<s>
Stress	B-Algorithm
–	I-Algorithm
strain	I-Algorithm
analysis	I-Algorithm
(	O
or	O
stress	B-Algorithm
analysis	I-Algorithm
)	O
is	O
an	O
engineering	O
discipline	O
that	O
uses	O
many	O
methods	O
to	O
determine	O
the	O
stresses	O
and	O
strains	O
in	O
materials	O
and	O
structures	O
subjected	O
to	O
forces	O
.	O
</s>
<s>
Stress	O
is	O
the	O
ratio	O
of	O
force	O
over	O
area	O
(	O
S	O
=	O
R/A	O
,	O
where	O
S	O
is	O
the	O
stress	O
,	O
R	O
is	O
the	O
internal	O
resisting	O
force	O
and	O
A	O
is	O
the	O
cross-sectional	B-Application
area	I-Application
)	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
is	O
a	O
primary	O
task	O
for	O
civil	O
,	O
mechanical	O
and	O
aerospace	O
engineers	O
involved	O
in	O
the	O
design	O
of	O
structures	O
of	O
all	O
sizes	O
,	O
such	O
as	O
tunnels	O
,	O
bridges	O
and	O
dams	O
,	O
aircraft	O
and	O
rocket	B-Application
bodies	O
,	O
mechanical	O
parts	O
,	O
and	O
even	O
plastic	O
cutlery	O
and	O
staples	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
is	O
also	O
used	O
in	O
the	O
maintenance	O
of	O
such	O
structures	O
,	O
and	O
to	O
investigate	O
the	O
causes	O
of	O
structural	O
failures	O
.	O
</s>
<s>
Typically	O
,	O
the	O
starting	O
point	O
for	O
stress	B-Algorithm
analysis	I-Algorithm
are	O
a	O
geometrical	O
description	O
of	O
the	O
structure	O
,	O
the	O
properties	O
of	O
the	O
materials	O
used	O
for	O
its	O
parts	O
,	O
how	O
the	O
parts	O
are	O
joined	O
,	O
and	O
the	O
maximum	O
or	O
typical	O
forces	O
that	O
are	O
expected	O
to	O
be	O
applied	O
to	O
the	O
structure	O
.	O
</s>
<s>
In	O
engineering	O
,	O
stress	B-Algorithm
analysis	I-Algorithm
is	O
often	O
a	O
tool	O
rather	O
than	O
a	O
goal	O
in	O
itself	O
;	O
the	O
ultimate	O
goal	O
being	O
the	O
design	O
of	O
structures	O
and	O
artifacts	O
that	O
can	O
withstand	O
a	O
specified	O
load	O
,	O
using	O
the	O
minimum	O
amount	O
of	O
material	O
or	O
that	O
satisfies	O
some	O
other	O
optimality	O
criterion	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
may	O
be	O
performed	O
through	O
classical	O
mathematical	O
techniques	O
,	O
analytic	O
mathematical	O
modelling	O
or	O
computational	O
simulation	O
,	O
experimental	O
testing	O
,	O
or	O
a	O
combination	O
of	O
methods	O
.	O
</s>
<s>
The	O
term	O
stress	B-Algorithm
analysis	I-Algorithm
is	O
used	O
throughout	O
this	O
article	O
for	O
the	O
sake	O
of	O
brevity	O
,	O
but	O
it	O
should	O
be	O
understood	O
that	O
the	O
strains	O
,	O
and	O
deflections	O
of	O
structures	O
are	O
of	O
equal	O
importance	O
and	O
in	O
fact	O
,	O
an	O
analysis	O
of	O
a	O
structure	O
may	O
begin	O
with	O
the	O
calculation	O
of	O
deflections	O
or	O
strains	O
and	O
end	O
with	O
calculation	O
of	O
the	O
stresses	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
is	O
specifically	O
concerned	O
with	O
solid	O
objects	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
adopts	O
the	O
macroscopic	O
view	O
of	O
materials	O
characteristic	O
of	O
continuum	O
mechanics	O
,	O
namely	O
that	O
all	O
properties	O
of	O
materials	O
are	O
homogeneous	O
at	O
small	O
enough	O
scales	O
.	O
</s>
<s>
Thus	O
,	O
even	O
the	O
smallest	O
particle	O
considered	O
in	O
stress	B-Algorithm
analysis	I-Algorithm
still	O
contains	O
an	O
enormous	O
number	O
of	O
atoms	O
,	O
and	O
its	O
properties	O
are	O
averages	O
of	O
the	O
properties	O
of	O
those	O
atoms	O
.	O
</s>
<s>
In	O
stress	B-Algorithm
analysis	I-Algorithm
one	O
normally	O
disregards	O
the	O
physical	O
causes	O
of	O
forces	O
or	O
the	O
precise	O
nature	O
of	O
the	O
materials	O
.	O
</s>
<s>
The	O
fundamental	O
problem	O
in	O
stress	B-Algorithm
analysis	I-Algorithm
is	O
to	O
determine	O
the	O
distribution	O
of	O
internal	O
stresses	O
throughout	O
the	O
system	O
,	O
given	O
the	O
external	O
forces	O
that	O
are	O
acting	O
on	O
it	O
.	O
</s>
<s>
In	O
civil	O
engineering	O
applications	O
,	O
one	O
typically	O
considers	O
structures	O
to	O
be	O
in	O
static	O
equilibrium	O
:	O
that	O
is	O
,	O
are	O
either	O
unchanging	O
with	O
time	O
,	O
or	O
are	O
changing	O
slowly	O
enough	O
for	O
viscous	B-Application
stresses	I-Application
to	O
be	O
unimportant	O
(	O
quasi-static	O
)	O
.	O
</s>
<s>
In	O
mechanical	O
and	O
aerospace	O
engineering	O
,	O
however	O
,	O
stress	B-Algorithm
analysis	I-Algorithm
must	O
often	O
be	O
performed	O
on	O
parts	O
that	O
are	O
far	O
from	O
equilibrium	O
,	O
such	O
as	O
vibrating	O
plates	O
or	O
rapidly	O
spinning	O
wheels	O
and	O
axles	O
.	O
</s>
<s>
However	O
,	O
these	O
concerns	O
lie	O
outside	O
the	O
scope	O
of	O
stress	B-Algorithm
analysis	I-Algorithm
proper	O
,	O
being	O
covered	O
in	O
materials	O
science	O
under	O
the	O
names	O
strength	O
of	O
materials	O
,	O
fatigue	O
analysis	O
,	O
stress	O
corrosion	O
,	O
creep	O
modeling	O
,	O
and	O
other	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
can	O
be	O
performed	O
experimentally	O
by	O
applying	O
forces	O
to	O
a	O
test	O
element	O
or	O
structure	O
and	O
then	O
determining	O
the	O
resulting	O
stress	O
using	O
sensors	O
.	O
</s>
<s>
Properties	O
that	O
are	O
directly	O
measured	O
via	O
a	O
tensile	O
test	O
are	O
the	O
ultimate	O
tensile	O
strength	O
,	O
maximum	O
elongation	O
and	O
reduction	O
in	O
cross-section	B-Application
area	O
.	O
</s>
<s>
Strain	B-Algorithm
gauges	I-Algorithm
can	O
be	O
used	O
to	O
experimentally	O
determine	O
the	O
deformation	O
of	O
a	O
physical	O
part	O
.	O
</s>
<s>
A	O
commonly	O
used	O
type	O
of	O
strain	B-Algorithm
gauge	I-Algorithm
is	O
a	O
thin	O
flat	O
resistor	O
that	O
is	O
affixed	O
to	O
the	O
surface	O
of	O
a	O
part	O
,	O
and	O
which	O
measures	O
the	O
strain	O
in	O
a	O
given	O
direction	O
.	O
</s>
<s>
While	O
experimental	O
techniques	O
are	O
widely	O
used	O
,	O
most	O
stress	B-Algorithm
analysis	I-Algorithm
is	O
done	O
by	O
mathematical	O
methods	O
,	O
especially	O
during	O
design	O
.	O
</s>
<s>
The	O
basic	O
stress	B-Algorithm
analysis	I-Algorithm
problem	O
can	O
be	O
formulated	O
by	O
Euler	O
's	O
equations	O
of	O
motion	O
for	O
continuous	O
bodies	O
(	O
which	O
are	O
consequences	O
of	O
Newton	O
's	O
laws	O
for	O
conservation	O
of	O
linear	O
momentum	O
and	O
angular	O
momentum	O
)	O
and	O
the	O
Euler-Cauchy	O
stress	O
principle	O
,	O
together	O
with	O
the	O
appropriate	O
constitutive	O
equations	O
.	O
</s>
<s>
Both	O
the	O
stress	B-Algorithm
and	I-Algorithm
strain	I-Algorithm
tensor	O
fields	O
will	O
normally	O
be	O
continuous	O
within	O
each	O
part	O
of	O
the	O
system	O
and	O
that	O
part	O
can	O
be	O
regarded	O
as	O
a	O
continuous	O
medium	O
with	O
smoothly	O
varying	O
constitutive	O
equations	O
.	O
</s>
<s>
The	O
basic	O
stress	B-Algorithm
analysis	I-Algorithm
problem	O
is	O
therefore	O
a	O
boundary-value	O
problem	O
.	O
</s>
<s>
The	O
calculation	O
of	O
the	O
stresses	O
(	O
stress	B-Algorithm
analysis	I-Algorithm
)	O
that	O
develop	O
within	O
such	O
systems	O
is	O
based	O
on	O
the	O
theory	O
of	O
elasticity	O
and	O
infinitesimal	O
strain	O
theory	O
.	O
</s>
<s>
When	O
the	O
applied	O
loads	O
cause	O
permanent	O
deformation	O
,	O
one	O
must	O
use	O
more	O
complicated	O
constitutive	O
equations	O
,	O
that	O
can	O
account	O
for	O
the	O
physical	O
processes	O
involved	O
(	O
plastic	O
flow	O
,	O
fracture	B-Application
,	O
phase	O
change	O
,	O
etc	O
.	O
)	O
</s>
<s>
The	O
mathematical	O
problem	O
represented	O
is	O
typically	O
ill-posed	B-Algorithm
because	O
it	O
has	O
an	O
infinitude	O
of	O
solutions	O
.	O
</s>
<s>
Stress	B-Algorithm
analysis	I-Algorithm
is	O
simplified	O
when	O
the	O
physical	O
dimensions	O
and	O
the	O
distribution	O
of	O
loads	O
allow	O
the	O
structure	O
to	O
be	O
treated	O
as	O
one	O
-	O
or	O
two-dimensional	O
.	O
</s>
<s>
Even	O
under	O
the	O
assumption	O
of	O
linear	O
elastic	O
behavior	O
of	O
the	O
material	O
,	O
the	O
relation	O
between	O
the	O
stress	B-Algorithm
and	I-Algorithm
strain	I-Algorithm
tensors	O
is	O
generally	O
expressed	O
by	O
a	O
fourth-order	O
stiffness	O
tensor	O
with	O
21	O
independent	O
coefficients	O
(	O
a	O
symmetric	O
6	O
×	O
6	O
stiffness	O
matrix	O
)	O
.	O
</s>
<s>
For	O
more	O
complicated	O
problems	O
one	O
must	O
generally	O
resort	O
to	O
numerical	O
approximations	O
such	O
as	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
,	O
and	O
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
factor	O
of	O
safety	O
on	O
ultimate	O
tensile	O
strength	O
is	O
to	O
prevent	O
sudden	O
fracture	B-Application
and	O
collapse	O
,	O
which	O
would	O
result	O
in	O
greater	O
economic	O
loss	O
and	O
possible	O
loss	O
of	O
life	O
.	O
</s>
<s>
Numerical	O
methods	O
include	O
direct	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
which	O
is	O
also	O
referred	O
to	O
as	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
When	O
a	O
structural	O
element	O
is	O
subjected	O
to	O
tension	O
or	O
compression	O
its	O
length	O
will	O
tend	O
to	O
elongate	O
or	O
shorten	O
,	O
and	O
its	O
cross-sectional	B-Application
area	I-Application
changes	O
by	O
an	O
amount	O
that	O
depends	O
on	O
the	O
Poisson	O
's	O
ratio	O
of	O
the	O
material	O
.	O
</s>
<s>
In	O
engineering	O
applications	O
,	O
structural	O
members	O
experience	O
small	O
deformations	O
and	O
the	O
reduction	O
in	O
cross-sectional	B-Application
area	I-Application
is	O
very	O
small	O
and	O
can	O
be	O
neglected	O
,	O
i.e.	O
,	O
the	O
cross-sectional	B-Application
area	I-Application
is	O
assumed	O
constant	O
during	O
deformation	O
.	O
</s>
<s>
For	O
this	O
case	O
,	O
the	O
stress	O
is	O
called	O
engineering	O
stress	O
or	O
nominal	O
stress	O
and	O
is	O
calculated	O
using	O
the	O
original	O
cross	B-Application
section	I-Application
.	O
</s>
<s>
where	O
P	O
is	O
the	O
applied	O
load	O
,	O
and	O
Ao	O
is	O
the	O
original	O
cross-sectional	B-Application
area	I-Application
.	O
</s>
<s>
In	O
some	O
other	O
cases	O
,	O
e.g.	O
,	O
elastomers	O
and	O
plastic	O
materials	O
,	O
the	O
change	O
in	O
cross-sectional	B-Application
area	I-Application
is	O
significant	O
.	O
</s>
<s>
Poisson	O
's	O
ratio	O
=	O
0.5	O
)	O
,	O
if	O
the	O
true	O
stress	O
is	O
desired	O
,	O
it	O
must	O
be	O
calculated	O
using	O
the	O
true	O
cross-sectional	B-Application
area	I-Application
instead	O
of	O
the	O
initial	O
cross-sectional	B-Application
area	I-Application
,	O
as	O
:	O
</s>
