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HCF (HIGH-CYCLE FATIGUE) The influence of mean stress two of these: can be taken into account in different ways. Goodman and Gerber are ’Our’ Haigh diagram can be considered as a rationalisation of the Gerber equation, where only 2 fatigue tests (‘alternating’ and ‘pulsating’ loads ) need be performed. Multiaxial stress state We first deal with stress states of the type , i.e., ‘in-phase’ stress components. The most-used hypothesis for such cases is by Sines (1959): In the above equations, is the stress deviator For a uniaxial case, we have I.e., a Goodman equation with Therefore, if we rewrite (3) with these values: One can also imagine a ‘Sines’ variant’ of the Gerber equation: or even using the Haigh diagram with the Sines’ equivalent stresses: Example As a simple example we look at a thin-walled cylindrical pressure vessel, loaded by a time-varying inner pressure The diameter is , it is assumed that Sines’ hypothesis can be used with data and a safety factor is required against fatigue. Compute the necessary wall thickness load cases (a) and (b) (a) We get , for the two and Sines’ hypothesis gives (b) In this case, Sines now gives With the given relations ----To compare cases (a) and (b), assume, for instance, : [Case (a) may be a little unrealistic, since it is unusual for a pressure vessel to be loaded by alternating positive and negative pressure.] General multiaxial stress states In this section, we will deal with stress states that cannot be expressed as Example: For such cases, knowledge is not so established, but one can, for instance, postulate a more general Sines’ expression: One simple way of illustrating this is by looking at a von Mises stress space on the deviatoric plane (i.e., looking along the space diagonal): The contour shown in the figure is the curve that the vector takes through the principal stress space. When this has been tested for all possible choices of , the minimum von Mises cylinder needed to contain all such curves is drawn (see the figure). The radius of this cylinder is then Example Study the the load cycle The definition of stress deviator gives which, in turn, gives Now search for the maximum of this expression! Both these solutions lead to the same expression for : and the fatigue initiation criterion becomes With numerical values inserted, e.g., 0 196.0 100 129.5 150 96.32 200 63.11 250 29.89 Critical-plane theories for fatigue failure In modern fatigue design, critical-plane theories have come to more frequent use. A critical plane is a plane on which a combination of shear stress amplitude (or shear amplitde ) and normal stress reaches a maximum value. As an example, Findley (1959) postulated the following critical-plane fatigue failure criterion: In this theory, life. and are material parameters, which are expected to be constants for a given fatigue The figure below shows an example of the definition of a critical plane. Figure 2. Definition of a critical plane. From Suresh S: Fatigue of materials