FORMULARIO PARA VIGAS Y PÓRTICOS

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
T ( x ) = −∑
A⋅ x − a
M( x ) = − ∑
( c− 2 )
( c − 2 )!
( c −1)
( c − 1) !
A⋅ x − a
c
c!
ecuaciones validas solo si n ≥ 0
en las expresiones
si
y si
n=0
n>0
x−a
n
x≤a
x−a
0
=0
x≥a
x−a
0
=1
x≤a
x−a
n
=0
x≥a
x−a
n
= ( 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
c=0
por lo tanto
P
Si
a
T(x)
x
1
x≤a
x−a = 0
x≥a
x − a = ( x − a)
1
1
entonces
T ( x) = − P x − a
M(x)
0
M( x ) = − P x − a
por lo tanto
1
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
M( x ) = −
x−a
2 ⋅1
T(x)
T ( x) = −
2
M(x)
por lo tanto
q
d
a
x
T(x)
M(x)
2
A=q
c=2
Si
x≤a
x−a
x≥a
3
x−a
3
=0
= ( x − a)
entonces
2
3
qd
1
x−a
1
qd
2
T ( x) = −
x−a
2 ⋅1
qd
M( x ) = −
x−a
3 ⋅ 2 ⋅1
q( x) =
por lo tanto
3
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! 
dM( x )
M(x) =
a
T ( x) =
b
dx
x
q/d
q
a
d
T ( x) =
b
q/d
q
-⟨ x-a⟩ 3 + ⟨ x-b⟩ 3  + ⟨ x-b⟩ 2
 2!
3! 
dM( x )
M( x ) =
dx
x
q/d
q
a
q
q/d
 ⟨ x-a⟩ 3 − ⟨ x-b⟩ 3 
⟨ x-a⟩ 2 +

2!
3! 
dM( x )
M(x) = −
d
T ( x) =
b
dx
x
qb
qa
M(x) = −
+
a
d
b
T ( x) =
x
qa
qb
b
x
T ( x) =
)
q b − q a /d
3!
qb
2!
⟨ x-b⟩ 2 +
 −⟨ x-a⟩ 3 + ⟨ x-b⟩ 3 


dx
M(x) = −
d
2!
⟨ x-a⟩ 2 +
dM( x )
+
a
(
qa
qa
2!
(q
a
⟨ x-a⟩ 2 +
)
− q b /d
3!
dM( x )
dx
qb
2!
⟨ x-b⟩ 2 +
 ⟨ x-a⟩ 3 − ⟨ x-b⟩ 3 


VIGA APOYADA EN LOS EXTREMOS
3.2.1
REACCIONES
P⋅b
RA =
L
RB =
B
C
P⋅a
L
x
ESFUERZOS CORTANTES
P⋅b
P⋅a
= cte ; QCB = −
= cte
QAC =
L
L
MOMENTOS FLECTORES
P⋅b
P⋅a
⋅ x ; MCB =
⋅ ( L − x)
MAC =
L
L
ANGULOS DE GIRO
P⋅a⋅b
⋅ ( L + b)
ϕA =
6⋅E⋅I⋅L
P
A
CARGA PUNTUAL EN LA VIGA
a
b
Formulario para vigas y pórticos
3.2
L
;
Mmax = MC =
P⋅a⋅b
⋅ ( L + a)
; ϕB = −
6⋅E⋅I⋅L
P⋅a⋅b
L
para
x0 = a
P⋅a⋅b
⋅ ( b − a)
; ϕC =
3⋅E⋅I⋅L
QB
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
3.5
M max
3.6
3.2.2
CARGA CONTÍNUA EN PARTE DE LA VIGA
REACCIONES
p⋅b⋅c
RA =
L
c
RB =
P
p⋅a⋅c
L
A
ESFUERZOS CORTANTES
p⋅b⋅c
p⋅b⋅c
c

− p⋅ − a + x
; QCD =
QAC =
2
L
L


;
QDB = −
C
p⋅a⋅c
L
a
x0 = a −
para
; ϕB = −
c b⋅c
+
L
2
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 



=

p⋅a⋅c L − x 
c2  
2
⋅
⋅ − ( L − x ) + b ⋅  L + a −

6⋅L
4 ⋅ a  
E ⋅ I 

y
CD
y
DB
M max
Prontuario para Cálculo de Estructuras
ANGULOS DE GIRO
p⋅a⋅b⋅c 
c2 
⋅L + b −
ϕA =

6⋅E⋅I⋅L 
4⋅a
b
L
MDB =
Mmax
D
x
MOMENTOS FLECTORES
p⋅b⋅c
p⋅b⋅c
p  
c 
⋅ x ; MCD =
⋅ x − ⋅  x −  a −  2
MAC =
2  
2 
L
L
p⋅ a⋅c
⋅ ( L − x)
L
p⋅b⋅c 
b⋅c
=
⋅ 2⋅a− c+
L 
2 ⋅ L 
B
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
1
RA = ( 2 ⋅ p1 + p2 )
6
;
RB =
1
( p1 + 2 ⋅ p2 ) .
6
P1
ESFUERZOS CORTANTES
p ( 3 ⋅ L − x ) + p2 ⋅ x 2
QA = RA ; Qx = RA − 1
⋅x
6⋅L
;
P2
QB = −RB
B
A
MOMENTOS FLECTORES
p ( 3L − x ) + p2 ⋅ x 2
Mx = RA ⋅ x − 1
⋅x
6⋅L
x
L2
L2
⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 )
2
2


1
1
para x 0 =
⋅  − p1 +
⋅ p12 + p22 + p1 ⋅ p2 
3
p2 − p1 

Formulario para vigas y pórticos
3.2.3
L
Mmax comprendido entre 0,125 ⋅
(
ANGULOS DE GIRO
L3
ϕA =
⋅ ( 8 ⋅ p1 + 7 ⋅ p2 )
360 ⋅ E ⋅ I
)
QA
QB
; ϕB = −
3
L
⋅ ( 7 ⋅ p1 + 8 ⋅ p2 )
360 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
x ( L − x ) 3 ( p1 − p2 ) x − 3 ( 4 p1 + p2 ) Lx

360EI ( 8 p1 + 7p2 ) L2 x + ( 8 p1 + 7p2 ) L3

3
yx =
0,01304 ⋅
+


( p1 + p2 ) ⋅ L4
2⋅E⋅I
x0
M max
3.7
FLECHA MAXIMA
( p + p2 ) ⋅ L4 y
entre 0,01302 ⋅ 1
2⋅E⋅I
2
3.8
3.2.4
MOMENTO FLECTOR
REACCIONES
R A = −R B = −
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
a
QA
QB
FLECHA
M⋅ a ⋅ b
fC =
⋅ ( b − a)
3⋅E⋅I⋅L
MC
M
MC
Prontuario para Cálculo de Estructuras
)
2
M ⋅ L ⋅ (L − x) 
a2  L − x  

⋅ 1− 3 ⋅ 2 − 


6⋅E⋅I
L  L  

B
M = MCizq + MCder
ECUACION DE LA ELASTICA
M⋅ L ⋅ x 
b2 x 2 
y AC = −
⋅ 1− 3 ⋅ 2 − 2 
6⋅E⋅I 
L
L 
yCB = −
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
3
3
ϕC =
⋅ a +b
3 ⋅ E ⋅ I ⋅ L2
(
+M
3.3.1
P
CARGA PUNTUAL EN LA VIGA
REACCIONES
P ⋅ b2
RA = 3 ⋅ ( L + 2 ⋅ a)
L
Formulario para vigas y pórticos
3.3 VIGA EMPOTRADA EN LOS EXTREMOS
C
;
RB =
ESFUERZOS CORTANTES
P ⋅ b2
QAC = 3 ⋅ ( L + 2 ⋅ a) = cte
L
P ⋅ a2
⋅ ( L + 2 ⋅ b)
L3
;
QCB = −
B
A
x
a
P ⋅ a2
⋅ ( L + 2 ⋅ b ) = cte
L3
b
L
MOMENTOS FLECTORES
P ⋅ a ⋅ b2
P ⋅ a2 ⋅ b
P ⋅ b2
M
=
−
M
=
⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L)
;
;
B
AC
L2
L2
L3
P ⋅ a2
2 ⋅ P ⋅ a2 ⋅ b2
= 3 ⋅ L ⋅ b + L2 − L ⋅ x − 2 ⋅ b ⋅ x ; MC =
para x0 = a
L
L3
MA = −
MBC
(
)
QB
QA
ECUACION DE LA ELASTICA
y AC =
P ⋅ b2 
2 ⋅ a ⋅ x  x2
⋅3 ⋅a − x −
⋅
6⋅E⋅I 
L  L2
y BC =
P ⋅ a2 
L − x ⋅  (L − x)
⋅  3 ⋅ b − (L − x) − 2 ⋅ b
⋅
6⋅E⋅I 
L 
L2
FLECHAS
P ⋅ a3 ⋅ b3
3 ⋅ E ⋅ I ⋅ L3
x=
fmax =
2 ⋅a⋅ L
L + 2⋅a
MC
2 ⋅ P ⋅ a3 ⋅ b2
3 ⋅ E ⋅ I ⋅ ( L + 2 ⋅ a)
2
0
3.9
para
;
MB
MA
x
fC =
2
3.10
3.3.2
CARGA CONTÍNUA EN PARTE DE LA VIGA
c
P
REACCIONES
p ⋅ b ⋅ c MA − MB
RA =
−
L
L
;
p ⋅ a ⋅ c MA − MB
RB =
+
L
L
C
c

; QBD = −RB = cte ; QCD = RA − p ⋅  x − a + 
a

x
a
MOMENTOS FLECTORES
MAC = RA ⋅ x + MA
;
MBD = RB ⋅ ( L − x ) + MB
MCD = RA ⋅ x + MA −
;
B
A
ESFUERZOS CORTANTES
QAC = RA = cte
D
MA = −
p ⋅ c3
12 ⋅ L2
p 
c
⋅x − a+ 
2 
2
b
2

12 ⋅ a ⋅ b2 
⋅L − 3⋅b +

c2


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 ⋅ x 3 − 12 ⋅ MA ⋅ x 3 
24 ⋅ E ⋅ I  
2

1
RB x 3 − 3 ( MB + LRB ) x 2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2 
y DB =
6EI 
y AC =
MA
MB
Prontuario para Cálculo de Estructuras
12 ⋅ a2 ⋅ b 
p ⋅ c3 
⋅ L − 3⋅a+
MB = −

2 
12 ⋅ L 
c2

L
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
L
M − MB
⋅ ( 2 ⋅ p1 + p2 ) − A
6
L
L
MA − MB
RB = ⋅ ( p1 + 2 ⋅ p2 ) +
6
L
P1
RA =
P2
ESFUERZOS CORTANTES
QA = RA
Qx = RA −
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x
2⋅L
B
A
Formulario para vigas y pórticos
3.3.3
x
⋅x
L
QB = −RB
MOMENTOS FLECTORES
L2
( 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
MA = −
Q
A
Q
B
ECUACION DE LA ELASTICA
yx =
 ( p − p1) 3

x2
⋅ 2
⋅ x + p1 ⋅ L ⋅ x 2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L 
24 ⋅ E ⋅ I ⋅ L 
5

MA
MB
3.11
3.12
3.3.4
MOMENTO FLECTOR
REACCIONES
6⋅M
RA = − 3 ⋅ a ⋅ b
L
;
RB =
6⋅M
⋅a⋅b
L3
C
A
ESFUERZOS CORTANTES
Qx = −
6⋅M
⋅ a ⋅ b = cte
L3
a
M⋅ a 
b
⋅2 − 3⋅ 
L 
L
MAC =
B
x
MOMENTOS FLECTORES
MA =
+M
MB = −
b
L
M⋅ b 
a
⋅2 − 3⋅ 
L 
L
M⋅ a  a 
x 
⋅ 3 ⋅ ⋅ 1− 2 ⋅  − 1
L  L 
L 
MCB = −
M⋅ b  b 
L−x 
⋅ 3 ⋅ ⋅ 1− 2 ⋅
 − 1
L  L 
L  
6⋅M 2
⋅a ⋅b
L3
;
MCder = MA +
M 3
⋅ L − 6 ⋅ a2 ⋅ b
L3
(
QB
)
ECUACION DE LA ELASTICA
y AC =
y BC =
M ⋅ b ⋅ x2
2⋅E⋅I⋅L
L− x b

⋅2⋅a⋅ 2 − 
L
L

M⋅ a ⋅ ( L − x )
2⋅E⋅I⋅L
2
MC
 b⋅ x a
⋅2 ⋅ 2 − 
L
L

FLECHA
M ⋅ a2 ⋅ b2
fC = −
⋅ ( a − b)
2 ⋅ E ⋅ I ⋅ L3
MA
MC
MB
Prontuario para Cálculo de Estructuras
MCizq = MA −
QA
3.4.1
P
CARGA PUNTUAL EN LA VIGA
REACCIONES
P ⋅ b2
P⋅a
RA =
⋅ ( 3 ⋅ L − b ) ; RB =
⋅ 3 ⋅ L2 − a2
2 ⋅ L3
2 ⋅ L3
ESFUERZOS CORTANTES
P ⋅ b2
P⋅a
QAC = −
⋅ ( 3 ⋅ L − b ) = cte ; QCB = −
⋅ 3 ⋅ L2 − a2 = const.
2 ⋅ L3
2 ⋅ L3
(
)
(
C
x
a
b
L
)
(
B
A
)
MOMENTOS FLECTORES
P⋅a 2
P⋅a 2
⋅ L − a2
⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b)
MB = −
; MC =
2 ⋅ L2
2 ⋅ L3
P⋅x 2
P⋅a
⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b ) ; MCB =
⋅ 2 ⋅ L3 − 3 ⋅ L2 ⋅ x + a2 ⋅ x
MAC =
2 ⋅ L3
2 ⋅ L3
(
)
Q
B
ANGULOS DE GIRO
ϕA =
P ⋅ a ( L − a)
2
; ϕC =
4⋅E⋅I⋅L
P ⋅ a ⋅ ( L − a)
2
3
4⋅E⋅I⋅L
(
⋅ L2 − 2 ⋅ a ⋅ L − a2
)
Q
A
ECUACION DE LA ELASTICA
P ⋅ b2 ⋅ x
y AC =
⋅ 3 ⋅ a ⋅ L2 − x 2 ⋅ ( 2 ⋅ L + a) 
12 ⋅ E ⋅ I ⋅ L3 
y BC =
P ⋅ a ⋅ ( L − x)
12 ⋅ E ⋅ I
2
MB
  a2  
a2   L − x  
⋅ 3 ⋅ 1− 2  −  3 − 2  ⋅ 

L  
L   L  
 
para x=L ⋅
a
2⋅L + a
MC
3.13
FLECHA MAXIMA
p ⋅ b2 ⋅ a
a
fmax =
⋅
6⋅E⋅I
2⋅L + a
Formulario para vigas y pórticos
3.4 VIGA APOYADA-EMPOTRADA
3.14
3.4.2
CARGA CONTÍNUA EN PARTE DE LA VIGA
REACCIONES
p ⋅ b ⋅ c MB
RA =
+
L
L
;
c
P
p ⋅ a ⋅ c MB
RB =
−
L
L
C
ESFUERZOS CORTANTES
QAC = RA = cte
;
QDB = −RB = cte
;
QCD
c

= RA − p ⋅  x − a + 
2

x
MOMENTOS FLECTORES
;
MAC = RA ⋅ x
MCD
MDB = RB ⋅ ( L − x ) + MB
a
p 
c
= RA ⋅ x − ⋅  x − a + 
2 
2
;
MB = −
2
(L − x)
2
6⋅E⋅I
⋅ RB ⋅ ( L − x ) + 3 ⋅ MB 
QB
QA
MB
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA


x
12 ⋅ a ⋅ b2  
y AC =
⋅  −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅  L − 3b +

48 ⋅ E ⋅ I ⋅ L 
c2

 
4


1
c
12ab2  

⋅  −8RALx 3 + 2 pL  x − a +  + pc3  L − 3b +
yCD =
 x
48 ⋅ E ⋅ I ⋅ L 
4
c2  


b
L
p⋅a⋅b⋅c 
c2 
L
a
⋅
+
−


4⋅b
2 ⋅ L2

ANGULOS DE GIRO

p ⋅ c3
12 ⋅ a ⋅ b2 
ϕA =
⋅  L − 3b +

48 ⋅ E ⋅ I ⋅ L 
c2

y DB = −
B
A
CARGA TRAPEZOIDAL EN TODA LA VIGA
RA =
P2
P1
REACCIONES
L
M
⋅ ( 2 ⋅ p1 + p2 ) + B
6
L
;
RB =
L
M
⋅ ( p1 + 2 ⋅ p2 ) − B
6
L
A
ESFUERZOS CORTANTES
Qx = RA −
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x
2⋅L
L
⋅x
;
QB = −RB
Q
A
MOMENTOS FLECTORES
Mx = RA ⋅ x −
p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x
6⋅L
B
x
⋅ x2
Formulario para vigas y pórticos
3.4.3
;
MB = −
Q
B
L2
⋅ ( 7 ⋅ p1 + 8 ⋅ p2 )
120
ANGULOS DE GIRO
L3
ϕA =
⋅ ( 3 ⋅ p1 + 2 ⋅ p2 )
240 ⋅ E ⋅ I
MB
ECUACION DE LA ELASTICA
yx =
3.15
x 
( p2 − p1) x4 + 5Lp1x3 − 20RALx2 + 5L 12RAL2 − ( 3p1 + p2 ) L3  
120EIL 
3.16
3.4.4
MOMENTO FLECTOR
REACCIONES
RA = −RB =
3 M 2
⋅ ⋅ L − a2
2 L3
(
)
ESFUERZOS CORTANTES
B
x
Qx = RA = cte
a
MOMENTOS FLECTORES
MCder = RA ⋅ a − M ;
MAC =
C +M
A
MCizq = RA ⋅ a
3 M⋅ x 2
⋅
⋅ L − a2
2 L3
(
)
;
MBC
M
⋅ L2 − 3 ⋅ a2
2 ⋅ L2

M  x  a2 
= ⋅ 3 ⋅ ⋅ 1− 2  − 2 
2  L 
L 

;
(
MB =
b
L
)
ANGULOS DE GIRO
 b  a 2

M
; ϕC =
⋅ b ⋅ 3 ⋅ ⋅ 1+  − 4 
4⋅E⋅I
 L  L 

ECUACION DE LA ELASTICA
M⋅ b ⋅ x 
⋅ −4 ⋅ L3 − x 2 − 3 ⋅ L2 ⋅ ( a + L ) 

4 ⋅ E ⋅ I ⋅ L3 
M
2
=
⋅ ( L − x ) ⋅ 2 ⋅ a2 ⋅ L − x ⋅ L2 − a2 


4 ⋅ E ⋅ I ⋅ L3
y AC =
y BC
(
QA
QB
MC
)
(
MB
)
MC
Prontuario para Cálculo de Estructuras
M
ϕA =
⋅ ( L − a) ⋅ ( 3 ⋅ a − L )
4⋅E⋅I⋅L
3.5.1
CARGA PUNTUAL EN LA VIGA
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
L
MOMENTOS FLECTORES
MAC = 0
;
b
MCB = −P ⋅ ( x − a)
;
MB = −P ⋅ b
ANGULOS DE GIRO
ϕ A = ϕC = −
P
⋅ b2
2⋅E⋅I
QB
ECUACION DE LA ELASTICA
y
AC
=
P ⋅ b2
⋅ ( 3 ⋅ ( L − x ) − b)
6⋅E⋅I
;
y
CB
=
MB
3.17
FLECHA MAXIMA
P ⋅ b3
P ⋅ b2
⋅ ( 2 ⋅ b + 3 ⋅ a)
fC =
; fA =
3⋅E⋅I
6⋅E⋅I
P
2
⋅ ( L − x ) ⋅ ( 2 ⋅ b + 3 ⋅ a)
6⋅E⋅I
3.18
3.5.2
CARGA CONTÍNUA EN PARTE DE LA VIGA
REACCIONES .
RB = p ⋅ c
c
ESFUERZOS CORTANTES .
c

QAC = 0 ; QCD = − p ⋅  x − a + 
2

P
; QDB = − p ⋅ c = cte
; ϕC = −
x
p ⋅ c2
MD = −
2
p ⋅ c  2 c2 
⋅b +

2⋅E⋅I 
12 
a
b
L
; ϕ A = ϕC
y DB =


p⋅c
p⋅c 
c2 
3
⋅ L − x 2 ⋅ ( 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 


)
Q
B
FLECHAS .
2
fD =
p⋅ c 
c
⋅ b− 
2
E ⋅ I 
b c 
⋅ + 
 3 12 
2


p ⋅ c 
c
p⋅ c  
c2 
fC =
⋅  b +  ⋅ ( 4 ⋅ b − c) + c3  ; fA =
⋅ a ⋅  3 ⋅ b2 +  + 2 ⋅ b3 
12 ⋅ E ⋅ I 
2
6
4
E
I
⋅
⋅

 


M
B
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA .
(
D
C
B
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 
⋅b −
ϕD = −

2⋅E⋅I 
4 
A
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
1
RB = ( p1 + p2 )
2
ESFUERZOS CORTANTES
p − p1 x 2
⋅ − p1 ⋅ x
Qx = − 2
L
2
A
;
L
QB = − ( p1 + p2 )
2
x2
⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1
6⋅L 
B
x
MOMENTOS FLECTORES
Mx = −
P2
P1
Formulario para vigas y pórticos
3.5.3
;
MB = −
L2
⋅ ( p2 + 2 ⋅ p1 )
6
L
ANGULOS DE GIRO
ϕA = −
L3 ⋅ ( 3 ⋅ p1 + p2 )
24 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
3

2  ( L − x)
2
L − x ) −
(
p2 − p1 ) + ( L − x ) p2 − 
(
yx =
5L


24EI 
2

L
L
x
p
p
L
p
p
2
2
2
−
−
+
+
+
(
)(
)
(
)
2
1
2
1


FLECHA
120 ⋅ E ⋅ I
MB
3.19
fA =
L4 ⋅ ( 4 ⋅ p2 + 11⋅ p1 )
QB
3.20
3.5.4
MOMENTO FLECTOR
REACCIONES
M
A
B
RB = 0
ESFUERZO CORTANTE
x
a
Qx = 0
L
MOMENTOS FLECTORES
MAC = 0
;
MCB = − M = cte
b
;
MAC = − M
ANGULOS DE GIRO
ϕC = ϕ A = −
ECUACION DE LA ELASTICA
y AC =
M
⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b)
2⋅E⋅I
;
y BC =
M
2
(L − x)
2⋅E⋅I
FLECHA
fC =
M ⋅ b2
2⋅E⋅I
;
fA =
M
⋅ b ⋅ ( 2 ⋅ L − b)
2⋅E⋅I
MB
Prontuario para Cálculo de Estructuras
M⋅ b
E⋅I
P
P
A
P
B
L/2
C
L/2
L
A
L
A
L/2
L
0,688 P
0,312 P
L
0,405 P
B
C
0,094 P
0,094 P
B
A
0,312 P
0,688 P
C
B
L/2
L/2
L/2
Formulario para vigas y pórticos
3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES
C
0,594 P
ESFUERZOS CORTANTES
ESFUERZOS CORTANTES
- 0,188 PL
- 0,094 PL
A
B
0,156 PL
C
0,156 PL
B
C
0,203 PL
MOMENTOS FLECTORES
3.21
MOMENTOS FLECTORES
A
3.22
Q
Q
A
L
B
Q
C
L
0,625 QL
A
0,375 L
B
L
L
C
0,437 QL
0,375 QL
0,063 QL
B
A
C
A
0,375 QL
0,375 L
B
0,563 QL
0,437 L
0,625 QL
ESFUERZOS CORTANTES
2
2
- 0,063 QL
- 0,125 QL
B
2
C
2
0,07 QL
0,07 QL
MOMENTOS FLECTORES
A
B
2
0,096 QL
MOMENTOS FLECTORES
C
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
A
C
Q
A
L
Q
B
C
k L
c QL
d L
a QL
A
C
B
b QL
d QL
a L
ESFUERZOS CORTANTES
2
f QL
A
B
2
g QL
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
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
3.23
MOMENTOS FLECTORES
MOMENTOS
FLECTORES
ESFUERZOS CORTANTES
C
2
e QL
Relación
entre
luces
Formulario para vigas y pórticos
3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES
3.24
Q
Q
A
B
L
Relación
entre
luces
C
k L
c QL
d L
A
C
B
a QL
d QL
b QL
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
2
f QL
k 2 − k +1
8
k f
d= −
2 k
f=
A
B
MOMENTOS FLECTORES
C
2
g QL
a = 0.5 − f
e=
a2
2
b = 0.5 + f
g=
d2
2
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
MOMENTOS
FLECTORES
ESFUERZOS CORTANTES
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
Formulario para vigas y pórticos
3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES
C
A
c QL
b QL
a L
a QL
ESFUERZOS CORTANTES
2
f QL
2
2
f QL
f=
k3 + 1
12 ⋅ k + 8
a = 0.5 − f
c=
k
2
e=
a2
2
b = 0.5 + f
g=
k2
−f
8
g QL
A
B
C
D
2
2
e QL
e QL
3.25
MOMENTOS FLECTORES
3.26
Q
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
2
f QL
B
2
e QL
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
ESFUERZOS
CORTANTES
C
2
g QL
MOMENTOS FLECTORES
f=
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
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
Relación
entre
luces
k=
I2 h
⋅
I1 l
3.9.1
y
N = 3 + 2k
a
s
p
CARGA REPARTIDA VERTICAL
B
C
I
2
x
REACCIONES
m
VA
VD
psn
=
l
psm
=
l
HA = HD =
Formulario para vigas y pórticos
3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
I
n
I
1
A
h
1
D
s 
3 ps 
 mn − 
2hlN 
12 
2
l
MOMENTOS FLECTORES
MB
MC
3 ps 
s2 
MB = MC = − ⋅
 mn − 
2 lN 
12 
En S
Mx = VA ⋅ x −
p(x − m)2
− HA ⋅ h
2
HA
HD
VD
3.27
VA
3.28
3.9.2
CARGA REPARTIDA HORIZONTAL
REACCIONES
VA = VD =
HD =
HA =
p
B
C
I
ph2
2l
ph ( 2N + k )
I
2
I
1
8N
h
1
y
ph ( 6N − k )
A
8N
D
l
MOMENTOS FLECTORES
MB
MY =
py(h − y) y
+ ⋅ MB
h
2
MC
MB
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
ph2
( 2N − k )
8N
ph2
MC = −
( 2N + k )
8N
En AB
MB =
Formulario para vigas y pórticos
3.9.3
CARGA PUNTUAL VERTICAL SOBRE DINTEL
REACCIONES
n
B
Pn
VA =
l
Pm
VD =
l
HA = HD =
P
m
C
I
I
2
I
1
h
1
3 Pmn
2 lhN
A
D
MOMENTOS FLECTORES
l
3 Pmn
MB = MC = − ⋅
2 lN
2N − 3
MP = Pmn
2lN
MB
MC
MP
HD
HA
VD
3.29
VA
3.30
3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
k1 =
I3 h1
⋅
I1 s
y
k2 =
I3 h 2
⋅
I2 s
p
3.10.1 CARGA REPARTIDA VERTICAL
C
s
REACCIONES
f
I
B
pl
2
h1 + h2
pl 2
HA = HD =
8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2
VA = VD =
3
I
x
I
h1
h2
2
1
A
D
l
MB = −
( h1 + h2 ) h1
pl2
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC = −
( h1 + h2 ) h2
pl 2
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
En
BC
MX =
px(l − x)
f

− HA  x + h1 
2
l


MC
MB
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
MOMENTOS FLECTORES
Formulario para vigas y pórticos
3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
C
REACCIONES
ph12
VA = VD =
2l
HA = ph1 − HD
s
f
p
I
3
B
I
h1
h1 ( 4 + 5k 1 ) + 2h2
ph12
HD =
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
I
1
h2
2
y
A
D
l
MOMENTOS FLECTORES
MC
h1 ( 4 + 5k 1 ) + 2h2
ph12
ph3
− 1 2
MB =
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC =
MB
h1 ( 4 + 5k 1 ) + 2h2
ph12 h2
2
8
h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
En AB
MY = HA y −
py 2
2
HD
HA
VD
3.31
VA
3.32
3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
p
REACCIONES
VA = VD =
pf ( h1 + h2 )
f
I
2l
B
HA = pf − HD
HD =
pf 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 )
8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2
2
1
2
1
C
s
I
h1
y
3
I
h2
2
1
A
D
l
MOMENTOS FLECTORES
MC = −
2
pfh1 8h1 (1+ k 1 ) + 4h1h2 + f ( h1 + h2 )
8
h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC
MB
ph2 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 )
8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2
2
1
2
1
En BC
l
py 2
MY = −VA y + HA ( y + h1 ) −
f
2
HA
HD
VA
VD
Prontuario para Cálculo de Estructuras
MB = pfh1 −
C
REACCIONES
Pb
VA =
l
Pa
VD =
l
h1(l + b) + h2 (l + a)
Pab
HA = HD = 2 2
2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
s
f
I
3
B
I
h1
I
a
b
h2
2
Formulario para vigas y pórticos
3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL
1
A
D
l
MOMENTOS FLECTORES
MB = −
h1 ( l + b ) + h2 ( l + a )
Pabh1
2
2
2l
h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC
MB
h1 ( l + b ) + h2 ( l + a )
Pabh2
MC = −
2
2
2l
h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MP =
Pab
 af

+ HA 
+ h1 
l
 l

MP
HD
HA
VA
VD
3.33
3.34
3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
k=
I2 h
⋅
I1 s
p
3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
C
s
REACCIONES
I
pl
2
8h + 5f
pl 2
HA = HE =
2
32 h ( 3 + k ) + f ( 3h + f )
B
VA = VE =
D
I
I
1
A
h
1
E
l
pl 2 h
8h + 5f
2
32 h ( 3 + k ) + f ( 3h + f )
MC
2
pl
f+h
MB
+
8
h
MB
MD
En BC y DC
MX = p
x (l − x)
2
+
MB 
2fx 
h+

h 
l 
HE
HA
VA
VE
Prontuario para Cálculo de Estructuras
MC =
f
2
x
MOMENTOS FLECTORES
MB = MD = −
I
2
p
REACCIONES
C
pl
VA = 3
8
pl
VE =
8
HA = HE =
s
I
B
I
2
I
I
1
A
l
pl 2 h
8h + 5f
2
64 h ( 3 + k ) + f ( 3h + f )
MC
pl 2 f + h
+
MC =
MB
16
h
En BC
x (l − x)
2
+
MB
h
h
1
E
MOMENTOS FLECTORES
MX = p
D
x
8h + 5f
pl 2
2
64 h ( 3 + k ) + f ( 3h + f )
MB = MD = −
f
2
Formulario para vigas y pórticos
3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
MB
MD
2fx 

h+ l 


HE
VA
VE
3.35
HA
3.36
3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
REACCIONES
C
ph2
VA = VE =
2l
HA = ph − HE
HE =
s
I
p
I
2
f
2
D
B
( 5k + 12 ) h + 6f
ph
2
16 h ( k + 3 ) + f ( f + 3h)
2
I
1
I
y
A
h
1
E
MOMENTOS FLECTORES
l
ph2
+ MD
MB =
2
ph2 f + h
+
MC =
MD
h
4
( 5k + 12 ) h + 6f
ph3
MD = −
2
16 h ( k + 3 ) + f ( f + 3h)
MC
MD
En AB
My = −
py 2
+ HA ⋅ y
2
HA
HE
VA
VE
Prontuario para Cálculo de Estructuras
MB
REACCIONES
p
pf
VA = VE =
( f + 2 h)
2l
HA = pf − HE
s
I
I
2
f
2
D
B
pf 8h ( k + 3 ) + 5f ( f + 4h)
16 h2 ( k + 3 ) + f ( f + 3h)
2
HE =
C
Formulario para vigas y pórticos
3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
y
I
I
1
A
MOMENTOS FLECTORES
x
h
1
E
l
MB = HA ⋅ h
MC = −
2
pf 2 4h ( k + 2 ) + f ( 5h + f )
⋅ 2
16 h ( k + 3 ) + f ( f + 3h)
MC
MD = −HE ⋅ h
MB
MD
En BC
Mx = HA ⋅ y − VA ⋅ x − p
2
2
HE
HA
VA
VE
3.37
f
siendo y = x + h
l
( y − h)
3.38
3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
p
REACCIONES
Pn
l
Pm
VA =
l
C
s
VA =
I
B
(
)
I
2
2
Pm 6hln+ f 3l − 4m
HA = HE = 2 2
4 l h ( k + 3 ) + f ( f + 3 h)
I
2
m
n
1
A
f
2
D
I
h
1
E
l
MOMENTOS FLECTORES
MB = MD = −HA ⋅ h
MC
MB
MD
HE
HA
VA
VE
Prontuario para Cálculo de Estructuras
Pm h + f
+
MC =
MB
h
2
hl + 2fm
MP = VA ⋅ m − HA
l
k1 =
I3 h1
⋅
I1 l
y
k2 =
I3 h 2
⋅
I2 l
3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p
B
C
REACCIONES
VA =
I
3
x
2
1
h −h
pl pl
+
2
8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2
2
1
I
2
2
h1
I
1
h2
2
Formulario para vigas y pórticos
3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL
D
2
2
h −h
pl pl
VD =
−
2
2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
A
h1 − h2
pl 2
HA = HD =
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
l
MOMENTOS FLECTORES
MB
( h1 + h2 ) h1
pl 2
MB = −
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC = −
MC
( h1 + h2 ) h2
pl 2
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
HD
En BC
VD
HA
VA
3.39
px 2
Mx = VA ⋅ x −
− HA ⋅ h1
2
3.40
3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
REACCIONES
p
B
C
2
1
ph
h − h2
VA = VD =
− HD 1
2l
l
HA = ph − HD
HD =
ph12
5k1h1 + 4h1 + 2h2
8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2
I
3
I
h1
I
1
h2
2
D
y
A
MOMENTOS FLECTORES
l
ph2 ph3
5k1h1 + 4h1 + 2h2
MB = − 1 − 1 2
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MB
ph12 h2
5k1h1 + 4h1 + 2h2
2
8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
MC
MB
En AB
My = HA ⋅ y −
py 2
2
HD
VD
HA
VA
Prontuario para Cálculo de Estructuras
MC = −
P
a
REACCIONES
( l + b ) h1 + ( l + a) h2
Pb Pab
VA =
h1 − h2
+ 3 2
l
2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
(
( l + b ) h1 + ( l + a) h2
Pa Pab
VD =
h1 − h2
− 3 2
l
2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
(
b
B
C
I
)
)
3
I
h1
I
1
h2
2
Formulario para vigas y pórticos
3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
D
A
HA = HD =
( l + b ) h1 + ( l + a) h2
Pab
2
2
2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
l
MOMENTOS FLECTORES
MB = −
MC = −
MB
( l + b) h1 + ( l + a) h2
Pabh1
2
2
h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
2l
MC
MP
HD
( l + b ) h1 + ( l + a) h2
Pabh2
2
2
h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2
2l
VD
HA
VA
3.41
MP = VA ⋅ a + MB
3.42
3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
k=
I2 h
⋅
I1 l
3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p
B
C
I
VA = VD =
pl
2
HA = HD =
pl 2
4h ( k + 2 )
I
I
1
h
1
A
MOMENTOS FLECTORES
D
l
pl2
6 ( k + 2)
MB
MC
En BC
Mx =
px ( l − x )
2
Mmáx pos =
−
pl 2
6( k + 2)
pl 2 3k + 2
l
para x =
24 k + 2
2
HA
MA
VA
MD
HD
VD
Prontuario para Cálculo de Estructuras
pl2
MA = MD =
12 ( k + 2 )
MB = MC = −
2
x
REACCIONES
p
REACCIONES
B
C
2
ph k
VA = VD =
l ( 6k + 1)
I
HA = ph − HD
HD =
I
ph ( 2k + 3 )
A
ph2
MC = −
24
MD =
ph2
24
l
2
2 

 3 − 6k + 1 − k + 2 


MC
MB
MB
2
1 

 3 + 6k + 1 − k + 2 


En AB
My = −
D
2
1 

 5 + 6k + 1 + k + 2 


ph2 
2
2 
1−
+

24  6k + 1 k + 2 
h
1
y
MOMENTOS FLECTORES
MB =
I
1
8 ( k + 2)
ph2
MA = −
24
2
Formulario para vigas y pórticos
3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
py 2
+ HA ⋅ y + MA
2
MA
HA
VA
MD
HD
VD
3.43
3.44
3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
REACCIONES
VA =
P
m
Pn  m ( n − m) 
1+ 2

l 
l ( 6k + 1) 
n
B
C
I
2
VD = P − VA
HA = HD =
3Pmn
2lh(k + 2)
I
MOMENTOS FLECTORES
MA =
I
1
A
D
Pmn  1
n− m 
−


2l  k + 2 l ( 6k + 1) 
Pmn  1
n− m 
+


l  k + 2 2l ( 6k + 1) 
MC = −
Pmn  1
n− m 
−


l  k + 2 2l ( 6k + 1) 
MD =
Pmn  1
n− m 
+


2l  k + 2 l ( 6k + 1) 
MP =
Pmn nMB mMC
+
+
l
l
l
l
MB
MC
MP
HA
MA
VA
MD
HD
VD
Prontuario para Cálculo de Estructuras
MB = −
h
1
Formulario para vigas y pórticos
3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR
REACCIONES
3Phk
VA = VD =
l(6k + 1)
P
HA = HD =
2
MOMENTOS FLECTORES
Ph 3k + 1
2 6k + 1
Ph 3k
MB = − MC =
2 6k + 1
Ph 3k + 1
MD =
2 6k + 1
P
B
C
I
I
2
I
1
h
1
A
D
MA = −
l
MB
MA
MC
HA
HD
VD
3.45
VA
MD
3.46
3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
k=
I2 h
⋅
I1 s
p
3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
C
s
REACCIONES
I
pl
2
k ( 4h + 5f ) + f
pl 2
HA = HE =
8 ( kh + f )2 + 4k h2 + hf + f 2
B
VA = VE =
(
f
2
D
x
I
)
I
1
h
1
A
E
l
MOMENTOS FLECTORES
pl2 kh ( 8h + 15f ) + f ( 6h − f )
48 ( kh + f )2 + 4k h2 + hf + f 2
(
kh (16h + 15f ) + f 2
pl2
MB = MD = −
48 ( kh + f )2 + 4k h2 + hf + f 2
(
pl 2
+ MA − HA ( h + f )
8
En BC
MC
)
MB
)
MD
MC =
2 xf  px

Mx = MA + VA ⋅ x − HA  h +
−
2
l 

2
HA
MA
VA
ME
HE
VE
Prontuario para Cálculo de Estructuras
MA = ME =
I
2
p
REACCIONES
pl
− VE
2
4k + 1
VE = 3 pl
32 ( 3k + 1)
k ( 4h + 5f ) + f
pl 2
HA = HE =
16 ( kh + f )2 + 4k h2 + hf + f 2
C
VA =
(
s
I
B
)
I
l
MD = −
kh (16h + 15f ) + f 2
pl 2
pl2
+
96 ( kh + f )2 + 4k f 2 + fh + h2
64 ( 3k + 1)
En BC
MC
)
kh (16h + 15f ) + f 2
pl 2
pl 2
−
96 ( kh + f )2 + 4k f 2 + fh + h2
64 ( 3k + 1)
(
(
MB
)
)
HA
MD
MA
VA
ME
HE
VE
3.47
2 xf  px 2

Mx = MA + VA ⋅ x − HA  h +
−
l 
2

l
MC = VE + ME − HE ( f + h)
2
h
1
E
pl 2 kh ( 8h + 15f ) + f ( 6h − f )
pl 2
+
96 ( kh + f )2 + 4k f 2 + fh + h2
64 ( 3k + 1)
MB = −
I
1
)
(
D
A
pl 2 kh ( 8h + 15f ) + f ( 6h − f )
pl 2
MA =
−
2
96 ( kh + f ) + 4k f 2 + fh + h2
64 ( 3k + 1)
ME =
f
2
x
MOMENTOS FLECTORES
(
I
2
Formulario para vigas y pórticos
3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
3.48
3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
REACCIONES
VA = VE =
ph2 k
2l ( 3k + 1)
C
s
HA = ph − HE
HE =
k 2 h + k ( 2 f + 3 h) + f
ph2
4 ( kh + f )2 + 4k f 2 + fh + h2
(
)
I
f
2
D
1
I
y
E
2
 2

ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f
2k + 1
+
6
24  ( kh + f )2 + 4k f 2 + fh + h2
3k + 1


(
l
)
MC
(
MD
)
MA
HA
VA
ME
HE
VE
Prontuario para Cálculo de Estructuras
MB
2
2


ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f
2k + 1
−
+
6
2
24 
3k + 1
( kh + f ) + 4k f 2 + fh + h2


En AB
py 2
My = MA + HA ⋅ y −
2
h
1
A
ph2
MB = MA + HA ⋅ h −
2
1
MC = ME − HE ( f + h) + VE
2
MD = ME − HE ⋅ h
ME =
I
2
B
MOMENTOS FLECTORES
MA = −
I
p
REACCIONES
p
3 pf 4k ( f + h) + f
8 l
3k + 1
HA = pf − HE
C
VA = VE =
HE =
s
I
2
pf 2k h ( k + 4 ) + f (10kh + 5kf + f )
4 ( kh + f )2 + 4k f 2 + fh + h2
(
)
I
MC = ME − HE ( h + f ) + VE
MD = ME − HE ⋅ h
y
I
(
E
(
l
)
l
2
h
1
A
MC
MB


kh ( 9f + 4h) + f ( 6h + f )
3 4h ( 3k + 2 ) + f 
pf 
−f
+

24  ( kh + f )2 + 4k f 2 + fh + h2
2
3k + 1


En BC
2
l ( y − h) p ( y − h)
−
My = MA + HA ⋅ y − VA
2f
2
ME =
D
1


kh ( 9f + 4h) + f ( 6h + f )
pf 
3 4h ( 3k + 2 ) + f 
+
f

24  ( kh + f )2 + 4k f 2 + fh + h2
2
3k + 1


MB = MA + HA ⋅ h
f
2
B
MOMENTOS FLECTORES
MA = −
I
2
Formulario para vigas y pórticos
3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
MD
)
MA
HA
HE
VE
3.49
VA
ME
3.50
3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
REACCIONES
p
VA = P − VE
2
Pm 3l ( kl + m) − 2m
VE = 3
3k + 1
l
2
2
Pm 3kl ( f + h) − 4fm ( k + 1) + 3lm ( f − kh)
HA = HE = 2
2
l
( kh + f ) + 4k f 2 + fh + h2
(
C
s
I
B
)
I
MOMENTOS FLECTORES
m
f
2
D
n
I
1
E
)
l
MC
MB
(
)
HA
MD
MA
VA
ME
HE
VE
Prontuario para Cálculo de Estructuras
MB = MA − HA ⋅ h
l
MC = ME + VE − HE ( h + f )
2
MD = ME − HE ⋅ h
 3flh ( kl + 2m) − 4fm2 ( kh + 2h + f ) + 2kh2 ln+ f 2 l ( 4m − l ) 


2
Pm 
kh + f ) + 4k f 2 + fh + h2

(
ME = 2 

2l  n n − m

(
)

+
3k + 1


En BC
2fm 

My = MA + VA ⋅ m − HA  h +
l 

h
1
A
 3flh ( kl + 2m) − 4fm2 ( kh + 2 h + f ) + 2kh2 ln+ f 2 l ( 4m − l ) 


2
Pm 
kh + f ) + 4k f 2 + fh + h2

(
MA = 2 

2l  n n − m

(
)
−

3k + 1


(
I
2