/
EJ
BD
b-1
00
1
GD
√2b
0F
b-√2
/2F
x0
00
0
tota
li25
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WB
D-5
/8F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.ptrm.031PROCEDIMENTO E RISULTATI 868523 Petronio Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(3 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoBA = ∫
o
b(3/2 -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoDC = ∫
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
LXoCD = ∫
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
IPERΣ01.ptrm.031PROCEDIMENTO E RISULTATI 868523 Petronio Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 1068. mm2
Ju = 262174. mm4
Jv = 116496. mm4
yg = 18.21 mmN = -118.8 NTy = -2850. NMx = 1567500. Nmmxm = 30. mmym = 53. mmum = 6. mmvm = 34.79 mmσm = N/A-Mv/Ju = -208.1 N/mm2
xc = 24. mmyc = 38. mmvc = 19.79 mmσc = N/A-Mv/Ju = -118.4 N/mm2
τc = 4.45 N/mm2
σo = √σ2+3τ2 = 118.7 N/mm2
S* = 4913. mm3mm 0 18 30 48x
0
12
53
y
38σc,τc
σm
u
v
IPERΣ01.ptrm.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.ptrm.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.pnzc.032REAZIONI 878201 Ponziani Camilla
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
11/40F11/40Fb
11/40F
A B
F1/2Fb
F1/2Fb
CD
F
91/40F2Fb
F
51/40F9/40Fb
EC
2F
11/40F2Fb
2F
11/40F
E
B
51/40F11/40Fb
51/40F11/40Fb
A
C
F
Fb
F
F
AF
G
F
F
F1/2Fb
F
D
G
IPERΣ01.pnzc.032AZIONI INTERNE 878201 Ponziani Camilla
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
0
1 11
11/4
0
-51/
40
-√2/2
11
1
F
11/40
0-91/40-51/40
20
-√2/2
0-1
0
F
-11/400
1/21/2 29/40
-20
11/4
011
/40
1
0
00
1/2
0
Fb
IPE
RΣ0
1.pn
zc.0
32P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7820
1 P
onzi
ani C
amill
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
CD
E
F
G
F
W
XX
q
q Sch
ema
di c
alco
lo ip
erst
atic
o
00
1/2
1/2
21/
2
-20
0 0
1
0
00 1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
-10
00
0-1
00
1 1
0
0
00 00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.pn
zc.0
32P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7820
1 P
onzi
ani C
amill
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
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
CD b01/2Fb0000
DC b0-1/2Fb00
EC b-x/b2Fb-2Fx+1/2qx2
-2Fx+2Fx2/b-1/2qx
3/bx
2/b
2
-11/24Fb2/EJ1/3Xb/EJ
CE b1-x/b-1/2Fb-Fx-1/2qx2
-1/2Fb-1/2Fx+1/2Fx2/b+1/2qx
3/b1-2x/b+x
2/b
2
EB b0-2Fb+2Fx0000
BE b02Fx00
AC b10010Xb/EJ
CA b-1001
FA √2b0Fb-√2/2Fx0000
GF b000000
FG b0000
DG b01/2Fb-Fx+1/2qx2
0000
GD b0-1/2qx2
00
totali-11/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAB11/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.pnzc.032PROCEDIMENTO E RISULTATI 878201 Ponziani Camilla
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-2 x/b +2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [- x2/b +2/3 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (- b +2/3 b -1/8 b ) Fb 1/EJ = -11/24 Fb2/EJ
LXoCE = ∫
o
b(-1/2 -1/2 x/b +1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx
= [-1/2 x -1/4 x2/b +1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (-1/2 b -1/4 b +1/6 b +1/8 b ) Fb 1/EJ = -11/24 Fb2/EJ
IPERΣ01.pnzc.032PROCEDIMENTO E RISULTATI 878201 Ponziani Camilla
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 534. mm2
Ju = 146122. mm4
Jv = 37890. mm4
yg = 16.99 mmN = 203.5 NTy = 1480. NMx = -888000. Nmmxm = 24. mmym = 53. mmum = 3. mmvm = 36.01 mmσm = N/A-Mv/Ju = 219.2 N/mm2
xc = 21. mmyc = 38. mmvc = 21.01 mmσc = N/A-Mv/Ju = 128. N/mm2
τc = 4.331 N/mm2
σo = √σ2+3τ2 = 128.3 N/mm2
S* = 2566. mm3mm 0 18 24 42x
0
6
53
y
38σc,τc
σm
u
v
IPERΣ01.pnzc.032
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.pnzc.032
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rclf.033REAZIONI 843694 Recalcati Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
19/40F
FFb
19/40F
F
A B
61/40F
FFb
61/40F
F21/40Fb
A
C
F1/2Fb
F1/2Fb
C
D
21/40F1/40Fb
19/40F
E
B
F
F1/2Fb
F
D F
FF G
FFb
F
G
E
61/40F1/40Fb
61/40F1/40Fb
EC
IPERΣ01.rclf.033AZIONI INTERNE 843694 Recalcati Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-19/40
-1-1
00
1 1 1
-√2/2
-61/40
F
1
-61/
400
21/4
0-1
9/40
1 0 0
-√2/2
0
F
-101
-21/
40-1
/2-1
/2
-1/4
00
-1/20 0 0
1
0
1/401/40
Fb
IPE
RΣ0
1.rc
lf.03
3P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4369
4 R
ecal
cati
Fra
nces
ca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
EF
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-10
1-1/2-1/2-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0100
1 0
00
00
0
0
-1-1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.rc
lf.03
3P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4369
4 R
ecal
cati
Fra
nces
ca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WE
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-Fb+
Fx
00
00
BA
b0
Fx
00
AC
bx/
bF
b-3/
2Fx
Fx-
3/2F
x2 /bx2 /b
2
01/
3Xb/
EJ
CA
b-1
+x/
b1/
2Fb-
3/2F
x-1
/2F
b+2F
x-3/
2Fx2 /b
1-2x
/b+
x2 /b2
CD
b0
-1/2
Fb
00
00
DC
b0
1/2F
b0
0
EB
b1-
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
BE
b-x
/b-1
/2F
x+1/
2qx2
1/2F
x2 /b-1
/2qx
3 /bx2 /b
2
DF
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0F
D b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b-1
00
10
Xb/
EJ
CE
b1
00
1
tota
li1/
24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WE
C-1
/40F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.rclf.033PROCEDIMENTO E RISULTATI 843694 Recalcati Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b( x/b -3/2 x2/b2 ) Fb 1/EJ dx = [1/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (1/2 b -1/2 b ) Fb 1/EJ = 0
LXoCA = ∫
o
b(-1/2 +2 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x + x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b + b -1/2 b ) Fb 1/EJ = 0
LXoEB = ∫
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
LXoBE = ∫
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
IPERΣ01.rclf.033PROCEDIMENTO E RISULTATI 843694 Recalcati Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 582. mm2
Ju = 166519. mm4
Jv = 44082. mm4
yg = 37.89 mmN = -712.5 NTy = 1500. NMx = -1005000. Nmmxm = 18. mmum = -3. mmvm = -37.89 mmσm = N/A-Mv/Ju = -229.9 N/mm2
xc = 21. mmyc = 16. mmvc = -21.89 mmσc = N/A-Mv/Ju = -133.3 N/mm2
τc = 4.308 N/mm2
σo = √σ2+3τ2 = 133.6 N/mm2
S* = 2870. mm3mm 0 18 24 42x
0
48
55
y
16σc,τc
σm
u
v
IPERΣ01.rclf.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rclf.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rzzm.034REAZIONI 833111 Rizzo Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
2/5F
1/2F2/5Fb
2/5F
1/2F
A
B
1/2F
1/2FFb
1/2F
1/2F1/2Fb
C
D
7/5F
1/2F2Fb
7/5F
1/2F3/5Fb
E
C
2/5F
3/2F2Fb
2/5F
5/2F
EB
9/10F2/5Fb
9/10F2/5Fb
A C
1/2F
1/2F
Fb
1/2F
1/2F
F
A
1/2F
1/2F
1/2F
1/2F
GF
1/2F
1/2F1/2Fb
1/2F
1/2F
DG
IPERΣ01.rzzm.034AZIONI INTERNE 833111 Rizzo Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2
-1/2
-1/2
-2/5-2/5
9/10
0
-1/2-1/2 -1/2
F
2/5
-1/2
-7/5
3/25/2
0
-√2/2
-1/21/2
-1/2
F
-2/5
0
11/
22
3/5
-20
2/5 2/5
1
0
00 1/20
Fb
IPE
RΣ0
1.rz
zm.0
34P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3311
1 R
izzo
Mic
hele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
C
D E
FG
F
W
XX
q
q
0 0
11/2 21
-200
0
1
000
1/2
0
Mo
fless
ione
da
caric
hi a
sseg
nati
-10
00 0-1
001
1
0
000
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.rz
zm.0
34P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3311
1 R
izzo
Mic
hele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
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
CD
b0
Fb-
1/2F
x0
00
0D
C b
0-1
/2F
b-1/
2Fx
00
EC
b-x
/b2F
b-F
x-2
Fx+
Fx2 /b
x2 /b2
-2/3
Fb2 /E
J1/
3Xb/
EJ
CE
b1-
x/b
-Fb-
Fx
-Fb+
Fx2 /b
1-2x
/b+
x2 /b2
EB
b0
-2F
b+3/
2Fx+
1/2q
x20
00
0B
E b
05/
2Fx-
1/2q
x20
0
AC
b1
00
10
Xb/
EJ
CA
b-1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li-2
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
B2/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.rzzm.034PROCEDIMENTO E RISULTATI 833111 Rizzo Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-2 x/b + x2/b2 ) Fb 1/EJ dx = [- x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
LXoCE = ∫
o
b(-1 + x2/b2 ) Fb 1/EJ dx = [- x +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
IPERΣ01.rzzm.034PROCEDIMENTO E RISULTATI 833111 Rizzo Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 798. mm2
Ju = 175127. mm4
Jv = 81018. mm4
yg = 39.82 mmN = -292. NTy = 1095. NMx = -1051200. Nmmxm = 18. mmum = -3. mmvm = -39.82 mmσm = N/A-Mv/Ju = -239.4 N/mm2
xc = 21. mmyc = 17. mmvc = -22.82 mmσc = N/A-Mv/Ju = -137.3 N/mm2
τc = 3.329 N/mm2
σo = √σ2+3τ2 = 137.4 N/mm2
S* = 3194. mm3mm 0 18 24 42x
0
42
55
y
17σc,τc
σm
u
v
IPERΣ01.rzzm.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rzzm.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rssl.035REAZIONI 866706 Rossi Luisa
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
2F
F3/2Fb
2F
F1/2Fb
A
B
23/40F23/40Fb
23/40F
C
D
23/40F
3F3Fb
23/40F
3F
ED
103/40F
F3Fb
63/40F
F37/40Fb
E
A
FFb
F
F
C
17/40F23/40Fb
17/40F23/40Fb
C A
F
F1/2Fb
F
BG
FGF
IPERΣ01.rssl.035AZIONI INTERNE 866706 Rossi Luisa
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
0
-23/40
11
-√2/2
-17/40
111
F
2
-23/
40
-3
103/
4063
/40
-√2/2
0
-100
F
-3/2
1/2
23/4
00 30
-3-3
7/40
1
0
-23/40 -23/40
1/2000
Fb
IPE
RΣ0
1.rs
sl.0
35P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6670
6 R
ossi
Lui
sa
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q
q
-3/21/2
00
30
-3 -3/2
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
-10
00
0-1
0
0
11
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.rs
sl.0
35P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6670
6 R
ossi
Lui
sa
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WAC
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b0-3/2Fb+2Fx0000
BA b0-1/2Fb+2Fx00
CD b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJDC bx/b00x
2/b
2
ED b03Fb-3Fx0000
DE b0-3Fx00
EA b-x/b-3Fb+2Fx-1/2qx2
3Fx-2Fx2/b+1/2qx
3/bx
2/b
2
23/24Fb2/EJ1/3Xb/EJ
AE b1-x/b3/2Fb+Fx+1/2qx2
3/2Fb-1/2Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
FC √2b0Fb-√2/2Fx0000
CA b10010Xb/EJ
AC b-1001
BG b01/2Fb-Fx+1/2qx2
0000
GB b0-1/2qx2
00
GF b000000
FG b0000
totali23/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAC-23/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.rssl.035PROCEDIMENTO E RISULTATI 866706 Rossi Luisa
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(3 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -2/3 b +1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
LXoAE = ∫
o
b(3/2 -1/2 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [3/2 x -1/4 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (3/2 b -1/4 b -1/6 b -1/8 b ) Fb 1/EJ = 23/24 Fb2/EJ
IPERΣ01.rssl.035PROCEDIMENTO E RISULTATI 866706 Rossi Luisa
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 912. mm2
Ju = 272448. mm4
Jv = 71424. mm4
yg = 34.13 mmN = -391. NTy = -2040. NMx = 1591200. Nmmxm = 18. mmum = -6. mmvm = -34.13 mmσm = N/A-Mv/Ju = 198.9 N/mm2
xc = 24. mmyc = 14. mmvc = -20.13 mmσc = N/A-Mv/Ju = 117.1 N/mm2
τc = 2.844 N/mm2
σo = √σ2+3τ2 = 117.3 N/mm2
S* = 4558. mm3mm 0 18 30 48x
0
48
55
y
14σc,τc
σm
u
v
IPERΣ01.rssl.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rssl.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rsss.036REAZIONI 877254 Russo Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
3F
1/8F3Fb
3F
1/8F
A
B
F
17/8F3Fb
F
17/8F7/8Fb
A C
F
2F3/2Fb
F
2F1/2Fb
C D
9/8F5/8Fb
1/8F
EB
F
F1/2Fb
F
D
F
F
F
G
F
Fb
F
G
E
1/8F5/8Fb
1/8F5/8Fb
E
C
IPERΣ01.rsss.036AZIONI INTERNE 877254 Russo Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/8
-1 -1
00
-1-1
-1
√2/2
-1/8
F
-3
17/8 2
-9/8-1/8
-10
0-√2/2
0
F
30
-3 -7/8 -3/21/2
5/80
1/2
00
01
0
-5/8
-5/8
Fb
IPE
RΣ0
1.rs
ss.0
36P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7725
4 R
usso
Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CD
EFG F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
30
-3-3
/2-3
/21/
2
00
1/2000
1
0 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
0000
0
0 1 1
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.rs
ss.0
36P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7725
4 R
usso
Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WE
B
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
3Fb-
3Fx
00
00
BA
b0
-3F
x0
0
AC
b-x
/b-3
Fb+
3/2F
x3F
x-3/
2Fx2 /b
x2 /b2
Fb2 /E
J1/
3Xb/
EJ
CA
b1-
x/b
3/2F
b+3/
2Fx
3/2F
b-3/
2Fx2 /b
1-2x
/b+
x2 /b2
CD
b0
-3/2
Fb+
2Fx
00
00
DC
b0
-1/2
Fb+
2Fx
00
EB
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
BE
bx/
b1/
2Fx-
1/2q
x21/
2Fx2 /b
-1/2
qx3 /b
x2 /b2
DF
b0
1/2F
b-F
x+1/
2qx2
00
00
FD
b0
-1/2
qx2
00
FG
b0
00
00
0G
F b
00
00
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li25
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WE
B-5
/8F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.rsss.036PROCEDIMENTO E RISULTATI 877254 Russo Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(3 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoCA = ∫
o
b(3/2 -3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -1/2 b ) Fb 1/EJ = Fb2/EJ
LXoEB = ∫
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
LXoBE = ∫
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
IPERΣ01.rsss.036PROCEDIMENTO E RISULTATI 877254 Russo Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 1128. mm2
Ju = 293725. mm4
Jv = 125856. mm4
yg = 36.21 mmN = 85. NTy = -2040. NMx = 1693200. Nmmxm = 18. mmum = -6. mmvm = -36.21 mmσm = N/A-Mv/Ju = 208.8 N/mm2
xc = 24. mmyc = 15. mmvc = -21.21 mmσc = N/A-Mv/Ju = 122.4 N/mm2
τc = 2.991 N/mm2
σo = √σ2+3τ2 = 122.5 N/mm2
S* = 5168. mm3mm 0 18 30 48x
0
42
55
y
15σc,τc
σm
u
v
IPERΣ01.rsss.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.rsss.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.slvg.037REAZIONI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
2/5F2/5Fb
1/2F
2/5F
A B
1/2F
1/2FFb
1/2F
1/2F1/2Fb
CD
1/2F
7/5F2Fb
1/2F
7/5F3/5Fb
EC
3/2F
2/5F2Fb
5/2F
2/5F
E
B
9/10F2/5Fb
9/10F2/5Fb
A
C
1/2F
1/2F
Fb
1/2F
1/2F
F
A
1/2F
1/2F
1/2F
1/2F
G
F
1/2F
1/2F1/2Fb
1/2F
1/2F
D
G
IPERΣ01.slvg.037AZIONI INTERNE 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2 1/2
2/5
2/5
-9/1
0
0
1/2
1/2
1/2
F
2/5
-1/2 -7/5
3/2
5/2
0
-√2/2
-1/2
1/2
-1/2
F
-2/50
11/2 23/5
-20
2/5
2/5
1
0
00
1/2
0
Fb
IPE
RΣ0
1.sl
vg.0
37P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3478
1 S
alva
tori
Gab
riele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
CD
E
F
G
F
W
X X
q
q Sch
ema
di c
alco
lo ip
erst
atic
o
00
11/
22
1
-20
0 0
1
0
00 1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
-10
00
0-1
00
1 1
0
0
00 00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sl
vg.0
37P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3478
1 S
alva
tori
Gab
riele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→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
CD
b0
Fb-
1/2F
x0
00
0D
C b
0-1
/2F
b-1/
2Fx
00
EC
b-x
/b2F
b-F
x-2
Fx+
Fx2 /b
x2 /b2
-2/3
Fb2 /E
J1/
3Xb/
EJ
CE
b1-
x/b
-Fb-
Fx
-Fb+
Fx2 /b
1-2x
/b+
x2 /b2
EB
b0
-2F
b+3/
2Fx+
1/2q
x20
00
0B
E b
05/
2Fx-
1/2q
x20
0
AC
b1
00
10
Xb/
EJ
CA
b-1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li-2
/3F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
A2/
5Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.slvg.037PROCEDIMENTO E RISULTATI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-2 x/b + x2/b2 ) Fb 1/EJ dx = [- x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
LXoCE = ∫
o
b(-1 + x2/b2 ) Fb 1/EJ dx = [- x +1/3 x3/b2 ]o
b Fb 1/EJ
= (- b +1/3 b ) Fb 1/EJ = -2/3 Fb2/EJ
IPERΣ01.slvg.037PROCEDIMENTO E RISULTATI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 498. mm2
Ju = 147997. mm4
Jv = 16614. mm4
yg = 35.6 mmN = 208. NTy = 780. NMx = -915200. Nmmxm = 12. mmum = -3. mmvm = -35.6 mmσm = N/A-Mv/Ju = -219.7 N/mm2
xc = 15. mmyc = 15. mmvc = -20.6 mmσc = N/A-Mv/Ju = -126.9 N/mm2
τc = 2.221 N/mm2
σo = √σ2+3τ2 = 127. N/mm2
S* = 2529. mm3mm 0 12 18 30x
0
48
55
y
15σc,τc
σm
u
v
IPERΣ01.slvg.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.slvg.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sblm.038REAZIONI 877022 Sblendido Maria Angela
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
F1/2Fb
F1/2Fb
AB
1/40F1/40Fb
1/40F
C D
F
1/40FFb
F
1/40F
E
D
F
79/40FFb
F
39/40F19/40Fb
EA
F
Fb
F
F
C
39/40F1/40Fb
39/40F1/40Fb
C
A
F
F1/2Fb
F
B
G
F
G
F
IPERΣ01.sblm.038AZIONI INTERNE 877022 Sblendido Maria Angela
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
0
1/40
11
√2/2
39/4
0
-1-1
-1
F
0
-1/40
1
-79/40-39/40
-√2/2
0
10
0
F
-1/2-1/2
1/40 0
-10
1-19/40
1
0
-1/4
0-1
/40-1/2
00
0
Fb
IPE
RΣ0
1.sb
lm.0
38P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7702
2 S
blen
dido
Mar
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
-1/2
-1/2
00
-10
1-1
/2
1
0
00
-1/20 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
10
0 0
01
0
0
-1-1
0 0 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sb
lm.0
38P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7702
2 S
blen
dido
Mar
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WCA
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b0-1/2Fb0000
BA b01/2Fb00
CD b1-x/b001-2x/b+x2/b
2
01/3Xb/EJDC b-x/b00x
2/b
2
ED b0-Fb+Fx0000
DE b0Fx00
EA bx/bFb-2Fx+1/2qx2
Fx-2Fx2/b+1/2qx
3/bx
2/b
2
-1/24Fb2/EJ1/3Xb/EJ
AE b-1+x/b1/2Fb-Fx-1/2qx2
-1/2Fb+3/2Fx-1/2Fx2/b-1/2qx
3/b1-2x/b+x
2/b
2
FC √2b0Fb-√2/2Fx0000
CA b-10010Xb/EJ
AC b1001
BG b0-1/2Fb+Fx-1/2qx2
0000
GB b01/2qx2
00
GF b000000
FG b0000
totali-1/24Fb2/EJ5/3Xb/EJ
iperstatica X=WCA1/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.sblm.038PROCEDIMENTO E RISULTATI 877022 Sblendido Maria
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b( x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [1/2 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (1/2 b -2/3 b +1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ
LXoAE = ∫
o
b(-1/2 +3/2 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx
= [-1/2 x +3/4 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (-1/2 b +3/4 b -1/6 b -1/8 b ) Fb 1/EJ = -1/24 Fb2/EJ
IPERΣ01.sblm.038PROCEDIMENTO E RISULTATI 877022 Sblendido Maria
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 642. mm2
Ju = 158306. mm4
Jv = 30006. mm4
yg = 37.71 mmN = 25.75 NTy = 1030. NMx = -957900. Nmmxm = 12. mmum = -3. mmvm = -37.71 mmσm = N/A-Mv/Ju = -228.1 N/mm2
xc = 15. mmyc = 16. mmvc = -21.71 mmσc = N/A-Mv/Ju = -131.3 N/mm2
τc = 3.092 N/mm2
σo = √σ2+3τ2 = 131.4 N/mm2
S* = 2852. mm3mm 0 12 18 30x
0
42
55
y
16σc,τc
σm
u
v
IPERΣ01.sblm.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sblm.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.scll.039REAZIONI 835340 Scalia Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
59/20F
1/2F9/2Fb
59/20F
1/2F31/20Fb
A
B
19/20F
9/2F9/2Fb
19/20F
9/2F
A C
19/20F
1/2F19/20Fb
19/20F
1/2F
D
C
5/2F
1/2F5/2Fb
3/2F
1/2F1/2Fb
B
E
3/2F
1/2F
3/2F
1/2F
F G
3/2F
1/2F1/2Fb
3/2F
1/2F
E F
9/20F19/20Fb
9/20F19/20Fb
DB
3/2F
1/2F
Fb
3/2F
1/2F
G
D
IPERΣ01.scll.039AZIONI INTERNE 835340 Scalia Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
19/20
1/2
1/2
1/2
-3/2 -3/2-3/2
-9/20
√2
F
59/2
0
-9/2
-19/
20
5/2
3/2
1/2-1/2
1/2
0
-√2/2
F
-9/2
-31/
20
9/2 0
19/2
00
-5/2
-1/2
0 0-1/2
0
-19/20-19/20
1
0
Fb
IPE
RΣ0
1.sc
ll.03
9P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3534
0 S
calia
Luc
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A B
C
D
EF
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-9/2 -5/2
9/2
0 00
-5/2 -1/2
00
-1/2
000
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
00
-10
0 0
00
001
1
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sc
ll.03
9P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3534
0 S
calia
Luc
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
D
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-9
/2F
b+2F
x9/
2Fx-
2Fx2 /b
x2 /b2
19/1
2Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
5/2F
b+2F
x5/
2Fb-
1/2F
x-2F
x2 /b1-
2x/b
+x2 /b
2
AC
b0
9/2F
b-9/
2Fx
00
00
CA
b0
-9/2
Fx
00
DC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CD
bx/
b0
0x2 /b
2
BE
b0
-5/2
Fb+
5/2F
x-1/
2qx2
00
00
EB
b0
1/2F
b+3/
2Fx+
1/2q
x20
0
FG
b0
1/2F
x-1/
2qx2
00
00
GF
b0
-1/2
Fx+
1/2q
x20
0
EF
b0
-1/2
Fb+
1/2F
x0
00
0F
E b
01/
2Fx
00
DB
b1
00
10
Xb/
EJ
BD
b-1
00
1
GD
√2b
0F
b-√2
/2F
x0
00
0
tota
li19
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WB
D-1
9/20
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.scll.039PROCEDIMENTO E RISULTATI 835340 Scalia Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(9/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [9/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (9/4 b -2/3 b ) Fb 1/EJ = 19/12 Fb2/EJ
LXoBA = ∫
o
b(5/2 -1/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [5/2 x -1/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (5/2 b -1/4 b -2/3 b ) Fb 1/EJ = 19/12 Fb2/EJ
IPERΣ01.scll.039PROCEDIMENTO E RISULTATI 835340 Scalia Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 828. mm2
Ju = 244195. mm4
Jv = 34128. mm4
yg = 32.37 mmN = 380. NTy = -1800. NMx = 1764000. Nmmxm = 12. mmum = -6. mmvm = -32.37 mmσm = N/A-Mv/Ju = 234.3 N/mm2
xc = 18. mmyc = 14. mmvc = -18.37 mmσc = N/A-Mv/Ju = 133.2 N/mm2
τc = 2.618 N/mm2
σo = √σ2+3τ2 = 133.2 N/mm2
S* = 4262. mm3mm 0 12 24 36x
0
48
55
y
14σc,τc
σm
u
v
IPERΣ01.scll.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.scll.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.shhe.040REAZIONI 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
3/2F
1/2FFb
3/2F
1/2F1/2Fb
A
B
17/40F
1/2F17/40Fb
17/40F
1/2F
C
D
17/40F
5/2F5/2Fb
17/40F
5/2F
ED
97/40F
1/2F5/2Fb
57/40F
1/2F23/40Fb
E
A
1/2F
1/2F
Fb
1/2F
1/2F
F
C
3/40F17/40Fb
3/40F17/40Fb
C A
1/2F
1/2F1/2Fb
1/2F
1/2F
BG
1/2F
1/2F
1/2F
1/2F
GF
IPERΣ01.shhe.040AZIONI INTERNE 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
-17/40
1/2
1/2
0
-3/40
1/21/21/2
F
3/2
-17/
40
-5/2
97/4
057
/40
-√2/2
0
-1/2-1/21/2
F
-11/
2
17/4
00 5/20
-5/2
-23/
40
1
0
-17/40 -17/40
1/2000
Fb
IPE
RΣ0
1.sh
he.0
40P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0982
8 S
hehu
Elto
n
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
CDE
FG
F
X
X
q
q
-11/2
00
5/2
0
-5/2 -1
1
0
00 1/
20
00
Mo
fless
ione
da
caric
hi a
sseg
nati
0 0
-10
00
0-1
0
0
11
00
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sh
he.0
40P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
0982
8 S
hehu
Elto
n
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WA
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-Fb+
3/2F
x0
00
0B
A b
0-1
/2F
b+3/
2Fx
00
CD
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
DC
bx/
b0
0x2 /b
2
ED
b0
5/2F
b-5/
2Fx
00
00
DE
b0
-5/2
Fx
00
EA
b-x
/b-5
/2F
b+2F
x-1/
2qx2
5/2F
x-2F
x2 /b+
1/2q
x3 /bx2 /b
2
17/2
4Fb2 /E
J1/
3Xb/
EJ
AE
b1-
x/b
Fb+
Fx+
1/2q
x2F
b-1/
2Fx2 /b
-1/2
qx3 /b
1-2x
/b+
x2 /b2
FC
√2b
0F
b-√2
/2F
x0
00
0
CA
b1
00
10
Xb/
EJ
AC
b-1
00
1
BG
b0
1/2F
b-1/
2Fx
00
00
GB
b0
-1/2
Fx
00
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
tota
li17
/24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
C-1
7/40
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.shhe.040PROCEDIMENTO E RISULTATI 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
o
b(5/2 x/b -2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [5/4 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (5/4 b -2/3 b +1/8 b ) Fb 1/EJ = 17/24 Fb2/EJ
LXoAE = ∫
o
b(1 -1/2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [ x -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= ( b -1/6 b -1/8 b ) Fb 1/EJ = 17/24 Fb2/EJ
IPERΣ01.shhe.040PROCEDIMENTO E RISULTATI 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 972. mm2
Ju = 264196. mm4
Jv = 56592. mm4
yg = 34.24 mmN = -505.8 NTy = -2975. NMx = 1547000. Nmmxm = 12. mmum = -6. mmvm = -34.24 mmσm = N/A-Mv/Ju = 200. N/mm2
xc = 18. mmyc = 14. mmvc = -20.24 mmσc = N/A-Mv/Ju = 118. N/mm2
τc = 4.294 N/mm2
σo = √σ2+3τ2 = 118.2 N/mm2
S* = 4576. mm3mm 0 12 24 36x
0
42
55
y
14σc,τc
σm
u
v
IPERΣ01.shhe.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.shhe.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.smnd.041REAZIONI 847315 Simonini Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
59/20F9/2Fb
1/2F
59/20F31/20Fb
A B
9/2F
19/20F9/2Fb
9/2F
19/20F
A
C
1/2F
19/20F19/20Fb
1/2F
19/20F
DC
1/2F
5/2F5/2Fb
1/2F
3/2F1/2Fb
B E
1/2F
3/2F
1/2F
3/2F
F
G
1/2F
3/2F1/2Fb
1/2F
3/2F
E
F
9/20F19/20Fb
9/20F19/20Fb
D
B
1/2F
3/2F
Fb
1/2F
3/2F
G
D
IPERΣ01.smnd.041AZIONI INTERNE 847315 Simonini Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2
-19/
20
-1/2
-1/2 -1/2
3/2
3/2
3/2
9/20
-√2
F
59/20
-9/2
-19/20
5/2 3/2
1/2
-1/2
1/20
-√2/2
F
-9/2 -31/20
9/2
0
19/200
-5/2 -1/2
00
-1/2
0-19/
20-1
9/20
1
0
Fb
IPE
RΣ0
1.sm
nd.0
41P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4731
5 S
imon
ini D
avid
e
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
XX
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-9/2
-5/2
9/2 0
00
-5/2
-1/2
0 0-1/2
0
00
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
0 0
-10
00
0 00 0
11
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sm
nd.0
41P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4731
5 S
imon
ini D
avid
e
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WD
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-9
/2F
b+2F
x9/
2Fx-
2Fx2 /b
x2 /b2
19/1
2Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
5/2F
b+2F
x5/
2Fb-
1/2F
x-2F
x2 /b1-
2x/b
+x2 /b
2
AC
b0
9/2F
b-9/
2Fx
00
00
CA
b0
-9/2
Fx
00
DC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CD
bx/
b0
0x2 /b
2
BE
b0
-5/2
Fb+
5/2F
x-1/
2qx2
00
00
EB
b0
1/2F
b+3/
2Fx+
1/2q
x20
0
FG
b0
1/2F
x-1/
2qx2
00
00
GF
b0
-1/2
Fx+
1/2q
x20
0
EF
b0
-1/2
Fb+
1/2F
x0
00
0F
E b
01/
2Fx
00
DB
b1
00
10
Xb/
EJ
BD
b-1
00
1
GD
√2b
0F
b-√2
/2F
x0
00
0
tota
li19
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WD
C-1
9/20
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.smnd.041PROCEDIMENTO E RISULTATI 847315 Simonini Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(9/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [9/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (9/4 b -2/3 b ) Fb 1/EJ = 19/12 Fb2/EJ
LXoBA = ∫
o
b(5/2 -1/2 x/b -2 x2/b2 ) Fb 1/EJ dx = [5/2 x -1/4 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (5/2 b -1/4 b -2/3 b ) Fb 1/EJ = 19/12 Fb2/EJ
IPERΣ01.smnd.041PROCEDIMENTO E RISULTATI 847315 Simonini Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 948. mm2
Ju = 262515. mm4
Jv = 52848. mm4
yg = 20.97 mmN = -589. NTy = -2790. NMx = 1590300. Nmmxm = 24. mmym = 55. mmum = 6. mmvm = 34.03 mmσm = N/A-Mv/Ju = -206.8 N/mm2
xc = 18. mmyc = 41. mmvc = 20.03 mmσc = N/A-Mv/Ju = -122. N/mm2
τc = 4.022 N/mm2
σo = √σ2+3τ2 = 122.2 N/mm2
S* = 4541. mm3mm 0 12 24 36x
0
12
55
y
41σc,τc
σm
u
v
IPERΣ01.smnd.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.smnd.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sldl.042REAZIONI 866259 Soldavini Luca Giovanni
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2F
1/2FFb
1/2F
1/2F1/2Fb
AB
1/2F
7/40F7/40Fb
1/2F
7/40F
C D
1/2F
7/40F1/2Fb
1/2F
7/40F
E
D
1/2F
73/40F1/2Fb
1/2F
33/40F33/40Fb
EA
1/2F
3/2F
Fb
1/2F
3/2F
F
C
53/40F7/40Fb
53/40F7/40Fb
C
A
1/2F
3/2F1/2Fb
1/2F
3/2F
B
G
1/2F
3/2F
1/2F
3/2F
G
F
IPERΣ01.sldl.042AZIONI INTERNE 866259 Soldavini Luca Giovanni
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/2
1/2
7/40
1/21/2
√2
53/4
0
-3/2
-3/2
-3/2
F
1/2
-7/40
1/2
-73/40-33/40
-√2/2
01/2
1/2
-1/2
F
-1-1/2
7/40 0
-1/2
0
1/2-33/40
1
0
-7/4
0-7
/40-1/2
00
0
Fb
IPE
RΣ0
1.sl
dl.0
42P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6625
9 S
olda
vini
Luc
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
-1-1
/2
00
-1/20
1/2
-1
1
0
00
-1/20 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
00
-10
0 0
0-1
0
0
11
0 0 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sl
dl.0
42P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6625
9 S
olda
vini
Luc
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WAC
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b0-Fb+1/2Fx0000
BA b01/2Fb+1/2Fx00
CD b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJDC bx/b00x
2/b
2
ED b0-1/2Fb+1/2Fx0000
DE b01/2Fx00
EA b-x/b1/2Fb-2Fx+1/2qx2
-1/2Fx+2Fx2/b-1/2qx
3/bx
2/b
2
7/24Fb2/EJ1/3Xb/EJ
AE b1-x/bFb-Fx-1/2qx2
Fb-2Fx+1/2Fx2/b+1/2qx
3/b1-2x/b+x
2/b
2
FC √2b0Fb-√2/2Fx0000
CA b10010Xb/EJ
AC b-1001
BG b0-1/2Fb+1/2Fx0000
GB b01/2Fx00
GF b01/2Fx-1/2qx2
0000
FG b0-1/2Fx+1/2qx2
00
totali7/24Fb2/EJ5/3Xb/EJ
iperstatica X=WAC-7/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.sldl.042PROCEDIMENTO E RISULTATI 866259 Soldavini Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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
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
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
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEA = ∫
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 = 7/24 Fb2/EJ
LXoAE = ∫
o
b(1 -2 x/b +1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [ x - x2/b +1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= ( b - b +1/6 b +1/8 b ) Fb 1/EJ = 7/24 Fb2/EJ
IPERΣ01.sldl.042PROCEDIMENTO E RISULTATI 866259 Soldavini Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 474. mm2
Ju = 143796. mm4
Jv = 14382. mm4
yg = 20.06 mmN = 2093. NTy = -1047. NMx = 917600. Nmmxm = 18. mmym = 55. mmum = 3. mmvm = 34.94 mmσm = N/A-Mv/Ju = -218.6 N/mm2
xc = 15. mmyc = 40. mmvc = 19.94 mmσc = N/A-Mv/Ju = -122.8 N/mm2
τc = 2.996 N/mm2
σo = √σ2+3τ2 = 123. N/mm2
S* = 2470. mm3mm 0 12 18 30x
0
6
55
y
40σc,τc
σm
u
v
IPERΣ01.sldl.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sldl.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.spgs.043REAZIONI 833411 Spagnolo Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
5/8F
1/2F1/2Fb
5/8F
1/2F
A B
11/8F
1/2F1/2Fb
11/8F
1/2F7/8Fb
A
C
1/2F
1/2FFb
1/2F
1/2F1/2Fb
C
D
3/8F
1/2F1/8Fb
5/8F
1/2F
E
B
3/2F
1/2F1/2Fb
3/2F
1/2F
D F
3/2F
1/2F
3/2F
1/2F
F G
3/2F
1/2F
Fb
3/2F
1/2F
G
E
15/8F1/8Fb
15/8F1/8Fb
EC
IPERΣ01.spgs.043AZIONI INTERNE 833411 Spagnolo Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-5/8
-1/2
-1/2
-1/2
-1/2
3/2 3/2 3/2
-√2
-15/8
F
1/2
-11/
81/
2
3/8
-5/8
1/2 1/2-1/2
-√2/2
0
F
-1/201/
2-7
/8-1
-1/2
1/8
0
-1/20 0 0
1
0
-1/8-1/8
Fb
IPE
RΣ0
1.sp
gs.0
43P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3341
1 S
pagn
olo
Silv
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-1/2
01/2
-1-1-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
00
00
0
0
11
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sp
gs.0
43P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3341
1 S
pagn
olo
Silv
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
E
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b0
-1/2
Fb+
1/2F
x0
00
0B
A b
01/
2Fx
00
AC
b-x
/b1/
2Fb-
3/2F
x-1
/2F
x+3/
2Fx2 /b
x2 /b2
1/4F
b2 /EJ
1/3X
b/E
JC
A b
1-x/
bF
b-3/
2Fx
Fb-
5/2F
x+3/
2Fx2 /b
1-2x
/b+
x2 /b2
CD
b0
-Fb+
1/2F
x0
00
0D
C b
01/
2Fb+
1/2F
x0
0
EB
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
BE
bx/
b-1
/2F
x+1/
2qx2
-1/2
Fx2 /b
+1/
2qx3 /b
x2 /b2
DF
b0
-1/2
Fb+
1/2F
x0
00
0F
D b
01/
2Fx
00
FG
b0
1/2F
x-1/
2qx2
00
00
GF
b0
-1/2
Fx+
1/2q
x20
0
GE
√2b
0F
b-√2
/2F
x0
00
0
EC
b1
00
10
Xb/
EJ
CE
b-1
00
1
tota
li5/
24F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WC
E-1
/8F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.spgs.043PROCEDIMENTO E RISULTATI 833411 Spagnolo Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
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
LXoCA = ∫
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
LXoEB = ∫
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
LXoBE = ∫
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
IPERΣ01.spgs.043PROCEDIMENTO E RISULTATI 833411 Spagnolo Silvia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 876. mm2
Ju = 264708. mm4
Jv = 62352. mm4
yg = 21.46 mmN = -3748. NTy = -1874. NMx = 1775500. Nmmxm = 30. mmym = 55. mmum = 6. mmvm = 33.54 mmσm = N/A-Mv/Ju = -229.3 N/mm2
xc = 24. mmyc = 41. mmvc = 19.54 mmσc = N/A-Mv/Ju = -135.3 N/mm2
τc = 2.63 N/mm2
σo = √σ2+3τ2 = 135.4 N/mm2
S* = 4459. mm3mm 0 18 30 48x
0
6
55
y
41σc,τc
σm
u
v
IPERΣ01.spgs.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.spgs.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sbsa.044REAZIONI 870941 Subasingha Arachchige Fernando Sh
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
14/5F
F4Fb
14/5F
F6/5Fb
A
B
4/5F
4F4Fb
4/5F
4F
A C
4/5F4/5Fb
4/5F
D
C
2F
F2Fb
F
F1/2Fb
B
E
FF G
F
F1/2Fb
F
E F
4/5F4/5Fb
4/5F4/5Fb
DB
FFb
F
G
D
IPERΣ01.sbsa.044AZIONI INTERNE 870941 Subasingha Arachchige Fernando
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1
4/5
0
11
-1-1 -1
-4/5
√2/2
F
14/5
-4
-4/5
21
01 0
0
-√2/2
F
-4-6
/5
4 0
4/5
0
-2-1
/2
0 0-1/2
0
-4/5-4/5
1
0
Fb
IPE
RΣ0
1.sb
sa.0
44P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7094
1 S
ubas
ingh
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A B
C
D
EF
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-4 -2
40 00
-2 -1/2
00
-1/2
000
1
0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
00
-10
0 0
00
001
1
0
0
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.sb
sa.0
44P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7094
1 S
ubas
ingh
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
D
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-4
Fb+
2Fx
4Fx-
2Fx2 /b
x2 /b2
4/3F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b2F
b+2F
x2F
b-2F
x2 /b1-
2x/b
+x2 /b
2
AC
b0
4Fb-
4Fx
00
00
CA
b0
-4F
x0
0
DC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CD
bx/
b0
0x2 /b
2
BE
b0
-2F
b+2F
x-1/
2qx2
00
00
EB
b0
1/2F
b+F
x+1/
2qx2
00
FG
b0
00
00
0G
F b
00
00
EF
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0F
E b
01/
2qx2
00
DB
b1
00
10
Xb/
EJ
BD
b-1
00
1
GD
√2b
0F
b-√2
/2F
x0
00
0
tota
li4/
3Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WB
D-4
/5F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.sbsa.044PROCEDIMENTO E RISULTATI 870941 Subasingha
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXBA = ∫
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
LXXDC = ∫
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
LXXCD = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXDB = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXBD = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAB = ∫
o
b(4 x/b -2 x2/b2 ) Fb 1/EJ dx = [2 x2/b -2/3 x3/b2 ]o
b Fb 1/EJ
= (2 b -2/3 b ) Fb 1/EJ = 4/3 Fb2/EJ
LXoBA = ∫
o
b(2 -2 x2/b2 ) Fb 1/EJ dx = [2 x -2/3 x3/b2 ]o
b Fb 1/EJ
= (2 b -2/3 b ) Fb 1/EJ = 4/3 Fb2/EJ
IPERΣ01.sbsa.044PROCEDIMENTO E RISULTATI 870941 Subasingha
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 618. mm2
Ju = 157731. mm4
Jv = 27774. mm4
yg = 17.48 mmN = 280. NTy = -1400. NMx = 1008000. Nmmxm = 18. mmym = 55. mmum = 3. mmvm = 37.52 mmσm = N/A-Mv/Ju = -239.3 N/mm2
xc = 15. mmyc = 39. mmvc = 21.52 mmσc = N/A-Mv/Ju = -137.1 N/mm2
τc = 4.192 N/mm2
σo = √σ2+3τ2 = 137.3 N/mm2
S* = 2834. mm3mm 0 12 18 30x
0
12
55
y
39σc,τc
σm
u
v
IPERΣ01.sbsa.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.sbsa.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.trns.045REAZIONI 843775 Tarantola Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/4F1/4Fb
1/4F
A
B
F1/2Fb
F1/2Fb
C
D
5/4F
F3/2Fb
5/4F
F1/4Fb
E
C
1/4F
F3/2Fb
1/4F
2F
EB
5/4F1/4Fb
5/4F1/4Fb
A C
FFb
F
F
A
FGF
F
F1/2Fb
F
DG
IPERΣ01.trns.045AZIONI INTERNE 843775 Tarantola Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
0
-1-1
-1/4-1/4
5/4
√2/2
-1 -1-1
F
1/4
0-5
/4
12
0
-√2/2
0-1
0
F
-1/4
0
1/2
1/2
3/2
1/4
-3/20
1/4 1/4
1
0
00 1/20
Fb
IPE
RΣ0
1.tr
ns.0
45P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4377
5 T
aran
tola
Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
A
B
C
D E
FG
F
W
XX
0 0
1/21/2 3/21/2
-3/2
000
1
000
1/2
0
Mo
fless
ione
da
caric
hi a
sseg
nati
1 0
00 01 00
-1-1
0
000
00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.tr
ns.0
45P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4377
5 T
aran
tola
Ste
fano
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WA
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b1-
x/b
00
1-2x
/b+
x2 /b2
01/
3Xb/
EJ
BA
b-x
/b0
0x2 /b
2
CD
b0
1/2F
b0
00
0D
C b
0-1
/2F
b0
0
EC
bx/
b3/
2Fb-
Fx
3/2F
x-F
x2 /bx2 /b
2
5/12
Fb2 /E
J1/
3Xb/
EJ
CE
b-1
+x/
b-1
/2F
b-F
x1/
2Fb+
1/2F
x-F
x2 /b1-
2x/b
+x2 /b
2
EB
b0
-3/2
Fb+
Fx+
1/2q
x20
00
0B
E b
02F
x-1/
2qx2
00
AC
b-1
00
10
Xb/
EJ
CA
b1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
00
00
0F
G b
00
00
DG
b0
1/2F
b-F
x+1/
2qx2
00
00
GD
b0
-1/2
qx2
00
tota
li5/
12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
C-1
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.trns.045PROCEDIMENTO E RISULTATI 843775 Tarantola Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(3/2 x/b - x2/b2 ) Fb 1/EJ dx = [3/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (3/4 b -1/3 b ) Fb 1/EJ = 5/12 Fb2/EJ
LXoCE = ∫
o
b(1/2 +1/2 x/b - x2/b2 ) Fb 1/EJ dx = [1/2 x +1/4 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ
= (1/2 b +1/4 b -1/3 b ) Fb 1/EJ = 5/12 Fb2/EJ
IPERΣ01.trns.045PROCEDIMENTO E RISULTATI 843775 Tarantola Stefano
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 762. mm2
Ju = 174852. mm4
Jv = 74862. mm4
yg = 15.31 mmN = -187.5 NTy = 750. NMx = -877500. Nmmxm = 24. mmym = 55. mmum = 3. mmvm = 39.69 mmσm = N/A-Mv/Ju = 198.9 N/mm2
xc = 21. mmyc = 38. mmvc = 22.69 mmσc = N/A-Mv/Ju = 113.6 N/mm2
τc = 2.274 N/mm2
σo = √σ2+3τ2 = 113.7 N/mm2
S* = 3181. mm3mm 0 18 24 42x
0
12
55
y
38σc,τc
σm
u
v
IPERΣ01.trns.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.trns.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vnls.048REAZIONI 878465 Vanoli Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/4F1/4Fb
1/4F
A B
F1/2Fb
F1/2Fb
CD
F
5/4F3/2Fb
F
5/4F1/4Fb
EC
F
1/4F3/2Fb
2F
1/4F
E
B
5/4F1/4Fb
5/4F1/4Fb
A
C
F
Fb
F
F
A
F
G
F
F
F1/2Fb
F
D
G
IPERΣ01.vnls.048AZIONI INTERNE 878465 Vanoli Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
0
1 1
1/4
1/4
-5/4
-√2/2
11
1
F
1/4
0-5/4
12
0
-√2/2
0-1
0
F
-1/40
1/21/2 3/21/4
-3/2
0
1/4
1/4
1
0
00
1/2
0
Fb
IPE
RΣ0
1.vn
ls.0
48P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7846
5 V
anol
i Sim
one
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
CD
E
F
G
F
W
XX
q
q Sch
ema
di c
alco
lo ip
erst
atic
o
00
1/2
1/2
3/2
1/2
-3/20
0 0
1
0
00 1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
-10
00
0-1
00
1 1
0
0
00 00
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.vn
ls.0
48P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7846
5 V
anol
i Sim
one
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
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
CD
b0
1/2F
b0
00
0D
C b
0-1
/2F
b0
0
EC
b-x
/b3/
2Fb-
Fx
-3/2
Fx+
Fx2 /b
x2 /b2
-5/1
2Fb2 /E
J1/
3Xb/
EJ
CE
b1-
x/b
-1/2
Fb-
Fx
-1/2
Fb-
1/2F
x+F
x2 /b1-
2x/b
+x2 /b
2
EB
b0
-3/2
Fb+
Fx+
1/2q
x20
00
0B
E b
02F
x-1/
2qx2
00
AC
b1
00
10
Xb/
EJ
CA
b-1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
00
00
0F
G b
00
00
DG
b0
1/2F
b-F
x+1/
2qx2
00
00
GD
b0
-1/2
qx2
00
tota
li-5
/12F
b2 /EJ
5/3X
b/E
J
iper
stat
ica
X=
WA
B1/
4Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.vnls.048PROCEDIMENTO E RISULTATI 878465 Vanoli Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-3/2 x/b + x2/b2 ) Fb 1/EJ dx = [-3/4 x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= (-3/4 b +1/3 b ) Fb 1/EJ = -5/12 Fb2/EJ
LXoCE = ∫
o
b(-1/2 -1/2 x/b + x2/b2 ) Fb 1/EJ dx = [-1/2 x -1/4 x2/b +1/3 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b -1/4 b +1/3 b ) Fb 1/EJ = -5/12 Fb2/EJ
IPERΣ01.vnls.048PROCEDIMENTO E RISULTATI 878465 Vanoli Simone
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 546. mm2
Ju = 162198. mm4
Jv = 37926. mm4
yg = 17.81 mmN = 177.5 NTy = 710. NMx = -990450. Nmmxm = 24. mmym = 55. mmum = 3. mmvm = 37.19 mmσm = N/A-Mv/Ju = 227.4 N/mm2
xc = 21. mmyc = 39. mmvc = 21.19 mmσc = N/A-Mv/Ju = 129.7 N/mm2
τc = 2.045 N/mm2
σo = √σ2+3τ2 = 129.8 N/mm2
S* = 2802. mm3mm 0 18 24 42x
0
6
55
y
39σc,τc
σm
u
v
IPERΣ01.vnls.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vnls.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vrga.049REAZIONI 843591 Virga Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
1/40F
FFb
1/40F
F
A B
79/40F
FFb
39/40F
F19/40Fb
A
C
F1/2Fb
F1/2Fb
C
D
1/40F1/40Fb
1/40F
E
B
F
F1/2Fb
F
D F
FF G
FFb
F
G
E
39/40F1/40Fb
39/40F1/40Fb
EC
IPERΣ01.vrga.049AZIONI INTERNE 843591 Virga Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/40
-1-1
-1
0
1 1 1
-√2/2
-39/40
F
1
-79/
40-3
9/40
0
-1/4
0
1 0 0
-√2/2
0
F
-101
-19/
40-1
/2-1
/2
1/40
0
-1/20 0 0
1
0
-1/40-1/40
Fb
IPE
RΣ0
1.vr
ga.0
49P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4359
1 V
irga
Ale
ssan
dro
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
AB
C
D
E
F
G
F
W
X
X
q
q
Sch
ema
di c
alco
lo ip
erst
atic
o
-10
1-1/2-1/2-1/2
0 0
-1/2
00
0
1
0
00
Mo
fless
ione
da
caric
hi a
sseg
nati
00
0-1
00
-10
00
00
0
0
11
Mx
fless
ione
da
iper
stat
ica
X=
1
IPE
RΣ0
1.vr
ga.0
49P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4359
1 V
irga
Ale
ssan
dro
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Quadro contributi PLV per iperstatica X=WCE
→Mx(x)Mo(x)MxMoMxMx∫MxMo/EJdx∫XMxMx/EJdx
AB b0-Fb+Fx0000
BA b0Fx00
AC b-x/bFb-2Fx+1/2qx2
-Fx+2Fx2/b-1/2qx
3/bx
2/b
2
1/24Fb2/EJ1/3Xb/EJ
CA b1-x/b1/2Fb-Fx-1/2qx2
1/2Fb-3/2Fx+1/2Fx2/b+1/2qx
3/b1-2x/b+x
2/b
2
CD b0-1/2Fb0000
DC b01/2Fb00
EB b-1+x/b001-2x/b+x2/b
2
01/3Xb/EJBE bx/b00x
2/b
2
DF b0-1/2Fb+Fx-1/2qx2
0000
FD b01/2qx2
00
FG b000000
GF b0000
GE √2b0Fb-√2/2Fx0000
EC b10010Xb/EJ
CE b-1001
totali1/24Fb2/EJ5/3Xb/EJ
iperstatica X=WCE-1/40Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
IPERΣ01.vrga.049PROCEDIMENTO E RISULTATI 843591 Virga Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCA = ∫
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
LXXEB = ∫
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
LXXBE = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCE = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoAC = ∫
o
b(- x/b +2 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [-1/2 x2/b +2/3 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (-1/2 b +2/3 b -1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ
LXoCA = ∫
o
b(1/2 -3/2 x/b +1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx
= [1/2 x -3/4 x2/b +1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (1/2 b -3/4 b +1/6 b +1/8 b ) Fb 1/EJ = 1/24 Fb2/EJ
IPERΣ01.vrga.049PROCEDIMENTO E RISULTATI 843591 Virga Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 456. mm2
Ju = 127247. mm4
Jv = 25560. mm4
yg = 33.58 mmN = -24.25 NTy = 970. NMx = -902100. Nmmxm = 18. mmum = -3. mmvm = -33.58 mmσm = N/A-Mv/Ju = -238.1 N/mm2
xc = 21. mmyc = 14. mmvc = -19.58 mmσc = N/A-Mv/Ju = -138.9 N/mm2
τc = 2.837 N/mm2
σo = √σ2+3τ2 = 138.9 N/mm2
S* = 2233. mm3mm 0 18 24 42x
0
48
52
y
14σc,τc
σm
u
v
IPERΣ01.vrga.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vrga.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vtlm.050REAZIONI 843551 Vitali Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
13/20F
1/2F13/20Fb
13/20F
1/2F
A
B
3/2F
1/2F3/2Fb
1/2F
1/2F1/2Fb
C
D
53/20F
1/2F7/2Fb
53/20F
1/2F17/20Fb
E
C
13/20F
7/2F7/2Fb
13/20F
7/2F
EB
23/20F13/20Fb
23/20F13/20Fb
A C
1/2F
1/2F
Fb
1/2F
1/2F
F
A
1/2F
1/2F
1/2F
1/2F
GF
1/2F
1/2F1/2Fb
1/2F
1/2F
DG
IPERΣ01.vtlm.050AZIONI INTERNE 843551 Vitali Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
-1/2
-1/2
-1/2
-1/2
-13/20
23/20
0
-1/2-1/2 -1/2
F
13/2
0
-3/2
-1/2
-53/
20
7/2
0
-√2/2
-1/21/2
-1/2
F
-13/
200
3/2
1/2
7/2
17/2
0
-7/20
13/20 13/20
1
0
00 1/20
Fb
IPE
RΣ0
1.vt
lm.0
50P
RO
CE
DIM
EN
TO
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TI 8
4355
1 V
itali
Mar
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
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Mila
no, v
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27.0
3.13
03.0
9.18
A
B
C
D E
FG
F
W
X
X
q
q
0 0
3/21/2 7/23/2
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4355
1 V
itali
Mar
tina
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
03.0
9.18
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WC
A
→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
CD
b0
3/2F
b-3/
2Fx+
1/2q
x20
00
0D
C b
0-1
/2F
b-1/
2Fx-
1/2q
x20
0
EC
b-x
/b7/
2Fb-
2Fx
-7/2
Fx+
2Fx2 /b
x2 /b2
-13/
12F
b2 /EJ
1/3X
b/E
JC
E b
1-x/
b-3
/2F
b-2F
x-3
/2F
b-1/
2Fx+
2Fx2 /b
1-2x
/b+
x2 /b2
EB
b0
-7/2
Fb+
7/2F
x0
00
0B
E b
07/
2Fx
00
AC
b1
00
10
Xb/
EJ
CA
b-1
00
1
FA
√2b
0F
b-√2
/2F
x0
00
0
GF
b0
-1/2
Fx+
1/2q
x20
00
0F
G b
01/
2Fx-
1/2q
x20
0
DG
b0
1/2F
b-1/
2Fx
00
00
GD
b0
-1/2
Fx
00
tota
li-1
3/12
Fb2 /E
J5/
3Xb/
EJ
iper
stat
ica
X=
WC
A13
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b
Svi
lupp
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erst
atic
a
IPERΣ01.vtlm.050PROCEDIMENTO E RISULTATI 843551 Vitali Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
LXXAB = ∫
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
LXXBA = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXEC = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXXCE = ∫
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
LXXAC = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXXCA = ∫
o
b(1 ) 1/EJ dx = [ x ]o
b 1/EJ
= ( b ) 1/EJ = b/EJ
LXoEC = ∫
o
b(-7/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [-7/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-7/4 b +2/3 b ) Fb 1/EJ = -13/12 Fb2/EJ
LXoCE = ∫
o
b(-3/2 -1/2 x/b +2 x2/b2 ) Fb 1/EJ dx = [-3/2 x -1/4 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (-3/2 b -1/4 b +2/3 b ) Fb 1/EJ = -13/12 Fb2/EJ
IPERΣ01.vtlm.050PROCEDIMENTO E RISULTATI 843551 Vitali Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
A = 672. mm2
Ju = 147014. mm4
Jv = 62496. mm4
yg = 37.25 mmN = -292.5 NTy = 1575. NMx = -771750. Nmmxm = 18. mmum = -3. mmvm = -37.25 mmσm = N/A-Mv/Ju = -196. N/mm2
xc = 21. mmyc = 16. mmvc = -21.25 mmσc = N/A-Mv/Ju = -112. N/mm2
τc = 5.014 N/mm2
σo = √σ2+3τ2 = 112.3 N/mm2
S* = 2808. mm3mm 0 18 24 42x
0
42
52
y
16σc,τc
σm
u
v
IPERΣ01.vtlm.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
IPERΣ01.vtlm.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 03.09.18
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