This article investigates the lateral vibration and resonance of bridges, crucial for transportation network integrity and traffic safety. It aims to understand the underlying principles and causes of these vibrations to enhance bridge design and maintenance. Utilizing Euler-Bernoulli beam theory and assumptions like linear elasticity and uniform material, the study simplifies complex bridge dynamics into a manageable model. It explores the bridge's flexural response under static and dynamic loads, focusing on resonance phenomena. Numerical simulations, including the finite difference method and analysis of nonlinear elastic responses, assess the bridge behavior under various load conditions, particularly periodic loads. The findings offer theoretical insights and practical guidelines for vibration control and safe bridge operation.

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