In the UK Advanced Gas Cooled reactors (AGRs) austenitic stainless steels are commonly used as piping material. It is known that complex piping networks can experience varied temperature gradients, changes in geometry or configuration and can be subjected to combined loading or multiaxial loading. These issues lead to elastic follow-up being present and full stress relaxation during the creep dwell is prevented in some locations. The accumulation of creep strain in those areas can cause creep damage to exceed allowable values and failure can occur. In components that are safety critical, FEA is commonly applied to ensure safe operation. Elastic follow-up is addressed in the EDF safety assessment code, under the name of R5 [1, 2]. The assessment procedure is divided into two main volumes, Vol. 2/3 and Vol. 4/5, which address uncracked and cracked components, respectively. Elastic follow-up in these volumes is represented by an elastic follow-up factor, Z or Zs. The value of elastic followup factor determines the rate of stress relaxation and the final value of the stress once relaxation is completed. It is therefore an important input to the calculation of creep damage. There are however difficulties in estimating the elastic follow-up due, for example, to changing stress triaxiality during relaxation and to the complex interaction between primary and secondary stresses. Therefore, FEA methods are commonly applied to obtain accurate elastic follow-up factors. These factors are then applied in simplified damage calculations. This thesis addresses a number of issues related to the evaluation and application of elastic follow-up. First, elastic follow-up under cyclic loading is examined analytically, demonstrating the importance of accurate calculations of elastic follow-up in such applications and how such calculations can be used not only to assess stress relaxation and creep damage but also plastic strain range and hence fatigue damage. The remainder of the thesis examines monotonic loading and for multiaxial loading, it is shown that limitations are needed in the application of simplified approaches for describing creep relaxation and the associated creep damage. It is shown that the limit on the amount of stress relaxation during a creep dwell, as currently used in codes, is dependent on the level of elastic followup and could be replaced by a limit on accumulated creep strain, independent of the level of elastic follow-up. Then, analyses of uncracked components under combined loading are used to develop an equation between different definitions of elastic follow-up, so that either definition can be used. The equation combines the definitions of elastic follow-up in R5 Vol. 2/3 and Vol. 4/5. The analysis of combined loading is extended to cracked structures and preliminary results suggest that the relationship between definitions of elastic follow-up for uncracked components still applies.