<s>
Structural	B-Algorithm
analysis	I-Algorithm
is	O
a	O
branch	O
of	O
solid	O
mechanics	O
which	O
uses	O
simplified	O
models	O
for	O
solids	O
like	O
bars	O
,	O
beams	O
and	O
shells	O
for	O
engineering	O
decision	O
making	O
.	O
</s>
<s>
Structural	B-Algorithm
analysis	I-Algorithm
uses	O
ideas	O
from	O
applied	O
mechanics	O
,	O
materials	O
science	O
and	O
applied	O
mathematics	O
to	O
compute	O
a	O
structure	O
's	O
deformations	O
,	O
internal	O
forces	O
,	O
stresses	O
,	O
support	O
reactions	O
,	O
velocity	O
,	O
accelerations	O
,	O
and	O
stability	O
.	O
</s>
<s>
Structural	B-Algorithm
analysis	I-Algorithm
is	O
thus	O
a	O
key	O
part	O
of	O
the	O
engineering	O
design	O
of	O
structures	O
.	O
</s>
<s>
In	O
the	O
context	O
to	O
structural	B-Algorithm
analysis	I-Algorithm
,	O
a	O
structure	O
refers	O
to	O
a	O
body	O
or	O
system	O
of	O
connected	O
parts	O
used	O
to	O
support	O
a	O
load	O
.	O
</s>
<s>
A	O
structural	O
system	O
is	O
the	O
combination	O
of	O
structural	B-Algorithm
elements	I-Algorithm
and	O
their	O
materials	O
.	O
</s>
<s>
The	O
structural	B-Algorithm
elements	I-Algorithm
guiding	O
the	O
systemic	O
forces	O
through	O
the	O
materials	O
are	O
not	O
only	O
such	O
as	O
a	O
connecting	O
rod	O
,	O
a	O
truss	B-Device
,	O
a	O
beam	O
,	O
or	O
a	O
column	O
,	O
but	O
also	O
a	O
cable	O
,	O
an	O
arch	O
,	O
a	O
cavity	O
or	O
channel	O
,	O
and	O
even	O
an	O
angle	O
,	O
a	O
surface	O
structure	O
,	O
or	O
a	O
frame	O
.	O
</s>
<s>
The	O
first	O
type	O
of	O
loads	O
are	O
dead	O
loads	O
that	O
consist	O
of	O
the	O
weights	O
of	O
the	O
various	O
structural	B-Algorithm
members	I-Algorithm
and	O
the	O
weights	O
of	O
any	O
objects	O
that	O
are	O
permanently	O
attached	O
to	O
the	O
structure	O
.	O
</s>
<s>
To	O
perform	O
an	O
accurate	O
analysis	O
a	O
structural	O
engineer	O
must	O
determine	O
information	O
such	O
as	O
structural	O
loads	O
,	O
geometry	B-Algorithm
,	O
support	O
conditions	O
,	O
and	O
material	O
properties	O
.	O
</s>
<s>
Advanced	O
structural	B-Algorithm
analysis	I-Algorithm
may	O
examine	O
dynamic	O
response	O
,	O
stability	O
and	O
non-linear	O
behavior	O
.	O
</s>
<s>
There	O
are	O
three	O
approaches	O
to	O
the	O
analysis	O
:	O
the	O
mechanics	O
of	O
materials	O
approach	O
(	O
also	O
known	O
as	O
strength	O
of	O
materials	O
)	O
,	O
the	O
elasticity	O
theory	O
approach	O
(	O
which	O
is	O
actually	O
a	O
special	O
case	O
of	O
the	O
more	O
general	O
field	O
of	O
continuum	O
mechanics	O
)	O
,	O
and	O
the	O
finite	B-Application
element	I-Application
approach	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
approach	O
is	O
actually	O
a	O
numerical	O
method	O
for	O
solving	O
differential	O
equations	O
generated	O
by	O
theories	O
of	O
mechanics	O
such	O
as	O
elasticity	O
theory	O
and	O
strength	O
of	O
materials	O
.	O
</s>
<s>
However	O
,	O
the	O
finite-element	B-Application
method	I-Application
depends	O
heavily	O
on	O
the	O
processing	O
power	O
of	O
computers	O
and	O
is	O
more	O
applicable	O
to	O
structures	O
of	O
arbitrary	O
size	O
and	O
complexity	O
.	O
</s>
<s>
The	O
method	O
of	O
mechanics	O
of	O
materials	O
is	O
limited	O
to	O
very	O
simple	O
structural	B-Algorithm
elements	I-Algorithm
under	O
relatively	O
simple	O
loading	O
conditions	O
.	O
</s>
<s>
The	O
structural	B-Algorithm
elements	I-Algorithm
and	O
loading	O
conditions	O
allowed	O
,	O
however	O
,	O
are	O
sufficient	O
to	O
solve	O
many	O
useful	O
engineering	O
problems	O
.	O
</s>
<s>
The	O
theory	O
of	O
elasticity	O
allows	O
the	O
solution	O
of	O
structural	B-Algorithm
elements	I-Algorithm
of	O
general	O
geometry	B-Algorithm
under	O
general	O
loading	O
conditions	O
,	O
in	O
principle	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
is	O
perhaps	O
the	O
most	O
restrictive	O
and	O
most	O
useful	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
It	O
does	O
,	O
however	O
,	O
make	O
it	O
generally	O
possible	O
to	O
solve	O
these	O
equations	O
,	O
even	O
with	O
highly	O
complex	O
geometry	B-Algorithm
and	O
loading	O
conditions	O
,	O
with	O
the	O
restriction	O
that	O
there	O
is	O
always	O
some	O
numerical	O
error	O
.	O
</s>
<s>
The	O
simplest	O
of	O
the	O
three	O
methods	O
here	O
discussed	O
,	O
the	O
mechanics	O
of	O
materials	O
method	O
is	O
available	O
for	O
simple	O
structural	B-Algorithm
members	I-Algorithm
subject	O
to	O
specific	O
loadings	O
such	O
as	O
axially	O
loaded	O
bars	O
,	O
prismatic	O
beams	O
in	O
a	O
state	O
of	O
pure	O
bending	O
,	O
and	O
circular	O
shafts	O
subject	O
to	O
torsion	O
.	O
</s>
<s>
For	O
the	O
analysis	O
of	O
entire	O
systems	O
,	O
this	O
approach	O
can	O
be	O
used	O
in	O
conjunction	O
with	O
statics	O
,	O
giving	O
rise	O
to	O
the	O
method	B-Algorithm
of	I-Algorithm
sections	I-Algorithm
and	O
method	O
of	O
joints	O
for	O
truss	B-Device
analysis	O
,	O
moment	B-Algorithm
distribution	I-Algorithm
method	I-Algorithm
for	O
small	O
rigid	O
frames	O
,	O
and	O
portal	O
frame	O
and	O
cantilever	O
method	O
for	O
large	O
rigid	O
frames	O
.	O
</s>
<s>
Except	O
for	O
moment	B-Algorithm
distribution	I-Algorithm
,	O
which	O
came	O
into	O
use	O
in	O
the	O
1930s	O
,	O
these	O
methods	O
were	O
developed	O
in	O
their	O
current	O
forms	O
in	O
the	O
second	O
half	O
of	O
the	O
nineteenth	O
century	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
they	O
contain	O
the	O
assumptions	O
(	O
among	O
others	O
)	O
that	O
the	O
materials	O
in	O
question	O
are	O
elastic	O
,	O
that	O
stress	B-Algorithm
is	O
related	O
linearly	O
to	O
strain	O
,	O
that	O
the	O
material	O
(	O
but	O
not	O
the	O
structure	O
)	O
behaves	O
identically	O
regardless	O
of	O
direction	O
of	O
the	O
applied	O
load	O
,	O
that	O
all	O
deformations	O
are	O
small	O
,	O
and	O
that	O
beams	O
are	O
long	O
relative	O
to	O
their	O
depth	O
.	O
</s>
<s>
There	O
are	O
2	O
commonly	O
used	O
methods	O
to	O
find	O
the	O
truss	B-Device
element	O
forces	O
,	O
namely	O
the	O
method	O
of	O
joints	O
and	O
the	O
method	B-Algorithm
of	I-Algorithm
sections	I-Algorithm
.	O
</s>
<s>
The	O
first	O
diagram	O
below	O
is	O
the	O
presented	O
problem	O
for	O
which	O
the	O
truss	B-Device
element	O
forces	O
have	O
to	O
be	O
found	O
.	O
</s>
<s>
This	O
type	O
of	O
method	O
uses	O
the	O
force	O
balance	O
in	O
the	O
x	O
and	O
y	O
directions	O
at	O
each	O
of	O
the	O
joints	O
in	O
the	O
truss	B-Device
structure	O
.	O
</s>
<s>
Although	O
the	O
forces	O
in	O
each	O
of	O
the	O
truss	B-Device
elements	O
are	O
found	O
,	O
it	O
is	O
a	O
good	O
practice	O
to	O
verify	O
the	O
results	O
by	O
completing	O
the	O
remaining	O
force	O
balances	O
.	O
</s>
<s>
This	O
method	O
can	O
be	O
used	O
when	O
the	O
truss	B-Device
element	O
forces	O
of	O
only	O
a	O
few	O
members	O
are	O
to	O
be	O
found	O
.	O
</s>
<s>
However	O
this	O
method	O
has	O
a	O
limit	O
in	O
that	O
the	O
cutting	O
line	O
can	O
pass	O
through	O
a	O
maximum	O
of	O
only	O
3	O
members	O
of	O
the	O
truss	B-Device
structure	O
.	O
</s>
<s>
This	O
restriction	O
is	O
because	O
this	O
method	O
uses	O
the	O
force	O
balances	O
in	O
the	O
x	O
and	O
y	O
direction	O
and	O
the	O
moment	O
balance	O
,	O
which	O
gives	O
a	O
maximum	O
of	O
3	O
equations	O
to	O
find	O
a	O
maximum	O
of	O
3	O
unknown	O
truss	B-Device
element	O
forces	O
through	O
which	O
this	O
cut	O
is	O
made	O
.	O
</s>
<s>
The	O
truss	B-Device
elements	O
forces	O
in	O
the	O
remaining	O
members	O
can	O
be	O
found	O
by	O
using	O
the	O
above	O
method	O
with	O
a	O
section	O
passing	O
through	O
the	O
remaining	O
members	O
.	O
</s>
<s>
For	O
complex	O
geometries	O
,	O
a	O
numerical	O
solution	O
method	O
such	O
as	O
the	O
finite	B-Application
element	I-Application
method	I-Application
is	O
necessary	O
.	O
</s>
<s>
It	O
is	O
common	O
practice	O
to	O
use	O
approximate	O
solutions	O
of	O
differential	O
equations	O
as	O
the	O
basis	O
for	O
structural	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
numerical	O
approximation	O
in	O
structural	B-Algorithm
analysis	I-Algorithm
is	O
the	O
Finite	B-Application
Element	I-Application
Method	I-Application
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
approximates	O
a	O
structure	O
as	O
an	O
assembly	O
of	O
elements	O
or	O
components	O
with	O
various	O
forms	O
of	O
connection	O
between	O
them	O
and	O
each	O
element	O
of	O
which	O
has	O
an	O
associated	O
stiffness	O
.	O
</s>
<s>
The	O
analytical	O
and	O
computational	O
development	O
are	O
best	O
effected	O
throughout	O
by	O
means	O
of	O
matrix	B-Architecture
algebra	I-Architecture
,	O
solving	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
Early	O
applications	O
of	O
matrix	O
methods	O
were	O
applied	O
to	O
articulated	O
frameworks	O
with	O
truss	B-Device
,	O
beam	O
and	O
column	O
elements	O
;	O
later	O
and	O
more	O
advanced	O
matrix	O
methods	O
,	O
referred	O
to	O
as	O
"	O
finite	B-Algorithm
element	I-Algorithm
analysis	I-Algorithm
"	O
,	O
model	O
an	O
entire	O
structure	O
with	O
one-	O
,	O
two-	O
,	O
and	O
three-dimensional	O
elements	O
and	O
can	O
be	O
used	O
for	O
articulated	O
systems	O
together	O
with	O
continuous	O
systems	O
such	O
as	O
a	O
pressure	O
vessel	O
,	O
plates	O
,	O
shells	O
,	O
and	O
three-dimensional	O
solids	O
.	O
</s>
<s>
Commercial	O
computer	O
software	O
for	O
structural	B-Algorithm
analysis	I-Algorithm
typically	O
uses	O
matrix	O
finite-element	B-Application
analysis	I-Application
,	O
which	O
can	O
be	O
further	O
classified	O
into	O
two	O
main	O
approaches	O
:	O
the	O
displacement	O
or	O
stiffness	B-Algorithm
method	I-Algorithm
and	O
the	O
force	O
or	O
flexibility	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
stiffness	B-Algorithm
method	I-Algorithm
is	O
the	O
most	O
popular	O
by	O
far	O
thanks	O
to	O
its	O
ease	O
of	O
implementation	O
as	O
well	O
as	O
of	O
formulation	O
for	O
advanced	O
applications	O
.	O
</s>
<s>
1873	O
:	O
Carlo	O
Alberto	O
Castigliano	O
presented	O
his	O
dissertation	O
"	O
Intorno	O
ai	O
sistemi	O
elastici	O
"	O
,	O
which	O
contains	O
his	B-Algorithm
theorem	I-Algorithm
for	O
computing	O
displacement	O
as	O
partial	O
derivative	O
of	O
the	O
strain	O
energy	O
.	O
</s>
