Force System Resultants. Engineering Mechanics: Statics

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1 Force System Resultants Engineering Mechanics: Statics

2 Chapter Objectives To discuss the concept of the moment of a force and show how to calculate it in 2-D and 3-D systems. Definition of the moment of a couple.

3 Chapter Objectives To present methods for determining the resultants of non-concurrent force systems in 2D and 3D systems. Reducing the given system of forces and couple moments into an equivalent force and couple moment at any point.

4 Chapter Outline Moment of a Force Force Couple Principles of Moments

5 Moment of a Force Carpenters often use a hammer in this way to pull a stubborn nail. Through what sort of action does the force F H at the handle pull the nail? How can you mathematically model the effect of force F H at point O?

6 Moment of a Force 2D System Moment of a force about a point (or an axis) is a measure of the tendency of the force to cause a body to rotate about the point or axis.

7 Case 1: Moment of a Force 2D System F x horizontal and acts perpendicular to the handle of the wrench and is located d y from the point O F x tends to turn the pipe about the z axis The larger the force or the distance d y the greater the turning effect

8 Moment of a Force 2D System The larger the force W or the distance D the greater the turning effect at point P.

9 Note that: Moment of a Force 2D System Moment axis (z) is perpendicular to shaded plane (x-y). i.e. The remaining third axis: z F x and d y lies on the shaded plane (x-y) Moment axis (z) intersects the plane at point O

Moment of a Force 2D System Case 2: Apply force F z to the wrench Pipe does not rotate about z axis The pipe not actually rotates but F z creates a tendency

10 Moment of a Force 2D System Case 2: Apply force F z to the wrench Pipe does not rotate about z axis The pipe not actually rotates but F z creates a tendency for rotation so causing (producing)moment along (M o ) x. Moment axis (x) is perpendicular to the shaded plane (y-z) F z and d y lies on the shaded plane (y-z)

11 Moment of a Force 2D System Case 3: Apply force F y to the wrench No moment is produced about point O Lack of tendency to rotate as line of action passes through O Note that, F y and d y both lies on the same line (and not forming any plane) hence No Moment is produced.

12 Moment of a force does not always cause rotation Force F; Moment of a Force 2D System tends to rotate the beam clockwise about A with moment M A = d A F tends to rotate the beam counterclockwise about B with moment M B = d B F

13 In General Moment of a Force 2D System Consider the force F and the point O which lies in the shaded plane The moment M O about point O, or about an axis passing through O and which is perpendicular to the plane, is a vector quantity.

14 Moment of a Force 2D System Moment M O is a vector having specified magnitude and direction.

15 Moment of a Force 2D System M = F x d M = Magnitude of the moment about point or axis [N.m] F= Magnitude of the force [N] d = perpendicular distance [m] Direction is determined by using the right hand rule

16 Moment of a Force 2D System Positive direction of the moment : Anti clockwise

17 Moment of a Force 2D System Positive moment Negative moment

18 Magnitude: Moment of a Force 2D System Scalar Formulation Use simple multiplication; For magnitude of M O : M O = F. d d = moment arm or perpendicular distance from the axis at point O to its line of action of the force. F = Magnitude of the force Units for moment is N.m, kn.m

19 Direction: Moment of a Force 2D System Scalar Formulation Direction of M O is specified by using right hand rule Fingers of the right hand are curled to follow the sense of rotation when force rotates about point O.

20 Direction: Moment of a Force 2D System Scalar Formulation Thumb points along the moment axis to give the direction and sense of the moment vector Moment vector is upwards and perpendicular to the shaded plane

21 Moment of a Force 2D System Scalar Formulation Direction M O is shown by a vector arrow with a curl to distinguish it from force vector Fig b. M O is represented by the counterclockwise curl, which indicates the action of F. Arrowhead shows the sense of rotation caused by F. Using the right hand rule, the direction and sense of the moment vector points out of the page.

22 Moment of a Force 2D System Vector Formulation In some two dimensional problems and most of the three dimensional problems, it is convenient to use a vector approach for moment calculations. The MOMENT of a force about point A may be represented by the cross product expression. M r F r Position vector which runs from the moment reference point to any point on the line of action of the force.

23 Moment of a Force 2D System Vector Formulation Without using the right hand rule directly apply the equation through cross product of vectors. M O = d X F The cross product directly gives the magnitude and the direction. Units for moment is N.m, kn.m

24 Sarrus Rule Moment of a Force 2D System Vector Formulation M o r F - k +k M o r x F y ( k) r y F x ( k)

25 Example: Moment of a Force 2D System For each case, determine the moment of the force about point O

26 Solution Line of action is extended as a dashed line to establish moment arm d Tendency to rotate is indicated and the orbit is shown as a colored curl (a)m (2m)(100N) N.m (CW) o (b)m (0.75m)(50N) N.m (CW) o Moment of a Force 2D System

27 Solution (c) M (4m 2cos30 m)(40n) N.m (CW) o (d) M (1sin 45 m)(60n) N.m (CCW) o (e) M (4m 1m)(7kN) kn.m (CCW) o Moment of a Force 2D System

28 Moment of a Force 2D System Example: Determine the moments of the 800 N force acting on the frame about points A, B, C and D.

29 Moment of a Force 2D System Solution (Scalar Analysis) M = (2.5m)(800N) = 2000 N.m (CW) A M = (1.5m)(800N) = 1200 N.m (CW) B M = (0m)(800N)= 0 N.m C Line of action of F passes through C M = (0.5m)(800N) = 400 N.m (CCW) D

30 Moment of a Force 2D System

31 Principles of Moments Principles of Moments Also known as Varignon s Theorem This principle states that the moment of a force about a point is equal to the sum of moments of the force s components about the point.

32 Principles of Moments Moment of a force about a point is equal to the sum of the moments of the forces components about the point

33 Principles of Moments

34 Principles of Moments Solution Method 1: From trigonometry using triangle BCD, CB = d = 100cos45 = mm Thus, M A =df= ( m) 200N = N.mm (CCW) = N.m (CCW) As a Cartesian vector, M A = { k} N.m

35 Principles of Moments Solution Method 2: Resolve 200 N force into x and y components Principle of Moments M A = df M A =(200)(200sin45 ) (100)(200cos45 ) = N.mm (CCW) = N.m (CCW)

36 Principles of Moments Example: Moment of a force about a point is equal to the sum of the moments of the forces components about the point d 3 sin m M O Fd 5 cos 45 3sin 30 5 sin 45 3cos kn m kn m M O F d x y F y d x

37 Moment of a Force 2D System Example: The force F acts at the end of the angle bracket. Determine the moment of the force about point O.

38 Solution Method 1: Resolve the given force into components and than apply the moment equation. M O = 400sin30 N(0.2m)-400cos30 N(0.4m) = N.m Moment of a Force 2D System As a Cartesian vector, M O = { k} N.m

39 Solution Method 2: Express as Cartesian vector r = {0.4i 0.2j} m F = {400sin30 i 400cos30 j} N = { i j}N For moment, O i j k M rxf Moment of a Force 2D System k N.m

40 Moment of a Force 2D System Example (T): Determine the moment of the 600 N force with respect to point O in both scalar and vector product approaches.

41 Resultant Moment of System of Coplanar Forces Resultant moment M Ro = addition of the moments produced by all the forces algebraically since all moment forces are collinear (for 2D case). M Ro = F.d taking counterclockwise (CCW), to be positive.

42 Resultant Moment of System of Coplanar Forces Resultant moment, M Ro = addition of the moments produced by all the forces algebraically, since all moment forces are collinear (for 2D case). M Ro =M 1 M 2 + M 3 = df= d 1 F 1 d 2 F 2 + d 3 F 3 taking counterclockwise (CCW) to be positive.

43 Resultant Moment of System of Coplanar Forces M + R O Fd Counterclockwise is positive

44 Resultant Moment of System of Coplanar Forces Example: Determine the resultant moment of the four forces acting on the rod about point O.

45 Solution: (by scalar analysis) Note that always positive moments acts in the +k direction, CCW M F.d M Ro Ro ( 50N)(2m) (60N)(0m) (20N)(3sin 30 (40N)(4m 3cos N.m N.m (CW) Resultant Moment of System of Coplanar Forces m) m)

46 Solution: (by vector analysis) M Ro M Ro dxf Resultant Moment of System of Coplanar Forces [2(i)X 50 (-j)] [0X 60 (i)] [3sin30 (-j)x 20 (i)] [ (4 3cos30 )(i)x 40 (-j)] [100( k)] [0] [30(k)] [ (-k)] N.m(-k) or(cw)

47 Moment (Revision)

48 Moment Moment force F about point O can be expressed using cross product M O = r X F where r represents position vector from O to any point lying on the line of action of F.

49 Moment Remember M = r (vector) X F (vector) Find the length r vectorially for each force F if not given, find the vectorial representation of F also.

50 Moment In some two dimensional problems and many three dimensional problems, it is convenient to use a vector approach for moment calculations. The MOMENT of a force about point A may be represented by the cross product expression M r F

51 Moment A M A r 3 r 2 r 1 F r F M r F Position vector which runs from the moment reference point to any point on the line of action of the force Due to the principle of transmissibility, can act at any point along its line of action and still create the same moment about point A. M A r1 F r2 F r3 F

52 The Moment Vector The result obtained from r X F doesn t depend on where the vector r intersects the line of action of F: r = r + u r F = (r + u) F = r F because the cross product of the parallel vectors u and F is zero.

53 Moment of a Couple Couple - two parallel forces - same magnitude but opposite direction - separated by perpendicular distance d Resultant force = 0 Tendency to rotate in specified direction Couple moment = sum of moments of both couple forces about any arbitrary point

54 Moment of a Couple The moment of a couple is defined as: A couple is defined as two parallel forces with the same magnitude but opposite in direction separated by a perpendicular distance d. M O = F. d (using a scalar analysis; right hand rule for direction), M O = d X F (using vector analysis).

55 Moment of a Couple The net external effect of a couple is zero since the net force equals zero and the magnitude of the net moment equals F.d The moment of a couple is a free vector. It can be moved anywhere on the body and have the same external effect on the body. Moments due to couples can be added using the same rules as adding any vectors.

56 Moment of a Couple 2D A O a d B F F C M O = F (a+d) F a = F d M O =M A =M B =M C Moment of a couple has the same value for all moment centers.

57 Moment of a Couple 2D M M 2D CCW couple M M 2D CW couple

58 Moment of a Couple 2D = The moment of a couple is a free vector. It can be moved anywhere on the body and have the same external effect on the body.

59 Moment of a Couple 2D APPLICATIONS

60 Moment of a Couple 2D APPLICATIONS (continued)

61 Moment of a Couple 2D Scalar Formulation Magnitude of couple moment M = F.d Direction and sense are determined by right hand rule In all cases, M acts perpendicular to plane containing the forces.

62 Moment of a Couple 2D Vectorial Formulation M = d X F In all cases, M acts perpendicular to plane containing the forces.

63 Moment of a Couple 2D Example: A couple acts on the gear teeth. Replace it by an equivalent couple having a pair of forces that act through points A and B. =

64 Moment of a Couple 2D Solution Magnitude of couple M = 24 N.m Direction out of the page since forces tend to rotate CCW M is a free vector and can be placed anywhere.

65 Solution Moment of a Couple 2D To preserve CCW motion, vertical forces acting through points A and B must be directed as shown For magnitude of each force, M = F.d 24 N.m = F (0.2m) F = N

66 Example (T): Moment of a Couple 2D

67 Moment of a Couple 2D Equivalent Couples Two different couples are equivalent if they produce the same moment, (magnitude as well as direction).

68 Moment of a Couple 2D Two couples are equivalent if they produce the same moment with magnitude and direction. = =

69 Example (T): Moment of a Couple 2D

Chapter -4- Force System Resultant

Ishik University / Sulaimani Civil Engineering Department Chapter -4- Force System Resultant 1 2 1 CHAPTER OBJECTIVES To discuss the concept of the moment of a force and show how to calculate it in two