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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