Emily Bondank
MAT 266 Brewer
Honors Project May 2011
Calculating Bending Moment for Beams
A structural beam is designed to support a number of forces over its span.
The sum of all of the forces on the beam is called the load. The individual forces that
add up to create the load can be distributed in a variation of ways but no matter the
distribution, the load applies the most stress to only one point in the beam. The
point that the stress is directed to is at the fixed end of the beam. In engineering, the
amount of stress applied to this part of the beam is called the ‘Bending Moment’ and
is used as an indicator to tell how likely the beam is to break. The Bending Moment
can be calculated for any variation of load distribution on a beam using the
knowledge of functions, substitution, differential equations, and integration.
Step One
The equation for the Bending Moment must be determined through finding its
variables and determining the relationship between the Bending Moment and those
variables.

The greater the load on the beam and the farther the distance the load is
from the beam denotes a greater stress applied to the fixed end and a larger
Bending Moment.

From this knowledge, it can be determined that the variables in the equation
of Bending moment are load and distance from the fixed end of the beam. It
can also be determined that the relationship between the Bending Moment
and both of these variables is linear.
Emily Bondank
MAT 266 Brewer
Honors Project May 2011
Therefore the equation for Bending Moment is:
M  l  d , where M denotes Bending Moment, l denotes load, and d denotes
distance from the load to the fixed end of the beam.

Step Two
It is necessary to determine the equation for load by finding the variables and
constants that determine load and the relationship those variables and constants
have with the load.

Load is represented as a constant with a value that depends on the
magnitude of the distributed load, with units of load divided by units of
distance, multiplied by the distance from the load to the fixed end of the
beam.

There is a relationship between the constant and the distance along the
beam defined by the function c(d).
Therefore the equation for load can be written as:
l(d)  c(d)  d
Step Three

The load equation now needs to be substituted into the Bending Moment Equation
and a differential equation needs to be written as the product of the constant
function and an infinitely small change in the independent variable, distance.
Two equations:
M ld
l(d)  c(d)  d
Through substitution one equation is made:

Emily Bondank
MAT 266 Brewer
Honors Project May 2011
M  (c(d)  d)  d
Now the small change in d is made in the substituted load function:

M  (c(d)  d)  d
Step Four

The function of change in Bending Moment is integrated in order to determine the
sum of Bending Moments and the sum of the individual forces that the load consists
of.
The equation is rearranged to prepare for integration:
M  c(d)  d  d

dM
 M and d  dd
dd


dM
 c(d)  d  dd
dd 
The equation is then integrated from length zero to length d, to determine the sum

of the forces:
d

0
dM

dd
d
 c(d)  d  dd
0
d

M
 c(d)  d  dd
0
Once the function for c(d) is imputed into the integral the exact value of the Bending

Moment can be found.
Emily Bondank
MAT 266 Brewer
Honors Project May 2011
Example
Determining the Bending
Moment for a Parabolic
type of load
Step One
M ld
Step Two

l(d)  c(d)  d
Step Three

M  (c(d)  d)  d
Step Four


M  c(d)  d  d
dM
 c(d)  d  dd
dd
d


0
dM

dd
d
 c(d)  d  dd
0
d

M
0


 c(d)  d  dd
A parabolic load is a load whose force varies quadratically along the beam's
length and which is zero at one of the beam's ends. The load of the beam is:
Emily Bondank
MAT 266 Brewer
Honors Project May 2011
c(d)  cd
 M 
d

cd  d  dd
0
d
M
 (cd)
1/ 2
 d  dd
0
d
M  c1/ 2  d 3 / 2  dd
0
M c
1/ 2
2 5 / 2 d
d 

5
0
2
M  c1/ 2  d 5 / 2
5
Reference:

http://www.understandingcalculus.com/chapters/11/11-4.php