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

58

6

) ( 

3

2

3 

A

)

( )   =

(

)

(

)

2 a c 2 log − −  +   −  −  − −  +   +      2 D D D D log 2

D

(

o

2 i

ia   + 

(

)

A ia  −

  

  

(

)

(

)

(

)

(

)

3

3

2 a c 2 log

log +  −  −  − −  + log

−  +   −  



D

D

D

D

(

)

2

2 i

ia

  +

(

)

2

A ia  −

 

3

(  − − −  −   − −   −  +  − −  +    ) ( ) ( ) 2 3 3 D D D D D 2 log 2 log log

(18)

(

)

3

3

4 i

ia

  +

    

  

 



(

)

(

)

(

)

(

)

2 −   +  2 ia

2 a c 2 ialog − −  −  −  +

ialog + − −  −

 

D

D

D

D

 − 

− − 

D

D

 

3

3

3

2

3 





(

) ( 2 i3a log 2

(

) ( 2 i3a log 2

)

)

3 − −  +  D

3 +  + 

 −  +

− −  −

D

D

 − 

 −  

D

D

The overall solution Φ ( ζ ), Ψ ( ζ ) follows by introducing the above expressions (Eqs.(17) and (18)) for Φ o ( ζ ) and Ψ o ( ζ ) in Eqs.(10). Instead of ζ = ξ +i η , one can use the variables z=x+iy by means of Eq.(5) or the transformations of Eqs.(7).

2.3. Notched beam: The stress field

Substituting from Eqs.(5), (17) and (18) in the following formulae (Muskhelishvili 1963):

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

    

    

           

    

( )  

( )

( )

2 e   =    −   + e

 



    

    

    

( )  

( )

( )

2 e   =    +   + e

 

(19)



( ) ( )

( ) ( )

    

   

     

    

( )   + 

( )

m

  = 

 

 the stress components are obtained at every point of the notched beam ABCDEFG, either in curvilinear or (through Eqs.(7)) in Cartesian coordinates. In this context, the critical axial stress component σ ξξ (or in Cartesian coordinates the σ xx -component) is plotted (red color) a long the height of the beam’s cross -section bisecting the notch (CH-locus), in juxtaposition to the respective component in case of the intact beam (blue color), for comparison reasons (Fig.4). For the specific figure to be drawn, a prismatic beam of dimen sions ℓ o =40 cm, b=10 cm, h=10 cm was considered, with a central parabolic notch of length equal to 2 cm and width BD=5 mm. The beam was loaded by two linearly distributed loads, P/2=6.5 kN each (i.e., an overall load of 13 kN), at sections F and G at distances d=20 cm with regard to the supporting cylinders at the lower side of the beam. As it was expected, a severe stress concentration (with stress concentration factor equal to about k ≈ 25), appears at the tip C of the notch. As it is, also, seen from Fig.4, the neutral axis of the beam shifts towards the notch. Moreover, it is observed that at point H on the upper side of the beam, the magnitude of the compressive axial stress is somehow higher in the case of the notched beam; that is to be attributed, also, to the approximate character of the present solution. Concerning the strain field, and for the same as above configuration, the beam is here assumed to be made of Dionysos marble (average Young’s modulus E=60 GPa and Poisson’s ratio ν=0.25 ). For this case the axial, ε xx , and transverse, ε yy , strains, are plotted along y-axis for plane strain conditions in Fig.5. These data were chosen in order to attempt a first comparison between the results of the present analytic solution and experimental data, available in a previously published study with notched marble beams under three-point bending (Kourkoulis et al. 1999).