PSI - Issue 28

Christos F. Markides et al. / Procedia Structural Integrity 28 (2020) 710–719 Christos F. Markides and Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2019) 000–000

718

9

30

0

Normal Shear

-30

Stress [MPa]

-60

-90

-0.010 -0.005

0.000 x [m]

0.005

0.010

Fig. 7. The distribution of normal and shear stresses along the interface of the FBD and the loading platen according to a preliminary numerical study.

acting at equally distanced points along two chords, supposed to stand as the two flat edges of the flattened disc. The idea behind this approach is that a point force can be conceived as the resultant of a stress distribution acting on the periphery of an eliminated hole. Then, considering an infinite number of such forces is equivalent to the introduction of an infinite number of small holes along the two parallel chords of the uniform disc, thus truncating it and transforming it into the flattened one. In this paper, and as a first step, these point forces were all assumed to be of the same magnitude. As it can be seen, such an assumption leads to a non-uniform pressure along the immediate vicinity of the loaded chords of the uniform disc, resulting to a distorted deformed shape of these chords. However, experimental reality demands the flat edges of the FBD to remain plane in the deformed state, thus rendering the original assumption of equal point forces along the chords of the uniform disc inadequate if those chords were to stand as the flat edges of the flattened disc. On the other hand, it is noticed that the quantitative results shown a moderate deviation from linearity of pressure (and the induced deformation) caused by the equal point forces. The above observations lead to the conclusion that the present approach may eventually offer an efficient analytic solution for the flattened disc provided that instead of equal point forces, ones of different magnitude would be con sidered, namely, such ones that the chords of the uniform disc would remain flat in the deformed state. In the direction of seeking such forces, a numerical model was developed, in parallel to the analytic solution, depicting properly the experimental reality, i.e., imposing uniform vertical displacements for the points on the chords of the uniform disc so that they could stand as the flat edges of the flattened disc. It was pointed out by the numerical analysis, that such a displacement field leads to a non-uniform, parabolic distribution of normal pressure along the chords of the uniform disc, in accordance with the analytic findings. It is exactly this parabolic pressure, provided by the numerical analysis that should be taken into account also in the analytic consideration of the uniform disc under point forces in order for the latter to provide an effective solution, efficiently describing the flattened disc configuration. This improvement of the present solution is the authors’ next step in the direction of providing an effective alternative for the FBD, taking, as much as possible, into account the actual boundary conditions prevailing along its flat edges, instead of assuming a uniform distribution of pressure. Concluding, it could be argued that the analytic procedure described here is somehow complicated and the formulae for the stresses are rather lengthy. Moreover, it could be anticipated that quite a few issues are left “open” (i.e., a more realistic loading scheme) and, also, that limitations exist (i.e., the load application points are singular and it is not permitted to closely approach them). In spite of the above limitations, it is to be accepted that the present analytic solution, which is in good qualitative agreement with the results of the respective numerical ones of the literature, offers a flexible tool in hands of scientists using the FBDT, since it provides closed expressions fot the stress-field components, which can be used for validation/calibration of numerical models, which, in turn, can be used for detailed parametric analyses of the problem.