Dynamical Bending of Rigid Plastic Annular Plates

7/29/2019 Dynamical Bending of Rigid Plastic Annular Plates

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DYNAMICAL BENDING OF RrGI~PLASTICANNULAR PLATESV. N. MAZALOV and Ju. V. NEMIROVSKY

Novosibirsk, Institute of Hydrodynamics, Siberian Branch of Academy of Sciences, U.S.S.R.(Received

MI, MzQ.4MO.MO.WY.. Yo6, arttoK, v. .Ktr K2E.6

NOTATlONradial and circumferential bending moments per unit length. respectively;shearing force per unit length;intensity of uniformly distributed load;limit bending moment per unit length for plates without or with holes, respectively;deflection;surface density of plate;external and internal radii of plate;radial co-ordinate;time;.some specific time;volume of plates with or without holes, of the same external radius;curvature rates in radial and circumferential directions, respectively;complete limit loads for plates with or without holes, respectively.

INTRODUCTIONComparatively few papers are devoted to investigations of the dynamic behavior of annularplates. The first investigation in this direction belongs to Shapiro[4], who considered anannular plate with a fixed interior boundary and an exterior boundary moving withconstant velocity during a short time. The dynamics of an annular plate, simply supportedalong the external boundary and subjected to the action of uniformly distributed transversepressure and shearing force along the internal boundary has been the object of [S] by Mroz.The load-time dependence was assumed to be rectangular in form. An approximatesoIution has been obtained for large values of load when the shearing force is absent at theinterior boundary of a plate.

The residual d~placement of a rigid-plastic annular plate, clamped along the interiorboundary and free afong the exterior and subjected to a transverse linear pulse along anarrow strip by the exterior edge ofthe plate has been investigated[6]. For large deflections,the solution has been constructed taking into account membrane forces.Exact and approximate analysis of dynamical bending of pulse loaded annular plates bytaking into account strain rate sensitivity has been given in [7]. The interior boundary wasfree, while the exterior one was hinge-supports. The plate was subjected to a lineardistribution of initial velocity which is equal to zero at the exterior boundary and takes itsmaximum value at the free interior one. The essential simplification of solution has beenachieved by identifying the flow mechanism with that for the analogous static problem.Mathematical difficulties of determination of validity limits of such a solution have beendiscussed.Finite deflections of dynamically loaded rigid-plastic and rigid-viscoplastic strain

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26 V. N. MZALOV and Jr. V. NEMIROVSKY

hardening annular plates have been determined in the papers [8,9] by Jones. Load andsupport conditions coincided with those from [7]. It was found that membrane forcesconsiderably reduce the residual deflections. It appears that the influence of material strainrate sensitivity is somewhat exaggerated in the paper [7].Recently Aggarwal and Ablow[ lo] have applied the general ideas which were developedfor circular plates by Wang[2], and Wang and Hopkins[3] to the case of annular rigid-plastic plates, free of load at the interior boundary and support conditions coincidingwith those in [2,3]. The condition of equality of radial bending moment to its limit valuealong the hinge circle, at which the derivative of deflection rate with respect to the radiusundergoes a break[ 11,121, has been left unsatisfied.The aim of the present paper is to determine the residual deflections of an angular rigid-plastic plate, rigidly fixed along the exterior boundary and free along the interior one, whenit is dynamically loaded under the action of uniformly distributed blast-type load[l]. Therectangular form of load-time characteristic considered in the paper[5] is only aparticular case of this more general type of load. The procedure of determining theresidual deflection at every point of a plate with the aid of an electronic computer is alsogiven here. This procedure allows further development of the analysis in order to examinearbitrary dependence of load on time.Usefulness of the solution of such a problem depends not only on our wish to give it, incontrast to the paper [lo] which attempts an exact solution of the problem in the spirit ofWang[2], Wang and Hopkins[3] for circular plates without cut-outs, but also on reasoningdeveloped later insection 2. Ifa hole with a radius exceeding some specific value is cut in thecenter of a circular plate, and the material which is removed is used to make the platethicker, then the limit load increases. In other words, the weight of an annular plate may beless for a given limit load. The radius of such a hole is comparatively small, it constitutesno more than 32.1% of the complete radius of a plate. Hence, our wish to obtain asolution for annular plates under dynamical loading is quite reasonable. The solution isuseful for examining the qualitative and quantitative aspects of the residual deflections ofcircular plates with or without a concentric hole when subjected to the same instantaneouslyapplied uniformly distributed transverse loads. Such comparisons are given later in section 4.The so-called boundary parameter for radial bending moment along the outsideboundary is introduced in this paper. It lets us not only solve the problem for hinge-supported and fully clamped plates, but allows us to investigate the dynamical bending ofa plate with a radial bending moment which corresponds to any value lying between anideal hinge and ideal clamping. In the process of solution, the effect of such boundaryconditions on residual deflection is evaluated. The qualitative picture (without anyformulae) of dynamical bending of an annular plate in the setting of Wang[2], and Wangand Hopkins[3] is given at the end of the paper, but in distinction to the work[lO], allnecessary conditions ofdynamical equilibrium along the hinge circle are satisfied accordingto the theory of Hopkins and Prager[ll, 121.Section 1. The setting of the problem and basic hypothesis

A circular ring-shaped plate fixed along the exterior boundary and load-free at theinterior one is loaded at some instant of time t = 0 by an instantaneously applieduniformly distributed transverse load q(t) of high intensity which decreases in time by somelaw, so that for any t the condition (1.1) is satisfied:sq(t)dt 3 tq(t), dqldt G 0, do) 2 qo0 (1.1)where q. is the limit load determined by the usual methods of limit equilibrium theory.The loading of the annular plate and typical plots of q(t) are presented in Fig. la.Loads of type (1.1) are known approximations for blast wave pressures, called loads ofblast type[l]. The simplest of these is a rectangular pulse of duration r. which may beobtained by taking the equality sign in the first two inequalities of (1.1). Just as in[2-6,10-123, thematerial ofthe plate is considered to be ideally-rigid-plastic and controlledby the Tresca yield condition and associated flow law. The plate moves, since q(0) 2 qo. If


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Dynamical bending of rigid-plastic annular plates 27

q(t))

onstr

Figs. la, b.

we take the co-ordinate origin at the centre of the plate, deflection axis W in the directionof the load, abscissa r along the radius, neglect rotatory inertia and membrane forces, thenthe equations of motion in nondimensional form[ 121 are

[XQ(X, T)]-mz(x, 5) = s x [6(x, T)- p(t)]x dxImi = MiMd, . . (i = 1,2), w = ~y~b2Mo~1t~2, p = qb2h&,QX=bQrM&, ~=tti, x=rb-, a=ab-...(O,


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18 V. N. MAZUALOV and Ju. V. NEMIROVSKY

and at x = 5, the plastic regime B controls the behavior (Fig. 1). Sincerni(5--O,?) = nli(


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Fig. 3comparatively small radius of hole. The intensivity of load p. begins to increase when theradius of a hole begins to exceed some specific value. It is easy to show that the completeload acting on the plate in the limit state is

P, = scb2qo( 1- a2) (2.8)which decreases along with growth of a for any value of fi. On the other hand the situationis quite the opposite, if the mass of material of an annular plate and a plate without a holeare made equal, viz. : v, = v (2.9)Under the condition (2.9) let us introduce the non-dimensional quantity A, = PJP,, wherePO is the value of P, at a = 0. Determine P, and PO from relations (2.7) having set in theformer 6, = 0 and 5 = const. Then under the condition (2.9) we have

Ai = (


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30 V. N. MAZALOV and Ju. V. NEMIROVSKY(p = 0) and clamped (fl = 1) along the exterior boundary. For other values of fi from (1.6)the results lie between these two limit cases. The specific values cc0 and a* are indexed1 and 2 for hinged and clamped boundaries, respectively. Values of A, = I, V-icomputed at P, = POare also given and expressed in a form using AP from (2.10) by theformula A = A;1:2. As is seen from Table 1, the load capacity of annular plates is higherthan that if continuous plates of the same weight, if a* c a < 1, where a* belongs to thesegment 0.158 < a* < 0.321 for any fi from (1.6).It should be remarked that the parameter a from (1.2) cannot be close to unity, sinceshearing stresses acting at the section r = const would then become comparable to theradial and circumferential bending stresses. Hence, instead of two-dimensional conditionsit is necessary to consider three-dimensional equilibrium and plasticity conditions at everypoint of the plate.Let us return now to the investigation of the dynamic solution determined by (2.5-2.7).Just as in the static case, for any t, /3, and a respectively from (1.3), (1.6) and (1.2),m,(x, T) in the interval x c x < 5 has a maximum at some point x0, at which m;(xo -0,~) =m;(xo+O, 7) = 0. Let p(O) monotonically increase from the value po. Thus


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,::0 02 0.4 0.6 0.6B

Fig. 5system (2.7) is performed by the help of the Rounge-Kutta method and using the initialconditions at t = 1

9(l) = 509 2w,( 1) = &( 1) = a, (2.13)For an arbitrary blast-like load (in the first two inequalities (1.1) the equality sign is

absent); then numerical integration of the system (2.7) is performed in the time interval0 < r < T/. Initial conditions for the functions W,(T),G,(r) are zero, and t(O) is determinedfrom the algebraic system of equations derived from (2.7)j(0) = lo = const, p = p(O) (2.14)

Secti on 3. The case of hi gh l oads (p(O ) 2 p*)For some x = q in the interval z < x < 5 in some initial segment of time 0 < T ,< r1 thepoint A is realized, so that the following stress profile holds (Fig. 1)

a


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32 V. N. MAZALOV and Ju. V. NEMIROVSKY

The quantity tQt is


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Dynamical bending of rigid-plastic annular plates 33the fourth equation, the

7 < fir where rl is determined from the conditionw(tl-0,7J = w(q+O, 7.1)

The initial conditions at 7 = 1 have the form (2.13) and(3.13)

q(l) = rlo, 2w,(l) = $(l) = a,At the instant T = TV he stress profile (3.1) is replaced by the profile (2.1), and velocityfield (3.4) is replaced by (2.4). In the final stage (pi < 7 < zJ) the motion is the same as thescheme of section 2 for the phase of inertial motion.In the first two relations (l.l), and also in (3.10) when the inequality holds, integrationof (3.9) should be performed on the segment 0 G T < rl, where tl, as earlier, is determinedfrom condition (3.13). The initial conditions for functions w,(7), c=(r), w,,(7), ti,(r) are zero,and ~(0) and t(O) are determined from the algebraic system of equations, derived from (3.9)under the conditions (3.11), (3.12) and (2.14). The final stage motion is described by thesystem of equations (2.7).

T

Fig. I.

Section 4. Numerical exampl es and discussion of resul t sNumerical integration of the systems (3.9) and (2.7) have been performed with the helpof an electronic computer M-220 using the method of Rounge-Kutta. In Figs. 7-8 theresults of integration of the system (3.9) for hinge-supported (j? = 0), and Figs. 9-lo-forclamped (B = 1) plates at a = 0.4 are plotted. The load-time parameters, and specific

values r1 are indicated on each of these figures. The plots in Figs. 9-10 have been derivedby integration of the system (2.7) in the interval TV< 7 < 7/. Increase of velocity G=(r) inthe interval 1 < T Q rl is caused not only by external load, but by internal moments,localized at the hinge circle x = 1. Until vanishing of this hinge these moments do nothelp tiit, o decrease. It is distinctly seen in Fig. 9, where after vanishing of the hinge (7 2 TV)the velocity tiit,decreases to zero at 7 = TV. As is shown in Figs. 8 and 10 at 7 = 7. therealization of maximum velocity &Jr) means that the beginning of deceleration is in thevicinity of x = r,i. n T = rl this deceleration propagates to the entire plate. Figures 11-12illustrate the effect of /I on the greatest residual deflections corresponding to low andhigh rectangular-type loads. Values of residual deflections w,(rJ for other B in the

NLM Vol. 11, No. 1-C


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34 V. N. MAZALOV and J u. V. NEMIROVSKY

uI 2 4 71 6 I2 18 24 30 36 I=,T

Fig. 9

~p(Ob53704...04f~l\ pI dO~era(I-rI...r.I

T, -0525 /~,=80.21 1Tng4.397 I

I0 I , igl , I I II32.J 7 I I 23 39 47 59 71 1

TFig. 10

.66


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0 20 40 60p(O)

Fig. 11

410-

310c-rpNb

p(O)=const OrThl0 . ..T=-0

p(O)Fig. 12

p(O)=const. .O~TSl60 _____--_-_--___-

40

f _---__- 30oe3

20

IO

0 01 02 03 04 05 06 07 08

Fig. 13interval 0


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36 V. N. MAZALOV and Ju. V. NEMIROVSKYduration to. Complete dynamical loads P: determined by formulae (2.8) where q. ischanged to q(0) do not change. Hence, the external energy applied to the plate remainsconstant. Under these conditions the residual deflection of an annular plate can be lessthan a plate without holes, provided the radius of the concentric hole is not too small. InFig. 13 are indicated specific values of r o, zI, r2 for P(0) = 4,3,2. The non-dimensionaldeflection w and complete non-dimensional load Pare expressed in terms of the parametersof a plate without a hole

w = Wy,bMo loI -2, P = P;/6nMoFrom Figs. 7-10 it follows that boundaries of regimes q and < move in the samedirection from the exterior boundary of a plate to the interior boundary for the caseswhen the load is absent (phase of inertial motion, see Figs. 7 and 9), and when it is

monotonically decreasing (Figs. 8, 10). Knowing this, formulae can be derived with thehelp of which the deflection function w(x, 7) can be computed at every point of a plateusing an electronic computer. Let us show this for a more general case of section 3. Firstof all, when integrating (3.4) we obtain

iIw(x, 7) = \+(x, 7)ds + Ii/(x) . . . ci < x < 1 (4.1)6

where 6 = 1 for a rectangular pulse, and 5 = 0 for any other load of type (l.l), and IL(x) isthe function that should be determined. Introduce the notation

i*s-u) . *Cl


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where I++,(X)s determined by (4.3) at 6 = 1 and $,(x) = 0 at 6 = 0. Analogously, from (4.7),(4.8) by the help of the second condition from (4.5) we get:

$:(x) = I&(X)+ J @to) [ 1 + (q - x)/w: + (j/o:) In x] d5.. . 5,


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38 V. N. and Jti.

p(O) >, p*, as it has been shown above in this paper. If to has meaning of duration of(l.l)-type dynamical load, then the initial velocity field (5.lf can be derived from the field(3.4) with the help of system (3.9) under limit transitions (l(O) + SC, to + 0, such that

sf0q(t) dt = constant (5.4)0

since under such limit transition


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Dynamical bending of rigid-plastic annular plates 39At the end of the article some problems are indicated, which are derived from the

above-mentioned by limiting transitions of various parameters.In this paper the dynamic behavior of a plate has been investigated when the externalloads are known. In reality they may be unknown. In this case one may consider variousinitial velocity fields as for example, in [2,3, lo]. But further distribution of loads in timein fact remains unknown, and choice of this or some other mechanism of dissipation ofsupplied external energy based on intuition can lead on to an error (as it happened, forexample, in the paper [lo]). Hence, the analysis of the corresponding problems whendynamical loads are taken into account, has evidently a methodical value, even if theexternal loads acting to the structure are not fully known.

REFERENCES1. P. S. Symonds, Large plastic deformations of beams under blast type loading, Proceedings of t he 2nd U .S.

Nat ional Congress of Appl ied Mechanics, pp. 505-515 (1954).2. A. J. Wang, The permanent deflection of a plastic plate under blast loading, J. Appl . M ech. 22.375-376 (1955).3. A. J. Wang and H. G. Hopkins, On the plastic deformation of built-in circular plates under impulsive

loading, J. Mech. Phy s. Solids 3, pp. 22-37 (1954).4. G. S. Shapiro, Shock on annular rigid plastic plates, Pri kl . Mat . i M ekh. 23, 172 (1959) (in Russian).5. Z. Mroz, Plastic deformations of annular plates under dynamic loads, Ar chs Mech. St osowanej 10,499 (1958).6. A. L. Florence, Annular plate under a transverse line impulse, Am. Inst.Aeronaut. Astronaut. J. 3, 1726 (1965).7. N. Perrone, Impulsively loaded strain-rate-sensitive plates, J. Appl. Mech. 34 (1967).8. N. Jones, Finite deflections of a simply supported rigid-plastic annular plate loaded dynamically, Inc. J.

Sol ids Str uct. 4, 593 (1968).9. N. Jones, Finite deflections of a rigid-visco-plastic strain-hardening annular plate loaded impulsively, J. Appl.Mech. 35,349 (1968).10. H. R. Aggarwal and C. M. Ablow, Plastic bending ofan annular plate by uniform impulse, Int. J. Non-LinearMech. 6, 69 (1971).11. H. G. Hopkins and W. Prager, The load carrying capacities of circular plates, J. M ech. Phys. Sol i ds 2, l(l953).12. H. G. Hopkins and W. Prager, On the dynamics of plastic circular plates, J. App l. M ath. Phy s. (ZAMP) 5,317 (1954).13. A. A: Gvozdev, Anal ysis of Structures by the M ethod ofLi mit Equikbrium . Stroiizdat (1949) (in Russian).14. A. S. Grigorev, The limit state of annular plates, Inzh. Sbornik 16, 177 (1953).15. A. L. Florence, Clamped circular rigid-plastic plates under blast loading, J. Appl . M ech. Series E, No. 2 1966).

Dans cet art i cl e on 6tudi e l a f l exi on dynamque depl aques annul ai res ci rcul ai res, ri gi despl asti ques, f i x&esa' l ' extgri eur et l i bres 2 l ' i nt&i eur, l orsqu' el l es sontsoumses i nstantan&nent 2 un effet de souf f l e t ransversedi stri bue' uni f orm $ment (1) .On montre que l es pl aques annul ai res sont p&f &abl esaux pl aques sans trous pui sque l eur capaci t i de chargeaugmente tandi s que l es d&f l exi ons r&i duel l es di mnuent.On i ntrodui t un param tre de l i m te pour esti mer l ' ef fetdes condi t i ons aux l i m tes sur l e moment du f l exi on radi al .On d6vel oppe une procgdure uti l i sabl e sur ordi nateurpour dgterm ner l es d6f l exi ons r&i duel l es en chaque poi ntde l a pl aque. On donne des exempl es num&i ques. Enfi n ondi scute l es parti cul ari t& de l a sol ut i on de notre probl smepour des pl aques annul ai res correspondant h l ' agencement deA. J . Wang (Z) , A. J . Wang et H. G. Hopki ns (3) pour despl aques snas trous.

Zusarnnenfassung:Di e vorl i egende Arbei t behandel t di e dynamschenBi egung der krei s- ri ngf srm gen Starr- pl asti schen Pl attenunter dem Ei nf l uss der queren unerwart et bei l i egenden


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40 V. N. MAZALOV and Ju. V. NEMIROVSKY

Bel astung sprenges Typus Ll J . Di e Bel astung i st gl ei ch-fb' rmg i i ber di e ganze Pl atte vertei l t . Der Aussenrandder Pl atte i st unbewegl i ch gel agert, wahrend der i nnereRand f rei i st.

Es i st ei n Vorzug der r i ngf orm gen Pl atten gegenuberden geschl ossenen Pl atten sowohl vom Gesi chtspunkt derErh' dhung der Tragl ast al s such vom Gesi chtspunkt derVerkl ei nerung der restl i chen Durchbi egungen behandel t.Der sogenannte grenze Parameter w rd f i i r ei ne Bewertungder grenzen Bedi ngungen auf das radi al e Bi egemoment amAussenrand ei gef i i hrt. Es i st di e Prozedur der Berechungenrest i l i cher Durchbi egung i n bel i ebi ger Punkte der Pl attebehandel t .

Es gi bt di e Azhl bei spi el en. Di e Aufgabe vonA. J . Wang [Z] , A. J . Wang und H. G. Hopki ns [3] f i i r di eri ngf orm gen Pl atten w rd i m Ende des Arti kel s behandel t.

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Dynamical Bending of Rigid Plastic Annular Plates