VIEWS: 27 PAGES: 6 POSTED ON: 5/13/2010
Brief summary This experiment is carried out to show the differing effects of axial and mid-span loading. The experiment highlights how theoretical buckling characteristics differ from actual results observed and from analysis of the results one can attempt to account for these differences. The experiments consists of three tests (as described in the procedure section) Test A) shows how a beam would behave with ‘normal’ mid-span loading. Test B) is similar to Test A) however it incorporates axial loads. By comparing the results with Test A) one can see how a single non-varying axial load affects the mid-span loading. Test C) investigates the effect of eccentricity on axial loading. Axial loads are applied with different eccentricities. Purpose of experiment The aim of this experiment is to investigate the buckling characteristics of a slender beam under axial and mid-span loading. The experiment can be used to predict how the properties of a beam in a real life application will differ from a theoretical beam. The experiment aims to answer the following 1. What are the effects of mid-span and axial loading? 2. How do mid-span and axial loads effect each other? 3. How close to theory are real life beams. (i.e. % error) 4. What is the effect of axial eccentricity 5. What needs to be considered when designing beams for ‘real-life’ applications? Procedure The apparatus is set-up as above. Three tests are performed; Test A) Investigate the characteristics of mid-span loading (W). Axial loading (Q) is set to zero (therefore Eccentricity (ε), is 0). The lateral mid-span load is varied. Increments of 0.2 LB are used. The lateral deflection δ is measured for every value of W. Care must be taken to calibrate the ruler to zero before the start of the experiment. Test B) Investigate the characteristics of mid-span loading, with different values of Q. Four values of Q are used (3,6,9,12 LB) and W is varied. Increments of 0.2LB (W) are used for Q = 3LB Increments of 0.2-0.05LB (W) are used for Q = 6LB Increments of 0.2-0.05LB (W) are used for Q = 9LB Increments of 0.05-0.01LB (W) are used for Q = 12LB The lateral deflection δ is measured for every value of W and Q. I.e. there are four sets of results, one for each value of Q. Care must be taken to calibrate the ruler to zero before the start of the experiment. Test C) Investigate the relationship between lateral mid-span deflection and eccentric applications of axial load. Three different eccentricities (0,3.18,6.35mm) are used and P( ‘true’ axial load) is varied P is calculated with the following formula; P = 0.75Q + 3.56 Increments of 0.1-0.05 are used for an eccentricity of 0 Increments of 0.1-0.02 are used for an eccentricity of 3.18mm Increments of 0.1-0.02 are used for an eccentricity of 6.35mm The lateral deflection δ is measured for every value of P and ε. I.e. there are three sets of results. Care must be taken to calibrate the ruler to zero before the start of the experiment. Observations Test A) The graph W Vs δ shows that up until a certain point a load W, causes a deflection that increases linearly with increasing weight. Then after a certain point a small increase in W causes a much larger deflection. Fig 1 shows how the mass of the scale pan can be calculated by the offset of the linear portion of the graph from zero. The y-intercept occurs at –2.4 therefore the mass of the scale pan is 2.4. This mass is added to Test B). 2.4N / 4.45 = 0.54 LB - added Test B) The beam behaves similarly to Test A). The W Vs δ graphs show that with addition of an axial load Q the beam buckles at a lower weight. As Q is increased the force W required to make the beam buckle is lessened. Test C) Again the beam has the same behaviour as in Test A), the beam deflection is proportional to the weight up until a point when it buckles. Then a large deflection is induced by a small increase in weight. By comparing the different eccentricities on a W Vs δ graph one can see that an eccentric load makes a beam buckle at a much lower weight and increasing eccentricity increases the effect. Error analysis 200 Identifiable sources of error include but are not limited to; Weights being dropped – causing swinging and resulting moments Weights not accurate (chipped and broken) Friction in the system particularly the pulley The beam not being supported by pin joints Movement in the notches and its associated friction Loading point Q does not represent a true axial load The theoretical Buckling load is Pc = π2EL(2L)-2 = π2EI/(4L)2 Where Pc = Critical buckling load E = Young’s modulus of elasticity (=200Gpa) L = Length I = Second moment of area = db3/12 When the assumption is made that; there is one free end and one supported round end Pc = π2EI/4L2 Pc = π2EI/4L2 Pc = (9.87 * 200*109 * I )/( 0.61)2 where I = ( (0.0254 * 0.00162 3) / 12) Pc = (9.87 * {200 *109}* {9*10-12} )/( 0.3721) I = 8.9991 *10-12 Pc = 47.75N From Test C) with zero eccentricity one can see that the Actual Pc = 50.0N (using P value, which represents true axial load) %error = (Theoretical – Experimental)/Average) *100 %error = [(47.75 – 50) /48.875] *100 %error = 4.6% Discussions Mid span and axial loading can both cause a beam to fail. However a single beam can support a much higher axial load then mid span load. It is easier to make a beam fail by mid-span loading then axial loading. The max deflection is going to be in the centre of the beam. The mid-span load acts directly at the point of max deflection whereas the axial load does not. The axial load is compressing the beam. The mid- span load acts parallel to the resulting deflection. After a critical point an axial load will induce failure much faster then a mid-span load. This effect is caused by moments. In the case of axial loading a small deflection of dx, causes a moment equal to dx multiplied by the distance to the force(which is perpendicular to the deflection), as the deflection increases so does the moment so the beam fails ‘quickly’. Mid span loading doesn’t have this induced moment since the force is in the same direction as the deflection (i.e. perpendicular distance =0). Any eccentricity in the axial load causes the beam to fail at lower weight, this is again due to moments. The load is not through the central axis of the beam. The eccentricity itself is a small distance perpendicular to the axial force being applied and therefore a moment is created. The mass of the scale pan is calculated in the observation section. The weight of the scale pan (2.4N) is not added in Test C) because it is insignificant compared to the weight required to buckle the beam i.e. it is less then the errors present in the experiment. There are errors associated with calculating the mass of the scale pan by this method too. It can be seen that there are errors in Test A) & Test B) since when the mass of the scale pan is added to Test B) it still does not go through the origin. The % error in Test C) is calculated = approx. 4.6% Conclusions One can see that this type of experimentation is prone to error. There is a proven error of 4.6% and this is not necessarily the max possible error, as positive and negative errors may have cancelled each other out. When designing beams for ‘real-life’ applications, if possible it is very important to ; Avoid eccentricity Use axial loads instead of mid-span loads Incorporate a safety factor This series of test demonstrates that real life situations cannot be predicted 100%accurately so it is important to make sure that a real life beam can withstand more then the design load. This safety factor insures that the beam is not going to buckle unpredictably. Theoretical buckling loads should be used as a guide. To model a real life system more accurately the following steps should be taken. Move Q to make it a true axial load Ensure that better more accurate weights are used Take measurements electronically Make sure the weights are not dropped and there is no swinging Avoid eccen Use axial loads Real life apps safety factor Refer to beginiing Imprve exp 14 12 10 8 6 y = 0.4802x - 2.4625 4 2 0 -10 -2 0 10 20 30 4 -4 -6 Series1 Series2 Linear (Series1)