AUTORES
Arianna Guardiola Víllora nació en Trieste (Italia), se graduó en la Escuela Técnica Superior de Ar-quitectura de Valencia donde se doctoró en 2006. Desde 1997 es profesora del Departamento de Mecánica de los Medios Conti-nuos y Teoría de Estructuras de la Universidad Politécnica de Va-lencia. Actualmente es Profesora Titular de Escuela Universitaria e imparte la asignatura Estructuras III de la ETSAV.
Agustín Pérez García nació en Alzira, estudió en la Escuela Téc-nica Superior de Arquitectura de Valencia donde se doctoró en 1986. Desde 1987 es profesor del Departamento de Mecánica de los Medios Continuos y Teoría de Estructuras de la Universidad Politécnica de Valencia. Actual-mente es Catedrático de Univer-sidad e imparte la asignatura In-troducción a las Estructuras de Edificación de la ETSAV.
INTRODUCCION
Esta publicación contiene materiales – documentos normativos y recomendaciones, hojas de cálculo y aplicaciones informáticas – útiles para llevar a término y justificar documentalmente muchos de los procesos de análisis asociados al diseño y el cálculo de estructuras tanto desde la perspectiva académica como desde la profesional. Con el objeto de facilitar el uso de la formulación incluida en sus diferentes capítulos se han diseñado una serie de hojas de cálculo que las implementan y que permiten obte-ner directamente resultados utilizando un ordenador. Todas ellas se encuentran en el CD que forma parte de esta obra. Así mismo, la lectura o consulta del material incluido en la publicación puede efectuarse mediante los ficheros en formato PDF incluidos en el CD. También se incorporan varios programas – EFCiD®, Architrave® y SigmaCAD – desarro-llados por los autores conjuntamente con otros profesores de la Universidad Politécnica de Valencia y diseñados para realizar cálculos en los ámbitos de la Resistencia de Ma-teriales y del Análisis de Estructuras. Se trata de versiones académicas que sólo pueden utilizarse en dicho ámbito. No obstante, existen versiones profesionales, bastante más completas y potentes, disponibles en el Centro de Transferencia de Tecnología de la Universidad Politécnica de Valencia. Originalmente, este libro fue diseñado para servir como material de apoyo para el desarrollo de los ejercicios y prácticas de la asignatura Introducción a las Estructuras de Edificación (Estructuras I) de la ETS de Arquitectura de Valencia pero progresivamente
ha ido incrementando sus contenidos con material útil para los alumnos de la asignatu-ra Estructuras III en la que se abordan, básicamente, el estudio de las estructuras de
acero. En general se ha tratado de presentar la información de manera sintética y del modo mas gráfico y claro que los autores han sido capaces de imaginar. Esperan con ello haber elaborado un material y unas herramientas informáticas realmente útiles. Los autores agradecen la colaboración del arquitecto Miguel Martínez Ausina en la elaboración de algunos de los contenidos de esta publicación. También quieren mos-trar su reconocimiento al profesor Ivan Cabrera Fausto autor del diseño de la portada y al resto de profesores de las asignaturas de Introducción a la Estructuras de Edificación y de Estructuras III de la ETSAV por la favorable acogida que le han prestado y por sus valiosas observaciones ya que han permitido depurar y ampliar su contenido. Por último desean dedicar este libro a sus hijos que han colaborado, con sus presencias y sus ausencias, a que esta tercera edición de la obra sea una realidad.
Los autores
1. ACCIONES EN LA EDIFICACION
2. MATERIALES ESTRUCTURALES
3. DISTRIBUCION DE SOLICITACIONES
4. DIAGRAMAS DE PREDIMENSIONADO
5. TABLAS Y DIAGRAMAS DIMENSIONADO
6. LIMITACION DE LAS DEFORMACIONES
7. ESTABILIDAD DE BARRAS A PANDEO
8. UNIONES EN ESTRUCTURAS DE ACERO
9. GEOMETRÍA DE MASAS
10. PROGRAMA EFCiD®
11. PROGRAMA Architrave®
12. PROGRAMA SigmaCAD
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 ecuación general previamente indicada.
3.2 Prontuario para Cálculo de Estructuras
( )
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
≤ − =
≥ − =
= − −
==
( )
( )( )
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
≤ − =
≥ − = −
= − −
= − −
==
M
M(x)
ax
P
M(x)
ax
T(x)
Limitación de las Deformaciones 3.3
( )
( )
( )
( )
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
qT x x a
qM x x a
A q
c
≤ − =
≥ − = −
= −
= − −
= − −⋅
==
( )
( )
( )
( )
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
qA
d
c
≤ − =
≥ − = −
= −
= − −⋅
= − −⋅ ⋅
=
=
2M(x)
q
x
a
T(x)
3
a
x
d
q
2T(x)
M(x)
3.4 Prontuario para Cálculo de Estructuras
Otros casos de carga que se resuelven por superposición de los anteriores
( )( ) [ ]
2 2qM x x-a x-b
2!q x-a x-bT x
= −⟨ ⟩ + ⟨ ⟩
= ⋅ −⟨ ⟩ + ⟨ ⟩
( )
( )
2 3 3
2 2
q q/dM x x-b - x-a x-b
2! 3!q/d
q x-b - x-a x-b2!
T x
= ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩
= ⋅ ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩
( )
( )
2 3 3
2 2
q q/dM x x-a x-a x-b
2! 3!q/d
q x-a x-a x-b2!
T x
= − ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩
= − ⋅ ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩
( ) ( )
( ) ( )
b aa b2 2 3 3
b a 2 2a b
q q /dq qM x x-a x-b x-a x-b
2! 2! 3!
q q /dq x-a q x-b x-a x-b
2!T x
− = − ⟨ ⟩ + ⟨ ⟩ + −⟨ ⟩ + ⟨ ⟩
− = − ⟨ ⟩ + ⟨ ⟩ + −⟨ ⟩ + ⟨ ⟩
( ) ( )
( ) ( )
a ba b2 2 3 3
a b 2 2a b
q q /dq qM x x-a x-b x-a x-b
2! 2! 3!
q q /dq x-a q x-b x-a x-b
2!T x
− = − ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩
− = − ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩
x
a
b
q
q
a
x
b
d
q/d
q
a
x
b
d
q/d
a
x
b
d
aq b
q
a
x
b
d
aq
bq
Form
ula
rio p
ara
viga
s y pórtic
os
3.5
3.2 VIGA APOYADA EN LOS EXTREMOS
3.2.1 CARGA PUNTUAL REACCIONES
A B
P b P aR R
L L
⋅ ⋅= =
ESFUERZOS CORTANTES
;AC CB
P b P aQ cte Q cte
L L
⋅ ⋅= = = − =
MOMENTOS FLECTORES
( )⋅ ⋅= ⋅ = ⋅ −
⋅ ⋅= = =max 0
;
para
AC CB
C
P b P aM x M L x
L L
P a bM M x a
L
ANGULOS DE GIRO
( ) ( ) ( ); ;6 6 3A B C
P a b P a b P a bL b L a b a
E I L E I L E I Lϕ ϕ ϕ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ + = − ⋅ + = ⋅ −
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) 22 2 2
2 2 21 ; 1
6 6AC CB
P L a L xP L b x b x a L xy y
E I E I LL L L
⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − = ⋅ − − = ⋅ − − ⋅ ⋅ ⋅ ⋅
FLECHA MAXIMA
( )2 23
2 2 2 para 39 3
C
P b L bf L b x
E I L
⋅ −= ⋅ − =⋅ ⋅ ⋅
x
a
A
L
b
C
PB
QA
BQ
maxM
3.6
Pro
ntu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.2.2 CARGA PUNTUAL CENTRAL REACCIONES
= =2A B
PR R
ESFUERZOS CORTANTES
= = − =2AC CB
PQ Q cte
MOMENTOS FLECTORES
( )= ⋅ = ⋅ −
⋅= = =max 0
;2 2
para 4 2
AC CB
C
P PM x M L x
P L LM M x
ANGULOS DE GIRO
ϕ ϕ ϕ⋅= − = =⋅ ⋅
2
; 016A B C
P L
E I
ECUACION DE LA ELASTICA
⋅ ⋅= ⋅ − ⋅ ⋅ ⋅
2 2
2
41
16 3AC
P L x xy
E I L
FLECHA MAXIMA
⋅=⋅ ⋅
3
48C
P Lf
E I
maxM
x
a
A
L
a
C
PB
QA
BQ
Form
ula
rio p
ara
viga
s y pórtic
os
3.7
3.2.3 CARGA CONTINUA EN PARTE DE LA VIGA REACCIONES
A B
p b c p a cR R
L L
⋅ ⋅ ⋅ ⋅= =
ESFUERZOS CORTANTES
; ;2AC CD DB
p b c p b c c p a cQ 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 cM x M x x a
L L
p a cM L x
L
p b c b c c b cM a c x a
L L L
⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅ − ⋅ − −
⋅ ⋅= ⋅ −
⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ − + = − + ⋅
ANGULOS DE GIRO 2 2
;6 4 6 4A B
p a b c c p a b c cL b L a
E I L a E I L bϕ 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 b
L E I a
p c cy L x a b c x a b c L b x
E I L a
p a c L x cy L x b L a
L E I a
⋅ ⋅= ⋅ − + ⋅ + − ⋅ ⋅ ⋅
= ⋅ ⋅ − − − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ −= ⋅ ⋅ − − + ⋅ + − ⋅ ⋅ ⋅
maxM
D
QA
BQ
x
a
A
L
b
C
p
B
c
3.8
Pro
ntu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.2.4 CARGA CONTINUA EN TODA LA VIGA REACCIONES
⋅= =2A B
p LR R
ESFUERZOS CORTANTES
⋅ = ⋅ − = − =
;2 2AB A B
L P LQ p x Q Q
MOMENTOS FLECTORES
( )⋅= ⋅ −
⋅= =2
m a x 0
2
p a ra 8 2
AB
p xM L x
p L LM x
ANGULOS DE GIRO
ϕ ϕ ⋅= − = −⋅ ⋅
3
24A B
p L
E I
ECUACION DE LA ELASTICA
( )⋅= ⋅ − ⋅ ⋅ +⋅ ⋅
3 2 3224AB
p xy x L x L
E I
⋅ ⋅= =⋅ ⋅
4
max 0
5para
384 2
p L Ly x
E I
VA
BV
A B
M max
p
x
L
Form
ula
rio p
ara
viga
s y pórtic
os
3.9
3.2.5 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( ) ( )1 2 1 2
1 12 ; 2
6 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
3
6
comprendido entre 0,125 y 0,1282 2
1 1para x
3
x A
p L x p xM R x x
L
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 B
L Lp p p p
E I E Iϕ ϕ= ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅
⋅ ⋅ ⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( ) ( )( ) ( )
3 21 2 1 2
2 31 2 1 2
3 3 4
360 8 7 8 7x
p p x p p Lxx L xy
EI p p L x p p L
− − + +− =
+ + +
FLECHA MAXIMA
( ) ( )4 41 2 1 2entre 0,01302 y 0,013042 2
p p L p p L
E I E I
+ ⋅ + ⋅⋅ ⋅
⋅ ⋅ ⋅ ⋅
QA
maxM
QB
p
A B
1 p2
0x
L
x
3.10
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.2.6 MOMENTO FLECTOR
REACCIONES
R RM
LA B
= − = −
ESFUERZOS CORTANTES
x
MQ cte
L= =
MOMENTOS FLECTORES
( )izq derC C M M M
AC CB
izq derC C
M MM x M L x
L L
M MM a M b
L L
= − ⋅ = − ⋅ −
= − ⋅ = − ⋅ = +
ANGULOS DE GIRO
( )
2 2
2 2
3 32
3 1 ; 3 16 6
3
A B
C
M L b M L a
E I E IL L
Ma b
E I L
ϕ ϕ
ϕ
⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅
= ⋅ +⋅ ⋅ ⋅
ECUACION DE LA ELASTICA 2 2
2 2
22
2
1 36
( )1 3
6
AC
CB
M L x b xy
E I L L
M L L x a L xy
E I LL
⋅ ⋅= − ⋅ − ⋅ − ⋅ ⋅
⋅ ⋅ − − = − ⋅ − ⋅ − ⋅ ⋅
FLECHA
( )3C
M a bf b a
E I L
⋅ ⋅= ⋅ −⋅ ⋅ ⋅
C
BA
M
QA QB
MC
MC
M
a
L
b
Form
ula
rio p
ara
viga
s y pórtic
os
3.11
3.3 VIGA EMPOTRADA EN LOS EXTREMOS
3.3.1 CARGA PUNTUAL
REACCIONES
( ) ( )2 2
3 32 ; 2A B
P b P aR L a R L b
L L
⋅ ⋅= ⋅ + ⋅ = ⋅ + ⋅
ESFUERZOS CORTANTES
( ) ( )2 2
3 32 ; 2AC CB
P b P aQ L a cte Q L b cte
L 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 L
L L L
P a P a bM L b L L x b x M x a
L L
⋅ ⋅ ⋅ ⋅ ⋅= − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅
⋅ ⋅ ⋅ ⋅= ⋅ ⋅ + − ⋅ − ⋅ ⋅ = =
ECUACION DE LA ELASTICA
( ) ( )
2 2
2
22
2
23
6
3 26
AC
BC
P b a x xy a x
E I L L
L xP a L xy b L x b
E I L L
⋅ ⋅ ⋅ = ⋅ ⋅ − − ⋅ ⋅ ⋅
−⋅ − ⋅ = ⋅ ⋅ − − − ⋅ ⋅ ⋅ ⋅
FLECHAS
( )3 3 3 2
max3 2
2;
3 3 2
2para
2
C
P a b P a bf f
E I L E I L a
a Lx
L a
⋅ ⋅ ⋅ ⋅ ⋅= =⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅
⋅ ⋅=+ ⋅
C
MC
MB
A B
a
L
x
b
P
B
AQ
Q
x
0
MA
3.12
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.3.2 CARGA PUNTUAL CENTRAL
REACCIONES
= =2A B
PR R
ESFUERZOS CORTANTES
= = = − =2 2AC CB
P PQ cte Q cte
MOMENTOS FLECTORES
⋅ = = − = ⋅ ⋅ −
⋅= = =max 0
4 18 8
8 2
A B AC
C
P L PL xM M M
L
P L LM M para x
ECUACION DE LA ELASTICA
⋅ ⋅ = − ⋅ − ⋅ ⋅ ⋅
2
3 448AC
P L x xy
E I L
FLECHAS
⋅=⋅ ⋅
3
192C
P Lf
E I
M maxx0
A B
a
L
x
a
P
C
B
AQ
Q
MBMA
Form
ula
rio p
ara
viga
s y pórtic
os
3.13
3.3.3 CARGA CONTINUA EN PARTE DE LA VIGA REACCIONES
;A B A BA B
p b c M M p a c M MR R
L L L L
⋅ ⋅ − ⋅ ⋅ −= − = +
ESFUERZOS CORTANTES
; ;AC A BD B CD A
cQ 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
123
12
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 b
L c
p c a bM L a
L c
= ⋅ + = ⋅ + − ⋅ − +
⋅ ⋅ ⋅= ⋅ − + = − ⋅ − ⋅ + ⋅
⋅ ⋅ ⋅= − ⋅ − ⋅ + ⋅
ECUACION DE LA ELASTICA
( )
( ) ( ) ( )
2
43 3
3 2 2
36
14 12
24 2
13 3 2 3
6
AC A A
CD A A
DB B B B A B B B
xy M R x
E I
cy p x a R x M x
E I
y R x M LR x M LR Lx M LR LEI
= ⋅ − ⋅ − ⋅⋅ ⋅
= ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅
= − + + + − +
C D
a
MA
QA
x
A
MB
BQ
L
b
B
p
c
3.14
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.3.4 CARGA CONTINUA EN TODA LA VIGA REACCIONES
⋅= =2A B
P LR R
ESFUERZOS CORTANTES
( )= = ⋅ − ⋅22AB x
PQ Q L x
MOMENTOS FLECTORES
( )
⋅= = −
= − ⋅ − ⋅ ⋅ + ⋅
⋅= =
= = ⋅
2
2 2
2
0
12
6 612
24 20 0,2113
A B
x
centro
x
P LM M
PM L L x x
P L LM para x
M para x L
ECUACION DE LA ELASTICA
⋅= ⋅ − ⋅ ⋅
⋅= =⋅ ⋅
24 2
2
4
max
24
384 2
x
P L x xy
E I L L
P L Ly para x
E I
x
L
QA
BQ
MBMA
p
A B
Form
ula
rio p
ara
viga
s y pórtic
os
3.15
3.3.5 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( )
( )
1 2
1 2
26
26
A BA
A BB
L M MR p p
L
L M MR p p
L
−= ⋅ ⋅ + −
−= ⋅ + ⋅ +
ESFUERZOS CORTANTES
( )1 22
2
A A
x A
B B
Q R
p L x p xQ R x
L
Q R
=⋅ ⋅ − + ⋅
= − ⋅⋅
= −
MOMENTOS FLECTORES
( )( )
( )
= − ⋅ ⋅ + ⋅
⋅ ⋅ − + ⋅= ⋅ + − ⋅
⋅
= − ⋅ + ⋅
2
1 2
1 2 2
2
1 2
3 260
3
6
2 360
A
x A A
B
LM p p
p L x p xM R x M x
L
LM 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 L
E I L
−= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅
p1p
MB
MA
AQ
QB
x
A
L
B
2
3.16
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.3.6 MOMENTO FLECTOR REACCIONES
3 3
6 6;A B
M MR a b R a b
L L
⋅ ⋅= − ⋅ ⋅ = ⋅ ⋅
ESFUERZOS CORTANTES
3
6x
MQ a b cte
L
⋅= − ⋅ ⋅ =
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 M
L L L L
M a a xM
L L L
M b b L xM
L L L
M MM M a b M M L a b
L L
ECUACION DE LA ELASTICA
( )
2
2
2
2
22
22
AC
BC
M b x L x by a
E I L LL
M a L x b x ay
E I L LL
⋅ ⋅ − = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅
⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅
FLECHA
( )2 2
32C
M a bf a b
E I L
⋅ ⋅= − ⋅ −⋅ ⋅ ⋅
MB
MC
MA
AQ QB
x
A
L
a b
B+M
CM
C
Form
ula
rio p
ara
viga
s y pórtic
os
3.17
3.4 VIGA APOYADA-EMPOTRADA
3.4.1 CARGA PUNTUAL
REACCIONES
( ) ( )2
2 23 3
3 ; 32 2A B
P b P aR L b R L a
L L
⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ −⋅ ⋅
ESFUERZOS CORTANTES
( ) ( )⋅ ⋅= − ⋅ ⋅ − = = − ⋅ ⋅ − =⋅ ⋅
22 2
3 33 ; 3
2 2AC CB
P b P aQ L b cte Q L a cte
L 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 b
L L
P x P aM b a b M L L x a x
L L
⋅ ⋅= − ⋅ − = ⋅ ⋅ ⋅ + ⋅⋅ ⋅⋅ ⋅= ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅⋅ ⋅
ANGULOS DE GIRO
( ) ( ) ( )2 2
2 23
; 24 4A C
P a L a P a L aL a L a
E I L E I Lϕ 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 a
E I L
P a L x a a L xy
E I LL L
⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ − − = ⋅ ⋅ − − − ⋅ ⋅ ⋅
FLECHA MAXIMA
2
para x=6 2 2max
p b a a af L
E I L a L a
⋅ ⋅= ⋅ ⋅⋅ ⋅ ⋅ + ⋅ +
P
C
x
a
L
b
A B
QA
BQ
MB
CM
3.18
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.4.2 CARGA PUNTUAL CENTRAL
REACCIONES
= ⋅ = ⋅5 11;
16 16A BR P R P
ESFUERZOS CORTANTES
= ⋅ = = − ⋅ =5 11;
16 16AC CBQ P cte Q P cte
MOMENTOS FLECTORES
= − ⋅ ⋅ = ⋅ ⋅
⋅ − = ⋅ ⋅ = ⋅ ⋅ −
3 5;
16 325
; 11 316 16
B C
AC CB
M P L M P L
P L L xM P x M
L
ANGULOS DE GIRO
ϕ ϕ⋅ ⋅= = −⋅ ⋅ ⋅ ⋅
2 2
;32 128A C
P L P L
E I E I
ECUACION DE LA ELASTICA
⋅= ⋅ ⋅ − ⋅ ⋅ ⋅
⋅ − = ⋅ − ⋅ − ⋅ ⋅ ⋅
2 2
2
2
3 596
( ) 9 1196
AC
BC
P L xy x
E I L
P L L xy L x
E I L
FLECHA MAXIMA
⋅ ⋅=⋅ ⋅
37;
768C
P Lf
E I
⋅=⋅ ⋅ ⋅
3
para x=48 5 5
max
P L Lf
E I
P
x
a
L
a
QA
BQ
M B
BM
C
A B
Form
ula
rio p
ara
viga
s y pórtic
os
3.19
3.4.3 CARGA CONTINUA EN PARTE DE LA VIGA REACCIONES
;B BA B
p b c M p a c MR R
L L L L
⋅ ⋅ ⋅ ⋅= + = −
ESFUERZOS CORTANTES
; ;2AC A DB B CD A
cQ 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 a
bL
= ⋅ = ⋅ − ⋅ − +
⋅ ⋅ ⋅= ⋅ − + = − ⋅ + − ⋅⋅
ANGULOS DE GIRO
ϕ ⋅ ⋅ ⋅= ⋅ − ⋅ + ⋅ ⋅ ⋅
3 2
2
123
48A
p c a bL b
E I L c
ECUACION DE LA ELASTICA
( ) ( )
⋅ ⋅= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − + ⋅ ⋅ ⋅
− ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ − + + = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ − ⋅ + ⋅
− = − ⋅ ⋅ − + ⋅ ⋅ ⋅
22 3
2
43
23
2
2
128 3
48
8 241
48 123
36
AC A
A
CD
DB B B
x a by R L x p c L b
E I L c
cR L x p L x a
yE I L a b
p c L b xc
L xy R L x M
E I
D
QA
QB
L
ax
C
b
c
p
BM
A B
3.20
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.4.4 CARGA CONTINUA EN TODA LA VIGA REACCIONES
= ⋅ ⋅ = ⋅ ⋅3 5;
8 8A BR P L R P L
ESFUERZOS CORTANTES
= ⋅ ⋅ − = ⋅ ⋅ = ⋅ ⋅
3 3 5; ;
8 8 8AB A B
xQ P L Q P L Q P L
L
MOMENTOS FLECTORES
⋅ ⋅= ⋅ ⋅ − ⋅ = −
= ⋅ ⋅ = ⋅
= = ⋅
2
2max
(3 4 );8 8
9 3;
128 83
04
AB B
rel
P x P LM L x M
M P L para x L
M para x L
ANGULOS DE GIRO
ϕ ⋅=⋅ ⋅
3
48A
P L
E I
ECUACION DE LA ELASTICA
⋅= ⋅ + ⋅ ⋅ −⋅ ⋅
2( 2 ) ( )48AB
P xy L x L x
E I
⋅ += = ⋅⋅ ⋅
4 1 33
185 16max
P Lf para x L
E I
QA
QB
BM
p
x
L
A B
Form
ula
rio p
ara
viga
s y pórtic
os
3.21
3.4.5 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES
( ) ( )1 2 1 22 ; 26 6
B BA B
L M L MR p p R p p
L L= ⋅ ⋅ + + = ⋅ + ⋅ −
ESFUERZOS CORTANTES
( )1 22;
2x A B B
p L x p xQ R x Q R
L
⋅ ⋅ − + ⋅= − ⋅ = −
⋅
MOMENTOS FLECTORES
( ) ( )2
1 2 21 2
3; 7 8
6 120x A B
p L x p x LM R x x M p p
L
⋅ ⋅ − + ⋅= ⋅ − ⋅ = − ⋅ ⋅ + ⋅
⋅
ANGULOS DE GIRO
( )3
1 23 2240A
Lp p
E Iϕ = ⋅ ⋅ + ⋅
⋅ ⋅
ECUACION DE LA ELASTICA
( )
( )
− ⋅ + = ⋅ + ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ +
⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅
42 1
3 21
2 31 2
5 20120
5 12 3
x A
A
p p xx
y L p x R L xE I L
L R L p p L
2
A
QA
QB
L
xB
p1
p
MB
3.22
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.4.6 MOMENTO FLECTOR REACCIONES
( )2 23
3
2A B
MR R L a
L= − = ⋅ ⋅ −
ESFUERZOS CORTANTES
x AQ R cte= =
MOMENTOS FLECTORES
( )
( )
2 22
22 2
3 2
; ; 32
3; 3 1 2
2 2
der izqC A C A B
AC BC
MM R a M M R a M L a
L
M x M x aM L a M
LL L
= ⋅ − = ⋅ = ⋅ − ⋅⋅
⋅= ⋅ ⋅ − = ⋅ ⋅ ⋅ − −
ANGULOS DE GIRO
( ) ( )ϕ
ϕ
= ⋅ − ⋅ ⋅ −⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅
2
34
3 1 44
A
C
ML a a L
E I L
M b ab
E I L L
ECUACION DE LA ELASTICA
( ) ( )
( ) ( )
3 2 23
2 2 2 23
4 34
24
AC
BC
M b xy L x L a L
E I L
My L x a L x L a
E I L
⋅ ⋅ = ⋅ − ⋅ − − ⋅ ⋅ + ⋅ ⋅ ⋅
= ⋅ − ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅
MC
MC
QA
x
a
L
b
A
M
C
+
BQ
B
MB
Form
ula
rio p
ara
viga
s y pórtic
os
3.23
3.5 VIGA EMPOTRADA EN UN EXTREMO
3.5.1 CARGA PUNTUAL 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 C
Pb
E Iϕ ϕ= = − ⋅
⋅ ⋅
ECUACION DE LA ELASTICA
( )( ) ( ) ( )2
23 ; 2 3
6 6AC CB
P b Py L x b y L x b a
E I E I
⋅= ⋅ ⋅ − − = ⋅ − ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅
FLECHA MAXIMA
( )3 2
; 2 33 6C A
P b P bf f b a
E I E I
⋅ ⋅= = ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅
BM
L
a
x
A
C
QB
b
B
P
3.24
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.5.2 CARGA PUNTUAL EN EL EXTREMO REACCIONES
BR P=
ESFUERZOS CORTANTES
= − =ABQ P cte
MOMENTOS FLECTORES
= − ⋅ ⋅; =-P L AB BM P x M
ANGULOS DE GIRO
ϕ ⋅= −⋅ ⋅
2
2A
P L
E I
ECUACION DE LA ELASTICA
= ⋅ − ⋅ ⋅ +⋅ ⋅
2( ) (2 )6AB
Py L x L x
E I
FLECHA MAXIMA
⋅=⋅ ⋅
3
3A
P Lf
E I
BM
QB
P
L
x
A B
Form
ula
rio p
ara
viga
s y pórtic
os
3.25
3.5.3 CARGA CONTINUA EN PARTE DEL VUELO
REACCIONES
BR p c= ⋅
ESFUERZOS CORTANTES
0 ; ;2AC CD DB
cQ Q p x a Q p c cte
= = − ⋅ − + = − ⋅ =
MOMENTOS FLECTORES
( )
2
220 ; ;
2 2;
AC CD D
DB B
cp x a
p 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 C
p c c p c cb b
E I E Iϕ 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 b
E I E I
p c cy x a c a x b b c
E I
⋅ ⋅= ⋅ − ⋅ ⋅ − + = ⋅ − ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ − + + ⋅ ⋅ − ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅ FLECHAS
( )
2
2 23 2 3
2 3 12
4 ; 3 212 2 6 4
D
C A
p c c b cf b
E I
p c c p c cf b b c c f a b b
E I E I
⋅ = ⋅ − ⋅ + ⋅
⋅ ⋅ = ⋅ + ⋅ ⋅ − + = ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅ ⋅
3.26
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.5.4 CARGA CONTINUA EN TODO EL VUELO REACCIONES
= ⋅BR p L
ESFUERZOS CORTANTES
= − ⋅ = − ⋅ =;AB BQ P x Q P L cte
MOMENTOS FLECTORES
⋅= − = − ⋅2 2
;2 2AB B
P x LM M P
ANGULOS DE GIRO
ϕ ⋅= −⋅ ⋅
3
6A
P L
E I
ECUACION DE LA ELASTICA
( ) ( )= ⋅ − ⋅ ⋅ + ⋅ ⋅ +⋅ ⋅
2 2 23 224AB
Py L x L L x x
E I
FLECHAS
⋅=⋅ ⋅
4
8A
P Lf
E I
x
MB
QB
L
p
AB
Form
ula
rio p
ara
viga
s y pórtic
os
3.27
3.5.5 CARGA TRAPEZOIDAL EN TODO EL VUELO REACCIONES
( )= ⋅ +1 22B
LR p p
ESFUERZOS CORTANTES
( )−= − ⋅ − ⋅ = − ⋅ +2
2 11 1 2;
2 2x B
p p x LQ p x Q p p
L
MOMENTOS FLECTORES
( ) ( )2 2
2 1 1 2 13 ; 26 6x B
x LM p p x L p M p p
L = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅
ANGULOS DE GIRO
( )31 23
24A
L p p
E Iϕ
⋅ ⋅ += −
⋅ ⋅
ECUACION DE LA ELASTICA
( ) ( ) ( ) ( )( ) ( ) ( )
−− − ⋅ − + − ⋅ −= ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ − ⋅ + + ⋅ ⋅ + ⋅
32 2
2 1 2
22 1 2 1
524
2 2 2x
L xL x p p L x p
y LE I
L L x p p L p p
FLECHA
( )42 14 11
120A
L p pf
E I
⋅ ⋅ + ⋅=
⋅ ⋅
MB
L
BQ
p
x
A
1
B
p2
3.28
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.5.6 MOMENTO FLECTOR REACCIONES
0BR =
ESFUERZO CORTANTE
0xQ =
MOMENTOS FLECTORES
0 ; ;AC CB ACM M M cte M M= = − = = −
ANGULOS DE GIRO
C A
M b
E Iϕ ϕ ⋅= = −
⋅
ECUACION DE LA ELASTICA
( ) ( )22 2 ;
2 2AC BC
M My b L x b y L x
E I E I= ⋅ ⋅ ⋅ − ⋅ − = −
⋅ ⋅ ⋅ ⋅
FLECHA
( )2
; 22 2C A
M b Mf f b L b
E I E I
⋅= = ⋅ ⋅ ⋅ −⋅ ⋅ ⋅ ⋅
a b
MB
L
x
AB
M
Form
ula
rio p
ara
viga
s y pórtic
os
3.29
3.5.7 MOMENTO FLECTOR EN EXTREMO DEL VUELO REACCIONES
0BR =
ESFUERZO CORTANTE
= 0xQ
MOMENTOS FLECTORES
= − =ABM M cte
ANGULOS DE GIRO
ϕ ⋅= −⋅A
M L
E I
ECUACION DE LA ELASTICA
( )= ⋅ ⋅ − ⋅ ⋅ +⋅ ⋅
2 222AC
My b x L x L
E I
FLECHA
⋅=⋅ ⋅
2
2A
M Lf
E I
A B
BM
L
x
M
3.30
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
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 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
Form
ula
rio p
ara
viga
s y pórtic
os
3.31
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
20,07 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 QL0,063 QL
A L
Q
L C
22
Q
3.32
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
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
10.5 0.5
8 2
2 2 2
k k k ff a f b f c
k
k f a dd e g
k
− += = − = + = +
= − = =
BC
A
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
Form
ula
rio p
ara
viga
s y pórtic
os
3.33
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
10.5 0.5
8 2
2 2 2
k k k ff a f b f c
k f a dd e g
k
− += = − = + = +
= − = =
A B C
2g QL
A
C
BL
Q
A C
Q
k L
B
f QL2
a QL
c QL
b QL d QL
d L
MOMENTOS FLECTORES
ESFUERZOS CORTANTES
3.34
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
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
10.5 0.5
12 8
2 2 8
kf a f b f
k
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
C
B
b QL
c QL
a L
D
a QL
ESFUERZOS CORTANTES
Form
ula
rio p
ara
viga
s y pórtic
os
3.35
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
10.5 0.5
12 8
2 2 8
kf a f b f
k
k a kc e g f
+= = − = +⋅ +
= = = −
k LB
LA
LDC
QQ Q
ESFUERZOS CORTANTES
MOMENTOS FLECTORES
e QL2
B
2g QL
f QL2
A
B
b QL
c QL
A
a QL
a L
e QL2
C D
2f QL
C
a L
a QL
D
b QL
c QL
3.36
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
3.9.1 CARGA REPARTIDA VERTICAL
= ⋅ = + ⋅2
1
3 2I h
k y N kI L
REACCIONES
⋅ ⋅=
⋅ ⋅=
⋅ ⋅= = ⋅ ⋅ − ⋅ ⋅ ⋅
23
2 12
A
D
A D
p s nV
L
p s mV
L
p s sH H m n
h L N
MOMENTOS FLECTORES
⋅= = − ⋅ ⋅ ⋅ − ⋅
⋅ −= ⋅ − − ⋅
2
2
3
2 12
( )
2
B C
x A A
p s sM M m n
L N
En S
p 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
Form
ula
rio p
ara
viga
s y pórtic
os
3.37
3.9.2 CARGA REPARTIDA VERTICAL UNIFORME
= ⋅ = + ⋅2
1
3 2I h
k y N kI L
REACCIONES
⋅= =
⋅= =⋅ ⋅
2
2
4
A D
A D
P LV V
P LH H
h N
MOMENTOS FLECTORES
( )
⋅= = −⋅
⋅ ⋅ − ⋅= −⋅
2
2
4
2 4
B C
x
P LM M
N
P x L x P LM
N
⋅ ⋅= − =⋅
2 2
max, 8 4 2pos
P L P L LM para x
N
h
p
A
B
D
C
L
I 2
I 1 1I
x
M B CM
A HD
VA VD
H
3.38
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.9.3 CARGA REPARTIDA HORIZONTAL
= ⋅ = + ⋅2
1
3 2I h
k y N kI L
REACCIONES
( )
( )
⋅= =⋅
⋅ ⋅ ⋅ +=
⋅⋅ ⋅ ⋅ −
=⋅
2
2
2
8
6
8
A D
D
A
p hV V
L
p h N kH
N
p h N kH
N
MOMENTOS FLECTORES
( )
( )
⋅= ⋅ ⋅ −⋅
⋅= − ⋅ +⋅
⋅ ⋅ −= + ⋅
2
2
28
28
( )
2
B
C
Y B
p hM N k
N
p hM N k
N
En AB
p y h y yM M
h
h
p
A
B
D
C
L
I 2
I 1 1I
y
MB
CM
HA HD
VA VD
MB
Form
ula
rio p
ara
viga
s y pórtic
os
3.39
3.9.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL
= ⋅ = + ⋅2
1
3 2I h
k y N kI L
REACCIONES
⋅=
⋅=
⋅ ⋅= =⋅ ⋅
3
2
A
D
A D
P nV
L
P mV
L
P m nH H
L h N
MOMENTOS FLECTORES
⋅ ⋅= = − ⋅⋅
⋅ −= ⋅ ⋅ ⋅⋅ ⋅
3
2
2 3
2
B C
P
P m nM M
L N
NM P m n
L N
h
A
B
D
C
L
I 2
I 1 1I
m nP
MB CM
HA HD
VA VD
MP
3.40
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
3.10.1 CARGA REPARTIDA VERTICAL
1 23 31 2
1 2
h hI Ik y k
I s I s= ⋅ = ⋅
REACCIONES
MOMENTOS FLECTORES
( )( ) ( )
+ ⋅⋅= −⋅ + + ⋅ + + ⋅
21 2 1
2 21 1 2 2 1 28 1 1B
h h hp LM
h k h k h h
( )( ) ( )
+ ⋅⋅= −⋅ + + ⋅ + + ⋅
⋅ ⋅ − = − ⋅ ⋅ +
21 2 2
2 21 1 2 2 1 2
1
8 1 1
( )
2
C
X A
h h hp LM
h k h k h h
En BC
p x L x fM H x h
L
( ) ( )
⋅= =
+⋅= =⋅ + + ⋅ + + ⋅
21 2
2 21 1 2 2 1 2
2
8 1 1
A D
A D
p LV V
h hp LH H
h k h k h h
h
p
A
B
D
C
l
I 3
I 1
2Ix
MB
CM
HA HD
VA VD
s
2
h 1
f
L
Form
ula
rio p
ara
viga
s y pórtic
os
3.41
3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
REACCIONES
( )( ) ( )
⋅= =
⋅= ⋅ −
⋅ + ⋅ + ⋅⋅=
⋅ + + ⋅ + + ⋅
21
1
21 1 21
2 21 1 2 2 1 2
2
4 5 2
8 1 1
A D
A D
D
p hV V
L
H p h H
h k hp hH
h k h k h h
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 2
2 8 1 1
4 5 2
8 1 1
2
B
C
Y A
h k hp h p hM
h k h k h h
h k hp h hM
h k h k h h
En AB
p yM H y
1 23 31 2
1 2
h hI Ik y k
I s I s= ⋅ = ⋅
1
h
A
B
D
C
L
I 3
I 1
2I 2
p
s
yh
f
MB
CM
HD
VD
HA
VA
3.42
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
1 23 31 2
1 2
h hI Ik y k
I s I s= ⋅ = ⋅
REACCIONES
( )
( ) ( )( ) ( )
⋅ ⋅ += =
⋅= ⋅ −
⋅ ⋅ + + ⋅ ⋅ + ⋅ +⋅= ⋅⋅ + + ⋅ + + ⋅
1 2
21 1 1 2 1 2
2 21 1 2 2 1 2
2
8 1 4
8 1 1
A D
A D
D
p f h hV V
L
H p f H
h k h h f h hp fH
h k h k h h
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 4
8 1 1
8 1 4
8 1 1
2
B
C
Y A A
h k h h f h hp f hM p f h
h k h k h h
h k h h f h hp hM
h k h k h h
En BC
L p yM V y H y h
f
1
h
A
B
D
C
L
I 3
I 1
2I 2
p
s
y
h
f
MB
CM
HD
VD
HA
VA
Form
ula
rio p
ara
viga
s y pórtic
os
3.43
3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL
1 23 31 2
1 2
h hI Ik y k
I s I s= ⋅ = ⋅
REACCIONES
( ) ( )
⋅=
⋅=
⋅ + + ⋅ +⋅ ⋅= = ⋅⋅ ⋅ + + ⋅ + + ⋅
1 22 2 2
1 1 2 2 1 2
( ) ( )
2 1 1
A
D
A D
P bV
L
P aV
L
h L b h L aP a bH H
L h k h k h h
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 aP a b hM
L h k h k h h
h L b h L aP a b hM
L h k h k h h
P a b a fM H h
L L
1
P
h
A
B
D
C
L
I 1
2I 2
s
h
f
a b
I 3
MB
CM
HD
VD
HA
VA
MP
3.44
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL Valor de la carga en proyección horizontal
2
1
I hk
I s= ⋅
REACCIONES
( ) ( )
⋅= =
⋅ ⋅ + ⋅= = ⋅⋅ + + ⋅ +
2
2
2
8 5
32 3 3
A E
A E
p LV V
p L h fH H
h k f h f
MOMENTOS FLECTORES
( ) ( )
( )
⋅ ⋅ ⋅ + ⋅= = − ⋅⋅ + + ⋅ ⋅ +
⋅ += + ⋅
⋅ − ⋅ ⋅ = ⋅ + ⋅ +
2
2
2
8 5
32 3 3
8
22
B D
C B
BX
p L h h fM M
h k f h f
p L f hM M
h
En BC y DC
x L x M f xM p h
h L
p
A
B
E
C
L
I 2
I 1
x
I 2
I 1
s
D
h
f
MB
CM
HA HE
VA VE
MD
Form
ula
rio p
ara
viga
s y pórtic
os
3.45
3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL Valor de la carga en proyección horizontal
2
1
I hk
I s= ⋅
REACCIONES
( ) ( )
⋅= ⋅
⋅=
⋅ ⋅ + ⋅= =⋅ + + ⋅ ⋅ +
2
2
38
8
8 5
64 3 3
A
E
A E
p LV
p LV
p L h fH H
h k f h f
MOMENTOS FLECTORES
( ) ( )
( )
⋅ ⋅ ⋅ + ⋅= = −⋅ + + ⋅ ⋅ +
⋅ += + ⋅
⋅ − ⋅ ⋅ = ⋅ + ⋅ +
2
2
2
8 5
64 3 3
16
2
2
B D
C B
BX
p L h h fM M
h k f h f
p L f hM M
h
En BC
x L x M f xM p h
h L
p
A
B
E
C
L
I 2
I 1
x
I 2
I 1
s
D
h
f
MB
CM
HA HE
VA VE
MD
3.46
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
2
1
I hk
I s= ⋅
REACCIONES
( )( ) ( )
⋅= =⋅
= ⋅ −
⋅ + ⋅ + ⋅⋅= ⋅⋅ + + ⋅ + ⋅
2
2
2
2
5 12 6
16 3 3
A E
A E
E
p hV V
L
H p h H
k h fp hH
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
p hM M
p h f hM M
h
k h fp hM
h k f f h
En AB
p yM H y
A
B
E
C
L
I 2
I 1
I 2
I 1
s
D
h
fp
y
MB
CM
HA HE
VA VE
MD
Form
ula
rio p
ara
viga
s y pórtic
os
3.47
3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL Valor de la carga en proyección vertical
2
1
I hk
I s= ⋅
REACCIONES
( )
( ) ( )( ) ( )
⋅= = ⋅ + ⋅⋅
= ⋅ −
⋅ ⋅ + + ⋅ ⋅ + ⋅⋅= ⋅⋅ + + ⋅ + ⋅
2
2
22
8 3 5 4
16 3 3
A E
A E
E
p fV V f h
L
H p f H
h k f f hp fH
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 fp fM
h k f f h
M H h
En BC
y hM H y V x p
fsiendo y x h
L
xA
B
E
C
L
I 2
I 1
I 2
I 1
s
D
h
f
p
y
MB
CM
HA HE
VA VE
MD
3.48
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
2
1
I hk
I s= ⋅
REACCIONES
( )( ) ( )
⋅=
⋅=
⋅ ⋅ ⋅ + ⋅ ⋅ − ⋅⋅= = ⋅⋅ ⋅ + + ⋅ + ⋅
2 2
2 2
6 n 3 4
4 3 3
A
A
A E
P nV
L
P mV
L
h L f L mP mH H
L h k f f h
MOMENTOS FLECTORES
= = − ⋅⋅ += + ⋅
⋅ + ⋅ ⋅= ⋅ − ⋅
22
B D A
C B
P A A
M M H h
P m h fM M
h
h L f mM V m H
L
P
A
B
E
C
L
I 2
I 1
I 2
I 1
s
D
h
f
m n
MB
CM
HA HE
VA VE
MD
Form
ula
rio p
ara
viga
s y pórtic
os
3.49
3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL
3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
= ⋅ = ⋅1 23 31 2
1 2
h hI Ik y k
I L I L
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 hp L p LV
h k h k h h
h hp L p LV
h k h k h h
h hp LH H
h k h k h h
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 hp LM
h k h k h h
h h hp LM
h k h k h h
En BC
p xM V x H h
h1
2h
A
B
D
C
L
I 3
I 1
2I
p
x
MB CM
HA
HD
VA
VD
3.50
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
= ⋅ = ⋅1 23 31 2
1 2
h hI Ik y k
I L I L
REACCIONES
( ) ( )
⋅ −= = − ⋅
⋅= ⋅ −
⋅ ⋅ ⋅ + ⋅ + ⋅= ⋅
⋅ + + ⋅ + + ⋅
21 1 2
21 1 1 1 2
2 21 1 2 2 1 2
2
5 4 2
8 1 1
A D D
A D
D
p h h hV V H
L L
H p h H
p h k h h hH
h k h k h h
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 2
2 8 1 1
5 4 2
8 1 1
2
B
C
y A
p h p h k h h hM
h k h k h h
p h h k h h hM
h k h k h h
En AB
p yM H y
h1
2h
p
A
B
D
C
L
I 3
I 1
2I
y
MB
CM
HA
HD
VA
VD
MB
Form
ula
rio p
ara
viga
s y pórtic
os
3.51
3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
= ⋅ = ⋅1 23 31 2
1 2
h hI Ik y k
I L I L
REACCIONES
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( ) ( )
+ ⋅ + + ⋅⋅ ⋅ ⋅= + ⋅ ⋅ −⋅ ⋅ + + ⋅ + + ⋅
+ ⋅ + + ⋅⋅ ⋅ ⋅= − ⋅ −⋅ ⋅ + + ⋅ + + ⋅
+ ⋅ + + ⋅⋅ ⋅= =⋅ ⋅ + + ⋅ + + ⋅
1 21 23 2 2
1 1 2 2 1 2
1 21 23 2 2
1 1 2 2 1 2
1 2
2 2 21 1 2 2 1 2
2 1 1
2 1 1
2 1 1
A
D
A D
L b h L a hP b P a bV h h
L L h k h k h h
L b h L a hP a P a bV h h
L L h k h k h h
L b h L a hP a bH H
L h k h k h h
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 hP a b hM
L h k h k h h
L b h L a hP a b hM
L h k h k h h
M V a M
h
h1
2
A
B
D
C
L
I3
I1
2I
a b
P
D
MP
MB CM
HA
HD
VA
V
3.52
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
= ⋅2
1
I hk
I L
REACCIONES
( )⋅ ⋅= = = =
⋅ ⋅ +
2
2 4 2A D A D
p L p LV V H H
h k
MOMENTOS FLECTORES
( )
( )
( )( )
⋅= =⋅ +
⋅= = −⋅ +
⋅ ⋅ − ⋅= −⋅ +
⋅ ⋅ += ⋅ =+
2
2
2
2
máx
12 2
6 2
2 6 2
3 2 para
24 2 2
A D
B C
x
positivo
p LM M
k
p LM M
k
En BC
p x L x p LM
k
p L k LM x
k
p
h
A
B
D
C
L
I 2
I 1 1I
x
MB CM
HA
VA
MA
HD
VD
MD
Form
ula
rio p
ara
viga
s y pórtic
os
3.53
3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
= ⋅2
1
I hk
I L
REACCIONES
( )
( )( )
⋅ ⋅= =⋅ ⋅ +
= ⋅ −
⋅ ⋅ ⋅ +=
⋅ +
2
6 1
2 3
8 2
A D
A D
D
p h kV V
L k
H p h H
p h kH
k
MOMENTOS FLECTORES
⋅ = − ⋅ + + ⋅ + +
⋅ = ⋅ − + ⋅ + +
⋅ = − ⋅ − − ⋅ + +
⋅ = ⋅ + − ⋅ + +
⋅= − + ⋅ +
2
2
2
2
2
2 15
24 6 1 2
2 21
24 6 1 2
2 23
24 6 1 2
2 13
24 6 1 2
2
A
B
C
D
y A A
p hM
k k
p hM
k k
p hM
k k
p hM
k k
En AB
p yM H y M
h
p
A
B
D
C
L
I 2
I 1 1I
y
MB
CMMB
HA
VA
MA
HD
VD
MD
3.54
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
= ⋅2
1
I hk
I L
REACCIONES
( )( )
⋅ −⋅= ⋅ + ⋅ ⋅ + = −
⋅ ⋅ ⋅= =⋅ ⋅ ⋅ +
21
6 1
3
2 ( 2)
A
D A
A D
m n mP nV
L L k
V P V
P m nH H
L h 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
CBP
P m n n mM
L k L k
P m n n mM
L k L k
P m n n mM
L k L k
P m n n mM
L k L k
m Mn MP m nM
L L L
h
A
B
D
C
L
I 2
I 1 1I
m nP
MB CM
MP
HA
VA
MA
HD
VD
MD
Form
ula
rio p
ara
viga
s y pórtic
os
3.55
3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR
= ⋅2
1
I hk
I L
REACCIONES
⋅ ⋅ ⋅= =⋅ ⋅ +
= =
3(6 1)
2
A D
A D
P h kV V
L k
PH H
MOMENTOS FLECTORES
⋅ ⋅ += − ⋅⋅ +⋅ ⋅= − = ⋅
⋅ +⋅ ⋅ += ⋅
⋅ +
3 1
2 6 13
2 6 13 1
2 6 1
A
B C
D
P h kM
k
P h kM M
k
P h kM
k
I 2
I 1 1I
P
h
A
B
D
C
L
MB CM
HD
VD
MD
HA
VA
MA
3.56
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS
3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL Valor de la carga en proyección horizontal
2
1
I hk
I s= ⋅
REACCIONES
( )( ) ( )
⋅= =
⋅ ⋅ + ⋅ +⋅= = ⋅⋅ + + ⋅ ⋅ + ⋅ +
2
2 2 2
24 5
8 4
A E
A E
p LV V
k h f fp LH H
k h f k h h f 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
k h h f f h fp LM M
k h f k h h f f
k h h f fp LM M
k h f k h h f f
p LM M H h f
En BC
x f p xM M V x H h
L
p
A
B
E
C
L
I 2
I 1
x
I 2
I 1
s
D
h
f
MB
CM
MD
HA
VA
MA
HE
VE
ME
Form
ula
rio p
ara
viga
s y pórtic
os
3.57
3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
Valor de la carga en proyección horizontal 2
1
I hk
I s= ⋅
REACCIONES
( )( )
( ) ( )
⋅= −⋅ += ⋅ ⋅ ⋅
⋅ ⋅ +⋅ ⋅ + ⋅ +⋅= = ⋅
⋅ + + ⋅ ⋅ + ⋅ +
2
2 2 2
24 1
332 3 1
4 5
16 4
A E
E
A E
p LV V
kV p L
k
k h f fp LH H
k h f k h h f f
MOMENTOS FLECTORES
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ −⋅ ⋅= ⋅ −⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ −⋅ ⋅= ⋅ +⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
⋅ ⋅ ⋅ + ⋅ +⋅ ⋅= − ⋅ −⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
⋅⋅= − ⋅
2 2
2 2 2
2 2
2 2 2
22 2
2 2 2
2
8 15 6
96 64 3 14
8 15 6
96 64 3 14
16 15
96 64 3 14
96
A
E
B
D
k h h f f h fp L p LM
kk h f k f f h h
k h h f f h fp L p LM
kk h f k f f h h
k h h f fp L p LM
kk h f k f f h h
k hp LM
( )( ) ( ) ( )
( )
⋅ ⋅ + ⋅ + ⋅+⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
⋅ ⋅ ⋅ = + ⋅ − ⋅ + −
= ⋅ + − +
2 2
2 2 2
2
16 15
64 3 14
2
2
2
x A A A
C E E E
h f f p L
kk h f k f f h h
x f p xEn BC M M V x H h
LL
M V M H f h
p
A
B
E
C
L
I 2
I 1
x
I 2
I 1
s
D
h
f
MB
CM
MD
HA
VA
MA
HE
VE
ME
3.58
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
2
1
I hk
I s= ⋅
REACCIONES
( )
( )( ) ( )
⋅ ⋅= =⋅ ⋅ ⋅ +
= ⋅ −
⋅ + ⋅ ⋅ + ⋅ +⋅= ⋅⋅ + + ⋅ ⋅ + ⋅ +
2
22
2 2 2
2 3 1
2 3
4 4
A E
A E
E
p h kV V
L k
H p h H
k h k f h fp hH
k h f k f f h h
MOMENTOS FLECTORES
( ) ( )( ) ( )
( )
( ) ( )( ) ( )
⋅ ⋅ + + ⋅ ⋅ ⋅ + ⋅ + ⋅⋅ ⋅ + = − ⋅ + ⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
⋅= + ⋅ −
= − ⋅ + + ⋅= − ⋅
⋅ ⋅ + + ⋅ ⋅ ⋅ + ⋅ + ⋅⋅ ⋅ += ⋅ − + ⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
2 22
2 2 2
2
2 22
2 2 2
6 15 16 6 2 16
24 3 14
2
2
6 15 16 6 2 16
24 3 14
A
B A A
C E E E
D E E
E
k h k k f h f fp h kM
kk h f k f f h h
p hM M H h
LM M H f h V
M M H h
k h k k f h f fp h kM
kk h f k f f h h
⋅= + ⋅ −2
2y A A
En ABp y
M M H y
A
B
E
C
L
I 2
I 1
I 2
I 1
s
D
h
fp
y
MB
CM
MD
HE
VE
ME
HA
VA
MA
Form
ula
rio p
ara
viga
s y pórtic
os
3.59
3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
Valor de la carga en proyección vertical
REACCIONES
( )
( ) ( )( ) ( )
⋅ ⋅ + +⋅= = ⋅ ⋅⋅ +
= ⋅ −
⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅ + ⋅ ⋅ +⋅= ⋅⋅ + + ⋅ ⋅ + ⋅ +
2
2 2 2
43
8 3 1
2 4 10 5
4 4
A E
A E
E
k f h fp fV V
L k
H p f H
k h k f k h k f fp fH
k h f k f f h h
MOMENTOS FLECTORES
( ) ( )( ) ( )
( )
( )
( ) ( )( ) ( )
( )
⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + +⋅ = − ⋅ ⋅ + ⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
= + ⋅
= − + + ⋅= − ⋅
⋅ ⋅ ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ + +⋅ = ⋅ − ⋅ + ⋅ ⋅ +⋅ + + ⋅ ⋅ + ⋅ +
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
A
B A A
C E E E
D E E
E
k h f h f h f h k fp fM f
kk h f k f f h h
M M H hL
M M H h f V
M M H h
k h f h f h f h k fp fM f
kk h f k f f h h
En BC
M( ) ( )⋅ − ⋅ −
= + ⋅ − ⋅ −⋅
2
2 2y A A A
L y h p y hM H y V
f
2
1
I hk
I s= ⋅
L
I 2
I 1
I 2
I 1
s
D
h
f
p
y
A
B
E
C
MB
CM
MD
HE
VE
ME
HA
VA
MA
3.60
P
rontu
ario
pa
ra C
álc
ulo
de Estru
ctu
ras
3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
2
1
I hk
I s= ⋅
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 k L m mP mV
kL
k L f h f m k L m f k hP mH H
L k h f k f f h h
MOMENTOS FLECTORES
( ) ( ) ( )( ) ( )
( )
( )
( ) ( )
⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ − ⋅ ⋅ ⋅ + ⋅ + + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ −
⋅ ⋅ + + ⋅ ⋅ + ⋅ + = ⋅ ⋅ ⋅ − − ⋅ +
= − ⋅ = + ⋅ − ⋅ + = − ⋅
⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ − ⋅ ⋅ ⋅ + ⋅ + + ⋅ ⋅ ⋅ ⋅
⋅= ⋅⋅
2 2 2
2 2 2
2
2 2
2
3 2 4 2 2 n 4
4
2
3 1
2
3 2 4 2 2 n
2
A
B A A C E E E D E E
E
f L h k L m f m k h h f k h L f L m L
P m k h f k f f h hM
L n n m
k
LM M H h M M V H h f M M H h
f L h k L m f m k h h f k h L
P mM
L
( )( ) ( )
( )
+ ⋅ ⋅ ⋅ −
⋅ + + ⋅ ⋅ + ⋅ +
⋅ − + ⋅ +
⋅ ⋅ = + ⋅ − ⋅ +
2
2 2 2
4
4
3 12
EnBC y A A A
f L m L
k h f k f f h h
n n m
kf m
M M V m H hL
P
A
B
E
C
L
I 2
I 1
I 2
I 1
s
D
h
f
m n
MB
CM
MD
HA
VA
MA
HE
VE
ME
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