Pasternak zemine oturan sonsuz bir kirişin hareketli tekil yük altındaki dinamik davranışının incelenmesi
Dynamic response of a infinite beam on a pasternak foundation and under a moving load
Yazar: NAGEHAN EVCAN
Danışman: PROF. DR. ABDUL HAYIR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / İnşaat Mühendisliği Ana Bilim Dalı / Yapı Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2012
Konu: Mühendislik Bilimleri = Engineering Sciences ; İnşaat Mühendisliği = Civil Engineering

Özet:
Çeşitli özellikteki zeminler ile temas halinde olan mühendislik yapılarının hesapları günümüzde giderek önem kazanan konulardandır. Hesaplarda genel olarak varsayımlara dayalı çeşitli tiplerde zemin modelleri kullanıldığı gibi, kullanılan analiz yöntemleri de çeşitli olabilmektedir. Bu tez çalışmasında birçok mühendislik uygulmasında karşılaşılan ve birçok pratik uygulamaya örnek olan elastik zemine oturan kiriş problemi ve dinamik davranışı incelenmektedir. Elastik zemin modeli olarak sonsuz kiriş için elastik zemin modellerinden iki parametreli olan ve kayma etkilerini de hesaba katan Pasternak zemin modeli seçilmiştir. Literatürde geniş bir kaplayan ve matematiksel açıdan daha kolay olan, tek parametreli zemin modeli olan Winkler zemin modelinin seçilmemiş olmasının sebebi iki parametreli zemin modellerine kıyasla gerçekçilikten daha uzak olmasıdır. Seçilen pasternak zemin üzerine oturan sonsuz kirişe hareketli tekil yük etkitilmiştir.Bu çalışmada tezin amacından bahsedilmiş, bu konuda çalışan bilim adamlarının konu hakkındaki çalışmalarına değinilmiş ve literatür çalışması yapılmıştır. Bu çalışmada kısaca geçmişte ve günümüzde hala kullanılmakta olan zemin modellerinden bahsedilmiş, sınıflara ayrılmış aralarındaki farklar ve benzerlikler; şekillerle ve formulasyonlarla belirtilmiştir. Tez konusu olarak neden iki parametreli olan Pasternak zemin seçildiğine dair açıklamalar yapılmış, bu zemin modelinin diğer modellere göre olumlu ve olumsuz yanları incelenmiştir. Çözümde kullanılan matematiksel yöntem olan Hızlı Fourier Dönüşümü, ters Fourier uygulaması ve zaman ortamından frekans ortamına geçiş hakkında açıklamalar yapılmış, tez konusu problemde şekil üzerinde uygulamalı olarak gösterilmiştir. Zemin üzerine oturan kiriş homojen, izotrop ve sonsuz uzunluktaki bir kiriş olarak düşünülmüştür. Buna ek olarak incelemenin konusu olan hareketli tekil yük altındaki sonsuz kiriş problemi günümüzde de hala en çok kullanılan zemin modellerinden olan Winkler zemin modeline de uygulanmış Pasternak zemin modeliyle kıyaslanmıştır.Hızlı Fourier dönüşümü uygulanmış analiz sonuçları frekans ortamına çevrilmiş, elde edilen sonuçlar bilgisayar ortamında Mathematica programında grafiklere dökülmüştür. Bilgisayar programı kullanılarak grafiklere dökülen sonuçların zemin tipine göre çeşitli verileri değiştirilerek kıyaslama ortamı sağlanmıştır.Son bölümde ise her iki zemin tipinede aynı veriler uygulanarak, karşılaştırmalı grafikler elde edilmiş olup, elde edilen grafiklere, sonuçlara ve kıyaslamalara dayanılarak zemin modellerinde gerçekçilik irdelenmiştir.

Abstract:
In recent years considerable attention has been given to the response of elastic beams on an elastic foundation which one of the structural engineering problems of theoreticl and practical interest. A large number of studies have been devoted to the subject. In these studies a number of foundation models having various degrees of sophistication have been used to capture the complex behaviour of the soil.Calculation of engineering structure, which is in contact with foundation has several features, has became more important in recent years. As different models of foundation are used, analysing method of calculation can be different.The purpose of this study is to analyze the dynamic response of beam on a elastik foundation, which has drawn a lot of attention due to its wide aplications in the engineering area.The concept of beams nd slabs resting on elastic foundations has been extensively used by geotechnical, pavement and railroad engineers for foundation design and analysis. The analysis of structures resting on elastic foundations is usually based on a relatively simple model of the foundation?s response to applied loads.Generally, the analysis of bending of beams resting on an elastic foundation is developed on the assumption that the reaction forces of the foundation are proportional, at every point, to the deflection of the beam at that point. The vertical deformation characteristics of the foundation are defined by means of continuous, closely spaced linear springs. The constant of proportionality of these springs is known as the modulus of subgrade reaction, k0. This simple representation of elastic foundation was introduced by Winkler in 1867. The Winkler?s model, which has been originally developed for the analysis of railrod tracks, is very simple but does not accurately represent the characteristics of many practical foundations. One of the most important deficiencies of the Winkler?s model is that a diplacement discontinuity appears between the loaded and the unloaded part of the foundation surface. In reality, the soil surface does not show any discontinuity.In order to eliminate the deficiency of Winkler?s model, improved theories have been introduced on refinement of Winkler?s model, by visualizinh various types of interconnections such as shear layers and bems along the Winkler springs (Filonenko-Borodich (1940), Hetenyi (1946), Pasternak (1954), and Vlasov (1960)). These theories have been attemped to find an applicable and simple model of representation of foundation medium.Although the Winkler?s model is a poor represention of the many practical subgrade or subbase materials, it is widely used in soil-structure problems for almost one and a half century. The foundation represented by Winkler model can not sustain shear stresses, and hence discontinuity of adjacent spring displacements can occur. This is the prime short-coming of this foundation model which in practical applications may result in significant inaccuracies in the evaluated structural response. İn order to overcome this problem many researchers have been propesed various mechanical foundation models considering interactions with the surroundings. Among them it is mentioned the class of two parameter foundations named like this because they have the second parameter which introduces interactions between adjacent springs, in addition to the first parameter from the ordinary Winkler?s model. This class of models includes Filonenko-Borodich, Pasternak, Hetenyi, and Vlasov foundations. Mathematically, the equations to describe the rection of the two parameter foundations are equilibrium ones, and the only difference is the definition of the parameters.Two parameter foundation models are more accurate than the one parameter foundation model. As a special case if the second parameter is neglected, the mechanical modeling of the foundation using the Pasternak?s formulation converges to the Winkler?s formulation. The simplest model for the soil is the one parameter Winkler model which represents the soil as a system of closely spaced but mutually independent linear springs. In the model, the foundation reaction is assumed to be proportional to the vertical displacement of the foundation at the same point. However, the Winkler model has various shortcomings due to the independence of the springs. Becuse the springs are assumed to be independent and unconnected to each other, no interaction exists between the springs. When loading displays a discontinuity, similar discontinuity will appear on the foundation surface as well. The soil outside the loading area does not contribute to the foundation response. In order to take care of these shortcomings and to improve the model, two parameter models have been proposed. Pasternak model is one of the simplest two parameter models used commonly.This model can be visualized as a system of closely spaced linear springs coupled to each other with elements which transmit a shar foce proportionl to the slope of the foundation surface. The model can be seen as a membrane having a surface tension laid on a system of elastic springs as well. Due to connection of spring the continuity of the foundation surface is maintained. However, a discontinuity in slope of the displacement can appear at a edge of the beam or the plate resting on the foundation. The anaytical aspects of the continuously supported structures and corresponding boundary conditions on various soil models have been discussed by Kerr (1964, 1976). İt is pointed out that the intuitive approach in the boundary conditions may lead to the incorrect formulationof the boundary conditions for the case of a two parameter foundation model.In this study; Pasternak model is used, in which shear interaction between the springs is considered, to represent the soil foundation. Up to now,the beams on two parameter foundations subjected to moving loads have received less attention, probably because of the model complexity and difficulties in estimating parameter values. The dynamic response of a beam subjected to moving harmonic load is investigated in this thesis. The beam is assumed as an infinite Bernoulli-Euler beam with constant cross section, and the soil is represented by a Pasternak foundation model.Static and dynamic responses of a completely free elastic beam resting on a two parameter tensionless Pasternak foundation are investigated by assumed that the beam is symetrically subjected to a uniformly distributed load and concentrted load at its middle. Governing equations of the problem are obtained and solved by paying attention on the boundary conditions of the problem including the concentrated edge foundation reaction in the case of complete contact and lift-off condition of the beam in a two parameter foundation. The nonlinear governing equation of the problem is evaluated numerically by adopting an iterative procedure. Numerical results are presented in figures to demonstrate the non linear behaviour of the beam foundation system for various values of the parameters of the problem compratively by considering the static and dynamic loading cases.The two parameter Pasternak foundation assumes the existence of shear interaction between the spring elements. This may be accomplished by connecting the ends of the springs with a beam consisting of incompressible vertical elements which deform only by transverse shear. The stiffness of the springs and the shear rigidity of this beam are the two parameters of the foundation.Of all available elastic foundation models, the Pasternak?s one is the most natural extension of the Winkler?s model for homogeneous foundation soil, when the second parameter, shear modulus, is considered in the analysis.Response of structural elements resting on the one and two parameter foundation is usually analyzed by assuming that the foundation supports compressive as well as tensile stresses. Athough this assumption simplifies the analysis considerably, it is questionable or not valid for many supporting media including the soil. In order to increse the validity of the model, tensionless foundation models which can support compressive reactions only are introduced. In this model separation between the foundation and the structural element takes place in order to avoid the tensile stresses. However, this assumption complicates the analysis makes it highly non-linear, since the region of contact and seperation is not known in advance. As a result, only a limited number of studies dealing with tensionless foundation are published.Response of structural elements such as, rings and beams on tensionless Winkler fondation is considered. There are various studies dealing with the static problems and the dynamic problems. The papers dealing with the rectangular and cicular plates on tensionless Winkler foundation can be considered as extension of the beam problem. There are various studies dealing with the plates subjected to static loads and to dynamic loads. Generally, in order to investigate lift-off occurence from the foundation and seperation condition, completely free beams and plates are considered in many studies and solutions are obtained by applying approximate numerical techniques to the nonlinear governing equation of the problem by employing the coordinate functions which satisfy the corresponding boundary conditions. In this way the nonlinear problem is reduced to the iterative solution of the system of the nonlinear algebraic equations, since the contact region is not known in advance. On the other hand, when the load is a function of time, then the contact region appears as a function of time as well. After the initial configuration of the contact region is found, the numerical analysis is carried out by adopting step-wise integration in the time domain by updating the contact region continuously.There are various studies dealing with the structural elements resting on the conventionl two-parameter foundation assuming continuous contact. When the boundary of the beam or the plate is fixed, then the boundary condition does not pose any new aspect that of the Winkler foundation. However, when a free end of the beam or plate is considered, then the corresponding boundary condition includes an additional concentrated load due to the membrane stiffness of the two-parameter foundation. When tensionless two-parameter Pasternak model is considered, the solution gets more complicated due to the free edge conditions even in the static problems. Another major difficulty lays in the definition of the contact zone. Due to these reasons the number of the publications on a two-parameter foundation model that reacts in compression only is very limited. Nonlinear oscillations take place, when the external loads depend on time.The beam and foundation were assumed to be homogeneous and isotropic. In the solution, the double Fourier transform technique is used to reduce the governing partial differantial equation to an algebraic equation and used the inverse Fourier transform to obtain the analytical solution of the integral form Fast Fourier Transform is applied to obtain explicit solution.