Failure assessment diagram under biaxial loading

Structural Safety, 12 (1993) 113-121

113

Elsevier

Failure assessment diagram under biaxial loading * Q. Shao Jiangsu Institute of Standards Research, Nanjing, China

D.M. Tan Technical University of Vienna, Institute for Applied and Technical Physics, Wiedner Hauptstrasse 8-10 / 137, A- 1040 Vienna, Austria

Abstract. The Failure Assessment Diagram (FAD) for evaluating the safety margin of cracked structure under biaxial loading is explored. The J-integral fracture parameters, i.e. Je (elastic) and Jep (elastic-plastic) are calculated by the finite element method and the instantaneous load of initiation of crack growth is examined by the method of local D.C. electric potential measurement. With the research work mentioned above, several failure assessment methods are compared in biaxial stress states. The modified methods of failure assessment of CEGB's R6 and EPRI's Engineering Approach for biaxial loading have been proposed respectively in the present work. The experimental results reveal that the J-integals based on the measured instantaneous load of initiation of crack growth, Ji, under different ratio of biaxial load are, in general, almost identically equal.

Key words: fracture; biaxial; failure assessment.

1. Introduction In recent years several proposals have been developed which aim to assess the safety of nuclear and chemical industries structures which might fail by ductile mechanism. The CEGB's R6 [1] and the EPRI's JFAD [2,3] are important methods of failure assessment. Biaxial and multiaxial stress states are common in pressure vessels and piping. A large number of investigations of failure assessment are based on uniaxial stress state. Experiments are carried out in uniaxial specimens such as CT, CCP, SECP, DECP. When the results of the investigations are used in engineering, some questions will be raised, e.g. whether the results in uniaxial stress states are completely availabe to the biaxial and multiaxial stress states in engineering, whether they are conservative or not, how large the errors are. When the biaxial loading conditions are introduced how the failure assessment diagram will change, which kind of FAD can give the most accurate assessment. To attempt to answer these questions, fracture experiments

in biaxial stress states were

carried

o u t in p r e s e n t

work. The

centre

cracked

* Discussion is open until November 1993 (please submit your discussion paper to the Editor, Ross B. Corotis). 0167-4730/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

114 TABLE 1 Mechanical properties and composition of 16MnR steel Material 16MnR

Mechanical properties trs ~u ~b MPa MPa % 384.5 564.0 57.0

Composition % ~ % 28.0

E MPa 2.05e5

3' 0.3

C 0.14

Si 0.47

Mn 1.46

P 0.01

S 0.17

cruciform specimens are made of common used mild steel for pressure vessels. Based on the results of numerical and theoretical analysis, the modified methods of failure assessment of classic C E G B ' s R6 and E P R I ' s J F A D for biaxial loading have been proposed respectively in the present work.

2. Experiment 2. 1. Material properties and specimens geometries The specimens material are plates with a thickness 10 mm, made of a Chinese Pressure Vessel Standard steel 16MnR. The mechanical properties and composition are given in Table 1. According to the cruciform specimen developed by Miller and Kfouri [4] and the present results of F E M analysis and the test machine in our lab, a new specimen is redesigned, see Fig. 1. The thickness is 5 mm, and the clamping part is 10 mm. The fracture experiments were undertaken with centre penetrated crack cruciform specimen. The total crack lengths 2a ranged from 30 mm to 56 mm, including length of spark machined notch and fatigue crack, W = 120 mm. Experiments were performed under load biaxial ratio k = 0, 0.5, 1 to investigate the effect of biaxial ratio on the initiation and growth of the crack. The reason to use these biaxial ratios is to simulate the common stress states in pressure vessels.

2.2. Fracture experiments Local D.C. potential m e a s u r e m e n t method was used to detect crack initiation. The initial loads are listed in Table 2.

t Y

J

o o

ol

ioo=[

/oOOOoO0o/ Fig. 1. Cruciform specimen geometries.

115 T A BLE 2

Initial load using the local D.C. potential method (k = Px/Py) Specimen no.

Biaxiality ratio k

a (mm)

a13 a14 al a3 a5 a12 b5 bl b3 b2 b4 c5 c3 c6 c7 c8 c4

0 0 0.5 0.5 0.5 1 0 0.5 0.5 1 1 0 0.5 0.5 0.5 0.5 1

18.55 17.14 17.79 17.46 17.46 17.45 22.50 22.35 22.41 22.15 22.30 27.93 26.53 27.53 28.00 27.41 27.56

Experimental loading Pi

Predicted loading Pip

error

20.50 20.63 21.50 23.38 22.75 24.35 16.15 20.63 20.88 21.55 20.75 16.75 17.50 18.00 17.63 17.60 19.38

19.53 19.35 23.71 23.71 23.71 24.06 16.89 20.45 20.45 21.14 21.14 15.07 17.45 17.45 17.45 17.45 19.06

5.9 6.3 9.3 1.39 4.04 0.79 2.30 0.88 1.22 0.52 1.84 11.10 0.28 3.15 1.03 0.86 1.68

%

By comparing these results with the data of traditional multi-specimens method, it is found that the initial loads are in good agreement with each other, maximum error is 3.15%, see Table 3. It shows the reliability of the local D.C. potential method.

3. Failure assessment diagram in biaxial stress states For plastic centre energy

failure assessment diagram under biaxial loading, two subjects will be discussed, the limit loads and the fracture parameters. The plastic limit load was discussed in [5]. For a cracked square thin plate, the limit load P0 can be derived by using the principle of equilibrium in plasticity theory. The results are given as follows:

Po=A(W- 2a)Oo

(1)

where A is the biaxial coefficient. A = [k 2 + (W/W- 2a) 2 - k(W/W- 2a)] 1/2. The crack initiation parameter Jc is calculated by elastic-plastic FEM according to the crack initial loads. The details are described in [5].

T A BLE 3

Comparison of the crack initial loads by using potential and multi-specimens methods Specmen no.

c3

c6

c7

c8

c9

Pi(T) (local DC) Pi(T) (multi-spec.) Error %

17.5 17.45 0.268

18.0 17.45 3.15

17.63 17.45 1.00

17.60 17.45 0.86

17.75 17.45 1.79

116

3. 1. Classic R6 FAD

The classic failure assessment diagram uses the non-dimensional load and stress intensity parameters Kr and Sr defined by Kr = K / K r = (K2/EJc) 1/2 and Sr = P/PL(O')= O.f/O.0" Kc is the material toughness and PL(O.) is the limit load defined for yield % or flow stress. For a certain structure, given loading conditions and defect sizes, the two parameters Kr and Sr are evaluated from the expressions above, if the assessment point (Sr, Kr) lies within the failure assessment curve, the defect is acceptable. T h e assessment curve is Sr Kr=

rr

Milne [6] modified the assessment curves in consideration of strain hardening as: Sr = o./{~r + (1

- Krl/2)(o.u/O-

0 -

(3)

1)}

where o.u is the ultimate streess and % the yield stress. Alpa [7] proposed an extensive model to analyse the influence of biaxial loading, and modified the restraining stress of Dugdale's model, by including the stress parallel to the crack plane in the following way: 6-

rr----E lnsec 2o'*

o'x = o.* -

+ o.;,

(4) o'y = o . *

(5)

where E is Young's modulus; o'~, o'~ are the stresses from infinitive place; and o'x, % are the local stresses, see Fig. 2. In case of the V o n Mises criterion, o'* is the restraining stress and expressed in the following way: o'* = 2

cry - o ' x ) +

4 ~ r 2 - 3(o', -o'x )

(6)

T h e expression for biaxial is not changed. Only ~r * is used instead of o-0 in the expression. T h e biaxially modified classic R6 curves and Milne's curves move outward for biaxial loading

cJ~° t

T

Sy

T

T

T OO

(3 (3* /.,4

-~S

X

(~*

Sx

....

S ~-

Fig. 2. Modified Dugdale model.

117 1.4

I

'

'

"

"

I

. . . .

I

~

,

I

. . . .

Modified R6 FAC k=l, k=0.5 = O

1.2 1

. . . .

k

o.8 0.6

Experimental Data

~

0.4

k=0.5

•

0.2

°

I I

k=0

I I

, , , t I ~ , , , I ~ , I, d I , , ,

0 0

0.5

Sr I

1.5

2

Fig. 3. Classic R6 curve and its biaxial modification. k = 0.5, 1 as V o n Mises yield criterion is used. Different curves are obtained for different k values. A unique curve is obtained for different crack lengths. By comparing the test data assessment points, it is found that the data change trends do not fit the assessment curves. The classic R6 curve gives a conservative prediction, see Fig. 3. The biaxially modified Milne's curve gives a good failure prediction, see Fig. 4.

3.2. Ainsworth's modified FAD Ainsworth [8] p r o p o s e d a modified m e t h o d (2nd approach of R6 Rev.3) by introducing reference stress Oref and strain Eref. The unique failure assessment curve is independent of the geometric condition. The whole true stress-strain curve is n e e d e d for this method. The assessment curve can be expressed in the following way: ] "~

~EEref =

- - +

O-ref

1

2/(

"2~ O-0 ] / ~

--I EEref

(7)

O'ref ]

.4

. . . .

I

. . . .

1.2

I

"

'

'

'

I

'

'

'

'

Milne's Modified FAC ---k=l, k=0.5

k~9

1

o.8 0.6

A

0.4

t, •

k'=l k=O_5

~,, --'~\_ ,,

a lc=0

0.2 0

I

0

I

i

I

,

0.5

,

\ x

,

=

Sr

I

=

1

i

i

i~l

o~*

1.5

,

2

Fig. 4. Milne's assessment curve and its biaxial modification.

118

I

cO

~

rettE f

eref

e

Fig. 5. Reference stress and strain.

where Sr = O'ref/O"0. The assessment curve is not continuous for typical mild steel 16MnR with a long yield platform. The assessment is unique for different biaxial ratios and crack lengths, see Fig. 5. Ainsworth's curve gives a conservative failure predition, see Fig. 6. 3.3. EPRI's JFAD

Bloom [3] derived JFAD (3rd approach of R6 Rev.3) using the J estimation scheme proposed by Shih [2]. The stress-strain curve may be described by e / e 0 = o-/o 0 + a(o/tro)". For a material obeying Ramberg-Osgood relation, Shih [2] suggested the value of J in eqn. (8). The JFAD can be expressed s in eqn. (9): J =Je +Lp

=

(8)

K2(ae)/E + a°oeoch(a/W, n)(P/Po) n+'

(9)

Kr = IL/]

1.4 1.2

Ainsworth's FAC

1

: o8 DataExperimental

0.6 0.4

,,

k=O

0.2

• o

k=0.5 k=0

J

0

0

i

,

,

I

0.5

,

l

:~

j' no

u

i

i

Sr

I

I

1

. . . .

1.5

Fig. 6. Ainsworth's assessment curve.

119

800

•

'

'

I

'

"

'

I

'

'

'

I

"

'

"

I

"

'

'

I

'

'

' 1

700 ~. 600

D., 500

"•

400

Exprimental StressStrain Curve

300

J~ 200

03

Ramberg-Osgood Fitting Curve

100 ,

0

0

,

0

I

,

,

,

0.02

I

,

~

,

I

•

,

,

I

,

,

,

I

,

,

,

0.04

0.06 0.08 0.1 0.12 Strain e Fig. 7. 16MnR stress-strain curve modification and Ramberg-Osgood fit. Several R a m b e r g - O s g o o d fitting methods are chosen to mininize the errors b e t w e e n the R a m b e r g - O s g o o d relation and F E M results. O n e best fit method is used, see Fig. 7. Biaxially modified failure assessment curves can be plotted for different crack lengths and biaxial ratios respectively, see Fig. 8(a)-8(c). By comparing these assessment curves with experimental results, the J F A D s give good agreement with experimental data plots.

4. Conclusions Experimental and theoretical analysis reveal that the biaxially modified classic C E G B ' s R6 failure assessment method based on Dugdale's model gives a conservative prediction, and the biaxial modified Milne's assessment curve gives a good failure prediction. Ainsworth's modified method gives a discontinuous assessment curve for the material with discontinuous stress-strain

1 . 4

1.2

"

'

'

'

I

(a)

0.6

'

'

'

I

'

'

'

'

I

'

'

'

'

EPPd's J FAD k=O

' o.8

'

.:=._

2a/w=0.417 \',= Experimental mental Data \X".N, " 0.4 = k=l ~'h"X • k=0.5 li; O.2 ° k=0 I :i 0 .... , ,, ,, t .... i,:l,~ , , , 0 0.5 1 1.5 2 Sr Fig. 8(a). JFAD and biaxial modification for a / W = 0.417.

120

1.2

EPRI's J FAD - k = l , k=0.5 k=0

(b)

- -

1

~

o.8

Experimental Data " k=l

0.4

0.2 0

X

2 w--o.33.L

0.6

" "

\ ~'J

k=0.5 k=O

. . . .

0

~

,

II

?,,,,

0.5

. . . .

Sr I

I

,

,

1.5

2

Fig. 8(b). JFAD and biaxial modification for a / W = 0.333.

1.4L

. . . .

,

. . . .

I

. . . .

1.2 1 (C) 1

I

. . . .

EPRI's J FAD k=0 - - - - k=l . . . . . . k=0.5

~'~'~

u

"~',x

~. 0.8

2a/w=0 25 ~,~, n~tal Da a k=l

0.6 0.4

•

0.2

o

0

. . . .

0

k=0.5

I!I

k---~'i

. . . .

0.5

,x "~-',fl -1 :

I

Sr I

. . . .

;If.l,

1.5

,,

,

2

Fig. 8(c). JFAD and biaxial modification for a / W = 0.25.

curves. T h e biaxial assessment curves (k > 0) move outward from the uniaxial case (k = 0). This means that the biaxial case is more safe than the uniaxial case in general, but the different assessment diagram can be obtained for different biaxial ratios. It is not appropriate to use uniaxial assessment curves simply for a biaxial case. T h e E P R I m e t h o d is not easy to use if the material has discontinuous stress-strain curves. T h e assessment curves will change not only for different crack lengths but also for biaxial ratios. In comparision, it is found that the proposed biaxially modifed failure assessment diagram of E P R I m e t h o d for biaxial loading agrees better with the experimental results than others.

References 1 R.P. Harrison, R.P. Loosemore, I. Milne and A.R. Dowling, Assessment of the integrity of strucutres containing defects, CEGB R / H / R 6 - R e v 3 , 1986. 2 C.F. Shih, M.D. German and V. Kumar, An engineering approach for examinig crack growth and stability in flawed structures, Int. J. Press. Vess. and Piping (1981).

121 3 J.M. Bloom, Procedure for the assessment of the integrity of nuclear pressure vessels and piping containing defects, EPRI NP-2431, 1982. 4 K.J. Miller and A.B. Kfouri, A comparision of elastic-plastic fracture parameters in biaxial stress states, Elastic-Plastic Fracture, ASTM ATP 668, 1979. 5 Dong-Ming Tan, Weng-Long Huang and Shu-Ho Dai, A study of failure assessment diagram (FAD) under biaxial loading, Proc. 3rd Int. Conf. on Biaxial /Multiaxial Fatigue, Stuttgart, Germany, 1989. 6 I. Milne, Failure assessment diagram and J-estimates, A comparision for Ferritic and Austenitic steel, Int. J. Press. Vess. and Piping, 13 (1983). 7 G. Alpa, Validity limits of the Dugdale model for thin cracked plates under biaxial loading, Eng. Fract. Mech., 12 (1979). 8 R.A. Ainsworth, The assessment of defects in structure of strain hardening material, Eng. Fract. Mech., 19 (1984) 633.