Force System Resultant

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CHAPTER 4 Force System Resultant
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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) ??
• Bending moment (M) ????

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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.

(2) Direction Right-Hard
rule A. Sense of rotation ( Force
rotates about Pt.O) Curled
fingers B. Direction and sense of
moment Thumb
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3.Resultant Moment of Coplanar Force
System
4.2 Cross Product
1. Definition
(1) magnitude of
(2)Direction of
perpendicular to the plane containing A B
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2. Law of operation (1)
(2)
(3)
3. Cartesian Vector Formulation (1) Cross
product of Cartesian unit vectors.
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(2) Cross product of vector A B in Cartesian
vector form
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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 .
(1) Magnitude MoMo r x F r F
sin?F r sin? 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.
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4.4 Principle of moments
Varignons theorem The moment of a force about
a point is equal to the sum of the moment of the
forces components about the point .
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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

Point O on axis aa
O
Axis of projection
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• 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
• MaMo cos?
• Mo ?ua( r F ) ? ua
• trip scalar product
• Here
• Mamagnitude of Ma
• ua unit vector define the direction of aa
  axis

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If
then
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
(Moua) ua
ua ( r x F )ua
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4.6 moment of a couple
1. Definition ( couple) ??? Two parallel
forces have the same magnitude, opposite
distances, and are separated by a perpendicular
distance d.
2. Scalar Formulation (1) Magnitude MFd
(2) Direction sense (Right-hand rule)
Thumb indicates the direction Curled
fingers indicates the sense of rotation
3. Vector Formulation
M r x F MMr x F r F sin? F d
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Remark
(1) The couple moment is equivalent to the sum
of the moment of both couple forces about any
arbitrary point 0 in space.
Mo rAx( -F ) rB x F (-rArB) x F r x
F M
(2) Couple moment is a free vector which can act
at any point in space.
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4. Equivalent Couples The forces of equal
couples lie either in the same plane or in
planes parallel to one another.
5. Resultant couple moment
Apply couple moment at any point p on a body and
add them vectorially.
MRSMS r x F
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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
Internal effect ?
Internal stresses are different.
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3. Point O is on the line of action of the force
A
A
A
equivalent
equivalent
o
o
o
Original system
Sliding vector
4. Point O is not on the line of action of the
force
Couple moment
F
F
Mc r x F
Mr x F
A
A
F
r
P
A
F
o
line of action
o
Original system
Force on Point A Force on point O couple
moment on any point p.
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Example
Point O is on the line of action of the force
Point O is not on the line of action of the force
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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
• FRF1F2SF
• (2)Moment summations
• MR0 SMCr1oF1r2oF2 SMC SM0
• MCCouple moment in the system
• MoCouple moment about pt.O of the
  force in the system.

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4.9 Further Reduction of a force couple system
1. Simplification to a single Resultant Force
(1)Condition FR MR0
or FRMR0 0 (2)Force system
A. Concurrent Force system
B.Coplanar Force System
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C. Parallel Force System 1. F1 // F2 ////
Fn 2. MR0 perpendicular to FR ,
MR0SM SrF
2. Reduction to a wrench (1) Condition FR
MR0 MR0M M// M
moment component FR M//
moment component // FR
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• (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
b
FR
FR
a
a
M//
a
b
M//
o
b
p
o
p
b
b
a
a