lHl I“ ll IN I l I \ I ‘ HIHH'I ANAWSES 03 {a CYLSNDRECAE: $91733; WEE EMMQ‘JAELE SEPFQRW fixed: {or E‘hé 90g?” (1% 55.- 8‘ MECHEGMQ 51‘s??? ENEVERSE‘Y‘E’ Ciaévge £31. Srmgam 32‘.— @633 harms LIBRARY Michigan State University This is to certifg that the thesis entitled ANALYSIS OF A CYLINDRICAL SHELL WITH IMMOVABLE SUPPORTS presented by George A. Granger, Jr. has been accepted towards fulfillment of the requirements for M. S. degree in Civil Engineering /;?:(%//{L;¢L/ Major professor Date //<:2 7 / 7 '/ [/[l‘j f/ 0-169 ABSTRACT ANALYSIS OF A CYLINDRICAL SHELL WITH IMMDVABLE SUPPORTS By George A. Granger, Jr. An analysis is made of a circular cylindrical shell with built- in ends subject to axially symmetrical loading. The two ends of the shell are not allowed to move toward each other. The shell considered has a constant radius and thickness. The analysis presented differs from that currently available in the technical literature in that the built-in ends cannot approach each other longitudinally. The purpose of this thesis is to analyze this problem and to determine quantitatively, for a shell of fixed pr0portion, the rela- tive magnitude of the membrane stresses and the bending stresses. Also a comparison is made with the case in which the ends of the shell can move toward each other. The usual thin-shell theory is applicable. The behavior of the structure is governed by a fourth—order differential equation and a constraint equation. Numerical results were obtained for a shell having a length-to-radius ratio of unity and a thickness—to— radius ratio of 0.01. These results are presented in the form of tables and figures. The results indicate that: l) The moment is about 7% lower with the immovable ends and longitudinal membrane forces acting, 2) the total stress is about 7% higher, the longitudinal membrane stress accounting for about lli% of the total, 3) the relationship between maximum stress and loading intensity is essentially linear, and 1;) the bending stress at midspan is negligibly small compared to that at the boundary. ANALYSIS OF A CYLINDRICAL SHELL WITH MDVABIE SUPPORTS By $wml.®w@mJL A THESIS submitted,to Michigan State university in partial fulfillment of the requirements for the degree of NESTER OF SCIENCE Department of Civil Engineering 1963 ACIWCNLEDCMENT The author wishes to acknowledge the contribution of Dr. Rdbert K.‘Wen for initiating the analysis presented in this thesis, for checking the final analysis, and for his many helpful suggestions and comments. ii TABLE OF CONTENTS Acmm MT 0 O O O O O O O O O O 0 LIST w T-ABIJES O O O O O O O O O O O 0 LIST OF FIGURE 0 O 0 O O O O O O O O O NOTATION I II III DITRODUCTION.......... ANALYSISCFPROBLEM . . . . . 2.1. Derivation of Governing Equations 2.2 Solution of Governing Equations 2.2.1 CaseI. (xi—4:0) 2.2.2 CaseII (°<‘- were) 2.2.3 Case III («P-ADO) NUMERICAL assume AND CONCLUDING REMARKS 3.1 General 3.2 Variation of Stresses and Deflections Along Shell 3.3 Variation of Combined Stresses at the Boundary and at the.Midspan 3.h Concluding Remarks iii Page ii \0 -<., B, Y Shell parameters 7\ Loading parameter ex Unit elongation in x direction 7) Poisson's ratio 63 II'm/h I INTRODUCTION This thesis deals with the analysis of a circular cylindrical shell with built-in ends subject to axially symmetrical loading. Solutions of this type of shell have been given by Timoshenko.1 'However, in those solutions the inter-relationship between the longitudinal membrane forces and.bending stresses has been neglected. In this study the analysis deals with the case in which the two ends of the shell are not allowed to move toward.each other. This constraint would cause tensile longitudinal membrane stresses to exist in the shell as it is deformed.by external load. On the other hand, it is expected that this membrane force mould decrease the deformation and the bending stresses in the shell. The purpose of this thesis is to analyse the preceding problem and to determine quantitatively, for a shell of fixed proportion, the relative magnitude of the membrane stresses and.the bending stresses. Also a comparison of the numerical results thus obtained is made with the case in which the ends of the shell can move to— 'ward each other and the effects of membrane forces on bending are neglected. In the analysis the usual small deflection theory of elastic thin shells is applied. The major assumptions are 1) that all do- formations are small, 2) the material obeys Hooke's Law under all 13. Timoshenko and S.‘Uoinowsky-Krieger, "Theory of Plates and Shells" 2nd ed, pp. h75-78, 1959. levels of loading, 3) any plane perpendicular to the neutral sur— face before bending remains a plane perpendicular to the deflected neutral surface after bending, and h) the radial normal stress is zero. The analysis yields a fourth-order differential equation with a parameter (representing the longitudinal membrane stress) that has to satisfy a constraint equation. Numerical results were obtained for a shell having a length- to—radius ratio of unity and a thickness-to-radius ratio of 0.01. In comparison with the numerical results for the case in which the effects of membrane forces are neglected, the results obtained from.the present analysis show that l) the bending moment is about 7% lower, 2) the total stress is about 7% higher, since the membrane stress amounts to about lh% of the total stress, and 3) the re- lationship between loading and stress is essentially linear. II ANALYSIS OF PROBLEM 2.1 Derivation of Governipggquations The cylinder considered in this thesis is shown in Figure I. It has length 9. which is maintained constant, radius a, thickness h, and is subjected to an internal pressure q. The coordinate system is as indicated. Shown in Figure II is a differential element cut from the cylinder by passing two planes containing the axis of the cylinder and two planes normal to it. Shown also in this figure are the internal forces acting in the shell. These forces are bending moments, Mx and Ma, membrane forces, Nx and Na, and shearing force, Qx' Their positive senses are shown in this figure. The dis- placements are depicted in Figure III, and are u in the x-direction, and w in, the z direction. Writing the equations for equilibrium of forces and moments, one obtains the following: Nx - constant (1) de N d3»: N .3}... + x3; .. :— + q a O (2) Q5. - i“: r 0 (3) dx The internal forces in terms of the displacements are: Nx=%:[fl-+§C%¥I+=> “all - (1*) dx N, =§3§ [BC + a) i:— +%(?:)1] (5) a. ll ;; _. VT X ”is? (6) 1. where E = Youngs modulus of elasticity a) e Poissons ratio 3 D = Flexur-al rigidity = REE—:37). Substituting Eqs. (3), (5), and (6) into (2) results in the follow— ing: _ SL1}: 1W __ ‘5‘“ mi- fl d'LU' 2‘ = j ‘5 I Wig: «Wu L733)“ “KENS 0“) in“ S:;:7xr.‘i.ng Eq. (1;) for du/dx and substituting into 13d. (8) one obtains &—m£fl-+Qm:-Lfl+i (0) av D 61- 2135 do D ' Equation (9) is a linc‘r Tcuxlt~rxr r diffcrrntial equation in terms of the radial deflection, external loading, and the normal fcrce. To facilitat: the solution it is wide dimensionless by in~ troducing the £0 lowing quantities: 9‘33}; 1/2;— [K By making the appropriate substitutions into Eq. (9) and multiply- . '3 '. .. . . ing by 9.) and 9/11 there emerges the dimens‘ionless equation 1+ fi, 1 5 v S‘ e __ make; #1356: _, U182 +334. (10) at? D A»? cat-D D we Din Introducing the notation 2. 0L 3 £131; B = EBB“; Y :: L1); >\ = 1&5 B 0.15 cm DH Equation (10) takes the form: ale._.ié§. +BeI=—-KY+7\ (11) “l" ch? Equation (11) is in the form of an ordinary differential equation with constant coefficients which may be solved by the usual methods. However, before a complete solution can be obtained, the value of the parameterecmust be known. In order to accomplish this one makes use of the conditions thatai, the change in lengthfl due to the loading, is equal to zero. +'\3- + ‘\L AQ- EX ix=-—:f\\-(& yuai - \\3_ *li A1¢£Q*¢1_§:A— 2h fail“ WISE) [m— +:)d&+_'_i)_ ”Ila-ll“ and now equating (12) Elia—m / MT~~ =1 W“ a. 1mm, it?) "‘7‘ * we)!“ it Take Eq. (12) and multiply through by—— E~\'-‘—Q and there emerges wk: ‘(Qéa-TM + Duff—eb- (NY (13) -\\L From Eqs. (11) and (13) one can solve foroQand e . 2.2 Solution of GoverningEquations To obtain the complementary solution of the differential e- quation, let Sc: 63"“? AAQ _ up. M 1 Substituting the above back into the homogeneous differential equation, we have TON-—-°ngeu+£$==() where Wm:- 3' W The complete solution is thus given by e 2:: Q, e‘N‘YJr QléMVY'F Qae‘ni‘nr Q‘\QTM1~Y—\- 99 (1)4) where 0,, 02, 03’ and C,4 are constants of integration to be deter- mined from the boundary conditions and G? is the particular solution given by the following: _ '11 99 "‘ “E— (15) Using the relations Emil: Cosh my 4-- 05mka 'W\ e Y= Coax. \MOY - th‘nmy and censiiering symmetry, Eq. (1h) may be written as ’\ -. a. b -- Ages“ rap-y) + B Kiosk WIN)*% (16) where Q. « -uo Yu;:_-+ 0k and bdrm The form of the solution depends on whether the quantity «ll-AA is less than, equal to, or greater than zero. 2.2.1 Case I Cali—saga) In this case :111 --\ F459, and me - -\ /§_—eg_ in which i 8%. letting %’=°¢=‘XU‘MQQ§§=—L§W\¢) in WhiCh the latter expression is the polar representation off? . = \ I ' NW ‘5. l; wane +,\—\/3g_°¢] =15: ngi] L . . M1? 1 EVA-est. -J\_\/)~‘ ac] = “‘52 [such] where s —\/&+.¢ , and t 8%. Therefore the solution has the following form: 1 g = Rhett “\wait Y3+Mfimhk in. ex) “\hkk Y) "r e? ( 7) Consider now the boundary conditions and external loading. As stated previously the loading is an internal uniform pressure and the boundary conditions are that the slope and deflection vanish at the support. Applying these conditions one obtains the follow- ing expressions: ens (cos 3 3 in 19p ‘3 A‘s? (18) “Him gas-am“ 4: A=-Lthe$A’Qe>LSm " Sh e-Ae a £s.m\‘i~«-A.s\n‘st ? A ? (19) The final form is therefore: 9 = RA‘ {CQSK‘RAcPchs‘X‘uPa- Agsmm M‘Xsm‘hmry Be? (20) Performing the necessary operations on Eq. (20) and substitut- ing into Eq. (13) yields: __ a oL—Jo e? \g; \‘lY 99K; (21) where a J. 1 >- < g IffiiDESsEEtiiD flisaicfiexé! 3)) Kf‘ %&(£A\p+1A$\‘(‘§ Afi‘fiAan} ‘9 3m *- +é~ “)1 +5:\<~‘: armada—e Amid} ecsmmi-Mém‘w) av * A3? L .‘Kl'K Q4”? A A 033A A £1~AKAD (0 (£65535- “M‘sw‘ '4‘wa 55. 3 >- ‘ l ‘ 1 ‘ ti‘a-n. Q. - W:- A \m’a A79 - fiAjz-‘x A‘, 0431‘” \ V «a: U4 x53???) k—k-bq—‘XJ‘ a) (amigfiw emf-k m) {mtg—rt] Solving; now f oreP gives: as“- --->r%$'~ "’- t: ”V‘r‘KEE "M \ lo but ‘: Z '._. -l-fi’T-k‘k 9V 2;- “3 Therefore solving for the loading function, >\ , gives the final form: >\=0\ , gives: ) =o< Y—Ys KY: 1'— %; VY‘Q+$§— (27) The moment and deflection equations are: Moment MK ‘= 4D}- [B‘MXCOSMMU‘D‘PB‘VA1V\.6\V\\M\T3 + 1%kM(an\AWW‘A] GP (28) Deflection may[g‘tawpareficmmgs] e? (29) 2.2.3 Case III (gm In this case m1 -"\/==L;Eg_ , and m2 ‘W/LESQ. . Therefore the solution has the following form: 9 = F\ Q&M‘W+E1Q.§b\hmln\,+e? (3o) 11 Applying the boundary conditions and external loading as previous- ly stated, one obtains the following expressions: : . _ \ Kmm@nnhdgngz§§§::§g§%§§:0(fin¢flkfiiY“;;]G}? FKCB €= 3‘ ‘°*f'"3°3 . _ ‘3. "‘3. m ima.‘ “\QasV‘MAi' MQQQb-KVV‘: “‘49? FAQ? The final form is therefore: 9 = [a task m‘xkfi-FkaasV. mpu‘a H] 99 (31) Performing the necessary operations on Eq. (31) and sub- stituting into Eq. (13) yields: 9‘ g G eekKS-N- \XYGPKG where 3x x. 3‘ K; it F‘ ”NM“: ‘3 “5““) “:4: (EN? *‘Fa mi) +Ffi-xwx‘wx M‘W —$A\-Q“_W\SM )] \<- l&$\nhLm+1_3$m\-Lm aw] WK Solving for the loading function,jK, gives the final form: = _ kn :31 ea (32) >\ 4 r we K: 12%“ 1+ The moment and deflection equations are now readily obtained. Performing the required operations yields: Moment 1 ”I M: Dfi‘f\mMm\w*F1N1Q-&M3 .199 ()3) Deflection v=h EF\W\M\0(+FA=.=QAM$\,+QG 9 (3h) 12 III NUMERICAL RESUDTS AND CONCLUDING-REMARKS 3.1. 932233; In order to have some quantitative idea about the'behavior of this type of structure, numerical solutions are obtained for a shell having the following proportions: h/a = 0.01 and 31/. - 1.0. It is further assumed.that E - 30 x 106psi and.z’- 0.3. With these assump- tions all the calculations fall into Case I (Art. 2.2.1). The procedure of computation consists of l) assume a value for Nx/h, 2) compute the corresponding value of q, and 3) compute the corresponding deflections, bending moments, and stresses. 3.2 variation of Stresses and.Def1ections Along Shell Table I lists the bending stresses and deflections for the case Mi/h - 3000 psi. Values were computed.for.fourteen different points along the half length of the shell. These same data are plotted in Figure IV (for deflections) and.Figure V (for stresses). In Figure V are shown also the normal stress and the combined stress along the shell. It may be seen from.Fig. V'that in the central portion of the shell the normal stress is predominate while in the close proximity of the support the bending stress makes an increasing contribution. At the support the bending stress accounts for about 86% of the total stress. This pattern of distribution of bending stresses is well- known for this type of structure in the absence of longitudinal meMbrane stresses. In this case the presence of the latter stresses does not change this distribution pattern but merely modifies the magnitude. From Fig. IV it may be seen that the deflection pattern is al- so similar to the case in which longitudinal membrane forces are ab- sent. 3.3 Variation of Combined Stresses at the Boundary and at the Mid: EEEE Regardless of whether longitudinal membrane stresses are pre- sent or not, the maximum stress for the shell always occurs at the boundary. In Table II are shown, for different values of loading intensity, the longitudinal.membrane stresses, the bending stresses, and the combined stresses at the support and at the midspan. From Table II Figures VI and VII are prepared. In Fig. VI are plotted.the variation of the bending stress with the load in- tensity at the support and at the midspan. It may‘be noted.that the variation is essentially linear. Furthermore, the value of the bending stress at the midspan is negligibly small as canpared to that at the boundary'(note that different scales apply to the two graphs in the Figure). In Fig. VII are plotted the variation of combined stress at the boundary as a function of load intensity. Again the variation is linear. At all levels of loading, the ratio of membrane stress to bending stress is about 1 : 6.h. In Table III are given a comparison of the bending stresses as obtained in this analysis with those given by Timoshenko.for the case in which the ends of the shell are allowed to move together and the effects of the longitudinal stresses are neglected. It may be seen that although the bending stress in the present analysis 1h is about 7% lower, the total stress is about 7% higher since the membrane stress accounts for about 116% of the total stress. Al- though these differences in percent of stresses are not exceeding- ly large, they are appreciable. 3.}4 Concluding Remarks In this thesis an analysis has been made of the problem in which a cylindrical shell with immovable ends is subjected to uni- form pressure and comparisons are made with the case in which the ends are allowed to move toward each other. It is found that the moment is about 7% lower with the case of immovable ends in which longitudinal tensile stresses are pre- sent. However, the total stress is about 7% higher, since the longitudinal membrane stress contributes about 11% to the total stress. The stress and loading relationship is found to be essentially linear. It should be noted that the above summary of results is based on the numerical calculation of one shell with a fixed proportion. But the example does show that the effects of immovable ends are not negligible. In regard to the linearity of the stress-load in- tensity relationship, it may be conjectured that this may not al- ways be the case if the R/a ratio is made much smaller than that used in the analysis, since in that case the structure basically approaches that of a flat plate. Timoshenko2 has shown that the latter structure behaves non-linearly under similar conditions. The numerical part of the present investigation should lend 2Ibid., pp. 13-17. 15 itself well to computer solution. The values presented herein were all computed long hand, and the work has been tedious and time consuming. For a more complete numerical investigation, further computations of the problem by use of the computer are suggested. 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