Bending Beam Concepts by Use of Beam Investigation - Lab Report Example

Structural Loading Elements and Materials 1.0 Introduction 3 1.1 Theory 3 1.1.2 Strain ( QUOTE

4 1.1.3 Young’s Modulus (E) 4 1.1.4 Stiffness, thickness and the constant for a beam 4 1.1.5 Beam deflection 5 1.1.6 Torsion Test 6 2.0 Procedure 7 2.1 Cantilever Test procedure 7 2.1 Cantilever Test procedure 7 2.2 Procedure for Torsion Test 8 2.2 Procedure for Torsion Test 8 3.0 Results 8 3.1 Cantilever test results 8 3.2 Calculation E 9 3.3 Torsion results 10 3.4 Calculation of G 11 4.0 Discussion 12 5.0 Conclusion 12 References 13 1.0 Introduction The student can be able to see and prove bending beam concepts by use of beam testing equipment. The cantilever bending and torsion test in solid shafts are some of the tests that can be performed. The desogn of the experioiments are in away that , they are made to be easy in their performance , they are clear with high level of accuaracy being achieved and they be undertaken by students working in groups or as individuals. Aim of study 1) Apply mechanical theory to simple static and dynamic engineering systems. 2) Identify basic material properties and select materials required to satisfy specific applications. 3) Identify problems, perform calculations and formulate solutions using models and measurement. 4) Perform practical/laboratory investigations. 1.1 Theory 1.1.1 Stress ( Stress refers to a force applied to a material and acting on specified area such that we have We have compressive or positive stress when the force acts in a way to compress the material while tensile stress or negative stress will be experienced when stress causes tension. 1.1.2 Strain () Strain () is what is obtained when the change in length is resulting from application of force is divided by the original length of material. Whether there is tensile or compressive strain is all dependants on the direction of force involved. 1.1.3 Young’s Modulus (E) The stiffness of a material is measured by its Young modulus which can be obtained through division of stress and strain with stiffer materials having a bigger E value. The expression of E is (Roylance, 2001). 1.1.4 Stiffness, thickness and the constant for a beam The stiffer the beam the smaller the deflection that will occur when the beam is subjected to a load (w). This may be analogous to Young’s Modulus (E) for the case where we have elastic strain, but for the case of E focus is on the property of material, but with regard to stiffness , the material of the beam as well as the dimension come into play.in case the bending of the cantilever takes place within the elastic region , then the stiffness can be obtained by dividing load applied and the deflection caused.  The stiffness of the beam will increase when its thickness is increased where the relationship is such that the stiffness of the beam is directly proportional to the thickness of the beam 1.1.5 Beam deflection We may have a lot of variation in beams both with regards to the dimension (geometry) and the composition. We may have beams that are straight or those which are curved , some may be tapered while others may have a uniform cross section. In terms of material composion, the beam may be made of homogeneous materials while others may be made of composite materials. With all this factors coming into play the analysis will be very complex , but luckily in many engineering situations we do not have such complicated situation . Things are made simple by Having straight non-tapering beams Deformation is strictly elastic Slander beams are looked at The deflection experienced are small With this assumptions we have the equation governing deflection  of the beam being given as Here we are looking at the second derivative for the deflected shape with respect to distance from fixed or anchoring   is described as the curvature, E  is Young modulus, I is the   area moment of inertia for the cross-section, while M gives the internal bending moment for the beam. A (weightless) cantilever beam, with an end load, can be calculated (at the free end B) using the expression 1.1.6 Torsion Testing Torsion will be experienced upon subjection of a shaft to a twisting moment, torque. This comes into play both cases involving rotating shafts for example in and motors and engines and stationary shafts for example in bolt and screw. Due to the torque applied, there is twisting of shaft from one end relative to the other end with shear stress being induced over the cross section. Failure in the shaft may come as a result of pure shear or it may be as a result of shear in combination with bending as well as stretching. The deflection in a shaft will vary in direct proportion with the torque causing it ie This is in agreement with theoretical formula for calculation of angular deflection courtesy of ( Boyer, 1987) :  . Where =constant where L is effective length of shaft, G is shear modulus of elasticity and J is polar moment of areas This matched to the theoretical formula for calculating angular deflection: 2.0 Procedure 2.1 Cantilever Test procedure This test was about performing bending test on a cantilever that was end loaded. First the bar was clamped in a horizontal axis and then hanging weights at the free end to cause defection deflection that was sensed by a dial test indicator (dti) at the point of application of the weight. The dimensions of the cantilever and the position of the hanging mass relative to built in end of the cantilever were taken. 2.2 Procedure for Torsion Test Bu use of the hand-operated testing machine, there was application of torque to the shaft that was being tested resulting in its deflection which was magnified with the figure being read from a protractor. There was increasing of the loading step by step so that five different values were obtained. The measurement were then taken in reverse direction as the masses were bveing downloaded so that five more values were recorded. There was then a repeat of loading and downloading so as to have another set of values. 3.0 Results 3.1 Cantilever test results After the test being undertaken for the cantilever the results that was obtained was as in table 1 and figure 1. It can be observed from the figure that there was a linear relationship between deflection and the force applied. The results in table1 shows that there is slight deviation in values in the process of loading and downloading but the deviation insignificant. Table 1 Mass (kg) Force (N) Deflection(1) Deflection(2) Average 1 9.81 0.74 0.90 0.82 2 19.62 2.50 2.61 2.56 3 29.43 4.19 4.29 4.24 4 39.24 5.90 6.01 5.96 5 49.05 7.70 7.67 7.69 6 58.86 9.21 9.32 9.27 7 68.67 10.46 10.92 10.69 8 78.48 12.66 12.90 12.78 Figure 2 3.2 Calculation E Using the expression Gradient of graph = Where  183263247900=183.2GP 3.3 Torsion results The shaft dimensions were as follows The overall length of shaft L=245mm =0.245m Effective length =210mm = 0.21m Diameter of shaft = 15.9=0.0159m The results of the shaft test are given in table 2 and the figure 4 gives the graph devived from the data. From the figure 2 the relationship between angular deflection vs torque for the test sample can be seen. An increase in the torque resulted to an increase in angular deflection. Table 2 Mass T (Nm) Loading Unloading Avg radians 5 24.9 0.9 0.74 0.82 0.01435 10 49.83 1.7 1.48 1.59 0.0278 15 74.75 2.6 2.21 2.41 0.0422 20 99.67 3.4 2.95 3.18 0.0557 25 124.59 4.3 3.69 4.0 0.070 Figure 4 3.4 Calculation of G From the equation associated with the graph the gradient of the graph is 0.558X10-3=0.000558 From the relationship G We have  0.000558 = gradient J= 4.0 Discussion For the first test involving cantilever it was found that the deflection with a linear relationship to the increase in the force applied. This conformed to the theoretical expectation. The calculated E value for the cantilever was found to be 183.2GPa which is relatively a small deviation from the theoretical E value of steel, the material which the cantilever is made from. The variation in the two values (calculated and theoretical) may be attributed to the fact that the actual steel that the specimen was made from had E value which was slightly lower. This could be attributed to problem with quality control during the manufacturing process. In the torsion experiment the relationship between the angle of twist and applied torque was found to be linear just as it is expected theoretically. The value of G obtained from the graph was 60Gpa, which was lower that the theoretical value for steel whose value is 80Gpa. This discrepancy in the value of G from the theoretical value could be attributed to the fact that the angles of twist are very small relative to the sensitivity of the instruments used to measure the deflection. 5.0 Conclusion From the experiment it has been shown that the the E value for a material can be obtained by cantilever experiment. The exprementy will be successful on condition that the delection involved as small. The E modulus of ridgidity can also be obtained by performing a tortion experiement. References Gere, James M.; Goodno, Barry J. Mechanics of Materials (Eighth ed.). pp. 1083–1087. ISBN 978-1-111-57773-5 Boyer, H.F., (1987).Atlas of Stress-Strain Curves, ASM International, Metals Park, Ohio,. Courtney, T.H., (1990). Mechanical Behavior of Materials, McGraw-Hill, New York, Hayden, H.W., Roylance D. (2001). Stress-strain curves. Cambridge Read More