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Thin-walledPressureVessels

Cylindrical Pressure Vessel withInternal Pressure

Both cylinderical and spherical pressurevessels are common structures that areused ranging from large gas storagestructures to small compressed air tanks inindustrial equipment. In this section, onlythin-walled pressure vessels will beanalyzed.

A pressure vessel is assumed to be thin-walled if the wall thickness is less than10% of the radius (r/t > 10). This conditionassumes that the pressure load will betransfered into the shell as pure tension (orcompression) without any bending. Thin-walled pressure vessels are also known asshell structures and are efficient storagestructures.

If the outside pressure is greater than theinside pressure, the shell could also faildue to buckling. This is an advanced topicand is not considered in this section.

Cylindrical Pressure Vessels

Cylindrical Vessels will ExpierenceBoth Hoop and Axial Stress in

the Mid-section

Only the middle cylindrical section of a

cylinder pressure vessel is examined inthis section. The joint between the endcaps and the mid-section will havecomplex stresses that are beyond thediscussion in this chapter.

In the mid-section, the pressure will causethe vessel to expand or strain in only theaxial (or longitudinal) and the hoop (orcircumferential) directions. There will be notwisting or shear strains. Thus, there will

only be the hoop stress, hand the axial

stress, a. as shown in the diagram at the

left.

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Cross Section Cut of

Cylindrical Vessel

Pressure vessels can be analyzed bycutting them into two sections, and thenequating the pressure load at the cut withthe stress load in the thin walls. In the axialdirection, the axial pressure from thediscarded sections will produce a total axial

force of p(

r

2

) which is simply the crosssection area times the internal pressure. Itis generally assumed that r is the insideradius.

The axial force is resisted by the axialstress in the vessel walls which have athickness of t. The total axial load in the

walls will be a(2rt). Since the cross

section is in equilbrium, the two axialforces must be equal, giving

p(r2) = a(2rt)

This can be simplified to

where r is the inside radius and t is the wallthickness.

Hoop Section Cut fromCylindrical Vessel

In addition to the axial stress, there will bea hoop stress around the circumference.

The hoop stress, h, can be determined by

taking a vertical hoop section that has awidth of dx. The total horizontal pressureload pushing against the section will bep(2r dx) as shown in the diagram.

The top and bottom edge section will resist

the pressure and exert a load of h(t dx)

(each edge). The edge loads have to equalthe pressure load, or

p(2r dx) =h(2t dx)

This can be simplified to

where r is the inside radius and t is the wallthickness.

Spherical Pressure Vessel

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Spherical Pressure VesselCut in Half

A spherical pressure vessel is really just aspecial case of a cylinderical vessel. Nomatter how the a sphere is cut in half, thepressure load perpendicular to the cutmust equal the shell stress load. This is the

same situation with the axial direction in acylindrical vessel. Equating the to loadsgive,

p(r2) = h(2rt)

This can be simplified to

Notice, the hoop and axial stress are the

same due to symmetry.