This technical article explores the behavior of drill string under bending vibration conditions, specifically targeting the unique challenges faced by the Bottom Hole Assembly (BHA). Undesirable oscillations during drilling operations, including axial, lateral, and torsional vibrations, can lead to complex and dynamic issues. The lateral vibration mode, with its multifaceted dynamics involving bending, buckling, and transverse motion, presents the most intricate challenges. Our research focuses on presenting three fundamental models to describe and predict drill string dynamics under bending effects. By simplifying the BHA as a cylindrical beam with fixed or rolling periodic touch points. In this article, we differentiate between three distinct cases: first, a static beam with fixed periodic touch points, excluding axial load influence; second, the inclusion of rolling touch points; and finally, fixed touch points under axial load, which will be the subject of further validation. Through simulations, we validate the novel contribution of this work, an ancient SLB expert's mathematical tool to effectively describe BHA dynamics under bending phenomena.