UNCLASSIFIED M 43^208 DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION. ALEXANDRIA. VIRGINIA UNCLASSIFIED

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1 UNCLASSIFIED M 43^208 DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION. ALEXANDRIA. VIRGINIA UNCLASSIFIED

2 NOTICE: When government or other drawings, specifications or other data are used for any purpose other than in connection with a definitely related government procurement operation, the U. S. Government thereby incurs nc responsibility, nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or othervise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use or sell any patented invention that may in any way be related thereto.,

3 00 X w ^O Technical Report No. 145 ICO 6*/-/j~ n ASYMMETRICAL CREEP BQCKLING OF CLAMPED SHALLOW SPHERICAL SHELLS Division of ENGINEERING MECHANICS (O N. C. Huang STANFORD UNIVERSITY 0D December 1963 (X Prepared under Office of Naval Research Contract Nonr 225(69) Contract Report No. 2

4 -* * AXISYMMETRICAL CREEP BUCKLING OF CLAMPED SHALLOW SPHERICAL SHELLS* N. C. Huang Stanford University Introduction The problem of axisymmetrical buckling of clamped shallow spherical elastic shells under uniform external pressure (Figure l) has been studied by many authors with different numerical methods. Consistent results for the buckling pressure for shells with a large range of geometrical parameters were obtained in their work. In this paper we shall consider the effect of viscoelastic properties of shells on the axisymmetrical buckling. For viscoelastic shells under constant uniform pressure, the average vertical deflection increases with time as shown In Figure 2. When time reaches a critical value t, the rate of Increase of the average vertical deflection becomes very high, which indicates the initiation of snapping of the shell at this Instant. After snapping, The results in this paper were obtained under Contract Ncnr 225(69) by Stanford University with the Office of Naval Resc?arch, Washington, D. C. Reproduction in whole or In part Is permitted for any purpose of the United States Government. Numbers in the raised square brackets designate References at the end of this paper. 1-

5 the shell continues to deform as shown In Figure 2. Our interest is to find the critical time t for different pressures and for shells with different geometrical parameters. The pressure on shells is considered to be applied either by pressurized air in a chamber of large volume or by an incompressible fluid. For the latter loading condition, the average deflection remains constant and the applied pressure relaxes. In this paper, the relaxation of external pressure of this type is analyzed, as well as prescribed pressure increases. Basic Equations Consider an elastic clamped shallow spherical shell as shown in Figure 1. For axisymmetrical deformation, the governing differential equations and boundary con- ditions are found in [1] as 12(l-v ) a -, 2EH r- r -r- r r 2 ß E r ß 2 (2) h L dr vx dr ; ^ a2 a ß = 0 at r = a (3) ag-v^/=0 atr=a (*) -2-

6 where ß j, w is the vertical deflection of the shell, ip is a stress function, and E and v are the Young's modulus and Poisson's ratio respectively. Let E o and v o be two constants and put r I - I 1 E o I = X 2 I, 1-v 2 x l 1 1 A = 2 [3(l-V*)]* ( H } 2 Ar Aa 2H ß 12(l-v 2 )a d> = 2- f AE o h J 32E ^ o H 3 h ' -w H o Equations (l- J +) can then be written as i 2 (x 2 e" + x 0" - e) = - x 2 d> - 2px 3 + x e $ (5) x *" + x * -» I, (x 2 e - I xe 2 ) (6) 3-

7 e(x) = o (7) AS>' (A) - v <t(a) = 0 (8) where ( ) = * ( ). The material properties are involved in the constants I, and I. In our viscoelastic theory, we use the following assumptions: (l) the strain components in the shell are small so the constitutive equations are linear; (2) for mathematical convenience, the bulk modulus of the shell, K, is assumed to be independent of time; (3) the Kirchhoff's hypotheses hold in the whole process of deformation; (4) the shell is assumed to deform quasi-statically, i.e. the effect due to its inertia force is neglected. In the viscoelastic problem, the Young's modulus E is replaced by an integral operator with a kernal function E(t). The kernal function equivalent to Poisson's ratio in elastic case, v(t), can be found directly from the relation»<t)-!-5 L ( 9) We use the notation E = E(0) v o and v = v(0), v ' o ' then q defined above is the classical buckling pressure of a complete spherical shell with the same thickness and radius of curvature. The constants I, and Ip are replaced by the integral operators, with kernal functions Ii(t) and Ip(t) which are determined by the following equations: -4

8 i x (t) -5^i 10 (l-v)* (1+v) * I 2 - (l-v 2 ) l x (11] where the starred notation represents the integral operation as shown in the following example: t f = f I(t-r) dt J dt Thus, equation (11) is a Volterra integral equation for I. Examining the derivation of equations (1-4) in [1], we are able to derive the governing equations for the deformation of viscoelastic shells under the above assumptions by applying the correspondence principle to equations (5-8): i* (x 2 e"+ xö'-e) = - x 2 $ - 2px 3 +xe$ (12; xc>" + x$' - * = 1 *(x 2 e - I x0 2 ) (13; 9(70 = 0 (14) A$'(A) - v**(x) =0 (15) -5

9 where 9, <t> and p are time dependent and ( ) ' = *HT ( ) Equations (12) and (13) are two nonlinear integrodifferential equations for the two unknown functions e(x,t) and <J>(x,t). These equations will be solved numerically. The average vertical deflection p is found in [1] as A p = \ [ x2 6dx (16; o Let M and M, be the meridional and circumferential moments respectively. Put 6(l-v 2 ) 2 m r = g 3 * H M r (17' 6(l-v v 2 ) 2 m o' a M, (i3; t E Ö ho J H r From the stress strain relations, we have m r = I 2 * 9 - I 3 ( ) (19) m t = I 2 * ( ) - I3 # 6 (20) where the operator I- is defined as I3 = I 2 * v (21) Hence, M r and M^- can be obtained from 0 by numerical differentiation and integrations.

10 Numerical Method The viscoelastic material in our investigation is polymethylmethacrylate. The nondimensional relaxation modulus is found in [5] as x ( t ) = (1 + erf[0.31(log 10 t - 3-6)]) (22) we take v =0.35 which is equivalent to K/E = Let t = t. We shall use the following approximation as shown in [6] for the time integration: n-1 i*f = Ko)f n - \ n i «fj + 1 +?j)[ut n -t. +1 )-i(t n -t.)]} J«= \ [f n G(l,t n ) - P(l,f n )] (23) where P(l,f v ' n' ) - f n-l,fl(0) L K ' - I (t v n -t n-l,)] y J n-2 + Z f(f.,,+f.) [I(t -t. 'i-l(t -t.)]} (24) o(i,t ) = i(o) + Kt n -t n _ 1 ) (25) From the above definition, F(I,f ) is linear with respect r to f, i.e. n

11 F(l,f n +g n ) = F(l,f n ) + F(l,g n ) F(l,kf ) = kf(l,f ) for constant k. Let (l-v)#ig = I, From equation (11) (1+v) * I 2 = (l-vj)l 2 Using equation (23)J we obtain U 2 ) " n 2(1 ) +F[l-v,(l ) ] G(l-v,t v ) 2 " ^ n' <*2> - (1-^) n v 2(I 2 ) +F[l+v,(l ) ] 2 (l + v,t ) < 2?> n' where the subscript n designates the values at t = t, Hence, I? (t) can be evaluated numerically. Let Ax =, where m is the total number of equal space steps. Define x. = (i-l)ax pj = P( V 0 ltj = e(x.,t.) and 1J -»(x 1.t. -8-

12 The first and second partial derivatives of 0 with respect to x can be approximated by the central differ- ence formulae: hi " 2Ex (" e i-l,j + 9 i+l, ii ).. ij -L» ( ) Ax)*-- *-**«3 1 J 1+1, J Equations (12)-(15) can be written as the following matrix difference equations: A. y., + B. y. + C. y,,, = D, in i-l, n in J in in i+l,n in (28; Ey + F y, -, +Gy, 0 =H mmn n J m+l,n n J m+2,n r (29) where in 'in - I* in (30) [(i-i) 2 - i^l]g(i 2,t n ) 0 in 2 i-l i-l) (31) ;2(i-i)'+i]o(i 2f t n ] 2x: B, in 4 x i 6 < i i' t J -[2(i-l)Vl] (32: -9

13 C in = ri(i-d 2 + ^lodj,,^) (i-d 2 +^ (33) D. in 3 in in - 1 (34) On (35) (36) Sl^-'nJ ro (37) (38) m+1 v ' m+l,n ; S ln = [(i-d 2 - i2 i ]F(I 2,0 i. ljn )-[2(i-D^l]F(l 2,e in ; + [(i-d 2 + ]F(I 2.» W(B )-\, 1 + 2x i 9 in*in W 10-

14 T ln - - J x.e^cd^tj - \ 4 P(l y e in )+ ^(i 15 e 2 in) (40) In our numerical analysis, one extra space station at x = A + Ax is added for the convenience of representation. Equations (39) and (40) contain terms nonlinear in 9. in and <t>. in. An interative method is used to solve this non-linear problem, i.e. we assume the values for 9. and <t>. and evaluate 9. and <D. by taking & in in in In * s. in and T. as known functions. Such iterations continue until in a certain convergence 0 criterion is satisfied. If D. in is known, we evaluate y. by the numerical technique given in [7] Let y in = P in *!+!,«+ ^in l* l > where P. is a two-by-two matrix and Q. is a column in 'in vector. From equation (28), we have Therefore, ( A in P i-l,n +B in) ^in + C in ^ + l,n = D in" A in «1-1,11 P. = - L. C. (42) in in in v ' Q. = L, (D. -A. Q., ) (43) ^in in v in in^i-l,n / v " where L. = (A, v in in P. l-l,n. + B. in' ) 1 (44) v ' Since y ln = 0-11-

15 P, - Qn =0 (45) In In v ' Equations (42) and (43) are the recurrence formulae for P. and Q., Therefore, the value of P, and Q ln can be determined for i = 1,2,...m+1. Substituting equation (4l) into equation (29) we can show that W,n = (Vm + l,n +G n)"^hn- E n^n- R n^m + l,n) ^ where R = E P + P (47) n n mn n v ' After y is determined by equations (46), y. (i=l,2,...m+l) 1111 C- s il 111 and p can be evaluated by equations (4l) and (l6) respectively. In the case where p is prescribed as a constant, it is found by the following procedure that the external pressure relaxes. Multiplying both sides of equation (12) by dx and integrating from x = 0 to x = A, we have A A 2 I 2 * 9'(A) = - \ h + [ x$(9-x)dx (48) o Again, if we multiply both sides of equation (12) by x dx and integrate, we obtain I» [A 2 9'(A) + 8p] = - E A 4 + K f x 3 O(0-x) dx (49) 2 3 X2 JQ A 12-

16 Eliminating I * ö'(a) from equations (48) and (49) and solving for p, we have P - ^ [8pl 2 + fx(l+f)(l- ) (e-x)dx] (50) A v hence, the relaxing external pressure at any time can be determined. In our numerical calculation, we computed I,(t), I 2 (t), I~(t) and v(t) by equations (22), (27), (21) and (9) respectively. These values are plotted as functions of t in Figure 3- The functions 6 and were evaluated by iterative procedure from equations (39)J (^0) (4l), (46), etc. If p was prescribed, the calculation continued until the absolute value of the difference between the value of p and the average value of p's found in the five previous iterative processes was less than If p was prescribed, the convergence criterion used here was that the absolute value of the difference between the value of p calculated by equation (50) and the average value of p's found in the five previous interative processes was less than In our computation, we used a log time step, therefore, the variations of 0 and $ at very early times could be taken into consideration. The calculated values of 6 and $ at each time step were stored for the sake of the history dependence of the function F [equation (24)]. For the purpose of comparison, numerical calculations were carried out for A = 6 and p = 0.8 (i.e. 80^ of the classical buckling pressure of a complete sphere) for different time intervals and space intervals. The results are shown in Figure 4. Although there are discrepancies between these curves for the -13-

17 different space intervals used, the critical times obtained from these curves show no noticeable difference. We, therefore, used Ax = and A(log,Qt) =0.5 in the subsequent computations. When t approaches t, p increases very rapidly. The iterative process becomes divergent as t exceeds a certain value t cr. This divergence ö of our iterative pro- r cedure arises from the fact that there does not exist a equilibrium state in the neighbourhood of the equilibrium state just before buckling defined by positive increaments of time. We define t er such that at t = t cr, the log 10 t cr iterative process converges and at t = 10 = t cr, the iterative * process diverges. The maximum relative error of t found from this definition is less cr than 1.2$. Numerical Results and Discussions A program for numerical computations was written in the BALGOL machine language and all calculations were made on the IBM 709O digital computer at Stanford University Computation Center. These facilities were in part made available under the National Science Foundation Grant NSP-GP 9^8. The average vertical deflection under different pressures are plotted against time for A = 6 and \ = 7 in Figures 5 and 6. The critical times under different pressures are shown in Table 1 and Figure J. From Figure 7, we find that the critical time decreases when the pressure increases. The rate of decrease becomes very rapid when the pressure is close to the elastic buckling pressure. As we know, the critical time is zero when the external pressure reaches the elastic buckling pressure based 14-

18 on the modv.li for instantaneous loading. Since the elastic buckling pressure for A = 7 is higher than that for X = 6"-, under the same external pressure, the shell with A = 7 possesses longer critical time than the shell with A = 6. Figures 8, 9 and 10 show the distributions of 0, m r and m, t at different times for A = 6 and p r The 9 curves in Figure 8 are the slope distributions before buckling. After buckling, the values of 9 are presumably positive in the whole range of x. Before buckling, the vertical deflection grows with time but the shape of the deflection curve remains approximately the same. From Figures 9 and 10, we find that the moments near the shell edge relax and that the moments in the region outside the boundary layer grow with time as a result of magnification of the slope curve (Figure 8). The case in which p = was evaluated and the corresponding relaxation curve of external pressure p is shown in Figure 11 for A = 6. Figures 12, 13 and 1^ represent the distribution of 9, m and m, at different time stages. When time increases, the deflections in the central portion of the shell decrease but the deflections elsewhere increase so the average vertical deflection remains a constant. In Figure 13 and lk, moments decrease with increasing time because of relaxation of external pressure. Acknowledgement The author is grateful to Professor E. H. Lee and Dr. E. Appleby for their generous assistance in the reading of proof and for valuable suggestions. 15-

19 REFERENCES [l] Bu^iansky, B.j "Buckling of Clamped Shallow Spherical Shells," Proceedings of the Symposium on the Theory of Thin Elastic Shells, Delft, August 1959, North Holland Publishing Company, Amsterdam, p. 64. [2] Weinitschke, H., "On the Stability Problem for Shallow Spherical Shells," Journal of Mathematics and Physics. Vol. 38, No. 4, p. 209, January i960. [3] Thurston, G. A., "A Numerical Solution of the Nonlinear Equations for Axisymmetric Bending of Shallow Spherical Shells," Journal of Applied Mechanics, Vol. 28, No. 4, p. 557, December I96I. [4] Archer, Ft. R., "On the Numerical Solution of the Nonlinear Equations for Shells of Revolution," Journal of Mathematics and Physics, Vol. 4, No. 3, p. 165, September [5] Bischoff, J., Catsiff, E. and Tobolsky, A. V., "Elastoviscous Properties of Amosphous Polymers in the Transition Region I," Journal of the American Chemical Society, Vol. 74, p. 3378, [6] Lee, E. H. and Rogers, T. G., "Solution of Viscoelastic Stress Analysis Problems Using Measured Creep or Relaxation Functions," Journal of Applied Mechanics, Vol. 30, No. 1, p. 127, March [7] Ralston, A. and Wilf, H., "Mathematical Methods for Digital Computers," Wiley and Sons, p. 121, i960. [8] Gurtin, M. E. and Sternberg, E., "On the Linear Theory of Viscoelasticity," Archive for Rational Mechanics and Analysis Vol. 11, p. 291,

20 log 10 It cr' p A = 6 A = Table 1 Critical Times for A = 6 and A = 7-17-

21 FIG. 1. Geometry of Clamped Shallow Spherical Shells

22 /" ÜJ O < (3 < CC UJ 3 '. er Log(time) FIQ. 2. Average Vertical Deflection of Clamped Shallow Spherical Shells

23 if) d loo to I? il 0) p CO o CO x: 0) i m oo 00 "6 N i I * + H c CO I * " o Ä x: P o 0-, o O e> CO 0) O c co P w C o Ü CM I 0! -p n co i o l t O B CO Ö Ö 2 CM Ö

24 2.0 X-6.0, p« AOog^t)«0.50; Ax»0.250 A(log l0 t)«0.50;ax A(log l0 t).0.25;ax»0.125 =- Ä(log 0 t)-0.50; Ax» (.2 </ t in hours I.I 1.0 J_ Log lo t -2 FIG. 4. Comparison of Average Vertical Deflections for Different Time Intervals and Different Space Intervals

25 X p p» ' p 0.85 I.I p»0.80 t in hours 1.0 J 1 1 L -3 Log*,! PIO. 5. Average Vertical Deflections for X - 6

26 2.0 FIG. 6. Average Vertical Deflections for X «7

27 FIG. 7. Critical Times for > = 6 and X =

28 LA 00 d II 1 CD ll u o c o H -p 3 H U -P 0} H P 00 o H I

29 m r U,t) 2h Loo t =» FIG. 9. m^ Distributions for X 6 and p - O.85

30 m t U,t) Log t = \=6.00 p =0.85 t in hours FIG. 10. m. Distributions for A = 6 and p = O.85

31 X p t in hours ' ' ' I 0 Log t *I FIG. 11. Relaxation of External Pressure When p Is Specified

32 ö(x.t) 2r kx=a PIG Distributions When p Is Specified

33 0.5 h Log, 0 t«-5.0 m r (x,t) FIG. 13- m Distributions When p Is Specified

34 0.5 mtu.o 0 0 X = 6.0 p t in hours PIG. Ik. m Distrlbjtions When p Is Specified