Eğri Eksenli Düzlemsel Kirişlerin Düzlem Dışı Statik Problemlerinin Analitik Çözümü

Abstract

Bu çalışmada, eğri eksenli düzlemsel kirişlerin düzlem dışı statik problemleri, kayma deformasyonu ve eksen uzaması etkileri gözönüne alınarak incelenmiştir. Mevcut çalışmaların çoğu, bu etkileri ihmal etmiş ve yaklaşık yöntemler kullanmıştır. Burada kiriş statik problemini ifade eden diferansiyel denklem takımlarının kesin çözümleri, farklı eksen eğrilerine haiz dairesel kesitli kirişler için verilmiştir. Diferansiyel denklem takımları bünyesinde yer alan integraller Mathematica programı yardımıyla çözülmüştür. Birinci bölümde, kiriş teorisi hakkında kısa bilgi verilmiş ve çalışmanın amacı belirtilmiştir. İkinci bölümde, eğri eksenli kirişlerin genel denklemleri verilmiştir. Statik problemi ifade eden genel denklemler, kayma deformasyonu ve eksen uzaması etkileri ihmal edilmeden verilmiştir. Üçüncü bölümde, düzlemsel eğri eksenli dairesel kesitli bir kirişin, düzlem dışı statiğini ifade eden genel denklemlerin, başlangıç değerleri yöntemi ile çözümüne yer verilmiştir. Ayrıca, bu bölümde konu ile ilgili üç farklı örneğin; - Çember eksenli dairesel sabit kesitli kiriş - Parabol eksenli dairesel değişken kesitli kiriş - Parabol eksenli dairesel sabit kesitli kiriş genel çözümü yer almaktadır. Her bir örneğe ilişkin genel asal matris elde edilmiş olup, denklemlerde yer alan integrallerin çözümü Mathematica programı yardımıyla yapılmıştır. Dördüncü bölümde; Ankastre-Serbest Mesnetli ve Serbest Ucundan Etkiyen F Tekil Yükünü Taşıyan, Dairesel Sabit Kesitli Çember Eksenli Kiriş ve İki Ucundan Ankastre Mesnetli, Orta Noktasından Etkiyen F Tekil Yükünü Taşıyan, Dikdörtgen Sabit Kesitli Kiriş olmak üzere iki farklı örneğin Mapple programı yardımıyla yapılan çözümü yer almaktadır. Ayrıca birinci örnekte, v, Qu, Clt, Mn, Mt ve Rb' nin, çubuk eksen eğrisi üzerinde belirlenen, çeşitli p açılarındaki yedi noktada değerleri hesaplanarak grafik üzerinde gösterilmiştir.

One of the structural elements most widely used in various engineering applications is the beams. Beams can be different forms like straight or curved. And also they can be variable cross-section. Curved beams have deep or shallow curvature and can be open or closed. Open beams are often called arches. In this work the stress resultants and displacements (due to out-of-plane loading) of planar arches with arbitrary boundary conditions are calculated by using initial values method. The method of initial values gives the values of the displacements and stress resultants throughout the rod once the initial displacements and initial stress resultants are known. Analytical solutions of static problems of planar arches with arbitrary boundary conditions of out-of-plane are given by considering axial and shear deformation effects. Static analysis of naturaly curved beams have many important applications in mechanical, aeronautical and civil engineering. Helicopter blades, bridges and flexible space structures are specific cases that have received considerable attention in recent years. There are structural forms, for instance in arch bridge and shell roof construction, in which curved members are incorporated and although curved beams have been treated in various ways by several authors, the presence of curvature still appears to be considered complex. They occur frequently in highway construction, particularly in urban freeways where multilevel interchanges and elevated roundabouts are designed and built withim tight geometric restrictions. There exist no elementary method in literature which provides analytical ready-to - use expressions of curved beams of out-of-plane. But we reed exact mathematical equations about curved elements. For this reasons, in many engineering applications which is about curved elements relate to out-of-plane problems is used approximate calculations. Most engineers are familiar with the displacement assumptions that allow the displacements at anypoint in a beam to be expressed in terms of the displacements along the centroidal and shear center axes. These assumptions allow a IX variety of different sets of elastic beam equations to be developed. The degree of complexity of these equations depends on the shape of the cross-section, type of loading and the specific phenomenon. The governing equations for the planar curved beam element in out-of-plane by formulating the equations by considering the initial values method. The formulation and solution of equations of planar curved beams relate to out-of- plane are fundamental to structural engineering. Most studies on the formulation of curved beam elements relate to in plane bending and describe mainly the field consistency requirements for membrane strains and shear strains. The purpose of this study to solve analytically the static problems of curved beam elements relate to out-of-plane by considering axial and shear deformation effects. In the literature, most of the studies uses the governing differential equations by neglecting these effects and solves them by numerical methods. Finite element method is the most common one. In the present study, the governing differential equations are simplified for the curved beams relate to out-of-plane. The exact solutions of the simplified equations are found for the boundary value problem with arbitrary boundary conditions. In the first chapter, a general view of curved beams and the aim of the present study are given. There are many investigations about curved beams in the literature. Some of them are rewieved in here. However, the beam theory is considered by several authors studiying on the strenght of materials, elasticity and structural analysis. But the theory is simplified for arches in which different form in plane. Most of the studies uses the governing differential equations given by Love [1], and neglects axial and shear deformations. These simplified governing differential equations are solved by using energy methods or numerical methods like finite element. The most important study on the theory of elasticity of curved beams belongs to Love [1]. In this study, the beam is represented by a space curve whose every point is coupled with a rigid orthonormal vector chad. Nagoranarayana and Prathap [2], have studied the extension of some concepts to the consistency requirements called for when the quadratic curved beam element is designed to be applied in situations where it must undergo out-of-plane bending and torsion under the action of shear forces. In this stud/there has three nodes, allowing a quadratic isoparametric representation and includes shear deformation according to the Timeshenko beam theory. Although the exactly integrated form of this element does not have severe locking problems, it does have a loss of accuracy and spurious force and moment oscillations if the transverse shear strain field is not modelled in a consistent fashion. Field-inconsistent representations of the out-of-plane transverse shear strain will result in a loss of efficiency and introduce spurious oscillations in the bending moment, torsional moment and shear force. The optimal field-consistent assumed strain interpolation for shear is derived. The issues involved in the formulation of this quadratic curved beam element are critically examined to show that two consistency conditions must be assured in describing the constrained out-of- plane transverse shear strain field and the term describing the curved centroidal axis appearing in the denominator of all the strain terms. Although the exactly integrated form of this element does not have severe locking problems, it does have a loss of accuracy and spurious force and moment occillations if the transverse shear strain field is not modelled in a consistent fashion. Field- inconsistent representations of the out-of-plane transverse shear strain will result in a loss of efficiency and introduce spurious occillations in the bending moment, torsional moment and shear force. Tüfekçi [3], has give the analytical solutions of static and dynamic problems of planar curved beams in plane. In that work, the stress resultants and displacements (due to in plane loading) of planar arches with arbitrary boundary conditions are calculated by using initial values method. Principal matrix and its inverse has been obtained for parabolic arches as well. İnan [4], has considered the beam as a space curve whose every point is coupled with a rigid orthonormal vector diad and the boundary value problem has been established in this curve. In Sundaramoorthy's [5] study, the thin-walled curved beam equations are formulated using the principle of virtual work. The formulation and solution of equations of thin- walled curved beams of open section are given related to in plane. The purpose of that study is to derive the consistent governing equations based on the large displacement theory for solving curved beam problems. This theory is well- established from a set of simple and realistic assumptions which are normally used in thin-walled straight beam analysis and based on the variational principles. Artan [6], have used the initial values method to obtain the stress resultants and displacements (due to in plane loading) of circular rods with variable cross-section. In this study, the principal matrix and its inverse required having for the applicability of the method of initial values are obtained. The expression of the solution of a circular rod loaded by both continuous and singular loads is given in a compact form. Irrespective of the number of singular loads and the degree of redundancy the system of algebraic equations to be solved contains no more than three variables as shown. Two examples exhibiting the advantages and priorities of the method are solved in full detail. And also the program of Mathematica is used throughout. Pantazopoulou [8], have used small deformation theory to derive polynomial interpolation functions for finite element analysis of three dimensional (3-D) curved beams. The performance of low-order polynomials combined with selectively reduced integration is evaluated under torsional and membrane 'locking" conditions. Low- order polynomials are also used in a three-field mixed formulation; in this approach, XI the constraint equations of the problems that result from the nonlinear geometry of the curved beam are enforced using collocation. The partial decoupling that stems from the assumption that one principal axis of the cross section lies in the plane of geometric curvature of the member was adopted. Based on this premise, sets of coupled interpolation functions were derived for FE formulation of two or three noded, three-dimensional curved beams; in addition to the variational form of the equilibrium equations, the proposed functions also satisfy the constraint equations of the problem at a number of collocation points, which are also referred to as discrete constraints. Results obtained using these functions are compared against analytical solutions and other alternative FE formulations that use low-order polynomials in combination with selectively reduced integration. The typical element considered is a circular arc; curved beams with a more complex geometry can be approximated with a number of such elements. Most Finite element models dealing with curved members analyze either the out-of- plane or the in-plane behaviour. Analysis of curved structures was typically carried out by idealizing the curved member as a sequence of short straight segments in the past. This approach has been attractive among bridge engineers, because it uses three dimensional beam frame elements which are commonly available in most structural analysis programs. This approach introduces approximations in the geometry of the beam end, in particular, it renders the normals to the centerline, at the nodes, indeterminate. In turn this produces jumps for some stress resultants at the nodes. Accordingly, the true value of such stress resultants, at the nodes, must be estimated by an averaging scheme. Such scheme provides another source of error. These errors are in addition to htose inherent in the derivation of the element equations. Errors due to approximations of geometry and those inherent in the finite element method combine algebrically. That is, in principle it is possible for geometric errors to compansate in patr for the errors in the finite element approximations. At the expense of increasing the number of elements, one can be minimized by developing elements which at least allow constant values for curvatures and torsion of the centerline. In this way the overall errors can be effectivelly reduced. The governing differantial equations of spatially curved and twisted beams are given for the static problems in the second chapter. The beam is represented by a space curve whose every point is coupled with a rigid orthonormal vector diad. The vectors are closen to be perpendicular to the tangent vector of the space curve in the initial state and they represent the cross-section of the beam. In deformed configuration, these directors still remain unit and perpendicular each other because of the assumption of a rigid cross-section. But, in contrast with Kirchhoff's beam theory, the restrictions of perpendicular cross- section and inextencible arc lenght are removed. The displacement of the point on the space curve and the rotation of the cross-section constitute the displacement field of the beam. The change in the tangent vector from the initial state and the rate of change vector of the deformed director diad are choosen to represent the state of strain. Xll In the third chapter, the governing of differential equations of static problems of planar curved beams are given. Exact solutions are found for the boundary value problem with arbitrary boundary conditions. A solution procedure is described to obtain fundamental matrix for circular cross-section and arbitrary boundary conditions. In this chapter, there are three examples for parabolic and circular curved beams. Fundamental matrix is obtained analytically for the circular and the parabolic beams. All the integrals are calculated and the element of the matrix are obtained as a function of the arc angle. In this chapter, three different examples with arbitrary boundary conditions are given. And also their fundamental matrixes are obtained analytically. The displacement and the stress resultants can be analytically calculated along the curved beam element by the method of initial values. The expressions contain only six initial values as the data to be known. In the fourth chapter, Two numerical examples are solved by using the program of Mapple. These examples are solved for the planar curved beam with circular and rectangular cross-section with cantilever concentrated load and its fundamental matrix is obtained. In the first example, the beam built-in at one end, loaded singular vertical force at the other end. In the second example, the beam built-in at two ends and loaded singular vertical force at the middle of the beam. The solution is describe the beams curvature and circular cross-section. The concentrated force acts at arbitrary direction at any point on the beam. Three different solutions and two numerical examples are given. All the integrals are calculated by using the program of mathematica.