Youngs Modulus

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Solid Mechanics Lab Report
Measurement of Young’s Modulus
Sherin Joseph
00549618

Measurement of Young’s Modulus of Aluminium using Cantilever loading

Abstract
The Young’s modulus of Aluminium was calculated using the measurements of the cantilever deflections. A beam of aluminium was clamped on one end and the other end was loaded with different weights, ranging from 1 to 15N. The deflection was then measured using a dial gauge. The slope of a load (abscissa) vs. deflection (ordinate) graph was then substituted into, E=4L3bd3(Slope). The Young’s modulus was calculated to be 67.3 GPa. The results were then compared with known Young’s Modulus value of Aluminium, 72.6 GPa.

A supplementary experiment was carried out to measure the deflection of a cantilever at different distances along the beam (from the origin) using a constant load. Measures were taken at regular intervals of 100mm from the origin when loaded with 15N. A deflection (ordinate) vs distance from origin (absicca) graph was then plotted. The known values of Young’s modulus and dimensions of the beam were used to deflection values at each interval, using the following formula yx= PL32EI xL2- 13xL3 .Both the curves were then plotted on the same graph to compare the uncertainties in the measured value.

Introduction
Young’s Modulus is a measure of the stiffness of a material. By Hooke’s law Young’s modulus is the ratio of stress to strain [where stress = forceCross sec. area and Strain =change in lengthinitial length ]. The beam is clamped to form a cantilever (fixed at one end). At the clamped end, there is no moment on the beam. When a load is applied to free end, it causes stress in the material, which leads to bending. The degree of bending is proportional to the load applied. As the load increases above the ultimate tensile strength the beam experiences plastic deformation (cannot go back to initial dimensions). When the load is increased further the material breaks. According to simple beam theory, deflection of a cantilever subject to load is given by, ∆ =PL32EI , where P is load/weight, L is the length of the rod, E is the Young’s modulus and I is the second moment of area, I= bd312 . From the Equation we can see that there is no y intercept as the equation is in the form y = mx. Therefore a plot of deflection against load will give a linear graph passing through the origin with a lope=4L3Ebd3 . From this equation Young’s modulus can be calculated as follows, E=4L3bd3(Slope)

Hypothesis
As the load increases the bending in the beam/deflection increases; the plot of deflection versus load will give a linear graph passing through the origin.

Materials & Method

Aluminium beam of known Young’s Modulus, E = 72.6 GPa
Rigid steel base, with clamp
Weight hanger, with 15× 1N weights
Dial gauge
Steel Rule, Vernier callipers
Exp 1
1) (APPENDIX A)
(APPENDIX A)
Set up the apparatus as shown for both the beams (Aluminium and steel). 2) Measure the dimensions of the beam using the callipers. 3) Zero the dial gauge with the weight hanger unhooked.
4) (APPENDIX B)
(APPENDIX B)
Slot a 1N weight onto the weight hanger and let the system equilibrate. Take the reading off the dial gauge, which gives the deflection. 5) Repeat the measurement by adding unit weights incrementing by 1N, from 1N till 15 N. 6) Repeat the experiment 3 times and calculate mean values.

Exp 2
1) Setup the cantilever as shown above, with the weight hanger hooked on. 2) Fix the dial gauge at/near the clamp and zero it.
3) Load the weight hanger with 15N and take deflection reading off the gauge. 4) (APPENDIX F)
(APPENDIX F)
Take a set of such deflection readings at regular intervals of 100mm from the origin till the end of the beam (1000mm). 5) Repeat experiment twice and calculate mean values.

(APPENDIX B)
(APPENDIX B)
Results
Exp1: The Results were tabulated; deflections against the load (which was...
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