Tablas de Kurt Beyer

Facultad de Ingeniería Universidad Nacional de La Plata

ESTRUCTURAS III Para alumnos de la carrera de Ingeniería Aeronáutica y Mecánica de la UNLP

TABLAS DE KURT – BEYER para el cálculo de solicitaciones en placas circulares y anulares simétricamente cargadas

Reeditadas por: Sr. Leonardo J. Marta Sr. Diego A. Massimino Srta. María de la Paz Alonso Sr. Julio. A. Guimmarra Revisado por: Ing. Juan P. Durruty

-2008-

Estructuras III

PLACAS CIRCULARES: Placa circular y placa anular sometidas a cargas con simetría central resultan según H. Marcus, las dos ecuaciones simultáneas de segundo orden: r ⎡ ⎤ d2M 1 = − − p p ⋅ r ⋅ dr ⎥ ⎢ 2 2 ∫ dr r 0 ⎣ ⎦

⎡M 1 r M ⎤ d 2ω = − ⎢ − 2 ⋅ ∫ r ⋅ dr ⎥ 2 dr ⎣N r 0 N ⎦

que nuevamente son análogas a las ecuaciones diferenciales de la curva funicular y a la elástica de la barra con rigidez a la flexión y que se prestan para el cálculo de los estados de tensión y deformación de la placa circular.

Tabla 63. Deformaciones y características de placas circulares y anulares simétricamente cargadas. Las siguientes igualdades y funciones son las que se utilizaran en la tabla. ρ=

φ0 = 1 − ρ4

r a

β=

b a

φ1 = 1 − ρ 2

N

E ⋅ h3 12 ⋅ (1 − μ )

φ 2 = ρ 2 ln ρ

ω' =

dω dr

φ 3 = ln ρ

φ4 =

1 −1 ρ2

Ahora se presentaran distintos casos con sus soluciones. 1) p

ω=

2a p⋅a2 (3 + μ ) ⋅ φ1 ; Mr = 16

p ⋅ a4 ⋅ [2 ⋅ (3 + μ ) ⋅ φ1 − (1 + μ ) ⋅ φ0 ] 64 N(1 + μ )

p⋅a2 Mt = [2(1 − μ ) + (1 + 3 ⋅ μ ) ⋅ φ1 ] ; 16

p⋅a4 5 + μ ⋅ ; 64 ⋅ N 1 + μ

Para ρ = 0

ω=

Para ρ = 1

ω' = −

p ⋅ a3 ; 8 ⋅ N(1 + μ )

Mr = Mt = Mt =

Qr = −

p⋅a ρ 2

Qr = −

p⋅a 2

p⋅a2 (3 + μ ) 16

p⋅a2 (1 − μ ) ; 8

Página 2 de 15

Estructuras III

2)

[ [

](

)

χ1 = (5 + μ ) − (7 + 3 ⋅ μ ) ⋅ β 2 ⋅ 1 − β 2 − 4(1 + μ ) ⋅ β 4 ⋅ ln β χ 2 = (3 + μ ) − (1 − μ ) ⋅ β 2 ⋅ 1 − β 2 + 4(1 + μ ) ⋅ β 2 ⋅ ln β

p 2b-2βa

](

)

2a

p⋅a4 (χ1 − 2χ 2 + 2χ 2 ⋅ φ1 ) ; ω= 64 N(1 + μ )

Para ρ ≤ β

p ⋅ a2 χ2 ; 16

Mr = Mt =

Qr = 0

Para ρ ≥ β

{[

]

p⋅a4 ω= 2 (3 + μ ) ⋅ 1 − 2β 2 + (1 − μ ) ⋅ β 4 ⋅ φ1 − (1 + μ ) ⋅ φ 0 − 4(1 + μ ) ⋅ β 4 φ 3 − 8(1 + μ ) ⋅ β 2 φ 2 64 N(1 + μ )

(

)

[

β2 ⎞ p⋅a ⎛ ⎜ ρ − ⎟⎟ Qr = − ρ⎠ 2 ⎜⎝

]

p⋅a2 (3 + μ )φ1 − (1 − μ ) ⋅ β 4 φ 4 + 4(1 + μ ) ⋅ β 2 φ 3 ; Mr = 16 Mt = Para ρ = 0

[

p⋅a2 (1 + 3μ ) ⋅ φ1 + (1 − μ ) ⋅β 4 φ 4 + 4(1 + μ ) ⋅ β 2 φ 3 + 2(1 − μ ) ⋅ 1 − β 2 16 p ⋅ a4 ω= χ1 64 N(1 + μ )

ω' = −

Para ρ = 1

3)

(

2 p ⋅ a3 1 − β2 ; 8N(1 + μ )

(

p

2a

Para ρ ≥ β

Mt =

p⋅a2 (1 − μ ) 1 − β 2 2 ; 8

(

)

2

Qr = −

p⋅a 1 − β2 2

(

)

[

]

χ2 ⎫ p⋅a4 ⎧ 2 2 2 ω= φ1 − φ 0 ⎬ ⎨1 + 4 − 5β + 4 2 + β ⋅ ln β ⋅ β + 2 64 N ⎩ 1+ μ ⎭ p⋅a Qr = − ρ 2 p⋅a2 p ⋅ a2 Mr = [χ 2 − (3 + μ ) + (3 + μ ) ⋅ φ1 ] ; Mt = [χ 2 − (1 + 3μ ) + (1 + 3μ ) ⋅ φ1 ] 16 16

ω=

(

)

⎤ p ⋅ a 4 2 ⎡ 2(3 + μ ) − (1 − μ )β2 2β ⎢ φ1 + 4φ2 + 2β2φ3 ⎥ 64 N 1+ μ ⎦ ⎣

[

]

p ⋅ a2 (1 − μ ) ⋅ β4φ4 − 4(1 + μ ) ⋅ β2φ3 ; Mr = 16 Mt =

Para ρ = 0

)]

χ1 = 4 − (1 − μ ) ⋅ β 2 ; χ 2 = [χ1 − 4(1 + μ ) ⋅ ln β] ⋅ β 2 ; χ 3 = 4(3 + μ ) − (7 + 3μ ) ⋅ β 2 + 4(1 + μ ) ⋅ β 2 ln β

2b-2βa

Para ρ ≤ β

)

}

ω=

Qr = −

[

p⋅b β 2 ρ

)]

p⋅a2 − (1 − μ ) ⋅ β 4 φ 4 − 4(1 + μ ) ⋅ β 2 φ 3 + 2(1 − μ ) ⋅ β 2 2 − β 2 ; 16

p ⋅ a 2 ⋅ b2 ⋅ χ3 ; 64 N(1 + μ )

(

Mt = Mr =

p ⋅ a2 χ2 16

Página 3 de 15

Estructuras III

Para ρ = β

[

]

p⋅a2 p⋅a2 χ 2 − (3 + μ ) ⋅ β 2 ; Mt = [χ 2 − (1 + 3μ ) ⋅ β] ; Q r = − p ⋅ b 2 16 16 2 2 p⋅a ⋅b p⋅b ω′ = − 2 − β2 ; (1 − μ ) 2 − β 2 ; Q r = − p ⋅ b β Mt = 8 N(1 + μ ) 2 8 Mr =

(

Para ρ = 1

)

(

)

4) P

(

)

(

)

χ1 = (3 + μ ) 1 − β2 + 2(1 + μ )β 2 ln β

2b-2βa

χ 2 = (1 − μ ) 1 − β2 − 2(1 + μ )ln β

2a Para ρ ≤ β

ω=

Para ρ ≥ β

ω=

P ⋅ a 2b [(χ1 − χ 2 ) + χ 2 φ1 ] ; 8 N(1 + μ )

[

Mr = Mt =

P⋅b χ2 ; 4

]

P ⋅ a2 ⋅ b (3 + μ ) − (1 − μ ) ⋅ β2 ⋅ φ1 + 2(1 + μ ) ⋅ β2φ3 + 2(1 + μ ) ⋅ φ2 } { 8 N(1 + μ ) β P⋅b (1 − μ ) ⋅ β2φ4 − 2(1 + μ ) ⋅ φ3 ; Q r = −P Mr = ρ 4 P⋅b Mt = − (1 − μ ) ⋅ β 2φ4 − 2(1 + μ ) ⋅ φ3 + 2(1 − μ )(1 − β 2 ) 4 P ⋅ a2 ⋅ b ω= χ1 8 N(1 + μ ) P⋅a ⋅b P⋅b Q r = −P ⋅ β ( ω′ = − 1 − β2 ); (1 − μ )(1 − β2 ) ; Mt = 2 N(1 + μ ) 2

[

]

[

Para ρ = 0 Para ρ = 1

Qr = 0

]

5) P a

ω= a

⎤ P ⋅ a2 ⎡3 + μ φ1 + 2φ2 ⎥ ⎢ 16πN ⎣ 1 + μ ⎦

2a Mr = −

Para ρ = 0 Para ρ = 1

P (1 + μ ) ⋅ φ3 ; 4π

Mt =

P [(1 − μ ) − (1 + μ ) ⋅ φ3 ] ; 4π

P ⋅ a2 3 + μ 16πN 1 + μ P⋅a P ω′ = − ; Mt = (1 − μ ) ; 4πN(1 + μ ) 4π

Qr = −

P 2πaρ

ω=

Qr = −

P 2πa

Página 4 de 15

Estructuras III

6) M

ω=

M ⋅ a2 φ1 ; 2 N(1 + μ )

Mr = Mt = M ;

Qr = 0

2a ω′ = −

Para ρ = 1

M⋅a N(1 + μ )

7) P

a

a ω=

⎞ P ⋅ a2 ⎛ 3 + μ ⋅ φ1 + φ0 + 8φ2 ⎟⎟ ⎜⎜ 2 ⋅ 64πN ⎝ 1 + μ ⎠

2a Observación: la carga P en este caso es:

P = ρ ⋅ π ⋅ a2 Para ρ = 0 Para ρ = 1

P [(3 + μ ) ⋅ φ1 + 4(1 + μ ) ⋅ φ3 ] ; 16π P Mt = − [(1 + 3μ ) ⋅ φ1 + 4(1 + μ ) ⋅ φ3 − 2(1 − μ )] ; 16π

Mr = −

P ⋅ a 2 7 + 3μ 64πN 1 + μ P⋅a ω′ = − ; 8πN(1 + μ )

Qr = −

⎞ P ⎛1 ⎜⎜ − ρ ⎟⎟ 2πa ⎝ ρ ⎠

ω=

Mt =

P (1 − μ ) 8π

8) p

p ⋅ a4 p ⋅ a2 (2 ⋅ φ1 − φ0 ) ; ω= Mr = [(3 + μ ) ⋅ φ1 − 2] 64 N 16 p ⋅ a2 2a Mt = [(1 + 3μ ) ⋅ φ1 − 2μ] ; Q r = − p ⋅ a ρ 16 2 4 2 p⋅a p⋅a (1 + μ ) Para ρ = 0 ω= ; Mr = Mt = 64 N 16 p ⋅ a2 p⋅a Para ρ = 1 M t = μM r = − μ; Qr = − 8 2

9) p

2b − 2βa 2a

χ1 = 1 − 4β 2 + β 4 (3 − 4 ⋅ ln β ) ; χ 2 = 1 − β2 β2 − 4 ⋅ ln β

(

)

Página 5 de 15

Estructuras III

Para ρ ≤ β Para ρ ≥ β

p ⋅ a4 p ⋅ a2 (1 + μ ) ⋅ χ 2 ; [(χ1 − 2χ 2 ) + 2χ 2φ1 ] ; Mr = Mt = 64 N 16 p ⋅ a4 ω= 2(1 − 2β2 − β4 )⋅ φ1 − φ0 − 4β4φ3 − 8β2φ2 ; 64 N 2 p ⋅ a2 Mr = − 2(1 − β2 ) + (3 + μ ) ⋅ φ1 − (1 − μ ) ⋅ β4φ4 + 4(1 + μ ) ⋅ β2φ3 ; 16 2 p ⋅ a2 Mt = − 2μ(1 − β 2 ) + (1 + 3μ ) ⋅ φ1 + (1 − μ ) ⋅ β4φ4 + 4(1 + μ ) ⋅ β2φ3 ; 16 β2 ⎞ p⋅a ⎛ ⎜⎜ ρ − ⎟⎟ ; Qr = − ρ⎠ 2 ⎝

ω=

[

]

[ [

Para ρ = 0 Para ρ = 1

Qr = 0

]

]

p ⋅ a4 χ1 64 N 2 p ⋅ a2 M t = μM r = − μ(1 − β2 ) ; 8 ω=

Qr = −

p⋅a 1 − β2 2

(

)

10) p

[

2b − 2β a 2a Para ρ ≤ β

Para ρ ≥ β

p ⋅ a4 [(χ1 − 2χ 2 + 1) + 2χ 2φ1 − φ0 ]; 64 N p ⋅ a2 {[(1 + μ ) ⋅ χ 2 − (3 + μ )] + (3 + μ ) ⋅ φ1} Mr = 16 p ⋅ a2 {[(1 + μ ) ⋅ χ 2 − (1 + 3μ )] + (1 + 3μ ) ⋅ φ1} Mt = 16 p ⋅ a 2 ⋅ b2 ω= ( 2 + β2 ) ⋅ φ1 + 2β2φ3 + 4φ2 ; 32 N p ⋅ b2 Mr = − 2(2 − β 2 ) + (1 − μ ) ⋅ β2φ4 − 4(1 + μ ) ⋅ φ3 ; 16 ω=

[

]

[

Para ρ = β

Para ρ = 1

]

[

Qr = −

p⋅b β 2 ρ

]

p ⋅ b2 − 2μ(2 − β 2 ) − (1 − μ ) ⋅ β2φ4 − 4(1 + μ ) ⋅ φ3 16 p ⋅ a2 p ⋅ a4 (1 + μ ) ⋅ χ 2 Mr = Mt = ω= χ1 ; 16 64 N p ⋅ a2 p ⋅ a2 (1 + μ ) ⋅ χ 2 − (3 + μ ) ⋅ β2 ; (1 + μ ) ⋅ χ 2 − (1 + 3μ ) ⋅ β2 ; Mr = Mt = 16 16 p⋅b Qr = − 2 p ⋅ b2 p⋅b M t = μM r = − μ(2 − β2 ); Qr = − β 8 2 Mt =

Para ρ = 0

]

χ1 = β2 4 − β2 (3 − 4 ⋅ ln β ) ; χ 2 = β2 (β2 − 4 ⋅ ln β) ;

[

]

[

]

Página 6 de 15

Estructuras III

11) P

χ1 = 1 − β2 (1 − 2 ⋅ ln β ) ; χ 2 = β2 − 1 − 2 ⋅ ln β ;

2b − 2βa 2a

P ⋅ a 2b P⋅b [(χ1 − χ 2 ) + χ 2φ1 ] ; (1 + μ ) ⋅ χ 2 ; Mr = Mt = 8N 4 P ⋅ a 2b ω= (1 + β2 )⋅ φ1 + 2β2φ3 + 2φ2 ; 8N P⋅b Mr = − 2(1 − β 2 ) − (1 − μ ) ⋅ β 2φ 4 + 2(1 + μ ) ⋅ φ3 ; 4 β P⋅b Q r = −P Mt = − 2μ(1 − β 2 ) + (1 − μ ) ⋅ β 2φ4 + 2(1 + μ ) ⋅ φ3 ; ρ 4

ω=

Para ρ ≤ β

[

Para ρ ≥ β

Qr = 0

]

[

]

[

]

P ⋅ a 2b χ1 8N P⋅b M t = μM r = − μ(1 − β 2 ) ; 2 ω=

Para ρ = 0 Para ρ = 1

Q r = −P ⋅ β

12) P

ω=

a

a 2a

Mr = −

P ⋅ a2 (φ1 + 2φ2 ) 16πN

P [1 + (1 + μ ) ⋅ φ3 ]; 4π

Para ρ = 0

P ⋅ a2 ω= 16πN

Para ρ = 1

M t = μM r = −

P μ; 4π

Mt = −

P [μ + (1 + μ ) ⋅ φ3 ] ; 4π

Qr = −

P 2πa

Qr = −

P 2πaρ

13) p

p

2b=2βa 2a 2b=2βa

χ1 = (3 + μ ) + 4(1 + μ ) ⋅

β2 ⋅ ln β , 1 − β2

χ 2 = (3 + μ ) − 4(1 + μ ) ⋅

β2 ⋅ ln β , 1 − β2

2a

Página 7 de 15

Estructuras III

ω=

[

]

p ⋅ a4 ⎧ 2 (3 + μ ) − β2χ 2 ⋅ φ1 − φ0 − 4 ⋅ β2χ1φ3 − 8 ⋅ β2φ2 ⎫⎬ , ⎨ 64 N ⎩1 + μ 1− μ ⎭

[

]

p⋅a ⎛ β2 ⎞ ⎜⎜ ρ − ⎟⎟ , 2 ⎝ ρ⎠

Mr =

p ⋅ a2 (3 + μ ) ⋅ φ1 − β2χ1φ4 + 4(1 + μ ) ⋅ β2φ3 , 16

Mt =

p ⋅ a2 (1 + 3μ ) ⋅ φ1 + β2χ1φ4 + 4(1 + μ ) ⋅ β2φ3 + 2(1 − μ ) − 2β2 [2(1 − μ ) − χ1 ]} { 16

Qr = −

2 ⎫ p⋅a2 ⎧ 4 2 1− β ( ) ( ) 5 + μ − 7 + 3 μ ⋅ β ⋅ − ⋅ β 2 χ1 ⋅ ln β⎬ ⎨ 64 N ⎩ 1+ μ 1− μ ⎭ 2 2 ⎞ p⋅a p ⋅ a b ⎛ χ1 ω′ = − − β2 ⎟⎟ , Mt = χ1 − (1 − μ ) ⋅ β2 ⎜⎜ 8 8N(1 + μ ) ⎝ 1 − μ ⎠

[

ω=

Para ρ = β

]

[

ω′ = −

Para ρ = 1

Mt =

]

p ⋅ a3 ⎡ χ1 ⎞⎤ 2⎛ ⎟⎥ ; ⎢1 − β ⎜⎜ 2 − 8 N(1 + μ ) ⎣ 1 − μ ⎟⎠⎦ ⎝

p ⋅ a2 (1 − μ ) − β2 [2(1 − μ ) − χ1 ]}; { 8

Qr = −

p⋅a 1 − β2 2

(

)

14) P

χ=

2b=2βa 2a 2b=2βa

P

β2 ⋅ ln β 1 − β2

⎤ ⎞ P ⋅ a 2 b ⎡⎛ 3 + μ 1+ μ ω= − 2χ ⎟⎟ ⋅ φ1 + 4 χφ3 + 2 ⋅ φ2 ⎥ ⎢⎜⎜ 8N ⎣⎝ 1 + μ 1− μ ⎠ ⎦

Mr = −

P⋅b (1 + μ )(− χφ4 + φ3 ) ; 2

Mt = −

⎡ ⎤ P⋅b (1 + μ ) ⋅ ⎢χφ4 + φ3 + ⎛⎜⎜ 2χ − 1 − μ ⎞⎟⎟⎥ 2 1 + μ ⎠⎦ ⎝ ⎣

Para ρ = β

ω=

β ρ

⎤ P ⋅ a 2b ⎡ 3 + μ 1+ μ 1 − β2 ) + 4 ⋅ χ ⋅ ln β⎥ ; ( ⎢ 8N ⎣ 1 + μ 1− μ ⎦

ω′ = −

P ⋅ a2 ⎛ 2 1+ μ ⎞ ⎜⎜ β − 2χ ⎟; 2 N(1 + μ ) ⎝ 1 − μ ⎟⎠

Mt = − Para ρ = 1

Q r = −P

ω′ = −

⎞ ⎛ P⋅b (1 + μ ) ⋅ ⎜⎜ 2 χ2 − 1 − μ ⎟⎟ ; Qr = −P ; 2 1+ μ ⎠ ⎝ β

P⋅a ⋅b ⎛ 1+ μ ⎞ ⎜⎜1 − 2χ ⎟; 2 N(1 + μ ) ⎝ 1 − μ ⎟⎠

Mt = −

Q r = −P ⋅ β ;

P⋅b (1 + μ ) ⋅ ⎛⎜⎜ 2χ − 1 − μ ⎞⎟⎟ 2 1+ μ ⎠ ⎝

Página 8 de 15

Estructuras III

15) M 2b=2βa 2a 2b=2βa

M

Mr = M

Para ρ = β

β2 ⋅ φ4 ; 1 − β2

ω=− ω′ =

Para ρ = 1

ω=−

M t = −M

M ⋅ b2 1 ⋅ 2 N(1 + μ ) 1 − β 2

β2 ⋅ (φ4 + 2) ; 1 − β2

⎛ ⎞ 1+ μ ⎜⎜ φ1 − 2 ⋅ φ 3 ⎟⎟ ; 1− μ ⎝ ⎠

Qr = 0 ;

M ⋅ b2 ⎛ 1 + μ ln β ⎞ ⋅ ⎜⎜1 − 2 ⋅ ⎟; 2 N(1 + μ ) ⎝ 1 − μ 1 − β2 ⎟⎠

M⋅b 1 ⋅ N(1 + μ ) 1 − β2

ω′ = 2

1 + β2 M t = −M ; 1 − β2

⎛ 1+ μ ⎞ ⋅ ⎜⎜ β2 + ⎟; 1 − μ ⎟⎠ ⎝

M⋅b β ; ⋅ 2 N (1 − μ ) 1 − β2

M t = −2M

β2 1 − β2

16) M 2b=2βa 2a 2b=2βa

ω=

M

⎞ ⎛ β2 M r = M⎜⎜1 − ⋅ φ4 ⎟⎟ ; 2 ⎠ ⎝ 1− β Para ρ = β

Para ρ = 1

ω=

⎛ M ⋅ a2 1+ μ 2 ⎞ ⋅ ⎜⎜ φ1 − 2 ⋅ β φ3 ⎟⎟ 2 2 N(1 + μ ) ⋅ (1 − β ) ⎝ 1− μ ⎠

⎞ ⎛ 1 + β2 β2 M t = M⎜⎜ + ⋅ φ4 ⎟⎟ ; 2 2 ⎠ ⎝1− β 1− β

Qr = 0

⎞ M ⋅ a2 ⎛ 1 + μ β2 ⋅ ⎜⎜1 − 2 ⋅ ⋅ ln β ⎟⎟ 2 2 N(1 + μ ) ⎝ 1− μ 1− β ⎠

ω′ = −

M⋅b 2 ; ⋅ 2 N (1 − μ ) 1 − β 2

ω′ = −

⎛ 1+ μ 2 ⎞ M⋅a ⋅ ⎜⎜1 + ⋅ β ⎟⎟ ; 2 N(1 + μ ) ⋅ (1 − β ) ⎝ 1 − μ ⎠

Mt = M

2 1 − β2

Mt = M

1 + β2 1 − β2

Página 9 de 15

Estructuras III

17) p 2b=2βa 2a 2b=2βa

p

χ1 = (1 + μ ) + (1 − μ ) ⋅ β2 ;

ψ1 = 4(1 + μ ) ⋅ β 2 ⋅ ln β ;

χ 2 = (1 − μ ) + (1 + μ ) ⋅ β2 ;

ψ=

ω=

2a

χ1 + ψ 1 2 ⋅β χ2

[

]

p ⋅ a4 2(1 − 2β 2 − ψ )⋅ φ1 − φ0 − 4ψφ3 − 8β 2φ2 ; 64 N

[

]

p ⋅ a2 2(1 − 2β2 + ψ ) − (3 + μ ) ⋅ φ1 + (1 − μ ) ⋅ ψφ 4 − 4(1 + μ ) ⋅ β2φ3 ; 16 p ⋅ a2 Mt = − 2μ(1 − 2β2 + ψ ) − (1 + 3μ ) ⋅ φ1 − (1 − μ ) ⋅ ψφ 4 − 4(1 + μ ) ⋅ β2φ3 ; 16 p⋅a ⎛ β2 ⎞ ⎜ ρ − ⎟⎟ ; Qr = − 2 ⎜⎝ ρ⎠ Mr = −

[

Para ρ = β

Para ρ = 1

]

[

]

2 p ⋅ a4 ( 1 − β2 ) − 2(1 − β2 ) ⋅ (ψ + 2β2 ) − 4(ψ + 2β4 ) ⋅ ln β ; 64 N p ⋅ a3 ψ − β4 p ⋅ a 2 1 − μ2 ω′ = − ⋅ ; Mt = ⋅ ⋅ (1 − β4 + 4β2 ⋅ ln β ) ; 8 N(1 + μ ) β 8 χ2

ω=

Mt = μ ⋅ Mr = −

p ⋅ a2 ⋅ μ ⋅ (1 − 2β2 + ψ ) ; 8

Qr = −

p⋅a ⋅ 1 − β2 2

(

)

18) P

χ = (1 + μ ) + (1 + μ ) ⋅ β 2 ;

2b=2βa 2a 2b=2βa

ψ = [1 + (1 + μ ) ⋅ ln β]⋅

2a

P

P ⋅ a 2b ω= [(1 + 2ψ ) ⋅ φ1 + 4ψφ 3 + 2φ 2 ] ; 8N

P⋅b [(1 − 2ψ ) − (1 − μ ) ⋅ φ 4 + (1 + μ ) ⋅ φ 3 ] , 2 P⋅b Mt = − [μ(1 − 2ψ ) − (1 − μ ) ⋅ φ 4 + (1 + μ ) ⋅ φ 3 ] . 2

Mr = −

Para ρ = β

ω=

[

β2 ; χ

β Q r = −P , ρ

]

P ⋅ a 2b (1 + 2ψ ) ⋅ 1 − β 2 + 2 β 2 + 2ψ ⋅ ln β , 8N

(

) (

)

Página 10 de 15

Estructuras III

ω′ =

Para ρ = 1

P ⋅ b2 1 − β 2 + 2 ⋅ ln β , 2 Nχ

(

)

Mt = μ ⋅ Mr = −

Mt = −

P ⋅ b 1− μ2 ⋅ 1 − β 2 + 2 ⋅ ln β , 2 χ

(

)

P⋅b μ(1 − 2ψ ) . 2

19) M

χ = (1 − μ ) + (1 + μ ) ⋅ β 2 ,

2b=2βa 2a 2b=2βa

ω=

Mr =

M ⋅ β2 [2 + (1 − μ ) ⋅ φ 4 ], χ

M ⋅ b2 1 − β 2 + 2 ⋅ ln β ; 2 Nχ M (1 − μ ) − (1 + μ ) ⋅ β 2 Mt = − χ

(

ω=

Mt = μ ⋅ Mr =

Mt =

)

[

Para ρ = 1

Qr = 0 ,

M

2a

Para ρ = β

M ⋅ b2 (φ1 + 2φ 3 ) , 2 Nχ

ω′ =

M ⋅ b2 [2μ − (1 − μ ) ⋅ φ 4 ] 2

M⋅b (1 − β 2 ) . Nχ

]

2M ⋅ β 2 μ. χ

20) p

χ1 = 2(1 − μ ) + (1 + 3μ ) ⋅ β 2 − 4(1 + μ ) ⋅ β 2 ln β , χ 2 = 2(1 − μ ) − (3 + μ ) ⋅ β 2 − 4(1 + μ ) ⋅ β 2 ln β ,

2a 2b=2βa Para ρ ≤ 1

ω=

p⋅a4 64 N

⎞ ⎛ 2χ 1 ⎜⎜ φ1 + φ 0 ⎟⎟ , ⎠ ⎝1 + μ

p⋅a2 [χ1 − (3 + μ ) + (3 + μ ) ⋅ φ1 ] , 16 p⋅a2 Mt = [χ1 − (1 + 3μ ) + (1 + 3μ ) ⋅ φ1 ], 16

Mr =

Para ρ ≥ 1

p⋅a4 ω= 64 N Mr =

Qr = −

p⋅a ρ 2

⎞ ⎞ ⎛ 2χ1 p ⋅ a ⎛ β2 2 2 ⎜⎜ − ρ ⎟⎟ , ⎜⎜ φ1 + φ 0 − 8β φ 3 − 8β φ 2 ⎟⎟ , Q r = 2 ⎝ ρ ⎠ ⎝1 + μ ⎠

[

]

p⋅a2 χ1 − (3 + μ ) + (3 + μ ) ⋅ φ1 − 2(1 − μ ) ⋅ β 2 φ 4 + 4(1 + μ ) ⋅ β 2 φ 3 , 16

Página 11 de 15

Estructuras III

[

Para ρ = 0

p⋅a4 ω= 64 N

Para ρ = 1

ω′ = −

Para ρ = β

]

p⋅a2 χ1 − (1 + 3μ ) + (1 + 3μ ) ⋅ φ1 + 2(1 − μ ) ⋅ β 2 φ 4 + 4(1 + μ ) ⋅ β 2 φ 3 , 16

Mt =

⎛ 2χ1 ⎞ ⎜⎜ − 1⎟⎟ , ⎝1+ μ ⎠

p ⋅a3 16 N

p⋅a2 Mr = Mt = χ1 , 16

⎛ χ1 ⎞ ⎜⎜ − 1⎟⎟ , ⎝1+ μ ⎠

Q r a =

p⋅a 2 β −1 2

(

)

Mr =

p⋅a2 [χ1 − (3 + μ )] ; 16

ω=−

p⋅a4 (3 − 5μ ) − (7 + 3μ ) ⋅ β 2 ⋅ β 2 − 1 + 16(1 + μ ) ⋅ β 2 ln β 64 N(1 + μ )

Mt =

{[

ω′ = −

p⋅a2 [χ1 − (1 + 3μ )]. 16

](

p ⋅ a 2b 2 − β2 ; 8 N(1 + μ )

(

)

Mt =

)

}

p⋅a (1 − μ ) ⋅ 2 − β 2 8

(

)

21)

[

]

1 (1 − μ ) + 4μβ 2 − (1 + 3μ ) ⋅ β 4 + 4(1 + μ ) ⋅ β 4 ln β 2 β 1 χ 2 = 2 (1 − μ ) ⋅ (1 − 2β 2 ) + (3 + μ ) ⋅ β 4 + 4(1 + μ ) ⋅ β 4 ln β β

p

χ1 =

[

2a 2b=2βa

]

Para ρ ≤ 1

ω=−

p⋅a4 χ 1 φ1 ; 32 N(1 + μ )

Para ρ ≥ 1

ω=−

p⋅a4 2χ 2 φ1 + (1 + μ ) ⋅ φ 0 + 4(1 + μ ) ⋅ 2β 2 − 1 ⋅ φ 3 + 8(1 + μ ) ⋅ β 2 φ 2 ; 64 N(1 + μ )

Mr = Mt = −

p ⋅a2 χ1 ; 16

[

(

[

]

)

]

p⋅a2 χ1 − (3 + μ ) ⋅ φ1 + (1 − μ ) ⋅ (2β 2 − 1) ⋅ φ 4 − 4(1 + μ ) ⋅ β 2 φ 3 ; 16 p⋅a2 Mt = − χ1 − (1 + 3μ ) ⋅ φ1 − (1 − μ ) ⋅ (2β 2 − 1)⋅ φ 4 − 4(1 + μ ) ⋅ β 2 φ 3 ; 16 Mr = −

Qr = 0 .

[

]

Para ρ = 0

ω=−

Para ρ = 1

ω′ =

Para ρ = β

ω=

p⋅a4 χ1 ; 32 N(1 + μ )

p ⋅ a3 χ1 ; 16 N(1 + μ )

Mr = Mt = −

⎞ p ⋅ a ⎛ β2 ⎜⎜ − ρ ⎟⎟ . Qr = 2 ⎝ ρ ⎠

p ⋅a2 χ1 . 16

p ⋅a2 Mr = Mt = − χ1 ; 16

Q r a =

p⋅a 2 β −1 . 2

(

)

2 ⎫ p⋅a4 ⎧ 2 4 β −1 ( ) ( ) ( ) − 4(1 + μ ) ⋅ 4β 2 − 1 ⋅ ln β⎬ 2 1 − μ − 3 − 5 μ ⋅ β + 7 + 3 μ ⋅ β ⎨ 2 64 N(1 + μ ) ⎩ β ⎭

[

]

(

)

Página 12 de 15

Estructuras III

22) p 2a 2b=2βa

Para ρ ≤ 1

ω=

p⋅a4 64 N

Mt =

Para ρ ≥ 1

⎛ χ ⎞ ⎜⎜ 2 φ1 − φ 0 ⎟⎟ ; ⎝ 1+ μ ⎠

ω=

Para ρ = 1

ω´=

Mr =

β

2

+ 2(1 + μ )

p⋅a2 [χ − (3 + μ ) + (3 + μ ) ⋅ φ1 ] 16 Qr = −

p⋅a4 32 N

p⋅a4 64 N

⎛ χ ⎞ ⎜⎜ 2 − 1⎟⎟ ; ⎝ 1+ μ ⎠

⎞ p ⋅a3 ⎛ χ ⎜⎜ − 1⎟⎟ ; 64 N ⎝ 1 + μ ⎠ 2 2 p⋅a (1 − μ ) β +2 1 ; Mt = 16 β

ω=−

1− μ

p⋅a2 [χ − (1 + 3μ ) + (1 + 3μ ) ⋅ φ1 ] ; 16

⎞ ⎛ 1− μ ⎜⎜ φ − 2φ 3 ⎟⎟ ; 2 1 ⎠ ⎝ (1 + μ) ⋅ β 2 2 ⎞ ⎛ p⋅a (1 − μ )⎜⎜ β 2− 1 + φ 4 ⎟⎟ ; Mr = − 16 ⎠ ⎝ β

ω=

Para ρ = 0

Para ρ = β

χ=

p⋅a4 32 N

⎡1 − μ β 2 − 1 ⎤ ⋅ 2 + 2 ⋅ ln β⎥ ; ⎢ ⎣1 + μ β ⎦

p⋅a ρ 2

Qr = 0 ⎡ β2 −1 ⎤ p⋅a2 (1 − μ )⎢− 2 − φ 4 ⎥ Mt = − 16 ⎣ β ⎦ Mr = Mt =

Mr = − Qr = −

p⋅a4 χ 16

2 p⋅a2 (1 − μ ) β −2 1 16 β

p⋅a 2

ω´= −

p ⋅a3 ; 8 N(1 + μ )β

Mt =

p ⋅ a 2 (1 − μ ) 8 β2

23) P 2a

⎛ 1⎞ χ = (1 − μ )⎜⎜ β − ⎟⎟ + 2(1 + μ) ⋅ β ⋅ ln β β⎠ ⎝

2b=2βa

Para ρ ≤ 1 Para ρ ≥ 1

P ⋅ a3 χ P⋅a Qr = 0 φ1 ; Mt = Mr = − χ; 8N 1 + μ 4 ⎫ ⎤ Pa 3 ⎧ ⎡ χ P ⋅ a3 2 2 2 ω= − + β φ − βφ − βφ ; M = − [χ + (1 − μ ) ⋅ βφ 4 − 2(1 + μ ) ⋅ βφ 3 ]; ⎨ 3 2⎬ r ⎥ 1 8 N ⎩ ⎢⎣1 + μ 4 ⎦ ⎭ β Qr = P ρ ω=−

Página 13 de 15

Estructuras III

Mt = −

P⋅a [χ − (1 − μ ) ⋅ βφ 4 − 2(1 + μ ) ⋅ βφ 3 ] . 4

P ⋅a3 χ 8 N(1 + μ )

Para ρ = 0

ω=−

Para ρ = 1

P⋅a2 ω´= χ; 4 N(1 + μ )

Para ρ = β

ω=

Mr = Mt = −

P ⋅ a3 ⎧ 2 ⎨ (1 − μ ) + (3 + μ ) ⋅ β 8N(1 + μ ) ⎩

[

Pa χ 4

]⋅ ⎛⎜⎜ β − β1 ⎞⎟⎟ − 2χ⎫⎬ ;

P⋅a2 ω´= (β 2 − 1) ; 2 N(1 + μ)

⎝

⎠

Mt =

P⋅a (1 − μ )(1 − β 2 ) 2β

⎭

24) P

χ = 2(1 + μ ) ⋅ β 2

2a 2b=2βa Para ρ ≤ 1

ω=

P⋅a2 8πN

⎡⎛ 1 − μ ⎞ ⎤ + 1⎟⎟ ⋅ φ1 + φ 2 ⎥ ; ⎢⎜⎜ ⎠ ⎣⎝ χ ⎦

[

]

P (1 − μ ) ⋅ (β 2 − 1) + χφ 3 ; 2 8πβ P Mt = − − (1 − μ ) ⋅ β 2 + 1 + χφ 3 ; 2 8πβ

Mr = −

[

Para ρ ≥ 1

(

)

]

Qr = −

P ; 2π.aδ

[

]

⎛1− μ ⎞ P ⎜⎜ φ1 − φ 3 ⎟⎟ ; M r = − (1 − μ ) ⋅ (β 2 − 1) + β 2 φ 4 ; 2 χ 8 πβ ⎝ ⎠ P Qr = 0 (1 − μ ) ⋅ − (β 2 − 1) − β 2 φ 4 ; Mr = − 2 8πβ

ω=

P⋅a2 8πN

[

]

⎛1− μ ⎞ ⎜⎜ + 1⎟⎟ ⎝ χ ⎠

Para ρ = 0

P⋅a2 ω= 8πN

Para ρ = 1

ω´= −

P ⋅ a ⎛ 1− μ ⎞ ⎜2 + 1⎟⎟ ; 8πN ⎜⎝ χ ⎠

Para ρ = β

ω=−

P⋅a2 8πN

Mr = −

⎡1 − μ 2 ⎤ ⎢ χ (β − 1) + ln β⎥ ; ⎣ ⎦

P (1 − μ ) ⋅ (β 2 − 1); 8πβ 2

ω´= −

P⋅a ; 4πN(1 + μ ) ⋅ β

Mt =

P (1 − μ ) ⋅ (β 2 + 1) 8πβ 2

Mt =

P (1 − μ ) 4πβ 2 Página 14 de 15

Estructuras III

25) M

ω= 2a

M ⋅a2 φ1 ; 2 N(1 + μ )

Mr = Mt = M ;

Qr = 0

2b=2βa

M ⋅a2 2 N(1 + μ )

Para ρ = 0

ω=

Para ρ = 1

ω´= −

M⋅a N(1 + μ )

Para ρ = β

ω=−

M ⋅a2 β2 − 1 ; 2 N(1 + μ )

(

)

ω´= −

M⋅b N(1 + μ )

26) M 2a

ψ=

2b=2βa Para ρ ≤ 1

Para ρ ≥ 1

Para ρ = 0

Para ρ = 1

ω=

M ⋅a2 χ φ1 ; 4N 1 + μ

Mr = Mt =

⎛ ψ ⎞ ⎜⎜ φ1 − 2φ 3 ⎟⎟ ; ⎝1+ μ ⎠ ⎡⎛ 1 ⎤ ⎞ M M r = (1 − μ ) ⋅ ⎢⎜⎜ 2 − 1⎟⎟ − φ 4 ⎥ ; 2 ⎠ ⎣⎝ β ⎦ 2 M⋅a χ ω= 4N 1 + μ M ⋅a2 ω= 4N

ω´= −

M⋅a ⎛ ψ ⎞ ⎜⎜1 + ⎟; 2 N ⎝ 1 + μ ⎟⎠

Para ρ = β

ω=−

M ⋅a2 4N

1− μ ; β2

M χ; 2

χ = (1 + μ ) + ψ

Qr = 0

Qr = 0

M χ; 2 M = χ; 2

Mr =

⎡ ⎤ M (1 − μ ) ⋅ ⎢⎛⎜⎜ 12 + 1⎞⎟⎟ − φ 4 ⎥ ; 2 ⎠ ⎣⎝ β ⎦

M (2 − χ ) ; 2 M = ψ (β 2 + 1) 2

M r a = −

M t a

⎡ ψ ⎤ M⋅a 2 ⎢1 + μ (β − 1) + 2 ⋅ ln β⎥ ; ω´= − N(1 + μ ) ⋅ β ; ⎣ ⎦

Mt = M ⋅ψ

Página 15 de 15