Mohrs Circle and Strain Gauge Rosette - Lab Report Example

Mohr's Circle and Strain Gauge Rosette Your Discipline (Mech, Aero, Ship) Summary Introduction Mohr's circle is a graphical method to determine the effect of a coordinate rotation on a tensor quantity. In the engineering sciences it is applied to analyse the effect of a coordinate rotation on stress, strain, second moment of area and moment of inertia. In this experiment, mechanical estimations of the principal strains as well as the direction of maximum strain shall be effected through the use of readings derived from a strain gauge rosette, and the resultant construction of the Mohr's Circle. The Mohr's Circle will provide the visual means by which the principal strains may be graphically estimated and assessed for accuracy by comparing the results derived therefrom to results computed by theoretical means. 2 Objectives The objectives of this experiment are: 2.1 To apply the 3-gauge rosette to an aluminum alloy beam subjected to biaxial strains; 2.2 To obtain readings from the three gauges, and to construct a Mohr's circle corresponding to the readings; 2.3 To set up and gather direct readings from two gauges set parallel to each of the axes (horizontal and vertical) along the test plane; 2.4 To compare the values for the principal strains derived from the Mohr's circle for the 3-gauge rosette, and those derived from the horizontal and vertical gauges; and 2.5 To compare the values derived mechanically from the gauge readings from those derived theoretically, as an indication of accuracy, and to explain the discrepancies between them. 3 Experimental 3.1 Apparatus An aluminium alloy beam, clamped at one end within a rig containing a cam whose full-range rotation leads to a repeatable tip deflection of = 0.5" = 12.7 mm, as shown in Figure1. The beam has dimensions as follows: breadth b = 25.4 mm, depth d = 6.35 mm and length (to the cam) L = 254 mm. Three strain gauges are mounted on the upper surface at 94 mm from the clamped end; these gauges are mounted at 15, 45, and 75 with respect to the longitudinal, x-axis, of the beam, as shown in Figure 2. Each of these gauges can be selected using a switchboard, to be one arm of a Wheatstone bridge arrangement [1 research and give reference ], with a dummy strain gauge providing temperature compensation. The Wheatstone bridge is a divided bridge circuit used to measure electrical resistance; these minute changes in resistance correspond to strain in a strain gage in what is known as a bonded resistance strain gage [2] Note: A reference was already provided by this, #1 below; should other references be provided Figure 1: Schematic of the clamped beam and cam Figure 2: Schematic of the strain gauges 1-5 as mounted on top of the beam 3.2 Procedure 1. The strain gauge amplifier should be warmed up, and set to the correct gauge factor of 2.1. 2. Select gauge 1 and balance (zero) the bridge with the beam unloaded. Then, rotate the lever to bend the beam and note the strain gauge reading. Rotate the lever back to its original position. 3. Repeat step 2. For gauges 2 and 3 and note the strain. 4. Results 4.1 Measurement Table 1: Strain measured from gauges 1-3 for unloaded and loaded beam. Beam condition Strain [] gauge 1 Strain [] gauge 2 Strain [] gauge 3 Strain [] gauge 4 Strain [] gauge 5 Unloaded 0 0 0 0 0 Loaded 4.2 Analysis On a graph paper, construct Mohr's circle for strain using: a) the geometrical method shown during the lab applying the reading from gauges 1-3 and b) using the general method applying the reading from gauges 4 and 5 assuming zero shear strain. From a), determine the principal strains I and II, as well as the direction of maximum shear strain. Also, determine the value of Poisson's ratio, , for the material. In this experiment I should theoretically be equal and in the direction of x. Inevitably there will be experimental errors so your result may be different. In the discussion section (6.), discuss possible reasons why the maximum strain obtained may not be predicted to occur in the x-direction, compare your experimental values obtained from gauges 1-3 with the ones from gauges 4 and 5 and also with values those from theory presented in the following section. 5. Theoretical Prediction For the rectangular beam cross-section, the second moment of area is give by . The tip deflection of a cantilever beam subject to a force W at the free end is [3]. Assuming a Young's modulus for aluminium of the required force to produce a deflection of 12.7 mm is given by Figure 3: Bending moment at strain gauge location At the strain gauge location, as shown in Figure 3, the bending moment M is Now employ the expression [3], and on the upper beam surface , to give . There is zero stress in the y- and z-directions, so Hooke's law reduces to , . Thus , . 6. Discussion The Mohr's circle constructed shows that values obtained through the use of the geometrical method applied to readings from the three-gauge rosette approximate those values obtained by the general method applied to readings from the 45-degree angled gauges aligned along the horizontal and vertical axes. However, the readings for the principal strains obtained from the gauges vary from those computed through the theoretical method by an error of 15% for x, and 7% for y. The errors are tolerable, as the units in microstrains are minute, and when rounded off would, for practical purposes, be useful in providing information of the beam's tensile strength. It must also be recalled that the Wheatstone bridge strain gage is dependent upon minute variations in current passed through a system of resistances, which again are subject to irregularities and imperfections of construction, as well as the physical limitations in the sensitivity of the gages. Furthermore, one cannot discount the compounding effect of human error in introducing additional discrepancies in the readings. It is common expected that manual systems would be prone to yield detectable errors, which is why the Mohr's circle is described as a method of estimation or approximation. It is possible that the maximum strain may not be predicted to occur in the x-direction, due to inconsistencies in the density of the material in the construction of the beam; some areas may be slightly more dense than others, or dimensions of thickness or breadth may not be perfect in some areas, causing irregularities in the distribution of the stressor. 7. Conclusion Based on the results obtained, it may be concluded that the mechanical method employing the use of the strain gage rosette and the Mohr's circle provide reasonably accurate estimations of principal strain measurements in a biaxial stress system, when compared to values obtained through theoretical prediction models. References 1. PP Benham, RJ Crawford and CG Armstrong, Mechanics of Engineering Materials, 1996, second edition, Harlow, Addison Wesley Longman 2. Omega Engineering, Inc. Accessed at: http://www.omega.com/Literature/Transactions/ volume3/strain2.html 3. WD Pilkey, Analysis and Design of Elastic Beams. John Wiley & Sons, Inc, 2002. Nomenclature In alphabetical order, Greek Symbols at end b breadth d depth E Young's modulus I Second moment of area L length M bending moment W load x, y, z Cartesian coordinates direct strain direct stress deflection Appendices Appendix 1: Schematic of a Wheatstone Bridge [National Instruments, accessed at National Instruments, accessed at http://zone.ni.com/devzone/cda/tut/p/id/4172#toc0] Wheatstone Bridge Assembly Notes 1. Use appropriate numbering of sections as outline (Introduction is numbered 1) 2. Use Word's equation editor (if you choose to use word) to insert equations via: Insert - Object - Microsoft Equation; Greek letters and mathematical symbols are available through buttons 3. Style: Avoid using "I", "me" or "we" Use past tense This is a formal report -be succinct but complete 4. The highlighted parts as well as missing sections need to be addressed by you! Read More