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Substituting
σ
r
p=−
i
in equation (D-121),
σ
σ
θ
r
D
t
D
t
D
t
≥
⎛
⎝
⎜
⎞
⎠
⎟
−+
−
1
2
1
1
2
(D-122)
For D/t = 15,
σ
θ
/
σ
r
= 7.0. For D/t = 25,
σ
θ
/
σ
r
= 12.0. For the large majority of
oil field casing (
15 25≤≤Dt ) the minimum value of
σ
θ
is quite a bit larger than
the maximum value of
σ
r
.
With the results of the above example in mind, and as a first step in the thin
cylinder approximation, let the radial stress be ignored. For future reference, the
simplified form of the effective stress resulting from this approximation will be:
[]
σσσσσ
θθ
ez
=+−
22
1
2
z
(D-123)
D.7.2 Using the Average Circumferential
Stress
Consider for a moment a cylinder loaded only by an internal pressure, .
equation (D-121) of the previous section gives the range of values of
p
i
σ
θ
between the inner and outer radius. Rearranging equation (D-121),
()
()
σ
σ
σ
σ
θ
θ
θ
θ
max
min
=
=
=
=
⎛
⎝
⎜
⎞
⎠
⎟
−+
⎛
⎝
⎜
⎞
⎠
⎟
−+
at r r
at r r
D
t
D
t
D
t
D
t
o
i
1
2
1
1
2
22
2
2
(D-124)
Using L’Hospital’s rule, it can be shown that the limit of this ratio as D/t
approaches infinity is unity. For
15 25
≤
≤
Dt , this ratio will vary between 1.17
and 1.09. This small variation in the value of
σ
θ
through the thickness of the
cylinder implies that the equations for a thick cylinder can be further simplified by
using a constant, average value of
σ
θ
. This average value may be determined
from:
()
σσ
θθ
ave
oi
r
r
ii oo
oi
rr
dr
pr pr
rr
i
o
=
−
=
−
−
∫
1
(D-125)
An even more convenient form of (
σ
θ
)
ave
can be written in terms of the mean
diameter,
, and the thickness. Recognizing that:
′
D
() (
rDtrD
oi
=
′
+=
′
−
1
2
1
2
,
)
t
(D-126)
equation (D-125) can be rewritten in the form:
D-42 Casing/Tubing Design Manual
October 2005
()
σ
θ
ave
io
D
t
p
D
t
p=
′
−
⎛
⎝
⎜
⎞
⎠
⎟
−
′
+
⎛
⎝
⎜
⎞
⎠
⎟
1
2
1
1
2
1 (D-127)
The usefulness of equation (D-127) is two-fold. First, if an equivalent external
pressure (see Figure D-14) is defined by:
p
i
p
o
r
i
r
o
(σ
θ
)
ave
(σ
θ
)
ave
Force Equilibrium:
(
)
()
() ()
22 2 0
11
2
1
2
2
pr pr t
pr pr
t
t
pDtpDt
QD
t
ii oo
ave
ave
ii oo
io
−− =
=
−
=′−−′
⎛
⎝
⎜
⎞
⎠
⎟
=−
′
σ
σ
θ
θ
+
Figure D-14. Simple derivation of
(
)
σ
θ
ave
as a function of
Q
Q
t
D
p
t
D
p
oi
=+
′
⎛
⎝
⎜
⎞
⎠
⎟
−−
′
⎛
⎝
⎜
⎞
⎠
⎟
11
(D-128)
then, by combining equations (D-127) and (D-128),
()
σ
θ
ave
QD
t
=−
′
2
(D-129)
which is identical in form to the familiar expression for the hoop stress [(
σ
θ
)
ave
] in
a very thin tube resulting from a uniform external pressure
.
Q
The second benefit to be derived from use of the average circumferential stress
has to do with the expression for the effective stress,
σ
e
. Substituting equation
(D-129) into equation (D-123),
σσσ
ezz
QD
t
QD
t
=
′
⎛
⎝
⎜
⎞
⎠
⎟
++
′
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
≤
22
2
2
1
2
σ
y
(D-130)
Casing/Tubing Design Manual D-43
October 2005
Not only does equation (D-130) have a simple form, but this expression also has
further benefits that can be discerned by solving equation (D-130) for the case of
initial yield when
σ
σ
=
y
,
′
=+−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
D
t
Q
y
z
y
z
y
2
1
2
1
3
4
2
1
2
σ
σ
σ
σ
σ
(D-131)
By comparing equation (D-131) with equation (D-119), either expression can be
written in the form:
YX X=±−
⎡
⎣
⎢
⎤
⎦
⎥
1
2
1
3
4
2
1
2
(D-132)
where:
()
XY
pr
rr
pp
thick cylinder
D
t
Q
thincylinder
zi
y
o
oi
io
y
z
yy
,
,
,
=
+
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−
′
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
σ
σσ
σ
σσ
2
2
2
22
(D-133)
and, therefore, Figure D-13 can also be considered a plot of equation (D-131)
provided the axes are changed according to equation (D-133). Notice from
equation (D-133) that one side benefit of the thin cylinder approximation is the
separation of pressure terms (i.e.,
) from the axial stress term,
Q
σ
z
.
The equivalence of the normalized forms of equations (D-119) and (D-131), as
expressed in equation (D-133), does not mean that the two expressions predict
yield identically. To compare the predictions for yield involves a comparison of
the expressions for effective stress. For thick cylinder theory, the effective stress
at the inner radius is defined by equation (D-80) with the values for
σ
r
and
σ
θ
determined from equations (D-110) and (D-111),
σσ
e
ii oo
oi
z
io
oi
o
pr p r
rr
pp
rr
r=
−
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
22
22
2
22
2
4
3
1
2
(D-134)
For thin cylinder theory, the effective stress is given by equation (D-130).
Consider the relation for
σ
e
given in equation (D-134) to be the standard. The
percent error associated with the approximations of ignoring radial stress and
using the average circumferential stress can be defined as:
()
()
ERROR x=−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 100%
130
134
σ
σ
.
where the subscripts refer to the appropriate equation numbers for
σ
e
from the
respective theories. Table D-5 illustrates calculated values of this error for typical
D-44 Casing/Tubing Design Manual
October 2005
values of , , and
p
i
p
o
σ
z
and a range of diameter:thickness ratios. Notice in the
table that for a wide range of parameters the error associated with the thin
cylinder approximation is less than 10%. The areas where the error exceeds 10%
are blocked for easy reference.
Table D-5. Error using Thin Cylinder Approximation (
σ
z
=
0
)
p
o
/p
i
D/t 0.01 0.1 1.0 10 100 1,000
10 22.2 23.6 0.0 8.9 9.9 10.0
15 14.4 15.3 0.0 5.9 6.6 6.7
20 10.6 11.3 0.0 4.5 4.9 5.0
25 8.4 8.9 0.0 3.6 4.0 4.0
30 7.0 7.4 0.0 3.0 3.3 3.3
The low error associated with the thin cylinder approximation is fortunate in view
of the usefulness of these simplified relations. This accuracy is particularly
important with regard to instances where
vanishes; that is, loading cases
where cross-sectional collapse of the tubular is of concern. Use of the thin
cylinder approximation permits one to analyze collapse from an entirely
theoretical posture that would be impossible if it were necessary to employ the
thick cylinder relations. Conversely, the poorer performance of the thin cylinder
approximation for burst conditions (
p
i
p
o
=
0) is of no great concern because burst
can be analyzed using the equations for a thick cylinder.
D.8 Changing Environmental Conditions
The stresses and strains for a cylindrical tube have been associated with a
specific set of pressure and axial loads. The pressure loads
and will, in
most cases, be computed explicitly from the fluid environment. Conversely, the
axial load will depend not only on the current environment, but may, according to
end constraints, also depend upon changes in the environment. The purpose of
this section is to consider axial loads generated by environmental changes.
p
i
p
o
D.8.1 General Comments
The base equation for the developments that follow is equation (D-56), which
relates axial strain to the three stresses,
σ
r
,
σ
θ
,
σ
z
, and the temperature
change
17
∆T
. Because equation (D-56) is linear in the stress and temperature
change,
17
Only elastic deformations will be considered in this development.
Casing/Tubing Design Manual D-45
October 2005
()
[]
()
()
[]
()
()
[]
∆
∆∆∆ ∆
εε ε σ µσ σ α
σµσσ α
σµσ σ α
θ
θ
θ
z z z z r base
zr base
zr
E
TT
E
TT
E
T
=−= − + + −
−−++−
=−++
21 2 2 2 2
1111
1
1
1
(D-135)
where
∆
σ
σ
σ
=−
21
for each component of stress and
∆
TT T
=
−
21
dz
. That is, the
change in axial strain is related to stress and temperature changes by an
equation identical in form to the stress-strain relation of equation (D-56). The total
length change due to the altered environment is then:
∆∆L
z
=
∫
ε
(D-136)
where
is the total length of the cylinder. The exact form of
L
∆
ε
z
can be
determined by first noting from equations (D-101) and (D-102) that:
∆∆ ∆ ∆
σσ
α
µ
θ
r
C
E
T+= −
−
⎧
⎨
⎩
⎫
⎬
⎭
2
12
1
(D-137)
Using equation (D-108) in equation (D-137):
∆∆
∆∆
σσ
θ
r
ii oo
oi
pr pr
rr
+=
−
−
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
2
22
22
(D-138)
where
∆ for each of the pressure terms. Now substituting equation (D-
138) into equation (D-135),
pp p=−
21
∆∆
∆∆
∆
εσµ α
zz
ii oo
oi
E
pr pr
rr
T=−
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
1
2
22
22
(D-139)
which relates the change in axial strain to changes in axial stress,
∆
σ
z
, internal
pressure, , external pressure, ∆p
i
∆
p
o
, and temperature
∆
T
. If these four
quantities are known functions of the axial coordinate,
, equation (D-139) can
then be substituted into equation (D-136) to determine the change in length of
the cylinder.
z
D.8.2 Change In Length
Assume that at least one end of the cylinder is free to move in an unrestricted
manner. Furthermore, let the functional forms of
σ
z
, , , and
T
be given by:
p
i
p
o
σ
γ
γ
γ
γ
zo z i io i o oo o o T
zp p zp p zTT+=+
=
+
=
+
,, , (D-140)
where the subscript, o, denotes conditions at
zz
=
0
, and
γ
z
,
γ
i
,
γ
o
, and
γ
T
are
constants representing gradients along the direction of the
-axis. z
Using equations (D-139) and (D-140) in equation (D-136),
D-46 Casing/Tubing Design Manual
October 2005
()
(
)
(
)
()
}
∆∆∆
∆∆ ∆ ∆
∆∆
∆∆
∆∆
∆∆
∆∆
L
E
z
E
pzrp z
rr
Tzdz
E
LL
E
pr p r
rr
L
rr
rr
LTL
zo z
io i i oo o o
oi
L
oT
zo z
io i oo o
oi
ii oo
oi
o
=+−
+−+
−
⎧
⎨
⎪
⎩
⎪
++
=+
⎛
⎝
⎜
⎞
⎠
⎟
−
−
−
⎛
⎝
⎜
⎜
+
−
−
⎞
⎠
⎟
⎟
++
∫
1
2
11
2
2
1
2
22
22
0
2
22
22
22
22
2
σγ
µ
γγ
αγ
σγ
µ
γγ
αγ
T
L
2
⎛
⎝
⎜
⎞
⎠
⎟
r
(D-141)
where
∆
γ
γ
γ
ii
=−
2i1
, and similarly for the other gradient terms.
Equation (D-141) can be simplified by computing average quantities along the
length
. For example, the average temperature change along the length is:
L L
()
∆∆ ∆∆ ∆∆T
L
Tdz
L
TzdzT
L
oT o T
L
==+=+
∫∫
11 1
2
00
γγ
L (D-142)
In a similar manner:
∆∆ ∆pp
i
io i
=+
1
2
γ
L
(D-143)
∆∆ ∆pp
o
oo o
=+
1
2
γ
L
(D-144)
Substituting equations (D-142) to (D-144) into (D-141):
()
∆∆ ∆ ∆∆
∆
L
E
LL
E
L
rr
pr pr
LT
zo z
oi
i
i
o
o
=+
⎛
⎝
⎜
⎞
⎠
⎟
−
−
−
+
11
2
2
2
22
22
σγ
µ
α
(D-145)
D.8.3 Tapered Strings
In the work of the previous two sections, it was assumed that the quantities
γ
z
,
γ
i
,
γ
o
,
γ
T
, r and r are constant over the length of the cylinder. Should any of
these quantities change, the integrations used to arrive at equation (D-145)
would have to either be recalculated or incremented into a series of integrals
where the above quantities are constant over each increment. Taking the latter
approach, equation (D-136) can be written,
i o
∆∆
∆∆ ∆
∆∆ ∆
∆
Ldz
dz dz dz
LL L
L
z
L
z
z
z
z
z
z
z
L
n
i
i
n
n
=
=+++
=+++
=
∫
∫∫ ∫
∑
−
=
ε
εε ε
0
0
12
1
1
1
2
1
K
K
(D-146)
Casing/Tubing Design Manual D-47
October 2005
where the quantities
γ
z
,
γ
i
,
γ
o
,
γ
T
, and are constant over each increment
, and each length change,
r
i
r
o
LL
ii
−
−1
∆
L
i
, is given by equation (D-145) with
appropriate subscripts on the pressure and temperature terms. Explicitly, for a
tapered string consisting of
n segments,
()
()
∆∆
∆∆
∆∆ ∆
LL
L
E
E
rr
pr p r T L
j
j
n
zo w
j
j
n
oi
j
i
i
o
o
j
j
j
=
=
+
⎛
⎝
⎜
⎞
⎠
⎟
⎧
⎨
⎪
⎪
⎩
⎪
⎪
−
−
−+
⎫
⎬
⎪
⎭
⎪
=
=
∑
∑
1
1
22
22
1
2
2
σγ
µ
α
(D-147)
where
, and
Lzz
jjj
=−
−1
z
n
= L z
1
0
=
.
D.8.4 Bending Stresses
All previous derivations have been centered on the assumption of axisymmetric
deformation. The addition of bending complicates the analysis in that the
deformation is no longer axisymmetric. However, as long as the displacements
and displacement gradients remain infinitesimal, and as long as the cylinder
material remains elastic, the effects of bending can be superimposed upon
previous results.
D-48 Casing/Tubing Design Manual
October 2005
A
A
B
B
C
C
D
D
E
E
F
F
A
B
C
D
E
F
Z
Y
ds
ds’
Neutral
Axis
a. Before bending
b. After bending
c. Exploded view of Segment
A-B-C-D-E-F. Radius of
curvature to F-C is R.
Table D-6. Deformation of a Tube during Bending
In analyzing bending, the assumptions of classic beam theory will be used to
describe the deformation of the tube. The primary restriction placed on the
deformation is that plane cross sections remain plane during bending. As
indicated in Figure D-15, this assumption implies that cross sections, such as A-
F-E or D-C-D, that are straight before bending will remain straight during
bending. However, the distance between cross sections will vary in the plane of
bending. As indicated in the figure, some longitudinal fibers (line segment A-B)
will elongate, while other longitudinal fibers such (line segment E-D) will contract.
This implies that there exists a plane perpendicular to the plane of bending for
which neither extension nor contraction occurs. This so-called neutral axis is
indicated in Figure D-15 by the line segment F-C. Placing a local
- coordinate
system on the neutral axis (see Figure D-15.c), the strain at any distance
from
the neutral axis is:
z y
y
(
)
ε
θθ
θ
z
ds ds
ds
Ryd Rd
Rd
y
R
=
′
−
=
+−
=
(D-148)
where
is the local radius of curvature of the neutral axis and d
θ
is the
infinitesimal angle subtended by the arc F-C. From equation (D-56),
R
σε
zb z
E
Ey
R
==
(D-149)
Casing/Tubing Design Manual D-49
October 2005
where the subscript
b
is used to emphasize that the axial stress is solely due to
bending effects. Because the resultant force on any cross section due entirely to
a bending moment will vanish,
FdA
E
R
ydA
E
R
Ay
zb
AA
== =
∫∫
σ
0= (D-150)
where
y
is the distance from the neutral axis to the centroid axis of the cross
section. The fact that equation (D-150) will vanish if and only if
y
=
0
indicates
that the neutral axis and centroid axis coincide.
The resultant moment acting on the cross section is:
MydA
E
R
ydA
EI
R
zb
AA
== =
∫∫
σ
2
(D-151)
where
I
is the moment of inertia of the cross section about the neutral axis.
Using equation (D-151) in equation (D-149),
σ
zb
Ey
R
Ey
M
EI
My
I
== =
1
(D-152)
which gives the axial stress due to bending at any position y in the cross-section.
Notice that
σ
zb
varies linearly through the cross-section having a maximum
magnitude at
. yr
o
=±
The fact that
σ
zb
has its maximum value at the outer radius alters one important
conclusion reached previously with regard to initial yield. First, it is important to
realize that all of the axisymmetric expressions derived previously in terms of
σ
z
remain valid in the presence of bending if one makes the substitution:
σ
σ
σ
zzazb
=
+
(D-153)
where
σ
za
is the contribution to axial stress from axisymmetric loads and
σ
zb
is
given by equation (D-152). In particular, equation (D-117) becomes:
() ()
()
d
dr
C
ET
C
r
C
ET
C
r
d
dr
C
ET
d
dr
za zb za zb
za zb
zb zb
1
2
2
2
4
2
1
2
2
5
1
12
32
12
12 2 2
12
−
−
⎛
⎝
⎜
⎞
⎠
⎟
+++ −−
−
⎛
⎝
⎜
⎞
⎠
⎟
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=− + + − −
−
⎛
⎝
⎜
⎞
⎠
⎟
α
µ
σσ
α
µ
σσ
σσ
σ
α
µ
σ
∆∆
∆
(D-154)
which, if one studies the plane of bending where
y
r
=
, becomes:
[]
()
d
dr
C
r
C
ET M
I
za zb
=− + + − −
−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
12 2
12
2
2
5
1
σσ
α
µ
∆
(D-155)
an expression which implies that the maximum value of
σ
y
is located at:
()
r
C
C
ET M
I
za zb
=+−−
−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
1
6
12
2
21
1
5
σσ
α
µ
∆
(D-156)
D-50 Casing/Tubing Design Manual
October 2005
In fact, according to the value of the external moment,
M
, initial yield can occur
at any radius between
and . r
i
r
o
D.9 References
1. American Petroleum Institute, Specification for Casing and Tubing, API
Specification 5CT (SPEC 5CT), Latest Edition.
2. Eringen, A. C.: Mechanics of Continua, John Wiley & Sons, Inc., New York,
(1967) 13-18.
3. Fung, Y. C.: Foundations of Solid Mechanics, Prentice-Hall, Englewood
Cliffs, New Jersey (1965) 131-153.
4. Kachanov, L. M.: Foundations of the Theory of Plasticity, North-Holland,
London (1971).
5. Needleman, A.: "Post-Bifurcation Behavior and Imperfection Sensitivity of
Elastic-Plastic Circular Plates," International Journal of Mechanical Sciences,
(1975) v. 17, 1-13.
6. Prager, W. and Hodge, P. G., Jr.: Theory of Perfectly Plastic Solids, John
Wiley & Sons, Inc., New York City (1951).
Casing/Tubing Design Manual D-51
October 2005