Mechanics of Materials Ii6

8/3/2019 Mechanics of Materials Ii6

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Mechanics of Materials II

UET, TaxilaLecture No. (6)


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Cylinders & Pressure vessels

Cylindrical or sphericalpressure vessels are

commonly used inindustry to carry bothliquids and gases underpressure.


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Classification of applicationsn Cylinders find many applications,

two of the most commoncategories being :

a- fluid containers such as :pressure vessels, hydraulic

cylinders, gun barrels, pipes,boilers and tanks.b- interference-fitted bearing

bushes, sleeves and the like.


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Other applications

n

Cylinders can act as beams orshafts eg. ( load buildingblocks) but in the presentchapter cylinders are loadedprimarily by internal andexternal pressures due toadjacent fluids or to contactingcylindrical surfaces.


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Pressure Loading

When the pressure vessel isexposed to this pressure, thematerial comprising thevessel is subjected topressure loading, and hencestresses, from all

directions.


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Factors that affect stresses

The normal stresses resulting fromthis pressure are function of :1- the radius of the element under

consideration ,2- the shape of the pressure vessel (i.e., open ended cylinder, closedend cylinder, or sphere)3- the applied pressure.


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Two types of analysis arecommonly applied to pressure

vessels.The most common method isbased on a simple mechanicsapproach and is applicable tothin wall pressure vessels which

by definition have a ratio of innerradius (r), to wall thickness (t) of r/t 10 .


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The second method is basedon elasticity solution and isalways applicableregardless of the r/t ratioand can be referred to asthe solution for thick wall pressure vessels.


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Limiting proportions (approx)

Thin Thick d/t > 20 d/t < 20

t/d < 1/20 t/d > 1/20

t/d < 0.05 t/d > 0.05

n Where d = Di = inner diametern t = Cylinder thickness


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Thin-Walled Pressure Assumptions

Several assumptions are made in thismethod.1) Plane sections remain plane

2) r/t 10 with t being uniform andconstant3) The applied pressure, p, is the gaugepressure (where p is the differencebetween the absolute pressure and theatmospheric pressure)


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n 4) Material is linear-elastic,isotropic and homogeneous.

n 5) Stress distributionsthroughout the wall thickness willnot vary

n 6) Element of interest is remotefrom the end of the cylinder and

other geometric discontinuities.n 7) Working fluid has negligible

weight.


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THIN CYLINDERS AND SHELLS

1- THIN CYLINDERS


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Thin cylinder representation


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Classifications of Cylinders

Cylinders are classed as beingeither :n open - in which there is no

axial component of wall stress ,or

n

closed - in which an axialstress must exist to equilibratethe fluid pressure.


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Different types of open & closed Cylinders


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n When a thin-walled

cylinder is subjected tointernal pressure, three

mutually perpendicularprincipal stresses will be

set up in the cylindermaterial.


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Types of stresses

Namely:1- The circumferential

or hoop stress.2- The longitudinal stress.

3- The radial stress


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n

Provided that the ratio of thickness to inside diameterof the cylinder is less than1/20, it is reasonablyaccurate to assume that the

hoop and longitudinalstresses are constant across

the wall thickness.


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n

Also, the magnitude of theradial stress set up is so

small in comparison withthe hoop and longitudinalstresses that it can beneglected.


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n

This is obviously anapproximation since, in

practice, it will vary fromzero at the outside surfaceto a value equal to theinternal pressure at theinside surface.


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n

For the purpose of the initialderivation of stress formulae itis also assumed that the ends

of the cylinder and any riveted joints present have no effect onthe stresses produced; inpractice they will have an effectand this will be discussed later.


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Thin cylinders under internalpressure


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Hoop or circumferential stress


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1- Hoop or circumferential stress

n

This is the stress which isset up in resisting thebursting effect of the appliedpressure and can be mostconveniently treated by

considering the equilibriumof half of the cylinder.


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n It is required to

calculate the hoopstress in terms of:n Pressure (p)n Inner diameter (d)n Thickness (t)

H lf f thi li d bj t d t


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Half of a thin cylinder subjected tointernal pressure showing the hoop and

longitudinal stresses acting on anyelement in the cylinder surface.


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n Consider the equilibrium of forces in the x-directionacting on the sectionedcylinder shown in figure 2.

It is assumed that thecircumferential stress H (or ( is constantthrough the thickness of the cylinder.


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Figure (2)


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Using the forceequilibrium to derivean equation for hoop

stress

C l l i h l f i i l


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Calculating the total force owing to internalpressure

n Total force on half-cylinder owing tointernal pressure


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Resisting force owing to hoop stress

n Total resisting force owing to hoop

stress H set up in the cylinderwalls=

Force =


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Final form of hoop stress


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Longitudinal stress


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Longitudinal stress

n Consider now the cylinder shown inNext Figure.

Cross-section of a thin cylinder.

d S i f C li d i l Thi W ll d P


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nd Section of Cylindrical Thin-Walled Pressur Vessel Showing Pressure and Internal Axial

Stresses


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Using the forceequilibrium to derive

an equation forlongitudinal stress


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n Now consider the

equilibrium of forcesin the z-direction

acting on the partcylinder shown innext figure .


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Force owing to internal pressure

n Total force on the end of the cylinderowing to internal pressure

Force on cylinder end :Force =


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For equilibrium of forces we need tocalculate the End

section area


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End Section Area

The cross-sectional area of the cylinder wall ischaracterized by the product of its wallthickness and the mean circumference

For the thin-wall pressure vessels whereD

>>t the cylindrical cross-section area may be

approximated by Dt .


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Longitudinal stress final form


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Changes indimensions:

(a) Change in length


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(a) Change in length

n

The change in length of the cylinder may be

determined from thelongitudinal strain by

neglecting the radialstress.


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From Hookes Law


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n

And change in length =longitudinal strain x originallength

Then change in length =


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(b) Change in diameter

As above, the change indiameter may be

determined from the strainon a diameter, i.e. thediametral strain .

n Now the change in diameter


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n Now the change in diameter may be found from a

consideration of thecircumferential change. n The stress acting around a

circumference is the hoop orcircumferential stress H

giving rise to the circumferentialstrain H.


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Ch i Di


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Change in Diameter

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Mechanics of Materials Ii6