The recurrence formula of fatigue damage model was derived under multilevel loading.
Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams.
The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP).
The number of partitions of n into k parts will be denoted by Pk(n), and a recurrence formula for it can be obtained from the definition This recurrence formula, together with the initial conditions Pk(n) = 0 if n < k, and Pk(k) = 1 determines Pk(n).
To calculate the integral of the convolution type, a recurrence formula is used that can be obtained by approximating the initial kernel with a linear combination of exponential functions (the truncated Prony's series).
New starting functions are proposed which are used in a recurrence formula to generate the higher terms in the set of orthogonal plate functions to approximate the mode shapes for circular and elliptical plates.
Using the fundamental solutions of each step rod and a recurrence formula developed in this paper, a new exact approach for determining the longitudinal natural frequencies and mode shapes of multi-step non-uniform rods is proposed.
A new exact approach which combines the basic solutions with unit matrix property and recurrence formula for determining natural frequencies and mode shapes of non-uniform shear beams is presented in this paper.
We use the n-ascending recurrence formula to calculate the polynomials in the domain of N − 1 − n ⩾ x ⩾ n ⩾ 0 and apply the n-descending recurrence relations in the domain of 0 ⩽ N − 1 − n ⩽ x ⩽ n.
The proposed model has been formulated to take into account the material damage evolution at different load levels and it allows the effect of the loading sequence to be included; by means of a recurrence formula derived for multilevel loading, complex load sequences can be considered.
We find the recurrence formula for m th-order moments.
