<s>
In	O
structural	O
engineering	O
,	O
deflection	B-Algorithm
is	O
the	O
degree	O
to	O
which	O
a	O
part	O
of	O
a	O
structural	B-Algorithm
element	I-Algorithm
is	O
displaced	O
under	O
a	O
load	O
(	O
because	O
it	O
deforms	O
)	O
.	O
</s>
<s>
The	O
deflection	B-Algorithm
distance	O
of	O
a	O
member	O
under	O
a	O
load	O
can	O
be	O
calculated	O
by	O
integrating	O
the	O
function	O
that	O
mathematically	O
describes	O
the	O
slope	O
of	O
the	O
deflected	O
shape	O
of	O
the	O
member	O
under	O
that	O
load	O
.	O
</s>
<s>
Standard	O
formulas	O
exist	O
for	O
the	O
deflection	B-Algorithm
of	O
common	O
beam	O
configurations	O
and	O
load	O
cases	O
at	O
discrete	O
locations	O
.	O
</s>
<s>
Otherwise	O
methods	O
such	O
as	O
virtual	O
work	O
,	O
direct	B-Algorithm
integration	I-Algorithm
,	O
Castigliano	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
Macaulay	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
the	O
direct	B-Algorithm
stiffness	I-Algorithm
method	I-Algorithm
are	O
used	O
.	O
</s>
<s>
The	O
deflection	B-Algorithm
of	O
beam	O
elements	O
is	O
usually	O
calculated	O
on	O
the	O
basis	O
of	O
the	O
Euler	O
–	O
Bernoulli	O
beam	O
equation	O
while	O
that	O
of	O
a	O
plate	O
or	O
shell	O
element	O
is	O
calculated	O
using	O
plate	O
or	O
shell	O
theory	O
.	O
</s>
<s>
An	O
example	O
of	O
the	O
use	O
of	O
deflection	B-Algorithm
in	O
this	O
context	O
is	O
in	O
building	O
construction	O
.	O
</s>
<s>
It	O
may	O
be	O
made	O
entirely	O
of	O
the	O
same	O
material	O
(	O
homogeneous	B-General_Concept
)	O
,	O
or	O
it	O
may	O
be	O
composed	O
of	O
different	O
materials	O
(	O
composite	O
)	O
.	O
</s>
<s>
Only	O
small	O
deflections	B-Algorithm
are	O
considered	O
(	O
max	O
deflection	B-Algorithm
less	O
than	O
1/10	O
of	O
the	O
span	O
)	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
equation	O
governing	O
the	O
beam	O
's	O
deflection	B-Algorithm
(	O
)	O
can	O
be	O
approximated	O
as	O
:	O
</s>
<s>
If	O
,	O
in	O
addition	O
,	O
the	O
beam	O
is	O
not	O
tapered	O
and	O
is	O
homogeneous	B-General_Concept
,	O
and	O
is	O
acted	O
upon	O
by	O
a	O
distributed	O
load	O
,	O
the	O
above	O
expression	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
The	O
formulas	O
expressed	O
are	O
approximations	O
developed	O
for	O
long	O
,	O
slender	O
,	O
homogeneous	B-General_Concept
,	O
prismatic	O
beams	O
with	O
small	O
deflections	B-Algorithm
,	O
and	O
linear	O
elastic	O
properties	O
.	O
</s>
<s>
Under	O
these	O
restrictions	O
,	O
the	O
approximations	O
should	O
give	O
results	O
within	O
5%	O
of	O
the	O
actual	O
deflection	B-Algorithm
.	O
</s>
<s>
Cantilever	O
beams	O
have	O
one	O
end	O
fixed	O
,	O
so	O
that	O
the	O
slope	O
and	O
deflection	B-Algorithm
at	O
that	O
end	O
must	O
be	O
zero	O
.	O
</s>
<s>
The	O
elastic	O
deflection	B-Algorithm
and	O
angle	O
of	O
deflection	B-Algorithm
(	O
in	O
radians	O
)	O
at	O
the	O
free	O
end	O
in	O
the	O
example	O
image	O
:	O
A	O
(	O
weightless	O
)	O
cantilever	O
beam	O
,	O
with	O
an	O
end	O
load	O
,	O
can	O
be	O
calculated	O
(	O
at	O
the	O
free	O
end	O
B	O
)	O
using	O
:	O
</s>
<s>
Note	O
that	O
if	O
the	O
span	O
doubles	O
,	O
the	O
deflection	B-Algorithm
increases	O
eightfold	O
.	O
</s>
<s>
The	O
deflection	B-Algorithm
at	O
any	O
point	O
,	O
,	O
along	O
the	O
span	O
of	O
an	O
end	O
loaded	O
cantilevered	O
beam	O
can	O
be	O
calculated	O
using	O
:	O
</s>
<s>
The	O
deflection	B-Algorithm
,	O
at	O
the	O
free	O
end	O
B	O
,	O
of	O
a	O
cantilevered	O
beam	O
under	O
a	O
uniform	O
load	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
deflection	B-Algorithm
at	O
any	O
point	O
,	O
,	O
along	O
the	O
span	O
of	O
a	O
uniformly	O
loaded	O
cantilevered	O
beam	O
can	O
be	O
calculated	O
using	O
:	O
</s>
<s>
Simply	O
supported	O
beams	O
have	O
supports	O
under	O
their	O
ends	O
which	O
allow	O
rotation	O
,	O
but	O
not	O
deflection	B-Algorithm
.	O
</s>
<s>
The	O
deflection	B-Algorithm
at	O
any	O
point	O
,	O
,	O
along	O
the	O
span	O
of	O
a	O
center	O
loaded	O
simply	O
supported	O
beam	O
can	O
be	O
calculated	O
using	O
:	O
</s>
<s>
The	O
special	O
case	O
of	O
elastic	O
deflection	B-Algorithm
at	O
the	O
midpoint	O
C	O
of	O
a	O
beam	O
,	O
loaded	O
at	O
its	O
center	O
,	O
supported	O
by	O
two	O
simple	O
supports	O
is	O
then	O
given	O
by	O
:	O
</s>
<s>
The	O
maximum	O
elastic	O
deflection	B-Algorithm
on	O
a	O
beam	O
supported	O
by	O
two	O
simple	O
supports	O
,	O
loaded	O
at	O
a	O
distance	O
from	O
the	O
closest	O
support	O
,	O
is	O
given	O
by	O
:	O
</s>
<s>
This	O
maximum	O
deflection	B-Algorithm
occurs	O
at	O
a	O
distance	O
from	O
the	O
closest	O
support	O
and	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
elastic	O
deflection	B-Algorithm
(	O
at	O
the	O
midpoint	O
C	O
)	O
on	O
a	O
beam	O
supported	O
by	O
two	O
simple	O
supports	O
,	O
under	O
a	O
uniform	O
load	O
(	O
as	O
pictured	O
)	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
deflection	B-Algorithm
at	O
any	O
point	O
,	O
,	O
along	O
the	O
span	O
of	O
a	O
uniformly	O
loaded	O
simply	O
supported	O
beam	O
can	O
be	O
calculated	O
using	O
:	O
</s>
<s>
The	O
change	O
in	O
length	O
of	O
the	O
beam	O
is	O
generally	O
negligible	O
in	O
structures	O
,	O
but	O
can	O
be	O
calculated	O
by	O
integrating	O
the	O
slope	O
function	O
,	O
if	O
the	O
deflection	B-Algorithm
function	O
is	O
known	O
for	O
all	O
.	O
</s>
<s>
If	O
the	O
beam	O
is	O
uniform	O
and	O
the	O
deflection	B-Algorithm
at	O
any	O
point	O
is	O
known	O
,	O
this	O
can	O
be	O
calculated	O
without	O
knowing	O
other	O
properties	O
of	O
the	O
beam	O
.	O
</s>
<s>
Building	O
codes	O
determine	O
the	O
maximum	O
deflection	B-Algorithm
,	O
usually	O
as	O
a	O
fraction	O
of	O
the	O
span	O
e.g.	O
</s>
<s>
Either	O
the	O
strength	O
limit	O
state	O
(	O
allowable	O
stress	O
)	O
or	O
the	O
serviceability	O
limit	O
state	O
(	O
deflection	B-Algorithm
considerations	O
among	O
others	O
)	O
may	O
govern	O
the	O
minimum	O
dimensions	O
of	O
the	O
member	O
required	O
.	O
</s>
<s>
The	O
deflection	B-Algorithm
must	O
be	O
considered	O
for	O
the	O
purpose	O
of	O
the	O
structure	O
.	O
</s>
<s>
When	O
designing	O
a	O
steel	O
frame	O
to	O
hold	O
a	O
glazed	O
panel	O
,	O
one	O
allows	O
only	O
minimal	O
deflection	B-Algorithm
to	O
prevent	O
fracture	B-Application
of	O
the	O
glass	O
.	O
</s>
<s>
The	O
deflected	O
shape	O
of	O
a	O
beam	O
can	O
be	O
represented	O
by	O
the	O
moment	B-Application
diagram	I-Application
,	O
integrated	O
(	O
twice	O
,	O
rotated	O
and	O
translated	O
to	O
enforce	O
support	O
conditions	O
)	O
.	O
</s>
