3D Linear Bifurcation Analysis of Steel Containment Vessel

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3D LINEAR BIFURCATION ANALYSIS OF STEEL CONTAINMENT VESSEL

Jay Schmidt Westinghouse Electric Company

Cranberry Twp, PA, USA

Dr. Bernd Laskewitz Westinghouse Electric Company

Cranberry Twp, PA, USA

ABSTRACT The AP1000

® Containment Vessel (CV) is a freestanding steel

containment designed to protect the public from radiation

release. The CV consists of 2 ellipsoidal heads connected by a

cylindrical shell and is constructed of carbon steel. The

AP1000 plant design has four large penetrations (two airlocks

and two equipment hatches) located in approximately the same

quadrant of the circumference of the shell which imposes

asymmetric effects in the shell structure. The CV is designed

and constructed in accordance with ASME Boiler and Pressure

Vessel Code, Section III, Subsection NE.

Traditionally, the local and global stability of freestanding steel

containments have been designed by use of formulae using

conservative assumptions based on an axisymmetric structure.

ASME Code Case N-284 “Metal Containment Shell Buckling

Design Methods, Class MC Section III, Division 1” outlines

methodology for satisfying the stability of the CV using two

approaches. Section 1710 provides a stress based buckling

approach using detailed formulae that assumes an axisymmetric

structure. The second approach provides guidance and

acceptability based on a linear bifurcation analysis (2D (1720)

or 3D (1730)). Due to the asymmetric structure of the CV, the

3D linear bifurcation method delivers the most accurate results.

The methodology and assumptions implemented by

Westinghouse to qualify stability of the CV via Code Case N-

284 are outlined. Also, the procedure to properly amplify the

stresses as required by N-284 is included as justification of the

methods used. This justification was thoroughly investigated by

the Nuclear Regulatory Commission (NRC) and deemed

acceptable.

INTRODUCTION The primary purpose of a nuclear power plant containment

vessel is to act as the final line of defense for radiation release

to the public. Previous generations of nuclear power plants

utilized concrete containments using steel liner plates on the

inside. The AP1000 plant CV is a free standing steel

containment vessel. A reason for this is the role the CV plays in

the passive cooling technology that is unique to the AP1000

plant design.

The AP1000 plant CV and other freestanding containments or

similarly shaped pressure vessels are susceptible to buckling

concerns. Large portions of unsupported sections of the CV

shell are areas most prone to large compressive stresses and

therefore of particular concern in this analysis.

Criteria for the AP1000 plant CV are taken from the American

Society of Mechanical Engineers (ASME) Boiler Pressure

Vessel Code (BPVC), Section NE, 2001 with 2002 Addenda.

This year of the code is the year used for licensing of the

AP1000 plant CV. The NE-3000 Article of this code details

design and qualification of the CV. Buckling criteria is also

contained in section NE-3000. The buckling rules outlined in

NE-3133 do not cover all geometries of pressure vessels and do

not provide reasonable safety margins. Non-uniform

compressive stresses are also not accounted for in NE-3133.

Because of these reasons, an ASME task force was created to

develop rules for the design of CVs under compressive stress

scenarios. More refined analysis based on testing and detailed

research of pressure vessel buckling led to Code Case N-284

“Metal Containment Shell Buckling Design Methods, Class

MC Section III Division 1”. Revision 1 of this code case is

used in the AP1000 plant licensing basis and the major

supporting criteria for the CV buckling analysis.

Proceedings of the 2016 24th International Conference on Nuclear Engineering ICONE24

June 26-30, 2016, Charlotte, North Carolina

ICONE24-60673

1 Copyright © 2016 by ASME

CODE CASE N-284-1 Scope and Analysis Methods

The following text is taken directly from Section 1110 of N-

284-1 to provide overall scope of the code case: “The purpose

of this case is to provide stability criteria for determining the

structural adequacy against buckling of containment shells with

more complex shell geometries and loading conditions than

those covered by NE-3133. Such effects as symmetrical or

unsymmetrical dynamic loading conditions, circumferential

and/or meridional stiffening for heads as well as cylindrical

shell, combined stress fields, discontinuity stresses and

secondary stresses are considered in the stability evaluation.”

Code Case N-284-1 outlines the qualification guidelines for

three different analysis methodologies and the required

procedure for each. The three methods can be separated into

two sub groups. The first being a stress based analysis using

interaction equations and the second being a finite element

linear bifurcation analysis. The linear bifurcation analysis can

be 2D or 3D. The 3D linear bifurcation analysis was chosen for

qualification of the AP1000 plant CV.

The stress based method and the 2D linear bifurcation analysis

are designed for axisymmetric structures. The 2D analyses do

not have the ability to capture the seismic response of the

AP1000 plant large penetrations (two personnel airlocks, and

two equipment hatches). Large dynamic loadings that are not

symmetric to the overall CV structures could lead to additional

buckling concerns. The 3D linear bifurcation analysis is able to

investigate the impact of the large penetrations to the CV

stability.

ANSYS Finite Element Analysis Models

Two detailed 3D ANSYS Finite Element (FE) models were

created for use in the stability assessment of the CV. The two

models were the same with one exception. The mesh size

varied between the two models. The first model had a mesh

size of 37 inches and the second mesh size was 18.5 inches.

Eight-node structural shell elements (Shell93) were used in the

analysis and the two final models can be seen in Figures 1 and

2 below.

The bottom ellipsoidal head is not explicitly modeled. The

bottom head is encased in concrete on both sides. For

simplicity the bottom of the ANSYS FE models are considered

fixed in all six degrees of freedom (three translations and three

rotations). In reality the CV bottom head would be able to slide

and rotate and would not act as a fixed boundary. For the

purpose of this analysis the assumed boundary condition is

conservative.

FIGURE1. AP1000 CV Model (Coarse Mesh)

FIGURE 2. AP1000 CV Model (Fine Mesh)


N-284-1 Analysis Procedure

The following steps are provided in Page 26 of N-284-1. This

is a high level procedure. More detailed information required to

complete the tasks outlined are discussed in greater detail in the

following sections.

1) Perform a static analysis of the basic loads and obtain

, , and .

2) Multiply the stresses by the Factor of Safety (FS) for

the corresponding Service Limit.

3) Calculate the capacity reduction factor ij for local and

general buckling.

4) Calculate any applicable plasticity reduction factors

i.

5) Compute amplified stresses for the imperfect is by

dividing the elastic stress by the capacity reduction

factors calculated in Step 3.

6) Divide the elastic stress components by the proper

plasticity reduction factors to obtain the amplified

inelastic stress component ip. If I is equal to 1, then

is = ip.

7) Apply amplified stresses to the FEA model and

perform eigenvalue buckling analysis*

Note*: ANSYS does not have a function that allows the

mapping of stresses onto the FE model, the loadings were

amplified until the average of the resulting stress components

were amplified to the amount specified by the process outlined

above.

Loading Conditions/ Service Levels

Code Case N-284-1 follows the same service level conditions

as NE-3000 because they are both used in the design and

qualification of MC components. Types of loadings (thermal,

seismic, pressure, etc.) were coupled together based on

applicable service loading conditions. Investigations of all

service loading conditions yielded two that would be the most

limiting case for buckling. These two cases consisted of

loadings, when applied simultaneously, would lead to

compressive stresses over unstiffened portions of the shell. The

applicable service levels investigated are shown in Table 1.

TABLE 1: Applicable Service Levels

Service Level Loadings

Design/Level A Dead Load

Pressure Load

(External)

Thermal Load

(Normal Operation

in cold weather)

Level D Dead Load

Pressure Load

(External)

Thermal Load

(Normal Operation

in cold weather)

Vessel Global

Seismic Load

Large Penetration

Seismic Load

Seismic Loadings

Service Level D load condition includes seismic loadings. The

CV is conservatively analyzed using equivalent static

acceleration data to represent the global seismic loading of the

vessel. The accelerations are provided via North-South and

East-West lateral directions as well as a Vertical component.

All components are provided at several elevations from the

bottom to the top of the CV. The values were interpolated

linearly if necessary to get more realistic loadings on

components between datum points. Additionally accidental and

added torque loadings were also applied to areas of the vessel

at the large penetrations. These components consisted of

acceleration in the radial direction (axial direction of the

penetration) and two rotational accelerations. One is about the

horizontal axis and the second is around the vertical axis. The

100-40-40 rule was used to create 21 load combinations to

analyze all permutations of seismic loading. The local

accelerations could act in-phase and out-of-phase with the CV

global seismic loading. To capture this loading possibility and

bound the analysis, four additional cases were created to

encompass all possible deviations of loading. This leads to a

total of 96 load combinations for the Level D evaluation.

Factors of Safety and Capacity Reduction Factors

Section 1400 of N-284-1 provides Factors of Safety (FS) per

service level. They are as follows:

a) For Design Conditions and Level A and B Service

Limits, FS = 2.0


b) For Level C Service Limits the allowable values are

120% of the values of (a). Use FS = 1.67.

c) For Level D Service Limits the allowable stress values

are 150% of the values of (a) Use FS = 1.34.

Section 1500 of N-284-1 provides guidelines for the Capacity

Reduction Factors per service level and is dependent upon what

type of geometry, loading direction, local/general buckling, etc.

Local buckling is defined as the buckling of the vessel shell

between stiffeners. The general buckling refers to an overall

collapse of the system. The capacity reduction factors used in

the evaluation are given below. Equations (1) and (2) correlate

to local buckling for axial and hoop loads respectively.

Equations (3) and (4) are the axial and hoop capacity reduction

factors for general instability, respectively. In order to solve

Equation (3), Equation 5 is needed.

t

RL 10: log473.052.1 (1)

80.0L (2)

LLG A

__

0.56.3 (3)

LG (4)

084.0

__

tl

AA

s

(5)

Where: L = Axial capacity reduction factor for local

buckling

L = Hoop capacity reduction factor for

local buckling

G = Axial capacity reduction factor for

general instability

G = Hoop capacity reduction factor for

general instability

R = the radius of the CV

t = the thickness of the CV shell

A = Cross sectional area of internal stiffener

ls = Length between stiffeners

Based on the mid-surface radius and thickness of the SCV, the

axial knockdown factor was calculated to be 0.267 and 0.457

for local buckling and general instability, respectively. The

hoop capacity reduction factor is defines as 0.8 for both local

and buckling and general instability.

For both Service Level A and D conditions, the external

pressure is a governing load. This loading also acts differently

on the structure, than dead weight, seismic, or thermal loadings.

The amplification of these loadings is accomplished via the

factors calculated in Equations (6) and (7) for hoop and axial

effects, respectively. The remaining load types are

conservatively amplified using the axial capacity reduction

factors shown in Equation (8).

L

extextshell

FSPP

_ (6)

L

extextTP

FSPP

_ (7)

L

iamplified

FSLL

(8)

Where: Pshell_ext = Amplified external pressure on the

side of the CV

PTP_ext = Amplified external pressure on the

top head of the CV

Pext = External pressure

Lamplified = Amplified load

Li = PC and CV acceleration loads

FS = Safety factor (2.0 for service Level

A/Design, 1.34 for service level D)

Two cases were created in ANSYS and compared. The first

with loadings as provided by the design specification

(unamplified) and the second with load amplification factors

calculated using Equations (6) - (8) applied to the FE model

(amplified). The resultants stress components were compared to

assure that the correct amplification occurred.

Thermal loading amplification is accomplished by increasing

the rate of thermal expansion. The axial amplification factor

calculated is multiplied to the original or base expansion

coefficient creating a higher rate of expansion which yields

higher stresses. These stresses were also checked for proper

amplification levels and were found to be correct.

ANALYSIS RESULTS

Table 4 shows the resulting Eigenvalues calculated for each

load combination. One combination exists for Design/Service

Level A and 96 combination results for Service Level D.


TABLE 4: Eigenvalues for Local and Global Instability

x

y z

Lev

el A

1/D

es2

Lev

el D

1 L

oca

l N

eg N

eg

Lev

el D

1 G

lob

al

Neg

Neg

Lev

el D

1 L

oca

l N

eg P

os

Lev

el D

1 G

lob

al

Neg

Po

s

Lev

el D

1 L

oca

l P

os

Neg

*

Lev

el D

1 L

oca

l P

os

Po

s*

1.44

0.4 0.4 1 1.75 2.21 1.70 2.18 1.65 1.61

-0.4 0.4 1 1.64 2.14 1.69 2.18 1.69 1.71

0.4 0.4 -1 1.62 2.13 1.61 2.10 1.78 1.78

-0.4 0.4 -1 1.63 2.16 1.57 2.14 1.79 1.80

0.4 1 0.4 1.17 1.73 1.18 1.70 1.33 1.29

-0.4 1 0.4 1.17 1.71 1.15 1.70 1.31 1.37

0.4 1 -0.4 1.15 1.71 1.16 1.68 1.39 1.39

0.4 -0.4 1 1.72 2.20 1.71 2.22 1.62 1.66

0.4 -0.4 -1 1.60 2.12 1.56 2.11 1.79 1.80

-0.4 1 -0.4 1.14 1.71 1.12 1.68 1.38 1.42

-0.4 -0.4 1 1.66 2.15 1.61 2.11 1.71 1.67

-0.4 -0.4 -1 1.61 2.15 1.65 2.13 1.81 1.80

0.4 -1 0.4 1.17 1.72 1.14 1.69 1.29 1.33

-0.4 -1 0.4 1.17 1.70 1.16 1.67 1.34 1.31

0.4 -1 -0.4 1.15 1.69 1.12 1.67 1.39 1.40

-0.4 -1 -0.4 1.14 1.71 1.16 1.68 1.42 1.40

1 0.4 0.4 1.20 1.80 1.22 1.77 1.45 1.41

-1 0.4 0.4 1.23 1.78 1.17 1.78 1.45 1.45

1 0.4 -0.4 1.16 1.77 1.18 1.75 1.54 1.51

-1 0.4 -0.4 1.19 1.78 1.14 1.74 1.54 1.52

1 -0.4 0.4 1.18 1.78 1.16 1.75 1.39 1.42

-1 -0.4 0.4 1.18 1.78 1.20 1.76 1.41 1.38

1 -0.4 -0.4 1.14 1.73 1.12 1.70 1.49 1.53

-1 -0.4 -0.4 1.14 1.74 1.17 1.77 1.51 1.47


* Global buckling analyses were not performed for load cases

in which the Eigenvalue exceeds 1.2 in the local buckling

analyses.

Note: The 1st “Negative” or “Positive” in the load cases

description refers to the applied global seismic acceleration

direction compared to the global coordinate system. The 2nd

“Negative” or “Positive” refers to the applied equivalent static

forces due to seismic acceleration of the large penetrations

according to their local coordinate systems. This approach is

extremely conservative since it combines in-phase and out-of-

phase movements of the shell and other seismic loadings.

Figures 3 and 4 below show the deformed shape of the critical

case of this analysis. Figure 3 shows the CV model from an

elevation view and Figure 4 shows the CV from the top as a

plan view. The buckling waves can be seen in the vicinity of

the shell away from the large penetrations but at similar

elevations.

FIGURE 3. Critical Eigenmode (Elevation View)

FIGURE 4. Critical Eigenmode (Plan View)

CONCLUSION

The stability of the Containment Vessel for all appropriate load

combinations per ASME NE-3000 are found to be within the

allowable values of ASME Code Case N-284-1. The lowest

Eigenvalue for local buckling is 1.12 (allowable > 1.0). For

global buckling, the lowest Eigenvalue is 1.67 (allowable

>1.2). Considering loading conditions that would add yield

compressive stresses in the vessel cylindrical wall, the overall

stability and the local buckling capacities are met with

conservative loading conditions applied.

REFERENCES

Laskewitz, B.R. and Schmidt J.E., (2011). APP-GW-GLR-

005, Rev 5. “Containment Vessel Design Adjacent to

Large Penetrations.

CASES OF ASME BOILER AND PRESSURE VESSEL

CODE: Case N-284-1 “Metal Containment Shell

Buckling Design Methods, Class MC Section III,

Division 1. March 14, 1995.

ASME Section III “Rules for Construction of Nuclear Power

Plant Components, Division 1 Subsection NE, Class MC

Components”, 2001 Edition with 2002 Addenda.


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3D Linear Bifurcation Analysis of Steel Containment Vessel