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