Discuss the Euler buckling formulae for each end condition considering theaccuracy of the theoretical ratios in comparison to the actual experimental ratios

The Euler Buckling formula is a theoretical approach used to predict the critical load at which a slender column under compression will buckle. The formula is given by:

Pcr = π²EI / (KL)²

where:

• Pcr is the critical load,
• E is the modulus of elasticity,
• I is the moment of inertia,
• K is the column effective length factor, and
• L is the length of the column.

The value of K depends on the end conditions of the column. For different end conditions, the values of K are as follows:

• Both ends pinned (hinged, free to rotate): K = 1
• Both ends fixed: K = 0.5
• One end fixed, one end pinned: K = 0.699
• One end fixed, one end free (cantilever): K = 2

The Euler Buckling formula assumes that the column is perfectly straight, has a constant cross-sectional area, and is made of a material with a uniform modulus of elasticity. However, in reality, these conditions are not always met, which can lead to discrepancies between the theoretical and experimental critical loads.

In practice, the actual experimental ratios may differ from the theoretical ratios due to factors such as imperfections in the column, non-uniform material properties, and load eccentricity. For example, a column that is not perfectly straight may buckle at a lower load than predicted by the Euler Buckling formula. Similarly, a column made of a material with a non-uniform modulus of elasticity may also buckle at a lower load.

In conclusion, while the Euler Buckling formula provides a useful theoretical prediction of the critical buckling load for slender columns, it is important to consider the potential discrepancies between the theoretical and experimental results due to the assumptions made in the formula. Therefore, it is often necessary to apply a factor of safety when using the Euler Buckling formula to ensure that the column will not buckle under the applied load.