Assumptions Made in Euler’s Column Theory – Columns and Struts – Strength of Materials


Hello Friends here in this video we are going to see euler’s theory for long columns with its assumptions let us get started now Euler’s Theory in 1757 a Swiss mathematician Leon Hard Euler gave the formula for stability of long columns in 1757 as I have written you’re a Swiss mathematician Leonhard Euler he gave the formula for stability of long columns now when user Siri was designed use Leo nodular he considered only the bending of column because he was analyzing long columns and in case of long columns the bending is more compared compared to the direct compression so Euler considered only bending of columns as the direct compression was negligible compared – bending so Euler while designing the formula widely riving the formula he considered only bending because when the column is long in that case the bending of the column is more as compared to the direct compression so here I am drawing a diagram of a long column subjected to load so when the under the action of this load the column will compress as well as it will bend but your since the length of the column is more the bending is very high as compared with the direct compression because load is acting in the downward direction column is fixed at the bottom so there will be direct compression but the direct compression is less compared to the bending value so you’ll are considered only the bending of columns and therefore I can say that Euler neglected the direct compression of columns so he neglected the direct compression and hence Euler’s theorem is applicable only for long columns so this was regarding the euler’s theory now I write down the assumptions euler made while deriving the formula for long columns so the assumptions which Euler made were first assumption is the column is subjected to axial loading it means the load which is acting on the column will pass only through its axis load will not be eccentric that is away from the axis it will pass only along the axis this is an assumption but in actual cases if we see columns may also be subjected to eccentric loading that is away from the axis so this is the first assumption next material of the column is homogenous and isotropic now the meaning of homogeneous and isotropic is that the material which Euler assumed homogeneous means having same composition throughout that is if a column is made up of steel that it will remain of steel throughout it means the theory is not applicable to composite columns that is having two different materials next isotropic means the material of the column will have same modulus of elasticity in all directions that is x y&z direction so this is also an assumption next the material of column is elastic and obeys Hookes law so Euler assumed that the material of the column being elastic it means when the load is applied the column will reflect as I have shown here but when the load is removed the column will come back to its original position so that is also a part of assumptions and for Hookes law here I’ll show the graph for oakes law Hookes law is basically a graph of stress on Y axis to strain on x axis so it is a graph of stress versus strain Hookes law means it is a graph of stress versus strain and within this Hookes law it states that within the elastic limit stress is directly proportional to strain that is here I will say that up to point a this is the elastic limit so within this elastic limit stress and strain they’re directly proportional that is at this point zero where the stress is zero strain is also zero but when stress will increase strain is going to increase again if we increase the stress value strain increases if we again increase the stress strain increases in that relation is linear up to the elastic limit so you will assume that the column is following Hookes law next the length of the column is very large compared to the other dimensions so it means that Euler assumed that the length of the column is large compared to the other dimensions which are called as lateral dimensions so he was analyzing just long columns then self weight of the column was neglected that is the own weight of the column is neglected and we are considering only the external loading on the column but in actual case even because of the self fit there is some amount of compression in the column then the column is straight before loading it means that the column is not initially bent as shown here initially it is not bent initially the column is straight and when the load is acting then only the column will rent so again this is an assumption made by Euler so by taking these assumptions he derived the formula which was later called as Euler’s formula or called as the calculation of Euler’s crippling load or Euler’s buckling load he gave the formula but before deriving that formula he assumed these points which I have written here so in this video we have seen assumptions made in ulis theory

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