VERIFICATION OF RECIPROCAL THEOREM AND YOUNG’S MODULUS OF DIFFERENT MATERIALS BY S.HARSHA M.SUMANTH G.SUMANTH KUMAR G.RAVI TEJA

OBJECTIVE The main objective of the project is to determine the Modulus of Elasticity for different materialsof same sizes using Simply Supported Beam and Cantilever Beam and also to verify the Maxwell’s Reciprocal Theorem for these materials.

INTRODUCTION An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non- permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress– strain curve in the elastic deformation region. As such, a stiffer material will have a higher elastic modulus.

LIST OF EXPERIMENTS 1.Deflection Test on Simply Supported Beam. 2.Deflection Test on Cantilever Beam. 3.Verification of Reciprocal Theorem.

Journal Results The ranges of Modulus of Elasticity from the journals are: 1.Aluminum : 69 Gpa 2.Glass : 50 – 90 Gpa 3.Wood : 11 Gpa

Procedure For Simply Supported Beam Initially take the length, breadth, thickness for the material which is going to be used. Then arrange the material on the supports and measure the span of material. Then fix the dail guage at the position where we require the deflection. Then calculate the deflections of the material for loading and unloading of load.

Procedure For Simply Supported Beam Then use the formula for Modulus of Elasticity and determine the value of E. E = [ (K*L^3)(dw)]/[(dy)*I] Where K = 1/48 when l = l/2 L = span of material dw = load dy = deflection I = B*D^3/12

Test Results For Simply Supported Beam After conducting the experiments on the 4 materials, the values of Modulus of Elasticity are : Aluminum : GPa Bison : 6.4 GPa Glass : GPa Wood : 7.74 GPa

Procedure For Reciprocal Theorem Theorem: The defection at A due to loading at B should be equal to deflection at B and loading at A. Keep the material on the supports and measure the span, breadth, thickness. Then in this we verify the theorem by choosing the deflections at ¼ distance from supports of total span.

Procedure For Reciprocal Theorem Then determine the deflections of material by loading at one point and noting deflection at other point. Then change the positions of dial gauge and load hanger and again do the same procedure. Compare the both deflections and see whether the both are same or not.

Test Results For Reciprocal Theorem Wood has an avg. deflection of when loading applied at A and deflection at B and has avg. deflection of when loading at B and deflection at A. In the case of Aluminum, has an avg. defection of at A and at B, when load applied at B and A.

Test Results For Reciprocal Theorem Whereas in the case of Bison, it has an avg. deflection of when load applied at A and deflection at B, and has an avg. deflection of when load applied at B and deflection taken at A. As in the case of Glass, the avg. deflection is at A and at B when load applied at B and A.

WOOD Load gms Dial Gauge ReadingDeflect ion Dial Gauge ReadingDeflect ion LoadingUnloadingMeanLoadingUnloadingMean Avg At Point A At Point B

ALUMINIUM Load gms Dial Gauge ReadingDeflect ion Dial Gauge ReadingDeflect ion LoadingUnloadingMeanLoadingUnloadingMean Avg At Point A At Point B

BISON Load gms Dial Gauge ReadingDeflect ion Dial Gauge ReadingDeflect ion LoadingUnloadingMeanLoadingUnloadingMean Avg At Point A At Point B

GLASS At Point A At Point B Load gms Dial Gauge ReadingDeflect ion Dial Gauge ReadingDeflect ion LoadingUnloadingMeanLoadingUnloadingMean Avg

Procedure For Cantilever Beam As in the case of Simply Supported Beam, the procedure will be the same but there we used two supports but here we use only one support fixed rigidly. There is a slight change in the calculation formula of Modulus of Elasticity. E = [L^3*dw]/[dy*(3*I)]

Test Results For Cantilever Beam After conducting experiments on wood and bison the values of MODULUS OF ELASTICITY are like following: WOOD : Gpa BISON : 8.95 GPa

WORK IN PROGRESS