PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 95–104

100

where b is the width of the bar cross-section. * U is obtained by summing the complementary strain energies in the different parts of the moving frame-like component, i.e.     p i n i i U U 1 * * , (23) where 5  p n since the component under consideration has 5 parts, namely bar, 1 2 DD , upper and lower arms of the lengthwise crack in portion, 2 3 D D , of the bar, 2 4 D D , the intact portion, 3 4 DD , of bar, 2 4 D D , and bar, 4 5 D D . First, the complementary strain energy, * 1 U , in bar, 1 2 DD , is found. The associated equation is

U u dV V   ( ) * 01 * 1 1 ,

(24)

where * 01 u is the specific complementary strain energy, 1 V is the volume of the bar. The equation for * 01 u is       d u * 01 .

(25)

The variation of strain,  , along the thickness is related to the curvature of the bar, 1  , as given below   n z z 1 1 1     , (26)

where

z h    .

2 h

(27)

1

2

Here, n z 1 is the distance of the neutral axis from the centre of the cross-section. Equations of equilibrium (28) and (29) can be applied to derive, 1  and n z 1 , in any cross-section of bar, 1 2 DD .   ( ) br A N dA  , (28)   ( ) 1 1 br A y z dA M  . (29) The notations used in Eqs. (28) and (29) are: N - axial force; 1 y M - bending moment; br A - area of the cross section. In order to determine N and 1 y M in any cross-section of bar, 1 2 DD , we carry-out reduction of the inertia