Distributed Loading
What is a Distributed Load?
• A load applied across
a length or area
instead of at one
point
Analyzing Distributed Loads
• A distributed load
can be equated with
a concentrated load
applied at a specific
point along the bar
Geometry Method
• The magnitude of the
resultant force is
equivalent to the
area under the curve
of the distributed
load
10 kN/m
1m
3m
F r  b * h  10 kN
F r  30 kN
2m
m
* 3m
Geometry Method
10 kN/m
• The location of the
resultant force is at
the center of mass of
the distributed load.
1m
3m
2m
30 kN
x  x0 
1
x  2 .5 m
2
b  1m 
1
* 3m
2
2.5 m
Geometry Method – Example
• For a triangular
distributed load, the
magnitude of the
resultant force is the
area of the triangle,
½*b*h
5 kN/m
4m
Fr 
1
4m
b * h  . 5 * 5 kN
2
F r  10 kN
m
* 4m
Geometry Method – Example
• For a triangular
distributed load, the
location of the
resultant force is 1/3
of the length of the
load, from the larger
end 1
1
x  x0 
x  4
3
b  0m 
3
m
5 kN/m
4m
10 kN
* 4m
3
1.33 m
4m
Integral Method
w( x)  x
• The magnitude of the
resultant force is
given by the integral
of the curve defining
the force, w(x)
5m
5m
Fr 

Fr 
2
x0
3
2m
w ( x ) dx  
[x
5m
x0
3
2
5m
]0
F r  7 . 45 kN
1
x
1/ 2
dx
2
Integral Method
w( x)  x
• The location of the
resultant force is
given by the centroid
of the area under the
curve
3
L
5m
2
x 

w ( x ) xdx

0

L

x
dx
0
7 . 45
w ( x ) dx
5m
7.45 kN
0
x 
2
5 * 7 . 45
x  3 .0 m
5
[x
2
5m
]0
3m
2m
1
2
kN / m
Integral Method – Example
• For a vertical bar,
simply integrate with
respect to y instead
of x
5m
Fr 

Fr 
1
y0
3
w ( y ) dy  
5m
y0
5m
2
y dy
w ( y )  y kN / m
2
3 5m
[ y ]0
F r  41 . 7 kN
2m
Integral Method – Example
• For a vertical bar,
simply integrate with
respect to y instead
of x
y 

L
w ( y ) ydy
0

L

1
4 * 41 . 7
y  3 . 75 m
3
y dy
4 5m
[ y ]0
3.75 m
0
41 . 7
w ( y ) dy
0
y 

5m
41.7 kN
References
• Vector Mechanics for Engineers: Statics (Beer, 9th
ed)
• Distributed Load on Beams
• www.engr.uky.edu/statics/Content/Chapter.../Chapter%205%20D.
doc
• Resolving Distributed Loads
• http://pages.uoregon.edu/struct/courseware/461/461_lectures/
461_lecture20/461_lecture20.html
• Distributed Load on a Beam (Centroid)
• http://www.youtube.com/watch?v=K4ocb9KwPjo