Factors Affecting The Weight a Column Can Withstand

Factors Affecting The Weight a Column Can Withstand Matthew Keeley   Physics EEI This extended experimental investigation explores the weight a paper column can withstand before it buckles and how changing the diameter, length and thickness of a column affects its critical load. Multiple columns with varying diameters, lengths and thicknesses were constructed and each one had masses added to it until it buckled. The hypotheses If the diameter of a paper column is increased, then the weight the paper column can withstand before buckling will also increase exponentially and If the length of a paper column is decreased, then the weight the paper column can withstand before buckling will increase exponentially were not supported while the hypothesis If the thickness of a paper column is increased, then the weight the paper column can withstand will also increase proportionally was supported. Columns are used in architecture and structural engineering, in the walls of houses and buildings, to transmit weight through compression from the structure above the column to the structural elements beneath (Merriam-webster.com, 2017). Objects are only referred to as columns when the force is applied axially; they are referred to as beams otherwise (Waddell, 1925). Column buckling is likely the only area of structural mechanics where failure is not due to the strength of the material, but the stiffness of the material and the shape of the column instead (McGinty, 2017). Buckling occurs in a column when its critical load is reached and this value can be determined by the Euler column formula, which is as follows: Where is the critical load (), is the modulus of elasticity (), is the area moment of inertia (, is the length of the column () and is the column effective length factor (Engineeringtoolbox.com, 2017). Engineers commonly use mm instead of regular SI unit, examples of the formula being used use mm (Critical Buckling Load (Example 1) Mechanics of Materials, 2013). This formula is used mainly to calculate the buckling load of steel and wooden columns so its application in the buckling of paper columns is questionable although it is the only method available. There are some unknown values in the equation without researching them using other sources, the value, the value and the value. The value, the modulus of elasticity (also known as youngs modulus, the elastic modulus or the tensile modulus) is a constant that is a measure of the stiffness of a material (Askeland et al., 1996). It is the slope of the stress-strain curve in the elastic region given by: A relationship known as Hookes Law, Hookes law states that the strain in a solid is proportional to the applied stress within the elastic limit of that solid (Encyclopedia Britannica, 2017). For example, if an object with a high modulus of elasticity had the same force applied to it as an object with a low modulus of elasticity there would be a greater change in dimension in the object with the smaller modulus of elasticity. The modulus of elasticity is represented in pascals () but the value is usually very large so it is found in gigapascals instead (. When calculating theoretical data to keep the units the same the modulus of elasticity was represented in as. The modulus of elasticity for paper is 2 (www-materials.eng.cam.ac.uk, 2017). The value represents area moment of inertia (also known as second moment of area). It is a geometrical property of an area representing how its points are distributed regarding an axis within the object (Beer and Johnston, 1990). It is calculated using multiple integral over the columns cross-section, but its easier to utilise an already existing formula for the second moment of area of the column in question. Since the column that will be used in the experiment is rolled up paper it will have a hollow cylindrical cross-section which will appear as: The formula for second moment of area for a hollow cylindrical cross-section is as follows: Where is the radius of the outside circle and is the radius of the inside circle (Efunda.com, 2017). The second moment of area also determines the way a column is most likely to buckle (towards the plane or the plane). Usually there would be multiple formulae for the second moment of area, one for buckling towards the plane and one for buckling toward the plane, but since the cross section in question is hollow cylindrical and the axis (where the weight will be applied) is in the centre of the cross-section the formulae are identical. If the cross section was a filled rectangular area, for instance, and appeared as: Then the formulae for second moment of area are as follows: One would have to solve for both and and find out on which plane the column is most likely to buckle along and use that value as the second moment of area in the Euler column formula (What is second moment of area?, 2015). The units for second moment of area are metres to the fourth power (, but since the units need to be kept the same and the radius will be represented in millimetres when doing theoretical data, it will be in millimetres to the fourth power () instead. The last unknown value is which is the column effective length factor (Wai-Fah and Duan, 1999). It is determined by the boundary conditions. The value changes depending on if the column is fixed on both ends, hinged on both ends, fixed on one end free on another, etc. The columns used in the experiment are free on both ends so the theoretical value is 1, but the actual value derived from various other experiments is 1.2, so that value will be used in theoretical data (Efunda.com, 2017). For this experiment to be a success many variables must be remain the same that were quite difficult to control. To attempt to control these variables some precautions were taken. For example, to keep the distribution of weight the same a transparent board was used so the weight could be placed in the centre of the column and distributed evenly. Also, the paper columns need to be made carefully so that there are no weaknesses in the column because weaknesses in the column arent factored into Eulers column formula. The dimensions for paper are 29.7mm x 21mm x 0.1mm (for 80gsm A4 paper). Theoretical Data Calculating second moment of area (). Substituting into Eulers column formula and solving to find critical load. Calculating the mass the column could withstand using . This value is very large and a paper column of the dimension used in the calculations would certainly crumble under this amount of force in real life applications, but this may be due to all the other variables that are difficult to control at play, such as weaknesses in the column geometrically and weight distribution rather than the formula being incorrect. Theoretical data results tables and graphs Changing Columns diameter Columns diameter (mm) Mass before column buckles (kg) 95 1063.45 90 904.06 85 761.45 80 634.69 75 522.84 70 424.96 65 340.14 60 267.42 55 205.89 50 154.60 Changing Column Length Column thickness (mm) Mass before column buckles (kg) 0.1 934.57 0.2 1862.98 0.3 2785.27 0.4 3701.46 0.5 4611.57 0.6 5515.63 0.7 6413.67 0.8 7305.72 0.9 8191.81 1.0 9071.95 Changing Columns Thickness Columns Length (mm) Mass before column buckles (kg) 210 934.56 200 1030.36 190 1141.68 180 1272.05 170 1426.11 160 1609.94 150 1831.76 140 2102.78 130 2438.73 120 2862.12 The following hypotheses that were prompted due to the background research are as follows: Changing Columns Diameter If the diameter of a paper column is increased, then the weight the paper column can withstand before buckling will also increase exponentially. Changing Columns Length If the length of a paper column is decreased, then the weight the paper column can withstand before buckling will increase exponentially. Changing Columns Thickness If the thickness of a paper column is increased, then the weight the paper column can withstand will also increase proportionally. Changing Columns Diameter Various paper columns were constructed carefully (as to reduce weak points in the column) with different diameters, starting at 9.5cm diameters reducing the diameter by 0.5cm for every column until 10 columns had been made, so that there was enough variation in the data to develop more accurate results. The column with the smallest diameter had a diameter of 5cm. The experiment was then set up like the diagram on the previous (without the weights). The board on the bottom of the column was set up to protect the bench from damage from the falling weights and a small transparent board was placed on top of the column so that the weights could be accurately placed in the centre of the column to keep the distribution of weight even.   50g masses were then added to the column until it buckled and the mass that is buckled at was graphed for later analysis. This process was completed for all the columns made beforehand and the experiment was repeated until 3 trials had been completed so the data discovered was more accurate. Changing Columns Length Paper columns with various lengths were constructed carefully, starting at a length of 21cm and reducing by 1cm until 10 columns had been made, so there was enough variation in the data to provide more accurate results. The column with the smallest length had a length of 12cm. The experiment was then set up like the diagram (without the weights). The board on the bottom of the column was set up to protect the bench from damage from the falling weights and a small transparent board was placed on top of the column so that the weights could be accurately placed in the centre of the column to keep the distribution of weight even.   50g masses were then added to the column until it buckled and the mass that is buckled at was graphed for later analysis. This process was completed for all the columns made beforehand and the experiment was repeated until 3 trials had been completed so the data discovered was more accurate. Changing Columns Thickness Paper columns with varying thicknesses were constructed by taping pieces of paper together (1 piece of paper has a thickness of 0.1mm, 2 taped together 0.2mm, etc.) until 10 columns had been made, so there was enough variation in the data to provide more accurate results. The experiment was then set up like the diagram (without the weights). The board on the bottom of the column was set up to protect the bench from damage from the falling weights and a small transparent board was placed on top of the column so that the weights could be accurately placed in the centre of the column to keep the distribution of weight even.   50g masses were then added to the column until it buckled and the mass that is buckled at was graphed for later analysis. This process was completed for all the columns made beforehand and the experiment was repeated until 3 trials had been completed so the data was more accurate. Variables Dependent Variable The independent variable is the mass the column can withstand before it buckles, as this is what the experiment is testing and what changes when the independent variables are manipulated. Independent Variables The independent variables in this experiment are the ones that get changed, the diameter, the length and the thickness. Changing these will affect the dependent variable. Controlled Variables The controlled variables are everything that was kept the same during the experiment, although these may have changed regardless of efforts to keep them the same during the experiment. They include: the temperature and pressure, brand of paper, consistency of columns, distribution of weight, wind conditions, material of column, weights that were used, elevation and the material experiment was performed on. Safety When the column buckles, the weights will fall off the column and potentially an injury could occur. To deal with this the falling weights must be avoided and people entering the area of the experiment should be careful walking through. A mechanism to catch the board so the weights dont fall could also be constructed. Scissors could potentially be used to cut someone. To deal with this the scissors were treated with caution and used appropriately. Wearing goggles will also protect the eyes. Changing Columns Diameter Diameter (mm) Mass before column buckled (kg) Trial 1 Trial 2 Trial 3 Average 95 2.0 1.4 1.7 1.7 90 1.7 1.6 2.0 1.8 85 1.7 1.1 1.5 1.4 80 1.2 1.8 2.0 1.7 75 1.3 2.4 1.5 1.7 70 1.5 1.4 2.0 1.6 65 1.5 1.5 1.6 1.5 60 1.2 1.6 1.7 1.5 55 0.9 1.3 1.0 1.1 50 0.8 1.0 0.6 0.8 Changing Columns Length Length (mm) Mass before column buckled (kg) Trial 1 Trial 2 Trial 3 Average 210 2.0 1.4 1.7 1.7 200 1.2 1.5 1.6 1.4 190 1.1 1.0 1.2 1.1 180 1.5 0.9 1.0 1.1 170 1.0 2.0 1.7 1.6 160 1.6 2.0 2.1 1.9 150 1.6 2.0 1.9 1.8 140 1.0 1.8 2.3 1.7 130 1.4 1.5 1.7 1.5 120 1.7 2.1 1.8 1.9 Changing Columns Thickness Thickness (mm) Mass before column buckled (kg) Trial 1 Trial 2 Trial 3 Average 0.1 2.0 1.4 1.7 1.7 0.2 2.1 1.8 2.3 2.1 0.3 2.8 3.0 1.7 2.5 0.4 3.3 4.2 2.6 3.4 0.5 4.2 3.4 4.8 4.1 0.6 5.1 5.4 4.5 5.0 0.7 5.9 6.3 5.7 6.0 0.8 7.6 6.6 7.8 7.3 0.9 8.0 8.5 9.0 8.5 1.0 10.0 9.0 8.9 9.3 The results for changing column diameter seem to have a pattern to them, the weight that the column can support increases with diameter, but while the mass the column could withstand changed with diameter the increments in which the value changed reduced every time the diameter increased (logarithmic relationship). The results for changing the length of the column provided results that were expected, the weight the column could withstand decreased as the length of the column was decreased though a proper relationship between the points was underivable. The results for the thickness of the column were as expected, the mass the column could withstand increased proportionally with the thickness of column. As evident by the graphs above the theoretical data differs greatly to the empirical data. The theoretical data shows an exponential relationship between the mass withstood and the diameter of the paper column while the empirical data shows a more logarithmic relationship (if the experiment was continued further the mass withstood would have continued to increase with diameter). The mass the column can withstand is also much larger in the theoretical data than the empirical data. This is because the theoretical calculations dont factor in the weaknesses in the column geometrically and its extremely unlikely that the distribution of mass was perfect, even if the mass was placed a millimetre off the axis the mass the column could withstand would decrease drastically. Therefore, it would be difficult to get empirical results the same as the theoretical data due to many variables that are nearly impossible to control when dealing with paper columns. As shown in the graphs above the mass the column can withstand does decrease as length increases in the empirical data but is hard to decipher a relationship when looking at the empirical data due to anomalies. These anomalies would yet again be caused by variables that are too difficult to control within the experiment and for the same reasons the mass the column can withstand in the theoretical data is much greater than the mass the column could withstand in reality. The relationship between these two sets of data is identical (both increasing proportionally) although the mass the column could withstand theoretically is much greater than the mass it could withstand empirically. A possible reason that the relationship was evident in the empirical data for changing the thickness of the column and not for changing the diameter and length could be that changing the thickness affects the mass the column can withstand much more than changing either the length of the column or its diameter (reducing anomalies), this is evident when comparing the theoretical data for the three variables. Due to the varying relationships found in the empirical data and the complexity of the formula used it is difficult to relate Eulers column formula to existing mathematical models when looking at changing the columns diameter or length because the relationship is either exponential () or logarithmic (). Eulers column formula can be related to the linear function that is found when changing the columns thickness though. because a column with 0 length, diameter or thickness

Creative Writing; Point of View from a Stuffed Animal Kitten

Sitting up high, I felt like the king of the world. All of my peers sat below me and some sat at my side like Louis the XIV’s assistants would sit by him. Because they were all below me, they all looked up to me. I would see that they leave my land happily with welcoming denizens who carefully picked them out. A plethora of young aliens would frequently skip back and forth along my lane, picking up and hugging my soft and furry subjects with glee. Occasionally some of my peers found themselves laying lonely on the ground, but help always came.A much taller, weathered-looking alien came by and swept the helpless up and put them back on their shelves. I never fell though, of course. Nothing could touch me at the time, until I met the day that would bring my demise as a king of my land. It was a normal day like every other. I was watching over my peers with full intention of protection when I noticed a peculiar little alien. This alien was different and very unlike all other litt le aliens that came through my land. He wore a frown on his round face and stomped his feet out of displeasure to the middle of the lane.After a quick perusal, his zealous eyes threw angry glances towards me as I remained patient while enduring utmost disrespect. Before I could register in my mind what was happening to me, I was being squeezed by the fractious alien’s hands. He squeezed and pinched every part of my body: ears, nose, whiskers, tail, and paws. I was violated over and over as I was scanned and thrown into a plastic bag. Leaving my kingdom, I felt uncomfortable warmth all over my body. I couldn’t see through the translucent bag, but the bright light which passed through blinded me.I was roughly tossed into what I now know is a transportation vehicle, hitting my head. Quickly, the vehicle turned on, made a loud noise, and began moving. It hit many bumps and I hit my head continuously leaving my entire body sore. I was being smothered by the plastic bag I wa s contained in and soon lost consciousness. Within moments, my life had gone from greatness to nothing. I was once the beloved king of my people, but quickly became an obsequious slave. I was thrown at walls, wailed into furniture, stomped on the floor, and dropped in dirt.My face would be drilled into the mud many times over the years. Sticky juice and soda pop would be spilled on my once soft, clean fur. My long whiskers were cut, and my white fur was dirty. I felt more ugly than I looked. The giant family dog and the alien boy often played tug-of-war with my sore body, pulling me back and forth ripping the stitches at my limbs. Thick dog slobber would drip down my snout, further dirtying my fur. All I wanted was to be loved. All I wanted was to be cared for. It wasn’t too long until the alien boy stopped playing with me.The dog forgot about my existence as well. The feeling was bitter-sweet. I wasn’t being mistreated anymore, and I wasn’t constantly exposed t o pain and discomfort, but I felt alone more than ever. Minutes felt like hours, hours felt like days, and days felt like weeks. I laid in a dusty corner for months facing the banal wall, only able to hear my surroundings. This feeling was common, common to anyone forgotten and mistreated. This is how I thought I would spend eternity. Soon life would change once more for me.One morning I found myself in a large bag with many others like me. There were teddy bears, toy horses, dolls, and cars. I was surprised to find out that they had been treated the same way I had been by the alien boy. Just like me, they all once had good lives which were ripped away from them by the alien. They were beaten and dirtied, but eventually forgotten. We were taken to the transportation vehicle and tossed into it. It started up, and within minutes we were at our destination. After being emptied out of the bag, my eyes met the faces of many little girl aliens.They rhapsodized over us, for their faces wer e filled with smiles as â€Å"oohs† and â€Å"ahhs† poured out of their mouths. They bounced up and down with excitement and anticipation. One alien gently grabbed my body and hugged me tightly to her chest. For the next several hours she swung me on the outside swings and took me everywhere she went. In the evening she would put me back on the top of my shelf – my new kingdom – for a respite and get me down again the next day. All of the new little aliens brought something to me I never had before. They brought me love that I had yearned for.

Analyse the cause and effects of the New Zealand Wars Essay

There have been many New Zealand Wars which have impacted the Maori. Throughout the history of New Zealand there have been several wars and all of them having different causes and effects. The Wars starting after the treaty, with wars like the Wairau incident, Kingitanga movement these wars have affected the Maori for a long time. The most important reason being the cause of the wars was the treaty and the different interpretations of the treaty being the Maori and English Versions of the Treaty where as there was a miscommunication that happened between the word ‘Kawanatanga’ and ‘Sovereignty’- an example being in Article One the Maori Version having: â€Å"Chiefs gave the queen ‘te Kawanatanga Katoa’ – the governance over the land.† And the English Version has: â€Å"Chiefs gave the queen ‘all rights and powers of sovereignty’ over the land.† This Miscommunication causing mass conflict between the Maori and English as they put faith in the treaty and were disappointed at the fact that the European Authority who ignored the conditions that both the Maori and English agreed on at the signing of the treaty. Conditions of Britain led to the Mass Migration and sky-rocket the demand for land by settlers as they were looking for a better life in a New World. The population in 1801 changed from 16 million to 26 million in 1841 as a wave of European settled in New Zealand. But the New Zealand Company overcame all of the barriers. Investors in the company were all promised 100 acres of farmland and one town acre; the 1,000 orders were taken within a month. To tackle the negatives of New Zealand were covered up by the company using pamphlets and broadsheets to promote the country of New Zealand ‘the Britain of the South’, a fertile land with a warm climate and free of activities of class war and starvation on the streets. The partially owned land made good by land purchases from Maori, one of which leading to the Wairau Incident in Marlborough. Gibbon Wakefield’s neat and fancy in it all plans didn’t work out as there was a lack of useable land. As an outcome of the New Zealand  Company’s policy the community of European people increased to 28,000. The New Zealand Company had established the outlines of immigration from Britain to New Zealand, setting in place the promotional pitch that were used by the region and in later years the government. Governor Grey attacked the British Colonial Office with proposals of a new plan that he thought would be appropriate for New Zealand’s situation. He established the principles that this vote should be extended that rapid colonisation combined with peace and prosperity would soon fuse the ‘two races into one nation’. The voting rights for the national government were property-based. Even with the qualifications required to be able to vote were put similarly low; it was very democratic at the time. While Maori owned land, it was owned collectively rather than individuals. The Maori charter was delayed by the court decision that communal ownership did not give the vote. The New Zealand Company’s Nelson settlement needed over 1000 allocations of 201 acres each, which had to be ‘fertile, productive land’. Over 200,000 acres were required, of which was almost a tenth of the ‘Native Tenths Reserves’. But there wasn’t enough suitable land, early 1843 company were sent to the unpurchased Wairau lands. Ngati Toa chiefs immediately asked that their land not be apart in the company’s ‘purchases’. After months of protests, the Ngati Toa chiefs evicted the surveyors and burnt down their shelters. Police Magistrate Henry Thompson responded to this by arming a militia in an attempt to arrest the chiefs at Wairau. On June 17 the whole situation fell apart as the confrontation ended in 20 Europeans and 4 Maori were killed.

Proposal for Starting a Music Program Essay - 616 Words

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