Evidence for beam loading by distributed injection of electrons in a Plasma Wakefield Accelerator.
Distributed Loading
Transcript of Distributed Loading
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