Background theory on BUCKLING and DFINITE ELEMENT ANALYSIS Dissertation

Background theory on BUCKLING and DFINITE ELEMENT ANALYSIS - Dissertation Example These underwent no major changes in the next 100 years. In 1845, A.H.E Lamarle proposed the theory that Eulers formula could only be effectively utilised if the slenderness ratio was beyond a certain prescribed limit. In 1889 Considere further made an addendum in the form that Eulers formula could not be used for inelastic buckling since the actual section modulus available on the concave and concave sides of a bend beam were different. This lead to the formulation of the Reduced Modulus theory for buckling and which is still undergoing lot of revisions. (Gere James & Timoshenko Stephen, 2004) Theory Euler by a series of experiments observed that the buckling stress generated in an axially loaded column is directly proportional to the Youngs Modulus, the moment of inertia of the material and inversely proportional to the effective length of the member. In other words he represented the formula by a simple equation. Pcr=Ð » x E x I/ Le^2 Here Pcr represents the critical load, E the Y oungs Modulus which is an inherent property of the material, I the moment of inertia is function of the dimensional values in terms of breadth and height of the material. Le represents the effective length of the column. ... Another combination is that of the column fixed at the base and pinned on top. Calculating this from a series of differential equation with known end conditions would provide an effective length of 0.7L. Hence the Euler’s equation for all the above commonly loaded conditions can be represented as Pcr=Ð »^2 x E x I/ (K x L)^2 where K=2 for fixed-free column, K=1 for pinned end columns, K=1/2 for columns with fixed ends and K=0.7 for column fixed at base and pinned at top. (Gere James & Timoshenko Stephen, 2004) Source: Gere James & Timoshenko Stephen, 2004 The Euler’s formula is used to calculate the corresponding critical stress that is generated due to this critical load Pcr. Here ?cr= Pcr/ A where A is the area of cross section of the member which could further written as ?cr=Ð »^2 x E/(L/r)^2. Here L/r can be together noted as the slenderness ratio. L as denoted earlier is the length of the column while r=v I/A is called the radius of gyration of the member. (Gere James & Timoshenko Stephen, 2004) Using Eulers Theory in Calculations For the analysis of simple beams using Eulers formula, slenderness ratios of columns should not surpass 180. For other members that absorb compression forces the L/r ratio is limited to 200. (Welded Tanks for Oil Storage, 2008) For checking whether the column provided for a section is safe, the actual compressive stress is calculated using the simple formula ? actual=P/A whether P is the external load and A is the cross section of the member. The L/r ratio of the selected member is checked and limited to 180. Thereafter maximum allowable compressive stress generated is found out by using the above formula ? allowable=Ð »^2 x E/(L/r)^2 for columns. If the actual stress calculated is less than the maximum