Crack Opening Displacement of the Plate Subjected to Bending Stress (G294)

Crack Opening Displacement of a Crack in a Plate Subjected to Bending Load

Hideo Machida1)_{, Yeon-Sik Yoo}2) 1)

TEPCO Systems Corporation, Tokyo, Japan

2)

Japan Nuclear Cycle Development Institute, Ibaraki, Japan ABSTRACT

This study was performed in order to clarify crack opening displacement (COD) of a through-wall crack in a plate subjected to bending load. The former COD evaluation method is mainly arranged only to the tensile load, and there is nothing that was arranged to the bending load. Then, authors evaluated COD of the through-wall crack in the plate which is subjected to the bending load using finite element method (FEM) analysis, and proposed simplified COD evaluation method for it. Proposal method is useful for a leakage evaluation; for example from the crack at an elbow crown or a vicinity of the coolant surface of a vessel in which the bending stress is relatively large.

KEY WORDS: crack opening displacement, LBB, leak rate, FINAS, finite element method, fracture mechanics, bending stress, simplified evaluation method

INTRODUCTION

Leak-Before-Break (LBB) concept has been applied to piping design of nuclear power plants (NPPs) for several years for the purpose of rationalization of pipe-supporting structures. In the application of the LBB concept to the aforementioned purpose, careful attention should be paid to leak detection on the structures. To prevent pipes from breaking, the leakage must be detected early and the plant must be stopped as safely as possible. For this reason, many studies on the COD for the through-wall crack were performed for many years. However, the former COD evaluation methods are mainly applied only to the tensile load, and there has been nothing that has been developed corresponding to the bending load. This is because, the dominant loads in light water reactor (LWR) plants—the mainstream of nuclear power plants—are internal pressure and thermal-expansion, and they react to the crack as a tensile load.

On the other hand, some severe parts in structural integrity evaluation in NPPs are subjected to large bending stress in addition to membrane stress: for example, the elbow crown, the vicinity of the sodium surface in a reactor vessel of a fast breeder reactor (FBR), etc.. When the bending load is applied to the through-wall crack, the crack will open in the tension side, but it will close in the compression side. If the effect of the bending stress on the COD is equivalent to that of the membrane stress, when the bending plus membrane stress is lower than zero, the crack will close and the leak detection will be impossible. However, it is hard to imagine that the effect of the bending stress which balances within wall thickness reacts on the through-wall crack greatly. Thus, it is very important to evaluate the COD suitably when the bending load is applied to the crack at the origin of leak detection.

In this study, the elastic CODs under bending load were investigated by FEM analyses, and they were expressed as the function of the crack size and the stress at the crack surface.

INFLUENCE OF ANALYTICAL PARAMETERS

FEM analyses on through-wall cracked plates were performed. Solid elements of 20 nodes were employed for these analyses and the analytical code was FINAS Ver.15.0 [1]. The material properties used in these analyses are shown in Table 1. The membrane and the bending stress are applied to the through-wall crack in the plate. Based on a superposition method, the membrane stress was modeled by a constant-distributing load and the bending stress was modeled by a linear-distributing load on the crack surface as shown in Fig. 1. The analyses consisted of the following two steps. The constant-distributing load is applied in the first step, and the linear-distributing load is added in the next step.

In order to evaluate the COD in the plate exactly, it is necessary to condition some pertinent parameters used in the FEM analysis. Typical parameters are the ratio of plate length to width (H/W) and the mesh size. The mesh size is classified into minimum mesh size at a crack front (amin),

Table 1 Material properties used in the analyses

Modulus of Elasticity Poisson’s Ratio

200 GPa 0.3

10M Pa

20M Pa

0MPa

Spet1 Step2

(membrane) (membrane + bending) Fig. 1 Loading condition applied to the crack surface Transactions of the 17th _{International Conference on}

Structural Mechanics in Reactor Technology (SMiRT 17)

Prague, Czech Republic, August 17 –22, 2003

node pitch (Np), and number of mesh in the wall thickness (Nt). In order to decide a set of parameters used for a specific analysis, the CODs were analyzed within the parameters shown in Table 2. The typical analysis model is shown in Fig. 2. The node pitch to the plate length direction was set to be equal to the crack length direction in the analysis model.

Table 2 Preliminary analysis conditions

Case Case

No. No.

P1 0.1 0.5 0.2 16 1.3 P17 0.7 0.5 0.2 16 1.3 P2 0.1 0.5 0.2 16 1.5 P18 0.7 0.5 0.2 16 1.5 P3 0.1 0.5 0.2 10 1.3 P19 0.7 0.5 0.2 10 1.3 P4 0.1 0.5 0.2 10 1.5 P20 0.7 0.5 0.2 10 1.5 P5 0.1 0.5 0.5 16 1.3 P21 0.7 0.5 0.5 16 1.3 P6 0.1 0.5 0.5 16 1.5 P22 0.7 0.5 0.5 16 1.5 P7 0.1 0.5 0.5 10 1.3 P23 0.7 0.5 0.5 10 1.3 P8 0.1 0.5 0.5 10 1.5 P24 0.7 0.5 0.5 10 1.5 P9 0.1 3 0.2 16 1.3 P25 0.7 3 0.2 16 1.3 P10 0.1 3 0.2 16 1.5 P26 0.7 3 0.2 16 1.5 P11 0.1 3 0.2 10 1.3 P27 0.7 3 0.2 10 1.3 P12 0.1 3 0.2 10 1.5 P28 0.7 3 0.2 10 1.5 P13 0.1 3 0.5 16 1.3 P29 0.7 3 0.5 16 1.3 P14 0.1 3 0.5 16 1.5 P30 0.7 3 0.5 16 1.5 P15 0.1 3 0.5 10 1.3 P31 0.7 3 0.5 10 1.3 P16 0.1 3 0.5 10 1.5 P32 0.7 3 0.5 10 1.5 a/W H/W amin Nt Np a/W H/W amin Nt Np

1 min

2/ 1 / -1

p i i

a a

N a a a a

=

= = ⋅⋅⋅ =

Crac k t

N

1

a

a

3

a

1

a

1

a

2 a

2 a 3 a

3

a

X

Y Z

Crack

Fig. 2 Example of analytical model

The CODs of each analytical case are shown in Table 3. The plate length has the largest effect on the COD, and the minimum mesh size at the crack front is the second. However, the effects of the node pitch or the number of mesh in the thickness on the COD are very little. The effect of plate length increases with the crack length, and that effect is larger in the membrane stress case than the bending stress case. Compared with the plate length, the effect of the minimum mesh size at the crack front is very small. In order to evaluate the effect of the plate length, the analyses for two crack length cases (a/W=0.05, 0.8) which have the parameters H/W=0.5 to 3.0 were carried out. The CODs of these cases are shown in Table 4. When H/W is 2.0 or more, the COD hardly changes in all cases. From these results, it can be deduced that the plate whose H/W is 2.0 or more satisfies an infinite plate condition on the FEM analysis. Consequently, the conditions for specific analyses were decided as shown in Table 5.

Table 3 COD of preliminary analyses

Mem. Bend. Mem. Bend. Mem. Bend. Mem. Bend.

P1 0.18175 0.1017 P9 0.17472 0.10124 P17 5.60509 0.74907 P25 1.74206 0.70512

P2 0.18173 0.10167 P10 0.1747 0.10122 P18 5.60458 0.74897 P26 1.74185 0.70471

P3 0.18175 0.10169 P11 0.17472 0.10124 P19 5.60498 0.74917 P27 1.74206 0.70512

P4 0.18173 0.10167 P12 0.1747 0.10121 P20 5.60448 0.74897 P28 1.74185 0.70471

P5 0.1814 0.10156 P13 0.17441 0.1011 P21 5.59999 0.74907 P29 1.74114 0.70482

P6 0.18137 0.10153 P14 0.17438 0.10108 P22 5.59897 0.74846 P30 1.74104 0.70431

P7 0.1814 0.10156 P15 0.17441 0.10109 P23 5.59999 0.74897 P31 1.74114 0.70482

P8 0.18137 0.10153 P16 0.17438 0.10107 P24 5.59897 0.74846 P32 1.74093 0.70441

Case COD (×10

-2

Table 4 Effect of plate length on COD Table 5 Conditions of the specific analyses

Mem. Bend. Mem. Bend.

0.5 1.010 1.001 3.909 1.057

1 1.002 1.000 1.288 1.002

1.5 1.000 1.000 1.028 1.000

2 1.000 1.000 1.002 1.000

2.5 1.000 1.000 1.000 1.000

3 1.000 1.000 1.000 1.000

H/W a/W=0.05 a/W=0.8 H/W amin. Nt Np

2 0.2 10 1.5

Note; Relative value over the case of a/W=3.0 is shown in the table.

CRACK OPENING DISPLACEMENT OF THE PLATE

Based on the specification of the analytical model shown in Table 5, the CODs were calculated corresponding to the parameters of the crack length.

First, the CODs due to a membrane stress were analyzed and compared with the former simplified evaluation method. The COD for membrane stress is given by Tada et al. [2]

4 ( ) ' m m m a V E σ

δ = ξ (1)

/

a W

ξ =

2 3 4 1

( ) 0.071 0.535 0.169 0.090 0.020 1.071 ln(1 )

m

V ξ ξ ξ ξ ξ ξ

ξ

= − − + − + − −

2

Plane Stress Condition '

Plane Strain Condition 1 E E E ν   =   − 

Where δm, a, σ m, W, E, ν and V_m( )ξ are COD, crack half length, membrane stress, plate half width, modulus of elasticity , Poisson’s ratio and COD influence function, respectively. The COD influence function is a sort of modification factor for the elastic COD. The COD influence functions calculated by Eq. (1) and the analyzed one (V_{m a−} ( )ξ ) in the both cases of plane strain condition (wall thickness center) and plane stress condition (on the surface) were plotted in Fig. 3. V_a( )ξ is calculated by the following function using the analyzed COD (δana).

' ( )

4

m a ana

m E V a ξ δ σ

− = (2)

At the plate surface (plane stress condition), the analytical results give good agreement with those of Eq. (1). At the center of thickness (plane strain condition), the analytical results are larger than those by Eq. (1). For the two cases that were analyzed, there is a difference from the case where a/W is smaller than 0.1, compared to the case where a/W is larger than 0.1. In this region, the analyzed COD at plate surface is larger than Eq. (1) and that at center of thickness is smaller than Eq. (1). Such a tendency might be influenced by the deformation at the crack tip The stress and strain distributions around the crack tip are shown in Fig. 4 with the deformation for the cases that a/W=0.005 and 0.1. At a/W=0.005, the crack curves like a horn only near the surface whereas the inside deforms uniformly. On the other hand, at

a/W=0.1, the crack curves from the center of thickness to the surface continuously. In the case of a/W=0.005, the influence of deformation of the crack edge appears in the X-stress at the surface, and the plane stress

1.0 1.5 2.0 2.5 3.0 3.5

0.0 0.2 0.4 0.6 0.8 1.0

Tada et al. [2] Plane Strain (Center of Thickness) Plane Stress

(Surface of Plate)

COD In flu en c e Fu n ct io n Fo r Mem b ra n e S tre ss ; V m (a /W )

Crack Length / Plate Width; a/W

condition is not satisfied. The X-strain at the center of thickness is almost zero, and the plane strain condition is satisfied. On the other hand, in the case of a/W=0.1, the X-stress at the crack surface is zero, but the X-strain at the center of thickness is not zero, so the plane stress condition is satisfied at the crack surface, but the plane strain condition is not satisfied at the center of thickness. Since it is pessimistic to evaluate the COD small in the origin of leak detection, it is recommended using Eq. (1) under the plane strain condition.

X Y Z 5.427 5.018 4.608 4.198 3.788 3.379 2.969 2.559 2.149 1.74 1.33 0.92 0.51 0.1 -0.309 -0.719 -1.129 V1

Output Set: FINAS STEP 1 Deformed(0.000471): Total Translation Contour: Node Stress-X

X Y Z 0.0000258 0.0000215 0.0000172 0.0000129 0.00000859 0.00000429 -3.987E-9 -0.0000043 -0.0000086 -0.0000129 -0.0000172 -0.0000215 -0.0000258 -0.0000301 -0.0000344 -0.0000387 -0.000043 V1

Output Set: FINAS STEP 1 Deformed(0.000471): Total Translation Contour: Node Strain-X

X-Stress X-Strain

a/w=0.1 X Y Z 2.104 1.928 1.751 1.575 1.399 1.223 1.047 0.871 0.695 0.519 0.343 0.167 -0.00945 -0.186 -0.362 -0.538 -0.714 V1

Output Set: FINAS STEP 1 Deformed(0.0000953): Total Translation Contour: Node Stress-X

X Y Z 0.0000231 0.0000209 0.0000188 0.0000167 0.0000145 0.0000124 0.0000103 0.00000814 0.00000601 0.00000388 0.00000174 -3.887E-7 -2.52E-6 -4.65E-6 -6.78E-6 -8.92E-6 -0.000011 V1

Output Set: FINAS STEP 1 Deformed(0.0000953): Total Translation Contour: Node Strain-X

X-Stress X-Strain a/W=0.005

Fig. 4 Displacement, stress and strain around crack

X

Y Z V1

Output Set: FINAS STEP 2 Deformed(0.0214): Total Translation

Fig. 5 COD subjected to membrane and bending loads

Next, the CODs due to a bending stress were evaluated. An example on the deformation of the crack under a membrane and a bending stress is shown in Fig. 5. The amplitudes of COD at the tension and compression side by a bending stress showed same absolute values with opposite signs. In the compression side, although a membrane plus bending stress is 0, the crack still opens here. From this result, even if the sum of membrane and bending stress has a negative value, the crack closure may not occur. The relationship between the crack length and the COD (tensile side) under the bending condition are shown in Fig. 6. From these analytical results, a simplified COD evaluation method for the bending stress using COD influence function (V_b( )ξ _{) fitted by a 9th order polynomial formula was proposed as} shown below. The error caused by fitting is 0.1% or less.

( )

4

'

b

b b

a V E

σ

δ = ξ (3)

( )

0 1 2 3 4 5 6 7 8 9

b

V ξ =A +Aξ+Aξ +Aξ +Aξ +Aξ +Aξ +Aξ +Aξ +Aξ

a W

ξ = , 0 a 0.9

W

< ≤

A0 A1 A2 A3 A4 A5 A6 A7 A8 A9

1.0549 -10.378 93.101 -490.58 1589.1 -3210.0 4008.3 -2954.0 1143.3 -168.25

The COD influence functions of Eq. (1) and Eq. (3) (V_m( )ξ and V_b( )ξ ) are shown in Fig. 7. Although the COD influence function for membrane stress increases monotonously, that for bending stress decreases till a/W=0.4 and increases gradually thereafter. This result shows apparently that the effect on the COD of the bending stress is smaller than that of the membrane stress, if their amplitudes are the same.

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0

C

ra

ck

O

p

en

in

g

D

is

p

lacem

en

t

(X

1

0

-2 mm

)

Crack Length / Plate Width; a/W

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.2 0.4 0.6 0.8 1.0

Membrane Stress[2] Bending Stress

C

OD In

fl

u

en

ce F

u

n

ct

io

n

,

V

(a

/W)

Crack Length / Plate Width, a/W

Fig. 6 COD by bending load Fig. 7 Comparison of COD influence function between membrane and bending stresses

APPLICATION IN THE ACTUAL PLANT

The proposed COD evaluation formula was applied to an advanced loop type FBR plant for which a design study was carried out by Japan Nuclear Cycle Development Institute. Evaluation of the COD for an axial crack at the elbow crown and a circumferential crack at the sodium surface of the reactor vessel are shown below.

Table 6 Piping specifications of advanced loop type FBR plant Hot Leg Cold Leg

Outer Diameter (mm) 1270 863.6

Thickness (mm) 15.9 12.7

Bend Radius (mm) 1270 863.6

Internal Pressure (MPa) 0.1 0.6

Through-wa ll Crack

θ

Crack Length = Rθ

R

Fig. 8 Axial crack at the elbow crown

The stress conditions when the crack at the elbow crown opens are evaluated as shown in Fig. 9. Since the stress by internal pressure is very small, then the circumferential bending stress to open the crack is limited corresponding to the membrane stress, it is difficult to propose the scenario that the crack at the elbow crown will open and will be detectable by leak monitors. Therefore, it will be necessary to detect the crack by the non-destructive test such as In-Service Inspection (ISI) at this portion.

0.0 10.0 20.0 30.0 40.0 50.0 60.0

0.0 0.2 0.4 0.6 0.8 1.0

Hot Leg Cold Leg

All

o

wa

bl

e

Circ

um

fe

re

nt

ia

l

Be

n

d

in

g

Stre

ss

(MP

a

)

Crack Length / Plate Width; a/W

Around the sodium level of the reactor vessel (I.D.=9600mm, t=30mm, r/t=160.5) of an advanced loop type FBR plant is a critical area to achieve structural integrity, because of changes of coolant temperature and surface level. The possibility of the leak detection, assuming a circumferential crack in this portion, was evaluated. The axial stresses at the sodium surface level in the reactor vessel are shown in Table 7.

Table 7 Stresses of the reactor vessel at the sodium level of advanced loop type FBR plant Axial Stress

(MPa) Internal Pressure 11.9

Dead Weight 19.1

Thermal Distribution 75 Factor of Stress

From Table 7, the axial membrane and bending stresses are as follows: 31.0 MPa

z m

σ ₋ = (4)

75.0 MPa

z b

σ ₋ = (5)

The relationship between crack length and maximum bending stress to open the crack is shown in Fig. 10. Leakage cannot be detected unless a crack length exceeds about 63% of the perimeter of the reactor vessel under the plate condition.

30.0 40.0 50.0 60.0 70.0 80.0 90.0

0.0 0.2 0.4 0.6 0.8 1.0

De

te

ct

ab

le

M

ax

im

u

m

Be

n

d

ing

S

tre

ss

(M

P

a)

Crack Length / Plate Width, a/W Actual Bending Stress

(75 MPa)

Detectable

Fig. 10 Detectable size of the crack at the sodium level of advanced loop type FBR plant

DISCUSSION

the cylinder that takes the bending stress into account, so a study on it will also be required. Following this study, authors are studying the COD evaluation method for cylinders subjected to bending load.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Plate Cylinder

C

rac

k O

p

e

n

in

g

D

is

p

la

ce

m

en

t (

m

m)

Crack Length / Width; a/W

Inner Surface

Outer Surface

Inner SurfaceOuter Surface

Fig. 11 COD of a plate and a cylinder; a/W=0.65

CONCLUSION

A simplified COD evaluation method was proposed for the through-wall cracked plates. This result is effective in leak evaluation on the through-wall crack at the part where the membrane and the bending stresses superimpose. However, since it gives too pessimistic result for circumferential crack of the cylinder, especially when a crack is large, one should be careful when using this application.

REFERENCES

1. Finite element nonlinear structural analysis system, PNC ZN9520 95-014, 1995.