Post on 05-Jul-2018
8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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8/16/2019 Matriz de Rigidez Para Edificios 3d1
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0
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 10/16
Es"'do e&e/en"'& -%e:
==
1
0
0
0
0
0
6eu
[ ][ ]
[ ]
[ ]
[ ]
[ ]
=
==
θθ
θθ
θ
θ
θ
θ
22
12
22
12
22
12
2
1
2
1
2
1
6
k
k
k
k
k k
M
M
P
P
P P
Ke P y
y
x
x
y
y
x
x
Piso
CM
[ ] [ ] 21222 l piso
k x
[ ] [ ]lB piso B k y
122
[ ] [ ] 11221 l piso
k x [ ] [ ]
lA piso A k y122
Des*&'z'/ien"osPiso
F-e!z'sPiso
A
$
#
[ ] [ ] [ ] [ ] [ ] lA piso AlB piso B x
x
x k yk yk P F 122122121
−−===∑ θ
Piso
[ ]233111
12482
242 L
EI
L
EI L
L
EI Lk P
x
x −=
−
−−
−
−==
θ
[ ] [ ] [ ] [ ] [ ] 1122121222121 l pisol piso y
y
y k xk xk P F +===∑ θ
x P
1
y P 1
[ ] [ ]lA piso A k y
122
[ ] A y [ ] [ ]lA piso A k y
222
[ ] [ ]lB piso B k y
222[ ] B y
[ ] [ ]lB piso B k y
122
Pó!"ico
APó!"ico
$
[ ]233121
12482
242 L
EI
L
EI L
L
EI Lk P
y
y =
−
−+
−
==
θ
1 M
#
x
P 1
A
$
y P
1
1 M
CM
[ ] 11 piso x [ ] 12 piso
x
0
[ ] 1 piso B y ⋅
A
$
[ ] 1 piso A y ⋅
0
00
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 11112211221222
112211221211
pisol piso pisol piso
piso AlA piso A piso BlB piso B piso
xk x xk x
yk y yk yk M M
+
++===∑ θθ
[ ]
L
EI L
L
EI L
L
L
EI L L
L
EI L L
L
EI Lk M
362482
224
2248
2224
2
3
333121
−=
−
−
−
+
−
+
−
−
−+
−
==
θθ
y P
2
x P
2
2 M
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 11/16
Es"'do e&e/en"'& -%e:
==
1
0
0
0
0
0
6eu
[ ][ ]
[ ]
[ ]
[ ]
[ ]
=
==
θθ
θθ
θ
θ
θ
θ
22
12
22
12
22
12
2
1
2
1
2
1
6
k
k
k
k
k k
M
M
P
P
P P
Ke P y
y
x
x
y
y
x
x
Piso#
CM
[ ] [ ] 22222 l piso
k x
[ ] [ ]lB piso B k y
222
[ ] [ ] 12221 l piso
k x [ ] [ ]
lA piso A k y222
Des*&'z'/ien"osPiso
F-e!z'sPiso
A
$
#
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x
x k yk yk P F 222222222
−−===∑ θ
Piso#
[ ]233222
24962
482 L
EI
L
EI L
L
EI Lk P
x
x =
−−
−==
θ
[ ] [ ] [ ] [ ] [ ] 1222122222222 l pisol piso y
y
y k xk xk P F +===∑ θ
x P
2
y P 2
Pó!"ico
#Pó!"ico
[ ]233222
24962
482 L
EI
L
EI L
L
EI Lk P
y
y −=
−+
==
θ
1 M
#
x
P 1
A
$
y P
1
1 M
y P
2
x P
2
2 M
[ ] [ ] 21222 l piso
k x
[ ]2
x [ ] [ ] 22222 l piso
k x[ ] [ ] 12221 l piso
k x [ ]1
x
[ ] [ ] 11221 l piso
k x
CM[ ] 21 piso x
[ ] 22 piso
x
[ ] 22 piso
x
[ ] 2 piso B y ⋅
A
$
[ ] 2 piso A y ⋅
[ ] 2 piso B y
[ ] 21 piso
x[ ]
2 piso A y
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 122
2
21222
2
2222
2
222
2
22222 l pisol pisolA piso AlB piso B piso k xk xk yk yk M M +++===∑ θθ
[ ] L
EI
L
EI L
L
EI L
L
EI L
L
EI Lk M 7296
248
296
248
2 3
2
3
2
3
2
3
2
222 =
−+
+
−+
==
θθ
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 12/16
Con &os dos es"'dos e&e/en"'&es -%e95 -%e: 5 es *osi(&e gene!'! &'s ec-'ciones *'!' &'s
s-( /'"!ices 6x75 6y7 677
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]∑−=−−=
−
−=
=
xen sresistente jlineas
lj jlA AlB B
lA piso AlA piso A
lA piso AlA piso A
lB piso BlB piso B
lB piso BlB piso B
x k Y k Y k Y k yk y
k yk y
k yk y
k yk y
k k
k k k
x x
x x
222211
122111
222211
122111
2221
1211
θ θ
θ θ
θ
[ ] [ ]233
1212
1224
4848
4896
20
02
2424
2448
20
02
L
EI
L
EI
L
L
L
EI
L
L
k Y k
xen sresistente jlineas
lj j x
−
−=
−
−
−
−−
−
−
−=−= ∑θ
[ ] [ ]∑−=
xen sresistente jlineas
ljlj x k Y k θ
- % e 9
- % e
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 111111111
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 211211212
−−== θ
[ ] [ ] [ ] [ ] [ ] 1111121112111 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1211122112212 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 111
2
11211
2
12
11
2
111
2
1111
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++==θθ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 21121112222112
22112211212
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 122122121
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 222222222
−−== θ
[ ] [ ] [ ] [ ] [ ] 1122121222121 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1222122222222 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 11112211221222
11221122121
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 122
2
21222
2
22
22
2
222
2
2222
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++== θθ
[ ] [ ][ ]
=
2
1
piso A
piso A
A y yY [ ] [ ]
[ ]
=
2
1
piso B
piso B
B y yY [ ] [ ] [ ]
=
2
1
piso j
piso j
j y yY
/'"!iz 6x7
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 13/16
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ][ ] [ ][ ] [ ] [ ]∑+=++=
+
+=
=
yen sresistente jlineas
ljljl l
l pisol piso
l pisol piso
l pisol piso
l pisol piso
y y
y y
y k X k xk xk xk x
k xk x
k xk x
k xk x
k k
k k k
1122
1222112111
1122111111
2222222112
2122221112
2221
1211
θ θ
θ θ
θ
∑+=
+
+=
yen sresistente jlineas
ljlj
l l l l
y k X k k
k k
x
x
k k
k k
x
xk
22221
1211
22
1
12221
1211
12
1
0
0
0
0
θ
2331212
1224
2424
2448
2
0
02
4848
4896
2
0
02
L
EI
L
EI
L
L
L
EI
L
L
k lB
lB
lA
lA
y
−
−=
−
−
+
−
−
−
−+=
θ
[ ] [ ]∑+=
yen sresistente jlineas
ljlj y k X k θ
Con &os dos es"'dos e&e/en"'&es -%e95 -%e: 5 es *osi(&e gene!'! &'s ec-'ciones *'!' &'s
s-( /'"!ices 6x75 6y7 677
- % e 9
- % e
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 111111111
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 211211212
−−== θ
[ ] [ ] [ ] [ ] [ ] 1111121112111 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1211122112212 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 111
2
11211
2
12
11
2
111
2
1111
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++==θθ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 21121112222112
22112211212
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 122122121
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 222222222
−−== θ
[ ] [ ] [ ] [ ] [ ] 1122121222121 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1222122222222 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 11112211221222
11221122121
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 122
2
21222
2
22
22
2
222
2
2222
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++== θθ
/'"!iz 6y7
[ ]
[ ]
[ ]
= 22
12
2
piso
piso
x
x
X [ ]
[ ]
[ ]
= 21
11
1
piso
piso
x
x
X [ ]
[ ]
[ ]
= 2
1
piso j
piso j
j x
x
X
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 14/16
Con &os dos es"'dos e&e/en"'&es -%e95 -%e: 5 es *osi(&e gene!'! &'s ec-'ciones *'!' &'s
s-( /'"!ices 6x75 6y7 677
- % e 9
- % e
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 111111111
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 211211212
−−== θ
[ ] [ ] [ ] [ ] [ ] 1111121112111 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1211122112212 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 111
2
11211
2
12
11
2
111
2
1111
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++==θθ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]21121112222112
22112211212
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 122122121
−−== θ
[ ] [ ] [ ] [ ] [ ]lA piso AlB piso B x
x k yk yk P 222222222
−−== θ
[ ] [ ] [ ] [ ] [ ] 1122121222121 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] 1222122222222 l pisol piso y
y k xk xk P +== θ
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 11112211221222
11221122121
pisol piso pisol piso
piso AlA piso A piso BlB piso B
xk x xk x
yk y yk yk M
+
++== θθ
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 122
2
21222
2
22
22
2
222
2
2222
l pisol piso
lA piso AlB piso B
k xk x
k yk yk M
+
++== θθ
/'"!iz 677
[ ] [ ]
[ ] [ ][ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] +
+
=
=
lA piso A piso AlA piso A
piso AlA piso AlA piso A
lB piso B piso BlB piso B
piso BlB piso BlB piso B
k y yk y
yk yk y
k y yk y
yk yk y
k k
k k k
11
2
22211
112211
2
1
11
2
22211
112211
2
1
2221
1211
θθ θθ
θθ θθ
θθ
=
−
−
+
−
−
−
−
−
−+
−
−
−
−
−
−+
−
−
=3
20
02
2424
2448
20
02
20
02
4848
4896
20
02
20
02
4848
4896
20
02
20
02
2424
2448
20
02
L
EI
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
k θθ
[ ] [ ] [ ] [ ] [ ] [ ]∑∑ +=
yen sresistente jlineas
ljljlj
xen sresistente jlineas
ljljlj X k X Y k Y k θθ
[ ] [ ]
[ ]
=
2
1
piso j
piso j
j x
x X
[ ] [ ]
[ ]
=
2
1
piso j
piso j
j y
yY
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] =
+
+
111
2
212112111
1111221111
2
11
211
2
222222112
1221222211
2
12
l piso pisol piso
pisol pisol piso
l piso pisol piso
pisol pisol piso
k x xk x
xk xk x
k x xk x
xk xk x
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] =+++= 2222211111 l l l l l l lA AlAlA AlB BlBlB B xk x xk x yk y yk y
L
EI
−
−
3636
3672
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 15/16
RES;ME)
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]
=
2
1
2
1
2
1
222122212221
121112111211
22212221
12111211
22212221
12111211
2
1
2
1
2
1
00
00
00
00
θ
θ
θθ θθ θ θ θ θ
θθ θθ θ θ θ θ
θ θ
θ θ
θ θ
θ θ
x
y
x
x
y y x x
y y x x
y y yy yy
y y yy yy
x x xx xx
x x xx xx
y
y
x
x
u
u
u
u
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
M
M
P
P
P
P
[ ]∑=
xen sresistente jlineas
lj xx k k
[ ] [ ]∑−=
xen sresistente jlineas
ljlj x k Y k θ
[ ]∑=
xen sresistente jlineas
lj yy k k
[ ] [ ]∑−=
yen sresistente jlineas
ljlj y k X k θ
[ ] [ ]∑+=
yen sresistente
jlineas
ljlj y k X k θ
[ ] [ ]∑−=
xen sresistente jlineas
ljlj x k Y k θ
[ ] [ ] [ ]
[ ] [ ] [ ]∑
∑ +=
yen sresistente jlineas
ljljlj
xen sresistente jlineas
ljljlj
X k X
Y k Y k θθ
[ ] [ ] [ ]
[ ] [ ][ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
+−
=
∑∑∑∑
∑∑
∑∑
2
1
2
1
2
1
2
1
2
1
2
1
0000
00
00
θ
θ
x
y
x
x
yen sresistente jlineas
ljljlj
xen sresistente jlineas
ljljlj
yen sresistente jlineas
ljlj
xen sresistente jlineas
ljlj
xen sresistente jlineas
lj j
xen sresistente jlineas
lj
yen sresistente jlineas
ljlj
xen sresistente jlineas
lj
y
y
x
x
uu
u
u
X k X Y k Y k X k Y
k yk
k X k
M
M
P P
P
P
[ ] u K P ⋅=
8/16/2019 Matriz de Rigidez Para Edificios 3d1
http://slidepdf.com/reader/full/matriz-de-rigidez-para-edificios-3d1 16/16
32
2424
2448
L
EI k k B
−
−==
l l l k Q ∆=L
L
Pó!"ico
#
A
$
# $% Pó!"ico A%
1∆
2∆
EI EI
EI EI
P&'n"'
31
4848
4896
L
EI k k A
−
−==
#EI #EI
#EI #EI
2221
1211
k k
k k E<EMPLO
2221
1211
k k
k k
[ ] [ ]
−
−==
20
02
1 L
L
Y X A[ ] [ ]
==
20
02
2 L
L
Y X B
3
2
3636
3672
1212
1224
1212
1224
1212
1224
7272
72144
00
00
1212
1224
00
00
7272
72144
L
EI
L L L
L
L
K
−
−
−
−
−
−
−
−
−
−
−
−
−
−
=