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

SHELLS HAVING THE FORM OF A SURFACE OF REVOLUTION AND LOADED SYMMETRICALLY

WITH RESPECT TO THEIR AXIS

127 Equations of Equilibrium Let us 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 9

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 nor-

mal to the shell, respectively

Assuming that the membrane forces

N» and N, do not approach their

critical values,2 we neglect the

change of curvature in deriving the Fic 267

equations of equilibrium and pro-

ceed as was shown in Art 105 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

1 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 8 Timoshenko, ‘‘ Theory

of Elastic Stability,’ p 491, 1936

533

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 + 7 e) d6 — M,rod8 — Myr: cos y dy dé

— Q,r2sin gridy dé = 0 After simplification, this equation, together with the two equations of Art 105, modified as explained above, gives us the following system of

= (Nyro) — Nori cos g — ToQ, + ToiY = 0

Nyro + Nori sin ¢ + “gee + Zriro = 0 (312)

x (M,ro) — 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 M, The number of unknowns can be reduced to three if we express the membrane forces N, and N» and the moments M, and Mg in terms of the components v and w of the displacement In the discussion in Art

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

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

nt inde t de (z+ reap) &

Hence the change of curvature of the meridian is!

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 com- ponent with respect to the y axis equal to

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 is obtained by dividing the angular change by the length r, dg of the are

(314)

established By substituting the displacements given by these equations

in expressions (314), the bending moments and bending stresses can be calculated These stresses were neglected in the membrane theory By comparing their magnitudes with those of the membrane stresses, a con- clusion can be drawn regarding the accuracy of the membrane theory

We take as a particular example a spherical shell under the action of its own weight (page 436) If the supports are as shown in Fig 215a, the displacements as given by the membrane theory from Eqs (f) and (0) (Art 108) are

g2+z 21 y°9%/ ha cose) ~ BA — nat tS) 608 ý / aq — 3+» h

The maximum value of this ratio is found at the top of the shell where

y = 0 and has a magnitude, for v = 0.3, of

h

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 shell can freely expand, as shown in Fig 215a Substituting expression (e) for the bending moments in Eqs (312), closer approximations for the membrane forces N, and N, can be obtained These results will differ from solutions (257) only by small quantities having the ratio h?/a? as a

_ From this discussion it follows that in the calculation of the stresses in

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 accu- rate 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

As the second variable we take the quantity

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 = 0

from which

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

2 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

and, observing that ro = 72 sin ¿, we obtain

1 aU

No = —- náo (Qạn) = — r, de (d)

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),

Te 7! = a Ne vN 6) (e)

v cot ¢ — w = 7 (No — »Ny) — (f)

Eliminating w from these equations, we find

d

ig —vcoty = a [((7a + vra)Ny — (re + vri)Nol (g)

Differentiation of Eq (f) gives?

1 We consider a general case by assuming in this derivation that the thickness h

of the shell is variable

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) oC) LL] ad (me)

kí = Ge tele (E) + Zoot °

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 (317) the

operation indicated by the symbol L( + + - ), which gives

LLU) + vL (6) = bhL(V) Substituting from the second of the equations (317),

L(V) =~ V — B= Eh ; |W) +2 u| - D

we obtain

LLU) + ob (2) ~ 2 4(u) - Fu = —_““y T1 T1 D (318)

In the same manner we also find the second —

LL(V) — »vL (*) +; + — ~ L(V) — — V = — 5 V (319)

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

simplification of Eqs (318) and (319) is possible Since in this case

which can be written in one of the two following forms:

LỊL(U) + iu?U] — tp [L(U) + +, 2U] = 9

These equations indicate that the solutions of the second-order equations

are also the solutions of Eq (320) By proceeding as was explained in

Art 118, it can be shown that the complete solution of Eq (320) can be obtained from the solution of one of the equations (821) The appli- cation of Eqs (321) to particular cases will be discussed in the two following articles

129 Spherical Shell of Constant Thickness In the case of a spherical shell of constant thickness r1 = 72 = a, and the symbol (z) of the pre- ceding article is

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

Jet + cot » GE — cot? vO, + 2ip?Q, = 0 (322)

A further simplification is obtained by introducing the new variables’

Nith these variables Eq (322) becomes

z(1 — z)” + [>z — (œ+ 8+ 1)zÌy' — aby = 0 (e)

tquations (d) and (e) coincide if we put

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

[his is the so-called hypergeometrical series It is convergent for all

values of x less than unity and can be used to represent one of the inte- srrals of Eq (d) Substituting for a, 8, and y their values (f) and using

a5 + Bit = 5+ 41 [PEE 2) _— ụa (i)

ve obtain as the solution of Eq (d)

_ 32 — 62 | (32 — 8%)(72 — 8%) ] | 2cm Ao[1+7g-T.3# + ig 1-2-2377 )

vhich contains one arbitrary constant Ao.

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| < 1 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

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

L(T: + tÏ¿) = —tu?(Tì + 21:)

Substituting expression (7) in the first of the equations (317) and apply- ing expressions (0), we then obtain

EhaV = aL(U) + »U = (Av — Bap?)I; + (Aap? + Bv)l; (p)

It is seen that the second unknown YV is also represented by the series

Having the expressions for U and V, we can obtain all the forces, moments, and displacements The forces N, and Nog are found from kgs (c) and (d) of the preceding article The bending moments M, and

M, are obtained from Eqs (314) Observing that in the case of a spheri-

cal shell r; = r2 = a and using notation (a), we obtain

M, = - 2 (GE + vest eV)

M, = - 2 (+9y + ot eV) a dy

In calculating the components » and w of displacement we use the

expressions for the strain in the middle surface:

éo = ay (No — »No) €9 = a7 (No — ¥Ny)

Substituting for N, and N, their expressions in U and V, we obtain

*xpressions for e, and e¢9 which can be used for calculating v and w as was

›xplained 1n Art 108

In practical applications the displacement ð in the planes of the parallel

ircles is usually important It can be obtained by projecting the com-

›onents v and w on that plane This gives (Fig 267)

6=vcosg — wsing

Uhe expression for this displacement in terms of the functions U and V is

eadily obtained if we observe that 6 represents the increase in the radius

‘0 of the parallel circle Thus

asin ¢

Eh

6 = asin ves = (Ne ~ »N,) = — 58 (SE — vot e) (r)

Thus all the quantities that define the bending of a spherical shell by orces and couples uniformly distributed along the edge can be repre- ented in terms of the two series J; and [o

The ease with which practical application of this analysis can be made epends on the rapidity of convergence of the series I; and I> This con-

ergence depends principally upon the magnitude of the quantity

Calculations show! that for p < 10 the convergence of the series is satis-

factory, and all necessary quantities can be found without much difhi-

culty for various edge conditions

As an example we shall take the case

of a spherical shell submitted to the action of uniform normal pressure p

(Fig 268) The membrane stresses in

this case are

obtained by using the general solutions (n) and (p) and determining the constants

A and B in these solutions so as to satisfy the boundary conditions

(Nena = Heosa (My)yna = 0

The stresses obtained in this way for a particular case in which a = 56.3 in., h = 2.36 in., a = 39°, p = 284 psi, and »y = 0.2 are shown in Fig 269

By superposing on the membrane forces (u) the horizontal forces H; and bending

moments M, uniformly distributed along the edge, we can also obtain the case of a shell with built-in edges (Fig 268b) The stresses in this case are obtained by super-

posing on the membrane stresses (t) the stresses produced in the shell by the forces

H, and the moments M, These latter stresses are obtained as before from the general solutions (n) and (p), the constants A and B being so determined as to satisfy

the boundary conditions

(oper =0 (V)gna = 0

The total stresses obtained in this way for the previously cited numerical example are shown in Fig 270

From the calculation of the maximum compressive and maximum tensile stresses

for various proportions of shells submitted to the action of a uniform normal pressure

p, it was found? that the magnitude of these stresses depends principally on the

1 Such calculations were made by L Bolle, Schweiz Bauzta., vol 66, p 105, 1915

magnitude of the quantity

a,

— sin? a

h and can be represented by comparatively simple formulas For the case represented

in Fig 268a these formulas for the numerically greatest stvess are as follows:

For i sin? a < 1.2 o = —1.24p (: — ? COS œ

For 1.9 < - sỉn? or l <3; sin œ < 12 ga = _ 5P 1.6 6 + 2.44 si + 2.4 sin “4h l cosa — 1

For the case represented in Fig 268b the formulas are:

For ———* <3 h o = —p(——*) | 0.75 — 0.038 (2 8P 2) sim h | h

n2

For 3 <———* < 12 e= 1.22

It was assumed in the foregoing discussion that the shell has no hole at the top If

there is such a hole, we must satisfy the boundary conditions on both the lower and

the upper edges of the shell This requires consideration of both the integrals (j) and (k) of Eq (d) (see p 541) and finally results in a solution of Eq (320) which con- tains four constants which must be adjusted in each particular case so as to satisfy the boundary conditions on both edges Calculations of this kind show! that, if the angle a is not small, the forces distributed along the upper edge have only a very small influence on the magnitude of stresses at the lower edge Thus, since these latter stresses are usually the most important, we can obtain the necessary information for the design of a shell with a hole by using for the calculation of the maximum stresses the formulas derived for shells without holes

The method of calculating stresses in spherical shells discussed in this article can also be applied in calculating thermal stresses Assume that the temperatures at the outer and at the inner surfaces of a spherical shell are constant but that there is a

linear variation of temperature in the radial direction If ¢ is the difference in the

temperatures of the outer and inner surfaces, the bending of the shell produced by the temperature difference is entirely arrested by constant bending moments

(op) max = (Ø6) max =

af we have only a portion of a sphere, supported as shown in Fig 268a, the edge is free to rotate, and the total thermal stresses are obtained by superposing on stresses

1 Ibid,

w) the stresses that are produced in the shell by the moments

atD(1 + vr)

Ma=- h

niformly distributed along the edge These latter stresses are obtained by using

he method discussed in this article In the case shown in Fig 268b the thermal tresses are given by formula (w), if the temperature of the middle surface always 2>mains the same Otherwise, on the stresses (w) must be superposed stresses pro- uced by forces H and moments M, which must be determined in each particular ase so as to satisfy the boundary conditions

130 Approximate Methods of Analyzing Stresses in Spherical Shells

n the preceding article it has already been indicated that the application

[ the rigorous solution for the stresses in spherical shells depends on the

pidity of convergence of the series entering into the solution The con- ergence becomes slower, and more and more terms of the series must be ilculated, as the ratio a/h increases, 2.e., as the thickness of the shell scomes smaller and smaller in comparison with its radius.? For such 1ells approximate methods of solution have been developed which give sry good accuracy for large values of a/h

One of the approximate methods for the solution of the problem is the

ethod of asymptotic integration.? Starting with Eq (320) and intro- icing, instead of the shearing force Q,, the quantity

e obtain the equation

z1Y + az + aye! + (64 + ao)z = 0 (b)

ll be interested in stresses near the edge where y = a@ (Fig 268) and Thermal stresses in shells have been discussed by G Eichelberg, Forschungsarb.,

263, 1923 For shells of arbitrary thickness see also E L McDowell and E rnberg, J Appl Mechanics, vol 24, p 376, 1957

Calculations by J E Ekstroém in Ing Vetenskaps Akad., vol 121, Stockholm,

3, show that for a/h = 62.5 it is necessary to consider not less than 18 terms of the

es

See O Blumenthal’s paper in Repts Fifth Intern Congr Math., Cambridge, 1912; also his paper in Z Math Physik, vol 62, p 343, 1914

a is not small, we can neglect the terms with the coefficients ae, ai, and

a2 in Eq (b) In this way we obtain the equation

This equation is similar to Eq (276), which we used in the investigation

of the symmetrical deformation of circular cylindrical shells Using the general solution of Eq (d) together with notation (a), we obtain

[e’*(C1 cos By + C2 sin By)

+ e8°(C; cos By + Cs sin Be)] (e)

From the previous investigation of the bending of cylindrical shells we know that the bending stresses produced by forces uniformly distributed

along the edge decrease rapidly as the distance from the edge increases

A similar condition also exists in the case of thin spherical shells Observ-

ing that the first two terms in solution (e) decrease while the second two

increase as the angle ¢ decreases, we conclude that in the case of a sphere

without a hole at the top it is permissible to take only the first two terms

in solution (e) and assume

Having this expression for Q, and using the relations (b), (c), and (d) of

Art 128 and the relations (p), (q), and (r) of Art 129, all the quantities defining the bending of the shell can be calculated, and the constants (,

and €; can be determined from the conditions at the edge This method

can be applied without any difficulty to particular cases and gives good accuracy for thin shells.!

Instead of working with the differential equation (320) of the fourth

order, we can take, as a basis for an approximate investigation of the

bending of a spherical shell, the two Eqs (317).? In our case these

equations can be written as follows:

de? + Cote Gy — (cot? e + v)Ÿ = —

1 An example of application of the method of asyniptotie integration is given by

S Timoshenko; see Bull Soc Eng Tech., St Petersburg, 1913 In the papers by

Blumenthal, previously mentioned, means are given for the improvement of the

approximate solution by the calculation of a further approximation

2 This method was proposed by J W Geckeler, Forschungsarb., no 276, Berlin,

1926, and also by I Y Staerman, Bull Polytech Inst Kiev, 1924; for a generalizu- tion see Y N Rabotnov, Doklady Akad Nauk S.S.S.R., n.s., vol 47, p 329, 1945

where Q, is the shearing force and V is the rotation of a tangent to a

meridian as defined by Eq (a) of Art 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 the same oscilla- ory character as has the function (f) Since @ is large in the case of thin shells, the derivative of the function (f) is large in comparison with the

‘unction itself, and the second derivative is large in comparison with the irst This indicates that a satisfactory approximation can be obtained

2y neglecting the terms containing the func-

ons Q, and V and their first derivatives

n the left-hand side of Eqs (g) In this |

way Eqs (g) can be replaced by the follow- |

ng simplified system of equations:!

vhere Af = 3(1 — py?) (;) (7) Fig 271

(he general solution of this equation is

Q, = Cie” cos Xp + Cre” sin Ay + Cze” cos Av + Cue” sin ro (k) onsidering the case in which there is no hole at the top (Fig 271a) and

he shell is bent by forces and moments uniformly distributed along the dge, we need consider from the general solution (k) only the first two erms, which decrease as the angle y decreases Thus

Q, = Cie” cos Av + Cre” sin Ay (L)

“he two constants C'; and C2 are to be determined in each particular case rom the conditions at the edge (g = a) In discussing the edge con-

itions it is advantageous to introduce the angle y = a — y (Fig 271)

ubstituting a — wy for ¢ in expression (J) and using the new constants

* This simpliication of the problem is equivalent to the replacement of the portion ' the shell near the edge by a tangent conical shell and application to this conical 1ell of the equation that was developed for a circular cylinder (Art 114); see E leissner, ‘A Stodola Festschrift,” p 406, Zurich, 1929

C and ¥, we can represent solution (/) in the form

Now, employing Eas (b), (c), and (d) of Art 128, we find

N, = —Q, cot ¢ = — cot (a — w)Ce” sin (Av + -y)

Ne = — a = —)\ V2 Ce-” sin (x +y— 1) (323)

Irom the first of the equations (h) we obtain the expression for the angle _ Of rotation

2

VY = Th de 7 i The bending moments can be determined from Eqs (gq) of the preceding article Neglecting the terms containing V in these equations, we find

ồ Eh de ap, Sin (œ — V)À V2 Ce” sin (xv + ¥ ï) (326)

With the aid of formulas (323) to (326) various particular cases can readily be treated

Take as an example the case shown in Fig 271b The boundary con-

ditions are

(M 5) pma = M, (Np) p=a = 0 (n)

By substituting y = 0 in the first of the equations (823), it can be con- cluded that the second of the boundary conditions (n) is satisfied by taking the constant y equal to zero Substituting y = 0 and y = 0 in

the first of the equations (325), we find that to satisfy the first of the

conditions (n) we must have