Elastik zemin üzerine oturan kirişler

Abstract

Bu çalışmada, elastik zemin üzerine oturan sabit kesitli, sonlu uzunlukta, ağırlıksız kirişlerin hareketli yük altındaki davranışı incelenmiştir. Birinci bölümde problemin analizi, kullanıldığı alanlar ve çözüm yöntemlerinden sözedilerek, genel tanımlar ve açıklamalar yapılmıştır. Literatürde bulunan konu ile ilgili çalışmalarla birlikte, yapılan çalışmanın amaç ve kapsamı verilmiştir. İkinci bölümde, ilk olarak genel elastik eğri denklemi verilmiş, bundan yola çıkarak sabit hızla hareket eden tekil kuvvet ele alınmış ve mod fonksiyonları yardımı ile çözümleri yapılmıştır. Çözümde rezonans durumu da ayrıca incelenmiştir. Bu halde problemin her bir mod için kesin çözümü yapılmıştır. Yük hızının sabit olmaması halinde benzer şekilde hareket edilmiş, ancak her bir mod için Runge - Kutta tekniği kullanılarak zamana göre sayısal çözüm yapılmıştır. Daha sonra kirişin üzerinde kendi kütlesine ilave olarak bir tekil kütlenin de olması hali ele alınmıştır. Bu kısımda ortogonallik koşulu tekil kütlenin bulunduğu terimde sağlanamadığından zamana bağlı kısım için bir 2. mertebeden diferansiyel denklem takımı elde edilmiştir. Bu denklem takımı da Runge - Kutta tekniği ile sayısal olarak çözülmüştür. Sayısal çözümde 5 terim almanın yeterli olduğu görülmüştür. Son olarak Winkler zemininin kübik nonlineerlik göstermesi hali ele alınmıştır. Zayıf nonlineerlik hali için pertürbasyon serisi kullanılarak, seride ilk iki terim alınmıştır. Zayıf nonlineerlik hali incelendiğinden 0. mertebe denklem kapalı çözülmüştür. 1. mertebe çözümde sayısal integrasyon kullanılması kolaylık sağlamaktadır. Sonuçlar bölümünde, ikinci bölümde incelenen durumlar için sayısal örnekler verilmiş, Fortran dilinde yazılan programlar yardımı ile elde edilen sonuçlar şekillerle gösterilmiş, kapalı çözüm ve sayısal integrasyon sonuçları karşılaştırılmıştır.

Beams on elastic foundations are widely used as structural elements in engineering aplication. In recent years, the increase on the use of soft filaments in airport structures and the building works on cold regions, intensified the need for the solutions of various problems of beams, plates and shells on elastic or viscoelastic media. The analysis of beams on elastic foundations can be defined in three steps; 1- Assumptions of the behaviour of the structure and type of the foundation 2- Selection of the foundation modulus, dimensions and materials of beam 3- Mathematical solutions of the problem by using the parameters of 1st and 2nd items. The study of beams on elastic foundations began in 1867. 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 P and the deflection of the foundation surface v is; P = kv where k is the foundation modulus. The corresponding deformations of the foundation surface for a uniform load, are shown in Figure 1. 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 but 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 satisfy the previous statements of the theory. ' * /\S A S X / Figure 1 The corresponding deformations of the foundation surface IX Figure 2 The displacement of the foundation surface Winkler hypothesis assumes that the foundation acts in tension as well as in compression. In practice, the beams does not behave like this. For example, a railroad rail actually lifts off its ballast in front of the moving train. Weitsman [20] made an extensive study of beams resting on tensionless foundation under static loading. In engineering applications, there are some important problems which can be handled successfully by means of Winkler hypothesis; the frames of ships, grid systems at plates and bridges, continuous foundations in one or two directions, rotational shells and perpendicular piles under the effect of the horizontal load. Beams on elastic foundations have been investigated by numerous researchers. Iyengar and Anantharamu [ll] studied beams on elastic foundations by using mathematical series. Weitsman [2l] made the first investigation of a beam on a tensionless foundation under a moving load, but restricted his study to determining the conditions under which separation would occur. A further investigation was made by Torby [22], but only a partial solution was obtained for certain conditions of load and speed. Choros and Adams [23] solved the steady problem of a single moving load acting on a tensionless foundation; their results include the determination of the location, magnitude and extent of the lift-off regions. In this work, the behaviour of weightless beam which has finite length, uniform cross- section, subjected to a dynamic concentrated load and supported by an elastic foundation is investigated. The problem is solved by using mode functions for the beam under the concentrated load. Firstly, the equation of weightless beam of finite length which has the same conditions with our example is given as; EI- âxA ? + pA â2v ât' = 0 (1) Due to linearity, below assumption is to be done; v = #r).T(t) Then; d4 d2 T,," El - jT + pA0- T=0 and (2) d4 1 d2T 1,,, are obtained. x Using the dimensionless quantity x = - ? and the solution 4> = eaoc the general solution of differential equation is as follows: (x)=A cosBx + B sinBx + C coshBx + D sinh/?x (4) The function (f> must satisfy the following boundary conditions: x = 0 ^"(0) = 0 0"'(O) = 0 x = 1 p'(\) = 0 f" (1) = 0 After derivating function, the equation cos /? cosh /?-l = 0 (5) is obtained. The natural frequencies are obtained by using this equation and the Method of Newton-Raphson. Thus, the nth mod function " corresponding to the nth natural frequency B" is as follows: fa =[cos/?"x + cosh/?"x + ^"(sin/?"x + sinhy9"x)] (6) The mod functions are orthogonal to each other and satisfy the conditions of orthogonality: J An <% ? m^n The equation (5) has infinite roots, for that reason the solution v^(Hx)T{t) can be written as; XI v(xj) = ^n(xn)Tn(t) (7) n=\ The governing equation of weightless, fixed dimensions and L lengthened beam under the concentrated load with constant velocity is: tfv d2v - /_ - _ \ EI - r + kv + pA - r = Pü6(x-b -v0t) âx4 y ât2 ° v ; The boundary conditions and initial conditions are: v"(0,f) = v'"(o,f) = v"(l,0 = v"*(l,/) = 0 f x avixfi) v(x,6) = - -1- = 0 ât If the expression m and Galerkin Method and orthogonality condition are used, the equation becomes as; Tm+Clm2 Tm =P0{cos[^(*+v0^)]+cosh[^(3+v0/)]+r"(sin[/?m(A + v0?)] + sinh[&,(6+v0/)])} Then the general solution is as; (8) Tm = Am cosClJ + Bm sinQj + P0< cos[4(fr+v0/)] cosh[fl"(ft+v0/)] n«2-A,2v0 n*2+A,V + r" -.MM sm[/3m(b +v0t)] sinh[/?m(fr +v0t)] 2,t 2 2rr 2 n;-/LV *V+A,'v0 (9) The governing equation of weightless, fixed dimensions and L lengthened beam under the concentrated load with accelerated velocity is: r, d*v d2v ( 1, (10) Here, if similar arrangements is done as in the above equation, the differential equation is obtained as; Xll Tm +(/C +k)Tm = PoT^-m(öy) OD Due to the terms of the right side, the closed-form solution can not be applied. The equation is written in matrix form; f+BT=C(â0r2) SB where; B,=tf+k C,=/>0rt(ff0r2) \tfdx After this, using Runge-Kutta Method, numerical solution can be done. In the case of the beam under a dynamic concentrated load with a concentrated mass, as well as mass of the beam, the governing equation is; â*v d2v â2v i \ i \ EI-^ + pA-^T + Mj^s(x-Ç0)+Kv=PQS(x-v0t) (12) Then, the differential equation is obtained as; ^+T-^İ;^(CoMM(4) + (A4+^)7;=P0-r-^-^(v0r) (13) lm2dx"=l \m2<^ M _ e pAL Orthogonality condition can not be obtained in the terms of concentrated mass, this equation can be written in matrix form; AT+BT = C(v0t) (14) where; xm ^(CoMCC) A:, = Oit + ? *ij ">J 1 o Bt=tf+k We obtained an equation system of second order and we carried out a numerical solution using Runge-Kutta method. Finally we assumed that Winkler-type foundation have cubic nonlinearity. The equation is; (fv â v i / \ EI r + pA - y+Kv + Ky =P0S(x-v0t) (15) dxA ât v ' By using the dimensionless quantities the equation is obtained as; ~ZT + -rT + kv + dcy =P0s(x-v0r) (16) âc ât Then we used the perturbation technics to solve the equation of motion. afr, °° °° " ' % + (/?/ +k)Tr +T^-^YJ$MnU^T"TmTl = JU(v0r) (17) *r ~ -*0r + Ehr pr4+k = nr _JL 00 00 CO 1 J $. ax In the zeroth step, the second order distinct equation is obtained. This equation have been solved exactly. XIV s0:f0r+n,2r0r=P0^(v0r) In the first step, non-homogeneous second order equations can be solved exactly too. K*i 00 00 CO el:Tlr +nr2Tlr = Y-3- HHJll AMr^T0nTOmTol In the end of this work, computer programs have been written by using Fortran programming language, and the solutions have been given in graphics.