Effects of Boundary Conditions and Initial 



Out-of-Roundness on the Strength of 



Thin- Walled Cylinders Subject to 



External Hydrostatic Pressure 



By G. D. GALLETLY' and R. BART' 



Using classical small-deflection theory, an investigation 

 was made of the effects of boundary conditions and initial 

 out-of-roundness on the strength of cylinders subject to 

 external hydrostatic pressure. The equations developed 

 in this paper for initially out-of-round cylinders with 

 clamped ends, and a slightly modified form of the equa- 

 tions previously derived by Bodner and Berks for simply 

 supported ends, were applied to some actual test results ob- 

 tained from nine steel cylinders which had been subjected 

 to external hydrostatic pressure. Three semiempirical 

 methods for determining the initial out-of-roundness of 

 the cylinders also were investigated and these are described 

 in the paper. The investigation indicates that if the 

 initial out-of-roundness is determined in a manner similar 

 to that suggested by Holt then the correlation between 

 the experimental and theoretical results is quite good. 

 The investigation also indicates that while the difference 

 in collapse pressures for clamped-end and simply sup- 

 ported perfect cylinders may be quite considerable, this 

 does not appear to be the case when initial out-of-round- 

 nesses of a practical magnitude are considered. 



Introduction 



SEVERAL analyses have appeared in the literature for the 

 elastic buckling of a thin cylindrical shell subject to external 

 hydrostatic pressure (1).' The majority of these analyses 

 have been based upon the classical small-deformation theory of 

 thin Bhells and have assumed a geometrically perfect, stress-free 

 structure prior to loading. The correlation obtained between 

 these theories and experimental results has been good for long 

 cylinders but rather poor for short cylinders. Efforts are cur- 

 rently being made by several investigators to explain the dis- 

 crepancy between theory and experiment in the short-cylinder 

 range by the use of large-deflection theory. At this date, how- 

 ever, it is not known by how much this discrepancy will be re- 



1 Shell Development Company, Emeryville, Calif. Assoc. Mem. 

 ASME; formerly Head, Plates and Shells Section, David Taylor 

 Model Basin, Washington, D. C. 



1 Structural Research Engineer, David Taylor Model Basin, Wash- 

 ington, D. C. 



1 Numbers in parentheses refer to the Bibliography at the end of the 

 paper. 



For presentation at the National Applied Mechanics Conference, 

 Urbana, 111., June 14-16, 1956, of The American Society or Me- 

 chanical Enoineers. 



Discussion of this paper should be addressed to the Secretary, 

 ASME, 29 West 39th Street, New York, N. Y., and will be accepted 

 until one month after final publication of the paper itself in the 

 Journal or Applied Mechanics. 



Note: Statements and opinions advanced in papers are to be 

 understood as individual expressions of their authors and not those of 

 the Society. Manuscript reoeived by ASME Applied Mechanics 

 Division, May 31, 1955. Paper No. 66— APM-9. 



duced as all the final reports on their work have not been pub- 

 lished. 



One possible cause for the discrepancy between theoretical and 

 experimental results can be ascribed to the initial out-of-round- 

 ness of the cylinders, and a number of investigations, using small- 

 defleotion theory, already have been made on the effect of initial 

 irregularities on the collapse pressure of cylinders subject to ex- 

 ternal hydrostatic pressure (2, 3, 4). As was to be expected, these 

 analyses showed that the initial irregularities reduced the failure 

 pressures below those of the perfect cylinders. However, when 

 these analyses were applied to some models which had been tested 

 experimentally, they predicted failure pressures which were less 

 than three quarters of those observed experimentally. Since 

 these analyses had assumed simple supports at the ends of the 

 cylinders and it was probable that the boundary conditions of the 

 models were Bomewhere between the extremes of simple supports 

 and clamped ends, it was of interest to investigate the reduction 

 in collapse pressure of clamped-end cylinders due to initial irregu- 

 larities, to see if the assumed boundary conditions would explain 

 the discrepancy between experiment and theory. Also, the 

 analyses assume that the initial out-of-roundness in the cylinders 

 is similar to one of the modes into which a perfectly circular 

 cylinder of the same dimensions would buckle, and actual shells 

 never satisfy this condition. It thus seemed desirable to in- 

 vestigate the various simplified methods that have been sug- 

 gested for determining the initial out-of-roundness of the cylinders 

 to see what effect these had upon the computed failure pressure. 

 These methods are described in the paper. 



It is also of interest to note that there are other limitations in 

 the existing linearized theories. These are: 



(a) The fact that in these problems there usually exist other 

 buckling pressures close to the minimum buckling pressure. 

 ThuB, use ol only that mode of initial out-of-roundneBs which 

 corresponds to the minimum buckling pressure is really only de- 

 fensible at pressures very close to the minimum buckling pressure. 



(6) The simple yield criterion used to predict failure. 



This point is discussed under the section entitled "Assumptions 

 Made in Analysis." 



The authors have not investigated the foregoing factors but 

 hope to do so in the future. 



The approach used in this paper is similar to that of Bodner and 

 Berks (3), except that instead of using a Donnell-type equation 

 Galerkin's method was employed in conjunction with a modified 

 Donnell-type equation. The initial out-of-roundness pattern 

 assumed by Bodner and Berks was of the form 



u>n =■ e sin m cos — 



(origin at mid-length) while that assumed by the present authors 

 was 



