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Deflection.

Abstract:

In this experiment, we were needed to find the deflection of Ring, Semicircle and quadrant made from the curved beam. The experiment is carry out by applied these beam with a load that weight 5N for circle and 2N for Semicircle and Quadrat. For the ring shape, the load is added 5N contiuosly until the load 40N and the dial reading is note down every time the load is added. Similar step is repeated using Semicircle and Quadrant that is we add 2N load continuously until 14N and take the dial reading. For theoretical value we use the Castigliano’s Theorem, theoretical value of deflection can be calculated. Through this experiment, the validity of the Castigliano’s Theorem in curved beam can be proven.Comparison is made between the measured deflection values and theoretical deflection values. Among reasons for discrepancies are likely the parallax errors when reading was being taken from the dial gauge due to the sensitivity of the dial gauge. A slight vibration or impact on the table will affect the reading on dial gauge. Generally, theoretical values exceed experimental values.

Introduction:

The proving of Deflection of Circular shape is based on the diameter deflection elastically under load. Applied load is known from its characteristic load-deflection curve. As far as studies are concerned, Read and Bell(Reid, S.R, and Bell, 1982) pointed out the fact that experiments in which metal rings are compressed to large deflections by a pair of opposed concentrated loads reveal a load-deflection characteristic which varies with the simple theory based upon rigid-perfectly behavior. Thus the influence of strain hardening on the deformation of thin rings subjected to opposed concentrated loads was investigated using a model in an approximate fashion and it is shown how the discrepancies between the experiments and the simple theory arise. O’Dogherty presented the fundamental formulaefor the moment and strain distributions in circular, octagonal and extended octagonal rings. Expressions were also given for calculating ring deflections or stiffnesses. A design equation for determining ring thickness is derived, based on maximum strain criteria for the ring material and on data from measurements of strain gauge bridge sensitivities of six orthogonal ring dynamometers(O’DoghertyM. J,1996). A procedure was given for the design of extended octagonal rings in terms of geometrical parameters. Design curves were presented for the determination of an appropriate mean ring radius and the calculation of ring thickness. Formulae were also presented for the calculation of the strain gauge bridge sensitivity to the applied orthogonal forces and moment.

In this experiment, the Castigliano’s theorem is involve for deflection of curved beam experiment. The Castigliano’s theorem is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the Energy principle structures, he is known for his two theorems. The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement (Wikipedia 2011).Though curved bars are not commonly found as structures by themselves, they are usually part of a mechanical member which has a combination of straight and curved elements. The study of how curved bars deflect is thus, important so as estimate the total mechanical displacement of structures that incorporate curved section. The calculation by Castigliano”s theorem of the deflection xi is simplified if the differentiation with respect to the load Pj is carried out before the integration or summation( Ferdinand , E. Russell and John, 2006). The general expression of Castigliona’s theorem is as follow: δ = 0sMEI ds*dMdW=1EI*0sMdMdW*ds--(1)

Where

strain energy U=M32EIdx

Where M is the moment induced by the force of loading, E is the elastic modulus of the beam material, I is the moment of inertia of the...