PSI - Issue 68

Victor Rizov et al. / Procedia Structural Integrity 68 (2025) 146–152 V. Rizov / Structural Integrity Procedia 00 (2025) 000–000

148

3

!

In order to determine the SERR,

, the static indeterminacy has to be resolved first. This is done via Eq. 4.

#

!

! = " !

,

(4)

"

!

!

!

where

is the strain energy in the beam,

is the torsion moment in the left and of the beam ( is a hyperstatic

! ! "

! ! "

!

unknown in the context of Eq. 4). The strain energy is given by

" # ! ! ! ! = + +

,

(5)

!

where

)

# $ # & ' '' ( ' = = ) " % " '' ( ' = = + ) $ # % " % " '' ( ' = = + $ # % & ( & $$ % & ! # & ' & ( ) ( ) & $$ % & " % # & ' & ( ) ( & $$ % &

!! " #

%

A !# !" !" &

=

,

(6)

"

!! " #

%

A !# !" !" &

=

,

(7)

"

!! " #

%

A !# !" !" &

=

.

(8)

"

&

&

! !

" ! ! !

" ! ! !

! !

! !

" ! !!

Here,

,

and

are the strain energies in portions

,

and

of the beam, respectively (Fig.

1). The quantities

and

are defined in Fig. 1,

is the layer number in the lower crack arm (because of the

! !

! !

! !

notch, the upper crack arm is free of stresses). Equations 9, 10 and 11 define the specific strain energies, ,

! " !"

! " !"

! " !"

and

, respectively.

# !

! " "!

=

! ! ! " !

,

(9)

!

! # # $

! " "!

=

! ! ! " !

,

(10)

!

! " "!

=

! ! ! " !

,

(11)

!

!" !

!" !

$" #"! = !

where

and

are the shear stress and strain induced by the support twisting (here,

).

!" !

The specific strain energies are defined by

as

"

!

#

=

" !

,

(12)

#"

!

!

!

"

!