The study of plate problems plays a crucial role in understanding the load-bearing capacity, deformation, stability, and vibration behavior of plate structures. Mathematically, such problems are typically formulated as the solution of biharmonic equations. In this work, we perform numerical simulations for clamped and simply supported plates using both conforming and non-conforming finite elements on rectangular and triangular meshes. The conforming elements adopted include the Bogner-Fox-Schmit (BFS) element and the Argyris element, while the non-conforming elements consist of the Adini and Morley elements. Numerical results demonstrate that all employed elements achieve the theoretically optimal error convergence rates. Finally, the proposed methods are successfully applied to a practical plate problem, yielding efficient and accurate simulation results.