An improved compounding method for calculating stress-intensity factors

Engineering Frurturr Mechunics Printed in Great Britain.

Vol. 23. No. 5. pp. 783-792.

AN IMPROVED CALCULATING

0013-7944186 $3.00 + .OO Pergamon Press Ltd.

1986

COMPOUNDING METHOD FOR STRESS-INTENSITY FACTORS

Royal Aircraft Establishment,

D. P. ROOKE Farnborough, Hants, GU 14 6TD, England

Abstract-The compounding methods for calculating stress-intensity factors for complex geometrical configurations are reexamined. It is shown that techniques which were developed specifically for problems involving localized loads, e.g. a pin-loaded hole with a crack at its edge, can also be used when the loading is remote from the crack. When these techniques are used for cracks at unloaded holes, and for cracks in stiffened sheets, the compounding equations are as simple as those derived previously. However, fewer additional calculations are now needed and numerical tests show that there is no loss of accuracy. These advantages make the new procedures preferable to those used in the earlier compounding method.

1. INTRODUCTION mechanics is now a well-established means of quantitatively assessing the behaviour of cracks in structural components. Such assessments enable engineers to establish inspection and maintenance schedules which are cost effective while, at the same time, ensuring the safety of the structure during its service life. The application of the principles of fracture mechanics requires a knowledge of the crack size, the service stress, the properties of the material and the stress-intensity factor. The first three quantities are usually available; however, in many practical problems, structural geometries and loadings are so complex that the available stressintensity-factor solutions are inadequate. Standard methods of evaluating stress-intensity factors are often costly and time consuming and many simpler methods have been developed[ I]: a particularly versatile method[2] is known as “compounding.” In the compounding method, stress-intensity factors for complex geometrical configurations are calculated from known results for simple ancillary configurations; it has wide applicability and is simple and cheap to use. It was originally developed[3] for cracks near to, but not touching, structural boundaries. In order to extend the method to cracks which intersect boundaries, for example edges, holes and stiffeners, it was necessary to introduce[4] the “equivalent crack” concept. The equivalent crack is defined as an isolated crack in the same stress tield, remote from the crack, with a stress-intensity factor equal to that of the original crack when all the boundaries except the one intersecting the crack are absent. In general, the equivalent crack has a different length from the original crack. This new concept was needed to facilitate the calculation of the contribution to the stressintensity factor from the presence of all the boundaries other than the one in contact with the crack. That boundary plus the crack are replaced by the equivalent crack before the effects of the other boundaries can be evaluated. The applicability of this model has been demonstrated [I, 21 and it has been used to solve problems of several cracks at a row of stiffeners[5] and at a row of unloaded holes[b]. However, the procedure involves substantial additional calculations: the length of the equivalent crack and the distances between the tip of the equivalent crack and the boundaries in the ancillary configurations must be calculated. If the problem being considered involves crack growth then the calculations must be done for every new crack length in the original configurations. In order to extend the method to cracks at loaded holes[2, 71 it was necessary to modify the equivalent crack model since it did not include any effects due to localized loads (or stresses) near the crack tip: thus the concept of the “equivalent load” was introduced. The equivalent load is the load required to act on an isolated crack to produce the same stress-intensity factor as the original crack at a loaded boundary in the absence of all other boundaries. A necessary addition to an equivalent load concept was an “equivalent stress” to represent the effects due FRACTURE

0 Controller,

Her Majesty’s Stationery Office, London 1986. 783

184

D. P. ROOKE

to remote loads. Both these concepts were used successfully[2, 71 to study the problem of cracked fastener holes near the edge of a sheet. The equivalent stress idea can, in fact, be used in compounding stress-intensity factors for cracks at unloaded boundaries, and replace the equivalent cracks used before[4, 61. The consequences of this change on the compounded solutions are examined in this paper. In section 2 the theory is developed and it is shown that the resulting equations are just as simple as those derived previously[4,6] and that less analysis is required since no equivalent-length calculations are needed. Some symmetries which exist in the original configuration can now be incorporated into the ancillary configurations. In section 3 stress-intensity factors obtained by the new procedure are compared with independent results for cracks from holes[8] and cracks at stiffeners[9, lo], and the accuracy is shown to be adequate for application to engineering structures. Further comparisons are made with earlier[4, 1I I work. Since the new procedures are simpler and the accuracy is as good, it is recommended that the equivalent stress be used in preference to the “equivalent crack” in the application of compounding to the calculation of stress-intensity factors for structural configurations. 2. THEORY The equivalent stress theory is developed in this section for a configuration consisting of a plane sheet containing two cracks at the edge of a hole that is located between two boundaries. The configuration is shown in Fig. 1, where the hole is circular of radius R; the two cracks, which lie along a diameter, are of lengths II and 1~; and the two boundaries are represented by B, and Bz. The sheet is subjected to a uniform, uniaxial, tensile stress u acting, remote from the hole, in a direction perpendicular to the crack line. The crack tips are A on the crack of length Ii and B on the crack of length 1~. For tip A, the resultant stress-intensity factor K, is given[2,41 by the following compounding equation: K, = Ko + 2 (K:, - Ko) + K,, all ” where K. is the stress-intensity

(1)

factor for tip A of the crack at the hole when all the other

(r

‘32 \

I t I

I I i u

Fig. 1. Cracked hole between two boundaries.

I

BI

Compounding

method for calculating stress-intensity

factors

785

I I I

81 I

--

“2

I

bl \

I I I

I I I

u

u

K=Kr

K=K,

Fig. 2. Ancillary configurations

I I I u‘

K=K;

for a cracked hole between two boundaries.

boundaries are absent and KC is the contribution due to boundary-boundary interactions. The factors KL are the stress-intensity factors for tip A of the equivalent crack in the presence of the nth boundary only. In previous work[2, 41 the equivalent crack was an isolated crack of length 2~2’ in a sheet with a remote stress a; the length 2~’ was defined so that the stressintensity factor of the equivalent crack was equal to &; that is,

uv% = Ko, In this new derivation the equivalent crack is an isolated crack of length 2a (= II + 2R + I*) in a sheet with a remote stress a’; the stress cr’ is defined so that the stress-intensity factor is again equal to Ko; therefore

uV’% =

Ko.

This definition of the equivalent crack is analogous to that used[7] in the application of compounding to cracks at loaded holes. The opening-mode stress-intensity factor for the configuration shown in Fig. 1 can be compounded from the known solution K. for two cracks at the edge of a hole in an infinite sheet which is subjected to a uniform stress u remote from the hole, and the solution K; for a crack between two boundaries B, and Bz in a sheet which is subjected to an equivalent stress u’ remote from the crack. The ancillary configurations required are shown in Fig. 2. Equation (1) for the resultant stress-intensity factor becomes, for tip A, K, = K,, + (K; - Ko) + Ke;

(4)

K, = K; + K,.

(5)

that is,

It is convenient to normalize eqn (5) with respect to K, the stress-intensity factor for an isolated crack of length 2a in a sheet subjected to a remote stress of (T. Therefore K=uVZ. Thus (5) becomes

(6)

D. P. ROOKE

786

where Qr = K,IK and Qr = K,IK. The term K;Ix can also be written, using eqn (3), in terms of normalized stress-intensity factors as follows:

K; -=i?

Ko it

x

K;

z

= Q,

0

$& =QoQ,,

(8)

where Q. = K,,IK and Q, is the normalized stress-intensity factor of tip A of the crack in the second ancillary configuration in Fig. 2. It is important to note that since K; will be proportional to CT’then it follows that Q, is independent of u’. Therefore eqn (7) becomes

Qr = QoQ, + Qe.

(9)

A similar equation exists for tip B where Q. and Q, are to be interpreted as the normalized stress-intensity factors for tip B in the ancillary configurations. Comparison of eqns (3) and (6) shows that the equivalent stress is related to the original stress by Q,; that is, (T’ = Q,,a.

(10)

Equation (9) is simpler than the corresponding equation that would result from the original compounding procedure[2, 41. The fact that the equivalent crack length 2~’ was different from the original crack length 2a meant that the position of each boundary had to be separately defined with respect to the equivalent crack in the ancillary configuration. There were, therefore, as many ancillary configurations as there were boundaries, and boundaries such as B, and Bz could not be considered together. Thus some geometrical symmetries that existed in the original configuration would not be present in the ancillary configurations. In the procedure described in this paper any symmetries will still be present and can often be used to reduce the number of ancillary configurations required. However, if Q, is not available for the crack between the boundaries B, and BZ, it can itself be compounded by considering B I and Bz separately. The ancillary configurations required are shown in Fig. 3 and consist of a crack of length 2a in the presence of one of the boundaries with a remote stress of u’. If K;, is the stress-intensity factor for tip A in the presence of B, and KL is the factor in the presence of Bz, then it follows[2, 31 that K; = Ko + (K;,

- Ko) + (K;z - Ko) + K;,

(11)

u’

u’

I 1I

I I t

at

a2

0

T

A

0

A

I I I

I I I

u’

u’

K: K;

w aI

a2

, K=K,,

K= K,?

Fig. 3. Ancillary configurations for a crack between two boundaries (u, = I, + R. (11= I2+ R).

Compounding

method for calculating

where K: is a boundary-boundary

Ql =

interaction

stress-intensity

factors

787

term. Therefore

2 =Q,, +Q,, (I

(12)

I + Q;,

where QI I = K; I/KU, Q12 = K;z/Ko and QL = KLIK. The normalized stress-intensity factors QI I and Q12 are independent of o’ since K;, . Ki2 and K. are proportional to u’. Thus eqn (9) becomes

Qr = QdQ,, +

Q,z

- I] + Qe,

(13)

where Qe now contains all the boundary-boundary interaction terms. The generalization of eqns (9) and (13) for an arbitrary number of boundaries

1 +

2 (Q,z - 1)I + Qe,

is given by

(14)

?l=I

where N is the total number of ancillary configurations, and may be less than the total number of boundaries. The Qn represent the normalized stress-intensity factors for the nth ancillary configuration; they are independent of cr’, the equivalent stress. Equation (14) is similar to that derived previously[2, 41, but now no calculations of equivalent crack lengths or equivalent distances to boundaries are required before the values of Qn can be determined from known solutions[l2]. The term Qe is determined, as before[2, 41, by representing the boundary-boundary interactions as two opposing forces P, acting on the edge of the hole in a direction through the hole centre and perpendicular to the crack line. The magnitude of P, is determined by the condition that the limiting value of K,, as the crack length 1 tends to zero, is given by lim {K,} = l.l2K,c~fl, /-+o where 1.12 is the edge correction factor and K, is the stress concentration site at the edge of the hole in the original untracked configuration.

(15) factor at the crack

3. TEST CASES The new procedure for compounding is tested in this section by calculating stress-intensity factors for several contigurations and comparing the results with earlier calculations. Two types of configurations are considered; cracks at the edges of circular holes in plane sheets and cracks at stiffeners in reinforced sheets. Stress-intensity factors are obtained, in section 3.1, for two cracks at a circular hole in a uniformly stressed strip and compared with the results of Newman[8] and previously compounded results[4]. In section 3.2 stress-intensity factors are obtained for a crack at the central stiffener in a periodic array of stiffeners in a uniformly stressed sheet. The results, for both a broken and an unbroken central stiffener, are compared with the results of Poe[9, lo] and previous compounded results[ll].

3.1

Cracks at a hole in a strip

The configuration shown in Fig. 4 consists of two cracks of equal length 1 at opposite ends of a diameter of a circular hole of radius R. The hole is located centrally in a strip of width 26 which is subjected to a uniform uniaxial tensile stress u remote from the hole; the stress acts in a direction parallel to the axis of the strip and perpendicular to the cracks. Also shown in Fig. 4 are the two ancillary configurations required for the determination of the stress-intensity factor. They are two cracks at the edge of a circular hole in an infinite sheet which is subjected to a uniform, uniaxial tensile stress remote from the hole, and a crack of length 2a ( = 21 + 2R)

788

D. P. ROOKE U’I

I i 1

I I I

2a 1

a

a

a

P R

A

A

R -

a

+

‘a

v

I I I

I t’ I

1I I

u

CT’

u

K:K,

K=K,

Fig. 4. Ancillary configurations

K= K;

for a cracked hole

in a strip.

centrally located in a strip of width 26; the crack is perpendicular to the axis of a strip and to the direction of the uniform, uniaxial tensile stress (T’ which acts remote from the hole. If the stress-intensity factor for crack tip A in the first ancillary configuration is KO and that for crack tip A in the second ancillary configuration is K; , then the resultant stress-intensity factor K, is given by (5). The normalized factor Qr is given by eqn (9), where QOcan be obtained from the work of Tweed and Rooke[ 131 and Qr from Case I. 1.1, Ref. (121. Thus if Qe can be determined then Qr can be calculated from (9). The term Q, is determined [2, 41 by considering the limiting behaviour of eqn (5) as the crack length tends to zero. The limits required are lim {K,} = 1 .12Ktafl,

(16)

l-+0

where Kt

is the stress concentration lim {K;} I-0

factor at the crack site in the untracked

= ‘iit {QIKo}

where Q,(R/b) is the normalized stress-intensity width 2b subjected to a uniform stress;

= Ql(R/b) ‘,i; {Ko},

(18)

1-O

factor for a hole in a uniformly stressed infinite sheet is 3; and lim {Ke} = 1.12 x s /+o

Substitution

(17)

factor for a crack of length 2R in a strip of

lim {Ko} = 1.12 x 3crfi, since the stress concentration

configuration;

(19)

m.

of eqns (16)-(19) into (5) gives Kg

= 3uQ, (R/b) + $$

;

WV

Compounding method for calculating stress-intensity factors Table

1. Parameters for boundary-boundary

789

interactions

2pe R/b

K,

Q(Rlb)

?lRU

0.25 0.50

3.24 4.32

I .02 1.18

0.18 0.78

that is, 2P, TRU

= K, -

3Q,(Rlb).

(21)

Thus P, can now be determined and Qe obtained from the work of Tweed and Rooke[ 131. For the two cases considered in this paper, namely R/b = 0.25 and 0.50, the parameters required in eqn (21) are given in Table 1 together with the values of 2P,l(nRcr). Values of the normalized stress-intensity factor Qr have been calculated for several crack lengths for the two test cases. They are compared in Table 2 with the numerical calculations of Newman[8] and the earlier[4] compounded results. The values of Q obtained in this paper are within 2% of those obtained by Newman[81 throughout the whole range of a/R and R/b. The differences are larger with the old compounding method[4], with a maximum of 4% for R/b = 0.25 and 9% for R/b = 0.5. The magnitude of the interaction term Qe is smaller with the new method than with the old. Previously Qe was approximately 7% and 30% of Qr for R/b = 0.25 and 0.5 respectively; the corresponding figures for the new method of compounding are 5% and 18%. 3.2 Single crack in a periodic array of stiffeners

The stress-intensity factor for a single crack located symmetrically about a stiffener in a periodically stiffened sheet has been obtained by Poe[9, lo] and by Rooke and Cartwright[l2], who used the original compounding method. The configuration studied is shown in Fig. 5. The parallel stiffeners are in a periodic array, a distance b apart, and are labelled with positive integers (+n) to the right of the crack and negative integers (-n) to the left. The stiffener across the crack is labelled n = 0. The crack of length 2a is perpendicular to the stiffeners and the sheet and stiffeners are subjected, remote from the crack, to a uniform, uniaxial tensile stress such that strain compatibility is maintained between the sheet and the stiffeners. The stiffeners which do not cross the crack are intact; the stiffener which crosses the crack (Fig. 5) may be broken or unbroken. Three ancillary configurations are required to calculate the stress-intensity factor for tip A; they are shown in Fig. 6. The tirst is a crack of length 2a located symmetrically about a single stiffener (So) in a sheet which is subjected to a uniform, uniaxial tensile stress u remote from the crack. A stress (&/Et)a acts on the stiffener, remote from the crack, where El and E2 are the Young’s modulus of the sheet and stiffener respectively. The stress-intensity factor for this configuration is K,,.

Table 2. Values of Q for two equal-length cracks at the central hole in a uniformly stressed strip

Rlb = 0.25

1.02 I .04 I .08 I .20 I .40 1.60 2.0

R/b = 0.5

Rooke[4]

This paper

Newman[8]

Rooke[4]

This paper

Newman[BI

0.66 0.86 I.11 I .22 1.25 I .28

0.67 0.87 1.09 I.18 1.22 1.27

0.66 0.85 I .08 I.18 I .22 1.28

0.66 0.90 I.19 1.63 1.95 -

0.65 0.88 I.16 I .52 I.81 -

0.65 0.88 1.14 1.50 I .82 -

D. P. ROOKE

790

E2 -0 El

E’* El

r

b

I IIr 2, El

I I IIu EZCT I IIo- 20 2D. El

El

El

Fig. 5. Crack at a stiffener in a periodically stiffened sheet

The second configuration is a crack of length 2a, whose centre is a distance b,, from a stiffener S,, to the right of the crack (i.e. on the side of tip A). The sheet is subjected to a stress u’ and the stiffener to a stress (EJEr)u’. The stress-intensity factor is K’+,,. The final ancillary configuration in Fig. 6 is a crack of length 2~2,whose centre is a distance b-, from a stiffener S_, to the left of the crack (i.e. on the opposite side to tip A). The sheet and stiffener are subjected to the same remote applied stresses as in the previous ancillary configuration; the stress-intensity factor is K’_ n.

lU1 Eitt -o-

El

u

b .”

+

Iiu’I K=

I L&y’ El K = K,:,

K, Fig. 6. Ancillary configurations

for a crack in a periodically stiffened sheet.

K : K_.,

Compounding

method for calculating stress-intensity

factors

791

Table 3. Values of Q for a crack at a stiffener in a periodically stiffened sheet 0 h 0.25

0.50 0.75 0.90

The resultant (1) as

Unbroken central stiffener

Broken central stiffener

Rookel I I]

This paper

Poe191

Rooke[ I I ]

This paper

Poe[ IO]

0.67 0.66 0.64 0.56

0.66 0.64 0.60 0.52

0.68 0.67 0.65 0.59

1.72 1.32 I.11 0.91

I .74

1.78 1.36 I.12 0.91

stress-intensity

I .36 I.13 0.94

factor for tip A in the original configuration

is given by eqn

x

2

Kr=Ko+ The normalized

(K:,,-Ko)+K,,

m # 0.

(22)

stress intensity factor is given by

1+

(Q,, - 1) + Qe,

i: ,n=

--x

m # 0,

(23)

1

where Q,, = KoIK and K,, = KhIKo which is independent of the stress u’. The boundaryboundary term Qe is expected to be small[ll] in stiffener configurations and is omitted from subsequent calculations. Values of Qr have been calculated from eqn (23) with Q0 and Qm obtained from Case 2.2.1 and 2.2.2 respectively[l2]. The results are compared in Table 3 with the numerical calculations of Poe[9, lo] and the earlier[l l] compounded results. As can be seen from Table 3 the values of Q,. obtained using the new compounding procedure differ by only a few percent from previous values[ 111. The agreement with Poe[9, 101 is slightly worse for an unbroken stiffener and slightly better for a broken stiffener. For crack lengths up to three-quarters of the bay width (a/b c 0.75) the maximum difference between the compounded results and Poe’s is 8%. This would normally be within the acceptable accuracy limits for most engineering applications, and would justify omitting the Qe correction terms. 4. DISCUSSION AND CONCLUSIONS A new compounding method using an “equivalent stress” concept has been shown (Tables 2 and 3) to give stress-intensity factors of acceptable accuracy for engineering structures. The equations in the new method are at least as simple as those in the previous method; and, if certain symmetries exist in the component geometry, they may be even simpler since fewer ancillary configurations will be required. The numerical procedures are easier to implement since the need is removed for additional calculations of equivalent crack lengths and equivalent distances to boundaries. Because of these advantages, the equivalent stress compounding technique is recommended in preference to the earlier “equivalent crack” technique for calculating stress-intensity factors. REFERENCES [I] D. P. Rooke, F. I. Baratta and D. J. Cartwright, Simple methods of determining stress intensity factors. Engng Fracture Mech. 14, 397-426 (1981). [2] D. P. Rooke, The compounding method of determining stress intensity factors for cracks in engineering structures. Ph.D. thesis, Department of Mechanical Engineering, University of Southampton (1982). [3] D. J. Cartwright and D. P. Rooke, Approximate stress intensity factors compounded from known solutions. Engng Fracture Mech. 6, 563-571 (1974). [4] D. P. Rooke, Stress intensity factors for cracked holes in the presence of other boundaries, in Fracture Mechanics in Engineering Practice (Ed. by P. Stanley), pp. 149-163. Applied Science Publishers, London (1977). [5] D. P. Rooke and D. J. Cartwright, Stress intensity factors for collinear cracks in a stiffened sheet. Int. J. Fracture 14, R237-240 (1978).

792

D. P. ROOKE

(61 D. P. Rooke, Stress intensity factors for cracks at a row of holes. Inr. J. Fracrure 18, R3l-36 (1982). [7] D. P. Rooke. Compounded stress intensity factors for cracks at fastener holes. Engng Fructure Mech. 19, 359374 (1984). [8] J. C. Newman Jr., An improved method of collocation for the stress analysis of cracked plates with various shaped boundaries. NASA TN D-6376 (I971 J. [9] C. C. Poe, Stress intensity factor for a cracked sheet with riveted and uniformly spaced stringers. NASA TR R358 (1971). [IO] C. C. Poe, The effect of broken stringers on the stress intensity factor for a uniformly stiffened sheet containing a crack. NASA TM X-71947 (1973). [I I] D. P. Rooke and D. J. Cartwright, The compounding method applied to cracks in stiffened sheets. Engn~ Fracture Mech. 8, 567-573 (1976). [I21 D. P. Rooke and D. J. Cartwright. Compendium of stress intensity factors. HMSO, London (1976). [I31 J. Tweed and D. P. Rooke, The elastic problem for an infinite solid containing a circular hole with a pair of radial edge cracks of different lengths. In!. J. Sci. 14, 925-933 (1976).