PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 97–105 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

99

3

loading consists of two transverse forces, F , applied at the free ends of beam. The upper surface of the beam is in contact with aggressive environment. As a result of this, a damage zone of depth,  , appears in the beam as shown in Fig. 1. The depth of the damage zone increases with the time, t , according to the following exponential law (Druyanov and Nepershin (1990)):

T t

t

   

   

− T T T 1

− T T T 2

−

−

T

e

e

= − 0 1  

+

,

(1)

1

2

1

2

1

2

where 0  is the ultimate depth of the damage zone, 1 T and 2 T are material properties. It is assumed that the mechanical response of the beam can be treated by the Hook’s law   E = , (2) where  is the stress,  is the strain, E is the modulus of elasticity. The distribution of the modulus of elasticity along the height of the beam cross-section is written as

2 3 +

h f h z

U E E e =

,

(3)

where

h z h −   3 .

(4)

In formula (3), U E is the value of the modulus of elasticity at the upper surface of the beam, f is a material property that controls the variation of the modulus of elasticity along the beam height, 3 z is the vertical centroidal axis of the cross-section. The distribution of U E along the beam length is expressed as

3 l l +

g x

UL U E E e =

,

(5)

1

where

1 3 0 x l l   + .

(6)

UL E is the value of U E in the left-hand end of the beam, g is a material property that controls the

In formula (5),

U E along the length of the beam. At ) 2( 1 1 3 l l l l x +   +

variation of

(7)

U E is written as

the distribution of

1 1 3 ) 2( l l x + + −

g l l

UL U E E e =

.

(8)

U E is distributed symmetrically with respect to the mid-span. Due to the symmetry,

Formulae (5) – (8) indicate that

2( l l l l x +   + 1 3

)

only half of the beam,

, is considered in the analysis.

1