##
Sonlu eleman programlama ile kiriş problemlerinin çözümü

Sonlu eleman programlama ile kiriş problemlerinin çözümü

##### Dosyalar

##### Tarih

1991

##### Yazarlar

Esi, Ufuk

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu yüksek lisans tezinde karmaşık ve zor fiziksel problemleri kabul edilebilir bir yaklaşıklıkla çözebilen bir sayısal yöntem olan "Sonlu Eleman Metodu" ndan yarar lanarak kiriş problemlerini çözen bir bilgisayar programı geliştirilmiştir. Tezde, teoriden programlamaya yumuşak bir geçiş ya pılmaya çalışılmıştır. Bu amaçla (Bölüm 2) 'de taşıyıcı sistemler, kiriş türleri ve teorisi hakkında genel bilgi ler verilmiştir. (Bölüm 3) 'de sonlu elemanlar metodu, metodun teorisi, sonlu eleman çeşitleri ve program yapı sından bahsedilmiştir. (Bölüm k) ' de sonlu elemanlarda pratik açıdan önem taşıyan giriş ve çıkış işlemlerine de ğinilmiş ve konu ile ilgili altprogramlar anlatılmıştır. (Bölüm 5) 'de ise, yöntem, izoparametrik kiriş elemanı için ele alınıp gerekli bağıntılar oluşturulmuş ve ilgili alt programlar tanıtılmıştır. Oluşturulan denklemlerin çözü müne ise (Bölüm 6)'da değinilmiş ve kullanılan cephesel çözüm yöntemi detaylı olarak tarif edilmiştir. Giriş bil gilerinin kontrolü ve hata teşhisleri (Bölüm 7) 'de anla tılmış ve ilgili programlar açıklanmıştır. Son bölüm ise, tüm altprogramların birleştirilmesine ve bilgisayar prog ramının kullanımının daha iyi kavranabilmesi amacıyla ye terli sayıda örnek uygulama için 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 in many branches of technology. One of the principal advan tages of the finite element method is the unifying app roach it offers to the solution of diverse engineering problems, so the method has attracted a uide variety of theoreticians and practitioners from various disciplines, including engineering, mathematics and computer science. During its early development for stress analysis problems the method relied heavily on a physical interp retation in which the structure was assumed to be composed of elements physically connected only at a number of discrete nodal points. Later the application of the method to structural mechanics problems was developed through the use of the principle of virtual work and energy methods. The method was then generalised and its wider mathematical roots were recognised; it was shown that finite elements could be applied to any mathematical problem for which a variational functional existed. More recently, finite element solutions have been developed which are based on the well known, classical techniques known as "weighted residual methods". In fact the method is now widely recognised a general numerical technique, for the solution of partial differential equation systems subject to appropriate boundary and initial conditions. In engineering, physics and applied mathematics three main areas of application of the finite element 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 XXll flow in porous media. The structure is first divided into distinct non-overlapping regions known as elements over which the main variables are interpolated. These elements are connected at a discrete number of points along their periphery knoun as nodal points. Eigenvalue problems are extensions of equilibrium problems in uihich specific or critical values of certain parameters must be determined. The stability of struc tures and the determination of the natural frequencies of linear elastic systems are examples of such problems. Propagation problems include problems in which some time-dependent phenomena takes place. Hydrodynamics and the dynamic transient analysis of elastic continua are two examples Df such problems. The success of the finite element method as a prac tical design aid depends on the availability of an ef ficient means of solving the resulting system of linear or non-linear simultaneous equations. Clearly the exis tence of the computer is vital tD the success of this. Increases in core storage capacities to the present level have allowed a wide variety of problems to be comfortably processed without the need for sophisticated data handling techniques. However the parallel development of auxiliary hardware, such as direct access discs to replace magnetic tape systems, has permitted the solution of a new magnitude of problem. This continuing trend is also revolutionising the position of non-linear appli cations. In the past, the economic limitations imposed by computer costs have restricted the general use of such techniques. However this barrier is being rapidly removed and the finite element solutions of such problems is already economically acceptable for selected indust rial applications. Such developments along with future enhancements in the element characteristics, equation solution techniques etc., suggest that the finite element method will play a major role in engineering design for many years to come. In this thesis we limit our attention to the appli cation of the finite element method to linear equilibrium XXlll problems of beams: Furthermore, detailed consideration will be given only to the finite element displacement method of structural analysis. The basic steps for deriving a finite element solution to an equilibrium problem can be summarised as, * Sub-division of the continuum into finite elements. * Evaluation of element stiffness and load terms. * Assembly of element stiffness and load terms, into an overall stiffness matrix and load vector. * Solution of the resulting linear simultaneous equations of subsidiary element quantities such as stresses in the displacement method. The layout of this thesis is such that as soon as a section of theory is completed, the programming con cerned with it is under-taken. In this way a series cf modular subroutines are initially presented which are later assembled to form the complete program. The application considered in this thesis is beam analysis. For the application a 3-noded parabolic isoparametric element is employed. Unlike the more usual types of beam' element, the isoparametric beam element can take account of transverse shear deformation since energy due to shear as well as bending is considered in the formulation. The element is quite versatile and can be used to analyse not only thin beams with negligible shear deformation but also thick beams and beams of sandwich construction in which shear effects are important. The main assumption we make is concerned with the cross-sectional behaviour of the beam. Usually, in beam theory we assume that normals to the neutral axis before deformation remain straight and normal to the neutral axis after deformation. In Chapter 2 some general informations about the carrier systems, the kinds of beams and beam theory are given. xxxv As yau mill see, in this thEsiB not only the finite element program but also ths finite element theory which will help any user to understand the structure of ths program is presented. Therefore in chapter 3 introduc tion and the basic expressions of the finite element method for structural applications, the kinds of finite elements and the structure of the program are presented. Chapter k deals with the problems associated with input and output. The input data. required for finite element analysis with isoparametric elements is discussed and subroutines for data assimilation are presented. Chapter 5 is devoted entirely to the development of specific expressions and subroutines for the isopara metric beam element it is at this stage that problems specifically associated with the isoparametric element concept are first encountered and the beam element serves as a convenient introductory vehicle in view of its relative simplicity. In particular the jacobian matrix, which enables transformation between local and global quantities to be made, is introduced and numerical inte gration techniques essential to isoparametric elements are discussed. The subroutines performing the standard steps, such as shape function and stiffness formulation, equivalent nodal force generation and stress resultant evaluation, are developed at this stage. These subrou tines are essentially of the same form as for more sop histicated applications and Chapter 5 therefore allows the reader to familiarise himself with the general struc ture of isoparametric element programs. The solution of equation systems by the frontal method is dealt with in Chapter 6 and a sophisticated subroutine is developed which can be employed as.a general purpose finite element solver. The frontal equation solution technique is described in detail and its advantages outlined. In Chapter 7 three subroutines are presented for input data checking. The data is checked in stages and if any errors are detected, appropriate diagnostic mes sages are printed and the remainder of the input data is echoed by lineprinter. The subroutines developed in Chapters h to 7 are assembled in Chapter B to form complete program which XXV can be employed fur beam analysis. Numerical examples for beam problems are also presented demonstrating the efficiency of the parabolic isoparametric element. The construction cf a finite element programs employing the displacement approach falls naturally into three phases. Phase 1. Semi-theoretical aspects such as input and output. Whilst being possibly the least technologically demanding aspect of a finite element analysis, from a practical engineering viewpoint, input/output is arguably the most important. The program subroutine controlling the input is considered in detail in Chapter k. The data input subroutine is named INPUT.. A separate subroutine is not employed to output the results. Instead the results are output as soon as they are obtained. The displacements are output in the equation solution subroutine FRONT and the stress com ponents are Dutput from the stress evaluation subroutine which is titled STREB. Phase 2. Stiffness and stress matrices and applied load vector generation. It is at this stage that the main use is made of the basic expressions of finite element theory. For structural analysis by the displacement approach, the process clearly follows the steps taken in the matrix methods of structural frame analysis. The stiffness and stress matrices for the beam are calculated in chapter 5. The subroutine which performs this task is named STIFB. After subsequent solution for the nodal displacements, the element stress matrices are employed in the evaluation of the stress components or stress resultants. This task is performed by subrou tine STREB. The displacement method of finite element analysis relies on all structural loading being interpreted as xxvi Equivalent nodal forces as described in sections 3.5.4. and 3,5,5. For example, pressures applied to element faces must be converted to equivalent discrete nodal forces. The subroutine which accomplishes this, as well as accepting the loading data is LOADS, described in detail in Chapter 5. Phase 3. Solution of the stiffness equations, The time spent on the solution of the stiffness equations of the structure represents a large percentage of the total computation time. Therefore the method of solving these equations is critical to an efficient solution. Whilst there are several mays in which the operations outlined in phaseB 1 and 2 can be performed, the optimal approach will only produce marginal savings in both computer core storage and solution costs. However, the equation solution scheme adopted can effect both factors considerably. In the preparation of a thesis such as this a dilemma immediately arises, since the simplest algorithms are generally inefficient with regard to either storage requirements or computational effort or both. In an attempt to provide programs which are of benefit to the elemantary student and also to interest the more sophisticated user, a relatively elemantary version of a frontal process equation solver is presented. In its present form the program is suitable for teaching or research purposes. Subroutine FRONT, described in Chapter 6, is the equation solution subroutine whose function is to assemble. the element stiffness equations and solve for the unknown displa cements and reactions using the frontal elimination technique. The function of the auxiliary subroutines is to carry out computations required by one or more of the primary subroutines and the task of each one is described in the thesis clearly. In this thesis all programs are written in FORTRAN In the programs, variable names are prepared logically according to English Language and all variables are chosen to be 5 characters in length. The basic aim of this thesis is that these programs should help the reader to take the painful step from theory to program, thus enabling him to develop programs for his own particular applications in his own environment. XXV11 BÖLÜM 1. GİRİŞ Sonlu elemenlar yöntemi, mühendislik, matematik ve bilgisayar bilimini içeren çeşitli bilim dallarından geniş bir kuramcı ve pratisyen topluluğunu cezbetmiştir. Halen piyasada bulunan sonlu eleman ders kitapları, esa sen tekniğin pratik doğasını izah etmek için sunulan uygulamalarla birlikte metodun temel teorik görünüşü ü- zerinde durur. Bununla beraber, bir sonlu eleman prog ramı geliştirmeyi denemiş herhangi birinin delil olacağı gibi, temel teori ile, çalışan bir bilgisayar şifresi arasında büyük bir uçurum vardır. Şu gerçektir ki, bazı sonlu eleman ders kitapları, hemen hemen sonradan akla gelen bir fikir olarak, genellikle bir ek şeklinde içe- rilen bir sonlu eleman bilgisayar programını sunmuşlar dır. Hiç bir ders kitabının tekniğin evvelâ programla ma görünüşlerine tahsis edilmiş olduğu görülmemektedir. Sonlu elemanlar yönteminden yola çıkılarak kiriş problemlerinin çözümü için geliştirilen bir sonlu eleman programının tarif edildiği bu yüksek lisans tezi zihin deki bu düşüncelere uygun düşmektedir. Bu programın, teoriden programa geçişteki meşakkatli adımı almasında okuyucuya yardımcı olması, böylece okuyucunun çevresin deki kendi özel uygulamaları için programlar geliştirme sini (veya en azından değerlendirmesini) mümkün kılması temel amaçtır. Programın geliştirilmesinde, aksan; sa delik, anlama kolaylığı ve uygulanabilme üzerine kurul muştur.

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 in many branches of technology. One of the principal advan tages of the finite element method is the unifying app roach it offers to the solution of diverse engineering problems, so the method has attracted a uide variety of theoreticians and practitioners from various disciplines, including engineering, mathematics and computer science. During its early development for stress analysis problems the method relied heavily on a physical interp retation in which the structure was assumed to be composed of elements physically connected only at a number of discrete nodal points. Later the application of the method to structural mechanics problems was developed through the use of the principle of virtual work and energy methods. The method was then generalised and its wider mathematical roots were recognised; it was shown that finite elements could be applied to any mathematical problem for which a variational functional existed. More recently, finite element solutions have been developed which are based on the well known, classical techniques known as "weighted residual methods". In fact the method is now widely recognised a general numerical technique, for the solution of partial differential equation systems subject to appropriate boundary and initial conditions. In engineering, physics and applied mathematics three main areas of application of the finite element 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 XXll flow in porous media. The structure is first divided into distinct non-overlapping regions known as elements over which the main variables are interpolated. These elements are connected at a discrete number of points along their periphery knoun as nodal points. Eigenvalue problems are extensions of equilibrium problems in uihich specific or critical values of certain parameters must be determined. The stability of struc tures and the determination of the natural frequencies of linear elastic systems are examples of such problems. Propagation problems include problems in which some time-dependent phenomena takes place. Hydrodynamics and the dynamic transient analysis of elastic continua are two examples Df such problems. The success of the finite element method as a prac tical design aid depends on the availability of an ef ficient means of solving the resulting system of linear or non-linear simultaneous equations. Clearly the exis tence of the computer is vital tD the success of this. Increases in core storage capacities to the present level have allowed a wide variety of problems to be comfortably processed without the need for sophisticated data handling techniques. However the parallel development of auxiliary hardware, such as direct access discs to replace magnetic tape systems, has permitted the solution of a new magnitude of problem. This continuing trend is also revolutionising the position of non-linear appli cations. In the past, the economic limitations imposed by computer costs have restricted the general use of such techniques. However this barrier is being rapidly removed and the finite element solutions of such problems is already economically acceptable for selected indust rial applications. Such developments along with future enhancements in the element characteristics, equation solution techniques etc., suggest that the finite element method will play a major role in engineering design for many years to come. In this thesis we limit our attention to the appli cation of the finite element method to linear equilibrium XXlll problems of beams: Furthermore, detailed consideration will be given only to the finite element displacement method of structural analysis. The basic steps for deriving a finite element solution to an equilibrium problem can be summarised as, * Sub-division of the continuum into finite elements. * Evaluation of element stiffness and load terms. * Assembly of element stiffness and load terms, into an overall stiffness matrix and load vector. * Solution of the resulting linear simultaneous equations of subsidiary element quantities such as stresses in the displacement method. The layout of this thesis is such that as soon as a section of theory is completed, the programming con cerned with it is under-taken. In this way a series cf modular subroutines are initially presented which are later assembled to form the complete program. The application considered in this thesis is beam analysis. For the application a 3-noded parabolic isoparametric element is employed. Unlike the more usual types of beam' element, the isoparametric beam element can take account of transverse shear deformation since energy due to shear as well as bending is considered in the formulation. The element is quite versatile and can be used to analyse not only thin beams with negligible shear deformation but also thick beams and beams of sandwich construction in which shear effects are important. The main assumption we make is concerned with the cross-sectional behaviour of the beam. Usually, in beam theory we assume that normals to the neutral axis before deformation remain straight and normal to the neutral axis after deformation. In Chapter 2 some general informations about the carrier systems, the kinds of beams and beam theory are given. xxxv As yau mill see, in this thEsiB not only the finite element program but also ths finite element theory which will help any user to understand the structure of ths program is presented. Therefore in chapter 3 introduc tion and the basic expressions of the finite element method for structural applications, the kinds of finite elements and the structure of the program are presented. Chapter k deals with the problems associated with input and output. The input data. required for finite element analysis with isoparametric elements is discussed and subroutines for data assimilation are presented. Chapter 5 is devoted entirely to the development of specific expressions and subroutines for the isopara metric beam element it is at this stage that problems specifically associated with the isoparametric element concept are first encountered and the beam element serves as a convenient introductory vehicle in view of its relative simplicity. In particular the jacobian matrix, which enables transformation between local and global quantities to be made, is introduced and numerical inte gration techniques essential to isoparametric elements are discussed. The subroutines performing the standard steps, such as shape function and stiffness formulation, equivalent nodal force generation and stress resultant evaluation, are developed at this stage. These subrou tines are essentially of the same form as for more sop histicated applications and Chapter 5 therefore allows the reader to familiarise himself with the general struc ture of isoparametric element programs. The solution of equation systems by the frontal method is dealt with in Chapter 6 and a sophisticated subroutine is developed which can be employed as.a general purpose finite element solver. The frontal equation solution technique is described in detail and its advantages outlined. In Chapter 7 three subroutines are presented for input data checking. The data is checked in stages and if any errors are detected, appropriate diagnostic mes sages are printed and the remainder of the input data is echoed by lineprinter. The subroutines developed in Chapters h to 7 are assembled in Chapter B to form complete program which XXV can be employed fur beam analysis. Numerical examples for beam problems are also presented demonstrating the efficiency of the parabolic isoparametric element. The construction cf a finite element programs employing the displacement approach falls naturally into three phases. Phase 1. Semi-theoretical aspects such as input and output. Whilst being possibly the least technologically demanding aspect of a finite element analysis, from a practical engineering viewpoint, input/output is arguably the most important. The program subroutine controlling the input is considered in detail in Chapter k. The data input subroutine is named INPUT.. A separate subroutine is not employed to output the results. Instead the results are output as soon as they are obtained. The displacements are output in the equation solution subroutine FRONT and the stress com ponents are Dutput from the stress evaluation subroutine which is titled STREB. Phase 2. Stiffness and stress matrices and applied load vector generation. It is at this stage that the main use is made of the basic expressions of finite element theory. For structural analysis by the displacement approach, the process clearly follows the steps taken in the matrix methods of structural frame analysis. The stiffness and stress matrices for the beam are calculated in chapter 5. The subroutine which performs this task is named STIFB. After subsequent solution for the nodal displacements, the element stress matrices are employed in the evaluation of the stress components or stress resultants. This task is performed by subrou tine STREB. The displacement method of finite element analysis relies on all structural loading being interpreted as xxvi Equivalent nodal forces as described in sections 3.5.4. and 3,5,5. For example, pressures applied to element faces must be converted to equivalent discrete nodal forces. The subroutine which accomplishes this, as well as accepting the loading data is LOADS, described in detail in Chapter 5. Phase 3. Solution of the stiffness equations, The time spent on the solution of the stiffness equations of the structure represents a large percentage of the total computation time. Therefore the method of solving these equations is critical to an efficient solution. Whilst there are several mays in which the operations outlined in phaseB 1 and 2 can be performed, the optimal approach will only produce marginal savings in both computer core storage and solution costs. However, the equation solution scheme adopted can effect both factors considerably. In the preparation of a thesis such as this a dilemma immediately arises, since the simplest algorithms are generally inefficient with regard to either storage requirements or computational effort or both. In an attempt to provide programs which are of benefit to the elemantary student and also to interest the more sophisticated user, a relatively elemantary version of a frontal process equation solver is presented. In its present form the program is suitable for teaching or research purposes. Subroutine FRONT, described in Chapter 6, is the equation solution subroutine whose function is to assemble. the element stiffness equations and solve for the unknown displa cements and reactions using the frontal elimination technique. The function of the auxiliary subroutines is to carry out computations required by one or more of the primary subroutines and the task of each one is described in the thesis clearly. In this thesis all programs are written in FORTRAN In the programs, variable names are prepared logically according to English Language and all variables are chosen to be 5 characters in length. The basic aim of this thesis is that these programs should help the reader to take the painful step from theory to program, thus enabling him to develop programs for his own particular applications in his own environment. XXV11 BÖLÜM 1. GİRİŞ Sonlu elemenlar yöntemi, mühendislik, matematik ve bilgisayar bilimini içeren çeşitli bilim dallarından geniş bir kuramcı ve pratisyen topluluğunu cezbetmiştir. Halen piyasada bulunan sonlu eleman ders kitapları, esa sen tekniğin pratik doğasını izah etmek için sunulan uygulamalarla birlikte metodun temel teorik görünüşü ü- zerinde durur. Bununla beraber, bir sonlu eleman prog ramı geliştirmeyi denemiş herhangi birinin delil olacağı gibi, temel teori ile, çalışan bir bilgisayar şifresi arasında büyük bir uçurum vardır. Şu gerçektir ki, bazı sonlu eleman ders kitapları, hemen hemen sonradan akla gelen bir fikir olarak, genellikle bir ek şeklinde içe- rilen bir sonlu eleman bilgisayar programını sunmuşlar dır. Hiç bir ders kitabının tekniğin evvelâ programla ma görünüşlerine tahsis edilmiş olduğu görülmemektedir. Sonlu elemanlar yönteminden yola çıkılarak kiriş problemlerinin çözümü için geliştirilen bir sonlu eleman programının tarif edildiği bu yüksek lisans tezi zihin deki bu düşüncelere uygun düşmektedir. Bu programın, teoriden programa geçişteki meşakkatli adımı almasında okuyucuya yardımcı olması, böylece okuyucunun çevresin deki kendi özel uygulamaları için programlar geliştirme sini (veya en azından değerlendirmesini) mümkün kılması temel amaçtır. Programın geliştirilmesinde, aksan; sa delik, anlama kolaylığı ve uygulanabilme üzerine kurul muştur.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1991

##### Anahtar kelimeler

Bilgisayar programları,
Kirişler,
Sonlu elemanlar yöntemi,
Finite element method,
Taşıyıcı sistemler,
Computer programs,
Beams,
Supporting systems