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

56

4

In this context, the points B, C, D of the parabolic notch correspond to the points ξ B , ξ C ≡O and ξ D , respectively, in the ζ -plane. Eqs.(5) to (7) may be, also, considered as introducing curvilinear coordinates (ξ,η) at any poi nt z=x+iy, outside and on the parabola BCD. Consider now the intersection of the region AEFG occupied by the intact beam and the region outside and on the parabola BCD. Then, assuming that the parabola is free from external stresses (regarding its right side as moving from B to D) it will be as if the wedged part formed by the parabola BCD and the lower side AF of the intact beam has been cut off from the beam, thus transforming it into the notched one.

0.04 0.03 0.02 0.01

-0.02

-0.04 0.02

-0.04

Fig. 3. The conformal mapping.

In addition, it will be assumed that the notch BCD will always be of small to moderate dimensions compared to the beam’s (AEFG) dimensions so that its presence will not si gnificantly affect the stress state at the sides of the beam. In that case, the solution of the intact beam (Eqs.(4)) will accurately enough describe the stress field of the notched one along its sides. That could be conceived as the introduction of the concept of an infinite strip (beam) containing the parabolic notch BCD, and in this context, the solution of the notched beam will be sought of in the form: ( ) ( ) ( ) ( ) ( ) ( ) I o I o z z z , z z z  =  +  =  + (8) where Φ I (z), Ψ I (z) will be given by Eqs.(4) while Φ o (z), Ψ o (z) are terms due to the presence of the notch BCD and they are subjected to determination; in accordance with the above assumptions, Φ o (z), Ψ o (z) will tend to zero for large |z|, having at the point at infinity the general forms (Muskhelishvili 1963):

X iY 1 1 +  

X iY 1 1 −  

( ) z  = −

( ) z  = o

o ,     z

o

+

+

(9)

z    

o

2 z 

2 z 

(X and Y being the components of the resultant force per unit thickness along BCD). Substituting from Eqs.(4) and (5) in Eqs.(8), and keeping the same symbols for the functions involved for ease, one takes:

A

Ac

A

Ac

( )

(

)

( )

( )

(

)

( )

2

2

o 2   =  + + +     = −  + − +   o ia , ia 4 4 4

(10)