A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION. Cheng Xiang-sheng (~ ~-~) ( Tongji University, Shanghai.)

Transcription

1 Applied Mamatics Mechanics (English Edition, Vol. 13, No. 10, Oct. 1992) Published by SUT, Shanghai, China A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION Cheng Xiang-sheng (~ ~-~) ( Tongji University, Shanghai.) (Received May 3, 1988; Communicated by Chien Wei-zang) This article will discuss bending problems rectangular plates with free boundaries on elastic foundations. Fee talk over two cases, that is, plate acted on its center by a concentrated force plate subjected t..o by a concentra'ted force equally at four corner points respectively. We select a fiexural which The one-dimensional satisfies not only problem all geometric motion boundary conditions a rigid flying on free plate edges under wholly explosive but also attack has an analytic solution boundary only conditions when polytropic total internal index forces. detonation We apt, ly products variational equals method to three. In general, a meanwhile numerical analysis n obtain is required. better approximate In this paper, solutions. however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by Key various words high explosives rectangular with thin polytropic plate, bending indices problem, or than Galerkin's but nearly variational equal to three. Final velocities flying plate method, obtained flexural agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic I. Introduction index) for estimation velocity flying plate is established. The bending problems rectangular plates possessed free edges all around on elastic foundations are very troublesome. 1. Introduction The accurate solutions found by partial differential equation elastic surface plate in fourth order must satisfy not only differential Explosive equation driven flying-plate bending technique surface ffmds its important plate but use also in study boundary behavior conditions materials geometrically under intense impulsive boundary loading, conditions shock synsis internal forces diamonds, free edges explosive wholly, welding which will cladding metals. The method estimation flyor velocity way raising it are questions be involed in great difficulties. The present problems were investigated by Vlasov, et al. in common interest. E1-83. Ref. [13 used ory elastic foundations about elastic semi-space, Under assumptions one-dimensional plane detonation rigid flying plate, normal approach namely, solving so-called ory problem elastic motion foundations flyor is concerning to solve two following parameters, system equations solved governing differential equation flow field detonation bending products surface behind plate. flyor Ref. (Fig.[23 I): applied Galerkin's method put forward a double cosine series as a flexural, which can be separated into different variables with its first --ff to third derivatives =o, every, is equal to zero on all free boundaries plate, ap but +u_~_xp coraputation + au is much more complicated. Later on Ref. E3] used principle superposition, au au resolved 1 y =0, system algebraic equations jointly in thirty-five orders in fifty-one orders respectively, consequently obtained (i.0 solutions. Ref. [4] applied variational as as method in solution. Using ory elastic a--t =o, foundations with regard to elastic semi-space, Ref. I6] found basic solutions in various boundary, conditions, n employing p =p(p, s), orem superposition, taking fifty terms in two groups series respectively, it found more general solutions. Ref. [8] applied where p, p, S, u are pressure, density, specific entropy particle velocity detonation products energy method (or variational method). Some methods mentioned above are simple respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory but most F m flyor are as anor more complicated. boundary. Both In are unknown; present article position we shall research R state bending parameters problems on it are governed rectangular by plates flow field with I free central edges rarefaction elastic wave foundations behind detonation by Galerkin's wave D by initial stage motion flyor also; 977 position F state parameters products

2 978 Cheng Xiang-sheng variational method. We choose a flexural that satisfies not only conditions free from resultant lateral shearing forces on all free edges but also conditions free from bending moments on teh whole free borders, finnaliy in flexural s re are only two independent coefficients, so calculation is simpler more convenient. Then we give two examples, one which is plate acted on its center by a concentrated force or is plate subjected to by a concentrated force equally at four corner points respectively. Finally we shall investigate result balance with regard to general loading acting on plate with whole reactional forces from foundations in order to examine reliance solutions obtained in this article. IL Method Solution Let four sides a rectangular elastic thin plate on elastic foundations be free, as shown in Fig.l. F a/2.-{ * For isotropic rectangular plate, its differential equation elastic surface is [9] The one-dimensional problem motion a rigid flying plate under explosive attack has r an analytic solution V'w.-}-kw/D-q/D=O only when polytropic index (2.1) detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior in which V' reflection is a biharmonic shock in operator, explosive D is products, flexural applying small parameter purterbation rigidity method, plates an analytic, D=EhZ/] first-order 2(1 approximate _pz). E,h, solution # are is obtained for problem flying plate moduli driven by various elasticity high explosives materials with polytropic plate, indices or than but nearly equal to three. Final thickness velocities plate, flying Poisson plate obtained ratio respectively agree very well k is with numerical results by computers. Thus Fig. I an basic analytic coefficient formula with elastic two parameters foundation, high explosive q is (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. intensity distributed load. If plate is orthotropic, n equation (2.1) must be replaced, by following t'~ 1. Introduction D1w D3w,,,,+ Dzw,,,,.+ /~w- q = 0 (2.2) For Explosive meaning driven D,,D2,D3, flying-plate see technique Ref.[ I0]. ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding Since four sides plate are free, it is more difficult to find accurate solutions cladding metals. The method estimation flyor velocity way raising it are questions bending surface plate by means partial differential equation elastic common interest. surface Under assumptions plate in fourth one-dimensional order, because plane detonation solutions rigid must flying satisfy plate, not only normal approach differential equation solving but problem also boundary motion conditions flyor is to solve geometry following system internal equations forces governing on all free borders. flow field detonation products behind flyor (Fig. I): If we can find a flexural bending surface thin plate, which may satisfy boundary conditions on --ff free sides wholly, =o, namely, boundary conditions ap +u_~_xp + au geometry internal forces, but doesn't gratify differential equations bending surface plate au probably, au n 1 y we may =0, apply, Galerkin's method find it from so-called Galerkin's system variational equations. For example, for (i.0 iso.tropic thin plate tg~ as as a--t =o, V w+-~-w (m=1,2,3,*-. where where p, w p, is S, u flexural are pressure, density, specific w,, entropy is form particle velocity which detonation is only products respectively, coordinates. with By trajectory system R reflected equation shock (2.3) detonation we may obtain wave D a as linear a boundary system Of trajectory algebraic F equations flyor as related anor to boundary. every independent Both are unknown; coefficient position in flexural R, state para- from meters which on we it can are find governed each coefficient. by flow field I central rarefaction wave behind detonation wave D For by initial orthotropic stage motion plates, Galerkin's flyor also; system position variational F state equations parameters becomes products t'~

3 A Free Rectangular Plate on Elastic Foundation 979 ~ (D,w,,,,+2Dswn,+D~w,wt+kw-q)w,,,dxdy=O (m---1,2,3,...) (2.4) In preceeding formulae all double integrations are all over region medium surface plate. The solutions found by Galerkin's system variational equations are approximate, because it satisfies differential equation bending surface plate "averagely" in a sense. In following we shall discuss determination flexural s. For plate shown in Fig.l, since whole borders are free, corresponding bending moments resultant internal shearing forces must be equal to zero. For instance, for case isotropic thin plate, boundary conditions are as follows. On sides x--o x=a, we ought to have [-w,,+ #w,,-i,.o,,.o =ffi 0 (2.5) l-w,,,+ (2 - #)w,,,],.,,,.~ = 0 (2.6) On The one-dimensional sides y=0 problem y=b, we ought motion to have a rigid flying plate under explosive attack has an analytic solution only when rw.,+#w,*31.0,,.b---0 polytropic index detonation products equals to three. (2.7) In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock [W,,, + (2 --/~) W,,,],-0,,- b = 0 (2.8) behavior reflection shock in explosive products, applying small parameter purterbation As mentioned method, an above, analytic, we first-order select approximate following flexural solution s is obtained for problem flying plate driven by various high explosives with.~ polytropic indices 2srx or than 2:ry but nearly equal to three. w-----f,cos 2~rx "4-fi cos "4-ft cos cost+ f, (2.9) Final velocities flying plate G obtained agree very well with a numerical results by computers. Thus an analytic In each formula with previous two parameters formulae, a high b explosive are lengths (i.e. detonation sides velocity thin polytropic plate along index) x for estimation y axes respectively, velocity flying f~-f4 plate is are established. undetermined parameters. It may be seen without any difficulty that deflections (2.9) on contour plate are not equal to 1. zero Introduction conditions free from resultant lateral shearing forces (2.6) with (2.8) can all be satisfied, but conditions free from bending Explosive driven flying-plate technique ffmds its important use in study behavior moments (2.5) with (2.7) can't be satisfied. If we desire to satisfy conditions (2.5) materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding (2.7), into which metals. we The may method substitute estimation flyor (2.9), velocity n we can way obtain raising it are questions common interest. A= Under assumptions one-dimensional plane detonation rigid flying plate, normal approach or write m solving as problem motion flyor is to solve following system equations governing flow field detonation products f,----fl,fa, behind fzffltf, flyor (Fig. I): (2.11) Thus if we have taken value coefficients f~ f~ as formulae (2.10), n conditions free from bending moments --ff on free =o, ap +u_~_xp + au sides wholly, can be satisfied. Hence flexural (2.9) now may be rewritten as au au 1 as a--t as =o, y =0, /, (2.12) In above expressions re are only two independent unknown parameters. Moreover, as corresponding bending moments resultant lateral shearing forces on all free edges where for this p, p, S, u are are pressure, equal to density, zero, specific problem entropy may be particle solved velocity from Galerkin's detonation system products respectively, variational equations with trajectory (2.3). R reflected shock detonation wave D as a boundary trajectory For F case flyor as square anor plates, boundary. having Both b=a, are unknown; taking #=0.167, position n R from (2.10) state we para- may meters obtain on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products (i.0

4 980 Cheng Xiang-sheng fl,=f12=fl (2.13) III. Numerical Examples Example 1 Let four sides a rectangular plate on elastic foundations be free acted on by a concentrated force P at center, as shown in Fig.1. Now we find solutions applying tlae system equation (2.3). Substituting (2.12) into (2.3) noticing that coordinates point on which force P acts are (a/2, b/2), we may obtain system which contains two algebraic equations pertaining to coefficients f~ f4 as follows: 9 fl~ fl kab,, 1 {8# ab[--ha-..i-~+ 1/ I I x"~ q-p (B,+B,-1) =0 (3.1) The kab], one-dimensional - P = 0 problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, For a numerical case analysis square plates is required. above In this system paper, equations however, by may utilizing be simplified "weak" as shock behavior reflection [1_~4 shock (flz in ka explosive ~ ~ products, 1 P applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic If set up formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. ka'/d=l O' (3.3) observing formula (2.13), in case 1. Introduction square plate, from (3.2) we may solve /8= X IO-'Pa~/D, f,=l XlO-'Pa=/D (3.4) Explosive driven flying-plate technique ffmds its important use in study behavior materials From under formula intense (2.12) impulsive we may loading, find shock maximum synsis flexion diamonds, center explosive plate welding cladding metals. The method Wm,x=wl=w(a/2,a/2) estimation flyor =4. velocity X lo-'paz/d way raising it are questions (3.5) common interest. The Under flexions assumptions at middle one-dimensional points frec edges plane detonation arc rigid flying plate, normal approach solving problem w~----w(o,a/2) motion =0. flyor is to solve IO-'Pa~/D following system equations (3.6) governing flow field detonation products behind flyor (Fig. I): The flexions for corner points are ws= w (0,0) 'Pa2/D (3.7) --ff =o, ap +u_~_xp + au The solutions (3.4)-(3.7) arc all same to results obtained by Rayleigh-Ritz method in Rcf.[8]. Since answers au obtain.cd au 1 y by using =0, Raylcigh-Ritz methods Galcrkin's method must be one same when flexural s satisfy all (i.0 boundary conditions geometry as as inner forces, which was proved in Rcf.[3] a--t =o, Wc may calculate bending moments twisting moments without any difficulty when expressions deflection p =p(p, (2.12) s), arc known. The author accomplished computation by finite difference method with rougher nets (2 obtained where p, p, S, u are pressure, density, specific entropy particle velocity detonation products maximum deflection at center plate. Its value is wt= , respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory which coincides F flyor with as anor that boundary. (3.5) in Both order are unknown; magnitude. position R state parameters Wc on now it are try governed to investigate by flow case field I central equilibrium rarefaction between wave behind whole detonation rcactional wave forces D by foundations initial stage with motion total flyor external also; loading position in order F to check state parameters up extent products

5 A Free Rectangtilar Plate on Elastic Foundation 981 reliance results calculated previously. For this purpose, substituting (2.12) into formula to calculate general reacting force foundations, applying (3.4) (3.3) we may obtain In such a manner it denotes that whole reacting force from foundations just maintains equilibrium with general exterior loading its error is equal to zero, hence reliance solutions obtained above is nudoubted. Example 2 Let four sides a square plate on elastic foundations be free acted by a concentrated x force equally at four comer points respectively, as shown in Fig.2. Now we still seek answers using system equation(2.3). o "-. Substituting (2.12) into (2.3) noticing that coordinates points on which The one-dimensional problem motion a rigid flying plate under explosive attack has four forces P act are: (0,0), (0,a), (a,0) (a,a), we may an analytic solution only when polytropic index detonation products equals to three. In general, obtain a numerical system which analysis contains is required. two algebraic In this paper, equations however, by utilizing Fig. 2 "weak" shock behavior in relation to coefficients reflection shock f3 in f4 as follows explosive products, applying small parameter purterbation method, an analytic, 16z 4 z first-order ka~ approximate z 1 solution 4 P is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. (3.9) Final velocities flying plate obtained agree very well with numerical results by computers. Thus kaz], - 4 P-~ 0 an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) from for estimation which we may find velocity flying plate is established. f s= - O X IO-'Pa2/D, /,----4 x IO-'PaZ/D (3.10) From (2.12) we may obtain maximum 1. Introduction deflections at comer points w~.x----ws=w(o,o) = IO-4Pa~/D (3.11) The Explosive deflection driven flying-plate center technique plate ffmds is its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding w I =w (a/2, a/2) = X IO-~Pa2/D (3.12) cladding metals. The method estimation flyor velocity way raising it are questions common The deflections interest. at middle points on free edges are Under assumptions wz----w( one-dimensional O,a/2) plane detonation X I O-4Pa~/D rigid flying plate, (3.13) normal approach We may solving calculate problem bending motion moments flyor is to twisting solve moments following when system expressions equations governing deflections flow (2,12) field are detonation known. The products author behind accomplished flyor (Fig. I): calculation by means finite difference method with rougher nets 2 x 2 obtained maximum deflections at corner points plate. Its value --ff is w3=l =o, x l O-~PaZ/D, which is closer to ap +u_~_xp + au value given by formula (3.11). au au 1 In order to verify conditions balance y between =0, general reacting forces foundations entire exterior loading, we substitute (2.12) into (i.0 as as computing formula whole reactional forces foundations, by applying (3.4) (3.10) we may obtain a--t =o, R,=kflwdxdy=ba2,f,-~kaZ 4X 10-'-~= 4P (3.14) where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, It may with be seen that trajectory whole R reflected reacting shock force detonation foundations wave D also as a just boundary keeps balance trajectory with total F external flyor as anor loading boundary. its error Both is are equal unknown; to zero. Therefore, position R preceding state solutions parameters are also on reliable. it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products

6 982 Cheng Xiang-sheng IV. Concluding Remarks 1. In this paper we investigated bending problems rectangular plate with free boundaries on elastic foundations, calculated deflections plates acted on by a concentrated force in center be subjected to by a concentrated forces equally at four corner points respectively, obtained better approximate solutions: 2. Since better flexural s are selected which satisfy not only all geometric boundary, conditions thin plate but also boundary conditions internal forces wholly, application Galerkin's method is suitable, but conditions free from bending moments on four free sides are approximately satisfied. 3. Owing to conditions free from bending moments on all free sides we obtained conditions constraints (2.10) between coefficients f~ f2 with coefficient f~ in flexural s, consequently we decreased number independent coefficients from four to two. In such a manner calculated process becomes much more convenient. The one-dimensional problem motion a rigid flying plate under explosive attack has 4. We pointed out that results obtained by Rayleigh-Ritz method Galerkin's an analytic solution only when polytropic index detonation products equals to three. In general, method a in numerical first numerical analysis is example required. in In this this paper paper, are however, all same, by utilizing which very "weak" well indicates shock behavior that flexural reflection s shock surely in satisfy explosive all products, boundary conditions applying small geometry parameter pur- terbation internal forces method, on an free analytic, borders first-order wholly approximate thin plates. solution is obtained for problem flying plate 5. driven The by general various reacting high explosives force with polytropic foundations indices just or keeps than balance but nearly with equal to exterior three. Final loading velocities totally, which flying also plate forcibly obtained indicates agree very well accuracy with numerical reliance results by results computers. calculated Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic above. index) for estimation velocity flying plate is established. References 1. Introduction [1 ] Vlasov, V.Z. N.N. Leontchiev, Beams,, Plates Shells on Elastic Foundation, Gosjechizdat (1960). (in Russian) Explosive driven flying-plate technique ffmds its important use in study behavior materials [2] Kononenko, under intense Y.S., impulsive On Approximate loading, shock calculation synsis diamonds, for rectangular explosive plates welding on elastic cladding foundation, metals. The Research method estimation Structural Theory, flyor velocity Collected Articles, way raising 9 it are I1 questions (1960).(in common Russian) interest. [ 3 ] Under Chang Fo-van, assumptions Elastic Thin one-dimensional Plates, Second plane Edition, detonation Science rigid Press flying (1984), plate, , normal (in approach Chinese) solving problem motion flyor is to solve following system equations governing [4] Cheng flow Shu-tao, field Rectangular detonation products plates behind with free flyor edges (Fig. on I): elastic foundations, Acta Mechanica Sinica, 4, 1 (1960). (in Chinese) [ 5 ] Fan Jia-shen Li Jia-you, --ff Research on rectangular =o, ap +u_~_xp + au plates with free edges on elastic foundations, Appl. Math. Mech. (English Ed.), 2, 4 (1981), au au 1 [ 6 ] Sheng Yao Huang Yih, A free rectangular y =0, plate on two-parameter elastic (i.0 foundation, Appl. Math. Mech. (English Ed.) 8, 4 (1987), as as [ 7 ] Wang Ke-lin Huang Yih, a--t Rectangular =o, plates with four free edges on elastic foundations, Computation Structural Mechanics Applications, 2 (1985). (in Chinese) [ 8 ] Cheng Xiang-sheng, The bending, stability vibrations rectangular plates with free where p, edges p, S, on u are elastic pressure, foundations, density, specific Appl. entropy Math. particle Mech. velocity (English detonation Ed.), 9, 6 products (1988), respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory [ 9 ] Timoshenko, F flyor S. as anor S. W0in0wslcy-Krieger, boundary. Both are unknown; Theory Plates position Shells, R Second.Edition, state parameters McGraw-HiU on it are governed Book by Comp., flow Inc. field (1959). I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products [10] Lechnitsky, S.G., Anisotropic Plates, Gostechizdat., M. (1957). (in Russian)