3 formulario para_vigas_y_porticos (1)

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3 formulario para_vigas_y_porticos (1)

  1. 1. 3 FORMULARIO PARA VIGAS Y PÓRTICOS
  2. 2. 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 a q x c A x a T 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 de carga y se indica el valor que deberían tomar los parámetros A y c en la ecua- ción general previamente indicada.
  3. 3. 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 ≤ − = ≥ − = − = − − = − − = =
  4. 4. Limitación de las Deformaciones 3.3 2 M(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 q T x x a q M x x a A q c ≤ − = ≥ − = − = − = − − = − − ⋅ = = 3 a x d q 2 T(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 d q x x a q d T x x a q d M x x a q A d c ≤ − = ≥ − = − = − = − − ⋅ = − − ⋅ ⋅ = =
  5. 5. 3.4 Prontuario para Cálculo de Estructuras Otros casos de carga que se resuelven por superposición de los anteriores x a b q ( ) ( ) ( ) 2 2q M x x-a x-b 2! dM x T x dx  = −〈 〉 + 〈 〉  = q a x b d q/d ( ) ( ) ( ) 3 3 2q/d q M x - x-a x-b x-b 3! 2! dM x T x dx  = 〈 〉 + 〈 〉 + 〈 〉  = q a x b d q/d ( ) ( ) ( ) 2 3 3q q/d M x x-a x-a x-b 2! 3! dM x T x dx  = − 〈 〉 + 〈 〉 − 〈 〉  = a x b d a q b q ( ) ( ) ( ) ( ) a b2 2 b a 3 3 q q M x x-a x-b 2! 2! q q /d x-a x-b 3! dM x T x dx = − 〈 〉 + 〈 〉 + −  + −〈 〉 + 〈 〉  = a x b d a q b q ( ) ( ) ( ) ( ) a b2 2 a b 3 3 q q M x x-a x-b 2! 2! q q /d x-a x-b 3! dM x T x dx = − 〈 〉 + 〈 〉 + −  + 〈 〉 − 〈 〉  =
  6. 6. Formularioparavigasypórticos3.5 3.2 VIGA APOYADA EN LOS EXTREMOS 3.2.1 CARGA PUNTUAL EN LA VIGA REACCIONES A B P b P a R R L L ⋅ ⋅ = = ESFUERZOS CORTANTES ;AC CB P b P a Q cte Q cte L L ⋅ ⋅ = = = − = MOMENTOS FLECTORES ( ) max 0; ; paraAC 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 b L b L a b a E I L E I L E I L ϕ ϕ ϕ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ + = − ⋅ + = ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA ( ) 22 2 2 2 2 2 1 ; 1 6 6 AC CB P L a L xP L b x b x a L x y y E I E I LL L L  ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ −   = ⋅ − − = ⋅ − −    ⋅ ⋅ ⋅ ⋅      FLECHA MAXIMA ( ) 2 23 2 2 2 para 39 3 C P b L b f L b x E I L ⋅ − = ⋅ − = ⋅ ⋅ ⋅ x a A L b C P B QA maxM B Q
  7. 7. 3.6ProntuarioparaCálculodeEstructuras 3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES A B p b c p a c R R L L ⋅ ⋅ ⋅ ⋅ = = ESFUERZOS CORTANTES ; ; 2 AC CD DB p b c p b c c p a c Q Q p a x Q L L L ⋅ ⋅ ⋅ ⋅ ⋅ ⋅  = = − ⋅ − + = −    MOMENTOS FLECTORES ( ) 2 max 0 ; 2 2 2 para 2 2 AC CD DB p b c p b c p 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 GIRO 2 2 ; 6 4 6 4 A B p a b c c p a b c c L b L a E I L a E I L b ϕ ϕ    ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ + − = − ⋅ + −    ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅    ECUACION DE LA ELASTICA ( ) 2 2 4 2 3 2 2 6 4 4 4 24 2 4 6 4 AC CD DB p b c x c y x a L b L E I a p c c y L x a b c x a b c L b x E I L a p a c L x c y L x b L a L E I a   ⋅ ⋅ = ⋅ − + ⋅ + −   ⋅ ⋅ ⋅          = ⋅ ⋅ − − − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + − ⋅    ⋅ ⋅ ⋅ ⋅        ⋅ ⋅ − = ⋅ ⋅ − − + ⋅ + −   ⋅ ⋅ ⋅    QA B Q x a A L b C P B c maxM D
  8. 8. Formularioparavigasypórticos3.7 3.2.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) ( )1 2 1 2 1 1 2 ; 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 R L ⋅ − + ⋅ = = − ⋅ = − ⋅ MOMENTOS FLECTORES ( ) ( ) ( ) ( ) 1 2 2 2 2 max 1 2 1 2 2 2 0 1 1 2 1 2 2 1 3 6 comprendido entre 0,125 y 0,128 2 2 1 1 para 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 p p p − + ⋅ = ⋅ − ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ +   = ⋅ − + ⋅ + + ⋅  −    ANGULOS DE GIRO ( ) ( ) 3 3 1 2 1 28 7 ; 7 8 360 360 A B L L p p p p E I E I ϕ ϕ= ⋅ ⋅ + ⋅ = − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA ( ) ( ) ( ) ( ) ( ) 3 2 1 2 1 2 2 3 1 2 1 2 3 3 4 360 8 7 8 7 x p p x p p Lxx L x y EI p p L x p p L  − − + +−  = + + +   FLECHA MAXIMA ( ) ( )4 4 1 2 1 2 entre 0,01302 y 0,01304 2 2 p p L p p L E I E I + ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ QA maxM QB P A B 1 P2 0x L x
  9. 9. 3.8ProntuarioparaCálculodeEstructuras 3.2.4 MOMENTO FLECTOR REACCIONES R R M L A B= − = − ESFUERZOS CORTANTES x M Q cte L = = 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 3 2 3 1 ; 3 1 6 6 3 A B C M L b M L a E I E IL L M a b E I L ϕ ϕ ϕ    ⋅ ⋅ = ⋅ ⋅ − = ⋅ ⋅ −    ⋅ ⋅ ⋅ ⋅    = ⋅ + ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA 2 2 2 2 22 2 1 3 6 ( ) 1 3 6 AC CB M L x b x y E I L L M L L x a L x y E I LL  ⋅ ⋅ = − ⋅ − ⋅ −  ⋅ ⋅    ⋅ ⋅ − −   = − ⋅ − ⋅ −   ⋅ ⋅    FLECHA ( ) 3 C M a b f b a E I L ⋅ ⋅ = ⋅ − ⋅ ⋅ ⋅ BA M+ QA QB MC MC M a L b C
  10. 10. Formularioparavigasypórticos3.9 3.3 VIGA EMPOTRADA EN LOS EXTREMOS 3.3.1 CARGA PUNTUAL EN LA VIGA REACCIONES ( ) ( ) 2 2 3 3 2 ; 2A B P b P a R L a R L b L L ⋅ ⋅ = ⋅ + ⋅ = ⋅ + ⋅ ESFUERZOS CORTANTES ( ) ( ) 2 2 3 3 2 ; 2AC CB P b P a Q L a cte Q L b cte L L ⋅ ⋅ = ⋅ + ⋅ = = − ⋅ + ⋅ = MOMENTOS FLECTORES ( ) ( ) 2 2 2 2 2 3 2 2 2 2 03 3 ; ; 2 2 2 ; 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 a L L ⋅ ⋅ ⋅ ⋅ ⋅ = − = − = ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ + − ⋅ − ⋅ ⋅ = = ECUACION DE LA ELASTICA ( ) ( ) 2 2 2 22 2 2 3 6 3 2 6 AC BC P b a x x y a x E I L L L xP a L x y b L x b E I L L ⋅ ⋅ ⋅  = ⋅ ⋅ − − ⋅ ⋅ ⋅   −⋅ − ⋅  = ⋅ ⋅ − − − ⋅ ⋅ ⋅ ⋅   FLECHAS ( ) 3 3 3 2 max3 2 2 ; 3 3 2 2 para 2 C P a b P a b f f E I L E I L a a L x L a ⋅ ⋅ ⋅ ⋅ ⋅ = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ = + ⋅ MC A B a L x b P B A Q Q x 0 MA C MB
  11. 11. 3.10ProntuarioparaCálculodeEstructuras 3.3.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES ;A B A B A B p b c M M p a c M M R R L L L L ⋅ ⋅ − ⋅ ⋅ − = − = + ESFUERZOS CORTANTES ; ;AC A BD B CD A c Q R cte Q R cte Q R p x a a   = = = − = = − ⋅ − +    MOMENTOS FLECTORES ( ) 2 3 2 2 2 3 2 2 2 ; 2 2 12 ; 3 12 12 3 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 b L c p c a b M L a L c   = ⋅ + = ⋅ + − ⋅ − +     ⋅ ⋅ ⋅ = ⋅ − + = − ⋅ − ⋅ +  ⋅    ⋅ ⋅ ⋅ = − ⋅ − ⋅ +  ⋅   ECUACION DE LA ELASTICA ( ) ( ) ( ) ( ) 2 4 3 3 3 2 2 3 6 1 4 12 24 2 1 3 3 2 3 6 AC A A CD A A DB B B B A B B B x y M R x E I c y p x a R x M x E I y R x M LR x M LR Lx M LR L EI = ⋅ − ⋅ − ⋅ ⋅ ⋅    = ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅  ⋅ ⋅      = − + + + − +  a MA Q A x A B Q L b B P c C D MB
  12. 12. Formularioparavigasypórticos3.11 3.3.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) ( ) 1 2 1 2 2 6 2 6 A B A A B B L M M R p p L L M M R p p L − = ⋅ ⋅ + − − = ⋅ + ⋅ + ESFUERZOS CORTANTES ( )1 22 2 A A x A B B Q R p L x p x Q R x L Q R = ⋅ ⋅ − + ⋅ = − ⋅ ⋅ = − MOMENTOS FLECTORES ( ) ( ) ( ) 2 1 2 1 2 2 2 1 2 3 2 60 3 6 2 3 60 A x A A B L M p p p L x p x M R x M x L L M p p = − ⋅ + ⋅ ⋅ ⋅ − + ⋅ = ⋅ + − ⋅ ⋅ = − ⋅ + ⋅ ECUACION DE LA ELASTICA ( )2 2 1 3 2 1 4 12 24 5 x A A p px y x p L x R L x M L E I L  − = ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅  ⋅ ⋅ ⋅    A Q Q B x A L B B MA M P2 P1
  13. 13. 3.12ProntuarioparaCálculodeEstructuras 3.3.4 MOMENTO FLECTOR REACCIONES 3 3 6 6 ;A B M M R a b R a b L L ⋅ ⋅ = − ⋅ ⋅ = ⋅ ⋅ ESFUERZOS CORTANTES 3 6 x M Q a b cte L ⋅ = − ⋅ ⋅ = 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 M L L L M M M M a b M M L a b L L ECUACION DE LA ELASTICA ( ) 2 2 2 2 2 2 2 2 AC BC M b x L x b y a E I L LL M a L x b x a y E I L LL ⋅ ⋅ −  = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅   ⋅ ⋅ − ⋅  = ⋅ ⋅ − ⋅ ⋅ ⋅   FLECHA ( ) 2 2 3 2 C M a b f a b E I L ⋅ ⋅ = − ⋅ − ⋅ ⋅ ⋅ MC A Q QB x A L a b B +M CM C A M BM
  14. 14. Formularioparavigasypórticos3.13 3.4 VIGA APOYADA-EMPOTRADA 3.4.1 CARGA PUNTUAL EN LA VIGA REACCIONES ( ) ( ) 2 2 2 3 3 3 ; 3 2 2 A B P b P a R L b R L a L L ⋅ ⋅ = ⋅ ⋅ − = ⋅ ⋅ − ⋅ ⋅ ESFUERZOS CORTANTES ( ) ( ) 2 2 2 3 3 3 ; 3 . 2 2 AC CB P b P a Q L b cte Q L a const L L ⋅ ⋅ = − ⋅ ⋅ − = = − ⋅ ⋅ − = ⋅ ⋅ MOMENTOS FLECTORES ( ) ( ) ( ) ( ) 2 2 2 2 3 2 3 2 2 3 3 ; 3 2 2 2 3 2 ; 2 3 2 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 2 3 ; 2 4 4 A C P a L a P a L a L a L a E I L E I L ϕ ϕ ⋅ − ⋅ ⋅ − = = ⋅ − ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA ( ) ( ) 2 2 2 3 2 2 2 2 2 3 2 12 3 1 3 12 AC BC P b x y a L x L a E I L P a L x a a L x y E I LL L ⋅ ⋅  = ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅  ⋅ ⋅ −     −  = ⋅ ⋅ − − − ⋅      ⋅ ⋅        FLECHA MAXIMA 2 para x= 6 2 2 max p b a a a f L E I L a L a ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ + Q Q MB A B x a L b A B C P MC
  15. 15. 3.14ProntuarioparaCálculodeEstructuras 3.4.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES ;B B A B p b c M p a c M R R L L L L ⋅ ⋅ ⋅ ⋅ = + = − ESFUERZOS CORTANTES ; ; 2 AC A DB B CD A c Q 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 GIRO 3 2 2 12 3 48 A p c a b L b E I L c ϕ  ⋅ ⋅ ⋅ = ⋅ − +  ⋅ ⋅ ⋅   ECUACION DE LA ELASTICA ( ) ( ) 2 2 3 2 4 2 3 3 2 2 12 8 3 48 1 12 8 2 3 48 4 3 6 AC A CD A DB B B x a b y R L x p c L b E I L c c ab y R Lx pL x a pc L b x E I L c L x y R L x M E I   ⋅ ⋅ = ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ − +   ⋅ ⋅ ⋅         = ⋅ − + − + + − +   ⋅ ⋅ ⋅      −  = − ⋅ ⋅ − + ⋅ ⋅ ⋅ Q Q a x C P M A A B B L b c B
  16. 16. Formularioparavigasypórticos3.15 3.4.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) ( )1 2 1 22 ; 2 6 6 B B A B L M L M R p p R p p L L = ⋅ ⋅ + + = ⋅ + ⋅ − ESFUERZOS CORTANTES ( )1 22 ; 2 x A B B p L x p x Q R x Q R L ⋅ ⋅ − + ⋅ = − ⋅ = − ⋅ MOMENTOS FLECTORES ( ) ( ) 2 1 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 2 240 A L p p E I ϕ = ⋅ ⋅ + ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA ( ) ( )4 3 2 2 3 2 1 1 1 25 20 5 12 3 120 x A A x y p p x Lp x R Lx L R L p p L EIL   = − + − + − +   2 A Q Q L x B BA 1P P MB
  17. 17. 3.16ProntuarioparaCálculodeEstructuras 3.4.4 MOMENTO FLECTOR REACCIONES ( )2 2 3 3 2 A B M R R L a L = − = ⋅ ⋅ − ESFUERZOS CORTANTES x AQ R cte= = MOMENTOS FLECTORES ( ) ( ) 2 2 2 2 2 2 3 2 ; ; 3 2 3 ; 3 1 2 2 2 der izq C 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 4 4 4 A C M M b a L a a L b E I L E I L L ϕ ϕ    = ⋅ − ⋅ ⋅ − = ⋅ ⋅ ⋅ ⋅ + −  ⋅ ⋅ ⋅ ⋅ ⋅     ECUACION DE LA ELASTICA ( ) ( ) ( ) ( ) 3 2 2 3 2 2 2 2 3 4 3 4 2 4 AC BC M b x y L x L a L E I L M y L x a L x L a E I L ⋅ ⋅  = ⋅ − ⋅ − − ⋅ ⋅ +  ⋅ ⋅ ⋅  = ⋅ − ⋅ ⋅ ⋅ − ⋅ −  ⋅ ⋅ ⋅ Q x a L b A M Q B A B B CM M MC C +
  18. 18. Formularioparavigasypórticos3.17 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 2 A C P b E I ϕ ϕ= = − ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA ( )( ) ( ) ( ) 2 2 3 ; 2 3 6 6 AC CB P b P y L x b y L x b a E I E I ⋅ = ⋅ ⋅ − − = ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ FLECHA MAXIMA ( ) 3 2 ; 2 3 3 6 C A P b P b f f b a E I E I ⋅ ⋅ = = ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ L a x A b B P Q MB B C
  19. 19. 3.18ProntuarioparaCálculodeEstructuras a x A C M Q L b P B D c B B 3.5.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES . BR p c= ⋅ ESFUERZOS CORTANTES . 0 ; ; 2 AC CD DB c Q Q p x a Q p c cte   = = − ⋅ − + = − ⋅ =    MOMENTOS FLECTORES . ( ) 2 2 2 0 ; ; 2 2 ; AC CD D DB B c p 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 12 D C A C p c c p c c b b E I E I ϕ ϕ ϕ ϕ    ⋅ ⋅ = − ⋅ − = − ⋅ + =    ⋅ ⋅ ⋅ ⋅    ECUACION DE LA ELASTICA . ( ) ( ) ( ) ( ) 2 2 2 3 4 2 2 3 2 ; 3 2 6 6 4 4 3 8 24 2 4 DB AC DC p c p c c y L x b a x y a x b b E I E I p c c y x a c a x b b c E I   ⋅ ⋅ = ⋅ − ⋅ ⋅ − + = ⋅ − ⋅ ⋅ + + ⋅   ⋅ ⋅ ⋅ ⋅         = ⋅ − + + ⋅ ⋅ − ⋅ ⋅ + + ⋅ ⋅   ⋅ ⋅      FLECHAS . ( ) 2 2 2 3 2 3 2 3 12 4 ; 3 2 12 2 6 4 D C A p c c b c f b E I p c c p c c f b b c c f a b b E I E I ⋅     = ⋅ − ⋅ +   ⋅         ⋅ ⋅  = ⋅ + ⋅ ⋅ − + = ⋅ ⋅ ⋅ + + ⋅      ⋅ ⋅ ⋅ ⋅        
  20. 20. Formularioparavigasypórticos3.19 L Q x A B B 1P 2 P B M 3.5.3 CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( )1 2 1 2 BR p p= + ESFUERZOS CORTANTES ( ) 2 2 1 1 1 2; 2 2 x B p p x L Q p x Q p p L − = − ⋅ − ⋅ = − + MOMENTOS FLECTORES ( ) ( ) 2 2 2 1 1 2 13 ; 2 6 6 x B x L M p p x L p M p p L  = − ⋅ − ⋅ + ⋅ ⋅ = − ⋅ + ⋅ ⋅ ANGULOS DE GIRO ( )3 1 23 24 A L p p E I ϕ ⋅ ⋅ + = − ⋅ ⋅ ECUACION DE LA ELASTICA ( ) ( ) ( ) ( ) ( )( ) ( ) 3 2 2 2 1 2 2 2 1 2 1 5 24 2 2 2 x L x L x p p L x p y L EI L L x p p L p p  − −  − − + − − =    − − + + +  FLECHA ( )4 2 14 11 120 A L p p f E I ⋅ ⋅ + ⋅ = ⋅ ⋅
  21. 21. 3.20ProntuarioparaCálculodeEstructuras M L x A B M a b B 3.5.4 MOMENTO FLECTOR REACCIONES 0BR = ESFUERZO CORTANTE 0xQ = MOMENTOS FLECTORES 0 ; ;AC CB ACM M M cte M M= = − = = − ANGULOS DE GIRO C A M b E I ϕ ϕ ⋅ = = − ⋅ ECUACION DE LA ELASTICA ( ) ( ) 2 2 2 ; 2 2 AC BC M M y b L x b y L x E I E I = ⋅ ⋅ ⋅ − ⋅ − = − ⋅ ⋅ ⋅ ⋅ FLECHA ( ) 2 ; 2 2 2 C A M b M f f b L b E I E I ⋅ = = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅
  22. 22. Formularioparavigasypórticos3.21 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 P 0,688 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
  23. 23. 3.22ProntuarioparaCálculodeEstructuras 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 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 QL 0,063 QL A L Q L C 22 Q
  24. 24. Formularioparavigasypórticos3.23 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 0.5 8 2 2 2 2 k k k f f a f b f c k k f a d d e g k − + = = − = + = + = − = = QQ B C A c QL f QL 2 A e QL L A a QL a L 2 C 2 g QL B C k L d L B d QLb QL MOMENTOS FLECTORES ESFUERZOS CORTANTES
  25. 25. 3.24ProntuarioparaCálculodeEstructuras Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES k a b c d f g 2,4 -0,045 1,045 1,427 0,973 0,545 0,473 2,5 -0,094 1,094 1,487 1,013 0,594 0,513 2,6 -0,145 1,145 1,548 1,051 0,645 0,553 2,7 -0,198 1,198 1,608 1,091 0,698 0,595 2,8 -0,255 1,255 1,669 1,130 0,755 0,638 2,9 -0,313 1,313 1,730 1,169 0,813 0,683 3,0 -0,375 1,375 1,791 1,208 0,875 0,730 2 2 2 1 0.5 0.5 8 2 2 2 k k f a f b f k f a d d e g k − + = = − = + = − = = A B C 2 g QL A C B L Q A C Q k L B f QL 2 a QL c QL b QL d QL d L MOMENTOS FLECTORES ESFUERZOS CORTANTES
  26. 26. Formularioparavigasypórticos3.25 3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES k a b c e f g 0,6 0,420 0,580 0,300 0,088 0,080 -0,035 0,7 0,418 0,582 0,350 0,087 0,081 -0,020 0,8 0,414 0,586 0,400 0,086 0,086 -0,006 0,9 0,408 0,592 0,450 0,083 0,091 -0,009 3 2 2 1 0.5 0.5 12 8 2 2 8 k f a f b f k k a k c e g f + = = − = + ⋅ + = = = − A CB k LL L D Q QQ MOMENTOS FLECTORES a L A A a QL 2 g QL 2 f QL C 2 2 e QL f QL B e QL 2 D c QL b QL C B b QL c QL a L D a QL ESFUERZOS CORTANTES
  27. 27. 3.26ProntuarioparaCálculodeEstructuras Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES k a b c e f g 1,0 0,400 0,600 0,500 0,080 0,100 0,025 1,1 0,390 0,610 0,550 0,076 0,110 0,041 1,2 0,378 0,622 0,600 0,072 0,122 0,058 1,3 0,365 0,635 0,650 0,066 0,135 0,076 1,4 0,349 0,651 0,700 0,061 0,151 0,094 1,5 0,322 0,668 0,750 0,055 0,168 0,113 1,6 0,313 0,687 0,800 0,049 0,187 0,133 1,7 0,292 0,708 0,850 0,043 0,208 0,153 1,8 0,269 0,731 0,900 0,036 0,231 0,174 1,9 0,245 0,755 0,950 0,030 0,255 0,196 2,0 0,219 0,781 1,000 0,024 0,281 0,219 3 2 2 1 0.5 0.5 12 8 2 2 8 k f a f b f k k a k c e g f + = = − = + ⋅ + = = = − k L B L A L DC QQ Q ESFUERZOS CORTANTES MOMENTOS FLECTORES e QL 2 B 2 g QL f QL 2 A B b QL c QL A a QL a L e QL 2 C D 2 f QL C a L a QL D b QL c QL
  28. 28. Formularioparavigasypórticos3.27 3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL 2 1 3 2 I h k y N k I l = ⋅ = + 3.9.1 CARGA REPARTIDA VERTICAL REACCIONES 2 3 2 12 A D A D psn V l psm V l ps s H H mn hlN = =   = = −    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 I 1 1I x sa m n MB CM HA HD VA VD
  29. 29. 3.28ProntuarioparaCálculodeEstructuras 3.9.2 CARGA REPARTIDA HORIZONTAL REACCIONES ( ) ( ) 2 2 2 8 6 8 A D D A ph V V l ph N k H N ph N k H N = = + = − = MOMENTOS FLECTORES ( ) ( ) 2 2 2 8 2 8 ( ) 2 B C Y B ph M N k N ph M N k N En AB py h y y M M h = − = − + − = + ⋅ h p A B D C l I 2 I 1 1I MB CM HA HD VA VD y MB
  30. 30. Formularioparavigasypórticos3.29 3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES 3 2 A D A D Pn V l Pm V l Pmn H H lhN = = = = MOMENTOS FLECTORES 3 2 2 3 2 B C P Pmn M M lN N M Pmn lN = = − ⋅ − = h A B D C l I 2 I 1 1I m n MB CM HA HD VA VD P MP
  31. 31. 3.30ProntuarioparaCálculodeEstructuras 3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO 1 23 3 1 2 1 2 h hI I k y k I s I s = ⋅ = ⋅ 3.10.1 CARGA REPARTIDA VERTICAL REACCIONES ( ) ( ) 2 1 2 2 2 1 1 2 2 1 2 2 8 1 1 A D A D pl V V h hpl H H h k h k hh = = + = = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) 2 1 2 1 2 2 1 1 2 2 1 28 1 1 B h h hpl M h k h k hh + = − + + + + ( ) ( ) ( ) 2 1 2 2 2 2 1 1 2 2 1 2 1 8 1 1 ( ) 2 C X A h h hpl M h k h k hh En BC px l x f M H x h l + = − + + + + −   = − +    h p A B D C l I 3 I 1 2 Ix MB CM HA HD VA VD s 2 h 1 f
  32. 32. Formularioparavigasypórticos3.31 3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES ( ) ( ) ( ) 2 1 1 2 1 1 21 2 2 1 1 2 2 1 2 2 4 5 2 8 1 1 A D A D D ph V V l H ph H h k hph H h k h k hh = = = − + + = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) 2 3 1 1 21 1 2 2 1 1 2 2 1 2 2 1 1 21 2 2 2 1 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 hh h k hph h M h k h k hh En AB py M H y + + = − + + + + + + = + + + + = − h A B D C l I 3 I 1 2 I MB CM HD VD 2 HA VA p s y h f 1
  33. 33. 3.32ProntuarioparaCálculodeEstructuras 3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 1 2 1 2 2 2 1 1 2 2 1 2 2 8 1 4 8 1 1 A D A D D pf h h V V l H pf H h k hh f h hpf H h k h k hh + = = = − + + + + = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 2 1 21 1 2 2 1 1 2 2 1 2 2 1 1 1 2 1 22 2 2 1 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 hh f h hpfh M pfh h k h k hh h k hh f h hph M h k h k hh En BC l py M V y H y h f + + + + = − + + + + + + + + = − + + + + = − + + − h A B D C l I 3 I 1 2 I MB CM HD VD 2 HA VA p s y h f 1
  34. 34. Formularioparavigasypórticos3.33 3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) 1 2 2 2 2 1 1 2 2 1 2 ( ) ( ) 2 1 1 A D A D Pb V l Pa V l h l b h l aPab H H l h k h k hh = = + + + = = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 21 2 2 2 1 1 2 2 1 2 1 22 2 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 hh h l b h l aPabh M l h k h k hh Pab af M H h l l + + + = − + + + + + + + = − + + + +   = + +    h A B D C l I 1 2 I MB C M HD VD 2 HA VA s a b I 3 MP 1 f h
  35. 35. 3.34ProntuarioparaCálculodeEstructuras 3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS 2 1 I h k I s = ⋅ 3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) 2 2 2 8 5 32 3 3 A E A E pl V 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 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
  36. 36. Formularioparavigasypórticos3.35 3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES ( ) ( ) 2 2 3 8 8 8 5 64 3 3 A E A E pl V pl V 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 I 2 I 1 x MB CM HA HE VA VE I 2 I 1 s D h f MD
  37. 37. 3.36ProntuarioparaCálculodeEstructuras 3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES ( ) ( ) ( ) 2 2 2 2 5 12 6 16 3 3 A E A E E ph V V l H ph H k h fph H h k f f h = = = − + + = + + + MOMENTOS FLECTORES ( ) ( ) ( ) 2 2 3 2 2 2 4 5 12 6 16 3 3 2 B D C D D y A ph M M ph f h M M h k h fph M h k f f h En AB py M 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
  38. 38. Formularioparavigasypórticos3.37 3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) 2 2 2 2 8 3 5 4 16 3 3 A E A E E pf V V f h l H 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 fpf 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 I 2 I 1 MB CM HA HE VA VE I 2 I 1 s D h f MD p x y
  39. 39. 3.38ProntuarioparaCálculodeEstructuras 3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) 2 2 2 2 6 ln 3 4 4 3 3 A A A E Pn V l Pm V l h f l mPm H H l h k f f h = = + − = = + + + MOMENTOS FLECTORES 2 2 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 I 2 I 1 MB CM HA HE VA VE I 2 I 1 s D h f MD m n
  40. 40. Formularioparavigasypórticos3.39 3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL 1 23 3 1 2 1 2 h hI I k y k I l I l = ⋅ = ⋅ 3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 2 2 1 1 2 2 1 2 2 2 1 2 2 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 2 1 2 2 8 1 1 2 8 1 1 8 1 1 A D A D h hpl pl V h k h k hh h hpl pl V h k h k hh h hpl H H h k h k hh − = + + + + + − = − + + + + − = = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 2 1 1 2 2 1 2 2 1 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 hh h h hpl M h k h k hh En BC px M V x H h + = − + + + + + = − + + + + = ⋅ − − ⋅ h A B D C l I 3 I 1 2 I MB CM HA HD VA VD p x h 2 1
  41. 41. 3.40ProntuarioparaCálculodeEstructuras 3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES ( ) ( ) 2 1 1 2 2 1 1 1 1 2 2 2 1 1 2 2 1 2 2 5 4 2 8 1 1 A D D A D D ph h h V V H l l H ph H ph k h h h H h k h k hh − = = − = − + + = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) 2 3 1 1 1 1 1 2 2 2 1 1 2 2 1 2 2 1 2 1 1 1 2 2 2 1 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 I 3 I 1 2 I MB CM HA HD VA VD y MB h1 2
  42. 42. Formularioparavigasypórticos3.41 3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 23 2 2 1 1 2 2 1 2 1 2 1 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 Pab V h h l l h k h k hh l b h l a hPa Pab V h h l l h k h k hh l b h l a hPab H H l h k h k hh + + + = + − + + + + + + + = − − + + + + + + + = = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 21 2 2 2 1 1 2 2 1 2 1 22 2 2 2 1 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 hh l b h l a hPabh M l h k h k hh M V a M + + + = − + + + + + + + = − + + + + = ⋅ + A B D C l I 3 I 1 2 I a b MB CM HA HD VA VD P MP h h1 2
  43. 43. 3.42ProntuarioparaCálculodeEstructuras 3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL 2 1 I h k I l = ⋅ 3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES ( ) 2 2 4 2 A D A D pl pl V V H H h k = = = = + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) 2 2 2 2 12 2 6 2 2 6 2 3 2 máx 24 2 2 A D B C x pl M M k 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 I 2 I 1 1I x MB CM HA VA p MA HD VD MD
  44. 44. Formularioparavigasypórticos3.43 3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES ( ) ( ) ( ) 2 6 1 2 3 8 2 A D A D D ph k V V l k H ph H ph k H k = = + = − + = + MOMENTOS FLECTORES 2 2 2 2 2 2 1 5 24 6 1 2 2 2 1 24 6 1 2 2 2 3 24 6 1 2 2 1 3 24 6 1 2 2 A B C D y A A ph M k k ph M k k ph M k k ph M k k En AB py M 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
  45. 45. 3.44ProntuarioparaCálculodeEstructuras 3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( )2 1 6 1 3 2 ( 2) A D A A D m n mPn V l l k V P V Pmn H H lh 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 CB P 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 I 2 I 1 1I m n MB CM P MP HA VA MA HD VD MD
  46. 46. Formularioparavigasypórticos3.45 3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES 3 (6 1) 2 A D A D Phk V V l k P H H = = + = = MOMENTOS FLECTORES 3 1 2 6 1 3 2 6 1 3 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 I 2 I 1 1I MB CM P HD VD MD HA VA MA
  47. 47. 3.46ProntuarioparaCálculodeEstructuras 3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS 2 1 I h k I s = ⋅ 3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) 2 2 2 2 2 4 5 8 4 A E A E pl V V k h f fpl H 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 fpl M M kh f k h hf f kh h f fpl 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 I 2 I 1 x MB CM I 2 I 1 s D h f MD HA VA MA HE VE ME
  48. 48. Formularioparavigasypórticos3.47 3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES ( ) ( ) ( ) ( ) 2 2 2 2 2 4 1 3 32 3 1 4 5 16 4 A E E A E pl V V k V pl k k h f fpl H 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 fpl pl M kkh f k f fh h kh h f f h fpl pl M kkh f k f fh h kh h f fpl pl M kkh f k f fh h kh h f fpl pl M kkh f k f fh h En BC M M + + − = − ++ + + + + + − = + ++ + + + + + = − − ++ + + + + + = − + ++ + + + = ( ) 2 2 2 2 A A C E E E xf px V x H h l l M V M H f h   + ⋅ − + −    = + − + p A B E C l I 2 I 1 x MB C M I 2 I 1 s D h f MD HA VA MA HE VE ME
  49. 49. 3.48ProntuarioparaCálculodeEstructuras 3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES ( ) ( ) ( ) ( ) 2 22 2 2 2 2 3 1 2 3 4 4 A E A E E ph k V V l k H ph H k h k f h fph H 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 1 6 24 3 14 2 1 2 6 15 16 6 2 1 6 24 3 14 2 A B A A C E E E D E E E y A A kh k kf h f fph 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 fph k M kkh f k f fh h En AB py M 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
  50. 50. Formularioparavigasypórticos3.49 3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) 2 2 2 2 43 8 3 1 2 4 10 5 4 4 A E A E E k f h fpf V V l k H pf H k h k f kh kf fpf H kh f k f fh h + + = = + = − + + + + = + + + + MOMENTOS FLECTORES ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 9 4 6 4 3 23 24 2 3 14 2 9 4 6 4 3 23 24 2 3 14 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 fpf M f kkh f k f fh h M M H h l M M H h f V M M H h kh f h f h f h k fpf M f kkh f k f fh h En BC l y h p y h M M H y V f  + + + + +  = − +  ++ + + +   = + ⋅ = − + + = − ⋅  + + + + +  = − +  ++ + + +   − − = + ⋅ − − 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
  51. 51. 3.50ProntuarioparaCálculodeEstructuras 3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 2 2 2 2 2 3 2 3 1 3 4 1 3 4 A E E A E V P V l kl m mPm V kl kl 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 2 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 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  + − + + + + −   + + + +  =   −  −  +  = − ⋅ = + − + = − ⋅  + − + + + + −   + + + +  =   − + +  2 y A A A En BC fm M M V m H h l     = + ⋅ − +    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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