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

1 m 3 m 2 m

kNF

mm

kNhbF

r

r

30

3*10*

Geometry Method

• The location of the resultant force is at the center of mass of the distributed load.

10 kN/m

1 m 3 m 2 m

mx

mmbxx

5.2

3*2

11

2

10

2.5 m

30 kN

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

kNF

mm

kNhbF

r

r

10

4*5*5.*2

1

4 m 4 m

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

5 kN/m

4 m 4 m

mx

mmbxx

34

4*3

10

3

10

1.33 m

10 kN

Integral Method

• The magnitude of the resultant force is given by the integral of the curve defining the force, w(x)

5 m 2 m

kNF

xF

dxxdxxwF

r

m

r

m

x

m

xr

45.7

][3

2

)(

5

0

23

5

0

2/15

0

21

)( xxw

Integral Method

• The location of the resultant force is given by the centroid of the area under the curve

5 m 2 m

mkNxxw /)( 21

mx

xx

dxx

dxxw

xdxxwx

m

m

L

L

0.3

][45.7*5

2

45.7)(

)(

5

0

25

5

0

23

0

0

7.45 kN

3 m

Integral Method – Example

• For a vertical bar, simply integrate with respect to y instead of x

mkNyyw /)(2

5 m

2 m

kNF

yF

dyydyywF

r

m

r

m

y

m

yr

7.41

][3

1

)(

5

0

3

5

0

25

0

Integral Method – Example

• For a vertical bar, simply integrate with respect to y instead of x

3.75 m

my

yy

dyy

dyyw

ydyywy

m

m

L

L

75.3

][7.41*4

1

7.41)(

)(

5

0

4

5

0

3

0

0

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