The Stiffness of Materials Provided in the Laboratory during the Experiment - Lab Report Example

Stiffness of materials Name: Date: University Affiliation PRACTICAL REPORT TITLE: STIFFNESS OF MATERIALS OBJECTIVE To determine the stiffness of various materials provided in the laboratory during the experiment ABSTRACT Fig 1: experimental set up Stiffness is very vital mechanical property of materials in engineering. It’s mostly predetermined by the constituent’s characteristics’ of the material under study. In this experiment a TE16 experimental set up as shown above was used to test the stiffness of several materials in the laboratory. The set up is also able to determine the effect of different materials and their geometry on the overall stiffness being measured in the lab. The materials that were selected had to have been of different materials and also different cross sectional areas to get wider scope of results. The specimen is loaded into the apparatus a measured force is applied with the dial gauge showing the deflection which is then recorded by the operator. The deflection will be used to calculate the stiffness value of the material using the following formulae: Where k- stiffness SI units N/M, F applied force, δ displacement caused by the applied force. The results will then be recorded analyzed discussed and conclusions made based on results that were obtained. (Bryan, 2001). INTRODUCTION Stiffness can be defined as the ability of any given body to resist being deformed under the application of any force given force. The more flexible an object is the less stiff it becomes and less flexible the object is the more stiff it is. Taking a body which can be defined as an elastic one its stiffness K can be defined or expressed by the following equation. Where: k- stiffness SI units N/M F applied force δ displacement caused by the applied force There are several types of stiffness which include: rotational stiffness, shear stiffness and torsion stiffness depending on direction of action of the force onto the object. ( Bryan,2001). APPARATUS Some of the apparatus used were Aluminium Mild steel Brass material Force loadings each 5N EXPERIMENTAL SETUP The apparatus for the experiment were set up as shown in the figure below Fig 1: experimental set up EXPERIMENTAL PROCEDURE Procedure a The following steps were repeated for all the three materials under study. 1. A tabulation for the results was developed as shown in the following table Load (W) Newton’s Deflection in mm 0 5 10 15 20 25 Experiment 1 a Material Beam size Table 1 2. The experiment was set up with the edges of the knife place 400 mm apart the specimen dimensions were then taken with aid of a vernier caliper. 3. A load of 5N was applied carefully and deflections noted until 25 N forces had been applied to the specimen. 4. The specimen was offloaded and a graph of deflection versus loading plotted. Procedure 1b. 1. A tabulation for the results was developed as shown in the following table Load (W) Newton’s Deflection in (d) mm 48d*L [48d*L]/ l3 0 5 10 15 20 25 Experiment 1 a material B= H= I= L= L3= Table 2 2. The experiment was set up with the edges of the knife place 400 mm apart the specimen dimensions were then taken with aid of a vernier caliper. 3. A load of 5N was applied carefully and deflections noted until 25 N forces had been applied to the specimen. 4. Columns three and four were then completed and a graph of load W was then plotted versus [48dl]/ l3 5. the gradient of the graph was then obtained from the g RESULTS Procedure 1 a Brass metal Load (W) Newton’s Deflection in mm 0 0 5 1.35 10 2.71 15 4.06 20 5.425 25 6.75 Experiment 1 a material Beam size Table 3 Aluminium metal Load (W) Newton’s Deflection in mm 0 0 5 1.673 10 3.316 15 4.943 20 6.605 25 8.333 Experiment 1 a Material Beam size Table 4 Mild steel Load (W) Newton’s Deflection in mm 0 0 5 0.683 10 1.368 15 2.035 20 2.748 25 3.48 Experiment 1 a material Beam size Table Figure 2 : graph of force versus deflection for all the three materials DISCUSSION From the graphs it was evident that the force applied was directly proportional to the deflection. A lower force cause lower deflection where as a higher force applied will cause more deflection to the material. In the experiment this was also noted at the least force application of 0 N no deflection occurred. The value of deflection kept on increasing up to the highest force applied of 25 N which recorded the highest deflection for all the three materials. From the initial equation of stiffness of a material: ( Bryan,2001). Where k- stiffness SI units N/M, F applied force, δ displacement caused by the applied force. When we make F the subject to the formulae we find that the force applied is directly proportional to the deflection on the specimen beam. This can therefore be concluded that the experiment results came out as they were expected before start of the experiment. A graph of force versus deflection provides a gradient which gives the stiffness value of the material under study. The relation is also derived from the above equation Where k- stiffness SI units N/M, F applied force, δ displacement caused by the applied force since the change in y axis gives the applied force F whereas the change in X axis gives the displacement value δ. ( Bryan,2001). Aluminium had a linear gradient with the equation of y =3.967 x - 0.007 in this equation the x coefficient value of 3.967 represented the stiffness value of aluminium. Brass material had a linear equation in the form y= 3.012 x + 0.013 in this equation the x coefficient value of 3.012 represented the stiffness value of brass material. Mild steel had a linear gradient with the equation of y =7.211 x + 0.014 in this equation the x coefficient value of 7.211 represented the stiffness value of brass material. From the experiment comparing the three materials mild steel had the highest stiffness this can be attributed to the stronger molecular bonding within its structure compared to the weaker bonding in molecules of brass material that recorded the least stiffness value. ( Bryan,2001). Procedure 1 b Brass material Load (W) Newton’s Deflection in (d) mm 48d*L [48d*L]/ l3 0 0 0 0 5 1.35 0.162 0.00000000253 10 2.71 0.3252 0.00000000508 15 4.06 0.4872 0.00000000761 20 5.425 0.651 0.00000001017 25 6.75 0.81 0.0000000127 Experiment 1 b material B= 18.05 H= 3.3 I= 64.50 L= 400 L3= 64000000 Table 6 The graph was the plotted as shown below Figure 3: a graph of load versus [48d*L]/ l3 Aluminium Load (W) Newton’s Deflection in (d) mm 48d*L [48d*L]/ l3 0 0 0 0 5 1.673 0.2 0.0000000031 10 3.316 0.398 0.0000000062 15 4.943 0.593 0.0000000092 20 6.605 0.793 0.000000013 25 8.333 0.9996 0.000000015 Experiment 1 b Material B= 19.11 H= 3.35 I=59.87 L=400 L3=64000000 Table 7 The following graph was plotted for aluminium material Figure 4: a graph of load versus [48d*L]/ l3 Mild steel Load (W) Newton’s Deflection in (d) mm 48d*L [48d*L]/ l3 0 0 0 0 5 0.683 0.082 0.0000000013 10 1.368 0.164 0.0000000026 15 2.035 0.2442 0.0000000038 20 2.748 0.329 0.0000000051 25 3.48 0.4176 0.0000000065 Experiment 1 b material B= 19.1 H= 3.18 I=51.18 L= 400 L3= 64000000 Table 8 The graph was then plotted as shown below Figure 5: a graph of load versus [48d*L]/ l3 The three graphs were combined into one to show the differences in the graph as shown below Figure 6: a graph of load versus [48d*L]/ l3 for all the three materials DISCUSSION From the graphs it’s evident that the stress applied to the material was directly proportional to the resultant strain on the x axis. Brass had an equation of the form Y=4E+09x+0.023 the coefficient of x gives the modulus of elasticity value of the material obtained from the experiment which is 40G Pa. Theoretically the modulus of elasticity value of brass is 97Gpa the deviation could have been caused by experimental sources of errors. The other reason for the very high deviation could have been a clear indicator of large number of impurities in the material which lower its modulus of elasticity ( Bryan,2001). Aluminium had an equation of the form Y=2E+09x+0.023 the coefficient of x gives the modulus of elasticity value of the material obtained from the experiment which is 20GPa. Theoretically the modulus of elasticity value of aluminium is 70GPa the deviation could have been caused by experimental sources of errors. . The other reason for the very high deviation could have been a clear indicator of large number of impurities in the material which lower its modulus of elasticity Mild steel had an equation of the form Y=2E+09x+0.016 the coefficient of x gives the modulus of elasticity value of the material obtained from the experiment which is 200GPa. Theoretically the modulus of elasticity value of aluminium is 210GPa the deviation could have been caused by experimental sources of errors. . The other reason for the very high deviation could have been a clear indicator of large number of impurities in the material which lower its modulus of elasticity ( Bryan,2001). Experimental sources of errors a. inaccuracy in taking recordings of deflection b. Errors in exact weights being used as loading c. Faulty apparatus used d. Errors during recording of results and computations CONCLUSSION At the end of the experiment we were able to identify experimental values for the modulus of elasticity of the three materials aluminium, brass and mild steel as well as their corresponding stiffness. It was concluded that mild steel had the best stiffness with brass having the least stiffness. There were deviations in experimental results when comparing to the theoretical more so when it came to determining the modulus of elasticity of the materials. This was attributed to experimental sources of errors discussed previously as well as impurities within the materials structure. ( Bryan,2001). REFERENCE Bryan, J. (2001). ‘Stiffness of materials, SFPE Handbook of mechanical properties of materials, 2nd edn Read More

Where k- stiffness SI units N/M, F applied force, δ displacement caused by the applied force.  When we make F the subject to the formulae we find that the force applied is directly proportional to the deflection on the specimen beam. This can therefore be concluded that the experiment results came out as they were expected before the start of the experiment. A graph of force versus deflection provides a gradient that gives the stiffness value of the material under study.

The relation is also derived from the above equation: k=F/δ

 Where k- stiffness SI units N/M, F applied force, δ displacement caused by the applied force since the change in y-axis gives the applied force F whereas the change in X-axis gives the displacement value δ.  ( Bryan,2001).

Aluminum had a linear gradient with the equation of y =3.967 x - 0.007 in this equation the x coefficient value of 3.967 represented the stiffness value of aluminum. Brass material had a linear equation in the form y= 3.012 x + 0.013 in this equation the x coefficient value of 3.012 represented the stiffness value of brass material. Mild steel had a linear gradient with the equation of y =7.211 x + 0.014 in this equation the x coefficient value of 7.211 represented the stiffness value of brass material.

 From the experiment comparing the three materials mild steel had the highest stiffness, this can be attributed to the stronger molecular bonding within its structure compared to the weaker bonding in molecules of brass material that recorded the least stiffness value. ( Bryan,2001).

 From the graphs, it’s evident that the stress applied to the material was directly proportional to the resultant strain on the x-axis.

Brass had an equation of the form Y=4E+09x+0.023 the coefficient of x   gives the modulus of elasticity value of the material obtained from the experiment which is 40G Pa. Theoretically the modulus of elasticity value of brass is 97Gpa the deviation could have been caused by experimental sources of errors. The other reason for the very high deviation could have been a clear indicator of a large number of impurities in the material which lower its modulus of elasticity ( Bryan,2001).

Aluminum had an equation of the form Y=2E+09x+0.023 the coefficient of x   gives the modulus of elasticity value of the material obtained from the experiment which is 20GPa. Theoretically, the modulus of elasticity value of aluminum is 70GPa the deviation could have been caused by experimental sources of errors. . The other reason for the very high deviation could have been a clear indicator of a large number of impurities in the material which lower its modulus of elasticity.

Mild steel had an equation of the form Y=2E+09x+0.016 the coefficient of x   gives the modulus of elasticity value of the material obtained from the experiment which is 200GPa. Theoretically, the modulus of elasticity value of aluminum is 210GPa the deviation could have been caused by experimental sources of errors. . The other reason for the very high deviation could have been a clear indicator of a large number of impurities in the material which lower its modulus of elasticity ( Bryan,2001).

 Experimental sources of errors:

1.  Inaccuracy in taking recordings of deflection
2. Errors in exact weights being used as loading
3. Faulty apparatus used
4. Errors during the recording of results and computations

 At the end of the experiment, we were able to identify experimental values for the modulus of elasticity of the three materials aluminum, brass, and mild steel as well as their corresponding stiffness. It was concluded that mild steel had the best stiffness with brass having the least stiffness. There were deviations in experimental results when comparing to the theoretical more so when it came to determining the modulus of elasticity of the materials. This was attributed to experimental sources of errors discussed previously as well as impurities within the structure of the material. ( Bryan,2001).

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