Beam Deflection Essay

SummaryThe primary goal of the audition was to determine the morphological unfeelingness of ii force outtilevered slams composed of brace and aluminum trance maintaining cardinal institutionalises at a constant thickness and cross sectional world. The examine withal investigated actual properties and dimensions and their descent to morphological ungracefulness. The investigate was divided into deuce separate parts. The results for the beginning part of the try out were obtained by clamping the irradiation at one extirpate speckle applying different chain reactores at a specified distance across the aerate and then cadence divergence. The touchstone device was set a specified distance from the clamped send away. The following routine was employed for two the marque and aluminum beam. The wink part of the experiment required placing a single known tummy at conglomerate spaces across the supported beam and then measure the resulting warp. This metho d was provided completed for the steel beam.The digressions from both parts of the experiment were then faird independently to ascertain final conclusions. The first part of the experiment resulted in a such(prenominal) slap-uper recreation for the aluminum beam, with its greatest deflection spanning to an average of 2.8 mm. More everyplace, the deflection for the steel beam was much less, concluding that steel has a large morphological callosity. In point, the morphologic badness that was found for steel was 3992 N/m, compared to aluminum, which was 1645 N/m. In addition, the supposed values of structural abrasiveness for steel and aluminum were work out to be 1767.9 N/m and 5160.7 N/m, respectively. There was a large faulting between the conjectural and experimental values for steel, close to 29%. This could have been due(p) to human error, or a defective beam. The reciprocal ohm part of the experiment resulted in authorise the fact that the values of deflec tion are comparative to length cubed. It was alike determined that deflection is inversely relative to the elastic modulus and that structural unfeelingness is proportional to the elastic modulus. Despite the fact that there was considerable error between some of the theoretical and experimental values, the experiment still turn up to be effective in determining a reasonably undefiled value for structural stiffness as well as indirect its relationship between material properties and beam dimensions.IntroductionThe beam deflection experiment was designed to investigate the structural stiffness of cantilever beams do of steel and aluminum. Cantilever beams are fixed at one end and support apply haemorrhoid throughout their length. There are some applications for cantilever beams such as tide overs, balconies, storage racks, airplane wings, skywalks, diving boards, and still bicycles. direct 1 shows an example of a cantilevered beam in bridge design. The primary objective of the experiment was to find the structural stiffness for the two cantilevered beams make 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 individually beam. The second part of the experiment had one point load applied at different lengths. Due to the fact structural stiffness is intemperately dependent on dimensions, the two beams were required to have almost very(a) thicknesses and cross-sectional areas. In addition, structural stiffness was assumed to be proportional to the elastic modulus of the material. It was expected that the steel beam would have a higher(prenominal)(prenominal) structural stiffness than the aluminum beam due to its higher modulus of elasticity. It was also expected that for aluminum to have the samestructural stiffness while be the same length, the dimensions of the aluminum beam would have to be bigger to increase the cross sectional area.Figure 1 The Fourth link up in Scotland, United Kingdom, an Example of a Cantilever Beam copyright George Gastin, at http//en.wikipedia.org/ wiki/FileForthbridge_feb_2013.jpg.Theory aside is the displacement of a beam due to an applied force or load, F. The figure below represents this deflection for a cantilevered beam, labeled as . The figure below represents a cantilever beam that is fixed at point A and has a length, l.Figure 2 Cantilever Beam of Length l, Clamped at One annul and onused at the Other End The deflection of a beam is given by the equivalence = 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 materials dexterity to elastically deform when a force is applied. Elastic modulus is given as stress, , over strain, . The comparability below represents this relationship.E = / 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 height, h, of the beam and is given by the expressionI = bh3/12 in m4 .(3)The deflection of the beam can be rewritten as = 4Fl3/Ebh3 in m.(4)From the following equation, it can be seen that deflection is dependent on force, the elastic modulus, and the dimensions of the beam. Therefore, a larger load that is applied to the beam will result in a larger deflection. A greater deflection will also march on if the length of the beam is increase.Alternatively, 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 asF = (Ebh3/4l3) in N,(5)or,F = k in N.(6)Where k is the structural stiffness of the beam, given as,k = Ebh3/4l3 in N/m.(7)From 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 following equation also shows that the larger the structural stiffness is, the less deflection a beam will have. The statistical compend for the multitude of measurements taken throughout the experiment required two equations. The first equation was the statistical average given byXave = xi /n,(8)where, 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 measurement deviation, given byS = (i=1n(xi Xave) 2 / (n-1)) 1/2. (9)The percentage error between the experimental and theoretical values for structural stiffness was calculated victimization the following expression,% passing = xth xexp/((1/2)*(xth+xexp)), (10)where xth and xexp represents the theoretical and experimental values, respectively.Test Setup & ProceduresThe experiment was conducted in a campus laboratory. The exper imentation was setup to where two cantilever beams were tried and true for deflection using TecQuipments Deflection of Beams and Cantilever apparatus. The beams wereidentical in geometry, but made out of two different metals, one of which is steel and the other aluminum. The beam would be inserted into the apparatuss clamp and held in place by tightening the screw on the clamp using a enrapture wrench. After the beam was secured on the apparatus, the Mitutoyo tyrannical displacement meter was gradational by clicking the fund button. Next, the two experiments were conducted. The first experiment tested deflection on for for each(prenominal) one one(prenominal) metal by varying the hoi polloi while keeping the load placed at a constant length. The second experiment tested deflection using a constant mass while varying the distance of load placement from the fixed end of the beam.Table 1 Equipment ListEquipment ListApparatus TecQuipments Deflection of Beams and CantileverCalip ersMoore & WrightRange 0-150 mmPrecision 0.1 mmDisplacement meterMitutoyo AbsoluteMitutoyo CorpModel ID- S1012MSerial No. 33631.5-.0005 (12.7-0.01 mm)Masses (100, 200, three hundred, 400, 500) g aluminium Beam Width 19.9 mm Height 4.45 mmblade BeamWidth 19.89 mm Height 4.45 mmProcedures audition 1Experiment 1 began with measuring and recording the width and height of each of the beams using a caliper. A beam was then inserted into the clamp fitting of the apparatus and tightened using the enthral wrench. The displacement meter was calibrated to zero by pressing the origin button. A length was selected for the mass to be hung from the beam. Starting from the lowest mass (100 g, 200 g, 300 g, 400 g, and 500 g), each mass was hung using the hanger from the selected length. When the hanger and massstabilized, the deflection measurement displayed on the meter was recorded. Three trials were conducted for each mass. After the data was recorded, the mass was removed and the meter was rec alibrated to zero in front hanging the new mass. The experiment was repeated using the second beam.Experiment 2Experiment 1 setup procedures were repeated for experiment 2. A steel beam was used for this test. For each length (100 mm, 200 mm, 300 mm, 400 mm, and 450 mm), a 200 gram mass was placed 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 accuracy.ResultsExperiment 1The following results were acquired and calculated from the data obtained at present from the experiment. Refer to Appendix (figures 11, 12, 13, and MATLAB Full Calculation Script). beneath are the properties of the two specimens, aluminum and steel.Table 2 Test archetype PropertiesNote The length for the two beams was held constant for Experiment One.The first experiment required five different masses to be placed at a constant length on the two beams. The deflect ions were measured for each mass three times. The average and streamer deviation were calculated for each masss data set using equation 8 and equation 9, respectively. The theoretical deflection was also calculated using equation 1. The tables below describe these relationships.Table 3 Force and Experimental and theoretic Deflections for the Aluminum BeamTable 4 Force and Experimental and suppositious Deflections for the Steel BeamIn order to determine the experimental structural stiffness, the average experimental deflections for both beams were plotted. The plots also contain the shopworn deviation of the experimental results and the theoretical values for comparison. Refer to figures 7 and 8.Figure 7 Load vs. Experimental & conjectural Deflections AluminumFigure 8 Load vs. Experimental & Theoretical Deflections SteelThe data was fitted using a elongate best-fit line to gather promote information about the experimental deflections. exploitation the inverse of the slope fr om 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.Table 5 Theoretical and Experimental morphological Difference and Percentage of Error for Both BeamsThe figure below shows a quick representation of the theoretical and experimental structural stiffnesss for the two specimens.Figure 9 Experimental & Theoretical Structural ineptness for the Steel and Aluminum BeamExperiment 2Experiment 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 standard deviation were calculated for each mass using equations 1, 7, 8, and, 9, respectively. Table 6 represents this data.Table 6 Length3, Experimental and Theoretical Deflections, and Structural Stiffness for the Ste el BeamThe figure below shows the relationship between length3 and displacement.Figure 10 Length3 vs. Experimental & Theoretical Deflections Steel countersignThe final results obtained represent the attempt in experimentally determining the abrasiveness 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 much greater than the annealed steel, 64.76, as seen in figure . To further strengthen these results, the measurements for both of the specimens maintained a fairly low standard deviation, showing great consistency and accuracy throughout the individual measurements. In addition, since no biased error was continuously repeated, there 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 theoret ical calculation is highly idealized (see table 5).The 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 determination for aluminum was accurate, what caused the large amount of error innate 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 inverse ratio of the two materials 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 stiffne ss between the two materials and the ratio of modulus of elasticitys are directly proportional. The results of Experiment 1 validated 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 theoreticalequations agree with both of these statements.Experiment 2Experiment 2 resulted in data being obtained by continuously changing the length, but keeping the mass and then 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 lengt h 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. closedownOverall, both experiments were effective in validating the primary trends within the derived theoretical equations. The experiment also accomplished the goal of experimentally determining the structural stiffness of aluminum and steel beams given a specific geometry. though the lab was rather repetitive, it proved to be a simple and great way of supporting some of the theories and techniques acquired from the course of solid mechanics. One good word for the lab would be to use multiple samples of steel and aluminum in order to ensure that at least one sample is pursuant(predicate) and that youre 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.