Euler's Theory and the Capacity of a Column

Introduction:
In the field of structural engineering, columns play a crucial role in supporting vertical loads. The capacity of a column depends on various factors, including its length, cross-sectional shape, and the support conditions at its ends. Euler's theory provides a mathematical framework to determine the critical load at which a column will buckle under compression.

Understanding Euler's Theory:
Euler's theory is based on the assumption that a column behaves as an ideal slender member subjected to axial compressive loads. According to this theory, a column will buckle when the applied load exceeds a critical value called the Euler buckling load. The Euler buckling load is given by the formula:

Pcr = (π^2 * E * I) / (K * L^2)

Where:
- Pcr is the critical buckling load
- E is the modulus of elasticity of the material
- I is the moment of inertia of the column's cross-section
- K is the effective length factor
- L is the length of the column

Effect of Support Conditions:
The support conditions at the ends of a column significantly influence its buckling behavior. In the given scenario, column 'AB' is fixed at end A and hinged at end B. This means that end A resists both translation and rotation, while end B allows rotation but not translation.

Replacing the Hinge with a Fixed Support:
If we replace the hinge at end B with a fixed support, it means that both ends of the column will now resist both translation and rotation. This change in support conditions will affect the effective length factor (K) in the Euler buckling load formula.

The effective length factor (K) depends on the support conditions and can have different values based on the column's end conditions. For a fixed-fixed column, the effective length factor is 0.5, while for a fixed-pinned column, the effective length factor is 1.0.

Percentage Increase in Capacity:
To determine the percentage increase in the capacity of the column when the hinge at end B is replaced by a fixed support, we need to compare the Euler buckling loads for both scenarios.

Let's assume that the Euler buckling load for the original hinged column (AB) is P_hinged and the Euler buckling load for the column with a fixed support at both ends is P_fixed.

The percentage increase in capacity can be calculated using the following formula:

Percentage Increase = ((P_fixed - P_hinged) / P_hinged) * 100

Since the effective length factor changes from 1.0 (hinged) to 0.5 (fixed-fixed), the Euler buckling load for the fixed-fixed column will be twice the Euler buckling load for the hinged column.

Therefore, the percentage increase in capacity will be:

Percentage Increase = ((2P_hinged - P_hinged) / P_hinged) * 100
Percentage Increase = 100%

Conclusion:
In conclusion, if the hinge at end B of column AB is replaced by a fixed support, the capacity of the column increases by 100%. This increase is due to the change in the effective length factor in the Euler buckling load formula. The fixed support at both ends provides additional resistance to buckling, resulting in a higher critical load for the