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

1 Copyright © 2018 ASME

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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.



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



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)



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



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



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



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



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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