3 FORMULARIO PARA VIGAS Y PÓRTICOS

3 FORMULARIO PARA VIGAS Y PÓRTICOS 3.1 Formulario para vigas y pórticos 3.1 Obtención de la Distribución de Solicitaciones mediante la Formulación de Macaulay Las Funciones de Macaulay permiten expresar tanto la distribución de cargas sobre una viga sometida a flexión como las leyes de Cortantes o Momentos Flectores generadas por dichas cargas. A continuación se muestra la expresión de tales funciones y las condiciones en las que deben aplicarse. q( x) = ∑ T ( x ) = −∑ M( x ) = − ∑ A⋅ x − a ( c− 2 ) ( c − 2 )! A⋅ x − a ( c −1) ( c − 1) ! A⋅ x − a c c! ecuaciones validas solo si n ≥ 0 en las expresiones si y si n=0 n>0 x−a n x≤a x−a 0 =0 x≥a x−a 0 =1 x≤a x−a n =0 x≥a x−a n = ( x − a) n En la siguientes tablas se particularizan estas funciones para cada caso de carga y se indica el valor que deberían tomar los parámetros A y c en la ecuación general previamente indicada. 3.2 Prontuario para Cálculo de Estructuras M Si x≤a a x≥a x x−a 0 =0 x−a 0 =1 entonces M(x) M( x ) = − M x − a 0 A=M c=0 por lo tanto P Si a x T(x) 1 x≤a x−a = 0 x≥a x − a = ( x − a) 1 1 entonces T ( x) = − P x − a M(x) 0 M( x ) = − P x − a por lo tanto 1 A=P c =1 3.3 Limitación de las Deformaciones Si x≤a q x≥a x−a x−a 2 2 =0 = ( x − a) 2 a entonces x q( x) = q x − a 0 q 1 x−a 1 q M( x ) = − x−a 2 ⋅1 T(x) T ( x) = − 2 M(x) A=q c=2 por lo tanto q d a x Si x≤a x−a x≥a 3 x−a 3 =0 = ( x − a) entonces 2 T(x) 3 M(x) 2 qd 1 x−a 1 qd 2 T ( x) = − x−a 2 ⋅1 qd M( x ) = − x−a 3 ⋅ 2 ⋅1 q( x) = por lo tanto 3 q d c=3 A= 3 3.4 Prontuario para Cálculo de Estructuras Otros casos de carga que se resuelven por superposición de los anteriores q q  −〈 x-a〉 2 + 〈 x-b〉 2   2!  dM( x ) M(x) = a T ( x) = b dx x q/d q d a T ( x) = b q/d q -〈 x-a〉 3 + 〈 x-b〉 3  + 〈 x-b〉 2  2! 3!  dM( x ) M( x ) = dx x q/d q a q q/d  〈 x-a〉 3 − 〈 x-b〉 3  〈 x-a〉 2 +  2! 3!  dM( x ) M(x) = − d T ( x) = b dx x qb qa M(x) = − + a d b T ( x) = x x T ( x) = 2! 〈 x-b〉 2 +  −〈 x-a〉 3 + 〈 x-b〉 3    dx + b ) q b − q a /d qb dM( x ) qb d 2! 〈 x-a〉 2 + 3! M(x) = − qa a ( qa qa 2! (q a 〈 x-a〉 2 + ) − q b /d 3! dM( x ) dx qb 2! 〈 x-b〉 2 +  〈 x-a〉 3 − 〈 x-b〉 3    VIGA APOYADA EN LOS EXTREMOS 3.2.1 REACCIONES P⋅b RA = L RB = B C P⋅a L x ESFUERZOS CORTANTES P⋅b P⋅a = cte ; QCB = − = cte QAC = L L MOMENTOS FLECTORES P⋅b P⋅a ⋅ x ; MCB = ⋅ ( L − x) MAC = L L ANGULOS DE GIRO P⋅a⋅b ⋅ ( L + b) ϕA = 6⋅E⋅I⋅L P A CARGA PUNTUAL EN LA VIGA a b Formulario para vigas y pórticos 3.2 L ; Mmax = MC = P⋅a⋅b ⋅ ( L + a) ; ϕB = − 6⋅E⋅I⋅L P⋅a⋅b L para x0 = a P⋅a⋅b ⋅ ( b − a) ; ϕC = 3⋅E⋅I⋅L QB QA ECUACION DE LA ELASTICA y AC = P ⋅ L ⋅ b ⋅ x  b2 x 2  ⋅ 1− 2 − 2  6⋅E⋅I  L L  ; y CB = 2 P ⋅ L ⋅ a ⋅ ( L − x )  a2  L − x   ⋅  1− 2 −     6⋅E⋅I L  L    FLECHA MAXIMA fC = P⋅b 9⋅ E ⋅I ⋅ L 3 ( ⋅ L2 − b2 ) 3 2 para x = L2 − b2 3 3.5 M max 3.6 3.2.2 CARGA CONTÍNUA EN PARTE DE LA VIGA c REACCIONES p⋅b⋅c RA = L RB = P p⋅a⋅c L A ESFUERZOS CORTANTES p⋅b⋅c p⋅b⋅c c  − p⋅ − a + x ; QCD = QAC = 2 L L   ; QDB = − C p⋅a⋅c L a x0 = a − para ; ϕB = − c b⋅c + L 2 p⋅a⋅b⋅c  c2  ⋅L + a−  6⋅E⋅I⋅L  4⋅b QB QA ECUACION DE LA ELASTICA  p⋅b⋅c x  2 c2   y AC = ⋅ − x + a ⋅  L + b −  6 ⋅ L E ⋅ I  4 ⋅ a    = 4     p c  c2   ⋅  L ⋅  x −  a −  − 4 ⋅ b ⋅ c ⋅ x3 + 4 ⋅ a ⋅ b ⋅ c ⋅  L + b − ⋅ x 24 ⋅ E ⋅ I ⋅ L    2  4⋅a     =  p⋅a⋅c L − x  c2   2 ⋅ ⋅ − ( L − x ) + b ⋅  L + a −  6⋅L 4 ⋅ a   E ⋅ I   y CD y DB M max Prontuario para Cálculo de Estructuras ANGULOS DE GIRO p⋅a⋅b⋅c  c2  ⋅L + b − ϕA =  6⋅E⋅I⋅L  4⋅a b L MDB = Mmax D x MOMENTOS FLECTORES p⋅b⋅c p⋅b⋅c p   c  ⋅ x ; MCD = ⋅ x − ⋅  x −  a −  2 MAC = 2   2  L L p⋅a⋅c ⋅ (L − x) L p⋅b⋅c  b⋅c = ⋅ 2⋅a− c+ L  2 ⋅ L  B CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES 1 RA = ( 2 ⋅ p1 + p2 ) 6 ; RB = 1 ( p1 + 2 ⋅ p2 ) . 6 P1 ESFUERZOS CORTANTES p ( 3 ⋅ L − x ) + p2 ⋅ x 2 ⋅x QA = RA ; Qx = RA − 1 6⋅L ; P2 QB = −RB B A MOMENTOS FLECTORES p ( 3L − x ) + p2 ⋅ x 2 Mx = RA ⋅ x − 1 ⋅x 6⋅L Formulario para vigas y pórticos 3.2.3 x L2 L2 ⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 ) 2 2   1 1 para x 0 = ⋅  − p1 + ⋅ p12 + p22 + p1 ⋅ p2  3 p2 − p1   L Mmax comprendido entre 0,125 ⋅ ( ANGULOS DE GIRO L3 ϕA = ⋅ ( 8 ⋅ p1 + 7 ⋅ p2 ) 360 ⋅ E ⋅ I ) QA QB ; ϕB = − 3 L ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 360 ⋅ E ⋅ I ECUACION DE LA ELASTICA x ( L − x ) 3 ( p1 − p2 ) x − 3 ( 4 p1 + p2 ) Lx  360EI ( 8 p1 + 7p2 ) L2 x + ( 8 p1 + 7p2 ) L3  3 yx = 0,01304 ⋅ +   ( p1 + p2 ) ⋅ L4 2⋅E⋅I x0 M max 3.7 FLECHA MAXIMA ( p + p2 ) ⋅ L4 y entre 0,01302 ⋅ 1 2⋅E⋅I 2 3.8 3.2.4 MOMENTO FLECTOR REACCIONES R A = −R B = − M L C ESFUERZOS CORTANTES M Qx = = cte L MOMENTOS FLECTORES M M MAC = − ⋅ x MCB = − ⋅ ( L − x ) L L M M izq der MC = − ⋅ a MC = − ⋅ b L L A QA QB MC M MC Prontuario para Cálculo de Estructuras ) 2 M ⋅ L ⋅ (L − x)  a2  L − x    ⋅ 1− 3 ⋅ 2 −    L  L   6⋅E⋅I  FLECHA M⋅ a ⋅ b fC = ⋅ ( b − a) 3⋅E⋅I⋅L b M = MCizq + MCder ECUACION DE LA ELASTICA M⋅ L ⋅ x  b2 x 2  y AC = − ⋅ 1− 3 ⋅ 2 − 2  6⋅E⋅I  L L  yCB = − B a L ANGULOS DE GIRO M ⋅ L  b2  M ⋅ L  a2  ϕA = ⋅  3 ⋅ 2 − 1 ; ϕ B = ⋅3⋅ − 1 6⋅E⋅I  L 6 ⋅ E ⋅ I  L2   M 3 3 ϕC = ⋅ a +b 3 ⋅ E ⋅ I ⋅ L2 ( +M Formulario para vigas y pórticos 3.3 VIGA EMPOTRADA EN LOS EXTREMOS P 3.3.1 CARGA PUNTUAL EN LA VIGA REACCIONES P ⋅ b2 RA = 3 ⋅ ( L + 2 ⋅ a) L C ; RB = ESFUERZOS CORTANTES P ⋅ b2 QAC = 3 ⋅ ( L + 2 ⋅ a) = cte L P ⋅ a2 ⋅ ( L + 2 ⋅ b) L3 ; QCB = − B A x a P ⋅ a2 ⋅ ( L + 2 ⋅ b ) = cte L3 b L MOMENTOS FLECTORES P ⋅ a ⋅ b2 P ⋅ a2 ⋅ b P ⋅ b2 ; ; M M = − = ⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L) B AC L2 L2 L3 2 ⋅ P ⋅ a2 ⋅ b2 P ⋅ a2 para x0 = a = 3 ⋅ L ⋅ b + L2 − L ⋅ x − 2 ⋅ b ⋅ x ; MC = L L3 MA = − MBC ( ) QB QA ECUACION DE LA ELASTICA y AC = P ⋅ b2  2 ⋅ a ⋅ x  x2 ⋅3 ⋅a − x − ⋅ L  L2 6⋅E⋅I  y BC = P ⋅ a2  L − x ⋅  (L − x) ⋅  3 ⋅ b − (L − x) − 2 ⋅ b ⋅ 6⋅E⋅I  L  L2 2 MB MA FLECHAS P ⋅ a3 ⋅ b3 3 ⋅ E ⋅ I ⋅ L3 x= fmax = 2 ⋅a⋅ L L + 2⋅a MC 2 ⋅ P ⋅ a3 ⋅ b2 3 ⋅ E ⋅ I ⋅ ( L + 2 ⋅ a) 2 0 3.9 para ; x fC = 3.10 3.3.2 CARGA CONTÍNUA EN PARTE DE LA VIGA c P REACCIONES p ⋅ b ⋅ c MA − MB RA = − L L ; p ⋅ a ⋅ c MA − MB RB = + L L C c  ; QBD = −RB = cte ; QCD = RA − p ⋅  x − a +  a  x a MOMENTOS FLECTORES MAC = RA ⋅ x + MA ; MBD = RB ⋅ ( L − x ) + MB MCD = RA ⋅ x + MA − ; B A ESFUERZOS CORTANTES QAC = RA = cte D MA = − p ⋅ c3 12 ⋅ L2 p  c ⋅x − a+  2  2 b 2  12 ⋅ a ⋅ b2  ⋅L − 3⋅b +  c2   Q A Q B ECUACION DE LA ELASTICA x2 ⋅ ( −3 ⋅ MA − RA ⋅ x ) 6⋅E⋅I 4    1 c yCD = ⋅  p ⋅  x − a +  − 4 ⋅ RA ⋅ x 3 − 12 ⋅ MA ⋅ x 3  24 ⋅ E ⋅ I   2  1 RB x 3 − 3 ( MB + LRB ) x 2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2  y DB = 6EI  y AC = MA MB Prontuario para Cálculo de Estructuras 12 ⋅ a2 ⋅ b  p ⋅ c3  ⋅ L − 3⋅a+ MB = −  2  12 ⋅ L  c2  L CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES L M − MB ⋅ ( 2 ⋅ p1 + p2 ) − A 6 L L MA − MB RB = ⋅ ( p1 + 2 ⋅ p2 ) + 6 L RA = P1 P2 ESFUERZOS CORTANTES Qx = RA − B A QA = RA p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L Formulario para vigas y pórticos 3.3.3 x ⋅x L QB = −RB MOMENTOS FLECTORES L2 ( 3 ⋅ p1 + 2 ⋅ p2 ) 60 p ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 2 Mx = RA ⋅ x + MA − 1 ⋅x 6⋅L L2 MB = − ( 2 ⋅ p1 + 3 ⋅ p2 ) 60 MA = − Q A Q B ECUACION DE LA ELASTICA yx =  ( p − p1) 3  x2 ⋅ 2 ⋅ x + p1 ⋅ L ⋅ x 2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L  24 ⋅ E ⋅ I ⋅ L  5  MA MB 3.11 3.12 3.3.4 MOMENTO FLECTOR REACCIONES 6⋅M RA = − 3 ⋅ a ⋅ b L ; RB = 6⋅M ⋅a⋅b L3 C Qx = − 6⋅M ⋅ a ⋅ b = cte L3 a M⋅ a  b ⋅2 − 3⋅  L  L MAC = B x MOMENTOS FLECTORES MA = +M A ESFUERZOS CORTANTES MB = − b L M⋅ b  a ⋅2 − 3⋅  L  L M⋅ a  a  x  ⋅ 3 ⋅ ⋅ 1− 2 ⋅  − 1 L  L  L  MCB = − M⋅ b  b  L−x  ⋅ 3 ⋅ ⋅ 1− 2 ⋅  − 1 L  L  L   6⋅M 2 ⋅a ⋅b L3 ; MCder = MA + M 3 ⋅ L − 6 ⋅ a2 ⋅ b L3 ( QB ) ECUACION DE LA ELASTICA y AC = y BC = M ⋅ b ⋅ x2 2⋅E⋅I⋅L L− x b  ⋅2⋅a⋅ 2 −  L L  M⋅ a ⋅ ( L − x ) 2⋅E⋅I⋅L 2 MC  b⋅ x a ⋅2 ⋅ 2 −  L L  FLECHA M ⋅ a2 ⋅ b2 fC = − ⋅ ( a − b) 2 ⋅ E ⋅ I ⋅ L3 MA MC MB Prontuario para Cálculo de Estructuras MCizq = MA − QA 3.4.1 P CARGA PUNTUAL EN LA VIGA REACCIONES P ⋅ b2 P⋅a RA = ⋅ ( 3 ⋅ L − b ) ; RB = ⋅ 3 ⋅ L2 − a2 2 ⋅ L3 2 ⋅ L3 ESFUERZOS CORTANTES P ⋅ b2 P⋅a ⋅ ( 3 ⋅ L − b ) = cte ; QCB = − ⋅ 3 ⋅ L2 − a2 = const. QAC = − 2 ⋅ L3 2 ⋅ L3 ( ) ( C B A x ) MOMENTOS FLECTORES P⋅a 2 P⋅a 2 ⋅ L − a2 ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b) ; MC = MB = − 2 ⋅ L2 2 ⋅ L3 P⋅x 2 P⋅a ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b ) ; MCB = ⋅ 2 ⋅ L3 − 3 ⋅ L2 ⋅ x + a2 ⋅ x MAC = 2 ⋅ L3 2 ⋅ L3 ( a b L ) ( ) Q B ANGULOS DE GIRO ϕA = P ⋅ a ( L − a) 2 ; ϕC = 4⋅E⋅I⋅L P ⋅ a ⋅ ( L − a) 2 4⋅E⋅I⋅L 3 ( ⋅ L2 − 2 ⋅ a ⋅ L − a2 ) Q A ECUACION DE LA ELASTICA P ⋅ b2 ⋅ x y AC = ⋅ 3 ⋅ a ⋅ L2 − x 2 ⋅ ( 2 ⋅ L + a)  12 ⋅ E ⋅ I ⋅ L3  y BC = P ⋅ a ⋅ ( L − x) 12 ⋅ E ⋅ I 2 MB   a2   a2   L − x   ⋅ 3 ⋅ 1− 2  −  3 − 2  ⋅   L   L   L     para x=L ⋅ a 2⋅L + a MC 3.13 FLECHA MAXIMA p ⋅ b2 ⋅ a a fmax = ⋅ 6⋅E⋅I 2⋅L + a Formulario para vigas y pórticos 3.4 VIGA APOYADA-EMPOTRADA 3.14 3.4.2 CARGA CONTÍNUA EN PARTE DE LA VIGA c REACCIONES p ⋅ b ⋅ c MB RA = + L L ; P p ⋅ a ⋅ c MB RB = − L L ESFUERZOS CORTANTES QAC = RA = cte ; C QDB = −RB = cte ; QCD c  = RA − p ⋅  x − a +  2  x MOMENTOS FLECTORES MAC = RA ⋅ x ; MCD MDB = RB ⋅ ( L − x ) + MB a p  c = RA ⋅ x − ⋅  x − a +  2  2 ; MB = − 2 (L − x) 2 6⋅E⋅I ⋅ RB ⋅ ( L − x ) + 3 ⋅ MB  QB QA MB Prontuario para Cálculo de Estructuras ECUACION DE LA ELASTICA   12 ⋅ a ⋅ b2   x y AC = ⋅  −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅  L − 3b +  48 ⋅ E ⋅ I ⋅ L  c2    4   1 12ab2   c  ⋅  −8RALx 3 + 2 pL  x − a +  + pc3  L − 3b + yCD =  x 48 ⋅ E ⋅ I ⋅ L  4 c2     b L p⋅a⋅b⋅c  c2  L a ⋅ + −   4⋅b 2 ⋅ L2  ANGULOS DE GIRO  p ⋅ c3 12 ⋅ a ⋅ b2  ⋅  L − 3b + ϕA =  48 ⋅ E ⋅ I ⋅ L  c2  y DB = − B A CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES RA = P2 P1 L M ⋅ ( 2 ⋅ p1 + p2 ) + B 6 L ; RB = L M ⋅ ( p1 + 2 ⋅ p2 ) − B 6 L B A x ESFUERZOS CORTANTES Qx = RA − p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L L ⋅x ; QB = −RB Q A MOMENTOS FLECTORES Mx = RA ⋅ x − p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 6⋅L Formulario para vigas y pórticos 3.4.3 ⋅ x2 ; MB = − Q B L2 ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 120 ANGULOS DE GIRO L3 ϕA = ⋅ ( 3 ⋅ p1 + 2 ⋅ p2 ) 240 ⋅ E ⋅ I MB ECUACION DE LA ELASTICA yx = 3.15 x  ( p2 − p1) x4 + 5Lp1x3 − 20RALx2 + 5L 12RAL2 − ( 3p1 + p2 ) L3   120EIL  3.16 3.4.4 MOMENTO FLECTOR REACCIONES RA = −RB = 3 M 2 ⋅ ⋅ L − a2 2 L3 ( ) ESFUERZOS CORTANTES B x Qx = RA = cte a MOMENTOS FLECTORES MCder = RA ⋅ a − M ; MAC = C +M A MCizq = RA ⋅ a 3 M⋅ x 2 ⋅ ⋅ L − a2 2 L3 ( ) ; MBC M ⋅ L2 − 3 ⋅ a2 2 ⋅ L2  M  x  a2  = ⋅ 3 ⋅ ⋅ 1− 2  − 2  2  L  L   ( MB = ; b L ) ANGULOS DE GIRO  b  a 2  M ; ϕC = ⋅ b ⋅ 3 ⋅ ⋅ 1+  − 4  4⋅E⋅I  L  L   ECUACION DE LA ELASTICA M⋅ b ⋅ x  ⋅ −4 ⋅ L3 − x 2 − 3 ⋅ L2 ⋅ ( a + L )   4 ⋅ E ⋅ I ⋅ L3  M 2 = ⋅ ( L − x ) ⋅ 2 ⋅ a2 ⋅ L − x ⋅ L2 − a2    4 ⋅ E ⋅ I ⋅ L3 y AC = y BC ( QA QB MC ) ( MB ) MC Prontuario para Cálculo de Estructuras M ϕA = ⋅ ( L − a) ⋅ ( 3 ⋅ a − L ) 4⋅E⋅I⋅L 3.5.1 CARGA PUNTUAL EN LA VIGA C Formulario para vigas y pórticos 3.5 VIGA EMPOTRADA EN UN EXTREMO P REACCIONES B A RB = P x ESFUERZOS CORTANTES QAC = 0 ; QCB = −P = cte a L MOMENTOS FLECTORES MAC = 0 ; b MCB = −P ⋅ ( x − a) ; MB = −P ⋅ b ANGULOS DE GIRO ϕ A = ϕC = − P ⋅ b2 2⋅E⋅I QB ECUACION DE LA ELASTICA y AC = P ⋅ b2 ⋅ ( 3 ⋅ ( L − x ) − b) 6⋅E⋅I ; y CB = MB 3.17 FLECHA MAXIMA P ⋅ b3 P ⋅ b2 ⋅ ( 2 ⋅ b + 3 ⋅ a) fC = ; fA = 3⋅E⋅I 6⋅E⋅I P 2 ⋅ ( L − x ) ⋅ ( 2 ⋅ b + 3 ⋅ a) 6⋅E⋅I 3.18 3.5.2 CARGA CONTÍNUA EN PARTE DE LA VIGA REACCIONES . RB = p ⋅ c c P ESFUERZOS CORTANTES . c  QAC = 0 ; QCD = − p ⋅  x − a +  2  ; QDB = − p ⋅ c = cte ; ϕC = − x p ⋅ c2 MD = − 2 p ⋅ c  2 c2  ⋅b +  2⋅E⋅I  12  a b L ; ϕ A = ϕC y DB =   p⋅c p⋅c  c2  3 ⋅ L − x 2 ⋅ ( 2 ⋅ b − a + x ) ; y AC = ⋅  ( a − x ) ⋅  3 ⋅ b2 +  + 2⋅b  6⋅E⋅I 6 ⋅ E ⋅ I  4    y DC = 4    p c c2  3 ⋅   x − a +  + 4 ⋅ c ⋅ ( a − x ) ⋅  3 ⋅ b2 +  + 8 ⋅ b ⋅ c 24 ⋅ E ⋅ I  2 4    ) Q B FLECHAS . 2 fD = p⋅ c  c ⋅ b−  E ⋅ I  2 b c  ⋅ +   3 12  2   p ⋅ c  c p⋅ c   c2  fC = ⋅  b +  ⋅ ( 4 ⋅ b − c) + c3  ; fA = ⋅ a ⋅  3 ⋅ b2 +  + 2 ⋅ b3  E I 12 ⋅ E ⋅ I  2 6 4 ⋅ ⋅      M B Prontuario para Cálculo de Estructuras ECUACION DE LA ELASTICA . ( D C B MOMENTOS FLECTORES . 2 c  p⋅ x − a+  2 MAC = 0 ; MCD = −  ; 2 MDB = − p ⋅ c ⋅ ( x − a) ; MB = − p ⋅ c ⋅ b ANGULOS DE GIRO . p ⋅ c  2 c2  ⋅b − ϕD = −  2⋅E⋅I  4  A CARGA TRAPEZOIDAL EN TODA LA VIGA REACCIONES 1 RB = ( p1 + p2 ) 2 ESFUERZOS CORTANTES p − p1 x 2 ⋅ − p1 ⋅ x Qx = − 2 L 2 A ; L QB = − ( p1 + p2 ) 2 x2 ⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1 6⋅L  B x MOMENTOS FLECTORES Mx = − P2 P1 Formulario para vigas y pórticos 3.5.3 ; MB = − L2 ⋅ ( p2 + 2 ⋅ p1 ) 6 L ANGULOS DE GIRO ϕA = − L3 ⋅ ( 3 ⋅ p1 + p2 ) 24 ⋅ E ⋅ I ECUACION DE LA ELASTICA 3  2  ( L − x) 2 L − x ) − ( p2 − p1 ) + ( L − x ) p2 −  ( yx = 5L   24EI  2  L L x p p L p p 2 2 2 − − + + + ( )( ) ( ) 2 1 2 1   FLECHA 120 ⋅ E ⋅ I MB 3.19 fA = L4 ⋅ ( 4 ⋅ p2 + 11⋅ p1 ) QB 3.20 3.5.4 MOMENTO FLECTOR REACCIONES M A B RB = 0 ESFUERZO CORTANTE x a Qx = 0 L MOMENTOS FLECTORES MAC = 0 MCB = − M = cte ; b MAC = − M ; ANGULOS DE GIRO ϕC = ϕ A = − ECUACION DE LA ELASTICA y AC = M ⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b) 2⋅E⋅I ; y BC = M 2 (L − x) 2⋅E⋅I FLECHA fC = M ⋅ b2 2⋅E⋅I ; fA = M ⋅ b ⋅ ( 2 ⋅ L − b) 2⋅E⋅I MB Prontuario para Cálculo de Estructuras M⋅ b E⋅I P P A P B L/2 C L/2 L/2 L A B L/2 L/2 L C L/2 L 0,688 P L 0,405 P 0,312 P A Formulario para vigas y pórticos 3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES B C 0,094 P A 0,312 P 0,094 P B C 0,594 P 0,688 P ESFUERZOS CORTANTES ESFUERZOS CORTANTES - 0,188 PL - 0,094 PL A B 0,156 PL C 0,156 PL B C 0,203 PL MOMENTOS FLECTORES 3.21 MOMENTOS FLECTORES A 3.22 Q Q A L B Q C L 0,625 QL A B L L C 0,375 L 0,437 QL 0,375 QL 0,063 QL B A C A B 0,375 QL 0,375 L 0,563 QL 0,437 L 0,625 QL ESFUERZOS CORTANTES 2 2 - 0,063 QL - 0,125 QL B C 2 2 0,07 QL 0,07 QL MOMENTOS FLECTORES B A 2 0,096 QL MOMENTOS FLECTORES C Prontuario para Cálculo de Estructuras ESFUERZOS CORTANTES A C Q A L Q B C k L c QL d L a QL A C B b QL d QL a L ESFUERZOS CORTANTES Relación entre luces MOMENTOS FLECTORES ESFUERZOS CORTANTES 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 Formulario para vigas y pórticos 3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES 2 f QL A B C 2 e QL 2 g QL a = 0.5 − f e= a2 2 b = 0.5 + f g= d2 2 c= k f + 2 k 3.23 MOMENTOS FLECTORES k 2 − k +1 8 k f d= − 2 k f= 3.24 Q Q A B L Relación entre luces C k L c QL d L MOMENTOS FLECTORES ESFUERZOS CORTANTES 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 A C B a QL d QL b QL 2 f QL k 2 − k +1 8 k f d= − 2 k f= A B MOMENTOS FLECTORES C 2 g QL a = 0.5 − f e= a2 2 b = 0.5 + f g= d2 2 Prontuario para Cálculo de Estructuras ESFUERZOS CORTANTES Q Q A Q B C D k L L a QL L a L b QL c QL D B Relación entre luces ESFUERZOS CORTANTES MOMENTOS FLECTORES k a b c e f g 0,6 0,420 0,580 0,300 0,088 0,080 -0,035 0,7 0,418 0,582 0,350 0,087 0,081 -0,020 0,8 0,414 0,586 0,400 0,086 0,086 -0,006 0,9 0,408 0,592 0,450 0,083 0,091 -0,009 Formulario para vigas y pórticos 3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES C A c QL b QL a L a QL ESFUERZOS CORTANTES 2 2 f QL f QL f= k3 + 1 12 ⋅ k + 8 a = 0.5 − f c= k 2 e= a2 2 b = 0.5 + f g= k2 −f 8 2 g QL A B D C 2 2 e QL e QL 3.25 MOMENTOS FLECTORES 3.26 Q Q A Q B C L D k L L a L a QL b QL c QL D C B A c QL b QL a QL a L 2 e QL 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 f QL B 2 MOMENTOS FLECTORES 2 f QL A ESFUERZOS CORTANTES C 2 g QL MOMENTOS FLECTORES f= k3 + 1 12 ⋅ k + 8 a = 0.5 − f c= k 2 e= b = 0.5 + f D 2 e QL a2 2 g= k2 −f 8 Prontuario para Cálculo de Estructuras ESFUERZOS CORTANTES Relación entre luces k= I2 h ⋅ I1 l 3.9.1 y N = 3 + 2k a s p CARGA REPARTIDA VERTICAL B C I 2 x REACCIONES m VA VD Formulario para vigas y pórticos 3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL psn = l psm = l I n I 1 A h 1 D s  3 ps   mn −  2hlN  12  2 HA = HD = l MOMENTOS FLECTORES MB MC 3 ps  s2  MB = MC = − ⋅  mn −  2 lN  12  En S Mx = VA ⋅ x − p(x − m)2 − HA ⋅ h 2 HA HD VD 3.27 VA 3.28 3.9.2 CARGA REPARTIDA HORIZONTAL REACCIONES p B C I VA = VD = HD = HA = 2 ph2 2l ph ( 2N + k ) I I 1 8N h 1 y ph ( 6N − k ) A 8N D l MOMENTOS FLECTORES MB MY = MC MB py(h − y) y + ⋅ MB h 2 HA HD VA VD Prontuario para Cálculo de Estructuras ph2 ( 2N − k ) 8N ph2 MC = − ( 2N + k ) 8N En AB MB = Formulario para vigas y pórticos 3.9.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL P m REACCIONES n B Pn VA = l Pm VD = l HA = HD = C I I 2 I 1 h 1 3 Pmn 2 lhN A D MOMENTOS FLECTORES l 3 Pmn MB = MC = − ⋅ 2 lN 2N − 3 MP = Pmn 2lN MB MC MP HD HA VD 3.29 VA 3.30 3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO k1 = I3 h1 ⋅ I1 s y k2 = I3 h 2 ⋅ I2 s p 3.10.1 CARGA REPARTIDA VERTICAL C s f REACCIONES I B pl 2 h1 + h2 pl 2 HA = HD = 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2 3 I x VA = VD = I h1 h2 2 1 A D l MB = − ( h1 + h2 ) h1 pl2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC = − ( h1 + h2 ) h2 pl 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 En BC MX = px(l − x) f  − HA  x + h1  2 l   MC MB HA HD VA VD Prontuario para Cálculo de Estructuras MOMENTOS FLECTORES Formulario para vigas y pórticos 3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR C REACCIONES s f p ph12 VA = VD = 2l HA = ph1 − HD I 3 B I h1 I h2 2 1 y h1 ( 4 + 5k 1 ) + 2h2 ph12 HD = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 A D l MOMENTOS FLECTORES MC h1 ( 4 + 5k 1 ) + 2h2 ph12 ph3 − 1 2 MB = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC = h1 ( 4 + 5k 1 ) + 2h2 ph12 h2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MB En AB MY = HA y − py 2 2 HD HA VD 3.31 VA 3.32 3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL p REACCIONES C s VA = VD = pf ( h1 + h2 ) f I 2l B HA = pf − HD HD = pf 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2 y 3 I h1 2 1 2 1 I h2 2 1 A D l MOMENTOS FLECTORES MC = − 2 pfh1 8h1 (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC MB ph2 8h (1+ k 1 ) + 4h1h2 + f ( h1 + h2 ) 8 h (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1 2 1 En BC l py 2 MY = −VA y + HA ( y + h1 ) − f 2 HA HD VA VD Prontuario para Cálculo de Estructuras MB = pfh1 − C REACCIONES Pb VA = l Pa VD = l h1(l + b) + h2 (l + a) Pab HA = HD = 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 s f I 3 B I h1 I a h2 2 Formulario para vigas y pórticos 3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL b 1 A D l MOMENTOS FLECTORES MB = − h1 ( l + b ) + h2 ( l + a ) Pabh1 2 2 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2l MC MB h1 ( l + b ) + h2 ( l + a ) Pabh2 MC = − 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MP = Pab  af  + HA  + h1  l  l  MP HD HA VA VD 3.33 3.34 3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS k= I2 h ⋅ I1 s p 3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL C s REACCIONES I pl 2 pl 2 8h + 5f HA = HE = 2 32 h ( 3 + k ) + f ( 3h + f ) I 2 B VA = VE = D x I I 1 A h 1 E l MOMENTOS FLECTORES 8h + 5f pl 2 h 2 32 h ( 3 + k ) + f ( 3h + f ) MC pl f+h MB + 8 h 2 MC = MB MD En BC y DC MX = p x (l − x) 2 + MB  2fx  h+  h  l  HE HA VA VE Prontuario para Cálculo de Estructuras MB = MD = − f 2 p REACCIONES C pl VA = 3 8 pl VE = 8 HA = HE = s I I 2 2 D B x 8h + 5f pl 2 2 64 h ( 3 + k ) + f ( 3h + f ) I I 1 A MB = MD = − l 8h + 5f pl 2 h 2 64 h ( 3 + k ) + f ( 3h + f ) MC pl 2 f + h MC = MB + 16 h En BC x (l − x) 2 + MB h h 1 E MOMENTOS FLECTORES MX = p f Formulario para vigas y pórticos 3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL MB MD 2fx   h+ l    HE VA VE 3.35 HA 3.36 3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES C ph2 VA = VE = 2l HA = ph − HE s I p I 2 f 2 D B ( 5k + 12 ) h + 6f ph 2 16 h ( k + 3 ) + f ( f + 3h) 2 HE = I I 1 h 1 y A E MOMENTOS FLECTORES l ph2 + MD MB = 2 ph2 f + h + MC = MD 4 h ( 5k + 12 ) h + 6f ph3 MD = − 2 16 h ( k + 3 ) + f ( f + 3h) MC MD En AB My = − py 2 + HA ⋅ y 2 HA HE VA VE Prontuario para Cálculo de Estructuras MB REACCIONES p pf VA = VE = ( f + 2 h) 2l HA = pf − HE C s I f 2 D B pf 8h ( k + 3 ) + 5f ( f + 4h) 16 h2 ( k + 3 ) + f ( f + 3h) 2 HE = I 2 Formulario para vigas y pórticos 3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL y I I 1 h 1 x A MOMENTOS FLECTORES E l MB = HA ⋅ h MC = − 2 pf 2 4h ( k + 2 ) + f ( 5h + f ) ⋅ 2 16 h ( k + 3 ) + f ( f + 3h) MC MD = −HE ⋅ h MB MD En BC Mx = HA ⋅ y − VA ⋅ x − p 2 2 HA HE VA VE 3.37 f siendo y = x + h l ( y − h) 3.38 3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL p REACCIONES Pn l Pm VA = l C s VA = I I 2 f 2 B ( ) D I 2 2 Pm 6hln+ f 3l − 4m HA = HE = 2 2 4 l h ( k + 3 ) + f ( f + 3 h) m n 1 A I h 1 E l MOMENTOS FLECTORES MB = MD = −HA ⋅ h MC MB MD HE HA VA VE Prontuario para Cálculo de Estructuras Pm h + f + MC = MB h 2 hl + 2fm MP = VA ⋅ m − HA l k1 = I3 h1 ⋅ I1 l y k2 = I3 h 2 ⋅ I2 l 3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL p B C REACCIONES I 3 x VA = h −h pl pl + 2 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1 I 2 2 h1 h −h pl pl VD = − 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2 1 I 1 h2 2 Formulario para vigas y pórticos 3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL D 2 2 A h1 − h2 pl 2 HA = HD = 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 l MOMENTOS FLECTORES MB ( h1 + h2 ) h1 pl 2 MB = − 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC = − MC ( h1 + h2 ) h2 pl 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 HD En BC VD HA VA 3.39 px 2 Mx = VA ⋅ x − − HA ⋅ h1 2 3.40 3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES p B C ph h − h2 VA = VD = − HD 1 2l l HA = ph − HD 2 1 HD = ph12 5k1h1 + 4h1 + 2h2 8 h12 (1+ k1 ) + h22 (1+ k2 ) + h1h2 I 3 I h1 I 1 h2 2 D y A MOMENTOS FLECTORES l 5k1h1 + 4h1 + 2h2 ph2 ph3 MB = − 1 − 1 2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MB ph12 h2 5k1h1 + 4h1 + 2h2 2 8 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 MC MB En AB My = HA ⋅ y − py 2 2 HD VD HA VA Prontuario para Cálculo de Estructuras MC = − P a REACCIONES ( l + b ) h1 + ( l + a) h2 Pb Pab VA = h1 − h2 + 3 2 l 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 ( ( l + b ) h1 + ( l + a) h2 Pa Pab VD = h1 − h2 − 3 2 l 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 ( b B C I ) 3 I ) h1 I 1 h2 2 Formulario para vigas y pórticos 3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL D A HA = HD = ( l + b ) h1 + ( l + a) h2 Pab 2 2 2l h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 l MOMENTOS FLECTORES MB = − MB MC ( l + b) h1 + ( l + a) h2 Pabh1 2 2 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2l MP MC = − HD ( l + b ) h1 + ( l + a) h2 Pabh2 2 2 h1 (1+ k1 ) + h22 (1+ k2 ) + h1h2 2l VD HA MP = VA ⋅ a + MB 3.41 VA 3.42 3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL k= I2 h ⋅ I1 l 3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL p B C I VA = VD = pl 2 HA = HD = pl 2 4h ( k + 2 ) I I 1 h 1 A MOMENTOS FLECTORES D l pl2 6 ( k + 2) MB MC En BC Mx = px ( l − x ) 2 Mmáx pos = − pl 2 6( k + 2) pl 2 3k + 2 l para x = 24 k + 2 2 HA HD MA VA MD VD Prontuario para Cálculo de Estructuras pl2 MA = MD = 12 ( k + 2 ) MB = MC = − 2 x REACCIONES p REACCIONES B C 2 ph k VA = VD = l ( 6k + 1) I HA = ph − HD HD = I ph ( 2k + 3 ) A ph2 MC = − 24 MD = ph2 24 D l 2 1    5 + 6k + 1 + k + 2    2 2  ph2  1− +  24  6k + 1 k + 2  2 2    3 − 6k + 1 − k + 2    h 1 y MOMENTOS FLECTORES MB = I 1 8 ( k + 2) ph2 MA = − 24 2 Formulario para vigas y pórticos 3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR MC MB MB 2 1    3 + 6k + 1 − k + 2    En AB My = − py 2 + HA ⋅ y + MA 2 HA MA VA HD MD VD 3.43 3.44 3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES VA = P m Pn  m ( n − m)  1+ 2  l  l ( 6k + 1)  n B C I 2 VD = P − VA HA = HD = 3Pmn 2lh(k + 2) I MOMENTOS FLECTORES MA = I 1 A D Pmn  1 n− m  −   2l  k + 2 l ( 6k + 1)  Pmn  1 n− m  +   l  k + 2 2l ( 6k + 1)  MC = − Pmn  1 n− m  −   l  k + 2 2l ( 6k + 1)  MD = Pmn  1 n− m  +   2l  k + 2 l ( 6k + 1)  MP = Pmn nMB mMC + + l l l l MB MC MP HA HD MA VA MD VD Prontuario para Cálculo de Estructuras MB = − h 1 Formulario para vigas y pórticos 3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR REACCIONES P B 3Phk VA = VD = l(6k + 1) P HA = HD = 2 MOMENTOS FLECTORES Ph 3k + 1 2 6k + 1 Ph 3k MB = − MC = 2 6k + 1 Ph 3k + 1 MD = 2 6k + 1 C I I 2 I 1 h 1 A D MA = − l MB MC HA MA VD 3.45 VA HD MD 3.46 3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS k= I2 h ⋅ I1 s p 3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL C s REACCIONES I pl 2 k ( 4h + 5f ) + f pl 2 HA = HE = 8 ( kh + f )2 + 4k h2 + hf + f 2 f 2 D B VA = VE = ( I 2 x I ) I 1 h 1 A E l MOMENTOS FLECTORES pl2 kh ( 8h + 15f ) + f ( 6h − f ) 48 ( kh + f )2 + 4k h2 + hf + f 2 ( kh (16h + 15f ) + f 2 pl2 MB = MD = − 48 ( kh + f )2 + 4k h2 + hf + f 2 ( pl 2 + MA − HA ( h + f ) 8 En BC MC ) MB ) MD MC = 2 xf  px  Mx = MA + VA ⋅ x − HA  h + − l  2  HA 2 HE MA VA ME VE Prontuario para Cálculo de Estructuras MA = ME = p REACCIONES pl − VE 2 4k + 1 VE = 3 pl 32 ( 3k + 1) k ( 4h + 5f ) + f pl 2 HA = HE = 16 ( kh + f )2 + 4k h2 + hf + f 2 C VA = ( s I I 2 f 2 B D x ) I I 1 h 1 Formulario para vigas y pórticos 3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL MOMENTOS FLECTORES A pl 2 kh ( 8h + 15f ) + f ( 6h − f ) pl 2 MA = − 2 96 ( kh + f ) + 4k f 2 + fh + h2 64 ( 3k + 1) ( ME = E ) l pl 2 kh ( 8h + 15f ) + f ( 6h − f ) pl 2 + 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1) ( MB = − kh (16h + 15f ) + f 2 pl 2 pl 2 − 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1) MD = − kh (16h + 15f ) + f 2 pl 2 pl2 + 96 ( kh + f )2 + 4k f 2 + fh + h2 64 ( 3k + 1) En BC MC ) ( ( MB ) ) HA HE MA VA ME VE 3.47 2 xf  px 2  Mx = MA + VA ⋅ x − HA  h + − l  2  l MC = VE + ME − HE ( f + h) 2 MD 3.48 3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR REACCIONES VA = VE = ph2 k 2l ( 3k + 1) C s HA = ph − HE HE = k 2 h + k ( 2 f + 3 h) + f ph2 4 ( kh + f )2 + 4k f 2 + fh + h2 ( ) I f 2 D I 1 h 1 y A E 2  2  2k + 1 ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f 6 + 24  ( kh + f )2 + 4k f 2 + fh + h2 3k + 1   ( l ) MC MB 2 2   2k + 1 ph2  kh ( k + 6 ) + kf (15h + 16f ) + 6f − + 6 2 24  3k + 1 ( kh + f ) + 4k f 2 + fh + h2   En AB py 2 My = MA + HA ⋅ y − 2 ( MD ) HA MA VA HE ME VE Prontuario para Cálculo de Estructuras ph2 MB = MA + HA ⋅ h − 2 1 MC = ME − HE ( f + h) + VE 2 MD = ME − HE ⋅ h ME = I 2 B MOMENTOS FLECTORES MA = − I p REACCIONES p 3 pf 4k ( f + h) + f 8 l 3k + 1 HA = pf − HE C VA = VE = HE = s I 2 pf 2k h ( k + 4 ) + f (10kh + 5kf + f ) 4 ( kh + f )2 + 4k f 2 + fh + h2 ( MC = ME − HE ( h + f ) + VE MD = ME − HE ⋅ h ) y I I 1 ( h 1 A E l ) MC l 2 MB   kh ( 9f + 4h) + f ( 6h + f ) 3 4h ( 3k + 2 ) + f  pf  −f +  24  ( kh + f )2 + 4k f 2 + fh + h2 2 3k + 1   En BC 2 l ( y − h) p ( y − h) − My = MA + HA ⋅ y − VA 2f 2 ME = D   kh ( 9f + 4h) + f ( 6h + f ) 3 4h ( 3k + 2 ) + f  pf  + f  24  ( kh + f )2 + 4k f 2 + fh + h2 2 3k + 1   MB = MA + HA ⋅ h f 2 B MOMENTOS FLECTORES MA = − I 2 Formulario para vigas y pórticos 3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL ( MD ) HA MA VE 3.49 VA HE ME 3.50 3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL REACCIONES p VA = P − VE 2 Pm 3l ( kl + m) − 2m VE = 3 3k + 1 l 2 2 Pm 3kl ( f + h) − 4fm ( k + 1) + 3lm ( f − kh) HA = HE = 2 2 l ( kh + f ) + 4k f 2 + fh + h2 ( C s I f 2 D B ) I MOMENTOS FLECTORES m n I 1 E ) l MC MB ( MD ) HA HE MA VA ME VE Prontuario para Cálculo de Estructuras MB = MA − HA ⋅ h l MC = ME + VE − HE ( h + f ) 2 MD = ME − HE ⋅ h  3flh ( kl + 2m) − 4fm2 ( kh + 2h + f ) + 2kh2 ln+ f 2 l ( 4m − l )    2 Pm  kh + f ) + 4k f 2 + fh + h2  ( ME = 2   2l  n n − m  ( )  + 3k + 1   En BC 2fm   My = MA + VA ⋅ m − HA  h + l   h 1 A  3flh ( kl + 2m) − 4fm2 ( kh + 2 h + f ) + 2kh2 ln+ f 2 l ( 4m − l )    2 Pm  kh + f ) + 4k f 2 + fh + h2  ( MA = 2   2l  n n − m  ( ) −  3k + 1   ( I 2