Ratio of Compressive Critical Load for Long Columns

To determine the ratio of the compressive critical load for a long column fixed at both ends and a column with one end fixed and the other end free, we need to analyze the behavior of each type of column.

1. Long column fixed at both ends:
- A long column fixed at both ends is also known as a "fixed-fixed" column.
- In this case, both ends of the column are rigidly fixed, preventing any lateral movement or rotation.
- The column is subjected to a compressive load along its length.
- Due to the fixed supports, the column can resist both axial and bending loads.
- The critical load for this type of column is generally higher than for other types.

2. Column with one end fixed and the other end free:
- This type of column is known as a "fixed-free" column.
- In this case, one end of the column is fixed, preventing lateral movement and rotation, while the other end is free to move and rotate.
- The column is subjected to a compressive load along its length.
- Due to the fixed support at one end, the column can resist axial loads but is unable to resist bending loads.
- The critical load for this type of column is generally lower than for the fixed-fixed column.

Ratio Calculation:
- Let's assume the compressive critical load for the fixed-fixed column is L1.
- Let's assume the compressive critical load for the fixed-free column is L2.
- The ratio of the compressive critical loads can be calculated as L1/L2.
- Since the critical load for the fixed-fixed column is generally higher, L1 > L2.
- Therefore, the ratio of L1/L2 will be greater than 1.
- From the given options, only option D) 1:16 satisfies this condition.

Conclusion:
The correct answer is option D) 1:16. The ratio of the compressive critical load for a long column fixed at both ends and a column with one end fixed and the other end free is 1:16. This indicates that the fixed-fixed column can withstand 16 times the compressive load compared to the fixed-free column.