Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of Newfoundlandspkenny@engr.mun.ca

ENGI 1313 Mechanics I

Lecture 21: Force and Couple Moment Systems

2 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

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.

3 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

Motivation for Equivalent Systems

Force and Couple System Number of loads acting on

body Simplify Loading

Reduce to equivalent system Examine Effects on Body

External equivalence only Internal effects (load-

displacement) are influence but not considered in this course

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Motivation for Equivalent Systems (cont.)

?

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Equivalent Force

Initial Single Force Principle of transmissibility

• Sliding vector External effect equivalence only

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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

External effect equivalence only

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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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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

kFjFiFFF zyxR

OCR MMM

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Illustration of Equivalent Force and Couple for Multiple Force System

Equivalent Force (Sliding Vector) and Couple Moment (Free Vector)

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Equivalent Single Resultant Force

Initial Multiple Force and Couple System FR and MRo must be perpendicular

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Equivalent Single Resultant Force (cont.)

Technique Valid for: Concurrent force systems

• Only forces

• No couple

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Equivalent Single Resultant Force (cont.)

Technique Valid for: Coplanar force systems

• Moment axis perpendicular to plane

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Equivalent Single Resultant Force (cont.)

Technique Valid for: Parallel force systems

• Couple moment acts only on two axes

• Couple moment perpendicular to force vector

14 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

Comprehension Quiz 21-01

A general system of forces and couple moments acting on a rigid body can be reduced to a ___ .

A) Single force B) Single moment C) Single force and two moments D) Single force and a single moment

Answer: D

15 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

Comprehension Quiz 21-02

The forces on the pole can be reduced to a single force and a single moment at point ____ .

A) P B) Q C) R D) S E) Any of these points.

Answer: E

P

Q

R

S

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Comprehension Quiz 21-03

Consider two couples acting on a body. The simplest possible equivalent system at any arbitraryarbitrary point on the body will have

A) one force and one couple moment B) one force C) one couple moment D) two couple moments

Answer: C

17 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

Comprehension Quiz 21-04

Three couples act on a body. The equivalent system will be ________ at different points on the body.

A) different when located B) the same even when located C) zero when located D) none of the above

Answer: B

18 ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng.

Example 21-01

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.

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Example 21-01 (cont.)

a) Equivalent resultant force and couple moment acting at Point A

FR MA

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Example 21-01 (cont.)

b) Equivalent single force location along the beam AB.

FR

2.10 ft

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Example 21-02 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.

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Example 21-02 (cont.)

Solution FR = 1400 N x = 3.00m y = 2.50 m

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References

Hibbeler (2007) http://wps.prenhall.com/

esm_hibbeler_engmech_1