3 formulario para_vigas_y_porticos (1)

3
FORMULARIO PARA
VIGAS Y PÓRTICOS

Formulario para vigas y pórticos 3.1
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 expre-
sión de tales funciones y las condiciones en las que deben aplicarse.
( )
( )
( )
( )
( )
( )
( )
( )
2
1
0
0
2 !
1 !
!
ecuaciones validas solo si 0
en las expresiones
si 0 0
1
y si 0 0
c
c
c
n
n
n n
A x a
q x
c
A x a
T x
c
A x a
M x
c
n
x a
n x a x a
x a x a
n x a x a
x a x a x a
−
−
⋅ −
=
−
⋅ −
= −
−
⋅ −
= −
≥
−
= ≤ − =
≥ − =
> ≤ − =
≥ − = −
∑
∑
∑
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 ecua-
ción general previamente indicada.

3.2 Prontuario para Cálculo de Estructuras
M
M(x)
a
x
( )
0
0
0
Si
0
1
entonces
por lo tanto
0
x a x a
x a x a
M x M x a
A M
c
≤ − =
≥ − =
= − −
=
=
P
M(x)
a
x
T(x)
( )
( )
( )
1
1 1
0
1
Si
0
entonces
por lo tanto
1
x a x a
x a x a x a
T x P x a
M x P x a
A P
c
≤ − =
≥ − = −
= − −
= − −
=
=

Limitación de las Deformaciones 3.3
2
M(x)
q
x
a
T(x)
( )
( )
( )
( )
2
2 2
0
1
2
Si
0
entonces
1
2 1
por lo tanto
2
x a x a
x a x a x a
q x q x a
q
T x x a
q
M x x a
A q
c
≤ − =
≥ − = −
= −
= − −
= − −
⋅
=
=
3
a
x
d
q
2
T(x)
M(x)
( )
( )
( )
( )
3
3 3
1
2
3
Si
0
entonces
1
2 1
3 2 1
por lo tanto
3
x a x a
x a x a x a
q d
q x x a
q d
T x x a
q d
M x x a
q
A
d
c
≤ − =
≥ − = −
= −
= − −
⋅
= − −
⋅ ⋅
=
=

3.4 Prontuario para Cálculo de Estructuras
Otros casos de carga que se resuelven por superposición de los anteriores
x
a
b
q
( )
( )
( )
2 2q
M x x-a x-b
2!
dM x
T x
dx
 = −〈 〉 + 〈 〉 
=
q
a
x
b
d
q/d
( )
( )
( )
3 3 2q/d q
M x - x-a x-b x-b
3! 2!
dM x
T x
dx
 = 〈 〉 + 〈 〉 + 〈 〉 
=
q
a
x
b
d
q/d
( )
( )
( )
2 3 3q q/d
M x x-a x-a x-b
2! 3!
dM x
T x
dx
 = − 〈 〉 + 〈 〉 − 〈 〉 
=
a
x
b
d
a
q b
q ( )
( )
( )
( )
a b2 2
b a 3 3
q q
M x x-a x-b
2! 2!
q q /d
x-a x-b
3!
dM x
T x
dx
= − 〈 〉 + 〈 〉 +
−
 + −〈 〉 + 〈 〉 
=
a
x
b
d
a
q
b
q
( )
( )
( )
( )
a b2 2
a b 3 3
q q
M x x-a x-b
2! 2!
q q /d
x-a x-b
3!
dM x
T x
dx
= − 〈 〉 + 〈 〉 +
−
 + 〈 〉 − 〈 〉 
=

Formularioparavigasypórticos3.5
3.2 VIGA APOYADA EN LOS EXTREMOS
3.2.1 CARGA PUNTUAL EN LA VIGA
REACCIONES
A B
P b P a
R R
L L
⋅ ⋅
= =
ESFUERZOS CORTANTES
;AC CB
P b P a
Q cte Q cte
L L
⋅ ⋅
= = = − =
MOMENTOS FLECTORES
( ) max 0; ; paraAC CB C
P b P a P a b
M x M L x M M x a
L L L
⋅ ⋅ ⋅ ⋅
= ⋅ = ⋅ − = = =
ANGULOS DE GIRO
( ) ( ) ( ); ;
6 6 3
A B C
P a b P a b P a b
L b L a b a
E I L E I L E I L
ϕ ϕ ϕ
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ + = − ⋅ + = ⋅ −
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) 22 2 2
2 2 2
1 ; 1
6 6
AC CB
P L a L xP L b x b x a L x
y y
E I E I LL L L
 ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − 
 = ⋅ − − = ⋅ − −    ⋅ ⋅ ⋅ ⋅     
FLECHA MAXIMA
( )
2 23
2 2 2
para
39 3
C
P b L b
f L b x
E I L
⋅ −
= ⋅ − =
⋅ ⋅ ⋅
x
a
A
L
b
C
P
B
QA
maxM
B
Q

3.6ProntuarioparaCálculodeEstructuras
3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA
REACCIONES
A B
p b c p a c
R R
L L
⋅ ⋅ ⋅ ⋅
= =
ESFUERZOS CORTANTES
; ;
2
AC CD DB
p b c p b c c p a c
Q Q p a x Q
L L L
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 
= = − ⋅ − + = − 
 
MOMENTOS FLECTORES
( )
2
max 0
;
2 2
2 para
2 2
AC CD
DB
p b c p b c p c
M x M x x a
L L
p a c
M L x
L
p b c b c c b c
M a c x a
L L L
 ⋅ ⋅ ⋅ ⋅  
= ⋅ = ⋅ − ⋅ − −  
  
⋅ ⋅
= ⋅ −
⋅ ⋅ ⋅ ⋅ 
= ⋅ ⋅ − + = − + ⋅  
ANGULOS DE GIRO
2 2
;
6 4 6 4
A B
p a b c c p a b c c
L b L a
E I L a E I L b
ϕ ϕ
   ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ + − = − ⋅ + −   
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅   
( )
2
2
4 2
3
2
2
6 4
4 4
24 2 4
6 4
AC
CD
DB
p b c x c
y x a L b
L E I a
p c c
y L x a b c x a b c L b x
E I L a
p a c L x c
y L x b L a
L E I a
  ⋅ ⋅
= ⋅ − + ⋅ + −  
⋅ ⋅ ⋅   
    
 = ⋅ ⋅ − − − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + − ⋅    ⋅ ⋅ ⋅ ⋅     
  ⋅ ⋅ −
= ⋅ ⋅ − − + ⋅ + −  
⋅ ⋅ ⋅   
QA
B
Q
x
a
A
L
b
C
P
B
c
maxM
D

3.2.3 CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
( ) ( )1 2 1 2
1 1
2 ; 2
6 6
A BR p p R p p= ⋅ + = + ⋅ .
ESFUERZOS CORTANTES
( )1 2 23
; ;
6
A A x A B B
p L x p x
Q R Q R x Q R
L
⋅ − + ⋅
= = − ⋅ = −
⋅
MOMENTOS FLECTORES
( )
( ) ( )
( )
1 2 2
2 2
max 1 2 1 2
2 2
0 1 1 2 1 2
2 1
3
6
comprendido entre 0,125 y 0,128
2 2
1 1
para x
3
x A
p L x p x
M R x x
L
L L
M p p p p
p p p p p
p p
− + ⋅
= ⋅ − ⋅
⋅
⋅ ⋅ + ⋅ ⋅ +
 
= ⋅ − + ⋅ + + ⋅ 
−   
ANGULOS DE GIRO
( ) ( )
3 3
1 2 1 28 7 ; 7 8
360 360
A B
L L
p p p p
E I E I
ϕ ϕ= ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅
⋅ ⋅ ⋅ ⋅
( ) ( ) ( )
( ) ( )
3 2
1 2 1 2
2 3
1 2 1 2
3 3 4
360 8 7 8 7
x
p p x p p Lxx L x
y
EI p p L x p p L
 − − + +−
 =
+ + +  
FLECHA MAXIMA
( ) ( )4 4
1 2 1 2
entre 0,01302 y 0,01304
2 2
p p L p p L
E I E I
+ ⋅ + ⋅
⋅ ⋅
⋅ ⋅ ⋅ ⋅
QA
maxM
QB
P
A B
1
P2
0x
L
x

3.2.4 MOMENTO FLECTOR
REACCIONES
R R
M
L
A B= − = −
ESFUERZOS CORTANTES
x
M
Q cte
L
= =
MOMENTOS FLECTORES
( )
izq der
C CM M M
AC CB
izq der
C C
M M
M x M L x
L L
M M
M a M b
L L
= − ⋅ = − ⋅ −
= − ⋅ = − ⋅ = +
ANGULOS DE GIRO
( )
2 2
2 2
3 3
2
3 1 ; 3 1
6 6
3
A B
C
M L b M L a
E I E IL L
M
a b
E I L
ϕ ϕ
ϕ
   ⋅ ⋅
= ⋅ ⋅ − = ⋅ ⋅ −   
⋅ ⋅ ⋅ ⋅   
= ⋅ +
⋅ ⋅ ⋅
2 2
2 2
22
2
1 3
6
( )
1 3
6
AC
CB
M L x b x
y
E I L L
M L L x a L x
y
E I LL
 ⋅ ⋅
= − ⋅ − ⋅ − 
⋅ ⋅  
 ⋅ ⋅ − − 
 = − ⋅ − ⋅ −   ⋅ ⋅   
FLECHA
( )
3
C
M a b
f b a
E I L
⋅ ⋅
= ⋅ −
⋅ ⋅ ⋅
BA
M+
QA QB
MC
MC
M
a
L
b
C

3.3 VIGA EMPOTRADA EN LOS EXTREMOS
REACCIONES
( ) ( )
2 2
3 3
2 ; 2A B
P b P a
R L a R L b
L L
⋅ ⋅
= ⋅ + ⋅ = ⋅ + ⋅
ESFUERZOS CORTANTES
( ) ( )
2 2
3 3
2 ; 2AC CB
P b P a
Q L a cte Q L b cte
L L
⋅ ⋅
= ⋅ + ⋅ = = − ⋅ + ⋅ =
MOMENTOS FLECTORES
( )
( )
2 2 2
2 2 3
2 2 2
2
03 3
; ; 2
2
2 ; para
A B AC
BC C
P a b P a b P b
M M M L x a x a L
L L L
P a P a b
M L b L L x b x M x a
L L
⋅ ⋅ ⋅ ⋅ ⋅
= − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅
⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ + − ⋅ − ⋅ ⋅ = =
( )
( )
2 2
2
22
2
2
3
6
3 2
6
AC
BC
P b a x x
y a x
E I L L
L xP a L x
y b L x b
E I L L
⋅ ⋅ ⋅ 
= ⋅ ⋅ − − ⋅ ⋅ ⋅  
−⋅ − ⋅ 
= ⋅ ⋅ − − − ⋅ ⋅ ⋅ ⋅  
FLECHAS
( )
3 3 3 2
max3 2
2
;
3 3 2
2
para
2
C
P a b P a b
f f
E I L E I L a
a L
x
L a
⋅ ⋅ ⋅ ⋅ ⋅
= =
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅
⋅ ⋅
=
+ ⋅
MC
A B
a
L
x
b
P
B
A
Q
Q
x
0
MA
C
MB

REACCIONES
;A B A B
A B
p b c M M p a c M M
R R
L L L L
⋅ ⋅ − ⋅ ⋅ −
= − = +
ESFUERZOS CORTANTES
; ;AC A BD B CD A
c
Q R cte Q R cte Q R p x a
a
 
= = = − = = − ⋅ − + 
 
MOMENTOS FLECTORES
( )
2
3 2
2 2
3 2
2 2
;
2 2
12
; 3
12
12
3
12
AC A A CD A A
BD B B A
B
p c
M R x M M R x M x a
p c a b
M R L x M M L b
L c
p c a b
M L a
L c
 
= ⋅ + = ⋅ + − ⋅ − + 
 
 ⋅ ⋅ ⋅
= ⋅ − + = − ⋅ − ⋅ + 
⋅  
 ⋅ ⋅ ⋅
= − ⋅ − ⋅ + 
⋅  
( )
( ) ( ) ( )
2
4
3 3
3 2 2
3
6
1
4 12
24 2
1
3 3 2 3
6
AC A A
CD A A
DB B B B A B B B
x
y M R x
E I
c
y p x a R x M x
E I
y R x M LR x M LR Lx M LR L
EI
= ⋅ − ⋅ − ⋅
⋅ ⋅
  
= ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅  ⋅ ⋅    
 = − + + + − + 
a
MA
Q
A
x
A
B
Q
L
b
B
P
c
C D
MB

REACCIONES
( )
( )
1 2
1 2
2
6
2
6
A B
A
A B
B
L M M
R p p
L
L M M
R p p
L
−
= ⋅ ⋅ + −
−
= ⋅ + ⋅ +
ESFUERZOS CORTANTES
( )1 22
2
A A
x A
B B
Q R
p L x p x
Q R x
L
Q R
=
⋅ ⋅ − + ⋅
= − ⋅
⋅
= −
MOMENTOS FLECTORES
( )
( )
( )
2
1 2
1 2 2
2
1 2
3 2
60
3
6
2 3
60
A
x A A
B
L
M p p
p L x p x
M R x M x
L
L
M p p
= − ⋅ + ⋅
⋅ ⋅ − + ⋅
= ⋅ + − ⋅
⋅
= − ⋅ + ⋅
( )2
2 1 3 2
1 4 12
24 5
x A A
p px
y x p L x R L x M L
E I L
 −
= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅ 
⋅ ⋅ ⋅   
A
Q
Q
B
x
A
L
B
B
MA
M
P2
P1

REACCIONES
3 3
6 6
;A B
M M
R a b R a b
L L
⋅ ⋅
= − ⋅ ⋅ = ⋅ ⋅
ESFUERZOS CORTANTES
3
6
x
M
Q a b cte
L
⋅
= − ⋅ ⋅ =
MOMENTOS FLECTORES
( )
⋅ ⋅   
= ⋅ − ⋅ = − ⋅ − ⋅   
   
 ⋅  
= ⋅ ⋅ ⋅ − ⋅ −  
  
 ⋅ − 
= − ⋅ ⋅ ⋅ − ⋅ −  
  
⋅
= − ⋅ ⋅ = + ⋅ − ⋅ ⋅2 3 2
3 3
2 3 2 3
3 1 2 1
3 1 2 1
6
; 6
A B
AC
CB
izq der
C A C A
M a b M b a
M M
L L L L
M a a x
M
L L L
M b b L x
M
L L L
M M
M M a b M M L a b
L L
( )
2
2
2
2
2
2
2
2
AC
BC
M b x L x b
y a
E I L LL
M a L x b x a
y
E I L LL
⋅ ⋅ − 
= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅  
⋅ ⋅ − ⋅ 
= ⋅ ⋅ − ⋅ ⋅ ⋅  
FLECHA
( )
2 2
3
2
C
M a b
f a b
E I L
⋅ ⋅
= − ⋅ −
⋅ ⋅ ⋅
MC
A
Q QB
x
A
L
a b
B
+M
CM
C
A
M
BM

3.4 VIGA APOYADA-EMPOTRADA
REACCIONES
( ) ( )
2
2 2
3 3
3 ; 3
2 2
A B
P b P a
R L b R L a
L L
⋅ ⋅
= ⋅ ⋅ − = ⋅ ⋅ −
⋅ ⋅
ESFUERZOS CORTANTES
( ) ( )
2
2 2
3 3
3 ; 3 .
2 2
AC CB
P b P a
Q L b cte Q L a const
L L
⋅ ⋅
= − ⋅ ⋅ − = = − ⋅ ⋅ − =
⋅ ⋅
MOMENTOS FLECTORES
( ) ( )
( ) ( )
2 2 2
2 3
2 3 2 2
3 3
; 3 2
2 2
3 2 ; 2 3
2 2
B C
AC CB
P a P a
M L a M b a b
L L
P x P a
M b a b M L L x a x
L L
⋅ ⋅
= − ⋅ − = ⋅ ⋅ ⋅ + ⋅
⋅ ⋅
⋅ ⋅
= ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅
⋅ ⋅
ANGULOS DE GIRO
( ) ( )
( )
2 2
2 2
3
; 2
4 4
A C
P a L a P a L a
L a L a
E I L E I L
ϕ ϕ
⋅ − ⋅ ⋅ −
= = ⋅ − ⋅ ⋅ −
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
( )
( )
2
2 2
3
2 2 2
2 2
3 2
12
3 1 3
12
AC
BC
P b x
y a L x L a
E I L
P a L x a a L x
y
E I LL L
⋅ ⋅
 = ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅
 ⋅ ⋅ −     − 
= ⋅ ⋅ − − − ⋅      ⋅ ⋅       
FLECHA MAXIMA
2
para x=
6 2 2
max
p b a a a
f L
E I L a L a
⋅ ⋅
= ⋅ ⋅
⋅ ⋅ ⋅ + ⋅ +
Q
Q
MB
A
B
x
a
L
b
A B
C
P
MC

REACCIONES
;B B
A B
p b c M p a c M
R R
L L L L
⋅ ⋅ ⋅ ⋅
= + = −
ESFUERZOS CORTANTES
; ;
2
AC A DB B CD A
c
Q R cte Q R cte Q R p x a
 
= = = − = = − ⋅ − + 
 
MOMENTOS FLECTORES
( )
2
2
2
;
2 2
;
42
AC A CD A
DB B B B
p c
M R x M R x x a
p a b c c
M R L x M M L a
bL
 
= ⋅ = ⋅ − ⋅ − + 
 
 ⋅ ⋅ ⋅
= ⋅ − + = − ⋅ + − 
⋅⋅  
ANGULOS DE GIRO
3 2
2
12
3
48
A
p c a b
L b
E I L c
ϕ
 ⋅ ⋅ ⋅
= ⋅ − + 
⋅ ⋅ ⋅  
( )
( )
2
2 3
2
4 2
3 3
2
2
12
8 3
48
1 12
8 2 3
48 4
3
6
AC A
CD A
DB B B
x a b
y R L x p c L b
E I L c
c ab
y R Lx pL x a pc L b x
E I L c
L x
y R L x M
E I
  ⋅ ⋅
= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − +  
⋅ ⋅ ⋅    
   
= ⋅ − + − + + − +   ⋅ ⋅ ⋅     
−
 = − ⋅ ⋅ − + ⋅ ⋅ ⋅
Q
Q
a
x
C
P
M
A
A
B
B
L
b
c
B

REACCIONES
( ) ( )1 2 1 22 ; 2
6 6
B B
A B
L M L M
R p p R p p
L L
= ⋅ ⋅ + + = ⋅ + ⋅ −
ESFUERZOS CORTANTES
( )1 22
;
2
x A B B
p L x p x
Q R x Q R
L
⋅ ⋅ − + ⋅
= − ⋅ = −
⋅
MOMENTOS FLECTORES
( )
( )
2
1 2 2
1 2
3
; 7 8
6 120
x A B
p L x p x L
M R x x M p p
L
⋅ ⋅ − + ⋅
= ⋅ − ⋅ = − ⋅ ⋅ + ⋅
⋅
ANGULOS DE GIRO
( )
3
1 23 2
240
A
L
p p
E I
ϕ = ⋅ ⋅ + ⋅
⋅ ⋅
( ) ( )4 3 2 2 3
2 1 1 1 25 20 5 12 3
120
x A A
x
y p p x Lp x R Lx L R L p p L
EIL
  = − + − + − +  
2
A
Q Q
L
x
B
BA
1P
P
MB

REACCIONES
( )2 2
3
3
2
A B
M
R R L a
L
= − = ⋅ ⋅ −
ESFUERZOS CORTANTES
x AQ R cte= =
MOMENTOS FLECTORES
( )
( )
2 2
2
2
2 2
3 2
; ; 3
2
3
; 3 1 2
2 2
der izq
C A C A B
AC BC
M
M R a M M R a M L a
L
M x M x a
M L a M
LL L
= ⋅ − = ⋅ = ⋅ − ⋅
⋅
  ⋅
= ⋅ ⋅ − = ⋅ ⋅ ⋅ − −  
   
ANGULOS DE GIRO
( ) ( )
2
3 ; 3 1 4
4 4
A C
M M b a
L a a L b
E I L E I L L
ϕ ϕ
  
= ⋅ − ⋅ ⋅ − = ⋅ ⋅ ⋅ ⋅ + −  ⋅ ⋅ ⋅ ⋅ ⋅    
( ) ( )
( ) ( )
3 2 2
3
2 2 2 2
3
4 3
4
2
4
AC
BC
M b x
y L x L a L
E I L
M
y L x a L x L a
E I L
⋅ ⋅  = ⋅ − ⋅ − − ⋅ ⋅ +
 ⋅ ⋅ ⋅
 = ⋅ − ⋅ ⋅ ⋅ − ⋅ −
 ⋅ ⋅ ⋅
Q
x
a
L
b
A
M
Q
B
A B
B
CM
M
MC
C +

3.5 VIGA EMPOTRADA EN UN EXTREMO
REACCIONES
BR P=
ESFUERZOS CORTANTES
0 ;AC CBQ Q P cte= = − =
MOMENTOS FLECTORES
( )0 ; ;AC CB BM M P x a M P b= = − ⋅ − = − ⋅
ANGULOS DE GIRO
2
2
A C
P
b
E I
ϕ ϕ= = − ⋅
⋅ ⋅
( )( ) ( ) ( )
2
2
3 ; 2 3
6 6
AC CB
P b P
y L x b y L x b a
E I E I
⋅
= ⋅ ⋅ − − = ⋅ − ⋅ ⋅ + ⋅
⋅ ⋅ ⋅ ⋅
FLECHA MAXIMA
( )
3 2
; 2 3
3 6
C A
P b P b
f f b a
E I E I
⋅ ⋅
= = ⋅ ⋅ + ⋅
⋅ ⋅ ⋅ ⋅
L
a
x
A
b
B
P
Q
MB
B
C

a
x
A
C
M
Q
L
b
P
B
D
c
B
B
REACCIONES .
BR p c= ⋅
ESFUERZOS CORTANTES .
0 ; ;
2
AC CD DB
c
Q Q p x a Q p c cte
 
= = − ⋅ − + = − ⋅ = 
 
MOMENTOS FLECTORES .
( )
2
2
2
0 ; ;
2 2
;
AC CD D
DB B
c
p x a
p c
M M M
M p c x a M p c b
 
⋅ − +  ⋅ = = − = −
= − ⋅ ⋅ − = − ⋅ ⋅
ANGULOS DE GIRO .
2 2
2 2
; ;
2 4 2 12
D C A C
p c c p c c
b b
E I E I
ϕ ϕ ϕ ϕ
   ⋅ ⋅
= − ⋅ − = − ⋅ + =   
⋅ ⋅ ⋅ ⋅   
ECUACION DE LA ELASTICA .
( ) ( ) ( )
( )
2
2 2 3
4 2
2 3
2 ; 3 2
6 6 4
4 3 8
24 2 4
DB AC
DC
p c p c c
y L x b a x y a x b b
E I E I
p c c
y x a c a x b b c
E I
  ⋅ ⋅
= ⋅ − ⋅ ⋅ − + = ⋅ − ⋅ ⋅ + + ⋅  
⋅ ⋅ ⋅ ⋅    
   
= ⋅ − + + ⋅ ⋅ − ⋅ ⋅ + + ⋅ ⋅   ⋅ ⋅     
FLECHAS .
( )
2
2 2
3 2 3
2 3 12
4 ; 3 2
12 2 6 4
D
C A
p c c b c
f b
E I
p c c p c c
f b b c c f a b b
E I E I
⋅    
= ⋅ − ⋅ +   ⋅    
    ⋅ ⋅ 
= ⋅ + ⋅ ⋅ − + = ⋅ ⋅ ⋅ + + ⋅     
⋅ ⋅ ⋅ ⋅        

L
Q
x
A
B
B
1P
2
P
B
M
REACCIONES
( )1 2
1
2
BR p p= +
ESFUERZOS CORTANTES
( )
2
2 1
1 1 2;
2 2
x B
p p x L
Q p x Q p p
L
−
= − ⋅ − ⋅ = − +
MOMENTOS FLECTORES
( ) ( )
2 2
2 1 1 2 13 ; 2
6 6
x B
x L
M p p x L p M p p
L
 = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅
ANGULOS DE GIRO
( )3
1 23
24
A
L p p
E I
ϕ
⋅ ⋅ +
= −
⋅ ⋅
( ) ( )
( ) ( )
( )( ) ( )
3
2 2
2 1 2
2
2 1 2 1
5
24
2 2 2
x
L x
L x p p L x p
y L
EI
L L x p p L p p
 −
−  − − + − −
=  
 − − + + + 
FLECHA
( )4
2 14 11
120
A
L p p
f
E I
⋅ ⋅ + ⋅
=
⋅ ⋅

M
L
x
A
B
M
a b
B
REACCIONES
0BR =
ESFUERZO CORTANTE
0xQ =
MOMENTOS FLECTORES
0 ; ;AC CB ACM M M cte M M= = − = = −
ANGULOS DE GIRO
C A
M b
E I
ϕ ϕ
⋅
= = −
⋅
( ) ( )
2
2 2 ;
2 2
AC BC
M M
y b L x b y L x
E I E I
= ⋅ ⋅ ⋅ − ⋅ − = −
⋅ ⋅ ⋅ ⋅
FLECHA
( )
2
; 2
2 2
C A
M b M
f f b L b
E I E I
⋅
= = ⋅ ⋅ ⋅ −
⋅ ⋅ ⋅ ⋅

3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
L/2 L/2
L
P P
L/2 L/2
L
0,312 P
0,312 P
0,688 P
0,688 P
- 0,188 PL
0,156 PL 0,156 PL
A B C
A B C
A B C B
- 0,094 PL
0,203 PL
A
A
0,594 P
C
B C
B
0,094 P
L/2L/2
0,405 P
L
A
P
L
C
0,094 P

ESFUERZOS CORTANTES
MOMENTOS FLECTORES MOMENTOS FLECTORES
ESFUERZOS CORTANTES
0,07 QL
- 0,125 QL
BA
A
0,625 QL
C
B C
B
0,375 QL
A
L
Q
L C
0,375 L
0,625 QL
0,375 QL
2
0,07 QL
2
0,375 L
- 0,063 QL
B
0,096 QL
A
2
A
0,437 L
C
0,563 QL
B C
B
0,437 QL
0,063 QL
A
L
Q
L C
22
Q

3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES
Relación
entre
luces
ESFUERZOS CORTANTES
MOMENTOS
FLECTORES
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
2
2 2
1
0.5 0.5
8 2
2 2 2
k k k f
f a f b f c
k
k f a d
d e g
k
− +
= = − = + = +
= − = =
QQ
B
C
A
c QL
f QL
2
A
e QL
L
A
a QL
a L
2
C
2
g QL
B
C
k L
d L
B
d QLb QL
MOMENTOS FLECTORES
ESFUERZOS CORTANTES

Relación
entre
luces
ESFUERZOS CORTANTES
MOMENTOS
FLECTORES
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
2 2
1
0.5 0.5
8
2 2 2
k k
f a f b f
k f a d
d e g
k
− +
= = − = +
= − = =
A B C
2
g QL
A
C
B
L
Q
A C
Q
k L
B
f QL
2
a QL
c QL
b QL
d QL
d L
MOMENTOS FLECTORES
ESFUERZOS CORTANTES

3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES
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
3
2 2
1
0.5 0.5
12 8
2 2 8
k
f a f b f
k
k a k
c e g f
+
= = − = +
⋅ +
= = = −
A CB
k LL L
D
Q QQ
MOMENTOS FLECTORES
a L
A
A
a QL
2
g QL
2
f QL
C
2
2
e QL
f QL
B
e QL
2
D
c QL
b QL
C
B
b QL
c QL
a L
D
a QL
ESFUERZOS CORTANTES

Relación
entre
luces
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
3
2 2
1
0.5 0.5
12 8
2 2 8
k
f a f b f
k
k a k
c e g f
+
= = − = +
⋅ +
= = = −
k L
B
L
A
L
DC
QQ Q
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
e QL
2
B
2
g QL
f QL
2
A
B
b QL
c QL
A
a QL
a L
e QL
2
C D
2
f QL
C
a L
a QL
D
b QL
c QL

3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
2
1
3 2
I h
k y N k
I l
= ⋅ = +
3.9.1 CARGA REPARTIDA VERTICAL
REACCIONES
2
3
2 12
A
D
A D
psn
V
l
psm
V
l
ps s
H H mn
hlN
=
=
 
= = − 
 
MOMENTOS FLECTORES
2
2
3
2 12
( )
2
B C
x A A
ps s
M M mn
lN
En S
p x m
M V x H h
 
= = − ⋅ − 
 
−
= ⋅ − − ⋅
h
p
A
B
D
C
l
I 2
I 1 1I
x
sa
m n
MB CM
HA HD
VA VD

3.9.2 CARGA REPARTIDA HORIZONTAL
REACCIONES
( )
( )
2
2
2
8
6
8
A D
D
A
ph
V V
l
ph N k
H
N
ph N k
H
N
= =
+
=
−
=
MOMENTOS FLECTORES
( )
( )
2
2
2
8
2
8
( )
2
B
C
Y B
ph
M N k
N
ph
M N k
N
En AB
py h y y
M M
h
= −
= − +
−
= + ⋅
h
p
A
B
D
C
l
I 2
I 1 1I
MB
CM
HA HD
VA VD
y
MB

3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
REACCIONES
3
2
A
D
A D
Pn
V
l
Pm
V
l
Pmn
H H
lhN
=
=
= =
MOMENTOS FLECTORES
3
2
2 3
2
B C
P
Pmn
M M
lN
N
M Pmn
lN
= = − ⋅
−
=
h
A
B
D
C
l
I 2
I 1 1I
m n
MB CM
HA HD
VA VD
P
MP

3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
1 23 3
1 2
1 2
h hI I
k y k
I s I s
= ⋅ = ⋅
3.10.1 CARGA REPARTIDA VERTICAL
REACCIONES
( ) ( )
2
1 2
2 2
1 1 2 2 1 2
2
8 1 1
A D
A D
pl
V V
h hpl
H H
h k h k hh
= =
+
= =
+ + + +
MOMENTOS FLECTORES
( )
( ) ( )
2
1 2 1
2 2
1 1 2 2 1 28 1 1
B
h h hpl
M
h k h k hh
+
= −
+ + + +
( )
( ) ( )
2
1 2 2
2 2
1 1 2 2 1 2
1
8 1 1
( )
2
C
X A
h h hpl
M
h k h k hh
En BC
px l x f
M H x h
l
+
= −
+ + + +
−  
= − + 
 
h
p
A
B
D
C
l
I 3
I 1
2
Ix
MB
CM
HA HD
VA VD
s
2
h 1
f

3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
REACCIONES
( )
( ) ( )
2
1
1
2
1 1 21
2 2
1 1 2 2 1 2
2
4 5 2
8 1 1
A D
A D
D
ph
V V
l
H ph H
h k hph
H
h k h k hh
= =
= −
+ +
=
+ + + +
MOMENTOS FLECTORES
( )
( ) ( )
( )
( ) ( )
2 3
1 1 21 1
2 2
1 1 2 2 1 2
2
1 1 21 2
2 2
1 1 2 2 1 2
2
4 5 2
2 8 1 1
4 5 2
8 1 1
2
B
C
Y A
h k hph ph
M
h k h k hh
h k hph h
M
h k h k hh
En AB
py
M H y
+ +
= −
+ + + +
+ +
=
+ + + +
= −
h
A
B
D
C
l
I 3
I 1
2
I
MB
CM
HD
VD
2
HA
VA
p
s
y
h
f
1

3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
REACCIONES
( )
( ) ( )
( ) ( )
1 2
2
1 1 1 2 1 2
2 2
1 1 2 2 1 2
2
8 1 4
8 1 1
A D
A D
D
pf h h
V V
l
H pf H
h k hh f h hpf
H
h k h k hh
+
= =
= −
+ + + +
=
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
2
1 1 1 2 1 21
1 2 2
1 1 2 2 1 2
2
1 1 1 2 1 22
2 2
1 1 2 2 1 2
2
1
8 1 4
8 1 1
8 1 4
8 1 1
2
B
C
Y A A
h k hh f h hpfh
M pfh
h k h k hh
h k hh f h hph
M
h k h k hh
En BC
l py
M V y H y h
f
+ + + +
= −
+ + + +
+ + + +
= −
+ + + +
= − + + −
h
A
B
D
C
l
I 3
I 1
2
I
MB
CM
HD
VD
2
HA
VA
p
s
y
h
f
1

REACCIONES
( ) ( )
1 2
2 2 2
1 1 2 2 1 2
( ) ( )
2 1 1
A
D
A D
Pb
V
l
Pa
V
l
h l b h l aPab
H H
l h k h k hh
=
=
+ + +
= =
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 21
2 2 2
1 1 2 2 1 2
1 22
2 2 2
1 1 2 2 1 2
1
2 1 1
2 1 1
B
C
P A
h l b h l aPabh
M
l h k h k hh
h l b h l aPabh
M
l h k h k hh
Pab af
M H h
l l
+ + +
= −
+ + + +
+ + +
= −
+ + + +
 
= + + 
 
h
A
B
D
C
l
I 1
2
I
MB
C
M
HD
VD
2
HA
VA
s
a b
I 3
MP
1
f
h

3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
2
1
I h
k
I s
= ⋅
3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
REACCIONES
( ) ( )
2
2
2
8 5
32 3 3
A E
A E
pl
V V
pl h f
H H
h k f h f
= =
+
= =
+ + +
MOMENTOS FLECTORES
( ) ( )
( )
2
2
2
8 5
32 3 3
8
2
2
B D
C B
B
X
pl h h f
M M
h k f h f
pl f h
M M
h
En BC y DC
x l x M fx
M p h
h l
+
= = −
+ + +
+
= +
−  
= + + 
 
p
A
B
E
C
l
I 2
I 1
x
MB
CM
HA HE
VA VE
I 2
I 1
s
D
h
f
MD

3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
REACCIONES
( ) ( )
2
2
3
8
8
8 5
64 3 3
A
E
A E
pl
V
pl
V
pl h f
H H
h k f h f
=
=
+
= =
+ + +
MOMENTOS FLECTORES
( ) ( )
( )
2
2
2
8 5
64 3 3
16
2
2
B D
C B
B
X
pl h h f
M M
h k f h f
pl f h
M M
h
En BC
x l x M fx
M p h
h l
+
= = −
+ + +
+
= +
−  
= + + 
 
p
A
B
E
C
l
I 2
I 1
x
MB
CM
HA HE
VA VE
I 2
I 1
s
D
h
f
MD

REACCIONES
( )
( ) ( )
2
2
2
2
5 12 6
16 3 3
A E
A E
E
ph
V V
l
H ph H
k h fph
H
h k f f h
= =
= −
+ +
=
+ + +
MOMENTOS FLECTORES
( )
( ) ( )
2
2
3
2
2
2
4
5 12 6
16 3 3
2
B D
C D
D
y A
ph
M M
ph f h
M M
h
k h fph
M
h k f f h
En AB
py
M H y
= +
+
= +
+ +
= −
+ + +
= − + ⋅
A
B
E
C
l
I 2
I 1
MB
CM
HA HE
VA VE
I 2
I 1
s
D
h
f
MD
p
y

REACCIONES
( )
( ) ( )
( ) ( )
2
2
2
2
8 3 5 4
16 3 3
A E
A E
E
pf
V V f h
l
H pf H
h k f f hpf
H
h k f f h
= = +
= −
+ + +
=
+ + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( )
22
2
2
4 2 5
16 3 3
2
B A
C
D E
x A A
M H h
h k f h fpf
M
h k f f h
M H h
En BC
y h
M H y V x p
f
siendo y x h
l
= ⋅
+ + +
= − ⋅
+ + +
= − ⋅
−
= ⋅ − ⋅ −
= +
A
B
E
C
l
I 2
I 1
MB
CM
HA HE
VA VE
I 2
I 1
s
D
h
f
MD
p
x
y

REACCIONES
( )
( ) ( )
2 2
2 2
6 ln 3 4
4 3 3
A
A
A E
Pn
V
l
Pm
V
l
h f l mPm
H H
l h k f f h
=
=
+ −
= =
+ + +
MOMENTOS FLECTORES
2
2
B D A
C B
P A A
M M H h
Pm h f
M M
h
hl fm
M V m H
l
= = − ⋅
+
= +
+
= ⋅ −
p
A
B
E
C
l
I 2
I 1
MB
CM
HA HE
VA VE
I 2
I 1
s
D
h
f
MD
m n

3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL
1 23 3
1 2
1 2
h hI I
k y k
I l I l
= ⋅ = ⋅
REACCIONES
( ) ( )
( ) ( )
( ) ( )
2 2
1 2
2 2
1 1 2 2 1 2
2 2
1 2
2 2
1 1 2 2 1 2
2
1 2
2 2
1 1 2 2 1 2
2 8 1 1
2 8 1 1
8 1 1
A
D
A D
h hpl pl
V
h k h k hh
h hpl pl
V
h k h k hh
h hpl
H H
h k h k hh
−
= +
+ + + +
−
= −
+ + + +
−
= =
+ + + +
MOMENTOS FLECTORES
( )
( ) ( )
( )
( ) ( )
2
1 2 1
2 2
1 1 2 2 1 2
2
1 2 2
2 2
1 1 2 2 1 2
2
1
8 1 1
8 1 1
2
B
C
x A A
h h hpl
M
h k h k hh
h h hpl
M
h k h k hh
En BC
px
M V x H h
+
= −
+ + + +
+
= −
+ + + +
= ⋅ − − ⋅
h
A
B
D
C
l
I 3
I 1
2
I
MB CM
HA
HD
VA
VD
p
x
h
2
1

REACCIONES
( ) ( )
2
1 1 2
2
1 1 1 1 2
2 2
1 1 2 2 1 2
2
5 4 2
8 1 1
A D D
A D
D
ph h h
V V H
l l
H ph H
ph k h h h
H
h k h k hh
−
= = −
= −
+ +
=
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
2 3
1 1 1 1 1 2
2 2
1 1 2 2 1 2
2
1 2 1 1 1 2
2 2
1 1 2 2 1 2
2
5 4 2
2 8 1 1
5 4 2
8 1 1
2
B
C
y A
ph ph k h h h
M
h k h k hh
ph h k h h h
M
h k h k hh
En AB
py
M H y
+ +
= − −
+ + + +
+ +
= −
+ + + +
= ⋅ −
h
p
A
B
D
C
l
I 3
I 1
2
I
MB
CM
HA
HD
VA
VD
y
MB
h1
2

REACCIONES
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
1 2
1 23 2 2
1 1 2 2 1 2
1 2
1 23 2 2
1 1 2 2 1 2
1 2
2 2 2
1 1 2 2 1 2
2 1 1
2 1 1
2 1 1
A
D
A D
l b h l a hPb Pab
V h h
l l h k h k hh
l b h l a hPa Pab
V h h
l l h k h k hh
l b h l a hPab
H H
l h k h k hh
+ + +
= + −
+ + + +
+ + +
= − −
+ + + +
+ + +
= =
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 21
2 2 2
1 1 2 2 1 2
1 22
2 2 2
1 1 2 2 1 2
2 1 1
2 1 1
B
C
P A B
l b h l a hPabh
M
l h k h k hh
l b h l a hPabh
M
l h k h k hh
M V a M
+ + +
= −
+ + + +
+ + +
= −
+ + + +
= ⋅ +
A
B
D
C
l
I 3
I 1
2
I
a b
MB CM
HA
HD
VA
VD
P
MP
h
h1
2

3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
2
1
I h
k
I l
= ⋅
REACCIONES
( )
2
2 4 2
A D A D
pl pl
V V H H
h k
= = = =
+
MOMENTOS FLECTORES
( )
( )
( )
( )
2
2
2
2
12 2
6 2
2 6 2
3 2
máx
24 2 2
A D
B C
x
pl
M M
k
pl
M M
k
En BC
px l x pl
M
k
pl k l
M pos para x
k
= =
+
= = −
+
−
= −
+
+
= =
+
h
A
B
D
C
l
I 2
I 1 1I
x
MB CM
HA
VA
p
MA
HD
VD
MD

REACCIONES
( )
( )
( )
2
6 1
2 3
8 2
A D
A D
D
ph k
V V
l k
H ph H
ph k
H
k
= =
+
= −
+
=
+
MOMENTOS FLECTORES
2
2
2
2
2
2 1
5
24 6 1 2
2 2
1
24 6 1 2
2 2
3
24 6 1 2
2 1
3
24 6 1 2
2
A
B
C
D
y A A
ph
M
k k
ph
M
k k
ph
M
k k
ph
M
k k
En AB
py
M H y M
 
= − + + + + 
 
= − + + + 
 
= − − − + + 
 
= + − + + 
= − + ⋅ +
h
p
A
B
D
C
l
I 2
I 1 1I
MB
CM
y
MB
HA
VA
MA
HD
VD
MD

REACCIONES
( )
( )2
1
6 1
3
2 ( 2)
A
D A
A D
m n mPn
V
l l k
V P V
Pmn
H H
lh k
 −
= + 
 + 
= −
= =
+
MOMENTOS FLECTORES
( )
( )
( )
( )
1
2 2 6 1
1
2 2 6 1
1
2 2 6 1
1
2 2 6 1
A
B
C
D
CB
P
Pmn n m
M
l k l k
Pmn n m
M
l k l k
Pmn n m
M
l k l k
Pmn n m
M
l k l k
mMnMPmn
M
l l l
 −
= −  + + 
 −
= − +  + + 
 −
= − −  + + 
 −
= +  + + 
= + +
h
A
B
D
C
l
I 2
I 1 1I
m n
MB CM
P
MP
HA
VA
MA
HD
VD
MD

3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR
REACCIONES
3
(6 1)
2
A D
A D
Phk
V V
l k
P
H H
= =
+
= =
MOMENTOS FLECTORES
3 1
2 6 1
3
2 6 1
3 1
2 6 1
A
B C
D
Ph k
M
k
Ph k
M M
k
Ph k
M
k
+
= −
+
= − =
+
+
=
+
h
A
B
D
C
l
I 2
I 1 1I
MB CM
P
HD
VD
MD
HA
VA
MA

3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
2
1
I h
k
I s
= ⋅
REACCIONES
( )
( ) ( )
2
2 2 2
2
4 5
8 4
A E
A E
pl
V V
k h f fpl
H H
kh f k h hf f
= =
+ +
= =
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( )
( ) ( )
( )
2
2 2 2
22
2 2 2
2
2
8 15 6
48 4
16 15
48 4
8
2
2
A E
B D
C A A
x A A A
kh h f f h fpl
M M
kh f k h hf f
kh h f fpl
M M
kh f k h hf f
pl
M M H h f
En BC
xf px
M M V x H h
l
+ + −
= =
+ + + +
+ +
= = −
+ + + +
= + − +
 
= + ⋅ − + − 
 
p
A
B
E
C
l
I 2
I 1
x
MB
CM
I 2
I 1
s
D
h
f
MD
HA
VA
MA
HE
VE
ME

3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
REACCIONES
( )
( )
( ) ( )
2
2 2 2
2
4 1
3
32 3 1
4 5
16 4
A E
E
A E
pl
V V
k
V pl
k
k h f fpl
H H
kh f k h hf f
= −
+
=
+
+ +
= =
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
2 2
2 2 2
2 2
2 2 2
22 2
2 2 2
22 2
2 2 2
8 15 6
96 64 3 14
8 15 6
96 64 3 14
16 15
96 64 3 14
16 15
96 64 3 14
A
E
B
D
x A
kh h f f h fpl pl
M
kkh f k f fh h
kh h f f h fpl pl
M
kkh f k f fh h
kh h f fpl pl
M
kkh f k f fh h
kh h f fpl pl
M
kkh f k f fh h
En BC M M
+ + −
= −
++ + + +
+ + −
= +
++ + + +
+ +
= − −
++ + + +
+ +
= − +
++ + + +
=
( )
2
2
2
2
A A
C E E E
xf px
V x H h
l
l
M V M H f h
 
+ ⋅ − + − 
 
= + − +
p
A
B
E
C
l
I 2
I 1
x
MB
C
M
I 2
I 1
s
D
h
f
MD
HA
VA
MA
HE
VE
ME

REACCIONES
( )
( )
( ) ( )
2
22
2 2 2
2 3 1
2 3
4 4
A E
A E
E
ph k
V V
l k
H ph H
k h k f h fph
H
kh f k f fh h
= =
+
= −
+ + +
=
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
2 22
2 2 2
2
2 22
2 2 2
2
6 15 16 6 2 1
6
24 3 14
2
1
2
6 15 16 6 2 1
6
24 3 14
2
A
B A A
C E E E
D E E
E
y A A
kh k kf h f fph k
M
kkh f k f fh h
ph
M M H h
M M H f h V
M M H h
kh k kf h f fph k
M
kkh f k f fh h
En AB
py
M M H y
 + + + + + = − +
 ++ + + +
 
= + ⋅ −
= − + +
= − ⋅
 + + + + + = − +
 ++ + + +
 
= + ⋅ −
A
B
E
C
l
I 2
I 1
MB
CM
I 2
I 1
s
D
h
f
MD
p
y
HE
VE
ME
HA
VA
MA

REACCIONES
( )
( ) ( )
( ) ( )
2
2 2 2
43
8 3 1
2 4 10 5
4 4
A E
A E
E
k f h fpf
V V
l k
H pf H
k h k f kh kf fpf
H
kh f k f fh h
+ +
= =
+
= −
+ + + +
=
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
2 2 2
2 2 2
2
9 4 6 4 3 23
24 2 3 14
2
9 4 6 4 3 23
24 2 3 14
2 2
A
B A A
C E E E
D E E
E
y A A A
kh f h f h f h k fpf
M f
kkh f k f fh h
M M H h
l
M M H h f V
M M H h
kh f h f h f h k fpf
M f
kkh f k f fh h
En BC
l y h p y h
M M H y V
f
 + + + + +
 = − +
 ++ + + +
 
= + ⋅
= − + +
= − ⋅
 + + + + +
 = − +
 ++ + + +
 
− −
= + ⋅ − −
A
B
E
C
l
I 2
I 1
MB
CM
I 2
I 1
s
D
h
f
MD
p
y
HE
VE
ME
HA
VA
MA

REACCIONES
( )
( ) ( ) ( )
( ) ( )
2
3
2 2
2 2 2 2
3 2
3 1
3 4 1 3
4
A E
E
A E
V P V
l kl m mPm
V
kl
kl f h fm k lm f khPm
H H
l kh f k f fh h
= −
+ −
=
+
+ − + + −
= =
+ + + +
MOMENTOS FLECTORES
( ) ( ) ( )
( ) ( )
( )
( )
( ) ( ) ( )
( ) ( )
( )
2 2 2
2 2 2
2
2 2 2
2 2 2
2
3 2 4 2 2 ln 4
4
2
3 1
2
3 2 4 2 2 ln 4
4
2
3 1
A
B A A
C E E E
D E E
E
flh kl m fm kh h f kh f l m l
Pm kh f k f fh h
M
l n n m
k
M M H h
l
M M V H h f
M M H h
flh kl m fm kh h f kh f l m l
Pm kh f k f fh h
M
l n n m
k
 + − + + + + −
 
+ + + + 
=  
− 
− 
+ 
= − ⋅
= + − +
= − ⋅
 + − + + + + −
 
+ + + + 
=  
−
+
+ 
2
y A A A
En BC
fm
M M V m H h
l


 
= + ⋅ − + 
 
p
A
B
E
C
l
I 2
I 1
MB
CM
I 2
I 1
s
D
h
f
MD
m n
HA
VA
MA
HE
VE
ME

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