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

55

3

zz M y I

(1)

xx  =

According to Muskhelishvili (1963), the complex potentials providing the above stress state in the region AEFG, in case it is considered lying in the z=x+iy complex plane, read as: ( ) ( ) 2 2 I I zz Aiz Aiz M * z , * z A 8 8 I    =  = −  =    (2) ( ) ( ) ( ) ( ) I I I I Aiz Aiz * z * z , * z * z 4 4    =  =  =  = − (3) where prime denotes the derivative and the (*) symbol will indicate reference to the centroidal coordinate system of axes (Fig.2). Subscript I indicates the intact beam.

Fig. 2. The intact beam under four-point bending.

Now consider a Cartesian reference, obtained from the previous one by a translation by a distance c from the old x- axis downward (Fig.1). In that system Φ I *(z) and Ψ I *(z) become: ( ) ( ) ( ) ( ) I I Ai Ai Ac z z ic , z z ic 4 4 4  = −  = − − − (4) In the last expressions, symbol (*) has been dropped to distinguish between Eqs.(4) and Eqs.(3). Eqs.(4) will be the solution of the intact beam constituting the basis for the solution of the notched beam in question. As a next step, a parabolic edge notch is introduced to the lower side of the intact beam. For this to be accomplished, one should first consider the following analytic function ω(ζ): ( ) ( ) 2 z i ia =   = −  + (5) which conformally maps the region of the z-plane outside the parabola BCD, on the infinite upper half plane in the mathematical ζ=ξ+iη complex plane (Fig.3); the focus of the parabola is at the origin of the coordinate axes whereas its vertex C lies at a distance a 2 upwards from it. Due to Eq.(5), every point z=x+iy in the z-pane, including the ABCDEFG region, corresponds to a unique point ζ=ξ+iη in the ζ -plane and vice-versa. The following transformations hold true: ( ) ( ) ( ) ( ) 2 2 x , 2 a , y , a   = +     = +  −  (6) 2.2. Notched beam: The complex potentials

2 y x y + +

2

x

( ) x, y

) ( ) , x, y

a



=

 =

−

(7)

(

2

2 2 y x y + +

2