Chapter 6 Equilibrium of Rigid Bodies - Drexel Universitycac542/L11.pdf · Equilibrium of Rigid...

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MEM202 Engineering Mechanics - Statics MEM

Chapter 6Equilibrium of Rigid Bodies

0

0

=++=

=++=

∑∑∑∑∑∑

kMjMiMC

kFjFiFR

zyx

zyxrrrr

rrrr

000000======

∑∑∑∑∑∑

zyx

zyx

MMMFFF

2


Equilibrium of Rigid Bodies

m 6m 51

N 300 N 450

m 42

m 4

N/m 100=w

A B

Roller Support

m 6m 51

N 300 N 450

m42

m4

N/m 100=w

A BFree endPin

SupportFixed

Support

Simply-Supported Cantilever

m 4.31=cx

N ,2502=R

N 909=yA N 341,1=yB

N 0=xA

m 4.31=cx

N ,2502=R

N 250,2=yA

N 0=xA

m-N 175,32=AM

Free-Body DiagramSupports and Connections

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6.2 Free-Body DiagramsIdealized Supports and Connections in 2-D

(Table 6-1, pp. 252-254)

Transmit a tensile force in the direction of the cable.

Transmit a force (tensile or compressive) in the direction of the axis of the link.

Transmit a compressive force perpendicular to the surface at the point of contact.

Transmit a compressive force perpendicular to the surface supporting the ball.

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(Table 6-1, pp. 252-254)

Transmit a force, usually expressed in terms of its components: a compressive normal force Rnand a tangential frictional force Rt. The angle θis related to the coefficient of frictional, µ, i.e., µ = tanθ = Rr/Rn

Transmit a force, usually expressed in terms of its components Rxand Ry.

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(Table 6-1, pp. 252-254)

Transmit a force (tensile or compressive) perpendicular to the surface of the guide.

Transmit a force (tensile or compressive) perpendicular to the axis of the shaft.

A smooth collar with a fixed connection.Transmit a force, usually expressed in terms of its components Rxand Ry, and a moment.

Transmit a force (tensile or compressive) perpendicular to the axis of the shaft, and a moment.

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(Table 6-1, pp. 252-254)

Transmit a force (tensile or compressive) along the axis of the spring.

Transmit and change the direction of a tensile force.For a frictionless pulley, T1 = T2.

More about pulleys

xR

yR

( )θpT

T

( )θpxR

X

Y

yR

MxR

yR

T

T

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Flexible cable

Rigid link

Ball, roller, or rocker

Fixed support

Smooth

pin

or hin

ge

Linear

elastic

spring

αLinkF

RollerFPinF

θ

FixY

FixXFixM

CableF

SpringF

LinkX

LinkYθtan=

Link

Link

XY

PinX

PinY

(3)

(1)(1)

(2)

(1)

(1)

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6.3 Equilibrium in Two Dimensions

( ) 0or 000 :2 ====− ∑∑∑∑ MMFFD zyx

000000

:3======

−∑∑∑∑∑∑

zyx

zyx

MMMFFF

D

Three equations, thus, can only solve for three unknownsAlternatively

000or

000or

000

===

===

===

∑∑∑∑∑∑∑∑∑

CBA

BAy

BAx

MMM

MMF

MMF

where A, B, and C are three points on the x-yplane which cannot be chosen arbitrarily

9


6.3 Equilibrium in Two Dimensions

022

0

0

41

4321

31

=−=

=−+−=

=+−=

∑∑∑

bFhFM

bFhFbFhFM

FFF

A

O

x

1Fr

2Fr

3Fr

4Fr

h2

b2

x

y

A B

CD

O

0

0

0

4321

42

31

=−+−=

=−=

=+−=

∑∑∑

bFhFbFhFM

FFF

FFF

O

y

x

Example

022

022

0

41

32

31

=−=

=+−=

=+−=

∑∑∑

bFhFM

hFbFM

FFF

A

C

x

( )⎪⎪⎩

⎪⎪⎨

⎧

====

==

hbFFFFhbFFFF

FFFF

2341

4321

42

31

or

:Solution

022

022

0

43

21

4321

=−=

=−=

=−+−=

∑∑∑

bFhFM

bFhFM

bFhFbFhFM

D

B

O

Choice Bad022

0220

21

32

31

⇐⎪⎭

⎪⎬

⎫

=−==+−=

=+−=

∑∑∑

bFhFMhFbFM

FFF

B

C

x

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6.3 Equilibrium in Two DimensionsThe Two-Force Body (Two-Force Members)

////0 BABA FFFF −=⇒==

⊥⊥

α

//BF

//AF

• The two forces acting on a two-force member must be equal, opposite, and collinear.• The weight of the member must be neglected.• The connections to the member must not support a moment about the connection (i.e., it

must be a pin connection).

BF

AF

BF

AF

BF

AF

00 =⇒=⋅=⊥⊥∑ BBA FLFM

00 =⇒=⋅=⊥⊥∑ AAB FLFM

α

AF

BF

//BF⊥BF

//AF⊥AF L

Examples of two-force members:

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6.3 Equilibrium in Two DimensionsThe Three-Force Body (Three-Force Members)

(or, Multiple-Force Members)

Concurrent force systems Parallel force systems

AF

BF

CF

BF

CF

AF

aL

0

0

=⇒

=++=

∑∑

MConcurrent

FFFF CBAr

rrrr

0

0

=⋅−⋅=

=−+=

∑∑

aFLFM

FFFF

cBA

CBA

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6.3 Equilibrium in Two DimensionsStatically Determinate, Statically Indeterminate, Partial Constraints

nu: number of unknowns, ne: number of equations

In general:nu = ne: Statically determinatenu > ne: Statically indeterminatenu < ne: Partial Constraints

In 2-D, ne = 3 (i.e., ΣFx = 0, ΣFy = 0, ΣM = 0), thereforenu = 3: Statically determinatenu > 3: Statically indeterminatenu < 3: Partial Constraints

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Statically Determinate Statically Indeterminate Partial Constraints

nu = 3, ne = 3 nu = 4, ne = 3 nu = 2, ne = 3

nu = 3, ne = 3 nu = 4, ne = 3 nu = 2, ne = 3

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Some Special CasesRigid link

Rigid link

Pin

nu = 3, ne = 3Yet, it is unstable!

(ΣM ≠ 0)

nu = 3, ne = 3Yet, it is unstable!

(No horizontal support)

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6.3 Equilibrium in Two DimensionsExample 6-11

Pin( ) o87.3643tan 1 == −φ

Tension in cable

( ) ( ) lb 75 03375 =⇒=−=∑ TTMC

(Tension in a cable remains constant)

0sin7575

0cos75

=−−=

=+=

∑∑

φ

φ

yy

xx

CF

CF

↑==

←=−=

lb 120lb 120

lb 60lb 60

y

x

C

CNOTE: As a rule of thumb, you should always assume unknown forces in positive directions.

⇒

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Pin Roller

Equal and opposite to forces acting on pulley

( ) ( )

0120

060

0186010

=−+=

=+=

=−=

∑∑∑

BAF

AF

BM

yy

xx

A

↑==

↑==

←=−=

lb 108lb 108

lb 12lb 12

lb 60lb 60

B

A

A

y

x

↑=

←==

lb 120

lb 60lb 75

y

x

C

CT

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N 4.8582 == TRA

( )( )N 8.171681.9175

=== mgW

N 4.8582

02

==

=−=∑WT

WTF

C

Cy

C

N 4.858=CT

B

T T

RA

A

N 2.4292

02

==

=−=∑C

Cy

TTTTF

08.17162.4294.8582.4294.858um EquilibriOverall :NOTE

=−−++=−−++ WTTTR CA

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Overall F.B.D.

xA

yA

xB

yB

4 unknowns, 3 equations!

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A

( ) ( ) ( )

( ) ( ) ( )[ ]01.060044.04006.0 or,

04007.33sin

06007.33cos

01.060016.04007.33sin6.0

=+−=

=+−−=

=−+=

=++−=

∑∑∑∑

yA

yy

xx

B

BM

BAF

BAF

AM

o

o

o

( ) o7.33300200tan 1 == −θ

A

AAC is a two-force member!

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6.3 Equilibrium in Two DimensionsProblem 6-54

T

W

R

75 kg40 kg

0=−+=∑ WRTFy

If the boy is standing on a scale, what would the reading on the scale be?

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6.3 Equilibrium in Two DimensionsProblem 6-55, 6-56

2P

23P

WP =29

Chapter 6 Equilibrium of Rigid Bodies - Drexel Universitycac542/L11.pdf · Equilibrium of Rigid...

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