ASME+Inner and Outer Cracks in Internally Pressurized Cylinders

A. S. K O B A Y A SH !Professor, M em . A SM E N . P O L V A N I C H Graduate Student. A. F. EMERY Professor, M em . A SM E W. J. LOVE Professor, M em . A SM E University of W ashi ngton, Departm ent of M echanical Engineering, Seattle, W ashington I nner and Outer C rack s in I nternally Pressuriz ed C ylinders Stress intensity factors of pressurized surface cracks at the internal surface and un- pressurized surface cracks at the external surface of an internally pressurized cylinder are estimated from stress intensity factors of a semi-elliptical crack in a finite-thickness fiat plate. Curvature effects of the cylinder are determined by comparing two-dimen- sional finite element solutions of fixed-grip, single edge-notched plates and single edge- notched cylinders. Stress intensity factors for semi-elliptical cracks with crack aspect ratios of b/a = 0.2 and 0.98 at crack depths up to 80 percent of the cylindrical wall thickness are shown for internally pressurized cylinders with outer to inner diameter ratios, R„/Ri, ranging from 10:9 to 5:4 f or ol ^ er surface cracks and to 3:2 for inner surface cracks. I ntroduction During the past two years, the authors have used stress in- tensity factor solutions of a semi-elliptical crack in fiat plate with a curvature correction to estimate the stress intensity factors of an unpressurized inner semi-elliptical crack in a pressurized cylinder [l] 1 and in a thermally shocked cylinder [2] as well as in a quarter-elliptical crack at the bore of a rotating disk [3]. Some uncertainty in modeling the curvature correction in these earlier papers led later to revisions in the curvature correction as well as in the format of presenting the final results [4], The purpose of this paper is to apply this numerical technique in estimating the stress intensity factors of a pressurized semi- elliptical crack at the internal surface (pressurized inner crack) and an unpressurized semi-elliptical crack at the external sur- face (outer crack) of an internally pressurized cylinder. Despite the practical significance of these two problems neither of these two surface crack configurations has yet been examined in detail. An approximate three-dimensional solution for a pressurized inner semi-elliptical crack of a pressurized cylinder was con- sidered by Underwood [5]. A nother approximate solution for unpressurized inner and outer-semi-elliptical crack in a pres- ^umber s in brackets designate References at end of paper. C ontributed by the P ressure V essels and P iping Division and presented at the P etroleum Mechanical Engineering and P ressure V essels and P iping C onference, Mexico C ity, Mexico, September 10-24, 1976, of TH E A MERI C A N SO C I ETY O F MEC H A N I C A L EN G I N EERS. Manuscript received at A SME H ead- quarters. May 6, 1976. P aper N o. 76-P V P -6. surized cylinder was obtained by Kobayashi [6]. N either Under- wood's nor Kobayashi's solutions studied cracks of a depth in which the effect of back surface had to be considered. The ap- plication of three-dimensional finite element method to these problems is still limited at this time due to limitation in computer capacity and computing costs, despite increasing availability of three-dimensional finite element codes. The only three-di- mensional finite element solution to either of the two problems under consideration is that of Blackburn and H ellen [7] who computed the stress intensity factor of one pressurized inner semi-elliptical crack and one unpressurized outer semi-elliptical crack in a pressurized cylinder with an outer-to-inner diameter ratio of R„/Ri = 1. 461. O n the other hand, the two-dimensional counterpart of these two sets of problems has been studied by several investigators. For example, Bowie and Freese [8] and more recently C lifton, et al. [9] studied pressurized inner cracks in a pressurized cylinder and Emery, et al. [10] considered outer cracks in a pressurized cylinder. The latter problem was also studied by two-dimen- sional finite element analysis [11]. The above brief review of available two- and three-dimensional solutions of the two practical problems considered by this paper indicates the need for more accurate three-dimensional analysis of surface cracks in a pressurized cylinder. Method of Approach The method of approach used in this paper is the previously described iterative procedure based on the alternating technique for solving three-dimensional problems in fracture mechanics. This procedure, with the exception of recent developments by Journal of Pressure Vessel Technology FEB R UA R Y 1977 / 83 Copyright © 1977 by ASME Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms r ^ M K •M cM K S-|7jj?-./^75(o Z 8in 2 e • b 2 cos 2 e) l/ « M KS' K i/{~hr ,/vs < a2s| n29 + b * mZ * > l/4 } /// W-J& (Ro/rr*l ^ " ' " " o W °b " Pi 2R;' ^R5 Fig. 1 Procedure for estimating stress intensity factor of an outer semi-elliptical crack in a pressurized cylinder W&, "•(Ro+Rj) R0/ Ri =5 / 4 R0/R| * | 0/ 9 CRACK LENGTH , b/(R0-Rj) Fig. 3 Curvature correction, M o(0), for an outer crack in a pressurized cylinder /?„//?, between 5/4 and 10/9 Smith and his coworkers [12, 13], is limited t o surface cracks in propriate fictitious pressures on the fictitious part of t he ellip- flat plates. The procedure has been used extensively and is well documented by t he authors [14, 15] and thus will not be re- peated here. The actual numerical procedure was improved substantially by Kobayashi and Enet anya [14] who used ap- tical crack which protrudes into t he empt y space to force the numerical convergence of t he iterative procedure. F urt her dis- cussions on this numerical convergence appear in reference [13]. The stress intensity magnification factors, MKS(0), for a semi- elliptical crack in flat plates were then converted to those in a cylinder by using a curvature correction, M c (6). The procedure r o£D~L/ "['"i+'v a. 1.4 5 1.2 Z o I - a . 1 (£ O o UJ 5 i.o I - 5 O 0. 9 .2 .4 .6 8 CRACK LENGTH, ^/ (Ro-Ri ) 1.0 Fig. 2 Curvature correction, Mc(0), for a pressurized inner crack in i pressurized cylinder of R0/Ri less than 3/2 "%»!'>• PI «V > (Ro/Ril*-! + 1 Pi • INNER PRESSURE °il(V)"Oee<r) 20 40 60 80 100 CIRCULAR ANGLE.fi DEGREES Fig. 4 Stress intensity magnification factor of an inner semi-elliptical crack in a pressurized cylinder 84 / FEB R UA R Y 1977 Transactions of the ASME Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms CIRCULAR ANGLE, 8 DEGREES Fig. 5 Stress intensity magnification factor of an inner semi-elliptical crack in a pressurized cylinder used to estimate the curvature correction factor is described in references [1-4]. The resultant stress intensity magnification factor, MK(6), for the two problems in this paper are not presented in final graphical form as in references [1 and 2]. Rather, the users are asked to multiply the flat plate solutions by the corresponding curvature correction in order to obtain the resultant stress in- tensity factor. This format of presenting the anal results allows for future upgrading of the curvature correction should better modeling of the curvature correction become available. Elliptical C rack in a Flat Plate The pressure profile prescribed on the surface of a semi-ellip- tical crack in a flat plate is restricted by the availability of these solutions in the following form <T„(X, y) = -Boo + Boi 0-0 + Bo + B« (1) CIRCULAR ANGLE, 8 DEGREES Fig. S Stress intensity magnification factor of an inner semi-ellip- tical crack in a pressurized cylinder where the cartesian coordinate system and the geometry for this outer semi-elliptical crack are given in Fig. 1. By's are the yet-to-be-determined pressure coefficients and b is the semi- minor diameter of the ellipse. This pressure profile is least square fitted to the appropriate hoop stress, 0t>&{r), generated by the prescribed internal pressure, pi, on the uncracked cylinder. For the pressurized inner semi-elliptical crack, the internal pres- sure, pi, is added to the hoop stress. The stress intensity factors for a semi-elliptical crack in a flat plate and subjected to one of the four components of the pressure distribution, (r„(x, y) = 1, (1 - („/&)), (1 - (y/b)f, ( 1 — i.y/b)f, shown in equation (1), are given in references [1 and 2], A simple superposition of these four solutions will then yield the flat plate solutions with prescribed crack pressures. Two-Dimensional Analogs The curvature correction, M„{d), previously derived for an unpressurized inner crack [4] is used for the pressurized inner semi-elliptical crack problem. The justification for such use is obvious from the fact that a single curvature correction curve appears to cover the hoop stress distributions in pressurized cylinders with external to internal radius ratios, Ro/Ri, varying from 10/9 to 3/ 2. This insensitivity of the curvature correction to the Ro/Ri values indicates that it should be applicable to the inner semi-elliptical crack problem despite the added constant crack pressure of pu For the unpressurized outer semi-elliptical crack problem, a new curvature correction, M e (6), was deter- mined following the procedure described .in reference [4]. Results C urvature Correction M c («). I n order to make this paper self- contained, the curvature correction, M c (6), reported in references [1 and 2] and which is also used here for the pressurized inner crack problem is reproduced in Fig. 2. Fig. 3 shows the curvature correction, M c (d), for an outer crack in a pressurized cylinder for Ro/Ri < 5/4. A comparison of Figs. 2 and 3 shows, as expected from geometric considerations, that the curvature correction for an outer crack is slightly larger than that for an inner crack. The crack depth, b, in Figs. 2 and 3 is the local crack depth along the periphery of the semi-elliptical crack. The curvature correction, M c (6), thus starts with no correction at the major axis of the semi-elliptical crack or 6 = 0 deg and reaches its extreme value at the deepest penetration of the elliptical crack surface at 6 = 90 deg. N ote that the curvature correction for the outer crack varies slightly with the cylindrical geometry and approaches that of the pressurized inner crack for Ro/Ri = 10/ 9. Pressurized Inner Semi-Elliptical Crack in a Pressurized Cylinder. For a pressurized inner semi-elliptical crack in a pressurized cylinder, the crack pressure represented by equation (1) must be least square fitted to the hoop stress with the superimposed prescribed inner pressure of Journal of Pressure Vessel Technology FEB R UA R Y 1977 / 85 Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms 20 40 60 80 100 C IRC ULAR A NGL E, 8 DEGREES Fig. 7 Stress i ntensi ty m agni fi cati on factor of an i nner sem i -ci rcul ar crack i n a pressuriz ed cyl i nder ffwfo y) = p< Ro*/(Ri + yf + l RMRi* - 1 (2) where p,- is the prescribed inner pressure. The coordinate system for the inner surface crack is identical to that shown in Fig. 2. The least square fitting was accomplished within 0.2 percent of the hoop stress represented by equation (2). Using the flat plate solutions in references [1 and 2], the stress intensity magnification factor, MKS{6), was obtained for a pres- surized semi-elliptical crack in a flat plate as shown in Figs. 4 i °eet'l -Pi (Rp/r) +1 (R 0/R i) 2 -I . 2 °ZI <Y >"<W > p, • INNER PRESSURE Ro/Rj =5/4 R0/R; = 7/6 R [/R j "10/9 20 40 60 80 100 C IRC ULAR A NGL E, 8 DEGREES Fig. 9 Stress i ntensi ty m agni fi cati on factor of an outer sem i -el l i p- ti cal crack i n a pressuriz ed cyl i nder through,8. I n order to show the peculiarities of the two boundary value problems considered in this paper, the . stress intensity magnification factors are normalized with the local stress intensity factor from the basic solution of a completely embedded elliptical crack. 2 As a result, despite a dip in stress intensity magnifica- tion factor, MKS(6), for crack aspect ratio of b/a = 0. 2, the actual stress intensity factor exhibits a minimum value at the front surface, 6 = 0 deg, and attains a maximum value at its deepest penetration, 6 = 90 deg. For a near semi-circular crack with crack aspect ratio of b/a = 0. 98, however, the maximum stress ! E(k) which appears in F igs. 4 through 16 is the complete elliptic integral of the second kind with i? = 1 — (.b/a)'. 20 40 60 80 C IRC ULAR A NGL E, 6 DEGREES Fig. 8 Stress i ntensi ty m agni fi cati on factor of an inner sem i-circular crack in a pressuriz ed cyl i nder R 0/R ;"3/2 0 20 40 60 80 C IRC ULAR A NGL E. fi DEGREES Fig. 10 Stress intensity m agni fi cati on factor of an outer sem i -el l i p- tical crack i n a pressuriz ed cylinder i 86 / FEB R UA R Y 1977 Transactions of the ASME Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms or o i - o < 7 O C A T u. 7 U) < s I T Y z CO . o + CO c N k ^ t ? „ . (Ro/rl'+l 9 9 '<R 0/R j) 2 -I 20 ' 40 60 80 C IRC ULAR A NGL E, 9 DEGREES R 0/R j=3/2 R 0/R | = 5/4 R 0/R j=7/6 Ro/Ri = I O/9 K O N F A C T T I O < z < 1 - co z H Z t o CO I d C •jr CO CY V ) o u + CD e U) "4 N ~ > | u h° \ iC Ji . CD 100 Fig. 11 Stress ti cal crack i n a i ntensi ty m agni fi cati on factor of an outer sem i -el l i p- pressuriz ed cyl i nder R 0 / R ; - 3/ 2 R 0/R ; • 5/4 R o/R j=I O /9 C IRC ULAR A NGL E, 9 DEGREES Fig. 13 Stress i ntensi ty m agni fi cati on factor of an outer sem i -ci rcul ar crack i n a pressuriz ed cyl i nder intensity factors occur near the free surface and the actual stress intensity factor continuously decreases with increasing 9 even at the maximum crack depth of b/(B„ — Bi) = 0. 8. A lso note that the differences in M K s{6) at the front surface, 6 = 0 deg, and the deepest penetration, 6 - 90 deg, are much larger than those at the flat tension plate problem due to the stress gradient in the hoop stress which attains a maximum value at the front surface. The stress intensity magnification factor, MK{6), for a pres- surized inner semi-elliptical crack in a pressurized cylinder is thus obtained from M K {8) = M c (d) • MKS{6) (3) V R | "3/ 2 Ro/Ri= 5/4 R o/ R i ' 7/ 6 Ro/R|> 10/9 l . 0 L O 20 40 60 80 100 C IRC ULAR ANGLE ,8 DEGREES Fig. 12 Stress i ntensi ty m agni fi cati on factor of an outer sem i -ci rcul ar crack in a pressuriz ed cyl i nder where the curvature correction, M c (d), which is taken from the Fig. 2 for this problem, is a function of 6 and varies with the local crack depth, b, along the crack periphery. Unpressurized O uter Semi-Elliptical Crack in a P ressurized Cyl- inder. For the unpressurized outer semi-elliptical crack in a pressurized cylinder, the crack pressure represented by equation (1) must be least square fitted to the hoop stress of c«(z, V) = Pi R° 2 /(R° ~ Vf + 1 Ro^fR? - 1 (4) where the origin of the coordinate system used in equation (4) is now located on the external surface of the cylinder as shown in Fig. 3. A gain by the use of the flat plate solutions in references [1 and 2], the stress intensity magnification factor, MKS(9), is obtained for an unpressurized outer semi-elliptical crack in a flat plate shown in Figs. 9 through 13. The notable differences between MRS values for the pressurized inner cracks and the outer cracks are the increased MRS values at the front surface, 6 = 0 deg. A lso the stacking sequences of the MKS curves for varying R 0 /Bi ratios are reversed in the pressurized inner and outer crack problems. The actual stress intensity factor, MK{9), for the unpres- surized outer semi-elliptical crack in a pressurized cylinder is obtained from equation (3) using the appropriate curvature cor- rection, M c {6), of Fig. 3 and the MKS(8) values in Figa. 9 through 13. Discussion The stress intensity magnification factors for a flat plate, MKS(6), reported in this paper are for an oblong crack of b/a = 0.2 and a nearly semicircular crack of b/a = 0. 98. For semi- elliptical cracks with a crack aspect ratio of 0.2 < b/a < 0. 98, a reasonable estimate of MKS value can be obtained by linear interpolation of these two extreme values [15]. Using this interpolation procedure, the stress intensity mag- nification factors, MK, can be compared with those obtained by finite element analysis by Blackburn, et al. [7], who computed Journal of Pressure Vessel Technology FEB R UA R Y 1977 / 87 Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms o V P| S?ii» p, = INNER PRESSURE R„/ Ri - 3/ 2 ^ " ) = P i L ( V R ^ " l 'J 20 40 60 80 CIRCULAR ANGLE,S DEGREES Fig. 14 Stress Intensity magnification factor of a pressurized inner semi-elliptical crack in a pressurized cylinder Itf-Ri* P| • INNER PRESSURE R„/Ri = 3/2 b/a • 0.6 b/Q • 0.98J THIS PAPER UNDERWOOD [ 5] 0 0.2 0.4 0.6 0.8 CRACK DEPTH , b/(R0-Rj) Fig. 16 Stress intensity magnification factor of a pressurized inner semi-elliptical crack in a pressurized cylinder the stress intensity factors along the crack periphery of a pres- surized inner semi-elliptical crack and an outer semi-elliptical crack with a crack aspect ratio of b/a = 0.6 and a crack depth b/(R„ — Ri) = 0.4 in a pressurized cylinder of R 0 /Rt = 1.461 Figs. 14 and 15 show the interpolated MK{6) values for semi- elliptical cracks with crack aspect ratios of b/a = 0. 2, 0. 4, 0. 6, 0. 8 and 0.98 at a crack depth of b/(R 0 — .Ri) = 0.4 in a pressurized cylinder of R„/Ri = 3/ 2. MK{6) values obtained by Blackburn, et al. [7], who considered a cylindrical geometry of R„/Ri = 1.4615, differs from the R 0 /Ri = 1.5 used in Figs. 14 and 15, (Rp/r) +1 Fig ica! 20 40 60 80 2* CIRCULAR ANGLE.fi DEGREES 15 Stress intensity magnification factor of an outer semi-ellip- crack in a pressurized cylinder respectively. The higher finite element results in Fig. 14 could be accounted for by this difference in cylindrical geometry. The 15 percent higher finite element results at 6 = 0 deg in Fig. 15 could also be accounted for by this large difference in cylin- drical geometries from which M C {B) was derived. For a pressurized inner crack, the results in this paper at the deepest crack penetration, i. e. B = 90 deg, can be compared with the results of Underwood [5]. For R 0 /Ri = 1.5, the stress intensity factor of the deepest penetration was computed follow- ing Underwood's approximate formula and plotted in Fig. 16 together with the results of this paper. A lthough reasonable agreement between the two results was expeoted for shallow cracks, the large differences between the two results at deeper crack depth demonstrates the importance of the back surface correction in these deep surface flaw problems. C onclusions 1 Using the procedure described in reference [4], the stress intensity magnification factors were derived for pressurized inner and unpressurized outer semi-elliptical cracks of b/a => 0.2 and 0.98 at crack depths of b/a = 0. 4, 0.6 and 0.8 in a pressurizod cylinder of RJRi = 10/9, 7/6, 5/4 and 3/2 (inner crack only). 2 Good correlations were obtained with the stress intensity factors obtained by Blackburn, et al. [7], who used three-dimen- sional finite element analysis to analyze a pressurized inner crack with b/a = 0.6 at a crack depth of b/(R„ —• Ri) = 0.4 in a pressurized cylinder with R a /Ri = 1.461. Ack nowledgment The work reported in this paper is sponsored by the Electric P ower Research I nstitute under C ontract N o. RP 231-0-0. The authors wish to thank Drs. C onway C han and A . Gopalakrish- nan of EP RI for their encouragement throughout the course of this research program. 88 / FEB R UA R Y 1977 Transactions of the ASME Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms References 1 Kobayashi, A . S. , P olvanich, N . , Emery, A . F . , and L ove, W. J. , "Stress I nt ensi t y F act or of a Surface C rack in a P res- surized C ylinder," Computational Fracture Mechanics, E. F . Rybicki and S. E. Bengley, eds. , A SME, 1975, pp. 121-132. 2 Kobayashi, A . S. , Emery, A . F . , P olvanich, N . , and L ove, W. J. , "Surface F law in a Thermally Shocked H ollow C ylinder, " Trans, of the 3rd Int. Conf. on Struc. Mech. in Reactor Tech., C EC A , C EE, C EEA , L uxembourg, 1975, P aper G 4/ 3. 3 Kobayashi, A . S. , Emery, A , F . , P olvanich, N . , and L ove, W. J. , "C orner F law at t he Bore of a Rotating Disk," Journal of Engineering for Power, TRA N S. A SME, V ol. 98, Ser. A , N o. 4, O ct. 1976, pp. 465-472. 4 Kobayashi, A . S. , Emery, A . F . , P olvanich, N . , and L ove, W. J. , "Surface F law in a P ressurized and Thermally Shocked H ollow C ylinder," to be published in t he Int. Journal of Pressure Vessels and Pipings. 5 Underwood, J. H . , "Stress I nt ensi t y F actors for I nternally P ressurized Thick-Wall C ylinder," Stress Analysis and Growth of Crack, A STM STP 513, 1972, pp. 59-70. 6 Kobayashi, A . S. , "A Simple P rocedure for Est i mat i ng Stress I nt ensi t y F actor in Region of H igh Stress G radi ent , " Significance of Defects in Welded Structures, T. Kanazawa and A . S. Kobayashi, eds. , University of Tokyo P ress, 1974, pp. 127-143. 7 Blackburn, W. S. , and H ellen, T. K. , "C alculation of Stress I ntensity F actors for Elliptical and Semi-Elliptical C racks in Blocks and C ylinders," C entral Electricity G enerating Board Report N o. RD/ B/ N 3103, Jul y 1974. (Abstracts cont'd from p. 82) Moment C apability of V alves Using Semi-A nalytical Finite Element A pproach (76-P et-38), by M. S. Kalsi, Manager, Design A nalysis A ssoc. Mem. A SME, and B. L . McDougal, Engineer, Design A nalysis, W-K-M V alve Division, A C F I ndustries, H ouston, Tex. C omparison of test results with analysis shows t hat t he semi- analytical finite-element technique can be used to accurately predict t he behavior of thick axisymmetric structures under non- axisymmetric loadings. The application of this method to a valve subjected to pipe bending moment is presented. High Notch Toughness Steel for Large LPG Tanks (76-Pet-57), by E. V . Bravenec, Supervising Metallurgist, Quality A ssurance, H ouston Works, Mem. A SME; and R. L . H artzell, Supervisor, Technical Services, Mem. A SME, A rmco Steel C orp. , H ouston, Tex. H igher C harpy V -notch impact (C V N ) energy t han is normally exhibited by A STM A 537 C lass 2 plate was needed to provide t he C V N energy of 35 ft-lb at -50 F (48 J at -46 C ) specified in the' heat-affected zones of t he weldments in large L P G tanks exceeding 127,000-m 3 capacity. Using special process- ing and rare eart h additions for sulfide shape control, plate was produced with C V N values more than twice those of conventional plate. The C V N values in the heat-affected zones were met and exceeded in both t he longitudinal and transverse directions using both t he shielded metal arc and t he submerged arc welding processes. Boreholes' Stability and I ts P rognosing in L edge Rock Masses (76- P et-58), by I . A . Turchaninov, P rofessor, Director of Mining I n- stitute of Kola Branch, of t he USSR A cademy of Sciences, R. V . Medvedev, and E. V . Kasparjan, Senior Research Workers, Mi n- ing I nstitute, S. M. Kirov Kola Branch of the USSR, A cademy of Sciences, Murmansk Region, USSR. We suggest t he method which permits to estimate and make prognosis of borehole stabil- 8 Bowie, O . L . , and F reese, C . E. , "Elastic A nalysis for Radial C rack in a C ircular Ri ng, " Engineering Fracture Mechan- , ics, V ol. 4, N o. 2, June 1972, pp. 315-322. 9 C lifton, R. J. , Simonson, E. R. , Jones, A . H . , and G reen, S. J. , "Det ermi nat i on of t he C ritical Stress-I ntensity F act or Kio from I nternally P ressurized Thick-Walled V essels," Ex- ; perimental Mechanics, V ol. 16, N o. 6, June 1976, pp. 233-238. 10 Emery, A . F . , and Segedin, C . M. , "The Eval uat i on of t he ' Stress I nt ensi t y F act or for C racks Subjected to Tension, Torsi on and F lexion by an Efficient N umerical Technique, " Journal of • Basic Engineering, TRAN Si A SME, V ol. 94, Series D. , N o. 2, June 1972, pp. 387-393. 11 Kobayashi, A . S. , "F ract ure Mechanics, " Experimental Techniques in Fracture Mechanics, A . S. Kobayashi, ed. , I owa J St at e University P ress, 1973, pp. 4-37. 6 12 Browning, W. M. , "The A nalysis of a Semi-C ircular C rack Emanat i ng from a H ole in a P l at e, " P hD thesis, C olorado r St at e University, Dec. 1974. i 13 G anong, G . C , "Quarter-Elliptical C racks Emanat i ng from H oles in P l at es, " P hD thesis, C olorado St at e University, , Jul y 1975. ' 14 Kobayashi, A . S. , and Enet anya, A . N . , "Stress I nt ensi t y 1 F act or of a C orner C rack, " Mechanics of Crack Growth, A STM . STP 590, 1975, pp. 477-495. 15 Kobayashi , A . S. , Enet anya, A . N . , and Shah, R. C , E "Stress I nt ensi t y F actors of Elliptical C racks, " Prospects of i Fracture Mechanics, G . C . Sih, H . C . van Elst, and D. Broek, 1 eds. , N oordhoff I nt ernat i onal P ublishing, L eyden, The N et her- lands, 1975, pp. 525-544. i t y in ledge rock masses on t he basis of complex determination o rock properties by core tests and calculations of destruction by t he theory of maxi mum equilibrium. Well-H ole Temperature Distribution in the P resence of A quifers (76- P et-59), by C . A . O ster, Battelle-N orthwest, and W. A . Scheffler, Joi nt C enter for G raduat e Study, Mem. A SME, Battelle, P acific N orthwest L aboratories, Richland, Wash. A method is described for determining t he t emperat ure distribution in a circulating drilling fluid when aquifers are present in t he formation. The dept h of an aquifer relative t o t he well dept h is shown to be an i mport ant parameter. A n aquifer near t he surface has much less influence on t he t emperat ure distributions t han one located near the well bot t om. I f t he drilling fluid has much greater density t han t he entering formation water, then t he temperature dis- tributions are altered significantly. V ibration Energy: A Quick A pproach to Rotor Dynamic O ptimization (76-P et-60), by H . R. Simmons, Senior Research Engineer, A pplied P hysics Div. , Southwest Research I nstitute, San A ntonio, Tex. Mem. A SME. A method is described for identifying the most significant rotor part s contributing to a vibration problem and for calculating critical speed changes due to structural modifica- tion of those part s. A pplication of t he method, which is based on t he principles used by Rayleigh, requires no more t han inspec- tion of or direct hand calculations from tabulated energy distribu- tion functions. Rotor optimization is accomplished by relating t he potential critical speed benefit of a proposed fix with its real cost and degree of difficulty. The computer technique for gen- erating t he baseline critical speeds and energy distribution tables has been adapt ed to a portable telephone computer terminal. Thus, this technique is highly suitable for troubleshooting and correcting field vibration problems. (Abstracts cont'd on p. 99) Journal of Pressure Vessel Technology FEB R UA R Y 1977 / 89 Downloaded From: http://pressurevesseltech.asmedigitalcollection.asme.org/ on 07/11/2014 Terms of Use: http://asme.org/terms Documents Similar To ASME+Inner and Outer Cracks in Internally Pressurized CylindersSkip carouselcarousel previouscarousel nexthtes tugFEM-Chapter-7.pdfansysconcrete technology homework 2.pdfPaper Crack GrowthExam114 - Theory and Mechanica Model Topics - Finite Element Analysis OverviewChapter 3- Self Assesment Truss StructureRr410104 Finite Element Methods124-TMT09-061Exploring Methods for Measuring Pipe WeldDamage Crack Rock CalculationFett(1998)ABSA FEA Reqt[IJCST-V5I3P29]:Mr. Shashikant Ashok Sandhan, Prof. V. L. 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