RESULTANTS

Force – Couple Systems = Reduction of a Force to an Equivalent

Force and Moment (Moving a Force to Another Point)

The force acting on a body has two effects:

the first one is the tendency to push or pull the body in the direction

of the force, and the second one is to rotate the body about any fixed

axis which does not intersect nor is parallel to the line of the force.

This dual effect can more easily be represented by replacing the

given force by an equal parallel force and a couple to compensate for

the change in the moment of the force.

F

Let us consider for acting at point A in a rigid body. It is possible

to slide force along its line of action, but it is not possible to

directly move it to point B without changing the external effect on the

rigid body.

F

F

In order to do this, two equal and opposite forces and are added to

point B without introducing any net external effects on the body. It is

seen that, the original force at A and and the equal and opposite one at

B constitute the couple M=Fd, which is counterclockwise for this case.

F

F

Therefore, we have replaced the original force at A by the same

force acting at a different point B and a couple, without altering

the external effects of the original force on the body. Since is

a free vector, its location is of no concern. The combination of

the force and couple is referred to as a force-couple system.

M

By reversing this process, we can combine a given couple and a

force which lies in the plane of the couple (normal to the couple

vector) to produce a single, equivalent force. Force can be moved

to a point by constructing a moment equal in magnitude and

opposite in direction . The magnitude and direction of

remains the same, but its new line of action will be distance

away from point B.

M

F

F

Md

1. A force F of magnitude 50 N is exerted on the automobile parking-

brake lever at the position x=250 mm. Replace the force by an

equivalent force-couple system at the pivot point O.

2. The device shown is part of an automobile seat-back-release

mechanism. The part is subjected to the 4 N force exerted at A and a

300 Nmm restoring moment exerted by a hidden torsional spring.

Determine the y-intercept of the line of action of the single equivalent

force.

3. A 50 N horizontal force is applied to the handle of the industrial

water valve as shown. The force is perpendicular to the vertical plane

containing line OA of the handle. Determine the equivalent force-

couple system at point O.

Simplification of Force Systems – Resultants

If two force systems are creating the same external effect on the

rigid body they are exerted on, they are said to be “equivalent”.

The resultant of a force system is the simplest combination that

they can be reduced without altering the external effects they

produce on the body.

Coplanar Force Systems

If the resultant of all forces lying in a single plane

such as xy is , this resultant is calculated by the vector sum of these

forces.

nFFFFR

...321

nFFFF

...,,,, 321

R

x

y

yx

yyxx

R

R

FFR

FRFR

1

22

tan

The location of the line of action of the resultant force to an

arbitrary point (such as point O is the origin of the xy coordinate

system) can be determined by using the Varignon’s theorem. The

moment of about point O will be equal the sum of the couple

moments constructed by moving its components to point O.

R

R

FdMMo

FR

R

Md o

160 mm

380 mm

150 mm

320 mm

C

B

A

800 N

800 N

4. The forces acting on the belts on a pulley system are equal with a

magnitude of 800 N. The pulley is secured to the steel column by means of

two bolts at A and B. Replace the two forces with a force-couple system, in

which the equivalent force will be at the midpoint of the bolts. Then,

determine the force each bolt will sustain by distributing this force and

couple to forces acting at points A and B.

5. Under nonuniform and slippery road conditions, the four forces shown are exerted

on the four drive wheels of the all-wheel-drive vehicle. Determine the resultant of

this system and the x- and y-intercepts of its line of action. Note that the front and

rear tracks are equal (i. e., ). CDAB

Three Dimensional Force Systems

The same principles can be enlarged to three dimensional force

systems. The resultant of forces acting on a body

can be obtained by moving them to a desired point. In this way, the

given force system will be converted to

1) Three dimensional, concurrent forces comprising the same

magnitudes and directions as the original forces,

2) Three dimensional couples.

nFFFF

...,,,, 321

By calculating the resultants of these forces and couples, a single

resultant force and a single couple can be obtained.

The resultant force,

222

321 ...

zyx

zzyyxx

n

FFFR

FRFRFR

FFFFFR

The resultant couple moment,

The selection of point O is arbitrary, but the magnitude and direction

of will depend on this point; whereas, the magnitude and

direction of are the same no matter which point is selected.

CFrM

M

R

C C

6. Determine the force-couple

system at O which is equivalent

to the two forces applied to the

shaft AOB.

Is perpendicular to ? R

OM

7. Represent the resultant of the force system acting on the pipe assembly by a

single force at A and a couple .

8. The special purpose milling-cutter is subjected to the force of 1200 N and a

couple of 240 N.m as shown. Replace this loading system by an equivalent

force-couple system at O.

9. The tension in cable AB is 450 N and the tension in cable CD is 270 N.

Suppose that you want to replace these two cables by a single cable EF so that the

force exerted on the wall at E is equivalent to the two forces exerted by cables AB

and CD on the walls at A and C. What is the tension in cable EF and what are the

coordinates of points E and F?

10. The threading dye is screwed onto the end of the fixed pipe which is bent

through an angle of 20°. Replace the two forces by an equivalent force at O and a

couple . Find and calculate the magnitude M of the moment which tends to screw

the pipe into the fixed block about its angled axis through O.

NmkikiMMM

kikie

kiM

jikki

jikkiM

jjjFR

oOC

OC

o

o

68.8594.034.08517

94.034.020cos20sin

8517

15025.02.020cos15.020sin15.0

20025.02.020cos15.020sin15.0

50200150

y

x

500 N

1700 N 3400 N

3

4

tan

30 cm 50 cm

34 cm

50 cm

50 cm z

800 N.m

tan 15

8

11. The pulleys and the gear are subjected to the loads shown. For these forces,

determine the equivalent force-couple system at point A.

15

8

jiR

jiijiFR

jijiF

iF

jijiF

35201040

800150050027202040

800150017

81700

17

151700

500

272020405

43400

5

33400

3

2

1

kjiCFrM

kC

kjijikjFr

kijFr

kjiFr

jikjijijikFr

CFrM

A

A

6.11391770960

800

75075040080015005.05.0

1505003.0

6.48910201360

272020405.016.03.02720204017

8

17

1534.05.0

33

22

11

11

12. The direction cosines of robot arm AB are cos x=0.6, cos y (y <90°) and cos z=0’dır.

For arm BC the direction cosines are, cos x=7/9, cos y=4/9 and cos z=4/9. A force of

magnitude F=250 N and a couple of magnitude C=27 Nm along the axis BC are applied to

the end C of arm BC. Determine the moment about line AD. Replace the force and couple

acting on the robot assembly with an equivalent force-couple at point A.

* Line MN lies in a plane parallel to the horizontal plane

* Line AD lies in the xz plane and makes a 37° angle with

the x axis

As a special case, if the resultant couple is perpendicular to

the resultant force , these two vectors can further be simplified to

obtain a single resultant force . The force can be slided a

distance d to form a moment , which is equal in magnitude

and opposite in direction , so that they will cancel each other

out. The distance d will be equal to d=M/R.

R

R

R

M

M

M

Wrench Resultants

When the resultant couple vector is parallel to the resultant

force , the resultant is called a “wrench”.

The wrench is the simplest form in which the resultant of a general

force system may be expressed.

By definition, a wrench is positive if the couple and force vectors

point in the same direction, and negative it they point in opposite

directions.

R

M

Wrench Resultants

A common example example of a wrench is found with the

application of a screw driver.

All force systems can be reduced to a wrench acting at a particular

line of action.

M R

//2

//1

RR

RRR

MMMM

eeMMM

Equivalent force-couple system at point O M is resolved into components M1 along the

direction of R and M2 normal to R.

Positive wrench

R

Md 2

Positive

Wrench

Negative

Wrench

13. In tightening a bolt whose center is at point O, a person exerts a 180 N force on the

ratchet handle with his right hand. In addition, with his left hand he exerts a 90-N force as

shown in order to secure the socket onto the bolt head. Determine the equivalent force-

couple system at O. The find the point in the x-y plane through which the line of action of

the resultant force of the wrench passes.

14.

C3 = 80 Nm (in plane ABCD) y > 90o

F1 = 30 N

F2 = 75 N

F3 = 40 N

C1 = 60 Nm

C2 = 100 Nm (in yz plane)

O (0, 0, 0) m A (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m

15. Replace the system comprising two forces, two couples and a positive wrench with an

equivalent force-couple acting at point O. Then, reduce the system further into a wrench and

determine the coordinates of point P, of which the line of action of the wrench cuts through

the yz plane.

A

B

C

O

D

E

G

37°

X Y

Z

45°

60° y

53°

30°

6 m

4 m

1C

1F

2C

3F

3C

2F

k78.063.0i015.0

52.9688.7795.1

k52.9688.77i95.1

R

Re

k52.9688.77i95.1FR

k64.3416i12k30cos4053sin60cos40i53cos60cos40

k88.4688.46i15.35

884

k)80()08(i181275

k5115i21.21kcos600cos6icos4530

601cos06cos45cos

04.124

222R

3

2222

1

222

jj

j

jjF

jj

F

jjF

oyy

Force:

k86756.841i86.847M

k23i64i100k3030i42.24

k2484.432i4.394k84.843562.56i04.375k16.6516.65M

k23i64

2432

k24i3280C

k24i324k8i6BDBA

i100C

k3030i42.24k60cos60cosicos4560C

k2484.432i4.394k64.3416i12k310i6

k84.843562.56i04.375k88.4688.46i15.358i12

k16.6516.65k5115i21.21k44i10

o

o

40

223

2

1

33

22

11

j

j

jjj

j

jj

jjjFr

jjjFr

jjjFr

CFrM o

Moment:

k86756.841i86.847M

k52.9677.88i95.1

o

j

jR

Equivalent force-couple system at point O

Reduction to a wrench in yz plane

k02.10402.84i2k78.063.0i015.035.133MM

35.133k78.063.0i015.0k86756.841i47.868M

////

//

jje

NmjjeM

R

Ro

Positive wrench

k02.10402.48i2M

k52.9677.88i95.1

//

j

jR

O

x y

z

k02.10402.48i2M//

j

k52.9677.88i95.1

jR

oM

M

Positive wrench

The coordinates of point P, of which the line of action of the wrench cuts through the yz plane:

myyk

mzzj

jizjziyky

jj

j

jj

Rr

27.39198.76295.1

66.47458.92595.1

k98.76258.259i49.8688.7745.152.9695.1

k98.76258.259i49.868k52.968.77i95.1kzjy

k98.76258.259i49.868M

k02.10402.48i2k867841.56i47.868MMM

M

//o

O x y

z

k02.10402.48i2M//

j

k52.9677.88i95.1

jR

Positive wrench

P(0;391.27;474.66) r