ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem...
Transcript of ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem...
![Page 1: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/1.jpg)
ME 201Engineering Mechanics: Statics
Unit 5.3
Reduction of a Simple Distributed
Loading
![Page 2: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/2.jpg)
Distributed Loads
Thus far we’ve been working with loads that
are concentrated at a point:
Many times in engineering we need to be
concerned with another type of loading
referred to as distributed loading:
10 N/m
6 kN/m
![Page 3: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/3.jpg)
Distributed Loads
Instead of being concentrated at a point, a
distributed load is spread out over a distance
It can be thought of as a collection of smaller
loads
To find FR, we need to sum an infinite
number of small forces
![Page 4: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/4.jpg)
Distributed Loads Consider a small differential element, dF
with a width of dx
and a height of w(x) w(x)
wdF
x dx x
The area of the element is
Since infinite number of forces, need to integrate to find FR
dxxwdF )(
AdAdxxwFAl
R )(
![Page 5: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/5.jpg)
Centroids for Simple Shapes
Where is the centroid for these common shapes?
b
h
b
h
2
bxc
2
hyc
3
bxc
3
hyc
![Page 6: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/6.jpg)
Example Problem SolutionGiven:
trapezoid
Find:
FR,
Solution:
FBD
FR
x
50 lb/ft
9 ftA B
100 lb/ft
9 ftA B
FrectFtri
ftftlbFtri 9/)50100(2
1
ftftlbFrect 9/50
lb225
lb450
lbFFF recttriR 675
9 ftA B
FR
x
![Page 7: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/7.jpg)
Example Problem Solution
5.44503225675 x
ftxtri 93
1
ftxrect 92
1
ft3
ft5.4
Given:
trapezoid
Find:
FR,
Solution:
FBD
FR
x
x
ft4
50 lb/ft
9 ftA B
100 lb/ft
9 ftA B
FrectFtri
9 ftA B
FR
x
lbFtri 225 lbFrect 450
lbFR 675
![Page 8: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/8.jpg)
Example Problem Solution
Given:
trapezoid
Find:
FR,
Solution:
FR integral
x
LR dxxwF )(
2
0
260 dxx
N160
|2
0
3
360
x
w=60x2 N/m
2 m
![Page 9: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/9.jpg)
Example Problem Solution
Given:
trapezoid
Find:
FR,
Solution:
FR integral
integral
x
w=60x2 N/m
2 m
L
L
dxxw
dxxxwx
)(
)(
x
2
0
2
2
0
2
60
60
dxx
dxxx
|
|2
0
3
2
0
4
360
460
x
x
160
240 m5.1
![Page 10: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/10.jpg)
![Page 11: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/11.jpg)
![Page 12: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/12.jpg)
![Page 13: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/13.jpg)
Distributed Loads
Thus far we’ve been working with loads that
are concentrated at a point:
Many times in engineering we need to be
concerned with another type of loading
referred to as distributed loading:
10 N/m
6 kN/m
![Page 14: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/14.jpg)
Distributed Loads
Instead of being concentrated at a point, a
distributed load is spread out over a distance
It can be thought of as a collection of smaller
loads
![Page 15: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/15.jpg)
Distributed Loads
To compute the resultant, FR, of a distributed load, consider the following:
To find FR, we need to sum an infinite number of small forces
![Page 16: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/16.jpg)
Distributed Loads
Consider a small differential element, dF
with a width of dx
and a height of w(x)
w(x)
wdF
x dx x
![Page 17: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/17.jpg)
Distributed Loads
The area of the element is
Since infinite number of forces, need to integrate to find FR
dxxwdF )(
AdAdxxwFAl
R )(
w(x)
wdF
x dx x
![Page 18: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/18.jpg)
Distributed Loads
Where does FR act?
Can be determined by equating the moments of
the force resultant and the force distribution
lR dxxwF )(
R
l
F
dxxxwx
)(
This is also the centroid of the area
lR dxxxwFx )(
l
l
dxxw
dxxxw
)(
)(
A
A
dA
xdA
![Page 19: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/19.jpg)
Centroids
For simple shapes, centroid can be found in a table
(see back cover of textbook)
Where is the centroid for these common shapes?
b
h
b
h
2
bxc
2
hyc
3
bxc
3
hyc
![Page 20: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/20.jpg)
Centroids of Simple Shapes
Where does FR act?
A. xc = b/2
B. xc > b/2
C. xc < b/2
D. Cannot tell from information given
b
h1
h2
![Page 21: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/21.jpg)
Example Problem
Given:
w=400 lb/ft
Find:
FR,
w lb/ft
10 ftA B
x
![Page 22: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/22.jpg)
Example Problem SolutionGiven:
w=400 lb/ft
Find:
FR, x
w lb/ft
10 ftA B
![Page 23: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/23.jpg)
Example Problem Solution
Given:
w=400 lb/ft
Find:
FR,
Solution:
FBD
FR
x
w lb/ft
10 ftA B
ftftlbFR 10/400
2
10 ftx
10 ftA B
FR
x
korlb 44000
ft5
x
![Page 24: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/24.jpg)
Example Problem
Given:
w=100x N/m
Find:
FR,
600 N/m
6 mA B
x
![Page 25: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/25.jpg)
Example Problem SolutionGiven:
w=100x N/m
Find:
FR,
600 N/m
6 mA Bx
![Page 26: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/26.jpg)
Example Problem Solution
Given:
w=100x N/m
Find:
FR,
Solution:
FBD
FR
600 N/m
6 mA B
mmNFR 6/6002
1
mx 63
2
x
x
6 mA B
FR
x
kNmorNm 8.11800
m4
![Page 27: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/27.jpg)
Example Problem
Given:
trapezoid
Find:
FR,
50 lb/ft
9 ftA B
100 lb/ft
x
![Page 28: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/28.jpg)
Example Problem SolutionGiven:
trapezoid
Find:
FR, x50 lb/ft
9 ftA B
100 lb/ft
![Page 29: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/29.jpg)
Example Problem SolutionGiven:
trapezoid
Find:
FR,
Solution:
FBD
FR
x
50 lb/ft
9 ftA B
100 lb/ft
9 ftA B
FrectFtri
ftftlbFtri 9/)50100(2
1
ftftlbFrect 9/50
lb225
lb450
lbFFF recttriR 675
9 ftA B
FR
x
![Page 30: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/30.jpg)
Example Problem Solution
5.44503225675 x
ftxtri 93
1
ftxrect 92
1
ft3
ft5.4
Given:
trapezoid
Find:
FR,
Solution:
FBD
FR
x
x
ft4
50 lb/ft
9 ftA B
100 lb/ft
9 ftA B
FrectFtri
9 ftA B
FR
x
lbFtri 225 lbFrect 450
lbFR 675
![Page 31: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/31.jpg)
Example Problem
Given:
function
Find:
FR,
w=60x2 N/m
2 mx
![Page 32: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/32.jpg)
Example Problem SolutionGiven:
function
Find:
FR, x
w=60x2 N/m
2 m
![Page 33: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/33.jpg)
Example Problem Solution
Given:
trapezoid
Find:
FR,
Solution:
FR integral
x
LR dxxwF )(
2
0
260 dxx
N160
|2
0
3
360
x
w=60x2 N/m
2 m
![Page 34: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/34.jpg)
Example Problem Solution
Given:
trapezoid
Find:
FR,
Solution:
FR integral
integral
x
w=60x2 N/m
2 m
L
L
dxxw
dxxxwx
)(
)(
x
2
0
2
2
0
2
60
60
dxx
dxxx
|
|2
0
3
2
0
4
360
460
x
x
160
240 m5.1
![Page 35: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/35.jpg)
In Class Exercise
![Page 36: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/36.jpg)
Solution
![Page 37: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/37.jpg)
In Class Exercise
![Page 38: ME 201 Engineering Mechanics: Statics 201/Supplemental Material/In Clas… · Example Problem Solution x 5 5 3 0 4.5 x tri 9 ft 3 1 x t 9 ft 2 1 3 ft 4.5 ft Given: trapezoid Find:](https://reader033.fdocuments.us/reader033/viewer/2022051815/603bdb67700ffd46e847f5ff/html5/thumbnails/38.jpg)
In Class Exercise