Keywords: random vibrations, damage, fatigue, derivative, sensitivity analysis, optimisation.

In this paper a damage sensitivity analysis for randomly excited elastic structures is presented. The frequency formulation of the Crossland's damage criterion is chosen as the damage assessment technique and the damage derivative computation technique is then applied to this frequency formulation. From the information provided by the damage derivatives several damage sensitivity criteria are formulated and compared through an optimisation procedure. The results are then discussed and conclusions given.

The study of sensitivity of structural systems is now acclaimed to be a very special area of interest by virtue of its utility in the field of computational structural mechanics. Sensitivity analyses are frequently used in order to obtain useful information for directing optimisation procedures. Sensitivity analysis considering fatigue life of structures subject to random excitations is rarely attempted due to the computational cost. The random vibration theory was developed to deal with random excitations and is based on the assumption of excitations fully described by their second order statistical properties: their mean value, their autocorrelation function and also their power spectral density function (PSD). A frequency domain approach is more suitable to predict the response of structures under random loading conditions than a time domain approach primarily for the computational cost. The spectral analysis is usually performed to deal with random vibration problem using the system response function to relate excitation to response PSD. The excitation processes are then assumed to be ergodic so the output response PSD of a linear structure can be easily evaluate from the input load PSD.

In the light of this, we need an efficient tool able to estimate fatigue life of a structure from the response PSD and more precisely from the stress PSD. Specific frequency formulations of commonly used damage criteria are chosen as fatigue life assessment techniques in the optimisation procedure. These frequency formulations proposed by Pitoiset [1] partly rely on the peak factor theory and can be applied directly after a spectral analysis, they are well suited to random vibration problems and give a fast and accurate estimation of the structural fatigue life from the response PSD (stress PSD). The structure is then supposed to be linear and subjected to stationary Gaussian loads of zero mean.

We present a computationally efficient method for damage derivative evaluation in order to evaluate the effects of material removal or addition related to the damage. Through the damage assessment procedure of structures under random load, it comes out that the structure fatigue life highly depends on the mean square random dynamic responses of the structure. That is why we first introduce the calculation of sensitivity quantities of the mean square dynamic responses based on the finite element method, the modal truncation method of eigenderivatives and some approximate process. Then the damage derivatives are explained from the frequency formulation of the Crossland's damage criterion. These derivatives allow a good estimation of the damage change in an element due to varying parameters of the other. Moreover the estimation strategy appears to be computationally efficient because only one modal analysis is required to obtain the overall damage derivatives.

From the damage derivatives several criteria for damage sensitivity quantities evaluation are given and adapted to an optimisation process. The optimisation procedure based on the finite element method involves the computation of these damage sensitivity quantities. An evolutionary structural optimisation algorithm [2] is used for its simplicity and effectiveness within application. The method uses the finite element analysis results and damage sensitivity quantity concept to select and remove the most inefficiently used material until reaching an optimal design. The optimisation procedure with fatigue life constraints is implemented and applied to a steel arch bridge subject to random loads represented by their PSD. Finally the different results obtained for each formulated damage sensitivity criteria are compared and discussed.

1

X. Pitoiset, A. Preumont, "Spectral methods for multiaxial random fatigue analysis of metallic structures", International Journal of Fatigue, vol. 22, p.541-550, 2000. doi:10.1016/S0142-1123(00)00038-4

2

Y.M. Xie, G.P. Steven, "Evolutionary structural optimisation", Berlin: Springer-Verlag, 1997

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