C.P. Andrasic; A.P. Parker -- Dimensionless Stress Intensity Factors for Cracked Thick Cylind

8/11/2019 C.P. Andrasic; A.P. Parker -- Dimensionless Stress Intensity Factors for Cracked Thick Cylind

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.556C-05

.4348-06

.O96C-07

.0211-W

1.152C+OO .174C-02 1.92DE-03

.643C-05

.0678-06

.736C-07

.504C-08

1.1598+00 .887C-02

.619C-03

.534C-04

.401C-06

.92OC-07

.797C-08


4/7


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6/7

192 Technical Note

The dimensionless K soluti ons presented of fer a way of obtaining accurate K values using the minimum of comput ing for crackedthick cylinder under polynomial crack face loading. The estimated errors associated with theK values in the tables are less than 1%.Consequently, the main source of error in their use will be in the least squares approximation of the crack face loading p(y).

EX MPLES

The followi ng examples demonstrate the use and accuracy of the tabulated K v alues. The first example is for a single radial crackin a thick cylinder (RI/RI = 2.0) subject to a uniform tension P on the outer surf ace. The hoop stress in the unflawed configu ration is

given by [9]:

00 = PRz2( It R,2/r2)/(Rz - RI), R, G r s R1,

which can be approximated by the follo wing polynom ial

(7)

P(Y)=P&r,] y/@-R,)], 0cy~(R~-R,), (8)

where the coefficients for the polynomi al a, are given by:

ao = 2.664, a, = - 2.568, a2 = 3.165,a3 = - 2.327, a4 = 0.735, a~ = aa = 0. (9)

The results are presented in Table 5 where comparisons have been made with th e work of Parker and Andrasic[B], and Bowie and

Freese[l]. Note that the y for the polynomial starts from the crack mo uth, hence I = RI t Y.The second example is for two external radial cracks in a thick cylinder subject to constant internal pressure P. For this case the

hoop stress in the unflawed configur ation is given by:

US = PR12(l t R22/r2)/(R22- RI), RI s r SRI IO)

which again can be approximated by a polynomi al (8) where

a,, = 0.2584, al = - 7.662, a2 = 1.668,a? = - 3.056, a4 = 2.439, a5 = ~6 = 0. (II)

Again note that i n this case r = R? - y. Results are presented in Table 6. Comparisons have been made wit h the results of Tracy (61 andParker and Andrasic[8].

RESULTS

For convenience of tabulation the real numbers in Tables I.142 have been written in the form:

nn nnEsnn (12)

where nn is a string of numeric characters and s is a sign, and the letterE represents a power of ten by which t he preceding number isto be mul tip lied. (e.g. l.E-6 represents the value 1.0 x 10-h.)

Table 5. Results of test calculations for a single internal radial crack in a thick cylinder loaded with uniformexternal tensi on P. KN = Pd(?ra)

i. Present ork

Q-R, )

0.010.020.030.040.050.060.070.030.090.1

: : :0.40.50.60.70.8

-r

K, fi.

2. 9562. 9312. 8932. 864

2. 8502. 8402. 8222. 804

2. 1912. 795

2. 7812. 8522. 9753. 1193. 3163. 5683. 967

-r

KI&2. 9492. 9202. 9012. 8722. 8452. 8322. 8242. 811

2. 1952. 781

2. 7192. 8592. 976

3. 1253. 3153. 5653. 955

6 t i f f .

-0.2-0.4

0.30.3

-0.2

-0.30.1

0.20.1

-0.3

-0.10.20.00.20.0

-0.1

-0.3i

2. 81 0.5

2. 78 0.02. 87 0.62. 99 0.53. 14 0.7

3. 32 0.1

3. 55 -0.5

3. 91 -1.4

1

Table 6. Results of test calculations for two external radial cracks in a thick cyli nder loaded with unifo rminternal pressure P. KM = Pd/ ?ra)

* Pmsent ork Parker d **draeic Tracyh-R, 1 h/h KXA $ Diff. Kr& %DIIP.

0.1 0., 03 0.3115 0. 4 0. 308 -0. 70. 2 0. 3482 0. 35w 0. 5 0. 343 - , . 5

0.3 0.3%8 0.3970 0. 1 0. 395 - 0. 5

I t: 0.4543. 5204 0.4529. 5188 - O. , 0. 3 0. 450. 515 - 0.9, .o0.6 0.5986 0.598? 0.1 0.590 - 1.40.7 0.7021 0.7015 - 0. 1 0. 688 - 2.0

0.8 0.8604 0.8632 0. 3


7/7

Department of Mechanical & Cioil Engineering,North Staflordshire Polytechnic,Beaconside,Staflord,England

Technical Note 193

C. P. ANDRASIC and A. P. PARKER

REFERENCES

[l] C. P. Andrasi c and A. P. Parker, Weight funct ions for crack ed, curv ed beams. Proc . 2nd Jnr. Conf. on Num.Methods in Fract.Mechanics , Swansea, pp. 67-78 (1980).

[2] A. F. Grandt, Stress intensity factors for cracked holes and rings loaded with polynomial crack face pressure di stributions. Inr. J.Fract ., 14, R221-R229. (1978).

[3] H. F. Bueckner A novel pri ncip le for the compu tation of stres s intens ity factor s. Z. Angew. Math. Mech. 50, 529-546 (1970).[4] J. R. Rice, Some remarks on Elastic Crack-Tip Stress fields. Jnt. J. SolidsStructures 8 751-758 (1972).[S] C. P. Andrasi c, Numerical Methods Appli ed to the Solutio n of Problems in Fracture Mechanic s. C.N.A.A., Ph.D., Thesis,

R.M.C.S., Shrivenh am, Wilts hire (198 ).[6] P. G. Tracy, Elastic analysis of radial cracks emanating from t he outer and inner surfaces of circular ring.Engng Frac. Mech. 11,

291-300 (1979).[7] 0. L. Bowi e and C. E. Freese, Elastic analysi s for a radial crack in a circ ular ring . Engng Fr ac. Mech. 4, 315-321 (1972).[E] A. P. Parker and C. P. Andrasic, Stress intensity factors for multiply-cracked thick cylind ers and cracked ring segments. Dept. of

Civil Engng Technote MAT/28, R.M.C.S., Shr ivenham, Wilts hire (1981).[9] S. P. Timosh enko and J. N. Goodier , Theory of Elastic ity. McGraw Hill, New York (1951).

(Received 12 November 1982; receiued for publication 17 January 1983)

Documents

C.P. Andrasic; A.P. Parker -- Dimensionless Stress Intensity Factors for Cracked Thick Cylind