SOME PROBLEMS ON BENDING AND VIBRATION OF BEAMS AND PLATES

Abstract

The thesis is divided into two parts. The first part deals with some problems of bending of beams aid plates of uniform and nonuniform depths. The second part deals with some problems of transverse vibration of beams and plates of uniform and stepped depths. The first part comprises of chapters I to V and the second part comprises of chapters VI to IX. A chapterwise summary is given as follows. PART I_ CHAPTER I- Effect of Secondary Terms on Bending of Beam* In this chapter instead of starting with stress equa tions of elasticity we have started with an assumption for displacement components. The displacement components are assumed to be infinite series in thickness coordinates and only a few terms are retained for our purpose. This gives rise to a cubical variation of normal stress and parabolic variation of shear stress. The equations of equilibrium are derived by energy principle. They are solved for a beam clamped at both the ends. The numerical results are compared with the shear theory as well as with classical theory. CHAPTER II- Effgct^f^gcondagL Terms on BendingnfCir.,^. Plates —"*~~~ta——~- -" ••- In this chapter a similar theory, as employed in chapter one, is employed for acisymmetric bending of a -11- circular plate. The coupled differential equations of equilibrium obtained by energy method are made uncoupled by elimination method. They are solved for a clamped edge plate. The numerical results are obtained for transverse deflection and normal stresses, and they are compared with shear theory and classical theory. CHAPTER III- Bending of Semi-infinite Plate of,.Linearly Varying. Depth with a~ Discontinuity in Variation. In this chapter bending of a uniformly loaded plate of infinite length and finite breadth is considered by using shear theory. Depth of the plate along the breadth varies linearly with a discontinuity in variation. The plate is assumed to be made up of two plates of different linear variations in depth and joined edge to edge. Both the plates have the same depth at the common edge. Ordinary differential equations of equilibrium for both the plates obtained by energy principle, can be easily solved. Arbitrary constants entering in the solutions are determined by the continuity conditions between the plates and the edge conditions for a clamped plate. Numerical results for maximum deflection and its position in the plate are computed for various values of rate of variation of depth and position of discontinuity which are taken in such a way that the average depth and breadth of the plate may remain constant. -IIICHAPTER IV- Bending of.aStepped Beam In this chapter bending of a rectangular beam, whose thickness along the length varies in steps of different length and depth, is considered. The beam is assumed to be made up of a number of small beam elements joined end to end having different lengths and different uniform depths. The equations of equilibrium for every element are derived by energy principle using shear theory. The beam is taken to be clamped at both the ends. Numerical results are computed for a beam made up of alternately thick and -fain beam elements. The alternate element have the same dimensions, but the consecutive elements, in general, have different dimensions. The variation of number of elements and their lengths and depths in a beam are chosen in such a way that the length and volume of the beam remain constant. CHAPTER V- AJtodULficatipn to Bending of Stepped Beams. In this chapter, bending of a beam made up of alter nately thick and thin beam elements, is considered. The elements at the ends of the beam are taken thick and the alternate elements are taken equal in depths. A modification taken here over the analysis of chapter IV is that the rotation of normal to the section of thick elements is not the same throughout the depth. The rotation for that part which is in continuation with thin elements is different than the rotation of the part out of it. It gives rise to five continuity conditions at each joint, instead of four -ivas taken in chapter IV. The numerical results for transverse deflection are compared with that of chapter IV. PART II CHAPTER VI- Effect of Secondary Terms on Transverse Vibration of—Beams ~""~* In this chapter, the displacement components are assumed to be an infinite series in thickness coordinates. But only two terms in longitudinal displacement and two terms in transverse displacement are retained. It can be considered as an improvement over Timoshenko theory. The equations of motion for free transverse vibrations are derived by Hamilton's energy principle. They are solved for harmonic vibrations. An eighth order frequency determinant is obtained for a beam clamped at both the ends. The frequencies are computed for first four normal modes of vibrations and they are compared with those of Timoshenko theory and classical theory. CHAPTER VII- Effect of Secondary Terms on Axisymmetric Transverse Vibration of Circular Plates. A theory similar to chapter VI is used here. The equations of motion obtained by Hamilton's energy principle are coupled in four variables. They are solved in terms of Bessel functions with the help of four auxiliary variables. A fourth order frequency determinant is obtained for a clamped edge plate. The frequency parameter is computed for first four normal modes of vibrations and compared with those -vof shear theory and classical theory. CHAPTER VIII- Transverse Vibration of,a Stepped Beam. The chapter deals with free transverse vibration of a beam of the same type as considered in chapter four. Equations of motion for every element are derived by Hamilton's energy principle on the basis of Timoshenko theory. Applying the clamped end conditions of the beam and the continuity conditions between the consecutive elements, a frequency determinant of order 4n is obtained for the beam consisting of n elements.The zeroes of the determinant give frequencies for various normal modes of vibration. Frequencies are computed for a beam made up of alternately thick and thin elements for first four normal modes of vibration. CHAPTER IX- A Modification to.Transverse. Vibration of Stepped Beamg_._ " In this chapter, free transverse vibration of a beamm of the type described in Chapter V, is considered. As in Chapter V, here also, a modification, over the analysis given in chapter VIII,is taken that the angle of shear of the part of a thick element in continuation with thin elements is different than that of the parts not in continuation. The frequency determinant obtained in this case is of order 5n+l, for a beam made up of n elements. The frequencies computed for a beam clamped at the ends are compared with the frequencies obtained in the previous chapter.