sin md 1 



[l-cos^] 



(origin at one end of the cylinder). Thus, in both cases, the 

 initial out-of-roundness satisfied the relevant boundary condi- 

 tions and was also similar in form to one of the assumed buckling 

 modes. The magnitude of the initial out-of-roundness at the ends 

 and center of the cylinder was also the same in both cases. 



The Bodner-Berks solution and that presented herein thus 

 represent lower and upper bounds for the effect of initial eccen- 

 tricities on the collapse pressures of elastically supported un- 

 stiffened cylinders, when the initial eccentricities have the same 

 ehape as one of the assumed buckling modes of the perfect 

 cylinder. While the two solutions are not exact, they should 

 provide good approximations to the exact solutions. One of the 

 limitations of using Donnell's equation is that the number of cir- 

 cumferential lobes should be fairly high, and thus the results will 

 be slightly in error for very long cylinders which buckle into two 

 or three circumferential lobeB. 



The final results of the investigation are given in Fig. 2 and in 

 Tables 2, 4, and 5. It can be seen that the correlation between 

 experiment and theory is quite good when method (c) is used to 

 determine the initial out-of-roundness of the models. [Method 

 (c) is similar to that suggested by Holt (5).] However, it is not 

 claimed that the results give a complete answer to the problem 

 and more work of both an experimental and theoretical nature is 

 required. 



Method of Analysis 



The modifications to Donnell's equation brought about by 

 initial eccentricities in the shell have been presented by Bodner 

 and Berks (3), and, prior to them, by Cicala (6). The equations 

 also have been derived by the authors in the Appendix, using a 

 somewhat different approach to that adopted by the previously 

 mentioned authors. For the case of uniform external hydrostatic 

 pressure applied on all sides of an imperfect cylinder the relevant 

 equations are, from Equations [19a], [196], [21], and [24] in the 

 Appendix 



DViw + lv V ' iw "" + ?# l/2(t0 + u>o)„ 



+ ^ (u> + w„] 



.] 



= 0. 



.(a) 



— [^(2 + v)w Iz e + — w m . 



■ (b) 



V*t> 



V'F = — 



. (c) 

 ■(d) 



.[1] 



where 



h = thickness of shell 



R = mean radius of shell 



p = applied hydrostatic pressure 



E = modulus of elasticity 



D = Eh'/12{\ — c ! ) 



v = Poisson's ratio 



F = stress function of the total membrane stresses 



v - {w + R'w)' ™-w 



u>j m initial radial out-of-roundness ( + inwards) 



u, v, w - elastic axial, tangential, and radial ( + inwards) 

 displacements of the imperfect cylinder minus the 

 uniform compression experienced by a perfect 

 cylinder (see Equation [18] in the Appendix). 



The subscripts x and 6 indicate partial differentiations with re- 

 spect to thoBe variables. 



The patterns assumed for w and w„, and which satisfy the 

 boundary conditions for clamped-end cylinders, were as follows 



B sin md 1 



;0 fl 



t] 



2ttx1 



■12] 



B 



half amplitude of ui-displacement 

 e - maximum value of initial radial out-of-roundness 

 m = number of circumferential waves 

 L •= Unsupported length of shell 

 x, 6 - axial and angular co-ordinates 



If the Expressions [2] happen to be an exact solution of the prob- 

 lem, then they will satisfy the differential equation of equilibrium, 

 Equation [la], exactly. However, as both w and w„ were chosen 

 to satisfy the boundary conditions rather than the equilibrium 

 equation, this, in general, will not be the case. The resulting ex- 

 pression will be a function of x and 6 which we shall denote by Q. 

 Galerkin's equation for determining the relations between the co- 

 efficients B and e is then 



£7>-.*[i-«?] 



Rdddx = 0. 



13) 



where t assumes the values 1 , 2, 3 



For i ^ m Equation [3] will be found to be zero identically. 

 For i = m the following relation between B and e, obtained from 

 Equation [3], will be found to hold 



[4] 









2 Por — p 







where 



p„, 



is given 



by the expression 











Hi) 



[3m 4 + 2m'A' + A*] 



( 



fH 





E 





12(1 — v') + 



V 



(m 



R ) > 





+ A')V 









(3m' + 1/2A') 













. 2irR 

 4._ 







[5] 



The smallest value of the buckling pressure of the perfect 

 cylinder p„ is found by minimizing Equation [5] with respect to 

 m. A relation similar to that expressed by Equation [5] has re- 

 cently been presented by Nash (7) using an energy method. A re- 

 lation similar to- Equation [4] was also obtained by Bodner and 

 Berks for simply supported imperfect cylinders. 



Thus, from Equations [2] and [4], we obtain the following ex- 

 pression for w 



2 p« — p 



"[*—?]■ 



[6] 



The bending moments in the shell can then be calculated from 

 the relations 



