Triaxial stress analysis

Casing Design

Prepared by: Tan Nguyen

The fundamental basis of casing design is that if stresses in the pipe

wall exceed the yield strength of the material, a failure condition

exists. Hence the yield strength is a measure of the maximum

allowable stress. To evaluate the pipe strength under combined

loading conditions, the uniaxial yield strength is compared to the

yielding condition.

The most widely accepted yielding criterion is based on the

maximum distortion energy theory, which is known as the

Huber-Von-Mises Theory. This theory states that if the triaxial stress

exceeds the yield strength, a yield failure is indicated. Note that the

triaxial stress is not a true stress. It is a theoretical value that allows a

generalized three-dimensional stress state to be compared with a

uniaxial failure criterion (the yield strength).

Casing Design

Combined Stress Effects (Triaxial stress analysis)

VME z t t r r z Ys s s2

s s2

s s2

s2

1

Where

sY – minimum yield stress, psi

sVME – triaxial stress, psi

VME: Von Mises Equivalent

sz, st, sr – axial tress, tangential

stress, and radial stress, psi

(1)

Casing Design

Setting the triaxial stress equal to the yield strength and solving

Y

z i

Y

z i

Y

t pi p p

s

s

s

s

s

s

2

1

4

31

2

This equation is for the ellipse of plasticity. Combining this eq. and

the equation of tangential stresses together and let r = ri, will give

the combinations of internal pressure, external pressure and axial

stress that will result in a yield strength mode of failure.

equation (1) give the results:

(2)

st – tangential stressespi –internal pressure

(3)

Casing Design

As axial tension increases, the critical burst-pressure

increases and the critical collapse-pressure decreases.

In contrast, as the axial compression increases, the

critical burst-pressure decreases and the critical

collapse-pressure increases.

Casing Design

Example

Compute the nominal collapse pressure rating for 5.5’’, N-80 casing

with a nominal wall thickness of 0.476’’ and a nominal weight per

foot of 26 lbf/ft. In addition, determine the collapse pressure for

in-service conditions in which the pipe is subjected to a 40,000 psi

axial tension stress and a 10,000 psi internal pressure. Assume a

yield strength mode of failure.

2 2

2 2 2 2

o i

i o i e o

tr r

p r r p r

s

Y

i

o i

i o i e o

Y

t i

pr r

p r r p r

p

s s

s

2 2

2 2 2 2

Y

i e

o i

o

Y

t i p p

r r

p r

s s

s2 2

2 2

5.5 4.548 80,000

2 5.52 2

2

i e

Y

t pi p p

s

s

12,649 12,649

i e e

Y

t pi p p p

s

s

Solution

For collapse pressure rating, r = ri then eq. (3) becomes

Casing Design

0

Y

z pi

s

s

1

Y

t pi

s

s

112,649

pe

pe 12,649 psi

From eq. (2) with we have

For in-service conditions of sz = 40,000 psi and pi = 10,000 psi

12,649

10,000 e

Y

t pi p

s

s

0.62580,000

40,000 10,000

Y

z pi

s

s

Solving eq. (14) gives

0.528412,649

10,000

e

Y

t pi p

s

s

pe 16,684 psi

Casing Design