CHAPTER 2: EQUILIBRIUM OF RIGID BODIES - Hacettepe · CHAPTER 2: EQUILIBRIUM OF RIGID BODIES Principle of Transmissibility This principle states that the conditions of equilibrium

CHAPTER 2: EQUILIBRIUM OF RIGID BODIES

We have studied the application of forces on particles.

What if do we have a rigid body having a finite size?

If we have a rigid body having finite size, there may be different forces acting on the rigid

body at different points of application. Therefore:

The objective of this section is:

Replace a given system of forces by a simpler equivalent system acting on a rigid body.

In order to achieve this: we need to learn the followings:

Moment of a force about a point

Vector product, cross product

Principles of moments (Varignon’s theorem)

Principle of transmissibility

Moment of a force along an axis

Moment of a couple


Principle of Transmissibility

This principle states that the conditions of equilibrium or motion of a rigid body remain

unchanged if a force acting at a given point of the rigid body is replaced by a force of the

same magnitude and direction but acting at a different point provided that they have the

same line of action.

This principle is based on experimental evidence and cannot be proven by statics. It can

be derived from the study of dynamics of rigid bodies.



It should be noted that the principle of transmissibility does not apply in the

analysis of deformations and internal forces


MOMENT OF A FORCE ABOUT A POINT

The moment of a force about a point provides a measure of tendency of rotation

about a fixed axis along a point. It is sometimes known as TORQUE. Moment is

vector.


Now consider a Fy acting on the same point. No moment is produced. If the line of action

of the force passes through the point 0, no moment is produced.

Now consider a Fz acting on the same point.

Magnitude: M0= Fd

[N.m] Moment arm (perpendicular distance

from 0 to the line of action F)

Fz


DIRECTION: we will use the right hand rule:

Fingers follow the sense of rotation. Right hand thumb points

along the direction of the moment (moment axis).



A 400N force is applied to the frame and

θ=20 . Find the moment of the force at A

HW: Please work on this example.


SOLUTION:


Let’s resolve F into its rectangular components:

Fx=400N cos 20 =375.9N

Fy=400N sin 20 =136.8N

Let’s assume + (usually counter-clockwise)

MA=136.8N 3m+375.9N 2m

MA=1162.2Nm (ANSWER)


MOMENTS IN 3-D:

Moment of a force about point O is defined as:


VECTOR CROSS PRODUCT:








GENERALIZING:


PRINCIPLES OF MOMENTS: (VARIGNON’S THEOREM)

The moment of a force about a point is equal to the sum of the moments of the force’s

components about the point.

Using the distribution law.


Let’s use the cross product for the example problem.


Moment of a Force about an axis:

We know that when the moment of a force is computed about a point, the moment will

be perpendicular to the plane containing the force and the moment arm. In some cases,

we need to find the component of this moment along a specified axis.

A force is applied to the tool to open a

gas valve. Find the magnitude of the

moment of this force about the z axis.



Moment of a Couple:


Equivalent Couples:

All three couples 1,2,3 have the

same moment M.

Couples which have the same

moment are called equivalent

couples. They have the same

effect on the rigid body.

Equivalent couples must be either

in the same or parallel lines.


When a couple acts on a rigid body, it does not matter where the two forces forming the

couple act or what magnitude and direction they have. The only thing that counts is the

moment of the couple. (magnitude + direction)

Resultant of Couple Moment:



EXAMPLE:

Determine the net moment due to the two couples

(d=8ft)

Let’s resolve the forces into x and y axis.



HW: Please work on this question:

Find the couple moment acting on the rod in

Cartesian vector notation:


Same result would have been obtained if one had found individual moments of two

forces about any point.

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CHAPTER 2: EQUILIBRIUM OF RIGID BODIES - Hacettepe · CHAPTER 2: EQUILIBRIUM OF RIGID BODIES Principle of Transmissibility This principle states that the conditions of equilibrium