Issue 53

V. Rizov et alii, Frattura ed Integrità Strutturale, 53 (2020) 38-50; DOI: 10.3221/IGF-ESIS.53.04

1 h h z    . (21) 1

1

2

2

Since beams of high length to height ratio are under consideration in the present paper, the distribution of the strains in the cross-section of the lower crack arm is treated in accordance with the Bernoulli’s hypothesis for the plane sections. Thus,  is expressed as

y      

z 

z

, (22)

C

y

1

1

1

1

1

1 C  is the strain in the centre of the lower crack arm cross-section,

1 y  and

1 z  are the curvatures of lower crack

where

arm in the 1 1 x y and 1 1 x z planes, respectively. The curvatures of the lower crack arm and the strain in the centre of the cross-section are found from the equations for equilibrium of the elementary forces in the cross-section of the lower crack arm

1 h b 2 2

1 h b N dy dz      1 1 1

, (23)

22

1 h b 2 2

1 h b M z dy dz      1 1 1 1 y

, (24)

22

1 h b 2 2

1 h b    

y dy dz 

M

, (25)

z

1 1 1

1

22

where 1 z M are the bending moments with respect to the centroidal axes, 1 y and 1 z ,  is the normal stress, b and 1 h are the width and height of the cross-section (Fig. 2). It is obvious that (Fig. 1) 1 0 N  , (26) 1 y M M  , (27) z M  . (28) After substituting of (16) in (23), (24) and (25) the equations for equilibrium are solved with respect to 1 C  , 1 y  and 1 z  by using the MatLab computer program. Then * 01 u is found by substituting of (19) and (22) in (18). The complementary strain energy in the un-cracked beam portion is written as 1 N is the axial force, 1 y M and 1 0

h b

l

2 2

b h a      

* 2

*

U

02 3 2 2 u dx dy dz

, (29)

2 2

where * 02 u is the strain energy density, 2 y and 2 z are the centroidal axes. Formula (18) is used to obtain *

02 u . For this

d  where d  is the distribution of the strains in the un-cracked beam portion. The

purpose,  is replaced with

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