formulario para vigas y porticos

7/15/2019 Formulario Para Vigas y Porticos

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3 FORMULARIO PARA

VIGAS Y PÓRTICOS



Formulario para vigas y pórticos 3.1

3.1

Obtención de la Distribución de Solicitaciones mediante laFormulación de Macaulay

Las Funciones de Macaulay permiten expresar tanto la distribución de cargassobre una viga sometida a flexión como las leyes de Cortantes o MomentosFlectores 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 a M 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 decarga y se indica el valor que deberían tomar los parámetros A y c en la ecua-



3.2 Prontuario para Cálculo de Estructuras

M

M(x)

a x

( )

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

≤ − =

≥ − =

= − −

=

=

P

M(x)

a x

T(x)

( )

( )

( )

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

≤ − =≥ − = −

= − −

= − −

=

=



Limitación de las Deformaciones 3.3

2M(x)

q

x

a

T(x)

( )

( )

( )

( )

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

q M x x a

A q

c

≤ − =

≥ − = −

= −

= − −

= − −⋅

=

=

3

a

x

d

q

2T(x)

M(x)

( )

( )

( )

( )

3

3 3

1

2

3

Si

0

entonces

1

2 1

3 2 1

por lo tanto

3

x a x a

x a x a x a

q dq x x a

q dT x x a

q d M x x a

q A

d

c

≤ − =

≥ − = −

= −

= − −⋅

= − −⋅ ⋅

=

=



3.4 Prontuario para Cálculo de Estructuras

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

x

a

b

q

( )

( )( )

2 2qM x x-a x-b

2!dM x

T xdx

= −⟨ ⟩ + ⟨ ⟩

=

q

a

x

b

d

q/d

( )

( )( )

3 3 2q/d qM x - x-a x-b x-b

3! 2!dM x

T xdx

= ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩

=

q

a

x

b

d

q/d

( )

( )( )

2 3 3q q/dM x x-a x-a x-b2! 3!

dM xT x

dx

= − ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩

=

a

x

b

d

aq b

q

( )

( )

( )( )

a b2 2

b a 3 3

q qM x x-a x-b

2! 2!q q /d

x-a x-b3!

dM xT x

dx

= − ⟨ ⟩ + ⟨ ⟩ +

− + −⟨ ⟩ + ⟨ ⟩

=

a

x

b

d

a

q

bq( )

( )

( )( )

a b2 2

a b 3 3

q qM x x-a x-b

2! 2!q q /d

x-a x-b3!

dM xT x

dx

= − ⟨ ⟩ + ⟨ ⟩ +

− + ⟨ ⟩ − ⟨ ⟩

=



F or m ul ar i o p ar avi g a s y p ó r t i c o s

3 . 5

3.2 VIGA APOYADA EN LOS EXTREMOS

3.2.1 CARGA PUNTUAL EN LA VIGAREACCIONES

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 a P a b M x M L x M M x a

L L L

⋅ ⋅ ⋅ ⋅= ⋅ = ⋅ − = = =

ANGULOS DE GIRO( ) ( ) ( ); ;

6 6 3 A 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 ; 16 6 AC CB

P L a L xP L b x b x a L xy y

E I E I LL L L

⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − = ⋅ − − = ⋅ − − ⋅ ⋅ ⋅ ⋅

FLECHA MAXIMA

( ) 2 232 2 2 para39 3

CP b L bf L b x

E I L⋅ −= ⋅ − =

⋅ ⋅ ⋅

x

a

A

L

b

C

PB

QA

maxM

BQ



3 . 6

P r ont u ar i o p ar a C á

l c ul o d eE s t r u ct ur a s

3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA

REACCIONES

A B

p b c p a cR R

L L

⋅ ⋅ ⋅ ⋅= =

ESFUERZOS CORTANTES

; ;2 AC CD DB

p b c p b c c p a cQ Q p a x QL L L

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = − ⋅ − + = −

MOMENTOS FLECTORES

( )

2

max 0

;2 2

2 para2 2

AC CD

DB

p b c p b c p c M x M x x a

L L

p a c M L x

L

p b c b c c b c M a c x a

L L L

⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅ − ⋅ − −

⋅ ⋅

= ⋅ −

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ − + = − +

⋅

ANGULOS DE GIRO2 2

;6 4 6 4 A B

p a b c c p a b c cL b L a

E I L a E I L bϕ ϕ

⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ + − = − ⋅ + −

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅


( )

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

⋅ ⋅= ⋅ − + ⋅ + −

⋅ ⋅ ⋅ = ⋅ ⋅ − − − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ −= ⋅ ⋅ − − + ⋅ + −

⋅ ⋅ ⋅

QA

BQ

x

a

A

L

b

C

P

B

c

maxM

D




3 . 7

3.2.3 CARGA TRAPEZOIDAL EN TODA LA VIGA

REACCIONES

( ) ( )1 2 1 2

1 12 ; 2

6 6 A BR p p R p p= ⋅ + = + ⋅ .

ESFUERZOS CORTANTES( )1 2 23

; ;6 A A x A B B

p L x p x

Q R Q R x Q RL

⋅ − + ⋅

= = − ⋅ = −⋅

MOMENTOS FLECTORES( )

( ) ( )

( )

1 2 2

2 2

max 1 2 1 2

2 2

0 1 1 2 1 22 1

3

6

comprendido entre 0,125 y 0,1282 2

1 1para x

3

x A

p L x p x M R x x

L

L L M p p p p

p p p p pp p

− + ⋅= ⋅ − ⋅

⋅

⋅ ⋅ + ⋅ ⋅ +

= ⋅ − + ⋅ + + ⋅

−

ANGULOS DE GIRO

( ) ( )3 3

1 2 1 28 7 ; 7 8360 360 A B

L Lp p p p

E I E Iϕ ϕ = ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅

⋅ ⋅ ⋅ ⋅


( ) ( ) ( )

( ) ( )

3 21 2 1 2

2 31 2 1 2

3 3 4

360 8 7 8 7 x

p p x p p Lx x 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

0 x

L

x



3 . 8



3.2.4 MOMENTO FLECTOR

REACCIONES

R R M

L A B

= − = −

ESFUERZOS CORTANTES

x

M

Q cteL= =

MOMENTOS FLECTORES

( )

izq der C CM M M

AC CB

izq der C C

M M M x M L x

L L

M M M 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 aE I E IL L

Ma b

E I L

ϕ ϕ

ϕ

⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅

= ⋅ +⋅ ⋅ ⋅

ECUACION DE LA ELASTICA2 2

2 2

22

2

1 36

( ) 1 36

AC

CB

M L x b xy

E I L L

M L L x a L xyE I LL

⋅ ⋅= − ⋅ − ⋅ −

⋅ ⋅

⋅ ⋅ − − = − ⋅ − ⋅ − ⋅ ⋅

FLECHA

( )3C

M a bf b a

E I L

⋅ ⋅= ⋅ −

⋅ ⋅ ⋅

BA

M+

QA Q

B

MC

MC

M

a

L

b

C



F or m ul ar i o p ar avi g a s y

p ó r t i c o s

3 . 9

3.3 VIGA EMPOTRADA EN LOS EXTREMOS

3.3.1 CARGA PUNTUAL EN LA VIGA

REACCIONES

( ) ( )2 2

3 32 ; 2 A B

P b P aR L a R L b

L L

⋅ ⋅= ⋅ + ⋅ = ⋅ + ⋅

ESFUERZOS CORTANTES

( ) ( )2 2

3 32 ; 2 AC CB

P b P aQ L a cte Q L b cte

L L

⋅ ⋅= ⋅ + ⋅ = = − ⋅ + ⋅ =

MOMENTOS FLECTORES

( )

( )

2 2 2

2 2 3

2 2 2

2 03 3

; ; 2

22 ; para

A B AC

BC C

P a b P a b P b M M M L x a x a L

L L L

P a P a b M L b L L x b x M x aL L

⋅ ⋅ ⋅ ⋅ ⋅= − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅

⋅ ⋅ ⋅ ⋅= ⋅ ⋅ + − ⋅ − ⋅ ⋅ = =


( )( )

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 L x

L a

⋅ ⋅ ⋅ ⋅ ⋅= =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅

⋅ ⋅=

+ ⋅

MC

A B

a

L

x

b

P

B

AQ

Q

x

0

MA

C

MB



3 .1 0




REACCIONES

; A B A B A 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; 312

123

12

AC A A CD A A

BD B B A

B

p c M R x M M R x M x a

p c a b M R L x M M L bL c

p c a b M L a

L c

= ⋅ + = ⋅ + − ⋅ − +

⋅ ⋅ ⋅= ⋅ − + = − ⋅ − ⋅ + ⋅

⋅ ⋅ ⋅= − ⋅ − ⋅ +

⋅


( )

( ) ( ) ( )

2

4

3 3

3 2 2

36

1 4 1224 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 xE I

y R x M LR x M LR Lx M LR LEI

= ⋅ − ⋅ − ⋅⋅ ⋅

= ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅

= − + + + − +

a

MA

QA

x

A

BQ

L

b

B

P

c

C D

MB




p ó r t i c o s

3 .1 1


REACCIONES

( )

( )

1 2

1 2

26

26

A B A

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

( )

( )

( )

21 2

1 2 2

2

1 2

3 260

3

6

2 360

A

x A A

B

L M p p

p L x p x M R x M x

L

L M p p

= − ⋅ + ⋅

⋅ ⋅ − + ⋅= ⋅ + − ⋅

⋅

= − ⋅ + ⋅


( )22 1 3 2

1 4 1224 5 x A A

p p xy x p L x R L x M L

E I L

−= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅

⋅ ⋅ ⋅

AQ

QB

x

A

L

B

BMAM

P2

P1



3 .1 2




REACCIONES

3 3

6 6; A B

M MR a b R a b

L L

⋅ ⋅= − ⋅ ⋅ = ⋅ ⋅

ESFUERZOS CORTANTES

36 x MQ a b cteL

⋅= − ⋅ ⋅ =

MOMENTOS FLECTORES

( )

⋅ ⋅ = ⋅ − ⋅ = − ⋅ − ⋅

⋅ = ⋅ ⋅ ⋅ − ⋅ −

⋅ − = − ⋅ ⋅ ⋅ − ⋅ −

⋅= − ⋅ ⋅ = + ⋅ − ⋅ ⋅2 3 2

3 3

2 3 2 3

3 1 2 1

3 1 2 1

6; 6

A B

AC

CB

izq der C A C A

M a b M b a M M

L L L L

M a a x M

L L L

M b b L x ML L L

M M M M a b M M L a b

L L


( )

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 ayE I L LL

⋅ ⋅ − = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅

⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅

FLECHA

( )2 2

32C

M a bf a b

E I L

⋅ ⋅= − ⋅ −

⋅ ⋅ ⋅

MC

AQ Q

B

x

A

L

a b

B+M

C

M

C

AM

BM




p ó r t i c o s

3 .1 3

3.4 VIGA APOYADA-EMPOTRADA


REACCIONES

( ) ( )2

2 23 33 ; 3

2 2 A B

P b P aR L b R L a

L L

⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ −

⋅ ⋅

ESFUERZOS CORTANTES

( ) ( )2

2 23 33 ; 3 .

2 2 AC CB

P b P aQ L b cte Q L a const

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 a M L a M b a b

L L

P x P a M b a b M L L x a x

L L

⋅ ⋅= − ⋅ − = ⋅ ⋅ ⋅ + ⋅

⋅ ⋅⋅ ⋅

= ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅⋅ ⋅

ANGULOS DE GIRO

( ) ( )( )

2 2

2 23; 2

4 4 A C

P a L a P a L aL a L a

E I L E I Lϕ ϕ

⋅ − ⋅ ⋅ −= = ⋅ − ⋅ ⋅ −

⋅ ⋅ ⋅ ⋅ ⋅ ⋅


( )

( )

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 MAXIMA2

para x=6 2 2max

p b a a af L

E I L a L a

⋅ ⋅= ⋅ ⋅

⋅ ⋅ ⋅ + ⋅ +

Q

Q

MB

A

B

x

a

L

b

A BC

P

MC



3 .1 4




REACCIONES

;B B A B

p b c M p a c MR R

L L L L

⋅ ⋅ ⋅ ⋅= + = −

ESFUERZOS CORTANTES

; ;2 AC 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 c M R x M R x x a

p a b c c M R L x M M L a

bL

= ⋅ = ⋅ − ⋅ − +

⋅ ⋅ ⋅= ⋅ − + = − ⋅ + −

⋅⋅

ANGULOS DE GIRO3 2

2

123

48 A

p c a bL b

E I L cϕ

⋅ ⋅ ⋅= ⋅ − +

⋅ ⋅ ⋅


( )( )

22 3

2

4 23 3

2

2

128 3

48

1 128 2 3

48 4

36

AC A

CD A

DB B B

x a by R L x p c L b

E I L c

c aby R Lx pL x a pc L b x

E I L c

L xy R L x M

E I

⋅ ⋅= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − +

⋅ ⋅ ⋅

= ⋅ − + − + + − + ⋅ ⋅ ⋅

− = − ⋅ ⋅ − + ⋅ ⋅ ⋅

Q

Q

a

x

C

P

M

A

A

B

B

L

b

c

B




p ó r t i c o s

3 .1 5


REACCIONES

( ) ( )1 2 1 22 ; 26 6

B B A B

L M L MR p p R p p

L L= ⋅ ⋅ + + = ⋅ + ⋅ −

ESFUERZOS CORTANTES

( )1 22;

2 x A B B

p L x p xQ R x Q R

L

⋅ ⋅ − + ⋅= − ⋅ = −

⋅

MOMENTOS FLECTORES

( )( )

21 2 2

1 2

3; 7 8

6 120 x A B

p L x p x L M R x x M p p

L

⋅ ⋅ − + ⋅= ⋅ − ⋅ = − ⋅ ⋅ + ⋅

⋅

ANGULOS DE GIRO

( )3

1 23 2240 A

Lp p

E Iϕ = ⋅ ⋅ + ⋅

⋅ ⋅


( ) ( )4 3 2 2 32 1 1 1 25 20 5 12 3

120 x A A

xy p p x Lp x R Lx L R L p p L

EIL = − + − + − +

2

A

Q Q

L

x

B

BA

1PP

MB



3 .1 6

P r ont u ar i o p ar a C á l c ul o d eE s t r u ct ur a s


REACCIONES

( )2 23

3

2 A 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

M M R a M M R a M L a

L

M x M x a M L a M

LL L

= ⋅ − = ⋅ = ⋅ − ⋅⋅

⋅= ⋅ ⋅ − = ⋅ ⋅ ⋅ − −

ANGULOS DE GIRO

( ) ( )2

3 ; 3 1 44 4 A C

M M b aL a a L b

E I L E I L Lϕ ϕ

= ⋅ − ⋅ ⋅ − = ⋅ ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅


( ) ( )

( ) ( )

3 2 23

2 2 2 23

4 34

24

AC

BC

M b xy L x L a LE I L

My L x a L x L a

E I L

⋅ ⋅ = ⋅ − ⋅ − − ⋅ ⋅ + ⋅ ⋅ ⋅

= ⋅ − ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅

Q

x

a

L

b

AM

Q

B

A B

B

CM

M

MC

C +




p ó r t i c o s

3 .1 7

3.5 VIGA EMPOTRADA EN UN EXTREMO


REACCIONES

BR P=

ESFUERZOS CORTANTES0 ; AC CBQ Q P cte= = − =

MOMENTOS FLECTORES

( )0 ; ; AC CB B M M P x a M P b= = − ⋅ − = − ⋅

ANGULOS DE GIRO

2

2 A C

Pb

E Iϕ ϕ = = − ⋅

⋅ ⋅


( )( ) ( ) ( )

22

3 ; 2 36 6 AC 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

⋅ ⋅= = ⋅ ⋅ + ⋅

⋅ ⋅ ⋅ ⋅

L

a

x

A

b

B

P

Q

MB

B

C



3 .1 8


a

x

AC

M

Q

L

b

P

B

D

c

B

B


REACCIONES .

BR p c= ⋅

ESFUERZOS CORTANTES .

0 ; ;2 AC 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 c M 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 c

b bE 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

⋅ = ⋅ − ⋅ + ⋅

⋅ ⋅ = ⋅ + ⋅ ⋅ − + = ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅ ⋅ ⋅




p ó r t i c o s

3 .1 9

L

Q

x

A

B

B

1P

2P

BM


REACCIONES

( )1 2

1

2BR p p= +

ESFUERZOS CORTANTES

( )2

2 11 1 2;

2 2 x B

p p x LQ p x Q p p

L

−= − ⋅ − ⋅ = − +

MOMENTOS FLECTORES

( ) ( )2 2

2 1 1 2 13 ; 26 6 x B

x L M p p x L p M p p

L = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅

ANGULOS DE GIRO

( )31 23

24 A

L p p

E Iϕ

⋅ ⋅ += −

⋅ ⋅


( )( )

( ) ( )

( ) ( ) ( )

32 2

2 1 2

22 1 2 1

524

2 2 2 x

L xL x p p L x p

y LEI

L L x p p L p p

−− − − + − −

= − − + + +

FLECHA

( )42 14 11

120 A

L p pf

E I

⋅ ⋅ + ⋅=

⋅ ⋅



3 .2 0


M

L

x

A

B

M

a b

B


REACCIONES

0BR =

ESFUERZO CORTANTE

0 xQ =

MOMENTOS FLECTORES

0 ; ; AC CB AC M M M cte M M= = − = = −

ANGULOS DE GIRO

C A

M b

E Iϕ ϕ

⋅= = −

⋅


( ) ( )2

2 2 ;2 2 AC 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

⋅= = ⋅ ⋅ ⋅ −

⋅ ⋅ ⋅ ⋅




p ó r t i c o s

3 .2 1

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



3 .2 2


ESFUERZOS CORTANTES

MOMENTOS FLECTORES MOMENTOS FLECTORES

ESFUERZOS CORTANTES

0,07 QL

- 0,125 QL

B A

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




p ó r t i c o s

3 .2 3

3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES

Relaciónentreluces

ESFUERZOS CORTANTESMOMENTOSFLECTORES

k a b c d e f g1 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 f f a f b f c

k

k f a dd e g

k

− += = − = + = +

= − = =

QQ

BC

A

c QL

f QL

2

A

e QL

L A

a QL

a L

2

C

2g QL

B

Ck L

d L

B

d QLb QL

MOMENTOS FLECTORES

ESFUERZOS CORTANTES



3

.2 4


Relaciónentreluces

ESFUERZOS CORTANTESMOMENTOSFLECTORES

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 0,6833,0 -0,375 1,375 1,791 1,208 0,875 0,730

2

2 2

10.5 0.5

8

2 2 2

k kf a f b f

k f a dd e g

k

− += = − = +

= − = =

A B C

2

g 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




p ó r t i c o s

3 .2 5

3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES

Relaciónentreluces

ESFUERZOSCORTANTES

MOMENTOSFLECTORES

k a b c e f g0,6 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 CB

k LL LD

Q QQ

MOMENTOS FLECTORES

a L

A

A

a QL

2

g QL

2

f 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



3

.2 6


Relaciónentreluces

ESFUERZOSCORTANTES

MOMENTOSFLECTORES

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

L A

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

2

f QL

C

a L

a QL

D

b QL

c QL




3 .2 7

3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

2

1

3 2I h

k y N kI l

= ⋅ = +

3.9.1 CARGA REPARTIDA VERTICAL

REACCIONES

23

2 12

A

D

A D

psnV

l

psmV

l

ps s

H H mnhlN

=

=

= = −

MOMENTOS FLECTORES

2

2

3

2 12

( )

2

B C

x A A

ps s M M mn

lN

En S

p x m M V x H h

= = − ⋅ −

−= ⋅ − − ⋅

h

p

A

B

D

C

l

I 2

I1 1

I

x

sa

m n

MB C

M

HA

HD

VA

VD



3

.2 8

P r ont u ar i o p ar a C á l c

ul o d eE s t r u ct ur a s

3.9.2 CARGA REPARTIDA HORIZONTAL

REACCIONES

( )

( )

2

2

2

8

6

8

A D

D

A

ph

V V l

ph N kH

N

ph N kH

N

= =

+=

−=

MOMENTOS FLECTORES

( )

( )

2

2

28

28

( )

2

B

C

Y B

ph M N k

N

ph M N k

N

En AB

py h y y

M Mh

= −

= − +

−

= + ⋅

h

p

A

B

D

C

l

I2

I1 1

I

MB

CM

HA

HD

VA

VD

y

MB




3 .2 9

3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL

REACCIONES

3

2

A

D

A D

PnV l

PmV

l

PmnH H

lhN

=

=

= =

MOMENTOS FLECTORES

32

2 32

B C

P

Pmn M M

lN

N M Pmn

lN

= = − ⋅

−=

h

A

B

D

C

l

I 2

I1 1

I

m n

MB C

M

HA

HD

VA

VD

P

MP



3

. 3 0



3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO

1 23 31 2

1 2

h hI Ik y k

I s I s= ⋅ = ⋅

3.10.1 CARGA REPARTIDA VERTICAL

REACCIONES

( ) ( )

21 2

2 21 1 2 2 1 2

2

8 1 1

A D

A D

plV V

h hplH H

h k h k hh

= =

+= =

+ + + +

MOMENTOS FLECTORES

( )

( ) ( )

21 2 1

2 21 1 2 2 1 28 1 1

B

h h hpl M

h k h k hh

+= −

+ + + +

( )

( ) ( )

21 2 2

2 21 1 2 2 1 2

1

8 1 1

( )2

C

X A

h h hpl M

h k h k hh

En BCpx l x f

M H x hl

+= −

+ + + +

− = − +

h

p

A

B

D

C

l

I3

I1

2I x

MB

CM

HA

HD

VA

VD

s

2

h1

f




3 . 3 1

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

phV V

l

H ph H

h k hphH

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 hph ph M

h k h k h h

h k hph h M

h k h k h h

En AB

py M H y

+ += −

+ + + +

+ +=

+ + + +

= −

h

A

B

D

C

l

I3

I1

2I

MB

CM

HD

VD

2

HA

VA

p

s

yh

f

1



3

. 3 2



3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL

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

pf h hV V

lH pf H

h k h h f h hpf H

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 hpfh M pfh

h k h k h h

h k h h f h hph M

h k h k h h

En BC

l py M V y H y h

f

+ + + += −

+ + + +

+ + + += −

+ + + +

= − + + −

h

A

B

D

C

l

I3

I 1

2I

MB

CM

HD

VD

2

HA

VA

p

s

y

h

f

1



F or m ul ar i o p ar avi g a s y p

ó r t i c o s

3 . 3 3


REACCIONES

( ) ( )1 2

2 2 21 1 2 2 1 2

( ) ( )

2 1 1

A

D

A D

PbV l

PaV

l

h l b h l aPabH 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 aPabh M

l h k h k h h

h l b h l aPabh M

l h k h k h h

Pab af M H h

l l

+ + += −

+ + + +

+ + += −

+ + + +

= + +

h

A

B

D

C

l

I1

2I

MB

CM

HD

VD

2

HA

VA

s

a b

I 3

MP

1

f

h



3

. 3 4



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

2

1

I hk

I s= ⋅

3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL

REACCIONES

( ) ( )

2

2

2

8 5

32 3 3

A E

A E

plV V

pl h f H H

h k f h f

= =

+= =

+ + +

MOMENTOS FLECTORES

( ) ( )

( )

2

2

2

8 5

32 3 3

8

2

2

B D

C B

B X

pl h h f M M

h k f h f

pl f h M M

h

En BC y DC x l x M fx

M p hh l

+= = −

+ + +

+= +

− = + +

p

A

B

E

C

l

I2

I1

x

MB

CM

HA

HE

VA

VE

I2

I1

s

D

h

f

MD




ó r t i c o s

3 . 3 5

3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

REACCIONES

( ) ( )

2

2

38

8

8 5

64 3 3

A

E

A E

plV

plV

pl h f H H

h k f h f

=

=

+= =

+ + +

MOMENTOS FLECTORES

( ) ( )

( )

2

2

2

8 5

64 3 3

16

2

2

B D

C B

B X

pl h h f M M

h k f h f

pl f h M M

h

En BC

x l x M fx M p h

h l

+= = −

+ + +

+= +

− = + +

p

A

B

E

C

l

I2

I1

x

MB

CM

HA

HE

VA

VE

I2

I1

s

D

h

f

MD



3

. 3 6




REACCIONES

( )

( ) ( )

2

2

2

2

5 12 6

16 3 3

A E

A E

E

phV V

l

H ph H

k h f phH

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

ph M M

ph f h M M

h

k h f ph M

h k f f h

En AB

py M H y

= +

+= +

+ += −

+ + +

= − + ⋅

A

B

E

C

l

I2

I1

MB

CM

HA H E

VA

VE

I2

I1

s

D

h

f

MD

p

y




ó r t i c o s

3 . 3 7


REACCIONES

( )

( ) ( )

( ) ( )

2

2

22

8 3 5 4

16 3 3

A E

A E

E

pf V V f h

lH pf H

h k f f hpf H

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 f pf M

h k f f h

M H h

En BC

y h M H y V x p

f siendo y x h

l

= ⋅+ + +

= − ⋅+ + +

= − ⋅

−= ⋅ − ⋅ −

= +

A

B

E

C

l

I2

I1

MB

CM

HA

HE

VA

VE

I2

I1

s

D

h

f

MD

p

x

y



3 . 3 8




REACCIONES

( )( ) ( )

2 2

2 2

6 ln 3 4

4 3 3

A

A

A E

PnV

l

Pm

V l

h f l mPmH H

l h k f f h

=

=+ −

= =+ + +

MOMENTOS FLECTORES

22

B D A

C B

P A A

M M H h

Pm h f M M

h

hl fm M V m H

l

= = − ⋅

+= +

+= ⋅ −

p

A

B

E

C

l

I2

I1

MB

CM

HA

HE

VA

VE

I2

I1

s

D

h

f

MD

m n




ó r t i c o s

3 . 3 9

3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL

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 hpl plV

h k h k h h

h hpl plV

h k h k h h

h hplH H

h k h k h h

−= +

+ + + +

−= −

+ + + +

−= =

+ + + +

MOMENTOS FLECTORES

( )

( ) ( )

( )

( ) ( )

21 2 1

2 21 1 2 2 1 2

21 2 2

2 2

1 1 2 2 1 2

2

1

8 1 1

8 1 1

2

B

C

x A A

h h hpl M

h k h k h h

h h hpl M

h k h k h hEn BC

px M V x H h

+= −

+ + + +

+= −

+ + + +

= ⋅ − − ⋅

h

A

B

D

C

l

I 3

I1

2I

MB C

M

HA

HD

VA

VD

p

x

h

2

1



3 .4 0



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

ph h hV V H

l l

H ph H

ph 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

ph ph k h h h M

h k h k hh

ph h k h h h M

h k h k hh

En AB

py M H y

+ += − −

+ + + +

+ += −

+ + + +

= ⋅ −

h

p

A

B

D

C

l

I3

I1

2I

MB

CM

HA

HD

VA

VD

y

MB

h1

2




ó r t i c o s

3 .4 1


REACCIONES

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )( )

( ) ( )

( ) ( )

1 21 2

3 2 21 1 2 2 1 2

1 21 23 2 2

1 1 2 2 1 2

1 2

2 2 2

1 1 2 2 1 2

2 1 1

2 1 1

2 1 1

A

D

A D

l b h l a hPb PabV h h

l l h k h k h h

l b h l a hPa PabV h h

l l h k h k h h

l b h l a hPabH H

l h k h k h h

+ + += + −

+ + + +

+ + += − −

+ + + +

+ + += =

+ + + +

MOMENTOS FLECTORES

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 212 2 2

1 1 2 2 1 2

1 22

2 2 21 1 2 2 1 2

2 1 1

2 1 1

B

C

P A B

l b h l a hPabh M

l h k h k h h

l b h l a hPabh

M l h k h k h h

M V a M

+ + += −

+ + + +

+ + +

= − + + + +

= ⋅ +

A

B

D

C

l

I3

I1

2I

a b

MB C

M

HA

HD

VA

VD

P

MP

h

h1

2



3 .4 2


3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

2

1

I hk

I l= ⋅


REACCIONES

( )

2

2 4 2 A D A D

pl plV V H H

h k= = = =

+

MOMENTOS FLECTORES

( )

( )

( )( )

2

2

2

2

12 2

6 2

2 6 2

3 2máx24 2 2

A D

B C

x

pl M Mk

pl M M

k

En BC

px l x pl M

k

pl k l M pos para x

k

= = +

= = −+

−= −

+

+= =+

h

A

B

D

C

l

I2

I1 1

I

x

MB C

M

HA

VA

p

MA

HD

VD

MD




ó r t i c o s

3 .4 3


REACCIONES

( )

( )( )

2

6 1

2 3

8 2

A D

A D

D

ph kV V

l k

H ph H

ph kH

k

= =+

= −

+=

+

MOMENTOS FLECTORES

2

2

2

2

2

2 15

24 6 1 22 2

124 6 1 2

2 23

24 6 1 2

2 13

24 6 1 2

2

A

B

C

D

y A A

ph M

k kph

Mk k

ph M

k k

ph M

k k

En AB

py M H y M

= − + +

+ +

= − + + +

= − − − + +

= + − + +

= − + ⋅ +

h

p

A

B

D

C

l

I2

I1 1

I

MB

CM

y

MB

HA

VA

MA

HD

VD

MD



3 .4 4



REACCIONES

( )

( )21

6 1

3

2 ( 2)

A

D A

A D

m n mPnV

l l k

V P V Pmn

H Hlh 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

Pmn n m M

l k l k

Pmn n m M

l k l k

Pmn n m M

l k l k

Pmn n m M

l k l k

mMnMPmn M

l l l

−= − + +

−= − + + +

−= − − + +

−= + + +

= + +

h

A

B

D

C

l

I2

I1 1

I

m n

MB C

M

P

MP

HA

VA

MA

HD

VD

MD




ó r t i c o s

3 .4 5

3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR

REACCIONES

3

(6 1)

2

A D

A D

PhkV V

l kP

H H

= =

+= =

MOMENTOS FLECTORES

3 12 6 1

3

2 6 13 1

2 6 1

A

B C

D

Ph k M

k

Ph k M M

k

Ph k M

k

+= −

+

= − =+

+=

+

h

A

B

D

C

l

I2

I1 1

I

MB C

M

P

HD

VD

MD

HA

VA

MA



3 .

4 6


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

2

1

I hk

I s= ⋅


REACCIONES

( )

( ) ( )

2

2 2 2

24 5

8 4

A E

A E

plV V

k h f f plH H

kh f k h hf 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

kh h f f h f pl M M

kh f k h hf f

kh h f f pl M M

kh f k h hf f

pl M M H h f

En BC

xf px M M V x H h

l

+ + −= =

+ + + +

+ += = −

+ + + +

= + − +

= + ⋅ − + −

p

A

B

E

C

l

I2

I1

x

MB

CM

I2

I1

s

D

h

f

MD

HA

VA

MA

HE

VE

ME




3 .4 7

3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL

REACCIONES

( )( )

( ) ( )

2

2 2 2

24 1

332 3 1

4 5

16 4

A E

E

A E

plV V

kV pl

k

k h f f plH H

kh f k h hf f

= −

+=

++ +

= = + + + +

MOMENTOS FLECTORES

( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( )

( )

( ) ( ) ( )

( )

( ) ( ) ( )

2 2

2 2 2

2 2

2 2 2

22 2

2 2 2

22 2

2 2 2

8 15 6

96 64 3 14

8 15 6

96 64 3 14

16 15

96 64 3 14

16 15

96 64 3 14

A

E

B

D

x A

kh h f f h f pl pl M

kkh f k f fh h

kh h f f h f pl pl M

kkh f k f fh h

kh h f f pl pl M

kkh f k f fh h

kh h f f pl pl M

kkh f k f fh h

En BC M M

+ + −= −

++ + + +

+ + −= + ++ + + +

+ += − −

++ + + +

+ += − +

++ + + +

=

( )

22

2

2

A A

C E E E

xf pxV x H h

ll

M V M H f h

+ ⋅ − + −

= + − +

p

A

B

E

C

l

I2

I1

x

MB

CM

I2

I1

s

D

h

f

MD

HA

VA

MA

HE

VE

ME



3 .4 8

P r ont u ar i o p ar a C á l c u

l o d eE s t r u ct ur a s


REACCIONES

( )

( )

( ) ( )

2

22

2 2 2

2 3 1

2 3

4 4

A E

A E

E

ph kV V

l k

H ph H

k h k f h f phH

kh f k f fh h

= =+

= −

+ + +=

+ + + +

MOMENTOS FLECTORES

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

2 22

2 2 2

2

2 22

2 2 2

2

6 15 16 6 2 16

24 3 14

21

2

6 15 16 6 2 16

24 3 14

2

A

B A A

C E E E

D E E

E

y A A

kh k kf h f f ph k M

kkh f k f fh h

ph M M H h

M M H f h V

M M H h

kh k kf h f f ph k M

kkh f k f fh h

En AB py M M H y

+ + + + + = − + ++ + + +

= + ⋅ −

= − + +

= − ⋅

+ + + + + = − + ++ + + +

= + ⋅ −

A

B

E

C

l

I2

I1

MB

CM

I2

I1

s

D

h

f

MD

p

y

HE

VE

ME

HA

VA

MA




3 .4 9


REACCIONES

( )

( ) ( )

( ) ( )

2

2 2 2

438 3 1

2 4 10 5

4 4

A E

A E

E

k f h f pf V V

l k

H pf H

k h k f kh kf f pf H

kh f k f fh h

+ += =

+

= −+ + + +

=+ + + +

MOMENTOS FLECTORES

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( )

( ) ( )

2 2 2

22 2

2

9 4 6 4 3 23

24 2 3 14

2

9 4 6 4 3 23

24 2 3 14

2 2

A

B A A

C E E E

D E E

E

y A A A

kh f h f h f h k f pf

M f kkh f k f fh h

M M H hl

M M H h f V

M M H h

kh f h f h f h k f pf M f

kkh f k f fh hEn BC

l y h p y h M M H y V

f

+ + + + + = − + ++ + + +

= + ⋅

= − + +

= − ⋅

+ + + + + = − +

++ + + +

− −= + ⋅ − −

A

B

E

C

l

I 2

I1

MB

CM

I 2

I1

s

D

h

f

MD

p

y

HE

VE

ME

HA

VA

MA



3 . 5 0

P r ont u ar i o p ar a C á l c u

l o d eE s t r u ct ur a s


REACCIONES

( )

( ) ( ) ( )

( ) ( )

2

3

2 2

2 2 2 2

3 2

3 13 4 1 3

4

A E

E

A E

V P V l kl m mPm

V klkl f h fm k lm f khPm

H H

l kh f k f fh h

= −+ −

=+

+ − + + −= =

+ + + +

MOMENTOS FLECTORES

( ) ( ) ( )

( ) ( )( )

( )

( ) ( ) ( )

( ) ( )

( )

2 2 2

2 2 2

2

2 2 2

2 2 2

2

3 2 4 2 2 ln 4

4

2

3 1

2

3 2 4 2 2 ln 4

4

23 1

A

B A A

C E E E

D E E

E

flh kl m fm kh h f kh f l m l

Pm kh f k f fh h M

l n n m

k M M H h

l M M V H h f

M M H h

flh kl m fm kh h f kh f l m l

Pm kh f k f fh h M

l n n m

k

+ − + + + + −

+ + + + =

− −

+

= − ⋅

= + − +

= − ⋅

+ − + + + + −

+ + + + =

− ++

2y A A A

En BCfm

M M V m H hl

= + ⋅ − +

p

A

B

E

C

l

I2

I1

MB

CM

I2

I1

s

D

h

f

MD

m n

HA

VA

MA

HE

VE

ME

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