PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 85–94

90

1 h    z

h

1

.

(22)

2

2

2

n z 2 - distance of the neutral axis from the centre of the

The notations in Eqs. (21) and (22) are: up  - curvature,

cross-section, 2 z - the vertical axis, 1 h - thickness of the lower arm of the crack. The quantities, up  and n z 2 , can be determined by making use of Eqs. (23) and (24).   ( ) up A N dA  ,

(23)

 

M

2 z dA



.

(24)

2

y

( ) up A

The notations in the above equations are as follows: N - axial force in the lower arm, up A - area of the cross section, 2 y M - bending moment. The axial force and the bending moment in any cross-section of the lower arm are

h

1



2



 1 1 q dz dx 1 2

0 h Rn   

N



,

(25)

2

h

1



2

 

  1 dx



 x dz

0    h

M

1 2 q z q Rn



1 Rs   

.

(26)

2

1 2

y

1

2

The notations in Eqs. (25) and (26) are:  - distance of the cross-section from the left end of the beam, b - width of the beam. The quantities, up  and n z 2 , are extracted from Eqs. (23) and (24) by making use of the MatLab. The complementary strain energies, * 2 U and * 3 U , in the other parts of the structure, i.e. in the upper arm of the crack and portion, 2 3 L L , of the beam are

U u dV up V up   ( ) * 0 * 2 , U u dV nc V nc   ( ) * 0 * 3 .

(27)

(28)

* 0 nc u - specific complementary strain energies,

In Eqs. (27) and (28) the following notations are used: * up V and nc V - volumes of corresponding parts of the structure. The quantities, * 1 U , * 2 U and * 3 U , are summed to obtain

0 up u and

* U (the integrals are solved by making use of the