Statics Ch3

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VECTOR MECHANICS FOR ENGINEERS:

STATICS

Tenth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

David F. MazurekLecture Notes:

John ChenCalifornia Polytechnic State University

CHAPTER

2013 The McGraw-Hill Companies, Inc. All rights reserved.

3Rigid Bodies:Equivalent Systems

of Forces


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Vector Mechanics for Engineers: StaticsTenth

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


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Edition

Introduction

3 - 3

Treatment of a body as a single particle is not always possible. Ingeneral, the size of the body and the specific points of application of the

forces must be considered.

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.

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.

First, we need to learn some new statics concepts, including:

moment of a force about a point

moment of a force about an axis

moment due to a couple

T


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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.

Internal forces, such as the force

between each wheel and the axel

it is mount on, are never shown

on a free body diagram.

TE


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Edition

Principle of Transmissibility: Equivalent Forces

3 - 5

Principle of Transmissibility -

Conditions of equilibrium or motion are

not affected by transmittinga force

along its line of action.

NOTE: F and F are equivalent forces.

Moving the point of application of

the force F to the rear bumperdoes not affect the motion or the

other forces acting on the truck.

TE


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Edition

Vector Product of Two Vectors

3 - 6

Concept of the moment of a force about a point

requires the understanding of the vector productorcross product.

Vector product of two vectors Pand Qis defined

as the vectorVwhich satisfies the following

conditions:

1. Line of action ofVis perpendicular to plane

containing Pand Q.

2. Magnitude ofVis

3. Direction ofVis obtained from the right-hand

rule.

sinQPV

Vector products:

- are not commutative,

- are distributive,

- are not associative,

QPPQ

2121 QPQPQQP

SQPSQP

TE


7/45 2013 The McGraw-Hill Companies, Inc. All rights reserved.


Edition

Vector Products: Rectangular Components

3 - 7

Vector products of Cartesian unit vectors,

0

0

0

kkikjjki

ijkjjkji

jikkijii

Vector products in terms of rectangularcoordinates

V Px

i Py

j Pz

k Qx

i Qy

j Qz

k

PyQz PzQy

i PzQx PxQz

j

PxQy PyQx r

k

i j k

Px Py Pz

Qx Qy Qz

TE




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 its point of application.

The momentofFabout O is defined as

FrMO

The moment vectorMOis perpendicular to theplane containing O and the force F.

Any force F that 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 ofMO, ,

measures 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 STE




Edition

Moment of a Force About a Point

3 - 9

Two-dimensional structures have length and breadth but

negligible depth and are subjected to forces containedonly in the plane of the structure.

The plane of the structure contains the point O and the

force F. MO, the moment of the force about O is

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 tiTE




Edition

Varignons Theorem

3 - 10

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

to the sum of the moments of the various

moments about the same point O.

Varignons Theorem makes it possible to

replace the direct determination of the

moment of a force Fby the moments of two

or more component forces ofF.

2121 FrFrFFr





Edition

Rectangular Components of the Moment of a Force

3 - 11

MO Mx

i My

j Mz

k

r

ir

jr

k

x y z

Fx Fy Fz

yFz zFy ri zFx xFz

rj xFy yFx

rk

The moment ofFabout O,

MO

r

F,

rx

i y

j z

kr

FFxr

i Fyr

j Fzr

k

The components of , Mx, My, and Mz, represent the

moments about the x-, y- and z-axis, respectively.

Mo





Edition


3 - 12

The moment ofFaboutB,

MB

rA /B

F

rA /B

rA

rB

xA xB r

i yA yB r

j zA zB r

kr

FFxr

i Fyr

j Fzr

k

rMB

i j k

xA xB yA yB zA zB Fx Fy Fz

V t M h i f E i St tiTe

E




Edition


3 - 13

For two-dimensional structures,

MO xFy yFz k

MO MZ

xFy yFz

MB xA xB Fy yA yB Fz

k

MB MZ

xA xB Fy yA yB Fz


E




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 (not shown) at O.

Determine:

a) the moment about O,

b) the horizontal force atA which creates the samemoment,

c) the smallest force at A which produces the same

moment,

d) the 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.


E




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 which, by our

sign convention, would be negative or

counterclockwise.

MO Fd

d 24in. cos60 12 in.

MO 100 lb

12 in.

MO

1200 lb in, or

1200 lb in


Ed




Edition

Sample Problem 3.1

3 - 16

b) Horizontal force atA that produces the same

moment,

d 24 in. sin60 20.8 in.

MO Fd

1200 lb in. F20.8 in.

F1200 lb in.

20.8 in.lb7.57F

Why must the direction of this F be to the right?


Ed



Vector Mechanics for Engineers: Staticsenth

Edition

Sample Problem 3.1

3 - 17

c) The smallest force atA to produce the same

moment occurs when the perpendicular distance is

a maximum or whenFis perpendicular to OA.

in.42

in.lb1200

in.42in.lb1200

F

F

FdMO

lb50F

What is the smallest force at A which produces thesame moment? Think about it and discuss with a

neighbor.

F(min) ?

Vector Mechanics for Engineers StaticsTe

Ed




dition

Sample Problem 3.1

3 - 18

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

force to produce the same moment,

MO Fd

1200 lb in. 240 lb d

d 1200 lb in.240 lb

5 in.

OBcos60 5 in. in.10OB

Vector Mechanics for Engineers: StaticsTeEd




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.





dition

Sample Problem 3.4

3 - 20

The rectangular plate is supported by

the brackets atA andB and by a wire

CD. Knowing that the tension in the

wire is 200 N, determine the moment

aboutA of the force exerted by the

wire at C.

The momentMA of the forceFexertedby the wire is obtained by evaluating

the vector product,

FrM ACA

Solution

How do you find the moment? What is

the equation for this moment? Discuss

this with a neighbor and describe in

detaileach variable in your equation,

including (a) how you would write it,

(b) what its units are, and (c) whether

or not there are any alternative

variables that could take its place.





dition

Sample Problem 3.4

3 - 21

SOLUTION:

12896120

08.003.0

kji

MA

MA 7.68 N m

i 28.8 N m

j 28.8 N m

k

rC A

rC

rA 0.3 m

i 0.08 m

j

FrM ACA

r

FFr

200 N

rC D

rC D

200 N 0.3 m

r

i 0.24 m r

j 0.32 m r

k

0.5 m

120 N r

i 96 N r

j 128 N r

k




Vector Mechanics for Engineers: Staticsnthdition

Scalar Product of Two Vectors

3 - 22

Thescalar productordot productbetween

two vectors Pand Qis defined as resultscalarcosPQQP

Scalar products:

- are commutative,

- are distributive,- are not associative,

PQQP

2121 QPQPQQP

undefine SQP

Scalar products with Cartesian unit components,

000111 ikkjjikkjjii

P

Q Px

i Py

j Pz

k Qx

i Qy

j Qz

k

2222PPPPPP

QPQPQPQP

zyx

zzyyxx





Scalar Product of Two Vectors: Applications

3 - 23

Angle between two vectors:

PQ

QPQPQP

QPQPQPPQQP

zzyyxx

zzyyxx

cos

cos

Projection of a vector on a given axis:

OL

OL

PPQ

QP

PQQP

OLPPP

cos

cos

alongofprojectioncos

zzyyxx

OL

PPP

PP

coscoscos

For an axis defined by a unit vector:

Vector Mechanics for Engineers: StaticsTen

Ed




Mixed Triple Product of Three Vectors

3 - 24

Mixed triple product of three vectors,

resultscalar QPS

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

r

Sr

Q P r

Pr

Sr

Q r

Qr

Pr

S

S

P

Q Sx PyQzPzQy Sy PzQx PxQz SzPxQy PyQx

Sx Sy Sz

Px Py Pz

Qx Qy Qz

Evaluating the mixed triple product,


Ed


25/45


Vector Mechanics for Engineers: Staticsnthition


3 - 25

Moment MOof a force Fapplied at the point A

about a point O,FrMO

Scalar momentMOL about an axis OL is the

projection of the moment vectorMOonto the

axis,

FrMM OOL

Moments ofFabout the coordinate axes,

xyz

zxy

yzx

yFxFM

xFzFMzFyFM


Ed


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3 - 26

Moment of a force about an arbitrary axis,

MBL

MB

r

rrA B

r

F

r

rA B

r

rA

r

rB

The result is independent of the pointB

along the given axis. That is, the same

result can be gotten using , for

example.

rA C


Edi


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Sample Problem 3.5

3 - 27

a) aboutA

b) about the edgeAB and

c) about the diagonalAG of the cube.d) Determine the perpendicular distance

betweenAG andFC.

A cube is acted on by a force Pas

shown. Determine the moment ofP


Edi


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Sample Problem 3.5

3 - 28

Moment ofPaboutA,

MA

rF A

PrrF A a

r

i ar

j ar

i r

j r

PP 2r

i 2r

j P 2r

i r

j r

MA ar

i r

j

P 2

r

i r

j

MA aP 2

i

j

k

Moment ofPaboutAB,MAB

i

MA

r

i aP 2 r

i r

j r

k2aPMAB

List an alternative to the position vector ,

and discuss your answer with a neighbor.

rF A


Edi


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Vector Mechanics for Engineers: Staticsnthtion

Sample Problem 3.5

3 - 29

Moment ofPabout the diagonalAG,

MAG

MA

r

rrG A

rG A

ar

i ar

j ar

k

a 3

1

3

r

i r

j r

kr

MA aP

2

r

i r

j r

k

MAG

1

3

r

i r

j r

k aP

2

r

i r

j r

k

aP

6111

6

aPMAG

What if, for , you had chosen instead?

How would that change the answer?

rA G


Edi


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Sample Problem 3.5

3 - 30

Perpendicular distance betweenAG andFC,

r

Pr

P

2

r

j r

k 1

3

r

i r

j r

k P

60 11

0 since by definitionr

P r

= P 1cos

Therefore,Pis perpendicular toAG.

PdaP

MAG 6

6

ad


Edit


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Moment of a Couple

3 - 31

Two forces Fand -Fhaving the same magnitude,

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

Moment of the couple,

FdrFM

Fr

Frr

FrFrM

BA

BA

sin

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.


Edit


32/45


Vector Mechanics for Engineers: Staticsthtion

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.

Vector Mechanics for Engineers: StaticsTent

Edit


33/45



Addition of Couples

3 - 33

Consider two intersecting planesP1 and

P2 with each containing a couple

M1

r

F1 in plane P1r

M2 rr

r

F2 in plane P2

Resultants of the vectors also form a

couple

M

r

R

r

F1

F2

By Varignons theorem

M

r

F1

r

F2

r

M1

r

M2

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 vectors obey the law of addition of vectors.

Couple vectors are free vectors, i.e., there is no point ofapplicationit simply acts on the body.

Couple vectors may be resolved into component vectors.


Edit


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Resolution of a Force Into a Force at O and a Couple

3 - 35

Force vectorFcan not be simply moved to O without modifying its

action on the body.

Attaching equal and opposite force vectors at O produces 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


36/45


Vector Mechanics for Engineers: Staticsthion

Resolution of a Force Into a Force at O and a Couple

3 - 36

Moving FfromA to a different point O requires the

addition of a different couple vectorMO

FrMO

'

The moments ofFabout O and Oare related,

FsM

FsFrFsrFrM

O

O

''

Moving the force-couple system from O to Orequires the

addition of the moment of the force at O about O.


Edit


37/45



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 inthex direction 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 pointD is a

good choice as only two of the forces will

produce non-zero moment contributions..


Editi


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Sample Problem 3.6

3 - 38

The three couples may be represented by

three couple vectors,

Mx 30 lb 18 in. 540 lb in.

My 20 lb 12 in. 240lb in.

Mz 20 lb 9 in. 180 lb in.

M 540 lb in.

i 240lb in.

j

180 lb in. r

k

Attach equal and opposite 20 lb forces in

thex direction atA


Editi


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Vector Mechanics for Engineers: Staticshion

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.

M

MD 18 in.

j 30 lb

k

9 in. rj 12 in. rk 20 lb ri

M 540 lb in.

i 240lb in.

j

180 lb in.

r

k


Editi


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Vector Mechanics for Engineers: Staticshon

System 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 at a given point O

The force and couple vectors may be combined into a

resultant force vector and a resultant couple vector,

R

F

MOR

rr

F

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

MO'R

MOR

s

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 O are 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


42/45


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.

ROMR

and

System can be reduced to a single force

by moving the line of action of until

its moment about O becomesR

OM

R

In terms of rectangular coordinates,R

Oxy MyRxR


Editio


43/45


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 atB based on the force-

couple system atA.

c) Determine the point of application

for the resultant force such that itsmoment aboutA is equal to the

resultant couple atA.


Editio


44/45



3 - 44

SOLUTION:

a) Compute the resultant force and theresultant couple atA.

R

F

150 N r

j 600 N r

j 100 N r

j 250 N r

j

R 600 N

j

MAR

rr

F 1.6

r

i 600r

j 2.8r

i 100r

j

4.8r

i 250r

j

MAR 1880 N m

k


Editio


45/45


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 fromA toB.

jR

N600

The couple atB is equal to the moment aboutB

of the force-couple system found atA.

MBR

MAR

rB A

R

1880 N m

r

k 4.8 m r

i 600 N r

j

1880 N m r

k 2880 N m r

k

MBR 1000 N m

k

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