Transcript Document
Stresses in Thin-walled Pressure Vessels (I)
1 (2t dy ) p (2 r dy )
1
pr
(Hoop Stress)
t
2 ( 2 rt ) p ( r )
2
2
pr
2t
(Longitudinal Stress)
Stresses in Thin-walled Pressure Vessels (II)
2 ( 2 rt ) p ( r )
2
1 2
pr
2t
Stress State under General Combined Loading
Plane Stress Transformation
x'
x
x
y
x
2
x 'y '
y'
y x' y '
x
2
y
2
y
x
y
2
x
cos 2 xy sin 2
sin 2 xy cos 2
y
2
cos 2 xy sin 2
Mohr’s Circle for Plane Stress
x'
x 'y '
x
y
2
x
y
2
x
y
2
cos 2 xy sin 2
sin 2 xy cos 2
x ' ave
ave
R
x
2
x 'y ' R
2
y
2
x y
2
2
xy
2
2
Principal Stresses
tan 2 p
1,2
x
y
2
2 xy
x y
x y
2
2
xy ave R
2
Maximum Shear Stress
tan 2 s
max
x y
2 xy
x y
2
2
xy R
2
Mohr’s Circle for 3-D Stress Analysis
max
1
2
max min
Mohr’s Circle for Plane Strain
ave
R
x
x
y
tan 2 p
2
y
2
2
xy
2
2
max 2 R
xy
x y
x
y
2
xy
2
Strain Analysis with Rosette
2
cos
1
1
2
cos
2
2
cos 2
3
3
sin 1
2
sin 2
2
sin 3
2
cos 1 sin 1 x
cos 2 sin 2 y
cos 3 sin 3
xy
Typical Rosette Analysis
εa = εx
ε = ε /2 + ε /2 + γ /2
y
xy
b x
ε =ε
c y
εa = εx
εb = εx/4 + 3εy/4 + 3 γxy/4
εc = εx/4 + 3εy/4 - 3 γxy/4
εmax
εmin
max
εmax
εmin
max
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
x y z
DV
p
p
2
E
E
(1 ) 1 3 p
3
3 (1 2 )
V
E
where DV/V is called Dilatation or
Volumetric Strain.
Define Bulk Modulus K as
K
p
DV / V
E
3 1 2
Failure Criterion for Ductile Materials
(Yielding Criterion)
σ1
σ1
|σ1| = σY
σ2
σ2
|σ2| = σY
Comparison of Yielding Criteria
Tresca Criterion
(Max. Shear Stress)
Von Mises Criterion
(Max. Distortion Energy)
1 1 2 2 Y
2
2
2
|σ1| = σY
|σ2| = σY
|σ1 – σ2| = σY