6-Moments Couples and Force Couple Systems_Partb.ppt

8/10/2019 6-Moments Couples and Force Couple Systems_Partb.ppt

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4.6 Moment due to Force

Couples

A Couple is defined as two Forces having the

same magnitude, parallel lines of action, and

opposite sense separated by a perpendicular

distance.

In this situation, the sum of the forces in each

direction is zero, so a couple does not affect

the sum of forces equations A force couple will however tend to rotate the

body it is acting on


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Moment Due to a Force Couple

By multiplying the magnitude of one Force by the

distance between the Forces in the Couple, the

moment due to the couple can be calculated.

M = Fdc The couple will create a moment around an axis

perpendicular to the plane that the couple falls in.

Pay attention to the sense of the Moment (Right

Hand Rule)


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Moment of a Couple

Two couples will have equal moments if

2211

dFdF

the two couples lie in parallel planes, and

the two couples have the same sense or

the tendency to cause rotation in the same

direction.

Do

example


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Why do we use Force Couples?

The reason we use Force Couples to analyze

Moments is that the location of the axis the

Moment is calculated about does not matter

The Moment of a Couple is constant over the

entire body it is acting on

Equivalent couplestwo couples that

produce the same magnitude and direction(example).


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Couples are Free Vectors

The point of action of a Couple does notmatter

The plane that the Couple is acting in does

not matter All that matters is the orientation of the plane

the Couple is acting in

Therefore, a Force Couple is said to be aFree Vector and can be applied at any pointon the body it is acting


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Example: Force Couple


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4.7 Simplification of a Force and

Couple System (Resolution of

Vectors)

The Moment due to the Force Couple is

normally placed at the Cartesian Coordinate

Origin and resolved into its x, y, and z

components (Mx, My, and Mz).


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Vector Addition of Couples

By applying Varignons Theorem to the Forces in the

Couple, it can be proven that couples can be added

and resolved as Vectors.


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Force Couple System

Two opposing force can be added to a rigid bodywithout affecting the equilibrium of it.

If there is a force acting at a distance from an axis,

two forces of equal magnitude and oppositedirection can be added at the axis with out affectingthe equilibrium of the rigid body.

The original force and its opposing force at the axismake a couple that equates to a moment on therigid body.

The other force at the axis results in the same forceacting on the body


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Force Couple Systems

As a result of this it can be stated that any

force (F) acting on a rigid body may be

moved to any given point on the rigid body as

long as a moment equal to moment of (F)

about the axis is added to the rigid body.


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Resolution of a System of

Forces in 3D

Each Force can be Resolved into a Force

and Moment at the point of interest using the

method just discussed.

The Resultant Force can then be found by

Vector Addition.

The Resultant Moment must also be found

using Vector Addition.


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SIMPLIFICATION OF FORCE AND COUPLE SYSTEMS

Objectives:

:

a) Determine the effect of moving a

force.

b) Find an equivalent force-couplesystem for a system of forces and

couples.


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READING QUIZ

1. A general systemof forces and couple moments acting on a

rigid body can be reduced to a ___ .

A) single force

B) single moment

C) single force and two moments

D) single force and a single moment

2. The original force and couple system and an equivalent

force-couple system have the same _____ effect on a body.

A) internal B) external

C) internal and external D) microscopic


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APPLICATIONS

What are the resultant effects on the persons handwhen the force is applied in these four different ways?

Why is understanding these difference important when

designing various load-bearing structures?


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APPLICATIONS (continued)

Several forces and a couple momentare acting on this vertical section of

an I-beam.

For the process of designing the I-

beam, it would be very helpful if

you could replace the various forces

and moment just one force and onecouple moment at point O with the

same external effect? How will

you do that?

| | ??


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SIMPLIFICATION OF FORCE AND COUPLE SYSTEM

(Section 4.7)

When a number of forces and couplemoments are acting on a body, it is

easier to understand their overall effect

on the body if they are combined into a

single force and couple moment having

the same external effect.

The two force and couple systems are

called equivalent systemssince they

have the same externaleffect on thebody.


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MOVING A FORCE ON ITS LINE OF ACTION

Moving a force from A to B, when both points are on the

vectors line of action, does not change the external effect.

Hence, a force vector is called a sliding vector. (But the

internal effect of the force on the body does depend on where

the force is applied).


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MOVING A FORCE OFF OF ITS LINE OF ACTION

When a force is moved, but not along its line of action, there is

a change in its external effect!Essentially, moving a force from point A to B (as shown above)

requires creating an additional couple moment. So moving a

force means you have to add a new couple.

Since this new couple moment is a free vector, it can beapplied at any point on the body.

B


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SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM

When several forces and couple moments

act on a body, you can move each forceand its associated couple moment to a

common point O.

Now you can add all the forces and

couple moments together and find oneresultant force-couple moment pair.


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If the force system lies in the x-y plane (a 2-D case), then the

reduced equivalent system can be obtained using the following

three scalar equations.

SIMPLIFICATION OF A FORCE AND COUPLE SYSTEM

(continued)

WR = W1 + W2(MR)o= W1 d1+ W2 d2


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EXAMPLE #1

1) Sum all the x and y components of the forces to find FRA.

2) Find and sum all the moments resulting from moving each

force component to A.

3) Shift FRAto a distance d such that d = MRA/FRy

Given: A 2-D force system

with geometry as shown.

Find: The equivalent resultant

force and couple

moment acting at A and

then the equivalentsingle force location

measured from A.

Plan:


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EXAMPLE #1

(continued)

+ FRx

= 150 (3/5) + 50100 (4/5)

= 60 lb

+ FRy = 150 (4/5) + 100 (3/5)

= 180 lb

+ MRA = 100 (4/5) 1100 (3/5) 6

150(4/5) 3 =640 lbft

FR= ( 602+ 1802)1/2 = 190 lb

= tan-1( 180/60) = 71.6

The equivalent single force FRcan be located at a distance d

measured from A.

d = MRA/FRy = 640 / 180 = 3.56 ft.

FR


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CONCEPT QUIZ

2. Consider two couplesacting on a body. The simplest possible

equivalent system at any arbitrary point on the body will have

A) One force and one couple moment.

B) One force.

C) One couple moment.

D) Two couple moments.

1. The forces on the pole can be reduced to

a single force and a single moment at

point ____ .

A) P B) Q C) R

D) S E) Any of these points.

R

Z

S

Q

P

X

Y


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GROUP PROBLEM SOLVING

1) Sum all the x and y components of the two forces to find FRA.

2) Find and sum all the moments resulting from moving each

force to A and add them to the 45 kN m free moment to find

the resultant MRA.

Given:A 2-D force and couple

system as shown.

Find: The equivalent resultant

force and couple

moment acting at A.

Plan:


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GROUP PROBLEM SOLVING (continued)

+ Fx = (5/13) 2630 sin 30

=5 kN

+ Fy =(12/13) 2630 cos 30

=49.98 kN

+ MRA = {30 sin 30(0.3m)30 cos 30(2m)(5/13) 26 (0.3m)

(12/13) 26 (6m)45 } = 239 kN m

Now find the magnitude and direction of the resultant.

FRA= ( 52+ 49.98 2)1/2= 50.2 kN and = tan-1(49.98/5)

= 84.3

Summing the force components:


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ATTENTION QUIZ

1. For this force system, the equivalent system at P is

___________ .

A) FRP= 40 lb (along +x-dir.) and MRP= +60 ft lb

B) FRP= 0 lb and MRP= +30 ft lb

C) FRP= 30 lb (along +y-dir.) and MRP= -30 ft lb

D) FRP= 40 lb (along +x-dir.) and MRP= +30 ft lb

P

1' 1'

30 lb40 lb

30 lb

x

y


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ATTENTION QUIZ

2. Consider three couples acting on a body. Equivalent

systems will be _______ at different points on the body.

A) Different when located

B) The same even when located

C) Zero when located

D) None of the above.


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Examples: Simplification of a Force and

Couple System:


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6-Moments Couples and Force Couple Systems_Partb.ppt

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