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

62 10

Fig. 7. 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 20 kN.

constitutive behavior of the material of the beam). In addition, the solution introduced must be considered as a first approximation of the exact one, since in its current form it ceases satisfying the assumptions imposed regarding the behavior of the complex potentials at infinity. For the exact solution to be obtained in closed form, four additional theorems, regarding the value of the complex potentials o o ,   and o o , ,   defined in the upper and lower ζ half plane respectively, should be first introduced and proven.

References

Bousinesq, J., 1892. Comptes Rendus de I' Academie des Sciences 114, 1510 – 1514. Carman,v.T., 1927. Abhandlungen Aerodynam. Inst., Techn. Hochschule, Aachen, 7, 33. Filon, L.N.G., 1903. Phil. Trans. 201A, 63 – 70. Flamant, M., 1892. Comptes Rendus de l' Academie des Sciences 114, 1465 – 1468. Kourkoulis, S.K., Exadaktylos, G.E., Vardoulakis, I., 1999. U-notched Dionysos-Pentelicon marble in three-point bending: The effect of non-linearity,

anisotropy and microstructure. Int. J. Fracture 98(3-4), 369 – 392. Lamb, H., 1909. Atti IV Cong. Intern. Matemat., Rome, 3, 12. Love, A.E.H., 1927. A Treatise on the Mathematical Theory of Elasticity, University Press, London. Muskhelishvili, N.I., 1963. Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen. Seewald, F., 1927. Abhandlungen Aerodynam. Inst., Techn. Hochschule, Aachen, 7, 11. Stokes, G.C., 1892. Mathematics and Physics Papers 5, 238. Timoshenko, S., 1934. Theory of Elasticity, Mc Graw Hill, New York. Wilson, C., 1891. The influence of surface loading on the flexure of beams, Phil. Mag. 32, 481 – 503.