Stability of compressed rods when their stiffness changes according to the law of the fourth power

Abstract

Abstract Based on the previously obtained representations for the state parameters of a rod with arbitrary continuous bending stiffness, the stability problem for a family of rods whose stiffness varies according to the law of the fourth power is solved. The spectrum of critical forces is determined and formulas for curved forms of equilibrium are derived. We propose to consider the stability coefficient as a function of the variable α, and having a set of values of this function corresponding to the values of the independent variable 0 < α ≤ 1, approximate the function K by a polynomial. The result is the expression K 2 = 0.0001α + 80.7626α 2 + 0.0002α 3. The power of the polynomial was chosen from the condition that the coefficient of determination should not be less than 0.9999. The directions for further research are presented: introduction of the results of this work into the practice of calculations for stability of real objects; application of general formulas for state parameters to the study of stability of rods with other laws of change in transverse stiffness encountered in practice.