ejercicio matriz< de reigidez condensada
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8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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＝++⋅m a ⋅c v ⋅k d ⋅− M A
≔masapiso1 =++⋅⋅750 4 10 ―――――⋅⋅⋅0.25 0.30 32 24
9.806
―――――⋅⋅⋅0.30 0.30 18 24
9.806
30009.839.
≔masapiso2 =++⋅550 4 10 ⋅⋅⋅0.25 0.30 32 24
9.806
⋅⋅⋅0.30 0.30 18 24
9.806 22009.839
≔m1 =masapiso1 30009.839
≔m2 =masapiso2 22009.839
≔ M =m2 00 m1
⎡
⎣
⎤
⎦
22009.839 00 30009.839
⎡
⎣
⎤
⎦
Matriz de rigidez de la viga
≔b 0.25 ≔hv 0.3
≔Le 4 ≔ I =――⋅b hv3
12 0.001
≔ Av =⋅b hv 0.075
≔ E =⋅⋅4700 ‾‾21 1000000 21538105766.292 ≔α 0
≔ ρ ⋅ E ― I
Le3 ≔ β ――
⋅ Av Le2
I ≔ss sin α ≔cc cos α
≔ Kv
+⋅ β cc2 ⋅12 ss2 ⋅⋅ss cc −12 β ⋅⋅6 Le ss −⋅− β cc2 ⋅12 ss2 ⋅⋅ss cc − β 12 ⋅⋅6 Le s
⋅⋅ss cc −12 β +⋅ β ss2 ⋅12 cc2 ⋅⋅6 Le cc ⋅⋅ss cc − β 12 −⋅− β ss2 ⋅12 cc2 ⋅⋅6 Le c
⋅⋅6 Le ss ⋅⋅6 Le cc ⋅4 Le2 ⋅⋅−6 Le ss ⋅⋅−6 Le cc ⋅2 Le
2
−⋅− β cc2 ⋅12 ss2 ⋅⋅ss cc − β 12 ⋅⋅−6 Le ss +⋅ β cc2 ⋅12 ss2 ⋅⋅ss cc −12 β ⋅⋅−6 Le s
⋅⋅ss cc − β 12 −⋅− β ss2 ⋅12 cc2 ⋅⋅−6 Le cc ⋅⋅ss cc −12 β +⋅ β ss2 ⋅12 cc2 ⋅⋅−6 Le
⋅⋅6 Le ss ⋅⋅6 Le cc ⋅2 Le2 ⋅⋅−6 Le ss ⋅⋅−6 Le cc ⋅4 Le
2
⎡
⎢⎢⎢⎢⎢⎢⎣
8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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= Kv
403839483.118 0 0 −403839483.118 0 00 2271597.093 4543194.185 0 −2271597.093 45431940 4543194.185 12115184.494 0 −4543194.185 6057592
−403839483.118 0 0 403839483.118 0 0
0 −2271597.093 −4543194.185 0 2271597.093 −45431940 4543194.185 6057592.247 0 −4543194.185 12115184
⎡⎢⎢⎢⎢
⎢⎣
Matriz de rigidez de la columna
≔b 0.30 ≔hv 0.3
≔Le 3 ≔ I =――⋅b hv3
12 0.001
≔ Av =⋅b hv 0.09
≔ E =⋅⋅4700 ‾‾21 1000000 21538105766.292 ≔α
2
≔ ρ ⋅ E ― I
Le3
≔ β ――⋅ Av Le
2
I
≔ss sin α ≔cc cos α
≔ Kc
+⋅ β cc2 ⋅12 ss2 ⋅⋅ss cc −12 β ⋅⋅6 Le ss −⋅− β cc2 ⋅12 ss2 ⋅⋅ss cc − β 12 ⋅6⋅⋅ss cc −12 β +⋅ β ss2 ⋅12 cc2 ⋅⋅6 Le cc ⋅⋅ss cc − β 12 −⋅− β ss2 ⋅12 cc2 ⋅6
⋅⋅6 Le ss ⋅⋅6 Le cc ⋅4 Le2 ⋅⋅−6 Le ss ⋅⋅−6 Le cc 2−⋅− β cc2 ⋅12 ss2 ⋅⋅ss cc − β 12 ⋅⋅−6 Le ss +⋅ β cc2 ⋅12 ss2 ⋅⋅ss cc −12 β −6
⋅⋅ss cc − β 12 −⋅− β ss2 ⋅12 cc2 ⋅⋅−6 Le cc ⋅⋅ss cc −12 β +⋅ β ss2 ⋅12 cc2 −6⋅⋅6 Le ss ⋅⋅6 Le cc ⋅2 Le2 ⋅⋅−6 Le ss ⋅⋅−6 Le cc 4
⎡⎢⎢⎢⎢⎢⎢⎣
= Kc
6461431.73 0 9692147.595 −6461431.73 0 960 646143172.989 0 0 −646143172.989
9692147.595 0 19384295.19 −9692147.595 0 96−6461431.73 0 −9692147.595 6461431.73 0 −96
0 −646143172.989 0 0 646143172.9899692147.595 0 9692147.595 −9692147.595 0 193
⎡
⎢⎢⎢⎢⎣
GENERANDO SUBMATRICES
8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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Kba3 0 ++ Kbb3 Kaa2 Kaa5 Kab2
0 Kba4 Kba2 ++ Kbb2 Kbb4 Kaa6⎣ ⎦
≔ X =+ Kaa1 Kaa3
410300914.848 0 9692147.595
0 648414770.081 4543194.1859692147.595 4543194.185 31499479.683
⎡
⎢⎣
⎤
⎥⎦
≔Y =+ Kbb1 Kaa4
410300914.848 0 9692147.5950 648414770.081 −4543194.185
9692147.595 −4543194.185 31499479.683
⎡⎢
⎣
⎤⎥
⎦
≔Z =++ Kbb3 Kaa2 Kaa5
416762346.578 0 00 1294557943.07 4543194.1850 4543194.185 50883774.873
⎡⎢
⎣
⎤⎥
⎦
≔T =++ Kbb2 Kbb4 Kaa6
416762346.578 0 00 1294557943.07 −4543194.1850 −4543194.185 50883774.873
⎡⎢
⎣
⎤⎥
⎦
u11 u12 u13 u21 u22 u23 u31 u32 u33 u41 u
≔ KP
X ,0 0
X ,0 1
X ,0 2
Kab1,0 0
Kab1,0 1
Kab1,0 2
Kab3,0 0
Kab3,0 1
Kab3,0 2
0 0
X ,1 0 X ,1 1 X ,1 2 Kab1 ,1 0 Kab1 ,1 1 Kab1 ,1 2 Kab3 ,1 0 Kab3 ,1 1 Kab3 ,1 2 0 0 X
,2 0 X
,2 1 X
,2 2 Kab1
,2 0 Kab1
,2 1 Kab1
,2 2 Kab3
,2 0 Kab3
,2 1 Kab3
,2 20 0
Kba1,0 0
Kba1,0 1
Kba1,0 2
Y ,0 0
Y ,0 1
Y ,0 2
0 0 0 Kab4,0 0
Kab
Kba1,1 0
Kba1,1 1
Kba1,1 2
Y ,1 0
Y ,1 1
Y ,1 2
0 0 0 Kab4,1 0
Kab
Kba1,2 0
Kba1,2 1
Kba1,2 2
Y ,2 0
Y ,2 1
Y ,2 2
0 0 0 Kab4,2 0
Kab
Kba3,0 0
Kba3,0 1
Kba3,0 2
0 0 0 Z ,0 0
Z ,0 1
Z ,0 2
Kab2,0 0
Kab
Kba3,1 0
Kba3,1 1
Kba3,1 2
0 0 0 Z ,1 0
Z ,1 1
Z ,1 2
Kab2,1 0
Kab
Kba3,2 0
Kba3,2 1
Kba3,2 2
0 0 0 Z ,2 0
Z ,2 1
Z ,2 2
Kab2,2 0
Kab
0 0 0 Kba4,0 0
Kba4,0 1
Kba4,0 2
Kba2,0 0
Kba2,0 1
Kba2,0 2
T ,0 0
T 0
0 0 0 Kba4,1 0
Kba4,1 1
Kba4,1 2
Kba2,1 0
Kba2,1 1
Kba2,1 2
T ,1 0
T 1
0 0 0 Kba4,2 0
Kba4,2 1
Kba4,2 2
Kba2,2 0
Kba2,2 1
Kba2,2 2
T ,2 0
T 2
⎡⎢
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎢⎢⎢⎢⎢⎢⎢
⎣
≔ filau11 =submatrix ,,,, KP 0 0 0 11 410300914.848 0 9692147.595 −403839483.118 0 0 −64614
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≔ filau21 =submatrix ,,,, KP 3 3 0 11 −403839483.118 0 0 410300914.848 0 9692147.595 0 0 0
≔Q =+ filau11 filau21 6461431.73 0 9692147.595 6461431.73 0 9692147.595 −6461431.73 0 96
≔ fila31 =submatrix ,,,, KP 6 6 0 11 −6461431.73 0 −9692147.595 0 0 0 416762346.578 0 0 −
≔ fila41 =submatrix ,,,, KP 9 9 0 11 0 0 0 −6461431.73 0 −9692147.595 −403839483.118 0 0
≔ A =+ fila31 fila41 −6461431.73 0 −9692147.595 −6461431.73 0 −9692147.595 12922863.46 0
1. SUMAR FILAS CON GRADOS DE LIBERTADO HORIZONTALES Y QUE ETSEN EN EL MISMO PLANO2. ELMINAR GRADOS VERTICALES POR ESBELTEZ
u11 u12 u13 u21 u22 u23 u31 u32 u33 u41 u
≔ KP
X ,0 0
X ,0 1
X ,0 2
Kab1,0 0
Kab1,0 1
Kab1,0 2
Kab3,0 0
Kab3,0 1
Kab3,0 2
0 0
X ,1 0
X ,1 1
X ,1 2
Kab1,1 0
Kab1,1 1
Kab1,1 2
Kab3,1 0
Kab3,1 1
Kab3,1 2
0 0
X ,2 0
X ,2 1
X ,2 2
Kab1,2 0
Kab1,2 1
Kab1,2 2
Kab3,2 0
Kab3,2 1
Kab3,2 2
0 0
Kba1 ,0 0 Kba1 ,0 1 Kba1 ,0 2 Y ,0 0 Y ,0 1 Y ,0 2 0 0 0 Kab4 ,0 0 Kab
Kba1,1 0
Kba1,1 1
Kba1,1 2
Y ,1 0
Y ,1 1
Y ,1 2
0 0 0 Kab4,1 0
Kab
Kba1,2 0
Kba1,2 1
Kba1,2 2
Y ,2 0
Y ,2 1
Y ,2 2
0 0 0 Kab4,2 0
Kab
Kba3,0 0
Kba3,0 1
Kba3,0 2
0 0 0 Z ,0 0
Z ,0 1
Z ,0 2
Kab2,0 0
Kab
Kba3,1 0
Kba3,1 1
Kba3,1 2
0 0 0 Z ,1 0
Z ,1 1
Z ,1 2
Kab2,1 0
Kab
Kba3,2 0
Kba3,2 1
Kba3,2 2
0 0 0 Z ,2 0
Z ,2 1
Z ,2 2
Kab2,2 0
Kab
0 0 0 Kba4,0 0
Kba4,0 1
Kba4,0 2
Kba2,0 0
Kba2,0 1
Kba2,0 2
T ,0 0
T 0
0 0 0 Kba4,1 0
Kba4,1 1
Kba4,1 2
Kba2,1 0
Kba2,1 1
Kba2,1 2
T ,1 0
T 1
0 0 0 Kba4,2 0
Kba4,2 1
Kba4,2 2
Kba2,2 0
Kba2,2 1
Kba2,2 2
T ,2 0
T 2
⎡⎢⎢⎢⎢⎢
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎢⎢⎢
⎣
8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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U11 U13 U21 U23 U31 U33 U41 U43
U11
U13
U23
U31
U33
U43
≔ KP1
Q,0 0
Q,0 1
Q,0 2
Q,0 3
Q,0 4
Q,0 5
Q,0 6
Q,0 7
X ,2 0
X ,2 2
Kab1,2 0
Kab1,2 2
Kab3,2 0
Kab3,2 2
0 0
Kba1,2 0
Kba1,2 2
Y ,2 0
Y ,2 2 0 0 Kab4
,2 0 Kab4
,2 2
A,0 0
A,0 1
A,0 2
A,0 3
A,0 4
A,0 5
A,0 6
A,0 7
Kba3,2 0
Kba3,2 2
0 0 Z ,2 0
Z ,2 2
Kab2,2 0
Kab2,2 2
0 0 Kba4,2 0
Kba4,2 2
Kba2,2 0
Kba2,2 2
T ,2 0
T ,2 2
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
3. SUMAR COLUMNAS DE GRADO DE LIBERTADO HORIZONTAL
≔COL11 =submatrix ,,,, KP1 0 5 0 0
⋅6.461 106
⋅9.692 106
0− ⋅6.461 106
⋅9.692 106
0
⎡
⎢⎢⎢⎢⎢⎢⎣
⎤
⎥⎥⎥⎥⎥⎥⎦
≔COL21 =submatrix ,,,, KP1 0 5 2 2
⋅9.692 106
0⋅9.692 106
− ⋅9.692 106
0⋅9.692 106
⎡⎢⎢⎢⎢
⎢⎣
⎤⎥⎥⎥⎥
⎥⎦
≔COL31 =submatrix ,,,, KP1 0 5 4 4
− ⋅3.917 10−8
− ⋅9.692 106
0⋅3.917 10−8
00
⎡⎢⎢⎢⎢⎢
⎣
⎤⎥⎥⎥⎥⎥
⎦
≔COL41 =submatrix ,,,, KP1 0 5 6 6
− ⋅6.461 106
0− ⋅9.692 106
⋅1.292 107
00
⎡
⎢⎢⎢⎢
⎣
⎤
⎥⎥⎥⎥
⎦
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≔ AQ =+COL11 COL21
⋅1.615 107
⋅9.692 106
⋅9.692 106
− ⋅1.615 107
⋅9.692 106
⋅9.692 106
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
≔ BQ =+COL31 COL41
− ⋅6.461 106
− ⋅9.692 106
− ⋅9.692 106
⋅1.292 107
00
⎡⎢⎢⎢⎢⎢
⎣
⎤⎥⎥⎥⎥⎥
⎦
≔ KP2
AQ,0 0
Q,0 1
Q,0 3
BQ,0 0
Q,0 5
Q,0 7
AQ,1 0
X ,2 2
Kab1,2 2
BQ,1 0
Kab3,2 2
0
AQ,2 0
Kba1,2 2
Y ,2 2
BQ,2 0
0 Kab4,2 2
AQ,3 0
A,0 1
A,0 3
BQ,3 0
A,0 5
A,0 7
AQ,4 0
Kba3,2 2
0 BQ,4 0
Z ,2 2
Kab2,2 2
AQ,5 0
0 Kba4,2 0
BQ,5 0
Kba2,2 2
T ,2 2
⎡⎢⎢⎢⎢⎢⎢
⎢⎢⎢
⎣
⎤⎥⎥⎥⎥⎥⎥
⎥⎥⎥
⎦
4. DETERMINAR LA MATRIZ DE RIGIDEZ CONDENSADA
≔ Kc − Kpp ⋅⋅ Kps Kss−1
Ksp
≔ Kpp =submatrix ,,,, KP2 0 1 0 1 ⋅1.615 107
− ⋅3.917 10−8
⋅9.692 106 ⋅3.15 107⎡⎣
⎤⎦
≔ Kps =submatrix ,,,, KP2 0 1 2 5 ⋅6.461 106 − ⋅6.461 106 ⋅9.692 106 ⋅3.917 10−8
⋅6.058 106 − ⋅9.692 106 ⋅9.692 106 0
⎡
⎣
⎤
⎦
≔ Kss =submatrix ,,,, KP2 2 5 2 5
⋅3.15 107 − ⋅9.692 106 0 ⋅9.692 106
− ⋅6.461 106 ⋅1.292 107 − ⋅9.692 106 − ⋅7.834 10−8
0 0 ⋅5.088 107 ⋅6.058 106
⋅9.692 10
6
0 ⋅6.058 10
6
⋅5.088 10
7
⎡⎢⎢⎢
⎣
⎤⎥⎥⎥
⎦
≔ Ksp =submatrix ,,,, KP2 2 5 0 1
⋅9.692 106 ⋅6.058 106
− ⋅1.615 107 ⋅3.917 10−8
⋅9.692 106 ⋅9.692 106
⋅9.692 106 0
⎡⎢⎢⎢
⎣
⎤⎥⎥⎥
⎦
≔ Kcon =− Kpp ⋅⋅ Kps Kss−1 Ksp ⋅7.637 106 − ⋅1.98 106
− ⋅2.689 106 ⋅3.064 107
⎡
⎣
⎤
⎦
8/17/2019 EJERCICIO MATRIZ< DE REIGIDEZ CONDENSADA
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5. rigidez de la estrucutra en el sentid !