3. Presentation of the calculation method#
3.1. Calculation framework and assumptions#
We place ourselves in the framework of a decomposition of the stress in the structure on a modal basis:
\({\sigma }_{\mathrm{total}}(t)={\sigma }_{\mathrm{stat}}+\sum _{i=1}^{\infty }{\alpha }_{i}{\sigma }_{\mathrm{mod}}^{i}\mathrm{cos}({\omega }_{i}t+{\phi }_{i})\),
by noting \({\sigma }_{\mathrm{stat}}\) the constraint related to static loading and \({\sigma }_{\mathrm{mod}}^{i}\) the constraints associated with the natural modes of the structure. \({\omega }_{i}\) and \({\phi }_{i}\) are respectively the pulsation (known) and the phase shift (unknown) of mode i.
The classical calculation would consist in evaluating the damage, the modal contributions and the phase differences being known. Here, conversely, the aim is to estimate the maximum modal contributions associated with an unlimited endurance of the structure.
It is not possible to identify in general terms the maximum contribution of each of the modes. It is therefore necessary to introduce three simplifying hypotheses: uniaxial fatigue criterion; a priory imposition of the relative weight of the modes; maximization of the amplitude.
Uniaxial fatigue criterion: the use of a uniaxial fatigue criterion (Wöhler method) is the same as assuming that the main directions of static and dynamic loading are the same. This hypothesis seems legitimate for the usual structures referred to (fins, pipe lines, etc.); it induces a conservatism that is undoubtedly excessive in the general case. Throughout the following, the notation \(\sigma\) will correspond to the constraint standard (von Mises or Tresca).
Note that a multiaxial approach (with Crossland or Dang Van criteria) would also be possible. However, these criteria require knowing, in addition to the tensile endurance limit of the material, the pure shear endurance limit. This data is often unavailable for power plant materials, and the functionality would then have been difficult to use.
Relative weight of the modes: it is assumed that the relative weight of the various natural modes considered is known. For example, this relative weight can be estimated from measurements on site. We also only consider a limited number of \(N\) modes (in practice, we will most often have \(N<4\) or 5). In other words: \({\sigma }_{\mathrm{total}}(t)={\sigma }_{\mathrm{stat}}+\alpha \sum _{i=1}^{N}{\beta }_{i}{\sigma }_{\mathrm{mod}}^{i}\mathrm{cos}({\omega }_{i}t+{\phi }_{i})\).
The coefficient \(\alpha\) is the parameter we are looking to calculate.
Amplitude maximization: in the stress expression above, \({\phi }_{i}\) phase differences are unknown. The alternating stress \({S}_{\mathrm{alt}}\), defined as the half-amplitude of variation of the stress, is then calculated as follows: \({S}_{\mathrm{alt}}=\alpha \sum _{i=1}^{N}{\beta }_{i}{\sigma }_{\mathrm{mod}}^{i}\). This definition of \({S}_{\mathrm{alt}}\) is conservative.
3.2. Calculation of allowable vibrations#
Under the hypotheses introduced above, the calculation boils down to identifying the coefficient \(\alpha\) associated with unlimited endurance of the structure.
The static constraint \({\sigma }_{\mathrm{stat}}\) corresponds here to an average constraint, which is generally not zero. It is necessary to take this stress into account in the Wöhler fatigue curve. This consideration is classically done using the Haigh diagram [R7.04.01], either with the Goodman line or with the Gerber parabola.
By noting \({S}_{l}\) the endurance limit and \({S}_{u}\) the material breakage limit, we have:
with Goodman’s right:
\({S}_{\mathit{alt}}^{\text{max}}\mathrm{=}{S}_{l}(1\mathrm{-}\frac{{\sigma }_{\mathit{stat}}}{{S}_{u}})\), which is \(\alpha ={S}_{l}(1-\frac{{\sigma }_{\mathrm{stat}}}{{S}_{u}})/\sum _{i=1}^{N}{\beta }_{i}{\sigma }_{\mathrm{mod}}^{i}\).
with Gerber’s parable:
\({S}_{\mathit{alt}}^{\text{max}}\mathrm{=}{S}_{l}(1\mathrm{-}\frac{{\sigma }_{\mathit{stat}}^{2}}{{S}_{u}^{2}})\), which is \(\alpha \mathrm{=}{S}_{l}(1\mathrm{-}\frac{{\sigma }_{\mathit{stat}}^{2}}{{{S}_{u}}^{2}})\mathrm{/}\mathrm{\sum }_{i\mathrm{=}1}^{N}{\beta }_{i}{\sigma }_{\mathit{mod}}^{i}\).
Figure 3.2-1: Haigh diagram
This calculation is done for all the nodes or Gauss points in the structure. The areas where \(\alpha\) is the weakest correspond to the areas that limit the life of the structure.
To pass from the coefficient \(\alpha\) to the admissible vibration amplitude, an additional operation has to be performed. This operation is described in § 4.4.