<s>
In	O
engineering	O
,	O
an	O
influence	B-Algorithm
line	I-Algorithm
graphs	O
the	O
variation	O
of	O
a	O
function	O
(	O
such	O
as	O
the	O
shear	O
,	O
moment	O
etc	O
.	O
</s>
<s>
felt	O
in	O
a	O
structural	O
member	O
)	O
at	O
a	O
specific	O
point	O
on	O
a	O
beam	O
or	O
truss	B-Device
caused	O
by	O
a	O
unit	O
load	O
placed	O
at	O
any	O
point	O
along	O
the	O
structure	O
.	O
</s>
<s>
Common	O
functions	O
studied	O
with	O
influence	B-Algorithm
lines	I-Algorithm
include	O
reactions	O
(	O
forces	O
that	O
the	O
structure	O
’s	O
supports	O
must	O
apply	O
for	O
the	O
structure	O
to	O
remain	O
static	O
)	O
,	O
shear	O
,	O
moment	O
,	O
and	O
deflection	B-Algorithm
(	O
Deformation	O
)	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
are	O
important	O
in	O
designing	O
beams	O
and	O
trusses	O
used	O
in	O
bridges	O
,	O
crane	O
rails	O
,	O
conveyor	O
belts	O
,	O
floor	O
girders	O
,	O
and	O
other	O
structures	O
where	O
loads	O
will	O
move	O
along	O
their	O
span	O
.	O
</s>
<s>
The	O
influence	B-Algorithm
lines	I-Algorithm
show	O
where	O
a	O
load	O
will	O
create	O
the	O
maximum	O
effect	O
for	O
any	O
of	O
the	O
functions	O
studied	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
are	O
both	O
scalar	O
and	O
additive	O
.	O
</s>
<s>
To	O
find	O
the	O
effect	O
of	O
any	O
non-unit	O
load	O
on	O
a	O
structure	O
,	O
the	O
ordinate	O
results	O
obtained	O
by	O
the	O
influence	B-Algorithm
line	I-Algorithm
are	O
multiplied	O
by	O
the	O
magnitude	O
of	O
the	O
actual	O
load	O
to	O
be	O
applied	O
.	O
</s>
<s>
The	O
entire	O
influence	B-Algorithm
line	I-Algorithm
can	O
be	O
scaled	O
,	O
or	O
just	O
the	O
maximum	O
and	O
minimum	O
effects	O
experienced	O
along	O
the	O
line	O
.	O
</s>
<s>
The	O
scaled	O
maximum	O
and	O
minimum	O
are	O
the	O
critical	O
magnitudes	O
that	O
must	O
be	O
designed	O
for	O
in	O
the	O
beam	O
or	O
truss	B-Device
.	O
</s>
<s>
In	O
cases	O
where	O
multiple	O
loads	O
may	O
be	O
in	O
effect	O
,	O
influence	B-Algorithm
lines	I-Algorithm
for	O
the	O
individual	O
loads	O
may	O
be	O
added	O
together	O
to	O
obtain	O
the	O
total	O
effect	O
felt	O
the	O
structure	O
bears	O
at	O
a	O
given	O
point	O
.	O
</s>
<s>
When	O
adding	O
the	O
influence	B-Algorithm
lines	I-Algorithm
together	O
,	O
it	O
is	O
necessary	O
to	O
include	O
the	O
appropriate	O
offsets	O
due	O
to	O
the	O
spacing	O
of	O
loads	O
across	O
the	O
structure	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
can	O
be	O
used	O
with	O
either	O
concentrated	O
or	O
distributed	O
loadings	O
.	O
</s>
<s>
The	O
effect	O
of	O
the	O
distributed	O
unit	O
load	O
can	O
also	O
be	O
obtained	O
by	O
integrating	O
the	O
point	O
load	O
’s	O
influence	B-Algorithm
line	I-Algorithm
over	O
the	O
corresponding	O
length	O
of	O
the	O
structures	O
.	O
</s>
<s>
The	O
Influence	B-Algorithm
lines	I-Algorithm
of	O
determinate	O
structures	O
becomes	O
a	O
mechanism	O
whereas	O
the	O
Influence	B-Algorithm
lines	I-Algorithm
of	O
indeterminate	O
structures	O
become	O
just	O
determinate	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
are	O
based	O
on	O
Betti	O
's	O
theorem	O
.	O
</s>
<s>
When	O
designing	O
a	O
beam	O
or	O
truss	B-Device
,	O
it	O
is	O
necessary	O
to	O
design	O
for	O
the	O
scenarios	O
causing	O
the	O
maximum	O
expected	O
reactions	O
,	O
shears	O
,	O
and	O
moments	O
within	O
the	O
structure	O
members	O
to	O
ensure	O
that	O
no	O
member	O
fails	O
during	O
the	O
life	O
of	O
the	O
structure	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
graph	O
the	O
response	O
of	O
a	O
beam	O
or	O
truss	B-Device
as	O
a	O
unit	O
load	O
travels	O
across	O
it	O
.	O
</s>
<s>
The	O
influence	B-Algorithm
line	I-Algorithm
helps	O
designers	O
find	O
where	O
to	O
place	O
a	O
live	O
load	O
in	O
order	O
to	O
calculate	O
the	O
maximum	O
resulting	O
response	O
for	O
each	O
of	O
the	O
following	O
functions	O
:	O
reaction	O
,	O
shear	O
,	O
or	O
moment	O
.	O
</s>
<s>
The	O
designer	O
can	O
then	O
scale	O
the	O
influence	B-Algorithm
line	I-Algorithm
by	O
the	O
greatest	O
expected	O
load	O
to	O
calculate	O
the	O
maximum	O
response	O
of	O
each	O
function	O
for	O
which	O
the	O
beam	O
or	O
truss	B-Device
must	O
be	O
designed	O
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
can	O
also	O
be	O
used	O
to	O
find	O
the	O
responses	O
of	O
other	O
functions	O
(	O
such	O
as	O
deflection	B-Algorithm
or	O
axial	O
force	O
)	O
to	O
the	O
applied	O
unit	O
load	O
,	O
but	O
these	O
uses	O
of	O
influence	B-Algorithm
lines	I-Algorithm
are	O
less	O
common	O
.	O
</s>
<s>
There	O
are	O
three	O
methods	O
used	O
for	O
constructing	O
the	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
The	O
first	O
is	O
to	O
tabulate	O
the	O
influence	O
values	O
for	O
multiple	O
points	O
along	O
the	O
structure	O
,	O
then	O
use	O
those	O
points	O
to	O
create	O
the	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
The	O
second	O
is	O
to	O
determine	O
the	O
influence-line	O
equations	O
that	O
apply	O
to	O
the	O
structure	O
,	O
thereby	O
solving	O
for	O
all	O
points	O
along	O
the	O
influence	B-Algorithm
line	I-Algorithm
in	O
terms	O
of	O
x	O
,	O
where	O
x	O
is	O
the	O
number	O
of	O
feet	O
from	O
the	O
start	O
of	O
the	O
structure	O
to	O
the	O
point	O
where	O
the	O
unit	O
load	O
is	O
applied	O
.	O
</s>
<s>
The	O
third	O
method	O
is	O
called	O
the	O
Müller-Breslau	B-Algorithm
'	I-Algorithm
s	I-Algorithm
principle	I-Algorithm
.	O
</s>
<s>
It	O
creates	O
a	O
qualitative	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
This	O
influence	B-Algorithm
line	I-Algorithm
will	O
still	O
provide	O
the	O
designer	O
with	O
an	O
accurate	O
idea	O
of	O
where	O
the	O
unit	O
load	O
will	O
produce	O
the	O
largest	O
response	O
of	O
a	O
function	O
at	O
the	O
point	O
being	O
studied	O
,	O
but	O
it	O
cannot	O
be	O
used	O
directly	O
to	O
calculate	O
what	O
the	O
magnitude	O
that	O
response	O
will	O
be	O
,	O
whereas	O
the	O
influence	B-Algorithm
lines	I-Algorithm
produced	O
by	O
the	O
first	O
two	O
methods	O
can	O
.	O
</s>
<s>
Shear	O
and	O
moments	O
are	O
given	O
positive	O
or	O
negative	O
values	O
according	O
to	O
the	O
same	O
conventions	O
used	O
for	O
shear	B-Application
and	I-Application
moment	I-Application
diagrams	I-Application
.	O
</s>
<s>
R	O
.	O
C	O
.	O
Hibbeler	O
states	O
,	O
in	O
his	O
book	O
Structural	O
Analysis	O
,	O
“	O
All	O
statically	O
determinate	O
beams	O
will	O
have	O
influence	B-Algorithm
lines	I-Algorithm
that	O
consist	O
of	O
straight	O
line	O
segments.	O
”	O
Therefore	O
,	O
it	O
is	O
possible	O
to	O
minimize	O
the	O
number	O
of	O
computations	O
by	O
recognizing	O
the	O
points	O
that	O
will	O
cause	O
a	O
change	O
in	O
the	O
slope	O
of	O
the	O
influence	B-Algorithm
line	I-Algorithm
and	O
only	O
calculating	O
the	O
values	O
at	O
those	O
points	O
.	O
</s>
<s>
An	O
influence	B-Algorithm
line	I-Algorithm
for	O
a	O
given	O
function	O
,	O
such	O
as	O
a	O
reaction	O
,	O
axial	O
force	O
,	O
shear	O
force	O
,	O
or	O
bending	O
moment	O
,	O
is	O
a	O
graph	O
that	O
shows	O
the	O
variation	O
of	O
that	O
function	O
at	O
any	O
given	O
point	O
on	O
a	O
structure	O
due	O
to	O
the	O
application	O
of	O
a	O
unit	O
load	O
at	O
any	O
point	O
on	O
the	O
structure	O
.	O
</s>
<s>
An	O
influence	B-Algorithm
line	I-Algorithm
for	O
a	O
function	O
differs	O
from	O
a	O
shear	O
,	O
axial	O
,	O
or	O
bending	O
moment	B-Application
diagram	I-Application
.	O
</s>
<s>
Influence	B-Algorithm
lines	I-Algorithm
can	O
be	O
generated	O
by	O
independently	O
applying	O
a	O
unit	O
load	O
at	O
several	O
points	O
on	O
a	O
structure	O
and	O
determining	O
the	O
value	O
of	O
the	O
function	O
due	O
to	O
this	O
load	O
,	O
i.e.	O
</s>
<s>
The	O
calculated	O
values	O
for	O
each	O
function	O
are	O
then	O
plotted	O
where	O
the	O
load	O
was	O
applied	O
and	O
then	O
connected	O
together	O
to	O
generate	O
the	O
influence	B-Algorithm
line	I-Algorithm
for	O
the	O
function	O
.	O
</s>
<s>
Once	O
the	O
influence	O
values	O
have	O
been	O
tabulated	O
,	O
the	O
influence	B-Algorithm
line	I-Algorithm
for	O
the	O
function	O
at	O
point	O
A	O
can	O
be	O
drawn	O
in	O
terms	O
of	O
x	O
.	O
</s>
<s>
Once	O
this	O
is	O
done	O
,	O
the	O
influence	B-Algorithm
line	I-Algorithm
is	O
complete	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
create	O
equations	O
defining	O
the	O
influence	B-Algorithm
line	I-Algorithm
across	O
the	O
entire	O
span	O
of	O
a	O
structure	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
understanding	O
where	O
the	O
slope	O
of	O
the	O
influence	B-Algorithm
line	I-Algorithm
changes	O
for	O
this	O
method	O
because	O
the	O
influence-line	O
equation	O
will	O
change	O
for	O
each	O
linear	O
section	O
of	O
the	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
the	O
complete	O
equation	O
is	O
a	O
piecewise	O
linear	O
function	O
with	O
a	O
separate	O
influence-line	O
equation	O
for	O
each	O
linear	O
section	O
of	O
the	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
According	O
to	O
www.public.iastate.edu	O
,	O
“	O
The	B-Algorithm
Müller-Breslau	I-Algorithm
Principle	I-Algorithm
can	O
be	O
utilized	O
to	O
draw	O
qualitative	O
influence	B-Algorithm
lines	I-Algorithm
,	O
which	O
are	O
directly	O
proportional	O
to	O
the	O
actual	O
influence	O
line.	O
”	O
Instead	O
of	O
moving	O
a	O
unit	O
load	O
along	O
a	O
beam	O
,	O
the	B-Algorithm
Müller-Breslau	I-Algorithm
Principle	I-Algorithm
finds	O
the	O
deflected	O
shape	O
of	O
the	O
beam	O
caused	O
by	O
first	O
releasing	O
the	O
beam	O
at	O
the	O
point	O
being	O
studied	O
,	O
and	O
then	O
applying	O
the	O
function	O
(	O
reaction	O
,	O
shear	O
,	O
or	O
moment	O
)	O
being	O
studied	O
to	O
that	O
point	O
.	O
</s>
<s>
The	O
principle	O
states	O
that	O
the	O
influence	B-Algorithm
line	I-Algorithm
of	O
a	O
function	O
will	O
have	O
a	O
scaled	O
shape	O
that	O
is	O
the	O
same	O
as	O
the	O
deflected	O
shape	O
of	O
the	O
beam	O
when	O
the	O
beam	O
is	O
acted	O
upon	O
by	O
the	O
function	O
.	O
</s>
<s>
Below	O
are	O
explanations	O
of	O
how	O
to	O
find	O
the	O
influence	B-Algorithm
lines	I-Algorithm
of	O
a	O
simply	O
supported	O
,	O
rigid	O
beam	O
(	O
such	O
as	O
the	O
one	O
displayed	O
in	O
Figure	O
1	O
)	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
reaction	O
’s	O
influence	B-Algorithm
line	I-Algorithm
will	O
be	O
a	O
straight	O
,	O
sloping	O
line	O
with	O
a	O
value	O
of	O
zero	O
at	O
the	O
location	O
of	O
the	O
second	O
support	O
.	O
</s>
<s>
This	O
creates	O
a	O
discontinuous	O
influence	B-Algorithm
line	I-Algorithm
that	O
reaches	O
zero	O
at	O
the	O
supports	O
and	O
whose	O
slope	O
is	O
equal	O
on	O
either	O
side	O
of	O
the	O
discontinuity	O
.	O
</s>
<s>
If	O
point	O
B	O
is	O
at	O
a	O
support	O
,	O
then	O
the	O
deflection	B-Algorithm
between	O
point	O
B	O
and	O
any	O
other	O
supports	O
will	O
still	O
create	O
a	O
triangle	O
,	O
but	O
if	O
the	O
beam	O
is	O
cantilevered	O
,	O
then	O
the	O
entire	O
cantilevered	O
side	O
will	O
move	O
up	O
or	O
down	O
creating	O
a	O
rectangle	O
.	O
</s>
<s>
This	O
will	O
create	O
a	O
continuous	O
influence	B-Algorithm
line	I-Algorithm
,	O
but	O
the	O
slopes	O
will	O
be	O
equal	O
and	O
opposite	O
on	O
either	O
side	O
of	O
the	O
hinge	O
at	O
point	O
B	O
.	O
</s>
<s>
The	B-Algorithm
Müller-Breslau	I-Algorithm
Principle	I-Algorithm
can	O
only	O
produce	O
qualitative	O
influence	B-Algorithm
lines	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
engineers	O
can	O
use	O
it	O
to	O
determine	O
where	O
to	O
place	O
a	O
load	O
to	O
incur	O
the	O
maximum	O
of	O
a	O
function	O
,	O
but	O
the	O
magnitude	O
of	O
that	O
maximum	O
cannot	O
be	O
calculated	O
from	O
the	O
influence	B-Algorithm
line	I-Algorithm
.	O
</s>
<s>
The	O
simplest	O
loading	O
case	O
is	O
a	O
single	O
point	O
load	O
,	O
but	O
influence	B-Algorithm
lines	I-Algorithm
can	O
also	O
be	O
used	O
to	O
determine	O
responses	O
due	O
to	O
multiple	O
loads	O
and	O
distributed	O
loads	O
.	O
</s>
<s>
To	O
calculate	O
the	O
response	O
of	O
a	O
function	O
to	O
all	O
these	O
point	O
loads	O
using	O
an	O
influence	B-Algorithm
line	I-Algorithm
,	O
the	O
results	O
found	O
with	O
the	O
influence	B-Algorithm
line	I-Algorithm
can	O
be	O
scaled	O
for	O
each	O
load	O
,	O
and	O
then	O
the	O
scaled	O
magnitudes	O
can	O
be	O
summed	O
to	O
find	O
the	O
total	O
response	O
that	O
the	O
structure	O
must	O
withstand	O
.	O
</s>
<s>
To	O
find	O
the	O
effect	O
of	O
a	O
distributed	O
load	O
,	O
the	O
designer	O
can	O
integrate	O
an	O
influence	B-Algorithm
line	I-Algorithm
,	O
found	O
using	O
a	O
point	O
load	O
,	O
over	O
the	O
affected	O
distance	O
of	O
the	O
structure	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
three-foot-long	O
track	O
acts	O
between	O
5	O
feet	O
and	O
8	O
feet	O
along	O
a	O
beam	O
,	O
the	O
influence	B-Algorithm
line	I-Algorithm
of	O
that	O
beam	O
must	O
be	O
integrated	O
between	O
5	O
and	O
8	O
feet	O
.	O
</s>
<s>
The	O
integration	O
of	O
the	O
influence	B-Algorithm
line	I-Algorithm
gives	O
the	O
effect	O
that	O
would	O
be	O
felt	O
if	O
the	O
distributed	O
load	O
had	O
a	O
unit	O
magnitude	O
.	O
</s>
<s>
While	O
the	O
influence	B-Algorithm
lines	I-Algorithm
of	O
statically	O
determinate	O
structures	O
(	O
as	O
mentioned	O
above	O
)	O
are	O
made	O
up	O
of	O
straight	O
line	O
segments	O
,	O
the	O
same	O
is	O
not	O
true	O
for	O
indeterminate	O
structures	O
.	O
</s>
<s>
Indeterminate	O
structures	O
are	O
not	O
considered	O
rigid	O
;	O
therefore	O
,	O
the	O
influence	B-Algorithm
lines	I-Algorithm
drawn	O
for	O
them	O
will	O
not	O
be	O
straight	O
lines	O
but	O
rather	O
curves	O
.	O
</s>
<s>
The	O
methods	O
above	O
can	O
still	O
be	O
used	O
to	O
determine	O
the	O
influence	B-Algorithm
lines	I-Algorithm
for	O
the	O
structure	O
,	O
but	O
the	O
work	O
becomes	O
much	O
more	O
complex	O
as	O
the	O
properties	O
of	O
the	O
beam	O
itself	O
must	O
be	O
taken	O
into	O
consideration	O
.	O
</s>
