Understanding of the Youngs Modulus Concept - Lab Report Example

LAB REPORT Name Institution Instructor Date Contents Introduction 2 Aim 2 Procedure 2 Experiment 1: 3 Experiment 2: 4 Experiment 3: 5 5 Experiment 1: 5 Experiment 1 Results 5 Experiment 1 Discussion 6 Experiment 1 calculations 8 Comments on results and calculations 9 Experiment 2: 9 Experiment 2 Results 9 Experiment 2 Discussion 10 11 Experiment 3: 11 Experiment 3 Results 11 Experiment 3 Discussion 13 Macaulay’s method 14 Calculations for experiment 3 14 Conclusion 16 References 17 Introduction This experiment is concerned with the determination and understanding of the Young’s Modulus concept through carrying out of three different experiments. These experiments include the one involving deflection of beams where the supports to the beam are different. This experiment involved the set-up of a simply supported beam, which was followed by addition of weights and taking records of the outcomes. The second experiment in this series of experiment involved the same procedure but one end of the beam used here had a fixed support (Czichos, Saito and Smith, 2006). However, in the third one, the same experimental set-up used in the second experiment was maintained but with slight changes that involved addition of a dial gauge at the centre of the beam to create a static indeterminate situation. Aim This laboratory experiment was aimed at obtaining a comprehensive and clear understanding of the concepts, which include Young’s modulus in differently supported deflection beams. Another aim of this experiment was to allow the mechanical engineering students to become familiar with various problems and cases requiring collection of experimental results and their study. Procedure The experiment carried out involved the consideration and use of several equipment and tools. These included the following: Indicator screw for support movement Allan Key Weights Steel Bream Calliper Load Cell Hook Experiment 1: 1. The beam was set-up was done on supports with knife-edges as a simulation for simply supported beam with a hanger and a dial gauge as shown below. 2. The beams code was noted and the thickness as well as the width of the beam was measured using a calliper. The measurement of the span L for the beam was measured and recorded then the dial gauge was initialised using the dial gauge through zeroing to ensure a firm seating of the dial gauge. Further, it was ensured that the ten weights (each measuring 0.5 Lg) were ready for use. 0.5 Kg weight was then added on the hanger. 3. Backlash was eliminated every time before taking any reading through frequent tapping on the dial gauge. This took place while the addition of weighs was happening carefully to avoid their removal. 4. The deflection of the beam was then recorded as the point in the middle 5. The addition of weights and taking of readings was performed repetitively until 3.5 Kg was attained. This was followed by the unloading of the 0.5 kg weights and ensuring that the recording of the dial gauge readings was done. The deflection obtained at the zero deflection was also recorded. 6. After the numbers were collected and recorded, the plotting of a graph of load against beam deflection in metres was done. 7. The gradient of the best fit straight line was determined through the plotted points 8. Proof of the beam material ‘s Young’s modulus was done to satisfy the formula indicated by: 9. The Young’s modulus value obtained in the first experiment was then used in the second experiment Experiment 2: Just like the commencement of the first experiment, the set-up of the second experiment was done. 1. The dial indicator was initialized through zeroing it 2. The cell indicator was zeroed and loaded 3. 0.5 kg was added to the hunger before the tip was returned of the dial gauge was made to return to zero through a slow turning of the knuckled ring in a particular direction. The tapping on of the dial gauge in the course of turning the knuckle ring was done to minimize backlash. 4. There was the addition of more 0.5 kg and repetition of the steps above until a mass of 3.5 kg was attained 5. 0.5 kg was unloaded in a stepwise procedure where the cell load reading was taken after zeroing the gauge 6. The unloading of 0.5 kg continued while recording of the cell load readings was being done until the load was zero Experiment 3: Before the conduction of the third experiment, the support at point A was changed and made fixed. The Allan key was then used in avoiding faking and slipping of the bolts and the point of contact 1 Similar to the first and the second experiments, the initial step involved the zeroing of the two gauges while tapping to eliminate striction 2 The cell load was then read 3 1 kg was added to the hunger and adjustment of the load done until the dial gauge was back to zero 4 Two readings were recorded for the dial gauge and the load cell 5 Weight addition and reading taking were repeated until 5 kg was attained Experiment 1: Experiment 1 Results The results obtained from the first experiment were recorded and tabulated as indicated in table one below Experiment 1 No. Load in Kg Load in Newton Loading (Beam Deflection in cm.) Loading (Beam Deflection in m.) Unloading (Beam Deflection in cm.) Unloading (Beam Deflection in m.) Average 1 0 0 0 0 6.12 0.0612 0.0306 2 0.5 4.905 7.68 0.0768 8.97 0.0897 0.08325 3 1 9.81 9.42 0.0942 9.62 0.0962 0.0952 4 1.5 14.715 11.12 0.1112 11.2 0.112 0.1116 5 2 19.62 12.84 0.1284 12.94 0.1294 0.1289 6 2.5 24.525 14.55 0.1455 14.65 0.1465 0.146 7 3 29.43 16.24 0.1624 16.26 0.1626 0.1625 8 3.5 34.335 17.99 0.1799 17.99 0.1799 0.1799 Table 1 Experiment 1 Discussion According to the information and the description given in the procedure, the dimensions of the beam were determined, recorded and used in the calculation of the moment of inertia as indicated in table 1 below. Beam A Beam Dimensions Conversion to SI Length 1.2 m 1.2 m width 25.7 mm 0.0257 m height 8.34 mm 0.00834 m Moment of Interia (I) 1.24237E-09 m^4 Slope 0.006829401 E 4.242968689 Table 2 The data obtained from the first experiment was computed using excel to allow for the plotting of the graph of Load against Deflection. The gradient determined after the plotting of the graph using data from the first experiment was 0.00682 N/m, which was later used, in the second experiment. Graph of Deflection (mm) against Load (N) Figure 1 The Slope obtained from Experiment 1 was found to be 0.00682 (N/m) and hence was used in experiment 2. Proof that deflection is given by is done through the Macaulay’s approach Experiment 1 calculations Considering a beam having a base length given as 0.0257 m and the height given as 0.00834 m, the determination of the moment of inertia employed the use of the formula: This was computed to obtain: = In this case, the length of the span was considered as 1.2 m (S = 1.2 m). From the graph of Deflection (mm) against Load (N) in figure, the calculation of the gradient of the line was done with the use of the relation; . This relation led to the determination of the slope for this particular graph as 0.006829. The Young’s modulus of the beam was then obtained through the use of the formula indicated as . The values were substituted and the Young’s modulus value computed and obtained as: . Comments on results and calculations The experimentally determined and calculated value of the Young’s modulus is usually considered inaccurate. This is due to a fact which is known and which suggests that the value of Young’s modulus for brass is given as 102 Gpa. Experiment 2: Experiment 2 Results The results obtained from the second experiment were recorded and tabulated as indicated in table one below Experiment 2 No. Load in Kg Load in Newton Newton/2 LCD loading LCD Unloading Average 1 0 0 0 0 0.7 0.35 2 0.5 4.905 2.4525 12.8 10.3 11.55 3 1 9.81 4.905 25.5 25.3 25.4 4 1.5 14.715 7.3575 38.2 37.8 38 5 2 19.62 9.81 51.4 50.7 51.05 6 2.5 24.525 12.2625 63.4 63.8 63.6 7 3 29.43 14.715 77.1 76.4 76.75 8 3.5 34.335 17.1675 94.8 94.8 94.8 Table 3 Experiment 2 Discussion The idea of the second experiment was similar to the one in the first experiment in the sense that it also involved the addition of loads and their conversion to Newtons. However, there were slight differences which included the use of expressions for Load Cell Deflection in the definition of spring extension (Guo and Shi, 2011). In this case, the plotting of graph was done and a gradient of the graph, which represented the spring constant, was obtained. The spring constant obtained in this particular experiment was later used in the third experiment. The data obtained from the outcome of the second experiment were used in the plotting of a graph of Force against the deflection of the beam. The slope of the graph was computed and found to be 0.19 N/m (k=0.19) which represented the load cell spring constant. Graph of Force in (N) against Deflection in (LC) Figure 2 Experiment 3: Experiment 3 Results The results obtained from the third experiment were recorded and tabulated as indicated in table one below Experiment 3 No. Weight Force Deflect at B in mm Deflect at B in m Unload Def at B in mm Unload Def at B in m Average 1 0 0 0 0 0.02 0.00002 0.00001 2 1 9.81 0.19 0.00019 0.22 0.00022 0.000205 3 2 19.62 0.35 0.00035 0.39 0.00039 0.00037 4 3 29.43 0.52 0.00052 0.54 0.00054 0.00053 5 4 39.24 0.78 0.00078 0.7 0.0007 0.00074 6 5 49.05 0.85 0.00085 0.85 0.00085 0.00085 Table 4 Experiment 3 No. Load LC at c Reaction Load at C Unload LC at c Reaction unload C Average 1 0 0 9 1.709626556 0.854813278 2 29.5 5.603775934 34.3 6.515576763 6.059676349 3 48.5 9.212987552 53.1 10.08679668 9.649892116 4 72.3 13.734 73.5 13.96195021 13.8479751 5 93.9 17.83710373 91.8 17.43819087 17.6376473 6 115.9 22.01619087 115.9 22.01619087 22.01619087 Table 5 Experiment 3 Discussion The third experiment involved the making and recording of two measurements; one of the measurements was taken for the dial gauge and the other one for the load cell. The spring constant determined in the second experiments was applied here in finding the reaction force exerted by the load cell. The data obtained in both cases were used in plotting of two separate graphs as indicated below. Graph of Deflection in (mm) against Load in (N) Figure 3 Graph of Reaction Force in (mm) against Load in (N) Figure 3 Macaulay’s method In the third experiment, the static state of the beam had been subjected to a change to static indeterminate from static indeterminate. MA, RA and RC are three forces, which are as a result of the nature of beam support. Where Rc represents the reaction force acting on the beam. Calculations for experiment 3 Applying the formula , which is an expression for the reaction force of the beam, enabled the comparison between the experimental values of RC and the theoretical ones. Considering that the additional external weights to the beam were 1 kg for every load, the formula was found to be useful in the computation of the gravitational acceleration. Therefore, the theoretical reaction for the initial load would be obtained as (Guo and Shi, 2011): The second load would then be computed as: The reaction forces were computed up to the fifth load a comparison made between the theoretical and experimental forces as indicated in the table 6 below. Reaction Force/ Load Theoretical values Experimental values 1 3.065 5.603 2 6.131 9.212 3 9.169 13.734 4 12.262 17.837 5 15.328 22 Average 9.191 13.6772 Similarity 67.1994268 Table 6 Table 6 above indicates a comparison between theoretical and experimental reaction forces values. It is evident that the similarity value of 67 is erroneous. This errors is attributable to several reasons which include: The errors encountered in the wrong usage of the formulaswhere the condition does not warrant their use Errors resulting from lab measurements and recording of the data and information as well as the associated in accuracies Improper or incorrect plotting of the values leading to the obtaining of a wrong value of the slope A wrong understanding of the given experimental values such as E=4.12 Gpa Conclusion In conclusion, the performance of this experimental exercise presented the students involved with an opportunity for participation and a practical interaction with the real life situation regarding beam deflection and Modulus of elasticity. In addition, the students who took part in the experiment were able to acquire useful skills and knowledge associated with setting up experiments and taking of experimental results. The experiment also offered a better opportunity for the experimental proving theoretical learned concepts such as the Macaulay’s method, which proved to be very useful and effective in obtaining solutions to the determinate and indeterminate static problems. References Czichos, H., Saito, T., & Smith, L. R. (2006). Springer handbook of materials measurement methods. Berlin, Germany, Springer. Guo, Z., & Shi, X. (2011). Experiment and calculation of reinforced concrete at elevated temperatures. Waltham, MA, Butterworth-Heinemann. Read More