PSI - Issue 26

Christos F. Markides et al. / Procedia Structural Integrity 26 (2020) 53–62 Ch. F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 6. The deformed notched beam and the detailed deformed configuration of the parabolic notch due to the present solution (red color) versus the relevant deformed intact beam according to Euler- Bernoulli’s solution (black color), for an overall load of 10 kN.

3. Discussion and concluding remarks

An attempt to provide an analytic solution for the stress- and displacement-field, developed in a prismatic beam with a central parabolic notch, subjected to four-point bending was described. The solution was achieved in terms of the complex potentials technique introduced by Muskhelishvili (1963) by adopting a suitable transformation. In accordance to previous studies (Kourkoulis et al. 1999), the presence of the notch is responsible for a shift of the neutral axis towards the crown of the notch. As it was expected, the notch creates severe stress concentration around its crown, which for the geometry considered here attains a value equal to about k ≈ 25. The results of the analysis concerning the strain field were considered in juxtaposition to respective experimental data available in literature. Although these data were obtained from a three-point bending test (rather a four-point one, which is the configuration solved analytically), the comparison is encouraging, especially close to the crown of the notch, where the field is expected to be insensitive to the exact position of the loading punch. Indeed, the analytically predicted distribution for the strains is both qualitatively and quantitatively very close to the ones obtained experimentally, providing a strong indication that the assumptions adopted for the solutions are, at least, sound. It could be here anticipated that the solution introduced is of limited practical importance (due to the rather lengthy algebraic expressions of the complex potentials, which, in turn, render determination of stresses and strains somehow complicated), especially considering the flexibility of numerical solutions. This argument sounds reasonable (although proper programming in a commercial package, like Mathematica or MATLAB, makes things rather straightforward) , however, one should always keep in mind that complicated numerical schemes, modelling notched beams of various geometries, must be somehow validated. In this direction the present solution could be proven valuable. Before concluding, it is to be mentioned that the present analysis is based on some critical assumptions (like, for example, the small to moderate length of the notch with respect to the height of the beam and, also, the linearity of the