Ch03-Rigid Bodi (Equivalent Systems of Forces)

8/14/2019 Ch03-Rigid Bodi (Equivalent Systems of Forces)

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VECTOR MECHANICS FOR ENGINEERS:

STATICS

Ninth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Lecture Notes:

J. Walt Oler

Texas Tech University

CHAPTER

2010 The McGraw-Hill Companies, Inc. All rights reserved.

3Rigid Bodies:Equivalent Systems

of Forces


2/48


Vector Mechanics for Engineers: StaticsNinth

Edition

Contents

3 - 2

Introduction

External and Internal Forces

Principle of Transmissibility: Equivalent

Forces

Vector Products of Two Vectors

Moment of a Force About a PointVarigons Theorem

Rectangular Components of the

Moment of a Force

Sample Problem 3.1

Scalar Product of Two Vectors

Scalar Product of Two Vectors:

Applications

Mixed Triple Product of Three Vectors

Moment of a Force About a Given Axis

Sample Problem 3.5

Moment of a Couple

Addition of Couples

Couples Can Be Represented By Vectors

Resolution of a Force Into a Force at Oand a Couple

Sample Problem 3.6

System of Forces: Reduction to a Force

and a Couple

Further Reduction of a System of Forces

Sample Problem 3.8

Sample Problem 3.10


3/48



Edition

Introduction

3 - 3

Treatment of a body as a single particle is not always possible. In

general, the size of the body and the specific points of application of theforces must be considered.

Most bodies in elementary mechanics are assumed to be rigid, i.e., the

actual deformations are small and do not affect the conditions of

equilibrium or motion of the body.

Current chapter describes the effect of forces exerted on a rigid body and

how to replace a given system of forces with a simpler equivalent system.

moment of a force about a point

moment of a force about an axis

moment due to a couple

Any system of forces acting on a rigid body can be replaced by an

equivalent system consisting of one force acting at a given point and one

couple.


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Edition

External and Internal Forces

3 - 4

Forces acting on rigid bodies are

divided into two groups:- External forces

- Internal forces

External forces are shown in a

free-body diagram.

If unopposed, each external force

can impart a motion of

translation or rotation, or both.

NE


5/48



Edition

Principle of Transmissibility: Equivalent Forces

3 - 5

Principle of Transmissibility-

Conditions of equilibrium or motion arenot affected by transmittinga force

along its line of action.

NOTE: Fand F are equivalent forces.

Moving the point of application of

the force Fto the rear bumper

does not affect the motion or the

other forces acting on the truck.

Principle of transmissibility may

not always apply in determining

internal forces and deformations.

NE


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Edition

Vector Product of Two Vectors

3 - 6

Concept of the moment of a force about a point is

more easily understood through applications ofthe vector product orcross product.

Vector product of two vectors Pand Qis defined

as the vector Vwhich satisfies the following

conditions:

1. Line of action of Vis perpendicular to plane

containing Pand Q.

2. Magnitude of Vis

3. Direction of Vis obtained from the right-hand

rule.

sinQPV

Vector products:

- are not commutative,

- are distributive,

- are not associative,

QPPQ

2121 QPQPQQP

SQPSQP

NE


7/48 2010 The McGraw-Hill Companies, Inc. All rights reserved.


Edition

Vector Products: Rectangular Components

3 - 7

Vector products of Cartesian unit vectors,

0

0

0

i i j i k k i j

i j k j j k j i

i k j j k i k k

r rr r r r r r

r r r r r r rv

Vector products in terms of rectangularcoordinates

x y z x y zV P i P j P k Q i Q j Q k r r r r r

y z z y z x x z

x y y x

P Q P Q i P Q P Q j

P Q P Q k

r

x y z

x y z

i j k

P P P

Q Q Q

r r

NE




Edition

Moment of a Force About a Point

3 - 8

A force vector is defined by its magnitude and

direction. Its effect on the rigid body also dependson it point of application.

The momentof Fabout Ois defined as

FrMO

The moment vector MOis perpendicular to theplane containing Oand the force F.

Any force Fthat has the same magnitude and

direction as F, is equivalentif it also has the same line

of action and therefore, produces the same moment.

Magnitude of MOmeasures the tendency of the force

to cause rotation of the body about an axis along MO.

The sense of the moment may be determined by the

right-hand rule.

FdrFMO sin

f SNE




Edition

Moment of a Force About a Point

3 - 9

Two-dimensional structureshave length and breadth but

negligible depth and are subjected to forces contained inthe plane of the structure.

The plane of the structure contains the point Oand the

force F. MO, the moment of the force about Ois

perpendicular to the plane.

If the force tends to rotate the structure clockwise, the

sense of the moment vector is out of the plane of the

structure and the magnitude of the moment is positive.

If the force tends to rotate the structure counterclockwise,

the sense of the moment vector is into the plane of the

structure and the magnitude of the moment is negative.

V t M h i f E i St tiNE




Edition

Varignons Theorem

3 - 10

The moment about a give point Oof theresultant of several concurrent forces is equal

to the sum of the moments of the various

moments about the same point O.

Varigons Theorem makes it possible to

replace the direct determination of the

moment of a force Fby the moments of two

or more component forces of F.

1 2 1 2r F F r F r F

r r r

L L





Edition

Rectangular Components of the Moment of a Force

3 - 11

O x y z

x y z

z y x z y x

M M i M j M k

i j k

x y z

F F F

yF zF i zF xF j xF yF k

rr r

rr r

The moment of Fabout O,

,O

x y z

M r F r xi yj zk

F F i F j F k

r r

rr r r





Edition


3 - 12

The moment of FaboutB,

/B A BM r F

r

/A B A B

A B A B A B

x y z

r r r

x x i y y j z z k

F F i F j F k

rr r

rr r r

B A B A B A Bx y z

i j k

M x x y y z zF F F

r r

r





Edition


3 - 13

For two-dimensional structures,

O y zO Z

y z

M xF yF k

M M

xF yF

O A B y A B z

O Z

A B y A B z

M x x F y y F k

M M

x x F y y F





Edition

Sample Problem 3.1

3 - 14

A 100-lb vertical force is applied to the end of a

lever which is attached to a shaft at O.

Determine:

a) moment about O,

b) horizontal force atA which creates the samemoment,

c) smallest force at A which produces the same

moment,

d) location for a 240-lb vertical force to produce

the same moment,

e) whether any of the forces from b, c, and d is

equivalent to the original force.





Edition

Sample Problem 3.1

3 - 15

a) Moment about O is equal to the product of the

force and the perpendicular distance between theline of action of the force and O. Since the force

tends to rotate the lever clockwise, the moment

vector is into the plane of the paper.

in.12lb100

in.1260cosin.24

O

O

M

dFdM

inlb1200 OM

V t M h i f E i St tiN

Ed




Edition

Sample Problem 3.1

3 - 16

c) Horizontal force atAthat produces the same

moment,

in.8.20

in.lb1200

in.8.20in.lb1200

in.8.2060sinin.24

F

F

FdM

d

O

lb7.57F

V t M h i f E i St tiNi

Ed




Edition

Sample Problem 3.1

3 - 17

c) The smallest forceAto produce the same moment

occurs when the perpendicular distance is amaximum or whenFis perpendicular to OA.

in.42

in.lb1200

in.42in.lb1200

F

F

FdMO

lb50F

Vector Mechanics for Engineers StaticsNi

Ed



Vector Mechanics for Engineers: Staticsinth

dition

Sample Problem 3.1

3 - 18

d) To determine the point of application of a 240 lb

force to produce the same moment,

in.5cos60

in.5lb402

in.lb1200

lb240in.lb1200

OB

d

d

FdMO

in.10OB

Vector Mechanics for Engineers: StaticsNi

Ed



Vector Mechanics for Engineers: Staticsinth

dition

Sample Problem 3.1

3 - 19

e) Although each of the forces in parts b), c), and d)produces the same moment as the 100 lb force, none

are of the same magnitude and sense, or on the same

line of action. None of the forces is equivalent to the

100 lb force.


Ed



Vector Mechanics for Engineers: Staticsnth

dition

Sample Problem 3.4

3 - 20

The rectangular plate is supported by

the brackets atAandBand by a wire

CD. Knowing that the tension in the

wire is 200 N, determine the moment

aboutAof the force exerted by the

wire at C.

SOLUTION:

The momentMAof the forceFexerted

by the wire is obtained by evaluating

the vector product,

A C A

M r F r


Ed




dition

Sample Problem 3.4

3 - 21

SOLUTION:

0.3 0 0.08

120 96 128

A

i j k

M

r r

r

7.68 N m 28.8 N m 28.8 N mAM i j k v

0.3 m 0.08 mC A C Ar r r i j r r r

A C AM r F

r

200 N

0.3 m 0.24 m 0.32 m200 N

0.5 m

120 N 96 N 128 N

C D

C D

rF F

r

i j k

i j k

rr

rr r

rr r

Vector Mechanics for Engineers: StaticsNin

Ed




dition

Scalar Product of Two Vectors

3 - 22

Thescalar productor dot productbetween

two vectorsP

andQ

is defined as cos scalar resultP Q PQ

Scalar products:

- are commutative,

- are distributive,- are not associative,

P Q Q P

1 2 1 2P Q Q P Q P Q

r r r

undefinedP Q S r rr

Scalar products with Cartesian unit components,

1 1 1 0 0 0i i j j k k i j j k k i v

x y z x y zP Q P i P j P k Q i Q j Q k r r r r r

2 2 2 2

x x y y z z

x y z

P Q P Q P Q P Q

P P P P P P

r r


Ed




dition

Scalar Product of Two Vectors: Applications

3 - 23

Angle between two vectors:

cos

cos

x x y y z z

x x y y z z

P Q PQ P Q P Q P Q

P Q P Q P Q

PQ

Projection of a vector on a given axis:

cos projection of along

cos

cos

OL

OL

P P P OL

P Q PQ

P QP P

Q

rr

rr

cos cos cos

OL

x x y y z z

P P

P P P

For an axis defined by a unit vector:


Ed




dition

Mixed Triple Product of Three Vectors

3 - 24

Mixed triple product of three vectors,

scalar resultS P Q r rr

The six mixed triple products formed from S, P, and

Qhave equal magnitudes but not the same sign,

S P Q P Q S Q S P

S Q P P S Q Q P S

r r r r r r r r

x y z z y y z x x z

z x y y x

x y z

x y z

x y z

S P Q S P Q PQ S PQ P Q

S P Q P Q

S S S

P P P

Q Q Q

Evaluating the mixed triple product,


Ed


25/48



ition


3 - 25

Moment MOof a force Fapplied at the point A

about a point O,

OM r F

r

Scalar momentMOLabout an axis OL is the

projection of the moment vector MOonto the

axis,

OL OM M r F r

Moments of Fabout the coordinate axes,

xyz

zxy

yzx

yFxFM

xFzFMzFyFM


Ed


26/48



ition


3 - 26

Moment of a force about an arbitrary axis,

BL B

A B

A B A B

M M

r F

r r r

r rr

r r r

The result is independent of the pointB

along the given axis.


Edi


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ition

Sample Problem 3.5

3 - 27

a) aboutA

b) about the edgeAB and

c) about the diagonalAGof the cube.d) Determine the perpendicular distance

betweenAGandFC.

A cube is acted on by a force Pas

shown. Determine the moment of P


Edi


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ition

Sample Problem 3.5

3 - 28

Moment of PaboutA,

2 2 2

2

A F A

F A

A

M r P

r ai a j a i j

P P i j P i j

M a i j P i j

r

r r r r r

r r r r r

r r r r r

2AM aP i j k r r r

Moment of PaboutAB,

2AB A

M i M

i aP i j k

rr r r

2aPMAB


Edi


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tion

Sample Problem 3.5

3 - 29

Moment of Pabout the diagonalAG,

1

3 3

2

1

3 2

1 1 1

6

AG A

A G

A G

A

AG

M M

r ai aj ak i j k

r a

aP

M i j k

aPM i j k i j k

aP

rr rrrr r r

rr r r

r rr r r r

6

aPMAG


Edi


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Vector Mechanics for Engineers: Staticsthtion

Sample Problem 3.5

3 - 30

Perpendicular distance betweenAGandFC,

1 0 1 12 3 6

0

P PP j k i j k

r rrr r r r

Therefore,Pis perpendicular toAG.

PdaP

MAG 6

6

ad


Edit


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Moment of a Couple

3 - 31

Two forces Fand -F having the same magnitude,

parallel lines of action, and opposite sense are saidto form a couple.

Moment of the couple,

sin

A B

A B

M r F r F

r r F

r F

M rF Fd

r r rr r

rr r

rr

The moment vector of the couple isindependent of the choice of the origin of the

coordinate axes, i.e., it is afree vectorthat can

be applied at any point with the same effect.

Vector Mechanics for Engineers: StaticsNint

Edit


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Moment of a Couple

3 - 32

Two couples will have equal moments if

2211 dFdF

the two couples lie in parallel planes, and

the two couples have the same sense orthe tendency to cause rotation in the same

direction.


Edit


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Addition of Couples

3 - 33

Consider two intersecting planesP1and

P2 with each containing a couple

1 1 1

2 2 2

in plane

in plane

M r F P

M r F P

r

r rr

Resultants of the vectors also form a

couple

1 2M r R r F F r r

By Varigons theorem

1 2

1 2

M r F r F

M M

r r

r r

Sum of two couples is also a couple that is equal

to the vector sum of the two couples


Edit


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Couples Can Be Represented by Vectors

3 - 34

A couple can be represented by a vector with magnitudeand direction equal to the moment of the couple.

Couple vectorsobey the law of addition of vectors.

Couple vectors are free vectors, i.e., the point of applicationis not significant.

Couple vectors may be resolved into component vectors.


Edit


35/48




3 - 35

Force vector Fcan not be simply moved to Owithout modifying its

action on the body.

Attaching equal and opposite force vectors at Oproduces no net

effect on the body.

The three forces may be replaced by an equivalent force vector and

couple vector, i.e, aforce-couple system.


Edit


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Vector Mechanics for Engineers: Staticsthion


3 - 36

Moving FfromAto a different point Orequires the

addition of a different couple vector MO

'OM r F

r

The moments of Fabout O and Oare related,

' 'O

O

M r F r s F r F s F

M s F

r r r r r

r rr

Moving the force-couple system from Oto Orequires the

addition of the moment of the force at Oabout O.


Edit


37/48


Vector Mechanics for Engineers: Staticshion

Sample Problem 3.6

3 - 37

Determine the components of thesingle couple equivalent to the

couples shown.

SOLUTION:

Attach equal and opposite 20 lb forces inthe +xdirection atA, thereby producing 3

couples for which the moment components

are easily computed.

Alternatively, compute the sum of the

moments of the four forces about an

arbitrary single point. The pointDis a

good choice as only two of the forces will

produce non-zero moment contributions..


Editi


38/48



Sample Problem 3.6

3 - 38

Attach equal and opposite 20 lb forces in

the +xdirection atA

The three couples may be represented by

three couple vectors,

in.lb180in.9lb20

in.lb240in.12lb20

in.lb540in.18lb30

z

y

x

M

M

M

540 lb in. 240lb in.

180 lb in.

M i j

k

r


Editi


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Sample Problem 3.6

3 - 39

Alternatively, compute the sum of the

moments of the four forces aboutD.

Only the forces at CandEcontribute to

the moment aboutD.

18 in. 30 lb

9 in. 12 in. 20 lb

DM M j k

j k i

rr r

540 lb in. 240lb in.

180 lb in.

M i j

k

r


Editi


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Vector Mechanics for Engineers: StaticshonSystem of Forces: Reduction to a Force and Couple

3 - 40

A system of forces may be replaced by a collection of

force-couple systems acting a given point O

The force and couple vectors may be combined into a

resultant force vector and a resultant couple vector,

ROR F M r F r

The force-couple system at Omay be moved to Owith the addition of the moment of Rabout O,

'

R R

O OM M s R

r

Two systems of forces are equivalent if they can be

reduced to the same force-couple system.


Editi


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Vector Mechanics for Engineers: StaticshonFurther Reduction of a System of Forces

3 - 41

If the resultant force and couple at Oare mutually

perpendicular, they can be replaced by a single force actingalong a new line of action.

The resultant force-couple system for a system of forces

will be mutually perpendicular if:

1) the forces are concurrent,

2) the forces are coplanar, or3) the forces are parallel.


Editio


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Vector Mechanics for Engineers: StaticshonFurther Reduction of a System of Forces

3 - 42

System of coplanar forces is reduced to a

force-couple system that ismutually perpendicular.

and ROR M

System can be reduced to a single force

by moving the line of action of until

its moment about Obecomes ROM

R

In terms of rectangular coordinates,R

Oxy MyRxR


Editio


43/48


Vector Mechanics for Engineers: StaticshonSample Problem 3.8

3 - 43

For the beam, reduce the system of

forces shown to (a) an equivalent

force-couple system atA, (b) an

equivalent force couple system atB,

and (c) a single force or resultant.

Note: Since the support reactions are

not included, the given system will

not maintain the beam in equilibrium.

SOLUTION:

a) Compute the resultant force for the

forces shown and the resultant

couple for the moments of the

forces aboutA.

b) Find an equivalent force-couple

system atBbased on the force-

couple system atA.

c) Determine the point of application

for the resultant force such that itsmoment aboutAis equal to the

resultant couple atA.


Editio


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3 - 44

SOLUTION:

a) Compute the resultant force and theresultant couple atA.

150 N 600 N 100 N 250 N

R F

j j j j

r r r r

600 NR j

1.6 600 2.8 100

4.8 250

R

AM r F

i j i j

i j

r

r r r r

r r

1880 N mRAM k


Editio


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3 - 45

b) Find an equivalent force-couple system atB

based on the force-couple system atA.The force is unchanged by the movement of the

force-couple system fromAtoB.

600 NR j

The couple atBis equal to the moment aboutB

of the force-couple system found atA.

1880 N m 4.8 m 600 N

1880 N m 2880 N m

R R

B A B AM M r R

k i j

k k

r

r r r

r r

1000 N mRBM k


Editio


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3 - 46

Three cables are attached to thebracket as shown. Replace the

forces with an equivalent force-

couple system atA.

SOLUTION:

Determine the relative position vectors

for the points of application of the

cable forces with respect toA.

Resolve the forces into rectangular

components.

Compute the equivalent force,

R F

Compute the equivalent couple,

RAM r F r


Editio


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Vector Mechanics for Engineers: StaticsonSample Problem 3.10

3 - 47

SOLUTION:

Determine the relative position

vectors with respect toA.

0.075 0.050 m

0.075 0.050 m

0.100 0.100 m

B A

C A

D A

r i k

r i k

r i j

r

rrr

r rr

Resolve the forces into rectangular

components.

700 N

75 150 50

175

0.429 0.857 0.289

300 600 200 N

B

E B

E B

B

F

r i j k

r

i j k

F i j k

rr rrr

rr r

rr r r

1200 N cos60 cos30

600 1039 N

DF i j

i j

r r

1000 N cos 45 cos 45

707 707 N

CF i j

i j

r r


Editio


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ecto ec a cs o g ee s Stat csonSample Problem 3.10 Compute the equivalent force,

300 707 600

600 1039

200 707

R F

i

j

k

r

r

r

1607 439 507 NR i j k

Compute the equivalent couple,

0.075 0 0.050 30 45

300 600 200

0.075 0 0.050 17.68

707 0 707

0.100 0.100 0 163.9

600 1039 0

RA

B A B

C A c

D A D

M r F

i j k

r F i k

i j k

r F j

i j k

r F k

v r

rr r

rr rr

rr r

r rr

rr r

rrr

30 17.68 118.9R

AM i j k

Ch03-Rigid Bodi (Equivalent Systems of Forces)

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