A partially prestressed member is a structural element that combines both prestressed steel and mild steel reinforcement to improve its performance under service loads. The correct answer is option 'A', which states that tensile stresses and cracking are permitted under service loads. Let's explain this in detail:

1. Definition of partially prestressed member:
A partially prestressed member refers to a structural element that is reinforced with both prestressed steel and mild steel reinforcement. The purpose of incorporating prestressed steel is to counteract the tensile stresses that occur due to the applied loads, while the mild steel reinforcement helps to control cracking and provide additional capacity to the member.

2. Tensile stresses and cracking:
Under service loads, a partially prestressed member allows for the occurrence of both tensile stresses and cracking. Tensile stresses may still be present in the member due to various reasons such as external loads, temperature effects, and shrinkage. However, the prestressed steel helps to limit the extent of these tensile stresses and prevent them from causing excessive deformation or failure.

Cracking is also permitted in a partially prestressed member under service loads. Cracks can develop due to the tensile stresses exceeding the capacity of the concrete or due to other factors such as chemical reactions or differential settlement. However, the presence of prestressed steel limits the width and propagation of cracks, maintaining the overall integrity of the member.

3. Benefits of partially prestressed members:
- Increased load-carrying capacity: The combination of prestressed steel and mild steel reinforcement allows the member to withstand higher loads and resist bending and deflection.
- Improved durability: By limiting the width and propagation of cracks, partially prestressed members offer enhanced durability and resistance to environmental factors such as corrosion.
- Cost-effectiveness: Partially prestressed members provide a balance between the cost of full prestressing and the performance of conventionally reinforced members, making them a cost-effective solution for various structural applications.

In conclusion, a partially prestressed member permits both tensile stresses and cracking under service loads. By incorporating prestressed steel and mild steel reinforcement, these members offer improved load-carrying capacity, durability, and cost-effectiveness compared to conventionally reinforced members.