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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 Spring '09
 ZHU
 Force, Stress

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