Formulario Vigas y Pórticos: Análisis Estructural

3
FORMULARIO PARA
VIGAS Y PÓRTICOS
3.1
Formulario para vigas y pórticos
3.1
Obtención de la Distribución de Solicitaciones mediante la
Formulación de Macaulay
Las Funciones de Macaulay permiten expresar tanto la distribución de cargas
sobre una viga sometida a flexión como las leyes de Cortantes o Momentos
Flectores generadas por dichas cargas. A continuación se muestra la expresión de tales funciones y las condiciones en las que deben aplicarse.
q( x) = ∑
A⋅ x − a
(c− 2)
T ( x ) = −∑
A⋅ x − a
( c −1)
M( x ) = − ∑
A⋅ x − a
( c − 2 )!
( c − 1) !
c
c!
ecuaciones validas solo si n ≥ 0
en las expresiones
si
y si
n=0
n>0
x−a
n
x−a
0
=0
x≥a
x−a
0
=1
x≤a
x−a
n
=0
x≥a
x−a
n
= ( x − a)
x≤a
n
En la siguientes tablas se particularizan estas funciones para cada caso de
carga y se indica el valor que deberían tomar los parámetros A y c en la
ecuación general previamente indicada.
3.2
Prontuario para Cálculo de Estructuras
M
Si
x≤a
a
x≥a
x
x−a
0
=0
x−a
0
=1
entonces
M(x)
M( x ) = − M x − a
0
A=M
por lo tanto
c=0
P
a
x
T(x)
M(x)
Si
x≤a
x−a = 0
x≥a
x − a = ( x − a)
1
1
1
entonces
T ( x) = − P x − a
0
M( x ) = − P x − a
1
por lo tanto
A=P
c =1
3.3
Limitación de las Deformaciones
Si
x≤a
q
x≥a
x−a
x−a
2
2
=0
= ( x − a)
2
a
entonces
x
q( x) = q x − a
0
q
1
x−a
1
q
2
M( x ) = −
x−a
2 ⋅1
T(x)
T ( x) = −
2
M(x)
A=q
por lo tanto
c=2
Si
q
x−a
x≥a
3
x−a
3
=0
= ( x − a)
d
a
entonces
x
2
T(x)
3
M(x)
x≤a
qd
1
x−a
1
qd
2
T ( x) = −
x−a
2 ⋅1
qd
3
M( x ) = −
x−a
3 ⋅ 2 ⋅1
q( x) =
por lo tanto
q
d
c=3
A=
3
3.4
Prontuario para Cálculo de Estructuras
Otros casos de carga que se resuelven por superposición de los anteriores
q
q
 −⟨ x-a⟩ 2 + ⟨ x-b⟩ 2 
2! 
T ( x ) = q ⋅ [ −⟨ x-a⟩ + ⟨ x-b⟩ ]
M(x ) =
a
b
x
q/d
q
a
d
b
q
q/d
 -⟨ x-a⟩ 3 + ⟨ x-b⟩ 3 
⟨ x-b⟩ 2 +
2!
3! 
q/d
-⟨ x-a⟩ 2 + ⟨ x-b⟩ 2 
T ( x ) = q ⋅ ⟨ x-b⟩ +

2! 
M(x) =
x
q/d
q
a
q
q/d
⟨ x-a⟩ 3 − ⟨ x-b⟩ 3 
⟨ x-a⟩ 2 +
2!
3! 
q/d
⟨ x-a⟩ 2 − ⟨ x-b⟩ 2 
T ( x ) = −q ⋅ ⟨ x-a⟩ +

2! 
M(x ) = −
d
b
x
qb
qa
a
M( x ) = −
qa
2!
⟨ x-a⟩ 2 +
qb
2!
⟨ x-b⟩ 2 +
T ( x ) = −q a ⟨ x-a⟩ + q b ⟨ x-b⟩ +
d
(q − q ) /d  −⟨ x-a⟩ + ⟨ x-b⟩ 
b
a
3!
3

3
( q − q ) /d  −⟨ x-a⟩ + ⟨ x-b⟩ 
b
a
2

2!
2

b
x
qa
a
qb
d
b
x
M( x ) = −
qa
2!
⟨ x-a⟩ 2 +
qb
2!
⟨ x-b⟩ 2 +
T ( x ) = −q a ⟨ x-a⟩ + q b ⟨ x-b⟩ +
(q − q ) /d ⟨ x-a⟩ − ⟨ x-b⟩ 
a
b
3!
3

3
(q − q ) /d ⟨ x-a⟩ − ⟨ x-b⟩ 
a
b
2!

2
2



VIGA APOYADA EN LOS EXTREMOS
3.2.1
CARGA PUNTUAL
P
A
REACCIONES
P⋅b
RA =
L
RB =
B
C
P⋅a
L
x
ESFUERZOS CORTANTES
P⋅b
P⋅a
QAC =
= cte ; QCB = −
= cte
L
L
a
b
Formulario para vigas y pórticos
3.2
L
MOMENTOS FLECTORES
P⋅b
P⋅a
MAC =
⋅ x ; MCB =
⋅ (L − x)
L
L
P ⋅ a⋅ b
para x0 = a
Mmax = MC =
L
ANGULOS DE GIRO
P⋅ a⋅ b
ϕA =
⋅ ( L + b)
6⋅E⋅I⋅L
; ϕB = −
QB
P⋅ a⋅ b
⋅ ( L + a)
6⋅ E⋅I ⋅ L
; ϕC =
P⋅ a⋅ b
⋅ ( b − a)
3⋅E⋅I⋅L
QA
ECUACION DE LA ELASTICA
y AC =
P ⋅ L ⋅ b ⋅ x  b2 x 2 
⋅ 1− 2 − 2 
6⋅ E⋅I 
L
L 
;
y CB =
2
P ⋅ L ⋅ a ⋅ ( L − x )  a2  L − x  
⋅ 1− 2 − 
 

6⋅ E⋅I
L  L  

FLECHA MAXIMA
fC =
P⋅b
(
9⋅E⋅I⋅L 3
⋅ L2 − b2
)
3
2
para x =
L2 − b2
3
M max
3.5
3.6
3.2.2
CARGA PUNTUAL CENTRAL
REACCIONES
RA = RB =
P
A
P
2
B
C
ESFUERZOS CORTANTES
QAC = QCB = −
x
P
= cte
2
a
MOMENTOS FLECTORES
a
L
P
MAC = ⋅ x
2
P
; MCB = ⋅ ( L − x )
2
P⋅L
L
Mmax = MC =
para x0 =
4
2
QB
ANGULOS DE GIRO
P ⋅ L2
16 ⋅ E ⋅ I
; ϕC = 0
QA
ECUACION DE LA ELASTICA
y AC =
P ⋅ L2 ⋅ x  4 x 2 
⋅ 1− ⋅ 
16 ⋅ E ⋅ I  3 L2 
FLECHA MAXIMA
fC =
P ⋅ L3
48 ⋅ E ⋅ I
M max
Prontuario para Cálculo de Estructuras
ϕ A = −ϕ B =
CARGA CONTINUA EN PARTE DE LA VIGA
REACCIONES
p⋅b⋅c
RA =
L
RB =
c
p⋅ a⋅ c
L
p
ESFUERZOS CORTANTES
p⋅b⋅c
p⋅b⋅c
c

QAC =
; QCD =
− p⋅ − a+ x
L
L
2


A
C
MOMENTOS FLECTORES
p⋅b⋅c
p⋅b⋅c
p  
c 
MAC =
⋅ x ; MCD =
⋅ x − ⋅  x −  a −  2
L
L
2  
2 
p⋅a⋅c
⋅ (L − x)
L
p⋅b⋅c 
b⋅c
Mmax =
⋅2⋅a− c +

2⋅L 
L 
D
x
a
MDB =
ANGULOS DE GIRO
p⋅a⋅b⋅c 
c2 
⋅L + b −
ϕA =

6⋅E⋅I⋅L 
4⋅a
B
p⋅a⋅c
; QDB = −
L
Formulario para vigas y pórticos
3.2.3
b
L
x0 = a −
para
; ϕB = −
c b⋅c
+
2
L
p⋅ a⋅ b⋅c 
c2 
⋅L + a −

6⋅E⋅I⋅L 
4⋅b
QB
QA
ECUACION DE LA ELASTICA

p⋅b⋅c x  2
c2  
y AC =
⋅
−x + a ⋅  L + b −

6 ⋅ L E ⋅ I 
4 ⋅ a  

4
  

p
c 
c2  
⋅  L ⋅  x −  a −   − 4 ⋅ b ⋅ c ⋅ x3 + 4 ⋅ a ⋅ b ⋅ c ⋅  L + b −
⋅ x
24 ⋅ E ⋅ I ⋅ L   
2 
4⋅a 



2



p⋅ a⋅ c L − x
c 
2
⋅
⋅ − ( L − x ) + b ⋅  L + a −
y DB =

6⋅L
4
⋅ a  
E ⋅ I 

y CD =
M max
3.7
3.8
3.2.4
CARGA CONTINUA EN TODA LA VIGA
p
REACCIONES
RA = RB =
p⋅L
2
A
B
ESFUERZOS CORTANTES
L

QAB = p ⋅  − x 
2

;
QA = −QB =
x
P⋅L
2
L
MOMENTOS FLECTORES
p⋅x
⋅ (L − x )
2
p ⋅ L2
Mm a x =
p a ra
8
MAB =
x0 =
ANGULOS DE GIRO
VB
VA
p ⋅ L3
24 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
y AB =
p⋅ x
⋅ x3 − 2 ⋅ L ⋅ x2 + L3
24 ⋅ E ⋅ I
)
5 ⋅ p ⋅ L4
384 ⋅ E ⋅ I
L
2
ymax =
(
para
x0 =
M max
Prontuario para Cálculo de Estructuras
ϕ A = −ϕ B = −
L
2
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
1
RA = ( 2 ⋅ p1 + p2 )
6
; RB =
1
( p1 + 2 ⋅ p2 ) .
6
p1
p
2
ESFUERZOS CORTANTES
p ( 3 ⋅ L − x ) + p2 ⋅ x 2
QA = RA ; Qx = RA − 1
⋅x
6⋅L
;
QB = −RB
B
A
MOMENTOS FLECTORES
p ( 3 L − x ) + p2 ⋅ x 2
Mx = RA ⋅ x − 1
⋅x
6⋅L
x
L
2
2
L
L
⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 )
2
2


1
1
para x 0 =
⋅  − p1 +
⋅ p12 + p22 + p1 ⋅ p2 
p2 − p1 
3

Mmax comprendido entre 0,125 ⋅
(
ANGULOS DE GIRO
L3
ϕA =
⋅ ( 8 ⋅ p1 + 7 ⋅ p2 )
360 ⋅ E ⋅ I
Formulario para vigas y pórticos
3.2.5
)
; ϕB = −
QA
QB
L3
⋅ ( 7 ⋅ p1 + 8 ⋅ p2 )
360 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
x ( L − x ) 3 ( p1 − p2 ) x − 3 ( 4p1 + p2 ) Lx + 


360EI ( 8 p1 + 7p2 ) L2 x + ( 8p1 + 7p2 ) L3 


3
yx =
0,01304 ⋅
( p1 + p2 ) ⋅ L4
2⋅E⋅I
x0
M max
3.9
FLECHA MAXIMA
( p + p2 ) ⋅ L4 y
entre 0,01302 ⋅ 1
2⋅E⋅I
2
3.10
3.2.6
MOMENTO FLECTOR
REACCIONES
R A = −R B = −
M
M
L
C
ESFUERZOS CORTANTES
M
Qx =
= cte
L
MOMENTOS FLECTORES
M
M
MAC = − ⋅ x
MCB = − ⋅ ( L − x )
L
L
M
M
izq
der
MC = − ⋅ a
MC = − ⋅ b
L
L
A
M = MCizq + Mder
C
QA
QB
2
M ⋅ L ⋅ (L − x) 
a2  L − x  
yCB = −
⋅ 1− 3 ⋅ 2 − 
 

6⋅E⋅I
L  L  

MC
M
MC
Prontuario para Cálculo de Estructuras
)
ECUACION DE LA ELASTICA
M⋅ L ⋅ x 
b2 x 2 
y AC = −
⋅ 1− 3 ⋅ 2 − 2 
6⋅ E⋅I 
L
L 
FLECHA
M⋅ a ⋅ b
fC =
⋅ ( b − a)
3⋅E⋅I⋅L
b
L
ANGULOS DE GIRO
M ⋅ L  b2 
M ⋅ L  a2 
ϕA =
⋅  3 ⋅ 2 − 1 ; ϕ B =
⋅3⋅
− 1
6⋅E⋅I  L
6 ⋅ E ⋅ I  L2


M
⋅ a3 + b3
ϕC =
3 ⋅ E ⋅ I ⋅ L2
(
B
a
3.3.1
P
CARGA PUNTUAL
REACCIONES
P ⋅ b2
RA = 3 ⋅ ( L + 2 ⋅ a)
L
;
RB =
Formulario para vigas y pórticos
3.3 VIGA EMPOTRADA EN LOS EXTREMOS
C
P ⋅ a2
⋅ ( L + 2 ⋅ b)
L3
B
A
x
ESFUERZOS CORTANTES
P ⋅ b2
QAC = 3 ⋅ ( L + 2 ⋅ a) = cte ;
L
QCB = −
P ⋅ a2
⋅ ( L + 2 ⋅ b ) = cte
L3
a
b
L
MOMENTOS FLECTORES
P ⋅ a ⋅ b2
P ⋅ a2 ⋅ b
P ⋅ b2
; MB = −
; MAC = 3 ⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L )
2
2
L
L
L
P ⋅ a2
2 ⋅ P ⋅ a2 ⋅ b2
MBC = 3 ⋅ L ⋅ b + L2 − L ⋅ x − 2 ⋅ b ⋅ x ; MC =
para x0 = a
L
L3
MA = −
(
)
QB
QA
ECUACION DE LA ELASTICA
P ⋅ b2 
2 ⋅ a ⋅ x  x2
⋅3 ⋅a − x −
⋅
6⋅E⋅I 
L  L2
P ⋅ a2 
L − x ⋅  (L − x)
yBC =
⋅  3 ⋅ b − ( L − x) − 2 ⋅ b
⋅
6⋅E⋅I 
L 
L2
P ⋅ a3 ⋅ b3
3 ⋅ E ⋅ I ⋅ L3
;
fmax =
2 ⋅a⋅ L
x=
L + 2⋅a
2 ⋅ P ⋅ a3 ⋅ b2
3 ⋅ E ⋅ I ⋅ ( L + 2 ⋅ a)
2
0
3.11
para
MB
MA
MC
FLECHAS
fC =
2
x
y AC =
3.12
3.3.2
CARGA PUNTUAL CENTRAL
REACCIONES
RA = RB =
P
P
2
C
P
QAC = = cte
2
B
A
ESFUERZOS CORTANTES
x
P
QCB = − = cte
2
a
a
L
MOMENTOS FLECTORES
MA = MB = −
P⋅L
8
Mmax = MC =
P⋅L
8
para
MAC =
PL  x 
⋅ 4 ⋅ −1
8  L 
x0 =
L
2
QB
QA
y AC = −
P ⋅ L ⋅ x2 
x
⋅3 − 4⋅ 
48 ⋅ E ⋅ I 
L
MB
MA
FLECHAS
M max
0
x
P ⋅ L3
fC =
192 ⋅ E ⋅ I
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA
CARGA CONTINUA EN PARTE DE LA VIGA
c
REACCIONES
p ⋅ b ⋅ c MA − MB
RA =
−
L
L
;
p
p ⋅ a ⋅ c MA − MB
RB =
+
L
L
C
ESFUERZOS CORTANTES
MBD = RB ⋅ ( L − x ) + MB
a
MCD = RA ⋅ x + MA −
;
B
x
MOMENTOS FLECTORES
;
D
A
c

QAC = RA = cte ; QBD = −RB = cte ; QCD = RA − p ⋅  x − a + 
a

MAC = RA ⋅ x + MA
MA = −
p 
c
⋅ x−a+ 
2 
2
Formulario para vigas y pórticos
3.3.3
b
2
12 ⋅ a ⋅ b2 
p ⋅ c3 
⋅ L − 3⋅b +

2 
c2
12 ⋅ L 

p ⋅ c3 
12 ⋅ a2 ⋅ b 
MB = −
L
a
⋅
−
3
⋅
+


12 ⋅ L2 
c2

L
Q
A
Q
B
ECUACION DE LA ELASTICA
x2
⋅ ( −3 ⋅ MA − RA ⋅ x )
6⋅E⋅I
4
 

1
c
yCD =
⋅  p ⋅  x − a +  − 4 ⋅ RA ⋅ x3 − 12 ⋅ MA ⋅ x3 
24 ⋅ E ⋅ I  
2

1
RB x3 − 3 ( MB + LRB ) x2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2 
y DB =

6EI 
y AC =
MA
MB
3.13
3.14
3.3.4
CARGA CONTINUA EN TODA LA VIGA
p
REACCIONES
RA = RB =
P⋅L
2
ESFUERZOS CORTANTES
B
A
P
QAB = Qx = ⋅ ( L − 2 ⋅ x )
2
x
L
MOMENTOS FLECTORES
MA = MB = −
P ⋅ L2
12
P
⋅ L2 − 6 ⋅ L ⋅ x + 6 ⋅ x 2
12
P ⋅ L2
L
Mcentro =
para x =
24
2
Mx = 0 para x0 = 0,2113 ⋅ L
Q
ECUACION DE LA ELASTICA
MA
Mx = −
)
P ⋅ L4  x x 2 
⋅ − 
24 ⋅ E ⋅ I  L L2 
ymax =
P ⋅ L4
384 ⋅ E ⋅ I
A
Q
B
2
para
x=
L
2
MB
Prontuario para Cálculo de Estructuras
yx =
(
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
p1
L
M − MB
⋅ ( 2 ⋅ p1 + p2 ) − A
6
L
L
MA − MB
RB = ⋅ ( p1 + 2 ⋅ p2 ) +
6
L
p2
RA =
ESFUERZOS CORTANTES
QA = RA
Qx = RA −
QB = −RB
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x
2⋅L
x
L
⋅x
MOMENTOS FLECTORES
L2
MA = −
⋅ ( 3 ⋅ p1 + 2 ⋅ p2 )
60
p ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 2
Mx = RA ⋅ x + MA − 1
⋅x
6⋅L
L2
MB = − ( 2 ⋅ p1 + 3 ⋅ p2 )
60
ECUACION DE LA ELASTICA
yx =
B
A
Formulario para vigas y pórticos
3.3.5
A
Q
B
MA
MB
3.15
 ( p − p1 ) 3

x2
⋅ 2
⋅ x + p1 ⋅ L ⋅ x2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L
24 ⋅ E ⋅ I ⋅ L 
5

Q
3.16
3.3.6
MOMENTO FLECTOR
REACCIONES
6⋅M
RA = − 3 ⋅ a ⋅ b
L
; RB =
6⋅ M
⋅ a⋅ b
L3
C
ESFUERZOS CORTANTES
Qx = −
+M
x
6⋅ M
⋅ a ⋅ b = cte
L3
a
b
MOMENTOS FLECTORES
M⋅ a 
b
MA =
⋅2 − 3⋅ 
L 
L
MAC =
B
A
L
M⋅ b 
a
MB = −
⋅2 − 3⋅ 
L 
L
M⋅ a  a 
x 
⋅  3 ⋅ ⋅ 1− 2 ⋅  − 1
L  L 
L 
MCB = −
QA
M⋅ b  b 
L− x 
⋅  3 ⋅ ⋅ 1− 2 ⋅
− 1
L  L 
L  
6⋅ M 2
⋅a ⋅b
L3
;
(
M
MCder = MA + 3 ⋅ L3 − 6 ⋅ a2 ⋅ b
L
)
ECUACION DE LA ELASTICA
y AC =
y BC =
M⋅ b ⋅ x2 
L− x b
⋅2 ⋅a⋅ 2 − 
2⋅E⋅I⋅L 
L
L
M⋅ a ⋅ ( L − x )
2⋅E⋅I⋅L
2
 b⋅ x a
⋅2⋅ 2 − 
L
L

FLECHA
M ⋅ a2 ⋅ b2
fC = −
⋅ ( a − b)
2 ⋅ E ⋅ I ⋅ L3
MC
MA
MC
MB
Prontuario para Cálculo de Estructuras
MCizq = MA −
QB
3.4.1
P
CARGA PUNTUAL
REACCIONES
P ⋅ b2
P⋅a
⋅ ( 3 ⋅ L − b ) ; RB =
⋅ 3 ⋅ L2 − a2
RA =
2 ⋅ L3
2 ⋅ L3
ESFUERZOS CORTANTES
P ⋅ b2
P⋅a
QAC = −
⋅ ( 3 ⋅ L − b ) = cte ; QCB = −
⋅ 3 ⋅ L2 − a2 = cte
2 ⋅ L3
2 ⋅ L3
(
x
a
b
L
)
(
2
; ϕC =
4⋅E⋅I⋅L
)
Q
B
Q
A
ANGULOS DE GIRO
P ⋅ a ( L − a)
B
A
)
MOMENTOS FLECTORES
P⋅a 2
P⋅a 2
MB = −
⋅ L − a2
⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b)
; MC =
2 ⋅ L2
2 ⋅ L3
P⋅ x 2
P⋅a
MAC =
⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b ) ; MCB =
⋅ 2 ⋅ L3 − 3 ⋅ L2 ⋅ x + a2 ⋅ x
2 ⋅ L3
2 ⋅ L3
ϕA =
C
)
(
(
P ⋅ a ⋅ ( L − a)
2
4 ⋅ E ⋅ I ⋅ L3
(
⋅ L2 − 2 ⋅ a ⋅ L − a2
)
ECUACION DE LA ELASTICA
P ⋅ b2 ⋅ x
y AC =
⋅ 3 ⋅ a ⋅ L2 − x2 ⋅ ( 2 ⋅ L + a) 
12 ⋅ E ⋅ I ⋅ L3 
y BC =
P ⋅ a ⋅ ( L − x)
2
12 ⋅ E ⋅ I
MB
  a2  
a2   L − x  
⋅ 3 ⋅ 1− 2  −  3 − 2  ⋅ 

L  
L   L  
 
FLECHA MAXIMA
p⋅b ⋅a
a
⋅
6⋅ E⋅I
2⋅L + a
MC
para x=L ⋅
a
2⋅L+a
3.17
fmax =
2
Formulario para vigas y pórticos
3.4 VIGA APOYADA-EMPOTRADA
3.18
3.4.2
CARGA PUNTUAL CENTRAL
REACCIONES
5
11
RA =
⋅ P ; RB =
⋅P
16
16
P
C
ESFUERZOS CORTANTES
5
11
QAC =
⋅ P = cte ; QCB = − ⋅ P = cte
16
16
B
A
x
MOMENTOS FLECTORES
a
3
5
MB = − ⋅ P ⋅ L ; MC =
⋅P⋅L
16
32
5
P⋅ L  L − x

MAC =
⋅ P ⋅ x ; MCB =
⋅ 11⋅
− 3
16
16 
L

a
L
ANGULOS DE GIRO
P⋅L
32 ⋅ E ⋅ I
QB
; ϕC = −
P⋅L
128 ⋅ E ⋅ I
2
QA
ECUACION DE LA ELASTICA
y AC =

P ⋅ L2
x2 
⋅ x ⋅ 3 − 5 ⋅ 2 
96 ⋅ E ⋅ I
L 

y BC =
P⋅L
L − x

⋅ (L − x)2 ⋅ 9 − 11⋅
96 ⋅ E ⋅ I
L 

MB
FLECHA MAXIMA
fC =
7 ⋅ P ⋅ L3
;
768 ⋅ E ⋅ I
fmax =
P ⋅ L3
48 ⋅ 5 ⋅E ⋅ I
para x=
L
5
MB
Prontuario para Cálculo de Estructuras
ϕA =
2
CARGA CONTINUA EN PARTE DE LA VIGA
REACCIONES
p ⋅ b ⋅ c MB
+
RA =
L
L
c
; RB =
p
p ⋅ a ⋅ c MB
−
L
L
ESFUERZOS CORTANTES
c

QAC = RA = cte ; QDB = −RB = cte ; QCD = RA − p ⋅  x − a + 
2

C
;
;
B
x
p 
c
MCD = RA ⋅ x − ⋅  x − a + 
2 
2
MDB = RB ⋅ ( L − x ) + MB
D
A
MOMENTOS FLECTORES
MAC = RA ⋅ x
Formulario para vigas y pórticos
3.4.3
MB = −
a
2
p⋅ a⋅ b⋅c 
c2 
⋅L + a−

2
2⋅L
4⋅b

ANGULOS DE GIRO

p ⋅ c3
12 ⋅ a ⋅ b2 
ϕA =
⋅L − 3⋅b +

48 ⋅ E ⋅ I ⋅ L 
c2

ECUACION DE LA ELASTICA


x
12 ⋅ a ⋅ b2  
y AC =
⋅  −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅  L − 3b +

48 ⋅ E ⋅ I ⋅ L 
c2

 
b
L
QB
QA
MB
4

c 

 −8 ⋅ RA ⋅ L ⋅ x 3 + 2 ⋅ p ⋅ L ⋅  x − a +  + 
4 
1


⋅
yCD =

2
48 ⋅ E ⋅ I ⋅ L
 + p ⋅ c3 ⋅  L − 3 ⋅ b + 12 ⋅ a ⋅ b  ⋅ x

2


c


( L − x ) ⋅ R ⋅ L − x + 3 ⋅ M 
y DB = −
)
B
 B (
2
3.19
6⋅E⋅I
3.20
3.4.4
CARGA CONTINUA EN TODA LA VIGA
REACCIONES
RA =
3
⋅P⋅L
8
;
RB =
p
5
⋅P⋅L
8
ESFUERZOS CORTANTES
3 x 
QAB = P ⋅ L ⋅  −  ;
8 L 
3
⋅ P ⋅ L;
8
QA =
QB =
5
⋅P⋅L
8
B
A
x
MOMENTOS FLECTORES
L
P⋅x
⋅ (3 ⋅ L − 4 ⋅ x);
8
9
Mmax rel =
⋅ P ⋅ L2 para
128
3
M = 0 para x = ⋅ L
4
MAB =
MB = −
x=
P⋅L
8
2
3
⋅ L;
8
QB
ANGULOS DE GIRO
ϕA =
P ⋅ L3
48 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
y AB =
P⋅x
⋅ (L + 2 ⋅ x) ⋅ (L − x)2
48 ⋅ E ⋅ I
fmax =
P ⋅ L4
185 ⋅ E ⋅ I
para
x=
1+ 33
⋅L
16
MB
Prontuario para Cálculo de Estructuras
QA
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
RA =
L
M
⋅ ( 2 ⋅ p1 + p2 ) + B
6
L
; RB =
L
M
⋅ ( p1 + 2 ⋅ p2 ) − B
6
L
p2
p1
Qx = RA −
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x
2⋅L
B
A
ESFUERZOS CORTANTES
x
⋅x
L
; QB = −RB
Formulario para vigas y pórticos
3.4.5
MOMENTOS FLECTORES
Mx = RA ⋅ x −
p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x
6⋅ L
Q
⋅x
2
;
L2
MB = −
⋅ ( 7 ⋅ p1 + 8 ⋅ p2 )
120
A
Q
B
ANGULOS DE GIRO
ϕA =
L3
⋅ ( 3 ⋅ p1 + 2 ⋅ p2 )
240 ⋅ E ⋅ I
MB
ECUACION DE LA ELASTICA
3.21
( p2 − p1) ⋅ x4 +



x
3
2

yx =
⋅  + 5 ⋅ L ⋅ p1 ⋅ x − 20 ⋅ RA ⋅ L ⋅ x +
120 ⋅ E ⋅ I ⋅ L 

2
3
 + 5 ⋅ L ⋅ 12 ⋅ RA ⋅ L − ( 3 ⋅ p1 + p2 ) ⋅ L  
3.22
3.4.6
MOMENTO FLECTOR
M+
REACCIONES
RA = −RB =
(
3 M 2
⋅ ⋅ L − a2
2 L3
)
ESFUERZOS CORTANTES
a
MOMENTOS FLECTORES
MCder = RA ⋅ a − M ;
(
b
L
) ;
(
M
⋅ L2 − 3 ⋅ a2
2 ⋅ L2

M  x  a2 
MBC = ⋅ 3 ⋅ ⋅ 1− 2  − 2 
2  L 
L 

MCizq = RA ⋅ a
3 M⋅ x 2
⋅
⋅ L − a2
2 L3
B
x
Qx = RA = cte
MAC =
C
A
MB =
;
ANGULOS DE GIRO
)
M
⋅ ( L − a) ⋅ ( 3 ⋅ a − L )
4⋅E⋅I⋅L
 b  a 2

M
ϕC =
⋅ b ⋅  3 ⋅ ⋅ 1+  − 4 
4⋅E⋅I
 L  L 

QB
ϕA =
MC
MB
ECUACION DE LA ELASTICA
M⋅ b ⋅ x 
⋅ −4 ⋅ L3 − x 2 − 3 ⋅ L2 ⋅ ( a + L ) 

4 ⋅ E ⋅ I ⋅ L3 
M
2
yBC =
⋅ ( L − x ) ⋅  2 ⋅ a2 ⋅ L − x ⋅ L2 − a2 


4 ⋅ E ⋅ I ⋅ L3
y AC =
(
)
(
)
MC
Prontuario para Cálculo de Estructuras
QA
3.5.1
CARGA PUNTUAL
C
Formulario para vigas y pórticos
3.5 VIGA EMPOTRADA EN UN EXTREMO
P
REACCIONES
B
A
RB = P
x
ESFUERZOS CORTANTES
QAC = 0 ; QCB = −P = cte
a
b
L
MOMENTOS FLECTORES
MAC = 0
;
MCB = −P ⋅ ( x − a)
;
MB = −P ⋅ b
QB
ANGULOS DE GIRO
ϕ A = ϕC = −
P
⋅ b2
2⋅E⋅I
ECUACION DE LA ELASTICA
y AC =
P ⋅ b2
⋅ ( 3 ⋅ ( L − x ) − b)
6⋅E⋅I
;
y CB =
MB
3.23
FLECHA MAXIMA
P ⋅ b3
P ⋅ b2
fC =
; fA =
⋅ ( 2 ⋅ b + 3 ⋅ a)
3⋅E⋅I
6⋅ E⋅I
P
2
⋅ ( L − x ) ⋅ ( 2 ⋅ b + 3 ⋅ a)
6⋅E⋅I
3.24
3.5.2
CARGA PUNTUAL EN EL EXTREMO
P
REACCIONES
RB = P
A
ESFUERZOS CORTANTES
QAB = −P = cte
B
x
L
MOMENTOS FLECTORES
MAB = −P ⋅ x;
MB =-P ⋅ L
QB
ECUACION DE LA ELASTICA
y AB =
P
⋅ (L − x)2 ⋅ (2 ⋅ L + x)
6⋅E⋅I
FLECHA MAXIMA
fA =
P ⋅ L3
3⋅E⋅I
MB
Prontuario para Cálculo de Estructuras
ANGULOS DE GIRO
P ⋅ L2
ϕA = −
2⋅E⋅I
CARGA CONTINUA EN PARTE DEL VUELO
REACCIONES
RB = p ⋅ c
ESFUERZOS CORTANTES
c

QAC = 0 ; QCD = −p ⋅  x − a + 
2

; QDB = −p ⋅ c = cte
MOMENTOS FLECTORES
2
c

p⋅ x − a+ 
2
MAC = 0 ; MCD = − 
;
2
MDB = − p ⋅ c ⋅ ( x − a) ; MB = − p ⋅ c ⋅ b
ANGULOS DE GIRO
p ⋅ c  2 c2 
ϕD = −
⋅b −

2⋅E⋅I 
4 
; ϕC = −
MD = −
p ⋅ c2
2
p ⋅ c  2 c2 
⋅b +

2⋅E⋅I 
12 
Formulario para vigas y pórticos
3.5.3
; ϕ A = ϕC
ECUACION DE LA ELASTICA
y DB =


p⋅c
p⋅c 
c2 
3
⋅ L − x2 ⋅ ( 2 ⋅ b − a + x ) ; y AC =
⋅ ( a − x ) ⋅  3 ⋅ b2 +
 + 2⋅b 
6⋅E⋅I
6 ⋅ E ⋅ I 
4 


y DC =
4



p
c
c2 
3
⋅  x − a +  + 4 ⋅ c ⋅ ( a − x ) ⋅  3 ⋅ b2 +
 + 8 ⋅ b ⋅ c
24 ⋅ E ⋅ I 
2
4 


(
)
FLECHAS
2
p⋅ c 
c  b c 
⋅ b −  ⋅  + 
E⋅I 
2   3 12 
fC =
2


p ⋅ c 
c
p⋅ c  
c2 
⋅  b +  ⋅ ( 4 ⋅ b − c) + c3  ; fA =
⋅ a ⋅  3 ⋅ b2 +  + 2 ⋅ b3 
12 ⋅ E ⋅ I 
2
6
⋅
E
⋅
I
4

 


3.25
fD =
3.26
3.5.4
CARGA CONTINUA EN TODO EL VUELO
p
REACCIONES
RB = p ⋅ L
B
ESFUERZOS CORTANTES
QAB = −P ⋅ x
A
QB = −P ⋅ L = cte
;
x
L
MOMENTOS FLECTORES
MAB = −
P ⋅ x2
;
2
MB = −P ⋅
L2
2
ANGULOS DE GIRO
QB
P ⋅ L3
6⋅E⋅I
ECUACION DE LA ELASTICA
y AB =
FLECHAS
fA =
(
P
2
⋅ ( L − x ) ⋅ 3 ⋅ L2 + 2 ⋅ L ⋅ x + x 2
24 ⋅ E ⋅ I
P ⋅ L4
8⋅E⋅I
)
MB
Prontuario para Cálculo de Estructuras
ϕA = −
CARGA TRAPEZOIDAL EN TODO EL VUELO
REACCIONES
L
RB = ⋅ ( p1 + p2 )
2
ESFUERZOS CORTANTES
Qx = −
p2 − p1 x2
⋅ − p1 ⋅ x
2
L
p2
p1
L
; QB = − ⋅ ( p1 + p2 )
2
MOMENTOS FLECTORES
x2
Mx = −
⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1
6⋅L 
A
B
Formulario para vigas y pórticos
3.5.5
x
;
L2
MB = − ⋅ ( p2 + 2 ⋅ p1 )
6
L
ANGULOS DE GIRO
ϕA = −
L3 ⋅ ( 3 ⋅ p1 + p2 )
24 ⋅ E ⋅ I
QB
ECUACION DE LA ELASTICA
( L − x ) ⋅ − (
y =
x

L − x)


5⋅ L


24 ⋅ E ⋅ I 
2

2
2
2
−
⋅
L
⋅
L
−
x
⋅
p
+
p
+
⋅
L
⋅
p
+
⋅
p
(
)
(
)
(
)
2
1
2
1 

2
FLECHA
L ⋅ ( 4 ⋅ p2 + 11⋅ p1)
3
⋅ ( p2 − p1 ) + ( L − x ) ⋅ p2 −
2
MB
4
fA =
120 ⋅ E ⋅ I
3.27
3.28
3.5.6
MOMENTO FLECTOR
REACCIONES
M
A
RB = 0
B
ESFUERZO CORTANTE
x
Qx = 0
a
MOMENTOS FLECTORES
MAC = 0
;
MCB = − M = cte
b
L
;
MAC = − M
ANGULOS DE GIRO
ϕC = ϕ A = −
M⋅ b
E⋅I
y AC =
M
⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b)
2⋅E⋅I
; yBC =
M
2
( L − x)
2⋅E⋅I
FLECHA
MB
fC =
M⋅ b
2⋅E⋅I
2
; fA =
M
⋅ b ⋅ ( 2 ⋅ L − b)
2⋅E⋅I
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA
Formulario para vigas y pórticos
3.5.7
MOMENTO FLECTOR EN EXTREMO DEL VUELO
REACCIONES
M
RB = 0
B
A
ESFUERZO CORTANTE
x
Qx = 0
L
MOMENTOS FLECTORES
MAB = − M = cte
ANGULOS DE GIRO
ϕA = −
M⋅ L
E⋅I
ECUACION DE LA ELASTICA
y AC =
(
M
⋅ b ⋅ x2 − 2 ⋅ L ⋅ x + L2
2⋅E⋅I
)
MB
FLECHA
fA =
3.29
M ⋅ L2
2⋅E⋅I
3.30
3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES
P
P
P
A
B
L/2
C
L/2
L
A
L/2
L/2
L
L/2
L
0,688 P
L
0,405 P
0,312 P
A
C
B
L/2
B
C
0,094 P
A
0,312 P
0,094 P
B
C
0,594 P
0,688 P
ESFUERZOS CORTANTES
- 0,188 PL
- 0,094 PL
A
B
0,156 PL
C
0,156 PL
MOMENTOS FLECTORES
A
B
0,203 PL
MOMENTOS FLECTORES
C
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
A
L
Q
B
Q
C
L
0,625 QL
Formulario para vigas y pórticos
Q
A
B
L
L
C
0,375 L
0,437 QL
0,375 QL
0,063 QL
A
B
C
A
B
0,375 QL
0,375 L
0,563 QL
0,437 L
0,625 QL
ESFUERZOS CORTANTES
ESFUERZOS CORTANTES
2
2
- 0,063 QL
- 0,125 QL
A
B
2
C
2
0,07 QL
0,07 QL
A
B
C
2
0,096 QL
MOMENTOS FLECTORES
3.31
MOMENTOS FLECTORES
C
3.32
3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES
Q
A
L
Q
B
C
k L
c QL
d L
a QL
A
Relación
entre
luces
MOMENTOS
FLECTORES
ESFUERZOS CORTANTES
k
a
b
c
d
e
f
g
1,1
0,361
0,639
0,676
0,424
0,065
0,139
0,09
1,2
0,345
0,655
0,729
0,471
0,060
0,155
0,111
1,3
0,326
0,674
0,784
0,516
0,053
0,174
0,133
1,4
0,305
0,695
0,840
0,560
0,047
0,195
0,157
1,5
0,281
0,719
0,896
0,604
0,040
0,219
0,183
1,6
0,255
0,745
0,953
0,647
0,033
0,245
0,209
1,7
0,226
0,774
1,011
0,689
0,026
0,274
0,237
1,8
0,195
0,805
1,070
0,730
0,019
0,305
0,267
1,9
0,161
0,839
1,128
0,772
0,013
0,339
0,298
2,0
0,125
0,875
1,128
0,812
0,008
0,375
0,330
2,1
0,086
0,914
1,247
0,853
0,004
0,414
0,364
2,2
0,045
0,954
1,308
0,892
0,001
0,455
0,399
2,3
0,001
0,999
1,367
0,933
0,000
0,499
0,435
C
B
b QL
d QL
a L
2
f QL
A
B
C
2
e QL
2
g QL
MOMENTOS FLECTORES
k 2 − k +1
8
k f
d= −
2 k
f=
a = 0.5 − f
e=
a2
2
b = 0.5 + f
g=
d2
2
c=
k f
+
2 k
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
A
Q
B
L
Relación
entre
luces
C
k L
c QL
d L
MOMENTOS
FLECTORES
ESFUERZOS CORTANTES
k
a
b
c
d
f
g
2,4
-0,045
1,045
1,427
0,973
0,545
0,473
2,5
-0,094
1,094
1,487
1,013
0,594
0,513
2,6
-0,145
1,145
1,548
1,051
0,645
0,553
2,7
-0,198
1,198
1,608
1,091
0,698
0,595
2,8
-0,255
1,255
1,669
1,130
0,755
0,638
2,9
-0,313
1,313
1,730
1,169
0,813
0,683
3,0
-0,375
1,375
1,791
1,208
0,875
0,730
Formulario para vigas y pórticos
Q
A
C
B
a QL
d QL
b QL
ESFUERZOS CORTANTES
2
f QL
k 2 − k +1
8
k f
d= −
2 k
f=
A
B
2
g QL
e=
a2
2
b = 0.5 + f
g=
c=
k f
+
2
d2
2
3.33
MOMENTOS FLECTORES
C
a = 0.5 − f
3.34
3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES
Q
Q
A
Q
B
C
L
D
k L
a QL
L
a L
b QL
c QL
D
B
Relación
entre
luces
ESFUERZOS
CORTANTES
MOMENTOS
FLECTORES
k
a
b
c
e
f
g
0,6
0,420
0,580
0,300
0,088
0,080
-0,035
0,7
0,418
0,582
0,350
0,087
0,081
-0,020
0,8
0,414
0,586
0,400
0,086
0,086
-0,006
0,9
0,408
0,592
0,450
0,083
0,091
-0,009
C
A
c QL
b QL
a QL
ESFUERZOS CORTANTES
2
2
f QL
f QL
2
g QL
A
B
C
D
2
2
e QL
e QL
MOMENTOS FLECTORES
f=
k3 + 1
12 ⋅ k + 8
a = 0.5 − f
c=
k
2
e=
a2
2
b = 0.5 + f
g=
k2
−f
8
Prontuario para Cálculo de Estructuras
a L
Q
A
Q
B
C
L
D
k L
L
a L
a QL
b QL
c QL
D
C
B
A
c QL
b QL
a QL
a L
ESFUERZOS CORTANTES
2
f QL
B
2
e QL
ESFUERZOS
CORTANTES
MOMENTOS
FLECTORES
k
a
b
c
e
f
g
1,0
0,400
0,600
0,500
0,080
0,100
0,025
1,1
0,390
0,610
0,550
0,076
0,110
0,041
1,2
0,378
0,622
0,600
0,072
0,122
0,058
1,3
0,365
0,635
0,650
0,066
0,135
0,076
1,4
0,349
0,651
0,700
0,061
0,151
0,094
1,5
0,322
0,668
0,750
0,055
0,168
0,113
1,6
0,313
0,687
0,800
0,049
0,187
0,133
1,7
0,292
0,708
0,850
0,043
0,208
0,153
1,8
0,269
0,731
0,900
0,036
0,231
0,174
1,9
0,245
0,755
0,950
0,030
0,255
0,196
2,0
0,219
0,781
1,000
0,024
0,281
0,219
2
f QL
A
Relación
entre
luces
Formulario para vigas y pórticos
Q
C
2
g QL
k3 + 1
12 ⋅ k + 8
a = 0.5 − f
c=
k
2
e=
b = 0.5 + f
D
2
e QL
a2
2
g=
k2
−f
8
3.35
MOMENTOS FLECTORES
f=
3.36
3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
a
3.9.1
s
CARGA REPARTIDA VERTICAL
I h
k= 2⋅
I1 L
y
p
B
N = 3+ 2⋅k
C
x
m
REACCIONES
n
I1
p⋅ s⋅n
VA =
L
p⋅ s⋅m
VD =
L
HA = HD =
I2
h
I1
A
D
3⋅p⋅ s 
s2 
⋅m⋅ n − 
2 ⋅ h⋅ L ⋅ N 
12 
L
MC
3 p⋅ s 
s2 
MB = MC = − ⋅
⋅  m⋅ n − 
2 L⋅N 
12 
En S
Mx = VA ⋅ x −
p ⋅ (x − m)2
− HA ⋅ h
2
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
MB
MOMENTOS FLECTORES
CARGA REPARTIDA VERTICAL UNIFORME
I h
k= 2⋅
I1 L
y
N = 3+ 2⋅k
p
B
REACCIONES
C
x
P⋅L
VA = VD =
2
P ⋅ L2
HA = HD =
4 ⋅ h⋅ N
I2
I1
h
I1
A
Formulario para vigas y pórticos
3.9.2
D
MOMENTOS FLECTORES
L
P ⋅ L2
4⋅N
P ⋅ x ⋅ ( L − x ) P ⋅ L2
Mx =
−
2
4⋅N
MB = MC = −
Mmax,pos =
P ⋅ L2 P ⋅ L2
−
8
4⋅N
MB
para
x=
MC
L
2
HA
HD
VA
VD
3.37
3.38
3.9.3
CARGA REPARTIDA HORIZONTAL
I h
k= 2⋅
I1 L
y
N = 3+ 2⋅k
p
B
C
I2
REACCIONES
VA = VD =
HD =
HA =
p ⋅ h2
2⋅L
I1
y
p ⋅ h ⋅ (2 ⋅ N + k)
8⋅N
A
p ⋅ h ⋅ (6 ⋅ N − k )
MB
MC
MB
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
p ⋅ h2
⋅ (2 ⋅ N − k)
8⋅N
p ⋅ h2
MC = −
(2 ⋅ N + k)
8⋅N
En AB
MB =
p ⋅ y ⋅ (h − y) y
+ ⋅ MB
2
h
D
L
8⋅N
MOMENTOS FLECTORES
MY =
h
I1
CARGA PUNTUAL VERTICAL SOBRE DINTEL
P
m
I h
k= 2⋅
I1 L
y
n
B
N = 3+ 2⋅k
C
I2
REACCIONES
I1
P⋅n
L
P⋅m
VD =
L
VA =
h
I1
A
Formulario para vigas y pórticos
3.9.4
D
3 P ⋅ m⋅ n
HA = HD =
2 L ⋅ h⋅ N
L
MOMENTOS FLECTORES
MB
MC
3 P ⋅ m⋅ n
MB = MC = − ⋅
2 L⋅N
MP
2⋅N−3
MP = P ⋅ m ⋅ n ⋅
2⋅L⋅N
HA
HD
VA
VD
3.39
3.40
3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
3.10.1 CARGA REPARTIDA VERTICAL
I h1
k1 = 3 ⋅
I1 s
y
I h2
k2 = 3 ⋅
I2 s
p
C
s
f
I3
B
REACCIONES
h2
I2
x
p⋅L
VA = VD =
2
h1 + h2
p ⋅ L2
HA = HD =
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
I1
h1
A
D
Ll
MOMENTOS FLECTORES
MC = −
( h1 + h2 ) ⋅ h2
p ⋅ L2
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
En
BC
MX =
p ⋅ x ⋅ (L − x)
f

− HA ⋅  ⋅ x + h1 
2
L


MB
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
MC
( h1 + h2 ) ⋅ h1
p ⋅ L2
MB = −
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
I h1
k1 = 3 ⋅
I1 s
I h2
k2 = 3 ⋅
I2 s
y
C
s
f
p
REACCIONES
I3
B
p⋅h
2⋅L
HA = p ⋅ h1 − HD
2
1
VA = VD =
I1
h1
y
A
h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2
p ⋅ h12
HD =
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB =
p⋅h
2
−
p⋅h
8
3
1
D
L
MOMENTOS FLECTORES
2
1
h2
I2
Formulario para vigas y pórticos
3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
MC
h1 ⋅ ( 4 + 5 ⋅ k 1 ) + 2 ⋅ h2
h ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
2
1
MB
h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2
p ⋅ h12 ⋅ h2
MC =
2
8
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
En AB
MY = HA ⋅ y −
p ⋅ y2
2
HA
HD
VA
VD
3.41
3.42
3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
I h1
k1 = 3 ⋅
I1 s
I h2
k2 = 3 ⋅
I2 s
y
y
I3
p ⋅ f ⋅ ( h1 + h2 )
HA = p ⋅ f − HD
C
s
f
REACCIONES
VA = VD =
p
B
2⋅L
h2
I2
I1
h1
A
2
p ⋅ f 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 )
HD =
⋅
8
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
D
L
MOMENTOS FLECTORES
p ⋅ f ⋅ h1
⋅
8
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB
p ⋅ h2 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 )
⋅
8
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
2
MC = −
En BC
L
p ⋅ y2
MY = −VA ⋅ ⋅ y + HA ⋅ ( y + h1 ) −
f
2
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
MB = p ⋅ f ⋅ h1 −
MC
8 ⋅ h12 ⋅ (1+ k 1) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 )
I h1
k1 = 3 ⋅
I1 s
I h2
k2 = 3 ⋅
I2 s
y
P
f
REACCIONES
I3
P⋅b
L
P⋅a
VD =
L
B
VA =
HA = HD =
C
s
b
I1
h1
A
h1 ⋅ (L + b) + h2 ⋅ (L + a)
P⋅a⋅b
⋅
2 ⋅ L2 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
h2
I2
a
Formulario para vigas y pórticos
3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL
D
L
MOMENTOS FLECTORES
MB = −
MC = −
MP =
P ⋅ a ⋅ b ⋅ h1
2⋅L
2
P ⋅ a ⋅ b ⋅ h2
2 ⋅ L2
⋅
h1 ⋅ ( L + b ) + h2 ⋅ ( L + a )
MC
h ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2
2
1
⋅
2
2
MB
h1 ⋅ ( L + b ) + h2 ⋅ ( L + a)
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
P⋅ a⋅ b
 a⋅f

+ HA ⋅ 
+ h1 
L
 L

MP
HA
HD
VA
VD
3.43
3.44
3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p
Valor de la carga en proyección horizontal
C
I h
k= 2⋅
I1 s
s
I2
REACCIONES
f
I2
B
D
x
p⋅L
2
p ⋅ L2
8⋅h+ 5⋅f
HA = HE =
⋅ 2
32 h ⋅ ( 3 + k ) + f ⋅ ( 3h + f )
VA = VE =
I1
I1
A
h
E
L
MOMENTOS FLECTORES
MC =
p ⋅ L2 ⋅ h
8⋅h+ 5⋅f
⋅ 2
32
h ⋅ (3 + k) + f ⋅ (3 ⋅ h + f )
MC
p ⋅ L2 f + h
+
⋅ MB
8
h
MB
MD
En BC y DC
MX = p ⋅
x ⋅ (L − x)
2
+
MB 
2⋅f ⋅ x
⋅h+
h 
L 
HA
HE
VA
VE
Prontuario para Cálculo de Estructuras
MB = MD = −
Valor de la carga en proyección horizontal
p
I h
k= 2⋅
I1 s
C
s
REACCIONES
I2
p⋅L
8
p⋅L
VE =
8
p ⋅ L2
8⋅h+ 5⋅f
HA = HE =
64 h2 ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )
f
I2
B
VA = 3 ⋅
D
x
I1
I1
A
Formulario para vigas y pórticos
3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
h
E
L
MOMENTOS FLECTORES
MB = MD = −
MC
p ⋅ L2 ⋅ h
8⋅h+ 5⋅f
2
64 h ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )
p ⋅ L2 f + h
+
⋅ MB
16
h
En BC
MC =
MX = p ⋅
x ⋅ (L − x)
2
+
MB
MD
MB 
2⋅f⋅ x
⋅h+
h 
L 
HA
HE
VE
3.45
VA
3.46
3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
I h
k= 2⋅
I1 s
C
s
I2
p
REACCIONES
f
I2
D
B
p⋅h
2⋅L
HA = p ⋅ h − HE
VA = VE =
HE =
2
I1
I1
h
y
( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f
p ⋅ h2
⋅ 2
16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
A
E
L
MOMENTOS FLECTORES
MC
MB
MD
En AB
My = −
p ⋅ y2
+ HA ⋅ y
2
HA
HE
VA
VE
Prontuario para Cálculo de Estructuras
p ⋅ h2
+ MD
2
p ⋅ h2 f + h
MC =
+
⋅ MD
h
4
( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f
p ⋅ h3
⋅ 2
MD = −
16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
MB =
Valor de la carga en proyección vertical
I h
k= 2⋅
I1 s
p
C
s
REACCIONES
I2
D
B
p⋅f
VA = VE =
⋅ ( f + 2 ⋅ h)
2⋅L
HA = p ⋅ f − HE
y
I1
p ⋅ f 8 ⋅ h ⋅ ( k + 3 ) + 5 ⋅ f ⋅ ( f + 4 ⋅ h)
⋅
16
h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
I1
h
x
2
HE =
f
I2
Formulario para vigas y pórticos
3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
A
E
L
MOMENTOS FLECTORES
MC
MB = HA ⋅ h
p ⋅ f2 4 ⋅ h ⋅ ( k + 2) + f ⋅ (5 ⋅ h + f )
⋅ 2
16
h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
2
MC = −
MB
MD = −HE ⋅ h
MD
En BC
Mx = HA ⋅ y − VA ⋅ x − p ⋅
2
2
HA
HE
VA
VE
3.47
f
siendo y = ⋅ x + h
L
( y − h)
3.48
P
3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
C
s
I h
k= 2⋅
I1 s
I2
B
D
REACCIONES
m
n
I1
P⋅n
VA =
L
P⋅m
VA =
L
HA = HE =
f
I2
I1
A
(
P ⋅ m 6 ⋅ h ⋅ L ⋅ n+ f ⋅ 3 ⋅ L − 4 ⋅ m
⋅
4 ⋅ L2
h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
2
2
h
E
L
)
MOMENTOS FLECTORES
MB
MB = MD = −HA ⋅ h
P⋅m h+ f
+
⋅ MB
h
2
h⋅ L + 2 ⋅ f ⋅ m
MP = VA ⋅ m − HA ⋅
L
MD
MC =
HA
HE
VA
VE
Prontuario para Cálculo de Estructuras
MC
3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
I h1
k1 = 3 ⋅
I1 L
y
p
I h2
k2 = 3 ⋅
I2 L
B
C
I3
x
REACCIONES
VA =
h −h
p⋅L p⋅L
+
⋅ 2
2
8 h1 ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2
VD =
h12 − h22
p⋅L p⋅L
−
⋅ 2
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
HA = HD =
h2
I2
2
1
2
2
2
2
h1
I1
Formulario para vigas y pórticos
3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL
D
A
h1 − h2
p ⋅ L2
⋅ 2
8
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
L
MOMENTOS FLECTORES
MB
p ⋅ L2
( h1 + h2 ) ⋅ h1
MB = −
2
8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MC = −
MC
p ⋅ L2
( h1 + h2 ) ⋅ h2
8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
HD
En BC
VD
HA
VA
3.49
p ⋅ x2
Mx = VA ⋅ x −
− HA ⋅ h1
2
3.50
3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
I h1
k1 = 3 ⋅
I1 L
y
I h2
k2 = 3 ⋅
I2 L
p
B
C
I3
REACCIONES
p⋅h
h − h2
− HD ⋅ 1
2⋅L
L
HA = p ⋅ h − HD
VA = VD =
HD =
2
1
h2
I2
h1
I1
D
y
p ⋅ h12
5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2
⋅ 2
8
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
A
L
MOMENTOS FLECTORES
p ⋅ h12 p ⋅ h13
5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2
−
⋅ 2
2
8
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB
MC
MB
p ⋅ h12 ⋅ h2
5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2
MC = −
⋅ 2
8
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
En AB
HD
p ⋅ y2
My = HA ⋅ y −
2
VD
HA
VA
Prontuario para Cálculo de Estructuras
MB = −
I h1
k1 = 3 ⋅
I1 L
P
I h2
k2 = 3 ⋅
I2 L
y
B
C
I3
REACCIONES
( L + b) ⋅ h1 + ( L + a) ⋅ h2
P⋅b P⋅a⋅b
VA =
+
⋅ 2
⋅ h1 − h2
3
L
2⋅L
h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
(
VD =
( L + b) ⋅ h1 + ( L + a) ⋅ h2
P⋅a P⋅a⋅b
−
⋅ h1 − h2
3
2
L
2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
(
a
)
I1
D
A
L
( L + b ) ⋅ h1 + ( L + a) ⋅ h2
P⋅a⋅b
HA = HD =
2
2
2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB
MOMENTOS FLECTORES
MB = −
P ⋅ a ⋅ b ⋅ h1
2⋅L
2
⋅
h2
I2
h1
)
b
Formulario para vigas y pórticos
3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
MC
( L + b ) ⋅ h1 + ( L + a) ⋅ h2
h ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MP
2
1
HD
MC = −
P ⋅ a ⋅ b ⋅ h2
2 ⋅ L2
VD
HA
VA
3.51
MP = VA ⋅ a + MB
( L + b ) ⋅ h1 + ( L + a) ⋅ h2
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
3.52
3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p
B
C
I h
k= 2⋅
I1 L
I2
x
REACCIONES
VA = VD =
I1
p⋅L
2
HA = HD =
p ⋅ L2
4 ⋅ h ⋅ (k + 2)
h
I1
A
D
MOMENTOS FLECTORES
L
MB = MC = −
p ⋅ L2
6 ⋅ ( k + 2)
MB
MC
En BC
Mx =
p ⋅ x ⋅ (L − x)
2
Mmáx positivo =
−
p ⋅ L2
6 ⋅ ( k + 2)
p ⋅ L2 3 ⋅ k + 2
L
⋅
para x =
24
k+2
2
HA
HD
MA
VA
MD
VD
Prontuario para Cálculo de Estructuras
p ⋅ L2
MA = MD =
12 ⋅ ( k + 2 )
I h
k= 2⋅
I1 L
p
B
C
I2
REACCIONES
VA = VD =
p ⋅ h2 ⋅ k
L ⋅ ( 6 ⋅ k + 1)
I1
HA = p ⋅ h − HD
HD =
p ⋅ h ⋅ (2 ⋅ k + 3)
y
8 ⋅ ( k + 2)
A
D
MOMENTOS FLECTORES
MA = −
MB =
MD =
L
p ⋅ h2 
2
1 
⋅5 +
+
24 
6 ⋅ k + 1 k + 2 
2
2 
p ⋅ h2 
⋅ 1−
+
24  6 ⋅ k + 1 k + 2 
MC = −
h
I1
Formulario para vigas y pórticos
3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
MC
MB
MB
p⋅h 
2
2 
⋅3 −
−
24 
6 ⋅ k + 1 k + 2 
2
p ⋅ h2 
2
1 
⋅3 +
−
24 
6 ⋅ k + 1 k + 2 
En AB
HA
HD
MD
MA
VA
VD
3.53
p ⋅ y2
My = −
+ HA ⋅ y + MA
2
3.54
3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
P
m
I h
k= 2⋅
I1 L
n
B
C
I2
REACCIONES
VA =
m ⋅ ( n − m) 
P⋅n 
⋅ 1+ 2

L  L ⋅ ( 6 ⋅ k + 1) 
I1
h
I1
VD = P − VA
HA = HD =
3 ⋅ P ⋅ m⋅ n
2 ⋅ L ⋅ h ⋅ (k + 2)
A
D
L
MOMENTOS FLECTORES
P ⋅ m⋅ n  1
n− m 
⋅
−


2 ⋅ L  k + 2 L ⋅ ( 6 ⋅ k + 1) 
MB = −

P ⋅ m⋅ n  1
n− m
⋅
+


L
 k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1) 
MC = −

P ⋅ m⋅ n  1
n− m
⋅
−
 k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1) 
L


MD =
P ⋅ m⋅ n  1
n− m 
⋅
+


2 ⋅ L  k + 2 L ⋅ ( 6 ⋅ k + 1) 
P ⋅ m ⋅ n n ⋅ MB m ⋅ MC
MP =
+
+
L
L
L
MB
MC
MP
HA
HD
MA
VA
MD
VD
Prontuario para Cálculo de Estructuras
MA =
I h
k= 2⋅
I1 L
P
B
C
I2
REACCIONES
VA = VD =
3 ⋅ P ⋅ h⋅ k
L ⋅ (6 ⋅ k + 1)
HA = HD =
P
2
I1
A
D
MOMENTOS FLECTORES
P ⋅ h 3 ⋅ k +1
⋅
2 6 ⋅ k +1
P⋅h 3⋅k
MB = − MC =
⋅
2 6⋅ k +1
P ⋅ h 3 ⋅ k +1
MD =
⋅
2 6 ⋅ k +1
MA = −
h
I1
Formulario para vigas y pórticos
3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR
L
MB
MC
HA
MA
MD
VD
3.55
VA
HD
3.56
3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
p
3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
Valor de la carga en proyección horizontal
C
I h
k= 2⋅
I1 s
s
I2
f
I2
B
REACCIONES
D
x
p⋅L
2
k ⋅ (4 ⋅ h + 5 ⋅ f ) + f
p ⋅ L2
HA = HE =
⋅
8 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
I1
VA = VE =
(
I1
A
h
E
)
L
MOMENTOS FLECTORES
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f )
⋅
48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
MB = MD = −
(
MC
)
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f
p⋅L
⋅
48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
2
2
(
p ⋅ L2
+ MA − HA ⋅ ( h + f )
8
En BC
MB
)
MD
MC =
2 ⋅ x ⋅ f  p ⋅ x2

−
Mx = MA + VA ⋅ x − HA ⋅  h +
2
L 

HA
HE
MA
VA
ME
VE
Prontuario para Cálculo de Estructuras
MA = ME =
Valor de la carga en proyección horizontal
I h
k= 2⋅
I1 s
p
C
REACCIONES
p⋅L
− VE
2
4 ⋅ k +1
VE = 3 ⋅ p ⋅ L ⋅
32 ⋅ ( 3 ⋅ k + 1)
k ⋅ (4 ⋅ h + 5 ⋅ f ) + f
p ⋅ L2
HA = HE =
⋅
16 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
s
VA =
(
I2
D
B
x
)
I1
I1
A
MOMENTOS FLECTORES
MA =
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f )
p ⋅ L2
⋅
−
2
2
2
96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h
64 ⋅ ( 3 ⋅ k + 1)
ME =
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f )
p ⋅ L2
⋅
+
2
2
2
96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h
64 ⋅ ( 3 ⋅ k + 1)
(
(
L
)
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2
p ⋅ L2
p ⋅ L2
⋅
−
2
2
2
96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h
64 ⋅ ( 3 ⋅ k + 1)
MD = −
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2
p ⋅ L2
p ⋅ L2
⋅
+
2
2
2
96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h
64 ⋅ ( 3 ⋅ k + 1)
(
(
h
E
)
MB = −
En BC
f
I2
Formulario para vigas y pórticos
3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
MC
)
MB
MD
)
HA
HE
MA
VA
ME
VE
3.57
2 ⋅ x ⋅ f  p ⋅ x2

Mx = MA + VA ⋅ x − HA ⋅  h +
−
L 
2

L
MC = VE ⋅ + ME − HE ( f + h)
2
3.58
3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
I h
k= 2⋅
I1 s
C
s
REACCIONES
VA = VE =
p⋅h ⋅k
2 ⋅ L ⋅ ( 3 ⋅ k + 1)
2
I2
p
f
I2
D
B
HA = p ⋅ h − HE
k2 ⋅ h + k ⋅ ( 2 ⋅ f + 3 ⋅ h) + f
p ⋅ h2
HE =
⋅
2
4
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
(
I1
I1
)
A
E
MOMENTOS FLECTORES
L


p ⋅ h2  k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f
2 ⋅ k + 1
⋅
+6⋅
2
24 
3 ⋅ k + 1
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2


2
MA = −
h
y
2
(
)
MB = MA + HA ⋅ h −
MB
2
2


p ⋅ h2  k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f
2 ⋅ k + 1
⋅ −
+
⋅
6
2
24 
3 ⋅ k + 1
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2


En AB
p ⋅ y2
My = MA + HA ⋅ y −
2
ME =
(
MD
)
HA
MA
HE
ME
VA
VE
Prontuario para Cálculo de Estructuras
p ⋅ h2
2
L
MC = ME − HE ⋅ ( f + h) + VE ⋅
2
MD = ME − HE ⋅ h
MC
Valor de la carga en proyección vertical
I h
k= 2⋅
I1 s
p
C
REACCIONES
s
3 p ⋅ f 4 ⋅ k ⋅ ( f + h) + f
VA = VE = ⋅
⋅
8 L
3 ⋅ k +1
HA = p ⋅ f − HE
I2
D
B
y
p ⋅ f 2 ⋅ k ⋅ h ⋅ ( k + 4 ) + f ⋅ (10 ⋅ k ⋅ h + 5 ⋅ k ⋅ f + f )
HE =
⋅
2
4
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
I1
2
(
)
I1
A
L


k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f )
3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f 
p⋅f 
⋅ f⋅
+
⋅

24  ( k ⋅ h + f )2 + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
2
3 ⋅ k +1


MB = MA + HA ⋅ h
MC = ME − HE ( h + f ) + VE ⋅
MD = ME − HE ⋅ h
(
)
L
2
MC
MB
(
MD
)
HA
MA
HE
ME
VA
VE
3.59


k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f )
3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f 
p⋅f 
⋅ −f ⋅
+
⋅
2

24 
3 ⋅ k +1
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2


En BC
2
L ⋅ ( y − h) p ⋅ ( y − h)
My = MA + HA ⋅ y − VA ⋅
−
2⋅f
2
ME =
h
E
MOMENTOS FLECTORES
MA = −
f
I2
Formulario para vigas y pórticos
3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
3.60
3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
P
I h
k= 2⋅
I1 s
C
s
REACCIONES
I2
B
VA = P − VE
2
P ⋅ m 3 ⋅ L ⋅ ( k ⋅ L + m) − 2 ⋅ m
VE = 3 ⋅
3 ⋅ k +1
L
D
m
n
I1
P ⋅ m 3 ⋅ k ⋅ L ⋅ ( f + h) − 4 ⋅ f ⋅ m ⋅ ( k + 1) + 3 ⋅ L ⋅ m ⋅ ( f − k ⋅ h)
⋅
2
L2
( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
2
HA = HE =
f
I2
I1
h
2
(
A
)
E
L
MOMENTOS FLECTORES
 3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 


2
P⋅m 

( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
⋅
MA =


2 ⋅ L2  n ⋅ n − m

(
)
−


3 ⋅ k +1


MB = MA − HA ⋅ h
L
MC = ME + VE ⋅ − HE ⋅ ( h + f )
2
MB
MD
MD = ME − HE ⋅ h
 3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 


2
P⋅m 

( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
ME =
⋅

2 ⋅ L2  n ⋅ n − m

(
)
+


3 ⋅ k +1


2⋅f ⋅m

En BC
My = MA + VA ⋅ m − HA ⋅  h +
L 

(
MC
)
)
HA
HE
MA
VA
ME
VE
Prontuario para Cálculo de Estructuras
(