Download pdf - 42 MPA-SEMINAR 2016, October 4 and 5 2016 · were included in the 2013 Editions of the German KTA 3201.2 and 3211.2. The background for these corrections is that the failure of thin-walled

42 MPA-SEMINAR 2016, October 4 and 5 2016

TÜV SÜD Energietechnik GmbH

Plastic buckling of thin-walled pipes with D/t > 50 and conclusions

for the nuclear design codesHenry Schau, Angelika Huber

TÜV SÜD Energietechnik GmbH, Mannheim, Germany

42nd MPA-Seminar

October 4th and 5th 2016


TÜV SÜD Energietechnik GmbH Folie 2

Abstract

The B2 stress indices of thin-walled straight pipes with D/t-ratios (D - outside diameter, t - wall thickness)

up to 140 are determined using nonlinear FE analyses. The analyses were performed for pipes made of

ideal elastic-plastic materials and for a few chosen steels. For the calculation of the B2 indices an earlier

derived equation is used. The analyses show that the type of imperfection, the D/t ratio, the modulus of

elasticity and the yield stress have significant influence on the B2 indices. Straight pipes with D/t ≤ 40 fail

with a simple kink. Pipes with D/t > 40 fail (mostly) by plastic buckling with some wrinkles. The transition

between both cases is not defined clearly. The numerical failure shapes are in good agreement with the

experimentally obtained shapes. For pipes with D/t > 60…80 and an out of roundness around of 0.5% the

nominal bending stress is for higher yield stresses at the point of instability smaller than the yield stress. In

these cases, the failure occurs by local (plastic) buckling and in the elastic range of the nominal bending

stress with small rotation angles. This is in agreement with the experimental results in the literature.

Curves for the dependence of B2 indices on the type of imperfection, the D/t ratio, the modulus of elasticity

and the yield stress are given. In general, there is a good formal agreement between the calculated values

and the values of the ASME Code. The analyses indicate a different temperature dependence of the B2

indices. Finally, the influence of the secondary stresses from disabled thermal expansion and anchor

movements on failure by buckling and on the safety of piping systems with D/t > 50 is discussed.



1. Introduction

The design of piping components for primary loads in modern nuclear codes, such as the ASME Boiler and

Pressure Vessel Code (further the ASME Code) [1], the French RCC-M [2] and the German KTA 3201.2 [3]

and 3211.2 [4], are based on the formula

B1 ∙D ∙ p

2 ∙ t+ B2 ∙

D ∙ M

2 ∙ I≤ σallow with I =

π

64∙ D4 − d4 (1)

B1 and B2 are the (primary) stress indices for the internal pressure p and the primary bending moment M

(d - inside diameter, I - moment of inertia, σallow - allowable stress). Initially, the stress indices were only

valid for piping components with dimensions D/t ≤ 50. With the 2008 Addendum to the 2007 Edition of the

ASME Code the range for the dimensions was extended to D/t ≤ 100. For the range 50 < D/t ≤ 100 the B1

indices in Table NB-3681(a)-1 are valid and the B2 indices should be multiplied by a correction factor

1/(X·Y) given in the NB-3683.2 (c) (T - design temperature in °C):

X = 1.3 − 0.006 ∙ ΤD t (2)

Y = 1.0224 − 0.000594 ∙ T for ferritic materials and

Y = 1.0 for other materials

The same corrections are given in the NC-3673.2 (class 2) und ND-3673.2 (class 3). These enhancements

were included in the 2013 Editions of the German KTA 3201.2 and 3211.2. The background for these

corrections is that the failure of thin-walled piping with D/t > 50 occurs predominantly by (local) plastic

buckling. Therefore, the failure and particularly the (plastic) buckling of straight thin-walled pipes are

studied using static and dynamic nonlinear FE analyses. The stress design formulas in the nuclear codes

will be discussed for the case of thin-walled pipes with D/t > 50.



2. Determination of the B2 indices and FE Analyses

The ASME Code does not give an appropriate method for determining the stress indices. A new method

to calculate the B2 stress index is given by Matzen and Tan [5]. In [6] this method was applied on thin-

walled straight pipes with D/t > 50. For an ideal elastic-plastic material the following equation for the

calculation of the B2 indices from the instability moment is given:

B2 =σy ∙ Zpl

MIL=Mpl

MILwith Zpl =

1

6∙ D3 − d3 (3)

(MIL - instability moment, σy - yield stress)

The instability moment MIL is defined in the NB-3213.26 of the ASME Code. For materials not exhibiting a

definite yield point the 0.2% offset yield stress (proof stress) is used. This yield stress is rather an

empirical approach and from a mechanical or theoretical point of view difficult to justify. To verify this

approach, the dependence of the B2 index on the D/t ratio is to be determined. The obtained B2 indices

must converge to 1.0 for small D/t ratios. Otherwise, the yield stress is to be corrected.

It is noted, that the ASME Code uses a similar method to derive the factor X in Equation (2). For this

purpose, the data for Mmax/Mpl from the General Electric Report [7] were used. The factor Y is based on

the temperature dependencies of the allowable compressive stresses in the Section II, Part D, Subpart 3

of the ASME Code and of the yield stresses.



The analyses were performed with the FE program Abaqus [8]. Geometric nonlinearities (large

displacement and strain) are taken into account. The FE models use S4 shell elements and are symmetric

to the plane of bending. The mesh is downsized axially to the centre. The considered pipes have an

outside diameter of 200 mm and lengths of 1000 mm, 2000 mm, 4000 mm and 8000 mm. The D/t ratio

varies from 20 to 140. The following imperfections are used in the analyses:

- scaled buckling modes from elastic buckling analyses

- analytically defined single dents

The single dents are defined by the following expression:

δr = α ∙ cos πz

λ+ 1 ∙ cos β ∙ φ (4)

r is the radial deviation from the ideal circular geometry (z - axial coordinate from the middle, -

circumferential angle to the plane of bending). The parameter is the half wavelength for elastic

buckling given by Timoshenko and Gere [9]. The geometric parameter is derived from the numerical

obtained elastic buckling modes. The size of the local deviation from the ideal circular cross section is

characterized by the out of roundness:

O = ΤDmax − Dmin D ∙ 100% (5)

(Dmin, Dmax - minimum and maximum outside diameter)

The ASME Code, Section I, PG-5.4.2 states “Pipe having a tolerance of ±1% on either the O.D. or the

I.D.”. The KTA 3201.2 and 3211.2 specify a maximal deviation of 2% for pipes. The ASTM A530 gives a

maximal out of roundness of 1.5% for thin-walled pipes with D/t > 33. The EN ISO 1127 specifies

tolerances for the outer diameter between 0.5 and 1.5%. Consequently, a local out of roundness of

about 0.5…1% is for pipes a usual technical tolerance.



For the following calculations the ideal elastic-plastic material model and the real material models for some

steels are used. Table 1 shows the material properties for these steels (E - modulus of elasticity, σu -

ultimate tensile strength).

Table 1: Important material properties of the steels

Material E in GPa σy in MPa σu in MPa Material data

X6CrNiNb18-10 200 240 588 Mutz [10]

AISI 304L 205 272 592 Wilkins [11]

22NiMoCr3-7 210 468 620 Bernauer et al. [12]

20MnMoNi5-5 205 506 666 Reicherter [13]



3. Results

For the first calculations pipes made of ideal elastic-plastic material are used. Figures 1 and 2 show

typical failure modes for pipes (length L = 2000 mm, E = 200 GPa, σy = 500 MPa, single dent with O =

0.5%) with D/t = 30 and 90. The pipe with D/t = 90 fails by plastic buckling with multiple wrinkles. These

shapes are in a good agreement with the shapes in [14-18].

Figure 1:

Failure mode for an imperfect, ideal elastic-plastic

pipe with D/t = 30

Figure 2:

Failure mode for an imperfect, ideal elastic-plastic pipe

with D/t = 90



Figure 3 displays experimental data (Mmax - maximum moment) for ferritic pipes and the linear approach

(green line) from the General Electric Report [7]. The yield stresses are between 336 and 586 MPa. The

linear fit of the data in the range of 39 ≤ D/t ≤ 114.6 gives:

ΤMmax Mpl = 1.3089 − 0.0053 ∙ ΤD t (6)

Figure 4 shows the numerical results for an ideal elastic-plastic pipe with a yield stress of 500 MPa for

various imperfections. The slopes from -0.0036 to -0.0041 for the curves in the range 40 ≤ D/t ≤ 120 with a

yield stress of 500 MPa and O = 0.5% are in a good agreement with the slope of -0.0053 of the

experimental data in Figure 3. Therefore, further studies are mostly performed for pipes having a single

dent O = 0.5% as an imperfection.

Figure 3:

Experimental data

for Mmax/Mpl for

some ferritic pipes

as a function of the

D/t ratio from [7]

Figure 4:

MIL/Mpl ratio for an

ideal elastic-plastic

pipe as a function of

the D/t-ratio for

various imperfections



Figure 5 demonstrates the MIL/Mpl ratio as a function of the D/t ratio for ideal elastic-plastic pipes (E = 200

GPa, σy = 500 MPa, single dent with O = 1.0%) with different pipe lengths.

There are no relevant differences between the results for pipes with lengths from 2000 to 8000 mm. For a

1000 mm long pipe the suppression of the ovalisation at the ends stabilizes the pipe and the boundary

conditions have a greater influence. Figure 6 shows the M/Mpl ratio for an ideal elastic-plastic pipe (E =

200 GPa, σy = 500 MPa) with D/t = 100 as a function of the rotation angle for various out of roundness.

The dashed blue lines are the curves for infinitely long pipes without an imperfection. The red line

represents the moment for the first yield.

Figure 5:

MIL/Mpl for an ideal elastic-plastic pipe

(E = 200 GPa, σy = 500 MPa) as a

function of the D/t-ratio for different lengths

Figure 6:

M/Mpl ratio for an ideal elastic-plastic

pipe with D/t = 100 as a function of the

rotation angle



The B2 indices for ideal elastic-plastic pipes depend also on the yield stress and the elastic modulus.

Figure 7 shows the B2 indices for ideal elastic-plastic pipes with D/t = 100 and O = 0.5% as a function of

the yield stress. Figure 8 gives the results for pipes with similar geometry as a function of the elastic

modulus. The B2 indices are larger with increasing yield stress and decreasing elastic modulus.

Figure 7:

B2 index for ideal elastic-plastic pipes (E = 200 GPa)

with D/t = 100 and O = 0.5% as a function of the yield

stress

Figure 8:

B2 index for ideal elastic-plastic pipes with

D/t = 100 and O = 0.5% as a function of the

elastic modulus



Figure 9:

B2 index for ideal elastic-plastic pipes with O = 0.5%

as a function of the D/t ratio for two yield stresses

Figure 10:

M/My ratio for an ideal elastic-plastic pipe

with O = 0.5% of the D/t ratio

Figure 10 shows the B2 indices for ideal elastic-plastic pipes (E = 200 GPa) with O = 0.5% as a function

of the D/t ratio for a yield stress of 250 and 500 MPa. The B2 indices depend (almost perfectly) linearly on

the D/t ratio in the range from 40 to 120. The slopes are 0.004 for the yield stress of 250 MPa and 0.0065

for 500 MPa. It should be noted, that yield stresses of about 250 MPa are typical for austenitic and 500

MPa for ferritic steels. Figure 11 indicates that for pipes with a yield stress of 250 MPa and D/t > 100 and

for pipes with a yield stress of 500 MPa and D/t > 70 the failure occurs in the elastic range of the nominal

bending stress.



Additional analyses were performed for pipes made of the austenitic steels X6CrNiNb18-10 and AISI

304L and of the ferritic steels 22NiMoCr3-7 and 20MnMoNi5-5. For pipes with D/t > 60…80 and higher

yield stresses (for ferritic pipes) the failure occurs in the elastic range of the nominal bending stress with

small rotation angles. This is in agreement with the experiments by Elchalakani et al. [14] and Sherman

[18]. A comparison for real and ideal elastic-plastic pipes is given in Table 2. There are only small

differences between the real and the ideal material properties.

Table 2: B2 indices for real and ideal elastic-plastic pipes with D/t = 100 and O = 0.5%

Material

B2 for D/t = 100

real ideal

X6CrNiNb18-10 1.465 1.490

AISI 304L 1.490 1.527

22NiMoCr3-7 1.721 1.745

20MnMoNi5-5 1.775 1.801



4. Conclusions for the design

For pipes with D/t > 60…80 (especially ferritic pipes) the failure occurs in the elastic range of the

(nominal) bending stress with small rotation angles. The B2 indices depend strongly from the D/t ratio, the

out of roundness, the elastic modulus und the yield stress. Previously obtained numerical results indicate

that the temperature dependency is small and the B2 indices at room temperature are conservative. The

yield stress and the elastic modulus decrease with increasing temperature. The B2 indices decrease with

lower yield stresses and increase with lower elastic moduli. The numerical analyses for austenitic pipes

give B2 = 1.24…1.34 for O = 0.5% and B2 = 1.47…1.53 for O = 1%. The ASME Code states B2 = 1.43.

The numerical analyses for ferritic pipes give B2 = 1.47…1.58 for O = 0.5% and B2 = 1.72…1.81 for O =

1%.

The ASME Code, Subsection NB gives the following limits for the primary stresses:

𝐷 ∙ 𝑝

4 ∙ 𝑡+ 𝐵2 ∙

𝐷 ∙ 𝑀

2 ∙ 𝐼≤ 1.5 ∙ 𝑆𝑚 ≤ 𝜎𝑦 for Level 0, A (9)

𝐷 ∙ 𝑝

4 ∙ 𝑡+ 𝐵2 ∙

𝐷 ∙ 𝑀

2 ∙ 𝐼≤ 𝑚𝑖𝑛 3 ∙ 𝑆𝑚; 2 ∙ 𝜎𝑦 ≤ 2 ∙ 𝜎𝑦 for Level D (10)

The allowable stress for Level A may be equal to σy. It has already been stated, that the failure for pipes

with yield stresses around 500 MPa, D/t ~ 100 and p ~ 0 occurs for nominal stresses smaller/equal the

yield stress (see also Figure 10). The pressure term and B2 index give always a particular margin

against failure. On the other hand, the safety margin in Level 0 /A against failure for a pipe with D/t ≤ 50

is higher than 2 (comparison with Level D). An additional (or possibly necessary) safety factor for D/t >

50 could be calculated with help of these considerations and depends primarily on the yield stresses.



The ASME Code, Subsection NB or NC gives the following limits for the secondary stresses from disabled

thermal expansion and anchor movements:

𝐶2 ∙𝐷 ∙ 𝑀

2 ∙ 𝐼≤ 3 ∙ 𝑆𝑚 ≤ 2 ∙ 𝜎𝑦 with 𝐶2 = 1.0 for straight pipes - NB-3653 (11)

𝑖 ∙𝐷 ∙ 𝑀

2 ∙ 𝐼≤ 𝑆𝐴 ≤ 1.5 ∙ 𝜎𝑦 with 𝑖 = 1.0 for straight pipes - NC-3653 (12)

The allowable moment for secondary stresses might be higher than the first yield moment. A failure by

plastic buckling only by secondary loads is therefore also possible.

The following points are under discussion:

The moments from disabled thermal expansion and anchor movements have the same effect in relation

to a failure due to (plastic) buckling as the primary moments and must be taken into account.

The B2 indices at room temperature are conservative also for higher temperatures. A correction factor

for higher temperatures is not necessary.

Calculation of an additional safety factor.

The corrections of the B2-indices are based only on straight pipes, therefore the B2 indices are also to

be determined for other components (especially elbows).



5. Summary

Ideal elastic-plastic pipes with D/t ≤ 40 fail with a simple kink. Pipes with D/t > 40…70 fail by (plastic)

buckling with some wrinkles. The numerical obtained failure shapes are in a good agreement with the

experimental shapes. The B2 index significantly depends on the D/t ratio, the elastic modulus and the

yield stress. The imperfections have a considerable influence on the B2 index. For ideal elastic-plastic

pipes with D/t = 100 and a usual out of roundness of 0.5% typical values for the B2 index are 1.3 for a

yield stress of 250 MPa and 1.5 for 500 MPa. The differences between the B2 indices for ideal and real

materials are small and therefore without practical importance. A comparison of the numerical obtained

B2 indices with experimental results indicates an out of roundness of 0.5%. Final statements are not

possible, because the required information about the experiments is not completely available.

For an out of roundness of 0.5% the obtained values of the B2 index are overall in a formal good

agreement with the values stated in the ASME Code. Some analyses show differences with respect to

the temperature dependence and the additional correction factor.

Some conclusions for the design or the design equations are given.



References

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Mechanical Engineers.

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