PLAC PLACAS AS DELG DELGAD ADAS AS MEDI MEDIAN ANTE TE MÉTOD MÉTODOS OS CLÁSIC CLÁSICOS OS
A NÁLISIS DE E STRUCTURAS II 4 O DE I.C.C.P.
Por R. Gallego Sevilla, G. Rus Carlborg y A. E. Martínez Castro
Departamento de Mecánica de Estructuras e Ingeniería Hidráulica Hidráulica , Universidad de Granada Edificio Politécnico Fuentenueva, C/ Severo Ochoa s/n, CP 18071 Granada
Octubre de 2007
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Placas delgadas (Teoría (Teoría de Kirchhoff). Resumen
Ecuación de gobierno: p( x, y) D
w, xxxx + 2 · w, xxyy + w, yy yy = Donde:
E h3 ; D= 12 ( 1 − ν 2 )
h3 E I ;D= I = 12 1 − ν 2
(0.1)
(0.2)
P(x, y)
x
y Q y
Qx
M y M yx
Mxy
Mx
A partir del campo de desplazamientos verticales, w( x, y), se obtienen: Giros:
∂w
θ x =
∂x
= w, x ;
θ y =
∂w ∂ y
= w, y
(0.3)
Momentos unitarios: Mx M y Mxy
= − D w, xx + ν w, yy = − D w, yy + ν w, xx = −2 G I w , xy = −D ( 1 − ν ) w, xy
(0.4)
E . 2 (1 + ν ) Cortantes unitarios: siendo G =
Qx Q y Cortante generalizado en bordes:
= − D w, xxx + w, xy y = − D w, yy y + w, yx x
(0.5)
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Índice general
Placas delgadas (Teoría de Kirchhoff). Resumen Capítulo 1. Placas delgadas rectangulares 1.1. 1.1. Plac Placas as delg delgad adas as recta ectang ngul ular ares es.. Méto Método do de Na Navi vier er . . . . . . . . . . . . . . . . . . . . . 1.1.1. Carga uniforme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. .1.2. Carg arga puntual. Función de Green. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3. Carga distribuida distribuida en una una linea linea y = η0 . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4. Carga distribuida distribuida en una una linea linea y = f ( x). . . . . . . . . . . . . . . . . . . . . . . 1.1.5. Momento Momento puntual puntual M y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6. Momento Momento distribuid distribuido o M y ( x) en una línea y = η0 . . . . . . . . . . . . . . . . . . 1.1.7. Superficie de carga lineal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8. Superficie de carga en un parche . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. 1.2. Plac Placas as delg delgad adas as recta ectang ngul ular ares es.. Méto Método do de Levy Levy . . . . . . . . . . . . . . . . . . . . . . . 1.2. 1.2.1. 1. Func Funció ión n de carg arga con coefi coefici cien enttes cons consta tant ntes es . . . . . . . . . . . . . . . . . . . . 1.2.2. 1.2.2. Placa Placa rectan rectangul gular ar someti sometida da a carga carga unifor uniforme. me. Placa Placa tetraa tetraapo poyada yada . . . . . . . . 1.2.3. Placa rectang rectangular ular tetraap tetraapoyada oyada sometida sometida a dos distribu distribucion ciones es de momento momento M y simétrico) . . . . . . . . . . . . . . . . . . . . . . . en dos bordes paralelos (caso simétrico) 1.2.4. Placa rectang rectangular ular tetraapoy tetraapoyada ada sometida sometida a dos distribucion distribuciones es de mom momento ento en dos bordes paralelos (caso antimétrico) antimétrico) . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5. 1.2.5. Placa Placa rectan rectangul gular ar tetraap tetraapoya oyada da someti sometida da a una ley de carga carga lineal lineal . . . . . . .
I
1 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16
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C APÍTULO 1 Placas delgadas rectangulares
1.1. Placas Placas delgadas delgadas rectangul rectangulares. ares. Método Método de Navier Navier El método de Navier es aplicable en las siguientes condiciones: 1. Placa rectangula rectangularr, de de dimensi dimensiones ones a × b. 2. Condición de apoyos simples en los cuatro cuatro bordes (placa tetraapoyada en bordes rectos). w = 0; w, nn = 0 Considérese la referencia R( O; x, y, z), situada en una esquina de la placa, con x ∈ [0, a] e y ∈ [0, b]. La ecuación de gobierno de flexión de placas delgadas es la siguiente: ∆
2
w( x , y ) =
p( x , y ) D
(1.1)
siendo: 2
= w, xxxx + 2 w, xxyy + w, yy yy w( x, y) ⇒ Campo de desplazamiento vertical, positivo en sentido z positivo. p( x, y) ⇒ Carga superficial, positiva en sentido z positivo. ∆
D ⇒ Rigidez de la placa de espesor h, y constantes elásticas E, ν , con D =
E h3 . 12 (1 − ν 2 )
La solución general es: ∞
w( x, y) =
∞
∑ ∑ wnm sen n 1m 1 =
donde n, m ∈
=
n π x m π y sen a b
(1.2)
,y wnm
1
pnm ; = 4 · π D Fnm
Fnm =
2 2
n a
2
+
m b
(1.3)
Los coeficient coeficientes es p nm corres correspo ponde nden n con con el desarr desarroll ollo o en serie serie de Fouri Fourier er doble doble con extens extensión ión impar impar para la carga: pnm
4 = ab
a
0
b
0
∞
p( x y) =
p( x, y) sen ∞
∑ ∑ p
n π x m π y sen d xd y a b
sen
n π x
sen
m π y
(1.4)
(1.5)
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1.1.1. 1.1.1. Carga Carga unifor uniforme me
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es constante, de valor p 0 .
z
p( x, y) = p0
y x a
b
Desplazamiento: ∞
∑
w( x , y ) =
n 1,3,5... m =
16 p0 sn ( x) sm ( y) n m π 6 D Fnm 1,3,5,... ∞
∑
(1.6)
=
con: Fnm =
2 2
n a
2
+
m b
n π x a m π y sm ( y) = sen b
(1.7)
sn ( x) = sen
(1.8)
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1.1.2. Carga Carga puntua puntual. l. Función Función de Green. Green.
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es una fuerza puntual, de valor p 0.
z p( x, y) = p0 δ( x − ξ ; y − η) η ξ
y
x a
b Desplazamiento:
w( x, y) = p 0 · K ( x, y; ξ , η) ∞
K ( x, y; ξ , η) =
4
∞
∑ ∑ a b π 4 D Fnm sn (ξ ) sm (η) sn ( x) sm ( y) n 1m 1 =
(1.9)
(1.10)
=
donde sn , sm vienen dadas en Eq. (1.8) y Fnm en Eq. (1.7). La función K ( x, y; ξ , η) es la función de Green (o solución fundamental) al problema de placas delgadas rectangulares con condiciones de contorno en apoyos simples.
La solución para una carga p( x, y) puede construirse a partir de la función de Green. w( x , y ) =
a
0
0
b
p(ξ , η) K ( x, y; ξ , η) dξ dη
(1.11)
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1.1.3. Carga Carga distr distribuida ibuida en una linea y = η 0 .
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es lineal, distribuida según la función q( x) en una línea de y constante, de valor η 0 .
z q( x ) η0
y x a
b Carga: p( x, y) = q( x) δ( y − η0 )
(1.12)
Desplazamiento: ∞
w( x , y ) =
∞
∑ ∑ n 1 m 1 =
=
4 · sm ( η0 ) sn ( x) sm ( y) γ n π 4 a b D Fnm
con: γ n =
0
a
sn (ξ ) q(ξ ) dξ
(1.13)
(1.14)
Si la función q( x) se expresa mediante su desarrollo en serie (en seno), se tiene: ∞
q( x ) = 2 qk = a
0
∑ qk sk (x); k 1 =
a
p( x) sk ( x)d x
(1.15)
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1.1.4. Carga Carga distr distribuida ibuida en una linea y = f ( x).
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es lineal, distribuida según la función q( x) en una línea definida en el plano x y según la función y = f ( x).
z q( x ) y = f ( x) y x a
b Carga:
p( x, y) = q( x) δ ( y − f ( x)) Coeficientes de la carga: pnm
4 = ab
0
a
(1.19)
q( x) sn ( x) sm ( f ( x)) d x
(1.20)
Desplazamiento: ∞
w ( x , y) =
∞
∑ ∑ n 1 m 1 =
Caso particular: y = c x. x. pnm =
4 ab
0
a
=
pnm sn ( x) sm ( y) π 4 D Fnm
(1.21)
m π c x dx b
(1.22)
q( x) sn ( x) sen
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1.1.5. Momento Momento puntual puntual M y
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . En el punto de coordenadas (ξ , η) actúa un momento M y .
z
η ξ
y
M y
x a
b Desplazamiento: 4 M y w( x , y ) = 3 2 π a b D
∞
∞
sn (ξ ) cm ( η) m sn ( x) sm ( y) Fnm 1
∑ ∑ n 1 m =
(1.25)
=
con: cm ( η) = cos
mπη b
(1.26)
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1.1.6. Momento Momento distribuido distribuido M y ( x) en una línea y = η0
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . En la línea ( My ( x) = ∑n 1 Mn sn ( x)) y = η0 se aplica un momento M y , distribuido ( My ∞
=
z
M y ( x) η0
y x a
b Desplazamiento: wnm llamando cm ( η0 ) = cos
mπ η0 b
2 m Mn mπ η 0 cos = 2 3 b b π D Fnm
se tiene: ∞
w( x , y ) =
(1.27)
∞
2 m Mn Cm ( η0 ) sn ( x) sm ( y) b2 π 3 D Fnm 1
∑ ∑ n 1 m =
=
(1.28)
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1.1.7. Superficie Superficie de carga carga lineal
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es de la forma p( x, y) = p 0 /b · y (triangular en y).
z
p0 y x a
b Carga: p( x, y) = p0
y b
(1.29)
Término wnm wnm =
−8 p 0 con n impar · (−1) m , con n m π 6 D Fnm
8p w( x , y ) = − 6 0 π D
( −1 )m ∑ ∑ n m Fnm sn (x) sm ( y) 1,3,5,... m 1 ∞
n
=
( 1 . 3 0)
∞
=
(1.31)
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1.1.8. Superficie Superficie de carga carga en un un parche parche
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es constante en un parche, con variable x ∈ [0, a] e y ∈ [b/2, b]. z
p0 y x
b/2 a
b
Desplazamiento:
w( x , y ) =
8 p0 π 6 D
sn ( x ) · n 1,3,5,... ∞
∑
n
=
sm ( y) − m Fnm m 1,3,5,... ∞
∞
∑
m
=
∑ 2,4,6,...
=
1 − (−1 ) m/ 2 m Fnm
sm ( y)
(1.32)
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1.2. Placas Placas delgadas delgadas rectangul rectangulares. ares. Método Método de Levy El método de Levy es aplicable en las siguientes condiciones: 1. Placa rectangula rectangularr, de de dimensi dimensiones ones a × b. 2. Condición Condición de apoyo apoyoss simples simples en dos bordes bordes paralelos. paralelos. w = 0, w , nn = 0. El método de Levy presenta ventajas sobre el método de Navier, en general: Se elimina elimina en parte parte el fenómeno fenómeno de Gibbs Gibbs para la la represent representación ación de cargas cargas con valor valores es no nulos nulos en los bordes perpendiculares a los simplemente apoyados. Las series convergen más rápido. Sólo hay 1 sumatorio. Considérese la siguiente figura: z
p( x, y)
Condiciones cualesquiera
? x
y
? a
b
La función de carga, p( x, y), se expresa en serie, serie, como sigue: ∞
p( x, y) =
∑ gn ( x) sen (λn y) n 1 =
con:
(1.33)
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La ecuación diferencial para w n ( x) es: 2 gn ( x) d 4 wn ( x ) 2 d wn ( x ) λ − 2 + λn4 wn ( x) = n 4 2 D dx dx
(1.38)
Esta ecuación se puede reescribir con una notación más compacta, wnIV ( x) − 2 λn2 wnI I ( x) + λn4 wn ( x) =
gn ( x) D
(1.39)
Esta ecuación es una Ecuación Diferencial Ordinaria (EDO), lineal, con coeficientes constantes. Su solución se obtiene sumando dos soluciones: la del problema homogéneo, w nh ( x), que es siempre p la misma, y depende de cuatro constantes ( A ( A n , Bn , Cn , Dn ) más una solución particular,w particular, wn ( x), que depende de la función g n ( x). p
wn ( x) = wnh ( x) + wn ( x) Solución del problema homogéneo: La E.D. a resolver es: (wnh ) IV ( x) − 2 λn2 (wnh ) I I ( x) + λn4 wnh ( x) = 0
(1.40)
( 1 .4 1 )
Su solución general es: wnh ( x) = ( An + Bn λn x) Sh (λn x) + (Cn + Dn λn x) Ch ( λn x) donde Ch = cosh y Sh = senh. Solución del problema particular p Se resuelve sustituyendo wn por wn en la ecuación 1.39.
(1.42)
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1.2.1. Función Función de carga carga con con coeficient coeficientes es consta constantes ntes
En este caso, g n ( x) = bn (constante). Así: ∞
∑ bn sen (λn y)
p( x, y) =
(1.43)
n 1 =
La solución particular es fácil de obtener. La ED para determinarla es la siguiente: p
p
p
(wn ) IV ( x) − 2 λn2 (wn ) I I ( x) + λn4 wn ( x) = p
bn D
(1.44)
p
Probando una solución de la forma wn ( x) = ωn , (una constante), se tiene: p
ωn =
bn D λn4
(1.45)
Y la solución general será:
∞
w( x , y ) =
∑ n 1 =
bn sen (λn y) ( An + Bn λn x) Sh (λn x) + (Cn + Dn λn x) Ch (λn x) + D λn4
(1.46)
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1.2.2. Placa rectangular rectangular sometida a carga uniforme. Placa tetraapoyada tetraapoyada
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . La carga es constante, de valor p 0 . z p0
x
y a
b
Se considera la referencia R( O; x, y, z) mostrada en la figura. Desplazamiento: w( x, y) =
con:
2 p0 b4 1 × ∑n 1,3,5,... D (n π ) 5 Ch (α n ) 2 Ch (α n ) + λn x Sh (λn x) − (2 + α n Th (α n ) ) Ch ( λn x) sen ( λn y) ∞
=
λn
=
n π b
(1.47)
(1.48)
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1.2.3. Placa rectangular rectangular tetraapoyada sometida a dos distribuciones de momento momento M y en dos bordes paralelos (caso simétrico)
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros parámetros del material material son E y ν . En dos bordes paralelos actúa una distribución de momentos simétrica, M y ( x).
z M y y
a x b
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1.2.4. Placa rectangular rectangular tetraapoyada tetraapoyada sometida a dos distribuciones distribuciones de momento en dos bordes paralelos (caso antimétrico)
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros parámetros del material material son E y ν . En dos bordes paralelos actúa una distribución de momentos antisimétrica, M y ( x).
z M y y
a x b
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1.2.5. Placa recta rectangular ngular tetr tetraapoy aapoyada ada sometida sometida a una ley de carga lineal lineal
Se considera una placa rectangular, de dimensiones a × b. La placa está simplemente apoyada en sus cuatro bordes. El espesor de la misma es h. Los parámetros del material son E y ν . Se aplica una carga distribuida, de valor máximo q.
q
y q b
x