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Bending of Corner-supported Rectangular Plate Under Concentrated Load

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Very Important-Bending of Corner-supported Rectangular Plate Under Concentrated Load
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  Applied Mathematics and Mechanics English Edition, Vol. 5 No. 3, Jun. 1984) Published by }~ST Press, .Wuhan, China ENDING OF CORNER-SUPPORTED RECTANGULAR PLATE UNDER CONCENTRATED LOAD Lin Peng-cheng ~r ) (Fuzhou University, Fujian) Received Oct. 17, 1982) ABSTRACT In this paper the solution for the bending of corner-sup- ported rectangular plate under concentrated load at any point (~/2, 7) of the middle line of the plate is given by means of a conception called modified simply supported edges and the me- thod of superposition. Some numerical example is presented. The solution obtained by this method checks very nicely with what was obtained by G.T. Shih [3] by means of spline finite element method when ~ ~ d/2 This shows that this method of solution is satisfactory~ I. Introduction In [I], Prof. Chang Fo-van considered the bending of corner-supported rec- tangular plate under distributed load, while in this paper, the solution for the bending of corner-supported rectangular plate under concentrated load at any point (a/2,rl) of the middle line x=a/2 of the plate is discussed. For this reason the problem is tackle-d as follows: Within the bounds of rectangular plate, the derlection w must satisfy the partial differential equation a,m a.w a w Pc) .x--a/2,1J--n) a~' + ~ a~'--r~u +-~-u c=---~- ~ in which O is flexural rigidity of the plate, <} x-a/z, u-,)=<} x-a/2)<~ u-~) 6 is Dirac function. Furthermore, it must satisfy the boundary condition o,o ,, o,:, . 1 Oy #,.a LOCi u~ay J,'. (1.2) 8iw . 8 w , ^ [ a va O~w ] ~u v~ .,., a~+ 2-~)~ l:l=o 1.3) The solution of this problem is given by means of a conception called modified simply supported edges and the method of superposition. II. The ConSit1 tuent Parts of Superposition ethod * Communicated by Chang Fo-van.  1410 Lin Peng-cheng (A) A rectang-alar plate with four simply supported edges is subject to con- centrated load at .any point (a/2,~) of the middle line x=a 2 of plate. In this case, the bending surface is/5]: Pc v ~ smh ?77 sinh ~Y'(I + a~.cotha,. -- ~t~:/'coth a/~ y'- w, = /)zr s ,~.,,s.--~ . mSsinhu,~ in which If ,cc:v in the exp. b b t a 2 mzb .,=5-- i and .tT'~ then the quantity y~ must be replaced by y , and the quantity ~ by b--~ (1.1). Therefore, '~ + (1 --p)a,~cotha=-- (1--#) ct~(b--~7) coth a.,(b--T1) V~Jv'~ u ~.,,~,... - b sinh .a,,, ( b-- rl) b m~ gt~/T,~ 9 sinha.~ sin ,-7~ sin- (2.2) (V,) v-b =-- ~ o+(1--1~)a~cotha~_(l_~t ) rl m-I 3 ... sinh a,.r/ b l/l:I DZ?I'X ;in-haT sin 2 sin a (2.3) Also the bending surface of the plate can be expressed by sin i~rr/ u Pb2 ,-,- i%oshbqC b 2 2 b b __ . --b- ( 2~ o which fl+=i:za/b. This equation of ber.,ding surface is satisfied for x~a/2. By symmetry., we can compute the deflections and internal component as bending mom- ents of the other half plate with this equation 2+ (1--U)~'tanh~ --_ ~V- ...... 2 2 sin ira? icty (V.) ._. -- 2blZ~ cosh/4 -b- sin -b- (2.4) ,) (b) Suppose the edges u=O, u=b 5f the rectangular plate are simply sup- ~orted edges, while the edges x=0, x----a are modified sLmply supported edges, the deflections along two edges x=0 , x=a are: (w). _ 0 ---- ~' b, sin irry x-a I-i b -- in this case, the bending surface of plate is: 2 i~rx Drx iz~y (Vv) ~-o = D l(l--ll) -n2 V~ X mSb~ ~rtx b 3 -.~ a2 2 sin + (2.6) (,n' ) a- m- ~,... ~ i i ~-+b: D4(1 - tL)~tr ~ \~ V~ mSb, mzx (I%) ,-0 = - a -., ,..., a z z c~ --- (2.7) +~ ~ m.I ~1.-. |.1 [ b~  Bending of Corner Supported Rectangular Plate 1411 coshfl~-- 1 3 + p (I-~) .=:y: ,~b, (~ ,.) ~i,,--'~ 2.8) V,)=., : D 2 -- b- . sinhfl l--u b (C) Suppose the edge y=O of rectangular plate is the modified simply sup- ported edge and the other three edges are simply supported edges, the deflections along edge y=O are: t~),. 0 = ~' C. sin 1 m.l~St... In this case, the bending surface of plate is: w~= c. cotha., sinh m~y 1--t~ m~YsinhmaY =-,, ,.. 2 sinhZa= a ~ 2 a a (2.9) (V,) ,.0 =--D (1-- )'~ ~ m ~ 3*. a.. ) ,,,~ ....... c,n(~fi~ ~zotham-~-si~a-hzh~- sin (2 10) 03 m.[ S . ~/ (V,),.~ =--D (I--#)'~ ~, m%..l_.( 3+# +a.cotha.)sin m~x (2 ll) 2a ..~,s,... stnna,~x ~--~ a (V.)..o: 2D (1 _p)~z Pc,.cosma i~y s ~ ~ /b z- ~z \-. sin ~- (2.12) --,,,, - ,- m L ~ + -~ - J (D) Suppose the edge y=b of rectangular p]~ is the modified simply sup- ported edge and the other three edges are simply :T~orted edges, the deflects along edge y=~ are : In this case, the bending su~face of plate is: III. w~:' 1--,u Y~' sinha.L~ 1 Z-p- + a~cotha. ) sinh mzy-- rrury coshrr~y ~ sin mzx 2 m-~ ... - a a a A -~ (2.13) (V,)~.0=D (1-t~)2w .m3a,. I 3+u . mzx m-ll$~-,- (] (V')'=~ 2 ..... s ... ) a (2.15) ~-,,s,... sinhZa= + otha. sin rnzx 2~_ V a~ v~ iScosin izry (V.)=..=D(1--#)' a ~ m.l s. m ~_~ CZ i2)2 sin ~-- 2.16) The Solution of the Problem of Corner-Supported Rectangular Plate Bending by Means of Superposition Method In order to satisfy the condition that the transverse forces along the free edge y=Oi must be equal to zero, by adding up the exps. (2.2), (2.6), (2.10), (2.14) and equating thblr sums to zero, we therefore obtain: _am_ {_3+~ +a.cotha,~ \ / 3+~ .. _ a. \_ 8 a' V b, sinuam 1--g \ 1--/~ sJnn am u o ~.~ 4a _rex -,Lb 2 t]:2y Pa z r ---~(1~) ~,~- L 2 + (1 -/~)a.cotha=  1412 Lin Peng cheng --(1--~) a=(b---Tr/) r ]sinh a ~(b-~) b-- - sinha= sin rt~__ 2 (m=l, 3, 5, ...) 3.11 In order to satisfy the condition that the transverse forces along the free edgey=b must be equal to zero, adding up the exps. (2.3), (2.7), (2,11), (2.15) and equating their sums to zero, we thus obtain: a=( a,. q_ 3+~ cotha '~__c~( 35F/z_+a.cotha~,. ) sinhZa= l--t~ / smna=~ l--~ 8 a s b~coshr 2Pa~ [2 +~-b' c-~1 (;: +'inzz)/=- D(1--#)2=Sn; + (1 ~ ~ ~ ~c~t h~ sinh a,.~, -- l--p)olcoth b j slnha,,, sin ,~ m=1,3,5,...) 3.2) In order to satisfy the transverse forces along the free edge x a to be equal to zero, by adding up the exps. 2.4), 2.8), 2.12), 2.16) and equating their sums to zero, we then obtain: COS ~ \ 'n am f b z -i,- \-= 'll3t''* )T/ ]2 + m2 m-l n ... t ~ +,;, a'a s coshfl,--I 3+I' , x +-4b s b, sinhfi, ( ..... fi Pa 1 l--i~ sinh fie) = Db 4(1--~)~'i , t4 fl, + (1 --tz)t~ e anh 2 coshfl --- sin b ~-- (i=1,2,3,...) (3.3) 2 By symmetry we can obtain just the same set of equations for edge x=0 as those for edge x=a . Hence, solving the above three sets of infinite sLmultaneous eqs. (3.1)- (3.3), we can find a=, b,, c, and thus the deflections, internal force component as bending moments, etc. at any point of plate and it will be possible to compute con- centrated reactive forces R at the corners of the plate according to the following expressions: (R),.,--2])(1--a),.~( 8'w, { a ~ imTtanhfl'2 -.~ exau )~:~= (1 -p) P._-z-b ,.,. si- b :osh fl, ~os i= 2 mzu., D[ 1 +/~ --(1--/~)~r* .~Y Pa z ~1--~ c~ sin~-~=a ) rn. ,~ 9 ,ZbjD . coshfle-- ~ l+i, fie / + (1-.)='~ -p~cos,~ ~inh); ~i-. - s~nh~ l +(1--a)n: V' mZc D ( I+P +a,cotha= )] (3.4) ,,.l'-'~,.. Pa=sinha,, ] --v tanh~-t { ~b. o 1 ~osh _p R) =~-o = -- (1 -- u) P -- sin-~-- ,~ + ( 1 -- ,,) ~' ~. m-|tsv*-. 2 m c~DX 1 + Po' t ] ,u. cotho=

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Jun 13, 2018
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