AUTORES
Arianna Guardiola Víllora nació en Trieste (Italia), se graduó en la Escuela Técnica Superior de Arquitectura de Valencia donde se doctoró en 2006. Desde 1997 es profesora del Departamento de Mecánica de los Medios Continuos y Teoría de Estructuras de la Universidad Politécnica de Valencia. Actualmente es Profesora Titular de Escuela Universitaria e imparte la asignatura Estructuras III de la ETSAV.
Agustín Pérez García nació en Alzira, estudió en la Escuela Técnica Superior de Arquitectura de Valencia donde se doctoró en 1986. Desde 1987 es profesor del Departamento de Mecánica de los Medios Continuos y Teoría de Estructuras de la Universidad Politécnica de Valencia. Actualmente es Catedrático de Universidad e imparte la asignatura Introducción a las Estructuras de Edificación de la ETSAV.
INTRODUCCION Esta publicación contiene materiales – documentos normativos y recomendaciones, hojas de cálculo y aplicaciones informáticas – útiles para llevar a término y justificar documentalmente muchos de los procesos de análisis asociados al diseño y el cálculo de estructuras tanto desde la perspectiva académica como desde la profesional. Con el objeto de facilitar el uso de la formulación incluida en sus diferentes capítulos se han diseñado una serie de hojas de cálculo que las implementan y que permiten obtener directamente resultados utilizando un ordenador. Todas ellas se encuentran en el CD que forma parte de esta obra. Así mismo, la lectura o consulta del material incluido en la publicación puede efectuarse mediante los ficheros en formato PDF incluidos en el CD. También se incorporan varios programas – EFCiD®, Architrave® y SigmaCAD – desarrollados por los autores conjuntamente con otros profesores de la Universidad Politécnica de Valencia y diseñados para realizar cálculos en los ámbitos de la Resistencia de Materiales y del Análisis de Estructuras. Se trata de versiones académicas que sólo pueden utilizarse en dicho ámbito. No obstante, existen versiones profesionales, bastante más completas y potentes, disponibles en el Centro de Transferencia de Tecnología de la Universidad Politécnica de Valencia. Originalmente, este libro fue diseñado para servir como material de apoyo para el desarrollo de los ejercicios y prácticas de la asignatura Introducción a las Estructuras de Edificación (Estructuras I) de la ETS de Arquitectura de Valencia pero progresivamente ha ido incrementando sus contenidos con material útil para los alumnos de la asignatura Estructuras III en la que se abordan, básicamente, el estudio de las estructuras de acero. En general se ha tratado de presentar la información de manera sintética y del modo mas gráfico y claro que los autores han sido capaces de imaginar. Esperan con ello haber elaborado un material y unas herramientas informáticas realmente útiles. Los autores agradecen la colaboración del arquitecto Miguel Martínez Ausina en la elaboración de algunos de los contenidos de esta publicación. También quieren mostrar su reconocimiento al profesor Ivan Cabrera Fausto autor del diseño de la portada y al resto de profesores de las asignaturas de Introducción a la Estructuras de Edificación y de Estructuras III de la ETSAV por la favorable acogida que le han prestado y por sus valiosas observaciones ya que han permitido depurar y ampliar su contenido. Por último desean dedicar este libro a sus hijos que han colaborado, con sus presencias y sus ausencias, a que esta tercera edición de la obra sea una realidad.
Los autores
1. ACCIONES EN LA EDIFICACION 2. MATERIALES ESTRUCTURALES 3. DISTRIBUCION DE SOLICITACIONES 4. DIAGRAMAS DE PREDIMENSIONADO 5. TABLAS Y DIAGRAMAS DIMENSIONADO 6. LIMITACION DE LAS DEFORMACIONES 7. ESTABILIDAD DE BARRAS A PANDEO 8. UNIONES EN ESTRUCTURAS DE ACERO 9. GEOMETRÍA DE MASAS 10. PROGRAMA EFCiD® 11. PROGRAMA Architrave® 12. PROGRAMA SigmaCAD
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) = ∑
A⋅ x − a
T ( x ) = −∑
A⋅ x − a
M( x ) = − ∑
(c− 2)
( c − 2 )! ( 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
0
=0
x≥a
x−a
0
=1
x≤a
x−a
n
=0
x≥a
x−a
n
= ( x − a)
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
por lo tanto
c=0
P a x
T(x)
Si x≤a
x−a = 0
x≥a
x − a = ( x − a)
1
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)
2
A=q
por lo tanto
c=2
Si
q
x−a
x≥a
3
x−a
3
=0
= ( x − a)
d
a
entonces
x 2
T(x) 3
M(x)
x≤a
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! T ( x ) = q ⋅ [ −〈 x-a〉 + 〈 x-b〉 ]
M(x ) =
a b x q/d q
a
d b
q q/d -〈 x-a〉 3 + 〈 x-b〉 3 〈 x-b〉 2 + 2! 3! q/d -〈 x-a〉 2 + 〈 x-b〉 2 T ( x ) = q ⋅ 〈 x-b〉 + 2!
M(x) =
x
q/d q
a
q q/d 〈 x-a〉 3 − 〈 x-b〉 3 〈 x-a〉 2 + 2! 3! q/d 〈 x-a〉 2 − 〈 x-b〉 2 T ( x ) = −q ⋅ 〈 x-a〉 + 2!
M(x ) = −
d b x
qb
qa a
M( x ) = −
qa 2!
〈 x-a〉 2 +
qb 2!
〈 x-b〉 2 +
T ( x ) = −q a 〈 x-a〉 + q b 〈 x-b〉 +
d
(q
b
(q
b
)
− q a /d 3!
)
− q a /d 2!
−〈 x-a〉 3 + 〈 x-b〉 3
−〈 x-a〉 2 + 〈 x-b〉 2
b x
qa
a
qb d b x
M( x ) = −
qa 2!
〈 x-a〉 2 +
qb 2!
〈 x-b〉 2 +
T ( x ) = −q a 〈 x-a〉 + q b 〈 x-b〉 +
(q
a
(q
a
)
− q b /d 3!
)
− q b /d 2!
〈 x-a〉 3 − 〈 x-b〉 3
2 2 〈 x-a〉 − 〈 x-b〉
VIGA APOYADA EN LOS EXTREMOS
3.2.1
CARGA PUNTUAL
P
A REACCIONES P⋅b RA = L
RB =
B
C
P⋅a L
x ESFUERZOS CORTANTES P⋅b P⋅a QAC = = cte ; QCB = − = cte L L
a
b
Formulario para vigas y pórticos
3.2
L MOMENTOS FLECTORES P⋅b P⋅a MAC = ⋅ x ; MCB = ⋅ (L − x) L L P ⋅ a⋅ b para x0 = a Mmax = MC = L ANGULOS DE GIRO P⋅ a⋅ b ϕA = ⋅ ( L + b) 6⋅E⋅I⋅L
; ϕB = −
QB
P⋅ a⋅ b ⋅ ( L + a) 6⋅ E⋅I ⋅ L
; ϕC =
P⋅ a⋅ b ⋅ ( b − a) 3⋅E⋅I⋅L
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
M max
3.5
3.6
3.2.2
CARGA PUNTUAL CENTRAL
REACCIONES
RA = RB =
P
A
P 2
B
C
ESFUERZOS CORTANTES
QAC = QCB = −
x
P = cte 2
a
MOMENTOS FLECTORES
MAC
P = ⋅x 2
Mmax = MC =
;
MCB
P⋅L 4
a L
P = ⋅ (L − x) 2 para
x0 =
L 2
QB
ANGULOS DE GIRO
P ⋅ L2 16 ⋅ E ⋅ I
; ϕC = 0
QA
ECUACION DE LA ELASTICA y AC =
P ⋅ L2 ⋅ x 4 x 2 ⋅ 1− ⋅ 16 ⋅ E ⋅ I 3 L2
FLECHA MAXIMA
fC =
P ⋅ L3 48 ⋅ E ⋅ I
M max
Prontuario para Cálculo de Estructuras
ϕ A = −ϕ B =
CARGA CONTINUA EN PARTE DE LA VIGA
REACCIONES p⋅b⋅c RA = L
RB =
c
p⋅ a⋅ c L
p
ESFUERZOS CORTANTES p⋅b⋅c p⋅b⋅c c QAC = ; QCD = − p⋅ − a+ x L L 2
A ; QDB
C
MOMENTOS FLECTORES p⋅b⋅c p⋅b⋅c p c MAC = ⋅ x ; MCD = ⋅ x − ⋅ x − a − 2 L L 2 2 p⋅a⋅c MDB = ⋅ (L − x) L p⋅b⋅c b⋅c c b⋅c Mmax = ⋅2⋅a− c + para x0 = a − + 2⋅L 2 L L ANGULOS DE GIRO p⋅a⋅b⋅c c2 ⋅L + b − ϕA = 6⋅E⋅I⋅L 4⋅a
; ϕB = −
B
p⋅a⋅c =− L
p⋅ a⋅ b⋅c c2 ⋅L + a − 6⋅E⋅I⋅L 4⋅b
D
x a
Formulario para vigas y pórticos
3.2.3
b L
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 2 p⋅ a⋅ c L − x c 2 ⋅ ⋅ − ( L − x ) + b ⋅ L + a − y DB = 6⋅L 4 ⋅ a E ⋅ I
y
CD =
M max
3.7
3.8
3.2.4
CARGA CONTINUA EN TODA LA VIGA p
REACCIONES RA = RB =
p⋅L 2
A
B
ESFUERZOS CORTANTES L QAB = p ⋅ − x 2
;
QA = −QB =
x
P⋅L 2
L
MOMENTOS FLECTORES
MAB = Mm a x
p⋅x ⋅ (L − x ) 2 p ⋅ L2 = p a ra 8
x0 =
ANGULOS DE GIRO
VB VA
p ⋅ L3 24 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
y AB =
p⋅ x ⋅ x3 − 2 ⋅ L ⋅ x2 + L3 24 ⋅ E ⋅ I
ymax =
(
5 ⋅ p ⋅ L4 384 ⋅ E ⋅ I
para
x0 =
) L 2
M max
Prontuario para Cálculo de Estructuras
ϕ A = −ϕ B = −
L 2
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES 1 RA = ( 2 ⋅ p1 + p2 ) 6
; RB =
1 ( p1 + 2 ⋅ p2 ) . 6
p1
p 2
ESFUERZOS CORTANTES p ( 3 ⋅ L − x ) + p2 ⋅ x 2 QA = RA ; Qx = RA − 1 ⋅x 6⋅L
;
QB = −RB
B
A MOMENTOS FLECTORES p ( 3 L − x ) + p2 ⋅ x 2 Mx = RA ⋅ x − 1 ⋅x 6⋅L
x L 2
2
L L ⋅ ( p1 + p2 ) y 0,128 ⋅ ⋅ ( p1 + p2 ) 2 2 1 1 para x 0 = ⋅ − p1 + ⋅ p12 + p22 + p1 ⋅ p2 p2 − p1 3 Mmax comprendido entre 0,125 ⋅
(
ANGULOS DE GIRO L3 ϕA = ⋅ ( 8 ⋅ p1 + 7 ⋅ p2 ) 360 ⋅ E ⋅ I
Formulario para vigas y pórticos
3.2.5
)
; ϕB = −
QA QB
L3 ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 360 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
yx =
3 2 x ( L − x ) 3 ( p1 − p2 ) x − 3 ( 4p1 + p2 ) Lx + 360EI ( 8 p1 + 7p2 ) L2 x + ( 8p1 + 7p2 ) L3
0,01304 ⋅
( p1 + p2 ) ⋅ L4 2⋅E⋅I
x0
M max
3.9
FLECHA MAXIMA ( p + p2 ) ⋅ L4 y entre 0,01302 ⋅ 1 2⋅E⋅I
3.10
3.2.6
MOMENTO FLECTOR
REACCIONES
R A = −R B = −
M
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
M = MCizq + Mder C
QB
MC M MC
Prontuario para Cálculo de Estructuras
2 M ⋅ L ⋅ (L − x) a2 L − x =− ⋅ 1− 3 ⋅ 2 − 6⋅E⋅I L L
FLECHA M⋅ a ⋅ b fC = ⋅ ( b − a) 3⋅E⋅I⋅L
QA
)
ECUACION DE LA ELASTICA M⋅ L ⋅ x b2 x 2 y AC = − ⋅ 1− 3 ⋅ 2 − 2 6⋅ E⋅I L L
yCB
b 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 ⋅ a3 + b3 ϕC = 3 ⋅ E ⋅ I ⋅ L2
(
B
a
3.3.1
P
CARGA PUNTUAL
REACCIONES P ⋅ b2 RA = 3 ⋅ ( L + 2 ⋅ a) L
;
RB =
Formulario para vigas y pórticos
3.3 VIGA EMPOTRADA EN LOS EXTREMOS
C
P ⋅ a2 ⋅ ( L + 2 ⋅ b) L3
B
A x
ESFUERZOS CORTANTES P ⋅ b2 QAC = 3 ⋅ ( L + 2 ⋅ a) = cte ; L
QCB = −
P ⋅ a2 ⋅ ( L + 2 ⋅ b ) = cte L3
a
b L
MOMENTOS FLECTORES
P ⋅ a ⋅ b2 P ⋅ a2 ⋅ b P ⋅ b2 ; MB = − ; MAC = 3 ⋅ ( L ⋅ x + 2 ⋅ a ⋅ x − a ⋅ L ) 2 2 L L L P ⋅ a2 2 ⋅ P ⋅ a2 ⋅ b2 = 3 ⋅ L ⋅ b + L2 − L ⋅ x − 2 ⋅ b ⋅ x ; MC = para x0 = a L L3
MA = − MBC
(
)
QB QA
ECUACION DE LA ELASTICA
yBC
P ⋅ b2 2 ⋅ a ⋅ x x2 ⋅3 ⋅a − x − ⋅ 6⋅E⋅I L L2
P ⋅ a2 L − x ⋅ (L − x) = ⋅ 3 ⋅ b − ( L − x) − 2 ⋅ b ⋅ 6⋅E⋅I L L2
P ⋅ a3 ⋅ b3 3 ⋅ E ⋅ I ⋅ L3
;
fmax =
2 ⋅a⋅ L x= L + 2⋅a
2 ⋅ P ⋅ a3 ⋅ b2 3 ⋅ E ⋅ I ⋅ ( L + 2 ⋅ a)
2
0
3.11
para
MB
MA
MC
FLECHAS fC =
2
x
y AC =
3.12
3.3.2
CARGA PUNTUAL CENTRAL
REACCIONES
RA = RB =
P
P 2
C
QAC
P = = cte 2
B
A
ESFUERZOS CORTANTES
QCB
x
P = − = cte 2
a
a L
MOMENTOS FLECTORES
MA = MB = −
P⋅L 8
Mmax = MC =
P⋅L 8
MAC = para
x0 =
PL x ⋅ 4 ⋅ −1 8 L L 2
QB QA
y AC = −
P ⋅ L ⋅ x2 48 ⋅ E ⋅ I
x ⋅3 − 4⋅ L
MB
MA
FLECHAS
M max 0
x
P ⋅ L3 fC = 192 ⋅ E ⋅ I
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA
CARGA CONTINUA EN PARTE DE LA VIGA
c
REACCIONES
p ⋅ b ⋅ c MA − MB RA = − L L
;
p
p ⋅ a ⋅ c MA − MB RB = + L L
C
ESFUERZOS CORTANTES
QAC = RA = cte ; QBD = −RB = cte ; QCD
;
MBD = RB ⋅ ( L − x ) + MB
c = RA − p ⋅ x − a + a
B x a
MCD = RA ⋅ x + MA − ;
D
A
MOMENTOS FLECTORES MAC = RA ⋅ x + MA
MA = −
p ⋅ c3 12 ⋅ L2
p c ⋅ x−a+ 2 2
Formulario para vigas y pórticos
3.3.3
b
2
12 ⋅ a ⋅ b2 ⋅L − 3⋅b + c2
p ⋅ c3 12 ⋅ a2 ⋅ b MB = − L a ⋅ − 3 ⋅ + 12 ⋅ L2 c2
L
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 ⋅ x3 − 12 ⋅ MA ⋅ x3 24 ⋅ E ⋅ I 2 1 RB x3 − 3 ( MB + LRB ) x2 + 3 ( 2 MA + LRB ) Lx − ( 3 MB + LRB ) L2 y DB = 6EI y AC =
MA
MB
3.13
3.14
3.3.4
CARGA CONTINUA EN TODA LA VIGA
p
REACCIONES
RA = RB =
P⋅L 2
ESFUERZOS CORTANTES
QAB
B
A
P = Qx = ⋅ ( L − 2 ⋅ x ) 2
x L
MOMENTOS FLECTORES
MA = MB = −
P ⋅ L2 12
P ⋅ L2 − 6 ⋅ L ⋅ x + 6 ⋅ x 2 12 P ⋅ L2 L Mcentro = para x = 24 2 Mx = 0 para x0 = 0,2113 ⋅ L
Q
ECUACION DE LA ELASTICA
MA
Mx = −
)
P ⋅ L4 x x 2 ⋅ − 24 ⋅ E ⋅ I L L2
ymax =
P ⋅ L4 384 ⋅ E ⋅ I
A
Q
B
2
para
x=
L 2
MB
Prontuario para Cálculo de Estructuras
yx =
(
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
p1
L M − MB ⋅ ( 2 ⋅ p1 + p2 ) − A 6 L L MA − MB RB = ⋅ ( p1 + 2 ⋅ p2 ) + 6 L
p2
RA =
ESFUERZOS CORTANTES
QA = RA Qx = RA − QB = −RB
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L
x L ⋅x
MOMENTOS FLECTORES
L2 MA = − ⋅ ( 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 ECUACION DE LA ELASTICA
yx =
B
A
Formulario para vigas y pórticos
3.3.5
A
Q
B
MA
MB
3.15
( p − p1 ) 3 x2 ⋅ 2 ⋅ x + p1 ⋅ L ⋅ x2 − 4 ⋅ RA ⋅ L ⋅ x − 12 ⋅ MA ⋅ L 24 ⋅ E ⋅ I ⋅ L 5
Q
3.16
3.3.6
MOMENTO FLECTOR
REACCIONES 6⋅M RA = − 3 ⋅ a ⋅ b L
; RB =
6⋅ M ⋅ a⋅ b L3
C
ESFUERZOS CORTANTES
Qx = −
+M
x
6⋅ M ⋅ a ⋅ b = cte L3
a
b
MOMENTOS FLECTORES M⋅ a b MA = ⋅2 − 3⋅ L L MAC =
B
A
L
M⋅ b a MB = − ⋅2 − 3⋅ L L
M⋅ a a x ⋅ 3 ⋅ ⋅ 1− 2 ⋅ − 1 L L L
MCB = −
QA
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
)
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
b⋅ x a ⋅2⋅ 2 − L L
FLECHA M ⋅ a2 ⋅ b2 fC = − ⋅ ( a − b) 2 ⋅ E ⋅ I ⋅ L3
MC MA MC
MB
Prontuario para Cálculo de Estructuras
MCizq = MA −
QB
3.4.1
P
CARGA PUNTUAL
REACCIONES P ⋅ b2 P⋅a ⋅ ( 3 ⋅ L − b ) ; RB = ⋅ 3 ⋅ L2 − a2 RA = 2 ⋅ L3 2 ⋅ L3 ESFUERZOS CORTANTES P ⋅ b2 P⋅a QAC = − ⋅ ( 3 ⋅ L − b ) = cte ; QCB = − ⋅ 3 ⋅ L2 − a2 = cte 2 ⋅ L3 2 ⋅ L3
(
x a
b L
)
(
2
; ϕC =
4⋅E⋅I⋅L
)
Q
B
Q
A
ANGULOS DE GIRO P ⋅ a ( L − a)
B
A
)
MOMENTOS FLECTORES P⋅a 2 P⋅a 2 MB = − ⋅ L − a2 ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b) ; MC = 2 ⋅ L2 2 ⋅ L3 P⋅ x 2 P⋅a MAC = ⋅ b ⋅ ( 3 ⋅ a + 2 ⋅ b ) ; MCB = ⋅ 2 ⋅ L3 − 3 ⋅ L2 ⋅ x + a2 ⋅ x 2 ⋅ L3 2 ⋅ L3
ϕA =
C
)
(
(
P ⋅ a ⋅ ( L − a)
2
4 ⋅ E ⋅ I ⋅ L3
(
⋅ L2 − 2 ⋅ a ⋅ L − a2
)
ECUACION DE LA ELASTICA P ⋅ b2 ⋅ x y AC = ⋅ 3 ⋅ a ⋅ L2 − x2 ⋅ ( 2 ⋅ L + a) 12 ⋅ E ⋅ I ⋅ L3
y BC =
P ⋅ a ⋅ ( L − x)
2
12 ⋅ E ⋅ I
MB
a2 a2 L − x ⋅ 3 ⋅ 1− 2 − 3 − 2 ⋅ L L L
FLECHA MAXIMA
p⋅b ⋅a a ⋅ 6⋅ E⋅I 2⋅L + a
MC para x=L ⋅
a 2⋅L+a
3.17
fmax =
2
Formulario para vigas y pórticos
3.4 VIGA APOYADA-EMPOTRADA
3.18
3.4.2
CARGA PUNTUAL CENTRAL
REACCIONES 5 11 RA = ⋅ P ; RB = ⋅P 16 16
P C
ESFUERZOS CORTANTES 5 11 QAC = ⋅ P = cte ; QCB = − ⋅ P = cte 16 16
B
A
x
MOMENTOS FLECTORES
a
3 5 MB = − ⋅ P ⋅ L ; MC = ⋅P⋅L 16 32 5 P⋅ L L − x MAC = ⋅ P ⋅ x ; MCB = ⋅ 11⋅ − 3 16 16 L
a L
ANGULOS DE GIRO
P⋅L 32 ⋅ E ⋅ I
QB
; ϕC = −
P⋅L 128 ⋅ E ⋅ I 2
QA
ECUACION DE LA ELASTICA y AC =
P ⋅ L2 x2 ⋅ x ⋅ 3 − 5 ⋅ 2 96 ⋅ E ⋅ I L
y BC =
P⋅L L − x ⋅ (L − x)2 ⋅ 9 − 11⋅ 96 ⋅ E ⋅ I L
MB
FLECHA MAXIMA
fC =
7 ⋅ P ⋅ L3 ; 768 ⋅ E ⋅ I
fmax =
P ⋅ L3 48 ⋅ 5 ⋅E ⋅ I
para x=
L 5
MB
Prontuario para Cálculo de Estructuras
ϕA =
2
CARGA CONTINUA EN PARTE DE LA VIGA
REACCIONES p ⋅ b ⋅ c MB + RA = L L
c ; RB =
p
p ⋅ a ⋅ c MB − L L
ESFUERZOS CORTANTES
c QAC = RA = cte ; QDB = −RB = cte ; QCD = RA − p ⋅ x − a + 2
C
;
MCD
MDB = RB ⋅ ( L − x ) + MB
MB = −
b L
p⋅ a⋅ b⋅c c2 ⋅L + a− 2 2⋅L 4⋅b
ECUACION DE LA ELASTICA x 12 ⋅ a ⋅ b2 y AC = ⋅ −8 ⋅ RA ⋅ L ⋅ x 2 + p ⋅ c3 ⋅ L − 3b + 48 ⋅ E ⋅ I ⋅ L c2
QB QA
MB
4 c −8 ⋅ RA ⋅ L ⋅ x 3 + 2 ⋅ p ⋅ L ⋅ x − a + + 4 1 = ⋅ 2 48 ⋅ E ⋅ I ⋅ L + p ⋅ c3 ⋅ L − 3 ⋅ b + 12 ⋅ a ⋅ b ⋅ x 2 c
(L − x) =−
2
6⋅E⋅I
⋅ RB ⋅ ( L − x ) + 3 ⋅ MB
3.19
y DB
a
2
ANGULOS DE GIRO p ⋅ c3 12 ⋅ a ⋅ b2 ϕA = ⋅L − 3⋅b + 48 ⋅ E ⋅ I ⋅ L c2
yCD
B x
p c = RA ⋅ x − ⋅ x − a + 2 2 ;
D
A
MOMENTOS FLECTORES MAC = RA ⋅ x
Formulario para vigas y pórticos
3.4.3
3.20
3.4.4
CARGA CONTINUA EN TODA LA VIGA
REACCIONES
RA =
3 ⋅P⋅L 8
;
RB =
p
5 ⋅P⋅L 8
ESFUERZOS CORTANTES 3 x QAB = P ⋅ L ⋅ − ; 8 L
3 ⋅ P ⋅ L; 8
QA =
QB =
5 ⋅P⋅L 8
B
A
x
MOMENTOS FLECTORES
L
P⋅x ⋅ (3 ⋅ L − 4 ⋅ x); 8 9 Mmax rel = ⋅ P ⋅ L2 para 128 3 M = 0 para x = ⋅ L 4 MAB =
MB = − x=
P⋅L 8
2
3 ⋅ L; 8
QB
ANGULOS DE GIRO
ϕA =
P ⋅ L3 48 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
y AB =
P⋅x ⋅ (L + 2 ⋅ x) ⋅ (L − x)2 48 ⋅ E ⋅ I
fmax =
P ⋅ L4 185 ⋅ E ⋅ I
para
x=
1+ 33 ⋅L 16
MB
Prontuario para Cálculo de Estructuras
QA
CARGA TRAPEZOIDAL EN TODA LA VIGA
REACCIONES
RA =
L M ⋅ ( 2 ⋅ p1 + p2 ) + B 6 L
; RB =
L M ⋅ ( p1 + 2 ⋅ p2 ) − B 6 L
p2
p1
Qx = RA −
p1 ⋅ ( 2 ⋅ L − x ) + p2 ⋅ x 2⋅L
B
A
ESFUERZOS CORTANTES
x ⋅x
L
; QB = −RB
Formulario para vigas y pórticos
3.4.5
MOMENTOS FLECTORES
Mx = RA ⋅ x −
p1 ⋅ ( 3 ⋅ L − x ) + p2 ⋅ x 6⋅ L
Q ⋅x
2
;
L2 MB = − ⋅ ( 7 ⋅ p1 + 8 ⋅ p2 ) 120
A
Q
B
ANGULOS DE GIRO
ϕA =
L3 ⋅ ( 3 ⋅ p1 + 2 ⋅ p2 ) 240 ⋅ E ⋅ I
MB
ECUACION DE LA ELASTICA
3.21
( p2 − p1) ⋅ x4 + x 3 2 yx = ⋅ + 5 ⋅ L ⋅ p1 ⋅ x − 20 ⋅ RA ⋅ L ⋅ x + 120 ⋅ E ⋅ I ⋅ L 2 3 + 5 ⋅ L ⋅ 12 ⋅ RA ⋅ L − ( 3 ⋅ p1 + p2 ) ⋅ L
3.22
3.4.6
MOMENTO FLECTOR
M+
REACCIONES
RA = −RB =
(
3 M 2 ⋅ ⋅ L − a2 2 L3
)
ESFUERZOS CORTANTES
a
MOMENTOS FLECTORES MCder = RA ⋅ a − M ;
(
b L
MCizq = RA ⋅ a
3 M⋅ x 2 ⋅ ⋅ L − a2 2 L3
B
x
Qx = RA = cte
MAC =
C
A
)
;
MBC
(
M ⋅ L2 − 3 ⋅ a2 2 ⋅ L2 M x a2 = ⋅ 3 ⋅ ⋅ 1 − 2 − 2 2 L L MB =
;
ANGULOS DE GIRO
)
M ⋅ ( L − a) ⋅ ( 3 ⋅ a − L ) 4⋅E⋅I⋅L b a 2 M ϕC = ⋅ b ⋅ 3 ⋅ ⋅ 1+ − 4 4⋅E⋅I L L
QB
ϕA =
MC MB
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 = yBC
(
)
(
)
MC
Prontuario para Cálculo de Estructuras
QA
3.5.1
CARGA PUNTUAL
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
b L
MOMENTOS FLECTORES
MAC = 0
;
MCB = −P ⋅ ( x − a)
;
MB = −P ⋅ b
QB
ANGULOS DE GIRO
ϕ A = ϕC = −
P ⋅ b2 2⋅E⋅I
ECUACION DE LA ELASTICA
y
AC
=
P ⋅ b2 ⋅ ( 3 ⋅ ( L − x ) − b) 6⋅E⋅I
;
y
CB
=
MB
3.23
FLECHA MAXIMA P ⋅ b3 P ⋅ b2 fC = ; fA = ⋅ ( 2 ⋅ b + 3 ⋅ a) 3⋅E⋅I 6⋅ E⋅I
P 2 ⋅ ( L − x ) ⋅ ( 2 ⋅ b + 3 ⋅ a) 6⋅E⋅I
3.24
3.5.2
CARGA PUNTUAL EN EL EXTREMO P
REACCIONES RB = P
A ESFUERZOS CORTANTES QAB = −P = cte
B
x L
MOMENTOS FLECTORES MAB = −P ⋅ x;
MB =-P ⋅ L
QB
ECUACION DE LA ELASTICA
y
AB
=
P ⋅ (L − x)2 ⋅ (2 ⋅ L + x) 6⋅E⋅I
FLECHA MAXIMA
fA =
P ⋅ L3 3⋅E⋅I
MB
Prontuario para Cálculo de Estructuras
ANGULOS DE GIRO P ⋅ L2 ϕA = − 2⋅E⋅I
CARGA CONTINUA EN PARTE DEL VUELO
REACCIONES RB = p ⋅ c ESFUERZOS CORTANTES c QAC = 0 ; QCD = −p ⋅ x − a + 2
; QDB = −p ⋅ c = cte
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 ϕD = − ⋅b − 2⋅E⋅I 4
; ϕC = −
MD = −
p ⋅ c2 2
p ⋅ c 2 c2 ⋅b + 2⋅E⋅I 12
Formulario para vigas y pórticos
3.5.3
; ϕ A = ϕC
ECUACION DE LA ELASTICA
y DB =
p⋅c p⋅c c2 3 ⋅ L − x2 ⋅ ( 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
(
)
FLECHAS 2
fD =
p⋅ c c ⋅ b − E⋅I 2
fC =
2 p ⋅ c c p⋅ c c2 ⋅ b + ⋅ ( 4 ⋅ b − c) + c3 ; fA = ⋅ a ⋅ 3 ⋅ b2 + + 2 ⋅ b3 12 ⋅ E ⋅ I 2 6 ⋅ E ⋅ I 4
b c ⋅ + 3 12
3.25
3.26
3.5.4
CARGA CONTINUA EN TODO EL VUELO p
REACCIONES RB = p ⋅ L
B ESFUERZOS CORTANTES
QAB = −P ⋅ x
A
QB = −P ⋅ L = cte
;
x L
MOMENTOS FLECTORES
MAB = −
P ⋅ x2 ; 2
MB = −P ⋅
L2 2
ANGULOS DE GIRO
QB
P ⋅ L3 6⋅E⋅I
ECUACION DE LA ELASTICA
y AB =
FLECHAS fA =
(
P 2 ⋅ ( L − x ) ⋅ 3 ⋅ L2 + 2 ⋅ L ⋅ x + x 2 24 ⋅ E ⋅ I
P ⋅ L4 8⋅E⋅I
) MB
Prontuario para Cálculo de Estructuras
ϕA = −
CARGA TRAPEZOIDAL EN TODO EL VUELO
REACCIONES L RB = ⋅ ( p1 + p2 ) 2 ESFUERZOS CORTANTES
Qx = −
p2 − p1 x2 ⋅ − p1 ⋅ x 2 L
p2
p1 L ; QB = − ⋅ ( p1 + p2 ) 2
MOMENTOS FLECTORES
x2 Mx = − ⋅ ( p2 − p1) ⋅ x + 3 ⋅ L ⋅ p1 6⋅L
A B
Formulario para vigas y pórticos
3.5.5
x ;
L2 MB = − ⋅ ( p2 + 2 ⋅ p1 ) 6
L
ANGULOS DE GIRO
ϕA = −
L3 ⋅ ( 3 ⋅ p1 + p2 ) 24 ⋅ E ⋅ I
ECUACION DE LA ELASTICA
yx
QB
( L − x )3 2 − ⋅ ( p2 − p1 ) + ( L − x ) ⋅ p2 − ⋅ 5⋅ L 24 ⋅ E ⋅ I 2 2 2 2 − ⋅ L ⋅ L − x ⋅ p + p + ⋅ L ⋅ p + ⋅ p ( ) ( ) ( ) 2 1 2 1
(L − x) =
2
FLECHA
L ⋅ ( 4 ⋅ p2 + 11⋅ p1)
MB
4
fA =
120 ⋅ E ⋅ I
3.27
3.28
3.5.6
MOMENTO FLECTOR
REACCIONES
M
A
RB = 0
B
ESFUERZO CORTANTE
x
Qx = 0
a
MOMENTOS FLECTORES MAC = 0
;
MCB = − M = cte
b L
;
MAC = − M
ANGULOS DE GIRO
ϕC = ϕ A = −
M⋅ b E⋅I
y AC =
M ⋅ b ⋅ ( 2 ⋅ L − 2 ⋅ x − b) 2⋅E⋅I
; yBC =
M 2 ( L − x) 2⋅E⋅I
FLECHA
MB fC =
M⋅ b 2⋅E⋅I 2
; fA =
M ⋅ b ⋅ ( 2 ⋅ L − b) 2⋅E⋅I
Prontuario para Cálculo de Estructuras
ECUACION DE LA ELASTICA
Formulario para vigas y pórticos
3.5.7
MOMENTO FLECTOR EN EXTREMO DEL VUELO
REACCIONES
M
RB = 0
B
A
ESFUERZO CORTANTE
x
Qx = 0
L
MOMENTOS FLECTORES MAB = − M = cte ANGULOS DE GIRO
ϕA = −
M⋅ L E⋅I
ECUACION DE LA ELASTICA
y AC =
(
M ⋅ b ⋅ x2 − 2 ⋅ L ⋅ x + L2 2⋅E⋅I
) MB
FLECHA
fA =
3.29
M ⋅ L2 2⋅E⋅I
3.30
3.6 VIGAS CONTINUAS DE DOS VANOS IGUALES P
P
P
A
B
L/2
C
L/2 L
A
L/2
L/2 L
L/2 L
0,688 P
L
0,405 P
0,312 P A
C
B
L/2
B
C
0,094 P
A
0,312 P
0,094 P
B
C
0,594 P
0,688 P
ESFUERZOS CORTANTES
- 0,188 PL - 0,094 PL
A
B
0,156 PL
C
0,156 PL
MOMENTOS FLECTORES
A
B
0,203 PL MOMENTOS FLECTORES
C
Prontuario para Cรกlculo de Estructuras
ESFUERZOS CORTANTES
A
L
Q
B
Q
C
L
0,625 QL
Formulario para vigas y p贸rticos
Q
A
B
L
L
C
0,375 L 0,437 QL
0,375 QL
0,063 QL
A
B
C
A
B
0,375 QL 0,375 L
0,563 QL
0,437 L
0,625 QL ESFUERZOS CORTANTES
ESFUERZOS CORTANTES
2
2
- 0,063 QL
- 0,125 QL
A
B 2
C 2
0,07 QL
0,07 QL
A
B
C
2
0,096 QL
MOMENTOS FLECTORES
3.31
MOMENTOS FLECTORES
C
3.32
3.7 VIGAS CONTINUAS DE DOS VANOS DESIGUALES Q
A
L
Q
B
C
k L
c QL d L
a QL
A
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
C
B
b QL
d QL
a L
2
f QL
A
B
C
2
e QL
2
g QL MOMENTOS FLECTORES
k 2 − k +1 8 k f d= − 2 k f=
a = 0.5 − f e=
a2 2
b = 0.5 + f g=
d2 2
c=
k f + 2 k
Prontuario para Cálculo de Estructuras
ESFUERZOS CORTANTES
A
Q
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
Formulario para vigas y pórticos
Q
A C
B
a QL d QL
b QL ESFUERZOS CORTANTES 2
f QL
k 2 − k +1 8 k f d= − 2 k f=
A
B
2
g QL
e=
a2 2
b = 0.5 + f g=
c=
k f + 2
d2 2
3.33
MOMENTOS FLECTORES
C
a = 0.5 − f
3.34
3.8 VIGAS CONTINUAS DE TRES VANOS CON SIMETRIA DE LUCES Q
Q
A
Q
B
C
L
D
k 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
C
A
c QL
b QL
a QL
ESFUERZOS CORTANTES
2
2
f QL
f QL 2
g QL A
B
C
D 2
2
e QL
e QL
MOMENTOS FLECTORES
f=
k3 + 1 12 ⋅ k + 8
a = 0.5 − f
c=
k 2
e=
a2 2
b = 0.5 + f
g=
k2 −f 8
Prontuario para Cálculo de Estructuras
a L
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 ESFUERZOS CORTANTES
2
f QL
B 2
e QL
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
2
f QL A
Relación entre luces
Formulario para vigas y pórticos
Q
C 2
g QL
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
3.35
MOMENTOS FLECTORES
f=
3.36
3.9 PORTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL HORIZONTAL a
3.9.1 k=
I2 h ⋅ I1 L
y
p B
N = 3+ 2⋅k
C x
VD
n
I1
p⋅ s⋅n = L p⋅ s⋅m = L
HA = HD =
I2
m
REACCIONES
VA
s
CARGA REPARTIDA VERTICAL
h
I1
A
D
3⋅p⋅ s s2 ⋅m⋅ n − 2 ⋅ h⋅ L ⋅ N 12
L
MC
3 p⋅ s s2 MB = MC = − ⋅ ⋅ m⋅ n − 2 L⋅N 12 En S Mx = VA ⋅ x −
p ⋅ (x − m)2 − HA ⋅ h 2 HA
HD VA
VD
Prontuario para Cálculo de Estructuras
MB
MOMENTOS FLECTORES
k=
CARGA REPARTIDA VERTICAL UNIFORME
I2 h ⋅ I1 L
y
N = 3+ 2⋅k
p B
REACCIONES
C I x
P⋅L VA = VD = 2 P ⋅ L2 HA = HD = 4 ⋅ h⋅ N
I
2
I
1
A
h
1
Formulario para vigas y pórticos
3.9.2
D
MOMENTOS FLECTORES
L
P ⋅ L2 4⋅N P ⋅ x ⋅ ( L − x ) P ⋅ L2 Mx = − 2 4⋅N
MB = MC = −
Mmax,pos =
P ⋅ L2 P ⋅ L2 − 8 4⋅N
M
para
x=
MC
B
L 2 HA
HD VA
VD
3.37
k=
3.38
3.9.3
CARGA REPARTIDA HORIZONTAL
I2 h ⋅ I1 L
y
N = 3+ 2⋅k
p
B
C I2
REACCIONES
VA = VD = HD = HA =
p ⋅ h2 2⋅L
I1 y
p ⋅ h ⋅ (2 ⋅ N + k) 8⋅N
A
p ⋅ h ⋅ (6 ⋅ N − k )
MB
MC
MB
HA
HD VA
VD
Prontuario para Cálculo de Estructuras
p ⋅ h2 ⋅ (2 ⋅ N − k) 8⋅N p ⋅ h2 MC = − (2 ⋅ N + k) 8⋅N En AB MB =
p ⋅ y ⋅ (h − y) y + ⋅ MB 2 h
D L
8⋅N
MOMENTOS FLECTORES
MY =
h
I1
CARGA PUNTUAL VERTICAL SOBRE DINTEL
P m
k=
I2 h ⋅ I1 L
y
n
B
N = 3+ 2⋅k
C I2
REACCIONES I1
P⋅n L P⋅m VD = L VA =
h
I1
A
Formulario para vigas y pórticos
3.9.4
D
3 P ⋅ m⋅ n HA = HD = 2 L ⋅ h⋅ N
L
MOMENTOS FLECTORES
MB
MC
3 P ⋅ m⋅ n MB = MC = − ⋅ 2 L⋅N MP
2⋅N−3 MP = P ⋅ m ⋅ n ⋅ 2⋅L⋅N HA
HD VA
VD
3.39
3.40
3.10 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL INCLINADO
3.10.1 CARGA REPARTIDA VERTICAL k1 =
I3 h1 ⋅ I1 s
y
k2 =
I3 h2 ⋅ I2 s
p C s
f I B
REACCIONES
3
I
x
p⋅L VA = VD = 2 h1 + h2 p ⋅ L2 HA = HD = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
I
h1
h2
2
1
A
D
Ll
MOMENTOS FLECTORES
MC = −
( h1 + h2 ) ⋅ h2 p ⋅ L2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
En
BC
MX =
p ⋅ x ⋅ (L − x) f − HA ⋅ ⋅ x + h1 2 L
MB
HA
HD VA
VD
Prontuario para Cálculo de Estructuras
MC
( h1 + h2 ) ⋅ h1 p ⋅ L2 MB = − 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
k1 =
I3 h1 ⋅ I1 s
k2 =
y
I3 h2 ⋅ I2 s
C s
f
p REACCIONES
I3 B
p⋅h 2⋅L HA = p ⋅ h1 − HD
2 1
VA = VD =
I1
h1
y A
h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2 p ⋅ h12 HD = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB =
p⋅h 2
−
p⋅h 8
3 1
D L
MOMENTOS FLECTORES 2 1
h2
I2
Formulario para vigas y pórticos
3.10.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
MC
h1 ⋅ ( 4 + 5 ⋅ k 1 ) + 2 ⋅ h2
h ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 2 1
MB
h1 ⋅ ( 4 + 5 ⋅ k 1) + 2 ⋅ h2 p ⋅ h12 ⋅ h2 MC = 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 En AB MY = HA ⋅ y −
p ⋅ y2 2
HA
HD VA
VD
3.41
3.42
3.10.3 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL k1 =
I3 h1 ⋅ I1 s
y
k2 =
I3 h2 ⋅ I2 s
y
I3
p ⋅ f ⋅ ( h1 + h2 )
HA = p ⋅ f − HD
C s
f
REACCIONES
VA = VD =
p
B
2⋅L
h2
I2
I1
h1
A
2 p ⋅ f 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 ) HD = ⋅ 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
D L
MOMENTOS FLECTORES
MC = −
p ⋅ f ⋅ h1 ⋅ 8
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB
2 p ⋅ h2 8 ⋅ h1 ⋅ (1+ k 1 ) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 ) ⋅ 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
En BC L p ⋅ y2 MY = −VA ⋅ ⋅ y + HA ⋅ ( y + h1 ) − f 2
HA
HD VA
VD
Prontuario para Cálculo de Estructuras
MB = p ⋅ f ⋅ h1 −
MC
8 ⋅ h12 ⋅ (1+ k 1) + 4 ⋅ h1 ⋅ h2 + f ⋅ ( h1 + h2 )
k1 =
I3 h1 ⋅ I1 s
k2 =
y
I3 h2 ⋅ I2 s
P f
REACCIONES
I3
P⋅b L P⋅a VD = L
B
VA =
HA = HD =
C s
b
I1
h1
A
h1 ⋅ (L + b) + h2 ⋅ (L + a) P⋅a⋅b ⋅ 2 ⋅ L2 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
h2
I2
a
Formulario para vigas y pórticos
3.10.4 CARGA PUNTUAL VERTICAL SOBRE DINTEL
D L
MOMENTOS FLECTORES
MB = −
MC = −
MP =
P ⋅ a ⋅ b ⋅ h1 2⋅L
2
P ⋅ a ⋅ b ⋅ h2 2 ⋅ L2
⋅
h1 ⋅ ( L + b ) + h2 ⋅ ( L + a )
MC
h ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2 2 1
⋅
2 2
MB
h1 ⋅ ( L + b ) + h2 ⋅ ( L + a)
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
P⋅ a⋅ b a⋅f + HA ⋅ + h1 L L
MP
HA
HD VA
VD
3.43
3.44
3.11 PÓRTICOS SIMPLES BIARTICULADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS 3.11.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p
Valor de la carga en proyección horizontal
k=
C
I2 h ⋅ I1 s
s
I2
REACCIONES
f
I2
B
D x
p⋅L 2 p ⋅ L2 8⋅h+ 5⋅f HA = HE = ⋅ 2 32 h ⋅ ( 3 + k ) + f ⋅ ( 3h + f )
VA = VE =
I1
I1
A
h E
L
MOMENTOS FLECTORES
MC =
p ⋅ L2 ⋅ h 8⋅h+ 5⋅f ⋅ 2 32 h ⋅ (3 + k) + f ⋅ (3 ⋅ h + f )
MC
p ⋅ L2 f + h + ⋅ MB 8 h
MB
MD
En BC y DC MX = p ⋅
x ⋅ (L − x) 2
+
MB h
2⋅f ⋅ x ⋅h+ L
HA
HE VA
VE
Prontuario para Cálculo de Estructuras
MB = MD = −
Valor de la carga en proyección horizontal
k=
p
I2 h ⋅ I1 s
C s
REACCIONES
I
p⋅L 8 p⋅L VE = 8 p ⋅ L2 8⋅h+ 5⋅f HA = HE = 64 h2 ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )
I
2
f 2
B
VA = 3 ⋅
D x I
I
1
A
Formulario para vigas y pórticos
3.11.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
h
1
E L
MOMENTOS FLECTORES
MB = MD = −
MC
p ⋅ L2 ⋅ h 8⋅h+ 5⋅f 2 64 h ⋅ ( 3 + k ) + f ⋅ ( 3 ⋅ h + f )
p ⋅ L2 f + h + ⋅ MB 16 h En BC MC =
MX = p ⋅
x ⋅ (L − x) 2
+
MB h
MB
MD
2⋅f⋅ x ⋅h+ L HA
HE VE
3.45
VA
3.46
3.11.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
k=
I2 h ⋅ I1 s
C s
I
p
REACCIONES
I
2
f 2
D
B
p⋅h 2⋅L HA = p ⋅ h − HE
VA = VE =
HE =
2
I
I
1
h
1
y
( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f p ⋅ h2 ⋅ 2 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
A
E L
MOMENTOS FLECTORES
MC
MB
MD
En AB My = −
p ⋅ y2 + HA ⋅ y 2
HA
HE VA
VE
Prontuario para Cálculo de Estructuras
p ⋅ h2 + MD 2 p ⋅ h2 f + h MC = + ⋅ MD h 4 ( 5 ⋅ k + 12 ) ⋅ h + 6 ⋅ f p ⋅ h3 ⋅ 2 MD = − 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h) MB =
Valor de la carga en proyección vertical
k=
I2 h ⋅ I1 s
p
C s
REACCIONES
I2
D
B
p⋅f VA = VE = ⋅ ( f + 2 ⋅ h) 2⋅L HA = p ⋅ f − HE HE =
f
I2 y
I1
2 p ⋅ f 8 ⋅ h ⋅ ( k + 3 ) + 5 ⋅ f ⋅ ( f + 4 ⋅ h) ⋅ 16 h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
I1
Formulario para vigas y pórticos
3.11.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
h
x A
E L
MOMENTOS FLECTORES
MC
MB = HA ⋅ h MC = −
2 p ⋅ f2 4 ⋅ h ⋅ ( k + 2) + f ⋅ (5 ⋅ h + f ) ⋅ 2 16 h ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h)
MB
MD = −HE ⋅ h
MD
En BC Mx = HA ⋅ y − VA ⋅ x − p ⋅
2
2
HA
HE VA
VE
3.47
f siendo y = ⋅ x + h L
( y − h)
3.48
P
3.11.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
C s
I h k= 2⋅ I1 s
I2 B
D
REACCIONES
m
n
I1
P⋅n VA = L P⋅m VA = L HA = HE =
f
I2
I1
A
(
P ⋅ m 6 ⋅ h ⋅ L ⋅ n+ f ⋅ 3 ⋅ L − 4 ⋅ m ⋅ 4 ⋅ L2 h2 ⋅ ( k + 3 ) + f ⋅ ( f + 3 ⋅ h) 2
2
h E
L
)
MOMENTOS FLECTORES MB
MB = MD = −HA ⋅ h P⋅m h+ f + ⋅ MB h 2 h⋅ L + 2 ⋅ f ⋅ m MP = VA ⋅ m − HA ⋅ L
MD
MC =
HA
HE VA
VE
Prontuario para Cálculo de Estructuras
MC
3.12.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL k1 =
I3 h1 ⋅ I1 L
y
k2 =
p
I3 h2 ⋅ I2 L
B
C I
I
VA =
h −h p⋅L p⋅L + ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h ⋅ (1+ k2 ) + h1 ⋅ h2
VD =
h12 − h22 p⋅L p⋅L − ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
2 1 2 2
HA = HD =
3
x
REACCIONES 2 2
h1
I
1
h2
2
Formulario para vigas y pórticos
3.12 PÓRTICOS SIMPLES BIARTICULADOS A DISTINTA ALTURA. DINTEL HORIZONTAL
D
A
h1 − h2 p ⋅ L2 ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
L
MOMENTOS FLECTORES MB
p ⋅ L2 ( h1 + h2 ) ⋅ h1 MB = − 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 MC = −
MC
p ⋅ L2 ( h1 + h2 ) ⋅ h2 8 h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
HD
En BC
VD HA VA
3.49
p ⋅ x2 Mx = VA ⋅ x − − HA ⋅ h1 2
3.50
3.12.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR k1 =
I3 h1 ⋅ I1 L
y
k2 =
I3 h2 ⋅ I2 L
p
B
C I
REACCIONES
3
I
p⋅h h − h2 − HD ⋅ 1 2⋅L L HA = p ⋅ h − HD
VA = VD =
HD =
2 1
h1
I
1
h2
2
D y
p ⋅ h12 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
A L
MOMENTOS FLECTORES
p ⋅ h12 p ⋅ h13 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 − ⋅ 2 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MB
MC
MB
p ⋅ h12 ⋅ h2 5 ⋅ k1 ⋅ h1 + 4 ⋅ h1 + 2 ⋅ h2 MC = − ⋅ 2 8 h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 En AB
HD
p ⋅ y2 My = HA ⋅ y − 2
VD HA VA
Prontuario para Cálculo de Estructuras
MB = −
k1 =
I3 h1 ⋅ I1 L
k2 =
y
P
I3 h2 ⋅ I2 L
B
C I3
REACCIONES
( L + b) ⋅ h1 + ( L + a) ⋅ h2 P⋅b P⋅a⋅b VA = + ⋅ 2 ⋅ h1 − h2 3 L 2⋅L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
(
VD =
( L + b) ⋅ h1 + ( L + a) ⋅ h2 P⋅a P⋅a⋅b − ⋅ h1 − h2 3 2 L 2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
(
a
)
I1
D
A L
( L + b ) ⋅ h1 + ( L + a) ⋅ h2 P⋅a⋅b HA = HD = 2 2 2 ⋅ L h1 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2 MB
MOMENTOS FLECTORES
MB = −
P ⋅ a ⋅ b ⋅ h1 2⋅L
2
⋅
2 1
h
h2
I2
h1
)
b
Formulario para vigas y pórticos
3.12.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
MC
( L + b ) ⋅ h1 + ( L + a) ⋅ h2 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
MP HD
MC = −
P ⋅ a ⋅ b ⋅ h2 2 ⋅ L2
VD HA VA
3.51
MP = VA ⋅ a + MB
( L + b ) ⋅ h1 + ( L + a) ⋅ h2
h12 ⋅ (1+ k1 ) + h22 ⋅ (1+ k2 ) + h1 ⋅ h2
3.52
3.13 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL HORIZONTAL
3.13.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL
p B
C
I h k= 2⋅ I1 L
I2 x
REACCIONES
VA = VD =
I1
p⋅L 2
HA = HD =
p ⋅ L2 4 ⋅ h ⋅ (k + 2)
h
I1
A
D
MOMENTOS FLECTORES L
MB = MC = −
p ⋅ L2 6 ⋅ ( k + 2)
MB
MC
En BC Mx =
p ⋅ x ⋅ (L − x) 2
Mmáx positivo =
−
p ⋅ L2 6 ⋅ ( k + 2)
p ⋅ L2 3 ⋅ k + 2 L ⋅ para x = 24 k+2 2
HA
HD MA VA
MD VD
Prontuario para Cálculo de Estructuras
p ⋅ L2 MA = MD = 12 ⋅ ( k + 2 )
k=
I2 h ⋅ I1 L
p
B
C I2
REACCIONES
VA = VD =
p ⋅ h2 ⋅ k L ⋅ ( 6 ⋅ k + 1)
I1
HA = p ⋅ h − HD HD =
p ⋅ h ⋅ (2 ⋅ k + 3)
y
8 ⋅ ( k + 2)
A
D
MOMENTOS FLECTORES
MA = − MB =
p ⋅ h2 24
p ⋅ h2 24
MC = −
h
I1
Formulario para vigas y pórticos
3.13.2 CARGA REPARTIDA HORIZONTAL SOBRE PILAR
L
2 1 ⋅5 + + 6 ⋅ k + 1 k + 2
2 2 ⋅ 1− + ⋅ + + 6 k 1 k 2
MC
MB MB
p⋅h 2 2 ⋅3 − − 24 6 ⋅ k + 1 k + 2 2
p ⋅ h2 2 1 ⋅3 + − 24 6 ⋅ k + 1 k + 2 En AB MD =
HA
HD MD
MA VA
VD
3.53
p ⋅ y2 My = − + HA ⋅ y + MA 2
3.54
3.13.3 CARGA PUNTUAL VERTICAL SOBRE DINTEL
P m
I h k= 2⋅ I1 L
n
B
C I2
REACCIONES
VA =
m ⋅ ( n − m) P⋅n ⋅ 1+ 2 L L ⋅ ( 6 ⋅ k + 1)
I1
h
I1
VD = P − VA HA = HD =
3 ⋅ P ⋅ m⋅ n 2 ⋅ L ⋅ h ⋅ (k + 2)
A
D L
MOMENTOS FLECTORES
P ⋅ m⋅ n 1 n− m ⋅ − 2 ⋅ L k + 2 L ⋅ ( 6 ⋅ k + 1)
MB = −
P ⋅ m⋅ n 1 n− m ⋅ + L k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1)
MC = −
P ⋅ m⋅ n 1 n− m ⋅ − k + 2 2 ⋅ L ⋅ ( 6 ⋅ k + 1) L
MD =
P ⋅ m⋅ n 1 n− m ⋅ + 2 ⋅ L k + 2 L ⋅ ( 6 ⋅ k + 1)
P ⋅ m ⋅ n n ⋅ MB m ⋅ MC MP = + + L L L
MB
MC
MP
HA
HD MA
VA
MD VD
Prontuario para Cálculo de Estructuras
MA =
k=
I2 h ⋅ I1 L
P
B
C I2
REACCIONES
VA = VD =
3 ⋅ P ⋅ h⋅ k L ⋅ (6 ⋅ k + 1)
HA = HD =
P 2
I1
A
D
MOMENTOS FLECTORES
P ⋅ h 3 ⋅ k +1 ⋅ 2 6 ⋅ k +1 P⋅h 3⋅k MB = − MC = ⋅ 2 6⋅ k +1 P ⋅ h 3 ⋅ k +1 MD = ⋅ 2 6 ⋅ k +1 MA = −
h
I1
Formulario para vigas y pórticos
3.13.4 CARGA PUNTUAL HORIZONTAL EN CABEZA DE PILAR
L
MB
MC
HA MA
MD VD
3.55
VA
HD
3.56
3.14 PÓRTICOS SIMPLES BIEMPOTRADOS A LA MISMA ALTURA. DINTEL A DOS AGUAS p
3.14.1 CARGA REPARTIDA VERTICAL SOBRE DINTEL Valor de la carga en proyección horizontal
k=
C
I2 h ⋅ I1 s
s
I
I
2
f 2
B
REACCIONES
D x
p⋅L 2 k ⋅ (4 ⋅ h + 5 ⋅ f ) + f p ⋅ L2 HA = HE = ⋅ 8 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
I
VA = VE =
(
I
1
h
1
A
E
)
L
MOMENTOS FLECTORES
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) ⋅ 48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
MB = MD = −
(
MC
)
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f p⋅L ⋅ 48 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2 2
2
(
p ⋅ L2 + MA − HA ⋅ ( h + f ) 8 En BC
MB
)
MD
MC =
2 ⋅ x ⋅ f p ⋅ x2 − Mx = MA + VA ⋅ x − HA ⋅ h + 2 L
HA
HE MA VA
ME VE
Prontuario para Cálculo de Estructuras
MA = ME =
Valor de la carga en proyección horizontal
k=
I2 h ⋅ I1 s
p C
REACCIONES
p⋅L − VE 2 4 ⋅ k +1 VE = 3 ⋅ p ⋅ L ⋅ 32 ⋅ ( 3 ⋅ k + 1) k ⋅ (4 ⋅ h + 5 ⋅ f ) + f p ⋅ L2 HA = HE = ⋅ 16 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ h2 + h ⋅ f + f 2
s
VA =
(
I
f 2
D
B x
)
I
I
1
A
MOMENTOS FLECTORES
MA =
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) p ⋅ L2 ⋅ − 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)
ME =
p ⋅ L2 k ⋅ h ⋅ ( 8 ⋅ h + 15 ⋅ f ) + f ⋅ ( 6 ⋅ h − f ) p ⋅ L2 ⋅ + 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)
(
(
L
)
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2 p ⋅ L2 p ⋅ L2 ⋅ − 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)
MD = −
k ⋅ h ⋅ (16 ⋅ h + 15 ⋅ f ) + f 2 p ⋅ L2 p ⋅ L2 ⋅ + 2 2 2 96 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f + f ⋅ h + h 64 ⋅ ( 3 ⋅ k + 1)
(
(
h
1
E
)
MB = −
En BC
I
2
Formulario para vigas y pórticos
3.14.2 CARGA REPARTIDA VERTICAL SOBRE MEDIO DINTEL
MC
)
MB
MD
)
HA
HE MA VA
ME VE
3.57
2 ⋅ x ⋅ f p ⋅ x2 Mx = MA + VA ⋅ x − HA ⋅ h + − L 2 L MC = VE ⋅ + ME − HE ( f + h) 2
3.58
3.14.3 CARGA REPARTIDA HORIZONTAL SOBRE PILAR k=
I2 h ⋅ I1 s
C s
REACCIONES
VA = VE =
p⋅h ⋅k 2 ⋅ L ⋅ ( 3 ⋅ k + 1) 2
I
p
I
2
f 2
D
B
HA = p ⋅ h − HE
k2 ⋅ h + k ⋅ ( 2 ⋅ f + 3 ⋅ h) + f p ⋅ h2 HE = ⋅ 2 4 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
(
I
I
1
)
A
E
MOMENTOS FLECTORES
L
p ⋅ h2 k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f 2 ⋅ k + 1 ⋅ +6⋅ 2 24 3 ⋅ k + 1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2
MA = −
h
1
y
2
(
)
MB = MA + HA ⋅ h −
MB
2 2 p ⋅ h2 k ⋅ h ⋅ ( k + 6 ) + k ⋅ f ⋅ (15 ⋅ h + 16 ⋅ f ) + 6 ⋅ f 2 ⋅ k + 1 ⋅ − + ⋅ 6 2 24 3 ⋅ k + 1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 En AB p ⋅ y2 My = MA + HA ⋅ y − 2
ME =
(
MD
)
HA MA
HE ME
VA
VE
Prontuario para Cálculo de Estructuras
p ⋅ h2 2 L MC = ME − HE ⋅ ( f + h) + VE ⋅ 2 MD = ME − HE ⋅ h
MC
Valor de la carga en proyección vertical
k=
I2 h ⋅ I1 s
p
C
REACCIONES s
3 p ⋅ f 4 ⋅ k ⋅ ( f + h) + f VA = VE = ⋅ ⋅ 8 L 3 ⋅ k +1 HA = p ⋅ f − HE
I
(
D y
I
)
I
1
A L
k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f ) 3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f p⋅f ⋅ f⋅ + ⋅ 24 ( k ⋅ h + f )2 + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2 3 ⋅ k +1
MC = ME − HE ( h + f ) + VE ⋅ MD = ME − HE ⋅ h
(
)
L 2
MC
MB
(
MD
)
HA MA
HE ME
VA
VE
3.59
k ⋅ h ⋅ ( 9 ⋅ f + 4 ⋅ h) + f ⋅ ( 6 ⋅ h + f ) 3 4 ⋅ h ⋅ (3 ⋅ k + 2) + f p⋅f ⋅ −f ⋅ + ⋅ 2 24 3 ⋅ k +1 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 2 En BC 2 L ⋅ ( y − h) p ⋅ ( y − h) My = MA + HA ⋅ y − VA ⋅ − 2⋅f 2 ME =
h
1
E
MOMENTOS FLECTORES
MB = MA + HA ⋅ h
f 2
B
2 p ⋅ f 2 ⋅ k ⋅ h ⋅ ( k + 4 ) + f ⋅ (10 ⋅ k ⋅ h + 5 ⋅ k ⋅ f + f ) HE = ⋅ 2 4 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
MA = −
I
2
Formulario para vigas y pórticos
3.14.4 CARGA REPARTIDA HORIZONTAL SOBRE DINTEL
3.60
3.14.5 CARGA PUNTUAL VERTICAL SOBRE DINTEL
P
I h k= 2⋅ I1 s
C s
REACCIONES
I
f 2
B
VA = P − VE 2 P ⋅ m 3 ⋅ L ⋅ ( k ⋅ L + m) − 2 ⋅ m VE = 3 ⋅ 3 ⋅ k +1 L HA = HE =
I
2
D m I
2 2 P ⋅ m 3 ⋅ k ⋅ L ⋅ ( f + h) − 4 ⋅ f ⋅ m ⋅ ( k + 1) + 3 ⋅ L ⋅ m ⋅ ( f − k ⋅ h) ⋅ 2 L2 ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2
(
n I
1
A
)
h
1
E L
MOMENTOS FLECTORES 3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 2 P⋅m ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 ⋅ MA = 2 ⋅ L2 n ⋅ n − m ( ) − 3 ⋅ k +1 MB = MA − HA ⋅ h
L MC = ME + VE ⋅ − HE ⋅ ( h + f ) 2
MB
MD
MD = ME − HE ⋅ h
3 ⋅ f ⋅ L ⋅ h ⋅ ( k ⋅ L + 2 ⋅ m) − 4 ⋅ f ⋅ m2 ( k ⋅ h + 2 ⋅ h + f ) + 2 ⋅ k ⋅ h2 ⋅ L ⋅ n+ f 2 ⋅ L ⋅ ( 4 ⋅ m − L ) 2 P⋅m ( k ⋅ h + f ) + 4 ⋅ k ⋅ f 2 + f ⋅ h + h2 ME = ⋅ 2 ⋅ L2 n ⋅ n − m ( ) + 3 ⋅ k +1 2⋅f ⋅m En BC My = MA + VA ⋅ m − HA ⋅ h + L
(
MC
)
)
HA
HE MA VA
ME VE
Prontuario para Cálculo de Estructuras
(