Introduction:
When designing members subject to fluctuating loads, it is important to consider the factor of safety to ensure that the member can withstand the applied loads without failure. Different design calculations can be used to estimate the strength of the member, each with its own level of conservatism. In this case, the most conservative estimates are obtained using the Goodman relation.

Explanation:
To understand why the Goodman relation provides the most conservative estimates, let's briefly discuss the three design calculations mentioned in the options:

1. Gerber Relation:
The Gerber relation is used to analyze the fatigue strength of a member by considering both tensile and bending stresses. It assumes that the failure occurs at the point of maximum tensile stress. While the Gerber relation provides a conservative estimate for tension-dominated fatigue failure, it is less conservative for members subjected to fluctuating loads.

2. Soderberg Relation:
The Soderberg relation is used to analyze the fatigue strength of a member by considering both tensile and bending stresses. It assumes that the failure occurs at the point of maximum shear stress. Similar to the Gerber relation, the Soderberg relation provides a conservative estimate for tension-dominated fatigue failure but is less conservative for members subjected to fluctuating loads.

3. Goodman Relation:
The Goodman relation is used to analyze the fatigue strength of a member by considering both tensile and alternating stresses. It assumes that the failure occurs at the point of maximum alternating stress. The Goodman relation is considered the most conservative among these three design calculations because it considers the alternating stress, which is a crucial factor in predicting fatigue failure for members subjected to fluctuating loads.

Conclusion:
In summary, when designing members subject to fluctuating loads with the same factor of safety, the most conservative estimates are obtained using the Goodman relation. This is because the Goodman relation considers the alternating stress, which is essential in predicting fatigue failure for such members. The Gerber and Soderberg relations are also commonly used but are less conservative for fluctuating loads compared to the Goodman relation.