shell at the center of oue of the bays in which lobes first appeared 

 in Model BR-4. If we adopt method (c) described earlier as our 

 criterion for initial out-of-roundnesa, then the stiffening rings of 

 thiB model had about one tenth the'out-of-roundnesB of the shell. 

 Similar results also were obtained for the other models. Thus, for 

 the problem of inter-ring collapse of stiffened cylinders, the as- 

 sumption of zero initial out-of-roundness of the stiffening rings 

 appears to be a reasonable one. 



It might be thought that a harmonic analysis of the initial out- 

 of-roundness would show that the amplitude of the harmonic 

 component corresponding to the value of m which minimized p„r 

 (Equation [11 ] in Table 1) was many times that of the other com- 

 ponents. To investigate this point a harmonic analysis, using 72 

 subdivisions of the circumference, was made of the initial circu- 

 larity contour at Station 4 of Model BR-4 using Filon's method 

 (15). The number of waves into which a perfect cylinder with 

 shell dimensions similar to BR-4 would buckle is, according to the 

 linear theory used herein, 11 for simply supported ends and 12 for 

 clamped ends. From the harmonic analysis it was found that 

 there were nine harmonics with amplitudes greater than the 

 eleventh and two greater than the twelfth. It also was found that 

 the amplitude of the largest harmonic (m — 8) was more than 

 twice that of the twelfth mode and more than five times that of 

 the eleventh mode. Further, even this largest amplitude was 

 much too small to effect a reasonable correlation between theory 

 and experiment. Thus the harmonic analysis of the BR-4 initial 

 out-of-roundness contour did not produce any results which 

 might be useful in practice. 



In conclusion, it might be of interest to mention that some at- 

 tempts have been made to determine experimentally the longi- 

 tudinal form of the buckling displacemnts in the multibay cylin- 

 ders. The results of these few investigations are summarized in 

 reference (14). However, more experimental work still remains to 

 be done before any conclusions regarding the shape of the buckling 

 displacement can be made. 



Summary 



In the preceding sections an attempt has been made to explain 

 some of the discrepancies that exist between experimental and 

 theoretical results for cylinders subjected to external hydrostatic 

 pressure. To do this it was assumed that (a) the actual boundary 

 conditions were Bomewhere between the extremes of simple sup- 

 ports and clamped ends; (b) the initial out-of-roundness was 

 similar in form to one of the modes into which a perfect cylinder 

 would buckle; (c) the Btress distribution in the equilibrium prob- 

 lem for the perfect cylinder could be represented by the membrane 

 stresses; and (d) the linear small-deformation equations of equi- 

 librium would describe the problem adequately. Any cold-work- 

 ing, residual, or welding stresses, or any elastic nonhomogeneity 

 that might have been present were neglected. It also was as- 

 sumed that failure (formation of a lobe) would occur when the 

 stresses at the most highly stressed point satisfied the octahedral 

 shear-stress criterion. Three simplified methods of measuring the 

 initial out-of-roundness were also investigated. The simplified 

 analyses, together with the different methods of measuring out-of- 

 roundness, were applied to nine steel welded cylinders with length- 

 diameter ratios of '/« to 2, thickness-diameter ratios of 0.0025 to 

 0.0065, and yield points of the steel of 30,000 to 60,000 psi. The 

 correlation between experimental and theoretical results was 

 quite good, when method (c) was used for determining the initial 

 out-of-roundness of the cylinders. 



Acknowledgment 



The authors wish to express their sincere appreciation to Taylor 

 Model Basin staff members Messrs. R. F. Keefe, A. F. Kirstein, 

 and R. C. Slankard for their able assistance in performing the 



numerical computations and determining the initial out-of- 

 roundness of the models, and to Mr. R. Stuekey in executing the 

 drafting work. 



They also wish to thank Dr. S. R. Bodner of the Polytechnic 

 Institute of Brooklyn for his constructive comments on the 

 authors' work. 



BIBLIOGRAPHY 



1 "Bupkling of Multiple-Bay Ring-Reinforced Cylindrical Shells 

 Subject to Hydrostatic Pressure," by W. A. Nash, Journal of Ap- 

 plied Mechanics, Trans. ASME, vol. 75, 1953, pp. 469-174. See 

 also reference (7) in Bibliography; (also, DTMB Report 785, April, 

 1954). 



2 "A Study of the Collapsing Pressure of Thin-Walled Cylin- 

 ders," by R. G. Sturm, University of Illinois Engineering Experiment 

 Station, Bulletin No. 329, 1941. 



3 "The Effect of Imperfections on the Stresses in a Circular 

 Cylindrical Shell Under Hydrostatic Pressure," by S. R. Bodner and 

 W. Berks, PIBAL Report No. 210, Polytechnic Institute of Brook- 

 lyn, Brooklyn, N. Y., December, 1952. 



4 "Effect of Small Initial Irregularities on the StreBseB in Cylindri- 

 cal Shells," by T. S. Wu, L. E. Goodman, and N. M. Newinark, Uni- 

 versity of Illinois, N6ori-071(06), T. O. VI, Structural Research Series 

 No. 50, April, 1953. 



5 "A Procedure for Determining the Allowable Out-of-Round- 

 ness for Vessels Under External Pressure," by M. Holt, Trans. ASME, 

 vol. 74, 1952, pp. 1225-1230. 



6 "The Effect of Initial Deformations on the Behavior of a 

 Cylindrical Shell Under Axial Compression," by P. Cicala, Quarterly 

 of Applied Mathematics, vol. 9, 1951, pp. 273-293. 



7 "Buckling of Thin Cylindrical Shells Subject to Hydrostatio 

 Pressure," by W. A. Nash, J ournal of the Aeronautical Sciences, vol. 21, 

 1954, pp. 354-355. 



8 "Stability of Thin-Walled Tubes Under Torsion," by L. H. 

 Donnell, NACA TR 479, 1933. 



9 "The Strength of Cylindrical Shells, Stiffened by Frames and 

 Bulkheads, Under Uniform External Pressure on All Sides," by K. 

 von Sanden and K. GUnther, Werft and Reederei, vol. 1, nos. 8, 9, and 

 10, 1920; vol. 2, no. 17, 1921; DTMB Translation 38, March, 1952. 



10 "Stress Distribution in a Circular Cylindrical Shell Under 

 Hydrostatic Pressure Supported by Equally Spaced Circular Ring 

 Frames, Part I— Theory," by V. L. Salerno and J. Pulos, PIBAL Re- 

 port No. 171-A, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 

 June, 1951. 



11 "The Use of Models in Determining the Strength of Thin- 

 Walled Structures," by H. E. Saunders and D. F. Windenburg, Trans. 

 ASME, vol. 54. Paper APM-54-25, 1932. 



12 "Collapse by Instability of Thin Cylindrical Shells Under Ex- 

 ternal Pressure," by D. F. Windenburg and C. Trilling, TranB. ASME, 

 vol. 56, Paper APM-56-20, 1934; DTMB Report 385, July, 1934. 



13 "Tests of the ElastitrStability of a Ring-Stiffened Cylindrical 

 Shell, Model BR-5 (X =» 1.705), 'Subjected to Hydrostatic Pressure," 

 by R. C. Slankard and W. A. Nash, DTMB Report No. 822, May, 

 1953. 



14 "Tests of the Elastic Stability of a Ring-Stiffened Cylindrical 

 Shell, Model BR-4 (X = 1.103), Subjected to Hydrostatic Pressure," 

 by R. C. Slankard, DTMB Report 876, February, 1955. 



15 "Integral Transforms in Mathematical Physics," by C. J. 

 Tranter, Methuen and Company Ltd., London, England, and John 

 Wiley A Sons, Inc., New York, N. Y., 1951, Bection 5.4. 



16 "An Expression for the Volume Change of a Cylindrical Shell," 

 by R. Bart, DTMB Memorandum to Head, Structures Division, 

 dated February 28, 1955. 



Appendix 



Approximate Equations of Equilibrium for an Initiallt 

 Out-of-Round Cylinder 



TheBe equations have been derived previously by Bodner and 

 Berks (3) using a different approach. Our reason for this new 

 derivation is to make clear what approximations and assumptions 

 are involved in the simplified equations of equilibrium. 



Consider a cylinder of mean radius R which has a small radial 

 initial out-of-roundness w . Selecting x, 8, and z as co-ordinate 

 axes and denoting the longitudinal, tangential, and radial elastic 

 displacements by U, V, and W, it is then possible to show that 

 the strains at the middle surface are given by 



