Section 4.7 Simplification of a Force and Couple System

Engineering Mechanics: Statics Fifteenth Edition Chapter 4 Force System Resultants Section 4.7 Simplification of a Force and Couple System Section 4.8 Further Simplification of a Force and Couple System Copyright © 2022 Pearson Education, Inc. All Rights Reserved Simplification of Force and Couple Systems & Their Further Simplification Today’s Objectives: a. Determine the effect of moving a force. b. Find an equivalent forcecouple system for a system of forces and couples. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Applications What are the resultant effects on the person’s hand when the force is applied in these four different ways? Why is understanding these differences important when designing various load-bearing structures? Copyright © 2022 Pearson Education, Inc. All Rights Reserved Section 4.7: Simplification of Force and Couple System When a number of forces and couple moments are acting on a body, it is easier to understand their overall effect on the body if they are combined into a single force and couple moment having the same external effect. The two force and couple systems are called equivalent systems since they have the same external effect on the body. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Moving a Force on Its Line of Action Moving a force from A to B, when both points are on the vector’s line of action, does not change the external effect. Hence, a force vector is called a sliding vector. (But the internal effect of the force on the body does depend on where the force is applied). Copyright © 2022 Pearson Education, Inc. All Rights Reserved Moving a Force off of its Line of Action When a force is moved, but not along its line of action, there is a change in its external effect! Essentially, moving a force from point A to B (as shown above) requires creating an additional couple moment. So moving a force means you have to “add” a new couple. Since this new couple moment is a “free” vector, it can be applied at any point on the body. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Simplification of a Force and Couple System When several forces and couple moments act on a body, you can move each force and its associated couple moment to a common point O. Now you can add all the forces and couple moments together and find one resultant force-couple moment pair. FR =  F MRO =  MC +  MO Copyright © 2022 Pearson Education, Inc. All Rights Reserved Section 4.8: Further Simplification of a Force and Couple System If FR and MRO are perpendicular to each other, then the system can be further reduced to a single force, FR , by simply moving FR from O to P. In three special cases, concurrent, coplanar, and parallel systems of forces, the system can always be reduced to a single force. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example I (1 of 3) Given: A 2-D force system with geometry as shown. Find: The magnitude of resultant force and couple moment acting at A and then the equivalent single force location measured from A. Plan: 1. Sum all the x and y components of the forces to find FRA. 2. Find and sum all the moments resulting from moving each force component to A. 3. Shift FRA to a distance d such that d= MRA FRy Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅰ (2 of 3) 3 + →  FRx = 50 ( sin 30 ) + 100   5 = 85lb 4 +   FRy = 200 + 50 ( cos 30 ) − 100   5 = 163.3lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅰ (3 of 3) FR = ( 852 + 163.3 1 2 2 ) = 184 lb The equivalent single force FR can be located at a distance d measured from A. D= 𝑀𝑅𝐴 𝐹𝑅 = 509.7 184 = 2.77 𝑓𝑡 Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅱ (1 of 3) Given: The slab is subjected to three parallel forces. Find: The equivalent resultant force and couple moment at the origin O. Also find the location (x, y) of the single equivalent resultant force. Plan: 1. Find FRO =  Fi = FRzo k 2. Find MRO =  ( ri  Fi ) = MRXO i + MRYO j 3. The location of the single equivalent resultant force is given as x=− MRyO FRzO and y = MRxO FRzO Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example Ⅱ (2 of 2) FRO = {100 k − 500 k − 400 k } = − 800 k N MRO = (3 i )  (100 k ) + (4 i + 4 j )  ( −500 k ) + (4 j )  ( −400 k ) = {−300 j + 2000 j − 2000 i – 1600 i } = { – 3600 i + 1700 j } N  m The location of the single equivalent resultant force is given as, ( −1700 ) = 2.13 m FRzo ( −800 ) ( −3600 ) = 4.5 m M y = Rxo = FRzo ( −800 ) x=− MRyo = Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (1 of 2) 1. 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. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Concept Quiz (2 of 2) 2. 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. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving Ⅰ (1 of 2) Given: A 2-D force and couple system as shown. Find: The equivalent resultant force and couple moment acting at A. Plan: 1. Sum all the x and y components of the two forces to find FRA. 2. Find and sum all the moments resulting from moving each force to A and add them to the 1500 N  m free moment to find the resultant MRA. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving Ⅰ (2 of 2) Summing the force components: + →  Fx = 450 ( cos 60 ) − 700 ( sin 30 ) = –125 N +   FY = – 450 ( sin 60 ) – 300 – 700 ( cos 30 ) = −1296 N Now find the magnitude and direction of the resultant. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving Ⅱ (1 of 3) Given: Forces and couple moments are applied to the pipe. Find: An equivalent resultant force and couple moment at point O. Plan: a. Find FRO =  Fi = F1 + F2 + F3 b. Find MRO =  Mc +  ( ri  Fi ) where, M C are any free couple moments. r i are the position vectors from the point O to any point on the line of action of F i . Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving Ⅱ (2 of 3) F1 = {300 k } N F2 = 200{cos45 i − sin 45 k } N = {141.4 i – 141.4 k } N F3 = {100 j } N r1 = {0.5 i } m, r2 = {1.1 i } m, r3 = {1.9 i } m Free couple moments are: MC1 = {100 k } N  m MC2 = 180 {cos45 i − sin 45 k } N  m = {127.3 i − 127.3 k } N  m Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving Ⅱ (3 of 3) Resultant force and couple moment at point O: FRO =  Fi = F1 + F2 + F3 FRO = {300 k } + {141.4 i − 141.4 k } + {100 j } = {141 i + 100 j + 159 k } N MRO =  MC +  ( ri  Fi ) MRO = {100 k } + {127.3 i – 127.3 k } i j k i j k i j k + 0 0.5 0 + 0 1.1 0 + 0 1.9 0 0 0 300 141.4 0 −141.4 0 100 0 MRO = {122 i − 183 k } N  m Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. For this force system, the equivalent system at P is ____________. A. FRP = 40 lb ( along + x - dir.) and MRP = + 60 ft · lb B. FRP = 0 lb and MRP = + 30 ft · lb C. FRP = 30 lb ( along + y - dir.) and MRP = − 30 ft · lb D. FRP = 40 lb ( along + x - dir.) and MRP = + 30 ft · lb Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. Consider three couples acting on a body. Equivalent systems 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. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. 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 Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. The original force and couple system and an equivalent force-couple system have the same ________ effect on a body. A. internal B. external C. internal and external D. microscopic Copyright © 2022 Pearson Education, Inc. All Rights Reserved