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
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