Winkler地基上四边自由正交各向异性矩形中厚板弯曲的有限积分变换解.pdf

应用数学MATHEMATICA APPLICATA2024,37(1):192-199Finite Integral Transform Solution ofthe Bending of Moderately ThickOrthotropic Rectangular Plate withFour Sides Free on WinklerFoundationNingjin(宁进),Eburilitu(额布日力吐)(School of Mathematical Sciences,Inner Mongolia University,Hohhot 010021,China)Abstract:The bending problem of a moderately thick orthotropic rectangular plate(ORP)with four sides free on Winkler foundation is studied by the finite integral transform(FIT)method.Based on the boundary conditions and basic equations of the bending ofmoderately thick ORP,the analytical bending solution of this plate is obtained by theFIT method and its corresponding inverse transform method.This analytical solutionis uniformly applicable to the calculation of bending problems of thin,moderately thickand thick isotropic and orthotropic rectangular plates.Then by specific examples thecorrectness of the obtained analytical solution is verified.Key words:Moderately thick orthotropic rectangular plate;Bending;Analyticalsolution;Finite integral transform methodCLC Number:O343AMS(2010)Subject Classification:74B05;74K20Document code:AArticle ID:1001-9847(2024)01-0192-081.IntroductionIn many engineering fields,such as civil engineering,aerospace engineering and mechan-ical engineering,elastic rectangular plates are common structural form.Therefore,exploringthe solutions of bending,buckling and vibration problems of elastic rectangular plates havealways been the focus of scholars.At present,many scholars have applied numerical methodsto study elastic rectangular plates,such as finite element method1,finite difference method2,discrete singular convolution method3and spline element method4.In addition to numericalmethods,there are also a few analytical methods for studying the elastic rectangular plates.The representative analytical methods include the inverse or the semi-inverse method5,theFourier series method6,the superposition method7and the FIT method8.Among the aboveanalytic methods,the FIT method is one of the best methods for solving partial differentialReceived date:2022-12-26Foundation item:Supported by the National Natural Science Foundation of China(12362001,11862019)and the Natural Science Foundation of Inner Mongolia(2023MS01008)Biography:Ningjin,female,Meng,Inner Mongolia,major in applied mathematics.Corresponding author:Eburilitu.No.1Ningjin,et al.：Finite Integral Transform Solution of the Bending193equations.Compared with the classical inverse method or semi-inverse method,this methodis not required to select any trail function in advance and has the advantages of good uni-versality.The FIT method has been applied to solve some practical problems in engineeringfields914.Especially,regarding the problems of plates and cylindrical shells,Lis group hasdone some very meaningful work by using the FIT method,for example,the bending prob-lem of cylindrical shell panels has been solved in 10,the buckling problems of thin plateshave been solved in 11,and the bending problems of rectangular plates have been solvedin 12-14.Since the basic equations of moderately thick ORP are more complex than theseof moderately thick isotropic rectangular plate,it becomes more difficult to solve the specificboundary condition problems analytically.As far as we know,moderately thick ORP withfour sides free on Winkler foundation has not been studied by any analytical methods so far.Therefore,based on the boundary conditions and basic equations of the bending of moderate-ly thick ORP on Winkler foundation,the analytical bending solution of this plate is studiedby using the FIT method and its corresponding inverse transform method.Finally,a numberof deflection values of isotropic and orthotropic rectangular plates with different thicknessesare calculated by using the obtained analytical solution and some of the calculated resultsare compared with those in 12,15.And the comparison results show that the obtained FITsolution is correct.2.Basic Equations and the Boundary ConditionsThe basic equations of moderately thick ORP on Winkler foundation16areC112Wx2+C222Wy2 C11xx C22yy+q KW=0,D112xx2+D662xy2+(D12+D66)2yxy+C11(Wx x)=0,D222yy2+D662yx2+(D12+D66)2xxy+C22(Wy y)=0,(2.1)where W is the deflection of the plate;q is the distributed transverse load;xand yare therotations in the xz and yz planes,respectively;D11=h3E112(112),D22=h3E212(112),D66=h3G1212and D12=h32E112(112)are the bending rigidities;C11=hE1(112)and C22=hE2(112)are theshear stiffness;E1and E2are the modulus of elasticity;G12is the shear modulus;1and2are the Poissons ratios and satisfy 1E2=2E1;K,h and k are the Winkler foundationmodulus,plate thickness,and shear correction factor,respectively.The bending moments Mxand My,the twisting moment Mxyand the shear forces Qxand Qycan be expressed asMx=D11xx D12yy,My=D12xx D22yy,Mxy=D66(xy+yx),Qx=C11(Wx x),Qy=C22(Wy y).The boundary conditions of moderately thick ORP with four sides free can be expressedasMx|x=0,a=Mxy|x=0,a=Qx|x=0,a=0;My|y=0,b=Mxy|y=0,b=Qy|y=0,b=0.(2.2)194MATHEMATICA APPLICATA20243.Solving Basic Equations by FITBy the double FIT,W(x,y),x(x,y)and y(x,y)can be expressed asW(m,n)=a0b0W(x,y)cos(mx)cos(ny)dxdy,(3.1)x(m,n)=a0b0 x(x,y)sin(mx)cos(ny)dxdy,(3.2)y(m,n)=a0b0y(x,y)cos(mx)sin(ny)dxdy,(3.3)where m=ma,n=nb,a and b are the length and width of the moderately thick ORP.Then by applying the corresponding inverse transform to Eqs.(3.1)-(3.3),we haveW(x,y)=1abm=0n=0mnW(m,n)cos(mx)cos(ny),(3.4)x(x,y)=2abm=1n=0nx(m,n)sin(mx)cos(ny),(3.5)y(x,y)=2abm=0n=1my(m,n)cos(mx)sin(ny),(3.6)wherem=1 if m=0,2 if m=1,2,n=1 if n=0,2 if n=1,2,.The integral transforms shown in Eqs.(3.1)-(3.3)are applied to the basic equations(2.1),and we obtainC11b0(1)mWx|x=aWx|x=0cos(ny)dy+C22a0(1)nWy|y=bWy|y=0cos(mx)dx C11b0(1)mx|x=a x|x=0cos(ny)dy C22a0(1)ny|y=b y|y=0cos(mx)dx C112mW C222nW C11mx C22ny+q KW=0,(3.7)D11mb0(1)mx|x=a x|x=0cos(ny)dy+D66a0(1)nxy|y=bxy|y=0sin(mx)dx+(D12+D66)a0(1)nyx|y=byx|y=0sin(mx)dx mny D112mx D662nx C11mW C11x=0,(3.8)D22na0(1)ny|y=b y|y=0cos(mx)dx+D66b0(1)myx|x=ayx|x=0sin(ny)dy+(D12+D66)b0(1)mxy|x=axy|x=0sin(ny)dy mnx D222ny D662my C22nW C22y=0.(3.9)From Eqs.(3.7)-(3.9)and boundary conditions(2.2),we have(C112m+C222n+K)W+C11mx+C22ny=q,(3.10)C11mW+(C11+D662n+D112m)x+(D12+D66)mny=(1)nD12ma0y|y=bcos(mx)dx+D12ma0y|y=0cos(mx)dxNo.1Ningjin,et al.：Finite Integral Transform Solution of the Bending195(1)mD11mb0 x|x=acos(ny)dy+D11mb0 x|x=0cos(ny)dy,(3.11)C22nW+(C22+D222n+D662m)y+(D12+D66)mnx=(1)nD22na0y|y=bcos(mx)dx+D22na0y|y=0cos(mx)dx(1)mD12nb0 x|x=acos(ny)dy+D12nb0 x|x=0cos(ny)dy.(3.12)The double FIT of the load q(x,y)is q(m,n)=a0b0q(x,y)cos(mx)cos(ny)dxdy.(3.13)Letb0 x|x=acos(ny)dy=Kn,b0 x|x=0cos(ny)dy=Ln,a0y|y=bcos(mx)dx=Im,a0y|y=0cos(mx)dx=Jm.Then Eqs.(3.10)-(3.12)can be written as:(C112m+C222n+K)W+C11mx+C22ny=q,(3.14)C11mW+(C11+D662n+D112m)x+(D12+D66)mny=(1)nD12mIm+D12mJm(1)mD11mKn+D11mLn,(3.15)C22nW+(C22+D222n+D662m)y+(D12+D66)mnx=(1)nD22nIm+D22nJm(1)mD12nKn+D12nLn.(3.16)According to Eqs.(3.14)-(3.16),W(m,n),x(m,n)andy(m,n)are represented byKn,Ln,Imand Jm,respectively,W(m,n)=P11 q+P12Im+P13Jm+P14Kn+P15Ln,(3.17)x(m,n)=P21 q+P22Im+P23Jm+P24Kn+P25Ln,(3.18)y(m,n)=P31 q+P32Im+P33Jm+P34Kn+P35Ln,(3.19)The coefficients Pij(i=1,2,3;j=1,2,5)are given in Appendix A.Substituting Eqs.(3.18)-(3.19)into Eqs.(3.5)-(3.6),and combining the boundary condi-tions Mx|x=0,a=My|y=0,b=0 and the Stokes transformation17,we obtainm=0mD11P22m+D12P32n+D12(1)nIm+m=0m(D11P23m+D12P33n D12)Jm+m=0mD11P24m+D12P34n+D11(1)mKn+m=0m(D11P25m+D12P35n D11)Ln=m=0m(D11P21m+D12P31n)q,n=0,1,2,(3.20)m=0m(1)mD11P22m+D12P32n+D12(1)nIm+m=0m(1)m(D11P23m+D12P33n D12)Jm+m=0m(1)mD11P24m+D12P34n+D11(1)mKn+m=0m(1)m(D11P25m196MATHEMATICA APPLICATA2024+D12P35n D11)Ln=m=0m(1)m(D11P21m+D12P31n)q,n=0,1,2,(3.21)n=0nD12P22m+D22P32n+D22(1)nIm+n=0n(D12P23m+D22P33n D22)Jm+n=0nD12P24m+D22P34n+D12(1)mKn+n=0n(D12P25m+D22P35n D12)Ln=n=0n(D12P21m+D22P31n)q,m=0,1,2,(3.22)n=0n(1)nD12P22m+D22P32n+D22(1)nIm+n=0n(1)n(D12P23m+D22P33n D22)Jm+n=0n(1)nD12P24m+D22P34n+D12(1)mKn+n=0n(1)n(D12P25m+D22P35n D12)Ln=n=0n(1)n(D12P21m+D22P31n)q,m=0,1,2,.(3.23)Eqs.(3.20)-(3.23)are four infinite equations about the undetermined constants Kn,Ln,Imand Jm(n,m=0,1,2,).Use Mathematica software to calculate Kn,Ln,Imand Jm,andthen substitute them into Eq.(3.17)to getW(m,n).By substituting the obtainedW(m,n)into Eq.(3.4),the analytical bending solution of moderately thick ORP with four sides freeon Winkler foundation can be obtained.It is worth noting that there are infinite numbersof undetermined constants Kn,Ln,Imand Jm(n,m=0,1,2,).However,when solvingpractical problems,we can obtain series expansion solutions with required accuracy by takingsufficiently large n and m.4.ExamplesHere we apply the FIT solution obtained in this paper to calculate the deflection valuesof isotropic and orthotropic square plates with different thicknesses on Winkler foundation.In order to achieve sufficient calculation accuracy,we take n and m to 1500 in the followingtwo examples.Example 4.1Here we calculate some deflection values of the four sides free thin,moderately thick and thick isotropic square plates on Winkler foundation subjected to aconcentrated load P at the center of the plates.The corresponding parameters are taken asK=100D/a4,D11=D22=D,D12=D,D66=(1 )D/2,C11=C22=C and =0.3.The calculated results are compared with those in 15,as shown in Tab.4.1(the numericalresults retain five significant digits).Tab.4.1Deflections W(Pa2/100D)of isotropic square plates with four free edgesh/aReferences(0,0)(0.125a,0.125a)(0.25a,0.25a)(0.375a,0.375a)0.05present0.411140.719921.04301.3726Ref.150.41080.71991.0431.3730.1present0.407630.713701.03971.3848Ref.150.40730.71371.0401.3850.2present0.388470.688541.02661.4323Ref.150.38820.68851.0271.432No.1Ningjin,et al.：Finite Integral Transform Solution of the Bending197Example 4.2Here we calculate some deflection values of the four sides free thin,moderately thick and thick orthotropic square plates on Winkler foundation subjected to aconcentrated load P at the center of the plates.When calculating thin plates,we take thecorresponding parameters as K=100D11/a4,1=0.1,2=0.4,D12=0.4D11,D22=4D11,D66=0.3D11,C11=12D11/h2,C22=48D11/h2and k=5/6;When calculating moderatelythick plates(h/a=0.1)and thick plates(h/a=0.2),the corresponding parameters are takenas K=100D11/a4,E1/E2=10,1=0.25,G12=0.5E2and k=5/6.The numerical resultsobtained from the calculation of the thin plate are compared with the results in 12,as shownin Tab.4.2(the numerical results retain five significant digits).Tab.4.2Deflections of orthotropic square plates with four free edgesh/aReferences(0,a/2)(a/8,a/2)(a/4,a/2)(3a/8,a/2)(a/2,a/2)(100D11W/Pa2)0.01present0.647220.861011.07731.27491.3868Ref.120.64710.86111.0771.2751.386(E2aW/100P)0.1present0.151780.203470.258410.313750.374150.2present0.0186630.0256300.0332050.0414780.059668RemarkFrom the calculation process of the FIT solution,we can know that the FITsolution obtained in this paper can calculate not only the concentrated load,but also theuniform load,linear load,continuous load and local load and other common loads.5.ConclusionThe analytical bending solution of moderately thick ORP with four sides free on Winklerfoundation is derived by the FIT method.Compared with the classical inverse or semi-inversemethod,the FIT method is not required to select any pre-determined trail function,and thesolution process has good generality.It can be seen from specific examples that the FITsolution obtained in this paper can uniformly calculate isotropic and orthotropic rectangularplates with different thicknesses.Furthermore,this FIT method can also be put to use instudying the bending,vibration and buckling of isotropic and orthotropic rectangular plateswith different thicknesses under more boundary conditions and different types of loads.Appendix AThe elements Pij(i=1,2,3;j=1,2,3,4,5)in Eqs.(3.17)-(3.19)are:T=2mC22(D12+D66)2n C11(C22+D662m+D222n)C11C22(D12+D66)(K+C112m+C222n)+C22(C11+D112m+D662n)C11(D12+D66)2mC2222n(K+C112m+C222n)(C22+D662m+D222n);P11=C22T(C11+D112m)(C22+D662m)+2nC22D66 D12(D12+2D66)2m+D22(C11+D112m)+D22D664n;P12=(1)n+1P13;P13=C22TD122mC22(D12+D66)2n C11(C22+D662m+D222n)D222nC22(C11+D112m+D662n)C11(D12+D66)2m;P14=(1)m+1P15;P15=C22TD112mC11(C22+D662m+D222n)C22D662n+D122nC11(C22+D122m+D662m)C22D662n;198MATHEMATICA APPLICATA2024P21=C22mTC11(C22+D662m+D222n)C22(D12+D66)2n;P22=(1)n+1P23;P23=C22mTD12(K+C112m)(C22+D662m)+C22D662m2n D222nD66(K+C112m+C222n)C11C22;P24=(1)m+1P25;P25=C22mTD122nC11C22(D12+D66)(K+C112m+C222n)+D11(K+C112m)(D662m+D222n)+C22(K+C112m+D662m2n+D224n);P31=C22nTC11C22(D12+D66)2m+C22(D112m+D662n),P32=(1)n+1P33;P33=C22nTD22(K+C222n)(D112m+D662n)+C11(K+D114m+2nC22+D662m2n)D122m(D12+D66)(K+C112m+C222n)C11C22;P34=(1)m+1P35;P35=nC22TD12C11(K+2nC22+D662m2n)+D662n(K+C222n)D112mD66(K+C112m+C222n)C11C22.References:1 BUCZKOWSKI R,TORBACKI W.Finite element modelling of thick plates on two-parameter elasticfoundationJ.International Journal for Numerical and Analytical Methods in Geomechanics,2001,25(14):1409-1427.2 HENWOOD D J,WHITEMAN J R,YETTRAM A L.Finite difference solution of a system of first-order partial differential equationsJ.International Journal for Numerical Methods in Engineering,2010,17(9):1385-1395.3 CIVALEKOMER.Three-dimensional vibration,buckling and bending analyses of thick rectangularplates based on discrete singular convolution methodJ.International Journal of Mechanical 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