<s>
Shear	O
force	O
and	O
bending	O
moment	B-Application
diagrams	I-Application
are	O
analytical	O
tools	O
used	O
in	O
conjunction	O
with	O
structural	B-Algorithm
analysis	I-Algorithm
to	O
help	O
perform	O
structural	O
design	O
by	O
determining	O
the	O
value	O
of	O
shear	O
forces	O
and	O
bending	O
moments	O
at	O
a	O
given	O
point	O
of	O
a	O
structural	B-Algorithm
element	I-Algorithm
such	O
as	O
a	O
beam	O
.	O
</s>
<s>
Another	O
application	O
of	O
shear	B-Application
and	I-Application
moment	I-Application
diagrams	I-Application
is	O
that	O
the	O
deflection	B-Algorithm
of	O
a	O
beam	O
can	O
be	O
easily	O
determined	O
using	O
either	O
the	O
moment	B-Algorithm
area	I-Algorithm
method	I-Algorithm
or	O
the	O
conjugate	B-Algorithm
beam	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
convention	O
of	O
placing	O
moment	B-Application
diagram	I-Application
on	O
the	O
tension	O
side	O
allows	O
for	O
frames	O
to	O
be	O
dealt	O
with	O
more	O
easily	O
and	O
clearly	O
.	O
</s>
<s>
The	O
second	O
drawing	O
is	O
the	O
loading	O
diagram	O
with	O
the	O
reaction	O
values	O
given	O
without	O
the	O
calculations	O
shown	O
or	O
what	O
most	O
people	O
call	O
a	O
free	B-Application
body	I-Application
diagram	I-Application
.	O
</s>
<s>
The	O
third	O
drawing	O
is	O
the	O
shear	O
force	B-Application
diagram	I-Application
and	O
the	O
fourth	O
drawing	O
is	O
the	O
bending	O
moment	B-Application
diagram	I-Application
.	O
</s>
<s>
For	O
the	O
bending	O
moment	B-Application
diagram	I-Application
the	O
normal	O
sign	O
convention	O
was	O
used	O
.	O
</s>
<s>
Below	O
the	O
moment	B-Application
diagram	I-Application
are	O
the	O
stepwise	O
functions	O
for	O
the	O
shear	O
force	O
and	O
bending	O
moment	O
with	O
the	O
functions	O
expanded	O
to	O
show	O
the	O
effects	O
of	O
each	O
load	O
on	O
the	O
shear	O
and	O
bending	O
functions	O
.	O
</s>
<s>
This	O
is	O
done	O
using	O
a	O
free	B-Application
body	I-Application
diagram	I-Application
of	O
the	O
entire	O
beam	O
.	O
</s>
<s>
Four	O
unknowns	O
cannot	O
be	O
found	O
given	O
two	O
independent	O
equations	O
in	O
these	O
unknown	O
variables	O
and	O
hence	O
the	O
beam	O
is	O
statically	B-Algorithm
indeterminate	I-Algorithm
.	O
</s>
<s>
One	O
way	O
of	O
solving	O
this	O
problem	O
is	O
to	O
use	O
the	O
principle	O
of	O
linear	O
superposition	O
and	O
break	O
the	O
problem	O
up	O
into	O
the	O
superposition	O
of	O
a	O
number	O
of	O
statically	B-Algorithm
determinate	I-Algorithm
problems	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
beam	O
is	O
statically	B-Algorithm
indeterminate	I-Algorithm
and	O
we	O
will	O
have	O
to	O
find	O
the	O
bending	O
moments	O
in	O
segments	O
of	O
the	O
beam	O
as	O
functions	O
of	O
Ra	O
and	O
Mc	O
.	O
</s>
<s>
By	O
plotting	O
each	O
of	O
these	O
equations	O
on	O
their	O
intended	O
intervals	O
,	O
you	O
get	O
the	O
bending	O
moment	O
and	O
shear	O
force	B-Application
diagrams	I-Application
for	O
this	O
beam	O
.	O
</s>
<s>
We	O
now	O
use	O
the	O
Euler	O
–	O
Bernoulli	O
beam	O
theory	O
to	O
compute	O
the	O
deflections	B-Algorithm
of	O
the	O
four	O
segments	O
.	O
</s>
<s>
Also	O
,	O
the	O
slopes	O
of	O
the	O
deflection	B-Algorithm
curves	O
at	O
this	O
point	O
are	O
the	O
same	O
,	O
i.e.	O
,	O
dw4/dx	O
=	O
dw3/dx	O
.	O
</s>
<s>
The	O
moment	B-Application
diagram	I-Application
is	O
a	O
visual	O
representation	O
of	O
the	O
area	O
under	O
the	O
shear	O
force	B-Application
diagram	I-Application
.	O
</s>
<s>
Another	O
note	O
on	O
the	O
shear	O
force	B-Application
diagrams	I-Application
is	O
that	O
they	O
show	O
where	O
external	O
force	O
and	O
moments	O
are	O
applied	O
.	O
</s>
<s>
For	O
example	O
,	O
at	O
x	O
=	O
10	O
on	O
the	O
shear	O
force	B-Application
diagram	I-Application
,	O
there	O
is	O
a	O
gap	O
between	O
the	O
two	O
equations	O
.	O
</s>
<s>
At	O
section	O
3	O
on	O
the	O
moment	B-Application
diagram	I-Application
,	O
there	O
is	O
a	O
discontinuity	O
of	O
50	O
.	O
</s>
<s>
Since	O
this	O
method	O
can	O
easily	O
become	O
unnecessarily	O
complicated	O
with	O
relatively	O
simple	O
problems	O
,	O
it	O
can	O
be	O
quite	O
helpful	O
to	O
understand	O
different	O
relations	O
between	O
the	O
loading	O
,	O
shear	O
,	O
and	O
moment	B-Application
diagram	I-Application
.	O
</s>
<s>
The	O
first	O
of	O
these	O
is	O
the	O
relationship	O
between	O
a	O
distributed	O
load	O
on	O
the	O
loading	O
diagram	O
and	O
the	O
shear	B-Application
diagram	I-Application
.	O
</s>
<s>
Since	O
a	O
distributed	O
load	O
varies	O
the	O
shear	O
load	O
according	O
to	O
its	O
magnitude	O
it	O
can	O
be	O
derived	O
that	O
the	O
slope	O
of	O
the	O
shear	B-Application
diagram	I-Application
is	O
equal	O
to	O
the	O
magnitude	O
of	O
the	O
distributed	O
load	O
.	O
</s>
<s>
Some	O
direct	O
results	O
of	O
this	O
is	O
that	O
a	O
shear	B-Application
diagram	I-Application
will	O
have	O
a	O
point	O
change	O
in	O
magnitude	O
if	O
a	O
point	O
load	O
is	O
applied	O
to	O
a	O
member	O
,	O
and	O
a	O
linearly	O
varying	O
shear	O
magnitude	O
as	O
a	O
result	O
of	O
a	O
constant	O
distributed	O
load	O
.	O
</s>
<s>
Similarly	O
it	O
can	O
be	O
shown	O
that	O
the	O
slope	O
of	O
the	O
moment	B-Application
diagram	I-Application
at	O
a	O
given	O
point	O
is	O
equal	O
to	O
the	O
magnitude	O
of	O
the	O
shear	B-Application
diagram	I-Application
at	O
that	O
distance	O
.	O
</s>
<s>
A	O
direct	O
result	O
of	O
this	O
is	O
that	O
at	O
every	O
point	O
the	O
shear	B-Application
diagram	I-Application
crosses	O
zero	O
the	O
moment	B-Application
diagram	I-Application
will	O
have	O
a	O
local	O
maximum	O
or	O
minimum	O
.	O
</s>
<s>
Also	O
if	O
the	O
shear	B-Application
diagram	I-Application
is	O
zero	O
over	O
a	O
length	O
of	O
the	O
member	O
,	O
the	O
moment	B-Application
diagram	I-Application
will	O
have	O
a	O
constant	O
value	O
over	O
that	O
length	O
.	O
</s>
<s>
By	O
calculus	O
it	O
can	O
be	O
shown	O
that	O
a	O
point	O
load	O
will	O
lead	O
to	O
a	O
linearly	O
varying	O
moment	B-Application
diagram	I-Application
,	O
and	O
a	O
constant	O
distributed	O
load	O
will	O
lead	O
to	O
a	O
quadratic	O
moment	B-Application
diagram	I-Application
.	O
</s>
<s>
The	O
only	O
parts	O
of	O
the	O
stepwise	O
function	O
that	O
would	O
be	O
written	O
out	O
are	O
the	O
moment	O
equations	O
in	O
a	O
nonlinear	O
portion	O
of	O
the	O
moment	B-Application
diagram	I-Application
;	O
this	O
occurs	O
whenever	O
a	O
distributed	O
load	O
is	O
applied	O
to	O
the	O
member	O
.	O
</s>
<s>
For	O
constant	O
portions	O
the	O
value	O
of	O
the	O
shear	O
and/or	O
moment	B-Application
diagram	I-Application
is	O
written	O
right	O
on	O
the	O
diagram	O
,	O
and	O
for	O
linearly	O
varying	O
portions	O
of	O
a	O
member	O
the	O
beginning	O
value	O
,	O
end	O
value	O
,	O
and	O
slope	O
or	O
the	O
portion	O
of	O
the	O
member	O
are	O
all	O
that	O
are	O
required	O
.	O
</s>
