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In this article, we study the stabilization of an Euler-Bernoulli beam equation with variable coefficients where boundary observation is subject to a time delay. To resolve the mathematical complexity of variable coefficients, we design an observer-predictor based on the well-posed open-loop system: the state of system is estimated with available observation and then predicted without observation. We show that the closed-loop system is stable exponentially under estimated state feedback by a numerical simulation illustrating our results.