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

717

8

The second locus considered is the radius towards the end-point z 1 of the straight edge of the FBD. The results (for the same as above numerical values) are plotted in Fig.5b, again in juxtaposition to the respective ones of the standard BD. As it was expected, shear stresses appear now all long this locus. What is to be stressed out for the specific locus, is that the radial stress develoed in the FBD appears increasing somehow abruptly as the end-point z 1 is approached. The specific behaviour of the radial stress should be obviously studied further. It seems that it could be attributed both: (i) to the simplified type of external load considered (namely, equal point forces all along the flat edges of the FBD), which, as it will be discussed in next sections, appears to be somehow unrealistic, and, also, (ii) to the singular nature of the stress field in the immediate vicinity of points z i at which the point force P is applied. 4. Discussion It is well known that the laboratory implememntation of the FBDT consists of squeezing the FBD between stiff plane loading platens, dictating that its straight edges must remain linear after the application of the load. On the other hand, the FBD was here considered to be acted by equal point forces all along its straight edges, an assumption resembling the uniform pressure, commonly adopted by all researchers dealing with the specific issue. In an attempt to check whether this assumption is compatible to experimental reality, the displacement field developed in the FBD is here determined taking advantage of the above obtained complex potentials. According to Muskhelishvili (1963), assuming that the complex potentials are known, displacements are determined by the well-known formulae:   2 ( ) ( ) ( ) u i v z z z z          (10) where μ = G is the shear modulus. The deformed shape of the disc (for an unrealistically high overall load P frame , assumed just to make the deformation clearly visible) is plotted in Fig.6, in juxtaposition to the undeformed one. It is clearly seen that, as long as the load induced is uniform, the flat edges of the FBD do not remain straight in the deformed state. Obviously, such a deformation of the flat edges of the FBD is incompatible to the experimental reality, indicating that the distribution of the externally induced load should be reconsidered. Preliminary study of the issue, by means of a numerical model, indicates that the stresses along the interface between the FBD and the loading platens is of parabolic nature (maximized at the end-points of the flat segments and minimized at their mid-point) (Fig.7, green continuous line). In addition the numerical solution indicates that a “sigmoeid” distribution of friction stresses appears (Fig.7, red dotted line). It is therefore clear that the problem should be resolved, assuming a more realistic distribution of the externally applied load, consisting of mutually compatible distributions of normal and shear (friction) forces. The authors’ team is working in this directions and preliminary results are encouraging. 5. Concluding remarks An attempt was described, in the present study to analytically confront the problem of the stress didtribution in a FBD under uniform external load. In this direction, the uniform disc was considered under a great number of point forces

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

P frame =200kN

P=const.

Deformed

Θ =80 o

y [m]

Undeformed

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

x [m]

Fig. 6. The deformed shape of a FBD made of a material with Young’s modulus E =3.2 GPa and Poisson’s ratio ν =0.36.