Dear Fatih Aruk,

first of all, what is the geometry of the structure you want to analyze and how to you measure the vibration (scanning optically or other)? For simple geometries like cantilevers it is sufficient to know the amplitude, e.g. as measured by AFM and thermal-noise excitation (check out papers by JE Sader), to reconstruct the vibration using eigenmode decomposition. For complex geometries you would need to measure the amplitude on each point of the surface. Then you can use FEM eigenmode analysis to find a matching modeshape.

Please provide more details on your subject.

Regards, Erwin

Random vibration? do you mean the vibration is induced by random excitation ?

Random Vibrations are not similar to Natural Frequency vibrations of a structure. Yes, they do have the harmonics of the natural frequencies engulfed when the structure is exposed to random vibrations. As Erwin has pointed out, which is correct, determine the displacement using optical or other tools to plot it. Then carry out also the FEM analysis of the structure experiencing random vibrations. Superimpose the mode shapes obtained from FEM analysis on what you obtained through the first method of testing to determine the stress levels at vulnerable points of the structure.

It depends on what you are going to do with the equivalent stress. If you are using it to predict the residual life of the structure then for random vibrations you can use the rms value of the maximum principal stress. That said you must remeber to allow for the threshold of the material. Note that you will generally get a family of life curves since the way in which naturally occuring initial discontinuities, from which cracks will grow, interact with the material is a function of the local metric (the local material threshold) which can vary from close to 0 to almost 1 MPa m^1/2. Similarly depending on the environment that your structure is exposed to the initiating discontinuity can vary from typically 10 microns to between 200-500 microns. This variation in the size of the initiating will make a huge difference in the life of your component. Nevertheless, for random vibrations acting on operational structures the rms approach usually works very well indeed, see the book by Barsom and Rolfe.

As such for lifing purposes mode interaction and load interaction in operational structures excited by random loads is irrelevant provided that the structure does not have artificial notches. This will not be true if you are interested in problems associated with yielding but it is true if you are concerned with the fatigue life of a structure under random loads.

The trick is to note that all real structures contain initiating discontinuities and the size of thess discontinuities is a function of the environment, the material and the manufacturing process.

Hope this helps

One more thing I am reminded of after reading Rhys comments. He has very aptly pointed out about the micro fractures/fault lines development which cause the ultimate failure of the structure. So kindly also see the standard fatigue results for this material and the specific cross section, if available, when exposed to continuous vibrations during its operations.

Fatih,

Good point about the difficulty to calculate an equivalent RMS value for Von Mises Stress. As you said the von Mises formula is only valid in its complex form and therefore the RMS calculation of the basic stress tensor is not correct.

But you'll find that this problem has actually been addressed before. You can check for example the paper of Pitoiset, Preumont, Kernilis : Tools for a Multiaxial Fatigue Analysis of Structures Submitted to Random Vibrations. ESA SP-428, February 1999. They propose a method to calculate an equivalent PSD for the von Mises Stress as a linear combination of the components of the PSD matrix of the stress tensor. The equivalent PSD then provides you with an equivalent Von Mises RMS value. The idea is pretty straight forward, and uses the matrix form definition of Von Mises Stress. The method can be derived for other stress invariants.

Also note that this derivation of PSD of Von Mises Stress has been implemented by MSC.Software in the Patran module MSC.Random, coupled with MSC.Nastran solver for Random Analysis. This interactive pre/post processor module will allow you to perform correct calculation of the equivalent Von Mises PSD and allows you to plot the PSD curves as well as RMS Von Mises Stress contour plots on your FE mesh.

If I remember correctly, in the random output the deformed shape is calculated and then for all the nodes the displacements are known. Then it is possible to load the structure with this pattern of displacements statically and so, with a statica analysis exstract von mises stresses and all the invariant needed.

Carlo is right.

I agree! It is a good old trick that works. But if the model has a large number of DOF, it can become quiet a cumbersome task to collect all the RMS displacements at nodes and then apply them as loads in a new static analysis.

Guillaume, thanks for your valueable comments. I also appreciate all other who comments on this.

I have been a user of Abaqus for Long time, but did not use it for about 2 years. Now, when I check, I see they implemented the method of Segalman, D. J., C. W. G. Fulcher, G. M. Reese, and R. V. Field, Jr., “An Efficient Method for Calculating RMS von Mises Stress in a Random Vibration Environment,” Sandia Report, SAND98-0260, 1998. At the time I worked, the Von Mises and other invariants were not an output.

Carlo, can you describe a little bit more how to extract von Mises with a statica? I think if we apply all the RMS displacement values to the structure, we will probably be doing a very conservative approach.

In a static analysis all the commercial Finite Element programs extrat the stress invariant as Von Mises and so on. I know that this is a conservative approach but it is also very simple. If all the checks on the strength pass, then the problem is solved; if a limited area doesn't pass a more sofisticated analysis can be executed using the approach suggested in ABAQUS documentation.