# Force System Resultant by mikesanye

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```									    CHAPTER 4
Force System Resultant
4.1 Moment of a Force - - - Scalar Formulation
1.Moment
A measure of the tendency of the force to cause a body to
rotate about the point or axis.
• Torque (T) 扭力
• Bending moment (M) 彎曲力矩

T     M                P            M
2. Vector quantity
(1) Magnitude ( N-m or lb-ft)
Mo = Fd
d = moment arm or perpendicular distance from point O
to the line of action of force.

d
o

Lime of action (sliding vector)

(2) Direction
Right-Hard rule
A. Sense of rotation ( Force rotates about Pt.O)
Curled fingers
B. Direction and sense of moment
Thumb
3.Resultant Moment of Coplanar Force System

Fn
M RO
don
do1           F3
do2     do3
F1
F2
4.2 Cross Product
1. Definition

(1) magnitude of

(2)Direction of
perpendicular to the plane containing A & B
2. Law of operation
(1)

(2)

(3)

3. Cartesian Vector Formulation
(1) Cross product of Cartesian unit vectors.
     
i i  j  j  k k  0
                  
i  j  i j sin 90 k  k


                                   j
j k  i ; k i  j             i
(2) Cross product of vector A & B in Cartesian vector form
                    
A  Ax i  Ay j  Az k
                    
B  Bx i  B y j  Bz k
                                              
A  B  ( Ax i  Ay j  Az k )  ( Bx i  B y j  Bz k )
                           
 Ax B y i  j  Ax Bz j  k  Ay Bx i  j
                           
 Ay Bz k  j  Az B y k  i  Az B y k  j
                     
 (Ay Bz -Az B y )i  ( Az Bx  Az Bx ) j 

 (Ax B y  Ay Bx )k
                  
A  Ax i  Ay j  Az k
         
i      j k
 
A  B  Ax Ay Az
Bx   By   Bz
Ay   Az  Ax    Az  Ax      Ay 
           i         j           k
By   Bz    Bx   Bz    Bx     By
4.3 Moment of a Force – Vector Formulation
1. Moment of a force F about pt. O
Mo = r x F
where r = A position vector from pt. O to any pt. on
the line of action of force F .

F   d
o

(1) Magnitude
Mo=|Mo|=| r x F | =| r|| F | sinθ=F r sinθ
=F d

(2) Direction
Curl the right-hand fingers from r toward F (r cross F ) and
the thumb is perpendicular to the plane containing r and F.
4.4 Principle of moments
Varignon’s theorem
The moment of a force about a point is equal to the sum of the
moment of the force’s components about the point .

F
F1
r
F2
o

Mo=r x F          Mo= r x (F1+F2)
F = F1+F2            = r x F1+ r x F2
= MO1+MO2
4.5 Moment of a force about a specified Axis
1. Objective
Find the component of this moment along a specified axis passes
through the point about which the moment of a force is computed.
2. Scalar analysis (See textbook)
3. Vector analysis
a

Point O on axis aa’
b

Ma Ө
Mo= r x F
O


b’
A           F

Moment
axis               Axis of projection
a’
(1) Moment of a force F about point 0
Mo = r × F
Here, we assume that bb’ axis is the moment axis of Mo

(2) Component of Mo onto aa´ axis

Ma = Ma ua

Ma=Mo cosθ
=Mo ●ua=( r × F ) ● ua
=trip scalar product

Here
Ma=magnitude of Ma
ua= unit vector define the direction of aa´ axis
               
If   U a  U ax i  U ay j  U azk
                  
r  rx i  ry j  rz k
                    
F  Fx i  Fy j  Fz k
 
then M a  Ua  ( r  F )

U ax U ay U az
 rx   ry   rz
Fx Fy    Fz

4.Method of Finding Moment about a specific axis
(1) Find the moment of the force about point O
Mo = r x F

(2) Resolving the moment along the specific axis

Ma = Maua
= (Mo•ua) ua
=[ua •( r x F )]ua
4.6 moment of a couple
1. Definition ( couple) 偶力矩
Two parallel forces have the same magnitude, opposite
distances, and are separated by a perpendicular distance d.

d

2. Scalar Formulation
(1) Magnitude     M=Fd
(2) Direction & sense (Right-hand rule)
•Thumb indicates the direction
•Curled fingers indicates the sense of rotation
3. Vector Formulation
M= r x F
|M|=M=|r x F |=r F sinθ                   θ d
=F d                            F                F
r
Remark:
(1) The couple moment is equivalent to the sum of the moment
of both couple forces about any arbitrary point 0 in space.

-F
Mo= rAx( -F )+ rB x F
=(-rA+rB) x F                         r B              A
=r x F= M                        F        rB       r
o   A

(2) Couple moment is a free vector which can act at any point in
space.

F         B
-F
r
A Mo=Mo’= r x F=M
o                 o’
4. Equivalent Couples
The forces of equal couples lie either in the same plane or in planes
parallel to one another.

A       F
F d                                      -F   plane A // plane B
-F
d
B -F                  F

5. Resultant couple moment
Apply couple moment at any point p on a body and add
them vectorially.
M2
M1

A
B                     M2
M1

MR=ΣM=Σ r x F
4.7 Equivalent system
1. Equivalent system
When the force and couple moment system produce the same
“external” effects of translation and rotation of the body as their
resultant , these two sets of loadings are said to be equivalent.
2. Principle of transmissibility
The external effects on a rigid body remain unchanged,when a
force, acting a given point on the body, is applied to another point
lying on line of action of the force.
line of action

Same external effect

F A
P         Internal effect ?         P
F         Internal stresses are different.
3. Point O is on the line of action of the force

F
A                                    A       equivalent             A
equivalent                                         F
F
o                                                                  o
-F        o
Original system
Sliding vector
4. Point O is not on the line of action of the force

F                                                      Couple moment
F                 Mc= r x F
M=r x F
A                    F               A
r                         P        A
o        line of                     o            F            o
action
Original system
-F
Force on Point A
=Force on point O + couple moment on any point p.
Example:

F                    F
o                                   o                    A
A

Point O is on the line of action of the force

F                        F
o                   A                o        X   P       A
d
M= F d (Free vector)

Mo= F d
Point O is not on the line of action of the force
4.8 Resultant of a force & couple system
1. Objective
Simplify a system of force and couple moments to
their resultants to study the external effects on the
body.

2. Procedures for Analysis

(1)Force summations
FR=F1+F2+……+ΣF

(2)Moment summations
MR0= ΣMC+r1o*F1+r2o*F2= ΣMC+ ΣM0

MC:Couple moment in the system
Mo:Couple moment about pt.O of the force in the system.
4.9 Further Reduction of a force & couple system
1. Simplification to a single Resultant Force
(1)Condition
FR MR0 or FR*MR0 = 0
(2)Force system
A. Concurrent Force system
F2                        F1                 FR

Equivalent
P                      =
System
Fn                             no couple moment

B.Coplanar Force System
y              F1,F2,F3 on xy plane
F3   M1&M2:z direction              MR0=ΣMC+ Σr * F    P        MR0
F2                    x          =>                           =>           d=
FR
F1                                              FR=ΣF
C. Parallel Force System
1. F1 // F2 //……// Fn
2. MR0 perpendicular to FR ,      MR0=ΣM+ Σr*F
z                                   z                          FR   z
F1              r2     F2             MR0         FR= ΣF

r1                     y     =                          y   =                              y
M1                                                                       p    o
r3      F3             x                              x                       MR0
x             M2                                                                   d = --------------

2. Reduction to a wrench                               |FR|d=|MR0|             FR

(1) Condition: FR       MR0
MR0=M +M//
M = moment component FR
M// = moment component // FR
(2) Wrench (or Screw)
An equivalent system reduces a simple resultant
force FR and couple moment MR0 at pt.0 to a
collinear force FR and couple moment M// at pt.

FR
MRo
a
b
o
FR                                  b       FR
M//           a a
a
b                    M//
b                                                     o
o p                                            p
b                                       b
a
a

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