Tail-equivalent linearization method for nonlinear random vibration

Nonlinear stochastic dynamic analysis for performance‐based earthquake engineering

“…Subsequent publications using this approach include Der Kiureghian and Li [20], Zhang and Der Kiureghian [21], Der Kiureghian [22], Franchin [23], Koo et al [24], Jensen and Capul [25], and Barbato and Conte [26]. In a recent paper [27], the authors formalized the use of FORM for solving nonlinear stochastic dynamic problems into a new ELM. It was shown that the method is equivalent to replacing the nonlinear system with a linear one, so that the tail probability of the linear response above a specified threshold matches the FORM approximation of the tail probability of the nonlinear response above the same threshold.…”

“…a given time t n is described in Fujimura and Der Kiureghian [27]. In test problems with closely spaced thresholds, the numbers of response and response gradient computations per threshold have been found to be around 5-10 and 2-5, respectively.…”

“…For the analysis described above, it is best, for both computational efficiency and accuracy, to compute the response gradients by the direct differentiation method, see Zhang and Der Kiureghian [30], Kleiber [31]. This capability is implemented in the OpenSees software [10,[32][33][34], as is the algorithm for finding the sequence of design points [27]. In short, computing the design points for a sequence of thresholds is computationally demanding, but feasible even for large dynamical systems, provided efficient algorithms are available for computing the response gradient.…”

Acceleration Response First Passage Failure Probability Analysis Method for Nonlinear Package

“…e mirror image excitation method presented by Koo et al [13] provided an approximate method for the nonlinear system failure probability estimation, and the solution of this method does not require the finding of the design point and response gradient; however, this method is restricted to the displacement first passage of single-degreeof-freedom problems. In the tail equivalent linearization method (TELM) [14,15], the nonlinear system was equivalently replaced by a linear system at the design point; this method can be adopted in many nonlinear cases with great accuracy, especially for high response threshold cases. In recent years, a promising method for the structural reliability problems is the response surface methodology [16,17]; in this method, the discrete points on the hypersurface around the design point were discretized and obtained, and the points were used to reconstruct a surface to replace the actual hypersurface relating to the real system.…”

An extended harmonic balance method based on incremental nonlinear control parameters

“…The Max-min [6], the parameter-expanding approach [7], frequency-amplitude formulation [8], the Variational Iteration [9], perturbation techniques [10,11,12], the iteration perturbation [13], the Homotopy Analysis [14,15], the Energy Balance analysis [16], the harmonic balance [17], the equivalent linearization method (ELM) [18,19] and 20 the Extended Lindstedt-Poincare approach [20] are some examples of these techniques. Each of these methods has some strong points and some weakness.…”