5. MOMENTS, COUPLES, FORCES SYSTEMS & FORCE RESOLUTION

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

Concept of a Moment

When the Force is applied at the CG

When the Force is not applied at he CG

When the Force is not applied at the CG, & the body is hinged at the CG

body

CG of the body

Objective: To explain the concept of a Moment

If a Force P is applied at the midpoint of the free, 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 will both 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

This tendency of a force to produce rotation about some point is called the Moment of a force

Moment of a Force

Objective: Definition of Moment in Statics

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

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

Objective: To explain the concept of Moment in Statics with everyday examples

Sign Convention for Moments

- +

Clockwise negative Anti-clockwise positive

Objective: To illustrate the sign conventions for Moment in Statics

F

d

M = - F d

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

A

B

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

Varignons Theorem

y

x

d

F

Fx

Fy

F

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

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

d

F

d

F

Fx

Fy

d cos

d sin

A A

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

F

d

F

d

M about A= -F x d

d sin

d cos

FdFd

dFdF

dFdFM xy

=+=

=

=

)sin(cossin.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 Varignonstheorem.

Proof Of Varignons Theorem

Objective: To prove Varignons Theorem

Plane of the couple

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 horizontal direction. (Because the sum of their horizontal and vertical components are zero)

d, arm of the couple

Objective: To explain the concept of a Couple in Statics

A A A5

10 15

B

C

D

10lb

10lb

10lb

10lb

10lb

10lb

2 2

2 2

2 2

lbftM A

.40210210

=+=

lbftM A

.40210210

=+=

lbftM A

.40210210

=+=

lbftM A

.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

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 above two 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

F FF Fd dd

=

FORCE SYSTEMS

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

y

xCollinear Force System

y

x

z

Coplanar Force System

y

x

z

Coplanar parallel

y

x

Coplanar Concurrent

x

y

z

Noncoplanar parallel

y

x

z

Noncoplanar concurrent

x

y

z

Noncoplanar nonconcurrent

FORCE SYSTEMS

Resolution of Forces into Rectangular Components

F

x

y

xF

yF

cosFFx =

sinFFy =

Sign convention for Forces

Forces towards right Positive

Forces upward Positive

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

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

RR

RR

RRR

xBCARBACR

=

=

+=

+==

x

y

x

y

sincos

AAAA

y

x

==

sincos

CCCC

y

x

==

sincos

BBBB

y

x

==

Vector Addition by the component method

x

y

1F

2F

xF1

yF1

cos11 FF x =sin11 FF y =

xF2

yF2

cos22 FF x =sin22 FF y =

yyy

xxx

FFRFFR

12

12

==

xR

yR

)( 22 yx RRR +=

x

y

RR1tan=

x

y

R

R

Objective: To add two vectors by the component method

FORCE SYSTEMS