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Cylinder stress

F

l

t σ

θ

Components of cylinder or circumferential stress .

In mechanics , a cylinder stress is a stress distributionwith rotational symmetry ; that is, which remains un-changed if the stressed object is rotated about some xedaxis.

Cylinder stress patterns include:

• Circumferential stress or hoop stress , a normalstress in the tangential ( azimuth ) direction;

• Axial stress , a normal stress parallel to the axis ofcylindrical symmetry;

• Radial stress , a stress in directions coplanar withbut perpendicular to the symmetry axis.

The classical example (and namesake) of hoop stress isthe tension applied to the iron bands, or hoops, of awooden barrel . In a straight, closed pipe , any force ap-plied to the cylindrical pipe wall by a pressure differen-tial will ultimately give rise to hoop stresses. Similarly, ifthis pipe has at end caps, any force applied to them bystatic pressure will induce a perpendicular axial stress onthe same pipe wall. Thin sections often have negligiblysmall radial stress , but accurate models of thicker-walledcylindrical shells require such stresses to be taken into ac-count.

1 Denitions

1.1 Hoop stress

The hoop stress is the force exerted circumferentially(perpendicular both to the axis and to the radius of theobject) in both directions on every particle in the cylin-der wall. It can be described as:

σ θ = F

tl

where:

• F is the force exerted circumferentially on an area ofthe cylinder wall that has the following two lengthsas sides:

• t is the radial thickness of the cylinder

• l is the axial length of the cylinder

An alternative to hoop stress in describing circumferentialstress is wall stress or wall tension (T ), which usually isdened as the total circumferential force exerted alongthe entire radial thickness: [1]

T = F

l

Along with axial stress and radial stress , circumferential

Cylindrical coordinates

stress is a component of the stress tensor in cylindricalcoordinates .

It is usually useful to decompose any force applied to anobject with rotational symmetry into components parallel

to the cylindrical coordinates r , z, and θ. These com-ponents of force induce corresponding stresses: radialstress, axial stress and hoop stress, respectively.

1

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2 3 PRACTICAL EFFECTS

2 Relation to internal pressure

2.1 Thin-walled assumption

For the thin-walled assumption to be valid the vessel must

havea wall thickness of no more than about one-tenth (of-ten cited as one twentieth) of its radius. This allows fortreating the wall as a surface, and subsequently using theYoung–Laplace equation for estimating the hoop stresscreated by an internal pressure on a thin-walled cylindri-cal pressure vessel:

σ θ = P r

t

σ θ = P r

2t

where

• P is the internal pressure

• t is the wall thickness

• r is the mean radius of the cylinder.

• σ θ is the hoop stress.

The hoop stress equation for thin shells is also approxi-mately valid for spherical vessels, including plant cells andbacteria in which the internal turgor pressure may reachseveral atmospheres.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t , and d are inches(in). SI units for P are pascals (Pa), while t and d =2r arein meters (m).

When the vessel has closed ends the internal pressure actson them to develop a force along the axis of the cylinder.This is known as the axial stress and is usually less thanthe hoop stress.

σ z = F

A =

P d 2

(d + 2 t )2 − d 2

Though this may be approximated to

σ z = P r

2t

Also in this situation a radial stress σ r is developed andmay be estimated in thin walled cylinders as:

σ r = − P

2

2.2 Thick-walled vessels

When the cylinder to be studied has a r /t ratio of less than10 (often cited as 20) the thin-walled cylinder equationsno longer hold since stresses vary signicantly betweeninside and outside surfaces and shear stress through thecross section can no longer be neglected.

These stresses and strains can be calculated using theLamé equations , a set of equations developed by Frenchmathematician Gabriel Lamé .

σ r = A −

B

r 2

σ θ = A + B

r 2

where

• A and B are constants of integration, which may bediscovered from the boundary conditions

• r is the radius at the point of interest (e.g., at theinside or outside walls)

A and B may be found by inspection of the boundary con-ditions. For example, the simplest case is a solid cylinder:

if R i = 0 then B = 0 and a solid cylinder cannot havean internal pressure so A = P o

3 Practical effects

3.1 Engineering

Fracture is governed by the hoop stress in the absenceof other external loads since it is the largest principalstress. Note that a hoop experiences the greatest stressat its inside (the outside and inside experience the sametotal strain which however is distributed over differentcircumferences), hence cracks in pipes should theoreti-cally start from inside the pipe. This is why pipe inspec-

tions after earthquakes usually involve sending a camerainside a pipe to inspect for cracks. Yielding is governedby an equivalent stress that includes hoop stress and thelongitudinal or radial stress when present.

3.2 Medicine

In the pathology of vascular or gastrointestinal walls , thewall tension represents the muscular tension on the wallof the vessel. As a result of the Law of Laplace , if ananeurysm forms in a blood vessel wall, the radius of thevessel has increased. This means that the inward force on

the vessel decreases, and therefore the aneurysm will con-tinue to expand until it ruptures. A similar logic appliesto the formation of diverticuli in the gut.[2]

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3

4 Historical development of thetheory

Cast iron pillar of Chepstow Railway Bridge , 1852. Pin-jointed wrought iron hoops (stronger in tension than cast iron) resist the

hoop stresses. [3]

The rst theoretical analysis of the stress in cylinderswas developed by the mid-19th century engineer WilliamFairbairn , assisted by his mathematical analyst EatonHodgkinson . Their rst interest was in studying the de-sign and failures of steam boilers .[4] Fairbairn realisedthat the hoop stress was twice the longitudinal stress, animportant factor in the assembly of boiler shells fromrolled sheets joined by riveting . Later work was appliedto bridge building, and the invention of the box girder .In the Chepstow Railway Bridge , the cast iron pillars are

strengthened by external bands of wrought iron . The ver-tical, longitudinal force is a compressive force, which castiron is well able to resist. The hoop stress is tensile and sowrought iron, a material with better tensile strength thancast iron, is added.

5 See also

• Can be caused by cylinder stress:

• Boston Molasses Disaster

• Boiler explosion

• Related engineering topics:

• Stress concentration• Hydrostatic test• Buckling• Blood pressure#Relation_to_wall_tension• Piping#Stress_analysis

• Designs very affected by this stress:

• Pressure vessel• Rocket engine

• Flywheel• The dome of Florence Cathedral

6 References

[1] Tension in Arterial Walls By R Nave. Department ofPhysics and Astronomy, Georgia State University. Re-trieved June 2011

[2] E. Goljan, Pathology, 2nd ed. Mosby Elsevier, Rapid Re-view Series.

[3] Jones, Stephen K. (2009). Brunel in South Wales . II:Communications and Coal. Stroud: The History Press.p. 247. ISBN 9780752449128 .

[4] Fairbairn, William (1851). “The Construction of Boil-ers”. Two Lectures: The Construction of Boilers, and OnBoiler Explosions, with the means of prevention . p. 6.

• Thin-walled Pressure Vessels . Engineering Funda-mentals . 19 June 2008.

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4 7 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

7 Text and image sources, contributors, and licenses

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