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

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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.

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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

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

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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

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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

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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)

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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

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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)

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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

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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

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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

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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

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

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ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 15 Comprehension Quiz 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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ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 16 Comprehension Quiz 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

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

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ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 18 Example 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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ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 19 Example (cont.) a) Equivalent resultant force and couple moment acting at Point A FRFR MAMA

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

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

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ENGI 1313 Statics I – Lecture 21© 2007 S. Kenny, Ph.D., P.Eng. 23 References Hibbeler (2007) mech_1

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