Structural Loading Elements and Materials - 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 11 5.0 Conclusion 12 References 12 1.0 Introduction By use of beam equipment the students has the ability of seeing and proving beam bending concepts. This includes cantilever and torsion testing. The beam apparatus allows the students to see and prove basic beam bending concepts. The experiments are designed in such away that they are easy, clear and accurate and are ideal for students when working in groups or when they are working in groups. 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 ( This is the force applied to a material for a certain area and given by the expression Stress may be compressive in case the force in question causes compression for the case where the force causes tension then the force will be considered as being tensile. 1.1.2 Strain () Strain () gives the ratio between change in length due to applied force and the original length the expression for strain is given as The strain can be compressive (positive) or tensile (negative) depending on the direction of force causing it. 1.1.3 Young’s Modulus (E) Young modulus is the obtained by dividing stress and strain and is a measure of the stillness of the material that is being investigated where a material that is stiffer will have higher E value. E is given as (Roylance, 2001). 1.1.4 Stiffness, thickness and the constant for a beam A beam that is stiffer will have smaller deflection for a load (w) compared to a beam with less stiffness. This may be seen as being similar to Young’s modulus (E) in elastic strain, only that E looks at property of the material, on the other hand stiffness depends on the beam materials as well as the dimension. While the beam (cantilever) bending is within elastic region, we have the stiffness (S) being given as the ratio of applied load and deflection  Increasing the thickness of a beam results to an increase in stiffness. Stiffness is directly proportional to the cube of thickness and these results to the relationship 1.1.5 Beam deflection In beams there can be a lot of variation in terms of geometry and composition. Some beams may be straight or curved; some beams may have uniform cross section or we may have a case where the beam is tapered. The materials used in the beam maybe homogeneous or the beam may be made of a combination of materials (composite). With all this factors to be put into consideration the analysis may become complex but for many engineering situations the cases involved are not complicated. There will be simplification in analysis when Beams are straight with no tapering Only elastic deformation is experienced in the beam The beam dealt with are slender Small deflections are dealt with With this simplification the equation that governs the deflection  of the beam is given by Here we the second derivative of the deflected shape with respect to  is described as the curvature, E  gives Young modulus, I represent   area moment of inertia for the cross-section, while M gives the internal bending moment for the beam. The elastic deflection  and angle of deflection  (in radians) at the free end in the example image: A (weightless) cantilever beam, with an end load, can be calculated (at the free end B) using 1.1.6 Torsion Test Torsion will always occur whenever a shaft is subjected to a twisting moment, torque. This principle applies in both rotating shafts such as those in engines and motors; and those which are stationary like in the case of bolt and screw. The torque results into twisting of the shaft in one end relative to the other end and as a result shear stress is induced in the cross section. Failure may come as pure shear or it may be shear stress combined with stretching and bending. The deflection in a shaft is proportional to the torque causing it ie This is in conformation with theoretical formula for calculating angular deflection:  ( 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 the theoretical formula for calculating angular deflection: 2.0 Procedure 2.1 Cantilever Test procedure The test involved performing bending test on an end-loaded cantilever using a trestle and hanging weights. First there was clamping of the bar in a horizontal axis and putting the hanging weights resulted to a deflection that was sensed by a dial test indicator (dti) at the point where the load is applied. 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 Using the hand-operated testing machine, torque was applied to load the rod and this resulted to a deflection which caused angular deflection or angle of twist from the large protractor. The loading was increased in step so that 5 five different force were applied to the shaft. After loading to a maximum value the next step involved downloading the masses and this also resulted to five measurements. The loading and downloading was repeated to have another set of values. 3.0 Results 3.1 Cantilever test results After performing the test the results were as shown in table 1 and figure 1. From the figure it can be seen that there was a linear relationship between deflection and the force applied. The results in table one indicates that there is slight deviation of values during the loading and downloading but the deviation is not significant. Table 1 Mass (kg) Force (N) Deflection(1) Deflection(2) Average 1 9.81 0.744 0.902 0.823 2 19.62 2.504 2.608 2.556 3 29.43 4.192 4.29 4.241 4 39.24 5.901 6.011 5.956 5 49.05 7.7 7.672 7.686 6 58.86 9.211 9.321 9.266 7 68.67 10.462 10.919 10.6905 8 78.48 12.66 12.9 12.78 Figure 2 3.2 Calculation E From the expression Gradient of graph = Where  183263247900=183.2GP 3.3 Torsion results The dimensions of the shafts were 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 was as shown in the table 2 below with the associated graph being as in figure 2 The figure 2 gives the relationship between angular deflection vs torque for the test sample. 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 In the cantilever test the deflection was found to increase linearly with the force applied. This was in agreement of what is expected theoretically. The value of E for the cantilever was calculated and was found to be 183.2GPa which is relatively close to the theoretical value of steel, the material which the cantilever is made from. The deviation could be attributed to the fact that the actual steel from which the cantilever was made from had lower E which is attributed to quality control defects in the manufacturing processing the torsion experiment there was also a linear relationship between angular twist and the applied torque just as 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 successfully been prooved the stiffness of a beam of the same type of material depends on their thickness. It has also been observed that steel will show high stiffness than brass of the same thickness. 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