Stresses and Strains in Cylindrical Shells Citation Williams, Harry Edwin (1956) Stresses and Strains in Cylindrical Shells. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1EB7-W116. https://resolver.caltech.edu/CaltechETD:etd-03272007-143555 Abstract As a consequence of assuming that the circumferential normal strain and in-plane shear strain of a cylindrical shell vanish, a relatively simple differential equation is derived which can be readily solved. An investigation of the frequencies of harmonic vibration of each Fourier component of the shell leads to a criterion determining a range of wave numbers within which the approximate method leads to satisfactory results. Experimental measurements show that solutions of the approximate equation are accurate except in the vicinity of a sharp variation of the load intensity, and so long as the length to diameter ratio of the shell is sufficiently large Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mechanical Engineering and Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Mechanical Engineering Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Housner, George W. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1956 Record Number: CaltechETD:etd-03272007-143555 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-03272007-143555 DOI: 10.7907/1EB7-W116 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1177 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 28 Mar 2007 Last Modified: 09 Aug 2023 18:43 Thesis Files Preview PDF (Williams_he_1956.pdf) - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

STRESSES AND STRADIS IN CYLINDRICAL SHELLS Thesis by Harry Edwin Willians In Partial Fulfillment of the Requirenents For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1996 CROW LEDGMEIES The auther wieahes to express his apprecinticn to srofessor George Vi. Housner for suggesting thie problem,and for his guidance throughout ite solution. The author also wishes to express his gratitude to the Shell Fellowship Comalttee and to the Alkegearch Manufacturing Company for the generous fellowships avarded the suthor during his residence at the Californie Institute of Technology. ABSTRACT * As & consequence of assuming that the circumferential normal strain and in-plane shear strain of a cylindrical shell vanish, a relatively simple differential equation is derived which can be veadily solved. An investigation of the frequencies of harmonic vibration of each Pourler component of the shell leads to a cri- terion determining a range of wave numbers within which the ap- proximate method leads to satisfactory resulta. Experimental meagurenents show that soluticns of the approximate equation are accurate except in the vicinity of a sharp variation of the load intensity, and so long as the length to diameter ratio of the shell is sufficiently large. CHAPTER ee a emeeen nal IE Iir Iv TABLE OF CONTENTS TITLE INTRODUCTION AND HISTORICAL BACKGROUND DESCRIPTION, ASSUMPTIONS AND DERIVATIONS i, Bagic Relations Siaple Beam Concept Derivation of the Approximate Differ- ential Equations - Reduction to a Total Differential Equation a. The Complete Cylinder a. The Curved Panel PARTICULAR INTEGRALS i. Ze at 4, De The Fundementsal Solution for the Pin- Ended Boundary Condition A Goneentrated Force at the Mid-Span Point A Concentrated Force at the Mid-Span Point with a Stiffening Ring 4& Concentrated Porce on a Curved Panel A Uniform Load on a Curved Panel APPLICABILITY OF TRE APPROXIMATION i. Fw * Derivation of the Frequency Factor Anelysis and Comparison of Resulte A Second Approximation Application to Statice Loads EXPERIMENTAL PROCEDURE AND RESULTS i. 2. 3. ~e Experinental Apparatus Radial Deflection Axial Menbrane Strain SUMMARY REFERENCES APPENDIX i APPENDIX IT TABLE I TABLE IT PAGE 21 2k 28 3G 50 33 at 62 &T ST 87 73 82 35 87 102 104 105 Chapter I INTRODUCTION AND HISTORICAL BACKGROUND A couron structural element is the thin-walled cylindrical shell which may be a simple load carrying member such as a monocoque air- fraue or a roof panel; or, it way be part of a system of members, as in the case of petroleum production apparatus. The general problem of the behavior of such a shell under the ac- tion of a system of radial and tangential loads has been formulated by many. Notably Love (1), Flugge (2), and Timoshenko (3) have, by the use of the thin shell approximation and linearization techniques, pro- duced the so-called "exact" equations. These equations consist of a system of three partial differential equations in the shell displacements u, Vy Wy and are of the second, third and fourth orders respectively. Recently Epstein (4) and Kennard (5) have undertaken a more funda- mental formulation of these general equations. They are concerned with the higher order terms which are usually neglected, and they show that these terms have appreciable Significance in obtaining the higher vi- brational frequencies, especially of torsional modes. Both these papers and the work of Chien (6) take, as a starting point, the basic equations of three-dimensional elasticity, and obtain the equations of the thin shell by a limiting process. Thus far, only two wethods of solution are available for these equations, and these have serious disadvantages. Timoshenko (3) shows that a system of Double Fourier Series will solve the equations in gener- al. However, from the difficulties associated with computing double ~ Be series, only solutions for the very slowly varying loading functions ean be obtained. On the other hand, Yuan (7) solved the Flugge equations for a concentrated force at an arbitrary point by transform methods. How- ever, this solution is not amenable to any generalization to other load- ing conditions because cf ite size and complexity. Other solutions for the general loading condition are available, but are approximations to the ™ exact” formulation of Love, Flugge and Timoshenko. These solutions resolve themselves into three classes, namely: i. The Inextensional Theory 2. The Finsterwalder Approximation 3. Donnell's Equations The theory of inextensional deformations of shells was derived by Lord Rayleigh(S, 9),and appears in his Theory of Sound (10). Love (1, 11) devotes his chapter 13 (1) to a formulation of the general inextensional theory; while Timoshenko (3) considers this theory as a method of solving the problem of concentrated radial loads acting on diametrically opposed points of the surface. Yuan (12) also considered this problem, but his method belongs in the third category. However, Hermes (13) has shown that, for the case of symmetrical line loads placed diametrically opposed, the inextensional theory results in unrealistic implications that fail to give a physically satisfactory representation of the response of the system. The second ciass of solutions represents the work of German and Scandanavian civil engineers. In an attempt to solve the problem of the response of long shell-type roof structures to distributed loads, they generally assumed that the effects of the axial bending moment (M.) and torque (M,.) were unimportant. The first solution in this mamer was pro- posed by Finsterwalder (14), who derived a single partial differential equation in the circumferential bending moment (M,). This approach was further simplified by Schorer (15) in 1935, when he reduced the differ- ential equation to two terms, again in the variable Moe by restricting the solution to long spans only. However, even with the simplifications introduced in this paper, any application requires considerable compu- tation. A paper by Odqvist (16) extends Schorer's solution, and presents results from which stresses can be obtained without extensive calculations. The applicebility of the above method to cases in which line or concen trated loada are applied to the structure, raised some doubts in the minds of Hoff (17) and Yuan (12). They replaced the differential equations arising from the condition of equilibrium as proposed by Love (1), Flugge (2) and Tinoshenko (3) by a simpler statement proposed by Donnell (18). The so-called Donnell's Equation is obtained from the “exact” equations of Love, etc. by considering certain terms as having less significance than the leading terms. Hoff and associates are able to cbtain a solution for the dieplacements, moments and mewbrane stresses arising from a sinu- goldally distributed line load applied along a generator. Yuan solved the problem of a concentrated radial load on the surface of a finite cylin- ger using the method of imges on a solution derived for a cylinder of infinite length. All the above solutions are applicable to the general problem by the method of superposition, if one assumes the accuracy of the differential equation to be sufficient, and thus represent the so- lution to the general problem for small displacements. . Therefore, given a distribution of loading on a thin eylindrical shell, one finds that applicable solutions are available only insofar ag Hoff's sOlution and Yuan's solution can be applied by the method of suserposition. fhe inextensional theory has been shown to be inapplicable on the basis - « of Hermes! work: and the method of Finsterwalder has been shown not to apply to shells in which there is considerable bending. However, the solutions proposed by Hoff and Yuan suffer from the same disadvantage as Yuan's more exact solution, in that they are of considersble complexi- ty, restricting their usefulness to the simplest of loading conditions. In addition, it is doubtful if Donnell's equation is applicable in the vicinity of a discontinuity in loading. It is the purpose of this paper to describe and develop a method of analysis that leads to solutions of an approximate differential equation. These solutions apply everywhere but in the immediate neighborhood of a discontinuity of loading, and give results in. this neighborhood that will have an accuracy equal to or greater than that cbtained with the classi- cal equations of Donnell. These solutions have the further advantage of being much simpler in form than the solutions of Hoff and Yuan, and thus have a greater applicebility to the general problem of radial loading. This work is an extension of a previous paper by Hayashi (19) in woaich the “simple beam" concept of Chapter II, Section 2 first appears. DESCRIPTION, ASSUMPTIONS AND DERIVATIONS 1. Basic Reletions: Consider the following representation of the circu- lar cylindrical shell shown in figure 1. Figure 1. oo . $ * 8 * * The displacements of the shell are given as u,v,w=y, inthe x,y, @ éirections respectively. The non-dimensional distances and displacements are defined by the equations al x= ¥/o @ = 47/0 zZ > Z*/o a Ku =o v= V% ars w/o (II-1.1) where u, v, w are considered fumetions of (x, # ) only. Considering an element of the cylindrical shell bounded by surfaces % x = const., 4d = const. shown in figure 2, the shear forces per unit length (Q) and bending moments per unit length (M) are indicated in their positive sense, along with the proper convention for normal and shear stresses. Figure ¢. If it ig assumed that the shell is thin, that is, h/a <<1, one can make the conventional assumption that the shell is in the condition of plane stress, or that so that, &, % ,%, 79 (II~1.2) = | os aa: (Ex + Vv <q) = (ey + ves) Be = SNe. G = E/ac+v) (II-1.3) Tf one assumes thet the membrane strains have negligible effect on the curvatures, then the strains (¢€.,€¢ »he ) are defined in terms of the membrane strains (¢, ,€2,% ) and the local curvatures (},, ky > Ky ) ag follows: wo Ey z ©, > Zz x, fgs 2 7 Fk (rr-1.4) Se = ¥ - at My where the curvatures are k, z GT tx? ky ~ 4, Cur? + 0 bq) (TX-1.5) ‘\ + J Yay a (urge + vie) ~v € ¥ “ €, = EQ 7 uw") (II-1.6) + Yy Rm + we + Vie The subscripts following a comma indicate differentiation. The bending moments per unit length are defined as M, = -D(k, +r ky) x Mm, ~DC ky +v x.) (IZ=1.7) E }* mM =-™M = DENG, D = 12 (i~ WY) Le ox The shear forces per unit length are given by QD * Mae? + oa Meow ¢ % = x Med - Myg,e’ (rr-1.8) In agreenent with Kirchhoff's suggestion, the total effective shear force per unit length is defined as (2341.9) 2. Simple Beam Concept: Consider the circular cylindrical shell show in figure 1 to tbe the limiting case of an n-sided polygonal shell shown in figure 3; that is, the limiting case of a structure composed of ele- ments of length 2 and width add , where A¢ = 27 /n, as n becomes very Consider one of these component elements shown in Figure 4. In general, Bs. - ", 2 %, e a * ad y - the element will be acted upon by radial loads in the z direction, anc 3b x 4 y ey t - oe * stresses distributed elong the sides y = + a Ap /2 ; = Oy XL. The % A stresses shown in figure ha, b are those acting on the surface a = 0, or the wenbrane stresses. Pigure 4b illustrates that for small A¢? one can take the axial meabrane stress 9, as being made up of a constant part QS plus a bending stress ¢, which varies linearly over the depth add. The simplest method of couputing these membrane stresses is to assume that the stresses in the elemental beam follow the simple beam theory. This theory, when applied with the assumption of plane atress, requires that EG *\F Ty, = cv, ~ LO ant ye Vout" 86 a result of assuming ° E , . , oF ; * : where ris the locel radius of curvature (s Lf Vyyrye) of the x -axis in * * 2 a2 CF 2 the x ,y -plane. Furthermore, if one differentiates 9, and 0, with re- % spect to y and uses the identity for ©, , it can be shown that this ~ TO «» requires OS = 0 in accordance with simple beam theory. Thus, applying the concepts of simple beam theory to cylindrical shells conceived as being made up of beam elements of infinitesimal depth, one obtains two relations between the three displaceuents, that is, * * . * ’ = Vit a tat? II-2.2 whi a ee a near Sa ey SE Se US a7 PR aan tt y — Po ALC BSULALE YEsSpectiverly whey =, = O Bhd = Ue 3 Derivation of the Approximate Differential Equations: In order to study the behavior of a shell under the restrictions ez 9 = 0, the differ- ential equations governing the behavior of u(x,#), v(x,¢), w(x,%) aust be Pormulated and then solved for the functions themselves. The usual manner of formulating these differential equations is to express the three equations of equillorium in terms of the three displacement func- tions. However, the restrictions €, 9 = 0 preclude this, for the forces ean no longer be expressed in terms of the displacements. The nethod that is applicable in this case is that of the Calculus of Variations with a Subsidiary Condition. This method leads to differential equations describing « shell acted upon by a general system of forces in which the deformation of the shell will only be an approximation to the exact solution, but the total energy of the system will be a minimun consistent with €. ,9 = 0. The total energsy of the syeten U is defined as the sum of the strain energy of the shell VY and the potential energy of any external forces. Thig potential energy le equal to the negative of the work (TD) done by these external forces acting on the shell es it deforme. Thus, U=V-T (II-3.1) The strain enersy per unit voluae V_, as given by Timoshenko (20), when i the plane stress approximation (eq. II-1.2) is applied, reduces to ta the V_= \ zu Vv i z or alternatively, using equation EI-1.3, as E 2 a A “seul: + €y +3vExey + 1s | (II-3.2) yen by the integral of the sbove expression Ce el we ots ea) 5 pots The total strain energ: throughout the volume of the shell, that is, V-> Ve adv'dd d2* (II-3.3) 2 where it is assumed that h/a << 1 so that the volume element of inte- gration is given by adr'ddda* d(vo.ume ) a Gm (\- % }du’dp d2’ Assuming that the only forces acting on the shell are radial loads over the surface of the shell given by the distribution a(x,¢), the external &S af = ah >) uw” advdd (TI-3.4) Axial and tangential components of load could be included in the same work becoues fashion. If equations II-1.4, II-1.5, II-1.6 are substituted into equation TI-+3,.2, taking into account equations II-2.1, II-2.2, then V, becomes a e+ # function of u, w only. Furthermore, if equation II-3.2 is substituted * into equation If-3.3, the integration over 2 can be carried out directly, x * % as u,w are functions of x , ts) only. Finally, substituting the ex- pression for V, T into equation TI-3.1, the total energy of the system U pecomes Eh [ 4°" he uy’ (usu 2c-v) G-v) We aan * 3 ne .- + * + G-v (ur) + 4%) Gee ar" 4 ay ~us'q pa dv'dd (11-3.5) # % where u ,w are related by the subsidiary condition that * iv + 7 Bigg = 9 (1I-3.6) % which is obtained by eliminating v from equations IT-2. 1, 11-2.2. - 13 - In order to obtein the differential equations of equilibrium from this expression for total energy U, it is necessary that the variation of U vanish subject to the subsidiary condition of equation IT-3.6. This is accouplished, according to the method outlined in Courant and Hilbert (21), by introducing the subsidiary condition into the ‘total energy ex- pression, thus forming a new function Ue, This new function ue is equal to U plus the result of first multiplying the subsidlary condition G by an arbitrary function A and then integrating this product over the surface of the shell, that is uv" =U “| Ae) Gdvadd % The function UY is then required to be a minimum for a suitable value of the parameter AL. Thus it follows that SU" = 0 or that, Eh »2 z ad 2 + 2 e,+r Tt ‘ “ff ee] * zee * qa (we * Wad ) * 2 % z + Vb uw "+o" I-v) he 5 - hue Gat WV (ar +o ieg) + ev fp. Seg u ) C4 & ~ > s A . 1 a, * XN 1 x .. fey > ~ar’g + AGry +E4de) bdvadp = 0 \EE~-367) This operation hag been carried out for two cases: 1) the complete cylin- % % * der, for which the limits of integration are x, <x < x, , $5444, where g, - $ 22% ; and 2) the curved panel, for which the limits of Es % * % wt integration are x, x <¥ x2, a8 ¢< $, where in both cases x, - x, =, ~ 1h . Due to the commutation between the variation and integration processes, and between the differentiation and variation processes, as shown in Po Hildebrand (22 the vejulrement that U be stationery reduces tet 3 a os tot (vg - 1%) wr] - |r, - a ) se" | > $2 dade fat aa + Co i4 ~ a u's ) Sur” | - (re ~ Lay) Sur" oA d= 4, Pe U4, be sks D * ost , ‘ + e{ aos Hye) Yep | 8674) [dv ae i uf F $2 * rs * ~ ¥ * Bs Sa * BP hag 7 ON gue Fn ‘| be" | oe ar 4 ron $2 YM pha cw, )- "| dv + [2 “DS, “9 ) A, | 32 dy a * * $, b us + + Eh ° id | |A $(u",) dy +] a, ky» du ad¢ u % $, “ ot re oh br , add 4 x t | |p| Wat * re Cos" + is) | 8 (wry) > v1? kT be VA-ve ~kKi-v 4? +} [ [Exc Bee + TS eg 7 Sete) uy + Agy | dd adt u ’ 2 * * : * {J [> Ca yy * oe ved t at ST 45b4 + ial uo, + * Rg. ZV * _~ gz. tv N O - (44 Ay)| Suma dg where all expressioug apply to the case of the curved panel, and only the latter Pive integrals apply to the case of the complete cylinder. The first four terme and the line integrals give rise to the boundary eonditions; while the surface integrals produce the required differential equations. Thus, the "natural" boundary conditions for a curved panel are, on the corners of the panel OF 5 uw” = 6 on the curved houndary (x = x,, xy ); “u*, 20 or, ua =o (IZ-3.9) iw, { ce | & ®e aft + els aN g + & ~~” ss —/ [ nn | \ fo oy. the straight boundary ¢= 6,1 4 ; \-v *, 1 boy” a =D Ceig a a4) + > L\, j eid \- a + & + » Zz E nN . ~ “ l 0) OF » oy = O esc 3615 AN = 0 oF, wy = 9 (f2-3.15). 4 The “natural” boundary conditions for a complete cylinder are equations bel zs Ei-3.0, If-3,.16, IT-3.11 as the other inmtegrais vanish frow the perlodici- Tt wust be noted the boundary conditions for a holonomic long are not ini the boundary conditions of w and lus derivatives, but as a result, the 5a ew boundary conditions on u and its derivatives are fixed. This will be lillustrated In later examples. Meking use of the definitions given in Chapter IT-l and the identities in equations II-2.1, Ii-2.2, the above boundary conditions expressed in phyelcal quantities are: on the corners of the panel, . e * »% on the curved boundary (x =x, , x, ); Ny R od2” = © OF 5 au = O M ~~. = ox ya 4 ° OY, ax ro) Sa a 3 or, Ww =0 * My = © or, wy, « 168 - where the value of M, $ at the corners of the panel measures the value of the concentrated reactions et the corners. Considering the surface integrals that result Prom the variational procedure, it is noted that the coefficient of the integrand of the first integral is Sw ; while the coefficient of the integrand of the second integral is Sur, Again, these variations ere not independent, and an argument different from that used for holonomic systems must be used to determine the differential equations. If AV. ig chosen so that the coef- ficient of tu vanishes, then, since the variation tw can be arbitrari- ly assigned inside x < x < x, ; $<P<4 , the coefficient of Swe must vanish. ‘Thus, the following differential equations are cbtalined: ~E * ~ irvR ” ¥ ~ La = “ am ee - We CO og oO 99) + = Asse = oO (II-3.16) 4 « " ¥ « 4. 2. ¥ - (11~3.17) ~ + ~ Avy = 0 in addition to the subsidiary condition (eq. II-3.6), making three differ- * & ential equations in the three dependent variables w , u A - Let the following notation be defined w” a : ne ey Vio = Zu” = art + 2.” + 1. id neter eet oF NT Vdd aX “ipo * If equation II-3.16 is operated on by ag, and equation II-3.17 is oper- Py og? independent of A. , that is, ated on by and the resulting equations added, one obtains an equation * A 2 , ? _ (-v un P Lv met as “144 acs wy 8 as 4 vag ~ Y/D ~ Eah s 1 + ; \-ve rw mk (a, de x Moa) | ; = 0 (II-3.18) rv This equation, plus the subsidiary condition (eg. IT-3.6) . L.y* =O ar t+ loo * Bs If equation ; # we :quations in the two dependent variables u,w. »y give two e cs II-3,10 is operated on by a and equation IT-3.5 is operated on by ses and the resulting equations added, one obtains wal "42 7 Loe ‘odo h a Ky. * * —_ a 4¢ + 12 ( ty os" vy 12 EE "+ Gigs eae 7° (@I-3.19) Introducing the dimensionless varlables (eq. TI-1.1), along with p~ye/D t= “lo (II-3.20) 4 woz _. Vu 2=V Vor = Fre 7 * Fede + Woood one obtains equation ITT-~3.19 in the form Vir 42V@ +ur~pP $v +aC-v)(w +65,,) (I2~3.21) T wee #V ried ve eed » 20 - Fe So pda ay @ rea S Feed ee fle 4 eI ‘gS fet pay qi a & Co] o & ce ia fot S cr iv] a 5B 3 et coupletely determine the displacements once tne loading function and poundary conditions are prescribed. Equation II-3.21 can be compared with the “exact” equations of Timoshenko (3) which are, in the present notation, I~v ~ R wy + = Urey + AY Vivg vary re) Vv I-vViov L+¥ _ 164 + we FAS 4x ur g 2 z + Ue vier, + Che (VV - VV) = 9 Vw - "4/42 (Vey + Ye ~ ur) + | vv + (Wr = f r¢ Donnell's Equations, simplified by Hoff end Yuan, are - 4 = Vow + EY Oras - Voip ze % x 2 Ve > VO Wed q < QAt+evy Wary t+ Weed 4, Reduction to a Total Differential Equation: In this section and in the following sections, we shall not consider the case of the complete cylinder or the curved panel in which one or more of the sides are free. It is only in these cases that the function A must be «omputed. Instead, attention will be vestricted to the cases in which a complete cylinder or eurved panel is supported such that the deflection w vanishes at the boundaries. a) The Complete Cylinder: Consider the case of the complete cylin- der shown in figure 5. _ =? in the interval -N<q¢s , the functions 4(v,$) , P@,p) can be expressed as >. Worn) Ga ng (11-4) M=O i) ur >) wm > P (em) Ga mg — (£I-4.2) Mz90 3 PO ¢) where w(x,f) must satisfy equation II-3.21, and B(x,n) is given by ® P (tim) * & { peud) Gea (IT-4.3) ~? ~ 22 Tf equations II-4).1, II-4.2 are substituted into equation IT~3.21, one q ? 3 obtains > Ga me [Co +124") Were ~ Axe Wau Cat Rv m+ a) .o) u + ont W Unt~ ih = on" P | 80 that, (m+ rR / v) W. 2 — 2% ~ 4 were 2 Wie (m2 ws ev) tM WOR) = Pp” (II-4, 4) Thus, the result of assuming €,{7=O is the reduction of the classical Sth order equation as obtained by Flugge (2) to a 4th order equation, Substituting equation II-4.1 into equations II-3.10, Ti-3.11 with the definitions contained in equation II-1.1, one cbtains the following boundary conditions on x = 0, &/a : either, W308 "Pixed End” (r-4.5) or W 2» Oo "Pin-Bnded” (£54.65) W = =0 a) at 4 It is apparent with these boundary conditions that for n = 0, W(x,0) = 0, so thet equation IT-4.1 reduces to: meat “37 & Introducing the dimensionless variables of equation FI-l.1 into & g equation TI-3.6, the subsidiary condition becones Oy o+ D sep = 9° and integrates to u(v,¢) ) ay W Gam +6 fi +9) (11-4. 8) M=6 Fron the requirement of periodicity of u(x,q¢), thet is, &UW,b) = UCL, Ptah) it follows that f = 6. If equation TI-4.5 is substituted into equation II-3.18, one finds thet ¢ must satisfy Sur = © 4@) = CV + Cou +3 Ags u = const. corresponds to translation of the cylinder as a rigid body, which produces no strain, C3 way be set equal to zero without loss of generality. Furthermore, since radial loads are the only external forces being considered, C, , Cz can be taken as equal to zero, as these constants correspond to a shell acting as a columo supporting its own weight and to a sheli in uniform axial compression respectively. Thus u becomes uo) = ). AW Cea.m (174.9) Mat ~ Ph With equation TI-4.9 it is evident that the boundary conditions on u cannot be prescribed independently of the boundary conditions on w. From equation TI-4.9, it follows that ‘r W,, = HE | RCUe) amp dg ~t Therefore, requiring u to vanish on the boundary implies that Wy vanishes; or, requiring Us, to vanish implies that Wy vanishes. Thus, for the case of the complete cylinder, the deflections w, u are given by equations II-4.7, II-4.9 respectively, and must satisfy the differ- ential equation II-4.4 with the boundary conditions arbitrarily chosen between uy = 0 a, = OG "Fixed End" kK = O and, w= 9 M,7 9 "Pin-Ended" Ky ° b) The Curved Panel: Consider the Case of the curved panel shown in figure 6. Figure 6 In the interval O<P<4¢, , the functions w(x,$ )»p(x,p) can be ex- pressed as pOG¢) = ry ». P(e my Sim mr /q, (II-.11) where w(x, ) must satisfy equation II-3.21, and p(x,n) is given by % PO”) -{ pv b): Sem ming /g - dp (II-4. 12) ce} However, as will be shown, only certain boundary conditions on p= °, 9g, can be satisfied. If equations II-4.10, II-4.11 are substituted into equation II-3.21, one obtains > Sim nk /¢., [a+ Wrens ~ 2m Wry (i ~ Vm 4 Hv) Az I tmi(ar-iyw - 2m 8H, | = Oo so that, Ca" +12 AW, MWUCU a2 ~4 He wpe Ne ~*~ : e in ne “ . 3 ° ; be Bon - where n = ri Substituting equation IT-4.10 into equatiors Ti-3.10, fo] II-3.11, the same boundary conditions result as for the complete cylinder, that is, either W=0 "Fixed-End" Wo =0 1% or, Wz0 "Pin-Ended" W =0 J Ve On x =t}-Ya, However, it follows that by substituting equation IT-4.10 into equations ITI-3.13, If-31.4, the boundary conditions w Mg = 0 are automatically satisfied on dp = O, $, Thus, the use of equation It-4.10 as a substitution is limited to the "Pin-Ended" condition and certain modifications, and as Brown (23) points out, is not applicable to the "Fixed-End" condition. For the "“Fixed-End" condition, w,p would have to be expanded in a series of orthogonal functions that automatically satis- fy the conditions w, w,g = 0 on p= 0,4. One finds on substituting equation II-4.10 into the subsidiary con- dition, that u becomes (ue) = ) ke Wy Stamdd lg, + $5@ + 9 (11-414) nazi a) on] Similar to the above result that the condition w, w,g. = 0 cannot be satisfied on ¢ = Q, ¢, by equation TI-k.10, it follows that this substi- tution cannot give Ug = Oo on d 2 Q, ¢, but, with f, ¢ = 0, it auto- matically satisfies the condition u = 6 GO, TherePore u becomes co ud) = ) nn) We en ab /4, mt (TI-4,15) Thus, for the case of the curved panel, the deflections w, u are given by equations IT-4.10, Ti-4.15 respectively, which wust satisfy the differential equation TI-4.13 subject to the boundary conditions arbi~ trarily chogen between a) ay,=0° “Pixed-End" (LI-4.16) “r= 0 and, jw = Oo M,=° "Pin-Endea" (1z-4.17) uve on x = + Rho ; dut Limited tc the conditions that og = O Me= 0 (I1-4.18) & = 0 ay gf Chapter ITI PARTICULAR THTEGRALS The analvsis of Chapter IT shows that the assumptions €,,2 = Q lead to the differential equations | vie 42 Vor +ur - P + Ayr W rvey + 2C- (or + Wig¢) = © rddhp ued YY For the case of the complete cylinder (fig. 5) acted upon by a distri- pution of forces such that * Pm) > 5 { PodGentas “© the deflections may be expressed as ax(v,o) = > WG, m7 Coa mp wm) oO 4 U(X, d) = » a Wy Ga mg mz where W(x,n) must satisfy (m + 2/_2) W, am Wo Cm - 2-V mie \-v) +m W (nt.1)° = P an Wee ex subject to the boundary conditions that either = 0 ay. = ° "Pixed-End" = O OF dar = My= © "Pin-Ended” Ue ° on x = 0, Le, For the case of the curved panel (fig. 6) acted upon by a distri- pution of forces such that $, Pr) = | P(x.) Sermo /q. dg fe) the deflections are w& (u,o) = > W Cem) Sew and 4, ma! % &(U, >) = ) (474) Wy Smads /p, mrt where W(x,n) must satisty ("+ 21) Wry ~4 mW CO -EV m t hv) + AC) W = amp /$ ° subject to the boundary conditions ur = 0 a =o "“Fixed-End" MR =z O or, Gr = GO MY=°e "Pin-Ended" Ya =o uy = 0 Meg = ° “= 0 On g = 0, 4, 8 In the following sections, some solutions of the above equations will be presented. 1. The Fundsmental Solution for the Pin-Ended Boundary Condition: The Fundamental Solution, that is, the solution for a unit concentrated force located at an arbitrary point, is of interest since the solution for any distribution of loading can be found from this solution by the method of superposition. Consider then the case of the complete cylinder (fig. 7), satisfying the “Pin-Ended" boundary conditions (eq. II-4.6), where the loading fune- * 4% ticn represents a concentrated unit radial force acting at (x =x, ,p= 0). Pigure 7. The concentrated force is expressed by 9 (Wg) = BCU) 8@) where &{x) ig the Dirac Delta Function. Introducing the dimensionless quantities as defined by equations II-1.1, II-3.20, and using the relation that Sar) = 4: 8) ene finds GCs) = Ge BCu-G)- 3@) ~ fs = &.3(v-z). S (} P «> 5 ¢ 6) dG ) substituting this dimensionless loading function into equation Ti-4,.3 16 pu») = <a O(&- %) (ITI-1.2) = 32 - With p (x, n) so defined, equation TI-k,4 becomes UNE ) (“ + 2A) W -2 w Wry ( >“ 2-¥ m + (~v)- + mi W C- i)* = OamM“/yD - S(u-%) This equation is equivalent to requiring that, for all points save x = x (m* +a Ar) W a Wry (ai vate by) +m WG) =O (1IT-1.2) YUCKY red subject to the condition of continuity of the function W and its first two derivatives at x = x, , plus the “Jump Condition" given by [ Baw | = MK, (IIE-1.5 . “US at Mere K a & /y D a Pad mt 412 (1? in addition, the boundary conditions given by equation IIT-4.6 must be oo see ay Ap a, Df. 204 ss satisfied on x = 0, */a, that is, Wo: W,,7 0 UX For a = 1, equation TII-1.2 reduces to W. = O WEN UY 3 tal is 4 The solution of this equation which setisfies the combinuity and "Juap" i is conditions” at x = x, , and the boundary conditions at x = 0, 4 fa is given by 3 44% Wow) = LEE) SEC ER eye] (IIT-1.4 Wut) =, ea ees BE (re ceye- eH | ( a 3 an° 6 For m> 1, it is assumed that the golution (W) is proportional to exp (mx). Mehing this substitution into equation IZI-1.2, ib is found that m must satisfy (n* + 2 /n2 um -~?2 mm (m*— a-Vm 4 Iv) an + am Oni) = 0 (III-1.5) if the roots of this equation are mis Bar (112-1.6) the solution takes the form WM m) = Gage (AGir xv + B Sunnae ) . a . i + SiaVr ‘Ce Surman + D Gin Xu) This solution for the above boundary and continuity conditions becomes WG,m) = B, Ga@e Surat + D, Ser Gor uy (III-1.7) (o<vg%) ~ 3h. Wm) = A, Ga Pu Garay + BLGr eu Sahay +O, Sar@u Suan ae + D, SinQu Grr wy (III-1.8) Cw <u da) where the coefficients are A_=- K (« SenQu, Gem ats - @ Gr@Gy Surr x4.) AxKB H+ QB) o B, = (- A, Garay, Sumeey + Cr Sm Gre ) Soe Ole 4 sma zene he ~ nea Ge Eh C K,. («Gs Qu, Smrau, +@ Singu Gad vu.) * LKB (a+ GF) D, = (<4 S Qe Ga -~ Gy See 04 Gah af Som Ql ye + Sind a O/, Sem Ma ee CHa, ~ Ca Sein 8 “) B= \ |- a, Sor G4, Ga Quy Garatc + ‘ - Gaer, Sane Ue San Qu, + Sumn® o Ue Ga Bt, Surmau, Gin aX SuneW/a ~ Geet Banat } Sew G&fq + Stk? «Ra -~ cf Son CU, Such wt, Gu ates Suet, + Sum ae, Goa 04, SUM Wt, SUL BEMQ | SunQg, Sonne us) | BAWCLG + Saw w/o Q > | - A (Smeta Greys ~ Swel/g + Sate Sonu, GaGa, x Ste @ha + Sinner Wf ) Sue@t, + Sua? xu, ~Cz (sanaey Cok “Qh - Seka Garwu, x SEM + Suah'w Ma ) (11I-1.9) Son Gte + Surat, 2. A Concentrated Force at the Mid-Syan Point: Consider the case of 2D the complete cylinder (fig. &), satisfying the "Pin-Ended” boundary con- ‘ditions (eq. IT-4.6), where the loading function represents a concen~ trated unit radial force acting at x = 0, @ = 0. This solution is identical to the previous solution if x, = X fea. However, rather than to make this substitution, it le more convenient for subsequent computation to take the point of application of the load as a new origin of ceordi- Pp. yea Y Ya. ’ le 3 i nates (fig. 8). Figure & In this case the leading funct becones , 49) = £- wv) 3@) so that 3) = oe S@)- $d) and PWg) = .. du) $C) with Pu») > OC) (LLZ~2.1) With p(x, 0) so defined, equation IT-4.4 reduces to Qn“ + Rv /e)W wn TRE Wa ay) + mi Cmte) Wo with the requirenent that W be an even Punction of x, and possess a4 "Juan" in the third derivative given by ms an | = K = & “ad mt + 2/2? With K,®@ defined by equations TII-1.5, IiZ-1.6, the solution satis- fying the boundary, "Jume" and continuity conditions is We) = K (KAS [ener] (o suk ao) (z1I-2.2) and Wm) = A Gr@y Garay +B Ga Gy Surv xy + Sonu Sunray + D’Gihay Wa@y (TIT-2.3) (o <u < <a) ~ 37 - where the coefficients are A =- B Suhaty Garods -~- ~ SurQ0s Ga’ Wag saras| ad B =-- Kn Vol (a? 4 @*) C = 5 ; : Ga QQ, + Seavey | S*C%ia OC + a Sinnaty, Ganay,] D = -(%)B (1I2-2.4) 3. A Concentrated Force at the Mid-Span Point with a Stiffening Ring: Consider the case of a "Pin-Euded” cylindrical shell with a stiffening ring (fig. 9), acted upon by a concentrated load applied at the uid-span point. The ring is idealized to have infinitesimal width, but finite praperties ctherwise. In order to etudy thie shell-ring combination, let us separate the system into three parts: 1) a ring acted upon by an arbitrarily distribu- ted radiel load; 2) a shell acted upon by the reaction to the distributed lead acting on the ring; and 3) a shell acted upon by a concentrated radi- |l force P. The total deflection of the shell is the sum of that produced by the distributed load and that produced by the concentrated force. The i Pa magnitude of the distributed loa obtained by requiring compatibility mw of the deflections of the ring-ghell combination. This method could be applied to any nuuber of stiffening wings, and algo extendec to include the effect of longeron stiffeners. Qt) Figure 9. Ring Congider a thin ring (fig. 9) of radius a, woment of inertia I and aodulus E,, acted upon by a distributed loading function @(¢). If the depth of the ring is smell compared to its radius, it can be shown thet only strain energy of bending is imoortant in the expression for the total etrain energy V, that is, » 2 2 * ¥ ¥t 2 Viz sai [co + Sve) a,d¢ na ata The external work T done on the ring is equal to bt Q(t) wa, de ~v tT a Thug, the total energy of the ring U becomes ® o * U = Exc + org) ~ os Jedd aas , ~\ st Requiring that w (¢$) be such that the total energy of the ayeten Is a ~ 39 « minimum, thet is $U r 0 % it follows that w must satisfy + . s _ A eade * RW 4 + or nye Introducing the dimensionless coordinates equation TII-3.1 becomes Paage TAM Wg + > 8 The substitution Gr = ) b Ga md MIO 19) Fee reduces equation IiL-3.1 a to Y came [b,C0taI~< | 2 0 a>O which requires bo = & he - yf (TIz-3.1) (III-3.2) (IIT-3.1 a) @Ir-3.3) (12T~3.4) c The loading function (eq. ITI-3.3) is completely arbitrary, however the condition of equilibrium of the ring as a rigid body reguires that the total force on the ring aust vanish, that is * Qo) Ga $ a,dd = 0 7 ¥ Jow Smq and¢d = 0 ~* The first equation requires C, = 0, while the second equation is identi- cally satiefied, as @ ig an even function of $ . Thus, the deflection of the ring under the arbitrary loading becomes: w"(4) = 4 wD aoe 7 Goa mg Gs 0) (III-3.5) MQ Consider the deflection of the shell under the reaction to the dig- tetbuted load acting on the ring, that is q@ie) = - 4. 8e) Q4) a where 1t is assumed that the effect of the ring on the shell can be satis- factorily approximated by a distribution of load over the line x = 0. It then follows that q Wid) =- 2. 80. Q@) fale so that p(w) = - S - 8&) Q@) E,a 1 Da; x PCam)> - C. 8Ce) (IIi-3.6) (“4 Rt) Wry ~2m Wy (es Bats Lv) + Cat YW to0 a fu pes st mn rh ede ir] m ce gubject to the condition of continuity of the function W an two derivatives at x = 0, and the “Jump Condition” given by Weer] * K = 7 Sat bf (r31-3.7) D a> V4 12 fy For n = 0, W, vanishes as Kj = 0. For n= 1, 4, vanishes as C, = 0. For n> 1, the solution satisfying the boundary, continulty and “Jump” conditions is Wu, r) = Geu CA Gib «% + B Simh « &) + Sm@u(C Seman + D’Gih ar) (o <u s faa) (LIr-3.8) where A/n =A =- Semmes Gabee — X SinGd Gi Ge a AO B = ‘ . Eb, L ; Con A of Cx? 4+Q@r) Da? \4 Bp Cc =C { [Se 8 CQ + K. Soh GirxQ [s Ga" ee + Sahae Ra. ose € “ aa ~ ke. Dig = D = - &e) (ZI1-3.9) Using the solution for the concentrated unit load given by equations Tii-2.2, IfI-2.3 and IIi-~2.4, the total deflection is obtained by super- posing on the above solution the result of multiplying the solution for the concentrated unit force by P, that is wcog) = PE BSH faces + ]oa4 \ + 2 (92 L \ _ _/eD Gand WRAL * ) NaC et+@?) i+ % [ge ‘ (P az Ln) * mre lage (A Gahug + Sunn wu) + Som Qe (E Sear 0 Boukw) | (III-3.10) Thug, the deflection of the shell is made up of a mode which represents the shell remaining circular (n = 1), plus higher modes which describe the local deviation from roundness. When applying the compatibility condition to the ringeshell coubi- nation, only the higher (n> 2) deflection modes must be equated, as the ving is free to move as a rigid body. Equating the deflection of the shell at x = 0 to the dePlection of the ring given by equation TII-3.5 leads to ov <} Cm @ 5 Gam = y _ A®@Ap Gand NE T (mn at ¥ ¢ "A Cats @*) “1 4e Ona? ; (® ~ Ly) m=o M22 se ibe gO that, as the coefficient of cach term 1~ AuCat+@?) Das A (mt ~ 0)? E,Tor (+ Raye) Finally, the ceflec Surface of the shell becoues w"(x,9) > PE AstD a [acegy -6ayts | Gag t+ R /y* + } [over (A Gahou + Scrmau) + SomQu (E Sutra, + BGanw)/ * mad Pat Ga und A (ott) E AGH Eat (123-3.12) \ 4% (ol? 4+ @*) Dax G4 i Po ba) A (oe ~ 1)* . E,ia* 4n order to evaluate the effecs of the atiffening ring on the radial deflections, equation EiT-3.1 BG GO be campared with equation Tif-2.% which represents the unetiffened ghell. Intriuducing the } Pa /eD P Aa(e4Q@*) f+ R2/Pm4 the deflection of the unstifvened shell becones wg) = PAs Da fa cany’ 6a r Joey (+ Rm / + ) [oxen (A Gahay + Send ocv) + Sem Gu (E Sona eu DGaher) MaQ x a By Ga mp a "(w,$) = ferpe [a (a)° - 6 (24) +1] Gaq V+ 2/7? >} [a Ry (A Garots Sahar) + SinQu G Sunes DGsdes)| a=A x aB Po WA (W.+)7 EL Ta Bp meh fi rmc ny ais T oy the behavior of Ag n becones layge, using the resulte of Appendia I for the behe a fad Wee Ags Gey bee i, “ fa & SOB om ce Bam pads ome By, ® becones lmrce, the Pinal Pacter A/S, B as n becomes large, the f4r aad dyad Freon pel Seibut lon af deh of Se 245 - ring is to give the deflected surface a smaller curvature, as & greater proportion of the load is carried in the lower modes. 4, A Concentrated Force on a Curved Panel: The results of Chapter Ii-kb can be applied to a curved panel (fig. 10), simply supported on all sides, with e concentrated force P applied at the mid-point. Figure 1O, The loading function «¢ is viven by ge 1e) = FP .8@):8(@- A) Also, pd) = TE. 8a): SCH - dora) go that, P (ua) = $2. 8a) Ste nb /a (172-401) With p(x,n) so defined, equation TI-k.13 reducea to _N 2 4 —. 4 - (a * 2A2)Wran “am Wry - e aes -v) +a) W * 9 (IiT-4.2) subject to the boundary conditions Woz Way ° u om x= & Y26,, and the requirement thet W be an even function of x and possess a "Jump" in the third derivative given by | ave KE De. 14 Re (Irr-4,3) Since, in most practical cases $<‘, the case h = 1 will not occur, W(x,n) can be taken proportional to exp (mx). With this substitution, m must satisfy GRA RG) a = RR EVRY Ye GY SO rth) If the roots of this equation are the assumed solution can be put in the form Wm) = Gren (A Ginat +B Sahat) (TIT-4.5) + SuwQe (c Suwamayu + D' Guar ar) Co st¥s Ma ) The coefficients which satisfy the above boundary, continuity and "Jump" -u7 - conditions are A > A/B = - [ Sones Couns ~ © Sem @2 Gr Oo Car®QQ 4 SeantxQ za Shes @ Za aa B = - . Ke Ax(OSGY RA R/y GC = C/B = | Sim @S Ga@Q 4 KX Suna Gahae Coa BS + Sea 2a aa @ ROA aq oO. an (TTT-4.6) 5. A Uniform Load on e Curved Panel: The results of Chapter II-).b can also be applied to a curved panel acted upon by a uniform vadial load of intensity Qs This could be an approximation to the case of a panel acted upon by its own weight. In this latter case, the loading function 4 7 > 7 &(¢-bé) = 9,7 LArG-ha)t > where there will be a corresponding tangentially distributed load given by | g Sin a4) = 9 (tea 8) ~ 9, yee #88 Thus the assumptions that q = constant (q.) and the effect of the tan- gential load is negligible, will be a good approximation to the eravity load problem providing ?, is suall. A(t) = 7ER°/meD (N= 13,~ ) (IxT-5.1) sO that, for simply supported sides, the assumption that wad) =) WGa) Stoned ig MAA leads to the reduced equation ("+ 18/2) Waray 2B Wry (EV as tv) + ROW a) = aK B /q, m = mt C /4) (1II-5.2) Taking W(x,n) to be proportional to exp (mx), it is found that m must satisfy the equation (a ae) -ain ( mm a-V AS 4K) am + Ri (AY = Oo If the roots of thie equation are the solution can be put in the forn toes oo ULF me W (um) = Wy + AGi eu Ganwy +B SurGu Surhau (o sus &/ac) Teposing the boundary conditions on x = * £ /2a that the coefficients become W, > 490 Lmtd (a ~1)* A =- We [Cor Sing Sean gt + aug Cou Gg Coan ee | / Z pr - QQ : Q z 2 2 - 2 [ce €) siege GrE8 Sunt R Gand +08 (Cor gO Coole 28 — Sun 8 Sah 2k | B =- wr [o£ eG Gane — Q2ae Sem GR Suing | / G G*) Son Gt Gaga Sunt Gahud +0@ (GEga Gured ~ SenGe Suns eh )| Cm = ',3, 4, ~---) Pat owed i va Chapter IV APPLICABILITY OF THE APPROXIMATION To establish some criterion for the range of approximate validity of he differential equation that has been derived, an investigation of the cr frequencies of harmonic vibration of each Fourier component of the de- formed surface was made. By comparison with more accurately determined values, ome can obtain a range of parameters within which the assumptions thet e,,9 = © give accurate resuite. This same criterion can be carried over to the static loading case. In the following section, the frequencies of harmonic vibration have been computed Including the effect of the longitudinal inertia terns, that is, body forces in the u, yedirections. These frequencies are only approximately equal to those cbtained from equation IT-3.81, aa the deri- vation of this equation did not provide for the inclusion of body forces in the x, y-directions. The body force in the z-direction ls cbtained by replacing v(x, pj) by - e hw}. where Q is the mass of the shell per unit volume. However, this effect was shown by Reisaner (24) to be unim- portant, that ie, the frequencies of transverse vibration were relatively uneffected by the presence of the longitudinal inertia terms. Thug, the conclusions drawn from the results of the general frequency equation ap- ply equally as well to equation Ii-3.21. 1. Derivation of Frequency Factor: Consider the harmonic vibration of a complete cylindrical shell, the ends of which are simply supported according to equation TI-l.17. In accordance with the boundary conditions, take the deflections to be of the following form: ™ a w* (2,6) = ) ) C3 Seon nite’ Gra md Gt we (Iv-1.2) fm m where = ut a/9 and m,n are wave numbers representing the nuuber of axial half-wavee and circumferential full waves respectively. Substituting equation IV-1.1 into equation TI-3.3 and using the relations given by equation II-3.2 and equation II-1.4, one obtains the Pollowing expression for the total strain energy of the shell Eni AEC, Crt z Ve = ) ) Enh {x(ays Bt] + Ges) + $3 =aa (*- ~«) + aves (AY (3 - C3) +Cp eS | + St [QG- Ca) + xe (Cs = ACQC3 +e/n)] Gs ot (IVv-2.2) Corresponding to the assumptions that € a7 vanish, the above expression recuces to Qb c* Al z 2 a VG) = yrs 2) ING or) + Tha Om) + XUV oti) + ee re ey | Gawe (rv-1.3) i 0 af Od as this assumption requires EP our attention ie restricted to frequencies below which the effect of rotary inertia is important, the kinetic energy per unit volume N, reduces to? x wh x N, * ae (uy * Vic + we ) (IV-1.4) Thus, the total kinetic energy of the shell N becomes N@) = Jf fn. adudgda* (IV-1.5) Substituting equation IV-1.4 inte equation IV-1.5 and using equetion iV-l.1, the total kinetic energy becomes Zz a . N= Tahoteayg (Cf a AC + A C)S or (IV-1.6) Corresponding to the assumptions thet ¢«, 9 veoish, this expression reduces to N@)= RR norea/y C7 4 AL) Sor (IV-2.7) The frequencies of vibration, as defined by a frequency factor V , are obtained by equating the maximum kinetic energy to the waximum strain energy. Hence VvzR/S where Vi= (veer ole A a Tyr a z R =X [ess at] + Ga-Ce)” +03 Tha Cat) 2 sale) (C3 ~C2) + Cs Tha Co) | Tt 2 2 Rz 2 2 2 + oY EGS e.) + 4 V/s: (<3 m ~ aC, + nce)| S = C3 4 Aey + (A) C, (1V¥-1.9) When €,,9 = 0, the equations in IV-1.9 reduce to M(B) + Bohs MO Gye eo eee Vv = ws 1a ta bo & on = 7 = = + me + A (Iv-1.10) 2. Analysis end Comparison of Results: As a result of the simplify ing assuaptions thab € / vanish, a single linear equation for the frequency factor was obtained. The exact method, as followed by Arnold and Warburton (25), involves substituting equation IV-1.1 into the three “exact” differential equations, thus obtaining three simultaneous, homo- geneous, linear equations in the variables C, , Cag » Cg. The require- ment that the determinant of the coefficients must vanish gives a cubic equation in the frequency factor. The ease vith which equation IV-1.10 is solved in comparison with the classical cubic equation indicates its possible shortconings. A cowparison of the results obteined by Arnold and Warburton, and those obteined from equation IV-1.10 is presented in figures 11, 12 for two values of h/a. It can be seen that for the lower circumferential modes (n-swall), equation IV-1.10 represents the actual physical system only in the neighborhood of lower axial modes (A~small). However, as the number of circunferential waves increases, equation IV-1.10 is satis- factory up to en increasing number of axial waves. This can be interpreted on the basis of the underlying assumptions derived in Chapter II-2. Consider an elemental panel (fig. 13) vibrating in such a way as to satisfy the boundary conditions of a simply supported beam, that is, vanishing of the deflection y and the bending mouwent v * * at the ends, and of the shear strain on the upper and lower gurfaces. To. >| » “| A t | LY 25g c pty wax =O PV Mee =0 es) fig Figure 15. (h/a = 0.010 ) ——- —- —Simple Beam Concept/ / / Arnold and / / 0.7, Warburton / OQ - Second / Approximation / / @ / ~ / % g / / ‘ “0.5 | / of a of S fo, n=12 . -~ [ oS > 0, 4 ~ ~ / o g Yaa o Ya 0.3 _ / / o 4 n= 10 lan m > ns ” ZLB 0.2 / / J La — n= 8 A iow ey a ae, DLL gg ne7 6 ULL n= 6 QO oy ma m= 8s b 0.5 1,0 1.5 2.0 2.5 3,0 3.5 Wave Number(>) Figure ll - Frequency Factor & ) Frequency Factor ( - 56 - ( h/a = 0,004 ) ------ Simple Beam Concept Arnold and Warburton O - Second Approximation 6 0.5 1.0 1.5 2.0. 2.5 3.0 3.5 Wave Number {X) Figure 12 - Frequency Factor - 57 - From the equations IV-1.1, one finds that the dimensions are such that the span~-to-depth or slenderness ratio is afd . The more exact theory of beamy, as given by Timoshenko (20), shows that neglecting €,,7 is justified only if the beam is slender, that is, n/X of the order of ten. Thus, by analogy, one should not expect agreement with "exact" theories when the model corresponds to a short-deep beam. Actually, the agreement extends satisfactorily to beams whose slenderness ratio is practically three. This is evident in the accompanying figures. These resulte cau be carried over to the static loading case. By analogy, if the loading function produces a deflected surface whose de- ecription by @ Double Fourier Series depends strongly on those modes which lie in the range n/~ < 4, one cannot expect Batisfactory agreement. Thus, the deflection due to a strongly varying loading function cannot be expected to be accurate, However, from Saint-Venant's Principle, this discrepancy will be localized in the vicinity of the strong variation. As will be shown in a later section, even for the ultimate case of a concen+ trated Load, the deflected surface can be described reasonably well even near the load. 3. A Second Approximations Certainly, in the exact treatment of de- formation, the strains ¢€, ‘Y do not vanish, though their effect may not be large. However, for computing natural frequencies, the restriction of movement caused by the vanishing of €,,Y gives rise to increasing error as n/X becomes small, as shown in figures 11, 12. In order to obtain this frequency range more accurately and yet retain some ease of calcue lation, an iterative scheme will be devised based on the results of neglecting €,f% . - 58 - As shown in Appendix II, one can apply to thin shells the classical twoudinensional equations of equilibrium. This means that the effect of bending on the in-plane equilibrium of thin shells is negligible. Thus, restricting attention to the net stresa acting over the thickness of the sheil, that is, the nenbrane stress, it follows that Te t hag > Cae (Iv~3.1) Nay + haw > 0 Vie (IV-3.2) where = an (€1+ ¥€a) (£V~3.3) C, = Fa (ea + ve,) (IV-3.4) wy GY (IV-3.5) Assuming, for a first approximation, that ¢€ a, yanish, equations LV-1.1 become a7 Caio) = DY S(A) Ga 28s" Ga ng Gtwe (1V-3.6) Vv. (¥,¢) = } ) &. Sam my Sung Gaet (Iv-3.7) a, (V4) = ». > C Sun AY Ga mp Gawt (1v-3.8) -“ 59 = *® Furthermore, u, way be congidered the exact deflection, as equation - % % ; * IV-3.6 is perfectly general, Hence YM , vw are derived from ug ag @ result of the assumption that €a,7% vanish. With this first approximation the steps to be followed are a) Compute €, frou equation IT-1.6. b) Coupute €a from equation IT-1.6 and substitute €..€a into IV-3.3 in order to evaluate @. c) With s. > coapube bY from equations TV-3.1, IV-3.5, wf Lio and similarly , a) Compute VO from equation IV-3.2, a e) With v, €, determine a new value for Fa from equation ¥ IV-3 * a e ES a % f) Finally, with u. determine a corrected v = v, from equation i 4 cy % » IT-1.6 using determined inc): anda neww = w, frou . #% % equation II-1.6 using €, computed ine) andv =v, . Following the above steps, one Pinds AY Sin mY Gon ng Grove hae Mm 7 z[™~1 3 [1 I p|e — Y'Sem mY Ga mp Gare > ase qd >t Ui ¥ M1 >] J \m F|? A yl 2 ‘ a . 69) ee) ~e4 (22) Jos mW" Sin mp Gare & 2 u 3™ 3M from Ye > a { (eres ~ Tw)dg » 60 « a) “D2. = {eos ca <a (3)-e Cot] snap ante fron Wy = f(¢ vi ~ Kaw J ec jJi-v’ ez) = S e(eey Qh] vQPay Summ Gan Gare from €4° nv Wy - VE, 34 3 |P E pay 2 2 - ¥ . I + Mo (3) 6 (22) Sem ahs Sim mg Grae im m4] 3] “WAY + AY] Semi ang Gor ¥ £ from Vi -[o- Luly dv and = Vig ~Q€Qq This iterative process could be repeated indefinitely, but on comparing the above results with equation IV-1.1, the displacement coefficients ave seen to correspond to (IV-3.9) « 61 = Tt hes been found that these mode shapes give sufficiently accurate frequencies so that further iterations are not warranted. Thus, defining That is, from equation IV-3.5, Azt+ & +, (¥-¥) . v Vv »\ B= A-Q)(v+ v)- nt +(4) the frequency factor expression (eq. Iv-1.5) becomes vV=R/S' where = R’= (AY + TZ, [Xs (mrt) sav XCmir) +a Chev) im [B* + (A-By vav(AyB-a) s AVA XP b-YAB + OTH (AAT i z S'= B+ 4a +(4) (IV-3.10) (iV-3.11) (IV-3.12) (IiV-1.13) Tt has been found convenient, in calculating frequencies with equation IV-1.13, to use the following procedure; - 62 ~ a) Compute VY from equation IV-1.10. b) Evaluate A,B. c) Compute V from equation IV-1.13. a) Repeat from b) until the process converges. This method has been followed to obtain the frequencies shown in figures 11, 12. Within the range of parameters presented, the frequency curves determined from equation IV-1.13 are not distinguishable from those of Arnold and Warburton. hy Application to Static Loads: In particular cases where the distri- bution of radial load is not a strongly varying function, it may be con- venient to use a double series solution rather than the methods of Chapter If. This method might be more applicable to automatic computing methods. Consider the following system (fig. 14) with an arbitrary distribution of radial load a. Figure 14. The displaceuents way be written wrvyd) = YPC, KCRG mae Gang VCE) DD ek Sin MH Sarmp wm ™@ ~ 63 1 or (Vid) = ” Jes im mile’ Ga md (IV=4.1) Corresponding to equation IV-1.2, the total energy V of the shell becomes . EA * m, j Vv 2) keer maces C8 VON) (TV-4.2) nN where gonad = MY (sia)+ RB] + ASBY tae? + av| B- War Got) | {IV~4.3) with A,B defined in equations 1V-3.10, IV-3.11 and IV-3.12. The total external work T corresponding to equation IT-23.4 is given by If one wakes the approximation that €a,Y vanish, the Principle of «~ 64 = Least Energy requires that $(V-T) = 0 (IVv-4.5) or —— “ar feD s - [ones f 7 0ud) Sum me da” (1v-4.6) where “4 ra 2 WT el 2,r (m1 \% Lom = x(4. +2) 4% (nt) 4 vAL (ni) + EY XY (=) (1v¥-2.7) This solution is essentially the one studied by Hayashi (19). In order to obtain a more accurate solution for a given loading con- dition, one can apply the results of Chapter IV-3. Equations IV-3.11 and IV-3.12 give C, as @ definite function of C, , Cy. If equations IV-h.2 and Iv-4.4 ave substituted into equation IV-4.5 and the variational pro- cedure carried out with C, ss the dependent variable, it follows that _ Bat Ci-v?) a * Sa Shot Gm) fortes fron soni as (Iv-4.8) For purposes of computation, it has been found that one can satisfactorily approximate the mode shapes by neglecting the inertial effect V , that is, by tesking B= A- vaya (Ay (IV-4.9) This is shown in figure 15. The preceding analysis consists of an expansion of the solution in terms of approximate mode shapes. It was seen in Chapter IV-3 that these modes led to frequencies thet were indistinguishable from those calcu- lated by more exact wethods. However, as the frequencies are cbtained from an expression involving the integral of the modes, one must expect the accuracy of the mode shapes to be less than that of the frequencies. This ig shown in figure 15. The “exact” mode shapes were obtained from Baron and Bleich (26). The mode shapes resulting from the assumptions thet €,,°Y vanish lie on the axis, as A, B are identically equal to one. Again, it 1s evident that this assumption is reasonable only in the vie cinity of » fn <1, that is, & model of the shell consisting of long slender elenenteal beaus. The second approximation given by equations IV-3.11 and IV-3.12 is applicable over @ much wider range. In fact, it is fortuitous that the error in approximating the mode shapes by equation IV-4.9 seems to be in the direction of closer correspondence with the re- sults of Baron and Bleich. Presumably, if the iterative procedure de- scribed in Chapter IV-3 were carried an additional step, it would result in mode shapes that would approximate those of Beron and Bleich more closely. - 66 7 1000 | B ( Eqn. 1V-3,12) 100 s A (Eqn. IV-3.11) rs) fa g ol ‘gS 3} o a 4 fal © - A(Baron and Bleich) 10 A * B(Baron and Bleich) & + A{Eqn. IV-4.9) 1.0 j Slenderness Parameter (/n) 0 1 2 3 4. 5 6 7 g 9 10 Figure 15 - Deflection Ratio - 67 - Chapter V EXPERIMENTAL PROCEDURE AND RESULTS It 49 the purpose of this chapter to examine the behavior of the shell surface in the neighborhood of a concentrated load, which is the ultiuate in discontinuous loading functions, in order to deterwine both how large is the region of discrepancy between the approximate theory end experiment, and to obtain a measure of this discrepancy in both deflection and strain. By extrapolating the resulte for a concentrated load to the more general loading functions, one can cbtain a meagure of the accuracy of the approxi- ‘mete method in the general case, 1. ; Experimental Apparatus; The cylindrical shell used in the experi- mental investigation was constructed by usual sheet metal technique with a welded longitudinal sean. The material was a low carbon steel that had “been cold-rolled. The dimensions are those given in Appendix I, Two rings soldered into the ends of the shell to retain the shape afforded a means of fastening the deflection-measuring fixture. | The load-applying device, shown in figure 16, consists of a rod ou which sandbags of any desired welght can be placed, and which Is guided by @ wooden frame. The load is actually applied to the shell through con- tact with the end of the rod on which a hemisphere of radius 1/4 inch has been ground. | 2. Radial Deflection: The apparatus used in the measurement of radi- al deflection was the same as that used by Hayashi (19). This fixture, shown in figure 17, consists of a long bar which rotates freely in the shell and on which igs mounted e slider holding a dial gage. It was found eonverient to use a l-inch Ames Dial Gage with a least reading of 0.001 in. a io oo i Figure 16 - Cylindrical Shell with Load Applying Apparatus - 69 - Figure 17 - Radial Deflection Measuring Fixture - 7O ~ The axial position of the slider eculd be measured by an attached steel tape (least reading 1/16 in.); while the angular orientation could be meagured by a protractor fastened to the shell (least reading 1°), Since the shell was constructed by the usual sheet metal technique, the cross-sections were not perfectly round, although the generators | were reasonably straight. The initial surface of the aid-span station is shown in figure 14, The scatter in the observed points is due to the effect of different orientations of the shell, that is, the deflection due to the weight of the shell itself. Deflection deta was taken at several points around the mid-span atetion with a concentrated load applied at this station, and is presented in figure 19. Each set of symbols applies tos different angular orlen- tation of the point of application of the load which are given in figure 18. It is to be noted that the initial curvature has @ warked effect on ‘ the final deflected surface. So-called high spots or peaks,as at g = 30° (fig. 18),give deflections which ere low; while at low spots or valleys, as at P = 150° (fig. 15), the deflections are high. This difference is essentially the result of applying the load alternatively along a wajor or a winor axis of a locally elliptic surface. The results chtained in the relatively constant-curvature regions, that is 20°S ¢@ < 90° ana 200°< d < 270° (fig. 18), were consistent in the peak readings. How- ever, it was generally found that the surface imperfections had a greater effect on the reaulting deflected surface in regions where the shape of the deflected surface was slowly varying, as at a comparatively large distance from the point of application of the load. A vepresentative set of data is shown in figures 20 and 21, where it is compared with the computed curves of Appondix I. For the circumferential - 71 - jo an a[Buy “sa uCT}eTZEq faoejins papeo[uN ~ BT aangtg wieag [\ | GO, Oy NG 4,0 4a a n | | | | | 1002 r g / a Q & g 1 OST o 3 / 5 S an: g S. * oor Ss WwW tos OSE Q00€ 9082 02 0st 9 00 09S 09| 4 [l2yug jo alsuy -~ 72 - Simple Beam Concept 5 3 w/P -~inch x 10° / lb. Figure 19 - Radial Deflection vs. Angle - Sta, 0 Comparison With Experiment ~ 73 - Simple Beam Q— Concept — Experiment w/P - inch x 10° / lb, | 4 Figure 20 - Radial Deflection vs. Angle - Sta. 0 Comparison With Experiment ~ Th - ydaou0y weag etdurtg — © ey ng “Tt peoy ay], NAYL r0yeL9Uay ev UO sdULIST [eIXY “SA UOTJIaTJaq [eIpey - IZ aInsty v "at / 01% your a/m quaurtzadxy — © 10 °T ug “0 us “O uO °T ng°T ——— r r aouezsiq [eIXy 0 variation (fig. 20), very good agreement is found in the region away from the load; while, im the region under the load, the curvatures that were calculated are found to be much emaller than those chserved, For the axial variation (fig. 21), the discrepency again is in an underesti- mation of the curvature in the nelghoorhood of the load, but there the effect extends to larger distemces from the load. \ From these vregults, it can be concluded that the approximate metuod underestimates the curvatures in the neighborhood of the load, but gives @ good representation eleewhere. This @iscrepancy lg more pronounced in the axial direction where the rate of change of curvature is not as large ae in the cireunferential éirecti on. .. Thus, gince the strain depends es- sentially on the local curvature, “and the discrepancy in the curvature extends from the load to a greater degree in the axial ction than in the circumferential direction, che can expect the discrepancy in the obe served strain to be in the region under the load, but to extend in the axial direction to a greater degree than in the clreumferential direction. 3. Axial Membrane Strain: In order to measure the axial strain, electric reaistance strain sages (Baldwi n, Type A-7) with a Gage length of 1/4 inch (width 1/8 inch) were attached to the surface of the shell. Two gages were Located at ench axial station, one on the inner wall and one on the outer wall. Measurements were taken at axial stations 9/16 inch and 1-3/3 2 inch from the mid-span station a& whieh the load was ape plied. The strain was measured by means of a Baldwin Strain Indicator. The epparatus is shown in figure 22. As the wall thickness of the shell was a nominal 0,015 inch, and the total thickness of the gages (paper + grid + paper) was about 0.005 inch, it was suspected that strain meagurements in bending might be greatly — Figure 2e - Cylindrical Shell with Strein Measuring Apparstus affected by this apparent increage in wall thickness due to the gage. In order to measure this effect, a strip of the same material was tested as a cantilever beam. Four gages of the same type were located on the beam in the same fashion as they were located on the shell. From the results snown in figure 23, it is evident that the reading of the gege can be as much ag 20-25 4 higher than that expected on the basis of simple bean theory. However, the most significant result is that, due to smali differ- ences, as in the amount of glue applied to the gage, 1+ was found that gages at the same station gave results that might differ as much ae 5-10 $. Thus, as the membrane strain is taken to be half the sum of the readings of the two gages at a given point, it is possible that the reaulting mem brane strain can be in error by an amount equal to approximately 2-5 % of cs the bending strain at thet point. Particularly in the neighborhood of the load this can amount to as much as 2-10 % of the membrane strain. The results of the membrane strain measurements are given in figures 24, 25 corresponding to the axial stations given above. It can be ob- served that within the Linits of the error in the gages mentioned above, the strain results are as predicted from the analysis of the deflection measuremente; that is, for any given axial station, the agreement is good away from the generator p = 0, but near this generator, the lack of curve- ture given ty the approximate theory results in strains that are not in @o0d agreement with those chserved. The strain in the imnediate vicinity of the point of application of the load was evidently above the yleld point of the material, as plastic flow was visible at the points where the load was applied. Several ate tempts were made to measure strain in this region, but due to the highly Localized behavior of the strain end its magnitude, the sages failed or Ast t ~j ee) t ~ 2 9 ‘oS a 4 _, + “a 200 * 3(Reversed),*4 t e 1100 _ Simple Beam Theory " x 3 a wv a” - 0 00 400 400 200 ZA 2 3 A 100 Deflection(y) - inch x 10 =. = a a3 ae y ~200 d ond tt ~~ ,300 ‘ 2 fF “oe l(Rever sed) a“ Ma g 1200 a a “ Simple Beam Theory ' a“ q 100 a“ i a“ wv -400 200 400 Deflection(y) - inch x 10° a 1-200 “4 < 2 4-300 Figure 23 - Strain Gage Calibration -79- €\/P - in, « 10°/ in, / lb, 1.5 tt 1,0" Q. gi 9 0,5" 1, 01 1,5" © * lbxperiment Oo: Calculated Figure 24 - Axial Membrane Strain vs. Arc Length at Sta, 1-3/32; Comparison With Experiment i 1,5" i, 0" 0,5" 0 0.5" 1,0" 1.5" —}. + © - Experiment Or Calculated Figure 25 - Axial Membrane Strain vs. Arc Length at Sta. 9/lé Comparison With Experiment indicated that possibly the adhesive had broken by exhibiting a changing null point. Thus the etrain in the vicinity of the point of application of the. load is divided into three regions. The first, closest to the load and extending 3 distance comparable to several shell thicknesses, is probably describable only in terms of a more complete three dimensional analysis and exhibits a plastic deformation. The second, extending about an inch from the load, is a region in which the strains €,,7 are important. The third, comprising the remainder of the shell, is a region in which the trains €2°Y are not significant and the approximate method is sufficient. As a result of considering a circular shell to be the limiting case of a polygonal shell, a method of analysis was developed that neglected the contribution of the in-plane shear and circumferential normal strains to the deformation. The classical eysten of three partial differential equations given by Timoshenko (3) has been reduced by this simplified approach to 6 less complex set, namely (viw 4 ave + w - P) + 18/9" Qrevy + ACH) (ar + Oy) rddhp Wud ~ © Vv = Oo ? hig 4 Yy = 0 For the case of a complete shell, the substitution uy (% > ) -)W (um) Gamd M>) reduces the equation for w to a totel differential equation of the fourth order given by (ms + ta/7*) Wivvu ~ an W) —n 2 Pico 2~¥ M+ wv) + (nt) W 20a This is to be compared with Donnell's equations which are 8 ~ ye —~ A Vow + Ee Svea = Vp a A vou VO ey 7 Direg Vv a“ (at V)ur 4 uy + Dodd With the same substitution for w, Donnell's equation for w reduces to a total differential equation of the eighth order. Aon anelysis was meade which establishes the conditions of loading under which the approximate golution gives satisfactory values for stresses throughout the shell. It was found that if the loading function could be expanded in a Double Fourier Series such that the principal coefficients corresponded to vave numbers having cireumferential wave numbers n greater than or equal to,about three times the axial wave number \ , the approxi- mate golution was in good agreeuent with the solutions obtained using 6 more exact approach. This is equivalent to requiring that the principal deflection modes of the shell give rise to elemental panels with a high slenderness ratio (length/depth). . Thus, if one has a strongly varying loading function, the approximate method will give more accurate results if a/R is small. For, with a/L smail, as » is proportional to af/2 , then for a given number of axial waves, @ greater range of n is within the desired range. When a/2 is large, good representation of a strongly varying loading function is not possible,as the principal modes lie outside the desired LANE A cowputed solution for a concentrated load was compared with experi- mentel measurements of radial deflection and axial membrane strain. I+ - Bh - was found that in the immediate neighborhood of the concentrated load, the approximete solution was in error. The cbserved deflection under the load wag about 20% larger than was calculated. Wo strain data could be obtained in this region. However, outside the region Immediately under the load, good correspondence was Pound between the observed and calcu- lated values of both deflection and strain. a) e 10. “ii. 5. »~ EF REFERENCES — Love, A-E.H. Mathematical Theory of Elasticity. New York: Dover Publications, 1944, 515-552, Flugge, W. Statice ud Dynamik der Shalen. Berlin: Julius Springer , 1934, 110-120, Timoshenko, 8. Theory of Plates and Shells. New York: McGraw-Hill Book Company, Inc., 1940, 433-440, Epstein, P.8. "On The Theory of Elastic Vibrations in Plates and Shelis," Journal of Mathematics and Physics, vol. 21, 1942, 198-209. Kennard, E.H. “The New Approach.to Shell Theory: Circular Cylinders," Journal of Applied Mechanics, vol. 20, 1993, 23-40. Chien, W.2. "The Intrinsic Theory of Thin Shells and Plates," Quarterly of Applied Mathematics, vol. 1, 1945, 297-327. Yuan, &.W. “Further Investigation of Thin Cylindrical Shells Sub- jected to Concentrated Loads,” Polytecinic Institute of Brooklyn, PIBAL Report No. 281, 1955. Lord Rayleigh, "On the Infinitesimal Bending of Surfaces of Revolution,” Proceedings of the London Mathematical Society, vol. 13, 1681, 4. Lord Rayleigh, "On the Bending and Vibration of Thin Elastic Shells,” Proceedings of the Royal Society of London, vol. 45, 1888, 105. Lord Rayleigh, Theory of Sound. New York: Dover Publications, 1945, 395-432, Love, A.E.H. "The Small Free Vibrations and Deformations of a Thin Elastic Shell," Philosophical Transactions of the Royal Society of London, vol. 179, 1085, 491. Yuan, &.W. “Thin Cylindrical Shells Subjected to Concentrated Loads," Quarterly of Applied Mathematics, vol. 4, 1946, 13~26, Hermes, R.M. "Oa the Inextensional Theory of Deformation of Right Circular Cylindrical Shells," Journal of Applied Mechanics, vol. 18, 1951, 341-344, Pinsterwalder, U. “Die Theorie der zylindrischen Schalengewolbe Systen ceiss-Dywidag und ihre Andwendung auf die Grosswarkthalle in Budapest," Publications of the International Association for Bridge and Structural Engineering, vol. 1, 1932, 127-152. Schorer, H. “Line Load Action of Thin Cylindrical Shells," Transactions of the American Society of Civil Engineers, vol. 101, 1936, 767-802. a) CN 18. ig. 256 26, ft Sdgvist, F.K.G. “Action of Forces and Monents Symmetrically Distribu- ted along a Generator of a Thin Cylindrical Shell," Journal of Applied Mechanics, vol. 13, 1946, 106-108. Hoff, W.J., Kempner, J., and Pohle, F.V. “Line Load Applied Along Generators of Thin-Walled Circular Cylindrical Shells of Finite Length,” Quarterly of Ayplied Mathematics, vol. 11, 1954, 411-425. Donnell, L.H. "Stability of Thin-Welled Tubes Under Torsion," National Advisory Committee for Aeronautics, Technical Report No. 479, 1933. : Hayashi, P.H. "A Study of a Laterally Loaded Thin Circular Cylindri- eal Shell,” Thesis for the Degree of Mechanical Engineer, California - Institute of Technology, 1953. Timoshenko, &., and Goodler, J.N. Theory of Elasticity. New York: McGraw-Hill Book Company, Inc., 1951, 145-178, Courent, R., and Hilbert, D. Methods of Mathematical Physics, Vol. i. New York: Interscience Publishers, Inc., 1953, 221-222. Hildebrand, F.B. Methods of Applied Mathematics. New York: Prentice- Hall, Inc., 1952, 120-200. Brown, H.K, "General Vibration of a Hinged Beam," Journal of Applied Physics, vol. 15, 1944, 410-413. Reissner, E. "On Transverse Vibrations of Thin, Shallow Elastic Shells," Quarterly of Applied Mathematics, vol. 13, 1955, 169-176. Arnold, R.N., and Warburton, G.B. "Phe Flexural Vibrations of Thin Cylinders," Proceedings of the Institution Of Mechanical Engineers, vol. 167, 1953, 62-74, Baron, i.L., and Bleich, H.H. "Tables for Frequencies and Modes of Free Vibration of Infinitely Long Thin Cylindrical Shells," Jouraal of Applied Mechanics, vol. 21, 1954, 178-184, Jolley, L.B.W.- Summation of Series. London: Chapman and Hall, Ltd., 1925, 186. Bromwich, T.J. An Intvoduction to the Theory of Infinite Series. London: Macuillen and Company, Ltd., 1926, 356. Dwight, H.B. Tables of Integrals and Other Mathematical Data. New York: The Macmillan Company, 1951, 141. - &F - Appendix Tf CALCULATIONS FOR A CONCENTRATED LOAD APPLIED AT THE MID-GPAN POINT OF A COMPLETE CYLINDER In order to carry out the indicated summation for the case presented in Chapter III-2, the roots «,@ of the following equation must be obtained. A =—T3 OA _ a — A- {-V where A= am. M7 avn m+ ra/ye 5 A (a ~ " B = Mm mn + ta it® It follows directly that in polar notation, m° can be written as i (to) where @ ~ ‘tan bac ~ (Ajay | 4 = VB This is shown graphically in figure A-l. Therefore, the roots wu are given by i Ct &a) (404+) Vx e@ Vn e so that w= VA(8 +2) @ - 418 ~ Ma) Figure A-l. For n> 10, n* is negligible in comparison with n‘ following simplifications can be made: A/a = me / (mt 4 iz /2) B 3 m°/ ( im’ + 12 /72) If N is defined as Na '/yi tia fem y 80 that the then (m 210) wu + @ = mn a? ~ @* = rt N* Using the following data, EB o= 30x 10 pei. & = 3,367 inch h = 0.015 inch HL = 5 inch Vo= 0,29 values of A, @ , A/S, B, C/B, D'/B have been computed and are presented in Table I for the first thirty-five terms. It is to be noted thet all the above items save B are general properties of the differential equation applied to a particular cylinder geometry for arbitrary loading function at the point x = 0 and "Pin-Ended" boundary conditions. Only B is charac- teristic of the loading function. For the case of the concentrated force of magnitude P Located at the mid-span point, the deflection w and the axial membrane strain ¢€, are given by os wu p>) > ) Wiiym)lorng + Ww )G¢ €.%,%) = > Fa Wray Geng + Wy CG) Gap where Wm) = B/ Gage (Suma + Ave Goh av) | + Sun @u (S73 Sueh 4 -~“@ Garor)| and For purposes of carrying out the calculations, the following functions will be introduced; ¢ (mu) = Gea bu (Sawn at + 4 Coun au) + SuQuCS Sunn we - @ Crhau) in A@u) = Guey|- (a+) Sembory + (0? A +aa@-S ~@°A) Gara] + SinGy [ces ~ ang - eS ) Suk au — ¢ Ca®s @°) Gen ual (A-I.1) Thus, the deflection and axiel weubrane strain relations are given by 4,4) =) BGamh x Fe) + WeGag mr) © (A-I.2) €.0,4)= ) Boinh xM@w +We Greg a4 maz! The functions v4 can be greatly simplified if the behavior of a 1@ is determined for n-large. For n such that N can be expanded as Z ! 2 a Therefore, a“, @ can be expressed ea R y so that “Ye = at 1+ ae) ‘ By: ~ |" Thus, as Lim mde \ 5 mow } Sun td Gane d it ts justifieble after a few terms to take an 5° Ve When this simplification is introduced, the functions # a become “Em,4) = -e “CGx@u + € Sin@u) 2 —*% Al Gn.) = We 2 (Gr@u - ¢ Sant) (m2 m ) (AeI.3) Por the values of L, & given previously, this simplification ean be ap- plied for n’ = 7. Defining the angle f as SHY = ed w+ @e GY = «fare it follows that i H 3 aa am e E(,m) = eo Sén(P 4 BY) } a, + mm rod Alum) = e | BG SumuCF-Rt) (m2 mt) Tre 80, for n-large, Lim |F| = ¢ fim | | = 0 nS ‘and, for x = 0, aim WF) = > 10 im (A) = N tT+o Uwe moat my oant The procedure in calculating w,<¢, is therefore to carry out explicit- ly the suumation indicated by equation A-I.2 using the relations of equation A-I.1 for n < n* and the relations of equation A-I.3 for mom . After a nuuber n" of terms, it becomes reasonable to make the approxi- mation that 3/p ee Mh (™ >") ~ 93 - & For an error in B/P less than 10%, n can be taken equal to thirty-five. Thus the remainder of the series can be taken to be of the fora R(t) == ” F my. ond Sue) 2-5 yc, n) aed m2 30 where 36 © Wah) =) BGanp x Pein) + Ry Gut) + WO IGS met €,(u,9) > ) BGang * A (um) +3n CAD) + Wry &,1) Ga > Mi From the behavior of (,. 4 described above, it follows that J Ge RAGo#)| aes | SL F629 - (A-I.4) Gand | At («, 35) (ss |? ms aE 5, C,%) AD while These latter functions can be determined explicitl: from a consider- ation of the-following function BO) =) Ay Gang (a-t.5) Mz In Jolley (29) and Bromwich (28), the following function is defined Gand. _ On Ca Sm $/a) (o<¢$ <ar) 7 aN i D1e Therefore, from the uniformity of | Fourier Series, it follows that \ = moo 1 San m ~ 6 and Figg = 2s (a Se qa) (A-I.7) Integrating once with respect to d results in om = $ Ina +a Qn ( Sen) 9% +C, This latter integral can be expressed in terms of a series given in Dwight (29), so that El, becomes “ap \ 3 Ss Eig = Pinas Git a] Band ~-$-48)-4(3) \ ry 2 ant (bia) , aC ami)! where the B, are Bernoulli Numbers. From equation A-I.6 it follows that go that C, =~ dim (ping) = 9 33 wt Integrating once more, the function B($) becomes rm 2 b\4 2 SC) = (SP Cona-0) 4(GF($) - 8 (E)'| 1 + (4) 2" B36 $v"? m (am+i)l ora ) From equation A-I.5 and the fact that Him ( o*In 4") = d+0 the constant Cc. becomes C,> S@) = @) where Xe) is the Riemann Zete Function of argument three. Finally, the function &@) can be evaluated in the following form: = f(s) + ($F (dma ~3) + (9)"on (8) - rr = (4y [ + L(y 1 (Of ge eB 36 4 na (A-I.7) The remainders R ; Sa are then calculated by evaluating equation A-I.7 for a@ particular value of p and subtracting from this the sum of the first thirty-five terms. The resulte of thie calculation are given in Table II for several values of d . = 96 » The calculations indicated above have been carried out for several exial stations for both axial membrane strain and deflection. The results are presented in figures A-I.2, A-I.3, AeI.4, Aw-I.5, A-I.6. The remainder terms for x # 0, using equation A-I.4, are found to be negligible compared to the contribution of the leading terns. - 97 - €,/p -in. x10°/ in. / Ib. _ ) Figure A-1,2 - Axial Membrane Strain vs, Angle at Sta. 1-3/32 Simple Beam Concept 60 - 98 - &/P-in, x 10° / in, / lb. -3 Figure A-1,3 - Axial Membrane Strain vs, Angle at Sta, 9/16 Simple Beam Concept 60° - 99 - /P -in, x 10° / in, /1b. 5 60° 40° 20° J 20° 40° 60 o | |-30 1|-40 1 -60 Figure A-1.4 - Axial Membrane Strain vs, Angle at Sta, 0 Simple Beam Concept - 100 - - «1 40 w/P - inch x 10?/ lb. Figure A-1,5 ~- Radial Deflection vs. Angle at Sta, 0 ss Simple Beam Concept 60 - 1Ol - ul jdasu0y ulesg aftduiig ‘peoy ay} nay z0yer1auary e uO asueysiq [eIxXy “SA UOT}IATJaq TeIpRYy - 9°[-Vy eansrz “qT / Ot x yout - q/m uz/1 u 2/1 ul ~ 108 « Appendix If APPROXIMATE EQUILIBRIUM EQUATIONS From Timoshenko (3), one obtains the following linearized differential equation for the static case: . i-v av = v Uwe rt Rae 2 bq * Zo. Vive a ye +. a Fa \ , i+v a-i&¥. Vis. +2 vy - = = Aa + a ek >> a. Lis , tw” wR inwy. - Lv _ * a0 ( virg * ga rae ) + \e ¥) Vivre ot ‘dd | =O If the dimensionless ratios given in equation II-1.1 and equation II-3.20 are introduced, the above equations become vue + Bee + UY Vg = Vay ~ Cha (© uy + Obed + EY Vin + “ea ) (A-II.1) if ness 1, the above equetions can be approximated as by \4v ~ yar Baw + Beg t Vag = Ve (A-II.2) & bai 2 tar & or equivalently, [a +7(%q- 0) | + (44 *Madig = 0 | mM (A-II.3) Iv.9 - ur ray + x (hy +Vy) N 0 ve Except for a factor of proportionality, these equations are just the differ- ential equations of equilibrium for the etatic two-dimensional theory of Glasticity. If the variables u, v in equation A-II.2 are separated, the equations become “4 Vw = Vary — Wig (A-II. hy) “~% It should be noted that equations A-II.4 are well suited for approxi- mate calculations. As the dependence on w can be placed on the right hand side of the equations, w can be treated ag a forcing function. If one ean determine an approximate form for w by some means, it can be substituted in equation A-II.4 in order to compute u, v. Thus, instead of the methods of Chapter II-4, one might calculate u, v from equation A-II.4 using as an approximate form for w the solution given by equation II-3,21. This method should lead to more accurate strains than the method of Chapter II-l, as, in this case, one obtains the exact a, v corresponding to an approximate w, whereas in the method of Chapter II-4, one obtains only approximate u, v corresponding to an approximate wv. a % ¢ x 10° D/B 2 0.08802 0.08765 ~566.5 ~1.00k. 3 0.2163 0.2140 2194.5 -1,011 4 0.3967 0.3890 ~100.4 “1.020 5 0.6305 0.6111 ~61.74 ~1.032 6 0.9195 0.8789 ~4i.76 ~1.046 7 1.266 1.190 ~30.07 “1.064 8 1.670 1.540 ~22.64 ~1.084 i) 2.138 1.928 “17.61 -1.109 10 2.680 2.357 ~14.08 -1.137 11 3.276 2.805 -11.28 “1.168 12 3.937 3.274 <9. 340 -1.203 13 4.664 3.759 ~7.63 ~1.2h1 14 5.459 4, 25h ~6.6h2 ~1.283 15 6.321 L751 ~5.682 ~1.330 16 7.2k7 5. 2hy =4..903 -1.382 17 8.237 5.727 =, 251 ~1.436 18 9.283 6.193 -3.716 -1.500 19 10.39 6.633 ~3.266 -1.566 20 11.55 7.046 -2.878 “1.639 al 12.74 742k ~2.553 ~1L.716 22 13.98 7.765 “2.271 -1.80¢ 23 15.25 8.068 ~2.027 «1.890 2h 16.54 8.331 “1.817 “1.985 25 17.84 8.551 1.634 “2.086 26 19.16 8.733 -1.473 ~2.194 27 20.49 8.877 “1.331 ~2. 308 28 21.61 8.985 -1.207 ~2.427 29 23.14 9.059 -1.097 #26554 30 24.45 9.103 -0.5998 -2,686 31 25.76 9,120 -0.9131 ~2.827 32 27.06 9.113 ~0.6358 ~2.965 33 28.35 9,084 ~0. 7665 ~3.122 34 29.62 9.039 ~0. 7043 ~3.277 35 30.85 8.976 ~O.6491 ~3.4kO TABLE II 3s eo g ). im Gang S- ) mm Gn mb Difference mai mi 9 1.201660 1. 202057 000397 5° 1.287174 1.187059 ~. 000116 10° 1.152567 1.152620 000052 20° 1.046473 1.046499 6000025 30° 0.907466 0.907486 000020 ho® 0. 748104 0.748119 000025 51.428° 9.552023 0.552015 000008 60° 0.400672 0.400686 .000103