Allowable Compressive Stress Rules in the ASME...
Transcript of Allowable Compressive Stress Rules in the ASME...
ALLOWABLE COMPRESSIVE STRESS RULES IN THE ASME BOILER AND PRESSURE VESSEL CODE, SECTION VIII, IN THE CREEP REGIME
Maan Jawad
Global Engineering & Technology
Camas, Washington, U.S.A.
ABSTRACT
This paper outlines several procedures for
developing allowable compressive stress rules in the
creep regime (time dependent regime). The rules are
intended for the ASME Boiler and Pressure Vessel
codes (Sections I and VIII). The proposed rules
extend the methodology presently outlined in
Sections I, II-D, and VIII of the ASME code for
temperatures below the creep regime into
temperatures where creep is a consideration.
1. INTRODUCTION
The 2017 edition of the ASME Boiler and Pressure
Vessel Code Sections I, III, and VIII limits the
calculations for allowable compressive stress to
temperatures below the creep regime (time-
independent regime). However, a number of research
articles recently published [1, 2, and 3] demonstrated
the feasibility of extending the allowable
compressive stress in the ASME code into the creep
regime (time-dependent regime). Presently the
ASME code is developing rules for allowable
compressive stress in the creep regime and this paper
summarizes some of the options available.
2. ISOCHRONOUS CURVES
The buckling equations at temperatures below the
creep regime are based on stress-strain curves.
However, at temperatures in the creep range the
stress-strain curves cannot be used since the stress
relaxes at any given strain. Accordingly, isochronous
curves form the bases for developing the buckling
equations. The isochronous curves at a given
temperature are obtained in a two – step process.
First, tensile specimens are stressed to a given level
and strain is measured as a function of time.
Different tensile specimens are used at various stress
levels. Curves are then plotted in terms of time
versus strain for various stress values. The second
step is to enter the chart at a given time and plot
stress versus strain. These pseudo stress – strain
curves [4, 5, and 6] are called isochronous curves.
Figure 1 shows Average Isochronous Curves [3] for
9Cr-1Mo-V steel at 1000oF. The isochronous curves
are used as pseudo stress-strain curves [2] to form
the basis for developing compressive stress needed in
the design of cylindrical and spherical shells under
axial compression and external pressure.
Proceedings of the ASME 2018 Symposium on Elevated Temperature Application of Materials for Fossil, Nuclear, and Petrochemical Industries
ETAM2018 April 3-5, 2018, Seattle, WA, USA
ETAM2018-6737
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Figure 1. Average isochronous curves for 9Cr-1Mo-
V at 1000oF [3].
Two methodologies are being considered by ASME
for obtaining equations and/or charts for use in
compressive stress calculations. These are the Strain
Method and the Remaining Life Method. Both of
these methods are discussed next.
3. STRAIN METHOD FOR CALCULATING
COMPRESSIVE STRESS
This method is based on isochronous curves found
originally in Section III-NH of the ASME code.
Section III-NH has been deleted from ASME as of
2017. The isochronous curves now reside in two
placed. The first is in Division 5 of Section III for
nuclear applications and the second is in Section II-D
for nonnuclear applications.
Calculations for allowable compressive stress require
either External Pressure Charts or External Pressure
Equations. There are three methods for developing
external pressure charts/curves in the creep regime
using the strain method. The first is plotting External
Pressure Charts from the isochronous curves. The
second is representing the charts with equations. And
the third method is by representing the isochronous
curves by equations then then taking the derivative of
equations to obtain tangent moduli from which
external pressure curves/equations are developed.
Details of these three methods are discussed below.
3.1 Method 1. External Pressure Charts (EPC).
The External Pressure Charts (EPC) are probably the
most widely used method for determining
compressive stress in shells. The EPC in the creep
regime are constructed from isochronous curves. The
ASME procedure for constructing the EPC consists
of reducing the Average stress-strain Isochronous
Curves by 20% to obtain Minimum Isochronous
Curves. The resulting minimum curves are then used
to plot external pressure curves by finding the
tangent moduli, Et, from the relationship d/d at
various stress values and then calculating
corresponding strains, called Factors A, from the
relationship A = /Et. The tangent modulus is
normally obtained from the isochronous curves by
either a graphical method or by a finite difference
procedure. The resulting curves are plotted as
compressive stress, Sc, versus Factor A. Figure 2
shows external pressure curves for 9Cr-1Mo-V steel
at 1000oF with various time periods.
One advantage of using the EPC is it gives the
designer an overall view of the trend of the stresses
which helps in the design process. One disadvantage
is the difficulty of interpolating between various
points on the log-log chart.
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Figure 2. External Pressure Chart at 1000
oF [3].
3.2. Method 2. Equations Representing EPC
The curves in the external pressure chart may be
represented by equations. The advantage of using
equations is the consistency of getting answers by
bypassing the process of reading the log-log charts.
As an example, the following equations are obtained
by regression analysis for the 100,000 hour curve in
Figure 2.
100,000 hour curve for 9Cr-1Mo-V steel
= E 0 < ≤ 0.000158 (1a)
= (G1 + G2 + G32)/(1 + G4 + G5
2)
0.000158 < ≤ 0.1 (1b)
Where,
G1 = 4.2211 G2 = 12,546.55
G3 = 540,532.9 G4 = 763.8523
G5 = 25,414.4
A plot of Eq.(1) is shown in Figure 3 together with
the actual curve for 100,000 hours.
Figure 3. Comparison of Eq.(1) and actual
compression curve for 9Cr-1Mo-V steel at 100F and
100,000 hours.
The knee in Figure 3 is due to merging of the
constant elastic modulus with the variable
isochronous curve at 6 ksi.
3.3. Method 3. Equations Representing
Isochronous Curves.
Individual External Pressure Curves correlating A to
Sc may also be obtained directly from available
Average Isochronous Curves. The procedure consists
of starting with an Average Isochronous stress-strain
Curve and reducing it by 20% to obtain a Minimum
Isochronous Curve. Then regression analysis is used
to find an equation simulating the minimum
isochronous curve. The d/d derivative of this
equation represents the tangent modulus Et from
which Factor A is determined from the relationship
A = /Et. An External Pressure Curve is then plotted
as a function of A versus Sc. This can be
demonstrated by taking the 100,000 hour
isochronous curve in Figure 1 and developing the
following equations which represent the 80% stress
level
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100,000 hour curve for 9Cr-1Mo-V steel
= E 0 < ≤ 0.000158 (2a)
= (G6 + G7 + G82)/(1 + G9 + G10
2)
0.000158 < ≤ 0.09 (2b)
= 50 + 15.5 0.09 < ≤ 0.1 (2c)
Where,
G6 = -1.6468 G7 = 46,186.61
G8 = 841,802.20 G9 = 2340.95
G10 = 26,013.74
A plot of Equations (2) results in a curve that is
identical to the curve in Figure 1 for 100,000 hours.
The derivatives of Equations (2) are
Et = E = 25,300 ksi 0 < ≤ 0.000158
(3a)
Et = (G13 / G12 )- (G11 G14 / G122)
0.000158 < ≤ 0.09 (3b)
Et = 50 0.09 < ≤ 0.1 (3c)
Where,
G11 = G6 + G7 + G82
G12 = 1 + G9 + G102
G13 = G7 + 2G8
G14 = G9 + 2G10
The factor A for the external pressure curve is
obtained from
A = /Et (4)
The external pressure curve correlating Factor A to
St is obtained by calculating Factor A for various
values of and plotting Factor A versus Sc on a log-
log graph. Figure 4 shows the result of the curve
obtained from Equation (3) together with the curve
from Figure 2 for 9Cr-1Mo-V steel at 1000oF and
100,000 hours.
One advantage of using this method is it bypasses the
laborious process of constructing an external
pressure chart.
Figure 4. Comparison of External Pressure Curves
for 9Cr-1Mo-V steel at 1000oF and 100,000 hours
obtained from Equation (3) and from Figure 2.
4. REMAINING – LIFE METHOD FOR
CALCULATING COMPRESSIVE STRESS
This methodology is based on Publication API 579-
1/ASME FFS-1 [6] which list equations for
constructing average isochronous curves for various
materials at any time and temperature in the creep
regime. One advantage of this method is the ability
to draw isochronous curves at any given time and
temperature compared to the Strain Method where
only specific times and temperatures are listed. A
brief description of this methodology is shown next.
4.1 Equations for Isochronous Curves
The isochronous curves are generated from the
following equations
t = e + c + p (5)
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Where, t is total strain, e is elastic strain, c is creep
strain, and p is plastic strain.
The equation for creep strain c may be written as [7]
= co e(m+p+c) (6) Where, m is a modified Norton’s component relating
stress to strain, p represents microstructural damage,
and c represents other factors. The quantity co is
initial strain rate.
Define
= m+p+c (7)
Then Equation (6) becomes
= co e (8)
This equation can be rearranged as
d/(e ) = co dT (9)
Integrating this equation using the limits 0 to and 0
to T gives
T = [1/ (co ( 1- e- ) (10)
or, in terms of creep strain,
c = - (1/) ln( 1- co T) (11)
Equation (8) is used by FFS1 to construct
isochronous curves. The quantities and co define
isochronous curves for various materials at various
temperatures and times. FFS1 uses the following
polynomials for and co to define the isochronous
curves Log10(co) = - [ (Ao +
sr) + (A1 + A2S1 + A3 S1
2 + A4 S13)/(460 + Te)] (12)
Log10() = [ (Bo + cd) + (B1 + B2 S1 +
B3 S12 + B4 S1
3)/(460 + Te)] (13)
S1 = log10() (14)
Constants Ai and Bi are based on actual data and are
tabulated in Reference [6] for various materials. It
must be noted that the data base for these materials
may be different than that in Section II-D of ASME.
The quantities sr and
cd are material scatter and
material ductility factors listed in Reference [6].
Various evaluations made by the author for 2.25Cr-
1Mo annealed steel at 1000oF have shown that the
external pressure curve obtained from API 579-
1/ASME FFS-1 [6] data base is essentially the same
as that obtained from the ASME III-NH [5] data base
when sr is taken as -0.50 and
cd is taken as 0.0.
The equation for elastic strain e is defined by
e = /Ey (15)
Where, Ey is modulus of elasticity and is stress.
The equation for plastic strain p is expressed as
p = 1 + 2 (16)
The expressions for 1 and 2 are fairly lengthy and
involve many constants as a function of yield and
tensile given in FFS1. Substituting the various
constants in the expression of 1 gives
1 = 0.5(t/3)(1/2){ 1 – tanh[2(t – 5)/6]} (17) Where,
1 = R = ys/ult
2 = m1 = [ln(R) + (’p – ys)]/{ln[(ln(1 +
’p))/(ln(1 + ys))]}
3 = A1 = [ys(1 + ys)]/[ ln(1+ys)]m1
4 = K = 1.5R1.5
– 0.5R2.5
– R3.5
5 = ys + K(ult – ys )
6 = K(ult – ys )
Similarly the expression for 2 is
2 = 0.5(t/8)(1/7){ 1 – tanh[2(t – 5)/6]} (18) Where,
7 = m2
8 = A2 = uts em2/(m2
m2)
There is a temperature limitation on ’p and m2. For
example, the values for ’p and m2 for ferrous steels
are limited to 900oF.
Hence, isochronous curves may be drawn from
Equation (5) where c is obtained from Equations
(11) – (14), e is obtained from Equation (15), and p
is obtained from Equations (16) – (18).
4.2 Equations for External Pressure Lines
The equations for external pressure lines are obtained by taking the derivative of Equation (5) to find the tangent modulus Et. The tangent modulus is obtained from Eq.(5) as follows
dt/d = de/d + dc/d + dp/d
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and
Et = d/dt = 1/( dt/d) = 1/( de/d +
dc/ddp/d ) (19)
The quantity de/d is calculated from Eq.(15) as
de/d = 1/Ey (20)
The quantity dc/d is obtained by taking the
derivative of Eq.(11) using a recursive computer
program with symbolic solution. The result is
dc/d = C10/C7 + C8/K1 (21) where,
A00 = A0 + sr
B00 = B0 + cd
K1 = 460 + Te
S1 = log10(0.8)
C1 = B00+(B1+ B2*S1 + B3*S12
+ B4*S13 )/K1
C2 =A00+(A1 + A2*S1+ A3*S12
+ A4*S13 )/K1
C3 = B2 + 2*B3*S1 + 3*B4*S12
C4 = A2 + 2*A3*S1 + 3*A4*S12
C5 = 10C1*(1/10C2)*T*ln(10)*C3 C6 = ln(10)*ln[ 1 - 10C1*(1/10C2)*T] C7 = 10C1*(1/10C2)*T - 1 C8 = (1/10C1)*C6*C4 C9 = C5/K1 -10C1*(1/10C2)*T*ln(10)*C4)/K1 C10 = (1/10C1)*C9
Where, Ci are terms of the strain derivative d/d.
The quantity dp/d is obtained by taking the
derivative of Eq.(16) using a recursive computer
program with symbolic solution. The result is
dp/d = d1/d + d2/d (22) where,
d1/d = (226t)-1{(t/3)(1/2) [
tanh((25 – 2t)/6)+ 1][ 6 – 22t +
22t tanh((25 – 2t)/6)]}
d2/d = (- 267t)-1{(t/8)(1/7) [
tanh((25 – 2t)/6) - 1][ 6 + 27t +
27t tanh((25 – 2t)/6)]} Equations (20) through (22) are substituted into
Equation (19) to determine the tangent modulus Et
for various t values. Factor A for constructing
external pressure line is obtained from Equation (4)
as
A = /Et
The external pressure curve correlating A to St is
obtained by calculating A for various values of and
plotting the result on log-log graph correlating Sc and
Factor A.
This remaining Life method has a great advantage
over the Strain Method in that the External Pressure
Curve can be drawn directly for any given time and
temperature in the creep range. For example, the
External Pressure Curve for 304 type stainless steel
at 87,600 hours (ten years) at 1328oF is obtained
from Equation (22) and is shown in Figure 5. And
while such a curve may be obtained from the Elastic
method, its development will require extensive time-
consuming interpolations and the results will not be
as accurate.
Figure 5. External pressure chart for 304 type
stainless steel at 1328oF and 87,600 hours.
5. PROPOSED PROCEDURE FOR CALCULATING ALLOWABLE AXIAL COMPRESSION IN CYLINDRICAL SHELLS IN THE CREEP RANGE
The elastic critical in-plane buckling equation for
long cylindrical shells [2] under axial compression is
given by
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crL = 0.63 E/(Ro/t) (23) Where, Ro is the outside radius.
Experimental data have shown [2] that buckling may
occur at a value of one-tenth of that calculated from
Equation (23) for large R/ t ratios. Accordingly, a
knock-down factor of 5.0 is used by ASME for the
effect of geometrical imperfections on axial
compression and Equation (23) becomes
= 0.125 E/(Ro/t) (24) The elastic strain is then given by
e = /E = 0.125 /(Ro/t) (25) For inelastic buckling, the classical equation for
buckling stress [2] is of the form
crL =[EsEt/(3(1 – 2))]0.5/(Ro/t) (26) Where,
= 0.5 – (0.5 –)(Es/E) For design purposes Equation (26) is simplified by
conservatively letting E = Es = Et. This results in the
equation
pcrL = 0.63 Et/(Ro/t) (27)
Using a knock-down factor of 5.0, Equation (27) can
be written as
p = 0.125 /(Ro/t) (28)
In addition to the knock-down factor of 5, a design
factor, DF, of 2.0 is used in order to take into account
such items as reduced modulus in the inelastic
regime and variation of material properties that cause
tests to deviate from theory. In temperatures below
the creep regime the design factor is set to 2.0. In the
creep regime it has been suggested to use a design
factor that varies with time [2] in accordance with
the following relationship
DF = 2.0 T ≤ 1.0 hour (29a)
2 DF = –––––––––––––– 1 + 0.0869 ln(T) 1 < T ≤ 100,000 hours (29b) DF = 1.0 T > 100,000 hours (29c)
The allowable compressive stress in long cylindrical
shells for a given temperature, outside radius Ro, and
operating time T is obtained by assuming a thickness
t and calculating factor A based on Equations (25)
and (28) from the equation
A = 0.125/(Ro/t) (30)
With a known value of A, compressive stress, Sc , is
calculated at a given temperature from any of the
methods given in Sections 3 and 4 for two time
periods. The first time period is at hot tensile (HT is
equal to or less than one hour) and the second is at
the operating time period specified. Then for each of
the two time periods calculated the allowable
compressive stress B using the smaller value
obtained from the equations
B = AE/(DF) (31)
and
B = Sc/(DF) (32)
The applied compressive stress shall be less than the
lower value of allowable compressive stress B. If
not, a new thickness and/or operating time are
chosen and the above procedure is repeated.
Equation (23) assumes a long cylinder. A more
accurate equation that takes length, L, into
consideration is presently in ASME Section VIII,
Division 2, for the time-independent regime. Various
ASME committees are presently evaluating this
equation for use in the creep regime.
6. PROPOSED PROCEDURE FOR
CALCULATING ALLOWABLE EXTERNAL
PRESSURE ON CYLINDRICAL SHELLS IN
THE CREEP RANGE
The allowable external pressure in the creep range can be
obtained for any cylindrical shells with a given
temperature, operating time T, outside diameter Do , and
given length L . The thickness t is assumed and Factor A
is obtained from the geometric Figure G of ASME BPV
II, Part D or from equations simulating the chart. With
Factor A known, the compressive stress Sc is determined
from any of the methods discussed in Sections 3 and 4
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above for two time periods. The first time period is at less
than one hour and the second is at the operating time
period specified. Then the design factor DF is determined
from the equations
DF = 3.0 T ≤ 1.0 hour (33a)
3
DF = ––––––––––––––
1 + 0.0434 ln(T)
1 < T ≤ 100,000 hours (33b)
DF = 2.0 T > 100,000 hours (33c)
For each of the two time periods mentioned above,
the allowable external pressure P is calculated from
Equations (34) and (35).
2AE
P = –––––––– (34)
(DF)(Do/t)
and,
2Sc
P = –––––––– (35)
(DF)(Do/t)
The actual external pressure shall be less than the
lowest value of calculated external pressure. If not, a
new thickness and/or operating time are chosen and
the above steps are repeated.
7. PROPOSED PROCEDURE FOR CALCULATING AXIAL COMPRESSION IN COLUMNS (EULER BUCKLING) IN THE CREEP RANGE.
The allowable axial stress on a column with an
effective length L, outside radius Ro , thickness t, and
operating time Te is obtained by first calculating
factor A from the equation
A = 2 / (L/r)
2 (36)
The compressive stress, Sc , is calculated from any of
the methods given in Sections 3 and 4 above for two
time periods. The first time period is for less than
one hour and the second at the operating time period
specified. The suggested design factor DF is taken as
1.90. For each of the two time periods above, the
allowable compressive stress c is calculated from
Eqs. (32) and (33) and the smaller value is used.
c = AE/(DF) (37)
and
c = Sc /(DF) (38)
The actual calculated compressive stress shall be less
than the lowest value of allowable compressive stress
c. If not, a new thickness and/or operating time are
chosen and the steps above are repeated.
8. EXAMPLE
A stainless steel cylindrical shell has Do = 100 inch,
L = 160 inch, and t = 0.25 inch. What is the
allowable external pressure for a life service of
87,600 hours at 1328oF?
Solution:
L/Do = 1.60 Do/t = 400
From the Geometric Chart in ASME Section II, Part
D, A = 0.0001
From Figure 5, Sc = 1000 psi
From Equation (33b),
3
DF = –––––––––––––––––– = 2.01
1 + 0.0434 ln(87,600)
From Equation (35), the allowable external pressure
is
2Sc 2(1000)
P = –––––––– = ––––––––––– = 2.5 psi
(DF)(Do/t) (2.01)(400)
It is of interest to note that existing EPC in ASME
Section II, Part D, based on Hot Tensile tests with no
consideration to operating life gives an allowable
external pressure value of about 3.3 psi at 1328oF.
Hence, about 30% reduction in the allowable
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external pressure must be considered when the shell
is operating in the creep regime for an extended
period of time.
9. ACKNOWLEDGEMENT Special thanks to Dr. Kevin Jawad for supplying the
derivatives in Section 4.2.
10. NOMENCLATURE
A = factor in External Pressure Charts
Ao – A4 = material factors
i = values of various stress and strain
functions of various materials
B = allowable compressive stress
Bo – B4 = material factors
Ci = terms of the creep strain derivative
dc/d. c = constant representing various factors in
the Norton equation.
Do = outside diameter
cd = material ductility factor. It ranges
from -0.50 to +0.50.
sr = material scatter factor. It ranges from
-0.50 to +0.50.
E = Young’s modulus of elasticity
Es = secant modulus, /
Et = tangent modulus, d/d
= strain
c = creep strain
e = elastic strain
p = inelastic strain
t = total strain
= strain rate
co = initial strain rate
1, 2 = components of plastic strain
L = length of cylinder
m = modified Norton’s component relating
stress to strain
=Poisson’s ratio
p = constant representing microstructural
damage in the Norton equation.
R = ys/ult Ro = outside radius
r = radius of gyration
S1 = log10()
Sc = compressive stress
= stress crL = elastic buckling stress
pcrL = inelastic buckling stress
ult = tensile strength
ys = yield stress T = time, hours
Te = temperature, F
t = thickness
11. REFERENCES
[1] P. Carter and D. Marriott, 2008. “Comparison and
validation of creep buckling analysis methods” STP-
PT-022. ASME Standards Technology, LLC.
[2] M. Jawad and D. Griffin, 2011. “External
pressure design in the creep range” STP-PT-029.
ASME Standards Technology, LLC.
[3] M. Jawad, R. Swindeman, M. Swindeman, and
D. Griffin, 2016. “Development of average
isochronous stress-strain curves and equations and
external pressure charts and equations for 9Cr-1Mo-
V steel” STP-PT-080. ASME Standards Technology,
LLC.
[4] M. Jawad and R. Jetter, 2011,” Design and
analysis of ASME boiler and pressure vessel
components in the creep range”. ASME Press.
[5] ASME Boiler and Pressure Vessel Code, Section
III, Subsection NH, 2015. ”Class 1 components in
Elevated Temperature Service”. American Society of
Mechanical Engineers.
[6]”Fitness-for-Service”, 2016. API 579-1/ASME
FFS-1. American Society of Mechanical Engineers.
[7] M. Prager, 1995. “Development of the MPC
Omega Method for Life Assessment in the Creep
Range” Journal of Pressure Vessel Technology,
Volume 117. American Society of Mechanical
Engineers.
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