Dynamic contact instability of spherical caps

Int. J. Impact Engno Vol. 13, No. 3, pp. 479-484, 1993

0734-'743X/93 $6.00 + 0.00 © 1993 Pergamon Press Ltd

Printed in Great Britain

DYNAMIC

CONTACT INSTABILITY SPHERICAL CAPS

OF

LONG-YUAN LI and T. C. K. MOLYNEAUX Dept. of Civil Engineering, University of Liverpool, Liverpool L69 3BX, U.K. (Received 23 N o v e m b e r 1992; and in revised f o r m 8 December 1992)

Summary--This paper presents a numerical study of the dynamic contact instability of shallow spherical shells. The problem considered is that of a spherical shell loaded through a massless rigid circular plate by a sudden step load acting in the direction normal to the loading plate and along the axis of symmetry of the shell. Thus, the spherical shell is subjected to a dynamic contact loading. The interaction characteristic of the circular plate and the spherical shell is described and the snap-through instability of the spherical shell due to the dynamic contact loading is discussed in detail. The numerical analysis demonstrates that the system may be dynamically stable even if local snap-through buckling occurs in the central area of the spherical shell. Dynamic instability occurs only in the case where the spherical shell exhibits an overall snap-through buckling.

NOTATION E P R a

H h P t w0 we wa

,u

Young's modulus ( =208 GPa for all numerical studies) Poisson's ratio ( = 0.3 for all numerical studies) mass density ( = 7.85 g c m -3 for all numerical studies) radius of the spherical shell ( = 10 cm for all numerical studies) base radius of the spherical shell ( = 7.07 cm for all numerical studies) rise of the spherical shell central point ( = 2.93 cm for all numerical studies) thickness of the spherical shell magnitude of sudden step load time = We~H, dimensionless normal displacement at spherical shell central point = W d H , dimensionless normal displacement of loading plate = W , / H , dimensionless average normal displacement of spherical shell a4 = R2h2, spherical shell geometry parameter PR - Eh S, dimensionless load

= R ~ / ~ ' dimensionless time

1. I N T R O D U C T I O N

Structures subjected to dynamic loading may exhibit large changes in response for small changes in loading. This phenomenon is usually called structural dynamic instability. It is possible to classify the problems of dynamic stability according to the types of loading and the characteristics of the instability [ 1 ]. Under such classification one group of problems is that where instability occurs under sudden step loadings--a typical example being the instability of shallow spherical shells. Both the axisymmetric and non-axisymmetric cases for shallow spherical .shells under various sudden step loadings have been studied by Budiansky and Roth [2]; Stricklin and Martinez [3]; Akkas and Bauld [4,5]; Ball and Burt [6]; Simitses [7,8]; Svalbonas and Kalnins [9]; Alwar and Reddy [ 10]. However, dynamic contact instability for such shallow spherical shells has received much less attention. This paper presents a numerical study of the axisymmetric snap-through instability problem of shallow spherical shells under dynamic contact loading. More specifically, the problem considered is that of a spherical shell loaded through a massless rigid planar plate by a sudden step load acting in the direction normal to the loading plate and along the 479

L.-Y. LI and T. C. K. MOLYNEAUX

480

axis of symmetry of the shell. The analysis of the behaviour is particularly complicated by the moving contact surface between the loading plate and the spherical shell which represents an essential characteristic of the dynamic interaction. The problem considered in this paper represents an idealized case of dynamic contact instability of shell structures. However, the behaviour of this idealized situation would be representative of many actual impact scenarios. Information learned from the present case may lead to a better understanding of practical dynamic contact instability problems. 2. ANALYSIS MODEL AND STABILITY CRITERIA The basic geometry of the model and notation adopted are shown in Fig. 1. The simply supported shallow spherical shell is assumed to be elastic. The circular loading plate, in contact with the spherical shell, shares an axis of symmetry with the shell and is assumed to be rigid and massless. The sudden step load is applied at the centre of the circular plate along the axis of symmetry as shown in Fig. 1. Analysis is performed by means of an explicit nonlinear finite element program--DYNA3D I11]. The contact surface between the circular plate and the spherical shell is treated as a separable and frictionless sliding interface. Because the circular plate is assumed to be massless, the applied load acting on the circular plate is transmitted fully to the spherical shell via the contact surface. The geometry of the contact between the loading plate and the spherical shell changes as the structure deforms. One of the objectives of studying such dynamic stability problems is to develop the methods to estimate the critical conditions for instability. For cases such as this where structures are subjected to a sudden step load the critical conditions are only related to the magnitude of the load, the so called critical dynamic load. Several definitions of critical conditions and the methods to determine corresponding critical dynamic loads have been suggested previously [1]. One such definition described by Budiansky and Roth [2] is adopted here. They suggest that the sudden step load is defined to be critical if the transient response increases suddenly with very little increase in the magnitude of the load. Mathematically, this definition is expressed as dP dWma,

-0

where Wr~a,represents the maximum response displacement. Budiansky and Roth's method is an easily understood concept that is favoured in many practical applications as it is straightforward to implement by observing the responses of the system to a range of loads.

I

I

~%,

i I %

I

I

0

FIG. 1. Geometryand notation of analysis model.

Dynamic contact instability of spherical caps 3. N U M E R I C A L

481

RESULTS

The basic equations relating to nonlinear finite element analysis of shell structures are well understood [12] and will not be presented herein. For convenience all results are presented in their dimensionless form in which the basic units for load, displacement and time are divided by corresponding factors, Eh3/R, H and Rx/p/E. Thus, the velocity, for example, presented in the following discussion corresponds to the true velocity divided by H and multiplied by Rx/p/E. Analyses were conducted for a range of sudden step loads. Figure 2 shows the transient responses in terms of average normal displacements of the spherical shell for three such values of sudden step loads. The results show that, when 2 < 8.81, the increases of the maximum response displacement are very small for small increases of the step load. However, when the load is increased from just below 2 = 8.81 to 2 = 8.81, the increase of the maximum response displacement becomes quite large. The load which corresponds to a sudden large increase in the response as shown in Fig. 3 is said to be the axisymmetric dynamic snap-through buckling load. Thus, we may conclude that the value, 2 = 8.81, is a critical dynamic load. To further verify this conclusion, the phase plane curves of the normal displacement of the shell central point are presented in Figs 4 and 5. Figure 4 shows the behaviour for 2 = 8.73 which is below the critical dynamic load and Fig. 5 is for 2 = 8.84 which is above the critical dynamic load. When the load is subcritical, it is seen from Fig. 4 that the spherical shell keeps vibrating about the steady-state position which corresponds to a static equilibrium position and the phase plane curve is thus bounded. However, when the load is supercritical, the phase plane curve shown in Fig. 5 escapes from the periodic

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Average normal displacement responses of shell (/t = 100).

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FIG. 3.

Load vs the maximum amplitude of average normal displacement response of shell (# = 1 0 0 ) .

L.-Y. LI and T. C. K. MOLYNEAUX

482

I

1.0

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0.8 0.6 0.4

e,

0.2

.~

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.~ -0.2 "o -0,4 -0.6

FIG. 4.

-1.2 -0,4 ' -0:8 ' dimensionless displacement

Phase plane curve for subcritical case (/,t = 100, 2 = 8.73).

1,0 0,8 0.6 >..

o

o.4

0.2

.§

o

~, -0.2

-0.4 -0.6 -2.5

FIG. 5.

i

-2.'0 -I .S -I .0 dimensionless displacement

-0.'5

0

Phase plane curve for supercritical case (/~ = 100, 2 = 8.84).

behaviour and becomes unbounded. This corresponds to the spherical shell suffering a large snap-through buckling displacement. After the snap-through, the spherical shell keeps vibrating about the new steady-state position which characterizes a postbuckling sfatic equilibrium position of the spherical shell. Note that the displacements are relative to the initial rise of the spherical shell central point and are presented in dimensionless form. Hence, Fig. 5 shows that the postbuckling equilibrium position corresponds to the case of the spherical shell failing by overall snap-through buckling. Figure 4 also exhibits several cusps as the spherical shell undergoes a local snap-through buckling. However, the local buckling of the spherical shell does not affect the stability of the whole system--the system is still dynamically stable after local buckling occurs in the central area of the shell. To clarify the interaction characteristics between the circular plate and the spherical shell, Fig. 6 presents the normal displacement responses of the loading plate and the spherical shell central point. It is observed that, for the case when the sudden step load is less than the critical dynamic load, the spherical shell central point and the rigid circular plate exhibit a contact-separate-contact periodic motion. In each period, the spherical shell snaps through, separating the central point from the loading plate and then snaps back to make contact. During this motion, the geometry of the contact between the loading plate and the shell changes from a point contact at the centre to a circle as the spherical shell locally buckles. However, when the load is larger than the critical dynamic load (see Fig. 7), the spherical shell central point and the circular plate do not contact again following the initial separation. Instead, each of them vibrates about their individual equilibrium position following an overall snap-through buckling deformation. Following this overall snap-through buckling

Dynamic contact instability of spherical caps I

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Normal displacement responses of circular plate and spherical shell (# = 100, 2 = 8.73).

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FIG. 7.

Normal displacement responses of circular plate and spherical shell (/a = 100, ,l = 8.84).

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1(~0 260 3(~0 4~0 spherical shell geometry parameter, p

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FiG. 8.. Critical dynamic load vs spherical shell geometry parameter.

the contact geometry between the circular plate and the spherical shell takes the form of an oscillating circle in the region of the shell boundary. A further investigation of the dynamic contact instability for different thicknesses of spherical shell in the range from h = 0.22 to h = 1.12 cm is shown in Fig. 8. As to be expected, the critical dynamic load increases with increasing shell thickness.

484

L.-Y. Ll and T. C. K. MOLYNEAUX 4. C O N C L U S I O N S

A s t u d y of the characteristics of the d y n a m i c c o n t a c t instability of shallow spherical shells is presented. The p r o b l e m is investigated by c o n s i d e r i n g a s u d d e n step load a p p l i e d radially to a spherical shell via a massless rigid l o a d i n g plate n o r m a l to the applied load. T h e e m p h a s i s of the present s t u d y is focused o n the i n t e r a c t i o n characteristic d u r i n g the b u c k l i n g process of the spherical shell. A d i s c u s s i o n of the n a t u r e of d y n a m i c stability is established t h r o u g h the use of B u d i a n s k y a n d R o t h ' s stability definition. T h e results o b t a i n e d from the present research show that the spherical shell is d y n a m i c a l l y stable even if local s n a p - t h r o u g h b u c k l i n g occurs. D y n a m i c instability occurs o n l y in the case where the spherical shell exhibits overall s n a p - t h r o u g h buckling. REFERENCES 1. G. J. SIMITSES,Instability of dynamically-loaded structures. Appl. Mech. Retd. 40(10), 1403-1408 (1987). 2. B. BUDIANSKYand R. S. ROTH, Axisymmetric dynamic buckling of clamped shallow spherical shells. Collected papers on instability of shell structures. NASA TN D-1510, 1962. 3. J. A. STRICKLINand J. E. MARTINEZ,Dynamic buckling ofclamped spherical shells under step pressure Ioadings. AIAA J. 7(6), 1212-1213 (1969). 4. N. AKKASand N. R. BAULD,JR, Axisymmetric dynamic buckling of clamped shallow spherical and conical shells under step loads. AIAA J. 8(12), 2276-2277 (1970). 5. N. AKKAS,Bifurcation and snap-through phenomena in asymmetric dynamic analysis of shallow spherical shells. Compat. Struct. 6(3), 241-251 (1976). 6. R.E. BALLand J. A. BURT,Dynamic buckling of shallow spherical shells. J. Appl. Mech. 40(2), 411-416 ( 1973). 7. G. J. StMITSES,On the dynamic buckling of shallow spherical caps. J. Appl. Mech. 41( 1), 299-300 (1974). 8. G. J. SIMITSESand C. M. BLACKMON,Snap-through buckling of eccentrically stiffened shallow spherical caps. Int. d. Solids Struct i i (9), 1035-1049 (1975). 9. V. SVALBONASand A. KALNINS,Dynamic buckling of shells: evaluation of various methods. Nucl. Engng Design 44, 331-356 (1977). 10. R. S. ALWARand B. S. REDDY,Dynamic buckling of isotropic and orthotropic shallow spherical caps with circular hole. Int. J. Mech. Sci. 21(I1 ), 681-688 (1979). I1. J. O. HALLQUIST,OASYS DYNA3D User's Manual Version 5.0. Oasys Limited, 13 Fitzroy St., London W1P 6BQ, November 1990. 12. T. J. R. HUGHESand W. K. LIu, Nonlinear finite element analysis of shells. Comp. Meth. Appl. Mechs. Part I, 26, 331-362 (1981); Part II, 27, 167-181 (1981).