Thin and Thick Cylinder

7/28/2019 Thin and Thick Cylinder

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Members Subjected to Axisymmetric Loads

Pressurized thin walled cylinder:

Preamble: Pressure vessels are exceedingly important in industry. Normally two types of pressure vesselare used in common practice such as cylindrical pressure vessel and spherical pressure vessel.

In the analysis of this walled cylinders subjected to internal pressures it is assumed that the radial plansremains radial and the wall thickness dose not change due to internal pressure. Although the internalpressure acting on the wall causes a local compressive stresses (equal to pressure) but its value isnegligibly small as compared to other stresses & hence the sate of stress of an element of a thin walledpressure is considered a biaxial one.

Further in the analysis of them walled cylinders, the weight of the fluid is considered negligible.

Let us consider a long cylinder of circular cross - section with an internal radius of R 2 and a constant wallthickness t' as showing fig.

This cylinder is subjected to a difference of hydrostatic pressure of ?p' between its inner and outer surfaces.In many cases, ?p' between gage pressure within the cylinder, taking outside pressure to be ambient.

By thin walled cylinder we mean that the thickness t' is very much smaller than the radius R i and we mayquantify this by stating than the ratio (t / R i )of thickness of radius should be less than 0.1.

An appropriate co-ordinate system to be used to describe such a system is the cylindrical polar one r, , zshown, where z axis lies along the axis of the cylinder, r is radial to it and is the angular co-ordinate aboutthe axis.

The small piece of the cylinder wall is shown in isolation, and stresses in respective direction have also beenshown.

Type of failure:

Such a component fails in since when subjected to an excessively high internal pressure. While it might failby bursting along a path following the circumference of the cylinder. Under normal circumstance it fails bycircumstances it fails by bursting along a path parallel to the axis. This suggests that the hoop stress issignificantly higher than the axial stress.

In order to derive the expressions for various stresses we make following


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Applications:

Liquid storage tanks and containers, water pipes, boilers, submarine hulls, and certain air plane componentsare common examples of thin walled cylinders and spheres, roof domes.

ANALYSIS: In order to analyse the thin walled cylinders, let us make the following assumptions:

There are no shear stresses acting in the wall.

The longitudinal and hoop stresses do not vary through the wall.

Radial stresses r which acts normal to the curved plane of the isolated element are negligibly small as

compared to other two stresses especially when

The state of tress for an element of a thin walled pressure vessel is considered to be biaxial, although theinternal pressure acting normal to the wall causes a local compressive stress equal to the internal pressure,

Actually a state of tri-axial stress exists on the inside of the vessel. However, for then walled pressure vesselthe third stress is much smaller than the other two stresses and for this reason in can be neglected.

Thin Cylinders Subjected to Internal Pressure:

When a thin walled cylinder is subjected to internal pressure, three mutually perpendicular principal stresseswill be set up in the cylinder materials, namely

Circumferential or hoop stress

The radial stress

Longitudinal stress

Now let us define these stresses and determine the expressions for them

Hoop or circumferential stress:

This is the stress which is set up in resisting the bursting effect of the applied pressure and can be mostconveniently treated by considering the equilibrium of the cylinder.


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In the figure we have shown a one half of the cylinder. This cylinder is subjected to an internal pressure p.

i.e. p = internal pressure

d = inside diameter

L = Length of the cylinder

t = thickness of the wall

Total force on one half of the cylinder owing to the internal pressure 'p'

= p x Projected Area

= p x d x L

= p .d. L ------- (1)

The total resisting force owing to hoop stresses H set up in the cylinder walls

= 2 . H .L.t --------- (2)

Because H.L.t. is the force in the one wall of the half cylinder.

the equations (1) & (2) we get

2 . H . L . t = p . d . L

H = (p . d) / 2t

Circumferential or hoop Stress ( H) = (p .d)/ 2t

Longitudinal Stress:

Consider now again the same figure and the vessel could be considered to have closed ends and contains afluid under a gage pressure p. Then the walls of the cylinder will have a longitudinal stress as well as acircumferential stress.

Total force on the end of the cylinder owing to internal pressure

= pressure x area

= p x d 2 /4


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Area of metal resisting this force = d.t. (approximately)

Because d is the circumference and this is multiplied by the wall thickness

OR

Change in Dimensions:

The change in length of the cylinder may be determined from the longitudinal strain.

Since whenever the cylinder will elongate in axial direction or longitudinal direction, this will also getdecreased in diameter or the lateral strain will also take place. Therefore we will have to also take intoconsideration the lateral strain. As we know that the poisson's ratio ( ) is


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where the -ve sign emphasized that the change is negative

Consider an element of cylinder wall which is subjected to two mutually r normal stresses L and H .

Let E = Young's modulus of elasticity


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Volumetric Strain or Change in the Internal Volume:

When the thin cylinder is subjected to the internal pressure as we have already calculated that there is achange in the cylinder dimensions i.e, longitudinal strain and hoop strains come into picture. As a result of which there will be change in capacity of the cylinder or there is a change in the volume of the cylinder hence it becomes imperative to determine the change in volume or the volumetric strain.

The capacity of a cylinder is defined as

V = Area X Length

= d 2/4 x L

Let there be a change in dimensions occurs, when the thin cylinder is subjected to an internal pressure.

(i) The diameter d changes to d + d

(ii) The length L changes to L + L


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Therefore, the change in volume = Final volume Original volume

Therefore to find but the increase in capacity or volume, multiply the volumetric strain by original volume.

Hence

Change in Capacity / Volume or

Cylindrical Vessel with Hemispherical Ends:

Let us now consider the vessel with hemispherical ends. The wall thickness of the cylindrical andhemispherical portion is different. While the internal diameter of both the portions is assumed to be equal

Let the cylindrical vassal is subjected to an internal pressure p.


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For the Cylindrical Portion

For The Hemispherical Ends:

Because of the symmetry of the sphere the stresses set up owing to internal pressure will be two mutuallyperpendicular hoops or circumferential stresses of equal values. Again the radial stresses are neglected incomparison to the hoop stresses as with this cylinder having thickness to diameter less than1:20 .

Consider the equilibrium of the half sphere

Force on half-sphere owing to internal pressure = pressure x projected Area


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= p. d2/4

Fig. Shown (by way of dotted lines) the tendency, for the cylindrical portion and the spherical ends toexpand by a different amount under the action of internal pressure. So owing to difference in stress, the twoportions (i.e. cylindrical and spherical ends) expand by a different amount. This incompatibly of deformationscauses a local bending and sheering stresses in the neighborhood of the joint. Since there must be physicalcontinuity between the ends and the cylindrical portion, for this reason, properly curved ends must be usedfor pressure vessels.

Thus equating the two strains in order that there shall be no distortion of the junction

But for general steel works = 0.3, therefore, the thickness ratios becomes

t2 / t 1 = 0.7/1.7 or

t 1 = 2.4 t 2

i.e. the thickness of the cylinder walls must be approximately 2.4 times that of the hemispheroid ends for nodistortion of the junction to occur.

Effect of end Plates and Joints:

In general, the strength of the components will be reduced by the presence of riveted joints. We mustintroduce a joint efficiency factor .

For thin cylinders:


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Fig: Thin ring rotating with constant angular velocity

Here the radial pressure p' is acting per unit length and is caused by the centrifugal effect if its own masswhen rotating.

Thus considering the equilibrium of half the ring shown in the figure,

2F = p x 2r (assuming unit length), as 2r is the projected area

F = pr

Where F is the hoop tension set up owing to rotation.

The cylinder wall is assumed to be so thin that the centrifugal effect can be assumed constant across thewall thickness.

F = mass x acceleration = m 2 r x r

This tension is transmitted through the complete circumference and therefore is resisted by the completecross -sectional area.

Hoop stress = F/A = m 2

r 2

/ A

Where A is the cross-sectional area of the ring.

Now with unit length assumed m/A is the mass of the material per unit volume, i.e. the density .

Hoop stress = 2 r 2, so H = 2 r 2

THICK CYLINDERS

Introduction

Thick cylinders are the cylindrical vessels, containing fluid under pressure and whose wallthickness is not smaller than twentieth portion of the diameter (i.e. D/20).

Unlike thin cylinders the radial stress in thick cylinders is not negligible ; rather it varies from the

inner surface where it is equal to the magnitude of the fluid pressure to the outer surface

where usually it is equal to zero if exposed to the atmosphere .

Circumferential (Hoop) stress also varies along the thickness.


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The variation of the radial stress and the circumferential stress along the thickness are obtained

with the help of the Lames Theory.

Difference in Treatment between Thin and Thick Cylinders In thin cylinders the hope stress is assumed to be constant across the thickness of the cylinder

wall.

In thin cylinders there is no pressure gradient across the wall. In thick cylinders neither of these assumptions can be used and the variation of hop andradial stress will be as shown

For any element in the wall of a thick cylinder the stresses will be radial stress, circumferential / hoop stresstangential, longitudinal axial stresses.

Lames Theory:

The assumptions made in Lames Theory are as:

i. The material is homogeneous and isotropic.

ii. Plane sections perpendicular to the longitudinal axis of the cylinder remain plane after the

application of internal pressure.

iii. The material is stressed within the elastic limit.

iv. All the fibers of the material are to expand or contract independently without being constrained by

the adjacent fibers.

A thick cylinder subjected to internal and external radial stresses (pressure) is shown in the figure.


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Consider an elemental ring of internal radius r, and thickness dr.Let R 1 = internal radius of the thick cylinder R 2 = external radius of the thick cylinder l = length of the cylinder P 1 = internal pressureP 2 = external pressure r = internal radial stress on the elemental ring

r + d r = external radial stress on the elemental ring c / H = circumferential / Hoop stress on the elemental ring

Bursting Force( r*2r.l) - [( r + d r)*2(r + dr)l ]= 2l [ - r dr - r.d r - dr.d r ]= - 2l ( r.dr + r.d r )

(Neglecting the product of the small quantities)

Restoring Force= 2 c .l.dr

Equating the resisting force to bursting force (for equilibrium), we get2 c .l.dr = -2l ( r.dr + rdr)or c = - r - rd r/dr

Now let us obtain another relation between the radial stress (pressure)and circumferential (hoop) stress byusing the condition that the longitudinal strain (e l) at any point I the section is the same.


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The Longitudinal Stress:

Hence at any point in the section of the elemental ring considered above , the following three principalstresses existi. The radial stress (pressure), r ii. The circumferential / hoop stress, c / Hiii. The longitudinal stress, l

Since the longitudinal strain (e l) is constant, we have

Where (1/m) is the Poissons ratio. But since l, m and E are constantsSo, ( c - r ) = constantLet ( c - r ) = 2A

Putting c = ( r + 2A) in the initial equation, we get :

Integrating both sides, we get log e ( r + A) = -2log e r + log e B (Where, log eB = constant of integration)So, log e ( r + A) = log e (B/r 2)

r + A = (B/r 2) , or, r =(B/r 2 A) / r =( A B/r 2 ). Similarly, c / H =( B/r 2 + A) .

These two Equations are called as Lames Equations

The constants A and B can be evaluated from the known internal and external radial pressure and radius.It should be noted that in the above expression r is compressive and c / H is tensile .

Sign ConventionsCircumferential or hoop stress ( c ) will be taken as positive if tensile and negative when compressive whileradial stress ( r ) will be taken positive if compressive and negative if tensile.

The curve shows the relation between stress and radius. It is evident from the graph that the maximumvalues of both c and r occur at the inner surface.

Further for the closed ends cylinder, the longitudinal stress is


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Case 1: Internal Pressure Only

At r = R 1 , r = - P and at r = R 2 , r = 0

Using these two conditions the following results will be obtained:

The hoop ( H) and radial ( r ) stresses are maximum at r =R 1 The maximum shear stress occurs at the inside radius (at r=R 1) and max = B/ r 2

Case 2: Internal and external pressures:

At r = R 1 , r = - P 1 and at r = R 2 , r = - P 2



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Change of Diameter:

Comparison with Thin Cylinder:

max is the limiting factor.

Maximum Shear Stress:

The stresses on an element at any point in the cylinder wall are principal stresses.

Half the difference between the greatest and least principle stresses.

[ H is normally tensile, r is compressive ]

Greatest max occurs at the inside radius where r = R 1.


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2) Thick Spheres:

For the hoop and radial stresses the following equations are used:

Case 1: Internal Pressure Only:

Case 2: Internal and external pressures:

At r = R 1, r = - P 1 and at r = R 2, r = - P 2


Factors Influencing the Design of Vessels:

A- Most Common Types of Vessels:

Open tanks. Flat-bottomed, vertical cylindrical tanks. Vertical cylindrical and horizontal vessels with formed ends. Spherical or modified spherical vessels

B- Factors Influencing the Choice of Vessel:

The function and location of the vessel The nature of the fluid Operating temperature and pressure Necessary volume for storage or capacity for processing.


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C- Usage:

Large volume of no hazardous liquids, such as brine and other aqueous solutions, may be stored inpounds or in open tanks.

If the fluid is toxic, combustible, or gaseous in the storage condition or if the pressure is greater than atmospheric, closed system is required.

For storage of fluids at atmospheric pressure, cylindrical tanks with flat bottom and conical or

domed roofs are commonly used. For small volumes under pressure, cylindrical tanks with formed heads are used. For large volumes under pressure, spheres are used.

D- Welding:

1- T welds:

2- Butt welds:


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3- Plug welds:

4- Lap welds:

5- Edge welds:

The plates of the shell may be butt or lap-welded depending upon the design and

economic consideration.

5/8 in is the maximum plate thickness for lap-welded horizontal joints.

3/8 in is the maximum plate thickness for lap-welded vertical joints.

Butt-welded joints may be used for shell plates for all thicknesses up to and including 1 in.

The plates for butt-welded joints must be squared.

Squaring of the plates for lap-welded joints is not necessary

Plates for lap-welded joints are less expensive.

Difference between Thin and Thick Cylinders:

Sr.No. Thin Shell Theory (Thin Cylinder) Thick Shell Theory (Thick Cylinder)1 Thickness (t) is less than or equal to

(d/20), where d = Inner Diameter

Thickness (t) is greater than (d/20), where

d = Inner Diameter 2 Hoop stress ( c) is assumed uniform over the

thickness.

For cylinder: c = (pd/2t) (Tensile in nature)

For sphere : c = (pd/4t) (Tensile in nature)

Hoop stress( c) is variable. It is maximum atinner surface, minimum at outer surface.

For cylinder : c =( B/r 2 + A)

For sphere : c =( B/r 3 + A)

Variation is hyperbolic.3 Longitudinal stress ( L) is assumed uniform

over the cross-section.

Longitudinal stress ( L) is assumed uniform

over the cross-section.


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For cylinder: L = (pd/4t) (Tensile in nature)

For sphere : L= (pd/2t) (Tensile in nature)

For cylinder: L = PR 12 / (R 22 R 12)

For sphere : L= PR 12 / (R 22 R 12)

(Tensile in nature)4 Radial stress is very small in case of thin

shells and hence, it is neglected.

Radial stress varies over thickness.

It is calculated from Lames formula.

For cylinder : r =(B/r 2 A)For sphere : r =(2B/r 3 A)

Both Compressive.5 Internal pressure is less. Internal pressure is more.

Compound Cylinder:

There is a large variation in hoop stress across the wall of cylinder to internal pressure. The

material of the cylinder is not therefore used to its best advantages. To obtain a more uniform

hoop stress distribution cylinders are often built up by shrinking one tube on to the outside of

another. This is called compound cylinder .

Advantages over a single cylinder: The advantages of compound cylinder are that when the

outer tube contracts on cooling, the inner tube is brought into a state of compression. The outer

tube will conversely be brought into a state of tension.

Shrinkage Allowance: In the design of compound cylinders it is important to relate the difference

in diameter of the mating cylinder to the stresses, this will produce. This difference in diameter at

the common surface is normally termed as shrinkage allowances.

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Thin and Thick Cylinder