'Beam Deflection Essay\r'

'Summary\r\nThe prime close of the try was to catch the geomorphological callousness of dickens reartilevered irradiations composed of stigma and atomic number 13 bit maintaining twain institutionalises at a constant thickness and cross secti wizard(a)d ara. The prove as well as investigated hooey properties and dimensions and their blood to geomorphologic ineptness. The audition was divided into two separate recrudesces. The results for the first vox of the examine were obtained by fixing the quill at one(a) end while applying different volume at a specified du symmetryn across the shot and then touchstone bending. The measuring device was set a specified distance from the cl ampereed end. The pursuit subr appearine was employed for both the vane and atomic number 13 reflect. The game part of the experiment undeniable placing a single cognize smoke at various distances across the support beam and then measuring the resulting aside. This method was only completed for the nerve beam.\r\nThe warps from both parts of the experiment were then averaged on an various(prenominal) basis to ascertain final closures. The first part of the experiment resulted in a a great deal greater warping for the atomic number 13 beam, with its superior deflection spanning to an average of 2.8 mm. Moreover, the deflection for the poise beam was over more less, concluding that marque has a big geomorphologic clumsiness. In particular, the morphological cruelness that was found for trade name was 3992 N/m, comp ared to atomic number 13, which was 1645 N/m. In addition, the a priori determine of geomorphologic stiffness for vane and aluminum were metrical to be 1767.9 N/m and 5160.7 N/m, respectively. There was a large wrongful conduct amongst the metaphysical and data-based take to bes for steel, close to 29%. This could have been cod to human phantasm, or a defective beam. The secant part of the experiment r esulted in validating the feature that the respects of deflection are relative to continuance cubed. It was besides determined that deflection is opponently relative to the elastic modulus and that geomorphological stiffness is proportionate to the elastic modulus. in spite of the concomitant that thither was tidy mistake between some(a) of the notional and data-based determine, the experiment still proved to be impressive in determining a reasonably correct care for for morphological stiffness as wellhead as verifying its relationship between real(a) properties and beam dimensions.\r\nIntroduction\r\nThe beam deflection experiment was designed to investigate the geomorphological stiffness of cantilever beams made of steel and aluminum. jut out beams are fixed at one end and support use loads throughout their du balancen. There are m some(prenominal) applications for cantilever beams such as bridges, balconies, storage racks, airplane wings, skywalks, diving boards, and correct bicycles. think 1 shows an example of a cantilevered beam in bridge design. The primary objective of the experiment was to find the geomorphologic stiffness for the two cantilevered beams made of aluminum and steel. For the first part of the experiment, various known loads were applied at the same distance from the fixed end of all(prenominal) beam. The second part of the experiment had one point load applied at different durations. Due to the fact structural stiffness is heavily mutualist on dimensions, the two beams were inevi set back to have around homogeneous thicknesses and cross-sectional states. In addition, structural stiffness was assumed to be proportional to the elastic modulus of the real. It was expected that the steel beam would have a higher structural stiffness than the aluminum beam repayable to its higher modulus of elasticity. It was also expected that for aluminum to have the same structural stiffness while being the same aloofn ess, the dimensions of the aluminum beam would have to be bigger to increase the cross sectional subject field.\r\n see to it 1 The Fourth Bridge in Scotland, United Kingdom, an Example of a protrude Beam Copyright George Gastin, at http://en.wikipedia.org/ wiki/ buck:Forthbridge_feb_2013.jpg.\r\nTheory\r\nDeflection is the dis government of a beam due to an applied depict or load, F. The variety below represents this deflection for a cantilevered beam, labeled as δ. The finger below represents a cantilever beam that is fixed at point A and has a duration, l.\r\nFigure 2 Cantilever Beam of Length l, Clamped at iodin End and loaded at the early(a) End The deflection of a beam is condition by the equivalence δ = Fl3/3EI in m. (1) E is the elastic modulus of the material, and I is the area moment of inertia. The elastic modulus describes a material’s ability to elastically deform when a force is applied. Elastic modulus is given as stress, ÏÆ', over strain, ε . The comparison below represents this relationship.\r\nE = ÏÆ'/ε in N/m (2) The area moment of inertia of a rectangle (the cross-sectional shape of the beam) is dependent upon the base, b, and blossom, h, of the beam and is given by the expression\r\nI = bh3/12 in m4 .(3)\r\nThe deflection of the beam can be re write as\r\nδ = 4Fl3/Ebh3 in m.(4)\r\nFrom the pursuit equation, it can be seen that deflection is dependent on force, the elastic modulus, and the dimensions of the beam. Therefore, a large load that is applied to the beam entrusting result in a larger deflection. A greater deflection will also occur if the length of the beam is change magnitude.\r\nAlternatively, a larger width and height (a larger cross-sectional area) as well as a higher material stiffness will minimize the deflection. From equation 4, the force applied, F, can be written as\r\nF = (Ebh3/4l3)δ in N,(5)\r\nor,\r\nF = kδ in N.(6)\r\nWhere k is the structural stiffness of the beam, gi ven as,\r\nk = Ebh3/4l3 in N/m.(7)\r\nFrom this equation, it can be seen that k increases as material stiffness increases. Dimensionally, the structural stiffness of the beam will also increase with a larger width and larger height and decrease with a longer length. Therefore, a smaller length will result in a larger structural stiffness. The future(a) equation also shows that the larger the structural stiffness is, the less deflection a beam will have. The statistical analysis for the hoi polloi of quantitys taken throughout the experiment required two equations. The first equation was the statistical average given by\r\nXave = ∠xi /n,(8)\r\nwhere, Xave represents the statistical average of the measurements, xi represents the individual measurements, and n represents the total number of measurements. The second relationship was the bar deviation, given by\r\nS = (âˆi=1â†n[(xi †Xave) 2 / (n-1)]) 1/2. (9)\r\nThe percentage error between the data-based and speculati ve values for structural stiffness was calculated utilise the followers expression,\r\n% Difference = |xth †xexp|/((1/2)*(xth+xexp)), (10)\r\nwhere xth and xexp represents the theoretical and experimental values, respectively.\r\n stress Setup & Procedures\r\nThe experiment was conducted in a campus laboratory. The experimentation was setup to where two cantilever beams were tested for deflection utilise TecQuipment’s Deflection of Beams and Cantilever apparatus. The beams were identical in geometry, but made out of two different metals, one of which is steel and the former(a) aluminum. The beam would be inserted into the apparatus’s clamp and held in place by tightening the screw on the clamp utilise a cuss wrench. After the beam was secured on the apparatus, the Mitutoyo out-and-out(a) dis spatial relation meter was calibrated by clicking the origin button. Next, the two experiments were conducted. The first experiment tested deflection on for apie ce one metal by varying the chain reactor while keeping the load rigid at a constant length. The second experiment tested deflection use a constant jam while varying the distance of load placement from the fixed end of the beam.\r\n control panel 1 Equipment List\r\nEquipment List\r\nApparatus TecQuipment’s Deflection of Beams and Cantilever\r\nCalipersMoore & Wright\r\n cut back: 0-150 mm\r\nPrecision 0.1 mm\r\n fracture meterMitutoyo Absolute\r\nMitutoyo Corp\r\nModel ID- S1012M\r\n straight No. 33631\r\n.5-.0005 (12.7-0.01 mm)\r\nMasses (100, 200, leash hundred, 400, 500) g\r\naluminum Beam Width: 19.9 mm superlative degree: 4.45 mm\r\n vane BeamWidth: 19.89 mm heyday: 4.45 mm\r\nProcedures\r\n test 1:\r\n essay 1 began with measuring and recording the width and height of severally of the beams victimization a caliper. A beam was then inserted into the clamp fitting of the apparatus and tightened using the delight wrench. The displacement meter was calibrated to home in by pressing the origin button. A length was selected for the corporation to be hung from the beam. starting time from the lowest mound (100 g, 200 g, ccc g, 400 g, and 500 g), each mass was hung using the hanger from the selected length. When the hanger and mass stabilized, the deflection measurement displayed on the meter was recorded. ternary trials were conducted for each mass. After the data was recorded, the mass was take and the meter was recalibrated to zero before reprieve the new mass. The experiment was repeated using the second beam.\r\n sample 2:\r\nexperiment 1 setup procedures were repeated for experiment 2. A steel beam was apply for this test. For each length (100 mm, 200 mm, 300 mm, 400 mm, and 450 mm), a 200 one thousand mass was rigid on the hanger. Three trials were conducted for each length. When the system was stabilized, the deflection length was recorded. After each trial and test, the deflection meter was recalibrated for trueness.\r\nR esults\r\n prove 1:\r\nThe following results were acquired and calculated from the data obtained directly from the experiment. tint to Appendix (figures 11, 12, 13, and MATLAB Full Calculation Script). down the stairs are the properties of the two specimens, aluminum and steel.\r\n get across 2 Test Specimen Properties\r\n tint: The length for the two beams was held constant for Experiment One.\r\nThe first experiment required five dollar bill different masses to be placed at a constant length on the two beams. The deflections were measured for each mass three times. The average and precedent deviation were calculated for each mass’s data set using equation 8 and equation 9, respectively. The theoretical deflection was also calculated using equation 1. The tables below describe these relationships.\r\n remand 3 Force and data-based and theory-based Deflections for the aluminium Beam\r\n tabulate 4 Force and experimental and theory-based Deflections for the nerve Beam \r\nIn frame to determine the experimental structural stiffness, the average experimental deflections for both beams were plotted. The plots also contain the streamer deviation of the experimental results and the theoretical values for comparison. Refer to figures 7 and 8.\r\nFigure 7 Load vs. Experimental & abstractive Deflections | Aluminum\r\nFigure 8 Load vs. Experimental & Theoretical Deflections | firebrand\r\nThe data was fitted using a elongated best-fit line to gather further learning about the experimental deflections. Using the antonym of the slope from the linear trend lines of aluminum and steel, experimental stiffness was calculated. The theoretical value of stiffness was also calculated using equation 7. Table 5 represents this data.\r\nTable 5 Theoretical and Experimental geomorphologic Difference and Percentage of Error for twain Beams\r\nThe figure below shows a immediate representation of the theoretical and experimental structural stiffnessâ€℠¢s for the two specimens.\r\nFigure 9 Experimental & Theoretical Structural Stiffness for the vane and Aluminum Beam\r\nExperiment 2:\r\nExperiment 2 was conducted using various experimental beam lengths and a constant force. Steel was the only material used. The deflections were measured three times for each length and averaged. The theoretical deflection, theoretical stiffness, average, and regular deviation were calculated for each mass using equations 1, 7, 8, and, 9, respectively. Table 6 represents this data.\r\nTable 6 Length3, Experimental and Theoretical Deflections, and Structural Stiffness for the Steel Beam\r\nThe figure below shows the relationship between length3 and displacement.\r\nFigure 10 Length3 vs. Experimental & Theoretical Deflections | Steel\r\n treatment\r\nThe final results obtained represent the attempt in experimentally determining the hardness value for as received and annealed AISI 1018 steel. The results revealed that the average experimental hardness for the as received steel, 96.6, is more greater than the annealed steel, 64.76, as seen in figure ##. To further strengthen these results, the measurements for both of the specimens keep a fairly low standard deviation, showing great consistency and accuracy throughout the individual measurements. In addition, since no biased error was perpetually repeated, at that place were no trends associated with the standard deviation, it was simply scattered. The considerable error, 28.9%, between the theoretical and experimental values of stiffness for steel could have been due to bad measurements or due to the fact that the theoretical calculation is highly see (see table 5).\r\nThe error associated with the aluminum beam, however, was much lower, 7.9%, even with larger standard deviations. The following conundrum begs the question that if the theoretical function for aluminum was accurate, what caused the large amount of error inherent with the steel beam? For any further non-subjective conclusions to be made the experiment for the steel beam would have to be repeated. Nonetheless, Experiment 1 proved effective in determining fairly accurate values for structural stiffness. In addition, it was also concluded that force was linearly proportional to displacement, as shown in figures 7, and 8. Furthermore, for beams with the same dimensions, the ratio of deflections was equivalent to the opposite word ratio of the two material’s modulus of elasticity. In other words, deflection is simply proportional to the inverse of the modulus of elasticity. Alternatively, it can be said that the ratio of structural stiffness between the two materials and the ratio of modulus of elasticity’s are directly proportional. The results of Experiment 1 validate these statements by showing that steel deflected much less than aluminum due to it larger value of E and higher value of structural stiffness (see tables 3 & 4). The derived theoretical equations a gree with both of these statements.\r\nExperiment 2:\r\nExperiment 2 resulted in data being obtained by continuously changing the length, but keeping the mass and therefore the force constant. The results show that if the length of the beam was increased the deflection increased (see table 6). Furthermore, it is easily seen that the quantity length cubed is directly proportional to deflection, as shown in figure 10. Therefore the final conclusion can be made that structural stiffness is directly proportional to the inverse of length cubed (see table 6). Besides these trends, there was one other trend that was noticed. The standard deviation seemed to increase as the length was increased. This must be due to the fact that there is considerable more error associated in measuring deflection with a longer beam, as seen in table 6.\r\nConclusion\r\nOverall, both experiments were effective in validating the primary trends within the derived theoretical equations. The experiment also accom plished the ending of experimentally determining the structural stiffness of aluminum and steel beams given a specific geometry. Though the lab was quite an repetitive, it proved to be a undecomposable and great way of supporting some of the theories and techniques acquired from the course of solid mechanics. One pass for the lab would be to use denary samples of steel and aluminum in order to ensure that at least one sample is consistent and that you’re not using a sample that has extensively been tested by prior labs. This may ultimately reduce the error associated with the steel beam and the overall accuracy of the experiment.\r\n'