Elastik zemine oturan doğru ve daire eksenli çubuklar

Abstract

Bu çalışmada Winkler Zemin tipi ele alınarak elastik zemin üzerine oturan doğru ve daire eksenli kirişlerin, çeşitli yüklemeler altındaki davranışları incelenmiştir. Elastik zemine oturan kirişler için fonksiyonel analizi yapılarak elde edilen fonksiyonele varyasyon tekniği uygulanmıştır. Ayrıca bu tür kirişler için kapalı formda rijitlik matrisleri ayrı ayrı elde edilmiştir. Bulunan rijitlik matrisleri, geliştirdiğimiz iki düğüm noktalı çubuk eleman üzerinde türetilmiştir. Bu yöntemle bilinmeyenlerde doğrudan elde edilmektedir. Birinci bölümde, problem tanıtılarak literatürde bulunan konu ile ilgili çalışmalarla, yapılan çalışmanın amaç ve kapsamı verilmiştir. İkinci bölümde ise elastik zemine oturan yatay düzlemdeki doğru ve daire eksenli, homojen, lineer elastik bir kiriş için temel denklemler verilmiştir. Üçüncü bölümde, fonksiyonel analiz yöntemleri kullanılarak daha önceden çubuklar için elde edilmiş fonksiyonellere benzer şekilde yeni fonksiyoneller, dinamik ve geometrik sınır koşulları ile birlikte elde edilmiştir. Bu fonksiyonellerin içine Lagrange çarpımı ile uygunluk koşulları ve yüklerin etkimesiyle oluşan süreksizlikler katılmıştır. Dördüncü bölümde, çözüm yolları anlatılmıştır. Burada, elde edilen fonksiyonele varyasyon tekniği uygulanarak deplasmanların, kuvvetlerin ve momentlerin bilinmeyen olarak tanımlandığı çubuk eleman ele alınarak bir sonlu eleman modeli geliştirilmiştir. Beşinci bölümde, uygulamalara yer verilip literatürde bulunan veya kesin çözümü bilinen problemler çözülmüş ve çözümler ile karşılaştırmalar yapılmıştır. Altıncı bölümde ise sonuçlar ile kullanılan yöntemin yararları hakkında bilgiler verilmiştir.

Beams on elastic foundations are widely used as structural elements in engineering application. Accordings to the beam axis geometry, beams can be either straight or curvlinear. In this research straight and circular beams on Winkler foundations under various loads are studied. Examples of the straight beams on elastic foundations are: continous foundations being in one or two directions, beams traversing beneath the railway, and vertical piles which are loaded horizontally in the ground. Likewise examples of the circular beams are circular foundations of water tanks and silos. Foundations could be classified into groups according to its responds. For examples Pasternak, Vlassov, Winkler, Filonenko-Borodich, Reissner foundations. Assuming that the base is consisted of closely independent linear springs Winkler provided the simplest representation of a continous elastic foundation. The relation between the pressure and the deflection of the foundation surface, both parallel to the z axis is given as p(y) = k u (y) where k is the foundation modulus. The corresponding deformations of the foundation surface for a uniform load, are shown in Figure 1. It can be seen that for this foundation model, the displacements of the loaded region will be constant whether the foundation is subjected to a rigid stamp or a uniform load. While for both types of loading the displacements are zero outside the loaded region. However for most media the displacement of the foundation surface shown in Figure 2, does not exactly follow the previous statements of the theory. I S7W a hi i) m h \ //////// /}i\//\ /)/ M Figure 1. The corresponding deformations of the foundation surface VI Figure 2. The displacement of the foundation surface This leads to the conclusion that the Winkler hypotesis is not exactly verified but provides sufficient convergence to the exact solution. Beams on elastic foundation have been investigated by numerous scientists and researchers. Heteny [3] studied beams on Winkler foundation and came up with an exact solution. No matter how simplified it was, this solution seems to be really time consuming. In 1960xs Iyengar and Anantharamu [4] studied beams on elastic foundation by means of series. Same problem was solved by using finite element method by Malter [5]. In 1967, a general solution method was developed using initial values method and transfer matrix given by İnan [9], Ting [14], in 1982, brought up a solution for the differential equation of an infinite beam with elastic supports on elastic foundation. In 1985 an exact stiffness matrix for a beam on a Winkler foundation was formulated by Eisenberger and Yankelevski [17]. In this study some new functionals are obtained for straight and circular bars on elastic foundations using functional analysis method. Parameters in this functional can be choosen with respect to necessity. The new finite element formulation for beams with straight and circular axis gives the rigidity matrices of beams. This new formulation is valid for the beam whether it is based on elastic foundation or not. In this study a homogeneous, linear elastic beam is considered into two groups: a) With straight axis b) With circular axis 1. Beams with straigth axis The beam equations are given as [9] vu d2M g+kv=0 dz _ dzv M dz2 EI =0 where M, q, v, EI and k represent bending moment, external distributive load, displacement, bending rigidity of beam and modulus of foundation, respectively. Geometric boundary conditions, -V =-tf v o v o Dynamic boundary conditions, -M =-M written in symbollic form. Quantities with hat have known values on the boundary. Field equations can be written in the operator form as, Q=P. U-f=0 <.+ rJ r*> <~» which is shown to be a potential operator given in equations (3.3). Using Gateaux differantial dQ(u,u), equations (3.1) and after few manipulations the following functional is obtained. Ku) =4 iv, v] - [q, v] - -J iJL, M] + [M\ v'] r-* 2 2 EI - [ (v-t?0), ro]c- W'.M0]m- [ (M-M0), v] "- [v0, f] " b- Beams with circular axis Field equations for circular beams are D+kRub-gR=0 d9 dMn -^-Mt+RTb-Rmn=0 dMt -^+Mn-Rmt=0 V1U dClt n Mt dQ n Dt dd t Dn dQ a Cb where Tb,MB,Mt, Q n, Q t, ub are shear force, bending moment, twisting moment, angle of bending, angle of twist, displacement of axis of beam in directions perpendicular to plane of beam, respectively. Those are unknowns to be obtained. DB,DC,Cb are bending rigidity, torsional rigidity and shearing rigidity of the beam, respectively. R is radius of the beam; Jc represents the modulus of the foundation, q, m,,,!^ are external distributive load and moments, respectively. Geometric boundary conditions, -Qt =-ût Dynamic boundary conditions, M =Af Tb-Tbn written in symbollic form. Quantities with hat have known values on the boundary. Using Gateaux differantial, after a few manipulations equations (3.6) gives the following functional. IX ?t(jj) =jy [Ui" Ubl ~R[q' Ub] +RtTb'QJ + lMn- a J - [Mt,Qn] - [Tb,ub] - [Mn, Q J - [M^, Û t] * Wt,Mt]-^-[Mn,Mj-^L{Tb,Tb] 2Dt L c 2Pn n n 2Cb + [ubl (Tb-fb)]a+[Cin, (Mn-Ûn)]a+IQC, (Mt-Ûc)]a + [ûb, Tb]e+ [ClB,Mn]e+ [û t,Mt), obtained for circular beam on elastic foundation. To obtain element rigidity matrices, variational method is applied to the given functionals of beam on elastic foundation. In the derivation of rigidity matrix of straight and circular beams on elastic foundation finite elements, finite element formulation is followed. The principle idea of finite element formulation is to achieve the relationship between the element unknowns at any point and the element nodal point unknowns directly through the use of interpolation functions, and the element matrices corresponding to the reguired degrees of freedom. Since the functinoals have only derivatives of first degree, linear shape functions for beams on elastic foundation would be necessary and sufficient. The bar element of beam with straight axis has two nodes. At each node the displacement and the bending moment are defined as a total of two unknowns. Likewise, the bar element of beam with circular axis has two nodes. At each node one displacement, two rotations, one shear force, one twisting, one bending moments are defined as a total of six unknowns. In this study, new functionals for straight and circular beams on elastic foundation with geometrical and dynamic boundary conditions are presented. In the literature survey the same functionals were not found. Also, these functionals have been proved to be potential and classical energy transformable equations. By using Fortran programming language, a computer program is written for beams on elastic foundation with both straight and circular axis. Comparision with results reported in the literature demonstrates the features of the proposed finite element formulation. Comparision of the results in terms of the different parameter shown that convergency the proposed method is excellent.