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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 aq x
c
A x aT x
c
A x aM 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)
ax
( )
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 Mc
≤ − =
≥ − =
= − −
==
P
M(x)
ax
T(x)
( )
( )( )
1
1 1
0
1
Si0
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 Pc
≤ − =
≥ − = −
= − −
= − −
==
Limitación de las Deformaciones 3.3
2M(x)
q
xa
T(x)
( )
( )
( )
( )
2
2 2
0
1
2
Si0
entonces
1
2 1
por lo tanto 2
x a x a
x a x a x a
q x q x aqT x x a
qM x x a
A qc
≤ − =
≥ − = −
= −
= − −
= − −⋅
==
3
ax
d
q
2T(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 dq x x a
q dT x x a
q dM x x a
qAd
c
≤ − =
≥ − = −
= −
= − −⋅
= − −⋅ ⋅
=
=
3.4 Prontuario para Cálculo de Estructuras
Otros casos de carga que se resuelven por superposición de los anteriores
x
ab
q
( )
( ) ( )
2 2qM x x-a x-b2!dM x
T xdx
= −⟨ ⟩ + ⟨ ⟩
=
q
a
xbd
q/d
( )
( ) ( )
3 3 2q/d qM x - x-a x-b x-b3! 2!
dM xT x
dx
= ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩
=
q
a
xbd
q/d
( )
( ) ( )
2 3 3q q/dM x x-a x-a x-b2! 3!
dM xT x
dx
= − ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩
=
a
xbd
aq b
q
( )
( )
( ) ( )
a b2 2
b a 3 3
q qM x x-a x-b
2! 2!q q /d
x-a x-b3!
dM xT x
dx
= − ⟨ ⟩ + ⟨ ⟩ +
− + −⟨ ⟩ + ⟨ ⟩
=
a
xbd
aq
bq
( )
( )
( ) ( )
a b2 2
a b 3 3
q qM x x-a x-b
2! 2!q q /d
x-a x-b3!
dM xT x
dx
= − ⟨ ⟩ + ⟨ ⟩ +
− + ⟨ ⟩ − ⟨ ⟩
=
Formula
rio para
vigas y pórticos
3.5
3.2 VIGA APOYADA EN LOS EXTREMOS
3.2.1 CARGA PUNTUAL EN LA VIGA REACCIONES
A BP b P aR RL L⋅ ⋅
= =
ESFUERZOS CORTANTES
;AC CBP b P aQ cte Q cteL L⋅ ⋅
= = = − =
MOMENTOS FLECTORES
( ) max 0; ; para AC CB CP b P a P a bM x M L x M M x aL L L⋅ ⋅ ⋅ ⋅
= ⋅ = ⋅ − = = =
ANGULOS DE GIRO
( ) ( ) ( ); ;6 6 3A B CP a b P a b P a bL b L a b aE I L E I L E I L
ϕ ϕ ϕ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ + = − ⋅ + = ⋅ −
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA ( ) 22 2 2
2 2 21 ; 16 6AC CB
P L a L xP L b x b x a L xy yE I E I LL L L
⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − = ⋅ − − = ⋅ − − ⋅ ⋅ ⋅ ⋅ FLECHA MAXIMA
( )2 23
2 2 2 para 39 3C
P b L bf L b xE I L⋅ −
= ⋅ − =⋅ ⋅ ⋅
x
a
A
L
b
C
PB
QA
maxM
BQ
3.6
Prontuario pa
ra C
álculo d
e Estructuras
3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES
A Bp b c p a cR RL L⋅ ⋅ ⋅ ⋅
= =
ESFUERZOS CORTANTES
; ;2AC CD DB
p b c p b c c p a cQ Q p a x QL L L⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = − ⋅ − + = −
MOMENTOS FLECTORES
( )
2
max 0
;2 2
2 para 2 2
AC CD
DB
p b c p b c p cM x M x x aL L
p a cM L xL
p b c b c c b cM a c x aL L L
⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅ − ⋅ − −
⋅ ⋅= ⋅ −
⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ − + = − + ⋅
ANGULOS DE GIRO 2 2
;6 4 6 4A Bp a b c c p a b c cL b L aE I L a E I L b
ϕ ϕ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ + − = − ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( )
22
4 23
22
6 4
4 424 2 4
6 4
AC
CD
DB
p b c x cy x a L bL E I a
p c cy L x a b c x a b c L b xE I L a
p a c L x cy L x b L aL E I a
⋅ ⋅= ⋅ − + ⋅ + −
⋅ ⋅ ⋅ = ⋅ ⋅ − − − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ −= ⋅ ⋅ − − + ⋅ + −
⋅ ⋅ ⋅
QA
BQ
x
a
A
L
b
C
PB
c
maxM
D
Formula
rio para
vigas y pórticos
3.7
3.2.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( ) ( )1 2 1 21 12 ; 26 6A BR p p R p p= ⋅ + = + ⋅ .
ESFUERZOS CORTANTES ( )1 2 23
; ;6A A x A B B
p L x p xQ R Q R x Q R
L⋅ − + ⋅
= = − ⋅ = −⋅
MOMENTOS FLECTORES ( )
( ) ( )
( )
1 2 2
2 2
max 1 2 1 2
2 20 1 1 2 1 2
2 1
36
comprendido entre 0,125 y 0,1282 2
1 1para x3
x Ap L x p x
M R x xL
L LM p p p p
p p p p pp p
− + ⋅= ⋅ − ⋅
⋅
⋅ ⋅ + ⋅ ⋅ +
= ⋅ − + ⋅ + + ⋅
−
ANGULOS DE GIRO
( ) ( )3 3
1 2 1 28 7 ; 7 8360 360A BL Lp p p pE I E I
ϕ ϕ= ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( ) ( )( ) ( )
3 21 2 1 2
2 31 2 1 2
3 3 4360 8 7 8 7x
p p x p p Lxx L xy
EI p p L x p p L
− − + +− =
+ + +
FLECHA MAXIMA ( ) ( )4 4
1 2 1 2entre 0,01302 y 0,013042 2
p p L p p LE I E I
+ ⋅ + ⋅⋅ ⋅
⋅ ⋅ ⋅ ⋅
QA
maxM
QB
P
A B
1 P2
0x
L
x
3.8
Prontuario pa
ra C
álculo d
e Estructuras
3.2.4 MOMENTO FLECTOR REACCIONES
R RMLA B= − = −
ESFUERZOS CORTANTES
xMQ cteL
= =
MOMENTOS FLECTORES
( )
izq derC C M M M
AC CB
izq derC C
M MM x M L xL LM MM a M bL L
= − ⋅ = − ⋅ −
= − ⋅ = − ⋅ = +
ANGULOS DE GIRO
( )
2 2
2 2
3 32
3 1 ; 3 16 6
3
A B
C
M L b M L aE I E IL LM a bE I L
ϕ ϕ
ϕ
⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ −
⋅ ⋅ ⋅ ⋅
= ⋅ +⋅ ⋅ ⋅
ECUACION DE LA ELASTICA 2 2
2 2
22
2
1 36
( ) 1 36
AC
CB
M L x b xyE I L L
M L L x a L xyE I LL
⋅ ⋅= − ⋅ − ⋅ −
⋅ ⋅ ⋅ ⋅ − − = − ⋅ − ⋅ − ⋅ ⋅
FLECHA
( )3CM a bf b aE I L⋅ ⋅
= ⋅ −⋅ ⋅ ⋅
BA
M+
QA QB
MC
MC
M
a
L
b
C
Formula
rio para
vigas y pórticos
3.9
3.3 VIGA EMPOTRADA EN LOS EXTREMOS
3.3.1 CARGA PUNTUAL EN LA VIGA REACCIONES
( ) ( )2 2
3 32 ; 2A BP b P aR L a R L bL L⋅ ⋅
= ⋅ + ⋅ = ⋅ + ⋅
ESFUERZOS CORTANTES
( ) ( )2 2
3 32 ; 2AC CBP b P aQ L a cte Q L b cteL L⋅ ⋅
= ⋅ + ⋅ = = − ⋅ + ⋅ =
MOMENTOS FLECTORES
( )
( )
2 2 2
2 2 3
2 2 22
03 3
; ; 2
22 ; para
A B AC
BC C
P a b P a b P bM M M L x a x a LL L L
P a P a bM L b L L x b x M x aL L
⋅ ⋅ ⋅ ⋅ ⋅= − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅
⋅ ⋅ ⋅ ⋅= ⋅ ⋅ + − ⋅ − ⋅ ⋅ = =
ECUACION DE LA ELASTICA
( ) ( )
2 2
2
22
2
236
3 26
AC
BC
P b a x xy a xE I L L
L xP a L xy b L x bE I L L
⋅ ⋅ ⋅ = ⋅ ⋅ − − ⋅ ⋅ ⋅
−⋅ − ⋅ = ⋅ ⋅ − − − ⋅ ⋅ ⋅ ⋅
FLECHAS
( )
3 3 3 2
max3 22;
3 3 22para
2
CP a b P a bf fE I L E I L a
a LxL a
⋅ ⋅ ⋅ ⋅ ⋅= =
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅
⋅ ⋅=
+ ⋅
MC
A B
a
L
x
b
P
B
AQ
Q
x
0
MA
C
MB
3.10
Prontuario pa
ra C
álculo d
e Estructuras
3.3.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES
;A B A BA B
p b c M M p a c M MR RL L L L⋅ ⋅ − ⋅ ⋅ −
= − = +
ESFUERZOS CORTANTES
; ;AC A BD B CD AcQ R cte Q R cte Q R p x aa
= = = − = = − ⋅ − +
MOMENTOS FLECTORES
( )
2
3 2
2 2
3 2
2 2
;2 2
12; 312
12312
AC A A CD A A
BD B B A
B
p cM R x M M R x M x a
p c a bM R L x M M L bL c
p c a bM L aL c
= ⋅ + = ⋅ + − ⋅ − +
⋅ ⋅ ⋅= ⋅ − + = − ⋅ − ⋅ +
⋅ ⋅ ⋅ ⋅
= − ⋅ − ⋅ + ⋅
ECUACION DE LA ELASTICA
( )
( ) ( ) ( )
2
43 3
3 2 2
36
1 4 1224 2
1 3 3 2 36
AC A A
CD A A
DB B B B A B B B
xy M R xE I
cy p x a R x M xE I
y R x M LR x M LR Lx M LR LEI
= ⋅ − ⋅ − ⋅⋅ ⋅
= ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅
= − + + + − +
a
MA
QA
x
A
BQ
L
b
B
Pc
C D
MB
Formula
rio para
vigas y pórticos
3.11
3.3.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( )
( )
1 2
1 2
26
26
A BA
A BB
L M MR p pL
L M MR p pL
−= ⋅ ⋅ + −
−= ⋅ + ⋅ +
ESFUERZOS CORTANTES
( )1 222
A A
x A
B B
Q Rp L x p x
Q R xL
Q R
=
⋅ ⋅ − + ⋅= − ⋅
⋅= −
MOMENTOS FLECTORES
( )
( )
( )
2
1 2
1 2 2
2
1 2
3 260
36
2 360
A
x A A
B
LM p p
p L x p xM R x M x
LLM p p
= − ⋅ + ⋅
⋅ ⋅ − + ⋅= ⋅ + − ⋅
⋅
= − ⋅ + ⋅
ECUACION DE LA ELASTICA
( )22 1 3 2
1 4 1224 5x A A
p pxy x p L x R L x M LE I L
−= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅
⋅ ⋅ ⋅
AQ
QB
x
A
L
B
BMAM
P2
P1
3.12
Prontuario pa
ra C
álculo d
e Estructuras
3.3.4 MOMENTO FLECTOR REACCIONES
3 36 6;A BM MR a b R a bL L⋅ ⋅
= − ⋅ ⋅ = ⋅ ⋅
ESFUERZOS CORTANTES
36
xMQ a b cteL⋅
= − ⋅ ⋅ =
MOMENTOS FLECTORES
( )
⋅ ⋅ = ⋅ − ⋅ = − ⋅ − ⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ −
⋅ − = − ⋅ ⋅ ⋅ − ⋅ −
⋅
= − ⋅ ⋅ = + ⋅ − ⋅ ⋅2 3 23 3
2 3 2 3
3 1 2 1
3 1 2 1
6 ; 6
A B
AC
CB
izq derC A C A
M a b M b aM ML L L LM a a xML L L
M b b L xML L L
M MM M a b M M L a bL L
ECUACION DE LA ELASTICA
( )
2
2
2
2
22
22
AC
BC
M b x L x by aE I L LL
M a L x b x ayE I L LL
⋅ ⋅ − = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅
⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅
FLECHA
( )2 2
32CM a bf a bE I L⋅ ⋅
= − ⋅ −⋅ ⋅ ⋅
MC
AQ QB
x
A
L
a b
B+M
CM
C
AM
BM
Formula
rio para
vigas y pórticos
3.13
3.4 VIGA APOYADA-EMPOTRADA
3.4.1 CARGA PUNTUAL EN LA VIGA REACCIONES
( ) ( )2
2 23 33 ; 3
2 2A BP b P aR L b R L aL L⋅ ⋅
= ⋅ ⋅ − = ⋅ ⋅ −⋅ ⋅
ESFUERZOS CORTANTES
( ) ( )2
2 23 33 ; 3 .
2 2AC CBP b P aQ L b cte Q L a constL L⋅ ⋅
= − ⋅ ⋅ − = = − ⋅ ⋅ − =⋅ ⋅
MOMENTOS FLECTORES
( ) ( )
( ) ( )
2 2 22 3
2 3 2 23 3
; 3 22 2
3 2 ; 2 32 2
B C
AC CB
P a P aM L a M b a bL L
P x P aM b a b M L L x a xL L
⋅ ⋅= − ⋅ − = ⋅ ⋅ ⋅ + ⋅
⋅ ⋅⋅ ⋅
= ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅⋅ ⋅
ANGULOS DE GIRO
( ) ( ) ( )2 2
2 23; 2
4 4A CP a L a P a L a
L a L aE I L E I L
ϕ ϕ⋅ − ⋅ ⋅ −
= = ⋅ − ⋅ ⋅ −⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( )
( )
22 2
3
2 2 2
2 2
3 212
3 1 312
AC
BC
P b xy a L x L aE I L
P a L x a a L xyE I LL L
⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅
⋅ ⋅ − − = ⋅ ⋅ − − − ⋅ ⋅ ⋅
FLECHA MAXIMA 2
para x=6 2 2maxp b a a af L
E I L a L a⋅ ⋅
= ⋅ ⋅⋅ ⋅ ⋅ + ⋅ +
Q
Q
MB
A
B
x
a
L
b
A BC
P
MC
3.14
Prontuario pa
ra C
álculo d
e Estructuras
3.4.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES
;B BA B
p b c M p a c MR RL L L L⋅ ⋅ ⋅ ⋅
= + = −
ESFUERZOS CORTANTES
; ;2AC A DB B CD AcQ 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 cM R x M R x x a
p a b c cM R L x M M L abL
= ⋅ = ⋅ − ⋅ − +
⋅ ⋅ ⋅= ⋅ − + = − ⋅ + −
⋅⋅
ANGULOS DE GIRO
3 2
2123
48Ap c a bL bE I L c
ϕ ⋅ ⋅ ⋅
= ⋅ − + ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( )
22 3
2
4 23 3
2
2
128 348
1 128 2 348 4
36
AC A
CD A
DB B B
x a by R L x p c L bE I L c
c aby R Lx pL x a pc L b xE I L c
L xy R L x M
E I
⋅ ⋅= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − +
⋅ ⋅ ⋅ = ⋅ − + − + + − + ⋅ ⋅ ⋅
− = − ⋅ ⋅ − + ⋅ ⋅ ⋅
Q
Q
ax
C
P
M
A
A
B
B
Lb
c
B
Formula
rio para
vigas y pórticos
3.15
3.4.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( ) ( )1 2 1 22 ; 26 6
B BA B
L M L MR p p R p pL L
= ⋅ ⋅ + + = ⋅ + ⋅ −
ESFUERZOS CORTANTES
( )1 22;
2x A B Bp L x p x
Q R x Q RL
⋅ ⋅ − + ⋅= − ⋅ = −
⋅
MOMENTOS FLECTORES
( ) ( )2
1 2 21 2
3; 7 8
6 120x A Bp L x p x LM R x x M p p
L⋅ ⋅ − + ⋅
= ⋅ − ⋅ = − ⋅ ⋅ + ⋅⋅
ANGULOS DE GIRO
( )3
1 23 2240AL p pE I
ϕ = ⋅ ⋅ + ⋅⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( )4 3 2 2 32 1 1 1 25 20 5 12 3
120x A Axy p p x Lp x R Lx L R L p p LEIL
= − + − + − +
2
A
Q Q
Lx
B
BA
1PP
MB
3.16
Prontuario pa
ra C
álculo d
e Estructuras
3.4.4 MOMENTO FLECTOR REACCIONES
( )2 23
32A BMR R L aL
= − = ⋅ ⋅ −
ESFUERZOS CORTANTES
x AQ R cte= =
MOMENTOS FLECTORES
( )
( )
2 22
22 2
3 2
; ; 32
3 ; 3 1 22 2
der izqC A C A B
AC BC
MM R a M M R a M L aL
M x M x aM L a MLL L
= ⋅ − = ⋅ = ⋅ − ⋅⋅
⋅= ⋅ ⋅ − = ⋅ ⋅ ⋅ − −
ANGULOS DE GIRO
( ) ( )2
3 ; 3 1 44 4A CM M b aL a a L bE I L E I L L
ϕ ϕ = ⋅ − ⋅ ⋅ − = ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( )
( ) ( )
3 2 23
2 2 2 23
4 34
24
AC
BC
M b xy L x L a LE I LMy L x a L x L aE I L
⋅ ⋅ = ⋅ − ⋅ − − ⋅ ⋅ + ⋅ ⋅ ⋅
= ⋅ − ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅
Q
x
a
L
b
AM
Q
B
A B
B
CM
M
MC
C +
Formula
rio para
vigas y pórticos
3.17
3.5 VIGA EMPOTRADA EN UN EXTREMO
3.5.1 CARGA PUNTUAL EN LA VIGA 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
2A CP bE I
ϕ ϕ= = − ⋅⋅ ⋅
ECUACION DE LA ELASTICA
( )( ) ( ) ( )2
23 ; 2 36 6AC CBP b Py L x b y L x b aE I E I⋅
= ⋅ ⋅ − − = ⋅ − ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅
FLECHA MAXIMA
( )3 2
; 2 33 6C AP b P bf f b aE I E I⋅ ⋅
= = ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅
L
a
x
A
b
B
P
Q
MB
B
C
3.18
Prontuario pa
ra C
álculo d
e Estructuras
a
x
AC
M
Q
L
b
P
BD
c
B
B
3.5.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES . BR p c= ⋅
ESFUERZOS CORTANTES .
0 ; ;2AC CD DBcQ Q p x a Q p c cte = = − ⋅ − + = − ⋅ =
MOMENTOS FLECTORES .
( )
2
220 ; ;2 2
;
AC CD D
DB B
cp x ap cM M M
M p c x a M p c b
⋅ − + ⋅ = = − = −
= − ⋅ ⋅ − = − ⋅ ⋅
ANGULOS DE GIRO . 2 2
2 2; ;2 4 2 12D C A Cp c c p c cb bE I E I
ϕ ϕ ϕ ϕ ⋅ ⋅
= − ⋅ − = − ⋅ + = ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA .
( ) ( ) ( )
( )
22 2 3
4 22 3
2 ; 3 26 6 4
4 3 824 2 4
DB AC
DC
p c p c cy L x b a x y a x b bE I E I
p c cy x a c a x b b cE I
⋅ ⋅= ⋅ − ⋅ ⋅ − + = ⋅ − ⋅ ⋅ + + ⋅
⋅ ⋅ ⋅ ⋅ = ⋅ − + + ⋅ ⋅ − ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅
FLECHAS .
( )
2
2 23 2 3
2 3 12
4 ; 3 212 2 6 4
D
C A
p c c b cf bE I
p c c p c cf b b c c f a b bE I E I
⋅ = ⋅ − ⋅ + ⋅ ⋅ ⋅ = ⋅ + ⋅ ⋅ − + = ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅ ⋅
Formula
rio para
vigas y pórticos
3.19
L
Q
x
A
B
B
1P2P
BM
3.5.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( )1 212BR p p= +
ESFUERZOS CORTANTES
( )2
2 11 1 2;
2 2x Bp p x LQ p x Q p p
L−
= − ⋅ − ⋅ = − +
MOMENTOS FLECTORES
( ) ( )2 2
2 1 1 2 13 ; 26 6x Bx LM p p x L p M p pL = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅
ANGULOS DE GIRO
( )31 23
24AL p p
E Iϕ
⋅ ⋅ += −
⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( ) ( ) ( )
( ) ( ) ( )
32 2
2 1 2
22 1 2 1
5242 2 2
x
L xL x p p L x py LEIL L x p p L p p
−− − − + − −
= − − + + +
FLECHA
( )42 14 11
120AL p p
fE I
⋅ ⋅ + ⋅=
⋅ ⋅
3.20
Prontuario pa
ra C
álculo d
e Estructuras
M
L
x
AB
M
a b
B
3.5.4 MOMENTO FLECTOR REACCIONES
0BR =
ESFUERZO CORTANTE
0xQ =
MOMENTOS FLECTORES
0 ; ;AC CB ACM M M cte M M= = − = = −
ANGULOS DE GIRO
C AM bE I
ϕ ϕ ⋅= = −
⋅
ECUACION DE LA ELASTICA
( ) ( )22 2 ;2 2AC BCM My b L x b y L xE I E I
= ⋅ ⋅ ⋅ − ⋅ − = −⋅ ⋅ ⋅ ⋅
FLECHA
( )2
; 22 2C AM b Mf f b L bE I E I⋅
= = ⋅ ⋅ ⋅ −⋅ ⋅ ⋅ ⋅
Formula
rio para
vigas y pórticos
3.21
3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
L/2 L/2L
P P
L/2 L/2L
0,312 P
0,312 P0,688 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
3.22
Prontuario pa
ra C
álculo d
e Estructuras
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 QL2
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 QL0,063 QL
A L
Q
L C
22
Q
Formula
rio para
vigas y pórticos
3.23
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 0.58 2
2 2 2
k k k ff a f b f ck
k f a dd e gk
− += = − = + = +
= − = =
B CA
c QL
f QL
2
A
e QL
LA
a QL
a L
2
C
2g QL
B
Ck L
d L
B
d QLb QL
MOMENTOS FLECTORES
ESFUERZOS CORTANTES
3.24
Prontuario pa
ra C
álculo d
e Estructuras
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 0.58
2 2 2
k kf a f b f
k f a dd e gk
− += = − = +
= − = =
A B C
2g QL
AC
BL
Q
A C
Q
k L
B
f QL2
a QL
c QL
b QL d QL
d L
MOMENTOS FLECTORES
ESFUERZOS CORTANTES
Formula
rio para
vigas y pórticos
3.25
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 0.512 8
2 2 8
kf a f b fk
k a kc e g f
+= = − = +
⋅ +
= = = −
A CBk LL L
D
Q QQ
MOMENTOS FLECTORES
a L
A
A
a QL
2g QL
2f QL
C
2
2e QL
f QL
B
e QL2
D
c QL
b QL
CB
b QL
c QLa L
D
a QL
ESFUERZOS CORTANTES
3.26
Prontuario pa
ra C
álculo d
e Estructuras
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 0.512 8
2 2 8
kf a f b fk
k a kc e g f
+= = − = +
⋅ +
= = = −
k LB
LA
LDC
QQ Q
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
e QL2B
2g QL
f QL2
A
B
b QL
c QL
A
a QL
a L
e QL2C D
2f QL
C
a L
a QL
D
b QL
c QL
Formula
rio para
vigas y pórticos
3.27
3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
2
1
3 2I hk y N kI l
= ⋅ = +
3.9.1 CARGA REPARTIDA VERTICAL REACCIONES
232 12
A
D
A D
psnVlpsmVl
ps sH H mnhlN
=
=
= = −
MOMENTOS FLECTORES
2
2
32 12
( )2
B C
x A A
ps sM M mnlN
En Sp x mM 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
3.28
Prontuario pa
ra C
álculo d
e Estructuras
3.9.2 CARGA REPARTIDA HORIZONTAL REACCIONES
( )
( )
2
228
68
A D
D
A
phV Vl
ph N kH
Nph N k
HN
= =
+=
−=
MOMENTOS FLECTORES
( )
( )
2
2
28
28
( )2
B
C
Y B
phM N kNphM N kN
En ABpy h y yM M
h
= −
= − +
−= + ⋅
h
p
A
B
D
C
l
I 2
I 1 1I
MB
CM
HA HD
VA VD
y
MB
Formula
rio para
vigas y pórticos
3.29
3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
32
A
D
A D
PnVlPmVl
PmnH HlhN
=
=
= =
MOMENTOS FLECTORES
32
2 32
B C
P
PmnM MlN
NM PmnlN
= = − ⋅
−=
h
A
B
D
C
l
I 2
I 1 1I
m n
MB CM
HA HD
VA VD
P
MP
3.30
Prontuario pa
ra C
álculo d
e Estructuras
3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
1 23 31 2
1 2
h hI Ik y k
I s I s= ⋅ = ⋅
3.10.1 CARGA REPARTIDA VERTICAL REACCIONES
( ) ( )2
1 22 21 1 2 2 1 2
2
8 1 1
A D
A D
plV V
h hplH Hh k h k hh
= =
+= =
+ + + +
MOMENTOS FLECTORES
( )( ) ( )
21 2 1
2 21 1 2 2 1 28 1 1B
h h hplMh k h k hh
+= −
+ + + +
( )( ) ( )
21 2 2
2 21 1 2 2 1 2
1
8 1 1
( )2
C
X A
h h hplMh k h k hh
En BCpx l x fM H x h
l
+= −
+ + + +
− = − +
h
p
A
B
D
C
l
I 3
I 1
2Ix
MB
CM
HA HD
VA VD
s
2
h 1
f
Formula
rio para
vigas y pórticos
3.31
3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES
( )( ) ( )
21
1
21 1 21
2 21 1 2 2 1 2
2
4 5 28 1 1
A D
A D
D
phV V
lH ph H
h k hphHh k h k hh
= =
= −
+ +=
+ + + +
MOMENTOS FLECTORES
( )( ) ( )( )
( ) ( )
2 31 1 21 1
2 21 1 2 2 1 2
21 1 21 2
2 21 1 2 2 1 2
2
4 5 22 8 1 1
4 5 28 1 1
2
B
C
Y A
h k hph phMh k h k hh
h k hph hMh k h k hh
En ABpyM H y
+ += −
+ + + +
+ +=
+ + + +
= −
h
A
B
D
C
l
I 3
I 1
2I
MB
CM
HD
VD
2
HA
VA
p
s
yh
f
1
3.32
Prontuario pa
ra C
álculo d
e Estructuras
3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES
( )
( ) ( )( ) ( )
1 2
21 1 1 2 1 22 21 1 2 2 1 2
2
8 1 48 1 1
A D
A D
D
pf h hV V
lH pf H
h k hh f h hpfHh k h k hh
+= =
= −
+ + + +=
+ + + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
21 1 1 2 1 21
1 2 21 1 2 2 1 2
21 1 1 2 1 222 21 1 2 2 1 2
2
1
8 1 48 1 1
8 1 48 1 1
2
B
C
Y A A
h k hh f h hpfhM pfhh k h k hh
h k hh f h hphMh k h k hh
En BCl pyM V y H y hf
+ + + += −
+ + + +
+ + + += −
+ + + +
= − + + −
h
A
B
D
C
l
I 3
I 1
2I
MB
CM
HD
VD
2
HA
VA
ps
y
h
f
1
Formula
rio para
vigas y pórticos
3.33
3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
( ) ( )1 2
2 2 21 1 2 2 1 2
( ) ( )2 1 1
A
D
A D
PbVlPaVl
h l b h l aPabH Hl h k h k hh
=
=
+ + += =
+ + + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( ) ( )( ) ( )
1 212 2 2
1 1 2 2 1 2
1 222 2 2
1 1 2 2 1 2
1
2 1 1
2 1 1
B
C
P A
h l b h l aPabhMl h k h k hh
h l b h l aPabhMl h k h k hh
Pab afM H hl l
+ + += −
+ + + +
+ + += −
+ + + +
= + +
h
A
B
D
C
l
I 1
2I
MB
CM
HD
VD
2
HA
VA
s
a b
I 3
MP
1
f
h
3.34
Prontuario pa
ra C
álculo d
e Estructuras
3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
2
1
I hkI s
= ⋅
3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES
( ) ( )2
2
28 5
32 3 3
A E
A E
plV V
pl h fH Hh k f h f
= =
+= =
+ + +
MOMENTOS FLECTORES
( ) ( )
( )
2
2
2
8 532 3 3
8
22
B D
C B
BX
pl h h fM Mh k f h f
pl f hM Mh
En BC y DCx l x M fxM 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
Formula
rio para
vigas y pórticos
3.35
3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES
( ) ( )2
2
38
88 5
64 3 3
A
E
A E
plV
plV
pl h fH Hh k f h f
=
=
+= =
+ + +
MOMENTOS FLECTORES
( ) ( )
( )
2
2
2
8 564 3 3
16
22
B D
C B
BX
pl h h fM Mh k f h f
pl f hM Mh
En BCx l x M fxM 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
3.36
Prontuario pa
ra C
álculo d
e Estructuras
3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES
( )( ) ( )
2
2
2
2
5 12 616 3 3
A E
A E
E
phV Vl
H ph Hk h fphH
h k f f h
= =
= −
+ +=
+ + +
MOMENTOS FLECTORES
( )( ) ( )
2
2
3
2
2
2
45 12 6
16 3 3
2
B D
C D
D
y A
phM M
ph f hM Mhk h fphM
h k f f hEn AB
pyM 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
Formula
rio para
vigas y pórticos
3.37
3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES
( )
( ) ( )( ) ( )
2
2
22
8 3 5 416 3 3
A E
A E
E
pfV V f hl
H pf Hh k f f hpfHh k f f h
= = +
= −
+ + +=
+ + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( )
22
2
2
4 2 516 3 3
2
B A
C
D E
x A A
M H hh k f h fpfMh k f f h
M H h
En BC
y hM H y V x p
fsiendo y x hl
= ⋅
+ + += − ⋅
+ + +
= − ⋅
−= ⋅ − ⋅ −
= +
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
3.38
Prontuario pa
ra C
álculo d
e Estructuras
3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
( )( ) ( )
2 2
2 2
6 ln 3 4
4 3 3
A
A
A E
PnVlPmVl
h f l mPmH Hl h k f f h
=
=
+ −= =
+ + +
MOMENTOS FLECTORES
22
B D A
C B
P A A
M M H hPm h fM M
hhl fmM 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
Formula
rio para
vigas y pórticos
3.39
3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL 1 23 3
1 21 2
h hI Ik y kI l I l
= ⋅ = ⋅
3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES
( ) ( )
( ) ( )
( ) ( )
2 21 2
2 21 1 2 2 1 2
2 21 2
2 21 1 2 2 1 2
21 2
2 21 1 2 2 1 2
2 8 1 1
2 8 1 1
8 1 1
A
D
A D
h hpl plVh k h k hh
h hpl plVh k h k hh
h hplH Hh k h k hh
−= +
+ + + +
−= −
+ + + +
−= =
+ + + +
MOMENTOS FLECTORES
( )( ) ( )
( )( ) ( )
21 2 1
2 21 1 2 2 1 2
21 2 2
2 21 1 2 2 1 2
2
1
8 1 1
8 1 1
2
B
C
x A A
h h hplMh k h k hh
h h hplMh k h k hh
En BCpxM V x H h
+= −
+ + + +
+= −
+ + + +
= ⋅ − − ⋅
h
A
B
D
C
l
I 3
I 1
2I
MB CM
HA
HD
VA
VD
p
x
h
2
1
3.40
Prontuario pa
ra C
álculo d
e Estructuras
3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES
( ) ( )
21 1 2
21 1 1 1 2
2 21 1 2 2 1 2
2
5 4 28 1 1
A D D
A D
D
ph h hV V Hl l
H ph Hph k h h hH
h k h k hh
−= = −
= −
+ +=
+ + + +
MOMENTOS FLECTORES
( ) ( )
( ) ( )
2 31 1 1 1 1 2
2 21 1 2 2 1 2
21 2 1 1 1 2
2 21 1 2 2 1 2
2
5 4 22 8 1 1
5 4 28 1 1
2
B
C
y A
ph ph k h h hMh k h k hh
ph h k h h hMh k h k hh
En ABpyM H y
+ += − −
+ + + +
+ += −
+ + + +
= ⋅ −
h
p
A
B
D
C
l
I 3
I 1
2I
MB
CM
HA
HD
VA
VD
y
MB
h1
2
Formula
rio para
vigas y pórticos
3.41
3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( ) ( )
1 21 23 2 2
1 1 2 2 1 2
1 21 23 2 2
1 1 2 2 1 2
1 22 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 PabV h hl l h k h k hh
l b h l a hPa PabV h hl l h k h k hh
l b h l a hPabH Hl h k h k hh
+ + += + −
+ + + +
+ + += − −
+ + + +
+ + += =
+ + + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( ) ( )( ) ( )
1 212 2 2
1 1 2 2 1 2
1 222 2 2
1 1 2 2 1 2
2 1 1
2 1 1
B
C
P A B
l b h l a hPabhMl h k h k hh
l b h l a hPabhMl h k h k hh
M V a M
+ + += −
+ + + +
+ + += −
+ + + +
= ⋅ +
A
B
D
C
l
I 3
I 1
2I
a b
MB CM
HA
HD
VA
VD
P
MP
h
h1
2
3.42
Prontuario pa
ra C
álculo d
e Estructuras
3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
2
1
I hkI l
= ⋅
3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES
( )2
2 4 2A D A Dpl plV V H H
h k= = = =
+
MOMENTOS FLECTORES
( )
( )
( )( )
2
2
2
2
12 2
6 2
2 6 2
3 2máx24 2 2
A D
B C
x
plM Mk
plM Mk
En BCpx l x plM
k
pl k lM pos para xk
= =+
= = −+
−= −
+
+= =
+
h
A
B
D
C
l
I 2
I 1 1I
x
MB CM
HA
VA
p
MA
HD
VD
MD
Formula
rio para
vigas y pórticos
3.43
3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES
( )
( )( )
2
6 1
2 38 2
A D
A D
D
ph kV Vl k
H ph Hph k
Hk
= =+
= −
+=
+
MOMENTOS FLECTORES
2
2
2
2
2
2 1524 6 1 2
2 2124 6 1 2
2 2324 6 1 2
2 1324 6 1 2
2
A
B
C
D
y A A
phMk k
phMk k
phMk k
phMk k
En ABpyM 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
3.44
Prontuario pa
ra C
álculo d
e Estructuras
3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
( )( )216 1
32 ( 2)
A
D A
A D
m n mPnVl l k
V P VPmnH Hlh k
−= + + = −
= =+
MOMENTOS FLECTORES
( )
( )
( )
( )
12 2 6 1
12 2 6 1
12 2 6 1
12 2 6 1
A
B
C
D
CBP
Pmn n mMl k l k
Pmn n mMl k l k
Pmn n mMl k l k
Pmn n mMl k l k
mMnMPmnMl 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
Formula
rio para
vigas y pórticos
3.45
3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES
3(6 1)
2
A D
A D
PhkV Vl kPH H
= =+
= =
MOMENTOS FLECTORES
3 12 6 1
32 6 1
3 12 6 1
A
B C
D
Ph kMkPh kM M
kPh kM
k
+= −
+
= − =+
+=
+
h
A
B
D
C
l
I 2
I 1 1I
MB CM
P
HD
VD
MD
HA
VA
MA
3.46
Prontuario pa
ra C
álculo d
e Estructuras
3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
2
1
I hkI s
= ⋅
3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES
( )( ) ( )
2
2 2 2
24 5
8 4
A E
A E
plV V
k h f fplH Hkh f k h hf f
= =
+ += =
+ + + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( )( ) ( )
( )
2
2 2 2
22
2 2 2
2
2
8 15 648 4
16 1548 4
8
22
A E
B D
C A A
x A A A
kh h f f h fplM Mkh f k h hf f
kh h f fplM Mkh f k h hf f
plM M H h f
En BCxf pxM M V x H hl
+ + −= =
+ + + +
+ += = −
+ + + +
= + − +
= + ⋅ − + −
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
Formula
rio para
vigas y pórticos
3.47
3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES
( )( )
( ) ( )2
2 2 2
24 13
32 3 14 5
16 4
A E
E
A E
plV VkV plk
k h f fplH Hkh 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 15 696 64 3 14
8 15 696 64 3 14
16 1596 64 3 14
16 1596 64 3 14
A
E
B
D
x A
kh h f f h fpl plMkkh f k f fh h
kh h f f h fpl plMkkh f k f fh h
kh h f fpl plMkkh f k f fh h
kh h f fpl plMkkh f k f fh h
En BC M M
+ + −= −
++ + + +
+ + −= +
++ + + +
+ += − −
++ + + +
+ += − +
++ + + +
=
( )
222
2
A A
C E E E
xf pxV x H hl
lM V M H f h
+ ⋅ − + −
= + − +
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
3.48
Prontuario pa
ra C
álculo d
e Estructuras
3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES
( )
( )( ) ( )
2
22
2 2 2
2 3 1
2 34 4
A E
A E
E
ph kV Vl k
H ph Hk h k f h fphH
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 1624 3 14
212
6 15 16 6 2 1624 3 14
2
A
B A A
C E E E
D E E
E
y A A
kh k kf h f fph kMkkh f k f fh h
phM M H h
M M H f h VM M H h
kh k kf h f fph kMkkh f k f fh h
En ABpyM 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
Formula
rio para
vigas y pórticos
3.49
3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES
( )
( ) ( )( ) ( )2
2 2 2
438 3 1
2 4 10 54 4
A E
A E
E
k f h fpfV Vl k
H pf Hk h k f kh kf fpfHkh f k f fh h
+ += =
+= −
+ + + +=
+ + + +
MOMENTOS FLECTORES
( ) ( )( ) ( )
( )
( )
( ) ( )( ) ( )
( )
( ) ( )
2 2 2
2 2 2
2
9 4 6 4 3 2324 2 3 14
2
9 4 6 4 3 2324 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 fpfM fkkh f k f fh h
M M H hlM M H h f V
M M H h
kh f h f h f h k fpfM fkkh f k f fh h
En BCl y h p y h
M M H y Vf
+ + + + + = − + ++ + + +
= + ⋅
= − + +
= − ⋅
+ + + + + = − + ++ + + +
− −= + ⋅ − −
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
3.50
Prontuario pa
ra C
álculo d
e Estructuras
3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES
( )
( ) ( ) ( )( ) ( )
2
32 2
2 2 2 2
3 23 13 4 1 3
4
A E
E
A E
V P Vl kl m mPmV
klkl f h fm k lm f khPmH 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
42
3 1
2
3 2 4 2 2 ln 4
42
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 hMl n n m
k
M M H hlM 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 hMl n n m
k
+ − + + + + −
+ + + + =
− − +
= − ⋅
= + − +
= − ⋅
+ − + + + + −
+ + + + =
−+ +
2y A A A
En BCfmM M V m H hl
= + ⋅ − +
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