Investigation of beam theories applied on tapered cantilever beams and free vibration applications

Beam theories consist of equations that help people understand the mechanics of beam type structures. The studies on this subject has been started with the works of Euler and Bernoulli, resulting the most renowned beam theory, Euler-Bernoulli Theory, late 19th century. The most important and comprehensive beam theory is named after Russian scientist, Stephen Timoshenko in early 20th century. Different from Euler-Bernoulli Theory, it includes shear and rotary inertia effects. This master's thesis is based on graduation project of "Investigation of Euler-Bernoulli and Timoshenko solutions of a tapered beam under thermal loads and free vibration analysis" which is also graduation project of same student. In this study, the beam theories consist of basic beam equations which provide an understanding for mechanics of beams. In this thesis, it is aimed to find natural frequencies of tapered beams with considered beam theories and their comparions with literature results. Euler-Bernoulli, Rayleigh, Shear and Timoshenko beam theories are the most common beam theories for analyzing the beam type structures. Most basic and simple theory, Euler-Bernoulli, and most detailed theory, Timoshenko, have been investigated in general, equations are provided and compared. As a result of these detailed examinations, the equations which make it possible to calculate natural frequencies are derived. During the study, in order to enlighten the next possible studies, Shear beam theory is also included and taken into account. In the same way, for Shear beam theory, general equations are provided and natural frequency calculations are explained. Afterwards, the comparisons of these beam theories are made and their pros and cons are investigated in a chapter. In this chapter, pros and cons of the shear theories have been tabulated to provide a better understanding for people to see the comparison. Besides, derivation of vibration equations and logic have been extensively mentioned in related sections for all the beam theories mentioned in this study. Their history, easiness and rate of employment have been explained. The calculation of the natural frequencies of the beam due to torsion have been explained in Euler-Bernoulli section, however, this application is not given among the results and discussion chapter. Vibration subject is told profoundly since it is the fundamental subject of this thesis study. Under vibration chapter, definition and classification of vibration, resonance phenomena are briefly explained. Especially, degrees of freedom is investigated deeply and several examples are provided. General information about the subject is provided for three different beam theories and their analytic solutions are mentioned. In this study, free vibration is chosen as the main subject since no external force is considered. Even though, there is no external force, the study has an utmost importance due to presentation of preliminary results for forced vibration, natural frequencies. The main goal of this thesis study is to apply these theories on a clamped beam. The beam is chosen is clamped since it has more application field than other support types in industry and it is more difficult to determine natural frequencies resulting from bending. In addition, these kinds of beams are regular beams and also they are explained in a detailed way. There are many studies in literature which comes in handy at the end of the thesis, comparing the results. Even though, the analyses are conducted with a software, MSC/Nastran, the employed finite element method is tried to be explained. In order to guide the future studies, utilized analyses methods for beams are recounted. From these methods, Finite Element Method (FEM) which is employed to analyze the structure is investigated and explained. This method is investigated deeply and the history is narrated. Using FEM, the natural frequency results are obtained for both regular clamped beams and tapered clamped beams. Implementation of finite element method to other support types apart from clamped support, is also mentioned. Obtained natural frequency results are expressed in Euler-Bernoulli beam theory results and tabulated. Moreover, for displacement calculations the effects of various forces and thermal loads and how they are going to be employed using FEM is mentioned. Tapered clamped beams are basically mentioned. Their cross section varies with length of the beam. As a result, varying cross section changes the area and moment of inertia, making it harder to solve the equations. The equations of tapered clamped beam and their nondimensionalizations are given profoundly. For these kinds of beams, some assumptions have been taken into account which does not exist in previous studies and different taper parameters are obtained. Using Euler-Bernoulli beam theory, this type of beams' natural frequency due to bending have been calculated and first eight modes have been tabulated for natural frequency values. Same assumptions are used in Timoshenko and Shear beam theories. In the continuous sections, how these kinds of approaches are applied is explained. For the employed software, another section has been created and they are introduced. Their application fields are investigated and some screenshots from software are provided in order to extend the coverage of the study. It is tried to supply enough knowledge for one to achieve free vibration analysis for beams. The analysis software is explained generally. Since there is no study for tapered beams in literature, MSC/Patran and MSC/Nastran software and how analyses can be conducted with them are explained in a very detailed way. This study has been done in order to create a simple tutorial for MSC/Patran software. All the screenshots starting from modeling to analyses have been provided. Afterwards, first eight mode shapes of the beam are also given by screenshots. The results chapter is tried to be composed as comprehensive as possible. Analyses were conducted with Euler-Bernoulli Beam Theory, Shear Beam Theory, Timoshenko Beam Theory, as well as MSC/Nastran software. Obtained results are compared with the results provided in the previous studies and interpreted. Furthermore, same processes are made for two different beams with two different materials providing a basis for future possible studies. In results section, eleven different beam configurations are taken into account. Seven of these are chosen from reference studies while remaining four are considered in order to extend the tables. Each of these configurations, all the results with different beam theories have been tabulated and output of analyses are given. Additionally, error functions are also plotted and discussed. Eventually, future studies are also mentioned in the results and discussion chapter. As future studies, it planned to extend the work to be applicable different boundary conditions, support types. Also as a solution method, differential transform method (DTM) can be used and compared with the other results.