Reduction factor and load carrying capacity of a long column:

A long column is a structural member that carries an axial compressive load. Due to the presence of various imperfections and eccentricities, the actual load carrying capacity of a long column is less than its theoretical load carrying capacity. Hence, a reduction factor is applied to the theoretical load carrying capacity to obtain the actual load carrying capacity.

The reduction factor Cr is given by the formula:

Cr = 1.25 - (Le/48b)

Where Le is the effective length of the column and b is the least radius of gyration of the cross-section. The effective length of the column depends on the end conditions and the slenderness ratio of the column.

Explanation of the given options:

a) Cr = 1.25 - (Le/24b)
This formula is incorrect as it overestimates the load carrying capacity of the column. The effective length of the column cannot be less than the actual length of the column.

b) Cr = 1 - (Le/48b)
This formula is also incorrect as it underestimates the load carrying capacity of the column. The factor of safety is too high and does not consider the various imperfections and eccentricities present in the column.

c) Cr = 1.25 - (Le/48b)
This formula is correct as it provides a reasonable estimate of the load carrying capacity of the column. The factor of safety is appropriate and considers the various imperfections and eccentricities present in the column.

d) Cr = 1.5 - (Le/48b)
This formula is incorrect as it provides a higher factor of safety than required. The load carrying capacity of the column is overestimated, and the design may be uneconomical.

Conclusion:

Option C is the correct answer as it provides a reasonable estimate of the load carrying capacity of a long column. The reduction factor considers the various imperfections and eccentricities present in the column and provides an appropriate factor of safety.