The simple beam theory can predict the deflection of beams that are fixed at both the ends, with a high level of accuracy if the length-to-diameter ratios for the beams are high. Hence the readily available formula (which is based on the simple beam theory) for beams may be used with confidence when the length-to-diameter ratios are high. However, studies on the accuracy of the results predicted by the simple beam theory are not common when the length-to-diameter ratios are small. Hence the present work compares the results obtained through the use of the well- known formula (based on the simple bending theory) with the results obtained by making use of the well-known commercial finite element software ANSYS, for beams of different length-to-diameter ratios and circular and square cross section, in order to get an idea of the accuracy of the results that are obtained by making use of the well-known formula. All these results are tabulated. The present study is restricted to beams that are fixed at both the ends and only those cases where a single concentrated load is applied at the midpoint between the two ends. The present study would be of use to designers who are concerned over the accuracy of the results that are obtained through the use of the well-known analytical formula. The present study is a continuation of the work presented in the author's earlier work "Kirana Kumara P, Studies on the Accuracy of the Results Predicted by the Simple Beam Theory in the case of Short Cantilever Beams, National Conference on: Role of Mechanical Engineering and Computational Science for a Sustainable Future (NCAME-2015), September 25th & 26th, 2015, Dayananda Sagar College of Engineering, Bengaluru, India", and hence the present work -- wherever useful -- borrows ideas and sentences from the said reference.