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Unformatted text preview: 3.1.E Cylindrical ThinWalled pressure vessels 125 126 3.2.E Cylindrical ThinWalled pressure vessels: Example Problems Problem 1: The walls of a cylindrical pressure vessel of radius R = 1m, and wall
thickness t=2cm are made of an isotropic linear elastic material with modulus E =10.0
GPa, Poisson’s ratio ν=0.2, and coefficient of thermal expansion α=1 x106 /ºC. When a
hot fluid at pressure p is introduced in the vessel, the
temperature of the wall increases by ∆T=100 ºC, and the
radius of the vessel increases by δR=1mm. What is the
value of the pressure p? 127 Problem 2 :A thin walled cylindrical tank has outer radius R, wall thickness t and length
L. The heads of the tank are flat and rigid, as shown, and the tank sits in a rigid well of
inner radius R. No friction acts between the tank and the well. The walls of the tank are
isotropic, linear elastic, with elastic constants E and ν. The tank is now subjected to an
internal pressure p and the top end of the tank rises in the well by a distance δ. obtain an
expression for δ in terms of the geometry of the problem, the material properties, and the
applied internal pressure.
R R
t
L δ p L 128 Problem 3
Part 1
A steel cylindrical thinwall vessel contains a
Circumferential
volatile fuel under pressure. For safety reasons,
gage (measures εθθ)
the pressure inside the tank should not exceed a
maximum value Pmax. A strain gage records the
circumferential strain in the tank and transmits this information to the control room (the gage
reads zero strain for zero pressure). At what value of strain, (εθθ)max should the operators take
action and open the relief valve on the tank (to reduce the pressure)? Express (εθθ)max in terms
of Pmax, the geometry of the vessel (radius R, thickness t), and the material properties of the
steel (Young’s modulus E, Poisson’s ratio ν).
Part 2
In part 1 we neglected the effects of the
constraints on deformation imposed by the
supports. The tank is attached to rigid supports
that prevent any elongation along the axis x of
the cylinder (but do not impose any
displacement constraint in the circumferential
direction at the location of the gage). x If you have to account for the effects of these supports does your relation for (εθθ)max change?
If so, what is the new expression? 129 130 Problem 4: A cylindrical tank of radius R,
length L, and wall thickness t is made of an
isotropic linear elastic material with modulus
E, Poisson’s ratio ν, and coefficient of thermal
expansion α. At temperature T0, with no
pressure in the tank, the tank fits without stress
between a rigid wall WB and a rigid plate C. A
spring of stiffness K fits exactly between the
rigid plate C and a second rigid wall WA. The
plate can only move in the horizontal direction,
as indicated in the figure. The tank is free to
expand radially. L C spring WA tank K 2R WB C WA uC WB 1) The internal pressure in the tank is brought to a value p. Obtain expressions for the
resulting displacement of the plate uC, and for the change in tank wall thickness δt.
What is the radial strain in the tank wall? What is the circumferential strain in the
tank wall?
2) The temperature of the wall of the pressurized tank is now increased by ∆T. What are
the new values for uP and δt? 131 132 133 134 135 136 137 Problem 4 (20 Pts)
Consider two thinwalled pressure
vessels between two frictionless fixed
walls at A and C, separated by a
movable frictionless rigid plate at B, as
shown in the figure. Both vessels have
an inner radius, R, and wall thickness,
t. Both are made out of the same
material with a Young’s modulus, E,
and Poisson’s ratio, ν. The left vessel
has length 2L, and the right vessel has
length L as indicated. Initially, the
pressure in both vessels is zero and
there is no stress in the walls of the
pressure vessels. If the left vessel is
pressurized to a pressure p, while the
pressure in the right vessel is kept to
zero, the rigid plate at B will displace
by uBx. Find an expression for uBx, the
distance the plate moves, in terms of the pressure p, and the known quantities, R, L, t, E,
and ν. 138 139 3.3.E Cylindrical ThinWalled pressure vessels: Conceptual Questions
For each of the following, cross the false (wrong) statements, circle the true (right) ones:
E1) The cylindrical walls of a vessel of outer radius R and wall thickness t (t<<R) are
made of an isotropic linear elastic material with modulus E and Poisson’s ratio ν. The
end caps are rigid. If the vessel is empty (zero gage pressure), the vessel fits precisely
(without any stress) inside a rigid frictionless well of inner diameter 2R.
With the vessel inside the rigid, frictionless well, a pressurized fluid is introduced in the
vessel, so that the internal gage pressure in the vessel is now p. For these conditions, in
the cylindrical walls of the vessel we will have:
a) No change in outer radius and therefore zero radial strain. pR
.
t
pR
c) σ xx =
.
2t
b) σ θθ = d) εθθ = –ν εxx. e) σθθ = ν σxx. x ρ θ 2R 2R 2R
p 140 E2) A rigid block of weight W rests over a soda can. The material of the can is isotropic
linear elastic with properties E, ν (0.5>ν > 0), α. The gage pressure inside the can is p.
Under these conditions, the height of the can is H, its radius is R, and the wall thickness is
t<<R.
(1) The weight is removed but the can is still closed, with internal pressure p:
(a) the axial stress changes but the circumferential stress stays the same
(b) the height of the can changes but its radius stays identical
(2) Then the can is opened, (so now the inside gage pressure is p=0), the height of the can
becomes H2 and its radius becomes R2:
(c) It is possible, for a special combination of W and p, to have H =H2
(d) It is possible, for a special combination of W and p, to have R =R2 W (0) p p p=0 H2 H1 H
2R (2) (1) 2R1 2R2 141 SOLUTIONS:
E1) The cylindrical walls of a vessel of outer radius R and wall thickness t (t<<R) are
made of an isotropic linear elastic material with modulus E and Poisson’s ratio ν. The
end caps are rigid. If the vessel is empty (zero gage pressure), the vessel fits precisely
(without any stress) inside a rigid frictionless well of inner diameter 2R.
With the vessel inside the rigid, frictionless well, a pressurized fluid is introduced in the
vessel, so that the internal gage pressure in the vessel is now p. For these conditions, in
the cylindrical walls of the vessel we will have:
a) No change in outer radius and therefore zero radial strain. pR
.
t
pR
c) σ xx =
.
2t
b) σ θθ = d) εθθ = –ν εxx. e) σθθ = ν σxx. x ρ θ 2R 2R 2R
p 142 E2) A rigid block of weight W rests over a soda can. The material of the can is isotropic
linear elastic with properties E, ν (0.5>ν > 0), α. The gage pressure inside the can is p.
Under these conditions, the height of the can is H, its radius is R, and the wall thickness is
t<<R.
(1) The weight is removed but the can is still closed, with internal pressure p:
(e) the axial stress changes but the circumferential stress stays the same
(f) the height of the can changes but its radius stays identical
(2) Then the can is opened, (so now the inside gage pressure is p=0), the height of the can
becomes H2 and its radius becomes R2:
(g) It is possible, for a special combination of W and p, to have H =H2
(h) It is possible, for a special combination of W and p, to have R =R2 W (0) p p p=0 H2 H1 H
2R (2) (1) 2R1 2R2 143 3.1.F Circular shafts in torsion
Axisymmetric loading and deformation: Stresses and Strains in cylindrical coordinates 144 145 Comparison and correspondences between rods in torsion and bars in tension 146 147 148 149 150 151 3.2.F Circular shafts in torsion. Example problems
Problem 1
The tapered AB shaft in the figure has a
variable cross section with a radius
defined by the equation R(x)=R0 eax,
whereR0 (units:[m]) and a (units:[1/m])
are constants, and the origin of the xaxis
is at A. The shaft is made of isotropic
linear elastic material with modulus G.
The shaft is fixed at B and subjected to a
torque T at the free end, A, as indicated in
the figure.
a) Determine the angle of twist at the
free end: ϕx(A).
b) Determine the maximum shear stress
in the shaft.
[Friendly hint: ∫ecx dx= (1/c) ecx for any
constant c] 152 153 Problem 2
The composite shaft of length L is
constructed from an inner core of
radius R and modulus Gc =3G*, and a
sleeve of outer radius 2 R and
modulus Gs =G*, bonded together.
One end of the shaft is fixed and the
other is free to rotate (see figure). A
uniform distributed torque, t
(t=constant), is applied to the shaft. B
t L A x (a) Graph the internal moment
(torque T ) as a function of x measured from the free end of the shaft.
(b) Determine the rotation angle of the shaft, ϕx , as a function of x. Determine the
angle of twist δϕ.
(c) Determine the maximum shear stress, σ θ x max , in both materials . 154 155 Problem 3 x The shaft ABCD in the figure is
2a
made from two segments: AB of
a
diameter a and length L/4, and
BD of diameter 2a and length 2L.
A
B
The shaft is confined by fixed
supports at the walls A and D. A
torque Q is applied to the shaft at
L
point C (at a distance L from the
4
fixed support D as indicated in
the figure). Please answer the
following questions [careful with the signs and conventions!] L
D C Q
2L (a) Obtain the reaction moments at A and D (MAX, MDX).
(b) For each of the three segments: AB, BC, CD :
determine the internal torque [T AB, T BC T CD]
determine the stress state along the surface of the shaft and draw
the stress components acting on a surface element aligned with
the x and θ axes as indicated to the right. x θ 156 157 158 159 160 161 3.3.F Circular shafts in torsion. Conceptual Questions
For each of the following, cross the false (wrong) statements, circle the true (right) ones: F1) The tapered shaft in the figure has a radius
linearly varying between rA and rB, and it is
subjected to a constant torque T as shown.
(a) The maximum shear stress for the shaft will
be at section A because it has the largest
radius.
(b) The shear strain at the outer radius will be
linearly varying (along x) between section A
and section B
(c) At each section x, the shear strain will be
linearly varying from the axis to the outer
radius
(d) Changes in temperature will only affect the shear stresses σxθ if an axial force is
superimposed to the torque. F2) The cylindrical shaft in the figure is
confined by fixed supports at the walls A
and B. A torque T is applied to the shaft at
point C as indicated in the figure, with LAC
> LBC.
(a) The peak magnitude of shear stress in the shaft will be in section BC.
(b) δϕAC = −δϕBC
AC (c) T = −T x CB (d) At any point of AC σxθ ≥0 ; At any point of BC σxθ ≤ 0. 162 F3) The homogeneous shaft of radius R
(known) and length L (known) in the
figure is composed of a linear isotropic
elastic material of known shear modulus
G. Under the effects of an unknown
distribution of twisting moments, the bar
is observed to deform. The value of the
xrotation along the bar is known, and it
is given as a function of the distance
from the fixed support, x, by the
relation: ϕx(x) = 2ax2–2aLx, where a is
a dimensional constant with units [1/m2], and L is the length of the bar. x a) End A is fixed, therefore the shear strain at any point of section A must be zero: εxθ (r,
x=0) = 0.
b) The angle of twist of shaft AB is zero: δϕ AB=0
c) At any distance (x) from the fixed support, the shear strain at the center (axis of
symmetry) of the bar is zero: εxθ (r=0, x) = 0.
d) The shear strain at any point of section B is zero: εxθ (r, x=L) = 0, therefore we can
conclude that there is no concentrated twisting moment acting at B.
e) Values of the shear stresses in the shaft cannot be determined because the distribution
of applied twisting moments is unknown.
F4) Shaft AB has outer radius R and is fixed between rigid walls at the two ends. A
twisting moment Q is applied at the midspan (section C) as indicated. The left half of the
shaft (AC) is solid, while the right half (CB) is hollow, (with an inner radius R/2). a) The magnitude of the maximum
shear strain is the same in AC and
CB. L/2
R b) The magnitude of the internal
twisting moment, T, is the same in
AC and CB. L/2
Q
x C c) The magnitude of the maximum shear stress is the same in AC and CB.
d) Equilibrium conditions yield Q = –TAC–TCB.
e) Equilibrium conditions yield Q = –TxA –TxB .
[TxA and TxB are the reaction moments at A and B] 163 SOLUTIONS F1) The tapered shaft in the figure has a radius
linearly varying between rA and rB, and it is
subjected to a constant torque T as shown.
(a) The maximum shear stress for the shaft will
be at section A because it has the largest
radius.
(b) The shear strain at the outer radius will be
linearly varying (along x) between section A
and section B
(c) At each section x, the shear strain will be
linearly varying from the axis to the outer
radius
(d) Changes in temperature will only affect the shear stresses σxθ if an axial force is
superimposed to the torque. F2) The cylindrical shaft in the figure is
confined by fixed supports at the walls A
and B. A torque T is applied to the shaft at
point C as indicated in the figure, with LAC
> LBC.
(a) The peak magnitude of shear stress in the shaft will be in section BC.
(b) δϕAC = −δϕBC
AC (c) T = −T x CB (d) At any point of AC σxθ ≥0 ; At any point of BC σxθ ≤ 0. 164 F3) The homogeneous shaft of radius R
(known) and length L (known) in the
figure is composed of a linear isotropic
elastic material of known shear modulus
G. Under the effects of an unknown
distribution of twisting moments, the bar
is observed to deform. The value of the
xrotation along the bar is known, and it
is given as a function of the distance
from the fixed support, x, by the
relation: ϕx(x) = 2ax2–2aLx, where a is
a dimensional constant with units [1/m2], and L is the length of the bar. x a) End A is fixed, therefore the shear strain at any point of section A must be zero: εxθ (r,
x=0) = 0.
b) The angle of twist of shaft AB is zero: δϕ AB=0
c) At any distance (x) from the fixed support, the shear strain at the center (axis of
symmetry) of the bar is zero: εxθ (r=0, x) = 0.
d) The shear strain at any point of section B is zero: εxθ (r, x=L) = 0, therefore we can
conclude that there is no concentrated twisting moment acting at B.
e) Values of the shear stresses in the shaft cannot be determined because the distribution
of applied twisting moments is unknown.
F4) Shaft AB has outer radius R and is fixed between rigid walls at the two ends. A
twisting moment Q is applied at the midspan (section C) as indicated. The left half of the
shaft (AC) is solid, while the right half (CB) is hollow, (with an inner radius R/2). a) The magnitude of the maximum
shear strain is the same in AC and
CB. L/2
R b) The magnitude of the internal
twisting moment, T, is the same in
AC and CB. L/2
Q
x C c) The magnitude of the maximum shear stress is the same in AC and CB.
d) Equilibrium conditions yield Q = –TAC–TCB.
e) Equilibrium conditions yield Q = –TxA –TxB .
[TxA and TxB are the reaction moments at A and B] 165 ...
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 Fall '11
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