The slenderness ratio of a column is a measure of its ability to resist buckling under a compressive load. It is defined as the ratio of the effective length of the column to its radius of gyration. The effective length is the unsupported length of the column between points of zero moments.

To understand why the correct answer is option 'D', let's break down the other options and explain why they are not correct:

a) If the length of the column is zero, it means there is no column at all. Therefore, this option is incorrect.

b) The radius of gyration is a measure of how the mass of a column is distributed around its axis. It is calculated as the square root of the ratio of the moment of inertia to the cross-sectional area. If the slenderness ratio is zero, it means that the length of the column is zero (as explained in option 'a'), which is not possible. Therefore, this option is incorrect.

c) If a column is supported on all sides throughout its length, it means that it is fully fixed or built into the structure. In this case, the points of zero moments are located at the supports. The slenderness ratio is not zero because it is the ratio of the unsupported length to the radius of gyration. Therefore, this option is incorrect.

Now let's explain why option 'D' is correct:

d) The slenderness ratio is zero when the column is between the points of zero moments. These points are located at the ends of the unsupported length of the column, where the bending moment is zero. In this case, the effective length of the column is equal to the unsupported length, and the slenderness ratio is zero. This means that the column is effectively restrained against buckling and can resist compressive loads without significant deformation. Therefore, option 'D' is the correct answer.

In summary, the slenderness ratio of a column is zero when it is between the points of zero moments, which means it is effectively restrained against buckling. This occurs when the column is unsupported over a certain length and can resist compressive loads without significant deformation.

Slenderness ratio means the ratio of the length of a column to the least radius of gyration of its cross-section. It is denoted by λ.