A Nonlinear Least Squares Logistic Fit Approach to Quantifying Uncertainty in Fatigue Stress-Life Models and an Application to Plain Concrete

A large number of fatigue life models for engineering materials such as concrete and steel are simply a linear or nonlinear relationship between the cyclic stress amplitude, σ_a, and the log of the number of cycles to failure, N_f. In the linear case, the relationship is a power-law relation between σ_a and N_f, with two constants determined by a linear least squares fit algorithm. The disadvantage of this simple linear fit of fatigue test data is that it fails to predict the existence of an endurance limit, which is defined as the cyclic stress amplitude at which the number of cycles is infinity. In this paper, we introduce a nonlinear least square fit based on a 4-parameter logistic function, where the curve of the y vs. x plot will have two horizontal asymptotes, namely, y₀, at the left infinity, and y₁, at the right infinity with y₁ < y₀ to simulate a fatigue model with a decreasing y for an increasing x. In addition, we need a third parameter, k, to denote the slope of the curve as it traverses from the left horizontal asymptote to the lower right horizontal asymptote, and a fourth parameter, x₀, to denote the center of the curve where it crosses a horizontal line half-way between y₀ and y₁. In this paper, the 4-parameter logistic function is simplified to a 3-parameter function as we apply it to model a fatigue sress-life relationship, because in a stress-log (life) plot, the left upper horizontal asymptote, y₀, can be assumed as a constant equal to the static ultimate strength of the material, U₀. This simplification reduces the logistic function to the following form:

y = U₀ − (U₀ − y₁) / (1 + exp(−k (x − x₀)),

where y = σ_a, and x = log(N_f). The fit algorithm allows us to quantify the uncertainty of the model and the estimation of an endurance limit, which is the parameter, y₁. An application of this nonlinear modeling technique is applied to fatigue data of plain concrete in the literature with excellent results. Significance and limitations of this new fit algorithm to the interpretation of fatigue stress-life data are presented and discussed.