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

x−a

0

=0

x≥a

x−a

0

=1

x≤a

x−a

n

=0

x≥a

x−a

n

= ( 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 c=0

por lo tanto

P Si

a

T(x)

x

1

x≤a

x−a = 0

x≥a

x − a = ( x − a) 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)

por lo tanto

q d

a x

T(x)

M(x)

2

A=q c=2

Si x≤a

x−a

x≥a

3

x−a

3

=0

= ( x − a)

entonces 2

3

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!  dM( x )

M(x) =

a

T ( x) =

b

dx

x q/d q a

d

T ( x) =

b

q/d q -〈 x-a〉 3 + 〈 x-b〉 3  + 〈 x-b〉 2  2! 3!  dM( x )

M( x ) =

dx

x

q/d q a

q q/d  〈 x-a〉 3 − 〈 x-b〉 3  〈 x-a〉 2 +  2! 3!  dM( x )

M(x) = − d

T ( x) =

b

dx

x

qb

qa

M(x) = − +

a

d b

T ( x) =

x

qa

qb

b x

T ( x) =

)

q b − q a /d 3!

qb 2!

〈 x-b〉 2 +

 −〈 x-a〉 3 + 〈 x-b〉 3   

dx

M(x) = −

d

2!

〈 x-a〉 2 +

dM( x )

+ a

(

qa

qa 2!

(q

a

〈 x-a〉 2 +

)

− q b /d 3!

dM( x ) dx

qb 2!

〈 x-b〉 2 +

 〈 x-a〉 3 − 〈 x-b〉 3   

VIGA APOYADA EN LOS EXTREMOS

3.2.1

REACCIONES P⋅b RA = L

RB =

B

C

P⋅a L

x

ESFUERZOS CORTANTES P⋅b P⋅a = cte ; QCB = − = cte QAC = L L MOMENTOS FLECTORES P⋅b P⋅a ⋅ x ; MCB = ⋅ ( L − x) MAC = L L ANGULOS DE GIRO P⋅a⋅b ⋅ ( L + b) ϕA = 6⋅E⋅I⋅L

P

A

CARGA PUNTUAL EN LA VIGA

a

b

Formulario para vigas y pórticos

3.2

L ;

Mmax = MC =

P⋅a⋅b ⋅ ( L + a) ; ϕB = − 6⋅E⋅I⋅L

P⋅a⋅b L

para

x0 = a

P⋅a⋅b ⋅ ( b − a) ; ϕC = 3⋅E⋅I⋅L

QB 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

3.5

M max

3.6

3.2.2

CARGA CONTÍNUA EN PARTE DE LA VIGA

REACCIONES p⋅b⋅c RA = L

c RB =

P

p⋅a⋅c L

A

ESFUERZOS CORTANTES p⋅b⋅c p⋅b⋅c c  − p⋅ − a + x ; QCD = QAC = 2 L L  

;

QDB = −

C

p⋅a⋅c L

a

x0 = a −

para

; ϕB = −

c b⋅c + L 2

p⋅a⋅b⋅c  c2  ⋅L + a−  6⋅E⋅I⋅L  4⋅b

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    

=

 p⋅a⋅c L − x  c2   2 ⋅ ⋅ − ( L − x ) + b ⋅  L + a −  6⋅L 4 ⋅ a   E ⋅ I  

y

CD

y

DB

M max

Prontuario para Cálculo de Estructuras

ANGULOS DE GIRO p⋅a⋅b⋅c  c2  ⋅L + b − ϕA =  6⋅E⋅I⋅L  4⋅a

b L

MDB = Mmax

D

x

MOMENTOS FLECTORES p⋅b⋅c p⋅b⋅c p   c  ⋅ x ; MCD = ⋅ x − ⋅  x −  a −  2 MAC = 2   2  L L

p⋅ a⋅c ⋅ ( L − x) L p⋅b⋅c  b⋅c = ⋅ 2⋅a− c+ L  2 ⋅ L 

B

CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES 1 RA = ( 2 ⋅ p1 + p2 ) 6

;

RB =

1 ( p1 + 2 ⋅ p2 ) . 6

P1

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

;

P2

QB = −RB

B

A

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

x

L2 L2 ⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 ) 2 2   1 1 para x 0 = ⋅  − p1 + ⋅ p12 + p22 + p1 ⋅ p2  3 p2 − p1  

Formulario para vigas y pórticos

3.2.3

L

Mmax comprendido entre 0,125 ⋅

(

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

)

QA QB

; ϕB = −

3

L ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 360 ⋅ E ⋅ I

ECUACION DE LA ELASTICA

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

yx =

0,01304 ⋅

+  

( p1 + p2 ) ⋅ L4 2⋅E⋅I

x0

M max

3.7

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

2

3.8

3.2.4

MOMENTO FLECTOR

REACCIONES

R A = −R B = −

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

a

QA

QB

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

MC M

MC

Prontuario para Cálculo de Estructuras

)

2 M ⋅ L ⋅ (L − x)  a2  L − x    ⋅ 1− 3 ⋅ 2 −    6⋅E⋅I L  L   

B

M = MCizq + MCder

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 3 3 ϕC = ⋅ a +b 3 ⋅ E ⋅ I ⋅ L2

(

+M

3.3.1

P

CARGA PUNTUAL EN LA VIGA

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

Formulario para vigas y pórticos

3.3 VIGA EMPOTRADA EN LOS EXTREMOS

C ;

RB =

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

P ⋅ a2 ⋅ ( L + 2 ⋅ b) L3

;

QCB = −

B

A x a

P ⋅ a2 ⋅ ( L + 2 ⋅ b ) = cte L3

b L

MOMENTOS FLECTORES

P ⋅ a ⋅ b2 P ⋅ a2 ⋅ b P ⋅ b2 M = − M = ⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L) ; ; B AC L2 L2 L3 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

y AC =

P ⋅ b2  2 ⋅ a ⋅ x  x2 ⋅3 ⋅a − x − ⋅ 6⋅E⋅I  L  L2

y BC =

P ⋅ a2  L − x ⋅  (L − x) ⋅  3 ⋅ b − (L − x) − 2 ⋅ b ⋅ 6⋅E⋅I  L  L2

FLECHAS P ⋅ a3 ⋅ b3 3 ⋅ E ⋅ I ⋅ L3 x=

fmax =

2 ⋅a⋅ L L + 2⋅a

MC

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

2

0

3.9

para

;

MB

MA

x

fC =

2

3.10

3.3.2

CARGA CONTÍNUA EN PARTE DE LA VIGA

c

P

REACCIONES p ⋅ b ⋅ c MA − MB RA = − L L

;

p ⋅ a ⋅ c MA − MB RB = + L L

C

c  ; QBD = −RB = cte ; QCD = RA − p ⋅  x − a +  a 

x a

MOMENTOS FLECTORES MAC = RA ⋅ x + MA

;

MBD = RB ⋅ ( L − x ) + MB

MCD = RA ⋅ x + MA − ;

B

A

ESFUERZOS CORTANTES

QAC = RA = cte

D

MA = −

p ⋅ c3 12 ⋅ L2

p  c ⋅x − a+  2  2

b

2

 12 ⋅ a ⋅ b2  ⋅L − 3⋅b +  c2  

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 ⋅ x 3 − 12 ⋅ MA ⋅ x 3  24 ⋅ E ⋅ I   2  1 RB x 3 − 3 ( MB + LRB ) x 2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2  y DB = 6EI  y AC =

MA

MB

Prontuario para Cálculo de Estructuras

12 ⋅ a2 ⋅ b  p ⋅ c3  ⋅ L − 3⋅a+ MB = −  2  12 ⋅ L  c2 

L

CARGA TRAPEZOIDAL EN TODA LA VIGA

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

P1

RA =

P2

ESFUERZOS CORTANTES

QA = RA Qx = RA −

p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L

B

A

Formulario para vigas y pórticos

3.3.3

x

⋅x

L

QB = −RB MOMENTOS FLECTORES L2 ( 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 MA = −

Q

A

Q

B

ECUACION DE LA ELASTICA

yx =

 ( p − p1) 3  x2 ⋅ 2 ⋅ x + p1 ⋅ L ⋅ x 2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L  24 ⋅ E ⋅ I ⋅ L  5 

MA

MB

3.11

3.12

3.3.4

MOMENTO FLECTOR

REACCIONES 6⋅M RA = − 3 ⋅ a ⋅ b L

;

RB =

6⋅M ⋅a⋅b L3

C A

ESFUERZOS CORTANTES

Qx = −

6⋅M ⋅ a ⋅ b = cte L3

a

M⋅ a  b ⋅2 − 3⋅  L  L

MAC =

B

x

MOMENTOS FLECTORES MA =

+M

MB = −

b L

M⋅ b  a ⋅2 − 3⋅  L  L

M⋅ a  a  x  ⋅ 3 ⋅ ⋅ 1− 2 ⋅  − 1 L  L  L 

MCB = −

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

(

QB

)

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

MC

 b⋅ x a ⋅2 ⋅ 2 −  L L 

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

MA MC

MB

Prontuario para Cálculo de Estructuras

MCizq = MA −

QA

3.4.1

P

CARGA PUNTUAL EN LA VIGA

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

(

)

(

C x a

b L

)

(

B

A

)

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

(

)

Q

B

ANGULOS DE GIRO

ϕA =

P ⋅ a ( L − a)

2

; ϕC =

4⋅E⋅I⋅L

P ⋅ a ⋅ ( L − a)

2

3

4⋅E⋅I⋅L

(

⋅ L2 − 2 ⋅ a ⋅ L − a2

)

Q

A

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

y BC =

P ⋅ a ⋅ ( L − x) 12 ⋅ E ⋅ I

2

MB

  a2   a2   L − x   ⋅ 3 ⋅ 1− 2  −  3 − 2  ⋅   L   L   L    

para x=L ⋅

a 2⋅L + a

MC

3.13

FLECHA MAXIMA p ⋅ b2 ⋅ a a fmax = ⋅ 6⋅E⋅I 2⋅L + a

Formulario para vigas y pórticos

3.4 VIGA APOYADA-EMPOTRADA

3.14

3.4.2

CARGA CONTÍNUA EN PARTE DE LA VIGA

REACCIONES p ⋅ b ⋅ c MB RA = + L L

;

c

P

p ⋅ a ⋅ c MB RB = − L L

C

ESFUERZOS CORTANTES

QAC = RA = cte

;

QDB = −RB = cte

;

QCD

c  = RA − p ⋅  x − a +  2 

x

MOMENTOS FLECTORES ;

MAC = RA ⋅ x

MCD

MDB = RB ⋅ ( L − x ) + MB

a

p  c = RA ⋅ x − ⋅  x − a +  2  2 ;

MB = −

2

(L − x)

2

6⋅E⋅I

⋅ RB ⋅ ( L − x ) + 3 ⋅ MB 

QB QA

MB

Prontuario para Cálculo de Estructuras

ECUACION DE LA ELASTICA   x 12 ⋅ a ⋅ b2   y AC = ⋅  −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅  L − 3b +  48 ⋅ E ⋅ I ⋅ L  c2    4   1 c 12ab2    ⋅  −8RALx 3 + 2 pL  x − a +  + pc3  L − 3b + yCD =  x 48 ⋅ E ⋅ I ⋅ L  4 c2    

b L

p⋅a⋅b⋅c  c2  L a ⋅ + −   4⋅b 2 ⋅ L2 

ANGULOS DE GIRO  p ⋅ c3 12 ⋅ a ⋅ b2  ϕA = ⋅  L − 3b +  48 ⋅ E ⋅ I ⋅ L  c2 

y DB = −

B

A

CARGA TRAPEZOIDAL EN TODA LA VIGA

RA =

P2

P1

REACCIONES L M ⋅ ( 2 ⋅ p1 + p2 ) + B 6 L

;

RB =

L M ⋅ ( p1 + 2 ⋅ p2 ) − B 6 L

A ESFUERZOS CORTANTES

Qx = RA −

p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L

L ⋅x

;

QB = −RB

Q

A

MOMENTOS FLECTORES

Mx = RA ⋅ x −

p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 6⋅L

B x

⋅ x2

Formulario para vigas y pórticos

3.4.3

;

MB = −

Q

B

L2 ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 120

ANGULOS DE GIRO

L3 ϕA = ⋅ ( 3 ⋅ p1 + 2 ⋅ p2 ) 240 ⋅ E ⋅ I

MB

ECUACION DE LA ELASTICA

yx =

3.15

x  ( p2 − p1) x4 + 5Lp1x3 − 20RALx2 + 5L 12RAL2 − ( 3p1 + p2 ) L3   120EIL 

3.16

3.4.4

MOMENTO FLECTOR

REACCIONES

RA = −RB =

3 M 2 ⋅ ⋅ L − a2 2 L3

(

)

ESFUERZOS CORTANTES

B

x

Qx = RA = cte

a

MOMENTOS FLECTORES MCder = RA ⋅ a − M ; MAC =

C +M

A

MCizq = RA ⋅ a

3 M⋅ x 2 ⋅ ⋅ L − a2 2 L3

(

)

;

MBC

M ⋅ L2 − 3 ⋅ a2 2 ⋅ L2  M  x  a2  = ⋅ 3 ⋅ ⋅ 1− 2  − 2  2  L  L   ;

(

MB =

b L

)

ANGULOS DE GIRO

 b  a 2  M ; ϕC = ⋅ b ⋅ 3 ⋅ ⋅ 1+  − 4  4⋅E⋅I  L  L  

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 = y BC

(

QA

QB

MC

)

(

MB

)

MC

Prontuario para Cálculo de Estructuras

M ϕA = ⋅ ( L − a) ⋅ ( 3 ⋅ a − L ) 4⋅E⋅I⋅L

3.5.1

CARGA PUNTUAL EN LA VIGA

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 L

MOMENTOS FLECTORES

MAC = 0

;

b

MCB = −P ⋅ ( x − a)

;

MB = −P ⋅ b

ANGULOS DE GIRO

ϕ A = ϕC = −

P ⋅ b2 2⋅E⋅I

QB

ECUACION DE LA ELASTICA

y

AC

=

P ⋅ b2 ⋅ ( 3 ⋅ ( L − x ) − b) 6⋅E⋅I

;

y

CB

=

MB

3.17

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

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

3.18

3.5.2

CARGA CONTÍNUA EN PARTE DE LA VIGA

REACCIONES . RB = p ⋅ c

c

ESFUERZOS CORTANTES . c  QAC = 0 ; QCD = − p ⋅  x − a +  2 

P ; QDB = − p ⋅ c = cte

; ϕC = −

x p ⋅ c2 MD = − 2

p ⋅ c  2 c2  ⋅b +  2⋅E⋅I  12 

a

b L

; ϕ A = ϕC

y DB =

  p⋅c p⋅c  c2  3 ⋅ L − x 2 ⋅ ( 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   

)

Q

B

FLECHAS . 2

fD =

p⋅ c  c ⋅ b−  2 E ⋅ I 

b c  ⋅ +   3 12 

2   p ⋅ c  c p⋅ c   c2  fC = ⋅  b +  ⋅ ( 4 ⋅ b − c) + c3  ; fA = ⋅ a ⋅  3 ⋅ b2 +  + 2 ⋅ b3  12 ⋅ E ⋅ I  2 6 4 E I ⋅ ⋅     

M

B

Prontuario para Cálculo de Estructuras

ECUACION DE LA ELASTICA .

(

D

C

B

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  ⋅b − ϕD = −  2⋅E⋅I  4 

A

CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES 1 RB = ( p1 + p2 ) 2 ESFUERZOS CORTANTES

p − p1 x 2 ⋅ − p1 ⋅ x Qx = − 2 L 2

A ;

L QB = − ( p1 + p2 ) 2

x2 ⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1 6⋅L 

B x

MOMENTOS FLECTORES

Mx = −

P2

P1

Formulario para vigas y pórticos

3.5.3

;

MB = −

L2 ⋅ ( p2 + 2 ⋅ p1 ) 6

L

ANGULOS DE GIRO

ϕA = −

L3 ⋅ ( 3 ⋅ p1 + p2 ) 24 ⋅ E ⋅ I

ECUACION DE LA ELASTICA 3  2  ( L − x) 2 L − x ) − ( p2 − p1 ) + ( L − x ) p2 −  ( yx = 5L   24EI  2  L L x p p L p p 2 2 2 − − + + + ( )( ) ( ) 2 1 2 1  

FLECHA

120 ⋅ E ⋅ I

MB

3.19

fA =

L4 ⋅ ( 4 ⋅ p2 + 11⋅ p1 )

QB

3.20

3.5.4

MOMENTO FLECTOR

REACCIONES

M

A

B

RB = 0 ESFUERZO CORTANTE

x a

Qx = 0

L

MOMENTOS FLECTORES

MAC = 0

;

MCB = − M = cte

b

;

MAC = − M

ANGULOS DE GIRO

ϕC = ϕ A = −

ECUACION DE LA ELASTICA y AC =

M ⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b) 2⋅E⋅I

;

y BC =

M 2 (L − x) 2⋅E⋅I

FLECHA

fC =

M ⋅ b2 2⋅E⋅I

;

fA =

M ⋅ b ⋅ ( 2 ⋅ L − b) 2⋅E⋅I

MB

Prontuario para Cálculo de Estructuras

M⋅ b E⋅I

P

P

A

P

B

L/2

C

L/2 L

A

L

A

L/2 L

0,688 P

0,312 P

L

0,405 P

B

C

0,094 P

0,094 P

B

A

0,312 P

0,688 P

C

B

L/2

L/2

L/2

Formulario para vigas y pórticos

3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES

C

0,594 P

ESFUERZOS CORTANTES

ESFUERZOS CORTANTES

- 0,188 PL - 0,094 PL A

B

0,156 PL

C

0,156 PL

B

C

0,203 PL MOMENTOS FLECTORES

3.21

MOMENTOS FLECTORES

A

3.22

Q

Q

A

L

B

Q

C

L

0,625 QL

A

0,375 L

B

L

L

C

0,437 QL

0,375 QL

0,063 QL B

A

C

A

0,375 QL

0,375 L

B

0,563 QL

0,437 L

0,625 QL

ESFUERZOS CORTANTES

2

2

- 0,063 QL

- 0,125 QL

B 2

C 2

0,07 QL

0,07 QL MOMENTOS FLECTORES

A

B 2

0,096 QL

MOMENTOS FLECTORES

C

Prontuario para Cálculo de Estructuras

ESFUERZOS CORTANTES

A

C

Q

A

L

Q

B

C

k L

c QL d L

a QL

A

C

B

b QL

d QL

a L ESFUERZOS CORTANTES

2

f QL

A

B

2

g QL

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

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

3.23

MOMENTOS FLECTORES

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

C

2

e QL

Relación entre luces

Formulario para vigas y pórticos

3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES

3.24

Q

Q

A

B

L

Relación entre luces

C

k L

c QL d L A C

B

a QL d QL

b QL

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

f QL

k 2 − k +1 8 k f d= − 2 k f=

A

B

MOMENTOS FLECTORES

C

2

g QL

a = 0.5 − f e=

a2 2

b = 0.5 + f g=

d2 2

Prontuario para Cálculo de Estructuras

ESFUERZOS CORTANTES

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

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

Formulario para vigas y pórticos

3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES

C

A

c QL

b QL

a L

a QL

ESFUERZOS CORTANTES

2

f QL

2

2

f QL

f=

k3 + 1 12 ⋅ k + 8

a = 0.5 − f

c=

k 2

e=

a2 2

b = 0.5 + f

g=

k2 −f 8

g QL A

B

C

D 2

2

e QL

e QL

3.25

MOMENTOS FLECTORES

3.26

Q

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

2

f QL

B 2

e QL

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

ESFUERZOS CORTANTES

C 2

g QL

MOMENTOS FLECTORES

f=

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

Prontuario para Cálculo de Estructuras

ESFUERZOS CORTANTES

Relación entre luces

k=

I2 h ⋅ I1 l

3.9.1

y

N = 3 + 2k a

s

p

CARGA REPARTIDA VERTICAL

B

C I

2

x

REACCIONES

m

VA VD

psn = l psm = l

HA = HD =

Formulario para vigas y pórticos

3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

I

n I

1

A

h

1

D

s  3 ps   mn −  2hlN  12  2

l

MOMENTOS FLECTORES

MB

MC

3 ps  s2  MB = MC = − ⋅  mn −  2 lN  12  En S Mx = VA ⋅ x −

p(x − m)2 − HA ⋅ h 2

HA

HD VD

3.27

VA

3.28

3.9.2

CARGA REPARTIDA HORIZONTAL

REACCIONES

VA = VD = HD = HA =

p

B

C I

ph2 2l

ph ( 2N + k )

I

2

I

1

8N

h

1

y

ph ( 6N − k )

A

8N

D l

MOMENTOS FLECTORES MB

MY =

py(h − y) y + ⋅ MB h 2

MC

MB

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

ph2 ( 2N − k ) 8N ph2 MC = − ( 2N + k ) 8N En AB MB =

Formulario para vigas y pórticos

3.9.3

CARGA PUNTUAL VERTICAL SOBRE DINTEL

REACCIONES

n

B

Pn VA = l Pm VD = l HA = HD =

P

m

C I

I

2

I

1

h

1

3 Pmn 2 lhN A

D

MOMENTOS FLECTORES

l

3 Pmn MB = MC = − ⋅ 2 lN 2N − 3 MP = Pmn 2lN

MB

MC

MP

HD

HA VD

3.29

VA

3.30

3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO k1 =

I3 h1 ⋅ I1 s

y

k2 =

I3 h 2 ⋅ I2 s

p

3.10.1 CARGA REPARTIDA VERTICAL

C s

REACCIONES

f I

B

pl 2 h1 + h2 pl 2 HA = HD = 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2

VA = VD =

3

I

x I

h1

h2

2

1

A

D l

MB = −

( h1 + h2 ) h1 pl2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

MC = −

( h1 + h2 ) h2 pl 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

En

BC

MX =

px(l − x) f  − HA  x + h1  2 l  

MC

MB

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

MOMENTOS FLECTORES

Formulario para vigas y pórticos

3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR

C

REACCIONES

ph12 VA = VD = 2l HA = ph1 − HD

s

f

p

I

3

B I

h1

h1 ( 4 + 5k 1 ) + 2h2 ph12 HD = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

I 1

h2

2

y

A

D l

MOMENTOS FLECTORES MC

h1 ( 4 + 5k 1 ) + 2h2 ph12 ph3 − 1 2 MB = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC =

MB

h1 ( 4 + 5k 1 ) + 2h2 ph12 h2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

En AB MY = HA y −

py 2 2

HD

HA VD

3.31

VA

3.32

3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

p

REACCIONES

VA = VD =

pf ( h1 + h2 )

f I

2l

B

HA = pf − HD

HD =

pf 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1 2 1

C s

I

h1

y

3

I

h2

2

1

A

D l

MOMENTOS FLECTORES

MC = −

2 pfh1 8h1 (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2

MC

MB

ph2 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1 2 1

En BC l py 2 MY = −VA y + HA ( y + h1 ) − f 2

HA

HD VA

VD

Prontuario para Cálculo de Estructuras

MB = pfh1 −

C

REACCIONES

Pb VA = l Pa VD = l h1(l + b) + h2 (l + a) Pab HA = HD = 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

s

f I

3

B I

h1

I a

b

h2

2

Formulario para vigas y pórticos

3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL

1

A

D l

MOMENTOS FLECTORES

MB = −

h1 ( l + b ) + h2 ( l + a ) Pabh1 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

MC

MB

h1 ( l + b ) + h2 ( l + a ) Pabh2 MC = − 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MP =

Pab  af  + HA  + h1  l  l 

MP

HD

HA VA

VD

3.33

3.34

3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS k=

I2 h ⋅ I1 s

p

3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

C s

REACCIONES

I

pl 2 8h + 5f pl 2 HA = HE = 2 32 h ( 3 + k ) + f ( 3h + f )

B

VA = VE =

D

I

I

1

A

h

1

E l

pl 2 h 8h + 5f 2 32 h ( 3 + k ) + f ( 3h + f )

MC

2

pl f+h MB + 8 h

MB

MD

En BC y DC MX = p

x (l − x) 2

+

MB  2fx  h+  h  l 

HE

HA VA

VE

Prontuario para Cálculo de Estructuras

MC =

f 2

x

MOMENTOS FLECTORES

MB = MD = −

I

2

p

REACCIONES

C

pl VA = 3 8 pl VE = 8 HA = HE =

s

I B

I

2

I

I

1

A l

pl 2 h 8h + 5f 2 64 h ( 3 + k ) + f ( 3h + f )

MC

pl 2 f + h + MC = MB 16 h En BC x (l − x) 2

+

MB h

h

1

E

MOMENTOS FLECTORES

MX = p

D

x

8h + 5f pl 2 2 64 h ( 3 + k ) + f ( 3h + f )

MB = MD = −

f 2

Formulario para vigas y pórticos

3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

MB

MD

2fx   h+ l    HE VA

VE

3.35

HA

3.36

3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES C

ph2 VA = VE = 2l HA = ph − HE HE =

s

I

p

I

2

f 2

D

B

( 5k + 12 ) h + 6f ph 2 16 h ( k + 3 ) + f ( f + 3h) 2

I

1

I y

A

h

1

E

MOMENTOS FLECTORES l

ph2 + MD MB = 2 ph2 f + h + MC = MD h 4 ( 5k + 12 ) h + 6f ph3 MD = − 2 16 h ( k + 3 ) + f ( f + 3h)

MC

MD

En AB My = −

py 2 + HA ⋅ y 2

HA

HE VA

VE

Prontuario para Cálculo de Estructuras

MB

REACCIONES

p

pf VA = VE = ( f + 2 h) 2l HA = pf − HE

s

I

I

2

f 2

D

B

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

HE =

C

Formulario para vigas y pórticos

3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

y I

I

1

A

MOMENTOS FLECTORES

x

h

1

E l

MB = HA ⋅ h MC = −

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

MC

MD = −HE ⋅ h MB

MD

En BC Mx = HA ⋅ y − VA ⋅ x − p

2

2

HE

HA VA

VE

3.37

f siendo y = x + h l

( y − h)

3.38

3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL p REACCIONES

Pn l Pm VA = l

C s

VA =

I B

(

)

I

2 2 Pm 6hln+ f 3l − 4m HA = HE = 2 2 4 l h ( k + 3 ) + f ( f + 3 h)

I

2

m

n

1

A

f 2

D I

h

1

E l

MOMENTOS FLECTORES

MB = MD = −HA ⋅ h

MC

MB

MD

HE

HA VA

VE

Prontuario para Cálculo de Estructuras

Pm h + f + MC = MB h 2 hl + 2fm MP = VA ⋅ m − HA l

k1 =

I3 h1 ⋅ I1 l

y

k2 =

I3 h 2 ⋅ I2 l

3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL p B

C

REACCIONES

VA =

I

3

x 2 1

h −h pl pl + 2 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1

I

2 2

h1

I

1

h2

2

Formulario para vigas y pórticos

3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL

D

2 2

h −h pl pl VD = − 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

A

h1 − h2 pl 2 HA = HD = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

l

MOMENTOS FLECTORES MB

( h1 + h2 ) h1 pl 2 MB = − 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC = −

MC

( h1 + h2 ) h2 pl 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

HD

En BC

VD HA VA

3.39

px 2 Mx = VA ⋅ x − − HA ⋅ h1 2

3.40

3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES

p

B

C

2 1

ph h − h2 VA = VD = − HD 1 2l l HA = ph − HD HD =

ph12 5k1h1 + 4h1 + 2h2 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2

I

3

I h1

I

1

h2

2

D y

A

MOMENTOS FLECTORES l

ph2 ph3 5k1h1 + 4h1 + 2h2 MB = − 1 − 1 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MB

ph12 h2 5k1h1 + 4h1 + 2h2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

MC

MB

En AB My = HA ⋅ y −

py 2 2

HD VD HA VA

Prontuario para Cálculo de Estructuras

MC = −

P

a

REACCIONES

( l + b ) h1 + ( l + a) h2 Pb Pab VA = h1 − h2 + 3 2 l 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

(

( l + b ) h1 + ( l + a) h2 Pa Pab VD = h1 − h2 − 3 2 l 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

(

b

B

C I

) )

3

I h1

I

1

h2

2

Formulario para vigas y pórticos

3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL

D

A

HA = HD =

( l + b ) h1 + ( l + a) h2 Pab 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2

l

MOMENTOS FLECTORES

MB = −

MC = −

MB

( l + b) h1 + ( l + a) h2 Pabh1 2 2 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2l

MC

MP HD

( l + b ) h1 + ( l + a) h2 Pabh2 2 2 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2l

VD HA VA

3.41

MP = VA ⋅ a + MB

3.42

3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL k=

I2 h ⋅ I1 l

3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

p B

C I

VA = VD =

pl 2

HA = HD =

pl 2 4h ( k + 2 )

I

I

1

h

1

A

MOMENTOS FLECTORES

D l

pl2 6 ( k + 2)

MB

MC

En BC Mx =

px ( l − x ) 2

Mmáx pos =

−

pl 2 6( k + 2)

pl 2 3k + 2 l para x = 24 k + 2 2

HA

MA

VA

MD

HD VD

Prontuario para Cálculo de Estructuras

pl2 MA = MD = 12 ( k + 2 ) MB = MC = −

2

x

REACCIONES

p

REACCIONES

B

C

2

ph k VA = VD = l ( 6k + 1)

I

HA = ph − HD HD =

I

ph ( 2k + 3 )

A

ph2 MC = − 24 MD =

ph2 24

l

2 2    3 − 6k + 1 − k + 2   

MC

MB MB

2 1    3 + 6k + 1 − k + 2   

En AB My = −

D

2 1    5 + 6k + 1 + k + 2   

ph2  2 2  1− +  24  6k + 1 k + 2 

h

1

y

MOMENTOS FLECTORES

MB =

I

1

8 ( k + 2)

ph2 MA = − 24

2

Formulario para vigas y pórticos

3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR

py 2 + HA ⋅ y + MA 2

MA

HA VA

MD

HD VD

3.43

3.44

3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES

VA =

P

m

Pn  m ( n − m)  1+ 2  l  l ( 6k + 1) 

n

B

C I

2

VD = P − VA HA = HD =

3Pmn 2lh(k + 2)

I

MOMENTOS FLECTORES

MA =

I

1

A

D

Pmn  1 n− m  −   2l  k + 2 l ( 6k + 1)  Pmn  1 n− m  +   l  k + 2 2l ( 6k + 1) 

MC = −

Pmn  1 n− m  −   l  k + 2 2l ( 6k + 1) 

MD =

Pmn  1 n− m  +   2l  k + 2 l ( 6k + 1) 

MP =

Pmn nMB mMC + + l l l

l

MB

MC

MP

HA

MA

VA

MD

HD VD

Prontuario para Cálculo de Estructuras

MB = −

h

1

Formulario para vigas y pórticos

3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES

3Phk VA = VD = l(6k + 1) P HA = HD = 2 MOMENTOS FLECTORES

Ph 3k + 1 2 6k + 1 Ph 3k MB = − MC = 2 6k + 1 Ph 3k + 1 MD = 2 6k + 1

P

B

C I

I

2

I

1

h

1

A

D

MA = −

l

MB

MA

MC

HA

HD VD

3.45

VA

MD

3.46

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

I2 h ⋅ I1 s

p

3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

C s

REACCIONES

I

pl 2 k ( 4h + 5f ) + f pl 2 HA = HE = 8 ( kh + f )2 + 4k h2 + hf + f 2

B

VA = VE =

(

f 2

D

x I

)

I

1

h

1

A

E l

MOMENTOS FLECTORES

pl2 kh ( 8h + 15f ) + f ( 6h − f ) 48 ( kh + f )2 + 4k h2 + hf + f 2

(

kh (16h + 15f ) + f 2 pl2 MB = MD = − 48 ( kh + f )2 + 4k h2 + hf + f 2

(

pl 2 + MA − HA ( h + f ) 8 En BC

MC

) MB

)

MD

MC =

2 xf  px  Mx = MA + VA ⋅ x − HA  h + − 2 l  

2

HA

MA

VA

ME

HE VE

Prontuario para Cálculo de Estructuras

MA = ME =

I

2

p

REACCIONES

pl − VE 2 4k + 1 VE = 3 pl 32 ( 3k + 1) k ( 4h + 5f ) + f pl 2 HA = HE = 16 ( kh + f )2 + 4k h2 + hf + f 2

C

VA =

(

s

I B

)

I

l

MD = −

kh (16h + 15f ) + f 2 pl 2 pl2 + 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1)

En BC

MC

)

kh (16h + 15f ) + f 2 pl 2 pl 2 − 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1)

(

(

MB

)

)

HA

MD

MA

VA

ME

HE VE

3.47

2 xf  px 2  Mx = MA + VA ⋅ x − HA  h + − l  2  l MC = VE + ME − HE ( f + h) 2

h

1

E

pl 2 kh ( 8h + 15f ) + f ( 6h − f ) pl 2 + 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1)

MB = −

I

1

)

(

D

A

pl 2 kh ( 8h + 15f ) + f ( 6h − f ) pl 2 MA = − 2 96 ( kh + f ) + 4k f 2 + fh + h2 64 ( 3k + 1) ME =

f 2

x

MOMENTOS FLECTORES

(

I

2

Formulario para vigas y pórticos

3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

3.48

3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES

VA = VE =

ph2 k 2l ( 3k + 1)

C s

HA = ph − HE HE =

k 2 h + k ( 2 f + 3 h) + f ph2 4 ( kh + f )2 + 4k f 2 + fh + h2

(

)

I

f 2

D

1

I y

E

2  2  ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f 2k + 1 + 6 24  ( kh + f )2 + 4k f 2 + fh + h2 3k + 1  

(

l

)

MC

(

MD

)

MA

HA VA

ME

HE VE

Prontuario para Cálculo de Estructuras

MB

2 2   ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f 2k + 1 − + 6 2 24  3k + 1 ( kh + f ) + 4k f 2 + fh + h2   En AB py 2 My = MA + HA ⋅ y − 2

h

1

A

ph2 MB = MA + HA ⋅ h − 2 1 MC = ME − HE ( f + h) + VE 2 MD = ME − HE ⋅ h ME =

I

2

B

MOMENTOS FLECTORES

MA = −

I

p

REACCIONES

p

3 pf 4k ( f + h) + f 8 l 3k + 1 HA = pf − HE

C

VA = VE =

HE =

s

I

2 pf 2k h ( k + 4 ) + f (10kh + 5kf + f ) 4 ( kh + f )2 + 4k f 2 + fh + h2

(

)

I

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

y

I

(

E

(

l

)

l 2

h

1

A

MC

MB

  kh ( 9f + 4h) + f ( 6h + f ) 3 4h ( 3k + 2 ) + f  pf  −f +  24  ( kh + f )2 + 4k f 2 + fh + h2 2 3k + 1   En BC 2 l ( y − h) p ( y − h) − My = MA + HA ⋅ y − VA 2f 2 ME =

D

1

  kh ( 9f + 4h) + f ( 6h + f ) pf  3 4h ( 3k + 2 ) + f  + f  24  ( kh + f )2 + 4k f 2 + fh + h2 2 3k + 1  

MB = MA + HA ⋅ h

f 2

B

MOMENTOS FLECTORES

MA = −

I

2

Formulario para vigas y pórticos

3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

MD

)

MA

HA

HE VE

3.49

VA

ME

3.50

3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES

p

VA = P − VE 2 Pm 3l ( kl + m) − 2m VE = 3 3k + 1 l 2 2 Pm 3kl ( f + h) − 4fm ( k + 1) + 3lm ( f − kh) HA = HE = 2 2 l ( kh + f ) + 4k f 2 + fh + h2

(

C s

I B

)

I

MOMENTOS FLECTORES

m

f 2

D

n

I

1

E

)

l

MC

MB

(

)

HA

MD

MA

VA

ME

HE VE

Prontuario para Cálculo de Estructuras

MB = MA − HA ⋅ h l MC = ME + VE − HE ( h + f ) 2 MD = ME − HE ⋅ h  3flh ( kl + 2m) − 4fm2 ( kh + 2h + f ) + 2kh2 ln+ f 2 l ( 4m − l )    2 Pm  kh + f ) + 4k f 2 + fh + h2  ( ME = 2   2l  n n − m  ( )  + 3k + 1   En BC 2fm   My = MA + VA ⋅ m − HA  h + l  

h

1

A

 3flh ( kl + 2m) − 4fm2 ( kh + 2 h + f ) + 2kh2 ln+ f 2 l ( 4m − l )    2 Pm  kh + f ) + 4k f 2 + fh + h2  ( MA = 2   2l  n n − m  ( ) −  3k + 1  

(

I

2