17. Shells having the forn of a surface of revolution and loaded symmetrically with respect to their axis

17. Shells having the forn of a surface of revolution and loaded symmetrically with respect to their axis

CHAPTER SHELLS HAVING REVOLUTION WITH 127 Equations THE AND 16 FORM OF LOADED RESPECT of Equilibrium A SURFACE OF SYMMETRICALLY TO THEIR Let us AXIS consider the conditions of equilibrium of an element cut from a shell by two adjacent meridian planes and two sections perpendicular to the meridians (Fig 267).!_ It can be concluded from the condition of symmetry that only normal stresses will act on the sides of the element lying in the meridian planes The stresses can be reduced to the resultant force Ngridy and the resultant moment M7, dy, Ne and M, being independent of the angle which defines the position of the meridians The side of the element perpendicular to the meridians which is defined by the angle » (Fig 267) is acted upon by normal stresses which result in the force Nyresin ¢ dé and the moment M, resin ¢ dé and by shearing forces which reduce the force Q, re sin ¢ dé normal to the shell The external load acting upon the element can be resolved, as before, into two components Yrr;sìn ¿ đọ đ0 and Zrr› sin ¿ đọ d6 tangent to the meridians and normal to the shell, respectively Assuming that the membrane forces N» and N, not approach their critical values,2 we neglect the change of curvature in deriving the equations of equilibrium and proceed as was shown in Art 105 Fic 267 In Eq (f) of that article, obtained by projecting the forces on the tangent to the meridian, the term — Q,ro must now be added to the left-hand side Also, in Eq (j), which was We use for radii of curvature and for angles the same notation as in Fig 213 * The question of buckling of spherical shells is discussed in Timoshenko, ‘‘ Theory of Elastic Stability,’ p 491, 1936 533 OF THEORY 534 PLATES SHELLS AND obtained by projecting the forces on the normal to the shell, an additional term d(Q,ro)/dy must be added to the left-hand side The third equation is obtained by considering the equilibrium of the moments with respect to the tangent to the parallel circle of all the forces acting on the element This gives? (a, + oa dc) (n + e) d6 — M,rod8 — Myr: cos y dy dé — Q,r2sin gridy dé = After simplification, this equation, together with the two equations of Art 105, modified as explained above, gives us the following system of | three equations of equilibrium: — Nori cos g — ToQ, + ToiY (Nyro) = (M,ro) (312) + Zriro = Nyro + Nori sin ¢ + “gee x = — Mori cos ¢ — Qyrito = O In these three equations of equilibrium are five unknown quantities, - three resultant forces Ny, No, and Q, and two resultant moments M, and The number of unknowns can be reduced to three if we express the M, membrane forces N, and N» and the moments M, and Mg in terms of In the discussion in Art the components v and w of the displacement 108 of the deformation produced by membrane stresses, we obtained for the strain components of the middle surface the expressions from which, by using Hooke’s law, we obtain Ne=7 Eh [1 fa ale (He) Eh => —1 +E woot y —w) hm Vo | @ oot e — 99 y {dv +2 (2 J { _ Nem - (313) ) To get similar expressions for the moments M, and Ms, let us consider Conthe changes of curvature of the shell element shown in Fig 267 the that sidering the upper and the lower sides of that element, we see initial angle between these two sides is dy Because of the displacement v along the meridian, the upper side of the element rotates with 1JIn this derivation we observe that the angle between the planes in which the moments Af¢ act is equal to cos ¢ dé SHELLS FORMING SURFACE OF REVOLUTION 535 respect to the perpendicular to the meridian plane by the amount U/T1 As a result of the displacement w, the same side further rotates about the same axis by the amount dw/(ridy) Hence the total rotation of the upper side of the element is v dw T1 T1 đọ (a) For the lower side of the element the rotation is Ụ dw d fv dw nt inde t de (z+ reap) & Hence the change of curvature of the meridian is! 1dfv dw = dela + nde) ) To find the change of curvature in the plane perpendicular to the meridian, we observe that because of symmetry of deformation each of the lateral sides of the shell element (Fig 267) rotates in its meridian plane by an angle given by expression (a) Since the normal to the right lateral side of the element makes an angle (1/2) — cos ¢ d6 with the tangent to the y axis, the rotation of the right side in its own plane has a component with respect to the y axis equal to — (2 v dw + ¬ 7] cos ¢ dé This results in a change of curvature xe _ fo = (2 dw + \cosg_ 25.) To [v dw \ cot¢ -(2+;5,) ` T2 _49) Using expressions (b) and (c), we then obtain Me Mo= — n|l = DÌ —D dđ fv xã; r, fv dw (3 +) dw | nde y Tự, fv dw rn) nde \cotgy , v d fv F2 T nao dw cot ý (314) ¬ r Substituting expressions (313) and (314) into Eqs (312), we obtain three equations with three unknown quantities v, w, and Q, Discussion of these equations will be left to the next article We can also use expressions (314) to establish an important conclusion regarding the accuracy of the membrane theory discussed in Chap 14 In Art 108 the equations for calculating the displacements v and w were ¿ The strain of the middle surface is neglected, and the change in curvature obtained by dividing the angular change by the length r, dg of the are is OF THEORY 536 SHELLS AND PLATES By substituting the displacements given by these equations established in expressions (314), the bending moments and bending stresses can be By These stresses were neglected in the membrane theory calculated comparing their magnitudes with those of the membrane stresses, a conclusion can be drawn regarding the accuracy of the membrane theory We take as a particular example a spherical shell under the action of its If the supports are as shown in Fig 215a, the own weight (page 436) displacements as given by the membrane theory from Eqs (f) and (0) (Art 108) are — 02q(1 + ») Eh - w= vcot¢ (+: _0øg( DI _ phase 1+y _ cos Ea Mo = gh? + Me = nang) th Ý (ở) ¢) formulas into Substituting these expressions moments, we obtain 1+ cos¢\ + 98 11 Ta T—, » (314) for the bending COS @ (e) S + COS Y S —_ bola to The corresponding bending stress at the surface of the shell is numerically equal to Taking the ratio of this stress to the compressive stress a given by the membrane theory [see Eqs (257)], we find g2+z 21 y°9%/ / 3+» h — nat tS) 608 ý BA cose) ~ aq — The maximum value of this ratio is found at the top of the shell where y = and has a magnitude, for v = 0.3, of h 3.29 — (f) It is seen that in the case of a thin shell the ratio (f) of bending stresses to membrane stresses is small, and the membrane theory gives satisfactory results provided that the conditions at the supports are such that the Substituting expression shell can freely expand, as shown in Fig 215a (e) for the bending moments in Eqs (312), closer approximations for the These results will differ membrane forces N, and N, can be obtained from solutions (257) only by small quantities having the ratio h?/a? as a | factor _ From this discussion it follows that in the calculation of the stresses in SHELLS FORMING SURFACE OF REVOLUTION 537 symmetrically loaded shells we can take as a first approximation the solution given by the membrane theory and calculate the corrections by means of Eqs (812) Such corrected values of the stresses will be accurate enough if the edges of the shell are free to expand If the edges are not free, additional forces must be so applied along the edge as to satisfy the boundary conditions The calculation of the stresses produced by these latter forces will be discussed in the next article 128 Reduction of the Equations of Equilibrium to Two Differential Equations of the Second Order From the discussion of the preceding article, it is seen that by using expressions (313) and (314) we can obtain from Eqs (312) three equations with the three unknowns 2, w, and Q, By using the third of these equations the shearing force Q, can be readily eliminated, and the three equations reduced to two equations with the unknowns v and w The resulting equations were used by the first investigators of the bending of shells.1_ Considerable simplification of the equations can be obtained by introducing new variables.2 As the first of the new variables we shall take the angle of rotation of a tangent toa meridian Denoting this angle by V, we obtain from Eq (a) of the preceding article dw As the second variable we take the quantity U = r2Q, (0) To simplify the transformation of the equations to the new variables we replace the first of the equations (312) by one similar to Eq (255) (see page 435), which can be obtained by considering the equilibrium of the portion of the shell above the parallel circle defined by the angle ¢ (Fig 267) Assuming that there is no load applied to the shell, this equation gives | 2rroN, sin ¢ + 2rroQ, cos gy = from which Ne = —Q, cot = ——U cot ¢ (c) Substituting in the second of the equations (312), we find, for Z = 0, d(Q,ro) riNg sin p = —Noro — “Ie 1See A Stodola, ‘Die Dampfturbinen,” 4th ed., p 597, 1910; H Keller, Mit¢ Forschungsarb., vol 124, 1912; E Fankhauser, dissertation, Zurich, 1913 This method of analyzing stresses in shells was developed for the case of a spherical shell by H Reissner, ‘‘ Miiller-Breslau-Festschrift,”’ p 181, Leipzig, 1912; it was generalized and applied to particular cases by E Meissner, Physik Z., vol 14, p 343, 1913; and Vierteljahrsschr naturforsch Ges Ziirich, vol 60, p 23, 1915 538 THEORY OF PLATES AND SHELLS and, observing that ro = 72 sin ¿, we obtain No = —- náo (Qạn) = — aU (d) r, de Thus the membrane forces N, and Ng are both represented in terms of the quantity U, which is, as we see from notation (b), dependent on the shearing force Qy To establish the first equation connecting V and U we use Eas (313), from which we readily obtain | dv Te 7! _= a1 Ne _ vN 6) v cot ¢ — w = (e) (No — »Ny) — (f) Eliminating w from these equations, we find d ig —vcoty = a [((7a + vra)Ny — (re + vri)Nol (g) d |7: ELT (h) Differentiation of Eq (f) gives? dv o cote v — đu — _ mv.) | The derivative dv/dy can be readily eliminated from Eqs to obtain (g) and (h) » + oe =r„V = “ae = [Œ + vra)Ne — (re + ori) Nol d Tạ ~ deg E (No — mv) | Substituting expressions (c) and (d) for N, and No, we finally obtain the following equation relating to U and V: T2 d?U d T2 To ri de +a (EZ) + oot» lini, r, E cot? Ta dh Tab @ — v dU _ vdh _ h de cot «| U=EhV 21E (815) The second equation for U and V is obtained by substituting expressions (314) for M, and Mg in the third of the equations (312) and using notations (a) and (b) In this way we find We consider a general case by assuming in this derivation that the thickness h of the shell is variable SHELLS r@V ild FORMING (r\ | tr SURFACE OF REVOLUTION 539 ro dhl dV fae ta [ae (q) + Hoty re ris Ge | (= ——|p T1 ate tte) V = -FU (316) ae — Thus the problem of bending of a shell having the form of a surface of revolution by forces and moments uniformly distributed along the parallel circle representing the edge is reduced to the integration of the two Eqs (315) and (316) of the second order If the thickness of the shell is constant, the terms containing dh/dg as a factor vanish, and the derivatives of the unknowns U and V in both equations have the same coefficients By introducing the notation pee) kí oC) LL] ad (me) tele Ge = đ( + ° Zoot + (E) + ) + Tịcot?@ _ a ( ae ) ‘ (@) the equations can be represented in the following simplified form: L(U) + ¬ U = EhV LW)—;.V=-=5 U (317) From this system of two simultaneous differential equations of the second order we readily obtain for each unknown an equation of the fourth order To accomplish this we perform on the first of the equations operation indicated by the symbol L( LLU) + vL (6) + + - ), which gives (317) the = bhL(V) Substituting from the second of the equations (317), L(V) =~ V — U B= Eh ; |W) +2 U u| - D we obtain LLU) + ob (2) ~ 4(u) - Fu T1 T1 = —_““y D (318) = — (319) In the same manner we also find the second — LL(V) — »vL (*) +; + —~ L(V) — —V V If the radius of curvature 7: is constant, as in the case of a spherical or a conical shell or in a ring shell such as is shown in Fig 220, a further PLATES OF THEORY 540 SHELLS AND Since in this case (318) and (319) is possible simplification of Eqs L (=)Tì = Tì= L(U) by using the notation ụ Eh DD y? r3 | (7) both equations can be reduced to the form LL(U) + p4U = (320) which can be written in one of the two following forms: LỊL(U) + iu?U] — [L(U) + +, 2U] = LỊL(U) or — tp?U] = — ?u?U] + ip {L(U) These equations indicate that the solutions of the second-order equations L(U) + tp?U = By proceeding as was explained in (320) are also the solutions of Eq (321) Art 118, it can be shown that the complete solution of Eq (320) can be The appliobtained from the solution of one of the equations (821) cation of Eqs (321) to particular cases will be discussed in the two following articles In the case of a spherical 129 Spherical Shell of Constant Thickness shell of constant thickness r1 = 72 = a, and the symbol (z) of the preceding article is LC: +: ) Foote [fa FC - cote =) (a) ++] a Considering the quantity aQ,, instead of U, as one of the unknowns in the further discussion and introducing, instead of the constant ,, a new constant p defined by the equation p "`.5 \ =) b5 (b) we can represent the first of the equations (321) in the following form: d?Q dq Jet + cot » GE — cot? vO, + 2ip?Q, = (322) A further simplification is obtained by introducing the new variables’ x= sin’? ¢ = sin ø@ _ Vs This solution of the equation was given by Meissner, op cit (6) SHELLS FORMING Nith these variables Eq d? sứ — 1) 3ã SURFACE OF REVOLUTION | 541 (322) becomes +(§z—?) [his equation belongs to a known econd order which has the form d — 2ip? ca + = (d) type of differential equation of the z(1 — z)” + [>z — (œ+ 8+ 1)zÌy' — aby = tquations (d) and (e) (e) coincide if we put x=9 a= + Vo V5 ++ 82? 1p >eV 8= + Bipp2 (f) A solution of Eq (e) can be taken in the form of a power series y = Apt Aww + Aor? + ¿+ + - - - (9) Substituting this series in Eq (e) and equating the coefficients for ‘ach power of x to zero, we obtain the following relations between the ›oeffioients: A,= 8ae4, rn An = An— A, so = 1) + ]) 2(y + 1) A (a +n—1)G+n—1) n(y +n — 1) With these relations series (g) becomes _ a8 , aa+l)B6+1), p= Aol t+ Set 1-2-y¥7+l) * + [his is the so-called afati(a+2)p6+1)6+2) I-2-3-+Œy + 17 hypergeometrical series 4+ 2) , mt It is convergent | 0) for all values of x less than unity and can be used to represent one of the intesrrals of Eq (d) Substituting for a, 8, and y their values (f) and using the notation a5 + Bit = 5+ 41 [PEE ve obtain as the solution of Eq (d) — +2 2) _— ụa _ 32 — 62 | (32— 8%)(72— 8%) 2cm Ao[1+7g-T.3# + ig 1-2-2377 vhich contains one arbitrary constant Ao (i) ] ) | 542 THEORY OF PLATES AND SHELLS The derivation of the second integral of Eq (d) is more complicated.! This integral can be written in the form 22 = 21 log x + = o(2) (k) where ¢(x) is a power series that is convergent for |z| < This second solution becomes infinite for z = 0, that is, at the top of the sphere (Fig 267), and should not be considered in those cases in which there is no hole at the top of the sphere If we limit our investigation to these latter cases, we need consider only solution (7) Substituting for 5? its value (7) and dividing series (7) into its real and imaginary parts, we obtain 21 => Si +- tŠs (L) where S; and S2 are power series that are convergent when |z| < corresponding solution of the first of the equations (321) is then U, = azsng = 1,4 71, The (m) where J; and J; are two series readily obtained from the series S,; and So The necessary integral of the second of the equations (321) can be represented by the case of a spherical of the differential represented in the same series J; and J, (see page 489) Thus, for the shell without a hole at the top, the general solution equation (320), which is of the fourth order, can be form U =aQ, = Ali + Bl, (7) where A and B are constants to be determined from the two conditions along the edge of the spherical shell Having expression (n) for U, we can readily find the second unknown V We begin by substituting expression (m) in the first of the equations (321), which gives Hence L(T: + tÏ¿) = LUI) = ple —tu?(Tì + 21:) L(I2) = —p?ly (0) Substituting expression (7) in the first of the equations (317) and applying expressions (0), we then obtain EhaV = aL(U) + »U = (Av — Bap?)I; + (Aap? + Bv)l; It is seen that the second unknown YV is I and (p) also represented by the series I Differential equations that are solved by hypergeometrical series are discussed in the book ‘‘Riemann-Weber, Die partiellen Differential-Gleichungen,” vol 2, pp 1-29, 1901 See also E Kamke, “‘Differentialgleichungen,”’ vol 1, 2d ed., p 465, Leipzig, 1943 ... FORMING SURFACE OF REVOLUTION 535 respect to the perpendicular to the meridian plane by the amount U/T1 As a result of the displacement w, the same side further rotates about the same axis by the amount... is the rotation of a tangent to a 128 In the case of very thin shells, if the angle ¢ is not small, the quantities Q, and V are damped out rapidly as the distance from the edge increases and have... comparing their magnitudes with those of the membrane stresses, a conclusion can be drawn regarding the accuracy of the membrane theory We take as a particular example a spherical shell under the action