Creep buckling of cylindrical shells under axial compression

Brief Communications Creep Buckling of Cylindrical Shells Under Axial Compression by

FRANCIS W. FRENCH, SHABAD A. PATEL AND B. VENKATRAMAN

Polytechnic Institute of Brooklyn, Brooklyn. New York

Introduction This paper analyses the creep buckling of a circular cylindrical shell under axial compression. It is assumed that the wall of the cylinder is of ideal sandwich construction. The analysis is based on a law which takes into account the instantaneous elastic and steady creep deformations. A numerical example illustrates the use of the analysis. The creep collapse of a long circular cylindrical shell under uniform lateral pressure has been previously analyzed by Hoff, Jahsman and Nachbar (1). A shell with ideal sandwich walls and having a slight initial deviation from circularity was considered in this analysis and it was assumed that the shell exhibited only steady creep deformations. Earlier, SundstrSm (2) advocated the creep buckling of a finite cylindrical shell under uniform radial pressure and axial compression. That analysis included the effects of elastic deformation and steady creep, and considered two types of shell wall construction, i.e., ideally sandwich and solid. However, the approximations involved in SundstrSm's analysis restrict the application of his results to cases where the lateral pressure is greater than the axial compression. In particular, SundstrSm's results cannot be used in the case of the creep buckling of a cylindrical shell subject to axial compression alone. This last problem is the subject of the present investigation. The basis for the analysis presented is that only instantaneous elastic and steady creep deformations are of importance and that these can be analyzed by the usual small deflection theory of shells• It is assumed that the shell walls are of ideal sandwich construction, with two concentric cylindrical thin sheets separated by an undeformable annular core. Thus, the sheets support only membrane stresses (bending and normal) considered constant across each sheet thickness, and the core supports the shear force• Further, it is assumed that the ends of the cylinder are simply supported and hence, the initial and later shapes of a generator can be represented by a half sine wave of increasing amplitude.

Governing Equations Based on the creep law proposed by Odqvist (3), the strain rate expression used in the present analysis is • ~i

-

E

~i

.

.

.

320

. ~kk ~j Jr CJ2 s..

.

(1)

Brief Communications

Here, ¢~. is the stress tensor, ¢~k is the first inv:~riant of ¢~i, 6~i is the KrSnecker delta, J2 = (1/2)s~js~¢ is the second invariant of the stress deviation tensor s~j = ¢i~ - ( 1 / 3 ) ~ - , v is Poisson's ratio, E is Young's modulus, C and m are creep constants and the dots denote differentiation with respect to time. For a constant stress, if it is required t h a t Eq. 1 reduce to a uniaxial law of the form, = (~/x)-,

(2)

it follows t h a t (4) m = (,~ -- 1)/2, C = 3("+')n/2~ ".

(3)

The constants ~, and n are usually determined from uniaxial creep tests under constant stress. The problem under consideration is a cylindrical shell of mean radius r and length L under an axial compressive load P per unit length of the shell circumference (Fig. 1 ). Since the analysis is based on the usual thin shell theory, it follows t h a t the stresses ~=, ~ , and ~ are all zero, where x and ~ are axial and circumferential coordinates in the median plane of the shell, and z is the radial coordinate. Further, due to the axial s y m m e t r y of the problem ~ = 0, hence the only stresses present are #~ and #~. The assumed ideal sandwich section of the shell wall is composed of two thin sheets, each of thickness h/2, separated b y an undeformable core of depth b -- h / 2 . As stated in the introduction, the sheets carry the m e m b r a n e stresses ¢~ and ~ and the core supports the shear force. The stress resultants for the problem are illustrated in Fig. 2. F r o m axial and circumferential equilibrium it follows that ~r= = P / h -- 2 M J b h , ¢~e = N~,/h - 2 M , , / b h ,

~x, = P / h -Jr- 2 M ~ / b h , ~ , = N ~ / h -Jr- 2 M ~ / b h ,

(4)

e

r'i

p

IV~ ~'~Mx

X

Mr 'I

L AX

I

h

[

.

_a ~---W'+AW'---t "

FIG. 1. Shell geometry.

Vo,. 2s, No. 5, N o v ~

,967

Fro. 2. Element of deformed shell.

32 1

Brief Communications

where subscripts e and i refer to the external and internal sheets, respectively. Thus, Eq. 1 together with Eq. 4 yield --(1/Ebh)[22VI~ + v(bfi[¢ -- 221)/¢)] + (C/3bh ) (G,/3b2h2) '~ X [ 2 ( b P - 2M~) - (bN¢ - 2 M r ) I , (1/Ebh ) [23"1~ - ~ (b~'~ + 23/I~) ] + (C/3bh) (G~/3b2h 2)'~ X [ 2 ( b P + 2M~) - (bN¢ + 2 M r ) I , (1/Ebh )[ (b~:¢ - 22l~/~) + 2~,2f/,-1 + (C/3bh )(G~/3b~h~-) " X [ 2 ( b N r - 2M¢) - (bP - 2M~)], (1/Ebh)[(b~:¢ + 2/1;/~) -- 2~3~r~ + (C/3bh)(G~/3b2h2) " × [ 2 ( b N r + 2M¢) - (bP + 2M~)]. (5)

~xi

In these equations G~ =

(bP-

2M~) ~ -

(bP--

2M~)(bN r-

2M¢) +

(bN~-

2Mr) 2,

G~ = (bP + 2M~) 2 -- (bP + 2M~)(bN~,+ 2M~,) + (bN~ + 2Mr) 2,

(6)

and from radial equilibrium N~ may be expressed as

(7)

N ~ = - r (O~M~/Ox ~ + PO~w/Ox ~),

where w is the radial displacement. On the ~ssumption that small deflection theory is valid, the axial mid circumferential curvature rates are bk~ = b(O2w/Ox2) = ~, -- ~ ,

bkr = bO2/r 2) = ~r~ -

~.

(8)

Hence, substitution of Eq. 5 into Eq. 8 yields b(O2tO/Ox~) = - ( 4 / E b h ) ( ~ l , - /O..r) + X , bO2/r 2) = - ( 4 / E b h ) ( 2 ~ r - v M , ) + ,~,

(9

where X = (C/3bh){(Go/3b'2h2)"[2(bP - 2M~) - (bN¢ - 221l¢)] - (G,/3b~h2)m[2(bP + 2M~) -- ( b Y r + 2 M r ) I } , = (C/3bh){ (G~/3b2h2)m[2(bN~ - 2Mr) - (bP - 2M~)] - (G,/3b2h2)m[2(bNr + 2Mr) - (DR + 2M~)]}.

(lo)

Since N r is known in terms of M~ and w from Eq. 7, Eq. 9 represent two equations in the three unknowns w, M~ and M r. Equations 9 can be solved only if one of the unknowns is related to the other two. In the absence of creep deformation, it is possible to relate the moments M~ and M r through the moment-curvature expressions, with the requirement that w be single valued. For the creep problem, however, it is extremely difficult to relate M~ and M r (see, Eqs. 9). Since M r acts to reduce the deflection, the time calculated to reach a given deflection by assuming M r = 0 will be less than the actual one. SundstrSm (2) makes a somewhat similar assumption on the basis of creep relaxation. Thus, assuming Mr = 0, Eqs. 9 can be solved for w and M~.

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Journal of The Franklin Institute

Brief Communications Solution The shell under consideration is a simply supported thin circular cylinder under uniform axial compression. In order to investigate the creep buckling of the shell, the initial elastic deformation of a generator under the given axial compression must be known. Further, this compression must be less than the elastic buckling load of the shell. The differential equation governing the displacement w of a longitudinal strip of such a shell with a solid wall of thickness h, is (5) D (dho/dx 4) + P (d2w/dx '2) + (Eh~/r ~)w = - uP~r,

(11 )

where D = Eh,a/12 (1 -- V-') is the bending rigidity of the shell. The solid wall cylinder will be equivalent to the ideal sandwich shell considered in this paper, if the bending and extensional rigidities of both shells are the same. This requirement shows that h~ = h = %/3b.

(12)

Subject to this relation, Eq. 11 applies to a sandwich shell. Now it is assumed that the initial elastic deformation of the shell before creep begins is given b y w (0) = A (0) sin Orx/L).

(13)

The assumption is justifiable, since the elastic deformations of the shell are similar to those of a bar under compression and on an elastic foundation [see, (5), pp. 108, 441 and (6)]. Substitution of this relation into Eq. 11, a Fourier expansion of the right hand side and the matching of coefficients of sin (Trx/L), yields

A (0) = 97r4/L 4

_

4uP / Trr p~r~/f2 + E h / r 2 •

(14)

The effect of creep is to increase the initial deformations, so that w(t) = A ( t ) sin Orx/L).

(15)

Further, since M~ (0) = Dd2w (O)/dx 2, it follows that M~(O) = - Or"D/L2)A (0) sir, Orz/L).

(16)

Therefore, it is assumed that M . ( t ) = B ( t ) sin @x/L).

(17)

Satisfying Eq. 9 at x = L / 2 with the use of Eqs. 15 and 17, the following

Vol, 284, No. 5, N . . . . ber 1967

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Brief Communications

equations in A (t) and B (t) are obtMned: fi, = L2r2(.X --}- ,~ ) / b ( L 2 - .~r~r2), [3 = E h b ( L Z X -[- 7r2r~ ) / 4 ( L ~ - u~r2r~).

(18)

I n these equations, the functions X and (I) have the values X = (C/3bh){ ( G , / 3 b 2 l : ) ' [ 2 ( b P -- 2B) - a ( A P + B)-] - (G~/3b2h2)'~[-2(bP + 2B) - a ( A P + B ) J l , ,P = (C/3bh ){ (G~/3b2h")"~2c~(AP + B ) -- (bP -- 2B)-] -- (G,/ab~h2)"[-2a(AP + B ) - (bP + 2B)-]}, where G~ = (bP - 2B) 2 -- a ( b P - 2 B ) ( A P + B ) + a 2 ( A P + B ) 2, G~ = (bP + 2B) ~ - a ( b P + 2 B ) ( A P + B ) + a 2 ( A P -{- B ) ~, a = 7r2rb/L '2.

(19)

(20)

The elimination of time between the equations for 2[ a n d / ~ (E(ts. 18) shows that

(21)

d B / d A = Ehb ~(L~X + 7r~r2,b)/4L2: ( . X + cp).

With the initial values of A and B known from Eqs. 14 and 16, Eq. 21 can be integrated numerically to obtain B as a function of A and hence, Eqs. 18 m a y be used to determine A (t) and B (t). E x a m p l e . To illustrate the application of the preceding analysis, a cylinder having a radius of 8 in. and a length of 40 in. is considered. I t is assumed t h a t the cylinder is made b y rolling a 5052 0 aluminum alloy sheet of thickness 0.040 in., maintained at 500°F and subjected to a uniform compression of 100 pds. per in. of the circumference. The elastic properties of the material at the assumed t e m p e r a t u r e are E = 8 X 106 psi a n d , = 0.3, and the creep properties are n = 4.1 and h, = 51,000 rain. 1/" psi. This value of h8 is obtained from tests oil solid wall cylinders (7). However, it can be related to the h of a sandwich wall

0.7

0.6 0.5 0.4 A,IN.o.zO.3 O. I

/

//

4 3 2 I Oh/Pc_

D 4

YIELD POINT---.-. /

.

J

8

e2o24283'

d

u0 10 20 30 4 0 .50 80 70 80 TIME,MIN. F[(~. 3. l)efle('tio~ vs. time.

324

Fw. 4. Stress vs. displacement.

Journal of The Franklin Institute

Brief Communications by enforcing the condition that the curvature rates of the two shells be equal under the same constant loading (8). Thus, ~ = [-3(~+:)/~/(2 + 1 / n ) ] M . Hence, with the use of the above properties of the material and Eqs. 3, m = 1.55 and C = 0.688 X 10-:s (psi)-" per min. The preceding constants are used in the numerical computations of the example considered. The results of deflections vs. time and stresses vs. deflections are plotted in Figs. 3 and 4, respectively. If the reaching of yield stress at any point in the cylinder is considered as its failure criterion, the computations show that failure occurs 80 min. after load application, with the corresponding maxinmm deflection being 32 times the thickness of the cylinder. On the other hand, the reaching of a given ultimate value of the deflection m a y also be considered as a failure criterion. Thus, alternatively assuming this value to be the sheet thickness and 10 times the sheet thickness, the failure times are found to be 58 rain. and 78 min., respectively.

This research was supported by the Air Force Office of Scientific Research of the Air l{esearch and DevelopInent Command under Contract AF 49(638)-302. The paper is part of a dissertation submitted by Francis W. French to the Polytechnic Institute of Brooklyn in partial fulfillment of the requirements for the degree of Doctor of Aeronautical Engineering. This author is presently with the AVCO Corp., Everett, Mass.

References (1) N. J. Hoff, W. E. Jahsman, and W. Nachbar, "A Study of Creep Collapse of a Long Circular Cylindrical Shell Under Uniform External Pressure," J. Aerospace Sci., Vol. 26, No. 10, Oct. 1959. (2) Erik Sundstr6m, "Creep Buckling of Cylindrical Shells," Trans. Royal Inst. Tech., Stockholm, Sweden, 1957. (3) F. K. G. Odqvist, "Recent Advances in Theories of Creep of Engineering Materials," Appl. Mech. Rev., Vol. 7, No. 12, Dec. 1954. (4) N. J. Hoff, "Stress Distribution in the Presence of Steady Creep," Proc. Conf. on High Speed Aeronautics, Polytechnic Inst. Brooklyn, 1955. (5) S. Timoshenko, "Theory of Elastic Stability," New York, McGraw-Hill Book Co., 1936. (6) George Gerard and A. C. Gilbert, "A Critical Strain Approach to Creep Buckling of Plates and Shells," J. Aerospace Sci., Vol. 25, No. 7, July 1958. (7) B. Erickson, F. W. French, S. A. Patel, N. J. Hoff, and J. Kempner, "Creep Bending and Buckling of Thin Circular Cylindrical Shells," NASA TN D-429, June 1960; also, PIBAL Rep. No. 477, July 1958. (8) S. A. Patel and B. Venkatraman, "Creep Behavior of Columns," PIBAL Rep. No. 422, May 1959.

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