Methodology to evaluate first transition foundation stiffness for columns on Winkler foundation

A methodology is developed in this paper to evaluate the first transition foundation stiffness for columns subjected to end concentrated loads, on a Winkler foundation, where the buckling mode shape changes. The method is based on earlier work of Naidu and Rao to predict the fundamental frequency of beam columns. wherein it was assumed that the mode shapes of vibration, initially loaded vibration, and buckling are the same. Very good predictions for the fundamental frequency of beam columns were obtained when the assumption of the same mode shapes is satisfied. In the case of the buckling of columns on Winkler foundations, at a certain value of the foundation stiffness the buckling mode shape totally changes while the mode shapes of vibration and initially loaded vibration remain the same, This led to a very large error in the predicted fundamental frequency parameter of the beam columns. This aspect is used in the present paper to evaluate the first transition foundation modulus of columns of Winkler foundations. The first transition foundation stiffness for simply supported columns on Winkler foundations, for which an exact solution is available, is obtained through the proposed methodology, and is found to be very accurate when compared to the literature values. The methodology is further used to find the first transition foundation modulus of clamped columns for which no solution is readily available. Very interesting and unexpected results were obtained for the transition stiffness and the corresponding mode shapes at transition for clamped columns of Winkler foundations.