On the estimation of von mises equivalent stress in random vibration analysis

Linear random vibration analysis aims to the probabilistic characterization of response functions; classical procedures are targeted to the time histories of output kinematic variables, such as displacement, velocity and acceleration, and to parameters which are linearly related to them, such as internal forces, deformations, stress components ecc. Much less attention has been devoted, by researcher and developers, to the case of response variables which are non-linearly dependent of the lagrangian coordinates or their derivatives; the most typical among these is the Von Mises equivalent stress, whose square value is quadratic in the components of the stress tensor, and thus in the lagrangian coordinates. The topic is of relevance especially for the structural safety of industrial equipment and systems, which is usually based on local stress integrity assessment. In the paper an innovative analytical procedure to deal with the probabilistic characterization of the VM stress is proposed as an extension of the classical approach adopted for linear output parameters. To this aim the matrix of the quadratic form delivering is decomposed into the sum of suitable factors; each of them deserves the same treatment as for linear parameters. The various features related to the numerical implementation of the procedure for stationary or non-stationary (evolutionary) random excitation are discussed, with reference to both the application via direct frequency domain treatment and to the modal superposition approach; an example is finally shown and commented.