Prontuario estructuras

3

FORMULARIO PARA VIGAS Y PÓRTICOS

Formulario para vigas y pórticos 3.1

3.1 Obtención de la Distribución de Solicitaciones mediante la Formulación de Macaulay

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

( )( )

( )

( )( )

( )

( )

( )

2

1

0

0

2 !

1 !

!

ecuaciones validas solo si 0

en las expresiones

si 0 0

1

y si 0 0

c

c

c

n

n

n n

A x aq x

c

A x aT x

c

A x aM x

c

n

x a

n x a x a

x a x a

n x a x a

x a x a x a

−

−

⋅ −=

−

⋅ −= −

−

⋅ −= −

≥

−

= ≤ − =

≥ − =

> ≤ − =

≥ − = −

∑

∑

∑

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

3.2 Prontuario para Cálculo de Estructuras

M

M(x)

ax

( )

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 Mc

≤ − =

≥ − =

= − −

==

P

M(x)

ax

T(x)

( )

( )( )

1

1 1

0

1

Si0

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 Pc

≤ − =

≥ − = −

= − −

= − −

==

Limitación de las Deformaciones 3.3

2M(x)

q

xa

T(x)

( )

( )

( )

( )

2

2 2

0

1

2

Si0

entonces

1

2 1

por lo tanto 2

x a x a

x a x a x a

q x q x aqT x x a

qM x x a

A qc

≤ − =

≥ − = −

= −

= − −

= − −⋅

==

3

ax

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 dM x x a

qAd

c

≤ − =

≥ − = −

= −

= − −⋅

= − −⋅ ⋅

=

=

3.4 Prontuario para Cálculo de Estructuras

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

x

ab

q

( )

( ) ( )

2 2qM x x-a x-b2!dM x

T xdx

= −⟨ ⟩ + ⟨ ⟩

=

q

a

xbd

q/d

( )

( ) ( )

3 3 2q/d qM x - x-a x-b x-b3! 2!

dM xT x

dx

= ⟨ ⟩ + ⟨ ⟩ + ⟨ ⟩

=

q

a

xbd

q/d

( )

( ) ( )

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

dM xT x

dx

= − ⟨ ⟩ + ⟨ ⟩ − ⟨ ⟩

=

a

xbd

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

xbd

aq

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

= − ⟨ ⟩ + ⟨ ⟩ +

− + ⟨ ⟩ − ⟨ ⟩

=

Formula

rio para

vigas y pórticos

3.5

3.2 VIGA APOYADA EN LOS EXTREMOS

3.2.1 CARGA PUNTUAL EN LA VIGA REACCIONES

A BP b P aR RL L⋅ ⋅

= =

ESFUERZOS CORTANTES

;AC CBP b P aQ cte Q cteL L⋅ ⋅

= = = − =

MOMENTOS FLECTORES

( ) max 0; ; para AC CB CP b P a P a bM x M L x M M x aL L L⋅ ⋅ ⋅ ⋅

= ⋅ = ⋅ − = = =

ANGULOS DE GIRO

( ) ( ) ( ); ;6 6 3A B CP a b P a b P a bL b L a b aE I L E I L E I L

ϕ ϕ ϕ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⋅ + = − ⋅ + = ⋅ −

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

ECUACION DE LA ELASTICA ( ) 22 2 2

2 2 21 ; 16 6AC CB

P L a L xP L b x b x a L xy yE I E I LL L L

⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − = ⋅ − − = ⋅ − − ⋅ ⋅ ⋅ ⋅ FLECHA MAXIMA

( )2 23

2 2 2 para 39 3C

P b L bf L b xE I L⋅ −

= ⋅ − =⋅ ⋅ ⋅

x

a

A

L

b

C

PB

QA

maxM

BQ

3.6

Prontuario pa

ra C

álculo d

e Estructuras

3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES

A Bp b c p a cR RL L⋅ ⋅ ⋅ ⋅

= =

ESFUERZOS CORTANTES

; ;2AC 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 para 2 2

AC CD

DB

p b c p b c p cM x M x x aL L

p a cM L xL

p b c b c c b cM a c x aL L L

⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅ − ⋅ − −

⋅ ⋅= ⋅ −

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ − + = − + ⋅

ANGULOS DE GIRO 2 2

;6 4 6 4A Bp a b c c p a b c cL b L aE I L a E I L b

ϕ ϕ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

= ⋅ + − = − ⋅ + − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

ECUACION DE LA ELASTICA

( )

22

4 23

22

6 4

4 424 2 4

6 4

AC

CD

DB

p b c x cy x a L bL E I a

p c cy L x a b c x a b c L b xE I L a

p a c L x cy L x b L aL E I a

⋅ ⋅= ⋅ − + ⋅ + −

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

⋅ ⋅ −= ⋅ ⋅ − − + ⋅ + −

⋅ ⋅ ⋅

QA

BQ

x

a

A

L

b

C

PB

c

maxM

D

Formula

rio para

vigas y pórticos

3.7

3.2.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES

( ) ( )1 2 1 21 12 ; 26 6A BR p p R p p= ⋅ + = + ⋅ .

ESFUERZOS CORTANTES ( )1 2 23

; ;6A A x A B B

p L x p xQ R Q R x Q R

L⋅ − + ⋅

= = − ⋅ = −⋅

MOMENTOS FLECTORES ( )

( ) ( )

( )

1 2 2

2 2

max 1 2 1 2

2 20 1 1 2 1 2

2 1

36

comprendido entre 0,125 y 0,1282 2

1 1para x3

x Ap L x p x

M R x xL

L LM p p p p

p p p p pp p

− + ⋅= ⋅ − ⋅

⋅

⋅ ⋅ + ⋅ ⋅ +

= ⋅ − + ⋅ + + ⋅

−

ANGULOS DE GIRO

( ) ( )3 3

1 2 1 28 7 ; 7 8360 360A BL Lp p p pE I E I

ϕ ϕ= ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅


( ) ( ) ( )( ) ( )

3 21 2 1 2

2 31 2 1 2

3 3 4360 8 7 8 7x

p p x p p Lxx L xy

EI p p L x p p L

− − + +− =

+ + +

FLECHA MAXIMA ( ) ( )4 4

1 2 1 2entre 0,01302 y 0,013042 2

p p L p p LE I E I

+ ⋅ + ⋅⋅ ⋅

⋅ ⋅ ⋅ ⋅

QA

maxM

QB

P

A B

1 P2

0x

L

x

3.8

Prontuario pa

ra C

álculo d

e Estructuras

3.2.4 MOMENTO FLECTOR REACCIONES

R RMLA B= − = −

ESFUERZOS CORTANTES

xMQ cteL

= =

MOMENTOS FLECTORES

( )

izq derC C M M M

AC CB

izq derC C

M MM x M L xL LM MM a M bL 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 LM a bE I L

ϕ ϕ

ϕ

⋅ ⋅= ⋅ ⋅ − = ⋅ ⋅ −

⋅ ⋅ ⋅ ⋅

= ⋅ +⋅ ⋅ ⋅

ECUACION DE LA ELASTICA 2 2

2 2

22

2

1 36

( ) 1 36

AC

CB

M L x b xyE I L L

M L L x a L xyE I LL

⋅ ⋅= − ⋅ − ⋅ −

⋅ ⋅ ⋅ ⋅ − − = − ⋅ − ⋅ − ⋅ ⋅

FLECHA

( )3CM a bf b aE I L⋅ ⋅

= ⋅ −⋅ ⋅ ⋅

BA

M+

QA QB

MC

MC

M

a

L

b

C

Formula

rio para

vigas y pórticos

3.9

3.3 VIGA EMPOTRADA EN LOS EXTREMOS


( ) ( )2 2

3 32 ; 2A BP b P aR L a R L bL L⋅ ⋅

= ⋅ + ⋅ = ⋅ + ⋅

ESFUERZOS CORTANTES

( ) ( )2 2

3 32 ; 2AC CBP b P aQ L a cte Q L b cteL L⋅ ⋅

= ⋅ + ⋅ = = − ⋅ + ⋅ =

MOMENTOS FLECTORES

( )

( )

2 2 2

2 2 3

2 2 22

03 3

; ; 2

22 ; para

A B AC

BC C

P a b P a b P bM M M L x a x a LL L L

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

⋅ ⋅ ⋅ ⋅ ⋅= − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅

⋅ ⋅ ⋅ ⋅= ⋅ ⋅ + − ⋅ − ⋅ ⋅ = =


( ) ( )

2 2

2

22

2

236

3 26

AC

BC

P b a x xy a xE I L L

L xP a L xy b L x bE I L L

⋅ ⋅ ⋅ = ⋅ ⋅ − − ⋅ ⋅ ⋅

−⋅ − ⋅ = ⋅ ⋅ − − − ⋅ ⋅ ⋅ ⋅

FLECHAS

( )

3 3 3 2

max3 22;

3 3 22para

2

CP a b P a bf fE I L E I L a

a LxL a

⋅ ⋅ ⋅ ⋅ ⋅= =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅

⋅ ⋅=

+ ⋅

MC

A B

a

L

x

b

P

B

AQ

Q

x

0

MA

C

MB

3.10

Prontuario pa

ra C

álculo d

e Estructuras


;A B A BA B

p b c M M p a c M MR RL L L L⋅ ⋅ − ⋅ ⋅ −

= − = +

ESFUERZOS CORTANTES

; ;AC A BD B CD AcQ R cte Q R cte Q R p x aa

= = = − = = − ⋅ − +

MOMENTOS FLECTORES

( )

2

3 2

2 2

3 2

2 2

;2 2

12; 312

12312

AC A A CD A A

BD B B A

B

p cM R x M M R x M x a

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

p c a bM L aL c

= ⋅ + = ⋅ + − ⋅ − +

⋅ ⋅ ⋅= ⋅ − + = − ⋅ − ⋅ +

⋅ ⋅ ⋅ ⋅

= − ⋅ − ⋅ + ⋅


( )

( ) ( ) ( )

2

43 3

3 2 2

36

1 4 1224 2

1 3 3 2 36

AC A A

CD A A

DB B B B A B B B

xy M R xE 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

Pc

C D

MB

Formula

rio para

vigas y pórticos

3.11


( )

( )

1 2

1 2

26

26

A BA

A BB

L M MR p pL

L M MR p pL

−= ⋅ ⋅ + −

−= ⋅ + ⋅ +

ESFUERZOS CORTANTES

( )1 222

A A

x A

B B

Q Rp L x p x

Q R xL

Q R

=

⋅ ⋅ − + ⋅= − ⋅

⋅= −

MOMENTOS FLECTORES

( )

( )

( )

2

1 2

1 2 2

2

1 2

3 260

36

2 360

A

x A A

B

LM p p

p L x p xM R x M x

LLM p p

= − ⋅ + ⋅

⋅ ⋅ − + ⋅= ⋅ + − ⋅

⋅

= − ⋅ + ⋅


( )22 1 3 2

1 4 1224 5x A A

p pxy x p L x R L x M LE I L

−= ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅

⋅ ⋅ ⋅

AQ

QB

x

A

L

B

BMAM

P2

P1

3.12

Prontuario pa

ra C

álculo d

e Estructuras


3 36 6;A BM MR a b R a bL L⋅ ⋅

= − ⋅ ⋅ = ⋅ ⋅

ESFUERZOS CORTANTES

36

xMQ a b cteL⋅

= − ⋅ ⋅ =

MOMENTOS FLECTORES

( )

⋅ ⋅ = ⋅ − ⋅ = − ⋅ − ⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ −

⋅ − = − ⋅ ⋅ ⋅ − ⋅ −

⋅

= − ⋅ ⋅ = + ⋅ − ⋅ ⋅2 3 23 3

2 3 2 3

3 1 2 1

3 1 2 1

6 ; 6

A B

AC

CB

izq derC A C A

M a b M b aM ML L L LM a a xML L L

M b b L xML L L

M MM M a b M M L a bL L


( )

2

2

2

2

22

22

AC

BC

M b x L x by aE I L LL

M a L x b x ayE I L LL

⋅ ⋅ − = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅

⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅

FLECHA

( )2 2

32CM a bf a bE I L⋅ ⋅

= − ⋅ −⋅ ⋅ ⋅

MC

AQ QB

x

A

L

a b

B+M

CM

C

AM

BM

Formula

rio para

vigas y pórticos

3.13

3.4 VIGA APOYADA-EMPOTRADA


( ) ( )2

2 23 33 ; 3

2 2A BP b P aR L b R L aL L⋅ ⋅

= ⋅ ⋅ − = ⋅ ⋅ −⋅ ⋅

ESFUERZOS CORTANTES

( ) ( )2

2 23 33 ; 3 .

2 2AC CBP b P aQ L b cte Q L a constL L⋅ ⋅

= − ⋅ ⋅ − = = − ⋅ ⋅ − =⋅ ⋅

MOMENTOS FLECTORES

( ) ( )

( ) ( )

2 2 22 3

2 3 2 23 3

; 3 22 2

3 2 ; 2 32 2

B C

AC CB

P a P aM L a M b a bL L

P x P aM b a b M L L x a xL L

⋅ ⋅= − ⋅ − = ⋅ ⋅ ⋅ + ⋅

⋅ ⋅⋅ ⋅

= ⋅ ⋅ ⋅ + ⋅ = ⋅ ⋅ − ⋅ ⋅ + ⋅⋅ ⋅

ANGULOS DE GIRO

( ) ( ) ( )2 2

2 23; 2

4 4A CP a L a P a L a

L a L aE I L E I L

ϕ ϕ⋅ − ⋅ ⋅ −

= = ⋅ − ⋅ ⋅ −⋅ ⋅ ⋅ ⋅ ⋅ ⋅


( )

( )

22 2

3

2 2 2

2 2

3 212

3 1 312

AC

BC

P b xy a L x L aE I L

P a L x a a L xyE I LL L

⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅

⋅ ⋅ − − = ⋅ ⋅ − − − ⋅ ⋅ ⋅

FLECHA MAXIMA 2

para x=6 2 2maxp b a a af L

E I L a L a⋅ ⋅

= ⋅ ⋅⋅ ⋅ ⋅ + ⋅ +

Q

Q

MB

A

B

x

a

L

b

A BC

P

MC

3.14

Prontuario pa

ra C

álculo d

e Estructuras


;B BA B

p b c M p a c MR RL L L L⋅ ⋅ ⋅ ⋅

= + = −

ESFUERZOS CORTANTES

; ;2AC A DB B CD AcQ R cte Q R cte Q R p x a = = = − = = − ⋅ − +

MOMENTOS FLECTORES

( )

2

2

2

;2 2

;42

AC A CD A

DB B B B

p cM R x M R x x a

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

= ⋅ = ⋅ − ⋅ − +

⋅ ⋅ ⋅= ⋅ − + = − ⋅ + −

⋅⋅

ANGULOS DE GIRO

3 2

2123

48Ap c a bL bE I L c

ϕ ⋅ ⋅ ⋅

= ⋅ − + ⋅ ⋅ ⋅


( ) ( )

22 3

2

4 23 3

2

2

128 348

1 128 2 348 4

36

AC A

CD A

DB B B

x a by R L x p c L bE I L c

c aby R Lx pL x a pc L b xE I L c

L xy R L x M

E I

⋅ ⋅= ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − +

⋅ ⋅ ⋅ = ⋅ − + − + + − + ⋅ ⋅ ⋅

− = − ⋅ ⋅ − + ⋅ ⋅ ⋅

Q

Q

ax

C

P

M

A

A

B

B

Lb

c

B

Formula

rio para

vigas y pórticos

3.15


( ) ( )1 2 1 22 ; 26 6

B BA B

L M L MR p p R p pL L

= ⋅ ⋅ + + = ⋅ + ⋅ −

ESFUERZOS CORTANTES

( )1 22;

2x A B Bp L x p x

Q R x Q RL

⋅ ⋅ − + ⋅= − ⋅ = −

⋅

MOMENTOS FLECTORES

( ) ( )2

1 2 21 2

3; 7 8

6 120x A Bp L x p x LM R x x M p p

L⋅ ⋅ − + ⋅

= ⋅ − ⋅ = − ⋅ ⋅ + ⋅⋅

ANGULOS DE GIRO

( )3

1 23 2240AL p pE I

ϕ = ⋅ ⋅ + ⋅⋅ ⋅


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

120x A Axy p p x Lp x R Lx L R L p p LEIL

= − + − + − +

2

A

Q Q

Lx

B

BA

1PP

MB

3.16

Prontuario pa

ra C

álculo d

e Estructuras


( )2 23

32A BMR R L aL

= − = ⋅ ⋅ −

ESFUERZOS CORTANTES

x AQ R cte= =

MOMENTOS FLECTORES

( )

( )

2 22

22 2

3 2

; ; 32

3 ; 3 1 22 2

der izqC A C A B

AC BC

MM R a M M R a M L aL

M x M x aM L a MLL L

= ⋅ − = ⋅ = ⋅ − ⋅⋅

⋅= ⋅ ⋅ − = ⋅ ⋅ ⋅ − −

ANGULOS DE GIRO

( ) ( )2

3 ; 3 1 44 4A CM M b aL a a L bE I L E I L L

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


( ) ( )

( ) ( )

3 2 23

2 2 2 23

4 34

24

AC

BC

M b xy L x L a LE I LMy L x a L x L aE I L

⋅ ⋅ = ⋅ − ⋅ − − ⋅ ⋅ + ⋅ ⋅ ⋅

= ⋅ − ⋅ ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅

Q

x

a

L

b

AM

Q

B

A B

B

CM

M

MC

C +

Formula

rio para

vigas y pórticos

3.17

3.5 VIGA EMPOTRADA EN UN EXTREMO

3.5.1 CARGA PUNTUAL EN LA VIGA REACCIONES BR P=

ESFUERZOS CORTANTES 0 ;AC CBQ Q P cte= = − =

MOMENTOS FLECTORES

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

ANGULOS DE GIRO

2

2A CP bE I

ϕ ϕ= = − ⋅⋅ ⋅


( )( ) ( ) ( )2

23 ; 2 36 6AC CBP b Py L x b y L x b aE I E I⋅

= ⋅ ⋅ − − = ⋅ − ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅

FLECHA MAXIMA

( )3 2

; 2 33 6C AP b P bf f b aE I E I⋅ ⋅

= = ⋅ ⋅ + ⋅⋅ ⋅ ⋅ ⋅

L

a

x

A

b

B

P

Q

MB

B

C

3.18

Prontuario pa

ra C

álculo d

e Estructuras

a

x

AC

M

Q

L

b

P

BD

c

B

B

3.5.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES . BR p c= ⋅

ESFUERZOS CORTANTES .

0 ; ;2AC CD DBcQ Q p x a Q p c cte = = − ⋅ − + = − ⋅ =

MOMENTOS FLECTORES .

( )

2

220 ; ;2 2

;

AC CD D

DB B

cp x ap cM M M

M p c x a M p c b

⋅ − + ⋅ = = − = −

= − ⋅ ⋅ − = − ⋅ ⋅

ANGULOS DE GIRO . 2 2

2 2; ;2 4 2 12D C A Cp c c p c cb bE I E I

ϕ 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 bE I E I

p c cy x a c a x b b cE I

⋅ ⋅= ⋅ − ⋅ ⋅ − + = ⋅ − ⋅ ⋅ + + ⋅

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

FLECHAS .

( )

2

2 23 2 3

2 3 12

4 ; 3 212 2 6 4

D

C A

p c c b cf bE I

p c c p c cf b b c c f a b bE I E I

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

Formula

rio para

vigas y pórticos

3.19

L

Q

x

A

B

B

1P2P

BM


( )1 212BR p p= +

ESFUERZOS CORTANTES

( )2

2 11 1 2;

2 2x Bp p x LQ p x Q p p

L−

= − ⋅ − ⋅ = − +

MOMENTOS FLECTORES

( ) ( )2 2

2 1 1 2 13 ; 26 6x Bx LM p p x L p M p pL = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅

ANGULOS DE GIRO

( )31 23

24AL p p

E Iϕ

⋅ ⋅ += −

⋅ ⋅


( ) ( ) ( ) ( )

( ) ( ) ( )

32 2

2 1 2

22 1 2 1

5242 2 2

x

L xL x p p L x py LEIL L x p p L p p

−− − − + − −

= − − + + +

FLECHA

( )42 14 11

120AL p p

fE I

⋅ ⋅ + ⋅=

⋅ ⋅

3.20

Prontuario pa

ra C

álculo d

e Estructuras

M

L

x

AB

M

a b

B


0BR =

ESFUERZO CORTANTE

0xQ =

MOMENTOS FLECTORES

0 ; ;AC CB ACM M M cte M M= = − = = −

ANGULOS DE GIRO

C AM bE I

ϕ ϕ ⋅= = −

⋅


( ) ( )22 2 ;2 2AC BCM My b L x b y L xE I E I

= ⋅ ⋅ ⋅ − ⋅ − = −⋅ ⋅ ⋅ ⋅

FLECHA

( )2

; 22 2C AM b Mf f b L bE I E I⋅

= = ⋅ ⋅ ⋅ −⋅ ⋅ ⋅ ⋅

Formula

rio para

vigas y pórticos

3.21

3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES

ESFUERZOS CORTANTES

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

MOMENTOS FLECTORES

L/2 L/2L

P P

L/2 L/2L

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

3.22

Prontuario pa

ra C

álculo d

e Estructuras

ESFUERZOS CORTANTES

MOMENTOS FLECTORES MOMENTOS FLECTORES

ESFUERZOS CORTANTES

0,07 QL

- 0,125 QL

BA

A

0,625 QL

C

B C

B

0,375 QL

A L

Q

L C

0,375 L

0,625 QL

0,375 QL

2 0,07 QL2

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

Formula

rio para

vigas y pórticos

3.23

3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES

Relación entre luces

ESFUERZOS CORTANTES MOMENTOS FLECTORES

k a b c d e f g

1,1 0,361 0,639 0,676 0,424 0,065 0,139 0,09

1,2 0,345 0,655 0,729 0,471 0,060 0,155 0,111

1,3 0,326 0,674 0,784 0,516 0,053 0,174 0,133

1,4 0,305 0,695 0,840 0,560 0,047 0,195 0,157

1,5 0,281 0,719 0,896 0,604 0,040 0,219 0,183

1,6 0,255 0,745 0,953 0,647 0,033 0,245 0,209

1,7 0,226 0,774 1,011 0,689 0,026 0,274 0,237

1,8 0,195 0,805 1,070 0,730 0,019 0,305 0,267

1,9 0,161 0,839 1,128 0,772 0,013 0,339 0,298

2,0 0,125 0,875 1,128 0,812 0,008 0,375 0,330

2,1 0,086 0,914 1,247 0,853 0,004 0,414 0,364

2,2 0,045 0,954 1,308 0,892 0,001 0,455 0,399

2,3 0,001 0,999 1,367 0,933 0,000 0,499 0,435

2

2 2

1 0.5 0.58 2

2 2 2

k k k ff a f b f ck

k f a dd e gk

− += = − = + = +

= − = =

QQ

B CA

c QL

f QL

2

A

e QL

LA

a QL

a L

2

C

2g QL

B

Ck L

d L

B

d QLb QL

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

3.24

Prontuario pa

ra C

álculo d

e Estructuras


ESFUERZOS CORTANTES MOMENTOS FLECTORES

k a b c d f g

2,4 -0,045 1,045 1,427 0,973 0,545 0,473

2,5 -0,094 1,094 1,487 1,013 0,594 0,513

2,6 -0,145 1,145 1,548 1,051 0,645 0,553

2,7 -0,198 1,198 1,608 1,091 0,698 0,595

2,8 -0,255 1,255 1,669 1,130 0,755 0,638

2,9 -0,313 1,313 1,730 1,169 0,813 0,683

3,0 -0,375 1,375 1,791 1,208 0,875 0,730

2

2 2

1 0.5 0.58

2 2 2

k kf a f b f

k f a dd e gk

− += = − = +

= − = =

A B C

2g QL

AC

BL

Q

A C

Q

k L

B

f QL2

a QL

c QL

b QL d QL

d L

MOMENTOS FLECTORES

ESFUERZOS CORTANTES

Formula

rio para

vigas y pórticos

3.25

3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES


ESFUERZOS CORTANTES

MOMENTOS FLECTORES

k a b c e f g

0,6 0,420 0,580 0,300 0,088 0,080 -0,035

0,7 0,418 0,582 0,350 0,087 0,081 -0,020

0,8 0,414 0,586 0,400 0,086 0,086 -0,006

0,9 0,408 0,592 0,450 0,083 0,091 -0,009

3

2 2

1 0.5 0.512 8

2 2 8

kf a f b fk

k a kc e g f

+= = − = +

⋅ +

= = = −

A CBk LL L

D

Q QQ

MOMENTOS FLECTORES

a L

A

A

a QL

2g QL

2f QL

C

2

2e QL

f QL

B

e QL2

D

c QL

b QL

CB

b QL

c QLa L

D

a QL

ESFUERZOS CORTANTES

3.26

Prontuario pa

ra C

álculo d

e Estructuras


ESFUERZOS CORTANTES

MOMENTOS FLECTORES

k a b c e f g

1,0 0,400 0,600 0,500 0,080 0,100 0,025

1,1 0,390 0,610 0,550 0,076 0,110 0,041

1,2 0,378 0,622 0,600 0,072 0,122 0,058

1,3 0,365 0,635 0,650 0,066 0,135 0,076

1,4 0,349 0,651 0,700 0,061 0,151 0,094

1,5 0,322 0,668 0,750 0,055 0,168 0,113

1,6 0,313 0,687 0,800 0,049 0,187 0,133

1,7 0,292 0,708 0,850 0,043 0,208 0,153

1,8 0,269 0,731 0,900 0,036 0,231 0,174

1,9 0,245 0,755 0,950 0,030 0,255 0,196

2,0 0,219 0,781 1,000 0,024 0,281 0,219

3

2 2

1 0.5 0.512 8

2 2 8

kf a f b fk

k a kc e g f

+= = − = +

⋅ +

= = = −

k LB

LA

LDC

QQ Q

ESFUERZOS CORTANTES

MOMENTOS FLECTORES

e QL2B

2g QL

f QL2

A

B

b QL

c QL

A

a QL

a L

e QL2C D

2f QL

C

a L

a QL

D

b QL

c QL

Formula

rio para

vigas y pórticos

3.27

3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

2

1

3 2I hk y N kI l

= ⋅ = +

3.9.1 CARGA REPARTIDA VERTICAL REACCIONES

232 12

A

D

A D

psnVlpsmVl

ps sH H mnhlN

=

=

= = −

MOMENTOS FLECTORES

2

2

32 12

( )2

B C

x A A

ps sM M mnlN

En Sp x mM V x H h

= = − ⋅ −

−= ⋅ − − ⋅

h

p

A

B

D

C

l

I 2

I 1 1I

x

sa

m n

MB CM

HA HD

VA VD

3.28

Prontuario pa

ra C

álculo d

e Estructuras

3.9.2 CARGA REPARTIDA HORIZONTAL REACCIONES

( )

( )

2

228

68

A D

D

A

phV Vl

ph N kH

Nph N k

HN

= =

+=

−=

MOMENTOS FLECTORES

( )

( )

2

2

28

28

( )2

B

C

Y B

phM N kNphM N kN

En ABpy h y yM M

h

= −

= − +

−= + ⋅

h

p

A

B

D

C

l

I 2

I 1 1I

MB

CM

HA HD

VA VD

y

MB

Formula

rio para

vigas y pórticos

3.29

3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES

32

A

D

A D

PnVlPmVl

PmnH HlhN

=

=

= =

MOMENTOS FLECTORES

32

2 32

B C

P

PmnM MlN

NM PmnlN

= = − ⋅

−=

h

A

B

D

C

l

I 2

I 1 1I

m n

MB CM

HA HD

VA VD

P

MP

3.30

Prontuario pa

ra C

álculo d

e Estructuras

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

( ) ( )2

1 22 21 1 2 2 1 2

2

8 1 1

A D

A D

plV V

h hplH Hh k h k hh

= =

+= =

+ + + +

MOMENTOS FLECTORES

( )( ) ( )

21 2 1

2 21 1 2 2 1 28 1 1B

h h hplMh k h k hh

+= −

+ + + +

( )( ) ( )

21 2 2

2 21 1 2 2 1 2

1

8 1 1

( )2

C

X A

h h hplMh k h k hh

En BCpx l x fM H x h

l

+= −

+ + + +

− = − +

h

p

A

B

D

C

l

I 3

I 1

2Ix

MB

CM

HA HD

VA VD

s

2

h 1

f

Formula

rio para

vigas y pórticos

3.31

3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES

( )( ) ( )

21

1

21 1 21

2 21 1 2 2 1 2

2

4 5 28 1 1

A D

A D

D

phV V

lH ph H

h k hphHh k h k hh

= =

= −

+ +=

+ + + +

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 22 8 1 1

4 5 28 1 1

2

B

C

Y A

h k hph phMh k h k hh

h k hph hMh k h k hh

En ABpyM H y

+ += −

+ + + +

+ +=

+ + + +

= −

h

A

B

D

C

l

I 3

I 1

2I

MB

CM

HD

VD

2

HA

VA

p

s

yh

f

1

3.32

Prontuario pa

ra C

álculo d

e Estructuras

3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES

( )

( ) ( )( ) ( )

1 2

21 1 1 2 1 22 21 1 2 2 1 2

2

8 1 48 1 1

A D

A D

D

pf h hV V

lH pf H

h k hh f h hpfHh k h k hh

+= =

= −

+ + + +=

+ + + +

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 48 1 1

8 1 48 1 1

2

B

C

Y A A

h k hh f h hpfhM pfhh k h k hh

h k hh f h hphMh k h k hh

En BCl pyM V y H y hf

+ + + += −

+ + + +

+ + + += −

+ + + +

= − + + −

h

A

B

D

C

l

I 3

I 1

2I

MB

CM

HD

VD

2

HA

VA

ps

y

h

f

1

Formula

rio para

vigas y pórticos

3.33


( ) ( )1 2

2 2 21 1 2 2 1 2

( ) ( )2 1 1

A

D

A D

PbVlPaVl

h l b h l aPabH Hl h k h k hh

=

=

+ + += =

+ + + +

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 aPabhMl h k h k hh

h l b h l aPabhMl h k h k hh

Pab afM H hl l

+ + += −

+ + + +

+ + += −

+ + + +

= + +

h

A

B

D

C

l

I 1

2I

MB

CM

HD

VD

2

HA

VA

s

a b

I 3

MP

1

f

h

3.34

Prontuario pa

ra C

álculo d

e Estructuras

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

2

1

I hkI s

= ⋅

3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES

( ) ( )2

2

28 5

32 3 3

A E

A E

plV V

pl h fH Hh k f h f

= =

+= =

+ + +

MOMENTOS FLECTORES

( ) ( )

( )

2

2

2

8 532 3 3

8

22

B D

C B

BX

pl h h fM Mh k f h f

pl f hM Mh

En BC y DCx l x M fxM p h

h l

+= = −

+ + +

+= +

− = + +

p

A

B

E

C

l

I 2

I 1

x

MB

CM

HA HE

VA VE

I 2

I 1

s

D

h

f

MD

Formula

rio para

vigas y pórticos

3.35

3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES

( ) ( )2

2

38

88 5

64 3 3

A

E

A E

plV

plV

pl h fH Hh k f h f

=

=

+= =

+ + +

MOMENTOS FLECTORES

( ) ( )

( )

2

2

2

8 564 3 3

16

22

B D

C B

BX

pl h h fM Mh k f h f

pl f hM Mh

En BCx l x M fxM p h

h l

+= = −

+ + +

+= +

− = + +

p

A

B

E

C

l

I 2

I 1

x

MB

CM

HA HE

VA VE

I 2

I 1

s

D

h

f

MD

3.36

Prontuario pa

ra C

álculo d

e Estructuras


( )( ) ( )

2

2

2

2

5 12 616 3 3

A E

A E

E

phV Vl

H ph Hk h fphH

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

phM M

ph f hM Mhk h fphM

h k f f hEn AB

pyM H y

= +

+= +

+ += −

+ + +

= − + ⋅

A

B

E

C

l

I 2

I 1

MB

CM

HA HE

VA VE

I 2

I 1

s

D

h

f

MD

p

y

Formula

rio para

vigas y pórticos

3.37


( )

( ) ( )( ) ( )

2

2

22

8 3 5 416 3 3

A E

A E

E

pfV V f hl

H pf Hh k f f hpfHh k f f h

= = +

= −

+ + +=

+ + +

MOMENTOS FLECTORES

( ) ( )( ) ( )

( )

22

2

2

4 2 516 3 3

2

B A

C

D E

x A A

M H hh k f h fpfMh k f f h

M H h

En BC

y hM H y V x p

fsiendo y x hl

= ⋅

+ + += − ⋅

+ + +

= − ⋅

−= ⋅ − ⋅ −

= +

A

B

E

C

l

I 2

I 1

MB

CM

HA HE

VA VE

I 2

I 1

s

D

h

f

MD

p

x

y

3.38

Prontuario pa

ra C

álculo d

e Estructuras


( )( ) ( )

2 2

2 2

6 ln 3 4

4 3 3

A

A

A E

PnVlPmVl

h f l mPmH Hl h k f f h

=

=

+ −= =

+ + +

MOMENTOS FLECTORES

22

B D A

C B

P A A

M M H hPm h fM M

hhl fmM V m H

l

= = − ⋅

+= +

+= ⋅ −

p

A

B

E

C

l

I 2

I 1

MB

CM

HA HE

VA VE

I 2

I 1

s

D

h

f

MD

m n

Formula

rio para

vigas y pórticos

3.39

3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL 1 23 3

1 21 2

h hI Ik y kI l I l

= ⋅ = ⋅


( ) ( )

( ) ( )

( ) ( )

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 plVh k h k hh

h hpl plVh k h k hh

h hplH Hh k h k hh

−= +

+ + + +

−= −

+ + + +

−= =

+ + + +

MOMENTOS FLECTORES

( )( ) ( )

( )( ) ( )

21 2 1

2 21 1 2 2 1 2

21 2 2

2 21 1 2 2 1 2

2

1

8 1 1

8 1 1

2

B

C

x A A

h h hplMh k h k hh

h h hplMh k h k hh

En BCpxM V x H h

+= −

+ + + +

+= −

+ + + +

= ⋅ − − ⋅

h

A

B

D

C

l

I 3

I 1

2I

MB CM

HA

HD

VA

VD

p

x

h

2

1

3.40

Prontuario pa

ra C

álculo d

e Estructuras


( ) ( )

21 1 2

21 1 1 1 2

2 21 1 2 2 1 2

2

5 4 28 1 1

A D D

A D

D

ph h hV V Hl l

H ph Hph k h h hH

h k h k hh

−= = −

= −

+ +=

+ + + +

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 22 8 1 1

5 4 28 1 1

2

B

C

y A

ph ph k h h hMh k h k hh

ph h k h h hMh k h k hh

En ABpyM H y

+ += − −

+ + + +

+ += −

+ + + +

= ⋅ −

h

p

A

B

D

C

l

I 3

I 1

2I

MB

CM

HA

HD

VA

VD

y

MB

h1

2

Formula

rio para

vigas y pórticos

3.41


( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

1 21 23 2 2

1 1 2 2 1 2

1 21 23 2 2

1 1 2 2 1 2

1 22 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 hl l h k h k hh

l b h l a hPa PabV h hl l h k h k hh

l b h l a hPabH Hl h k h k hh

+ + += + −

+ + + +

+ + += − −

+ + + +

+ + += =

+ + + +

MOMENTOS FLECTORES

( ) ( )( ) ( )

( ) ( )( ) ( )

1 212 2 2

1 1 2 2 1 2

1 222 2 2

1 1 2 2 1 2

2 1 1

2 1 1

B

C

P A B

l b h l a hPabhMl h k h k hh

l b h l a hPabhMl h k h k hh

M V a M

+ + += −

+ + + +

+ + += −

+ + + +

= ⋅ +

A

B

D

C

l

I 3

I 1

2I

a b

MB CM

HA

HD

VA

VD

P

MP

h

h1

2

3.42

Prontuario pa

ra C

álculo d

e Estructuras

3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL

2

1

I hkI l

= ⋅


( )2

2 4 2A D A Dpl 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

plM Mk

plM Mk

En BCpx l x plM

k

pl k lM pos para xk

= =+

= = −+

−= −

+

+= =

+

h

A

B

D

C

l

I 2

I 1 1I

x

MB CM

HA

VA

p

MA

HD

VD

MD

Formula

rio para

vigas y pórticos

3.43


( )

( )( )

2

6 1

2 38 2

A D

A D

D

ph kV Vl k

H ph Hph k

Hk

= =+

= −

+=

+

MOMENTOS FLECTORES

2

2

2

2

2

2 1524 6 1 2

2 2124 6 1 2

2 2324 6 1 2

2 1324 6 1 2

2

A

B

C

D

y A A

phMk k

phMk k

phMk k

phMk k

En ABpyM H y M

= − + + + +

= − + + +

= − − − + +

= + − + +

= − + ⋅ +

h

p

A

B

D

C

l

I 2

I 1 1I

MB

CM

y

MB

HA

VA

MA

HD

VD

MD

3.44

Prontuario pa

ra C

álculo d

e Estructuras


( )( )216 1

32 ( 2)

A

D A

A D

m n mPnVl l k

V P VPmnH Hlh k

−= + + = −

= =+

MOMENTOS FLECTORES

( )

( )

( )

( )

12 2 6 1

12 2 6 1

12 2 6 1

12 2 6 1

A

B

C

D

CBP

Pmn n mMl k l k

Pmn n mMl k l k

Pmn n mMl k l k

Pmn n mMl k l k

mMnMPmnMl l l

−= − + +

−= − + + +

−= − − + +

−= + + +

= + +

h

A

B

D

C

l

I 2

I 1 1I

m n

MB CM

P

MP

HA

VA

MA

HD

VD

MD

Formula

rio para

vigas y pórticos

3.45

3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES

3(6 1)

2

A D

A D

PhkV Vl kPH H

= =+

= =

MOMENTOS FLECTORES

3 12 6 1

32 6 1

3 12 6 1

A

B C

D

Ph kMkPh kM M

kPh kM

k

+= −

+

= − =+

+=

+

h

A

B

D

C

l

I 2

I 1 1I

MB CM

P

HD

VD

MD

HA

VA

MA

3.46

Prontuario pa

ra C

álculo d

e Estructuras

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

2

1

I hkI s

= ⋅


( )( ) ( )

2

2 2 2

24 5

8 4

A E

A E

plV V

k h f fplH Hkh f k h hf f

= =

+ += =

+ + + +

MOMENTOS FLECTORES

( ) ( )( ) ( )

( )( ) ( )

( )

2

2 2 2

22

2 2 2

2

2

8 15 648 4

16 1548 4

8

22

A E

B D

C A A

x A A A

kh h f f h fplM Mkh f k h hf f

kh h f fplM Mkh f k h hf f

plM M H h f

En BCxf pxM M V x H hl

+ + −= =

+ + + +

+ += = −

+ + + +

= + − +

= + ⋅ − + −

p

A

B

E

C

l

I 2

I 1

x

MB

CM

I 2

I 1

s

D

h

f

MD

HA

VA

MA

HE

VE

ME

Formula

rio para

vigas y pórticos

3.47

3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES

( )( )

( ) ( )2

2 2 2

24 13

32 3 14 5

16 4

A E

E

A E

plV VkV plk

k h f fplH Hkh 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 15 696 64 3 14

8 15 696 64 3 14

16 1596 64 3 14

16 1596 64 3 14

A

E

B

D

x A

kh h f f h fpl plMkkh f k f fh h

kh h f f h fpl plMkkh f k f fh h

kh h f fpl plMkkh f k f fh h

kh h f fpl plMkkh f k f fh h

En BC M M

+ + −= −

++ + + +

+ + −= +

++ + + +

+ += − −

++ + + +

+ += − +

++ + + +

=

( )

222

2

A A

C E E E

xf pxV x H hl

lM V M H f h

+ ⋅ − + −

= + − +

p

A

B

E

C

l

I 2

I 1

x

MB

CM

I 2

I 1

s

D

h

f

MD

HA

VA

MA

HE

VE

ME

3.48

Prontuario pa

ra C

álculo d

e Estructuras


( )

( )( ) ( )

2

22

2 2 2

2 3 1

2 34 4

A E

A E

E

ph kV Vl k

H ph Hk h k f h fphH

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 1624 3 14

212

6 15 16 6 2 1624 3 14

2

A

B A A

C E E E

D E E

E

y A A

kh k kf h f fph kMkkh f k f fh h

phM M H h

M M H f h VM M H h

kh k kf h f fph kMkkh f k f fh h

En ABpyM M H y

+ + + + + = − + ++ + + +

= + ⋅ −

= − + +

= − ⋅

+ + + + + = − + ++ + + +

= + ⋅ −

A

B

E

C

l

I 2

I 1

MB

CM

I 2

I 1

s

D

h

f

MD

p

y

HE

VE

ME

HA

VA

MA

Formula

rio para

vigas y pórticos

3.49


( )

( ) ( )( ) ( )2

2 2 2

438 3 1

2 4 10 54 4

A E

A E

E

k f h fpfV Vl k

H pf Hk h k f kh kf fpfHkh f k f fh h

+ += =

+= −

+ + + +=

+ + + +

MOMENTOS FLECTORES

( ) ( )( ) ( )

( )

( )

( ) ( )( ) ( )

( )

( ) ( )

2 2 2

2 2 2

2

9 4 6 4 3 2324 2 3 14

2

9 4 6 4 3 2324 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 fpfM fkkh f k f fh h

M M H hlM M H h f V

M M H h

kh f h f h f h k fpfM fkkh f k f fh h

En BCl y h p y h

M M H y Vf

+ + + + + = − + ++ + + +

= + ⋅

= − + +

= − ⋅

+ + + + + = − + ++ + + +

− −= + ⋅ − −

A

B

E

C

l

I 2

I 1

MB

CM

I 2

I 1

s

D

h

f

MD

p

y

HE

VE

ME

HA

VA

MA

3.50

Prontuario pa

ra C

álculo d

e Estructuras


( )

( ) ( ) ( )( ) ( )

2

32 2

2 2 2 2

3 23 13 4 1 3

4

A E

E

A E

V P Vl kl m mPmV

klkl f h fm k lm f khPmH 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

42

3 1

2

3 2 4 2 2 ln 4

42

3 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 hMl n n m

k

M M H hlM 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 hMl n n m

k

+ − + + + + −

+ + + + =

− − +

= − ⋅

= + − +

= − ⋅

+ − + + + + −

+ + + + =

−+ +

2y A A A

En BCfmM M V m H hl

= + ⋅ − +

p

A

B

E

C

l

I 2

I 1

MB

CM

I 2

I 1

s

D

h

f

MD

m n

HA

VA

MA

HE

VE

ME

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Prontuario estructuras