r. — d[ w „ +£*m] 



f, D — w„ + W0„ 



[7] 



The maximum bending stresses are then given by 



Cj, ~ ± — (Af,) mu ; <r^ - ± — - (Miinix. 



[8] 



To obtain the total normal stresses we now have to add the 

 membrane stresses to Equation [8]. To determine these latter we 

 solve Equations [Id) and [6] for the stress function F (periodic 

 terms only). The total membrane stresses are then given by 



2ft R> ' 



<r~> - 



pR 



+ F„. 



[91 



The total normal stresses are obtained by adding algebraically 

 Equations [8] and [9]. The greatest normal stresses occur at 

 mid-leDgth of the cylinder (z — L/2) and where sin md — ±1 

 (trough and crest points of the lobes). At these points the twist- 

 ing moment A/,j is zero and thus the normal stresses are principal 

 stresses. The absolute maximum normal stresses occur at the 

 outer shell wall for the trough points. 



Having obtained the maximum principal stresses a t and oj in 

 terms of the initial out-of-roundness, the geometric parameters 

 and the applied external hydrostatic pressure, we now employ the 

 octahedral shear-stress criterion of failure (which gives the same 

 results as the Hencky-von Mises criterion of failure), vis. 



<T(' + <r.* 



Oil, ■ 



[10] 



where <7, is the yield point of the material. Substitution of the 

 maximum principal stresses <7, and erg in Equation [10] then gives 

 an equation relating the initial out-of-roundness, the geometric 

 parameters of the shell, the yield point of the material and the 

 pressure at which the shell begins to yield p t . 



It should be noted that instead of using the yielding criterion 

 given by Equation [10] where <r, and <7» are principal stresses, it 

 is more accurate to use the expression 



<r$' + <?,* — "»", + 3r, s ». 



..[10a] 



where now <r„ o-j, and t,j are the normal and shear stresses at any 

 point and which are functions of x and 6. Yielding will first occur 

 in the cylinder for those values of x and 6 which maximize the 

 right-hand side of Equation [10a ]. However, to compute these 

 values of x and 8 by differentiation of equation [10a] involves 

 more complications than seem warranted. Trials indicate that the 

 stress condition at the outer shell wall for trough points of a lobe is 

 probably as unfavorable as anywhere else. As mentioned earlier 

 the twisting moment M^ is zero at these points and thus Equa- 

 tion [10a] reduces to Equation [10]. 



As we shall later present curves of p„ the pressure at which shell 

 yielding commences, versus e/ft, the initial eccentricity- 

 Bhell thickness ratio, for both simply supported and clamped-end 

 cylinders, we have summarized the results obtained in this paper 

 and those obtained by Bodner and Berks in Table 1. (We have 

 added a few terms to the latter solution, as Bodner and Berks 

 neglected the periodic terms in Equation [9]). Also, we used 

 the expression w^/R' for the circumferential change in curva- 

 ture instead of (ww/R* + vs/R) which was used by Bodner 

 and Berks when computing the bending stresses due to initial 

 out-of-roundness. Our expression is consistent with the expres- 

 sion used in deriving the approximate equations of equilibrium 

 for an initially out-of-round cylinder (see Appendix) and also is the 

 same as that used by Donnell in his work on perfect cylinders (8). 

 The effect of using ww/R' instead of (wm/R* + vg/R) is to 

 eliminate the quantity / which appears in the equations de- 

 veloped by Bodner and BerkB.) It can be seen that the final 

 equation (Equation [12] in Table 1) relating the initial out-of- 

 roundness and the pressure to cause first yielding is essentially 

 the same for both clamped-end and simply supported cylinders, 

 except for the factor of 2 required by the definition of e as the 

 maximum initial out>-of-roundness and the slight changes in 

 definition of the quantities K and (9 appearing in that equation. 



AN INITIAL OUT-OF-ROUNDNESS SIMILAR IN SHAPE TO ONE OF THE ASSUMED BUCKLING MODES 



-Simple supports- 



-Clamped ends- 



Equa- 

 tion 



AsBUfned w 

 and u* 



w — A sin mS i 



tci ■ e sin m9 cos — 



(Origin at mid-length) 



[2ti"I 

 1 — cos -r— ; 



«>»"■« /2 sin md 1 — cos — — 

 (Origin at one end) 



1 12(1 — ,') [m» + J']') . tR I ! 



m' + 2m , A 1 + A'] 

 12(1 — »') 



A' 



Equation relating p, 

 and e 



2p„(l — 



ft " V Per / I 4 



m 



[m« + A']' 



[8m- + \ A'] 



■4(1 — g + fl«)ff)'/i 



4(1 — + fi')K 



HW* 



m' + ►«« + 2(1 — ,') — 



( 1 for simple supports 

 1 2 for clamped ends 



R A 4 



2tR [11] 



L 



[12] 



h [m 1 + «']' 



I rm* + 8> + 2(1 — f 



"1 



[m» + «•]• 



m> + ,J' + 2(1 — , 



"" 



[m» + !»]« 



2m» — 1 + >A> + 2(1 — ►") 

 L[2m'— 1] + A> + 2(1 — ,') 



h [m' + A']» 



R m»A' 



2m' — 1 + rA 1 + 2(1 — »•) 



h [m> + A'] 

 R A' 



h [m> + A']' 



[-G&H 



