Elastik çubukların titreşim frekanslarının ve burkulma yüklerinin sonlu elemanlar yöntemiyle analizi

Abstract

Bu çalışmada, Sonlu Elemanlar Yöntemi kullanılarak elastik çubukların değişik sınır şartlarında titreşim karekteristikleri, statik ve dinamik burkulma yükleri incelenmiştir. Çubukların titreşim karakteristiklerinin incelenmesi ankastre-serbest, ankastre-ankastre ve basit-basit sınır şartlarında yapılmıştir.Ankastre-serbest halde çubuğun serbest ucunda nokta kütle etkiside hesaplamalara dahil edilmiştir. Yine çubukların burkulma yükleri ankastre-serbest, ankastre-basit, basit-basit sınır şartlarında statik ve dinamik olarak incelenmiştir. Teorik ve deneysel çalışmaları yapabilmek için birisi metal diğeri plastik olmak üzere iki çubuk kullamlmıştır.Bu çubukların elastiktik modülleri yapılan eğilme deneyi ile elde edilmiştir. Deney sonucunda elastiktik modülleri beklenen aralıklarda hesaplanmıştır. Çubukların ankastre-serbest halde serbest frekansları ve serbest uçta kütle bulunması durumunda bunun frekansa etkisi deneysel olarak incelenmiştir. Ayrıca plastik çubuğun burkulma yükleri ankastre-serbest sınır şartlarında statik ve dinamik olarak ölçülmüştür.Değişik sınır şartları için teorik olarak elde edilen serbest frekanslar ve burkulma yükleri ile deneysel olarak elde edilen değerler birbirlerine yakınlık göstermektedir.

In this study, vibrational characteristics and buckling loads of rectangular cross section beams are calculated by the finite element method. The Finite Element Method approximates a structure in two distinct ways. The first approximation made in finite element modelling is to divide the structure into number of small simple parts. These small parts are called finite elements and the procedure of dividing the structure is called discretization. Each element has an equation of motion that can easily be solved or approximated.Each element has endpoints called nodes, which connect it to the next elements.The equation of vibration for each individual finite element is then determined and solved.This forms the second level of approximation in the finite element method. In the analysis of free vibration, effect of longitudinal displacement of the beam is considered. In addition, it is taken into account that fixed- free beam which has a lumped mass effect on the beam's free end node.In the buckling analysis, both static and dynamic calculations are presented. For this purpose, beam is considered as a rectangular beam. The beam is of rectangular cross section area A with width b, thickness h and length L.Also associated with the beam is a flexural bending stiffness EI, where E is Young's Modulus for the beam, I is cross-sectional area moment of inertia about the z axis. In the middle of xz plane, the deflection of the beam in the y direction is noted as w(x,t) and in the x direction the deflection of the beam is noted as u(x,t). These deflections are defined by 2nd and 3rd degree polynomials. Ci,,Cö are real constants of integration with respect to x. w(x,t) = CiX3 + C2X2 +C3X1 +C4 (1) IX u(x,t) = c5x + c6 (2) In order to analyze, the equation of motion of the system, using Hamilton's principle can be written as follows: 5j(T-U)dt+JTv t, t U)dt+ J8WV = 0 1 (3) where T is kinetic energy, U is potential energy and 8WV is virtual work done by active forces which affect the system. After required mathematical steps the equation of motion of the system becomes as follows: [[KI-Mcd^S^O (4) The mass and stiffness matrices are found trough the expressions of potential and kinetic energy and by writing the virtual work expression in equation of Hamilton Principle. For a non-rotating beam undergoing transverse vibration, the potential energy is given by the expression. V,4ei3t fd2v/y 0\dx j dx (5) In the finite element formulation, the beam is divided into elements and the above expression becomes 1 «f f \2 ^2 d2w 2 0Kdx2J dx (6) where w = [N]w{8} W v 'W (7) The equation (6) yields the following. X Vl=^{5}^[Ke]w{5}w (8) If the beam stiffhed due to additional stresses created by axial force, as it happens in the buckling problem, this axial force creates stress in the neutral surface and the additional strain energy stored in the element can be written as follows: V, =-A 1. f (dw f (dw ^2 2 2"T\dxJ dx (9) This equation ( 9 ) yields the following. V2=|{5}^[KG]{8}W (10) where [K^J is called the geometrik stiffness matrix. Because of longitudial displacement, the stored potential energy in the element is, Vboy ~J EA(§)2dx (11) where u = [N]u{5}u (12) and the equation (11) yields the following: Vboy=^{8}I[Ke]u{8}u (13) The total stiffness matrices for bending vibration is as follows; [Ke] = [Ke]u+[Ke]w (14) XI In order to obtain the mass matrix, the kinetic energy which is equal to work done by inertia forces of the beam mass is written as equation (15) for transverse vibrations in xz plane using Euler theory. t4pa( 'duY fdwv At) dx (15) In this equation, the first term is the kinetic energy due to longitudial displacement and the second term is bending energy respectively. In the case of a lumped mass on the beam's free end node, the kinetic energy due to this mass is calculated as follows: -M^J T* = 2"\ AT) <16> The element matrices given above are combined to form the global stiffness and mass matrices according to FEM formulation.Using these matrices the system equations are formed as follows: [[K]-[MW](8)pO (17) and solved numerically. In the buckling problem, determining the critical buckling load of an elastic structure is investigate. This corresponds to the calculations of eigenvalues of the generalized eigenvalue problem*; The buckling loads of the beams, can be calculated by using equation (18), The critical buckling Pcr, also called the stability limit, is the minimum of the equation (18). When P is equal to Pcr, the first eigenfrequency becomes zero.In that case, the beam takes a non-zero but static deflected shape;it corresponds to global buckling of the beam, also called Euler buckling. XII In the case of a structure subjected to an initial stress field ox, the equation (9) must be added to the strain energy of the system.The equation (9) yields equation (10) with the geometric stiffness matrix component [KqJ. Since the geometric stiffness expression possesses the same form as the linear stiffness one, it is immediately obvious that the linear equations of motion are replaced in the prestressed case by [AİKGİ+lKlJtSJi+lMHSj^O (19) and the associated eigenvalue problem takes the form. [*[KG] + [K]]{5}, - n2[M]{8}, = 0 (20) When H=0,we get an eigenproblem with eigenvalues X\ corresponding to the critical loads. [[Kl + AfKoUW^O (21) The equation (21) consists an eigenvalue problem in which we seek to determine the eigenvalues X-, (i= 1,2,...). The eigenvalue AT" yields the prestress state in which the system buckles. For theoretical calculations, two kinds of beams which one is a plastic beam and the other is a metal beam, are used. Young's Modulus of these materials are calculated by carrying out the bending experiments. In the bending experiments, the beam rests on two supports and is loaded by means of a loading nose midway between supports. The load is applied to the specimen at the specified crosshead rate, and simultaneous load-deflection data is taken. The load-deflection curve is plotted,after required mathematical steps, Young's modulus of these materials are obtained. In conclusion, the free vibration frequencies and the buckling loads of the elastic beams are investigated by the finite element method. Both buckling and vibration analysises are implemented for different boundary conditions XIII In vibration analysis, the free vibrations of fixed-free beam are obtained by theoretical calculations and carrying out experiments. The free eigenfrequencies of the beams are on the decrease, as the quantity of the mass on the free end node of fixed-free beam increases. At the definite quantity of mass, first eigenfrequency is equal to zero In the case of a beam subjected to an axial force, the geometric stiffness matrix must be added to the equation of the motion of the beam. When the compression force is equal to the critical buckling load of the beam, the first eigenfrequency vanishes.The eigenfrequencies are systematically lowered by a compression load and increased by a traction force. In buckling analysis, the critical buckling loads of the beams are obtained by calculations and by experiments for the fixed-free beam. In dynamic analysis, the first buckling load is equal to zero, when the beam is excited at the first eigenfrequency.