5._momentcouples

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5. MOMENTS, COUPLES, FORCES SYSTEMS

& FORCE RESOLUTION


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(a) Translation (b) Translation & Rotation (c) Rotation

Concept of a Moment

When the Force is

applied at the CGWhen the Force is not

applied at he CG

When the Force is not

applied at the CG, & thebody is hinged at the CG

body

CG

of the

body

Objective: To explain the concept of a Moment


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If a Force P is applied at the midpoint of thefree, rigid, uniform object, it will slide the

object such that every point moves an equal

distance. The object is said to translate.

If the same force is applied at some other

point as in second figure, then the object willboth translate and rotate.

If the point on the object is fixed against

translation, (third figure) then the applied

force causes the object to rotate only.

Objective: Explanation of the Concept of Moment - continued


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This tendency of a

force to produce

rotation about some

point is called theMoment of a force

Moment of a Force

Objective: Definition of Moment in Statics


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Moment of a Force

F

d

The tendency of a force to produce

rotation of a body about some reference

axis or point is called the MOMENT OF A

FORCE

M=Fxd

Objective: An example to illustrate the definition of Moment in Statics


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F= 25#

15

Lever arm

M= - F x d

= -25 x 15

= - 375 #-in

90 deg

d

F

Moment = Force x Perpendicular Distance = Fxd

Example One: Closing the Door

Example Two:

Tightening the

NUT

Common Examples in the Application of the Concept of Moment

Ob ective: To ex lain the conce t of Moment in Statics with ever da exam les


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Sign Convention for Moments

- +

Clockwise negative Anti-clockwise positive

Objective: To illustrate the sign conventions for Moment in Statics


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F

d

M = - F d

What is the moment at A for the Noodle Beam fixed at A andloaded by Force F at B?

A

B

Objective: To illustrate that Moment is always Force x Distance, irrespective of the shape

of the structure


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Varignons Theorem

y

x

d

F

Fx

Fy

F

M=-F.d M= -Fy.x + Fx.y

AA

According to Varignons Theorem, a Force can be resolved into its

components and multiplied by the perpendicular distances for easy

calculation of the Moment

=

Objective: To explain Varignons Theorem


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d

F

d

F

Fx

Fy

d cos

d sin

AA

M about A= F x d )sin()cos()( dFdFdF xy +=

Substitute for Fx and Fy

F x d =

)sin(sin)cos(cos)( dFdFdF +=

22

sincos)( FdFddF +=

)sin(cos 22 +=FdFd

Fd

Proof of Varignons Theorem


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F

d

F

d

M about A= -F x d

d sin

d cos

Fd

Fd

dFdF

dFdFM xy

=

+=

=

=

)sin(cos

sin.sincos.cos

sincos

22

yF

xF

On the Left hand side the Moment is got directly by multiplying F times d.

On the Right hand side it is proved the Moment is F.d using Varignons

theorem.

Proof Of Varignons Theorem

Ob ective: To rove Vari nons Theorem


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Plane of thecouple

dF F

FF

Concept of a Couple

When you grasp the opposite side of

the steering wheel and turn it, you are

applying a couple to the wheel.

A couple is defined as two forces (coplanar) having the same magnitude, parallel

lines of action, but opposite sense. Couples have pure rotational effects on the

body with no capacity to translate the body in the vertical or horizontaldirection. (Because the sum of their horizontal and vertical components are zero)

d, arm of the couple

Ob ective: To ex lain the conce t of a Cou le in Statics


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A A A5

10 15

B

C

D

10lb

10lb

10lb

10lb

10lb

10lb

2 2

2 2

2 2

lbft

MA

.40

210210

=

+=

lbft

MA

.40

210210

=

+=

lbft

MA

.40210210

=

+=

lbftMA

.40210210

=

+=

Effect of Couple applied at different points at the base

of a Cantilever

Thus it is clear that the effect of a couple at the base of the Cantilever is

independent of its (couples) point of application.

Objective: To explain that the effect of a Couple is independent of its point of application

F


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d

1. Introduce two equal

and opposite forces at

B (which does not alter

the equilibrium of the

structure)

FF

F

d

REPLACING A FORCE

WITH A FORCE & A

COUPLE

2. Replace

the abovetwo Forces

with a

Couple= F.d Hence a Force can be replaced with an Equivalent

Fore and a Couple at another point.

Objective: To explain how a Force can be replaced by a Force and couple at another

point


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FF

F F

d dd

=


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FORCE SYSTEMS

Objective: To explain various types of Force systems which occur in Construction


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y

x

Collinear Force System


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y

x

z

Coplanar Force System


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y

x

z

Coplanar parallel


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y

x

Coplanar Concurrent


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x

y

z

Noncoplanar parallel


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y

x

z

Noncoplanar concurrent


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x

y

z

Noncoplanar nonconcurrent

FORCE SYSTEMS


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FORCE SYSTEMS

Resolution of Forces into Rectangular Components

F

x

y

xF

yF

cosFFx =

sinFFy =


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Sign convention for Forces

Forces towards rightPositive

Forces upwardPositive


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Resolution of a Force How to Apply cos and sin

cosF

cosF

cosF

cosF

cosF

cosFsinF

sinF

sinF

sinF

sinF

sinF

Fy

x

F

FF

F F


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Vector Addition By Component Method

A

B

C

Ax

Ay

Cx

Cy

Bx

By

R

Rx

Ry

)(tan

tan

))()((

1_

22

x

y

x

y

yx

yyyy

xxxx

R

R

R

R

RRR

xBCAR

BACR

=

=

+=

+=

=

x

y

x

y

sin

cos

AA

AA

y

x

=

=

sin

cos

CC

CC

y

x

=

=

sin

cos

BB

BB

y

x

=

=

Vector Addit ion by the component method


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x

y

1F

2

F

xF1

yF1

cos11

FFx =

sin11 FFy =

xF2

yF2

cos22 FFx =

sin22 FF y =

yyy

xxx

FFR

FFR

12

12

=

=

xR

yR

)( 22

yx RRR +=

x

y

R

R1

tan

=

x

y

R

R

Objective: To add two vectors by the component method

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