The effect of axial constraint on the instability of thin circular cylindrical shells under uniform axial compression

1111. J. Jlech. Sci. Pergamon l'ress

r.u.

1962. vot, 4, PI'. 253-258. l'rinted in Great Britain

THE EFFECT OF AXIAL CONSTRAINT ON THE INSTABILITY OF THIN CIRCULAR CYLINDRICAL SHELLS UNDER UNIFORM AXIAL COMPRESSION* J.

SJXGER

Department of Aeronautical Engineering Technion, Israel Institute of Technology, Haifa, Israel (Received 3 Nocember 1961)

Summary-The effect of axial elastic restraint on the instability of a circular cylindrical shell under uniform axial compression is analysed by a Rayleigh-Ritz approach within the bounds of linear theory. The effect is calculated for a wide range of parameters, and design curves are presented for the percentage increase in critical load. NOTATION

Am,Bm,Gm displacement coefficients defined by equation (9) D diameter of shell E modulus of elasticity h thickness of shell k spring constant of axial elastic restraint non-dimensional critical axial load coefficient (P jhDE) [6(1- V 2)j7T 3] (LjD)2 (Djh)

u, L

Z

length of shell integer axial load critical axial load in the presence of axial constraints critical axial load in the absence of axial constraints symmetric buckling load in the presence of axial constraints symmetric buckling load in the absence of axial constraints radius of shell number of circumferential waves axial displacement pre buckling axial displacement circumferential displacement radial displacement non-dimensional geometrie parameter 2~(1-v2) (LjD)2

f3

(tjR)

1n

P Pk Po

Pk* p*o

R

t u ii

v w

Y constant defined by equation (3) ,\

(1Il7TjL)

v

Poisson's ratio mem brane stresses prior to buckling circumferential co-ordinate

iix,ci 9' TX9 c/>

* This work was sponsored in part by the Directorate of Aerospace Sciences, AFOSR, through the European Office, Aerospace Research, United States Air Force. 253

254

J.

SI:->OER

THE usual linear analysis of the stability of simply supported thin cylindrical shells under uniform axial compression implies free warping normal to the curved edges (see, for example, Ref. 1). When there is some axial restraint during buckling it is generally assumed that its effect on the buckling stresses can be neglected (see, for example, Ref. 2). This assumption, however, is based only on physical considerations presented clearly by Batdorf3, and on inference from analyses for the cases of external pressure! and of torsion." Though the linear theory has been found inadequate for the investigation of instability under axial compression, it may stiII be useful to estimate the effect of axial elastic restraint on the buckling loads. This effect is therefore analysed by a Rayleigh-Ritz approach within the bounds of linear theory.

~

,

It

I

z,w p

L

ace

x,u

2"

"~

"If,\}-

0

ELASTIC RESTRAINT

OFCUM:D EDGES) k FIG. 1. Cylindrical shell under uniform axial compression with elastic restraint along curved edges.

Two types of axial elastic restraints can be envisaged: (a) restraints which are active from the beginning of loading and (b) restraints which come into action only at the onset of buckling. Re~traints of type (a) first reduce the axial stress applied to the cylinder by the axial load, and then at the onset of buckling their effect is identical to that of restraints of type (b). The stability analysis therefore actually considers the effect of axial elastic restraints which come into action at the onset of buckling for two states of prebuckling stress. For a thin cylindrical shell under uniform axial compression (Fig. 1) with restraints of type (b) the stress state prior to buckling is given by the usual membrane stresses ax = - (PJ21TRh)

ao> = 0 7"0>=0

(1)

For restraints of type (a), which arc active also during loading, the hoop stress and shear stress remain zero as above, but the axial prebuckling stress changes to (2) ax = - (yPJ21TRh) where (3) y = 1- [1 + (2EJkL)]-1

Effcct of axial constraint on instability of thin circular cylindrical shells

255

To derive equation (2), one notes that the uniform axial stress ofthe cylinder before buckling is (4)

where (ii)L/2 is the prcbuckling displacement at the boundary x = (L/2). A positive end displacement would give rise to a compressive spring force in the restraints; but since the axial load causes negative end displacement, the restraints produce a relieving tensile force. Now, (ii)Ll2 =

(Ll2

Jo

(ux/E)dx

or (~t>U2 = -

= - (L/2E!l)[(P/27TR)+k!l(~t)L/2J

(PL/47TREh) [1 + (kL/2E)]-1

(5) (6)

Substitution of (ii)L12 from equation (6) into equation (4) yields the axial prebuckling stress of equation (2). It may be pointed out that here, contrary to the case of external pressure where the hoop stress is the prime cause of buckling;' the difference between the critical axial load for restraints of type (a) and those of type (b) is very pronounced. The analysis is now carried out for restraints of type (a) bearing in mind that the results reduce to those of type (b) if one sets y = 1. The displacement boundary conditions of the problem are 10=0

at x

= ±L/2

(7)

v=O A cylinder under uniform axial compression may buckle in an axisymmetrical pattern or in a more general form. For the more general buckled shape, the displacement functions 1t

= AmsintePsin (m7Tx/L)

v = B mcosteP cos (m7Tx/L) 10 = Cmsin teP cos (m7Tx/L)

where 1n = 1,3,5,

.

(8)

t = 1,2,3, .

satisfy the boundary conditions, equation (7), and are admissible. Substitution of the assumed displacement functions, equation (8), into the appropriate total potential-energy expresssion (equation 7 of Ref. 6), which includes the prebuckling stresses, and minimization with respect to the free parameters yields the usual stability determinant. Hence the critical axial load in the presence of axial constraints is

(9)

where and

f3 =

(l/R)

A = (m7T/L)

(10)

(ll)

J.

256

SINGER

In the absence of axial constraints k = 0, y = I and the critical load becomes Po = [2rrRhE/(I- v2)]{(h2/I2)[(,\2 + ,82}2/,\2J + (I/R2)(I- v2)[,\2/(,V+,82)2J} (12) which reduces for short and moderately long shells, when the critical load can be treated as a continuous function of [(,\2 + ,82)2/,\2J, to the usual Po = 2rrRhE[I/3(I-v2)JI/2 (h/R) (13) (sec, for example, Ref. I). In equations (9) and (12) the integral values of t (the number of lobes) and odd integral values of m (the number of longitudinal half waves) which make P a minimum must be used. For the case without axial constraint, a method for determination of the correct value of t and rn is given in Ref. 7. Though the curves there were calculated from slightly different equations, they can be used to obtain first approximations. In the presence of axial constraint both t and rn depend also on k, but should not differ much from those of the unrestrained shell. For axisymmetrieal buckling, the corresponding displacement functions arc u

= Amsin (1Jlrrx/L)

V

= O.

tv =

Om cos (mrrx/L)

(14)

and a similar analysis yields for the critical load in the presence of axial constraints PZ = [2rrRhE/y(I-v2)J (Z2/ I 2) ,\2 (I/R2,\2) (I - v2) [,\2 + (4k/EL)J} (15) { x t + ,\2+(I-v2)(4k/EL) Equation (15) may also be obtained directly from equation (9) when,8 = O. In the absence of axial constraints I: = 0, y = I and the critical load becomes (16) Pt = [2rrRhE/(I-v 2)]{(h2/I2),\2+ [(I-v2)/R2,\2J} which reduces for shells of moderate length to the usual

Pt

=

2rrRltE[I/3(I- v 2 )]l / 2 (h/R)

(17)

Equations (9) and (15) represent restraints of type (a), which are active from the beginning of the loading. For restraints of typc (b), which come into action only at the onset of buckling, the same formulae apply but with y, defined by equation (3), replaced by y = 1. The critical load for restraints of type (a) is (I/y) times that for restraints of type (b). Hence the numerical work was carried out for restraints of type (b), and if restraints of type (a) are to be considered the results have only to be multiplied by (I/y). The critical load in the presence or absence of axial elastic restraint has been calculated from equations (9), (12), (15) and (16) for an extensive range of relative length (L/D), for three values of elastic restraint of type (b),

k

0·oo5E k = 0·01 E k = 0·05 E =

Effect of axial constraint on instability of thin circ ular cy lindr ical shells

257

and three values of relative thi ckness

lill) = 0·001 hiD = 0·003

hll) = 0·005 All the calculations indicate that effect of axial elastic restraints for an axisymmetrical buckling pattern is mu ch smaller than for the more general pa t tern of equati on is}. Th e effect for the axisymmetrical ease is always less t ha n (1/100) that for t he more general form. Hence the numerical results for t he axi symmetric buckli ng pattern arc not given in det ail. 1·25

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GEOMETRIC PARAMETER

I~IG.

<,

I I

~k=OOIE

~(/rk,"l d±toJJJl 23456810

1\

Z =2Vj:V1

(l/D)~

2. Incr ca se in cri t ica l load du e to axi al clas tic restraint.

Genera l curves are obtained by transformation of the results to a parametric form similar to that suggested by Batdorf". All relative thicknesses arc then included in one curve. Aft er a number of cases have been computed from equati ons (f) and (12) and th e typical ripples are smoot hed out, the ratio of axial load coefficients ]{Xk to K x • are plotted vs. the geometric parameter Z (Fig. 2), where K x = (PlltDE) [6(I-v 2)/7T 3] (LID)2(Dlh) (IS) and Z = 2,/(I-v 2 )(L ID )2 (I V) Note that the non-dimensional parameters of equations (IS) and (If) differ from those of Batdorf by a multiplying factor (hID) . These curves represen t design cur ves for the estimation of the effect of axi al elastic restraints (which come into action at t he onset of bu cklin g) on cylindrical shells under uniform axi al compression. The results of Fig. 2 have to be multiplied by (I/y) for restraints which arc active from the beginning of loading. For increasing spring constants 1:, y varies from I to O. For large values of 1:, y is very small and hence the amplificati on of the critical load is very large for restrai nt s of ty pe (a) which are active

258

J. SnWER

from the beginning of loading. For example, for the largest spring constant considered in the calculations, k: = O·05E, y = 0,286, which results in an amplification of 3·50, whereas the increase of the critical load for restraints of type (b) is at most 25 per cent. AcknOlcledllement-The author wishes to express his appreciation to Messrs 1'£. Klajn and R. Irom for their help with the computational work. REFERENCES 1. S. TDIOSIIEN"KO, Theory of Elastic Stability p. 453. l'IcGraw.HiII, New York (1936).

2. G. GERARD and H. BECKER, Handbook of Structural Stability Part III. Buckling of curved plates and shells. NACA TN 3783 (1957). 3. S. B. BATDORF, A simplified JIethod of Elastic Stability Analysis for Thin Cylindrical Shells. NACA Report 874 (1947). 4. J. SINGER, J. Appl. 11Iech. 27, 737 (1960); Technical Note No.1, Contract No. AF 61 (052) 123. Technion Research and Development Foundation, Haifa, Israel, September (1959). 5. L. 1'1. DOKN"ELL, Stability of Thin-Walled Tubes Under Torsion. NACA Report No. 479 (1933). 6. J. SeWER, The Effect of Axial Constraint on the Instability of Thin Circular Cylindrical Shells Under Uniform Axial Compression. Technical Notc No.4, Contract No. AF (052) 339. Technion Research and Development Foundation, Haifa, Israel, September (1961). 7. W. FLUGGE, Stresses 'in Shells p. 428. Springer-Verlag, Berlin (1960).