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)
Stresses in Thin-walled Pressure Vessels (II)
t
pr
221
)r()rt2( 22 p
Stress State under General Combined Loading
Plane Stress Transformation
2sin2cos22 xy
yxyx'x
2cos2sin2 xy
yx'y'x
2sin2cos22 xy
yxyx'y
'y'xyx
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
Principal Stresses
yx
xyp
22tan
R
22 ave2xy
2
yxyx2,1
Maximum Shear Stress
xy
yxs 2
2tan
R
22xy
2
yxmax
Mohr’s Circle for 3-D Stress Analysis
minmaxmax 2
1
Mohr’s Circle for Plane Strain
2
yxave
2
xy
2
yx
22R
yx
xyp2tan
2xy
2yxmax R2
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 Analysis on a Cross-section of Beams
Stress Field in Beams
Stress trajectories indicating the direction of principal stress of the same magnitude.
Re-visit of Pressure Vessel Stress Analysis
Relations among Elastic Constants
Constitutive Relations under Tri-axial Loading
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
Failure Criterion for Ductile Materials(Yielding Criterion)
σ2
|σ2| = σYσ2
|σ1| = σY
σ1 σ1
Comparison of Yielding Criteria
Von Mises Criterion
(Max. Distortion Energy)
22221
21 Y
Tresca Criterion
(Max. Shear Stress)
|σ1 – σ2| = σY
|σ1| = σY
|σ2| = σY