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Deflection of Cantilever Beam
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TABLE OF CONTENTS Page (i) ABSTRACT 1 1. INTRODUCTION 1 2. OBJECTIVE 1 3. THEORY 1 3.1 Torsion of Circular Elastic Bars 1 3.2 Bending of Circular Elastic Bars 2 3.3 Combined Bending and Torsion of Circular Elastic Bars 3 4. EXPERIMENTAL PROCEDURE 6 4.1 Apparatus and Equipment 6 4.2 Procedure 8 4.2.1 Variation of Torques Versus Torsional Strain 4.2.2 Variation of Bending Moment Versus Bending Strain 5. RESULTS 10 5.1 Variation of Torques Versus Torsional Strain 10 5.2 Variation of Bending Moment Versus Bending Strain 11 6.. DISCUSSIONS 14 7. CONCLUSIONS 16 8. REFERENCES 16 9. APPENDIX and LOG SHEETS (i) Abstract The circular shaft is mainly use to transmit torque in rotary machines and acts as a beam in the bending theory. The aim of this experiment is to verify the simple bending and torsion theories. Torsion and bending of circular elastic bars were carried out with different weights and equal weights respectively. The readings for strain measurement are then obtained from the strain gauge. 1. Introduction The circular shaft is one of the most common mechanical engineering components. It is mainly used to transmit torque in rotary machines. Sometimes the same shaft is also subjected to bending due to loading transverse to the shaft, in which case the shaft acts as a beam in the bending theory. The torsion of the shaft is governed by the theory of circular. The torsion theory is accurate when stresses remain elastic and the solution provided (the well-known torsion formula) is known to be ¡®exact¡¯. The bending of the shaft as beam is governed by the bending theory, which also requires the materials of the shaft to be loaded within the elastic range and the deflection be small. When both torsion and bending are applied to the shaft, the total stresses in the shaft can be decomposed into two parts, one is produced by bending, and the other produced by torsion. The total stress obeys the Principle of Superposition for elastic problem. 2. Objectives 2.1 To review and verify the torsion theory with its limitation. 2.2 To review and verify the bending theory with its limitation. 2.3 To understand the combined torsional and bending stress. 3. Theory 3.1 Torsion of Circular Elastic Bars To establish a relation between the internal torque and the stress it sets up in members with circular solid cross sections, it is necessary to make two assumptions. These, in addition to the homogeneity of the material, are as follows: 1: A plane section of material perpendicular to the axis of a circular member remains plane after the torques are applied, i.e. no warpage or distortion of parallel planes are normal to the axis of a member takes place. For small deformations it is assumed that parallel planes perpendicular to the axis remain a constant distant apart. This is not true if deformations are large. However, since the usual deformations are very small, stresses not considered here are negligible. 2: In a circular member subjected to torque, shear strains g vary linearly from the central axis reaching maximum at the periphery (the outside surface). 3: If attention is confined to the linearly elastic material, Hook¡¯s law applies, and follows that shear stress t is proportional to shear strain g. In the elastic case, on the basis of the previous assumptions, and at any give section a relationship can be written as follows, (tmax/c)¨°A r2dA = T, where tmax and c are constants.
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