stresses in thin-walled pressure vessels (i) (hoop stress) (longitudinal stress)
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Stresses in Thin-walled Pressure Vessels (I)
)y 2()y 2(1 drdt p
t
pr1 (Hoop Stress)
)r()rt2( 22 p
t
pr
22 (Longitudinal Stress)
Mohr’s Circle for Plane Stress
2sin2cos22 xy
yxyx'x
2cos2sin2 xy
yx'y'x
22'y'x
2ave'x R
2
yxave
2xy
2
yx
2R
Strain Analysis with Rosette
xy
y
x
3332
32
2222
22
1112
12
3
2
1
sincossincos
sincossincos
sincossincos
Typical Rosette Analysis
εa = εx
εb = εx/2 + εy/2 + γxy/2
εc = εy
εmax
εmin
max
εmax
εmin
max
εa = εx
εb = εx/4 + 3εy/4 + γxy/4
εc = εx/4 + 3εy/4 - γxy/4
3
3
Stress Field in Beams
Stress trajectories indicating the direction of principal stress of the same magnitude.
Dilatation and Bulk Modulus
For the special case of “hydrostatic” loading -----
σx = σy = σz = –p
where V/V is called Dilatation or Volumetric Strain.
E
p
E
pzyx 2
Ep
V
V )21(331)1( 3
213/
E
VV
pK
Define Bulk Modulus K as
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