PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

78 4

m

n

2

1

1

+

+

n nH







2 m u B mD a * 0 1 = −

−

.

(7)

1

1

+

+

The distribution of the strains in the height direction is treated by applying the Bernoulli’s hypothesis for plane sections since beams of high length to height ratio are analyzed in the present paper. Thus, the distribution of  is written as ( ) n z z 1 1 1 = −   , (8) where

1 1 h z h −  

1

.

(9)

2

2

In formula (8), 1  is the curvature of the upper crack arm, n z 1 is the coordinate of the neutral axis, 1 z is the vertical centroidal axis of the cross-section of the upper crack arm. The following equations for equilibrium of the elementary forces in the cross-section of the upper crack arm are used to determine the curvature and the coordinate of the neutral axis:

h

1

2

h  

N b =

dz

,

(10)

1

1

2

−

2

h

1

2

h  

1 1 M b z dz = 1

,

(11)

2

−

2

where 1 N and 1 M are the axial force and the bending moment in the upper crack arm. In order to derive 1 N and 1 M , the beam is treated as a structure of two degrees of internal static indeterminacy ( 1 N and 1 M are taken as internal redundants). The static indeterminacy is resolved by applying the theorem of Castigliano for structures which exhibit material non-linearity

*

N U



0

=

,

(12)



1

*

M U

 

0

=

,

(13)

1

where * U is the complementary strain energy in the beam. Since the complementary strain energies in the un-cracked parts, 3 1 0 x l   and l a x l +   3 1 1 , of the beam do not depend on 1 N and 1 M , only the complementary strain energy in the crack arms is involved in (12) and (13). Therefore, * U is written as

* 1 * U U U = + , 2 *

(14)