thin cylinders
DESCRIPTION
In this lecture - Thin CylindersTRANSCRIPT
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STRUCTURAL ANALYSIS – 1 Chandigarh University
1D r. O m P r a k a s h
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Structural Analysis-ICode of the subject : CET-225
Lecture – 2
Dr.Omprakash
Department of Civil Engineering
Chandigarh University
2Dr. OmParkash
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Thin Cylinders&
Spherical Shells
Analysis of above under Pressures
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Thin Cylinders• Shape– Use of Shape (Tanks, Boilers, pipelines, Vault)
– Thin and Thick
– Remember Thin!
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Thin Cylinders• Pressure– Internal and External atmospheric
• Stress– Failure • Hoop type failure• Longitudinal failure
– Strain change in• diameter• length• volume
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THIN CYLINDERS
INTRODUCTION: In many engineering applications, cylinders are frequently used for
transporting or storing of liquids, gases or fluids.
Eg: Pipes, Boilers, storage tanks etc.
These cylinders are subjected to fluid pressures.
When a cylinder is subjected to a internal pressure, at any point on the
cylinder wall, three types of stresses are induced on three mutually
perpendicular planes.
They are ,
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Three types of stresses :
1. Hoop or Circumferential Stress (σC) or (σ1) – This is
directed along the tangent to the circumference and
tensile in nature. Thus, there will be increase in
diameter.
2. 2. Longitudinal Stress (σL) or (σ2 ) – This stress is
directed along the length of the cylinder. This is also
tensile in nature and tends to increase the length.
3. 3. Radial pressure ( pr ) – It is compressive in nature. Its
magnitude is equal to fluid pressure on the inside wall
and zero on the outer wall if it is open to atmosphere
THIN CYLINDERS
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THIN CYLINDERS
σ C σ L
1. Hoop Stress (C) 2. Longitudinal Stress (L) 3. Radial Stress (pr)
Element on the cylinder wall subjected to these three stresses
σ Cσ C
σC
p
σ L
σ L
σ L
p ppr
σ Lσ L
σ C
σ C
pr
pr
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THIN CYLINDERS
A cylinder or spherical shell is considered to be thin when the metal
thickness is small compared to internal diameter.
i. e., when the wall thickness, ‘t’ is equal to or less than ‘d/20’, where
‘d’ is the internal diameter of the cylinder or shell, we consider the
cylinder or shell to be thin, otherwise thick
Magnitude of radial pressure is very small compared to other two
stresses in case of thin cylinders and hence neglected
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THIN CYLINDERS
Longitudinal
axisLongitudinal stress
Circumferential stress
t
The stress acting along the circumference of the cylinder is called circumferential stresses whereas the stress acting along the length of the cylinder (i.e., in the longitudinal direction ) is known as longitudinal stress
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The bursting will take place if the force due to internal (fluid) pressure (acting vertically upwards and downwards) is more than the resisting force due to circumferential stress set up in the material.
p
σc σc
P - internal pressure (stress)
σc –circumferential stress
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THIN CYLINDERS
P - internal pressure (stress)
σc – circumferential stress
dL
σc
p
t
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EVALUATION OF CIRCUMFERENTIAL or HOOP STRESS (σC):
Consider a thin cylinder closed at both ends and subjected to internal
pressure ‘p’ as shown in the figure.
Let d=Internal diameter, t = Thickness of the wall
L = Length of the cylinder.
p d
t
σcσc
dltp
d
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THIN CYLINDERS
To determine the Bursting force across the diameter:Consider a small length ‘dl’ of the cylinder and an elementary area ‘dA’ as
shown in the figure.
rpp dθdldAdF
dθdldFx θcos2
dp
dA
σcσc
dlt
p
d
dθ
θ
Force on the elementary area,
Horizontal component of this force
dθdl 2
dp
dθdldFy θsin2
dp
Vertical component of this force
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THIN CYLINDERS
The horizontal components cancel out when integrated over semi-circular portion as there will be another equal and opposite horizontal component on the other side of the vertical axis.
sin2
dpforce bursting ldiametrica Total
0
dθdl
dA
σcσc
dlt
pθ
d
dθ
surface. curved theof area projectedp
dp cosdl2
dp 0
dl
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THIN CYLINDERS
dlcc tσ 2)σ stress ntialcircumfere to(due force Resisting
dldl dptσ2 i.e., c
)1....(....................t2
dpσ stress, ntialCircumfere c
dL
σc
p
t
force Burstingforce Resisting um,equillibriUnder
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THIN CYLINDERS
)1....(....................t2
dpσ stress, ntialCircumfere c
Force due to fluid pressure = p × area on which p is acting = p ×(d ×L)
(bursting force)
Force due to circumferential stress = σc × area on which σc is acting
(resisting force) = σc × ( L × t + L ×t ) = σc × 2 L × t
Under equilibrium bursting force = resisting force
p ×(d ×L) = σc × 2 L × t
Assumed as rectangular
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LONGITUDINAL STRESS (σL):
p
σL
The force, due to pressure of the fluid, acting at the ends of the thin cylinder, tends to burst the cylinder as shown in figure
P
A
B
The bursting of the cylinder takes place along the section AB
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EVALUATION OF LONGITUDINAL STRESS (σL):
d4
πp cylinder) of end (on the force bursting alLongitudin 2
p
t
σL
tdπσ force Resisting L
td π force thisresistingsection cross of Area
cylinder. theof material theof stress alLongitudinσLet L
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THIN CYLINDERS
tdπσd4
πp i.e., L
2
)2 .........(..........t4
dpσ stress, alLongitudin L
LC σ2σ (2), & (1) eqs From
force resisting force bursting um,equillibriUnder
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THIN CYLINDERS
tdπσd4
πp i.e.,
force resisting force bursting um,equillibriUnder
L2
)2 .........(..........t4
dpσ stress, alLongitudin L
tdπσ
σσRe
d4
πp
acting is pon which area pressure fluid todue Force
L
LL
2
actingiswhichonareaforcesisting
p
circumference
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EVALUATION OF STRAINS
A point on the surface of thin cylinder is subjected to biaxial stress system, (Hoop stress and Longitudinal stress) mutually perpendicular to each other, as shown in the figure. The strains due to these stresses i.e., circumferential and longitudinal are obtained by applying Hooke’s law and Poisson’s theory for elastic materials.
σ C=(pd)/(2t)σ C=(pd)/(2t)
σL=(pd)/(4t)
σ L=(pd)/(4t)