# Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland ENGI.

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Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 21:Force and Couple Moment Systems

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

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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 4 Motivation for Equivalent Systems (cont.) ?

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 5 Equivalent Force Initial Single Force  Principle of transmissibility Sliding vector  External effect equivalence only

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 6 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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 7 Illustration of Equivalent Force and Couple for a Single Force System Equivalent Force (Sliding Vector) Equivalent Force (Sliding Vector) and Couple Moment (Free Vector)

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 8 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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 9 Illustration of Equivalent Force and Couple for Multiple Force System Equivalent Force (Sliding Vector) and Couple Moment (Free Vector)

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 10 Equivalent Single Resultant Force Initial Multiple Force and Couple System  F R and M Ro must be perpendicular

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 11 Equivalent Single Resultant Force (cont.) Technique Valid for:  Concurrent force systems Only forces No couple

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 12 Equivalent Single Resultant Force (cont.) Technique Valid for:  Coplanar force systems Moment axis perpendicular to plane

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 13 Equivalent Single Resultant Force (cont.) Technique Valid for:  Parallel force systems Couple moment acts only on two axes Couple moment perpendicular to force vector

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

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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 16 Comprehension Quiz 21-03 arbitrary Consider two couples acting on a body. The simplest possible equivalent system at any arbitrary 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

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

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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 19 Example 21-01 (cont.) a) Equivalent resultant force and couple moment acting at Point A  FRFR MAMA

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 20 Example 21-01 (cont.) b) Equivalent single force location along the beam AB.  FRFR 2.10 ft

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

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 22 Example 21-02 (cont.) Solution  F R = 1400 N   x = 3.00m  y = 2.50 m

ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 23 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1

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