<p>In this paper, we establish the exponential stability of the one-dimensional Euler-Bernoulli beam equation with local viscosity. On the one hand, we demonstrate that the Euler-Bernoulli beam equation system is exponentially stable by employing the multiplier method, which relies on an appropriately constructed Lyapunov function. On the other hand, we discretize the Euler-Bernoulli beam equation system using the finite volume difference method. For the resulting semi-discrete system, we construct a discretized multiplier based on the discretized Lyapunov function. Finally, we prove that the semi-discrete Euler-Bernoulli beam equation system is also uniformly exponentially stable.</p>