In this paper, an analytical method to estimate the deformation strains that can quantify the severity of bulges, as it applies to coke drums, is presented. The proposed method is based on classical shell theory and API 579-1/ASME FFS-1 (2016) procedures involving triaxiality limits. In this first part of the work, only the theoretical development is presented along with the comparison of the results from this theoretical approach with two case studies that emulate the bulging due to different loading scenarios. The developed approach is then applied to a deformed coke drum. In the next part of this paper, the application of this approach on selected in-service coke drums that are equipped with strain gages will be presented. The authors would like to emphasize the well-known fact that the coke drum is a complex pressure vessel for which any single simplified assessment technique may not be sufficient to quantify the life or fitness-for-service (FFS) of a coke drum due to the complexities associated with the various parameters that affect the mechanical integrity of the coke drum. This paper is an attempt to advance the assessment techniques that are currently utilized in the industry.