Formulas Torsion PDF

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Resumen completo de formulas de torsion....

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TORSIÓN: RESUMEN DE FÓRMULAS Rueda dentada: Mt = P . cos α . R

Momento torsor: Polea y correa: Mt = (T2-T1 ).R Ec. de equivalencia: ∫ ρ ⋅τ ⋅ dF = Mt = 0 [6] F

Sección circular maciza : τ =

Mt ⋅ ρ [13] Io

τ max =

Mt [15] Wt

π ⋅ d 3 [14´] π ⋅ d 4 [10] Io = W0 = 16 32 Mt Mt Mt dϕ = ⋅ dz ϕ = ⋅ L [16] θ = [17]

I W = W0 = o = [14] t d/2

G ⋅ Io G ⋅ Io Sección circular hueca (anular) di π ⋅d4 Io = 1 − η 4 [18] η = 32 d

(

)

G⋅ I o

[19] Wo =

Dimensionado: por resistencia W o =

(

π ⋅ d3 1 −η 4 16

)

[20]

Mt Mt [15´] por deformación Io = [17´] τ adm G ⋅ θ adm

π rad  [21] ⋅ 18000  cm 

π  rad  ° [22] ó θ adm = θ adm ⋅ ⋅ 180  m  Capacidad de carga: p/resistencia: Mt = W o ⋅ τ adm [23] p/deformación: Mt = G ⋅ I o ⋅θ adm [24] °

con θ °adm[° / m] : θ adm = θ adm ⋅

Evaluación del momento torsor Mt en función de potencia N y velocidad de giro n. N [CV ] [25] n [rpm ] N[W ] Si N[W], n[rps] ó [Hz]: Mt [N .m ] = [27] 2π ⋅ n[ Hz ]

Si N[CV], n[rpm]: Mt [kgf ⋅ cm ] = 71620 ⋅

Si N[HP]: Mt [kgf ⋅ cm ] = 72575 ⋅

N [HP ] [26] n[rpm]

Sección no circular tubular de pared delgada (Bredt)

τ ⋅ t = q = cte [28] τ = Mt [30] 2⋅ A⋅ t

El máximo “ τ ” ocurre para el mínimo valor de “t”

Mt ⋅ L S dS Mt ⋅ L ⋅ S [44] Espesor constante: ϕ = [45] 2 ∫0 4 ⋅G⋅ A 4 ⋅ G ⋅ A2 ⋅ t t Mt ⋅ L n Si Espesor constante por tramos: ϕ = ∑ [46] 4 ⋅ G ⋅ A 2 i=1 t i

Espesor variable continuo:ϕ =

Barras de sección rectangular: (a>b)

τ máx =

Mt Mt [31] ó τ máx = β ⋅G ⋅ θ ⋅b [32] = 2 α α ⋅ a ⋅ b Wt

a/b

α β a/b

α β a/b δ

1

1,1

1,2

1,25

=

Mt Mt [33] = 3 G ⋅β ⋅ a ⋅b C

Inercia ficticia I f = β ⋅ a ⋅ b 3

C = G ⋅ β ⋅ a ⋅b 3 = G ⋅ I f

Wt = α ⋅ a ⋅ b 2

siendo: θ

1,3

1,4

1,5

1,6

1,7

τ ´máx = δ ⋅τ máx 1,75

1,8

0,208

0,214

0,219

0,221

0,223

0,227

0,231

0,234

0,237

0,239

0,240

0,141 2

0,154 2,25

0,166 2,5

0,172 3

0,177 4

0,187 5

0,196 6

0,204 8

0,211 10

0,214 20

0,217 8

0,248

0,252

0,258

0,267

0,282

0,292

0,299

0,307

0,313

0,332

0,333

0,229

0,240

0,249

0,263

0,281

0,292

0,299

0,307

0,313

0,332

0,333

1

1,5

1,75

2

2,5

3

4

6

8

10

1

0,859

0,825

0,795

0,766

0,753

0,745

0,743

0,742

0,742

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Sección rectangular muy alargada (a >> b)

τ máx ≅

3⋅ Mt a ⋅b

[34]

2

τmáx = G⋅ θ ⋅ b [35]

Módulo resist. a torsión:Wt =

a .b 2 3

θ=

3 ⋅ Mt G ⋅ a ⋅b 3

Rigidez a torsión: C =

[36]

G .a.b 3 3

Inercia ficticia If =

a.b 3 3

Otras formas de secciones de paredes delgadas abiertas Secciones abiertas simples de espesor uniforme:

τ max =

3 ⋅ Mt Sm ⋅ e

[37] τmax = G ⋅ θ ⋅ e [37´] θ =

2

3 ⋅ Mt G ⋅ Sm ⋅ e3

[38]

Secciones abiertas simples y compuestas con tramos de distintos espesores:

3 ⋅ Mt

θ=

n

G⋅ ∑ Smi ⋅ ei

3

3 ⋅ Mt

[39] τ = G ⋅θ ⋅ e = i i

n

∑ Smi ⋅ ei

i =1

s/ Foppl: θ =

If =

3

⋅ ei =

Mt [40] τ max = Mt ⋅ emax [41] .e If If i

i =1

Mt [39´] Valores de η : 1(p/perf.ángulo), 1,1(p/perfil U y T), 1,3(p/perfil doble T”) G ⋅η ⋅ I f

1 n 3 Smi ⋅ ei ∑ 3 i =1

Concentración de tensiones: k = 1,74 ⋅ 3

c [42] r

Mt ⋅ ϕ [43] Energía en torsión: Te = 2

Mt 2 ⋅ L S dS U= [43´ ] 8 ⋅ G ⋅ A 2 ∫0 t

Sección circular maciza o tubular:

U =

Sección maciza rectangular:

U =

Pared delgada sección abierta:

U=

Mt2 ⋅ L [47] 2 ⋅ G ⋅ Io Mt 2 ⋅ L 2 ⋅ β ⋅G ⋅ a ⋅b 3 3⋅ Mt 2 ⋅ L n

2 ⋅ G ⋅ ∑ Smi ⋅ ei3

U=

U =

[49] U =

2 ⋅G ⋅ A2 ⋅ϕ 2 [43´´ ] S dS L⋅∫ 0 t

⋅ ϕ 2 ⋅ G ⋅ Io [48] 2⋅ L

ϕ 2 ⋅ β ⋅ G ⋅ a ⋅ b3 [50] 2⋅ L 2

ϕ ⋅G n [51] U = ⋅ ∑ Smi ⋅ e3i [52] 6 ⋅ L i= 1

i= 1

Este material de apoyo didáctico está destinado exclusivamente para el uso interno en las asignaturas Estabilidad de la carrera Ingeniería Ingeniería Eléctrica y Resistencia de Materiales de la Carrera Ingeniería Civil, Facultad Regional Santa Fe de la U.T.N. Profesor: Docente Auxiliar:

Ing. Hugo A. Tosone. Ing. Federico Cavalieri

Marzo de 2010.

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