<s>
In	O
structural	O
engineering	O
,	O
the	O
flexibility	B-Algorithm
method	I-Algorithm
,	O
also	O
called	O
the	O
method	O
of	O
consistent	O
deformations	O
,	O
is	O
the	O
traditional	O
method	O
for	O
computing	O
member	O
forces	O
and	O
displacements	O
in	O
structural	O
systems	O
.	O
</s>
<s>
Its	O
modern	O
version	O
formulated	O
in	O
terms	O
of	O
the	O
members	O
 '	O
flexibility	O
matrices	B-Architecture
also	O
has	O
the	O
name	O
the	O
matrix	O
force	O
method	O
due	O
to	O
its	O
use	O
of	O
member	O
forces	O
as	O
the	O
primary	O
unknowns	O
.	O
</s>
<s>
Flexibility	O
is	O
the	O
inverse	O
of	O
stiffness	B-Algorithm
.	O
</s>
<s>
The	O
spring	O
stiffness	B-Algorithm
relation	O
is	O
Q	O
=	O
k	O
q	O
where	O
k	O
is	O
the	O
spring	O
stiffness	B-Algorithm
.	O
</s>
<s>
For	O
a	O
system	O
composed	O
of	O
many	O
members	O
interconnected	O
at	O
points	O
called	O
nodes	O
,	O
the	O
members	O
 '	O
flexibility	O
relations	O
can	O
be	O
put	O
together	O
into	O
a	O
single	O
matrix	B-Architecture
equation	I-Architecture
,	O
dropping	O
the	O
superscript	O
m	O
:	O
</s>
<s>
Unlike	O
the	O
matrix	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
,	O
where	O
the	O
members	O
 '	O
stiffness	B-Algorithm
relations	O
can	O
be	O
readily	O
integrated	O
via	O
nodal	O
equilibrium	O
and	O
compatibility	O
conditions	O
,	O
the	O
present	O
flexibility	O
form	O
of	O
equation	O
(	O
)	O
poses	O
serious	O
difficulty	O
.	O
</s>
<s>
With	O
member	O
forces	O
as	O
the	O
primary	O
unknowns	O
,	O
the	O
number	O
of	O
nodal	O
equilibrium	O
equations	O
is	O
insufficient	O
for	O
solution	O
,	O
in	O
general	O
—	O
unless	O
the	O
system	O
is	O
statically	B-Algorithm
determinate	I-Algorithm
.	O
</s>
<s>
For	O
statically	B-Algorithm
indeterminate	I-Algorithm
systems	O
,	O
M	O
>	O
N	O
,	O
and	O
hence	O
,	O
we	O
can	O
augment	O
(	O
)	O
with	O
I	O
=	O
M−N	O
equations	O
of	O
the	O
form	O
:	O
</s>
<s>
Equations	O
(	O
)	O
and	O
(	O
)	O
are	O
the	O
solution	O
for	O
the	O
primary	O
system	O
which	O
is	O
the	O
original	O
system	O
that	O
has	O
been	O
rendered	O
statically	B-Algorithm
determinate	I-Algorithm
by	O
cuts	O
that	O
expose	O
the	O
redundant	O
forces	O
.	O
</s>
<s>
That	O
is	O
,	O
using	O
the	O
unit	B-Algorithm
dummy	I-Algorithm
force	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
While	O
the	O
choice	O
of	O
redundant	O
forces	O
in	O
(	O
)	O
appears	O
to	O
be	O
arbitrary	O
and	O
troublesome	O
for	O
automatic	O
computation	O
,	O
this	O
objection	O
can	O
be	O
overcome	O
by	O
proceeding	O
from	O
(	O
)	O
directly	O
to	O
(	O
)	O
using	O
a	O
modified	O
Gauss	B-Algorithm
–	I-Algorithm
Jordan	I-Algorithm
elimination	I-Algorithm
process	O
.	O
</s>
<s>
It	O
is	O
apparent	O
from	O
the	O
above	O
process	O
that	O
the	O
matrix	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
is	O
easier	O
to	O
comprehend	O
and	O
to	O
implement	O
for	O
automatic	O
computation	O
.	O
</s>
<s>
For	O
these	O
reasons	O
,	O
the	O
matrix	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
is	O
the	O
method	O
of	O
choice	O
for	O
use	O
in	O
general	O
purpose	O
structural	B-Algorithm
analysis	I-Algorithm
software	O
packages	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
for	O
linear	O
systems	O
with	O
a	O
low	O
degree	O
of	O
statical	O
indeterminacy	O
,	O
the	O
flexibility	B-Algorithm
method	I-Algorithm
has	O
the	O
advantage	O
of	O
being	O
computationally	O
less	O
intensive	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
procedure	O
of	O
the	O
direct	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
is	O
so	O
mechanical	O
that	O
it	O
risks	O
being	O
used	O
without	O
much	O
understanding	O
of	O
the	O
structural	O
behaviors	O
.	O
</s>
<s>
The	O
main	O
advantages	O
of	O
the	O
flexibility	B-Algorithm
method	I-Algorithm
is	O
that	O
the	O
result	O
error	O
is	O
independent	O
of	O
the	O
discretization	O
of	O
the	O
model	O
and	O
that	O
it	O
is	O
indeed	O
a	O
very	O
fast	O
method	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
elastic-plastic	O
solution	O
of	O
a	O
continuous	O
beam	O
using	O
the	O
force	O
method	O
requires	O
only	O
4	O
beam	O
elements	O
whereas	O
a	O
commercial	O
"	O
stiffness	B-Algorithm
based	O
"	O
FEM	B-Application
code	O
requires	O
500	O
elements	O
in	O
order	O
to	O
give	O
results	O
with	O
the	O
same	O
accuracy	O
.	O
</s>
<s>
To	O
conclude	O
,	O
one	O
can	O
say	O
that	O
in	O
the	O
case	O
where	O
the	O
solution	O
of	O
the	O
problem	O
requires	O
recursive	O
evaluations	O
of	O
the	O
force	O
field	O
like	O
in	O
the	O
case	O
of	O
structural	O
optimization	O
or	O
system	O
identification	O
,	O
the	O
efficiency	O
of	O
the	O
flexibility	B-Algorithm
method	I-Algorithm
is	O
indisputable	O
.	O
</s>
