Elastik zemine oturan daire eksenli çubuklar ve taşıma matrisi

Abstract

Çalışmada Winkler Zemin tipi ele alınarak, elastik zemin üzerine oturan daire eksenli çubukların farklı yüklemeler altındaki davranışları başlangıç değerleri metodu ile incelenmiştir. Başlangıç değerleri metodu mühendislik problemlerinde sayıların fazlalığının meydana getirdiği zorluklan ortadan kaldırmak için sınır değer problemlerini; başlangıç değerleri problemlerine dönüştüren bir metoddur. Bu metod uygulanarak daire eksenli çubukların ve elastik zemine oturması hali için farklı yollarla kapalı şekilde taşıma matrisleri elde edilmiştir. Birinci bölümde, incelenen problem ana hatlarıyla tanıtılarak, bu konu ile ilgili literatürde yapılan çalışmalara yer verilmiştir. İkinci bölümde, elastik zemine oturan homojen, lineer lineer elastik çubuklar için temel alan denklemleri verilmiştir. Üçüncü bölümde, daire eksenli çubuklar için literatürde bulunan taşıma matrisi, idempotent ve nilpotentlerin yardımı ile ve Cayley-Hamilton Teoremi yardımı ile elde edilmesi şeklinde faklı iki yolla elde edilmiştir. Daire eksenli çubukların elastik zemine oturması durumunda da taşıma matrisi benzer şekilde Sylvester Teoremi ve Cayley-Hamilton Teoremi yardımı ile farklı yollardan çözüm yoluna gidilmiştir. Dördüncü bölümde, daire eksenli çubukların dinamik olarak serbest titreşim problemi ele alınmış; serbest titreşim frekanslarına kesin çözüm olan Başlangıç Değerleri Metodu ile Frekans Determinantı kullanılmış ve yaklaşık çözüm olan Ayrık Kütle Metodu ile çözüm yoluna gidilmiştir. Beşinci bölümde, sayısal uygulamalara yer verilmiş, kesin çözümleriyle karşılaştırma yapılmıştır. Altıncı bölümde, yapılan çalışmadan elde edilen sonuçlar açıklanmaktadır.

Circular Beams On Elastic Foundation and Transfer Matrix Beams on elastic foundations are widely used as structurel elements in engineering aplication. According to the beam axis geometry, beams can be either straight and cicvlinear. 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 examles of the circular beams are circular foundations of water tanks and silos. Foundation could be clafssified 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 continuous elastic foundation. The relation between the pressure and the deflection of the foundation surface, both parallel to the z axes is given as; P(y) = ku(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 displacements of the foundation surface shown in Figure 2., dose not exactly follow the previous statements of the theory. ^jhİmhİ?,^,^^** >^d »' '^: (a) (b) Figure 1. The coresponding deformations of the foundation surface VI '//?//r f'/S rs 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 behaviour. Beams on elastic foundation have been investigated by numerous scientist and researchers. Heteny [3] studied bems on Winkler foundations and came up with an exact solutions. No matter how simplified it was, this solution seems to be really time consuming. In 1967, a general solution method was developed using initial values method and transfer matrix given by înan [6] etc. In this study linear elastic beams is cosidered into two groups; a) With Straight axis b) With circular axis but only circular axis beams are studied. Beams With Circular Axis Field equations for circular beams are d0 dM d9 dMt de dnt de -qR + kRUb = 0 2- - Mt + R Tb - R mn = 0 + M" - R mt =0 - nn - r - i- = o n Dt a- + a -R - ^ = 0 d6 * Dn ^Hk+Rn r i=0 de n cb Vll where Tb, Mn, Mt, Qn, Qt, Ub are shear force, bending moment, twisting moment, angle of bending, angle of twisting, displacement of axis of beam in directions perpendicular to plane of beam, respectively. Those are unknows to be obtained. Dn, Dt, Cb are bending rigidity, torsional rigidity and shearing rigidity of the beam, respectively. R is radius of the beam; k represent the madulus of the foundation, q, mn, mt are external distrubituve load and moments, respectively. In this study, circular beams on elastic foundation and their transfer matrix is tried to be obtained. Static and dynamic analysis of circular beams on elastic foundation is done. To solve circular beams on elastic foundation by using initial values method and transfer matrix. Transfer matrix is a matrix which obtains any x, from an initial x=0 with step by step integration and transfer matrix can be axpressed as follows: S(x) = F(x).S(0) In this equation, S(0) shows initial (x=0) boundary conditions and S(x) shows any x points' boundary conditions. Transfer matrix, F(x) is an exponential matrix form as F(x) = e*D If transfer matrix F(x) is shown in a form of matrix polynomial series the equation accurs as below;,xD xk-k F(x) = ew=I + xD+-D+... +- D V ; 2! k! here D is coefficient matrix. In this study, loads which apply verticaly to the axis direction of circular beams is taken as a column matrix; Ub(cp)' S(q>) = nt(cp) Mn(cp) Mt(q>) Also, from diferantial equation of circular beams o on elastic foundations, the below characteristic equation can be aptained; m6 + 2m4 + 1 + kR4 Dn 2 kR* m =0 Dt vni At first, transfer matrix is obtained for k=0 by using two diferent theorems. First one is Nilpotent and Idempotents theorem. In this theorem, if the square of an endomorphism is equal to itself such an endomorphism is called an Idempotent and if a k' th power of an endomorphism is zero such an endomorphism is called a Nilpotent. The resolvent of an endomorphism can be expressed as m. "m i i i "b I RU,D)= Z\r x +, x2+L + *\fr-K) (x-xj (x~xj« J where the sum is over i < n distinct eigenvalues of D. In this theorem; Spectral expansion can be written as, where, I is unit matrix, i are numbers of the disting value, Xt are eigenvalues of D, Sk are the indices of the eigenvalues, P; are idempotent, Q; are nilpotents. Nilpotents and Idempotents can be obtained by using mathematical functions of D matrix. Then transfer matrix is obtained by inserting nilpotents, idempotents and X^s in the below equation (See (3.38));.w^w^ + T^ql} i i r! Secondly transfer matrix can be obtained by using Cayley-Hamilton's theorem. In this theorem transfer matrix can be expressed as; O(d) = a0 1 + a, D + % D2+ -+an_, Dn_1 and to obtained this equation, a; values must be calculated by using eigenvalues of D matrix. By using either Nilpotents and Idempotents or Cayley-Hamilton Theorems, the same transfer matrix obtained. (See (3.47)) The most important aim of this study is to obtain the transfer matrix of cicular beams on elastic foundations. As known the eigenvalues of the characteristic equation of circular beams on elastic foundation are six different roots. Therefore, there are any nilpotents; so Sylvester's Theorem must be used. [22] IX Sylvester's Theorem: An Idempotent of a diagonable endomorphism can be given as; n (v-d) fc'.j*' n (v*J By using Sylvester's Theorem, six idempotents are obtained, and by these idempotents in exponential matrix equation. With this theorem transfer matrix of circular beams on elastic foundations is obtained as in (3.97). Cayley-Hamilton Theorem gives the same transfer matrix as well.(See (3.103)) With the obtained transfer matrix of circular beams problems are solved for some static and Dynamic loads. In dynamic analysis for free vibration, frequency determinant of circular beams are obtained and explained. Also for this problem, a computer program is developed by using both Mathematica [23] programing language nd Basic programing language with sweeping matrix. In this thesis Mathematica Programing Language is used where needed. Comparison of the results in terms of different parameter shown that convergency the proposed method and obtained tranfer matrix of circular beams is excellent.