LECTURE+33+COE-3001-A

Have x fl l l 2 ri t 2 p i ri 0 p

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Unformatted text preview: a circumferenJal seam. M. Mello/Georgia Tech Aerospace 6 WE HAVE A Maximum in- plane Shear stress at the outer surface of a thin- walled cylinder 1, in the plane of rotation 2 •  Recall how in the case of the spherical pressure vessel there is no disNncNon between 1 , and 2 on the surface of the sphere •  If the z axis is normal to the spherical surface, then there is no maximum in- plane shear stress for any the rotaNons of the material element coordinate frame about the z axis. Indeed 1 = 2 = , and so: 1 2 (⌧max )z = =0 2 6= •  The opposite holds true in the case of the thin- walled cylinder since 1 2. In fact we have 1 = 2 2 and so: ( 1 /2) 1 2 1 1 (⌧max )z = = = 2 2 4 pr pr •  i.e., ( ⌧ max ) = (7- 8) (maximum in- plane shear stress) … since 1 = t z 4t (result is technically +/- since we can rotate 45° to either side) 4/3/13 M. Mello/Georgia Tech Aerospace 7 WE ALSO HAVE Maximum out- of- plane Shear Stress at the outer surface of a thin- walled cylinder •  In order to analyze and understand maximum shear stress it is criNcal that we not forgot about the 3rd dimension! •  We may not have normal stress in the z- direcNon or shear stress in the xy plane 2 , 3 in the •  but that doesn’t mean we can’t have maximum shear stress out- of- plane . plane of rotation = pr , t 1, 1 3 2 = pr , 2t 3 =0 45 in the plane of rotation (⌧max )x = ± aver = 4/3/13 y + 2 1 3 2 z = =± 1 2 = 1 2 pr 2t =± pr 2t (7- 9a) Absolute max shear stress !! 45 (⌧max )y = ± aver M. Mello/Georgia Tech Aerospace = 2 3 2 x + 2 =± z = 2 2 2 2 =± = pr 4t (7- 9b) pr 4t 8 EXAMPLE 7- 2(a) GIVEN: •  Cylindrical pressure vessel constructed by wrapping long narrow steel plate around a mandrel and then welding along the edges to form a helical joint. •  α=55° with respect to the longitudinal axis •  Inner radius: r=1.8m •  wall thickness: t=20mm •  ElasJc modulus: E=200GPa •  Poisson raJo: ν=0.30 •  Internal pressure: p = 800kPa x Problem 7- 1(a): •  Calculate the hoop (circumferenJal) and longitudinal stresses, σ1, and σ2 “Hoop stress” SoluJon 7- 1(a): 1 = 2 = 4/3/13...
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