Assumptions Madb in Analysis 

 Besides the approximate strain and change in curvature-dis- 

 placement relations UBed in the derivation of the Donnell equa- 

 tion for an initially out-of -round cylinder, and the assumption 

 that it is initially stress free, it might be useful to list some of the 

 other assumptions that have been made both in the analysis of 

 this paper and that of Bodner and Berks. These are as followB; 



1 That the circumferential membrane stress in the shell is 

 constant along its length and equal to — pR/h, whereas this is 

 actually not the case. 



2 The assumption that the initial out-of-roundness is small 

 (with an order of magnitude of one shell thickness) is symmetric 

 with respect to the center line of the shell, and has the same form 

 as one of the buckling modes of a perfect cylinder with the same 

 shell dimensions. Actual shells rarely, if ever, satisfy the last two 

 requirements and so the question arises as to how the initial out- 

 of-roundness should be measured. 



3 The assumption that failure ooours (appearanoe of visible 

 lobes) when the most highly stressed points in the cylinder start 

 to yield. Actually, failure does not occur until plastic regions 

 form at the trough and crest points of the lobes. The pressure re- 

 quired to produce these yield zones is greater than that at which 

 the most highly stressed points begin to yield and neglect of this 

 effect therefore underestimates the strength of the shells. An 

 adequate theory to take this effect into account has not been de- 

 veloped as yet, but, as for beams, presumably the ratio of the 

 pressure to cause first yielding to the pressure required for the 

 formation of plastic regions depends on the relative magnitudes 

 of the direct stresses in the cylinder wall and the bending stresses 

 resulting from initial out-of-roundness. 



4 The assumption that Poisson's ratio is a constant and equals 

 0.3. 



As it is intended to apply the analyses developed for unstiffened 

 oylinderB to stiffened cylinders which failed by buckling between 

 ring stiffeners, it would be well if we enumerated the additional 

 assumptions that were made. These are: 



5 That the stiffening rings at the ends of any bay are per- 

 fectly circular; i.e., they do not have any initial out-of-roundness. 

 This never occurs in practice, of course, but should not be too 

 serious if the ciroularity of the stiffening rings is very much better 

 than that of the shell, or if the predominant mode of initial out-of- 

 roundness in the rings is very different from the predominant 

 mode in the shell. 



6 As for unstiffened cylinders, the circumferential membrane 

 stress in the perfect cylinder is assumed to be —pR/h, whereas 

 it actually varies along the length of the shell. A more correct 

 representation of the stress distribution would be obtained by 

 using the analysis of von Sanden and Giinther (9), or more ac- 

 curately still, that of Salerno and Pulos (10). 



Methods or Detebmininq Initial Out-of-Roundness 

 As mentioned hitherto, the analyses assume that the initial 

 out-of-roundness is symmetrical about the mid-length of the shell 

 and that its circumferential variation is in the shape of one of the 

 buckling modes of a perfect cylinder. Actual shells do not meet 

 either of these requirements. If we do not make a harmonic 

 analysis of the initial out-of-roundness, and also extend the theory 

 to account for the various harmonic components, the question 

 arises as to how we shall measure the quantity e, which is defined 

 in the analysis as the maximum initial out-of-roundness when its 

 shape is similar to one of the buckling modes. As far as the 

 authors are aware three simplified, semiempiricai methods for de- 

 termining the initial out-of-roundness have been proposed in the 

 literature so far. These are (see Fig. 1): 

 (o) The oentroid of the initial circularity contour is first deter- 



ARC LENGTH 

 »/m Rm 



•/h^O.247' 180 



METHOD (c) 



Fio. 1 Illustrations of Three Methods for Determining 



Initial Oot-of-Roundnxss of Cylinders at Station 4 of Model 



BR-6 



mined. Then the angle r/m, where m is the number of lobes into 

 which the perfect cylinder would buckle, is calculated. A sector 

 of a circle, subtending this angle r/m is then drawn on transparent 

 paper and placed with its apex at the centroid of the initial circu- 

 larity contour. The sector is then rotated so that it traverses the 

 entire circumference of the circularity contour until the location is 

 found at which the maximum difference between the two sector 

 radii occurs. The initial out-of-roundness is then taken as this 

 maximum difference. This method of determining the initial 

 out-of-roundness is essentially that proposed by Saunders, Tril- 

 ling, and Windenburg (11, 12). 



(6) Both the centroid and the area of the initial circularity con- 

 tour are determined. The radius R m of the circle whose 

 area is the same as that of the initial circularity contour 

 is then determined. A circle, with center at the centroid of 

 the initial circularity contour and of radius R n , is then drawn. 

 The initial out-of-roundness is then taken as the maximum value 

 of [R, — R n ] and [R„ — R t ], where R, and ft, are the radius vec- 

 tors from the centroid to points on the initial circularity contour 

 which are exterior and interior, respectively, to the circle of radius 

 ft m . A method for determining the initial out-of-roundness simi- 

 lar to the foregoing has been suggested, among others, by Bodner 

 and Berks (3). 



(c) As in (6) both the centroid of the initial circularity contour 

 and the radius ft„ of the mean circle are determined. AIbo, as in 

 (a), the angle r/m is calculated. The arc length of one half-lobe 

 is then obtained as (w/m)R m . This arc is then moved around the 

 initial circularity contour with its end points always in contact 

 with the contour. The initial out-of-roundness is then taken as 

 the maximum radial distance between the circularity contour and 

 the arc. This method for determining the initial out-of-roundness 

 is somewhat similar to the method proposed by Holt (5). 



Numerical Results 



In this section we present the resultB obtained by applying the 

 analysis developed for simply supported imperfect cylinders (3), 

 and its extension to clamped ends, to nine steel welded cylinders 

 that have been tested at the Taylor Model Basin. At failure, all 

 these models had lobes which partially covered the circumference 



