THE C A L C U L A T I O N OF A X I A L L Y S Y M M E T R I C A L V E S S E L S * Placido

Cicala **

Whenever possible, metal vessels are given an axially symmetrical structure (shell o f revolution). In this case, calculating in accordance with the membrane theory what we may call a " principal " configuration o f stresses, which balances the loads and relates the reactions to the restraints, is simple, if the set of forces at work complies with axial symmetry ([1], par. 657-667). Assuming an axially symmetrical support, it is also easy to compute the deviations from the membrane regime arising out of the parallel o f iunction of the vessel with the supporting waU when the relevant elastic coefficients are known ([1], par. 699). The values of these coefficients are known for given shapes of the midsurface (cylinder, cone, sphere). It has been found ([2] Part II, p. 20) that these coefficients may be expressed in simple formulae valid for meridians of any shape. The procedure of systematic approximation for this, as for other investigations, has made it possible to reduce the calculation to the simplest form consistent with a set order of error in the results (1). The more general problem of the junction of several walls of revolution along a given parallel will be dealt with in the following. The problem becomes more complex when the reaction of the supporting elements is concentrated along arcs of equidistant meridians or, generally speaking, along the lines of iunction with several supports at intervals. To the stresses that we shall compute as being due to forces evenly distributed over a parallel (Fig. la) we must add the stresses due to a balanced system of loads (Fig. lb) which carries the distributed forces on to the supporting elements S. This latter problem has not yet been studied sufficiently thoroughly for us to be able to deduce generally applicable results. The analysis will have to be pushed further using the shallow shell theory. So here consideration will be confined to the problem of multiple axially symmetrical junctions. We consider the case o f an axially symmetrical load and that of a load antimetric to the meridian plane; the latter case is of interest for its bearing on the determination of the effects o f wind or o f seismic actions on a vessel with a vertical axis or for the effects of weight in the event of a horizontal axis. On parallel of junction g, with radius r, converge several * First published in Italian in "Costruzioni Metalliche", N,~. 2, 1965. ** Professore di Scienza delle Costruzioni, Politecnico di Torino. (1) Previous investigations on this subject are synthetized in the more recent monograph: P. Cicala, Systematic approximation approach to linear shell theory, Levrotto e Bella, Torino, 1965. 58

arcs of meridian, numbered 1 to 4 in the case of Fig. 2. For i-th branch let 9~ be the angle that the tangent to the meridian, facing the junction, forms with the radial direction r issuing from axis Z: thus, for example, 0 < 9 < =/2 for a cap with an angle < 1 8 0 o, whereas =/2 < 9 < = for the remainder of the spherical shell. Let Rt = r/sin 9 , ,

z!

a)

[

Zl

b)

i

Fig. 1.

,I

14

i

Fig. 2. the radius of curvature of the section perpendicular to the middle surface tangent to the parallelgand let R , , be the radius of curvature of the meridian, assumed to be positive if the centre of curvature lies on the side o f the axis Z with reference to g. Let b be the thickness of wall, and sn = b / V / 1 2 ( 1 - I;) where ,, is the Poisson coefficient. Further, let L = w/R-~,/2. It is assumed that the thickness is constant or at least that its variation in an interval L is negligible. H i marks the force acting on branch i in the plane of the parallel g in a centrifugal direction, M, the moment around the axis tangent to the parallel: H , and Mt refer, as usual, to the arc of the parallel. To establish

the signs of moment Mi and of rotation/3 around the tangent at g select any common positive direction: for example, the one that turns direction ~ upward. Lastly, let E be the modulus of elasticity of the material and ~ the dilatation of the fibre lying in the direction of g, common to the various branches. The theory supplies the following equations: 0e

r

OH = fl EbL ' Oe 0/3 r r aM = OH = ±f2--E~s~ ' 0/3 = f a

2L

aM

Ehs~

(1)

where L f1 = 1 - - 2v - - cos q~. r

Z3R L

.f~--fa =fl -q-(+ + z ~ , , / - r

(2) cos

The indices characterising the quantities proper to each branch have been omitted here. Of the two signs given for the second elastic coefficient the upper one applies to branches lying above parallel g and the lower to those lying under it. Assuming f l = f ~ = f a = 1, (1) supplies the expressions given by Belluzzi, who, postulating equivalence on intuitive grounds, extended the known results for the spherical shell to any form o f meridian. The theory of systematic approximation confirms the validity of those results in the order of approximation tSi/z, (5 being a factor of the order of thickness. Equations (2) reduce the error to the order of 6: for the spherical shell they coincide with those given by Het6nyi. In general terms, the effect of the curvature of the meridian affects the factor in parentheses in (2), which is 1 for a sphere, 0.5 for a cone and 0 on the minimum section for a hyperboloid with an asymptotic cone having an angle of 90 ° . The condition o f congruence of the deformations o f the various branches results in the equations

+ OH H + ~ M ) t = e (3)

[3

Off H Off +-if-H--- + ~ M ) t =

where ~, fl are the values of e, /3 calculated for the prind p a l (membrane) solution. By means o f equations (3) one can express the values of H i and Mt for each o f the branches as a function o f the common deformations e and /3. These unknowns are then obtained by substituting those expressions in the equations of equilibrium.

Y~+ ~H, =

0,

VM, = 0

(4)

The summations must be extended to the converging branches. The value of H is deduced from the main solution, as resultant of the membrane stresses N~ in the direction o f the meridians. Fig. 2a shows f / constructed on the assumption that branches 1 and 3 are subject to tensile and 2 and 4 to compression stresses. The vertical compo-

nents o f vectors N , , multiplied by 2nr represent the resulting forces for the various branches. A situation of this kind may occur in a nuclear reactor containment building

[31. Once the values H and M have been calculated for each branch, the corrective stresses to be added to the principal stresses are determined easily. If the resultant H of the principal stresses is not zero, the corrective stresses, compared with these, are of the order 6-½: there is thus, properly speaking, a local concentration of stresses. In the order of approximation permitted by (2), ]~ is to be neglected. On the other hand, if errors of the order 6½ are admitted in the calculation of the corrective stresses, factors f = 1 may be posited and then terms can be ignored. If far = 0, as occurs for the effects of internal pressure in a vessel that presents on parallel g two continuing branches, the stresses due to the effect of the junction are of the same order as the principal stresses. For a calculation with errors:of relative order ,~ account

q~

3

I

4

Fig. 3. must be taken here of fl (zero for a spherical vessel anyway) whilst fl can be ignored when working with an approximation of ~ : terms ~ must be taken into account in both cases.

Corrections for a s h a l l o w shell. Expressions (2) are valid provided that angle 9 is not too acute. When this condition does not hold and hence, whilst R / L has a high value (2), the ratio x = r/~/2L is low, one must resort to the shaUow shell theory. The results to which this leads (Ref. 2, Part II, p. 22) are presented in Figs. 3 and 4. Fig. 3 gives, as a function o f x, the values for the factors f for equations (1) relative to the cap ( 9 - - 0 ) . Fig. 4 gives the same factors for the remainder of the shell ( 9 - - n ) . In the calculation it has been assumed that ~ = 0.3. For small values of the argument x the following equations may be used: A = (1 - - v)/A/r-2x, h = xZ]4( 1 + ")

f s = x[%/2(1 + v)

(5)

(2) If the R/L ratio were not high, the whole theory, including the shallow shell theory, would lose validity. 59

s~ = 0 + , , ) / ~ / ~ ,

Antimetric load.

j3 -- : , s ( o , ~ s g - - ln~)/0 - - ,,),

j3 = ~ / V ' ~ 0 - - ")

(6)

Equations (6) yield an error of under 4 ~ for x ~< 0.2; equations (5) still yield an error of under 4% for x = 0.8.

,,,

\

Consider a shell of revolution subjected to loads whose components perpendicular to the wall and tangent to the meridian are proportional to sin 0 and whose component tangent to the parallel is proportional to cos ~, e being the angle of longitude measured from the meridian plane with reference to which the loads are antimetric (Ref. 4). The calculation of the membrane stresses is reduced to that of a beam whose axis is axis Z of the shell (Ref. 2, Part II, p. 24). The stresses o along the meridians are proportional to sin 0, that is to say, on a section perpendicular to the axis Z the normal components of the meridional o have a distribution proportional to the distance from the axis ~ = 0: they can therefore be calculated by Navier's formula on the basis of the bending moment M f on the section. The tangential stresses are distributed according to a law proportional to cos 0, in accordance with the approximate shear theory for a circular hollow beam. The resultant of these stresses differs, however, from the shearing force on the section for the resultant of the meridional ~ which is M1/r tan ~0.

Fig. 4.

The dash lines in Figs. 3 and 4 correspond to the asymptotic hyperbolas that you get from (2) with cos ~0= 1 for factorsf and cos ~0= - - 1 for factors .7. It is impossible to say for what range of x and 9 factors the values indicated by the curves approximate closer to the true value than those yielded by (2) (3) : but the smallness of the difference between the dash lines and the continuous lines for x > 3 deprives this uncertainty of importance. Note that for ~ 0, even if the r / L values are high, it is necessity to resort to a more complex theory. This possibility is ruled out at the design stage, however, because of the high value reached by the membrane stresses in that case. (s) To decide on this point one would have to push both theories to a further approximation to find which of the two results in the smaller correction.

The theory that leads to (1)-(4) shows that the same relations apply to this case too, it being understood that the quantities H , M, ~ and t / a l l vary proportionally to sin 0. The same (3) will therefore be used to express/--/t and Ms as functions o f , and p. These quantities are calculated by means of (4), in which the known term/-~ is missing unless external forces are applied to parallel g. Lastly, note that this treatment can be extended considerably, by way of approximation. Thus, for a vessel with a horizontal axis resting on saddles that transmit a radial load Q, it may be considered that the wall on the two sides of the saddle assume forces H = Q / 2 and moments M = H L . The maximum value of Q, always with reference to the arc of parallel, is deduced from the resultant force on the basis of a plausible law of distribution, let us say proportional to cos 0, 0 being the angle between the meridian and the vertical planes. Received 14 March 1966.

REFERENCES

[3] R.W. BAILEY and R. HicKs, Stress analysis problems associated

[1 ] O. BELLUZZI,Scienza delle Costruzioni, Vol. uI, Chap. XXVII, XXVIII, XXlX, Zanichelli, Bologna. [2] P. CXCALA,Calcolo delle strufture a parete sotlile, Parts I and II, Levrotto e Bella, Torino, 1964.

with the design of reactor pressure vessels, Nuclear Reactor Containment Buildings and Pressure Vessels, p. 134-149, Butterworths, London, 1960. [4] F. A. LECKIE and R. K. LIVESLmr, The stress analysis of a containment vessel subject to lateral loads, Ibidem, p. 150-163.

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