Abstract
This experiment aims at determining the Young’s Modulus (E) by using the strain gauge technique. It is determined for a simply supported beam by measuring the strain in the beam when it is loaded in 3 point bending. i.e. load is applied exactly at the center of the beam. The bending of a beam is an example of Hooke’s Law, which states that, when an elastic body is strained in any way, the deformation is directly proportional to the applied stress provided the body is not strained beyond its elastic limit. In this experiment, Young’s Modulus is studied for three different materials namely: steel, aluminium and brass with solid sections using a strain gauge. The beam is simply supported on two knife-edges, which is subjected to a vertical load applied midway between the supports. A series of observations will be made by varying the load and keeping the distance between knife-edges constant. Slope will be calculated from a graph between load and strain, which will be used to calculate the Young’s modulus. These values will then be compared with the known data values to examine the levels of accuracy and possible percentage of error affecting the results. Sources of errors will also be identified and discussed. These evaluations can then be used to determine the factors, which must be concentrated on while conducting research, collecting data and statistical techniques that can be employed.
Introduction
Young’s Modulus (E) which is otherwise commonly termed as Tensile Modulus or elastic modulus of any material, may be defined as the ratio of stress along an axis to the strain along the same axis, when the range of stress is within which Hooke’s law holds. The slope of stress-strain curve at any point is called as Tangent Modulus in solid mechanics. The initial, linear portion of the stress-strain curve is known as Young’s Modulus. It is the measure of stiffness of a material.
In this experiment, Young’s Modulus of the materials is determined using strain gauge technique, where a simply supported beam is loaded at its center. The outline of the experiment is as follows.
Three beams: steel, aluminium and brass with solid sections are provided. The dimensions of these beams must be measured and recorded in Table 1. When supported at the ends and loaded with weights at the center, the strain on the surface of the beam at a quarter spans can be measured using electrical resistance strain gauges. The stiffness of the material (E) can then be calculated using the equations:
Figure 1: Experimental Setup
Figure 2: Geometry of Test Beam.
Where W is the centrally applied load (in N = W×g) =5
L is the span (distance between the supports) =0.5
d is the depth of the beam (in metres)
b is the breadth (width) of the beam (in metres)
g is the acceleration due to gravity (9.81 ms-2)
e is the strain at the quarter span position recorded by the meter. (The meter reading is in micro-strain: use reading x 10-6 in the equation, and ignore any leading zeros in the reading, e.g. –0020 = -20 x 10-6)
Experimental Procedure
Follow the following steps to conduct the experiment.
- Measure the dimensions of the provided beams at different points and record in Table 1.
- Place the supports 0.5m apart and place the beam on the supports, hang the weights exactly on the center point of the beam.
- Fix the strain gauges to the beam at 0.25L.
- Keep adding weights to the weight hanger and record the corresponding strain values from the strain gauge.
- Record strain values even while removing weights one by one. Ensure that the reading is 0 or close to 0 on removing all weights.
- Turn over the beam such that the strain gauge is now facing downwards. Record strain values again while adding and removing loads.
Calculations:
In an Excel sheet, plot the values of strain and load along x and y-axis respectively. Draw a best-fit line through the points and find its slope. This can be used to calculate E.
Therefore the calculation is,
Units GNm-2 or GPa
Discussion and Results
Aluminium:
The above graph shows the stress- strain diagram of Aluminium simply supported beam. Behavior of steel and Aluminium are similar under the application of load. They have tensile strengths in the range of 70-700 MPa. It has about one-third of the density and stiffness of steel. It is a very soft, ductile and malleable material. Hence characterized to have more strain rate than steel. It can be noticed from the graph that, unlike steel, Aluminium tends to strain more with the application of initial load of 5 N. The Young’s modulus of Aluminium lies in the range of 69x109 N/m2, GPa. From the above graph, slope value is found to be 0.4472
Steel:
Steel is an alloy of iron and small amount of carbon. This graph shows the stress-strain behavior of a simply supported steel beam. Stress or load is in units of Newton (N). It is clear from the above graph that, the strain has increased with increase in the load added to the weight hanger. Similarly, on removing load, the strain rate on the steel beam would reduce. This straight line is originally the elastic zone of the steel beam. Elastic zone is the range within which the material withstands the load without deforming, and returns to its original state on removal of load. In other words, the material is said to obey Hooke’s law within this range. Young’s modulus of steel lies within the range of 190-210 GPa. Maximum slope is calculated from any point of the graph as 0.5556.
Brass:
Brass is an alloy of copper and zinc. It is more malleable. The tensile strength of brass is in the range of 338-469 MPa and the modulus of elasticity (Young’s Modulus) lies in the range of 100-125 GPa. From the graph, it is observed that the strain rate of brass is much more compared to that of steel and Aluminium. With addition of load, the strain rate in brass beam is seen to rise significantly, unlike steel and Aluminium. The slope value is calculated as 0.1923.
Conclusion
The percentage of error in the results generated lies within the acceptable limits. Despite the quality of equipment used and effort of the people performing experiment, there will be some uncertainties associated with the collected data. Some of them are random errors, which always occur and can be rectified to an extent by repeating the experiment and statistical analysis of collected data. Other type of error is the systematic error, which occurs when the system adopted to perform the experiment is not correct. This pushes the results in one direction and often difficult to identify. Possible causes for error in our experiment may include manual error during the measurement of beam dimensions, which influences the result largely, but can be tried and improved to improve reliability of the result. Other source for error might be with the systematic error of the equipment. This can also be minimized by carrying out the experiment in appropriate environmental conditions, periodical calibration of the equipment and zero setting before using the equipment.
Of the three materials tested in the experiment, it is found that steel is cheaper, stronger and more suitable for structural purposes, when compared to that of aluminium and brass. It is heavier and more durable. Steel’s high modulus of elasticity makes it suitable for engineering applications like railroad tracks, structural applications, wires, and reinforcement with concrete, major appliances, magnetic cores, internal and external body of automobiles, trains, ships, etc. It becomes brittle at low temperatures and hence not a good option to use in extreme weather conditions.
Aluminium, though cheaper than steel, it is comparatively softer, durable, ductile and malleable metal. It is easier to work with. Despite of the softness, aluminium has a large range of applications in the fields of transportation, packaging, construction, household items, street lighting poles, etc. Aluminium has high thermal conductivity and electrical conductivity. It is capable of being a superconductor. Another advantage of aluminium is that, theoretically it is 100% recyclable without the loss of any properties.
Brass is a tensile metal and has great ability to bend. It has greater efficiency with machines compared to steel. Machinability of brass can be improved by adding lead to it. It is used in decoration purposes because of its gold color, in musical instruments, valves, bearings and moving parts since it does not break easily. Due to high anti-corrosive nature of brass, it has some significant advantages in the fields of industrial, agricultural applications and petroleum products. Brass can withstand extreme weather conditions.
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Pearson, B. (2009, August 21). Deflection of a Beam – Young’s Modulus.
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