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

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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

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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

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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

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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

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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

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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

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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

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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

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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

− += = − = + = +

= − = =

QQ

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

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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

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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

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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

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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

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de Estru

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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

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ario

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ra C

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de Estru

ctu

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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

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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

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ario

pa

ra C

álc

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de Estru

ctu

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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

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pa

ra C

álc

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de Estru

ctu

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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

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pa

ra C

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de Estru

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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

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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

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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

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ario

pa

ra C

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de Estru

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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