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Schema.agsr.001REAZIONI 877947 Agosti Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
3/4F5/4Fb
5/4F
3/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2F
B D
3/2F
1/2F
3/2F
1/2F3/2Fb
D
E
3/2F
1/2F1/2Fb
3/2F
1/2F
EF
3/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.agsr.001AZIONI INTERNE 877947 Agosti Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
3/4
-7/4
1/2
3/2
-3/2
3/23/2
0
F
7/4
-5/4
-1/2
-3/2
1/2
0
-10
1/2
F
0 7/4
5/4
0
1/2 0
0-3
/2
-1/200
0
0-1/2
-1/2
0
Fb
Sch
ema.
agsr
.001
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
947
Ago
sti R
icca
rdo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-3/2
-1/2
0 0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
agsr
.001
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
947
Ago
sti R
icca
rdo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx
-1/2
Fx2 /b
x2 /b2
-1/6
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fb+
1/2F
x-1
/2F
b+F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-1/2
Fb+
1/2F
x0
00
0F
E b
01/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li5/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-5
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.agsr.001PROCEDIMENTO E RISULTATI 877947 Agosti Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb 1/EJ dx = [-1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
A = 540. mm2
Ju = 154030. mm4
Jv = 37908. mm4
yg = 36.6 mmN = 500. NTy = -1500. NMx = -885000. Nmmxm = 18. mmum = -3. mmvm = -36.6 mmσm = N/A-Mv/Ju = -209.4 N/mm2
xc = 21. mmyc = 15. mmvc = -21.6 mmσc = N/A-Mv/Ju = -123.2 N/mm2
τc = 4.251 N/mm2
σo = √σ2+3τ2 = 123.4 N/mm2
S* = 2619. mm3mm 0 18 24 42x
0
48
54
y
15σc,τc
σm
u
v
Schema.agsr.001
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.agsr.001
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.agsr.001
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bldd.002REAZIONI 912028 Baldin Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2F
B D
F
1/2F
F
1/2FFb
D
E
F
1/2F
F
1/2F
EF
1/2F
F
B
FFG
G
A
Schema.bldd.002AZIONI INTERNE 912028 Baldin Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4
-1/2
11
-1/2
1
0
F
5/4
-7/4
1/2
-1
-1/21/2
0
0
0
F
0 5/4
7/4
0
-1/20
0-1000
0
0000
Fb
Sch
ema.
bldd
.002
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 912
028
Bal
din
Dan
iele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-1
/2
0 0-1/2
0
0-1
00 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
bldd
.002
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 912
028
Bal
din
Dan
iele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-1
/2F
x1/
2Fx2 /b
x2 /b2
1/6F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b1/
2Fb-
1/2F
x1/
2Fb-
Fx+
1/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-1/2
Fb+
1/2F
x0
00
0D
B b
01/
2Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-1/2
Fx+
1/2q
x20
00
0F
E b
01/
2Fx-
1/2q
x20
0
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li7/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-7
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.bldd.002PROCEDIMENTO E RISULTATI 912028 Baldin Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o
b Fb 1/EJ
= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
A = 756. mm2
Ju = 165564. mm4
Jv = 74844. mm4
yg = 39. mmN = -725. NTy = -1450. NMx = -928000. Nmmxm = 18. mmum = -3. mmvm = -39. mmσm = N/A-Mv/Ju = -219.6 N/mm2
xc = 21. mmyc = 17. mmvc = -22. mmσc = N/A-Mv/Ju = -124.3 N/mm2
τc = 4.541 N/mm2
σo = √σ2+3τ2 = 124.5 N/mm2
S* = 3111. mm3mm 0 18 24 42x
0
42
54
y
17σc,τc
σm
u
v
Schema.bldd.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bldd.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bldd.002
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bltt.003REAZIONI 893013 Beltran Toledo Italo Jose
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
3/4F
1/2F1/2Fb
3/4F
1/2F
B D
3/2F
1/2F
3/2F
1/2F3/2Fb
D
E
3/2F
1/2F1/2Fb
3/2F
1/2F
EF
1/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.bltt.003AZIONI INTERNE 893013 Beltran Toledo Italo Jose
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
7/4
-3/4
1/2
3/2
-1/2
3/200
F
7/4
-5/4
-1/2
-3/2
1/2
0
0
1/2
-1/2
F
0 7/4
5/4
0
1/2 0
0-3
/2
-1/200
0
0000
Fb
Sch
ema.
bltt.
003
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 893
013
Bel
tran
Tol
edo
Italo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-3/2
-1/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
bltt.
003
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 893
013
Bel
tran
Tol
edo
Italo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx
-1/2
Fx2 /b
x2 /b2
-1/6
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fb+
1/2F
x-1
/2F
b+F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-1/2
Fb+
1/2F
x0
00
0F
E b
01/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li5/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-5
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.bltt.003PROCEDIMENTO E RISULTATI 893013 Beltran Toledo Italo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb 1/EJ dx = [-1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
A = 864. mm2
Ju = 251424. mm4
Jv = 62208. mm4
yg = 33. mmN = 835. NTy = -2505. NMx = -1753500. Nmmxm = 18. mmum = -6. mmvm = -33. mmσm = N/A-Mv/Ju = -229.2 N/mm2
xc = 24. mmyc = 14. mmvc = -19. mmσc = N/A-Mv/Ju = -131.5 N/mm2
τc = 3.627 N/mm2
σo = √σ2+3τ2 = 131.7 N/mm2
S* = 4368. mm3mm 0 18 30 48x
0
48
54
y
14σc,τc
σm
u
v
Schema.bltt.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bltt.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bltt.003
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bnda.004REAZIONI 888197 Bendo Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
1/2F1/2Fb
5/4F
1/2F
B D
2F
1/2F
2F
1/2F2Fb
D
E
2F
1/2FFb
2F
1/2F1/2Fb
EF
F
1/2F1/2Fb
1/2F
F
B
FFG
G
A
Schema.bnda.004AZIONI INTERNE 888197 Bendo Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
1/2
2
-1/2
-1/2
1
0
F
7/4
-5/4
-1/2
-2
1/210
0
0
F
0 7/4
5/4
0
1/2 0
0-2
-1-1/2
-1/2
0
0000
Fb
Sch
ema.
bnda
.004
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 888
197
Ben
do A
less
andr
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
EF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
-1/201/
20
0-2
-1-1
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
bnda
.004
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 888
197
Ben
do A
less
andr
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
00
x2
01/
3Xb3 /E
JB
A b
-b+
x0
0b2 -2
bx+
x2
BC
bb-
x-1
/2F
b+1/
2Fx
-1/2
Fb2 +
Fbx
-1/2
Fx2
b2 -2bx
+x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x1/
2Fx
-1/2
Fx2
x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-Fb+
1/2F
x0
00
0F
E b
01/
2Fb+
1/2F
x0
0
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-7
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VA
7/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.bnda.004PROCEDIMENTO E RISULTATI 888197 Bendo Alessandro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
A = 1080. mm2
Ju = 276955. mm4
Jv = 116640. mm4
yg = 35.4 mmN = 625. NTy = -2500. NMx = -1875000. Nmmxm = 18. mmum = -6. mmvm = -35.4 mmσm = N/A-Mv/Ju = -239.1 N/mm2
xc = 24. mmyc = 15. mmvc = -20.4 mmσc = N/A-Mv/Ju = -137.5 N/mm2
τc = 3.778 N/mm2
σo = √σ2+3τ2 = 137.7 N/mm2
S* = 5022. mm3mm 0 18 30 48x
0
42
54
y
15σc,τc
σm
u
v
Schema.bnda.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bnda.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.bnda.004
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnr.005REAZIONI 868395 Bernasconi Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
1/2F1/2Fb
5/4F
1/2F
B D
2F
1/2F
F
1/2F3/2Fb
D
E
F
1/2F1/2Fb
F
1/2F
EF
1/2F
F
B
FFG
G
A
Schema.brnr.005AZIONI INTERNE 868395 Bernasconi Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
1/2
1/21
-1/2
1
0
F
7/4
-5/4
-1/2
-2-11/2
0
0
0
F
0 7/4
5/4
0
1/2 0
0-3
/2
-1/200
0
0000
Fb
Sch
ema.
brnr
.005
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 868
395
Ber
nasc
oni R
icca
rdo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-3/2
-1/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brnr
.005
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 868
395
Ber
nasc
oni R
icca
rdo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-1/2
Fb+
1/2F
x0
00
0F
E b
01/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brnr.005PROCEDIMENTO E RISULTATI 868395 Bernasconi Riccardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 468. mm2
Ju = 136587. mm4
Jv = 14364. mm4
yg = 34.38 mmN = 325. NTy = -650. NMx = -789750. Nmmxm = 12. mmum = -3. mmvm = -34.38 mmσm = N/A-Mv/Ju = -198.1 N/mm2
xc = 15. mmyc = 14. mmvc = -20.38 mmσc = N/A-Mv/Ju = -117.2 N/mm2
τc = 1.824 N/mm2
σo = √σ2+3τ2 = 117.2 N/mm2
S* = 2300. mm3mm 0 12 18 30x
0
48
54
y
14σc,τc
σm
u
v
Schema.brnr.005
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnr.005
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnr.005
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brss.006REAZIONI 887236 Berselli Samuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
31/16F 15/16F23/16Fb
A B
23/16F
15/16F23/16Fb
23/16F
15/16F
B
C
23/16FB D
F
FFb
D
EF
EF
F
B
FFGG
A
Schema.brss.006AZIONI INTERNE 887236 Berselli Samuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
15/1
6
-23/16
0
1
0
1
0
F
31/16 15/16
-23/
16
0
-1
0
0
0
0
F
0 23/16
23/1
60
0 0
0-1000
0
0000
Fb
Sch
ema.
brss
.006
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 887
236
Ber
selli
Sam
uele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0 000
0-1
00 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brss
.006
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 887
236
Ber
selli
Sam
uele
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx-
1/2q
x2-1
/2F
x2 /b+
1/2q
x3 /bx2 /b
2
-1/2
4Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fx+
1/2q
x2-1
/2F
x+F
x2 /b-1
/2qx
3 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
00
00
0D
B b
00
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
00
00
0F
E b
00
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li23
/24F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-2
3/16
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brss.006PROCEDIMENTO E RISULTATI 887236 Berselli Samuele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
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
LXoBA = ∫
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
A = 612. mm2
Ju = 149427. mm4
Jv = 27756. mm4
yg = 36.88 mmTy = -980. NMx = -842800. Nmmxm = 12. mmum = -3. mmvm = -36.88 mmσm = -Mv/Ju = -208. N/mm2
xc = 15. mmyc = 16. mmvc = -20.88 mmσc = -Mv/Ju = -117.8 N/mm2
τc = 3.031 N/mm2
σo = √σ2+3τ2 = 117.9 N/mm2
S* = 2773. mm3mm 0 12 18 30x
0
42
54
y
16σc,τc
σm
u
v
Schema.brss.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brss.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brss.006
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brmm.007REAZIONI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
F1/2Fb
7/4F
B D
F
FFb
D
EF
EF
F
B
FFG
G
A
Schema.brmm.007AZIONI INTERNE 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4 -7/4
0
1
0
1
0
F
5/4
-7/4
1 0
-1
0
0
0
0
F
0 5/4
7/4
0
-1/20
0-1000
0
0000
Fb
Sch
ema.
brm
m.0
07P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
2983
7 B
orm
olin
i Mat
teo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-1
/2
0 0-1/2
0
0-1
00 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brm
m.0
07P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
2983
7 B
orm
olin
i Mat
teo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-1
/2F
x1/
2Fx2 /b
x2 /b2
1/6F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b1/
2Fb-
1/2F
x1/
2Fb-
Fx+
1/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0D
B b
01/
2qx2
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
00
00
0F
E b
00
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li7/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-7
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brmm.007PROCEDIMENTO E RISULTATI 829837 Bormolini Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o
b Fb 1/EJ
= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
A = 792. mm2
Ju = 225759. mm4
Jv = 30240. mm4
yg = 31.36 mmTy = -1740. NMx = -1583400. Nmmxm = 12. mmum = -6. mmvm = -31.36 mmσm = -Mv/Ju = -220. N/mm2
xc = 18. mmyc = 13. mmvc = -18.36 mmσc = -Mv/Ju = -128.8 N/mm2
τc = 2.491 N/mm2
σo = √σ2+3τ2 = 128.9 N/mm2
S* = 3879. mm3mm 0 12 24 36x
0
48
54
y
13σc,τc
σm
u
v
Schema.brmm.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brmm.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brmm.007
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brsa.008REAZIONI 891317 Borsani Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
23/16F 23/16F23/16Fb
A B
15/16F
23/16F23/16Fb
31/16F
23/16F
B
C
15/16FB D
F
FFb
D
EF
EF
F
B
FFGG
A
Schema.brsa.008AZIONI INTERNE 891317 Borsani Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
23/1
623
/16
-15/16
0
1
0
1
0
F
23/16
-15/
16-3
1/16
0
-1
0
0
0
0
F
0 23/16
23/1
60
0 0
0-1000
0
0000
Fb
Sch
ema.
brsa
.008
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 891
317
Bor
sani
Ale
ssio
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-1
/2
-1/200
0
0-1
00 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brsa
.008
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 891
317
Bor
sani
Ale
ssio
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
-1/2
Fx
-1/2
Fx2
x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JB
A b
-b+
x1/
2Fb-
1/2F
x-1
/2F
b2 +F
bx-1
/2F
x2b2 -2
bx+
x2
BC
bb-
x-1
/2F
b+F
x-1/
2qx2
-1/2
Fb2 +
3/2F
bx-3
/2F
x2 +1/
2qx3
b2 -2bx
+x2
-1/8
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x1/
2qx2
-1/2
qx3
x2
BD
b0
00
00
0D
B b
00
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
00
00
0F
E b
00
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-3
1/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
HC
31/1
6F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brsa.008PROCEDIMENTO E RISULTATI 891317 Borsani Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx = [-1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBC = ∫
o
b(-1/2 +3/2 x/b -3/2 x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [-1/2 x +3/4 x2/b -1/2 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b +3/4 b -1/2 b +1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -9/8 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -9/8 Fb3/EJ
Schema.brsa.008PROCEDIMENTO E RISULTATI 891317 Borsani Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 936. mm2
Ju = 248849. mm4
Jv = 52704. mm4
yg = 33.46 mmTy = -1760. NMx = -1707200. Nmmxm = 12. mmum = -6. mmvm = -33.46 mmσm = -Mv/Ju = -229.6 N/mm2
xc = 18. mmyc = 14. mmvc = -19.46 mmσc = -Mv/Ju = -133.5 N/mm2
τc = 2.62 N/mm2
σo = √σ2+3τ2 = 133.6 N/mm2
S* = 4446. mm3mm 0 12 24 36x
0
42
54
y
14σc,τc
σm
u
v
Schema.brsa.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brsa.008
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnl.009REAZIONI 889394 Brandizi Leonardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
3/4F5/4Fb
5/4F
3/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2F
B D
3/2F
1/2F
3/2F
1/2F3/2Fb
D
E
3/2F
1/2F3/2Fb
3/2F
1/2FFb
EF
3/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.brnl.009AZIONI INTERNE 889394 Brandizi Leonardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
3/4
-7/4
1/2
3/2
-3/2
3/23/2
0
F
7/4
-5/4
-1/2
-3/2
1/2
0
-10
1/2
F
0 7/4
5/4
0
1/2 0
0-3
/2
-3/2-1
00
0-1/2
-1/2
0
Fb
Sch
ema.
brnl
.009
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 889
394
Bra
ndiz
i Leo
nard
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-3/2
-3/2
-1
0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brnl
.009
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 889
394
Bra
ndiz
i Leo
nard
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
1/2F
x0
00
0F
E b
0F
b+1/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brnl.009PROCEDIMENTO E RISULTATI 889394 Brandizi Leonardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 936. mm2
Ju = 248849. mm4
Jv = 52704. mm4
yg = 20.54 mmN = 580. NTy = -1740. NMx = -1774800. Nmmxm = 24. mmym = 54. mmum = 6. mmvm = 33.46 mmσm = N/A-Mv/Ju = 239.3 N/mm2
xc = 18. mmyc = 40. mmvc = 19.46 mmσc = N/A-Mv/Ju = 139.4 N/mm2
τc = 2.59 N/mm2
σo = √σ2+3τ2 = 139.5 N/mm2
S* = 4446. mm3mm 0 12 24 36x
0
12
54
y
40σc,τc
σm
u
v
Schema.brnl.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnl.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brnl.009
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brga.010REAZIONI 881254 Broggi Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2F
B D
F
1/2F
F
1/2FFb
D
E
F
1/2FFb
F
1/2FFb
EF
1/2F
F
B
FFG
G
A
Schema.brga.010AZIONI INTERNE 881254 Broggi Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4
-1/2
11
-1/2
1
0
F
5/4
-7/4
1/2
-1
-1/21/2
0
0
0
F
0 5/4
7/4
0
-1/20
0-1
-1-1
00
0000
Fb
Sch
ema.
brga
.010
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 881
254
Bro
ggi A
ndre
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
E
F
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
1/2 0-1/2
0
0-1
-1-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
brga
.010
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 881
254
Bro
ggi A
ndre
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
00
x2
01/
3Xb3 /E
JB
A b
-b+
x0
0b2 -2
bx+
x2
BC
bb-
x1/
2Fb-
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
CB
b-x
-1/2
Fx
1/2F
x2x2
BD
b0
-1/2
Fb+
1/2F
x0
00
0D
B b
01/
2Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb-
1/2F
x+1/
2qx2
00
00
FE
b0
Fb+
1/2F
x-1/
2qx2
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VA
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.brga.010PROCEDIMENTO E RISULTATI 881254 Broggi Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -5/6 Fb3/EJ
LXoCB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -5/6 Fb3/EJ
A = 468. mm2
Ju = 136587. mm4
Jv = 14364. mm4
yg = 19.62 mmN = -740. NTy = -1480. NMx = -799200. Nmmxm = 18. mmym = 54. mmum = 3. mmvm = 34.38 mmσm = N/A-Mv/Ju = 199.6 N/mm2
xc = 15. mmyc = 40. mmvc = 20.38 mmσc = N/A-Mv/Ju = 117.7 N/mm2
τc = 4.154 N/mm2
σo = √σ2+3τ2 = 117.9 N/mm2
S* = 2300. mm3mm 0 12 18 30x
0
6
54
y
40σc,τc
σm
u
v
Schema.brga.010
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brga.010
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.brga.010
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.crlm.011REAZIONI 893348 Carlino Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
3/4F
1/2F1/2Fb
3/4F
1/2F
B D
3/2F
1/2F
3/2F
1/2F3/2Fb
D
E
3/2F
1/2F3/2Fb
3/2F
1/2FFb
EF
1/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.crlm.011AZIONI INTERNE 893348 Carlino Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
7/4
-3/4
1/2
3/2
-1/2
3/200
F
7/4
-5/4
-1/2
-3/2
1/2
0
0
1/2
-1/2
F
0 7/4
5/4
0
1/2 0
0-3
/2
-3/2-1
00
0000
Fb
Sch
ema.
crlm
.011
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 893
348
Car
lino
Mau
ro
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
DE
F
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
05/
2
2 01/2
0
0-3/2
-3/2
-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
crlm
.011
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 893
348
Car
lino
Mau
ro
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HD
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
5/2F
x-5
/2F
x2x2
-5/6
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-5/2
Fb+
5/2F
x-5
/2F
b2 +5F
bx-5
/2F
x2b2 -2
bx+
x2
BC
b-b
+x
2Fb-
2Fx
-2F
b2 +4F
bx-2
Fx2
b2 -2bx
+x2
-2/3
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-2
Fx
-2F
x2x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
1/2F
x0
00
0F
E b
0F
b+1/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li-1
/2F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HD
3/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.crlm.011PROCEDIMENTO E RISULTATI 893348 Carlino Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-5/2 x2/b2 ) Fb2 1/EJ dx = [-5/6 x3/b2 ]o
b Fb2 1/EJ
= (-5/6 b ) Fb2 1/EJ = -5/6 Fb3/EJ
LXoBA = ∫
o
b(-5/2 +5 x/b -5/2 x2/b2 ) Fb2 1/EJ dx = [-5/2 x +5/2 x2/b -5/6 x3/b2 ]o
b Fb2 1/EJ
= (-5/2 b +5/2 b -5/6 b ) Fb2 1/EJ = -5/6 Fb3/EJ
LXoBC = ∫
o
b(-2 +4 x/b -2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-2 x +2 x2/b -2/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-2 b +2 b -2/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/3 Fb3/EJ
LXoCB = ∫
o
b(-2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-2/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-2/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/3 Fb3/EJ
Schema.crlm.011PROCEDIMENTO E RISULTATI 893348 Carlino Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 864. mm2
Ju = 251424. mm4
Jv = 62208. mm4
yg = 21. mmN = 895. NTy = -2685. NMx = -1584150. Nmmxm = 30. mmym = 54. mmum = 6. mmvm = 33. mmσm = N/A-Mv/Ju = 209. N/mm2
xc = 24. mmyc = 40. mmvc = 19. mmσc = N/A-Mv/Ju = 120.7 N/mm2
τc = 3.887 N/mm2
σo = √σ2+3τ2 = 120.9 N/mm2
S* = 4368. mm3mm 0 18 30 48x
0
6
54
y
40σc,τc
σm
u
v
Schema.crlm.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.crlm.011
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.csrf.012REAZIONI 879105 Caserta Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
1/2F1/2Fb
5/4F
1/2F
B D
2F
1/2F
2F
1/2F2Fb
D
E
2F
1/2F2Fb
2F
1/2F3/2Fb
EF
F
1/2F1/2Fb
1/2F
F
B
FFG
G
A
Schema.csrf.012AZIONI INTERNE 879105 Caserta Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
1/2
2
-1/2
-1/2
1
0
F
7/4
-5/4
-1/2
-2
1/210
0
0
F
0 7/4
5/4
0
1/2 0
0-2
-2-3/2
-1/2
0
0000
Fb
Sch
ema.
csrf
.012
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 879
105
Cas
erta
Fra
nces
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-2
-2-3
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
csrf
.012
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 879
105
Cas
erta
Fra
nces
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-2F
b+1/
2Fx
00
00
FE
b0
3/2F
b+1/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.csrf.012PROCEDIMENTO E RISULTATI 879105 Caserta Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 612. mm2
Ju = 149428. mm4
Jv = 27756. mm4
yg = 17.12 mmN = 345. NTy = -1380. NMx = -883200. Nmmxm = 18. mmym = 54. mmum = 3. mmvm = 36.88 mmσm = N/A-Mv/Ju = 218.6 N/mm2
xc = 15. mmyc = 38. mmvc = 20.88 mmσc = N/A-Mv/Ju = 124. N/mm2
τc = 4.268 N/mm2
σo = √σ2+3τ2 = 124.2 N/mm2
S* = 2773. mm3mm 0 12 18 30x
0
12
54
y
38σc,τc
σm
u
v
Schema.csrf.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.csrf.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.csrf.012
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.cste.013REAZIONI 877793 Castiglione Ettore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
1/2F1/2Fb
5/4F
1/2F
B D
2F
1/2F
F
1/2F3/2Fb
D
E
F
1/2F3/2Fb
F
1/2FFb
EF
1/2F
F
B
FFG
G
A
Schema.cste.013AZIONI INTERNE 877793 Castiglione Ettore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
1/2
1/21
-1/2
1
0
F
7/4
-5/4
-1/2
-2-11/2
0
0
0
F
0 7/4
5/4
0
1/2 0
0-3
/2
-3/2-1
00
0000
Fb
Sch
ema.
cste
.013
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
793
Cas
tiglio
ne E
ttore
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
0
0-3/2
-3/2
-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
cste
.013
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
793
Cas
tiglio
ne E
ttore
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-3/2
Fb+
1/2F
x0
00
0F
E b
0F
b+1/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.cste.013PROCEDIMENTO E RISULTATI 877793 Castiglione Ettore
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 756. mm2
Ju = 165564. mm4
Jv = 74844. mm4
yg = 15. mmN = 460. NTy = -920. NMx = -966000. Nmmxm = 24. mmym = 54. mmum = 3. mmvm = 39. mmσm = N/A-Mv/Ju = 228.2 N/mm2
xc = 21. mmyc = 37. mmvc = 22. mmσc = N/A-Mv/Ju = 129. N/mm2
τc = 2.881 N/mm2
σo = √σ2+3τ2 = 129.1 N/mm2
S* = 3111. mm3mm 0 18 24 42x
0
12
54
y
37σc,τc
σm
u
v
Schema.cste.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.cste.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.cste.013
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.crra.014REAZIONI 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
31/16F 15/16F23/16Fb
A B
23/16F
15/16F23/16Fb
23/16F
15/16F
B
C
23/16FB D
F
FFb
D
E
FFb
FFb
EF
F
B
FFGG
A
Schema.crra.014AZIONI INTERNE 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
15/1
6
-23/16
0
1
0
1
0
F
31/16 15/16
-23/
16
0
-1
0
0
0
0
F
0 23/16
23/1
60
0 0
0-1
-1-1
00
0000
Fb
Sch
ema.
crra
.014
PR
OC
ED
IME
NT
O E
RIS
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I 914
406
Cor
radi
no A
ndre
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0 000
0-1
-1-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
crra
.014
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 914
406
Cor
radi
no A
ndre
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx-
1/2q
x2-1
/2F
x2 /b+
1/2q
x3 /bx2 /b
2
-1/2
4Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fx+
1/2q
x2-1
/2F
x+F
x2 /b-1
/2qx
3 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
00
00
0D
B b
00
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb
00
00
FE
b0
Fb
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li23
/24F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-2
3/16
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.crra.014PROCEDIMENTO E RISULTATI 914406 Corradino Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
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
LXoBA = ∫
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
A = 792. mm2
Ju = 225759. mm4
Jv = 30240. mm4
yg = 22.64 mmTy = -2300. NMx = -1725000. Nmmxm = 24. mmym = 54. mmum = 6. mmvm = 31.36 mmσm = -Mv/Ju = 239.6 N/mm2
xc = 18. mmyc = 41. mmvc = 18.36 mmσc = -Mv/Ju = 140.3 N/mm2
τc = 3.293 N/mm2
σo = √σ2+3τ2 = 140.4 N/mm2
S* = 3879. mm3mm 0 12 24 36x
0
6
54
y
41σc,τc
σm
u
v
Schema.crra.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.crra.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.crra.014
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dlsf.015REAZIONI 772937 D’Alessio Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
F1/2Fb
7/4F
B D
F
FFb
D
E
FFb
FFb
EF
F
B
FFG
G
A
Schema.dlsf.015AZIONI INTERNE 772937 D’Alessio Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4 -7/4
0
1
0
1
0
F
5/4
-7/4
1 0
-1
0
0
0
0
F
0 5/4
7/4
0
-1/20
0-1
-1-1
00
0000
Fb
Sch
ema.
dlsf
.015
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 772
937
D’A
less
io F
ranc
esca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-1
/2
0 0-1/2
0
0-1
-1-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
dlsf
.015
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 772
937
D’A
less
io F
ranc
esca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-1
/2F
x1/
2Fx2 /b
x2 /b2
1/6F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b1/
2Fb-
1/2F
x1/
2Fb-
Fx+
1/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0D
B b
01/
2qx2
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb
00
00
FE
b0
Fb
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li7/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-7
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.dlsf.015PROCEDIMENTO E RISULTATI 772937 D’Alessio Francesca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb 1/EJ dx = [1/6 x3/b2 ]o
b Fb 1/EJ
= (1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb 1/EJ = 1/6 Fb2/EJ
A = 1080. mm2
Ju = 276955. mm4
Jv = 116640. mm4
yg = 18.6 mmTy = -1930. NMx = -1563300. Nmmxm = 30. mmym = 54. mmum = 6. mmvm = 35.4 mmσm = -Mv/Ju = 199.8 N/mm2
xc = 24. mmyc = 39. mmvc = 20.4 mmσc = -Mv/Ju = 115.2 N/mm2
τc = 2.916 N/mm2
σo = √σ2+3τ2 = 115.3 N/mm2
S* = 5022. mm3mm 0 18 30 48x
0
12
54
y
39σc,τc
σm
u
v
Schema.dlsf.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dlsf.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dlsf.015
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dnsg.016REAZIONI 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
23/16F 23/16F23/16Fb
A B
15/16F
23/16F23/16Fb
31/16F
23/16F
B
C
15/16FB D
F
FFb
D
E
FFb
FFb
EF
F
B
FFGG
A
Schema.dnsg.016AZIONI INTERNE 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
23/1
623
/16
-15/16
0
1
0
1
0
F
23/16
-15/
16-3
1/16
0
-1
0
0
0
0
F
0 23/16
23/1
60
0 0
0-1
-1-1
00
0000
Fb
Sch
ema.
dnsg
.016
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 845
411
Dan
esi G
abrie
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
E
F
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0 000
0-1
-1-1
0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
dnsg
.016
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 845
411
Dan
esi G
abrie
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
00
x2
01/
3Xb3 /E
JB
A b
b-x
00
b2 -2bx
+x2
BC
b-b
+x
1/2F
x-1/
2qx2
-1/2
Fbx
+F
x2 -1/2
qx3
b2 -2bx
+x2
-1/2
4Fb3 /E
J1/
3Xb3 /E
JC
B b
x-1
/2F
x+1/
2qx2
-1/2
Fx2 +
1/2q
x3x2
BD
b0
00
00
0D
B b
00
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb
00
00
FE
b0
Fb
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li23
/24F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VC
-23/
16F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.dnsg.016PROCEDIMENTO E RISULTATI 845411 Danesi Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(-1/2 x/b + x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-1/4 x2/b +1/3 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/4 b +1/3 b -1/8 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 23/24 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-1/6 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/6 b +1/8 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 23/24 Fb3/EJ
A = 540. mm2
Ju = 154030. mm4
Jv = 37908. mm4
yg = 17.4 mmTy = -1020. NMx = -877200. Nmmxm = 24. mmym = 54. mmum = 3. mmvm = 36.6 mmσm = -Mv/Ju = 208.4 N/mm2
xc = 21. mmyc = 39. mmvc = 21.6 mmσc = -Mv/Ju = 123. N/mm2
τc = 2.891 N/mm2
σo = √σ2+3τ2 = 123.1 N/mm2
S* = 2619. mm3mm 0 18 24 42x
0
6
54
y
39σc,τc
σm
u
v
Schema.dnsg.016
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dnsg.016
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dnsg.016
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.grts.017REAZIONI 917200 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
5/2F
1/2F
5/2F5/2Fb
A B
2F
3/2FFb
2F
3/2FFb
B
C
5/2F
3/2F3/2Fb
5/2F
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
5/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.grts.017AZIONI INTERNE 917200 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
3/2
-5/2
3/2
3/2
-5/2
3/23/2
0
F
5/2
-2
-3/2
-3/2
3/2
0
-10
1/2
F
0 5/2
1-1
3/2 0
0-3
/2
-3/200
0
0-1/2
-1/2
0
Fb
Sch
ema.
grts
.017
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 917
200
Grit
cul S
ergh
ei
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
03/
2
0-1
3/2
0
0-3/2
-3/2
0 0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
grts
.017
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 917
200
Grit
cul S
ergh
ei
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b3/
2Fx
-3/2
Fx2 /b
x2 /b2
-1/2
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-3/2
Fb+
3/2F
x-3
/2F
b+3F
x-3/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b-F
xF
x-F
x2 /b1-
2x/b
+x2 /b
2
1/6F
b2 /EJ
1/3X
b/E
JC
B b
x/b
Fb-
Fx
Fx-
Fx2 /b
x2 /b2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li2/
3Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.grts.017PROCEDIMENTO E RISULTATI 917200 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(-3/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ
LXoBA = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb 1/EJ dx = [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb 1/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb 1/EJ = -1/2 Fb2/EJ
LXoBC = ∫
o
b( x/b - x2/b2 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [1/2 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (1/2 b -1/3 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 7/6 Fb2/EJ
LXoCB = ∫
o
b( x/b - x2/b2 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [1/2 x2/b -1/3 x3/b2 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (1/2 b -1/3 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 7/6 Fb2/EJ
Schema.grts.017PROCEDIMENTO E RISULTATI 917200 Gritcul Serghei
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 498. mm2
Ju = 141019. mm4
Jv = 31734. mm4
yg = 35.17 mmN = 975. NTy = -975. NMx = -877500. Nmmxm = 18. mmum = -3. mmvm = -35.17 mmσm = N/A-Mv/Ju = -216.9 N/mm2
xc = 21. mmyc = 15. mmvc = -20.17 mmσc = N/A-Mv/Ju = -123.6 N/mm2
τc = 2.87 N/mm2
σo = √σ2+3τ2 = 123.7 N/mm2
S* = 2491. mm3mm 0 18 24 42x
0
48
53
y
15σc,τc
σm
u
v
Schema.grts.017
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.grts.017
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dmcm.018REAZIONI 870485 D’Amico Michele Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
2F 2F2Fb
A B
5/2F
F3/2Fb
5/2F
FFb
B
C
5/2F
1/2F1/2Fb
5/2F
1/2F
B D
F
1/2F
F
1/2FFb
D
E
F
1/2FFb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.dmcm.018AZIONI INTERNE 870485 D’Amico Michele Francesco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1
-5/2
1/2
11
-3/2
1
0
F
2
-5/2
-1/2
-1
1/23/2
0
0
0
F
0 2
3/2
-1
1/2 0
0-1
-100
0
0000
Fb
Sch
ema.
dmcm
.018
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 870
485
D’A
mic
o M
iche
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-1
/2
-1 -1
1/2
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
dmcm
.018
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 870
485
D’A
mic
o M
iche
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
-1/2
Fx
-1/2
Fx2
x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JB
A b
-b+
x1/
2Fb-
1/2F
x-1
/2F
b2 +F
bx-1
/2F
x2b2 -2
bx+
x2
BC
bb-
x-F
b-F
b2 +F
bxb2 -2
bx+
x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JC
B b
-xF
b-F
bxx2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
1/2F
x+1/
2qx2
00
00
FE
b0
3/2F
x-1/
2qx2
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.dmcm.018PROCEDIMENTO E RISULTATI 870485 D’Amico Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx = [-1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBC = ∫
o
b(-1 + x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [- x +1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/E
= (- b +1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
LXoCB = ∫
o
b(- x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
Schema.dmcm.018PROCEDIMENTO E RISULTATI 870485 D’Amico Michele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 714. mm2
Ju = 156210. mm4
Jv = 68670. mm4
yg = 38.15 mmN = 495. NTy = -990. NMx = -940500. Nmmxm = 18. mmum = -3. mmvm = -38.15 mmσm = N/A-Mv/Ju = -229. N/mm2
xc = 21. mmyc = 16. mmvc = -22.15 mmσc = N/A-Mv/Ju = -132.6 N/mm2
τc = 3.057 N/mm2
σo = √σ2+3τ2 = 132.8 N/mm2
S* = 2894. mm3mm 0 18 24 42x
0
42
53
y
16σc,τc
σm
u
v
Schema.dmcm.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dmcm.018
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dcml.019REAZIONI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
5/2F
1/2F
5/2F5/2Fb
A B
2F
5/2FFb
2F
5/2FFb
B
C
3/2F
3/2F3/2Fb
3/2F
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
3/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.dcml.019AZIONI INTERNE 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
5/2
-3/2
3/2
3/2
-3/2
3/200
F
5/2
-2
-3/2
-3/2
3/2
0
0
1/2
-1/2
F
0 5/2
1-1
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
dcm
l.019
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
976
Di C
amill
o Lo
renz
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
-1 -1
3/2
0
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
dcm
l.019
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 877
976
Di C
amill
o Lo
renz
o
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x-F
b-F
b2 +F
bxb2 -2
bx+
x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JC
B b
-xF
b-F
bxx2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-4
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b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.dcml.019PROCEDIMENTO E RISULTATI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBC = ∫
o
b(-1 + x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [- x +1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/E
= (- b +1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
LXoCB = ∫
o
b(- x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
Schema.dcml.019PROCEDIMENTO E RISULTATI 877976 Di Camillo Lorenzo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 816. mm2
Ju = 230061. mm4
Jv = 52992. mm4
yg = 31.79 mmN = -1740. NTy = -1740. NMx = 1740000. Nmmxm = 18. mmum = -6. mmvm = -31.79 mmσm = N/A-Mv/Ju = 238.3 N/mm2
xc = 24. mmyc = 13. mmvc = -18.79 mmσc = N/A-Mv/Ju = 140. N/mm2
τc = 2.487 N/mm2
σo = √σ2+3τ2 = 140.1 N/mm2
S* = 3946. mm3mm 0 18 30 48x
0
48
53
y
13σc,τc
σm
u
v
Schema.dcml.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.dcml.019
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.lbtt.020REAZIONI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/2F 5/2F5/2Fb
A B
2F
5/2FFb
2F
5/2FFb
B
C
2F
3/2F3/2Fb
2F
3/2F
B D
2F
3/2F
2F
3/2F2Fb
D
E
2F
3/2F2Fb
2F
3/2F1/2Fb
EF
F
3/2F1/2Fb
3/2F
F
B
FFG
G
A
Schema.lbtt.020AZIONI INTERNE 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
5/2
-2
3/2
2
-3/2
-3/2
1
0
F
5/2
-2
-3/2
-2
3/210
0
0
F
0 5/2
1-1
3/2 0
0-2
-2-1/2
-1/2
0
0000
Fb
Sch
ema.
lbtt.
020
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 850
296
Laba
te T
omm
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
-1 -1
3/2
0
0-2
-2-1
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
lbtt.
020
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 850
296
Laba
te T
omm
aso
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x-F
b-F
b2 +F
bxb2 -2
bx+
x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JC
B b
-xF
b-F
bxx2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-2F
b+3/
2Fx
00
00
FE
b0
1/2F
b+3/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-4
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.lbtt.020PROCEDIMENTO E RISULTATI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBC = ∫
o
b(-1 + x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [- x +1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/E
= (- b +1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
LXoCB = ∫
o
b(- x/b ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/2 x2/b ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -3/2 Fb3/EJ
Schema.lbtt.020PROCEDIMENTO E RISULTATI 850296 Labate Tommaso
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 1032. mm2
Ju = 260495. mm4
Jv = 107424. mm4
yg = 34.56 mmN = 2145. NTy = -2860. NMx = -1515800. Nmmxm = 18. mmum = -6. mmvm = -34.56 mmσm = N/A-Mv/Ju = -199. N/mm2
xc = 24. mmyc = 14. mmvc = -20.56 mmσc = N/A-Mv/Ju = -117.5 N/mm2
τc = 4.236 N/mm2
σo = √σ2+3τ2 = 117.8 N/mm2
S* = 4630. mm3mm 0 18 30 48x
0
42
53
y
14σc,τc
σm
u
v
Schema.lbtt.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.lbtt.020
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.lgml.021REAZIONI 835162 Ligammari Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/2F 5/2F5/2Fb
A B
2F
5/2FFb
2F
5/2FFb
B
C
2F
3/2F3/2Fb
2F
3/2F
B D
2F
3/2F
F
3/2F3/2Fb
D
E
F
3/2F3/2Fb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.lgml.021AZIONI INTERNE 835162 Ligammari Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
5/2
-2
3/2
3/21
-3/2
1
0
F
5/2
-2
-3/2
-2-13/2
0
0
0
F
0 5/2
1-1
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
lgm
l.021
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 835
162
Liga
mm
ari L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
-3/2 -1
3/2
0
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
lgm
l.021
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 835
162
Liga
mm
ari L
uca
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
00
x2
01/
3Xb3 /E
JB
A b
b-x
00
b2 -2bx
+x2
BC
b-b
+x
-3/2
Fb+
1/2F
x3/
2Fb2 -2
Fbx
+1/
2Fx2
b2 -2bx
+x2
2/3F
b3 /EJ
1/3X
b3 /EJ
CB
bx
Fb+
1/2F
xF
bx+
1/2F
x2x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li5/
3Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-5/2
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.lgml.021PROCEDIMENTO E RISULTATI 835162 Ligammari Luca
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(3/2 -2 x/b +1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [3/2 x - x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (3/2 b - b +1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 5/3 Fb3/EJ
LXoCB = ∫
o
b( x/b +1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b +1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 5/3 Fb3/EJ
A = 438. mm2
Ju = 124871. mm4
Jv = 12114. mm4
yg = 33.08 mmN = 920. NTy = 1380. NMx = -800400. Nmmxm = 12. mmum = -3. mmvm = -33.08 mmσm = N/A-Mv/Ju = -209.9 N/mm2
xc = 15. mmyc = 14. mmvc = -19.08 mmσc = N/A-Mv/Ju = -120.2 N/mm2
τc = 4.034 N/mm2
σo = √σ2+3τ2 = 120.4 N/mm2
S* = 2190. mm3mm 0 12 18 30x
0
48
53
y
14σc,τc
σm
u
v
Schema.lgml.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.lgml.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.lgml.021
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mnrs.022REAZIONI 878035 Manrique Silvera Gerald
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
43/16F 27/16F35/16Fb
A B
35/16F
27/16F19/16Fb
35/16F
27/16FFb
B
C
35/16F
FFb
35/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.mnrs.022AZIONI INTERNE 878035 Manrique Silvera Gerald
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
27/1
6
-35/16
1
1
-1
1
0
F
43/16 27/16
-35/
16
-1
-1
1
0
0
0
F
0 35/16
19/1
6-1
1 0
0-1
-100
0
0000
Fb
Sch
ema.
mnr
s.02
2P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7803
5 M
anriq
ue S
ilver
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
-1/2 -1
10
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mnr
s.02
2P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
7803
5 M
anriq
ue S
ilver
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
Fx-
1/2q
x2-F
x2 +1/
2qx3
x2
-5/2
4Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-1/2
Fb+
1/2q
x2-1
/2F
b2 +1/
2Fbx
+1/
2Fx2 -1
/2qx
3b2 -2
bx+
x2
BC
b-b
+x
-1/2
Fb-
1/2F
x1/
2Fb2 -1
/2F
x2b2 -2
bx+
x2
1/3F
b3 /EJ
1/3X
b3 /EJ
CB
bx
Fb-
1/2F
xF
bx-1
/2F
x2x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li9/
8Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-27/
16F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mnrs.022PROCEDIMENTO E RISULTATI 878035 Manrique Silvera
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(- x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx = [-1/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ
= (-1/3 b +1/8 b ) Fb2 1/EJ = -5/24 Fb3/EJ
LXoBA = ∫
o
b(-1/2 +1/2 x/b +1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx
= [-1/2 x +1/4 x2/b +1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ
= (-1/2 b +1/4 b +1/6 b -1/8 b ) Fb2 1/EJ = -5/24 Fb3/EJ
LXoBC = ∫
o
b(1/2 -1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x -1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 4/3 Fb3/EJ
LXoCB = ∫
o
b( x/b -1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 4/3 Fb3/EJ
Schema.mnrs.022PROCEDIMENTO E RISULTATI 878035 Manrique Silvera
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 582. mm2
Ju = 140714. mm4
Jv = 25506. mm4
yg = 36.03 mmN = 1370. NTy = -1370. NMx = -863100. Nmmxm = 12. mmum = -3. mmvm = -36.03 mmσm = N/A-Mv/Ju = -218.6 N/mm2
xc = 15. mmyc = 15. mmvc = -21.03 mmσc = N/A-Mv/Ju = -126.6 N/mm2
τc = 4.166 N/mm2
σo = √σ2+3τ2 = 126.8 N/mm2
S* = 2567. mm3mm 0 12 18 30x
0
42
53
y
15σc,τc
σm
u
v
Schema.mnrs.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mnrs.022
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mrna.023REAZIONI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
2F 2F2Fb
A B
5/2F
F3/2Fb
5/2F
FFb
B
C
5/2F1/2Fb
5/2F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.mrna.023AZIONI INTERNE 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1
-5/2 -5/2
1
1
-1
1
0
F
2
-5/2
0-1
-1
1
0
0
0
F
0 2
3/2
-1
1/2 0
0-1
-100
0
0000
Fb
Sch
ema.
mrn
a.02
3P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3547
7 M
aran
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@ A
dolfo
Zav
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si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01
1/2-1
1/2
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mrn
a.02
3P
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CE
DIM
EN
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ISU
LTA
TI 8
3547
7 M
aran
ga A
ndre
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
Fx
-Fx2
x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-Fb+
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
BC
b-b
+x
1/2F
b-3/
2Fx
-1/2
Fb2 +
2Fbx
-3/2
Fx2
b2 -2bx
+x2
01/
3Xb3 /E
JC
B b
xF
b-3/
2Fx
Fbx
-3/2
Fx2
x2
BD
b0
1/2F
b-1/
2qx2
00
00
DB
b0
-Fx+
1/2q
x20
0
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li2/
3Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mrna.023PROCEDIMENTO E RISULTATI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx = [-1/3 x3/b2 ]o
b Fb2 1/EJ
= (-1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBA = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx = [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ
= (- b + b -1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBC = ∫
o
b(-1/2 +2 x/b -3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-1/2 x + x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/2 b + b -1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = Fb3/EJ
LXoCB = ∫
o
b( x/b -3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = Fb3/EJ
Schema.mrna.023PROCEDIMENTO E RISULTATI 835477 Maranga Andrea
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 756. mm2
Ju = 207276. mm4
Jv = 26352. mm4
yg = 30.31 mmN = 2340. NTy = -2340. NMx = -1591200. Nmmxm = 12. mmum = -6. mmvm = -30.31 mmσm = N/A-Mv/Ju = -229.6 N/mm2
xc = 18. mmyc = 47. mmvc = 16.69 mmσc = N/A-Mv/Ju = 131.2 N/mm2
τc = 3.613 N/mm2
σo = √σ2+3τ2 = 131.4 N/mm2
S* = 3841. mm3mm 0 12 24 36x
0
48
53
y
47σc,τc
σm
u
v
Schema.mrna.023
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mrna.023
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mrtm.024REAZIONI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
35/16F 35/16F35/16Fb
A B
27/16F
35/16F19/16Fb
43/16F
35/16FFb
B
C
27/16F
FFb
27/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.mrtm.024AZIONI INTERNE 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
35/1
635
/16
-27/16
1
1
-1
1
0
F
35/16
-27/
16-4
3/16
-1
-1
1
0
0
0
F
0 35/16
19/1
6-1
1 0
0-1
-100
0
0000
Fb
Sch
ema.
mrt
m.0
24P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4621
9 M
artig
noni
Mat
teo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
DEF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
03/
2
1/2-1
10
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mrt
m.0
24P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4621
9 M
artig
noni
Mat
teo
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HD
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
3/2F
x-3
/2F
x2x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-3/2
Fb+
3/2F
x-3
/2F
b2 +3F
bx-3
/2F
x2b2 -2
bx+
x2
BC
b-b
+x
1/2F
b-F
x-1/
2qx2
-1/2
Fb2 +
3/2F
bx-1
/2F
x2 -1/2
qx3
b2 -2bx
+x2
-1/2
4Fb3 /E
J1/
3Xb3 /E
JC
B b
xF
b-2F
x+1/
2qx2
Fbx
-2F
x2 +1/
2qx3
x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li11
/24F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HD
-11/
16F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mrtm.024PROCEDIMENTO E RISULTATI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-3/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
LXoBA = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb2 1/EJ dx = [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
LXoBC = ∫
o
b(-1/2 +3/2 x/b -1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-1/2 x +3/4 x2/b -1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/2 b +3/4 b -1/6 b -1/8 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 23/24 Fb3/EJ
LXoCB = ∫
o
b( x/b -2 x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x2/b -2/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b -2/3 b +1/8 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 23/24 Fb3/EJ
Schema.mrtm.024PROCEDIMENTO E RISULTATI 846219 Martignoni Matteo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 900. mm2
Ju = 233812. mm4
Jv = 48816. mm4
yg = 32.66 mmN = 2340. NTy = -2340. NMx = -1731600. Nmmxm = 12. mmum = -6. mmvm = -32.66 mmσm = N/A-Mv/Ju = -239.3 N/mm2
xc = 18. mmyc = 14. mmvc = -18.66 mmσc = N/A-Mv/Ju = -135.6 N/mm2
τc = 3.595 N/mm2
σo = √σ2+3τ2 = 135.7 N/mm2
S* = 4311. mm3mm 0 12 24 36x
0
42
53
y
14σc,τc
σm
u
v
Schema.mrtm.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mrtm.024
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mccg.025REAZIONI 867110 Mecacci Giorgia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
FFb
1/2F
F2Fb
A B
1/2F1/2Fb
1/2F
B
C
F
3/2F3/2Fb
F
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
5/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.mccg.025AZIONI INTERNE 867110 Mecacci Giorgia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
0
-1
3/2
3/2
-5/2
3/23/2
0
F
1
-1/2
-3/2
-3/2
3/2
0
-10
1/2
F
1 2
1/2
0
3/2 0
0-3
/2
-3/200
0
0-1/2
-1/2
0
Fb
Sch
ema.
mcc
g.02
5P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6711
0 M
ecac
ci G
iorg
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
EF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
11
-1/203/
20
0-3/2
-3/2
0 0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mcc
g.02
5P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
6711
0 M
ecac
ci G
iorg
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb
Fbx
x2
1/2F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-Fb
Fb2 -F
bxb2 -2
bx+
x2
BC
bb-
x-1
/2F
b+1/
2Fx
-1/2
Fb2 +
Fbx
-1/2
Fx2
b2 -2bx
+x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x1/
2Fx
-1/2
Fx2
x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-2
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VA
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mccg.025PROCEDIMENTO E RISULTATI 867110 Mecacci Giorgia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b ) Fb2 1/EJ dx = [1/2 x2/b ]o
b Fb2 1/EJ
= (1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBA = ∫
o
b(1 - x/b ) Fb2 1/EJ dx = [ x -1/2 x2/b ]o
b Fb2 1/EJ
= ( b -1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBC = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
Schema.mccg.025PROCEDIMENTO E RISULTATI 867110 Mecacci Giorgia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 924. mm2
Ju = 235641. mm4
Jv = 52560. mm4
yg = 20.11 mmN = 1785. NTy = -1785. NMx = -1410150. Nmmxm = 24. mmym = 53. mmum = 6. mmvm = 32.89 mmσm = N/A-Mv/Ju = 198.8 N/mm2
xc = 18. mmyc = 39. mmvc = 18.89 mmσc = N/A-Mv/Ju = 115. N/mm2
τc = 2.746 N/mm2
σo = √σ2+3τ2 = 115.1 N/mm2
S* = 4349. mm3mm 0 12 24 36x
0
12
53
y
39σc,τc
σm
u
v
Schema.mccg.025
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mccg.025
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mssf.026REAZIONI 889516 Messa Federico
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2FFb
1/2F3/2Fb
A B
F
1/2FFb
F
1/2F
B
C
F
1/2F1/2Fb
F
1/2F
B D
F
1/2F
F
1/2FFb
D
E
F
1/2FFb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.mssf.026AZIONI INTERNE 889516 Messa Federico
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
-1/2
-1
1/2
11
-3/2
1
0
F
1/2
-1
-1/2
-1
1/23/2
0
0
0
F
1 3/2
10
1/2 0
0-1
-100
0
0000
Fb
Sch
ema.
mss
f.026
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 889
516
Mes
sa F
eder
ico
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
EF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
11
1/2 01/2
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mss
f.026
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 889
516
Mes
sa F
eder
ico
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb
Fbx
x2
1/2F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-Fb
Fb2 -F
bxb2 -2
bx+
x2
BC
bb-
x1/
2Fb-
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
CB
b-x
-1/2
Fx
1/2F
x2x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
1/2F
x+1/
2qx2
00
00
FE
b0
3/2F
x-1/
2qx2
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-1
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VA
1/2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mssf.026PROCEDIMENTO E RISULTATI 889516 Messa Federico
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b ) Fb2 1/EJ dx = [1/2 x2/b ]o
b Fb2 1/EJ
= (1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBA = ∫
o
b(1 - x/b ) Fb2 1/EJ dx = [ x -1/2 x2/b ]o
b Fb2 1/EJ
= ( b -1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBC = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -5/6 Fb3/EJ
LXoCB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -5/6 Fb3/EJ
Schema.mssf.026PROCEDIMENTO E RISULTATI 889516 Messa Federico
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 462. mm2
Ju = 129608. mm4
Jv = 14346. mm4
yg = 19.18 mmN = 475. NTy = -950. NMx = -798000. Nmmxm = 18. mmym = 53. mmum = 3. mmvm = 33.82 mmσm = N/A-Mv/Ju = 209.3 N/mm2
xc = 15. mmyc = 39. mmvc = 19.82 mmσc = N/A-Mv/Ju = 123.1 N/mm2
τc = 2.753 N/mm2
σo = √σ2+3τ2 = 123.2 N/mm2
S* = 2253. mm3mm 0 12 18 30x
0
6
53
y
39σc,τc
σm
u
v
Schema.mssf.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mssf.026
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mnta.027REAZIONI 887897 Montagner Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
FFb
1/2F
F2Fb
A B
1/2F
F1/2Fb
1/2F
F
B
C
3/2F3/2Fb
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
3/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.mnta.027AZIONI INTERNE 887897 Montagner Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
1
0
3/2
3/2
-3/2
3/200
F
1
-1/2
-3/2
-3/2
3/2
0
0
1/2
-1/2
F
1 2
1/2
0
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
mnt
a.02
7P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
8789
7 M
onta
gner
Ale
ssio
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
13/
2
0 03/2
0
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mnt
a.02
7P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
8789
7 M
onta
gner
Ale
ssio
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb+
1/2F
xF
bx+
1/2F
x2x2
2/3F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-3/2
Fb+
1/2F
x3/
2Fb2 -2
Fbx
+1/
2Fx2
b2 -2bx
+x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-1
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
1/2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mnta.027PROCEDIMENTO E RISULTATI 887897 Montagner Alessio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b +1/6 b ) Fb2 1/EJ = 2/3 Fb3/EJ
LXoBA = ∫
o
b(3/2 -2 x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [3/2 x - x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (3/2 b - b +1/6 b ) Fb2 1/EJ = 2/3 Fb3/EJ
A = 852. mm2
Ju = 238569. mm4
Jv = 62064. mm4
yg = 20.54 mmN = 1770. NTy = -1770. NMx = -1593000. Nmmxm = 30. mmym = 53. mmum = 6. mmvm = 32.46 mmσm = N/A-Mv/Ju = 218.8 N/mm2
xc = 24. mmyc = 39. mmvc = 18.46 mmσc = N/A-Mv/Ju = 125.3 N/mm2
τc = 2.644 N/mm2
σo = √σ2+3τ2 = 125.4 N/mm2
S* = 4277. mm3mm 0 18 30 48x
0
6
53
y
39σc,τc
σm
u
v
Schema.mnta.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mnta.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mnta.027
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mntm.028REAZIONI 844334 Montemarano Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
FFb
F2Fb
A B
1/2F
F1/2Fb
1/2F
F
B
C
1/2F
3/2F3/2Fb
1/2F
3/2F
B D
2F
3/2F
2F
3/2F2Fb
D
E
2F
3/2F2Fb
2F
3/2F1/2Fb
EF
F
3/2F1/2Fb
3/2F
F
B
FFG
G
A
Schema.mntm.028AZIONI INTERNE 844334 Montemarano Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1
-1/2
3/2
2
-3/2
-3/2
1
0
F
1
-1/2
-3/2
-2
3/210
0
0
F
1 2
1/2
0
3/2 0
0-2
-2-1/2
-1/2
0
0000
Fb
Sch
ema.
mnt
m.0
28P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
4433
4 M
onte
mar
ano
Mau
ro
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
13/
2
0 03/2
0
0-2
-2-1
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mnt
m.0
28P
RO
CE
DIM
EN
TO
E R
ISU
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TI 8
4433
4 M
onte
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ano
Mau
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/bF
b+1/
2Fx
-Fx-
1/2F
x2 /bx2 /b
2
-2/3
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-3/2
Fb+
1/2F
x-3
/2F
b+2F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-2F
b+3/
2Fx
00
00
FE
b0
1/2F
b+3/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li1/
3Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-1
/2F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mntm.028PROCEDIMENTO E RISULTATI 844334 Montemarano Mauro
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(- x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b -1/6 b ) Fb 1/EJ = -2/3 Fb2/EJ
LXoBA = ∫
o
b(-3/2 +2 x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-3/2 x + x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-3/2 b + b -1/6 b ) Fb 1/EJ = -2/3 Fb2/EJ
A = 606. mm2
Ju = 141406. mm4
Jv = 27738. mm4
yg = 16.76 mmN = 690. NTy = -920. NMx = -874000. Nmmxm = 18. mmym = 53. mmum = 3. mmvm = 36.24 mmσm = N/A-Mv/Ju = 225.1 N/mm2
xc = 15. mmyc = 38. mmvc = 21.24 mmσc = N/A-Mv/Ju = 132.4 N/mm2
τc = 2.805 N/mm2
σo = √σ2+3τ2 = 132.5 N/mm2
S* = 2587. mm3mm 0 12 18 30x
0
12
53
y
38σc,τc
σm
u
v
Schema.mntm.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mntm.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mntm.028
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mttv.029REAZIONI 853842 Motta Valentina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
FFb
F2Fb
A B
1/2F
F1/2Fb
1/2F
F
B
C
1/2F
3/2F3/2Fb
1/2F
3/2F
B D
2F
3/2F
F
3/2F3/2Fb
D
E
F
3/2F3/2Fb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.mttv.029AZIONI INTERNE 853842 Motta Valentina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1
-1/2
3/2
3/21
-3/2
1
0
F
1
-1/2
-3/2
-2-13/2
0
0
0
F
1 2
1/2
0
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
mttv
.029
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
842
Mot
ta V
alen
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@ A
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Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
13/
2
0 03/2
0
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
mttv
.029
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
842
Mot
ta V
alen
tina
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb+
1/2F
xF
bx+
1/2F
x2x2
2/3F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-3/2
Fb+
1/2F
x3/
2Fb2 -2
Fbx
+1/
2Fx2
b2 -2bx
+x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-1
/3F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
1/2F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.mttv.029PROCEDIMENTO E RISULTATI 853842 Motta Valentina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b +1/6 b ) Fb2 1/EJ = 2/3 Fb3/EJ
LXoBA = ∫
o
b(3/2 -2 x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [3/2 x - x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (3/2 b - b +1/6 b ) Fb2 1/EJ = 2/3 Fb3/EJ
A = 750. mm2
Ju = 156599. mm4
Jv = 74826. mm4
yg = 14.69 mmN = 650. NTy = 975. NMx = -975000. Nmmxm = 24. mmym = 53. mmum = 3. mmvm = 38.31 mmσm = N/A-Mv/Ju = 239.4 N/mm2
xc = 21. mmyc = 37. mmvc = 22.31 mmσc = N/A-Mv/Ju = 139.8 N/mm2
τc = 3.019 N/mm2
σo = √σ2+3τ2 = 139.9 N/mm2
S* = 2910. mm3mm 0 18 24 42x
0
12
53
y
37σc,τc
σm
u
v
Schema.mttv.029
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mttv.029
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.mttv.029
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.nzza.030REAZIONI 881418 Nuzzo Arianna
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
19/16FFb
3/16F27/16Fb
A B
11/16F
3/16F11/16Fb
11/16F
3/16F
B
C
11/16F
FFb
11/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.nzza.030AZIONI INTERNE 881418 Nuzzo Arianna
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
3/16
-11/16
1
1
-1
1
0
F
19/16 3/16
-11/
16
-1
-1
1
0
0
0
F
1 27/16
11/1
60
1 0
0-1
-100
0
0000
Fb
Sch
ema.
nzza
.030
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 881
418
Nuz
zo A
riann
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
EF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
11/
2
-1/201
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
nzza
.030
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 881
418
Nuz
zo A
riann
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb-
1/2q
x2F
bx-1
/2qx
3x2
3/8F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb-
Fx+
1/2q
x21/
2Fb2 +
1/2F
bx-3
/2F
x2 +1/
2qx3
b2 -2bx
+x2
BC
bb-
x-1
/2F
b+1/
2Fx
-1/2
Fb2 +
Fbx
-1/2
Fx2
b2 -2bx
+x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x1/
2Fx
-1/2
Fx2
x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-1
9/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VA
19/1
6F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.nzza.030PROCEDIMENTO E RISULTATI 881418 Nuzzo Arianna
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b -1/2 x3/b3 ) Fb2 1/EJ dx = [1/2 x2/b -1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/2 b -1/8 b ) Fb2 1/EJ = 3/8 Fb3/EJ
LXoBA = ∫
o
b(1/2 +1/2 x/b -3/2 x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx
= [1/2 x +1/4 x2/b -1/2 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/2 b +1/4 b -1/2 b +1/8 b ) Fb2 1/EJ = 3/8 Fb3/EJ
LXoBC = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
Schema.nzza.030PROCEDIMENTO E RISULTATI 881418 Nuzzo Arianna
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 780. mm2
Ju = 214152. mm4
Jv = 30096. mm4
yg = 22.16 mmN = 2570. NTy = -2570. NMx = -1362100. Nmmxm = 24. mmym = 53. mmum = 6. mmvm = 30.84 mmσm = N/A-Mv/Ju = 199.4 N/mm2
xc = 18. mmyc = 40. mmvc = 17.84 mmσc = N/A-Mv/Ju = 116.8 N/mm2
τc = 3.797 N/mm2
σo = √σ2+3τ2 = 116.9 N/mm2
S* = 3797. mm3mm 0 12 24 36x
0
6
53
y
40σc,τc
σm
u
v
Schema.nzza.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.nzza.030
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ptre.031REAZIONI 888074 Pietrobelli Eleonora
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2FFb
1/2F3/2Fb
A B
F
1/2FFb
F
1/2F
B
C
F1/2Fb
F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.ptre.031AZIONI INTERNE 888074 Pietrobelli Eleonora
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
-1/2
-1 -1
1
1
-1
1
0
F
1/2
-1
0-1
-1
1
0
0
0
F
1 3/2
10
1/2 0
0-1
-100
0
0000
Fb
Sch
ema.
ptre
.031
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 888
074
Pie
trob
elli
Ele
onor
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
11/
2
0 01/2
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
ptre
.031
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 888
074
Pie
trob
elli
Ele
onor
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/bF
b-1/
2Fx
-Fx+
1/2F
x2 /bx2 /b
2
-1/3
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fb-
1/2F
x-1
/2F
b+1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
1/2F
b-1/
2qx2
00
00
DB
b0
-Fx+
1/2q
x20
0
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li2/
3Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.ptre.031PROCEDIMENTO E RISULTATI 888074 Pietrobelli Eleonora
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(- x/b +1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x2/b +1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/6 b ) Fb 1/EJ = -1/3 Fb2/EJ
LXoBA = ∫
o
b(-1/2 +1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x +1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/6 b ) Fb 1/EJ = -1/3 Fb2/EJ
A = 1068. mm2
Ju = 262174. mm4
Jv = 116496. mm4
yg = 18.21 mmN = 2690. NTy = -2690. NMx = -1560200. Nmmxm = 30. mmym = 53. mmum = 6. mmvm = 34.79 mmσm = N/A-Mv/Ju = 209.6 N/mm2
xc = 24. mmyc = 38. mmvc = 19.79 mmσc = N/A-Mv/Ju = 120.3 N/mm2
τc = 4.2 N/mm2
σo = √σ2+3τ2 = 120.5 N/mm2
S* = 4913. mm3mm 0 18 30 48x
0
12
53
y
38σc,τc
σm
u
v
Schema.ptre.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ptre.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ptre.031
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ppvy.032REAZIONI 849430 Popova Yulia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
11/16FFb
11/16F27/16Fb
A B
3/16F
11/16F11/16Fb
19/16F
11/16F
B
C
3/16F
FFb
3/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.ppvy.032AZIONI INTERNE 849430 Popova Yulia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
11/1
611
/16
-3/16
1
1
-1
1
0
F
11/16
-3/1
6-1
9/16
-1
-1
1
0
0
0
F
1 27/16
11/1
60
1 0
0-1
-100
0
0000
Fb
Sch
ema.
ppvy
.032
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 849
430
Pop
ova
Yul
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
AB C
D
EF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
11
0 010
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
ppvy
.032
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 849
430
Pop
ova
Yul
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
Fb
Fbx
x2
1/2F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-Fb
Fb2 -F
bxb2 -2
bx+
x2
BC
bb-
x1/
2Fx-
1/2q
x21/
2Fbx
-Fx2 +
1/2q
x3b2 -2
bx+
x2
1/24
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x-1
/2F
x+1/
2qx2
1/2F
x2 -1/2
qx3
x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-1
1/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VA
11/1
6F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.ppvy.032PROCEDIMENTO E RISULTATI 849430 Popova Yulia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b( x/b ) Fb2 1/EJ dx = [1/2 x2/b ]o
b Fb2 1/EJ
= (1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBA = ∫
o
b(1 - x/b ) Fb2 1/EJ dx = [ x -1/2 x2/b ]o
b Fb2 1/EJ
= ( b -1/2 b ) Fb2 1/EJ = 1/2 Fb3/EJ
LXoBC = ∫
o
b(1/2 x/b - x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/4 x2/b -1/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/4 b -1/3 b +1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -23/24 Fb3/EJ
LXoCB = ∫
o
b(1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/6 b -1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -23/24 Fb3/EJ
Schema.ppvy.032PROCEDIMENTO E RISULTATI 849430 Popova Yulia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 534. mm2
Ju = 146122. mm4
Jv = 37890. mm4
yg = 16.99 mmN = 1400. NTy = -1400. NMx = -882000. Nmmxm = 24. mmym = 53. mmum = 3. mmvm = 36.01 mmσm = N/A-Mv/Ju = 220. N/mm2
xc = 21. mmyc = 38. mmvc = 21.01 mmσc = N/A-Mv/Ju = 129.4 N/mm2
τc = 4.097 N/mm2
σo = √σ2+3τ2 = 129.6 N/mm2
S* = 2566. mm3mm 0 18 24 42x
0
6
53
y
38σc,τc
σm
u
v
Schema.ppvy.032
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ppvy.032
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.pzzm.033REAZIONI 891221 Pozzoni Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
3/4F5/4Fb
5/4F
3/4F
B
C
7/4F
3/2F1/2Fb
7/4F
3/2FFb
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
5/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.pzzm.033AZIONI INTERNE 891221 Pozzoni Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
3/4
-7/4
3/2
3/2
-5/2
3/23/2
0
F
7/4
-5/4
-3/2
-3/2
3/2
0
-10
1/2
F
0 7/4
5/4
0
1/2-1
0-3
/2
-3/200
0
0-1/2
-1/2
0
Fb
Sch
ema.
pzzm
.033
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 891
221
Poz
zoni
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
03/
2
1 01/2
-1
0-3/2
-3/2
0 0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
pzzm
.033
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 891
221
Poz
zoni
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HD
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
3/2F
x-3
/2F
x2x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-3/2
Fb+
3/2F
x-3
/2F
b2 +3F
bx-3
/2F
x2b2 -2
bx+
x2
BC
b-b
+x
Fb-
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-F
x-F
x2x2
BD
b0
1/2F
b-3/
2Fx
00
00
DB
b0
Fb-
3/2F
x0
0
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li1/
6Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
HD
-1/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.pzzm.033PROCEDIMENTO E RISULTATI 891221 Pozzoni Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-3/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
LXoBA = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb2 1/EJ dx = [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
LXoBC = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (- b + b -1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
LXoCB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
Schema.pzzm.033PROCEDIMENTO E RISULTATI 891221 Pozzoni Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 582. mm2
Ju = 166519. mm4
Jv = 44082. mm4
yg = 37.89 mmN = 1425. NTy = -1425. NMx = -1011750. Nmmxm = 18. mmum = -3. mmvm = -37.89 mmσm = N/A-Mv/Ju = -227.8 N/mm2
xc = 21. mmyc = 16. mmvc = -21.89 mmσc = N/A-Mv/Ju = -130.6 N/mm2
τc = 4.093 N/mm2
σo = √σ2+3τ2 = 130.8 N/mm2
S* = 2870. mm3mm 0 18 24 42x
0
48
55
y
16σc,τc
σm
u
v
Schema.pzzm.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.pzzm.033
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.prva.034REAZIONI 853194 Previtali Alessia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2FFb
B D
F
1/2F
F
1/2FFb
D
E
F
1/2FFb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.prva.034AZIONI INTERNE 853194 Previtali Alessia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4
1/2
11
-3/2
1
0
F
5/4
-7/4
-1/2
-1
1/23/2
0
0
0
F
0 5/4
7/4
0
-1/2 -1
0-1
-100
0
0000
Fb
Sch
ema.
prva
.034
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
194
Pre
vita
li A
less
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
1 0-1/2
-1
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
prva
.034
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
194
Pre
vita
li A
less
ia
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HD
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
1/2F
x-1
/2F
x2x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-1/2
Fb+
1/2F
x-1
/2F
b2 +F
bx-1
/2F
x2b2 -2
bx+
x2
BC
b-b
+x
Fb-
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-F
x-F
x2x2
BD
b0
-1/2
Fb-
1/2F
x0
00
0D
B b
0F
b-1/
2Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
1/2F
x+1/
2qx2
00
00
FE
b0
3/2F
x-1/
2qx2
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li1/
2Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
HD
-3/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.prva.034PROCEDIMENTO E RISULTATI 853194 Previtali Alessia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx = [-1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBC = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (- b + b -1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
LXoCB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
Schema.prva.034PROCEDIMENTO E RISULTATI 853194 Previtali Alessia
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 798. mm2
Ju = 175127. mm4
Jv = 81018. mm4
yg = 39.82 mmN = 685. NTy = -1370. NMx = -1054900. Nmmxm = 18. mmum = -3. mmvm = -39.82 mmσm = N/A-Mv/Ju = -239. N/mm2
xc = 21. mmyc = 17. mmvc = -22.82 mmσc = N/A-Mv/Ju = -136.6 N/mm2
τc = 4.165 N/mm2
σo = √σ2+3τ2 = 136.8 N/mm2
S* = 3194. mm3mm 0 18 24 42x
0
42
55
y
17σc,τc
σm
u
v
Schema.prva.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.prva.034
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.prft.035REAZIONI 867435 Prifti Tomas
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
3/4F
3/2F1/2Fb
3/4F
3/2FFb
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
3/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.prft.035AZIONI INTERNE 867435 Prifti Tomas
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
7/4
-3/4
3/2
3/2
-3/2
3/200
F
7/4
-5/4
-3/2
-3/2
3/2
0
0
1/2
-1/2
F
0 7/4
5/4
0
1/2-1
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
prft.
035
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 867
435
Prif
ti T
omas
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
DEF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
-1
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
prft.
035
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 867
435
Prif
ti T
omas
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx
-1/2
Fx2 /b
x2 /b2
-1/6
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fb+
1/2F
x-1
/2F
b+F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
1/2F
b-3/
2Fx
00
00
DB
b0
Fb-
3/2F
x0
0
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li5/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-5
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.prft.035PROCEDIMENTO E RISULTATI 867435 Prifti Tomas
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb 1/EJ dx = [-1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
A = 912. mm2
Ju = 272448. mm4
Jv = 71424. mm4
yg = 34.13 mmN = 1965. NTy = -1965. NMx = -1611300. Nmmxm = 18. mmum = -6. mmvm = -34.13 mmσm = N/A-Mv/Ju = -199.7 N/mm2
xc = 24. mmyc = 14. mmvc = -20.13 mmσc = N/A-Mv/Ju = -116.9 N/mm2
τc = 2.74 N/mm2
σo = √σ2+3τ2 = 117. N/mm2
S* = 4558. mm3mm 0 18 30 48x
0
48
55
y
14σc,τc
σm
u
v
Schema.prft.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.prft.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.prft.035
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rttl.036REAZIONI 892357 Rattaggi Lorenza
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
3/2F1/2Fb
5/4F
3/2FFb
B D
2F
3/2F
2F
3/2F2Fb
D
E
2F
3/2F2Fb
2F
3/2F1/2Fb
EF
F
3/2F1/2Fb
3/2F
F
B
FFG
G
A
Schema.rttl.036AZIONI INTERNE 892357 Rattaggi Lorenza
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
3/2
2
-3/2
-3/2
1
0
F
7/4
-5/4
-3/2
-2
3/210
0
0
F
0 7/4
5/4
0
1/2-1
0-2
-2-1/2
-1/2
0
0000
Fb
Sch
ema.
rttl.
036
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 892
357
Rat
tagg
i Lor
enza
@ A
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elan
i Ros
si, P
olite
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o di
Mila
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ers.
27.0
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9.19
AB C
DEF
G
W
FX
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
-1/201/
2-1
0-2
-2-1
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
rttl.
036
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 892
357
Rat
tagg
i Lor
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
00
x2
01/
3Xb3 /E
JB
A b
-b+
x0
0b2 -2
bx+
x2
BC
bb-
x-1
/2F
b+1/
2Fx
-1/2
Fb2 +
Fbx
-1/2
Fx2
b2 -2bx
+x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x1/
2Fx
-1/2
Fx2
x2
BD
b0
1/2F
b-3/
2Fx
00
00
DB
b0
Fb-
3/2F
x0
0
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-2F
b+3/
2Fx
00
00
FE
b0
1/2F
b+3/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-7
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VA
7/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.rttl.036PROCEDIMENTO E RISULTATI 892357 Rattaggi Lorenza
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ = [-1/6 x3/b2 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (-1/6 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -7/6 Fb3/EJ
A = 1128. mm2
Ju = 293725. mm4
Jv = 125856. mm4
yg = 36.21 mmN = 1455. NTy = -1940. NMx = -1707200. Nmmxm = 18. mmum = -6. mmvm = -36.21 mmσm = N/A-Mv/Ju = -209.2 N/mm2
xc = 24. mmyc = 15. mmvc = -21.21 mmσc = N/A-Mv/Ju = -122. N/mm2
τc = 2.845 N/mm2
σo = √σ2+3τ2 = 122.1 N/mm2
S* = 5168. mm3mm 0 18 30 48x
0
42
55
y
15σc,τc
σm
u
v
Schema.rttl.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rttl.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rttl.036
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rsse.037REAZIONI 917955 Rossi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
3/2F1/2Fb
5/4F
3/2FFb
B D
2F
3/2F
F
3/2F3/2Fb
D
E
F
3/2F3/2Fb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.rsse.037AZIONI INTERNE 917955 Rossi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
3/2
3/21
-3/2
1
0
F
7/4
-5/4
-3/2
-2-13/2
0
0
0
F
0 7/4
5/4
0
1/2-1
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
rsse
.037
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 917
955
Ros
si E
doar
do
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
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06.0
9.19
A
B C
DEF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 01/2
-1
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
rsse
.037
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 917
955
Ros
si E
doar
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
1/2F
b-3/
2Fx
00
00
DB
b0
Fb-
3/2F
x0
0
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.rsse.037PROCEDIMENTO E RISULTATI 917955 Rossi Edoardo
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 498. mm2
Ju = 147997. mm4
Jv = 16614. mm4
yg = 35.6 mmN = 650. NTy = 975. NMx = -906750. Nmmxm = 12. mmum = -3. mmvm = -35.6 mmσm = N/A-Mv/Ju = -216.8 N/mm2
xc = 15. mmyc = 15. mmvc = -20.6 mmσc = N/A-Mv/Ju = -124.9 N/mm2
τc = 2.776 N/mm2
σo = √σ2+3τ2 = 125. N/mm2
S* = 2529. mm3mm 0 12 18 30x
0
48
55
y
15σc,τc
σm
u
v
Schema.rsse.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rsse.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.rsse.037
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.slrm.038REAZIONI 896150 Salari Mehdi
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
31/16F 15/16F23/16Fb
A B
23/16F
15/16F23/16Fb
23/16F
15/16F
B
C
23/16F
F
23/16F
FFb
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.slrm.038AZIONI INTERNE 896150 Salari Mehdi
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
15/1
6
-23/16
1
1
-1
1
0
F
31/16 15/16
-23/
16
-1
-1
1
0
0
0
F
0 23/16
23/1
60
0-1
0-1
-100
0
0000
Fb
Sch
ema.
slrm
.038
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 896
150
Sal
ari M
ehdi
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
DEF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0 00-1
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
slrm
.038
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 896
150
Sal
ari M
ehdi
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x-1/
2qx2
1/2F
x2 -1/2
qx3
x2
1/24
Fb3 /E
J1/
3Xb3 /E
JB
A b
-b+
x-1
/2F
x+1/
2qx2
1/2F
bx-F
x2 +1/
2qx3
b2 -2bx
+x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
-Fx
00
00
DB
b0
Fb-
Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-2
3/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
HC
23/1
6F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.slrm.038PROCEDIMENTO E RISULTATI 896150 Salari Mehdi
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx = [1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/6 b -1/8 b ) Fb2 1/EJ = 1/24 Fb3/EJ
LXoBA = ∫
o
b(1/2 x/b - x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx = [1/4 x2/b -1/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/4 b -1/3 b +1/8 b ) Fb2 1/EJ = 1/24 Fb3/EJ
A = 642. mm2
Ju = 158306. mm4
Jv = 30006. mm4
yg = 37.71 mmN = -1380. NTy = -960. NMx = -950400. Nmmxm = 12. mmum = -3. mmvm = -37.71 mmσm = N/A-Mv/Ju = -228.5 N/mm2
xc = 15. mmyc = 16. mmvc = -21.71 mmσc = N/A-Mv/Ju = -132.5 N/mm2
τc = 2.882 N/mm2
σo = √σ2+3τ2 = 132.6 N/mm2
S* = 2852. mm3mm 0 12 18 30x
0
42
55
y
16σc,τc
σm
u
v
Schema.slrm.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.slrm.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.slrm.038
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.slvg.039REAZIONI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F1/2Fb
7/4F
FFb
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.slvg.039AZIONI INTERNE 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4 -7/4
1
1
-1
1
0
F
5/4
-7/4
0-1
-1
1
0
0
0
F
0 5/4
7/4
0
-1/2 -1
0-1
-100
0
0000
Fb
Sch
ema.
slvg
.039
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 834
781
Sal
vato
ri G
abrie
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
DEF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01
3/2 0-1/2
-1
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
slvg
.039
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 834
781
Sal
vato
ri G
abrie
le
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
Fx
-Fx2
x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-Fb+
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
BC
b-b
+x
3/2F
b-3/
2Fx
-3/2
Fb2 +
3Fbx
-3/2
Fx2
b2 -2bx
+x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-3
/2F
x-3
/2F
x2x2
BD
b0
-1/2
Fb-
1/2q
x20
00
0D
B b
0F
b-F
x+1/
2qx2
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li1/
6Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-1/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.slvg.039PROCEDIMENTO E RISULTATI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx = [-1/3 x3/b2 ]o
b Fb2 1/EJ
= (-1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBA = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx = [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ
= (- b + b -1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBC = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/2 Fb3/EJ
LXoCB = ∫
o
b(-3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/2 Fb3/EJ
Schema.slvg.039PROCEDIMENTO E RISULTATI 834781 Salvatori Gabriele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 828. mm2
Ju = 244195. mm4
Jv = 34128. mm4
yg = 32.37 mmN = -2993. NTy = -1710. NMx = -1778400. Nmmxm = 12. mmum = -6. mmvm = -32.37 mmσm = N/A-Mv/Ju = -239.4 N/mm2
xc = 18. mmyc = 14. mmvc = -18.37 mmσc = N/A-Mv/Ju = -137.4 N/mm2
τc = 2.487 N/mm2
σo = √σ2+3τ2 = 137.5 N/mm2
S* = 4262. mm3mm 0 12 24 36x
0
48
55
y
14σc,τc
σm
u
v
Schema.slvg.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.slvg.039
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.sccm.040REAZIONI 853403 Scoccimarro Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
23/16F 23/16F23/16Fb
A B
15/16F
23/16F23/16Fb
31/16F
23/16F
B
C
15/16F
F
15/16F
FFb
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.sccm.040AZIONI INTERNE 853403 Scoccimarro Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
23/1
623
/16
-15/16
1
1
-1
1
0
F
23/16
-15/
16-3
1/16
-1
-1
1
0
0
0
F
0 23/16
23/1
60
0-1
0-1
-100
0
0000
Fb
Sch
ema.
sccm
.040
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
403
Sco
ccim
arro
Mar
tina
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
DEF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
0 00-1
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
sccm
.040
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 853
403
Sco
ccim
arro
Mar
tina
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b0
0x2 /b
2
01/
3Xb/
EJ
BA
b1-
x/b
00
1-2x
/b+
x2 /b2
BC
b-1
+x/
b1/
2Fx-
1/2q
x2-1
/2F
x+F
x2 /b-1
/2qx
3 /b1-
2x/b
+x2 /b
2
-1/2
4Fb2 /E
J1/
3Xb/
EJ
CB
bx/
b-1
/2F
x+1/
2qx2
-1/2
Fx2 /b
+1/
2qx3 /b
x2 /b2
BD
b0
-Fx
00
00
DB
b0
Fb-
Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li23
/24F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-2
3/16
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.sccm.040PROCEDIMENTO E RISULTATI 853403 Scoccimarro Martina
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoBC = ∫
o
b(-1/2 x/b + x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [-1/4 x2/b +1/3 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 23/24 Fb2/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [-1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (-1/6 b +1/8 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 23/24 Fb2/EJ
A = 972. mm2
Ju = 264196. mm4
Jv = 56592. mm4
yg = 34.24 mmN = -2588. NTy = -2760. NMx = -1518000. Nmmxm = 12. mmum = -6. mmvm = -34.24 mmσm = N/A-Mv/Ju = -199.4 N/mm2
xc = 18. mmyc = 15. mmvc = -19.24 mmσc = N/A-Mv/Ju = -113.2 N/mm2
τc = 4.19 N/mm2
σo = √σ2+3τ2 = 113.4 N/mm2
S* = 4813. mm3mm 0 12 24 36x
0
42
55
y
15σc,τc
σm
u
v
Schema.sccm.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.sccm.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.sccm.040
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.shhe.041REAZIONI 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
3/4F5/4Fb
5/4F
3/4F
B
C
7/4F
3/2F3/2Fb
7/4F
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
5/2F
F
B
3/2F
F
3/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.shhe.041AZIONI INTERNE 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
3/4
-7/4
3/2
3/2
-5/2
3/23/2
0
F
7/4
-5/4
-3/2
-3/2
3/2
0
-10
1/2
F
0 7/4
5/4
0
3/2 0
0-3
/2
-3/200
0
0-1/2
-1/2
0
Fb
Sch
ema.
shhe
.041
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 809
828
She
hu E
lton
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 03/2
0
0-3/2
-3/2
0 0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
shhe
.041
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 809
828
She
hu E
lton
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
x1/
2Fx2
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
BA
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-5
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
5/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.shhe.041PROCEDIMENTO E RISULTATI 809828 Shehu Elton
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx = [1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
LXoBA = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx = [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ = 1/6 Fb3/EJ
A = 948. mm2
Ju = 262515. mm4
Jv = 52848. mm4
yg = 20.97 mmN = -3098. NTy = -2655. NMx = 1593000. Nmmxm = 24. mmym = 55. mmum = 6. mmvm = 34.03 mmσm = N/A-Mv/Ju = -209.8 N/mm2
xc = 18. mmyc = 40. mmvc = 19.03 mmσc = N/A-Mv/Ju = -118.8 N/mm2
τc = 4.025 N/mm2
σo = √σ2+3τ2 = 119. N/mm2
S* = 4776. mm3mm 0 12 24 36x
0
12
55
y
40σc,τc
σm
u
v
Schema.shhe.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.shhe.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.shhe.041
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.smbm.042REAZIONI 886588 Simbula Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F
1/2F1/2Fb
7/4F
1/2F
B D
F
1/2F
F
1/2FFb
D
E
F
1/2FFb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.smbm.042AZIONI INTERNE 886588 Simbula Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4
1/2
11
-3/2
1
0
F
5/4
-7/4
-1/2
-1
1/23/2
0
0
0
F
0 5/4
7/4
0
1/2 0
0-1
-100
0
0000
Fb
Sch
ema.
smbm
.042
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 886
588
Sim
bula
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
DEF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
1 01/2
0
0-1
-10 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
smbm
.042
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 886
588
Sim
bula
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HD
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
1/2F
x-1
/2F
x2x2
-1/6
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-1/2
Fb+
1/2F
x-1
/2F
b2 +F
bx-1
/2F
x2b2 -2
bx+
x2
BC
b-b
+x
Fb-
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-F
x-F
x2x2
BD
b0
1/2F
b-1/
2Fx
00
00
DB
b0
-1/2
Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
1/2F
x+1/
2qx2
00
00
FE
b0
3/2F
x-1/
2qx2
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li1/
2Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
HD
-3/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.smbm.042PROCEDIMENTO E RISULTATI 886588 Simbula Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb2 1/EJ dx = [-1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb2 1/EJ = -1/6 Fb3/EJ
LXoBC = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (- b + b -1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
LXoCB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-1/3 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/3 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 2/3 Fb3/EJ
Schema.smbm.042PROCEDIMENTO E RISULTATI 886588 Simbula Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 474. mm2
Ju = 143796. mm4
Jv = 14382. mm4
yg = 20.06 mmN = 680. NTy = -1360. NMx = -897600. Nmmxm = 18. mmym = 55. mmum = 3. mmvm = 34.94 mmσm = N/A-Mv/Ju = 219.6 N/mm2
xc = 15. mmyc = 40. mmvc = 19.94 mmσc = N/A-Mv/Ju = 125.9 N/mm2
τc = 3.893 N/mm2
σo = √σ2+3τ2 = 126.1 N/mm2
S* = 2470. mm3mm 0 12 18 30x
0
6
55
y
40σc,τc
σm
u
v
Schema.smbm.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.smbm.042
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.stcm.043REAZIONI 886709 Steccanella Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
7/4F
1/2F
7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
3/4F
3/2F3/2Fb
3/4F
3/2F
B D
3/2F
3/2F
3/2F
3/2F3/2Fb
D
E
3/2F
3/2F3/2Fb
3/2F
3/2F
EF
3/2F
F
B
3/2FFG
1/2F
1/2F
G
A
Schema.stcm.043AZIONI INTERNE 886709 Steccanella Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
7/4
-3/4
3/2
3/2
-3/2
3/200
F
7/4
-5/4
-3/2
-3/2
3/2
0
0
1/2
-1/2
F
0 7/4
5/4
0
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
stcm
.043
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 886
709
Ste
ccan
ella
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
-1/203/
20
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
01
1 000
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
stcm
.043
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 886
709
Ste
ccan
ella
Mar
co
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
A
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx/
b0
0x2 /b
2
01/
3Xb/
EJ
BA
b-1
+x/
b0
01-
2x/b
+x2 /b
2
BC
b1-
x/b
-1/2
Fb+
1/2F
x-1
/2F
b+F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
-1/6
Fb2 /E
J1/
3Xb/
EJ
CB
b-x
/b1/
2Fx
-1/2
Fx2 /b
x2 /b2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-3/2
Fx
00
00
ED
b0
3/2F
b-3/
2Fx
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
1/2F
x-1/
2qx2
00
00
AG
b0
-1/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b2 /EJ
tota
li-7
/6F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
A7/
4Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.stcm.043PROCEDIMENTO E RISULTATI 886709 Steccanella Marco
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoBC = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb 1/EJ dx - 1 (-1) (-1) Fb2/EJ
= [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ - 1 (-1) (-1) Fb2/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb 1/EJ - 1 (-1) (-1) Fb2/EJ = -7/6 Fb2/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 ) Fb 1/EJ dx - 1 (-1) (-1) Fb2/EJ = [-1/6 x3/b2 ]o
b Fb 1/EJ - 1 (-1) (-1) Fb2/EJ
= (-1/6 b ) Fb 1/EJ - 1 (-1) (-1) Fb2/EJ = -7/6 Fb2/EJ
A = 876. mm2
Ju = 264708. mm4
Jv = 62352. mm4
yg = 21.46 mmN = 2520. NTy = -2520. NMx = -1789200. Nmmxm = 30. mmym = 55. mmum = 6. mmvm = 33.54 mmσm = N/A-Mv/Ju = 229.6 N/mm2
xc = 24. mmyc = 41. mmvc = 19.54 mmσc = N/A-Mv/Ju = 135. N/mm2
τc = 3.537 N/mm2
σo = √σ2+3τ2 = 135.1 N/mm2
S* = 4459. mm3mm 0 18 30 48x
0
6
55
y
41σc,τc
σm
u
v
Schema.stcm.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.stcm.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.stcm.043
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.srzp.044REAZIONI 919311 Suarez Pena Bryam Abraham
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
3/2F3/2Fb
5/4F
3/2F
B D
2F
3/2F
2F
3/2F2Fb
D
E
2F
3/2F2Fb
2F
3/2F1/2Fb
EF
F
3/2F1/2Fb
3/2F
F
B
FFG
G
A
Schema.srzp.044AZIONI INTERNE 919311 Suarez Pena Bryam Abraham
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
3/2
2
-3/2
-3/2
1
0
F
7/4
-5/4
-3/2
-2
3/210
0
0
F
0 7/4
5/4
0
3/2 0
0-2
-2-1/2
-1/2
0
0000
Fb
Sch
ema.
srzp
.044
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 919
311
Sua
rez
Pen
a B
ryam
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01/
2
0 03/2
0
0-2
-2-1
/2
-1/20
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
srzp
.044
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 919
311
Sua
rez
Pen
a B
ryam
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b1/
2Fx
-1/2
Fx2 /b
x2 /b2
-1/6
Fb2 /E
J1/
3Xb/
EJ
BA
b1-
x/b
-1/2
Fb+
1/2F
x-1
/2F
b+F
x-1/
2Fx2 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x0
00
0E
D b
02F
b-2F
x0
0
EF
b0
-2F
b+3/
2Fx
00
00
FE
b0
1/2F
b+3/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li5/
6Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-5
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.srzp.044PROCEDIMENTO E RISULTATI 919311 Suarez Pena Bryam
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(-1/2 x2/b2 ) Fb 1/EJ dx = [-1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
LXoBA = ∫
o
b(-1/2 + x/b -1/2 x2/b2 ) Fb 1/EJ dx = [-1/2 x +1/2 x2/b -1/6 x3/b2 ]o
b Fb 1/EJ
= (-1/2 b +1/2 b -1/6 b ) Fb 1/EJ = -1/6 Fb2/EJ
A = 618. mm2
Ju = 157731. mm4
Jv = 27774. mm4
yg = 17.48 mmN = 975. NTy = -1300. NMx = -1001000. Nmmxm = 18. mmym = 55. mmum = 3. mmvm = 37.52 mmσm = N/A-Mv/Ju = 239.7 N/mm2
xc = 15. mmyc = 39. mmvc = 21.52 mmσc = N/A-Mv/Ju = 138.1 N/mm2
τc = 3.893 N/mm2
σo = √σ2+3τ2 = 138.3 N/mm2
S* = 2834. mm3mm 0 12 18 30x
0
12
55
y
39σc,τc
σm
u
v
Schema.srzp.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.srzp.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.srzp.044
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.thid.045REAZIONI 868406 Taha Iad
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
7/4F 7/4F7/4Fb
A B
5/4F
7/4F5/4Fb
5/4F
7/4F
B
C
5/4F
3/2F3/2Fb
5/4F
3/2F
B D
2F
3/2F
F
3/2F3/2Fb
D
E
F
3/2F3/2Fb
F
3/2F
EF
3/2F
F
B
FFG
G
A
Schema.thid.045AZIONI INTERNE 868406 Taha Iad
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
7/4
-5/4
3/2
3/21
-3/2
1
0
F
7/4
-5/4
-3/2
-2-13/2
0
0
0
F
0 7/4
5/4
0
3/2 0
0-3
/2
-3/200
0
0000
Fb
Sch
ema.
thid
.045
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 868
406
Tah
a Ia
d
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B
C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
00
-1/203/
20
0-3/2
-3/2
0 0 0
00 0 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
thid
.045
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I 868
406
Tah
a Ia
d
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
00
x2
01/
3Xb3 /E
JB
A b
b-x
00
b2 -2bx
+x2
BC
b-b
+x
-1/2
Fb+
1/2F
x1/
2Fb2 -F
bx+
1/2F
x2b2 -2
bx+
x2
1/6F
b3 /EJ
1/3X
b3 /EJ
CB
bx
1/2F
x1/
2Fx2
x2
BD
b0
3/2F
b-3/
2Fx
00
00
DB
b0
-3/2
Fx
00
DE
b0
-2F
x+1/
2qx2
00
00
ED
b0
3/2F
b-F
x-1/
2qx2
00
EF
b0
-3/2
Fb+
3/2F
x0
00
0F
E b
03/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li7/
6Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-7/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.thid.045PROCEDIMENTO E RISULTATI 868406 Taha Iad
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(1/2 - x/b +1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [1/2 x -1/2 x2/b +1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/2 b -1/2 b +1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 7/6 Fb3/EJ
LXoCB = ∫
o
b(1/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [1/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (1/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 7/6 Fb3/EJ
A = 762. mm2
Ju = 174852. mm4
Jv = 74862. mm4
yg = 15.31 mmN = -887.5 NTy = -1065. NMx = 873300. Nmmxm = 24. mmym = 55. mmum = 3. mmvm = 39.69 mmσm = N/A-Mv/Ju = -199.4 N/mm2
xc = 21. mmyc = 38. mmvc = 22.69 mmσc = N/A-Mv/Ju = -114.5 N/mm2
τc = 3.229 N/mm2
σo = √σ2+3τ2 = 114.6 N/mm2
S* = 3181. mm3mm 0 18 24 42x
0
12
55
y
38σc,τc
σm
u
v
Schema.thid.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.thid.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.thid.045
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.tscr.046REAZIONI 829283 Tascau Radu Alexandru
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
31/16F 15/16F23/16Fb
A B
23/16F
15/16F23/16Fb
23/16F
15/16F
B
C
23/16F
FFb
23/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.tscr.046AZIONI INTERNE 829283 Tascau Radu Alexandru
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0 0
15/1
6
-23/16
1
1
-1
1
0
F
31/16 15/16
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0
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PR
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@ A
dolfo
Zav
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olite
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Mila
no, v
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27.0
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06.0
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PR
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
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cont
ribut
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per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
1/2F
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qx3
x2
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J1/
3Xb3 /E
JB
A b
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x+1/
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bx-F
x2 +1/
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b2 -2bx
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BC
bb-
x0
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bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-2
3/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
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X=
HC
23/1
6F
Svi
lupp
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alco
lo ip
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a
Schema.tscr.046PROCEDIMENTO E RISULTATI 829283 Tascau Radu
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx = [1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/6 b -1/8 b ) Fb2 1/EJ = 1/24 Fb3/EJ
LXoBA = ∫
o
b(1/2 x/b - x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx = [1/4 x2/b -1/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ
= (1/4 b -1/3 b +1/8 b ) Fb2 1/EJ = 1/24 Fb3/EJ
A = 804. mm2
Ju = 237762. mm4
Jv = 30384. mm4
yg = 23.11 mmN = -2516. NTy = -1750. NMx = 1540000. Nmmxm = 24. mmym = 55. mmum = 6. mmvm = 31.89 mmσm = N/A-Mv/Ju = -209.7 N/mm2
xc = 18. mmyc = 41. mmvc = 17.89 mmσc = N/A-Mv/Ju = -119. N/mm2
τc = 2.565 N/mm2
σo = √σ2+3τ2 = 119.1 N/mm2
S* = 4181. mm3mm 0 12 24 36x
0
6
55
y
41σc,τc
σm
u
v
Schema.tscr.046
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.tscr.046
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.tscr.046
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ttrm.047REAZIONI 876617 Totaro Mario
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
5/4F 5/4F5/4Fb
A B
7/4F
1/4F7/4Fb
7/4F
1/4F
B
C
7/4F1/2Fb
7/4F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.ttrm.047AZIONI INTERNE 876617 Totaro Mario
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
1/4
-7/4 -7/4
1
1
-1
1
0
F
5/4
-7/4
0-1
-1
1
0
0
0
F
0 5/4
7/4
0
1/2 0
0-1
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0
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Fb
Sch
ema.
ttrm
.047
PR
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O E
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ULT
AT
I 876
617
Tot
aro
Mar
io
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B
C
D
EF
G
W
F
X
q
Sch
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di c
alco
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01
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Mo
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sseg
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0-1
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0
00
00 0 0
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Mx
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X=
1
Sch
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ttrm
.047
PR
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Tot
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
Fx
-Fx2
x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-Fb+
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
BC
b-b
+x
3/2F
b-3/
2Fx
-3/2
Fb2 +
3Fbx
-3/2
Fx2
b2 -2bx
+x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-3
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x-3
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x2x2
BD
b0
1/2F
b-1/
2qx2
00
00
DB
b0
-Fx+
1/2q
x20
0
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li1/
6Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VC
-1/4
F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.ttrm.047PROCEDIMENTO E RISULTATI 876617 Totaro Mario
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx = [-1/3 x3/b2 ]o
b Fb2 1/EJ
= (-1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBA = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx = [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ
= (- b + b -1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBC = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/2 Fb3/EJ
LXoCB = ∫
o
b(-3/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-1/2 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-1/2 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/2 Fb3/EJ
Schema.ttrm.047PROCEDIMENTO E RISULTATI 876617 Totaro Mario
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 1092. mm2
Ju = 292252. mm4
Jv = 116784. mm4
yg = 18.99 mmN = 1900. NTy = -1900. NMx = -1767000. Nmmxm = 30. mmym = 55. mmum = 6. mmvm = 36.01 mmσm = N/A-Mv/Ju = 219.4 N/mm2
xc = 24. mmyc = 40. mmvc = 21.01 mmσc = N/A-Mv/Ju = 128.7 N/mm2
τc = 2.78 N/mm2
σo = √σ2+3τ2 = 128.8 N/mm2
S* = 5131. mm3mm 0 18 30 48x
0
12
55
y
40σc,τc
σm
u
v
Schema.ttrm.047
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.ttrm.047
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vnnv.048REAZIONI 877057 Vennettilli Vittorio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
23/16F 23/16F23/16Fb
A B
15/16F
23/16F23/16Fb
31/16F
23/16F
B
C
15/16F
FFb
15/16F
F
B D
F
F
F
FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
G
A
Schema.vnnv.048AZIONI INTERNE 877057 Vennettilli Vittorio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
0
23/1
623
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-15/16
1
1
-1
1
0
F
23/16
-15/
16-3
1/16
-1
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1
0
0
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F
0 23/16
23/1
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1 0
0-1
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0
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Fb
Sch
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vnnv
.048
PR
OC
ED
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NT
O E
RIS
ULT
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I 877
057
Ven
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
AB
C
D
EF
G
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FX
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Sch
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di c
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0 010
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01
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00
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Sch
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vnnv
.048
PR
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NT
O E
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I 877
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Ven
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VA
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
00
x2
01/
3Xb3 /E
JB
A b
-b+
x0
0b2 -2
bx+
x2
BC
bb-
x1/
2Fx-
1/2q
x21/
2Fbx
-Fx2 +
1/2q
x3b2 -2
bx+
x2
1/24
Fb3 /E
J1/
3Xb3 /E
JC
B b
-x-1
/2F
x+1/
2qx2
1/2F
x2 -1/2
qx3
x2
BD
b0
Fb-
Fx
00
00
DB
b0
-Fx
00
DE
b0
-Fx
00
00
ED
b0
Fb-
Fx
00
EF
b0
-Fb+
Fx
00
00
FE
b0
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
00
00
0A
G b
00
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-2
3/24
Fb3 /E
J2/
3Xb3 /E
J
iper
stat
ica
X=
VA
23/1
6F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.vnnv.048PROCEDIMENTO E RISULTATI 877057 Vennettilli Vittorio
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoBC = ∫
o
b(1/2 x/b - x2/b2 +1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/4 x2/b -1/3 x3/b2 +1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/4 b -1/3 b +1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -23/24 Fb3/EJ
LXoCB = ∫
o
b(1/2 x2/b2 -1/2 x3/b3 ) Fb2 1/EJ dx - 1 (-1) (-1) Fb3/EJ
= [1/6 x3/b2 -1/8 x4/b3 ]o
b Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ
= (1/6 b -1/8 b ) Fb2 1/EJ - 1 (-1) (-1) Fb3/EJ = -23/24 Fb3/EJ
A = 546. mm2
Ju = 162198. mm4
Jv = 37926. mm4
yg = 17.81 mmN = 1000. NTy = -1000. NMx = -990000. Nmmxm = 24. mmym = 55. mmum = 3. mmvm = 37.19 mmσm = N/A-Mv/Ju = 228.8 N/mm2
xc = 21. mmyc = 39. mmvc = 21.19 mmσc = N/A-Mv/Ju = 131.2 N/mm2
τc = 2.88 N/mm2
σo = √σ2+3τ2 = 131.3 N/mm2
S* = 2802. mm3mm 0 18 24 42x
0
6
55
y
39σc,τc
σm
u
v
Schema.vnnv.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vnnv.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vnnv.048
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vgnd.049REAZIONI 890834 Vignati Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
3/4F
1/2F
3/4F3/4Fb
A B
9/4F
5/4F9/4Fb
9/4F
5/4F
B
C
7/4F
3/2F3/2Fb
7/4F
3/2F
B D
1/2F
3/2F
1/2F
3/2F1/2Fb
D
E
1/2F
1/2F1/2Fb
1/2F
1/2F
EF
1/2F
F
B
1/2F
F
1/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.vgnd.049AZIONI INTERNE 890834 Vignati Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2
-5/4
-7/4
-3/2
-1/2
-1/2
-1/2-1/2
0
F
3/4
-9/4
3/2
1/2
-1/2
0
-10
-1/2
F
0 3/4
9/4
0
-3/20
01/
21/2000
0-1/2
1/2
0
Fb
Sch
ema.
vgnd
.049
PR
OC
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IME
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ULT
AT
I 890
834
Vig
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Dav
ide
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-3
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0 0-3/2
0
01/2
1/2
0 0 0
0-1
/2 1/2 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
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ione
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iper
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ica
X=
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Sch
ema.
vgnd
.049
PR
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I 890
834
Vig
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Dav
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@ A
dolfo
Zav
elan
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si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
/2F
x3/
2Fx2 /b
x2 /b2
1/2F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b3/
2Fb-
3/2F
x3/
2Fb-
3Fx+
3/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-3/2
Fb+
3/2F
x0
00
0D
B b
03/
2Fx
00
DE
b0
1/2F
x0
00
0E
D b
0-1
/2F
b+1/
2Fx
00
EF
b0
1/2F
b-1/
2Fx
00
00
FE
b0
-1/2
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
1/2F
b-1/
2Fx
00
00
AG
b0
-1/2
Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li3/
2Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-9
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.vgnd.049PROCEDIMENTO E RISULTATI 890834 Vignati Davide
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(3/2 x2/b2 ) Fb 1/EJ dx = [1/2 x3/b2 ]o
b Fb 1/EJ
= (1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
LXoBA = ∫
o
b(3/2 -3 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -3/2 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -3/2 b +1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
A = 456. mm2
Ju = 127247. mm4
Jv = 25560. mm4
yg = 33.58 mmN = -1068. NTy = 915. NMx = -896700. Nmmxm = 18. mmum = -3. mmvm = -33.58 mmσm = N/A-Mv/Ju = -239. N/mm2
xc = 21. mmyc = 14. mmvc = -19.58 mmσc = N/A-Mv/Ju = -140.3 N/mm2
τc = 2.676 N/mm2
σo = √σ2+3τ2 = 140.4 N/mm2
S* = 2233. mm3mm 0 18 24 42x
0
48
52
y
14σc,τc
σm
u
v
Schema.vgnd.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vgnd.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vgnd.049
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vrtd.050REAZIONI 898022 Vretenar Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
1/4F
F
1/4F1/4Fb
A B
11/4F
7/4F11/4Fb
11/4F
7/4F
B
C
7/4F
5/2F5/2Fb
7/4F
5/2F
B D
F
5/2F
F
5/2FFb
D
E
F
3/2FFb
F
1/2F
EF
1/2F
F
B
FFG
FFb
F
G
A
Schema.vrtd.050AZIONI INTERNE 898022 Vretenar Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1
-7/4
-7/4
-5/2
-1-1
1/2
-1
0
F
1/4
-11/
4
5/2
1
-3/2-1/2
0
0
-1
F
0 1/4
11/4
0
-5/20
01100
0
0010
Fb
Sch
ema.
vrtd
.050
PR
OC
ED
IME
NT
O E
RIS
ULT
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I 898
022
Vre
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-5
/2
0 0-5/2
0
01
10 0 0
00 1 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
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ione
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iper
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ica
X=
1
Sch
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PR
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Vre
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@ A
dolfo
Zav
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si, P
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cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-5
/2F
x5/
2Fx2 /b
x2 /b2
5/6F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b5/
2Fb-
5/2F
x5/
2Fb-
5Fx+
5/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-5/2
Fb+
5/2F
x0
00
0D
B b
05/
2Fx
00
DE
b0
Fx
00
00
ED
b0
-Fb+
Fx
00
EF
b0
Fb-
3/2F
x+1/
2qx2
00
00
FE
b0
-1/2
Fx-
1/2q
x20
0
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li11
/6F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-1
1/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.vrtd.050PROCEDIMENTO E RISULTATI 898022 Vretenar Daniele
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(5/2 x2/b2 ) Fb 1/EJ dx = [5/6 x3/b2 ]o
b Fb 1/EJ
= (5/6 b ) Fb 1/EJ = 5/6 Fb2/EJ
LXoBA = ∫
o
b(5/2 -5 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [5/2 x -5/2 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (5/2 b -5/2 b +5/6 b ) Fb 1/EJ = 5/6 Fb2/EJ
A = 672. mm2
Ju = 147014. mm4
Jv = 62496. mm4
yg = 37.25 mmN = -1050. NTy = 1500. NMx = -780000. Nmmxm = 18. mmum = -3. mmvm = -37.25 mmσm = N/A-Mv/Ju = -199.2 N/mm2
xc = 21. mmyc = 16. mmvc = -21.25 mmσc = N/A-Mv/Ju = -114.3 N/mm2
τc = 4.775 N/mm2
σo = √σ2+3τ2 = 114.6 N/mm2
S* = 2808. mm3mm 0 18 24 42x
0
42
52
y
16σc,τc
σm
u
v
Schema.vrtd.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vrtd.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.vrtd.050
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.wxng.051REAZIONI 832033 Wu Xiangyu
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
3/2F
3/4F
3/2F
3/4F3/4Fb
A B
9/4F
1/4F9/4Fb
9/4F
1/4F
B
C
3/4F
3/2F3/2Fb
3/4F
3/2F
B D
1/2F
3/2F
1/2F
3/2F1/2Fb
D
E
1/2F
1/2F1/2Fb
1/2F
1/2F
EF
1/2F
F
B
1/2FFG
1/2FFb
3/2F
G
A
Schema.wxng.051AZIONI INTERNE 832033 Wu Xiangyu
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
3/2
-1/4
-3/4
-3/2
-1/2
1/2
-1/2
00
F
3/4
-9/4
3/2
1/2
-1/2
0
0
-1/2
-3/2
F
0 3/4
9/4
0
-3/20
01/
21/2000
0010
Fb
Sch
ema.
wxn
g.05
1P
RO
CE
DIM
EN
TO
E R
ISU
LTA
TI 8
3203
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q Sch
ema
di c
alco
lo ip
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atic
o
0-3
/2
0 0-3/2
0
01/2
1/2
0 0 0
00 1 0 M
o fle
ssio
ne d
a ca
richi
ass
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ti
01
1 000
00
00 0 0
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x fle
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atic
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Sch
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
HC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
bx
-3/2
Fx
-3/2
Fx2
x2
-1/2
Fb3 /E
J1/
3Xb3 /E
JB
A b
-b+
x3/
2Fb-
3/2F
x-3
/2F
b2 +3F
bx-3
/2F
x2b2 -2
bx+
x2
BC
bb-
x0
0b2 -2
bx+
x2
01/
3Xb3 /E
JC
B b
-x0
0x2
BD
b0
-3/2
Fb+
3/2F
x0
00
0D
B b
03/
2Fx
00
DE
b0
1/2F
x0
00
0E
D b
0-1
/2F
b+1/
2Fx
00
EF
b0
1/2F
b-1/
2Fx
00
00
FE
b0
-1/2
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
1/2F
x-1/
2qx2
00
00
AG
b0
-3/2
Fx+
1/2q
x20
0
BC
elon
gazi
one
asta
N1B
Cε B
CL B
C-F
b3 /EJ
tota
li-3
/2F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
HC
9/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.wxng.051PROCEDIMENTO E RISULTATI 832033 Wu Xiangyu
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(-3/2 x2/b2 ) Fb2 1/EJ dx = [-1/2 x3/b2 ]o
b Fb2 1/EJ
= (-1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
LXoBA = ∫
o
b(-3/2 +3 x/b -3/2 x2/b2 ) Fb2 1/EJ dx = [-3/2 x +3/2 x2/b -1/2 x3/b2 ]o
b Fb2 1/EJ
= (-3/2 b +3/2 b -1/2 b ) Fb2 1/EJ = -1/2 Fb3/EJ
A = 768. mm2
Ju = 208192. mm4
Jv = 43776. mm4
yg = 30.5 mmN = -1245. NTy = 2490. NMx = -1419300. Nmmxm = 18. mmum = -6. mmvm = -30.5 mmσm = N/A-Mv/Ju = -209.5 N/mm2
xc = 24. mmyc = 13. mmvc = -17.5 mmσc = N/A-Mv/Ju = -120.9 N/mm2
τc = 3.732 N/mm2
σo = √σ2+3τ2 = 121.1 N/mm2
S* = 3744. mm3mm 0 18 30 48x
0
48
52
y
13σc,τc
σm
u
v
Schema.wxng.051
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.wxng.051
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.wxng.051
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.zhny.052REAZIONI 846641 Zhan Yuchen
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
3/4F
F
3/4F3/4Fb
A B
9/4F
1/4F9/4Fb
9/4F
1/4F
B
C
5/4F
3/2F3/2Fb
5/4F
3/2F
B D
3/2F
D
E
1/2F1/2F1/2Fb
EF
F
1/2F1/2Fb
1/2F
F
B
FFG
FFb
F
G
A
Schema.zhny.052AZIONI INTERNE 846641 Zhan Yuchen
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1
-1/4
-5/4
-3/2
0
1/2
1/2
-1
0
F
3/4
-9/4
3/2
0
-1/2
10
0
-1
F
0 3/4
9/4
0
-3/20
000
-1/2
-1/2
0
0010
Fb
Sch
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zhny
.052
PR
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O E
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I 846
641
Zha
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@ A
dolfo
Zav
elan
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si, P
olite
cnic
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Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
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F
X
X
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Sch
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Sch
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PR
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I 846
641
Zha
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@ A
dolfo
Zav
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i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
/2F
x3/
2Fx2 /b
x2 /b2
1/2F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b3/
2Fb-
3/2F
x3/
2Fb-
3Fx+
3/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-3/2
Fb+
3/2F
x0
00
0D
B b
03/
2Fx
00
DE
b0
00
00
0E
D b
00
00
EF
b0
-1/2
Fx
00
00
FE
b0
1/2F
b-1/
2Fx
00
FB
b0
-1/2
Fb+
Fx-
1/2q
x20
00
0B
F b
01/
2qx2
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li3/
2Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-9
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.zhny.052PROCEDIMENTO E RISULTATI 846641 Zhan Yuchen
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(3/2 x2/b2 ) Fb 1/EJ dx = [1/2 x3/b2 ]o
b Fb 1/EJ
= (1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
LXoBA = ∫
o
b(3/2 -3 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -3/2 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -3/2 b +1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
A = 984. mm2
Ju = 244285. mm4
Jv = 98208. mm4
yg = 33.68 mmN = -2113. NTy = 2535. NMx = -1571700. Nmmxm = 18. mmum = -6. mmvm = -33.68 mmσm = N/A-Mv/Ju = -218.9 N/mm2
xc = 24. mmyc = 14. mmvc = -19.68 mmσc = N/A-Mv/Ju = -128.8 N/mm2
τc = 3.877 N/mm2
σo = √σ2+3τ2 = 129. N/mm2
S* = 4483. mm3mm 0 18 30 48x
0
42
52
y
14σc,τc
σm
u
v
Schema.zhny.052
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.zhny.052
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.zhny.052
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.zlln.053REAZIONI 878043 Zullo Nicola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
3/4F
F
3/4F3/4Fb
A B
9/4F
1/4F9/4Fb
9/4F
1/4F
B
C
5/4F
3/2F3/2Fb
5/4F
3/2F
B D
3/2F
F
3/2F1/2Fb
D
E
F
1/2F1/2Fb
F
1/2F
EF
1/2F
F
B
FFG
FFb
F
G
A
Schema.zlln.053AZIONI INTERNE 878043 Zullo Nicola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1
-1/4
-5/4
-3/2
-3/2
-1
1/2
-1
0
F
3/4
-9/4
3/2
01
-1/2
0
0
-1
F
0 3/4
9/4
0
-3/20
01/
21/2000
0010
Fb
Sch
ema.
zlln
.053
PR
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ED
IME
NT
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I 878
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Zul
lo N
icol
a
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
01
5/2 0-3/2
0
01/2
1/2
0 0 0
00 1 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
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iper
stat
ica
X=
1
Sch
ema.
zlln
.053
PR
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ED
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NT
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Zul
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@ A
dolfo
Zav
elan
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si, P
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cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
VC
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
Fx
-Fx2
x2
-1/3
Fb3 /E
J1/
3Xb3 /E
JB
A b
b-x
-Fb+
Fx
-Fb2 +
2Fbx
-Fx2
b2 -2bx
+x2
BC
b-b
+x
5/2F
b-5/
2Fx
-5/2
Fb2 +
5Fbx
-5/2
Fx2
b2 -2bx
+x2
-5/6
Fb3 /E
J1/
3Xb3 /E
JC
B b
x-5
/2F
x-5
/2F
x2x2
BD
b0
-3/2
Fb+
3/2F
x0
00
0D
B b
03/
2Fx
00
DE
b0
1/2q
x20
00
0E
D b
0-1
/2F
b+F
x-1/
2qx2
00
EF
b0
1/2F
b-1/
2Fx
00
00
FE
b0
-1/2
Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b3 /EJ
tota
li-1
/6F
b3 /EJ
2/3X
b3 /EJ
iper
stat
ica
X=
VC
1/4F
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.zlln.053PROCEDIMENTO E RISULTATI 878043 Zullo Nicola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
LXXAB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBA = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXBC = ∫
o
b(1 -2 x/b + x2/b2 ) b2 1/EJ dx = [ x - x2/b +1/3 x3/b2 ]o
b b2 1/EJ
= ( b - b +1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXXCB = ∫
o
b( x2/b2 ) b2 1/EJ dx = [1/3 x3/b2 ]o
b b2 1/EJ
= (1/3 b ) b2 1/EJ = 1/3 b3/EJ
LXoAB = ∫
o
b(- x2/b2 ) Fb2 1/EJ dx = [-1/3 x3/b2 ]o
b Fb2 1/EJ
= (-1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBA = ∫
o
b(-1 +2 x/b - x2/b2 ) Fb2 1/EJ dx = [- x + x2/b -1/3 x3/b2 ]o
b Fb2 1/EJ
= (- b + b -1/3 b ) Fb2 1/EJ = -1/3 Fb3/EJ
LXoBC = ∫
o
b(-5/2 +5 x/b -5/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ
= [-5/2 x +5/2 x2/b -5/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-5/2 b +5/2 b -5/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/6 Fb3/EJ
LXoCB = ∫
o
b(-5/2 x2/b2 ) Fb2 1/EJ dx + 1 (-1) (-1) Fb3/EJ = [-5/6 x3/b2 ]o
b Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ
= (-5/6 b ) Fb2 1/EJ + 1 (-1) (-1) Fb3/EJ = 1/6 Fb3/EJ
Schema.zlln.053PROCEDIMENTO E RISULTATI 878043 Zullo Nicola
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 408. mm2
Ju = 112717. mm4
Jv = 9864. mm4
yg = 31.65 mmN = -1000. NTy = 1200. NMx = -804000. Nmmxm = 12. mmum = -3. mmvm = -31.65 mmσm = N/A-Mv/Ju = -228.2 N/mm2
xc = 15. mmyc = 13. mmvc = -18.65 mmσc = N/A-Mv/Ju = -135.5 N/mm2
τc = 3.48 N/mm2
σo = √σ2+3τ2 = 135.6 N/mm2
S* = 1961. mm3mm 0 12 18 30x
0
48
52
y
13σc,τc
σm
u
v
Schema.zlln.053
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.zlln.053
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.054REAZIONI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
15/16F
F
1/16F7/16Fb
A B
39/16F
17/16F39/16Fb
39/16F
17/16F
B
C
23/16F
2F2Fb
23/16F
2F
B D
F
2F
F
2FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
FFb
F
G
A
Schema.xxxx.054AZIONI INTERNE Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1 1
-17/
16
-23/16
-2
-1
1
-1
0
F
15/16-1/16
-39/
16
2
1
-1
0
0
-1
F
0 7/16
39/1
60
-20
01100
0
0010
Fb
Sch
ema.
xxxx
.054
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
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27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
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0-2
0 0-20
01
10 0 0
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Mo
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caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
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Mx
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ione
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ica
X=
1
Sch
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.054
PR
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O E
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
/2F
x-1/
2qx2
3/2F
x2 /b+
1/2q
x3 /bx2 /b
2
5/8F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b2F
b-5/
2Fx+
1/2q
x22F
b-9/
2Fx+
3Fx2 /b
-1/2
qx3 /b
1-2x
/b+
x2 /b2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-2F
b+2F
x0
00
0D
B b
02F
x0
0
DE
b0
Fx
00
00
ED
b0
-Fb+
Fx
00
EF
b0
Fb-
Fx
00
00
FE
b0
-Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li13
/8F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-3
9/16
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.xxxx.054PROCEDIMENTO E RISULTATI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(3/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx = [1/2 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ
= (1/2 b +1/8 b ) Fb 1/EJ = 5/8 Fb2/EJ
LXoBA = ∫
o
b(2 -9/2 x/b +3 x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx = [2 x -9/4 x2/b + x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ
= (2 b -9/4 b + b -1/8 b ) Fb 1/EJ = 5/8 Fb2/EJ
A = 552. mm2
Ju = 132127. mm4
Jv = 23256. mm4
yg = 35.13 mmN = -891.3 NTy = 1240. NMx = -892800. Nmmxm = 12. mmum = -3. mmvm = -35.13 mmσm = N/A-Mv/Ju = -239. N/mm2
xc = 15. mmyc = 15. mmvc = -20.13 mmσc = N/A-Mv/Ju = -137.6 N/mm2
τc = 3.89 N/mm2
σo = √σ2+3τ2 = 137.8 N/mm2
S* = 2487. mm3mm 0 12 18 30x
0
42
52
y
15σc,τc
σm
u
v
Schema.xxxx.054
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.054
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.054
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.055REAZIONI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
1/4F
F
1/4F1/4Fb
A B
11/4F
7/4F11/4Fb
11/4F
7/4F
B
C
7/4F
3F5/2Fb
7/4F
2F
B D
F
2F
F
2FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
FFb
F
G
A
Schema.xxxx.055AZIONI INTERNE Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1
-7/4
-7/4 -7/4
-2
-1
1
-1
0
F
1/4
-11/
4
3 2
1
-1
0
0
-1
F
0 1/4
11/4
0
-5/20
01100
0
0010
Fb
Sch
ema.
xxxx
.055
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
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@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
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atic
o
0-5
/2
0 0-5/2
0
01
10 0 0
00 1 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
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.055
PR
OC
ED
IME
NT
O E
RIS
ULT
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I Nom
e:
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-5
/2F
x5/
2Fx2 /b
x2 /b2
5/6F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b5/
2Fb-
5/2F
x5/
2Fb-
5Fx+
5/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-5/2
Fb+
3Fx-
1/2q
x20
00
0D
B b
02F
x+1/
2qx2
00
DE
b0
Fx
00
00
ED
b0
-Fb+
Fx
00
EF
b0
Fb-
Fx
00
00
FE
b0
-Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li11
/6F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-1
1/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.xxxx.055PROCEDIMENTO E RISULTATI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(5/2 x2/b2 ) Fb 1/EJ dx = [5/6 x3/b2 ]o
b Fb 1/EJ
= (5/6 b ) Fb 1/EJ = 5/6 Fb2/EJ
LXoBA = ∫
o
b(5/2 -5 x/b +5/2 x2/b2 ) Fb 1/EJ dx = [5/2 x -5/2 x2/b +5/6 x3/b2 ]o
b Fb 1/EJ
= (5/2 b -5/2 b +5/6 b ) Fb 1/EJ = 5/6 Fb2/EJ
A = 720. mm2
Ju = 188659. mm4
Jv = 22464. mm4
yg = 29.2 mmN = -1138. NTy = 1950. NMx = -1267500. Nmmxm = 12. mmum = -6. mmvm = -29.2 mmσm = N/A-Mv/Ju = -197.8 N/mm2
xc = 18. mmyc = 12. mmvc = -17.2 mmσc = N/A-Mv/Ju = -117.1 N/mm2
τc = 2.878 N/mm2
σo = √σ2+3τ2 = 117.2 N/mm2
S* = 3341. mm3mm 0 12 24 36x
0
48
52
y
12σc,τc
σm
u
v
Schema.xxxx.055
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.055
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.055
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.056REAZIONI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
F
7/16F
F
7/16F7/16Fb
A B
31/16F
9/16F39/16Fb
47/16F
9/16F
B
C
15/16F
2F2Fb
15/16F
2F
B D
F
2F
F
2FFb
D
E
F
FFb
F
F
EF
F
F
B
FFG
FFb
F
G
A
Schema.xxxx.056AZIONI INTERNE Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1
-9/1
6-9
/16
-15/16
-2
-1
1
-1
0
F
7/16
-31/
16-4
7/16
2
1
-1
0
0
-1
F
0 7/16
39/1
60
-20
01100
0
0010
Fb
Sch
ema.
xxxx
.056
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
e:
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-2
0 0-20
01
10 0 0
00 1 0
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
xxxx
.056
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
e:
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-2
Fx
2Fx2 /b
x2 /b2
2/3F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b2F
b-2F
x2F
b-4F
x+2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b1/
2Fx-
1/2q
x2-1
/2F
x+F
x2 /b-1
/2qx
3 /b1-
2x/b
+x2 /b
2
-1/2
4Fb2 /E
J1/
3Xb/
EJ
CB
bx/
b-1
/2F
x+1/
2qx2
-1/2
Fx2 /b
+1/
2qx3 /b
x2 /b2
BD
b0
-2F
b+2F
x0
00
0D
B b
02F
x0
0
DE
b0
Fx
00
00
ED
b0
-Fb+
Fx
00
EF
b0
Fb-
Fx
00
00
FE
b0
-Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
00
00
0G
F b
00
00
GA
b0
Fb-
Fx
00
00
AG
b0
-Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li13
/8F
b2 /EJ
2/3X
b/E
J
iper
stat
ica
X=
WB
C-3
9/16
Fb
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.xxxx.056PROCEDIMENTO E RISULTATI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(2 x2/b2 ) Fb 1/EJ dx = [2/3 x3/b2 ]o
b Fb 1/EJ
= (2/3 b ) Fb 1/EJ = 2/3 Fb2/EJ
LXoBA = ∫
o
b(2 -4 x/b +2 x2/b2 ) Fb 1/EJ dx = [2 x -2 x2/b +2/3 x3/b2 ]o
b Fb 1/EJ
= (2 b -2 b +2/3 b ) Fb 1/EJ = 2/3 Fb2/EJ
LXoBC = ∫
o
b(-1/2 x/b + x2/b2 -1/2 x3/b3 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [-1/4 x2/b +1/3 x3/b2 -1/8 x4/b3 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (-1/4 b +1/3 b -1/8 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 23/24 Fb2/EJ
LXoCB = ∫
o
b(-1/2 x2/b2 +1/2 x3/b3 ) Fb 1/EJ dx + 1 (-1) (-1) Fb2/EJ
= [-1/6 x3/b2 +1/8 x4/b3 ]o
b Fb 1/EJ + 1 (-1) (-1) Fb2/EJ
= (-1/6 b +1/8 b ) Fb 1/EJ + 1 (-1) (-1) Fb2/EJ = 23/24 Fb2/EJ
Schema.xxxx.056PROCEDIMENTO E RISULTATI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
A = 864. mm2
Ju = 219048. mm4
Jv = 44928. mm4
yg = 31.83 mmN = -806.3 NTy = 1720. NMx = -1427600. Nmmxm = 12. mmum = -6. mmvm = -31.83 mmσm = N/A-Mv/Ju = -208.4 N/mm2
xc = 18. mmyc = 14. mmvc = -17.83 mmσc = N/A-Mv/Ju = -117.2 N/mm2
τc = 2.73 N/mm2
σo = √σ2+3τ2 = 117.3 N/mm2
S* = 4172. mm3mm 0 12 24 36x
0
42
52
y
14σc,τc
σm
u
v
Schema.xxxx.056
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.056
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.057REAZIONI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
1/2F
3/4F
1/2F
3/4F3/4Fb
A B
9/4F
5/4F9/4Fb
9/4F
5/4F
B
C
11/4F
3/2F3/2Fb
11/4F
3/2F
B D
1/2F
3/2F
1/2F
3/2F1/2Fb
D
E
1/2F
1/2F1/2Fb
1/2F
1/2FFb
EF
1/2F
F
B
1/2F
F
1/2F1/2Fb
FG
1/2F1/2Fb
1/2F
G
A
Schema.xxxx.057AZIONI INTERNE Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
-1/2
-5/4
-11/4
-3/2
1/2
-1/2
1/21/2
0
F
3/4
-9/4
3/2
-1/2
-1/2
0
-10
1/2
F
0 3/4
9/4
0
-3/20
0-1
/2
-1/2-1
00
0-1/2
-1/2
0
Fb
Sch
ema.
xxxx
.057
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
e:
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
A
B C
D
EF
G
W
F
X
X
q
Sch
ema
di c
alco
lo ip
erst
atic
o
0-3
/2
0 0-3/2
0
0-1/2
-1/2
-1
0 0
0-1
/2
-1/20
Mo
fless
ione
da
caric
hi a
sseg
nati
0-1
-100
0
00
00 0 0
00 0 0
Mx
fless
ione
da
iper
stat
ica
X=
1
Sch
ema.
xxxx
.057
PR
OC
ED
IME
NT
O E
RIS
ULT
AT
I Nom
e:
@ A
dolfo
Zav
elan
i Ros
si, P
olite
cnic
o di
Mila
no, v
ers.
27.0
3.13
06.0
9.19
Qua
dro
cont
ribut
i PLV
per
iper
stat
ica
X=
WB
C
→M
x(x)
Mo(
x)M
xMo
MxM
x∫M
xMo/
EJd
x∫X
MxM
x/E
Jdx
AB
b-x
/b-3
/2F
x3/
2Fx2 /b
x2 /b2
1/2F
b2 /EJ
1/3X
b/E
JB
A b
1-x/
b3/
2Fb-
3/2F
x3/
2Fb-
3Fx+
3/2F
x2 /b1-
2x/b
+x2 /b
2
BC
b-1
+x/
b0
01-
2x/b
+x2 /b
2
01/
3Xb/
EJ
CB
bx/
b0
0x2 /b
2
BD
b0
-3/2
Fb+
3/2F
x0
00
0D
B b
03/
2Fx
00
DE
b0
-1/2
Fx
00
00
ED
b0
1/2F
b-1/
2Fx
00
EF
b0
-1/2
Fb-
1/2F
x0
00
0F
E b
0F
b-1/
2Fx
00
FB
b0
00
00
0B
F b
00
00
FG
b0
-Fx+
1/2q
x20
00
0G
F b
01/
2Fb-
1/2q
x20
0
GA
b0
-1/2
Fb+
1/2F
x0
00
0A
G b
01/
2Fx
00
BC
elon
gazi
one
asta
N1B
Cε B
CL B
CF
b2 /EJ
tota
li3/
2Fb2 /E
J2/
3Xb/
EJ
iper
stat
ica
X=
WB
C-9
/4F
b
Svi
lupp
i di c
alco
lo ip
erst
atic
a
Schema.xxxx.057PROCEDIMENTO E RISULTATI Nome:
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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
LXXBC = ∫
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
LXXCB = ∫
o
b( x2/b2 ) 1/EJ dx = [1/3 x3/b2 ]o
b 1/EJ
= (1/3 b ) 1/EJ = 1/3 b/EJ
LXoAB = ∫
o
b(3/2 x2/b2 ) Fb 1/EJ dx = [1/2 x3/b2 ]o
b Fb 1/EJ
= (1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
LXoBA = ∫
o
b(3/2 -3 x/b +3/2 x2/b2 ) Fb 1/EJ dx = [3/2 x -3/2 x2/b +1/2 x3/b2 ]o
b Fb 1/EJ
= (3/2 b -3/2 b +1/2 b ) Fb 1/EJ = 1/2 Fb2/EJ
A = 912. mm2
Ju = 222885. mm4
Jv = 52416. mm4
yg = 19.68 mmN = -3190. NTy = 1740. NMx = -1531200. Nmmxm = 24. mmym = 52. mmum = 6. mmvm = 32.32 mmσm = N/A-Mv/Ju = 218.5 N/mm2
xc = 18. mmyc = 39. mmvc = 19.32 mmσc = N/A-Mv/Ju = 129.2 N/mm2
τc = 2.62 N/mm2
σo = √σ2+3τ2 = 129.3 N/mm2
S* = 4027. mm3mm 0 12 24 36x
0
12
52
y
39σc,τc
σm
u
v
Schema.xxxx.057
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.057
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
Schema.xxxx.057
@ Adolfo Zavelani Rossi, Politecnico di Milano, vers.27.03.13 06.09.19
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