Lecture Problem 136

E106 Section 2Corina Tom

3-7-07

Problem Statement

Calculate the maximum radius and wall thickness of a spherical pressure vessel made of Ti-6A1-4V Titanium Alloy periodically pressurized to 500 kPa, so that it will leak before breaking.

Use data on Table 9.1 on pg. 298 of Callister.

Pressure Vessels!

Designed to contain a significant pressure

Force is distributed evenly over the entire surface

Usually spherical or cylindrical (curvy!)Sharp angles = stress concentrationsPressure results in membrane stresses

It’s a stressful state

Stress does not depend on direction

Thin-wall t ≤ 0.1ri

ri = ro = r

Balance forces:

P σxx = σyy

€

σ xx =rP

2t

Leak-Before-Break

Want crack through entire thickness before rapid crack propagation

€

K Ic =Yσ πa

€

ac = t → leakage

€

ac = a 2a

t

Leak-Before-Break

Vessel must contain pressure without yielding

€

K Ic =Yσ y πt

€

σ =σ y

Assumptions

Thin-walled pressure vesselPlane StrainY = 1One half internal crack length

equal to thickness will ensure leakage

Max t

Kic = 55 MPa m1/2

σy = 910 MPa Y = 1

€

t =K Ic

2

πY 2σ y2

= 0.001163€

K Ic =Yσ y πt

t = 1.16 mm

Max r

σy = 910 MPaP = 500 kPa = .5

MPaT = 0.00116 m

€

r =2tσ yP

= 4.2325

r = 4.23 m

€

σ y =rP

2t

Design Review

Leak-before-breakNo plastic deformationWant critical crack length = thickness

Yielding before failurePlastic deformation of wallsWant large critical crack lengths

€

K Ic =Yσ y πt

€

σ y =rP

2t

€

K Ic =Yσ yN

⎛

⎝ ⎜

⎞

⎠ ⎟ πac

Great Molasses FloodNorth End - BostonJan. 15, 1919Purity Distilling

CompanyNeglected safety

precautions21 people and

many horses killedOver 150 injured Aftermath - wikipedia.org

Applications

Poor pressure vessel design may lead to catastrophic failure

Methods to catch safety hazardsLeakingVessel Distortion

References

Callister, W. Fundamentals of Materials Science and Engineering. 2nd Edition. John Wiley & Sons, Inc., 2005.

“Great Molasses Flood.” Massachusetts Foundation for the Humanities. <massmoments.org>

Rossman and Dym. Continuum Mechanics: Mechanics of Solids and Fluids. Harvey Mudd College, 2006.

Questions???