BENDING OF UNIFORMLY LOADED RECTANGULAR PLATES WITH TWO ADJACENT EDGES CLAMPED, ONE EDGE SIMPLY SUPPORTED AND THE OTHER EDGE FREE ~

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1 Applied Mthemtics nd Mechnics (English Edition, Vol. 21, No. 1, Jn 2000) Published by Shnghi University, Shnghi, Chin Article ID: (2000) BENDING OF UNIFORMLY LOADED RECTANGULAR PLATES WITH TWO ADJACENT EDGES CLAMPED, ONE EDGE SIMPLY SUPPORTED AND THE OTHER EDGE FREE ~ Zho Fngxin (jt~j:fl~) t, Zhng Yingjie (.~t~-~) ~, Zho Zuxin (~dlft~) 2, Zhng Song (,~ Abstrct ~z}) 3, Yu Bingyi (r 3 (1. Shenyng Reserch Institute Foundry, Shenyng , P R Chin; The one-dimensionl 2. Hudong University problem Science the motion nd Technology, rigid Shnghi flying plte , under P explosive R Chin; ttck hs n nlytic solution only when the polytropic index detontion products equls to three. In 3. Shenyng Polytechnic University, Shenyng , P R Chin) generl, numericl nlysis is required. In this pper, however, by utilizing the "wek" shock behvior the reflection shock in (Communicted the explosive by products, Shen Huishen) nd pplying the smll prmeter purterbtion method, n nlytic, first-order pproximte solution is obtined for the problem flying plte driven Abstrct: by vrious In this high pper, explosives n exct with solution polytropic for indices n uniformly other thn loded but rectngulr nerly equl plte to three. Finl velocities flying plte obtined gree very well with numericl results by computers. Thus with two djcent edges clmped, one edge simply supported nd the other edge free, n nlytic formul with two prmeters high explosive (i.e. detontion velocity nd polytropic ws given by using the concept generlized simply supported edges nd superposition index) for estimtion the velocity flying plte is estblished. method. The numericl results were given for the deflections long the free edge nd bending moments long the clmped edges squre plte. Key words: bending rectngulr pltes; generlized simply supported edges; Explosive driven superposition flying-plte technique ffmds its importnt use in the study behvior mterils CLC under number: intense impulsive ;O175.2 loding, shock Document synthesis code: dimonds, A nd explosive welding nd cldding metls. The method estimtion flyor velocity nd the wy rising it re questions Introduction Under the ssumptions one-dimensionl plne detontion nd rigid flying plte, the norml pproch solving the problem motion flyor is to solve the following system equtions governing During the the flow lte field seventies, detontion Zhng products Fufn behind obtined the the flyor exct (Fig. solutions I): to the bending problem rectngulr cntilever pltes nd rectngulr pltes which hve two djcent edges clmped nd the other two djcent edges free nd ber with concentrted lod nd uniform lod [1-3] In this p +u_~_xp + u pper, n exct solution ws given for n uniformly loded rectngulr plte with two djcent edges clmped, one edge simply supported u u nd the 1 other edge free by using Zhng' s method. The numericl results were given for the deflections long the free edge nd bending moments S s long the clmped edges squre plte. This plte my be suitble for some engineering --T pplictions such s projects brn, wter-storehouse, sensor-bsed chip nd civil construction. 1 Concept Generlized Simply-Supported Edges nd Boundry Conditions where p, p, S, u re pressure, density, specific entropy nd prticle velocity detontion products respectively, Boundry with condition the trjectory simply-supported R reflected edges shock is bending detontion moment wve D M s = boundry 0, nd deflection nd the trjectory IV = 0 while F tht flyor s generlized nother boundry. simply supported Both re unknown; edges is M the = position 0, nd 1V R ~ nd 0. In the both stte cses, prmeters shering on force it re exists governed long by the the edges. flow field If deflection I centrl exists rrefction long wve simply behind supported the detontion edge, wve this D nd by initil stge motion flyor lso; the position F nd the stte prmeters products * Received dte: ; Revised dte: Biogrphy: Zho Fngxin ( ), Mster, Senior engineer 117

2 118 Zho Fngxin, Zhng Yingjie nd Zho Zuxin et l edge is defined s generlized simply-supported edge. By mens the concept the generlized simply-supported edge nd superposition method, the problem uniformly loded rectngulr plte with two djcent edges x = 0 nd y = 0clmped, the edge y = simply supported nd the edge y = b free cn be solved (Fig. 1 ). The boundry conditions re given s follows clmped edge (~)~=o = = 0, x=0 edge supported (W)y=o = = 0, y=o Abstrct The one-dimensionl \l problem the motion (V/)x=o rigid = flying -g~sj.=o plte under = o, explosive ttck hs free edge n nlytic solution only when the polytropic index detontion products equls to three. In )'1 generl, numericl nlysis is required. In this O W pper, however, 0 2 W / by utilizing the "wek" shock -- +/~ ~-r~ / behvior the reflection Fig. 1 shock in the explosive OY products, 2 nd pplying y= b the smll prmeter purterbtion method, n nlytic, first-order pproximte solution is obtined for the problem flying Iv 3 w ] plte driven by vrious high explosives with polytropic -- + indices (2- fz) other ~ thn ] but nerly = O. equl to three. (1) Finl velocities flying plte obtined gree very 8Y well 3 with numericl results r= b by computers. Thus n nlytic formul with two prmeters high explosive (i.e. detontion velocity nd polytropic index) 2 Composition for estimtion the Superposition velocity flying plte is estblished. 1 ) For n uniformly loded rectngulr plte with simply supported edges, the boundry beingx = O, x =, y = Ondy = b, the skew cmber the plte, ngulrity nd shering force Explosive long the driven edges cn flying-plte be respectively technique described ffmds its s importnt follows use in the study behvior mterils 4q 4 1 [1 retry 1 rnrcy W - under 773-_5 intense ~ impulsive ms[ - ch loding, shock + rnrcysh synthesis dimonds, nd explosive welding nd cldding metls. Drr.,=,,3,... The method estimtion 2 flyor velocity nd the wy rising it re questions Under the 2sh= ssumptions x ~ one-dimensionl " 2 1 plne detontion 2sh.~ sh ~ nd rigid sin flying plte, the norml pproch solving the problem motion flyor is to solve the following system equtions governing the flow field mrcb detontion products behind the flyor (Fig. I): = -, (2) 0 W 2q 3 th u u 1 sin m~x (3) y=o - D~ ' e S s --T 11 pi p +u_~_xp + P~ 1 I itry _ 2fl,, sin---b--,~,- where D is the bending rigidity nd/1 is the Poisson's rtio. u itr b, x=o- D~ 4 ~ 7 th 2 i= 1,3,"" ch2 ~-J where p, p, S, u re pressure, density, specific entropy nd prticle velocity detontion products respectively, with the trjectory R reflected shock detontion wve D s boundry nd the trjectory F flyor s nother boundry. Both re unknown; the position R nd the stte prmt~x meters (Vy)r=~ on it re =- governed ~z... by the...~-~ flow (3- field ~)th-~-- I centrl rrefction (1 - f z ) wve -2~-~ - behind 1 sin -- the detontion wve (5) D nd by initil stge motion flyor lso; the position F nd the stte prmeters products ch TJ (4)

3 Bending Uniformly Loded Rectngulr Plte 119 2) For rectngulr plte with simply supported edges, the bending moment long the edge y -= 0 is given s it gives M(x) = ~ E.,sin mn xx m=l W - 2 E m [ m rnny ~ -- -ST---sin 2 Dn z m= ~- sh., rnnysh rnnr m2y ch '==xl sin rnnx cth,~ --, (6) Abstrct " + The one-dimensionl y=o - 2riD ~-~.,=, problem ethm the - motion --sh z,. sin rigid --, flying plte under explosive ttck hs (7) n nlytic solution only when the polytropic index detontion products equls to three. In generl, ~ ie.~ iny 8 W numericl 2b2 nlysis ~ is ~ required. b 2 In this i 2 2sin pper, b however, by utilizing the "wek" shock (8) behvior the reflection shock in the explosive products, nd pplying the smll prmeter purterbtion method, n nlytic, first-order pproximte solution is obtined for the problem flying plte driven by vrious high explosives with polytropic indices other thn but nerly equl to three. Finl velocities flying plte obtined 7r me[ gree 1 very + 1 well -,u with m cthm numericl ] sin -- mnx results by computers. Thus (9) ~)y=b = - (1 + /1) ~T== sh.~ L ~ n nlytic formul with two prmeters high explosive (i.e. detontion velocity nd polytropic index) 3) for estimtion For rectngulr the velocity plte with flying simply plte supported is estblished. edges, the bending moment long the edgex = Ois given s M(y) = ~F/sin iny i=1 b ' Explosive driven flying-plte technique ffmds its importnt use in the study behvior mterils it gives under intense impulsive loding, shock synthesis dimonds, nd explosive welding nd cldding metls. The method estimtion flyor velocity nd the wy rising it re questions b 2 Fi[ 19i, ~nx inx. inx W - 2Dr: 2 7~[ - ~sn b ~ sh -g-- + Under the ssumptions i= sh'fll one-dimensionl plne detontion nd rigid flying plte, the norml pproch solving inx the. problem inx iny motion flyor in is to solve the following system equtions governing the cth/?i flow field ~--ch detontion ] sin products fli - behind, the flyor (Fig. I): 7- --U' b 0 W~ 2 2 Fire. ) _----.,:~,:~ ( z r2) zsinm~x, (11) p +u_~_xp + u,-o ~'- bz) ~ p' i' j + 7 u u 1 O W) b F, [ ethfl, ~, ] iny.=o - 2riD ~ -~ S sh2fl s i sin -- b ' (12) --T f 2 in 2 J 2 ri., V p =p(p, + (2- s), ~) U.eosir~ sin rnnx (13) V;)y=b = To, where p, p, S, u re pressure, density, specific entropy nd prticle velocity detontion products respectively, with the trjectory R reflected shock detontion wve D s boundry nd the trjectory F flyor s nother boundry. Both re unknown; the position R nd the stte pr- 4) For rectngulr plte with the three edges x = 0, x = nd y = 0 simply supported meters on it re governed by the flow field I centrl rrefction wve behind the detontion wve D nd nd the by edge initil y stge = b generlized motion simply flyor lso; supported, the position the bending F nd moment the stte is prmeters given s products ( IV)y= b = ~.~sin mn xx,.,= I (lo)

4 Zho Fngxin, Zhng Yingjie nd Zho Zuxin et l it gives o (. ) _ :if-;..[ rn.ych m.y.]sin r. ~x, +,,cth,~ sh m~y (14) (_~W) = 1 2/ZTt~ t - sh,, m,,,{1 tl +,u /1 + o~cthz] sin -. y=o.,= -- i z b z m----y + (2 - /.z) -- 2, Abstrct rt~ 7~2; (15) i~y (16) b ' The one-dimensionl problem the motion rigid flying plte under explosive ttck hs n nlytic (vy)r= solution b : only (1 _ when ~)2 ~.~.., the ~'~[1 polytropic index -/~/~cth"~ detontion + products sin ~. x_x m equls to three. (17) In generl, numericl nlysis is required. In this pper, however, by utilizing the "wek" shock behvior the reflection shock in the explosive products, nd pplying the smll prmeter purterbtion 3 The method, Method n nlytic, Superposition first-order pproximte solution is obtined for the problem flying plte driven by vrious high explosives with polytropic indices other thn but nerly equl to three. Finl velocities In order to obtin flying zero plte ngulrity obtined (3-5~W.W/ gree very well = 0 long with numericl the clmped results edge by y computers. = 0, the sum Thus \ v), / y=0 n nlytic formul with two prmeters high explosive (i.e. detontion velocity nd polytropic Eqs. (3), (7), (11) nd (15) re mde to be zero, nd then the first eqution is deduced. index) for estimtion the velocity flying plte is estblished. After tht, in order to obtin zero ngulrity (0O--~)) = 0long the clmped edge x = 0, the sum Eqs. (4), (8), (12) nd (16) re mde to be zero, nd then the second eqution is deduced. Explosive driven flying-plte technique ffmds its importnt use in the study behvior mterils Finlly, under in intense order to impulsive stisfy tht loding, the shering shock synthesis force long dimonds, the free edge nd y explosive = b is equl welding to nd zero, cldding i.e. ( Vy)y= metls. b = 0, The let method the sum estimtion Eqs. (5), flyor (9), velocity (13) nd the (17) wy be equl rising to it zero, re questions thus the third common eqution interest. is obtined. Under the ssumptions one-dimensionl plne detontion nd rigid flying plte, 2 the norml b 2 pproch From the solving bove the three problem simultneous motion equtions, flyor the is to unknown solve the number following m, system ~-E,~ nd equtions -~Fi cn governing the flow field detontion products behind the flyor (Fig. I): be solved. 4 An Clculting Exmple p +u_~_xp + u In the cse /z = 0.3, for n uniformly loded squre plte with two djcent edges u u 1 x = 0, y = 0 clmped, the edge x = simply y supported =0, nd the edge y = free, the clculting results re expressed s follows. S s 4.1 Deflections long the free --T edge Prt vlues deflections long the free edge y = re listed in Tble 1. The skew curve long the free edge y = is shown in Fig. 2. where p, p, S, u re pressure, density, specific entropy nd prticle velocity detontion products respectively, Tble 1 Prt with vlues the trjectory deflections R reflected long the shock free edge detontion wve D s boundry nd the trjectory F flyor s nother boundry. Both re unknown; the position R nd the stte prx( x ) meters on it re governed by the flow field I centrl rrefction wve behind the detontion wve D nd by initil stge motion flyor lso; the position F nd the stte prmeters products (W)y.~(x 10-3~q-~_4~ , k /J / x=0

5 Bending Uniformly Loded Rectngulr Plte Moments long the clmped edges Prt vlues moments long the clmped edges y = 0nd x = 0 re listed in Tbles 2 nd 3 nd the corresponding moments distribution curves re shown in Figs. 3 nd 4 respectively. It is evident tht long the clmped edge x = 0, the moments re lrger ner the free edge I i I I ~ 0.08 W(q4/D) Fig. 2 The skew curve long the free edge y = The one-dimensionl problem the motion rigid flying plte under explosive ttck hs n nlytic solution only when the polytropic index detontion products equls to three. In generl, numericl nlysis is required. In this pper, -0.l however, by utilizing the "wek" shock behvior the reflection shock in the explosive products, nd pplying the smll prmeter purterbtion -0.1 method, n nlytic, first-order pproximte solution is obtined for the problem flying plte driven M( by ~ 2) vrious high explosives with polytropic indices other thn but nerly equl to three. Finl velocities flying plte obtined gree very well with numericl results by computers. Thus n nlytic formul with two prmeters high explosive -0.2 (i.e. detontion velocity nd polytropic M(q 2) index) for estimtion the velocity flying plte is estblished. Fig. 3 Abstrct Distribution moment long Fig. 4 clmped edge y = 0 Distribution moment long the the clmped edge x = 0 Explosive driven flying-plte technique ffmds its importnt use in the study behvior mterils Tble under 2 intense Prt vlues impulsive moments loding, long shock synthesis clmped dimonds, edge y = nd 0 explosive welding nd cldding metls. The method estimtion flyor velocity nd the wy rising it re questions x(x ) Under M(x)(x the 10-Zq ssumptions 2) 0 one-dimensionl plne detontion nd rigid flying plte, the norml pproch solving the problem motion flyor is to solve the following system equtions governing the x(x flow ) field detontion 0.6 products 0.65 behind 0.75 the flyor (Fig I): M(x)(x 10-2q z) p +u_~_xp + Tble 3 Prt vlues moments u long u the clmped 1 edge x = 0 y(x ) S s M(y)( x 10 -~ q') 0 --T y( x ) where M(y)(x p, p, S, 10-1q u re -') pressure, density, specific entropy nd prticle velocity 0 detontion products respectively, with the trjectory R reflected shock detontion wve D s boundry nd the trjectory F flyor s nother boundry. Both re unknown; the position R nd the stte prmeters on it re governed by the flow field I centrl rrefction wve behind the detontion wve References : D nd by initil stge motion flyor lso; the position F nd the stte prmeters products [ 1 ] Zhng Fufn. Bending rectngulr cntilever pltes[ J]. Journl Tsinghu University, 1979,19 (2) : (in Chinese) u

6 122 Zho Fngxin, Zhng Yingjie nd Zho Zuxin et l [2] Zhng Fufn. Bending uniformly loded cntilever rectngulr pltes[j]. Applied Mthemtics nd Mechnics ( English Edition) 1980,1(3) : [3] Zhng Fufn. Rectngulr pltes with two djcent edges clmped nd other two djcent edges free [J]. Act Mechnic Solid Sinic, 1981, (4) : (in Chinese) Abstrct The one-dimensionl problem the motion rigid flying plte under explosive ttck hs n nlytic solution only when the polytropic index detontion products equls to three. In generl, numericl nlysis is required. In this pper, however, by utilizing the "wek" shock behvior the reflection shock in the explosive products, nd pplying the smll prmeter purterbtion method, n nlytic, first-order pproximte solution is obtined for the problem flying plte driven by vrious high explosives with polytropic indices other thn but nerly equl to three. Finl velocities flying plte obtined gree very well with numericl results by computers. Thus n nlytic formul with two prmeters high explosive (i.e. detontion velocity nd polytropic index) for estimtion the velocity flying plte is estblished. Explosive driven flying-plte technique ffmds its importnt use in the study behvior mterils under intense impulsive loding, shock synthesis dimonds, nd explosive welding nd cldding metls. The method estimtion flyor velocity nd the wy rising it re questions Under the ssumptions one-dimensionl plne detontion nd rigid flying plte, the norml pproch solving the problem motion flyor is to solve the following system equtions governing the flow field detontion products behind the flyor (Fig. I): p +u_~_xp + u u u 1 S --T s where p, p, S, u re pressure, density, specific entropy nd prticle velocity detontion products respectively, with the trjectory R reflected shock detontion wve D s boundry nd the trjectory F flyor s nother boundry. Both re unknown; the position R nd the stte prmeters on it re governed by the flow field I centrl rrefction wve behind the detontion wve D nd by initil stge motion flyor lso; the position F nd the stte prmeters products