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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exist. Only for relatively long beams (the length of which significantly exceeds the remaining dimensions) and at sections far from the loaded ones the classical Bernoulli-Euler technical theory provides satisfactory results. On the other hand, as one approaches the loaded cross sections serious perturbations appear (Love 1927). The problem concerns engineers already from the end of the 19 th century. Among the pioneering relative studies, one should mention the experimental ones by Wilson (1891) and Flamant (1892) and the analytical ones by Bousinesq (1892), Stokes (1892), Filon (1903) and Lamb (1909). Later the problem was revisited by v. Carman (1927), Seewald (1927), Timoshenko (1934) and others. Unfortunately, things become far more complicated in case the beam is notched, even under the simplest loading scheme of four-point bending. For this configuration the stress concentration around the tip of the notch seriously alters the stress field, rendering the classical solution definitely inapplicable, even as a rough approximation. Closed form analytic solutions do not as yet exist, in spite of the increased practical importance of the specific test. In this context, an attempt is here presented to analytically solve the problem of a rectilinear beam under four-point bending, assuming that the beam is parabolically notched at its central section. Adopt ing Muskhelishvili’s (1963) complex potentials technique an approximate closed-form solution for the stress- and displacement-field is obtained. The goal is achieved by making use of specific properties of Cauchy integrals taken over infinite straight lines, in conjunction with a suitable transformation in terms of curvilinear coordinates. The solution introduced, is somehow cumbersome (due to the lengthy mathematical expressions) and, also, is valid under specific conditions and restrictions, which limit its applicability. However, it can be considered as a useful tool in the direction of validating complicated numerical schemes modelling notched beams of various geometries.

2. Analytic considerations

The elastic equilibrium of a notched beam under four-point bending is here confronted as a 1 st fundamental problem of plane linear elasticity. Namely, stresses and displacements will be obtained in the longitudinal mid-section of the beam and particularly in the part ABCDEFG of it, subjected to pure bending by moments M=Pd/2>0 (Fig.1). The solution proposed for the notched beam is based on the solution of the relevant intact one under similar loading conditions. In this context, the solution of the intact beam under pure bending is briefly presented below.

Fig. 1. The notched beam under four-point bending.

2.1. The intact beam

It is here recalled that in the case of the intact beamAEFG under pure bending (shaded area in Fig.2), Euler- Bernoulli’s simplifying assumptions lead to a linear distribution of axial stresses σ xx along the height of the beam’s cross -section, as follows (where I zz is the second moment of area with respect to the z-coordinate axis):