Introduction:
In structural engineering, it is important to consider the effective length factor for the design of compression members. The effective length factor takes into account the end conditions of the member and provides a factor that is used to determine the effective length of the member. In the case where one end of the member has both translation and rotation restrained, while the other end has only translation restrained, the effective length factor recommended by IS 456:2000 is 0.80.

Explanation:
To understand why the effective length factor of 0.80 is recommended for this particular situation, let's break down the concept step by step:

1. End Conditions:
The end conditions of a structural member play a crucial role in determining its behavior under load. In this case, one end of the member is fully restrained, meaning that both translation (movement in the direction of the applied load) and rotation are prevented. The other end, however, is only restrained in translation, allowing rotation.

2. Buckling and Lateral-Torsional Buckling:
When a compression member is subjected to an axial load, it can fail through two main modes: buckling and lateral-torsional buckling. Buckling occurs when the member bends out of shape due to the applied load, while lateral-torsional buckling is a combination of lateral bending and twisting.

3. Effective Length:
The effective length of a compression member is a measure of how much it can bend or buckle before it is restrained by the end conditions. It is crucial to determine the effective length correctly to ensure the stability and safety of the structure.

4. Effective Length Factor:
The effective length factor, denoted as K, is used to determine the effective length of the member. It is defined as the ratio of the effective length to the actual length of the member. A higher effective length factor implies a shorter effective length, meaning that the member is more restrained and less likely to buckle or fail.

5. Recommended Effective Length Factor:
According to IS 456:2000, for a compression member with one end having both translation and rotation restrained, while the other end has only translation restrained, the recommended effective length factor is 0.80. This means that the effective length of the member will be 0.80 times the actual length.

Conclusion:
In summary, the effective length factor recommended by IS 456:2000 for a compression member with one end having both translation and rotation restrained, while the other end has only translation restrained, is 0.80. This factor takes into account the end conditions and helps determine the effective length of the member, which is crucial for its stability and safety.