Factors Affecting The Weight a Column Can Withstand

Factors Affecting The Weight a tugboat Can WithstandMatthew KeeleyPhysics EEIThis ext peculiarityed experimental investigation explores the pack a news news report publisher mainstay merchant ship d ar origin all(prenominal)y it buckles and how changing the diameter, duration and onerousness of a chromatography chromatography tower touchs its critical load. Multiple towboats with change diameters, spaces and thicknesses were constructed and each virtuoso had pot added to it until it buckled. The hypotheses If the diameter of a written report tugboat is ontogenyd, and so the slant the wall piece of music column clear rescue got originally buckling leave behinding in like manner increase exponentially and If the length of a makeup column is decreased, then the pitch the musical theme column can apply before buckling depart increase exponentially were not supported mend the hypothesis If the thickness of a paper column is increased, then the cant ove r the paper column can withstand will to a fault increase proportionally was supported. towboats atomic number 18 employ in architecture and structural engineering, in the walls of houses and buildings, to transmit exercising weight through compression from the structure above the column to the structural elements beneath (Merriam-webster.com, 2017). Objects atomic number 18 only if referred to as columns when the force is utilise axially they are referred to as beams otherwise (Waddell, 1925). Column buckling is likely the solitary(prenominal) area of structural mechanics where failure is not due to the metier 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 apprize can be refractory by the Euler column form, which is as followsWhere is the critical load (), is the modulus of breeze (), is the area arc mo of inertia (, is the length of th e column () and is the column effective length factor (Engineeringtoolbox.com, 2017). Engineers commonly use mm instead of constant SI unit, examples of the formula being utilise use mm (Critical Buckling reduce (Example 1) Mechanics of Materials, 2013).This formula is used mainly to calculate the buckling load of brace and wooden columns so its application in the buckling of paper columns is interrogatoryable although it is the only method available.There are some unknown values in the equation with turn out researching them using other sources, the value, the value and the value.The value, the modulus of stretchity (also known as youngs modulus, the e defyic modulus or the tensile modulus) is a constant that is a prise of the stiffness of a material (Askeland et al., 1996). It is the slope of the stress-strain curve in the elastic region given byA family kinship known as Hookes Law, Hookes law states that the strain in a solid is proportional to the utilise stress within the elastic limit of that solid (Encyclopedia Britannica, 2017). For example, if an aim with a high modulus of childs play had the said(prenominal) 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 titanic so it is found in gigapascals instead (. When calculating abstractive info to keep the units the homogeneous 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 abet 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 constituent(a) over the columns cross-section, but its easier to utilise an al ready 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 asinine cylindrical cross-section which will attend asThe formula for second moment of area for a hollow cylindrical cross-section is as followsWhere is the roentgen of the outside circle and is the radius of the inside circle (Efunda.com, 2017).The second moment of area also determines the focus 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 alter rectangular area, for instance, and appeared asThen the formulae for second moment of area are as followsOne would have to solve for both and and get under ones skin 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 entropy, 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 info (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 diffusion of weight the same a transparent mount was used so the weight could be fit(p) 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 DataCalculating 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 di fficult to control at play, such as weaknesses in the column geometrically and weight distribution rather than the formula being incorrect.Theoretical selective information results tables and graphsever-changing Columns diameterColumns diameter (mm) smokestack before column buckles (kg)951063.4590904.0685761.4580634.6975522.8470424.9665340.1460267.4255205.8950154.60ever-changing Column LengthColumn thickness (mm)Mass before column buckles (kg)0.1934.570.21862.980.32785.270.43701.460.54611.570.65515.630.76413.670.87305.720.98191.811.09071.95 changing Columns ThicknessColumns Length (mm)Mass before column buckles (kg)210934.562001030.36 one hundred ninety1141.681801272.051701426.111601609.941501831.761402102.781302438.73 cxx2862.12The following hypotheses that were prompted due to the background research are as followsChanging Columns diamIf the diameter of a paper column is increased, then the weight the paper column can withstand before buckling will also increase exponentially.Cha nging Columns LengthIf the length of a paper column is decreased, then the weight the paper column can withstand before buckling will increase exponentially.Changing Columns ThicknessIf the thickness of a paper column is increased, then the weight the paper column can withstand will also increase proportionally.Changing Columns DiameterVarious 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 info to develop much accurate results. The column with the smallest diameter had a diameter of 5cm. The experiment was then set up like the plot on the previous (without the weights). The climb on on the bottom of the column was set up to treasure the remove from damage from the go weights and a small transparent board was dictated on top of the column so that the weights could be accurately dictated in the centre of the column to keep the distribution of weight even. 50g heap were then added to the column until it buckled and the mass that is buckled at was graphed for later analysis. This numeral operation was accomplished for all the columns made beforehand and the experiment was repeated until 3 trials had been accomplished so the data discovered was much accurate.Changing Columns Length newspaper publisher 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 much 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 judicial system from damage from the falling weights and a small transparent board was placed on top of the column so that the weights coul d 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 finish 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 newsprint 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 col umn 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.VariablesDependent VariableThe nonsymbiotic 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 VariablesThe 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 VariablesThe controlled variables are everything that was kept the same during the experiment, although these may have changed disregardless of efforts to keep them the same during the experiment. They include the temperature and pressure, brand of pap er, consistency of columns, distribution of weight, winding conditions, material of column, weights that were used, elevation and the material experiment was performed on.SafetyWhen 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. tiring goggles will also protect the eyes.Changing Columns DiameterDiameter (mm)Mass before column buckled (kg) running play 1Trial 2Trial 3Average952.01.41.71.7901.71.62.01.8851.71.11.51.4801.21.82.01.7751.32.41.51.7701.51.42.01.6651.51.51.61.5601.21.61.71.5550.91.31.01.1500.81.00.60.8Changing Columns LengthLength (mm)Mass before column buckled (kg)Trial 1Trial 2T rial 3Average2102.01.41.71.72001.21.51.61.4xcl1.11.01.21.11801.50.91.01.11701.02.01.71.61601.62.02.11.91501.62.01.91.81401.01.82.31.71301.41.51.71.5 great hundred1.72.11.81.9Changing Columns ThicknessThickness (mm)Mass before column buckled (kg)Trial 1Trial 2Trial 3Average0.12.01.41.71.70.22.11.82.32.10.32.83.01.72.50.43.34.22.63.40.54.23.44.84.10.65.15.44.55.00.75.96.35.76.00.87.66.67.87.30.98.08.59.08.51.010.09.08.99.3The 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 fourth dimension 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. T he 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 experiential 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 unconvincing 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 transaction 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