1 2
3 4
5
6
F
F
P P
hh
L/4 L/2 L/4
L
EI
cEA
c= ∞
EI
tEI
tEI
cEA
c= ∞
Risoluzione del “sistema a nodi fissi”:
La soluzione di tale sistema si basa sull’individuazione delle azioni che le travi caricare dalle azioni esterne “Q” e “P” trasferiscono ai nodi di estremità. Si noti che i carichi “F” essendo direttamente agenti sui nodi non rientrano nelle soluzioni del sistema zero.
3 4
F
P P
Q
h
QL²/12 QL²/12
SISTEMA 0 - Momenti
3 4
F
P P
Q
h
SISTEMA 0 - Tagli
QL/2 QL/2
Risoluzione dei “sistemi a nodi spostabili”:
1 2
3 4
5
6 ϕ1ϕ1ϕ1
ϕ1
ϕ1 ϕ1ϕ1 ϕ1
SISTEMA 1 - Rotazione del nodo 2
2
3 4
5
SISTEMA 1 - Momenti SISTEMA 1 - Tagli
2
3 4
5
(2EIc/h)ϕ1
(4EIc/h)ϕ1
(3EIc/h)ϕ1
(6EIc/h²)ϕ1
(6EIc/h²)ϕ1
(3EIc/h²)ϕ1
1
3 4
5
6 2
∆1∆1
∆1∆1
1 2
3 4
5
6
SISTEMA 2 - Momenti SISTEMA 2 - Tagli
1 2
3 4
5
6
(6EIc/h²)∆1
(6EIc/h²)∆1
(3EIc/h²)∆1
(12EIc/h³)∆1
(12EIc/h³)∆1
(3EIc/h³)∆1
(3EIc/h³)∆1
(EAt/L)∆1 (EAt/L)∆1
1
3 4
5
6
SISTEMA 3 - Rotazione del nodo 3
2 ϕ2 ϕ2 ϕ2
ϕ2 ϕ2ϕ2ϕ2ϕ2
1 2
3 4
5
6
SISTEMA 3 - Momenti SISTEMA 3 - Tagli
1 2
3 4
5
6
(4EIc/h)ϕ2
(2EIc/h)ϕ2
(4EIt/L)ϕ2 (2EIt/L)ϕ2
(6EIc/h²)ϕ2
(6EIc/h²)ϕ2
(6EIt/L²)ϕ2 (6EIt/L²)ϕ2
1
4
5
6 2
3
∆2
∆2∆2
∆2
1 2
3 4
5
6
SISTEMA 4 - Momenti SISTEMA 4 - Tagli
1 2
3 4
5
6
(6EIc/h²)∆2
(6EIc/h²)∆2
(12EIc/h³)∆2
(12EIc/h³)∆2
(EAt/L)∆2 (EAt/L)∆2
1
3 4
5
6
SISTEMA 5 - Rotazione del nodo 5
2
ϕ3 ϕ3 ϕ3 ϕ3
ϕ3ϕ3 ϕ3ϕ3
2
3 4
5
SISTEMA 5 - Momenti SISTEMA 5 - Tagli
2
3 4
5
(2EIc/h)ϕ3
(4EIc/h)ϕ3
(3EIc/h)ϕ3
(6EIc/h²)ϕ3
(6EIc/h²)ϕ3
(3EIc/h²)ϕ3
(3EIc/h²)ϕ3
1
3 4
5
6 2
∆3
∆3
∆3
∆3
1 2
3 4
5
6
SISTEMA 6 - Momenti SISTEMA 6 - Tagli
1 2
3 4
5
6
(6EIc/h²)∆3
(6EIc/h²)∆3
(3EIc/h²)∆3
(12EIc/h³)∆3
(12EIc/h³)∆3
(3EIc/h³)∆3
(3EIc/h³)∆3
(EAt/L)∆3 (EAt/L)∆3
1
4 4
5
6
SISTEMA 7 - Rotazione del nodo 4
2
ϕ4ϕ4 ϕ4ϕ4 ϕ4ϕ4ϕ4ϕ4
2
3 4
5
SISTEMA 7 - Momenti SISTEMA 7 - Tagli
2
3 4
5
(2EIt/L)ϕ4 (4EIt/L)ϕ4
(4EIc/h)ϕ4
(2EIc/h)ϕ4
(6EIc/h²)ϕ4
(6EIc/h²)ϕ4
(6EIt/L²)ϕ4 (6EIt/L²)ϕ4
1
4
5
6 2
3
∆4
∆4∆4
∆4
1 2
3 4
5
6
SISTEMA 8 - Momenti SISTEMA 8 - Tagli
1 2
3 4
5
6
(6EIc/h²)∆4
(6EIc/h²)∆4
(12EIc/h³)∆4
(12EIc/h³)∆4
(EAt/L)∆4 (EAt/L)∆4
QL²/12
3
(2EIc/h)ϕ1
2
4
5
(6EIc/h²)∆1 (4EIc/h)ϕ2 (6EIc/h²)∆2
(4EIt/L)ϕ2
(2EIt/L)ϕ4 QL²/12
(2EIt/L)ϕ2
(2EIc/h)ϕ3 (6EIc/h²)∆3 (4EIt/L)ϕ4
(4EIc/h)ϕ4 (6EIc/h²)∆4
(4EIc/h)ϕ1
(3EIc/h)ϕ1 (6EIc/h²)∆1
(3EIc/h²)∆1 (2EIc/h)ϕ2 (6EIc/h²)∆2
1 6
(4EIc/h)ϕ3
(3EIc/h)ϕ3 (6EIc/h²)∆3
(3EIc/h²)∆3 (2EIc/h)ϕ4 (6EIc/h²)∆4
3
2
4
5
1 6
F
F
(6EIc/h²)ϕ1 (12EIc/h³)∆1 (6EIc/h²)ϕ2 (12EIc/h³)∆2 (EAt/L)∆2
(6EIc/h²)ϕ1
(3EIc/h²)ϕ1
(3EIc/h²)ϕ1 (12EIc/h³)∆1
(3EIc/h³)∆1
(3EIc/h³)∆1 (EAt/L)∆1 (6EIc/h²)ϕ2 (12EIc/h³)∆2
(EAt/L)∆3 (EAt/L)∆4
(EAt/L)∆1 (EAt/L)∆2
(6EIc/h²)ϕ3
(6EIc/h²)ϕ3
(3EIc/h²)ϕ3
(3EIc/h²)ϕ3 (12EIc/h³)∆3
(12EIc/h³)∆3
(3EIc/h³)∆3 (EAt/L)∆3
(EAt/L)∆4
(6EIc/h²)ϕ4
(6EIc/h²)ϕ4
(3EIc/h³)∆3 (12EIc/h³)∆4
(12EIc/h³)∆4
Si definiscono le seguenti relazioni:
h K
1= EI
c2 2
h K = EI
cL K
3= EI
t4 3
h K = EI
cL K
5= EA
tQL²/12
3
2K1ϕ1
4
4K1ϕ2
4K3ϕ2
QL²/12
4K1ϕ1 6K2∆1 2K1ϕ2
3 4
F
6K2ϕ1 12K4∆1
K5∆2
6K2∆2 6K2∆1 6K2∆2
2K1ϕ3 6K2∆3 4K1ϕ4 6K2∆4
6K2∆4 2K1ϕ4 6K2∆3 4K1ϕ3 2K3ϕ4 4K3ϕ4
2K3ϕ2
6K2ϕ3 12K4∆3
6K2ϕ2 6K2ϕ4
12K4∆2 12K4∆4
K5∆4 K5∆4 K5∆2
12K4∆2 6K2ϕ2 12K4∆1 6K2ϕ1
12K4∆4 6K2ϕ4 12K4∆3 6K2ϕ3
seguente:
+
[ ] [ ]
[ ] [ ]
[ ]
[ ]
[ ] [ ]
[ ] [ ]
[ ]
[ ]
∆
⋅ +
−
⋅
−
∆
⋅ +
⋅ +
∆
⋅
=
∆
⋅
−
⋅ +
−
∆
⋅ +
⋅ +
⋅
⋅ +
=
∆
⋅ +
⋅ +
∆
⋅ + +
−
⋅ +
− +
∆
⋅
=
∆
⋅
−
⋅
−
∆
⋅
− +
⋅ +
=
∆
⋅ +
∆
⋅ +
−
⋅ +
∆
⋅ +
⋅ +
=
⋅
−
∆
⋅
−
⋅ + +
∆
⋅ +
⋅
⋅ +
−
=
∆
⋅ +
∆
⋅ +
⋅
−
∆
⋅ + +
−
⋅ +
− +
=
∆
⋅
−
⋅ +
∆
⋅
− +
⋅ +
=
4 5 4 4 2 3 4 3 2 2 5
4 2 4 3 1 3 2 3 1 2 3 2
4 4 4 2 3 5 4 4 3 2 2 1
5
4 2 4 1 3 2 2 3 1 1
4 5 2 5 4 2 2 1 4 1 2
4 3 2 2 2 3 1 1 2 1 1 2
3 5 2 4 2 2 1 5 4 4 1 2 2
2 2 2 1 1 2 2 1 1 1
12 6
12 6
0
6 4
6 2
12 2 0
12 6
3 12 3
6 0
6 2
3 6 4
3 0
12 6
12 6
0
2 6
4 6
12 2 0
12 6
3 12 3
6 0
6 2
3 6 4
3 0
K K K
K K
K
K K
K K
K L K
Q
K K
K K K K
K K
K K
K K K
K
K K
K K
K K
F
K K
K K K
L K Q
K K
K K
K K K
K F
K K
K K K
K
ϕ ϕ
ϕ ϕ
ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ ϕ
ϕ
ϕ ϕ
ϕ ϕ
4 _
4 _
5 _
5 _
3 _
3 _
2 _
2 _
Nodo Nodo Nodo Nodo Nodo Nodo Nodo Nodo
[ ] [ ] F
ij= K
ijhk⋅ [ ] δ
hk[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
∆
∆
∆
∆
⋅
+
−
−
− +
− +
−
−
−
− +
−
−
− +
− +
−
−
−
=
− ⋅
−
− ⋅
4 4 3 3 2 2 1 1
5 4 2
4 2
5
2 3
1 2
1 3
4 2
5 4 2
5
2 1
2 1
5 5
4 2
4 2
3 2
3 1 2
1
5 4
2 5
4 2
2 1
2 1
2 2
12 6
12 6
0 0
0
6 4
6 2
0 2
0 0
12 6
15 3
0 0
0
6 2
3 7
0 0
0 0
0 0
0 12
6 12
6
0 2
0 0
6 4
6 2
0 0
0 12
6 15
3
0 0
0 0
6 2
3 7
0 12 0 0 12
0
ϕ ϕ ϕ ϕ
K K K
K K
K
K K
K K
K K
K K
K K K
K
K K
K K
K K
K K
K K
K K
K K K
K
K K
K K
K K
K K
K K
L Q
F L Q
F
La soluzione risulta:
[ ] δ
hk= [ ] [ ] K
ijhk −1⋅ F
ijvalida se e solo se det [ ] K
ijhk≠ 0 ovvero se la matrice di rigidezza è invertibile.
Per la soluzione si considerino i seguenti valori geometrici e meccanici:
Modulo di elasticità del calcestruzzo: E = 3 ⋅ 10
7[kN/m
2] Sezione trasversale dei pilastri: A
c= 0 . 4
2[m
2] Sezione trasversale delle travi: A
t= 0 . 3 ⋅ 0 . 6 [m
2]
21333 3
12 4 . 10 0 3
4 7
1
=
⋅
⋅
=
= h
K EI
c[kNm]
9 7111 12
4 . 10 0 3
4 7
2 2
=
⋅
⋅
=
= h
K EI
c[kN]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
∆
∆
∆
∆
⋅
− − +
− +
− +
−
−
−
− +
−
−
− +
− +
−
−
=
− ⋅
−
⋅
4 4 3 3 2 2 1
5 4 2
4 2
5
2 3
1 2
1 3
4 2
5 4 2
5
2 1
2 1
5 5
4 2
4 2
3 2
3 1 2
1
5 4
2 5
4 2
2 2
12 6
12 6
0 0
0
6 4
6 2
0 2
0 0
12 6
15 3
0 0
0
6 2
3 7
0 0
0 0
0 0
0 12
6 12
6
0 2
0 0
6 4
6 2
0 0
0 12
6 15
3
0 12 0 0 12
ϕ ϕ ϕ
K K K
K K
K
K K
K K
K K
K K
K K K
K
K K
K K
K K
K K
K K
K K
K K K
K
K K
K K
K K
L Q
F L Q
∆
∆
∆
∆
⋅
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
=
−
−
−
4 4 3 3 2 2 1 1
703440 42666
28440 42666
675000 0
0 0
42666 166332
42666 42666
0 40500
0 0
28440 42666
710550 21333
0 0
675000 0
42666 42666
21333 149331
0 0
0 0
675000 0
0 0
703440 42666
28440 42666
0 40500 0
0 42666 166332
42666 42666
0 0
675000 0
28440 42666
710550 21333
0 0
0 0
42666 42666
21333 149331
0 34 . 53
0 0 10
34 . 53
10 0
ϕ ϕ ϕ ϕ
La soluzione del sistema si può ottenere con il Metodo di Cramer:
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
=
∆
∆
∆
∆
−
−
−
−
−
−
−
−
4 4 3 3 2 4 3 3
4 4 3 3 2 2 1 1
10 172 . 1
10 051 . 1
10 809 . 7
10 262 . 2
10 174 . 1
10 469 . 8
10 792 . 7
10 000 . 2
ϕ ϕ ϕ
ϕ [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] m rad
m rad m rad m rad
Sostituendo il valore degli spostamenti nell’equilibrio dei nodi, si ottiene il sistema equilibrato con le azioni esterne:
QL²/12
3
(2EIc/h)ϕ1
2
4
5
(6EIc/h²)∆1 (4EIc/h)ϕ2 (6EIc/h²)∆2
(4EIt/L)ϕ2
(2EIt/L)ϕ4 QL²/12
(2EIt/L)ϕ2
(2EIc/h)ϕ3 (6EIc/h²)∆3 (4EIt/L)ϕ4
(4EIc/h)ϕ4 (6EIc/h²)∆4
(4EIc/h)ϕ1
(3EIc/h)ϕ1 (6EIc/h²)∆1 (2EIc/h)ϕ2 (6EIc/h²)∆2
(4EIc/h)ϕ3
(3EIc/h)ϕ3 (6EIc/h²)∆3 (2EIc/h)ϕ4 (6EIc/h²)∆4
3
2
4
5
F
F
(6EIc/h²)ϕ1 (12EIc/h³)∆1 (6EIc/h²)ϕ2 (12EIc/h³)∆2 (EAt/L)∆2
(6EIc/h²)ϕ1
(3EIc/h²)ϕ1 (12EIc/h³)∆1
(EAt/L)∆1 (6EIc/h²)ϕ2 (12EIc/h³)∆2
(EAt/L)∆3 (EAt/L)∆4
(EAt/L)∆1 (EAt/L)∆2
(6EIc/h²)ϕ3
(6EIc/h²)ϕ3
(3EIc/h²)ϕ3 (12EIc/h³)∆3
(12EIc/h³)∆3
(EAt/L)∆3 (EAt/L)∆4
(6EIc/h²)ϕ4
(6EIc/h²)ϕ4 (12EIc/h³)∆4
(12EIc/h³)∆4
3
2
4
5
(6EIt/L²)ϕ2
QL/2 QL/2
P P
(6EIt/L²)ϕ2 (6EIt/L²)ϕ4 (6EIt/L²)ϕ4
1 2
3 4
5
6
Momenti
10.9 [kNm]
10.9 [kNm]
38.2 [kNm]
38.2 [kNm]
79.1[kNm]
79.1[kNm]
21.8[kNm]
21.8[kNm]
10
20 20
10
10
Tagli
2
3 4
5
9.1 [kN]
9.1 [kN]
10
20 20
12.7 [kN]
19.1 [kN]
19.1 [kN]
7.3 [kN]
28.8 [kN]
20 [kN] 20 [kN]
51.3 [kN]
* *
* *
10
10
Normali
1 2
3 4
5
6
