Approximate Large Deflection Analysis of Thin Rectangular Plates under Distributed Lateral Line Load

Transcription

1 Second Interntionl Conference on Advnces in Engineering nd Technolog Approite Lrge eflection Anlsis of Thin Rectngulr Pltes under istriuted Lterl Line Lod Alln Okodi, Y N. Zir, Jckson A. Mkli Grdute Student, Fcult of Technolog, Mkerere Universit, P. O. Bo 76, Kpl, Ugnd Senior Lecturer, Fcult of Technolog, Mkerere Universit, P. O. Bo 76, Kpl, Ugnd Corresponding uthor eil: nzir@tech.k.c.ug Professor, Fcult of Technolog, Mkerere Universit, P. O. Bo 76, Kpl, Ugnd ABSTRACT Reserch on lrge deflection of thin rectngulr pltes to dte hs focused on pltes under uniforl distriuted lod. There is need to etend the theor to other fors of lods nd to get sipler pproite solutions tht re esier to use. An pproite nlsis of lrge-deflection of thin rectngulr pltes under distriuted lterl line-lod is presented in this pper. The nlsis is sed on solving Von Kárán equtions. The lod, deflection nd stress re represented gle trigonoetric series in the -direction nd truncted coe function in the -direction is ssued. The functions re sustituted into the Von Kárán equtions to get third degree polnoils descriing reltionships eteen lod nd deflection coefficients. The polnoils re solved ug scientific orkplce function polnoilsroots, to get defections cused different lods. Results re copred ginst the ect solution nd it is seen tht the to results sho siilrit in the trends of reltionships eteen lods nd deflections, nd deflections nd nuer of coefficients, ut deflections otined ug these pproitions re higher thn of the ect ethod for the se lod. The proposed pproite ethod is seen to e sipler nd could e dopted here ccurc is not ver criticl. Keords: Approite, Ect, Lrge deflection; Line-lod; Stress; Von Kárán equtions.. INTROUCTION Oing to the copleit of ect solutions for lrge deflection nlsis, severl pproite solutions for uniforl-loded pltes of siple, regulr shpes hve een suggested, notl Tioshenko, ut the led to significnt loss in ccurc (Tioshenko nd Woinsk-Krieger, 99). Wng nd El-Sheikh (), suggested n pproite theticl solution for pltes under uniforl distriuted lod cting lterll. Their solution is sed on solving Von Kárán equtions (equtions, ), ut the used the Microsoft ecel solver for solving the finl loddeflection equtions. Rdoslv () suggested n pproite solution for plte strips under tension in the longitudinl direction. The tension is in the for of in-plne forces pulling the plte in opposite directions. F F F E () q t F F F () is the deflection of the plte, E is Young s odulus, t is the plte thickness, q is the distriuted lod, nd is the plte fleurl rigidit such tht: 6

2 Okodi, Zir nd Mkli Et v F =, F, F sher stress in - plne., v is Poisson s constnt, nd F is Air stress function such tht., re the norl stresses in nd directions, is the F p p f n, n, p denotes in-plne lods. p = for lterl lods. cos cos n (c) Z Y q t X Figure : Tpicl sipl supported plte under distriuted line lod, (dpted uthors fro literture). METHOOLOGY This pper presents the findings of stud tht nlzed the ehviour of thin rectngulr pltes sujected to uniforl distriuted lterl line-lods ug n pproite ethod sed on ssuption of truncted coe function (Okodi, ). The specific ojectives ere to derive nd solve pproite lod-deflection reltionships for thin rectngulr pltes ith siple supports, sujected to distriuted lterl line lods nd to copre the outcoe ith results otined ug the ect ethod. The folloing steps ere folloed to chieve the foreentioned ojectives: i) erivtion of theticl reltionships eteen lod nd deflection coefficients for pltes under lterl line-lods, ii) Solution of the derived reltionships to otin the deflections cused vrious lods ug proprietr softre (Scientific ork plce), iii) Grphicl presenttion of results. The reserch s liited to thin rectngulr pltes sujected to distriuted lterl line-lods ith sipl supported edges. 7

3 Second Interntionl Conference on Advnces in Engineering nd Technolog Consider the sipl supported rectngulr plte shon in Figure.. X. Y 8 Figure : Sipl supported plte, ith Lev s es (Adopted uthors fro literture) For lod q cting long the -is, lod intensit is given q q () Lod is lso represented Fourier series in direction s follos q ( ) q () B integrtion, it cn e shon tht coefficients q re defined the equtions elo, q q,,,,... Sustituting for q in the lod series ields the epression elo q q ( ) (c) The generl deflected shpe ould e s follos: ( ) Y () here Y is the coefficient of deflection nd is function of onl. It ust e such tht it stisfies the oundr conditions. Tht is to s,,, nd,. The coe function is considered in this pper nd iu of five ters re used in deterining the deflected shpe of the plte. Choog ver n ters significntl increses the volue of ork to e done lthough ccurc is iproved. Onl odd vlues of stisf the oundr conditions nd therefore even vlues re ignored. Y cos () So, ( ) cos, (c) Leding to the folloing deflected shpe equtions

4 Okodi, Zir nd Mkli ( ) cos for one deflection ter ( ) cos cos for to deflection ters ( ) cos cos cos for three deflection ters ( ) 7 cos cos cos 7 cos for four deflection ters ( ) 7 9 cos cos cos 7 cos 9 cos for five deflection ters Epressions for Air stress re siilrl derived, nd re shon elo. F ( ) f cos for one stress ters F ( ) f cos f cos for to stress ters F ( ) f cos f cos f cos for three stress ters F ( ) 7 f cos f cos f cos f7 cos for four stress ters F ( ) 7 9 f cos f cos f cos f7 cos f9 cos for five stress ters ifferentition nd sustitution of the epressions for deflection nd stress function into Von Kárán equtions result in equtions ith f nd s unknons. B tking onl the first ters for eple, the folloing equtions ( nd ) re otined. f cos cos cos * E () cos q f t cos cos () These re to epressions in to unknons ( f, ). Thus e re le to deterine the unknons t n loction (, ) on plte of knon diensions. To deonstrte the solution procedure, consider the cse of deflection t the centre of squre plte. Plte diensions, nd loction, is,. For one ter, q t f * Sustitute for f : q Et 9

5 Second Interntionl Conference on Advnces in Engineering nd Technolog On further siplifiction, the ove epression ecoes, q. () Et Et This is the lod-deflection reltionship used to get the deflection coefficients for n lod. Consider one deflection coefficient; the deflection t the centre is the se s the coefficients. B folloing the se procedure, reltionships for,, nd deflection ters hve een derived for sipl supported pltes. These epressions hve een solved coputer softre for * steel plte of thickness.. RESULTS Fro Tle, the deflections re seen to increse ith lod s ould e epected. This is siilr to results of the ect ethod. Tle results hve een plotted to sho the trend of deflection ith nuer of deflection ters (Figure ), nd to sho the trend of deflection ith lod (Figure ). An dditionl plot (Figure ) of lod ginst deflection hs een de to copre the results of the ect nd pproite ethods for thick plte. Tle : Shoing pproite deflections for vrious nuers of ters Lod eflection () for different nuers of ters (N) ter ters ters ters ters Plot of deflection ginst nuer of coefficients 6 Nuer of coefficients Figure : Plot of pproite deflection ginst nuer of ters The plot of deflection ginst nuer of coefficients or ters (ug results of N lod) in Figure ove shos the folloing: ) The vlue of deflection is highest for one ter (7.8% higher thn the loest), ) The vlue is loest t to ters (7.8% loer thn vlue t one ter), c) The vlue t three ters is.7% loer thn the highest ut.% higher thn the vlue t to ters, hich is the loest, d) The vlue t four ters is.% loer thn the highest ut.% loer thn the vlue t three ters tht precedes it. e) The vlue t five ters is.8% loer thn the highest vlue ut.% higher thn the vlue t four ters.

6 Okodi, Zir nd Mkli The plot in figure further indictes tht the deflections fll steepl hen the nuer of ters is incresed fro one to to. Hoever the deflection there-on generll reduces ith increg nuer of ters leit ith etreel lo grdient. The fll in the vlue of deflection hen the nuer of ters is incresed fro one to to, the rise there-on nd the tendenc to stgnte on gle vlue (ppro. in this cse) is in-keeping ith the results otined other reserchers ho used ect ethod to stud uniforl loded pltes, notl Wng nd El- Sheikh (). This shos tht convergence is fster for the pproition chosen. Hoever the deflections re ver high nd the difference eteen the deflections otined ug one ter is etreel high copred to deflections otined ug higher ters. This is t vrince ith results of the ect ethod nd is likel to e cused the ssued theticl reltionship eteen coefficients of deflection nd coefficients of Air stress. Lod-deflection curves for vrious nuer of coefficients Five coeficients Four coefficients Three coefficients To coefficients One coefficient 6 8 eflection () Figure : Plot of lod ginst pproite deflections The plot of lod ginst deflection (Figure ) indictes tht the deflections increse non-linerl ith increg lod, hich is in-keeping ith the theor of lrge deflection. Hoever there is rked vrition in vlues s the nuer of deflection coefficients (ters) is incresed. As eplined ove, this lrge vrince is likel to e cused the ssued theticl reltionship eteen coefficients of deflection nd coefficients of Air stress. Figure lso shos tht the plots for one ter nd to ters re seprte lines ith cler distnces prt. The plots for three, four nd five ters re close together (nded together into unch). This iplies tht one coefficient (ter) over-estites the deflection, to coefficients under-estites the deflection, hile the deflection converges the third coefficient (ter) to vlue, hich vries ith lod. Coprison of Ect ethod to reserch results Ect ethod This reserch eflection () Figure : Plot of lod ginst deflection for results of ect nd pproite ethods The plot in Figure, of lod ginst deflection (for results otined ug one coefficients ith oth ect nd pproite ethods) shos tht oth ethods intin non-liner reltionship eteen lod nd deflection of thin pltes, hich llos the theor of lrge deflection to hold.