Introduction:
In this question, we are given that vertical columns of the same material, height, and weight have the same end conditions. We need to determine which cross-section will carry the maximum load.

Analysis:
To determine the cross-section that can carry the maximum load, we need to consider the strength and stiffness of different cross-sections. The load-carrying capacity of a column is influenced by its moment of inertia, which is a measure of its resistance to bending.

Options:
Let's analyze each option one by one:

a) Solid circular section:
- The solid circular section has a symmetrical shape, which provides uniform resistance to bending.
- It has a higher moment of inertia compared to other sections.
- Due to its symmetrical shape, it can efficiently distribute the load along its circumference.
- Therefore, the solid circular section is a good choice for load-carrying capacity.

b) Thin hollow circular section:
- The thin hollow circular section consists of an outer and inner circular shape.
- It has a larger moment of inertia compared to other sections, including the solid circular section.
- The distribution of material in the hollow section adds extra stiffness and strength.
- Due to its larger moment of inertia and added stiffness, the thin hollow circular section can carry a higher load compared to other sections.
- Therefore, the thin hollow circular section is the correct answer for maximum load-carrying capacity.

c) I-section:
- The I-section consists of a vertical web and horizontal flanges.
- It provides high strength and stiffness, especially in the vertical direction.
- However, the moment of inertia of the I-section is usually smaller compared to the circular sections.
- Therefore, the I-section may not be able to carry as much load as the circular sections.

d) Solid square section:
- The solid square section has a symmetrical shape, but it has a smaller moment of inertia compared to circular sections.
- The distribution of material in the square section is not as efficient as in the circular sections.
- Therefore, the solid square section may not be able to carry as much load as the circular sections.

Conclusion:
Among the given options, the thin hollow circular section is the cross-section that can carry the maximum load. It has a larger moment of inertia and added stiffness due to the distribution of material in the hollow section.