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Plak denkleminin genel çözümü için bir metod

Plak denkleminin genel çözümü için bir metod

##### Dosyalar

##### Tarih

1992

##### Yazarlar

Kütüğ, Zafer

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Plak Diferansiyel Denklemini çözmek için birçok metod mevcuttur. Daha genel olan Integral metodlannın yanında Fourier serileri ile çözüm pratiğe daha yatkındır ve oldukça hızlıdır. Bugüne kadar çeşitli bilim adamları seriler yardımıyla, çeşitli sınır koşullarına sahip plakların çökme ifadelerini araştırmışlardır. Bunların en ünlüleri Navier ve Levy dir.Her iki bilim adamı da plak diferansiyel denklemini homojen ve özel çözüm olarak ayrı ayrı ele alıp, daha sonra bunları süperpoze etmişlerdir. Bu çalışmada ise, plak genel denkleminin çözümüne homojen ve özel çözüm olarak değil, bir bütün olarak yaklaşılmış ve en genel çözümün elde edilmesine çalışılmıştır. Bunun için Değişkenlerin Ayrılması Metodu ve Fourier Serileri kullanılmıştır. Sonuçta, çeşitli tip yüklemeler için en genel çökme ifadeleri bulunmuştur. Bu çökme ifadeleri çeşitli sınır koşullarına sahip plaklara kolaylıkla uygulanabilir. Buna ait birkaç örnek çalışmanın sonunda verilmiştir.

Plates are straight,plane (flat,noncurved) surface structures whose thickness Is slight compared to their other dimensions.Geometrically, they are bound either by straight or curved lines.Statically,plates have free,simply supported,and fixed boundary conditions,including elastic supports and elastic restraints,or,in some cases,point supports.The static or dynamic loads carried by planes are predominantly perpendicular to the plate surface/The load-carrying action of plates resembles, to a certain extent, that of beams or cables; thus plates can be approximated by a gridwork of beams or by a network of cables, depending on the flexural rigidity of the structures. Such a gridwork analogy for lateral bending of plates, for instance, is merely an approximation of the actual plate behavior, since it arbitrarily breaks the continuity of the structure and usually neglects the torsional stiffness of the original plate, which can significantly contribute to its load-carrying capacity. The two-dimensional structural action of plates results in lighter structures and therefore offers numerous economic advantages.Both have contributed considerably to the wide use of plates in all fields of engineering. Many structures,such as containers, ships, etc., require complete enclosure that can be accomplished easily by plates, without -VII- use of additional covering; consequently, further saving in material and labor Is achieved. Although thin shell structures also offer the above- mentioned advantages to an even greater extent, there are numerous structural members requiring plane surfaces which bar the use of single- or double-curved surface structures.lt is customary to subdivide plates into the following.major categories based on their structural action: 1- Stiff plates.which are thin plates with flexural r1gidity,carry1ng loads two-dimensional ly.mostly by internal (bending and torsional) moments and by transverse shears,generally In a manner similar to beams.in engineering practice, plate is understood to mean stiff plate, unless otherwise specified. 2- Membranes,which are thin plates whithout flexural rigid1ty,carry the lateral loads by axial and central shear forces.This load-carriying action can be approximated by a network of stressed cables since, because of their extreme thinness, their moment resistance is of a negligible order of magnitude. 3- Flexible plates,which represent a combination of stiff plates and membranes and carry external loads by the combined action of internal moments,transverse and central shear forces,and axial forces. Such plates,because of their favorable weight-to-load ratio, are widely used by the aerospace industry 4- Thick plates,whose internal stress condition resembles that of three-dimensional continua. -VIII- All structural theories distinguish sharply between structures having small deflections and those having large deflections.For the former,the law of superposition is generally applicable,while for the latter, the so-called "structural theories of the second order" must be used The plate which will be dealt with in this study is having small thickness with regard to other dimensions,flexib1e,elastic,having small deflection and,the boundaries are straight and perpendicular to each other/The material of the plate is homogeneous and isotropic. The equation of the plate,developed by Lagrange in 181 1, is a non- homogeneous biharmonic linear differantial equation of the forth order: V4w=V2V2w^+2J^+^M^ 3x4 8x23y2 3y4 D where, ?4=72?2=a4_+2_J4_+34_ 3x4 BxV 8y4 There exist lost of solution methods for that equatioaSome of them are finite diffecerences method, finite elements methodjntegral methods, single and double Fourier series etc... First two solution methods are numerical solutions and rather convinient for computers. The integral methods are univerce but for the practical applications, solution with the single and double Fourier series is much more useful than the integral methods.So in this study we will examine "Solution of the Plate Equation" for several p(x,y) loading cases. -IX- Untill today, some scientists have researched the deflection function w(x,y) of the plate,having different boundries conditions,by using the series Navier (1785-1836) derived the correct differential equal i on of rectangular plates with f lexural resistance. For the solution of certain boundary value problems he introduced an "exact" method which transforms the differential equations into algebraic equations. Navier's method is based on the use of the trigonometric series introduced by Fouurler in the same decade. This so-called forced solution of differential equations yields mathematically exact solutions with relative ease,provided that the boundary conditions of the plate are of the Navier type, i.e., simply supportedNavier seperated the solution of the equation of the plate into two parts: Homogeneous solution and particular solution The former is obtained from the homogeneous of the plate 74wh=0 It expresses the deflection shape of the plate due to the loads acting on the plate boundries.The latter, obtaind from the non-homogenous equation of the plate is fallowing: V Wp-- - The particular solution expresses the deflection of the plate due to the loadings, perpendicularly acted on the plate surface. The particular solution is independent from the boundary conditions of the plate.Navier superposed the homogeneous solution and the particular solution to obtain the solution for the whole plate.Navier thought the deflection function as a double odd trigonometric series and took a simple supported rectengular plate so that the series converges rapidly he also had obtained the load function p(x,y) as a double odd trigonometric series by the similar way of the deflection function.And at last, he set up a relationship between coeff iciens of the deflection series Wmn and that of the lood series ^mn for the some m and n. -X- Ifî 1899,Levy developed another method for the exact solution of the plate equation : solution with the single Fourier series. Also Levy, like Navier,divided the solutions into two parts:homogeneous and particular solutions As different from Navier, he took a plate which two opposite sides were too long and simply supported to get the particular solution of the deflection function. Thus he reduced the parameter of the deflection equation of the plate being from two to one so he obtained the particular solution of the deflection function as a single sinus series which depends on the remaining parameter. As Navier did,he set up a relationship between the coefficients of the deflection function series wm and that of the loading series pm for the same m. In our study, however, the diffarential equation of the plate is considered as a whole, not seperated to homogeneous and particular parts.And it is tried to obtain the most general solution for the plate under different loadings. For this, DV4w(x,y)~pz(x,y) non- homogeneous linear biharmonic equation of the fourth order of plate is reduced to two second order non homogenous harmonic equations; ?2 w(x,y)=> $ (x,y) 724> (x,y)= pz (x,y) -XI- in those equations ?$,VÎ and w(x,y) is taken as following multiplication of two functions $(x,y)=X{x)Y(y) w(x,y)=X(x) Y(y) depending on x and y seperately. And substituting those Into the equation V24>(x,y)=XHY+XY"=p(x,y) we obtain the functions X(x) and Y(y). But for the different p(x,y) functions, it can be easily seen that we found different X(x) and Y(y) functions, and of course, different X(x) and Y(y) functions. It Is taken four different double Fourier series for p(x,y) functions as following, CO 4» P(x,y)=ZZ pmnS1n^xSin^y m n P(x,y)-ZZ PmnSin^xCos^y m n P(x,y)-^]T p^Cos^x Cos^y m n <» CO P(x,y)=ZZ PmSin^x m n -XII- In the f jrst chapter It is Indicated the sign rule and to obtain the differential equation of the plate.ln the second chapter It is mentioned about Fourier Analysis. For the some loading, commonly faced in practice, functions are obtained as double series by using Fourier analysis Some of those results are used in further chapters.ln the third chapter our solution method is introduced and appllcated on several types of loading and obtained the most general results for each type. In the fourth chapter our results are applicated and compared with several problems which were solved before in different references.

Plates are straight,plane (flat,noncurved) surface structures whose thickness Is slight compared to their other dimensions.Geometrically, they are bound either by straight or curved lines.Statically,plates have free,simply supported,and fixed boundary conditions,including elastic supports and elastic restraints,or,in some cases,point supports.The static or dynamic loads carried by planes are predominantly perpendicular to the plate surface/The load-carrying action of plates resembles, to a certain extent, that of beams or cables; thus plates can be approximated by a gridwork of beams or by a network of cables, depending on the flexural rigidity of the structures. Such a gridwork analogy for lateral bending of plates, for instance, is merely an approximation of the actual plate behavior, since it arbitrarily breaks the continuity of the structure and usually neglects the torsional stiffness of the original plate, which can significantly contribute to its load-carrying capacity. The two-dimensional structural action of plates results in lighter structures and therefore offers numerous economic advantages.Both have contributed considerably to the wide use of plates in all fields of engineering. Many structures,such as containers, ships, etc., require complete enclosure that can be accomplished easily by plates, without -VII- use of additional covering; consequently, further saving in material and labor Is achieved. Although thin shell structures also offer the above- mentioned advantages to an even greater extent, there are numerous structural members requiring plane surfaces which bar the use of single- or double-curved surface structures.lt is customary to subdivide plates into the following.major categories based on their structural action: 1- Stiff plates.which are thin plates with flexural r1gidity,carry1ng loads two-dimensional ly.mostly by internal (bending and torsional) moments and by transverse shears,generally In a manner similar to beams.in engineering practice, plate is understood to mean stiff plate, unless otherwise specified. 2- Membranes,which are thin plates whithout flexural rigid1ty,carry the lateral loads by axial and central shear forces.This load-carriying action can be approximated by a network of stressed cables since, because of their extreme thinness, their moment resistance is of a negligible order of magnitude. 3- Flexible plates,which represent a combination of stiff plates and membranes and carry external loads by the combined action of internal moments,transverse and central shear forces,and axial forces. Such plates,because of their favorable weight-to-load ratio, are widely used by the aerospace industry 4- Thick plates,whose internal stress condition resembles that of three-dimensional continua. -VIII- All structural theories distinguish sharply between structures having small deflections and those having large deflections.For the former,the law of superposition is generally applicable,while for the latter, the so-called "structural theories of the second order" must be used The plate which will be dealt with in this study is having small thickness with regard to other dimensions,flexib1e,elastic,having small deflection and,the boundaries are straight and perpendicular to each other/The material of the plate is homogeneous and isotropic. The equation of the plate,developed by Lagrange in 181 1, is a non- homogeneous biharmonic linear differantial equation of the forth order: V4w=V2V2w^+2J^+^M^ 3x4 8x23y2 3y4 D where, ?4=72?2=a4_+2_J4_+34_ 3x4 BxV 8y4 There exist lost of solution methods for that equatioaSome of them are finite diffecerences method, finite elements methodjntegral methods, single and double Fourier series etc... First two solution methods are numerical solutions and rather convinient for computers. The integral methods are univerce but for the practical applications, solution with the single and double Fourier series is much more useful than the integral methods.So in this study we will examine "Solution of the Plate Equation" for several p(x,y) loading cases. -IX- Untill today, some scientists have researched the deflection function w(x,y) of the plate,having different boundries conditions,by using the series Navier (1785-1836) derived the correct differential equal i on of rectangular plates with f lexural resistance. For the solution of certain boundary value problems he introduced an "exact" method which transforms the differential equations into algebraic equations. Navier's method is based on the use of the trigonometric series introduced by Fouurler in the same decade. This so-called forced solution of differential equations yields mathematically exact solutions with relative ease,provided that the boundary conditions of the plate are of the Navier type, i.e., simply supportedNavier seperated the solution of the equation of the plate into two parts: Homogeneous solution and particular solution The former is obtained from the homogeneous of the plate 74wh=0 It expresses the deflection shape of the plate due to the loads acting on the plate boundries.The latter, obtaind from the non-homogenous equation of the plate is fallowing: V Wp-- - The particular solution expresses the deflection of the plate due to the loadings, perpendicularly acted on the plate surface. The particular solution is independent from the boundary conditions of the plate.Navier superposed the homogeneous solution and the particular solution to obtain the solution for the whole plate.Navier thought the deflection function as a double odd trigonometric series and took a simple supported rectengular plate so that the series converges rapidly he also had obtained the load function p(x,y) as a double odd trigonometric series by the similar way of the deflection function.And at last, he set up a relationship between coeff iciens of the deflection series Wmn and that of the lood series ^mn for the some m and n. -X- Ifî 1899,Levy developed another method for the exact solution of the plate equation : solution with the single Fourier series. Also Levy, like Navier,divided the solutions into two parts:homogeneous and particular solutions As different from Navier, he took a plate which two opposite sides were too long and simply supported to get the particular solution of the deflection function. Thus he reduced the parameter of the deflection equation of the plate being from two to one so he obtained the particular solution of the deflection function as a single sinus series which depends on the remaining parameter. As Navier did,he set up a relationship between the coefficients of the deflection function series wm and that of the loading series pm for the same m. In our study, however, the diffarential equation of the plate is considered as a whole, not seperated to homogeneous and particular parts.And it is tried to obtain the most general solution for the plate under different loadings. For this, DV4w(x,y)~pz(x,y) non- homogeneous linear biharmonic equation of the fourth order of plate is reduced to two second order non homogenous harmonic equations; ?2 w(x,y)=> $ (x,y) 724> (x,y)= pz (x,y) -XI- in those equations ?$,VÎ and w(x,y) is taken as following multiplication of two functions $(x,y)=X{x)Y(y) w(x,y)=X(x) Y(y) depending on x and y seperately. And substituting those Into the equation V24>(x,y)=XHY+XY"=p(x,y) we obtain the functions X(x) and Y(y). But for the different p(x,y) functions, it can be easily seen that we found different X(x) and Y(y) functions, and of course, different X(x) and Y(y) functions. It Is taken four different double Fourier series for p(x,y) functions as following, CO 4» P(x,y)=ZZ pmnS1n^xSin^y m n P(x,y)-ZZ PmnSin^xCos^y m n P(x,y)-^]T p^Cos^x Cos^y m n <» CO P(x,y)=ZZ PmSin^x m n -XII- In the f jrst chapter It is Indicated the sign rule and to obtain the differential equation of the plate.ln the second chapter It is mentioned about Fourier Analysis. For the some loading, commonly faced in practice, functions are obtained as double series by using Fourier analysis Some of those results are used in further chapters.ln the third chapter our solution method is introduced and appllcated on several types of loading and obtained the most general results for each type. In the fourth chapter our results are applicated and compared with several problems which were solved before in different references.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1992

##### Anahtar kelimeler

Değişkenlerin ayrılması yöntemi,
Diferensiyel denklemler,
Fourier seriler,
Variables separation method,
Differential equations,
Fourier series