PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 397–404 Author name / Structural Integrity Procedia 00 (2026) 000–000

400 4

where m t is the temperature in the beam on the level of the crack. Equation (10) is applied for obtaining m t .

h t t −

h

t m

1 t = +

2 1

,

(10)

2

where h is the beam thickness, 2 h is the thickness of the lower arm of the crack (Fig. 1). The longitudinal displacement, 1 D u , of the left-hand end of the beam is also zero, i.e. 0 1 = D u .

(11)

1 D u .

Equation (12) is applied for determining

l a + 1

l a − 2

u u D g + 1

h h − + − 1 2 2

  

  

∫ 0

∫

D f 1

+ − (

z dx n )

u

z dx

κ

κ

=

+

,

(12)

1

1 3

3 4

1

D

D D

D D

n

2

l

1

where n z and n z 1 are the coordinates of the neutral axis in portions, 1 3 DD and 3 4 DD , of the beam, respectively. The axial forces in the left-hand and right-hand sides of section, 3 D , of the beam are equal (this is so because the lower arm of the crack is free of stresses). Thus, we can write D rght D lft N N 3 3 = , (13) where ∫ − = 2 2 3 1 1 h h D lft N b dz σ , (14) ∫ − = 2 2 3 4 1 3 h h D D D rght dz b N σ , (15) where σ and 3 4 D D σ are the stresses, b is the beam width. σ is related to the strain, ε , via Eq. (6). The distribution of ε is presented by the law written in Eq. (16). ( ) n D D z z − = 1 3 ε κ , (16) where

1 h − ≤ ≤ z

h

1

.

(17)

2

2

3 4 D D ε , in the right-hand side of section, 3 D . The

3 4 D D σ and the strain,

Equation (6) is applied also for relating

3 4 D D ε is written in Eq. (18). ( ) n

distribution of

D D D D z z 3 4 1 1 3 4 − = ε κ ,

(18)

where

z h − ≤ ≤ .

2 h

(19)

1

2

3 D , of the beam are also equal, i.e.

The bending moments in the left-hand and right-hand sides of section,

D rght D lft M M 3 3 = .

(20)

These bending moments are related to the stresses via Eqs. (21) and (22), respectively.