Increase in radius of gyration and its effect on allowable stress in a long column:

Radius of gyration is an important property of a structural element and it is defined as the distance from the axis of rotation to a point where the total mass of the object can be assumed to be concentrated. In a long column, the radius of gyration plays a significant role in determining the allowable stress that can be applied to the column. Let us discuss how increasing the radius of gyration can help in increasing the allowable stress in a long column.

What is a long column?

A long column is a structural element which is designed to withstand compressive loads. A column is considered long when its length is several times its cross-sectional dimension. A long column is subjected to buckling, which is the lateral deflection of the column due to compressive loads. Buckling is a critical failure mode for long columns and it can result in the collapse of the entire structure.

How does radius of gyration affect the allowable stress in a long column?

The allowable stress in a long column is determined by the Euler's formula, which states that the critical buckling stress is proportional to the modulus of elasticity of the material, the moment of inertia of the cross-section, and the square of the effective length of the column. The effective length of the column is a function of the slenderness ratio, which is the ratio of the length of the column to the radius of gyration.

The slenderness ratio is inversely proportional to the radius of gyration. Therefore, by increasing the radius of gyration, the slenderness ratio decreases, and the effective length of the column increases. This results in an increase in the allowable stress that can be applied to the column.

Why is increasing the length of the column not a viable option?

Increasing the length of the column will increase the effective length of the column and decrease the allowable stress. Therefore, increasing the length of the column is not a viable option to increase the allowable stress in a long column.

Conclusion:

In conclusion, increasing the radius of gyration is an effective way to increase the allowable stress in a long column. This can be achieved by using a more compact cross-section or by using a material with a higher modulus of elasticity. Designers should always consider the slenderness ratio and the radius of gyration while designing long columns to ensure their stability and safety.