Sonlu eleman programlama ile plak eğilme problemlerinin çözülmesi

Abstract

Bu tezde karmaşık ve zor fiziksel problemleri kabul edi lebilir bir yaklaşıklıkla çözebilen bir sayısal yöntem olan "Sonlu elemanlar" yönteminden yararlanarak plak eğilme problemi çözebilen bir bilgisayar programı hazırlanmıştır. Tezde sadece programlamayı anlatmakla kalmayıp teoride anlatılmaya çalışılmıştır. Bu nedenle bölüm 2 ' de sonlu ele manlar yöntemi ve bu yöntemin temel ifadeleri hakkında bilgi verilmiştir. Daha sonraki bölümde yöntem plak eğilme durumu için ele alınıp bağıntılar oluşturulmuştur. Dördüncü bölüm de yayılı yüklerin noktasal eşdeğerini hesaplayan alt program tanıtılmıştır. Eşitlik çözümlerinde kullanılan frontal eşit lik çözümü yönteminden beşinci bölümde bahsedilmiştir. Daha sonraki bölümde sonlu elemanlarda pratik açıdan çok büyük ö- nem taşıyan giriş ve çıkış işlemleri anlatılmış ve konu ile ilgili altprogramlar anlatılmıştır. Yedinci bölümde hata ta nımlamada kullanılan altprogramlardan kısaca söz edilmiştir. Son bölüm ise tüm program yapısı ve bu programın kullanılma sıyla çözülmüş örnek uygulamalara ayrılmıştır.

The finite element method is now firmly established as an engineering tool of wide applicability. No Longer is it regarded as the sole province of the researcher or academic but it is now employed for design purposes xn many branches of technology. One of the principal advantages of the finite element method is the unifying approach it offers to the solution of diverse engineering problems. This's why, the metod has attrached a wide variety of theoreticians and prac titioners from various branches, including engineering mat hematics and computer science. During its early development for stress analysis prob lems the method relied heavily on a physical interpretation in which the structure was assumed to be composed of elements physically connected only at a number of discrete nodal po ints. Later the application of the method to structural mec hanics problems was developed through the use of the prin ciple of virtual work and energy methods. Three main areas of application of finite elemant method can be identified. These are: Equilibrium problems in which the system does not vary with time. Examples of such problems include the stress analysis of linear elastic systems, electrostatics, magneto statics, steady - State thermal conduction and fluid flow in porous media. The structure is divided into distinct non- overlapping regions known as elements which are connected at a discrete number of points along their periphery known as nodal points. Eigenvalue problems are extensions of equilibrium prob lems in which specific or critical values of certain parame ters must be determined. The stability of structures and determination of the natural frequencies of linear elastic systems are examples of such problems. Propagation problems include problems in which some VI time dependent phenomena takes place. Hydrodynamics and the dynamic transient analysis of elastic continua are two examples of such problems. Since the basic aim of this text to take the painful step from theory to program, we limit here our attention to the application of the finite element method to linear equilib rium problems. Furthermore, detailed consi deration will be given only to the finite element displacement method of structural analysis. The basic steps for deriving a finite solution to an equilibrium can be summarises as. - Sub-division of the continua into finite elements - Eva luation of elements stiffness and load terms - Assembly of element stiffness and load terms into an overall stiffness matrix and load vector. - Solution of the resulting linear simul taneous equa tions for the unknown nodal forces. - Evaluation of subsidiary element quantities such as stresses in the displacement method. The layout of this text is such that as soon as a sec tion of theory is completed, the programming concerned with it is under taken. In this way a series of modular subrou tines are initially presented which are later assembled to form the complete program. The application considered in this text is plate bending problems. Because the classical problems of plate flexure were amongst the first subjects for the application of the finite element method. Primary attention has been given to solving problems based on the so called 'thin plate' theory in which shear deformations are neglected. When shear deformation is of importance - as it is in thick plates, cellular plates and plates of sand wich con struction - the solution to the problem becomes significan tly more complex and much effort has recently been expended in finding an adequate approach. In this text, an approach is presented which not only permits all types of plates to be occommodated in the solution scheme but also avoids the continuity requirements which have made the solution of thin plates so difficult. In the formulation of the plate element we use the assumptions adopted by midline. - The deflections of the plate (w) are small Vll - Normal to the midsurface before deformation remain straight but not necessarily normal to the midsurface after deformation. - Stresses normal to the midsurface are negligible irrespective of the loading. For this application an 8-noded parabolic isoparamet ric elements is used. Parabolic isoparametric elements are extre mely versatile, good performers and well-tested The construction of a finite element program employing the displacement approach naturally falls into three phases Phase 1. Semi-theoretical aspects such as input and output. The program subroutine controlling the input is described in detail in chapter 6, this data input subroutine is named INPUT. A Seperate subroutine is n' t employed to output the results. Instead the results are output as soonas they are obtained. The displacement are output in the equation solu tion subroutine FRONT and the stress components are output- front the stress evalution subroutine called STREPB. Phase 2. Stiffness and stresses matrices and applied load vector generation. The stiffness and stress matrices for plate bending condition are calculated at this stage. The subroutines which performs these task are called STIFPB and STREPB. The displacement method of finite element anal ysis relies on all structural loading being interpreted as equivalent nodal forces. The subroutine which occomplish this as well ac accepting the loading data for plate element is LOADPB described in chapter 4. Phase 3. Solution of the stiffness equations. Subroutine FRONT mentioned in chapter 5 is the equation solution subroutine whose function is to assemble the element stiffness equations and reactions using the frontal elimina tion technique. In addition to these primary subroutines mentioned above three phases the auxi liary subroutines carry out the computation required by oneormore of the primary subroutines These auxiliary subroutines are described in text clearly. vin In this text all programs are written in FORTRAN. In the programs, variable names are prepared logically occor- ding to English language. All variables are chosen to be 5 characters in lenght and are abbreviations of long names. Such as NDOFN The Number of Degrees Of Freedom Per Node NNODE The Number Of NODes per Element PROPS The array Of Material PRORerties 5 As you will see, in this text not only the finite elem- ment program but also the finite element theory which will help any user to understand structure of program is presen ted. Therefor m chapter 2 the basic expressions of the finite element method for structural applications are presen ted. Chapter 3 present the basic expressions and subroutines for plate berding applications. In particular all the subrou tines necessary for the construction of the element stiffness matrices and for the evaluation of the stresses from the known displacements are developed. In the element method all loading must be applied to the structure as equivalent nodal loads. Subroutine is necessary to compute the nodal force equivelents and chapter 4 is devo ted to the development of such subroutines. The solutions of equation systems by the frontal method is dealt with in chapters and a sophisticated subroutine is developed which can be employed as a general purpose finite element solver. The frontal equation solutiontechnique is described in detail and its advantages outlined. Alarge proportion of time spent in any finite element analysis is that required for checking the input data. So, In chapter 6 and 7, The subroutines are developed for these purposes to check data. The subroutines developed in the chapters are assembled in chapter 8 to form complete program which can be employed for plate bending situations. Numerical examples for each application are also presented demonstrating the efficiency of the parabolic isoparametric element. IX The basic aim of this text is that these programs should help the reader to take the painful step from theory to prog ram, thus enabling him to develop (or at least apprecieve) programs for his own particular applications in his own envi ronment.