ANS:D - All the above.

The correct assumption for the design of a prestressed concrete member from the options provided is: All the above. Let's break down each assumption:

1. A transverse plane section remains a plane after bending: This assumption is essential in the design of reinforced and prestressed concrete members. It implies that the plane sections before and after bending remain plane, which allows for simplification in the analysis and design process. This assumption is commonly applied in both reinforced and prestressed concrete design.
2. During deformation limits, Hook's law is equally applicable to concrete as well as to steel: Hook's law states that stress is directly proportional to strain within the elastic limit of a material. This assumption implies that both concrete and steel behave linearly within their respective elastic limits. While concrete is non-linear and exhibits a curved stress-strain relationship after reaching its elastic limit, within the deformation limits considered in design, it can be assumed to behave linearly. This assumption simplifies the analysis and allows for the use of linear elastic principles in the design of prestressed concrete members.
3. Variation of stress in reinforcement due to changes in external loading is negligible: This assumption implies that the change in stress in the reinforcement due to variations in external loading is insignificant. In other words, it assumes that the stress in the reinforcement remains relatively constant under different loading conditions within the design limits. This simplifies the analysis and design process by allowing for the assumption of a constant stress in the reinforcement.