Engineering Analysis relies heavily on Computer Modelling without much recourse to Mathematical Analysis. The objective of this booklet is to demonstrate how mathematical analysis helps in giving an insight to the problem. The benefit of developing a simple theory that backs up computer modelling and quantifies the effect of design parameters, is illustrated. Another objective of the booklet is to demonstrate how the use of Standards, Codes and in particular Guidelines without analysis back-up, can lead to undue conservatism. These two themes are typical of the Oil and Gas Industry. The author is not aware of practice applied in other major safety driven industries. The three case studies presented in here are related to Vibration Induced Fatigue Failure and are given only to demonstrate the concept.