AUTORES

Arianna Guardiola Víllora nació en Trieste (Italia), se graduó en la Escuela Técnica Superior de Arquitectura de Valencia donde se doctoró en 2006. Desde 1997 es profesora del Departamento de Mecánica de los Medios Continuos y Teoría de Estructuras de la Universidad Politécnica de Valencia. 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 Técnica 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. Actualmente es Catedrático de Universidad e imparte la asignatura Introducción 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 obtener 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 – desarrollados 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 Materiales 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 asignatura 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 mostrar 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

3.1

Formulario para vigas y pórticos

3.1

Obtención de la Distribución de Solicitaciones mediante la Formulación de Macaulay

Las Funciones de Macaulay permiten expresar tanto la distribución de cargas sobre una viga sometida a flexión como las leyes de Cortantes o Momentos Flectores generadas por dichas cargas. A continuación se muestra la expresión de tales funciones y las condiciones en las que deben aplicarse.

q( x) = ∑

A⋅ x − a

T ( x ) = −∑

A⋅ x − a

M( x ) = − ∑

(c− 2)

( c − 2 )! ( c −1)

( c − 1) !

A⋅ x − a

c

c!

ecuaciones validas solo si n ≥ 0 en las expresiones

si

y si

n=0

n>0

x−a

n

x−a

0

=0

x≥a

x−a

0

=1

x≤a

x−a

n

=0

x≥a

x−a

n

= ( x − a)

x≤a

n

En la siguientes tablas se particularizan estas funciones para cada caso de carga y se indica el valor que deberían tomar los parámetros A y c en la ecuación general previamente indicada.

3.2

Prontuario para Cálculo de Estructuras

M

Si x≤a

a

x≥a

x

x−a

0

=0

x−a

0

=1

entonces

M(x)

M( x ) = − M x − a

0

A=M

por lo tanto

c=0

P a x

T(x)

Si x≤a

x−a = 0

x≥a

x − a = ( x − a)

1

1

1

entonces T ( x) = − P x − a

M(x)

0

M( x ) = − P x − a por lo tanto

1

A=P c =1

3.3

Limitación de las Deformaciones

Si x≤a

q

x≥a

x−a x−a

2

2

=0

= ( x − a)

2

a entonces

x

q( x) = q x − a

0

q 1 x−a 1 q M( x ) = − x−a 2 ⋅1

T(x)

T ( x) = − 2

M(x)

2

A=q

por lo tanto

c=2

Si

q

x−a

x≥a

3

x−a

3

=0

= ( x − a)

d

a

entonces

x 2

T(x) 3

M(x)

x≤a

qd 1 x−a 1 qd 2 T ( x) = − x−a 2 ⋅1 qd M( x ) = − x−a 3 ⋅ 2 ⋅1

q( x) =

por lo tanto

3

q d c=3 A=

3

3.4

Prontuario para Cálculo de Estructuras

Otros casos de carga que se resuelven por superposición de los anteriores

q q −〈 x-a〉 2 + 〈 x-b〉 2 2! T ( x ) = q ⋅ [ −〈 x-a〉 + 〈 x-b〉 ]

M(x ) =

a b x q/d q

a

d b

q q/d -〈 x-a〉 3 + 〈 x-b〉 3 〈 x-b〉 2 + 2! 3! q/d -〈 x-a〉 2 + 〈 x-b〉 2 T ( x ) = q ⋅ 〈 x-b〉 + 2!

M(x) =

x

q/d q

a

q q/d 〈 x-a〉 3 − 〈 x-b〉 3 〈 x-a〉 2 + 2! 3! q/d 〈 x-a〉 2 − 〈 x-b〉 2 T ( x ) = −q ⋅ 〈 x-a〉 + 2!

M(x ) = −

d b x

qb

qa a

M( x ) = −

qa 2!

〈 x-a〉 2 +

qb 2!

〈 x-b〉 2 +

T ( x ) = −q a 〈 x-a〉 + q b 〈 x-b〉 +

d

(q

b

(q

b

)

− q a /d 3!

)

− q a /d 2!

−〈 x-a〉 3 + 〈 x-b〉 3

−〈 x-a〉 2 + 〈 x-b〉 2

b x

qa

a

qb d b x

M( x ) = −

qa 2!

〈 x-a〉 2 +

qb 2!

〈 x-b〉 2 +

T ( x ) = −q a 〈 x-a〉 + q b 〈 x-b〉 +

(q

a

(q

a

)

− q b /d 3!

)

− q b /d 2!

〈 x-a〉 3 − 〈 x-b〉 3

2 2 〈 x-a〉 − 〈 x-b〉

VIGA APOYADA EN LOS EXTREMOS

3.2.1

CARGA PUNTUAL

P

A REACCIONES P⋅b RA = L

RB =

B

C

P⋅a L

x ESFUERZOS CORTANTES P⋅b P⋅a QAC = = cte ; QCB = − = cte L L

a

b

Formulario para vigas y pórticos

3.2

L MOMENTOS FLECTORES P⋅b P⋅a MAC = ⋅ x ; MCB = ⋅ (L − x) L L P ⋅ a⋅ b para x0 = a Mmax = MC = L ANGULOS DE GIRO P⋅ a⋅ b ϕA = ⋅ ( L + b) 6⋅E⋅I⋅L

; ϕB = −

QB

P⋅ a⋅ b ⋅ ( L + a) 6⋅ E⋅I ⋅ L

; ϕC =

P⋅ a⋅ b ⋅ ( b − a) 3⋅E⋅I⋅L

QA

ECUACION DE LA ELASTICA

y

AC

=

P ⋅ L ⋅ b ⋅ x b2 x 2 ⋅ 1− 2 − 2 6⋅ E⋅I L L

;

y

CB

=

2 P ⋅ L ⋅ a ⋅ ( L − x ) a2 L − x ⋅ 1− 2 − 6⋅ E⋅I L L

FLECHA MAXIMA fC =

P⋅b 9⋅E⋅I⋅L 3

(

⋅ L2 − b2

)

3

2

para x =

L2 − b2 3

M max

3.5

3.6

3.2.2

CARGA PUNTUAL CENTRAL

REACCIONES

RA = RB =

P

A

P 2

B

C

ESFUERZOS CORTANTES

QAC = QCB = −

x

P = cte 2

a

MOMENTOS FLECTORES

MAC

P = ⋅x 2

Mmax = MC =

;

MCB

P⋅L 4

a L

P = ⋅ (L − x) 2 para

x0 =

L 2

QB

ANGULOS DE GIRO

P ⋅ L2 16 ⋅ E ⋅ I

; ϕC = 0

QA

ECUACION DE LA ELASTICA y AC =

P ⋅ L2 ⋅ x 4 x 2 ⋅ 1− ⋅ 16 ⋅ E ⋅ I 3 L2

FLECHA MAXIMA

fC =

P ⋅ L3 48 ⋅ E ⋅ I

M max

Prontuario para Cálculo de Estructuras

ϕ A = −ϕ B =

CARGA CONTINUA EN PARTE DE LA VIGA

REACCIONES p⋅b⋅c RA = L

RB =

c

p⋅ a⋅ c L

p

ESFUERZOS CORTANTES p⋅b⋅c p⋅b⋅c c QAC = ; QCD = − p⋅ − a+ x L L 2

A ; QDB

C

MOMENTOS FLECTORES p⋅b⋅c p⋅b⋅c p c MAC = ⋅ x ; MCD = ⋅ x − ⋅ x − a − 2 L L 2 2 p⋅a⋅c MDB = ⋅ (L − x) L p⋅b⋅c b⋅c c b⋅c Mmax = ⋅2⋅a− c + para x0 = a − + 2⋅L 2 L L ANGULOS DE GIRO p⋅a⋅b⋅c c2 ⋅L + b − ϕA = 6⋅E⋅I⋅L 4⋅a

; ϕB = −

B

p⋅a⋅c =− L

p⋅ a⋅ b⋅c c2 ⋅L + a − 6⋅E⋅I⋅L 4⋅b

D

x a

Formulario para vigas y pórticos

3.2.3

b L

QB QA

ECUACION DE LA ELASTICA p⋅b⋅c x 2 c2 y AC = ⋅ −x + a ⋅ L + b − 6 ⋅ L E ⋅ I 4 ⋅ a 4 p c c2 ⋅ L ⋅ x − a − − 4 ⋅ b ⋅ c ⋅ x3 + 4 ⋅ a ⋅ b ⋅ c ⋅ L + b − ⋅ x 24 ⋅ E ⋅ I ⋅ L 2 4⋅a 2 p⋅ a⋅ c L − x c 2 ⋅ ⋅ − ( L − x ) + b ⋅ L + a − y DB = 6⋅L 4 ⋅ a E ⋅ I

y

CD =

M max

3.7

3.8

3.2.4

CARGA CONTINUA EN TODA LA VIGA p

REACCIONES RA = RB =

p⋅L 2

A

B

ESFUERZOS CORTANTES L QAB = p ⋅ − x 2

;

QA = −QB =

x

P⋅L 2

L

MOMENTOS FLECTORES

MAB = Mm a x

p⋅x ⋅ (L − x ) 2 p ⋅ L2 = p a ra 8

x0 =

ANGULOS DE GIRO

VB VA

p ⋅ L3 24 ⋅ E ⋅ I

ECUACION DE LA ELASTICA

y AB =

p⋅ x ⋅ x3 − 2 ⋅ L ⋅ x2 + L3 24 ⋅ E ⋅ I

ymax =

(

5 ⋅ p ⋅ L4 384 ⋅ E ⋅ I

para

x0 =

) L 2

M max

Prontuario para Cálculo de Estructuras

ϕ A = −ϕ B = −

L 2

CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES 1 RA = ( 2 ⋅ p1 + p2 ) 6

; RB =

1 ( p1 + 2 ⋅ p2 ) . 6

p1

p 2

ESFUERZOS CORTANTES p ( 3 ⋅ L − x ) + p2 ⋅ x 2 QA = RA ; Qx = RA − 1 ⋅x 6⋅L

;

QB = −RB

B

A MOMENTOS FLECTORES p ( 3 L − x ) + p2 ⋅ x 2 Mx = RA ⋅ x − 1 ⋅x 6⋅L

x L 2

2

L L ⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 ) 2 2 1 1 para x 0 = ⋅ − p1 + ⋅ p12 + p22 + p1 ⋅ p2 p2 − p1 3 Mmax comprendido entre 0,125 ⋅

(

ANGULOS DE GIRO L3 ϕA = ⋅ ( 8 ⋅ p1 + 7 ⋅ p2 ) 360 ⋅ E ⋅ I

Formulario para vigas y pórticos

3.2.5

)

; ϕB = −

QA QB

L3 ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 360 ⋅ E ⋅ I

ECUACION DE LA ELASTICA

yx =

3 2 x ( L − x ) 3 ( p1 − p2 ) x − 3 ( 4p1 + p2 ) Lx + 360EI ( 8 p1 + 7p2 ) L2 x + ( 8p1 + 7p2 ) L3

0,01304 ⋅

( p1 + p2 ) ⋅ L4 2⋅E⋅I

x0

M max

3.9

FLECHA MAXIMA ( p + p2 ) ⋅ L4 y entre 0,01302 ⋅ 1 2⋅E⋅I

3.10

3.2.6

MOMENTO FLECTOR

REACCIONES

R A = −R B = −

M

M L

C

ESFUERZOS CORTANTES M Qx = = cte L MOMENTOS FLECTORES M M MAC = − ⋅ x MCB = − ⋅ ( L − x ) L L M M izq der MC = − ⋅ a MC = − ⋅ b L L

A

M = MCizq + Mder C

QB

MC M MC

Prontuario para Cálculo de Estructuras

2 M ⋅ L ⋅ (L − x) a2 L − x =− ⋅ 1− 3 ⋅ 2 − 6⋅E⋅I L L

FLECHA M⋅ a ⋅ b fC = ⋅ ( b − a) 3⋅E⋅I⋅L

QA

)

ECUACION DE LA ELASTICA M⋅ L ⋅ x b2 x 2 y AC = − ⋅ 1− 3 ⋅ 2 − 2 6⋅ E⋅I L L

yCB

b L

ANGULOS DE GIRO M ⋅ L b2 M ⋅ L a2 ϕA = ⋅ 3 ⋅ 2 − 1 ; ϕ B = ⋅3⋅ − 1 6⋅E⋅I L 6 ⋅ E ⋅ I L2 M ⋅ a3 + b3 ϕC = 3 ⋅ E ⋅ I ⋅ L2

(

B

a

3.3.1

P

CARGA PUNTUAL

REACCIONES P ⋅ b2 RA = 3 ⋅ ( L + 2 ⋅ a) L

;

RB =

Formulario para vigas y pórticos

3.3 VIGA EMPOTRADA EN LOS EXTREMOS

C

P ⋅ a2 ⋅ ( L + 2 ⋅ b) L3

B

A x

ESFUERZOS CORTANTES P ⋅ b2 QAC = 3 ⋅ ( L + 2 ⋅ a) = cte ; L

QCB = −

P ⋅ a2 ⋅ ( L + 2 ⋅ b ) = cte L3

a

b L

MOMENTOS FLECTORES

P ⋅ a ⋅ b2 P ⋅ a2 ⋅ b P ⋅ b2 ; MB = − ; MAC = 3 ⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L ) 2 2 L L L P ⋅ a2 2 ⋅ P ⋅ a2 ⋅ b2 = 3 ⋅ L ⋅ b + L2 − L ⋅ x − 2 ⋅ b ⋅ x ; MC = para x0 = a L L3

MA = − MBC

(

)

QB QA

ECUACION DE LA ELASTICA

yBC

P ⋅ b2 2 ⋅ a ⋅ x x2 ⋅3 ⋅a − x − ⋅ 6⋅E⋅I L L2

P ⋅ a2 L − x ⋅ (L − x) = ⋅ 3 ⋅ b − ( L − x) − 2 ⋅ b ⋅ 6⋅E⋅I L L2

P ⋅ a3 ⋅ b3 3 ⋅ E ⋅ I ⋅ L3

;

fmax =

2 ⋅a⋅ L x= L + 2⋅a

2 ⋅ P ⋅ a3 ⋅ b2 3 ⋅ E ⋅ I ⋅ ( L + 2 ⋅ a)

2

0

3.11

para

MB

MA

MC

FLECHAS fC =

2

x

y AC =

3.12

3.3.2

CARGA PUNTUAL CENTRAL

REACCIONES

RA = RB =

P

P 2

C

QAC

P = = cte 2

B

A

ESFUERZOS CORTANTES

QCB

x

P = − = cte 2

a

a L

MOMENTOS FLECTORES

MA = MB = −

P⋅L 8

Mmax = MC =

P⋅L 8

MAC = para

x0 =

PL x ⋅ 4 ⋅ −1 8 L L 2

QB QA

y AC = −

P ⋅ L ⋅ x2 48 ⋅ E ⋅ I

x ⋅3 − 4⋅ L

MB

MA

FLECHAS

M max 0

x

P ⋅ L3 fC = 192 ⋅ E ⋅ I

Prontuario para Cálculo de Estructuras

ECUACION DE LA ELASTICA

CARGA CONTINUA EN PARTE DE LA VIGA

c

REACCIONES

p ⋅ b ⋅ c MA − MB RA = − L L

;

p

p ⋅ a ⋅ c MA − MB RB = + L L

C

ESFUERZOS CORTANTES

QAC = RA = cte ; QBD = −RB = cte ; QCD

;

MBD = RB ⋅ ( L − x ) + MB

c = RA − p ⋅ x − a + a

B x a

MCD = RA ⋅ x + MA − ;

D

A

MOMENTOS FLECTORES MAC = RA ⋅ x + MA

MA = −

p ⋅ c3 12 ⋅ L2

p c ⋅ x−a+ 2 2

Formulario para vigas y pórticos

3.3.3

b

2

12 ⋅ a ⋅ b2 ⋅L − 3⋅b + c2

p ⋅ c3 12 ⋅ a2 ⋅ b MB = − L a ⋅ − 3 ⋅ + 12 ⋅ L2 c2

L

Q

A

Q

B

ECUACION DE LA ELASTICA

x2 ⋅ ( −3 ⋅ MA − RA ⋅ x ) 6⋅E⋅I 4 1 c yCD = ⋅ p ⋅ x − a + − 4 ⋅ RA ⋅ x3 − 12 ⋅ MA ⋅ x3 24 ⋅ E ⋅ I 2 1 RB x3 − 3 ( MB + LRB ) x2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2 y DB = 6EI y AC =

MA

MB

3.13

3.14

3.3.4

CARGA CONTINUA EN TODA LA VIGA

p

REACCIONES

RA = RB =

P⋅L 2

ESFUERZOS CORTANTES

QAB

B

A

P = Qx = ⋅ ( L − 2 ⋅ x ) 2

x L

MOMENTOS FLECTORES

MA = MB = −

P ⋅ L2 12

P ⋅ L2 − 6 ⋅ L ⋅ x + 6 ⋅ x 2 12 P ⋅ L2 L Mcentro = para x = 24 2 Mx = 0 para x0 = 0,2113 ⋅ L

Q

ECUACION DE LA ELASTICA

MA

Mx = −

)

P ⋅ L4 x x 2 ⋅ − 24 ⋅ E ⋅ I L L2

ymax =

P ⋅ L4 384 ⋅ E ⋅ I

A

Q

B

2

para

x=

L 2

MB

Prontuario para Cálculo de Estructuras

yx =

(

CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES

p1

L M − MB ⋅ ( 2 ⋅ p1 + p2 ) − A 6 L L MA − MB RB = ⋅ ( p1 + 2 ⋅ p2 ) + 6 L

p2

RA =

ESFUERZOS CORTANTES

QA = RA Qx = RA − QB = −RB

p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L

x L ⋅x

MOMENTOS FLECTORES

L2 MA = − ⋅ ( 3 ⋅ p1 + 2 ⋅ p2 ) 60 p ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 2 Mx = RA ⋅ x + MA − 1 ⋅x 6⋅L L2 MB = − ( 2 ⋅ p1 + 3 ⋅ p2 ) 60 ECUACION DE LA ELASTICA

yx =

B

A

Formulario para vigas y pórticos

3.3.5

A

Q

B

MA

MB

3.15

( p − p1 ) 3 x2 ⋅ 2 ⋅ x + p1 ⋅ L ⋅ x2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L 24 ⋅ E ⋅ I ⋅ L 5

Q

3.16

3.3.6

MOMENTO FLECTOR

REACCIONES 6⋅M RA = − 3 ⋅ a ⋅ b L

; RB =

6⋅ M ⋅ a⋅ b L3

C

ESFUERZOS CORTANTES

Qx = −

+M

x

6⋅ M ⋅ a ⋅ b = cte L3

a

b

MOMENTOS FLECTORES M⋅ a b MA = ⋅2 − 3⋅ L L MAC =

B

A

L

M⋅ b a MB = − ⋅2 − 3⋅ L L

M⋅ a a x ⋅ 3 ⋅ ⋅ 1− 2 ⋅ − 1 L L L

MCB = −

QA

M⋅ b b L− x ⋅ 3 ⋅ ⋅ 1− 2 ⋅ − 1 L L L 6⋅ M 2 ⋅a ⋅b L3

;

MCder = MA +

(

M 3 ⋅ L − 6 ⋅ a2 ⋅ b L3

)

ECUACION DE LA ELASTICA

y AC = y BC =

M⋅ b ⋅ x2 2⋅E⋅I⋅L

L− x b ⋅2 ⋅a⋅ 2 − L L

M⋅ a ⋅ ( L − x ) 2⋅E⋅I⋅L

2

b⋅ x a ⋅2⋅ 2 − L L

FLECHA M ⋅ a2 ⋅ b2 fC = − ⋅ ( a − b) 2 ⋅ E ⋅ I ⋅ L3

MC MA MC

MB

Prontuario para Cálculo de Estructuras

MCizq = MA −

QB

3.4.1

P

CARGA PUNTUAL

REACCIONES P ⋅ b2 P⋅a ⋅ ( 3 ⋅ L − b ) ; RB = ⋅ 3 ⋅ L2 − a2 RA = 2 ⋅ L3 2 ⋅ L3 ESFUERZOS CORTANTES P ⋅ b2 P⋅a QAC = − ⋅ ( 3 ⋅ L − b ) = cte ; QCB = − ⋅ 3 ⋅ L2 − a2 = cte 2 ⋅ L3 2 ⋅ L3

(

x a

b L

)

(

2

; ϕC =

4⋅E⋅I⋅L

)

Q

B

Q

A

ANGULOS DE GIRO P ⋅ a ( L − a)

B

A

)

MOMENTOS FLECTORES P⋅a 2 P⋅a 2 MB = − ⋅ L − a2 ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b) ; MC = 2 ⋅ L2 2 ⋅ L3 P⋅ x 2 P⋅a MAC = ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b ) ; MCB = ⋅ 2 ⋅ L3 − 3 ⋅ L2 ⋅ x + a2 ⋅ x 2 ⋅ L3 2 ⋅ L3

ϕA =

C

)

(

(

P ⋅ a ⋅ ( L − a)

2

4 ⋅ E ⋅ I ⋅ L3

(

⋅ L2 − 2 ⋅ a ⋅ L − a2

)

ECUACION DE LA ELASTICA P ⋅ b2 ⋅ x y AC = ⋅ 3 ⋅ a ⋅ L2 − x2 ⋅ ( 2 ⋅ L + a) 12 ⋅ E ⋅ I ⋅ L3

y BC =

P ⋅ a ⋅ ( L − x)

2

12 ⋅ E ⋅ I

MB

a2 a2 L − x ⋅ 3 ⋅ 1− 2 − 3 − 2 ⋅ L L L

FLECHA MAXIMA

p⋅b ⋅a a ⋅ 6⋅ E⋅I 2⋅L + a

MC para x=L ⋅

a 2⋅L+a

3.17

fmax =

2

Formulario para vigas y pórticos

3.4 VIGA APOYADA-EMPOTRADA

3.18

3.4.2

CARGA PUNTUAL CENTRAL

REACCIONES 5 11 RA = ⋅ P ; RB = ⋅P 16 16

P C

ESFUERZOS CORTANTES 5 11 QAC = ⋅ P = cte ; QCB = − ⋅ P = cte 16 16

B

A

x

MOMENTOS FLECTORES

a

3 5 MB = − ⋅ P ⋅ L ; MC = ⋅P⋅L 16 32 5 P⋅ L L − x MAC = ⋅ P ⋅ x ; MCB = ⋅ 11⋅ − 3 16 16 L

a L

ANGULOS DE GIRO

P⋅L 32 ⋅ E ⋅ I

QB

; ϕC = −

P⋅L 128 ⋅ E ⋅ I 2

QA

ECUACION DE LA ELASTICA y AC =

P ⋅ L2 x2 ⋅ x ⋅ 3 − 5 ⋅ 2 96 ⋅ E ⋅ I L

y BC =

P⋅L L − x ⋅ (L − x)2 ⋅ 9 − 11⋅ 96 ⋅ E ⋅ I L

MB

FLECHA MAXIMA

fC =

7 ⋅ P ⋅ L3 ; 768 ⋅ E ⋅ I

fmax =

P ⋅ L3 48 ⋅ 5 ⋅E ⋅ I

para x=

L 5

MB

Prontuario para Cálculo de Estructuras

ϕA =

2

CARGA CONTINUA EN PARTE DE LA VIGA

REACCIONES p ⋅ b ⋅ c MB + RA = L L

c ; RB =

p

p ⋅ a ⋅ c MB − L L

ESFUERZOS CORTANTES

c QAC = RA = cte ; QDB = −RB = cte ; QCD = RA − p ⋅ x − a + 2

C

;

MCD

MDB = RB ⋅ ( L − x ) + MB

MB = −

b L

p⋅ a⋅ b⋅c c2 ⋅L + a− 2 2⋅L 4⋅b

ECUACION DE LA ELASTICA x 12 ⋅ a ⋅ b2 y AC = ⋅ −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅ L − 3b + 48 ⋅ E ⋅ I ⋅ L c2

QB QA

MB

4 c −8 ⋅ RA ⋅ L ⋅ x 3 + 2 ⋅ p ⋅ L ⋅ x − a + + 4 1 = ⋅ 2 48 ⋅ E ⋅ I ⋅ L + p ⋅ c3 ⋅ L − 3 ⋅ b + 12 ⋅ a ⋅ b ⋅ x 2 c

(L − x) =−

2

6⋅E⋅I

⋅ RB ⋅ ( L − x ) + 3 ⋅ MB

3.19

y DB

a

2

ANGULOS DE GIRO p ⋅ c3 12 ⋅ a ⋅ b2 ϕA = ⋅L − 3⋅b + 48 ⋅ E ⋅ I ⋅ L c2

yCD

B x

p c = RA ⋅ x − ⋅ x − a + 2 2 ;

D

A

MOMENTOS FLECTORES MAC = RA ⋅ x

Formulario para vigas y pórticos

3.4.3

3.20

3.4.4

CARGA CONTINUA EN TODA LA VIGA

REACCIONES

RA =

3 ⋅P⋅L 8

;

RB =

p

5 ⋅P⋅L 8

ESFUERZOS CORTANTES 3 x QAB = P ⋅ L ⋅ − ; 8 L

3 ⋅ P ⋅ L; 8

QA =

QB =

5 ⋅P⋅L 8

B

A

x

MOMENTOS FLECTORES

L

P⋅x ⋅ (3 ⋅ L − 4 ⋅ x); 8 9 Mmax rel = ⋅ P ⋅ L2 para 128 3 M = 0 para x = ⋅ L 4 MAB =

MB = − x=

P⋅L 8

2

3 ⋅ L; 8

QB

ANGULOS DE GIRO

ϕA =

P ⋅ L3 48 ⋅ E ⋅ I

ECUACION DE LA ELASTICA

y AB =

P⋅x ⋅ (L + 2 ⋅ x) ⋅ (L − x)2 48 ⋅ E ⋅ I

fmax =

P ⋅ L4 185 ⋅ E ⋅ I

para

x=

1+ 33 ⋅L 16

MB

Prontuario para Cálculo de Estructuras

QA

CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES

RA =

L M ⋅ ( 2 ⋅ p1 + p2 ) + B 6 L

; RB =

L M ⋅ ( p1 + 2 ⋅ p2 ) − B 6 L

p2

p1

Qx = RA −

p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L

B

A

ESFUERZOS CORTANTES

x ⋅x

L

; QB = −RB

Formulario para vigas y pórticos

3.4.5

MOMENTOS FLECTORES

Mx = RA ⋅ x −

p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 6⋅ L

Q ⋅x

2

;

L2 MB = − ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 120

A

Q

B

ANGULOS DE GIRO

ϕA =

L3 ⋅ ( 3 ⋅ p1 + 2 ⋅ p2 ) 240 ⋅ E ⋅ I

MB

ECUACION DE LA ELASTICA

3.21

( p2 − p1) ⋅ x4 + x 3 2 yx = ⋅ + 5 ⋅ L ⋅ p1 ⋅ x − 20 ⋅ RA ⋅ L ⋅ x + 120 ⋅ E ⋅ I ⋅ L 2 3 + 5 ⋅ L ⋅ 12 ⋅ RA ⋅ L − ( 3 ⋅ p1 + p2 ) ⋅ L

3.22

3.4.6

MOMENTO FLECTOR

M+

REACCIONES

RA = −RB =

(

3 M 2 ⋅ ⋅ L − a2 2 L3

)

ESFUERZOS CORTANTES

a

MOMENTOS FLECTORES MCder = RA ⋅ a − M ;

(

b L

MCizq = RA ⋅ a

3 M⋅ x 2 ⋅ ⋅ L − a2 2 L3

B

x

Qx = RA = cte

MAC =

C

A

)

;

MBC

(

M ⋅ L2 − 3 ⋅ a2 2 ⋅ L2 M x a2 = ⋅ 3 ⋅ ⋅ 1 − 2 − 2 2 L L MB =

;

ANGULOS DE GIRO

)

M ⋅ ( L − a) ⋅ ( 3 ⋅ a − L ) 4⋅E⋅I⋅L b a 2 M ϕC = ⋅ b ⋅ 3 ⋅ ⋅ 1+ − 4 4⋅E⋅I L L

QB

ϕA =

MC MB

ECUACION DE LA ELASTICA M⋅ b ⋅ x ⋅ −4 ⋅ L3 − x 2 − 3 ⋅ L2 ⋅ ( a + L ) 4 ⋅ E ⋅ I ⋅ L3 M 2 = ⋅ ( L − x ) ⋅ 2 ⋅ a2 ⋅ L − x ⋅ L2 − a2 4 ⋅ E ⋅ I ⋅ L3

y AC = yBC

(

)

(

)

MC

Prontuario para Cálculo de Estructuras

QA

3.5.1

CARGA PUNTUAL

C

Formulario para vigas y pórticos

3.5 VIGA EMPOTRADA EN UN EXTREMO

P

REACCIONES

B

A

RB = P

x ESFUERZOS CORTANTES QAC = 0 ; QCB = −P = cte

a

b L

MOMENTOS FLECTORES

MAC = 0

;

MCB = −P ⋅ ( x − a)

;

MB = −P ⋅ b

QB

ANGULOS DE GIRO

ϕ A = ϕC = −

P ⋅ b2 2⋅E⋅I

ECUACION DE LA ELASTICA

y

AC

=

P ⋅ b2 ⋅ ( 3 ⋅ ( L − x ) − b) 6⋅E⋅I

;

y

CB

=

MB

3.23

FLECHA MAXIMA P ⋅ b3 P ⋅ b2 fC = ; fA = ⋅ ( 2 ⋅ b + 3 ⋅ a) 3⋅E⋅I 6⋅ E⋅I

P 2 ⋅ ( L − x ) ⋅ ( 2 ⋅ b + 3 ⋅ a) 6⋅E⋅I

3.24

3.5.2

CARGA PUNTUAL EN EL EXTREMO P

REACCIONES RB = P

A ESFUERZOS CORTANTES QAB = −P = cte

B

x L

MOMENTOS FLECTORES MAB = −P ⋅ x;

MB =-P ⋅ L

QB

ECUACION DE LA ELASTICA

y

AB

=

P ⋅ (L − x)2 ⋅ (2 ⋅ L + x) 6⋅E⋅I

FLECHA MAXIMA

fA =

P ⋅ L3 3⋅E⋅I

MB

Prontuario para Cálculo de Estructuras

ANGULOS DE GIRO P ⋅ L2 ϕA = − 2⋅E⋅I

CARGA CONTINUA EN PARTE DEL VUELO

REACCIONES RB = p ⋅ c ESFUERZOS CORTANTES c QAC = 0 ; QCD = −p ⋅ x − a + 2

; QDB = −p ⋅ c = cte

MOMENTOS FLECTORES 2 c p⋅ x − a+ 2 MAC = 0 ; MCD = − ; 2 MDB = − p ⋅ c ⋅ ( x − a) ; MB = − p ⋅ c ⋅ b ANGULOS DE GIRO p ⋅ c 2 c2 ϕD = − ⋅b − 2⋅E⋅I 4

; ϕC = −

MD = −

p ⋅ c2 2

p ⋅ c 2 c2 ⋅b + 2⋅E⋅I 12

Formulario para vigas y pórticos

3.5.3

; ϕ A = ϕC

ECUACION DE LA ELASTICA

y DB =

p⋅c p⋅c c2 3 ⋅ L − x2 ⋅ ( 2 ⋅ b − a + x ) ; y AC = ⋅ ( a − x ) ⋅ 3 ⋅ b2 + + 2⋅b 6⋅E⋅I 6 ⋅ E ⋅ I 4

y DC =

4 p c c2 3 ⋅ x − a + + 4 ⋅ c ⋅ ( a − x ) ⋅ 3 ⋅ b2 + + 8 ⋅ b ⋅ c 24 ⋅ E ⋅ I 2 4

(

)

FLECHAS 2

fD =

p⋅ c c ⋅ b − E⋅I 2

fC =

2 p ⋅ c c p⋅ c c2 ⋅ b + ⋅ ( 4 ⋅ b − c) + c3 ; fA = ⋅ a ⋅ 3 ⋅ b2 + + 2 ⋅ b3 12 ⋅ E ⋅ I 2 6 ⋅ E ⋅ I 4

b c ⋅ + 3 12

3.25

3.26

3.5.4

CARGA CONTINUA EN TODO EL VUELO p

REACCIONES RB = p ⋅ L

B ESFUERZOS CORTANTES

QAB = −P ⋅ x

A

QB = −P ⋅ L = cte

;

x L

MOMENTOS FLECTORES

MAB = −

P ⋅ x2 ; 2

MB = −P ⋅

L2 2

ANGULOS DE GIRO

QB

P ⋅ L3 6⋅E⋅I

ECUACION DE LA ELASTICA

y AB =

FLECHAS fA =

(

P 2 ⋅ ( L − x ) ⋅ 3 ⋅ L2 + 2 ⋅ L ⋅ x + x 2 24 ⋅ E ⋅ I

P ⋅ L4 8⋅E⋅I

) MB

Prontuario para Cálculo de Estructuras

ϕA = −

CARGA TRAPEZOIDAL EN TODO EL VUELO

REACCIONES L RB = ⋅ ( p1 + p2 ) 2 ESFUERZOS CORTANTES

Qx = −

p2 − p1 x2 ⋅ − p1 ⋅ x 2 L

p2

p1 L ; QB = − ⋅ ( p1 + p2 ) 2

MOMENTOS FLECTORES

x2 Mx = − ⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1 6⋅L

A B

Formulario para vigas y pórticos

3.5.5

x ;

L2 MB = − ⋅ ( p2 + 2 ⋅ p1 ) 6

L

ANGULOS DE GIRO

ϕA = −

L3 ⋅ ( 3 ⋅ p1 + p2 ) 24 ⋅ E ⋅ I

ECUACION DE LA ELASTICA

yx

QB

( L − x )3 2 − ⋅ ( p2 − p1 ) + ( L − x ) ⋅ p2 − ⋅ 5⋅ L 24 ⋅ E ⋅ I 2 2 2 2 − ⋅ L ⋅ L − x ⋅ p + p + ⋅ L ⋅ p + ⋅ p ( ) ( ) ( ) 2 1 2 1

(L − x) =

2

FLECHA

L ⋅ ( 4 ⋅ p2 + 11⋅ p1)

MB

4

fA =

120 ⋅ E ⋅ I

3.27

3.28

3.5.6

MOMENTO FLECTOR

REACCIONES

M

A

RB = 0

B

ESFUERZO CORTANTE

x

Qx = 0

a

MOMENTOS FLECTORES MAC = 0

;

MCB = − M = cte

b L

;

MAC = − M

ANGULOS DE GIRO

ϕC = ϕ A = −

M⋅ b E⋅I

y AC =

M ⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b) 2⋅E⋅I

; yBC =

M 2 ( L − x) 2⋅E⋅I

FLECHA

MB fC =

M⋅ b 2⋅E⋅I 2

; fA =

M ⋅ b ⋅ ( 2 ⋅ L − b) 2⋅E⋅I

Prontuario para Cálculo de Estructuras

ECUACION DE LA ELASTICA

Formulario para vigas y pórticos

3.5.7

MOMENTO FLECTOR EN EXTREMO DEL VUELO

REACCIONES

M

RB = 0

B

A

ESFUERZO CORTANTE

x

Qx = 0

L

MOMENTOS FLECTORES MAB = − M = cte ANGULOS DE GIRO

ϕA = −

M⋅ L E⋅I

ECUACION DE LA ELASTICA

y AC =

(

M ⋅ b ⋅ x2 − 2 ⋅ L ⋅ x + L2 2⋅E⋅I

) MB

FLECHA

fA =

3.29

M ⋅ L2 2⋅E⋅I

3.30

3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES P

P

P

A

B

L/2

C

L/2 L

A

L/2

L/2 L

L/2 L

0,688 P

L

0,405 P

0,312 P A

C

B

L/2

B

C

0,094 P

A

0,312 P

0,094 P

B

C

0,594 P

0,688 P

ESFUERZOS CORTANTES

- 0,188 PL - 0,094 PL

A

B

0,156 PL

C

0,156 PL

MOMENTOS FLECTORES

A

B

0,203 PL MOMENTOS FLECTORES

C

Prontuario para Cรกlculo de Estructuras

ESFUERZOS CORTANTES

A

L

Q

B

Q

C

L

0,625 QL

Formulario para vigas y p贸rticos

Q

A

B

L

L

C

0,375 L 0,437 QL

0,375 QL

0,063 QL

A

B

C

A

B

0,375 QL 0,375 L

0,563 QL

0,437 L

0,625 QL ESFUERZOS CORTANTES

ESFUERZOS CORTANTES

2

2

- 0,063 QL

- 0,125 QL

A

B 2

C 2

0,07 QL

0,07 QL

A

B

C

2

0,096 QL

MOMENTOS FLECTORES

3.31

MOMENTOS FLECTORES

C

3.32

3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES Q

A

L

Q

B

C

k L

c QL d L

a QL

A

Relación entre luces

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

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

C

B

b QL

d QL

a L

2

f QL

A

B

C

2

e QL

2

g QL MOMENTOS FLECTORES

k 2 − k +1 8 k f d= − 2 k f=

a = 0.5 − f e=

a2 2

b = 0.5 + f g=

d2 2

c=

k f + 2 k

Prontuario para Cálculo de Estructuras

ESFUERZOS CORTANTES

A

Q

B

L

Relación entre luces

C

k L

c QL d L

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

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

Formulario para vigas y pórticos

Q

A C

B

a QL d QL

b QL ESFUERZOS CORTANTES 2

f QL

k 2 − k +1 8 k f d= − 2 k f=

A

B

2

g QL

e=

a2 2

b = 0.5 + f g=

c=

k f + 2

d2 2

3.33

MOMENTOS FLECTORES

C

a = 0.5 − f

3.34

3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES Q

Q

A

Q

B

C

L

D

k L

a QL

L

a L

b QL

c QL

D

B

Relación entre luces

ESFUERZOS CORTANTES

MOMENTOS FLECTORES

k

a

b

c

e

f

g

0,6

0,420

0,580

0,300

0,088

0,080

-0,035

0,7

0,418

0,582

0,350

0,087

0,081

-0,020

0,8

0,414

0,586

0,400

0,086

0,086

-0,006

0,9

0,408

0,592

0,450

0,083

0,091

-0,009

C

A

c QL

b QL

a QL

ESFUERZOS CORTANTES

2

2

f QL

f QL 2

g QL A

B

C

D 2

2

e QL

e QL

MOMENTOS FLECTORES

f=

k3 + 1 12 ⋅ k + 8

a = 0.5 − f

c=

k 2

e=

a2 2

b = 0.5 + f

g=

k2 −f 8

Prontuario para Cálculo de Estructuras

a L

Q

A

Q

B

C

L

D

k L

L

a L a QL

b QL

c QL

D C

B

A

c QL

b QL

a QL

a L ESFUERZOS CORTANTES

2

f QL

B 2

e QL

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

2

f QL A

Relación entre luces

Formulario para vigas y pórticos

Q

C 2

g QL

k3 + 1 12 ⋅ k + 8

a = 0.5 − f

c=

k 2

e=

b = 0.5 + f

D 2

e QL

a2 2

g=

k2 −f 8

3.35

MOMENTOS FLECTORES

f=

3.36

3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL a

3.9.1 k=

I2 h ⋅ I1 L

y

p B

N = 3+ 2⋅k

C x

VD

n

I1

p⋅ s⋅n = L p⋅ s⋅m = L

HA = HD =

I2

m

REACCIONES

VA

s

CARGA REPARTIDA VERTICAL

h

I1

A

D

3⋅p⋅ s s2 ⋅m⋅ n − 2 ⋅ h⋅ L ⋅ N 12

L

MC

3 p⋅ s s2 MB = MC = − ⋅ ⋅ m⋅ n − 2 L⋅N 12 En S Mx = VA ⋅ x −

p ⋅ (x − m)2 − HA ⋅ h 2 HA

HD VA

VD

Prontuario para Cálculo de Estructuras

MB

MOMENTOS FLECTORES

k=

CARGA REPARTIDA VERTICAL UNIFORME

I2 h ⋅ I1 L

y

N = 3+ 2⋅k

p B

REACCIONES

C I x

P⋅L VA = VD = 2 P ⋅ L2 HA = HD = 4 ⋅ h⋅ N

I

2

I

1

A

h

1

Formulario para vigas y pórticos

3.9.2

D

MOMENTOS FLECTORES

L

P ⋅ L2 4⋅N P ⋅ x ⋅ ( L − x ) P ⋅ L2 Mx = − 2 4⋅N

MB = MC = −

Mmax,pos =

P ⋅ L2 P ⋅ L2 − 8 4⋅N

M

para

x=

MC

B

L 2 HA

HD VA

VD

3.37

k=

3.38

3.9.3

CARGA REPARTIDA HORIZONTAL

I2 h ⋅ I1 L

y

N = 3+ 2⋅k

p

B

C I2

REACCIONES

VA = VD = HD = HA =

p ⋅ h2 2⋅L

I1 y

p ⋅ h ⋅ (2 ⋅ N + k) 8⋅N

A

p ⋅ h ⋅ (6 ⋅ N − k )

MB

MC

MB

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

p ⋅ h2 ⋅ (2 ⋅ N − k) 8⋅N p ⋅ h2 MC = − (2 ⋅ N + k) 8⋅N En AB MB =

p ⋅ y ⋅ (h − y) y + ⋅ MB 2 h

D L

8⋅N

MOMENTOS FLECTORES

MY =

h

I1

CARGA PUNTUAL VERTICAL SOBRE DINTEL

P m

k=

I2 h ⋅ I1 L

y

n

B

N = 3+ 2⋅k

C I2

REACCIONES I1

P⋅n L P⋅m VD = L VA =

h

I1

A

Formulario para vigas y pórticos

3.9.4

D

3 P ⋅ m⋅ n HA = HD = 2 L ⋅ h⋅ N

L

MOMENTOS FLECTORES

MB

MC

3 P ⋅ m⋅ n MB = MC = − ⋅ 2 L⋅N MP

2⋅N−3 MP = P ⋅ m ⋅ n ⋅ 2⋅L⋅N HA

HD VA

VD

3.39

3.40

3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO

3.10.1 CARGA REPARTIDA VERTICAL k1 =

I3 h1 ⋅ I1 s

y

k2 =

I3 h2 ⋅ I2 s

p C s

f I B

REACCIONES

3

I

x

p⋅L VA = VD = 2 h1 + h2 p ⋅ L2 HA = HD = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

I

h1

h2

2

1

A

D

Ll

MOMENTOS FLECTORES

MC = −

( h1 + h2 ) ⋅ h2 p ⋅ L2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

En

BC

MX =

p ⋅ x ⋅ (L − x) f − HA ⋅ ⋅ x + h1 2 L

MB

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

MC

( h1 + h2 ) ⋅ h1 p ⋅ L2 MB = − 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

k1 =

I3 h1 ⋅ I1 s

k2 =

y

I3 h2 ⋅ I2 s

C s

f

p REACCIONES

I3 B

p⋅h 2⋅L HA = p ⋅ h1 − HD

2 1

VA = VD =

I1

h1

y A

h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2 p ⋅ h12 HD = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

MB =

p⋅h 2

−

p⋅h 8

3 1

D L

MOMENTOS FLECTORES 2 1

h2

I2

Formulario para vigas y pórticos

3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR

MC

h1 ⋅ ( 4 + 5 ⋅ k 1 ) + 2 ⋅ h2

h ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 2 1

MB

h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2 p ⋅ h12 ⋅ h2 MC = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 En AB MY = HA ⋅ y −

p ⋅ y2 2

HA

HD VA

VD

3.41

3.42

3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL k1 =

I3 h1 ⋅ I1 s

y

k2 =

I3 h2 ⋅ I2 s

y

I3

p ⋅ f ⋅ ( h1 + h2 )

HA = p ⋅ f − HD

C s

f

REACCIONES

VA = VD =

p

B

2⋅L

h2

I2

I1

h1

A

2 p ⋅ f 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 ) HD = ⋅ 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

D L

MOMENTOS FLECTORES

MC = −

p ⋅ f ⋅ h1 ⋅ 8

h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

MB

2 p ⋅ h2 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 ) ⋅ 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

En BC L p ⋅ y2 MY = −VA ⋅ ⋅ y + HA ⋅ ( y + h1 ) − f 2

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

MB = p ⋅ f ⋅ h1 −

MC

8 ⋅ h12 ⋅ (1+ k 1) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 )

k1 =

I3 h1 ⋅ I1 s

k2 =

y

I3 h2 ⋅ I2 s

P f

REACCIONES

I3

P⋅b L P⋅a VD = L

B

VA =

HA = HD =

C s

b

I1

h1

A

h1 ⋅ (L + b) + h2 ⋅ (L + a) P⋅a⋅b ⋅ 2 ⋅ L2 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

h2

I2

a

Formulario para vigas y pórticos

3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL

D L

MOMENTOS FLECTORES

MB = −

MC = −

MP =

P ⋅ a ⋅ b ⋅ h1 2⋅L

2

P ⋅ a ⋅ b ⋅ h2 2 ⋅ L2

⋅

h1 ⋅ ( L + b ) + h2 ⋅ ( L + a )

MC

h ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2 2 1

⋅

2 2

MB

h1 ⋅ ( L + b ) + h2 ⋅ ( L + a)

h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

P⋅ a⋅ b a⋅f + HA ⋅ + h1 L L

MP

HA

HD VA

VD

3.43

3.44

3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS 3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

p

Valor de la carga en proyección horizontal

k=

C

I2 h ⋅ I1 s

s

I2

REACCIONES

f

I2

B

D x

p⋅L 2 p ⋅ L2 8⋅h+ 5⋅f HA = HE = ⋅ 2 32 h ⋅ ( 3 + k ) + f ⋅ ( 3h + f )

VA = VE =

I1

I1

A

h E

L

MOMENTOS FLECTORES

MC =

p ⋅ L2 ⋅ h 8⋅h+ 5⋅f ⋅ 2 32 h ⋅ (3 + k) + f ⋅ (3 ⋅ h + f )

MC

p ⋅ L2 f + h + ⋅ MB 8 h

MB

MD

En BC y DC MX = p ⋅

x ⋅ (L − x) 2

+

MB h

2⋅f ⋅ x ⋅h+ L

HA

HE VA

VE

Prontuario para Cálculo de Estructuras

MB = MD = −

Valor de la carga en proyección horizontal

k=

p

I2 h ⋅ I1 s

C s

REACCIONES

I

p⋅L 8 p⋅L VE = 8 p ⋅ L2 8⋅h+ 5⋅f HA = HE = 64 h2 ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )

I

2

f 2

B

VA = 3 ⋅

D x I

I

1

A

Formulario para vigas y pórticos

3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

h

1

E L

MOMENTOS FLECTORES

MB = MD = −

MC

p ⋅ L2 ⋅ h 8⋅h+ 5⋅f 2 64 h ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )

p ⋅ L2 f + h + ⋅ MB 16 h En BC MC =

MX = p ⋅

x ⋅ (L − x) 2

+

MB h

MB

MD

2⋅f⋅ x ⋅h+ L HA

HE VE

3.45

VA

3.46

3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR

k=

I2 h ⋅ I1 s

C s

I

p

REACCIONES

I

2

f 2

D

B

p⋅h 2⋅L HA = p ⋅ h − HE

VA = VE =

HE =

2

I

I

1

h

1

y

( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f p ⋅ h2 ⋅ 2 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)

A

E L

MOMENTOS FLECTORES

MC

MB

MD

En AB My = −

p ⋅ y2 + HA ⋅ y 2

HA

HE VA

VE

Prontuario para Cálculo de Estructuras

p ⋅ h2 + MD 2 p ⋅ h2 f + h MC = + ⋅ MD h 4 ( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f p ⋅ h3 ⋅ 2 MD = − 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h) MB =

Valor de la carga en proyección vertical

k=

I2 h ⋅ I1 s

p

C s

REACCIONES

I2

D

B

p⋅f VA = VE = ⋅ ( f + 2 ⋅ h) 2⋅L HA = p ⋅ f − HE HE =

f

I2 y

I1

2 p ⋅ f 8 ⋅ h ⋅ ( k + 3 ) + 5 ⋅ f ⋅ ( f + 4 ⋅ h) ⋅ 16 h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)

I1

Formulario para vigas y pórticos

3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

h

x A

E L

MOMENTOS FLECTORES

MC

MB = HA ⋅ h MC = −

2 p ⋅ f2 4 ⋅ h ⋅ ( k + 2) + f ⋅ (5 ⋅ h + f ) ⋅ 2 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)

MB

MD = −HE ⋅ h

MD

En BC Mx = HA ⋅ y − VA ⋅ x − p ⋅

2

2

HA

HE VA

VE

3.47

f siendo y = ⋅ x + h L

( y − h)

3.48

P

3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL

C s

I h k= 2⋅ I1 s

I2 B

D

REACCIONES

m

n

I1

P⋅n VA = L P⋅m VA = L HA = HE =

f

I2

I1

A

(

P ⋅ m 6 ⋅ h ⋅ L ⋅ n+ f ⋅ 3 ⋅ L − 4 ⋅ m ⋅ 4 ⋅ L2 h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h) 2

2

h E

L

)

MOMENTOS FLECTORES MB

MB = MD = −HA ⋅ h P⋅m h+ f + ⋅ MB h 2 h⋅ L + 2 ⋅ f ⋅ m MP = VA ⋅ m − HA ⋅ L

MD

MC =

HA

HE VA

VE

Prontuario para Cálculo de Estructuras

MC

3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL k1 =

I3 h1 ⋅ I1 L

y

k2 =

p

I3 h2 ⋅ I2 L

B

C I

I

VA =

h −h p⋅L p⋅L + ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2

VD =

h12 − h22 p⋅L p⋅L − ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

2 1 2 2

HA = HD =

3

x

REACCIONES 2 2

h1

I

1

h2

2

Formulario para vigas y pórticos

3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL

D

A

h1 − h2 p ⋅ L2 ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

L

MOMENTOS FLECTORES MB

p ⋅ L2 ( h1 + h2 ) ⋅ h1 MB = − 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 MC = −

MC

p ⋅ L2 ( h1 + h2 ) ⋅ h2 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

HD

En BC

VD HA VA

3.49

p ⋅ x2 Mx = VA ⋅ x − − HA ⋅ h1 2

3.50

3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR k1 =

I3 h1 ⋅ I1 L

y

k2 =

I3 h2 ⋅ I2 L

p

B

C I

REACCIONES

3

I

p⋅h h − h2 − HD ⋅ 1 2⋅L L HA = p ⋅ h − HD

VA = VD =

HD =

2 1

h1

I

1

h2

2

D y

p ⋅ h12 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

A L

MOMENTOS FLECTORES

p ⋅ h12 p ⋅ h13 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 − ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

MB

MC

MB

p ⋅ h12 ⋅ h2 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 MC = − ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 En AB

HD

p ⋅ y2 My = HA ⋅ y − 2

VD HA VA

Prontuario para Cálculo de Estructuras

MB = −

k1 =

I3 h1 ⋅ I1 L

k2 =

y

P

I3 h2 ⋅ I2 L

B

C I3

REACCIONES

( L + b) ⋅ h1 + ( L + a) ⋅ h2 P⋅b P⋅a⋅b VA = + ⋅ 2 ⋅ h1 − h2 3 L 2⋅L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

(

VD =

( L + b) ⋅ h1 + ( L + a) ⋅ h2 P⋅a P⋅a⋅b − ⋅ h1 − h2 3 2 L 2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

(

a

)

I1

D

A L

( L + b ) ⋅ h1 + ( L + a) ⋅ h2 P⋅a⋅b HA = HD = 2 2 2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 MB

MOMENTOS FLECTORES

MB = −

P ⋅ a ⋅ b ⋅ h1 2⋅L

2

⋅

2 1

h

h2

I2

h1

)

b

Formulario para vigas y pórticos

3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL

MC

( L + b ) ⋅ h1 + ( L + a) ⋅ h2 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

MP HD

MC = −

P ⋅ a ⋅ b ⋅ h2 2 ⋅ L2

VD HA VA

3.51

MP = VA ⋅ a + MB

( L + b ) ⋅ h1 + ( L + a) ⋅ h2

h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2

3.52

3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

p B

C

I h k= 2⋅ I1 L

I2 x

REACCIONES

VA = VD =

I1

p⋅L 2

HA = HD =

p ⋅ L2 4 ⋅ h ⋅ (k + 2)

h

I1

A

D

MOMENTOS FLECTORES L

MB = MC = −

p ⋅ L2 6 ⋅ ( k + 2)

MB

MC

En BC Mx =

p ⋅ x ⋅ (L − x) 2

Mmáx positivo =

−

p ⋅ L2 6 ⋅ ( k + 2)

p ⋅ L2 3 ⋅ k + 2 L ⋅ para x = 24 k+2 2

HA

HD MA VA

MD VD

Prontuario para Cálculo de Estructuras

p ⋅ L2 MA = MD = 12 ⋅ ( k + 2 )

k=

I2 h ⋅ I1 L

p

B

C I2

REACCIONES

VA = VD =

p ⋅ h2 ⋅ k L ⋅ ( 6 ⋅ k + 1)

I1

HA = p ⋅ h − HD HD =

p ⋅ h ⋅ (2 ⋅ k + 3)

y

8 ⋅ ( k + 2)

A

D

MOMENTOS FLECTORES

MA = − MB =

p ⋅ h2 24

p ⋅ h2 24

MC = −

h

I1

Formulario para vigas y pórticos

3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR

L

2 1 ⋅5 + + 6 ⋅ k + 1 k + 2

2 2 ⋅ 1− + ⋅ + + 6 k 1 k 2

MC

MB MB

p⋅h 2 2 ⋅3 − − 24 6 ⋅ k + 1 k + 2 2

p ⋅ h2 2 1 ⋅3 + − 24 6 ⋅ k + 1 k + 2 En AB MD =

HA

HD MD

MA VA

VD

3.53

p ⋅ y2 My = − + HA ⋅ y + MA 2

3.54

3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL

P m

I h k= 2⋅ I1 L

n

B

C I2

REACCIONES

VA =

m ⋅ ( n − m) P⋅n ⋅ 1+ 2 L L ⋅ ( 6 ⋅ k + 1)

I1

h

I1

VD = P − VA HA = HD =

3 ⋅ P ⋅ m⋅ n 2 ⋅ L ⋅ h ⋅ (k + 2)

A

D L

MOMENTOS FLECTORES

P ⋅ m⋅ n 1 n− m ⋅ − 2 ⋅ L k + 2 L ⋅ ( 6 ⋅ k + 1)

MB = −

P ⋅ m⋅ n 1 n− m ⋅ + L k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1)

MC = −

P ⋅ m⋅ n 1 n− m ⋅ − k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1) L

MD =

P ⋅ m⋅ n 1 n− m ⋅ + 2 ⋅ L k + 2 L ⋅ ( 6 ⋅ k + 1)

P ⋅ m ⋅ n n ⋅ MB m ⋅ MC MP = + + L L L

MB

MC

MP

HA

HD MA

VA

MD VD

Prontuario para Cálculo de Estructuras

MA =

k=

I2 h ⋅ I1 L

P

B

C I2

REACCIONES

VA = VD =

3 ⋅ P ⋅ h⋅ k L ⋅ (6 ⋅ k + 1)

HA = HD =

P 2

I1

A

D

MOMENTOS FLECTORES

P ⋅ h 3 ⋅ k +1 ⋅ 2 6 ⋅ k +1 P⋅h 3⋅k MB = − MC = ⋅ 2 6⋅ k +1 P ⋅ h 3 ⋅ k +1 MD = ⋅ 2 6 ⋅ k +1 MA = −

h

I1

Formulario para vigas y pórticos

3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR

L

MB

MC

HA MA

MD VD

3.55

VA

HD

3.56

3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS p

3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL Valor de la carga en proyección horizontal

k=

C

I2 h ⋅ I1 s

s

I

I

2

f 2

B

REACCIONES

D x

p⋅L 2 k ⋅ (4 ⋅ h + 5 ⋅ f ) + f p ⋅ L2 HA = HE = ⋅ 8 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2

I

VA = VE =

(

I

1

h

1

A

E

)

L

MOMENTOS FLECTORES

p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) ⋅ 48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2

MB = MD = −

(

MC

)

k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f p⋅L ⋅ 48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2 2

2

(

p ⋅ L2 + MA − HA ⋅ ( h + f ) 8 En BC

MB

)

MD

MC =

2 ⋅ x ⋅ f p ⋅ x2 − Mx = MA + VA ⋅ x − HA ⋅ h + 2 L

HA

HE MA VA

ME VE

Prontuario para Cálculo de Estructuras

MA = ME =

Valor de la carga en proyección horizontal

k=

I2 h ⋅ I1 s

p C

REACCIONES

p⋅L − VE 2 4 ⋅ k +1 VE = 3 ⋅ p ⋅ L ⋅ 32 ⋅ ( 3 ⋅ k + 1) k ⋅ (4 ⋅ h + 5 ⋅ f ) + f p ⋅ L2 HA = HE = ⋅ 16 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2

s

VA =

(

I

f 2

D

B x

)

I

I

1

A

MOMENTOS FLECTORES

MA =

p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) p ⋅ L2 ⋅ − 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)

ME =

p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) p ⋅ L2 ⋅ + 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)

(

(

L

)

k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2 p ⋅ L2 p ⋅ L2 ⋅ − 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)

MD = −

k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2 p ⋅ L2 p ⋅ L2 ⋅ + 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)

(

(

h

1

E

)

MB = −

En BC

I

2

Formulario para vigas y pórticos

3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

MC

)

MB

MD

)

HA

HE MA VA

ME VE

3.57

2 ⋅ x ⋅ f p ⋅ x2 Mx = MA + VA ⋅ x − HA ⋅ h + − L 2 L MC = VE ⋅ + ME − HE ( f + h) 2

3.58

3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR k=

I2 h ⋅ I1 s

C s

REACCIONES

VA = VE =

p⋅h ⋅k 2 ⋅ L ⋅ ( 3 ⋅ k + 1) 2

I

p

I

2

f 2

D

B

HA = p ⋅ h − HE

k2 ⋅ h + k ⋅ ( 2 ⋅ f + 3 ⋅ h) + f p ⋅ h2 HE = ⋅ 2 4 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2

(

I

I

1

)

A

E

MOMENTOS FLECTORES

L

p ⋅ h2 k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f 2 ⋅ k + 1 ⋅ +6⋅ 2 24 3 ⋅ k + 1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2

MA = −

h

1

y

2

(

)

MB = MA + HA ⋅ h −

MB

2 2 p ⋅ h2 k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f 2 ⋅ k + 1 ⋅ − + ⋅ 6 2 24 3 ⋅ k + 1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 En AB p ⋅ y2 My = MA + HA ⋅ y − 2

ME =

(

MD

)

HA MA

HE ME

VA

VE

Prontuario para Cálculo de Estructuras

p ⋅ h2 2 L MC = ME − HE ⋅ ( f + h) + VE ⋅ 2 MD = ME − HE ⋅ h

MC

Valor de la carga en proyección vertical

k=

I2 h ⋅ I1 s

p

C

REACCIONES s

3 p ⋅ f 4 ⋅ k ⋅ ( f + h) + f VA = VE = ⋅ ⋅ 8 L 3 ⋅ k +1 HA = p ⋅ f − HE

I

(

D y

I

)

I

1

A L

k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f ) 3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f p⋅f ⋅ f⋅ + ⋅ 24 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2 3 ⋅ k +1

MC = ME − HE ( h + f ) + VE ⋅ MD = ME − HE ⋅ h

(

)

L 2

MC

MB

(

MD

)

HA MA

HE ME

VA

VE

3.59

k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f ) 3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f p⋅f ⋅ −f ⋅ + ⋅ 2 24 3 ⋅ k +1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2 En BC 2 L ⋅ ( y − h) p ⋅ ( y − h) My = MA + HA ⋅ y − VA ⋅ − 2⋅f 2 ME =

h

1

E

MOMENTOS FLECTORES

MB = MA + HA ⋅ h

f 2

B

2 p ⋅ f 2 ⋅ k ⋅ h ⋅ ( k + 4 ) + f ⋅ (10 ⋅ k ⋅ h + 5 ⋅ k ⋅ f + f ) HE = ⋅ 2 4 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2

MA = −

I

2

Formulario para vigas y pórticos

3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

3.60

3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL

P

I h k= 2⋅ I1 s

C s

REACCIONES

I

f 2

B

VA = P − VE 2 P ⋅ m 3 ⋅ L ⋅ ( k ⋅ L + m) − 2 ⋅ m VE = 3 ⋅ 3 ⋅ k +1 L HA = HE =

I

2

D m I

2 2 P ⋅ m 3 ⋅ k ⋅ L ⋅ ( f + h) − 4 ⋅ f ⋅ m ⋅ ( k + 1) + 3 ⋅ L ⋅ m ⋅ ( f − k ⋅ h) ⋅ 2 L2 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2

(

n I

1

A

)

h

1

E L

MOMENTOS FLECTORES 3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 2 P⋅m ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 ⋅ MA = 2 ⋅ L2 n ⋅ n − m ( ) − 3 ⋅ k +1 MB = MA − HA ⋅ h

L MC = ME + VE ⋅ − HE ⋅ ( h + f ) 2

MB

MD

MD = ME − HE ⋅ h

3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 2 P⋅m ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 ME = ⋅ 2 ⋅ L2 n ⋅ n − m ( ) + 3 ⋅ k +1 2⋅f ⋅m En BC My = MA + VA ⋅ m − HA ⋅ h + L

(

MC

)

)

HA

HE MA VA

ME VE

Prontuario para Cálculo de Estructuras

(

Solicitaciones en vigas y pórticos

Published on May 16, 2012

Capítulo 3 del libro "Prontuario y Herramientas Informáticas para Cálculo de Estructuras". Autores: Arianna Guardiola-Villora y Agustin Pére...

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