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1. 3 FORMULARIO PARA VIGAS Y PÓRTICOS

3. 3.1 Formulario para vigas y pórticos 3.1Obtención 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. ( c A x a q x c − ) − 2 ⋅ − ( ) ∑ = ( ) 2 ! ( ) c − 1 A x c a ⋅ − − ( ) T x ∑ = − ( ) 1 ! c A x c a ⋅ − ( ) ∑ M x = − ! ecuaciones validas solo si n 0 ≥ n en las expresiones x a − 0 si n 0 x a x a 0 = ≤ − = 0 x a x a 1 ≥ − = n y si n 0 x a x a 0 > ≤ − = ( ) n n 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.

4. 3.2 Prontuario para Cálculo de Estructuras M Si 0 x a x a 0 ≤ − = a 0 x a x a 1 ≥ − = x entonces M(x) ( ) 0 M x M x a = − − por lo tanto A c M = = 0 P Si x 1 ≤ a x − a = 0 a ( ) 1 1 x ≥ a x − a = x − a x entonces T(x) ( ) ( ) M x 0 T x = − P x − a 1 = − P x − a M(x) por lo tanto A c = = P 1

5. 3.3 Limitación de las Deformaciones Si x q 2 ≤ a x a 0 − = ( ) 2 2 x a x a x a ≥ − = − a entonces x ( ) 0 q x q x q a = − T(x) ( ) T x 1 x a = − − 1 q ( ) 2 M x x a = − − 2 2 1 ⋅ M(x) por lo tanto A c q = = 2 Si q 3 x a x a 0 ≤ − = ( ) 3 3 x a x a x a ≥ − = − d a x entonces 2 q d ( ) 1 T(x) q x x a = − 1 q d ( ) T x 2 x a = − − 2 1 q d ⋅ ⋅ ⋅ 3 M(x) ( ) 3 M x x a = − − 3 2 1 q d por lo tanto A = c 3 =

6. 3.4 Prontuario para Cálculo de Estructuras Otros casos de carga que se resuelven por superposición de los anteriores q q 2! dM x dx ( )     2 2 M x x-a x-b = −〈 〉 + 〈 〉 ( ) a ( ) T x = b x q/d q/d 3! dM x dx q 2! q ( )     3 3 2 M x = - x-a 〈 〉 + 〈 x-b 〉 + 〈 x-b 〉 ( ) a d ( ) T x = b x q/d q 2! q/d 3! q ( )     2 3 3 M x x-a x-a x-b = − 〈 〉 + 〈 〉 −〈 〉 ( ) dM x dx a d ( ) T x = b x q 2! q q 2! ( ) a b 2 2 M x x-a x-b = − 〈 〉 + 〈 〉 + q q b ( ) a q 3! /d − b a     3 3 x-a x-b + −〈 〉 + 〈 〉 a d ( ) dM x dx b ( ) T x = x q 2! q q 2! ( ) a b 2 2 M x x-a x-b = − 〈 〉 + 〈 〉 + q a q ( ) b q 3! /d − a b     3 3 x-a x-b + 〈 〉 − 〈 〉 a d ( ) dM x dx b ( ) T x = x

7. Formulario para vigas y pórticos 3.2 VIGA APOYADA EN LOS EXTREMOS P A B 3.2.1CARGA PUNTUAL EN LA VIGA REACCIONES P b P a R R L L ESFUERZOS CORTANTES P b Q cte Q L MOMENTOS FLECTORES P b M x M L ANGULOS DE GIRO ( ) ; 6 E I L ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA P L b x b y E I L ⋅ ⋅  FLECHA MAXIMA ( 9 3 E I L ⋅ ⋅ ⋅ C ⋅ ⋅ = = A B x P a L ⋅ ⋅ a b ; cte = = = − = AC CB L P a L P a b L ⋅ ⋅ ⋅ ⋅ ( ) ; L x ; M M para x a = ⋅ = ⋅ − = = = AC CB max C 0 Q B P a b ⋅ P a b E I L ⋅ ⋅ ⋅ P a b E I L ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( ) ( ) L b L a ; b a ϕ = ⋅ + ϕ = − ⋅ + ϕ = ⋅ − A B C QA 6 3 ( )         2 P L a L ⋅ ⋅ ⋅ − x      2 2 2 ⋅ ⋅ ⋅ x L a L L − x       = ⋅ 1 − − ; y = ⋅ 1 − − AC CB 2 2 2 6 6 E I ⋅ ⋅ L 2 2 P b ⋅ L b − 3 ) 2 2 2 C f L b para x = ⋅ − = 3 M 3.5 max

8. 3.6 3.2.2CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES p b c p a c R R L L ESFUERZOS CORTANTES p b c p b c Q Q L L MOMENTOS FLECTORES p b c p b c M x M L L p a c M L x L p b c b c M a c L L ⋅   ANGULOS DE GIRO 2 ; 6 4 E I L a ⋅ ⋅ ⋅ ⋅   ECUACION DE LA ELASTICA p b c x y x a L L E I ⋅ ⋅         = ⋅ ⋅ − −     ⋅ ⋅ ⋅      ⋅ ⋅ − = ⋅ ⋅ − − +  ⋅ ⋅   c P ⋅ ⋅ ⋅ ⋅ = = A B A B C D c p a c L ⋅ ⋅ ⋅ ⋅ ⋅ ⋅       = ; = − p ⋅ − a + x ; Q = − AC CD DB 2 x       ⋅ ⋅ ⋅ ⋅ p c       a b 2 = ⋅ ; = ⋅ − x ⋅ x − a − AC CD 2 2 ⋅ ⋅ L ( ) = ⋅ − DB ⋅ ⋅ ⋅ c b c L ⋅     = ⋅ 2 ⋅ − + para x = a − + max 0 2 2 Q Prontuario para Cálculo de Estructuras B QA           2 p a b c ⋅ ⋅ ⋅ c p a b c E I L ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ c = ⋅ L + b − = − ⋅ L + a − ϕ ϕ A B 6 4 ⋅ b            2 c ⋅ ⋅ 2 b = ⋅ − + ⋅ + − AC 6 4 a ⋅     4       2    p E I L c c 3 y L x a 4 b c x ⋅ ⋅ 4 a b c ⋅ L b x − ⋅ + ⋅ ⋅ ⋅ + − ⋅ CD 24 2 4 ⋅ a           2 p a c L L x c ( ) 2 y L x b ⋅ L + a − M DB 6 E I 4 a ⋅ max

9. 3.2.3CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) ( 1 2 2 ; 6 6 ESFUERZOS CORTANTES ( 1 3 ; 6 MOMENTOS FLECTORES ( ) 1 2 3 6 L ⋅ Formulario para vigas y pórticos 1 1 ) . R p p R p 2 p = ⋅ + = + ⋅ P 1 A B 1 2 P 2 ) p L x ⋅ p x ⋅ − + ⋅ 2 2 Q R Q R x ; Q R = = − ⋅ = − A A x A B B L B A x p L x p x − + ⋅ 2 M R x x = ⋅ − ⋅ x A L 2 2 L L ( ) ( ) M comprendido entre 0,125 p p y 0,128 p p ⋅ ⋅ + ⋅ ⋅ + max 1 2 1 2 2 2 QA         1 − 1 3 ( ) 2 1 2 2 para x p p p p p ⋅ = ⋅ − + ⋅ + + 0 1 1 2 p p 2 1 ANGULOS DE GIRO 3 L ϕ = ⋅ ⋅ ECUACION DE LA ELASTICA ( ) (   FLECHA MAXIMA QB 3 L ( ) ( ) 8 p 7 p ; 7 p 8 p ⋅ ⋅ + ⋅ ϕ = − ⋅ ⋅ + ⋅ A 1 2 B 1 2 360 E I 360 E I ⋅ ⋅ ( ) ( ( )       3 2 3 p − p x − 3 4 p + p Lx + x L − x 1 2 1 2 p y = ) ) x 360 EI 2 3 8 p + 7 p L x + 8 p + 7 L 1 2 1 2 ( ) ( ) 4 4 p + p E I ⋅ ⋅ ⋅ L p + p E I ⋅ ⋅ ⋅ L 1 2 1 2 entre 0,01302 ⋅ y 0,01304 ⋅ 2 2 M max x 3.7 0

10. 3.8 3.2.4MOMENTO FLECTOR REACCIONES M L R R = − = − A B M + C ESFUERZOS CORTANTES M Q cte L MOMENTOS FLECTORES M M x L M M a L ANGULOS DE GIRO M L E I M E I L ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA M L x y E I ⋅ ⋅ A B a = = b x L M L ( ) M L x = − ⋅ = − ⋅ − AC CB M L izq C der C izq C der C M b M M M = − ⋅ = − ⋅ = +           2 2 b L M L E I ⋅ ⋅ a L ⋅ ⋅ Prontuario para Cálculo de Estructuras 3 1 ;   3 1 ϕ = ⋅ ⋅ − ϕ = ⋅ ⋅ − QA QB A B 2 2 6 6 ⋅ ⋅ ( ) 3 3 a b ϕ = ⋅ + C 2 3       2 2 b L x L ⋅ ⋅ 1 3 − ⋅ = − ⋅ − MC AC 2 2 6         2 2 M L L ⋅ ⋅ ( E I ⋅ ⋅ x ) a L L x − −      y 1 3 − ⋅ = − ⋅ − M CB 2 6 L MC FLECHA M a b E I L ⋅ ⋅ ⋅ ⋅ ⋅ ( ) C f b a = ⋅ − 3

11. 3.3VIGA EMPOTRADA EN LOS EXTREMOS Formulario para vigas y pórticos P 3.3.1CARGA PUNTUAL EN LA VIGA REACCIONES P b R L ESFUERZOS CORTANTES 2 P b Q L MOMENTOS FLECTORES P a b M L P a M L ECUACION DE LA ELASTICA 2 3 6 E I ⋅ ⋅  C B 2 2 A P a L ⋅ ⋅ ( ) ( ) L 2 a ; R L 2 b = ⋅ + ⋅ = ⋅ + ⋅ A B 3 3 x a b 2 P a L ⋅ ⋅ ( ) ( ) L 2 a cte ; Q L 2 b cte = ⋅ + ⋅ = = − ⋅ + ⋅ = AC CB 3 3 L 2 2 2 ⋅ ⋅ P a ⋅ ⋅ b P b L ⋅ ( ) = − ; M = − ; M = ⋅ L x ⋅ + ⋅ 2 a x ⋅ − a L ⋅ A B AC 2 2 3 L Q B 2 2 2 ⋅ 2 P a ⋅ ⋅ ⋅ b ( ) 2 = ⋅ L b ⋅ + L L x − ⋅ − ⋅ 2 b x ⋅ ; M = para x = a BC C 0 Q 3 3 L A 2 P b ⋅ 2 a x L x L ⋅ ⋅      y a x = ⋅ ⋅ − − ⋅ AC 2 ( ) MB 2 L x − 2 P a ⋅ ⋅ ⋅ L − ⋅ x       ( ) MA 3 2 y = ⋅ ⋅ b − L − x − ⋅ b ⋅ BC 2 6 E I L L FLECHAS P a f = MC 3 3 3 2 b 2 E I ⋅ ⋅ ⋅ P a ⋅ ⋅ b ⋅ ⋅ ⋅ ; max f = C ( ) 3 2 3 E I L ⋅ ⋅ ⋅ 3 L 2 a + ⋅ x 0 2 L a L ⋅ ⋅ ⋅ para x = 3.9 2 a +

12. 3.10 3.3.2CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES p b c M M R R L L ESFUERZOS CORTANTES c P ⋅ ⋅ − p a c L ⋅ ⋅ M − M = − ; = + A B A B A B L C D B A c a       Q R cte ; Q R cte ; Q R p x a = = = − = = − ⋅ − + x AC A BD B CD A MOMENTOS FLECTORES a b 2 p c       L M = R ⋅ + x M ; M = R ⋅ + x M − ⋅ x − a + AC A A CD A A 2 2       3 2 p c ⋅ 12 ⋅ a b c ⋅ ( ) M = R ⋅ L − x + M ; M = − ⋅ L − ⋅ 3 b + Prontuario para Cálculo de Estructuras BD B B A 2 2 12 ⋅ L QA       3 2 p c ⋅ 12 ⋅ a c ⋅ b M = − ⋅ L − ⋅ 3 a + Q B 2 2 12 ⋅ L B ECUACION DE LA ELASTICA ( 6 1 24 E I ⋅ ⋅   2 x ⋅ ⋅ ) y 3 M R x = ⋅ − ⋅ − ⋅ AC A A E I MA       4 c       MB 3 3 y p x a 4 R x 12 M x = ⋅ ⋅ − + − ⋅ ⋅ − ⋅ ⋅ CD A A 2 1 EI ( ) ( ) ( )     3 2 2 y R x 3 M LR x 3 2 M LR Lx 3 M LR L = − + + + − + DB B B B A B B B 6

13. 3.3.3CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) 1 2 2 6 L L M M R p p L ESFUERZOS CORTANTES Q R p L x p x Q R x L Q R = − Formulario para vigas y pórticos L M − M P1 R = ⋅ ⋅ p + p − A B A P2 − ( ) = ⋅ + ⋅ 2 + A B B 1 2 6 B A = A A ( ) x 2 ⋅ ⋅ − + ⋅ 1 2 = − ⋅ x A 2 ⋅ L B B MOMENTOS FLECTORES ( 3 60 2 L ) M p 2 p = − ⋅ + ⋅ Q A 1 2 A ( ) p 3 L x L p x ⋅ ⋅ − + ⋅ 1 2 QB 2 M R x M x = ⋅ + − ⋅ x A A 6 ⋅ 2 L ( ) M 2 p 3 p = − ⋅ + ⋅ B 1 2 60 ECUACION DE LA ELASTICA 2 x y E I L ⋅ ⋅ ⋅ M ( )         p p − A M 2 1 3 2 x p L x ⋅ ⋅ 4 R L x ⋅ ⋅ − 12 M L = ⋅ ⋅ + − ⋅ ⋅ ⋅ B x 1 A A 24 5 3.11

14. 3.12 3.3.4MOMENTO FLECTOR REACCIONES 6 A R a b L ESFUERZOS CORTANTES 6 x Q a b L MOMENTOS FLECTORES ⋅  = ⋅ − ⋅    ⋅ = ⋅ ⋅    ⋅ = − ⋅ ⋅   ⋅ = − ⋅ 3 C A M M L ECUACION DE LA ELASTICA 2 2 2 E I L ⋅ ⋅ ⋅  ⋅ ⋅ − = ⋅ ⋅ ⋅ FLECHA ( 3 2 E I L ⋅ ⋅ ⋅ M 6 M ⋅ ⋅ ; R a b ⋅ = − ⋅ ⋅ = ⋅ B 3 3 L C +M B A ⋅ M x = − ⋅ ⋅ = cte 3 a b L M a L M a L b L M b L ⋅ a L          M 2 3 M = − ⋅ 2 − ⋅ 3 A B    a L x L       L M 3 ⋅ − ⋅ 1 2 − 1 AC    M b L b L − L x       Prontuario para Cálculo de Estructuras M 3 ⋅ − ⋅ 1 2 − 1 Q QB CB A 6 M M L ( ) izq 2 der C 3 2 a ⋅ b ; M = M + ⋅ L − ⋅ 6 a ⋅ b A 3 M b x ⋅ L x b L ⋅ −      y a = ⋅ ⋅ ⋅ − AC 2 L M ( ) 2 M C M a L x b x L ⋅ a L       B y ⋅ 2 ⋅ − BC 2 2 E I L M A MC 2 2 M a ⋅ b ⋅ ) C f = − ⋅ a b −

15. 3.4VIGA APOYADA-EMPOTRADA Formulario para vigas y pórticos P 3.4.1CARGA PUNTUAL EN LA VIGA REACCIONES P b R = C 2 P a ⋅ ⋅ ⋅ ( ) ( ) 2 2 ⋅ 3 ⋅ − L b ; R = ⋅ 3 ⋅ L − a B A A B 3 3 2 2 ⋅ L L ESFUERZOS CORTANTES P b Q L ⋅ MOMENTOS FLECTORES P a M L P x M L ⋅ ANGULOS DE GIRO ( 4 E I L ⋅ ⋅ ⋅ ECUACION DE LA ELASTICA 2 P b x y E I L P a L y E I ⋅ ⋅ x 2 ⋅ P a ⋅ ⋅ ( ) ( ) 2 2 = − ⋅ 3 ⋅ − L b = cte ; Q = − ⋅ 3 ⋅ L − a = const . AC CB 3 3 2 2 L a b L ⋅ ⋅ ⋅ P a ⋅ ⋅ ( ) ( ) 2 2 2 = − ⋅ L − a ; M = ⋅ b ⋅ 3 ⋅ a + ⋅ 2 b B C 2 3 2 2 L P a ⋅ ⋅ ( ) ( ) 2 3 2 2 = ⋅ b ⋅ 3 ⋅ a + 2 ⋅ b ; M = ⋅ 2 ⋅ L − ⋅ 3 L ⋅ + x a ⋅ x AC CB 3 3 2 2 L Q B Q ) ( ) 2 2 P a L ⋅ − a P a L ⋅ ⋅ − a ( ) A 2 2 = ; = ⋅ L − ⋅ 2 a L ⋅ − a ϕ ϕ A C 3 4 E I L ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( )     2 2 3 a L ⋅ x 2 L a = ⋅ ⋅ − ⋅ ⋅ + AC 3 12 MB ( ) 2         x ⋅ −              2 2 a L a L L x −      = ⋅ 3 ⋅ 1 − − 3 − ⋅ BC 2 2 12 L FLECHA MAXIMA p b f = ⋅ ⋅ 2 ⋅ ⋅ a a L ⋅ + a L ⋅ + para x= ⋅ L ⋅ max 3.13 6 E I 2 a 2 a MC

16. 3.14 3.4.2CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES p b c M p a c R R L L L ESFUERZOS CORTANTES c P M L ⋅ ⋅ ⋅ ⋅ ; = + = − B B A B C B c       A Q R cte ; Q R cte ; Q R p x a = = = − = = − ⋅ − + AC A DB B CD A 2 x MOMENTOS FLECTORES a b 2 p c       L M R x ; M R x x a = ⋅ = ⋅ − ⋅ − + AC A CD A 2 2       2 p a b c ⋅ ⋅ ⋅ c ⋅ ( ) M R L x M ; M L a = ⋅ − + = − ⋅ + − DB B B B 2 4 ⋅ b 2 L Prontuario para Cálculo de Estructuras Q B ANGULOS DE GIRO p c E I L ⋅ ⋅ ⋅ Q A       3 2 12 a b c ⋅ ⋅ ⋅ ϕ = ⋅ L − 3 b + A 2 48 ECUACION DE LA ELASTICA x y E I L ⋅ ⋅ ⋅ M                   2 12 a b c ⋅ ⋅ B 2 3 8 R L x ⋅ ⋅ p c ⋅ L 3 b = ⋅ − ⋅ + ⋅ − + AC A 2 48     4       2 1 E I L ⋅ ⋅ ⋅ c 12 ab c       3 3 y = ⋅ − 8 R Lx + 2 pL x − a + + pc L − 3 b + x CD A 2 48 4 ( ) 2 L − x ( )     y = − ⋅ R ⋅ L − x + 3 ⋅ M DB B B 6 E I ⋅ ⋅

17. Formulario para vigas y pórticos 3.4.3CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) 1 2 2 ; 6 L ESFUERZOS CORTANTES ( ) 1 2 2 2 L ⋅ MOMENTOS FLECTORES ( ) 1 2 3 6 L ⋅ ANGULOS DE GIRO ( ) 1 2 3 2 240 E I ⋅ ⋅ ECUACION DE LA ELASTICA ( ) 2 1 1 5 120 EIL P 2 1P L M L M L ( ) R = ⋅ ⋅ p + p + R = ⋅ p + 2 ⋅ p − B B A B 1 2 6 B A x L p L x p x ⋅ ⋅ − + ⋅ Q = R − ⋅ x ; Q = − R x A B B Q Q B A p L x p x ⋅ ⋅ − + ⋅ 2 L ( ) 2 M = R ⋅ − x ⋅ x ; M = − ⋅ 7 ⋅ p + ⋅ 8 p x A B 1 2 120 MB 3 L ϕ = ⋅ ⋅ p + ⋅ p A x ( )     12     4 3 2 2 3 y p p x Lpx 20 R Lx 5 L R L 3 p p L = − + − + − + x A A 1 2 3.15

18. 3.16 3.4.4MOMENTO FLECTOR REACCIONES ( 3 2 L ESFUERZOS CORTANTES Q R cte = = 3 M ) 2 2 R R L a = − = ⋅ ⋅ − M A B C+ B A x x A a MOMENTOS FLECTORES b L M ⋅    ( ) der C izq C 2 2 M = R ⋅ a − M ; M = R ⋅ a ; M = ⋅ L − ⋅ 3 a A A B 2 2 L            2 3 2 M x L ⋅ M x L a L ( ) 2 2 M = ⋅ ⋅ L − a ; M = ⋅ 3 ⋅ ⋅ 1 − − 2 AC BC 3 2 2 Prontuario para Cálculo de Estructuras ANGULOS DE GIRO M E I L ⋅ ⋅ ⋅ Q Q A B         2 M E I ⋅ ⋅ b L a L       ( ) ( ⋅ ) L a a L b ϕ = ⋅ − 3 ⋅ − ; ϕ = ⋅ ⋅ 3 ⋅ ⋅ 1 + − 4 A C 4 4 ECUACION DE LA ELASTICA M b x y E I L M y E I L ⋅ ⋅ ⋅ MC ⋅ ⋅ ( ) ( )     3 2 2 = ⋅ − ⋅ 4 L − x − ⋅ 3 L ⋅ a + L AC 3 4 ⋅ ⋅ ⋅ M B ( ) ( ) 2     2 2 2 = ⋅ L − x ⋅ 2 ⋅ a ⋅ − ⋅ L x L − a BC 3 4 M C

19. 3.5VIGA EMPOTRADA EN UN EXTREMO Formulario para vigas y pórticos P 3.5.1CARGA PUNTUAL EN LA VIGA REACCIONES R P = C B A B x ESFUERZOS CORTANTES 0 ; AC Q = MOMENTOS FLECTORES 0 ; AC M = ANGULOS DE GIRO a b Q = − = P cte CB L ( ) M = − ⋅ P x − a ; M = − ⋅ P b CB B P E I ⋅ ⋅ Q 2 ϕ = ϕ = − ⋅ b B A C 2 ECUACION DE LA ELASTICA ( 3 6 E I ⋅ ⋅ FLECHA MAXIMA 3 ; 3 E I ⋅ ⋅ 2 P b ⋅ P E I ⋅ ⋅ ) ( ) ( ) ( ) 2 y = ⋅ ⋅ L − x − b ; y = ⋅ L − x ⋅ 2 ⋅ b + 3 ⋅ a AC CB 6 MB 2 P b ⋅ P b ⋅ ⋅ ⋅ ( ) C f = f = ⋅ 2 ⋅ b + ⋅ 3 a A 6 E I 3.17

20. 3.18 3.5.2CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES . R p c = ⋅ c B P ESFUERZOS CORTANTES . A c       Q 0 ; Q p x a ; Q p c cte = = − ⋅ − + = − ⋅ = C D AC CD DB 2 B MOMENTOS FLECTORES . 2 x c       p ⋅ x − a + 2 p c ⋅ 2 M = 0 ; M = − ; M = − a b AC CD D 2 = − ⋅ 2 ( ) M = − ⋅ p c ⋅ x − a ; M p c b ⋅ DB B L ANGULOS DE GIRO . p c E I ⋅ ⋅             2 2 ⋅ c p c E I ⋅ ⋅ ⋅ c 2 2 ϕ = − ⋅ b − ; ϕ = − ⋅ b + ; ϕ = ϕ Prontuario para Cálculo de Estructuras D C A C 2 4 2 12 ECUACION DE LA ELASTICA . ( 6 E I ⋅ ⋅ Q               2 p c ⋅ p c E I ⋅ ⋅ ⋅ c ) ( B ) ( ) 2 2 3 y = ⋅ L − x ⋅ 2 ⋅ b − a + x ; y = ⋅ a − x ⋅ 3 ⋅ b + + ⋅ 2 b DB AC 6 4         4       2 p ⋅ ⋅ c c       ( ) 2 3 y = ⋅ x − a + + 4 ⋅ c a ⋅ − x ⋅ 3 ⋅ b + + 8 ⋅ b ⋅ c DC 24 E I 2 4 FLECHAS . p c f E I 2 ⋅ ⋅ c b c             = ⋅ b − ⋅ + M D 2 3 12 B         2               2 p c ⋅ ⋅ ⋅ c p c E I ⋅ ⋅ ⋅ c       ( ) 3 2 3 C f = ⋅ b + ⋅ 4 b c ⋅ − + c ; A f = ⋅ a ⋅ 3 ⋅ b + + ⋅ 2 b 12 E I 2 6 4

21. Formulario para vigas y pórticos 3.5.3CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES ( ) 1 2 2 ESFUERZOS CORTANTES 2 2 1 1 ; 2 L MOMENTOS FLECTORES 2 3 6 L ⋅ ANGULOS DE GIRO ( ) 1 2 3 24 E I ⋅ ⋅ ECUACION DE LA ELASTICA ( )( ) ( ( )( 2 1 2 L L x p p − − +  FLECHA ( ) 2 1 4 11 120 E I ⋅ ⋅ 1 2P R = p + p B 1P A p − p x L ( ) Q = − ⋅ − p x ⋅ Q = − p + p 1 2 x B 2 B x 2 x L ( ) ( ) L     M = − ⋅ p − p ⋅ + x L p ⋅ ⋅ ; M = − ⋅ p + ⋅ 2 p 2 1 1 2 1 x B 6 3 L ⋅ ⋅ p + p ϕ = − A Q B          3 L x L − ( ) 2 ) 2 L x EI − p p L x p − − + − − y = 2 1 2 5 x 24 ) ( ) 2 2 L p 2 p + + 2 1 M B 4 L p p ⋅ ⋅ + ⋅ A f = 3.19

22. 3.20 3.5.4MOMENTO FLECTOR REACCIONES 0 B R = ESFUERZO CORTANTE 0 x Q = MOMENTOS FLECTORES 0 ; AC CB M M = ANGULOS DE GIRO M b E I ⋅ ECUACION DE LA ELASTICA ( 2 2 E I ⋅ ⋅ FLECHA 2 ; 2 E I ⋅ ⋅ M A B x a b L M cte ; M M = − = = − AC Prontuario para Cálculo de Estructuras ⋅ ϕ = ϕ = − C A M M E I ⋅ ⋅ ) ( ) 2 y = ⋅ b ⋅ ⋅ − ⋅ − L 2 x b ; y = L − x AC BC 2 M b ⋅ M E I ⋅ ⋅ ( ) C f f b 2 L b = = ⋅ ⋅ ⋅ − M A 2 B

23. 3.6VIGAS CONTINUAS DE DOS VANOS IGUALES Formulario para vigas y pórticos P P P A B C A B C L/2 L/2 L/2 L/2 L/2 L/2 L L L L 0,688 P 0,405 P 0,312 P 0,094 P 0,094 P A B C A B C 0,312 P 0,594 P 0,688 P ESFUERZOS CORTANTES ESFUERZOS CORTANTES - 0,188 PL - 0,094 PL A B C A B C 0,156 PL 0,156 PL 0,203 PL MOMENTOS FLECTORES MOMENTOS FLECTORES 3.21

24. 3.22 Q Q Q A B C A B C L L L L 0,625 QL 0,375 L 0,437 QL 0,375 QL 0,063 QL A B C A B C 0,375 QL 0,563 QL 0,375 L 0,437 L Prontuario para Cálculo de Estructuras 0,625 QL ESFUERZOS CORTANTES ESFUERZOS CORTANTES 2 2 - 0,125 QL - 0,063 QL A B C A B C 2 2 2 0,07 QL 0,07 QL 0,096 QL MOMENTOS FLECTORES MOMENTOS FLECTORES

25. 3.7VIGAS CONTINUAS DE DOS VANOS DESIGUALES Formulario para vigas y pórticos Q Q Relación entre luces MOMENTOS FLECTORES ESFUERZOS CORTANTES A B C L k L k a b c d e f g 0,361 0,639 0,676 0,424 0,065 0,139 0,09 1,1 c QL 0,345 0,655 0,729 0,471 0,060 0,155 0,111 1,2 d L a QL 0,326 0,674 0,784 0,516 0,053 0,174 0,133 1,3 0,305 0,695 0,840 0,560 0,047 0,195 0,157 1,4 C A B 0,281 0,719 0,896 0,604 0,040 0,219 0,183 1,5 0,255 0,745 0,953 0,647 0,033 0,245 0,209 1,6 b QL d QL a L 0,226 0,774 1,011 0,689 0,026 0,274 0,237 1,7 0,195 0,805 1,070 0,730 0,019 0,305 0,267 1,8 ESFUERZOS CORTANTES 0,161 0,839 1,128 0,772 0,013 0,339 0,298 1,9 0,125 0,875 1,128 0,812 0,008 0,375 0,330 2,0 2 f QL 0,086 0,914 1,247 0,853 0,004 0,414 0,364 2,1 0,045 0,954 1,308 0,892 0,001 0,455 0,399 2,2 0,001 0,999 1,367 0,933 0,000 0,499 0,435 2,3 A B C 2 k k 1 k f k − + 2 f a 0.5 f b 0.5 f c = = − = + = + e QL 8 f k 2 2 g QL 2 2 k a d 3.23 d e g = − = = MOMENTOS FLECTORES 2 2 2

26. 3.24 Q Q Relación entre luces MOMENTOS FLECTORES ESFUERZOS CORTANTES A B C L k L k a b c d f g c QL -0,045 1,045 1,427 0,973 0,545 0,473 2,4 -0,094 1,094 1,487 1,013 0,594 0,513 2,5 d L -0,145 1,145 1,548 1,051 0,645 0,553 2,6 A -0,198 1,198 1,608 1,091 0,698 0,595 2,7 C B a QL -0,255 1,255 1,669 1,130 0,755 0,638 2,8 -0,313 1,313 1,730 1,169 0,813 0,683 2,9 Prontuario para Cálculo de Estructuras d QL b QL -0,375 1,375 1,791 1,208 0,875 0,730 3,0 ESFUERZOS CORTANTES 2 f QL 2 k k 1 − + f a 0.5 f b 0.5 f = = − = + 8 f k 2 2 k a d d e g = − = = 2 2 2 A B C 2 MOMENTOS FLECTORES g QL

27. 3.8VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES Formulario para vigas y pórticos Q Q Q Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES A B C D L k L L k a b c e f g 0,420 0,580 0,300 0,088 0,080 -0,035 0,6 0,418 0,582 0,350 0,087 0,081 -0,020 0,7 a L a QL b QL 0,414 0,586 0,400 0,086 0,086 -0,006 0,8 c QL 0,408 0,592 0,450 0,083 0,091 -0,009 0,9 D B C A c QL a QL b QL a L 3 k 1 + f a 0.5 f b 0.5 f = = − = + ESFUERZOS CORTANTES 12 k 8 ⋅ + 2 2 k a k c e g f = = = − 2 2 2 2 8 f QL f QL 2 g QL A B C D 2 2 e QL e QL 3.25 MOMENTOS FLECTORES

28. 3.26 Q Q Q Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES A B C D L k L L k a b c e f g 0,400 0,600 0,500 0,080 0,100 0,025 1,0 0,390 0,610 0,550 0,076 0,110 0,041 1,1 a L a QL b QL 0,378 0,622 0,600 0,072 0,122 0,058 1,2 c QL 0,365 0,635 0,650 0,066 0,135 0,076 1,3 D 0,349 0,651 0,700 0,061 0,151 0,094 1,4 C B A 0,322 0,668 0,750 0,055 0,168 0,113 1,5 c QL 0,313 0,687 0,800 0,049 0,187 0,133 1,6 b QL Prontuario para Cálculo de Estructuras a QL a L 0,292 0,708 0,850 0,043 0,208 0,153 1,7 ESFUERZOS CORTANTES 0,269 0,731 0,900 0,036 0,231 0,174 1,8 0,245 0,755 0,950 0,030 0,255 0,196 1,9 0,219 0,781 1,000 0,024 0,281 0,219 2,0 2 2 f QL f QL 3 k 1 + f a 0.5 f b 0.5 f = = − = + 12 k 8 ⋅ + A B C D 2 2 2 g QL e QL 2 2 e QL k a k c e g f = = = − 2 2 8 MOMENTOS FLECTORES

29. 3.9PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL Formulario para vigas y pórticos I h l k y N 3 2 k = ⋅ = + 2 I a s 1 p 3.9.1CARGA REPARTIDA VERTICAL REACCIONES psn V l psm V l ps H H mn hlN  MOMENTOS FLECTORES 3 2 lN En S p x m M V x = ⋅ − B C I2 x m n h I1 I 1 = A = D A D      2 3 2 s = = − l A D 12 MB M C       2 ps s M = M = − ⋅ mn − B C 12 2 ( − ) − H ⋅ h x A A 2 HA HD 3.27 VA VD

30. 3.28 3.9.2CARGA REPARTIDA HORIZONTAL REACCIONES 2 ph V V l ph N k H N ph N k H N MOMENTOS FLECTORES ( ) 2 8 N ph M N k N En AB py h y y M M h p B C I2 = = A D 2 ( ) h I1 I 2 8 6 8 + 1 = D y ( ) − = A D A l Prontuario para Cálculo de Estructuras 2 ph M MB C M = N − k B MB 2 ( ) = − 2 + C 8 ( − ) = + ⋅ Y B 2 HA HD VA VD

31. Formulario para vigas y pórticos 3.9.3CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES Pn V l Pm V l Pmn H H lhN MOMENTOS FLECTORES 3 2 2 3 2 lN P m n B C I2 = A = D h I1 I 1 3 2 = = A D A D l Pmn lN M = M = − ⋅ B C MB M N − C M = Pmn P MP HA HD 3.29 VA VD

32. 3.30 3.10PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO h s h I I I 1 2 3 I 3 k = ⋅ y k = ⋅ 1 2 s 1 2 p 3.10.1CARGA REPARTIDA VERTICAL REACCIONES pl V V = = C s f I3 B h I x 2 2 A D 2 pl I1 h1 2 h + + h H = H = 1 h 2 ( ) ( ) A D A D 2 1 2 2 8 1 1 h + k + k + 1 2 hh 1 2 MOMENTOS FLECTORES 2 pl M h Prontuario para Cálculo de Estructuras l ( ) 1 ( k ) h + h h h + M 1 2 1 1 k = − C ( ( ) B 2 1 2 2 8 1 + k h + + ) 1 + 1 2 hh 2 h h 2 pl 1 + 2 2 k M = − MB ( ) ( ) C 2 1 2 2 8 1 h + h + + 1 2 hh 1 2 En BC px l ( ) − x f l       M = − H x + h X A 1 2 HA HD VA VD

33. Formulario para vigas y pórticos 3.10.2CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES 2 1 2 l H ph H = − C s f p I3 ph B V V = = h I 2 2 A D I1 h 1 A 1 D y ( ) h + 4 5 h k 2 k h + + + + 2 1 ph A D 1 k 1 1 2 H = ( ) ( ) D 2 1 2 2 8 h 1 1 2 hh + 1 2 l MOMENTOS FLECTORES ph M = − M ( ) C h + 4 5 h k 2 k h + + + + 2 1 3 1 ph 1 k 1 1 2 ( ( ) k ( ) B 2 1 2 2 2 8 h 1 1 2 hh + 1 2 MB ) h + 4 5 h 2 k h + + + + 2 1 8 ph h 1 k 1 1 2 M = 2 ( ) ( ) C 2 1 2 2 h 1 + 1 2 hh 1 2 En AB 2 py M = H y − HA HD Y A 2 VA VD 3.31

34. 3.32 3.10.3CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES ( ) 1 2 2 l H pf H = − p C s f pf h h + V V = = y I3 A D B h I 2 2 A D I1 h 1 ( ( ) ) ( ) ) 2 1 2 1 8 h h 1 1 k k 4 h hh f h h + + + + + + pf 1 1 2 1 1 2 H = A D ( D 2 2 8 k 1 2 hh + + 1 2 MOMENTOS FLECTORES l Prontuario para Cálculo de Estructuras ( ( ) ) ( ) ) 2 1 2 1 M 8 h h 1 1 k k 4 h hh f h h + + + + + + pfh C 1 1 2 1 1 2 M pfh 1 = − ( B 1 2 2 8 k 1 2 hh + + 1 2 MB ( ( ) ) ( ) ) 2 1 2 1 8 h h 1 1 k k 4 h hh f h h + + + + + + ph 1 1 2 1 1 2 M = − 2 ( C 2 2 8 + k + 1 2 hh 1 2 HA HD En BC 2 l f py VA VD ( ) M = − V y + H y + h − Y A A 1 2

35. Formulario para vigas y pórticos 3.10.4CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES Pb V l Pa V l h l b Pab H H l h k h + + MOMENTOS FLECTORES ( ) ( ) ( 1 1 2 2 1 1 l h k h + + C s f I3 = A B h I 2 2 a = b D I1 h 1 ( ) h l ( a ) + + + + = = 1 2 + ( ) ( ) A D A D 2 2 1 2 2 2 1 1 k 1 2 hh 1 2 l ( ) h l + b + h l + a + Pabh M 1 2 + C M = − 1 ) B 2 2 2 k 1 2 hh 2 MB ( k ) h ( ) h l + b + + h l + a + Pabh MP 1 + 2 + M = − 2 ( ) ( ) C 2 2 1 2 2 2 l h 1 1 k 1 2 hh 1 2 Pab l af l       HA HD M = + H + h P A 1 VA VD 3.33

36. 3.34 3.11PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS I h s k = ⋅ 2 I 1 p 3.11.1CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES pl V V = = C s f I2 I2 B D A E 2 pl x 2 8 k h + + 5 f f I1 I1 h H = H = ( ) ( ) A E 2 32 h 3 + 3 h + f A E MOMENTOS FLECTORES Prontuario para Cálculo de Estructuras l 2 pl h 8 k h + + 5 f f M = M = − ( ) ( ) B D 2 32 h 3 + 3 h + f M C 2 pl f + h M = + M C B 8 h MB MD En BC y DC ( ) x l − x M h 2 fx l       B M = p + h + X 2 HA HE VA VE

37. Formulario para vigas y pórticos 3.11.2CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL REACCIONES pl V = p C 38 pl s A f I2 I2 V = B D E 8 x 2 pl 8 k h + + 5 f f I1 I1 H = H = h ( ) ( ) A E 2 64 h 3 + 3 h + f A E MOMENTOS FLECTORES l 2 pl h 8 k h 5 f f + + M M = = − ( ) ( ) B D 2 64 h 3 3 h f + + M C 2 pl f h + M M = + C B 16 h MB MD En BC ( ) x l x − M h 2 fx l       M p h = + B + X 2 HA HE 3.35 VA VE

38. 3.36 3.11.3CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES 2 ph V V l H ph H k h f ph H h k f f h + + + MOMENTOS FLECTORES 2 ph M M = + C s = = A E f 2 I2 I2 p = − D A E B ( ( ) 5 12 3 6 + + 2 = ) ( ) E 2 16 3 I1 I1 h y A E l B D 2 Prontuario para Cálculo de Estructuras M C 2 ph f h + M M = + C D 4 ph h ( ( ) 5 k k + 12 3 h f f 6 + f + + 3 M = − MB MD ) ( ) D 2 16 h 3 h + En AB 2 py M H y = − + ⋅ y A 2 HA HE VA VE

39. Formulario para vigas y pórticos 3.11.4CARGA REPARTIDA HORIZONTAL SOBRE DINTEL REACCIONES ( ) 2 2 l H pf H h k f f h pf H h k f f h + + + MOMENTOS FLECTORES M H h h k f h pf M h k f f h M H h = − ⋅ p C pf s V V f h = = + f A E I2 I2 = − D A E B ( ( ) ) ( ) 2 8 3 3 5 4 + + + y = ( ) E I1 I1 2 16 3 h x A E l = ⋅ B A ( ) ( ) 2 4 2 5 + f + + + M 2 C = − ⋅ ( ) ( ) C 2 16 3 3 + + D E MB MD En BC ( ) 2 y h − M H y V x p = ⋅ − ⋅ − x A A 2 HA HE f l siendo y x h = + VA VE 3.37

40. 3.38 3.11.5CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES Pn V l Pm V l h f l Pm H H l h k f f + + MOMENTOS FLECTORES M M H h Pm h f M M h hl fm M V m H l p C s = f A I2 I2 B D = A m n ( ) I1 I1 2 2 6 ln + 3 − 4 m h = = ( ) ( ) A E 2 2 4 3 + 3 h A E l = = − ⋅ Prontuario para Cálculo de Estructuras B D A + M C = + C B 2 + 2 MB MD = ⋅ − P A A HA HE VA VE

41. 3.12PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL Formulario para vigas y pórticos h h I I I 1 2 3 I 3 k = ⋅ y k = ⋅ 1 2 l l 1 2 3.12.1CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES 2 2 1 2 2 2 1 1 2 2 8 1 1 h k h h h pl pl V h k h + + + p B C I3 x h I h + h − pl pl 2 2 V = + ( ) ( ) A k 1 2 hh + + + h I1 1 D 2 2 1 2 2 − = − ( ) ( ) D 2 1 2 2 h + 2 8 1 1 k 1 2 hh + 1 2 A 2 h − pl H H = = 1 h 2 + ( ) ( ) A D 2 1 2 2 8 h 1 k 1 k 1 2 hh + + l 1 2 MOMENTOS FLECTORES 2 pl M h MB M ( ) ( ) ) + C h + h h + 1 2 1 1 k = − ( ( ) B 2 1 2 2 8 1 k h 1 2 hh + + 1 2 ) + h + h h + 2 pl 1 2 1 2 k M = − ( ( ) C 2 1 2 2 8 h 1 k h 1 2 hh + + HD 1 2 En BC VD 2 px M V x H h = ⋅ − − ⋅ HA x A A 1 2 3.39 VA

42. 3.40 3.12.2CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES 2 1 1 2 2 l l H ph H ph kh h h H h k h k + + + + MOMENTOS FLECTORES 2 3 1 1 1 1 2 1 1 2 8 1 5 4 8 1 1 h k h En AB py M H y = ⋅ − p B C ph h h − I3 V V H = = − A D D h I = − 2 2 A D h1 I1 2 1 5 4 2 + + D = 1 1 1 2 ( ) ( ) D 2 1 2 2 8 1 1 1 2 hh y 1 2 A l ph ph 5 + kh k 4 h 2 k h + + + M = − − 1 1 2 ( ) ( 2 k ) B 2 2 h h 1 2 hh + + Prontuario para Cálculo de Estructuras 2 M MB 2 1 ph h 1 1 kh h h + + + C M = − 2 1 2 ( ) ( ) C 2 1 2 2 MB 1 2 hh + + + 1 2 2 HD y A 2 VD HA VA

43. Formulario para vigas y pórticos 3.12.3CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) 1 1 2 2 1 l l h k h + + P a b B C ( 1 ) ) l b h l a h k + + + Pb Pab I3 ( ) 1 2 V h h = + − ( A 1 2 3 2 2 1 2 hh + + h I 2 2 2 h1 I1 ( ) ( 1 ) ) D l + b h k l a h k + + + Pa l Pab l ( ) 1 h 2 V h h = − − ( ) ( D 1 2 3 2 1 2 2 2 h 1 1 2 hh + + + 1 2 A ( ) ( 1 ) ) l + b h k l a h k + + + Pab l 1 h 2 H H = = l ( ) ( A D 2 2 1 2 2 2 h 1 1 2 hh + + + 1 2 MOMENTOS FLECTORES Pabh M l MB M C ( ) ( 1 ) ) l + b h k l a h k + + + + 1 h 2 = − 1 ( ) ( B 2 2 1 2 2 2 h 1 1 2 hh + + 1 2 MP HD ( ) ( 1 ) ) l + b h k l a h k + + + + Pabh 1 h 2 M = − 2 ( ) ( C 2 2 1 2 2 2 l h 1 1 2 hh + + VD 1 2 HA M V a M = ⋅ + VA P A B 3.41

44. 3.42 3.13PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL I h l k = ⋅ 2 I 1 3.13.1CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES pl V V H = = = p B C I2 x 2 pl h I1 I H = 1 ( ) A D A D 2 4 h k 2 + MOMENTOS FLECTORES A D l Prontuario para Cálculo de Estructuras 2 pl k M M = = ( ) A D 12 2 + 2 pl k MB M M M = = − C ( ) B C 6 2 + En BC ( ) px l x − 2 pl k M = − ( 3 ) x 2 6 2 + 2 pl k + 2 l máx M pos para x = = 24 k 2 2 + HA HD MA MD VA VD

45. Formulario para vigas y pórticos 3.13.2CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES 2 ph k V V l k H ph H ph k H k + MOMENTOS FLECTORES 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 k k En AB py M H y M = − + ⋅ + p B C I2 = = ( ) A D 6 1 + = − A D h I1 I ( ( ) 1 2 3 + = y ) D 8 2 A D l ph       M = − + + A k k + + 2 ph       M MB M = − + C B k k + + MB 2 ph       M = − − − C k k + + 2 ph       M = + − D + + HA HD 2 MA MD y A A 2 VA VD 3.43

46. 3.44 3.13.3CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES ( ) ( ) 6 1 l l k V P V Pmn H H lh k + MOMENTOS FLECTORES 1 2 2 6 1 l k l k   + +    − = − +   + +   − = − −   + +    − = +     + +   P m n         m n m + − Pn B C V 1 = + I2 A 2 = − D A 3 h I1 I = = 1 A D 2 ( 2) A D     Pmn n − m l M = − ( ) A Prontuario para Cálculo de Estructuras         Pmn l 1 n m M MB M ( ) B k 2 2 6 l k 1 C Pmn l 1 n m M ( ) C k 2 2 6 l k 1 MP Pmn 1 n m M ( ) D 2 l k 2 l 6 k 1 mM nM Pmn l C M B = + + HA HD P l l MA MD VA VD

47. Formulario para vigas y pórticos 3.13.4CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES 3 (6 1 ) l k P H H = = P B C Phk + I2 V = V = A D A D 2 h I1 I 1 MOMENTOS FLECTORES 3 2 6 A D Ph k + + 1 1 M = − l A k Ph 3 k k + M = − M = B C 2 6 1 1 1 MB M Ph k 3 + + C M = D 2 6 k HA HD MA MD VA VD 3.45

48. 3.46 3.14PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS I h s k = ⋅ 2 I 1 p 3.14.1CARGA REPARTIDA VERTICAL SOBRE DINTEL REACCIONES pl V V = = C s f I2 I2 B D x A E 2 pl ( ) I1 I1 k 4 h 5 f f + + 2 h H = H = ( ) A E ( ) 2 8 2 2 kh + f + 4 k h + hf + f A E MOMENTOS FLECTORES Prontuario para Cálculo de Estructuras l ( ) ( + ( + ) kh 8 h + 15 f + f 6 h − f 2 pl M = M = M ) C A E ( ) 2 48 2 2 kh + f + 4 k h hf + f ( 2 ) + 2 kh 16 h 15 f + f 2 pl M = M = − MB MD ( ) B D ( ) 48 2 2 kh + f + 4 k h hf + f 2 pl ( ) M = + M − H h + f C A A 8 En BC HA HE 2 MA ME 2 xf l px       M = M + V ⋅ x − H h + − VA VE x A A A 2

49. 3.14.2CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL Formulario para vigas y pórticos REACCIONES pl V = p C − V A E 2 pl s 4 k + 1 + f V = 3 ( ) I2 I2 E 32 3 pl = k 1 ( ) B D k 4 h + 5 f + f 2 H = H x ( ) A E ( ) 2 16 2 2 kh + f + 4 k h + hf + f I1 I1 h MOMENTOS FLECTORES 2 kh pl M kh A E ( + ) ( ( ) 8 h + 15 f + f 6 h − f 2 pl = − ) ( ) A ( ) l 2 96 64 3 k + 1 2 2 f + 4 k f + fh + h ( + ) ( ( ) kh 8 h + 15 f + f 6 h − f 2 2 pl pl M = + M ) ( ) E ( ) C 2 96 64 3 k + 1 2 2 kh f + 4 k f + fh + h ( ) ) + 2 kh 16 h + 15 f + f 2 2 pl pl MB MD M = − − ( ) ( ) B ( 2 96 64 3 k + 1 2 2 kh + f + 4 k f fh + h ( ) ) + 2 kh 16 h + 15 f + f 2 2 pl pl M = − + ( ) ( ) D ( 2 96 64 3 k + 1 2 2 kh + f + 4 k f fh + h HA HE MA ME 2 2 xf l px       VA VE En BC M = M + V ⋅ − x H h + − x A A A 2 l ( ) M = V + M − H f + h 3.47 C E E E 2

50. 3.48 3.14.3CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES 2 ph k V V l k H ph H k h k f h ph H kh f k f fh + + + + C = = ( ) A E 2 3 1 + s f = − I2 I2 A E p ( 4 ) 2 2 3 f + + + 2 D B = ( ) E ( ) 2 4 2 2 h I1 I1 h MOMENTOS FLECTORES y A E         ( ) 2 ( k f ) 2 2 kh k 6 kf 15 h 16 f 6 f + + + + 2 ph 2 3 k k 1 1 + + l M 6 = − + ( ) A ( ) 24 2 2 kh f 4 fh h + + + + M Prontuario para Cálculo de Estructuras C 2 ph M M H h = + ⋅ − B A A 2 + 1 2 ( ⋅ ) M M M M H H f h h V = = − − + C E E E MB MD D E E         ( ) 2 ( k f ) 2 2 kh k 6 kf 15 h 16 f 6 f + + + + 2 ph 2 3 k k 1 1 + + M 6 = − + ( ) E ( ) 24 2 2 kh f 4 fh h + + + + HA HE En AB MA ME 2 py VA VE M M H y = + ⋅ − y A A 2