<s>
Unified	B-Algorithm
framework	I-Algorithm
is	O
a	O
general	O
formulation	O
which	O
yields	O
nth	O
-	O
order	O
expressions	O
giving	O
mode	O
shapes	O
and	O
natural	O
frequencies	O
for	O
damaged	O
elastic	O
structures	O
such	O
as	O
rods	O
,	O
beams	O
,	O
plates	O
,	O
and	O
shells	O
.	O
</s>
<s>
Christides	O
and	O
Barr	O
(	O
1984	O
)	O
used	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
,	O
Shen	O
and	O
Pierre	O
(	O
1990	O
)	O
used	O
the	O
Galerkin	B-Algorithm
Method	I-Algorithm
,	O
and	O
Qian	O
et	O
al	O
.	O
</s>
<s>
(	O
1991	O
)	O
used	O
a	O
finite	B-Application
element	I-Application
method	I-Application
to	O
predict	O
the	O
behavior	O
of	O
a	O
beam	O
with	O
an	O
edge	O
crack	O
.	O
</s>
<s>
They	O
were	O
developed	O
in	O
a	O
few	O
cases	O
for	O
Timoshenko	B-Algorithm
beam	I-Algorithm
theory	I-Algorithm
or	O
plate	O
theories	O
with	O
expressions	O
provided	O
only	O
for	O
particular	O
boundary	O
conditions	O
and	O
beam	O
or	O
plate	O
shapes	O
;	O
</s>
<s>
The	O
lack	O
of	O
generality	O
of	O
damage	O
models	O
is	O
addressed	O
by	O
proposing	O
a	O
‘	O
unified	O
framework’	O
which	O
is	O
valid	O
for	O
self-adjoint	O
systems	O
using	O
beam	O
theories	O
like	O
Euler	O
–	O
Bernoulli	O
beam	O
theory	O
,	O
Timoshenko	B-Algorithm
,	O
plate	O
theories	O
like	O
Kirchhoff	O
and	O
Mindlin	O
and	O
shell	O
theories	O
.	O
</s>
<s>
The	O
model	O
was	O
presented	O
and	O
verified	O
for	O
a	O
damaged	O
beam	O
with	O
notch	O
type	O
damage	O
,	O
using	O
first-order	O
perturbation	O
only	O
,	O
for	O
the	O
Euler	O
–	O
Bernoulli	O
beam	O
theory	O
in	O
the	O
paper	O
by	O
Dixit	O
and	O
Hanagud	O
(	O
2011	O
)	O
and	O
using	O
Timoshenko	B-Algorithm
beam	I-Algorithm
theory	I-Algorithm
in	O
the	O
paper	O
by	O
Dixit	O
and	O
Hanagud	O
(	O
2009	O
)	O
.	O
</s>
<s>
The	O
inertia	O
effect	O
(	O
kinetic	O
energy	O
)	O
,	O
which	O
,	O
unlike	O
the	O
stiffness	O
effect	O
(	O
strain	B-Algorithm
energy	I-Algorithm
)	O
,	O
of	O
the	O
damage	O
has	O
been	O
neglected	O
by	O
researchers	O
,	O
is	O
included	O
in	O
it	O
.	O
</s>
