Equivalent Force Systems

Transcription

1 Equivalent Force Systems EQUIVALENT SYSTEMS for SINGLE FORCE Determining the effect of moving a force. 1. MOVING A FORCE ON ITS LINE OF ACTION 2. MOVING A FORCE OFF OF ITS LINE OF ACTION

2 Equivalent Force Systems 1. MOVING A FORCE ON ITS LINE OF ACTION Moving a force from A to O, when both points are on the vectors 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). the two systems are equivalent

3 Equivalent Force Systems 2. MOVING A FORCE OFF OF ITS LINE OF ACTION Moving a force from point A to B (as shown above) requires creating an additional couple moment. Since this new couple moment is a free vector, it can be applied at any point P on the body. the two systems are equivalent Use this process repeatedly for systems of forces

4 Equivalent Force Systems 2. MOVING A FORCE OFF OF ITS LINE OF ACTION Example: Moving a force from point A to B requires creating an additional couple moment. M B = (80+40)(150 kn) = kn.mm = kn.m

5 Equivalent Force Systems 2. MOVING A FORCE OFF OF ITS LINE OF ACTION Example: Determine the equivalent system for the 400 N force acting at point O? M o = (200 sin60 )(400 N) = N.mm = N.m Opposite direction Equal in magnitude Having perpendicular distance

6 Equivalent Force Systems Example (T): Determine equivalent force and couple at A?

7 Equivalent Force Systems 2D Replace by Equivalent System

8 RESULTANT OF PARALLEL FORCES AT A POINT 2D A system of forces and moments can be simplify into a single resultant force and moment acting at a specified point. Parallel system of forces

9 RESULTANT OF PARALLEL FORCES AT A POINT 2D Parallel system of forces If the force system lies in the x-y plane (2-D case), then the reduced equivalent system can be obtained using the following two scalar equations. F R M RA F y M A

10 RESULTANT OF PARALLEL FORCES AT A POINT 2D Parallel system of forces F R M RA F y M A

11 RESULTANT OF PARALLEL FORCES AT A POINT 2D Parallel system of forces Example: P F 1 F 2 F h 1 h 2 h 3 h 2 h 3 Given that: F 1 = 20 kn, F 2 = 30 kn, F 3 = 40 kn and h 1 = 2 m h 2 = 3 m h 3 = 1 m Replace the given system of parallel forces by a force-couple system acting at the point P?

12 Further Reduction of Parallel Forces 2D Parallel system of forces Several parallel forces acting on the stick can be replaced by a single resultant force F R acting at a distance d from the point of grip. The equivalent Force: F R = F 1 + F F N To find distance d use: F R d = F 1 d 1 + F 2 d F N d N F N d N

13 Further Reduction of Parallel Forces 2D Parallel system of forces

14 Further Reduction of Parallel Forces 2D Parallel system of forces Example: Replace the force system by: 1- a single force couple resultant at point P, 2- a single force resultant along the x-axis.

15 RESULTANT OF PARALLEL FORCES AT A POINT 2D Example (T): y F 1 F 2 F 3 F 4 F 5 Parallel system of forces Find the equivalent Force - couple system at O? F 1 = 100 N, F 2 = 90 N, F 3 = 80 N, F 4 = 70 N, F 5 = 60 N and a = 0.2 m x O a a a a

16 Further Reduction of Parallel Forces 2D Example (T): Parallel system of forces y F 1 F 2 F 3 F 4 F 5 O a a a a x For the given force system find a representative single force and its location on the x-axis? F 1 = 100 N; F 2 = 90 N; F 3 = 80 N; F 4 = 70 N; F 5 = 60 N; a = 0.2 m

17 Equivalent Force Systems 2D Replace by Equivalent System

18 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D A system of forces and moments can be simplified into a single resultant force and moment acting at a specified point. General system of forces and moments

19 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D General system of forces and moments When several forces and couple moments act on a body, each force should be move with its associated couple moment to that common point O. Add all the forces and couple moments together and find one resultant force-couple moment pair.

20 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D General system of forces and moments If the force system lies in the x-y plane (2-D case), then the reduced equivalent system can be obtained using the following three scalar equations.

21 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D Reducing the given system of forces and couple moments into equivalent resultant force and couple moment: 1- add all the forces algebraically, 2- determine the moments of each of these forces with respect to that point 3- add all the existing moments the system.

22 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D For resultant moment calculations if the system contains forces and moment then, M Ro = M + (r X F)

23 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D General system of forces and moments When several forces acting on a given system with couple moments, a single resultant force F R can be replaced to such a distance d along the specified line so that, it will have the same external effect on the given system. i.e. Locate the resultant force F R at such a distance d so that, this resultant force F R will overcome the couple-moment effect also. Determine: F R = F 1 + F F N Find distance d: F R d = (F 1 d 1 + F 2 d F N d N ) + (M 1 +M M N )

24 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D General system of forces and moments M 2 M 1

25 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D Example: General system of forces and moments C

26 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D 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.

27 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D Example: Determine the magnitude and directional sense of the resultant moment of the forces about point O and point P.

28 Reducing the given system of forces and couple moments into an equivalent SINGLE force ALONG A LINE 2D General system of forces and moments M 2 M 1

29 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANT POINT 2D General system of forces and moments Exercise: Replace the given system of forces and couple by a single [1] 20 points force and its location on the line XX? x E x 2m 2m

30 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANT POINT 2D Exercise : General system of forces and moments Replace the given system of forces and couple by a single force and its location on the line XX? [1] 20 points x 2m E 2m m 66 N 38 N x

31 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANY POINT 2D Example (T): Replace the forces acting on the brace by an equivalent resultant force and couple moment acting at point A. i.e. Determine equivalent single force and couple (moment) at A.

32 Reducing the given system of forces and couple moments into an equivalent SINGLE force and couple moment at ANT POINT 2D General system of forces and moments Example (T): Determine equivalent force and couple (moment) at O.

33 Reducing the given system of forces and couple moments into an equivalent SINGLE force ALONG A LINE 2D Example: Determine equivalent single force along x-axis.

34 Moment of a Force About A POINT 3D Example (T): Replace the given 4 forces by: a) an equivalent resultant force and couple moment acting at point O. b) an equivalent single force with proper location.

35 Moment of a Force About A POINT 3D Solution (T):