Moments

1
Moments

• ENGR 221
• February 3, 2003

2
Lecture Goals

• 4.3 Vector Representation of a Moment
• 4.4 Couples
• 4.6.1 Simplification of a Force System
  Resultants

3
Quiz Problem
The barge B is pulled by two tugboats A and C.
At a given instant the tension in cable AB is
4500-lb and the tension in cable BC is 2000-lb.
Determine the magnitude and direction of the
resultant of the two forces applied at B at that
instant. (Draw the FBD!)
4
Moment Characteristics
A moment of a force about a point or axis
action. The magnitude is defined as the magnitude
of the force F and perpendicular distance d from
the line of action of force to the axis.
5
Moment Characteristics
The magnitude is defined as
6
Moment Characteristics
The magnitude is defined as Point 0 is the
moment center Distance d is the perpendicular
moment arm q is the angle between the two vectors
7
Moment Characteristics
The magnitude is defined as Counter clockwise
moment is positive Clockwise moment is negative
8
Moment Characteristics
The line of action follows the right hand rule to
compute the direction of the vector Use the
index finger as the direction of r vector and
middle finger as the direction of the F vector
and the thumb is the resulting direction.
9
Varigons Theorem
As with the summation of force combining to get
resultant force Similar resultant comes from
the addition of moments
10
Moment Characteristics
Resultant of Coplanar Forces that act on a Rigid
Body
Engineering Mechanics ,Boresi Schmidt
11
Example - Problem
The jib crane is oriented so that the boom DA is
parallel to the x axis. At the instant shown the
tension in the cable AB is 13 kN. Determine the
moment about each of the coordinate axes of the
force exerted on D and moment along DA by the
cable at A.
12
Moment along an Axis?
How does one compute the value of the moment
along an axis? To find component of a vector
we take a dot product with the magnitude and use
the vector to compute the components
13
Moment Magnitude of Line
Using the dot product of the line AB Using
the lAB to find the components of the moment
14
Example Crane Problem
The jib crane is oriented so that the boom DA is
parallel to the x axis. At the instant shown the
tension in the cable AB is 13 kN. Determine the
moment about each of the coordinate axes of the
force exerted on D and moment along DA by the
cable at A.
15
Example Crane Problem
Find the moment about D so that the vector DA
16
Example Crane Problem
Find the tension force in AB is A(3.2m, 4.8m, 0
m) and B(3.2m, 0m, 2m)
17
Example Crane Problem
The force vector is along AB with a magnitude of
13 kN
18
Example Crane Problem
The moment at D is
19
Example Crane Problem

The cross product of two vectors is
20
Example Crane Problem

The cross product of two vectors is The dot
product of DA
21
Example Crane Problem

So there would be no moment on bar AB. What
would it have been if there was a bar DE going
off at 45o angle The dot product of DE
22
Example Crane Problem

Moment would be
23
Moment about Lines?
Why do we need to know the moment about a line?

• Intellectual Exercise?
• The whim of the instructor ?
• Others?

24
Answer Moment about Lines
The answer is others. The moment about axes on a
beam or a shaft is used to calculate the stresses
in the three directions
25
Class - Example
A vertical force P of magnitude 60 lb is applied
to the crank at A. Knowing that q 75o,
determine the moment P alone each of the
coordinate axes.
26
Couple Properties
A couple is a set of free vectors. Two forces F
and -F having the same magnitude, parallel line
of action and opposite sense are said to form a
couple.
27
Couple - Properties
A couple. The resulting moment is a function of
its magnitude, direction and orientation of the
moment.
28
Couple - Properties
Couples are found in simple beams to compute the
stresses in the body.
29
Couple - Properties
Couples have the following properties

• Couples can be translated to a parallel
  position in its plane.
• Couple can be rotated about its plane.
• The magnitude of two force of a couple the
  distance between them can be changed provided
  the product of Fd remains constant.

30
Couples - Properties
The dynamical effect of a couple on a rigid body
is not changed if the couple is displaced or
rotated in it plane or if the couple is displaced
from its plane to a parallel plane The magnitude
of the moment of a couple is product Fd, where F
is the magnitude of either force of the couple
and d is the length of the arm of the couple.
31
Three Dimensional Moments
Moment Components
As two dimensions, a moment can be described
using a magnitude M and three angles, qx, qy,
and qz. The components of the vector are Mx, My,
and Mz.
32
Three Dimensional Moments
Moment Components
The three angles, qx, qy, and qz are defined as
33
Three Dimensional Moments
Moment Components
These moment cosines are
34
Three Dimensional Moments
Moment Components
Substitute into the moment
The magnitude is M and unit vector is
35
Example - Drive Shaft
The axles and drive shaft of an automobile are
acted upon by the three couples. Replace these
couples by a single equivalent couple.
36
Example - Drive Shaft
The individual components can be moved to the
origin and add together
37
Example - Drive Shaft
Add the three components together
38
Example - Drive Shaft
The magnitude of the resulting moment is
39
Example - Drive Shaft
The directional cosines of the moments
40
Example - Column
A crane column supports a 16-kip load. Reduce
the load to an axial force along AB and a couple.
41
Example - Column
The couple of 16-kip forces results in a moment
16 kips
16 kips
42
Class - Example
A 400 N force is applied to a bent plate.
Determine an equivalent force couple system a) at
A b) at B
43
Example Column II
A 100 kN load is applied eccentrically on a
column. Determine the components of the force
and couple at G which are equivalent to the 100
kN load.
44
Example Column II
The force can be broken into 2 separated parts
45
Example Column II
The first part is
46
Example Column II
The second moment component is
47
Example Column II
The second part is
48
Example Column II
Add the components together
49
Example Column II
The magnitude is
50
Example - Drive Shaft
The directional cosines of the moments
51
Class Example
Replace the two forces by a force couple system
at the origin.
52
Homework (Due 2/10/03)
Problems
4-31, 4-36, 4-38, 4-40, 4-46, 4-57, 4-73, 4-76
53
Knowledge - Example
Why do we study free-body diagrams and use
equilibrium to find the forces?
Results of the St. Louis hotel disaster
54
Knowledge -Example
There was an on-site modification of the bridge
design.
55
Knowledge - Example
Notice the free-body diagram shows that the force
at B and E doubles in the modified structure.