Abstract: Based on the mathematical similarity in the eigenvalue problem of Euler-Bernoulli beam theory, Timoshenko beam theory and Reddy's third-order beam theory, relationships of the eigenvalues of the three theories for simply-supported beams are investigated. Solving of the eigenvalue problem is converted into an algebra equation to be solved and the analytical relationships of the three theories are expressed explicitly. These relationships enable the conversion of the classical (Euler-Bernoulli) beam solutions to their shear deformable counterparts using the Timoshenko beam theory and Reddy's third-order beam theory. The shear deformable results obtained from these relationships may be used to check the validity, convergence and accuracy of numerical results of the Timoshenko beam theory and Reddy's third-order beam theory.