The tail equivalent linearization method for nonlinear stochastic processes, genesis and developments

This chapter aims to provide a general prospective of the Tail Equivalent Linearization Method, TELM, by offering a review that starts with the original idea and covers a broad array of developments, including a selection of the most recent developments. The TELM is a linearization method that uses the first-order reliability method (FORM) to define a tail-equivalent linear system (TELS) and estimate the tail of the response distribution for nonlinear systems under stochastic inputs. In comparison with conventional linearization methods, TELM has a superior accuracy in estimating the response distribution in the tail regions; therefore, it is suitable for high reliability problems. Moreover, TELM is a non-parametric method and it does not require the Gaussian assumption of the response. The TELS is numerically defined by a discretized impulse-response function (IRF) or frequency-response function (FRF), thus allowing higher flexibility in linearizing nonlinear structural systems. The first part of the chapter focuses on the original idea inspiring TELM. The second part offers fourth developments of the method, which were studied by the authors of this chapter. These developments include: TELM in frequency domain, TELM with sinc expansion formula, TELM for multi-supported structures, and the secant hyperplane method giving rise to an improved TELM.