In case of long columns, the stress due to direct load is very small in comparison with the stress due to buckling.
Assumptions made in the Euler’s Column theory:
The following assumptions are made in the Euier's column theory:
1. The column is initially perfectly straight and the load is applied axially.
2. The cross-section of the column is uniform throughout its length.
3. The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke's law.
4. The length of the column is very large as compared to its lateral dimensions.
5. The direct stress is very small as compared to the bending stress.
6. The column will fail by buckling alone.
7. The self-weight of column is negligible.
End conditions for Long Columns:
In case of long columns, the stress due to direct load is very small in comparison with the stress due to buckling. Hence the failure of long columns takes place entirely due to buckling (or bending). The following four types of end conditions of the columns are important:
1. Both the ends of the column are hinged (or pinned).
2. One end is frxed and the other end is free.
3. Both the ends of the column are frxed.
4. One end is frxed and the other is pinned.
For a hinged end, the deflection is zero.
For a fixed end the deflection and slope are zero.
For a free end the deflection is not zero.
The following sign conventions for the bending of the columns will be used :
1. A moment which will bend the column with its convexity towards its initial central line as shown in Fig. (a) is taken as positive. In Fig (a), AB represents the initial centre line of a column. Whether the column bends taking the shape AB' or AB", the moment producing this type of curvature is positive.
2. A moment which will tend to bend the column with its concavity towards its initial centre line as shown in Fig. (b) is taken as negative.