The concept to increase the fundamental natural frequency (below which no vibration could occur) of a structure as high as possible is commonly adopted to make the structure better in dynamic environment. Due to design limitations, the fundamental frequency of a beam can be increased via adding additional intermediate point supports. If the intermediate supports are rigid, the optimum locations of the supports should be at the nodal points of a higher vibration mode without the supports, and the fundamental frequency is correspondingly raised. For elastic supports, which often occurs in practice, the optimum locations are still the same as the case of rigid supports with no decrease in fundamental frequency provided that the support stiffness exceeds a certain minimum value. Indeed, the minimum stiffness prediction is very important in the design of beams, since the bracing or support materials can be reduced without any loss of performance. This paper investigates and discusses the minimum stiffness of an intermediate support through the span of a beam for maximum value of its fundamental frequency. In this study it is assumed that the intermediate support of the beam is provided through the span of the beam (at an interval of 0.05 of the span). The finite element technique is used in the analysis of a beam model with different end conditions. It is found that when the intermediate support is not at the optimum location, there exists a minimum stiffness of the support to give the fundamental frequency of the rigidly supported beam (such minimum stiffness phenomenon also occurs in the buckling of beams). Design curves are obtained to estimate the minimum stiffness of an intermediate support through the span of a beam and the corresponding fundamental frequency.