Transcript Lecture 21 - Faculty of Engineering and Applied Science

ENGI 1313 Mechanics I
Lecture 21:
Force and Couple Moment Systems
Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]
Lecture 21 Objective
to determine the effect of moving a force,
 to find an equivalent single force-couple
system for a system of forces and couples.

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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Motivation for Equivalent Systems

Force and Couple System


Simplify Loading


Reduce to equivalent system
Examine Effects on Body


3
Number of loads acting on
body
External equivalence only
Internal effects (loaddisplacement) are influence
but not considered in this
course
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Motivation for Equivalent Systems (cont.)
?
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Force

Initial Single Force

Principle of transmissibility
• Sliding vector

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External effect equivalence only
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Force and Couple

Initial Single Force

Couple moment is a free vector
• Equivalent system force not on the line of action of
initial system force

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External effect equivalence only
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Illustration of Equivalent Force and
Couple for a Single Force System
Equivalent Force (Sliding Vector)
Equivalent Force (Sliding Vector) and Couple Moment (Free Vector)
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Force and Couple

Initial Multiple Force System
Equivalent system force not on
the line of action of initial
system force
 Couple moment is a free vector
 External effect equivalence only






FR  F   Fx ˆi   Fy ˆj   Fz kˆ

MR  MC  MO
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Illustration of Equivalent Force and
Couple for Multiple Force System
Equivalent Force (Sliding Vector) and Couple Moment (Free Vector)
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Single Resultant Force

Initial Multiple Force and Couple System

10
FR and MRo must be perpendicular
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Single Resultant Force (cont.)

Technique Valid for:

Concurrent force systems
• Only forces
• No couple
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Single Resultant Force (cont.)

Technique Valid for:

Coplanar force systems
• Moment axis perpendicular to plane
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Equivalent Single Resultant Force (cont.)

Technique Valid for:

Parallel force systems
• Couple moment acts only on two axes
• Couple moment perpendicular to force vector
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Comprehension Quiz 21-01

A general system of forces and couple moments
acting on a rigid body can be reduced to a ___ .





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A) Single force
B) Single moment
C) Single force and two moments
D) Single force and a single moment
Answer: D
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Comprehension Quiz 21-02

The forces on the pole can be reduced to a
single force and a single moment at point ____ .
S





A) P
B) Q
C) R
D) S
E) Any of these points.
R
Q
P

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Answer: E
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Comprehension Quiz 21-03

Consider two couples acting on a body. The
simplest possible equivalent system at any
arbitrary point on the body will have





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A) one force and one couple moment
B) one force
C) one couple moment
D) two couple moments
Answer: C
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Comprehension Quiz 21-04

Three couples act on a body. The equivalent
system will be ________ at different points on
the body.





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A) different when located
B) the same even when located
C) zero when located
D) none of the above
Answer: B
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Example 21-01

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For the force and
couple system, find
(a) the equivalent
resultant force and
couple moment
acting at Point A and
(b) the equivalent
single force location
along the beam AB.
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Example 21-01 (cont.)

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a) Equivalent
resultant force
and couple
moment acting
at Point A
© 2007 S. Kenny, Ph.D., P.Eng.
FR

MA
ENGI 1313 Statics I – Lecture 21
Example 21-01 (cont.)

b) Equivalent single
force location along
the beam AB.
FR

2.10 ft
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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Example 21-02

21
Example Problem
4.18. The slab is
subjected to four
parallel forces.
Determine the
magnitude and
direction of a resultant
force equivalent to the
given force system
and locate its point of
application on the
slab.
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
Example 21-02 (cont.)

Solution
FR = 1400 N 
 x = 3.00m
 y = 2.50 m

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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21
References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1

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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 21