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