<s>
The	O
Timoshenko	B-Algorithm
–	I-Algorithm
Ehrenfest	I-Algorithm
beam	I-Algorithm
theory	I-Algorithm
was	O
developed	O
by	O
Stephen	O
Timoshenko	O
and	O
Paul	O
Ehrenfest	O
early	O
in	O
the	O
20th	O
century	O
.	O
</s>
<s>
If	O
the	O
shear	O
modulus	O
of	O
the	O
beam	O
material	O
approaches	O
infinity	O
—	O
and	O
thus	O
the	O
beam	O
becomes	O
rigid	O
in	O
shear	O
—	O
and	O
if	O
rotational	O
inertia	O
effects	O
are	O
neglected	O
,	O
Timoshenko	B-Algorithm
beam	I-Algorithm
theory	I-Algorithm
converges	O
towards	O
ordinary	O
beam	O
theory	O
.	O
</s>
<s>
Four	O
boundary	O
conditions	O
are	O
needed	O
for	O
the	O
problem	O
to	O
be	O
well-posed	B-Algorithm
.	O
</s>
<s>
Starting	O
from	O
the	O
above	O
assumption	O
,	O
the	O
Timoshenko	B-Algorithm
beam	I-Algorithm
theory	I-Algorithm
,	O
allowing	O
for	O
vibrations	O
,	O
may	O
be	O
described	O
with	O
the	O
coupled	O
linear	O
partial	O
differential	O
equations	O
:	O
</s>
<s>
(	O
see	O
also	O
the	O
derivation	O
of	O
the	O
Timoshenko	B-Algorithm
beam	I-Algorithm
theory	I-Algorithm
as	O
a	O
refined	O
beam	O
theory	O
based	O
on	O
the	O
variational-asymptotic	O
method	O
in	O
the	O
book	O
by	O
Khanh	O
C	O
.	O
Le	O
leading	O
to	O
different	O
shear	O
coefficients	O
in	O
the	O
static	O
and	O
dynamic	O
cases	O
)	O
.	O
</s>
