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 forces can be reduced without

altering the external effect they produce on the body.

Equilibrium of a body is the condition in which the resultant

of all the forces acting on the body is zero. This condition

is studies in Statics.

When the resultant of all forces acting on a body is not

zero, the acceleration of the body is obtained by equating

the force resultant to the product of the mass and

acceleration of the body ( ). This condition is

studied in Dynamics. Thus, the determination of resultants

is basic to both Statics and Dynamics.

For some systems the resultant will only be a force, for

others it will only be a couple, but in general the resultant

will comprise both a force and a couple.

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Coplanar Force Systems

If the resultant of all forces

lying in a single plane

such as the xy plane is , then, this

resultant is calculated by the vector

sum of these forces.

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..., , , ,321

R

R

R

R

R

jin

RRFRFRFR

nRFFFFRF

y

yx

x

yxR

yxyyxx

Rn

cos cos

coscos

,

...

22

321

2F

1F

3F

nF

The location of the line of action of the resultant force

to an arbitrary point (such as point O) can be

determined by using the Varignon theorem. The moment of

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

moments constructed by moving its components to point O.

R

R

R

Md

MRddFdFdF

O

O

332211

This calculation can be defined by an equation as:

If the resultant force of coplanar forces

is zero, but their couple moment is different from

zero, the resultant will be a couple moment

perpendicular to the plane containing the forces.

RdMM

FR

O

nFFFF

..., , , , 321

M

The same principles can be applied 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

Three Dimensional Force Systems

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..., , , ,321

1) Three dimensional, concurrent forces comprising

the same magnitudes and directions as the original

forces,

2) Three dimensional couples.

By calculating the resultants of these forces and

couples, a single resultant force and a single resultant

couple can be obtained.

The resultant force,

R

R

R

R

R

R

kjin

RRRR

FRFRFR

nRFFFFRF

zz

y

yx

x

zyxR

zyx

zzyyxx

Rn

cos cos cos

coscoscos

, ,

...

222

321

The resultant couple moment,

M

M

M

M

M

M

kjin

MMMM

FrMFrMFrM

nMMMMMM

zz

y

yx

x

zyxM

zyx

zyyxx

Mn

cos cos cos

coscoscos

, ,

...

222

z

321

The selection of point O is arbitrary, but the

magnitude and direction of will depend on this

point; whereas, the magnitude and direction of

will remain the same irrespective of the point it is

moved to.

M

R

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 .

M

R

R

The force can be shifted a distance d to form a

moment , which is equal in magnitude and opposite

in direction to , so that they will cancel each other

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

M

R

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.

M

R

By definition, a wrench is positive if the couple and

force vectors point in the same direction, and negative

if they point in opposite directions. A common example

of a positive wrench is found with the application of a

screwdriver. All force systems can be reduced to a

wrench acting at a particular line of action.

M

R

M

R

Positive wrench Negative wrench

Positive wrench

//M

R

x y

z

y

//M

//M

M

R

Md

R

R

R

resolve into two components

M

move to delete

R

M

move to line of application of R

//M

angle between and less than 90° R

M

M

: sliding vector, : free vector R

//M

Negative wrench

//M

R

x y

z

//M

//M

M

R

Md

R

R

R

//M

resolve into two components

M

move to delete

R

M

move to line of application of R

//M

angle between and greater than 90°

R

M

: sliding vector, : free vector R

//M