engi 1313 mechanics i
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ENGI 1313 Mechanics I. Lecture 22:Equivalent Force Systems and Distributed Loading. Lecture 22 Objective. to demonstrate by example equivalent force systems to determine an equivalent force for a distributed load. Example 22-01. - PowerPoint PPT PresentationTRANSCRIPT
Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of [email protected]
ENGI 1313 Mechanics I
Lecture 22: Equivalent Force Systems and Distributed Loading
2 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Lecture 22 Objective
to demonstrate by example equivalent force systems
to determine an equivalent force for a distributed load
3 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Example 22-01
Handle forces F1 and F2 are applied to the drill. Replace this system by an equivalent resultant force and couple moment acting at point O. Express the results in Cartesian vector form.
a 0.15 m
b 0.25 m
c 0.3 m
F1
6
3
10
N
F2
0
2
4
N
4 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Equivalent Distributed Loads
Reduce to Single Equivalent Load Magnitude and position
Applications Environment
• Wind• Fluid
Dead load• Weight• Snow, sand• Objects
5 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Distributed Loads
Load Intensity Pressure (Pa)
• Force per unit area
• N/m2 or lb/ft2
Infinite # parallel vertical loads
Reduce from Area to Line Load
m
Nxwm
m
Naxpw
2
6 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Distributed Loads (cont.)
Consider Element Length (dx) Element force (dF) acts
on element length (dx) The line load w(x)
represents force per unit length
dxxwdF
7 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Distributed Loads (cont.)
Magnitude Net Force (FR) Summation of all
element forces
Area under line load curve
dxxwdFFR
AdAdxxwFAL
R
8 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Distributed Loads (cont.)
Resultant Moment (MRo) Element force moment
Total moment
Equivalent force
dFxMdF
O
LL
ORo dxxwxdFxMMdF
L
RRo dxxwxxFM
9 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Distributed Loads (cont.)
Centroid Geometric center of area Resultant force line of
action
L
RRo dxxwxxFM
AdAdxxwFAL
R
A
A
L
L
dA
dAx
dxxw
dxxwx
x
10 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Example 22-02
Determine equivalent resultant force and location for the following distributed load.
2
Lxft5
lb4000
lbft2x
400
lb4000
dxxwx
lb4000
dAx
dA
dAx
x
10
0
2
10
0
10
0
A
A
wLFlb4000ft
lbx400dx400dAdxxwF R
10
0
10
0AL
R
L/2
11 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Example 22-03
Determine equivalent resultant force and location for the following distributed load.
3
L,
3
L2xm4
N1800
mN3x
100
N1800
dxxwx
N1800
dAx
dA
dAx
x
6
0
3
6
0
6
0
A
A
2
wLFN1800
m
Nx50dxx100dAdxxwF R
6
0
26
0AL
R
L/3
12 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Comprehension Quiz 22-01
What is the location of FR or distance d?
A) 2 m B) 3 m C) 4 m D) 5 m E) 6 m
Answer: D
FR
3 m 3 m
d
13 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
Example 22-04
The beam is subjected to the distributed loading. Determine the length b of the uniform load and its position a on the beam such that the resultant force and couple moment acting on the beam are zero.
w1 40lb
ft
w2 60lb
ft
c 10ft
d 6 ft
14 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.
References
Hibbeler (2007) http://wps.prenhall.com/
esm_hibbeler_engmech_1