In this paper, a nonlinear beam equation containing an integral term of the deformation energy, which is unknown before the solution is found, is investigated under different boundary conditions. First, we set the unknown integral term as a scalar variable and then develop a weak-form integral equation to solve the integral. By using the sinusoidal functions as test functions and bases of the numerical solution, we obtain a fast-convergence iterative scheme. Due to the orthogonality of the sinusoidal functions, the expansion coefficients of the numerical solution are in the closed form. The proposed iterative algorithms converge quickly and provide highly accurate numerical solutions of the nonlinear beam equation containing the integral term, as confirmed using five numerical examples

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