Lecture7 (37)

Equivalent Force Systems
EQUIVALENT SYSTEMS for SINGLE FORCEEQUIVALENT SYSTEMS for SINGLE FORCE
Determining the effect of moving a force.
1. MOVING1. MOVING A FORCEA FORCE ONON ITS LINE OF ACTIONITS LINE OF ACTION
2. MOVING2. MOVING A FORCEA FORCE OFFOFF OF ITS LINE OF ACTIONOF ITS LINE OF ACTION

1. MOVING A FORCE1. MOVING A FORCE ONON ITS LINE OF ACTIONITS 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 equivalentthe two systems are equivalent

2. MOVING A FORCE2. MOVING A FORCE OFFOFF OF ITS LINE OF ACTIONOF 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” vectorfree” vector, it can be applied at any point P on the
body.
the two systems are equivalentthe two systems are equivalent
Use this process repeatedly for systems of forcesUse this process repeatedly for systems of forces

Moving a force from point A to B requires creating an additional
couple moment.
≡
MB= (80+40)(-150 kN)
= -18000 kN.mm
= -18.000 kN.m
Example:

Mo = (200 sin60°)(400 N)
= 69282 N.mm
= 69.282 N.m
Example:
Determine the equivalent system
for the 400 N force acting at point O?

Determine equivalent force and couple at A?
Example:

Equivalent Force Systems ‘2D’
Replace by
Equivalent
System

RESULTANT OF PARALLEL FORCES
AT A POINT ‘2D’
AA system of forcessystem of forces andand momentsmoments cancan bebe
simplify intosimplify into a single resultanta single resultant forceforce andand
momentmoment acting at a specified pointacting at a specified point..
Parallel system of forcesParallel system of forces

If the force system lies in the x-y plane (2-D case), thenIf the force system lies in the x-y plane (2-D case), then
the reduced equivalent system can be obtained usingthe reduced equivalent system can be obtained using
the following two scalar equations.the following two scalar equations.
ARA
yR
MM
FF
∑
∑
=
=
AT A POINT ‘2D’

ARA
yR
MM
FF
∑
∑
=
=
AT A POINT ‘2D’

a a a a
O
F1 F2
F3 F4 F5
y
x
Find the equivalent Force couple system at
O?
F1 = 100 N, F2 = 90 N,
F3 = 80 N, F4 = 70 N,
F5 = 60 N and a = 0.2 m
Parallel system of forcesParallel system of forcesExample:
AT A POINT ‘2D’

4 4 4
333
F1 F2 F3
h1 h2 h2h3
h3
•
P
Given that:
F1 = 20 kN, F2 = 30 kN, F3 = 40 kN and
h1 = 2 m h2 = 3 m h3 = 1 m
Replace the given system of parallel forces by a force-couple system
acting at the point P?
Example:
AT A POINT ‘2D’

Further Reduction ofFurther Reduction of
ParallelParallel ForceForcess ‘2D’‘2D’
Several parallel forces acting on the stick can be
replaced by a single resultant force FR acting at a
distance d from the point of grip.
The equivalent Force:
FR = F1 + F2 + .... + FN
To find distance d use:
FRd = F1d1 + F2d2 + ..... + FNdN
dN
FN

≡≡ ≡≡

a a a a
O
F1 F2
F3 F4 F5
y
x
For the given force system find a
representative single force and its
location on the x-axis?
F1 = 100 N; F2 = 90 N; F3 = 80 N;
F4 = 70 N; F5 = 60 N; a = 0.2 m
Example:

Example:
Replace the force system by:
1- a single force – couple resultant at point P,
2- a single force resultant along the x-axis.
ParallelParallel ForceForcess ‘2D’

AA system of forcessystem of forces andand momentsmoments cancan bebe
simplifsimplifiedied into a single resultantinto a single resultant forceforce andand
moment acting at a specified pointmoment acting at a specified point..
General system of forces and
moments
Reducing the given system of forces and couple
moments into an equivalent SINGLE force and
couple moment at ANT POINTat ANT POINT ‘2D’

WhenWhen several forces and couple momentsseveral forces and couple moments act on aact on a
body, each forcebody, each force should beshould be movemove withwith its associatedits associated
couple moment tocouple moment to thatthat common point O.common point O.
AAdd all the forces and couple moments togetherdd all the forces and couple moments together
and findand find one resultant force-couple moment pone resultant force-couple moment pairair..
moments

When several forces acting on a given system with
couple moments, a single resultant force FR 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 FR at such a distance
d so that, this resultant force FR will overcome the
couple-moment effect also.
Determine: FR = F1 + F2 + .... + FN
Find distance d:
F d = (F d + F d + .... + F d ) + (M +M +...+M )
moments

M2
M1
≡≡≡≡
moments

[1] 20 points
Replace the given system of
forces and couple by a single
force and its location on the
line OE?
•
E
2m 2m
Example:
moments

Example:
The beam AE is subjected to a system of coplanar forces.
Determine the magnitude, direction and the location on the
beam of a resultant force which is equivalent to the given
system of forces measured from E. i.e. Determine the
equivalent force system measured from point E.
moments

C
Example: General system of forces and
moments

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.

M2
M1
moments
≡≡

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.

For resultant moment calculations if the system contains
forces and moment then,
MMRoRo = ∑M + ∑(r X F)= ∑M + ∑(r X F)

Determine equivalent force and couple (moment) at O.
Example:
moments

[1] 20 points
•E
2m 2m
Example:
moments
Replace the given system of forces
and couple by equivalent force-
couple system acting at the point E.

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.

Example: Determine the magnitude and directional sense of the
resultant moment of the forces about point O and point P.

Determine equivalent single force and couple at O.
Example:

Example:
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.
57.2°

Example:Example:
For the given force – couple system, obtain an equivalent (SINGLE)
force along the line OP?
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.

Example:
Replace the given forces by:
a) an equivalent resultant force and couple moment acting at point H.
b) a single force and its location acting along the line A-D-B.
10 mm
•
10 mm
• •
•
•
•
A
B
C
D
E
H
F1
F2
F3
M1
M2
F2 = 42.806 N
F1 = 50.215 N
F3 = 97.317 N
M1 = 2171.575 Nmm
M2 = 820.679 Nmm
moments

Lecture7 (37)

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Lecture7 (37)