Engineering Mechanics I (Statics) (Part 2/2)

Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
INTERNAL FORCES AND MOMENTS
Chapter Objectives
248
CHAPTER 6
Students will be able to
Use the method of sections for determining internal forces
in 2-D load cases
6.1 Internal Loadings Developed in Structural
Members
6.2 Shear and Moment equations and Diagrams
6.3 Relations between Distributed Load, Shear,
and Moment
6.4 Cables
Chapter Outline

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6.1INTERNALLOADINGSDEVELOPEDINSTRUCTURALMEMBERS
Overview of Internal Loadings
249
Chapter 6 Internal Forces and Moments
Internal loadings can be determined by using the method of sections.
− The force component NB that acts perpendicular to the cross section is termed the normal force.
− The force component VB that is tangent to the cross section is called the shear force.
− The couple moment MB is referred to as the bending moment.

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6.1INTERNALLOADINGSDEVELOPEDINSTRUCTURALMEMBERS
Sign Convention
250

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6.2 SHEAR AND MOMENT EQUATIONS AND DIAGRAMS
257
Beams are structural members designed to support loadings applied perpendicular to their axes.
A simply supported beam is pinned at one end and roller supported at the other, whereas a cantilevered
beam is fixed at one end and free at the other.
If the resulting functions of x are plotted, the graphs are termed the shear diagram and bending-moment
diagram, respectively.

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6.3RELATIONSBETWEENDISTRIBUTEDLOAD,SHEAR,ANDMOMENT
Distributed Load
262
If a beam is subjected to several concentrated forces, couple moments, and distributed loads, the method
of constructing the shear and bending-moment may become quite tedious.
Here, a simpler method for constructing these diagrams is discussed—a method based on differential
relations that exist between the load, shear, and bending moment.
• Distributed Load
The distributed load will be considered positive when the loading acts upward as shown.
The internal shear force and bending moment shown on the free-body diagram are assumed to act in
the positive sense according to the established sign convention in Chapter 6.1.

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Relation between the Distributed Load and Shear
263
The relationship between the distributed load and shear is differentiation and integration.

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Relation between the Shear and Moment
264
The relationship between the shear and moment is also differentiation and integration.

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6.4 CABLES
Cable Subjected to Concentrated Loads
271
Flexible cables and chains combine strength with lightness and often are used in structures for support
and to transmit loads (tension) from one member to another.
• Assumption for Cables
The weight of the cables is neglected, and therefore, it subjects to a constant tensile force.
Generally, we will make the assumption that the cable is perfectly flexible (no moment) and
inextensible (constant length).
• Cable Subjected to Concentrated Loads

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6.4 CABLES
Cable Subjected to a Distributed Load
272
Consider weightless cable subjected to a loading function measured in the x direction.
( )
w w x
=
2
0
2
( )
H
d y w x
dx F
= 

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6.4 CABLES
Cable Subjected to Its own Weight
273
When the weight of a cable becomes important in the force analysis, the loading function along the cable
will be a function of the arc length s rather than the projected length x.
2
1
dy ds
dx dx
 
= − 
 
 

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FRICTION
Chapter Objectives
277
CHAPTER 7
1. Understand the characteristics of dry friction
2. Draw a FBD including friction
3. Solve problems involving friction
4. Determine the forces on a wedge
5. Determine tension in a belt
7.1 Characteristics of Dry Friction
7.2 Problems Involving Dry Friction
7.3 Wedges
7.4 Frictional Forces on Screws
7.5 Frictional Forces on Flat Belts
7.6 Frictional Forces on Collar Bearings,
7.7 Frictional Forces on Journal Bearings
7.8 Rolling Resistance
Chapter Outline

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7.1 CHARACTERISTICS OF DRY FRICTION
Theory of Dry Friction
278
Chapter 7 Friction
Friction is a force that resists the movement of two contacting surfaces that slide relative to one another.
Two types of friction, fluid and dry friction, are common; however, in this chapter, we will study the
effects of dry friction, which is sometimes called Coulomb friction since its characteristics were studied
extensively by the French physicist Charles-Augustin de Coulomb in 1781.
This force always acts tangent to the surface at the points of contact and is directed so as to oppose the
possible or existing motion between the surfaces.
• Theory of Dry Friction
As shown on the free-body diagram of the block, the floor exerts an uneven distribution of both normal
force DNn and frictional force DFn along the contacting surface. As the result,
The Angle fs that Rs makes with N is called the angle of static friction.
s
x h
= = =
W N P F W P
1 1 1
tan tan tan
s s
s s
F N
N N

f 
− − −
   
= = =
   
   

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Characteristics of Dry Friction
279
Chapter 7 Friction
• Characteristics of Dry Friction
As a result of experiments that pertain to the foregoing discussion, we can state the following rules
which apply to bodies subjected to dry friction.
− The frictional force acts tangent to the contacting surfaces in a direction opposed to the motion or
tendency for motion of one surface relative to another.
− When slipping at the surface of contact is about to occur, called impending motion, the maximum
static frictional force is proportional to the normal force, such that
where the constant of proportionality, s, is called the coefficient of static friction.
− When slipping at the surface of contact is occurring, the kinetic frictional force is proportional to the
normal force, such that
Here the constant of proportionality, k, is called the coefficient of kinetic friction.
− The maximum static frictional force Fs that can be developed is independent of the area of contact,
provided the normal pressure is not very low nor great enough to severely deform or crush the
contacting surfaces of the bodies.
− The maximum static frictional force is generally greater than
the kinetic frictional force for any two surfaces of contact.
However, if one of the bodies is moving with a very low velocity
over the surface of another, Fk becomes approximately equal
to Fs, i.e., .
s s
F N

=
s k
F N

=
s k
 


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Impending Tipping or Slipping
280
Chapter 7 Friction
• Impending Tipping or Slipping
For a given W and h of the box, how can we determine if the block will slide or tip first?
In this case, we have four unknowns and only three equation of equilibrium.
Hence, we have to make an assumption to give us another equation (the friction equation) and solve
for the unknowns.
Finally, we need to check if our assumption was correct.

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7.2 PROBLEMS INVOLVING DRY FRICTION
Types of Friction Problems
281
Chapter 7 Friction
If a rigid body is in equilibrium when it is subjected to a system of forces that includes the effect of friction,
the force system must satisfy not only the equations of equilibrium but also the laws that govern the
frictional forces.
• Types of Friction Problems
In general, the problems involving dry friction can easily be classified once free-body diagrams are
drawn since the total number of unknowns can be identified and compared with the total number of
available equilibrium equations.
− No Apparent Impending Motion (Equilibrium)
Problems in this category are strictly equilibrium problems, which require the number of unknowns
to be equal to the number of available equilibrium equations.
− Impending Motion at All Points of Contact
In this case, the total number of unknowns will equal the total number of available equilibrium
equations plus the total number of available frictional equations.
− Impending Motion at Some Points of Contact
Here the number of unknowns will be less than the number of available equilibrium equations plus
the number of available frictional equations or conditional equations for tipping.

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7.2 PROBLEMS INVOLVING DRY FRICTION
Equilibrium Versus Frictional Equations
282
Chapter 7 Friction

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7.3 WEDGES
289
Chapter 7 Friction
A wedge is a simple machine that is often used to transform an applied force into much larger forces,
directed at approximately right angles to the applied force.
We have excluded the weight of the wedge since it is usually small compared to the weight W of the block.
Also, note that the frictional forces F1 and F2 must oppose the motion of the wedge.
Likewise, the frictional force F3 of the wall on the block must act downward so as to oppose the block’s
upward motion.
If the block is to be lowered, then the frictional forces will all act in a sense opposite to original one.
If P is not applied and friction forces hold the block in place, then the wedge is referred to as self-locking.

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7.4 FRICTIONAL FORCES ON SCREWS
Upward Impending Motion
292
Chapter 7 Friction
In most cases, screws are used as fasteners; however, in many types of machines they are incorporated
to transmit power or motion from one part of the machine to another.
• Upward Impending Motion
Applying the force equations of equilibrium along the horizontal and vertical axes, we have the applied
torsional moment as
where q is the slope or the lead angle, fs is the angle of static friction ( )
r is the mean radius of the thread.
l is the lead of the screw and it is equivalent to the distance the screw advances when it turns
one revolution.
tan( ) where
2
s
l
M rW
r
q f q

= + =
1
tan ( )
s
F N
f −
=
0 sin( )
0 cos( ) 0
x s
y s
M
F R
r
F R W
q f
q f
= = − +
= = + − =



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Self-Locking Screw
293
Chapter 7 Friction
• Self-Locking Screw
A screw is said to be self-locking if it remains in place under any axial load W
when the moment M is removed.
Here the angle of static friction fs becomes greater than or equal to q.
0 sin( )
0 cos( ) 0
x s
y s
F R
F R W
q f
q f
= = −
= = − − =



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Downward Impending Motion
294
Chapter 7 Friction
• Downward Impending Motion ( )
If the screw is not self-locking, it is necessary to apply a moment to prevent the screw
from winding downward.
• Downward Impending Motion ( )
If a screw is self-locking, a couple moment must be applied to the screw
in the opposite direction to wind the screw downward.
s
q f

s
f q

' tan( )
s
M rW q f
= −
" tan( )
s
M rW f q
= −

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295
Chapter 7 Friction
Upward Impending
Motion
Stay
Downward
Impending Motion
s
q f

s
f q
 s
f q
=
tan( )
s
M rW q f
= +
0
M = ' tan( )
s
M rW q f
= −
" tan( )
s
M rW f q
= − any M 0
M =

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7.5 FRICTIONAL FORCES ON FLAT BELTS
298
Chapter 7 Friction
Whenever belt drives or band brakes are designed, it is necessary to determine the frictional forces
developed between the belt and its contacting surface.
• Frictional Analysis
Since dq is of infinitesimal size, and .
Also, the product of the two infinitesimals dT and dq /2 may be neglected when compared to
infinitesimals of the first order.
sin( 2) 2
d d
q q
= cos( 2) 1
dq =
2 1
T Te
=

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7.6 FRICTIONAL FORCES ON COLLAR BEARINGS
301
Chapter 7 Friction
Pivot and collar bearings are commonly used in machines to support an axial load on a rotating shaft.
Provided the bearing is new and evenly supported,
then the applied moment needed to overcome all
the frictional forces
3 3
2 1
2 2
2 1
2
( )
3
s
R R
M P
R R

−
=
−
2 1
3 3
2 1
2 2
2 1
and 0
2 2
( )
3 3
s s
R R R
R R
M P PR
R R
 
→ →
−
= →
−

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7.7 FRICTIONAL FORCES ON JOURNAL BEARINGS
302
Chapter 7 Friction
When a shaft or axle is subjected to lateral loads, a journal bearing is commonly used for support.
If the bearing is partially lubricated, k is small, and therefore . Under these conditions,
a reasonable approximation to the moment needed to overcome the frictional resistance becomes
sin( )
if 0 then sin( ) tan( )
k
k k k k
k
M Rr
M Rr
f
f f f 

=
→ = =

sin tan
k k k
f f 
 

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7.8 ROLLING RESISTANCE
305
Chapter 7 Friction
The magnitude of the force of deformation, Nd, and its horizontal component is always greater than that
of restoration, Nr, and consequently a horizontal driving force P must be applied to the cylinder to
maintain the motion, called rolling resistance.
The distance a is termed the coefficient of rolling resistance, which has the dimension of length.
if 0 then cos 1
cos
Wa Wa
P P
r r
q q
q
=  → =  =

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CENTER OF GRAVITY AND CENTROID
Chapter Objectives
307
CHAPTER 8
1. Understand the concepts of center of gravity, center of mass,
and centroid
2. Determine the location of the center of gravity (CG)
3. Determine the location of the center of mass
4. Determine the location of the centroid using the method of
composite bodies
8.1 Center of Gravity and the Centroid
8.2 Composite Bodies
8.3 Theorems of Pappus and Guldinus
8.4 Resultant of a General Distributed Loading
Chapter Outline

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8.1 CENTER OF GRAVITY AND THE CENTROID
Center of Gravity
308
Chapter 8 Center of Gravity and Centroid
Knowing the resultant or total weight of a body and its location is important when considering the effect
this force produces on the body.
• Center of Gravity
An infinite number of particles in a body will form a parallel force system, and the resultant of this
system is the total weight, which passes through a single point called the center of gravity
where are the coordinates of the center of gravity
are the coordinates of an arbitrary particle in the body.
, ,
x y z
, ,
x y z
xdW ydW zdW
x y z
dW dW dW
= = =
  
  

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Center of Mass
309
• Center of Mass
In order to study the dynamic response or accelerated motion of a body, it becomes important to
locate the body’s center of mass
This location can be determined by substituting .
Actually, particles have their weight only when under the influence of gravitational attraction, whereas
the center of mass is independent of gravity.
dW gdm
=
xdm ydm zdm
x y z
dm dm dm
= = =
  
  

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Centroid of a Volume
310
• Centroid of a Volume
If the body is made from a homogeneous material, then its density r will be constant and the centroid
coincides with the center of mass or the center of gravity
This location can be determined by substituting .
V V V
V V V
xdV ydV zdV
x y z
dV dV dV
= = =
  
  
dm dV
r
=

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Centroid of an Area
311
• Centroid of an Area
If an area lies in the x–y plane and is bounded by a curve, then its centroid will be in this plane and
can be determined from integrals similar to centroid of a volume, namely,
A A
A A
xdA ydA
x y
dA dA
= =
 
 

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Centroid of a Line
312
• Centroid of a Line
If a line segment (or rod) lies within the x–y plane and it can be described by a thin curve, then its
centroid is determined from
Here, the length of the differential element is given by the Pythagorean theorem
L L
L L
xdL ydL
x y
dL dL
= =
 
 
2
2
2 2
( ) ( ) 1 or 1
dy dx
dL dx dy dx dy
dx dy
 
   
   
 
= + = + +
   
 
 
   
   

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313
• List of Center (or Centroid)
2D Problems 3D Problems
Center of Gravity
Center of Mass of a Body
Centroid of a Volume
Centroid of an Area
Centroid of a Line
xdW ydW
x y
dW dW
= =
 
 
additional eq.
zdW
z
dW
= 

xdm ydm
x y
dm dm
= =
 
 
additional eq.
zdm
z
dm
= 

V V
V V
xdV ydV
x y
dV dV
= =
 
 
additional eq. V
V
zdV
z
dV
=


A A
A A
xdA ydA
x y
dA dA
= =
 
 
additional eq. V
V
zdA
z
dA
=


L L
L L
xdL ydL
x y
dL dL
= =
 
 
additional eq. V
V
zdL
z
dL
=



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Procedure for Calculating
314

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Centroids of Common Shapes of Areas and Lines
325

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Centroids of Common Shapes of Areas and Lines
326

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Centroids of Common Shapes of Volumn
327

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8.2 COMPOSITE BODIES
328
A composite body consists of a series of connected “simpler” shaped bodies, which may be rectangular,
triangular, semicircular, etc.
where are the coordinates of the center of
gravity of the composite body
are the coordinates of the center of
gravity of each composite part of the body
SW is the sum of the weights of all the
composite parts of the body, or simply the
total weight of the body.
When the body has a constant density or specific
weight, the center of gravity coincides with the
centroid of the body.
, ,
x y z
, ,
x y z
xW yW zW
x y z
W W W
= = =
  
  

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8.3 THEOREMS OF PAPPUS AND GULDINUS
Surface Area
335
The two theorems of Pappus and Guldinus are used to find the surface area and volume of any body of
revolution.
They were first developed by Pappus of Alexandria during the fourth century a.d. and then restated at a
later time by the Swiss mathematician Paul Guldin or Guldinus (1577–1643).
• Surface Area
If we revolve a plane curve about an axis that does not intersect the curve we will generate a surface
area of revolution.
Therefore the first theorem of Pappus and Guldinus states that the area of a surface of revolution
equals the product of the length of the generating curve and the distance traveled by the centroid of
the curve in generating the surface area.
A r L
q
=
2 2
2 2
L L
L
L
dA r dL dA r dL
A r dL A r L
rdL rdL
r r rdL r L
L
dL
 
 
=  =
 =  =
=  =  =
 

 



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8.3 THEOREMS OF PAPPUS AND GULDINUS
Volume
336
• Volume
A volume can be generated by revolving a plane area about an axis that does not intersect the area.
Therefore the second theorem of Pappus and Guldinus states that the volume of a body of revolution
equals the product of the generating area and the distance traveled by the centroid of the area in
generating the volume.
• Composite Shapes
We may also apply the above two theorems to lines or areas that are composed of a series of
composite parts.
V r A
q
=
A r L
V r A
q
q
=
=


2 2
2 2
A A
A
A
dV r dA dV r dA
V r dA V r A
rdA rdA
r r rdA r A
A
dA
 
 
=  =
 =  =
=  =  =
 

 



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8.4 RESULTANT OF A GENERAL DISTRIBUTED LOADING
Magnitude of Resultant Force
341
Here, we will generalize the method in Chapter 3.9 to include flat surfaces that have an arbitrary shape
and are subjected to a variable load distribution.
• Magnitude of Resultant Force
The force acting on the differential area dA of the plate can be determined under the loading curve.
The magnitude is the sum of the differential forces acting over the plate’s entire surface area A.
This result indicates that the magnitude of the resultant force is equal to the total volume under the
distributed-loading diagram.
( , )
R A V
F p x y dA dV V
= = =
 

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8.4 RESULTANT OF A GENERAL DISTRIBUTED LOADING
Location of Resultant Force
342
• Location of Resultant Force
The location is determined by setting the moments of FR equal to the moments of all the differential
forces dF about the respective y and x axes.
Hence, the line of action of the resultant force passes through the geometric center or centroid of the
volume under the distributed-loading diagram.
( , ) ( , )
( , ) ( , )
V V
A A
A V A V
xdV ydV
xp x y dA yp x y dA
x y
p x y dA dV p x y dA dV
= = = =
 
 
   

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MOMENTS OF INERTIA
Chapter Objectives
343
CHAPTER 9
1. Define the moments of inertia for an area
2. Determine the moments of inertia for an area by integration
3. Apply the parallel-axis theorem
4. Determine the moment of inertia for a composite
5. Explain the concept of the mass moment of inertia
6. Determine the mass moment of inertia of a composite body
9.1 Definition of Moments of Inertia for Areas
Moment of Inertia
9.2 Parallel-Axis Theorem for an Area
9.3 Radius of Gyration of an Area
9.4 Moments of Inertia for Composite Areas
9.5 Product of Inertia for an Area
9.6 Moments of Inertia for an Area about Inclined
Axes
9.7 Mohr’s Circle for Moments of Inertia
9.8 Mass Moment of Inertia
Chapter Outline

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9.1 DEFINITION OF MOMENTS OF INERTIA FOR AREAS
344
Chapter 9 Moments of Inertia
Whenever a distributed load acts perpendicular to an area and its intensity varies linearly, the calculation
of the moment of the loading about an axis will involve an integral of the form .
The integral is sometimes referred to as the “second moment” of the area about an axis (the x axis), but
more often it is called the moment of inertia of the area.
• Moment of Inertia
For the entire area A the moments of inertia are determined by integration
We can also formulate this quantity for a differential area dA about the “pole” O or z axis. This is
referred to as the polar moment of inertia.
From the above formulations it is seen that Ix, Iy, and JO will
always be positive since they involve the product of distance
squared and area.
Centroid of an area is determined by the first moment of an area
about an axis.
2
y dA

2 2
x x y y
A A A A
I dI y dA I dI x dA
= = = =
   
2
O O x y
A A
J dJ r dA I I
= = = +
 

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9.2 PARALLEL-AXIS THEOREM FOR AN AREA
345
The parallel-axis theorem can be used to find the moment of inertia of an area about any axis that is
parallel to an axis passing through the centroid C and about which the moment of inertia is known.
• Moment of Inertia
To start, we choose a differential element dA located at an arbitrary distance from the centroidal axis.
And finally, for the polar moment of inertia
The form of each of these three equations states that the moment of
inertia for an area about an axis is equal to its moment of inertia about a
parallel axis passing through the area’s centroid plus the product of the
area and the square of the perpendicular distance between the axes.
2 2
' '
x x y y y x
I I Ad I I Ad
= + = +
2
O c
J J Ad
= +

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9.2 PARALLEL-AXIS THEOREM FOR AN AREA
346

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9.3 RADIUS OF GYRATION OF AN AREA
347
The radius of gyration of an area about an axis has units of length and is a quantity that is often used for
the design of columns in structural mechanics.
Provided the areas and moments of inertia are known, the radius of gyration can be determined
Similar to finding moment of inertia of a differential area about an axis
y
x
x y
O
O
I
I
k k
A A
J
k
A
= =
=
2 2
2 2
x x x
y y y
I k A dI y dA
I k A dI x dA
=  =
=  =

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9.4 MOMENTS OF INERTIA FOR COMPOSITE AREAS
354
A composite area consists of a series of connected “simpler” parts or shapes, such as rectangles,
triangles, and circles.
Provided the moment of inertia of each of these parts is known or can be determined about a common
axis, then the moment of inertia for the composite area about this axis equals the algebraic sum of the
moments of inertia of all its parts.

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356

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9.5 PRODUCT OF INERTIA FOR AN AREA
361
It will be shown in the next section that the property of an area, called the product of inertia, is required in
order to determine the maximum and minimum moments of inertia for the area.
• Product of Inertia
The product of inertia of the area with respect to the x and y axes is defined as
• Parallel-Axis Theorem
The parallel-axis theorem also apply to the product of inertia
xy xy
A A
I dI xydA
= =
 
' '
xy x y x y
I I Ad d
= +

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Huang
9.6MOMENTSOFINERTIAFORANAREAABOUTINCLINEDAXES
366
In structural and mechanical design, it is sometimes necessary to calculate the moments and product of
inertia Iu, Iv, and Iuv for an area with respect to a set of inclined u and v axes when the values for q, Ix, Iy,
and Ixy are known.
• Principal Moments of Inertia
The axis with the maximum or minimum moment of inertia is called the principal axes of the area, and
the corresponding moments of inertia are called the principal moments of inertia.

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9.7 MOHR’S CIRCLE FOR MOMENTS OF INERTIA
369
The moments of inertia for an area about inclined axes have a graphical solution that is convenient to use
and generally easy to remember, called Mohr’s circle, named after the German engineer Otto Mohr
(1835–1918).
2 2
2 2
2 2
x y x y
u uv xy
I I I I
I I I
+ +
   
− + = +
   
   

Shieh-Kung
Huang
9.7 MOHR’S CIRCLE FOR MOMENTS OF INERTIA
370

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Huang
9.8 MASS MOMENT OF INERTIA
Parallel-Axis Theorem
373
The mass moment of inertia of a body is a measure of the body’s resistance to angular acceleration.
Considering a rigid body, we define the mass moment of inertia of the body about the z axis as
For most applications, r will be a constant, and the body’s moment of inertia is then
computed using volume elements for integration
• Parallel-Axis Theorem
If the moment of inertia of the body about an axis passing through the body’s mass center
is known, then the moment of inertia about any other parallel axis can be
determined by using the parallel-axis theorem.
where IG represents the moment of inertia about the z’ axis passing through
the mass center G
m represents the mass of the body
d represents the distance between the parallel axes
2
z z m
I dI r dm
= =
 
2
z z V
I dI r dV
r
= =
 
2
G
I I md
= +

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Huang
9.8 MASS MOMENT OF INERTIA
Radius of Gyration
374
• Radius of Gyration
Occasionally, the moment of inertia of a body about a specified axis is reported in handbooks using
the radius of gyration, k.
• Composite Bodies
If a body is constructed from a number of simple shapes such as disks, spheres, and rods, the
moment of inertia of the body about any axis z can be determined by adding algebraically the
moments of inertia of all the composite shapes calculated about the same axis.
I
k
m
=

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VIRTUAL WORK
Chapter Objectives
379
CHAPTER 10
Understand and use the method of virtual work
10.1 Definition of Work
10.2 Principle of Virtual Work
10.3 Principle of Virtual Work for a System of
Connected Rigid Bodies
10.4 Conservative Forces
10.5 Potential Energy
10.6 Potential-Energy Criterion for Equilibrium
10.7 Stability of Equilibrium Configuration
Chapter Outline

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10.1 DEFINITION OF WORK
380
Chapter 10 Virtual Work
The principle of virtual work was proposed by the Swiss mathematician Jean Bernoulli in the eighteenth
century.
• Work of a Force
If we use the definition of the dot product the work can also be written as
As the above equations indicate, work is a scalar, and like other scalar
quantities, it has a magnitude that can either be positive or negative.
In the SI system, the unit of work is a joule (J, ), and The unit
of work in the FPS system is the foot-pound ( ).
• Work of a Couple Moment
Since , the work of the couple moment M is therefore
• Virtual Work
The definitions of the work of a force and a couple have been presented in terms of actual movements
expressed by differential displacements having magnitudes of dr and dq.
Consider now an imaginary or virtual movement of a body in static equilibrium, which indicates a
displacement or rotation that is assumed and does not actually exist, and the virtual work done by
dU d
= 
F r
1 1
J N m
= 
ft lb

M Fr
=
dU d
= 
M θ
U U
   
=  = 
F r M θ

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10.2 PRINCIPLE OF VIRTUAL WORK
381
The principle of virtual work states that if a body is in equilibrium, then the algebraic sum of the virtual
work done by all the forces and couple moments acting on the body is zero for any virtual displacement
of the body. Thus,
When using the principle of virtual work, it is not necessary to
include the work done by the internal forces acting within the
body since a rigid body does not deform when subjected to
an external loading, and furthermore, when the body moves
through a virtual displacement, the internal forces occur in
equal but opposite collinear pairs, so that the corresponding
work done by each pair of forces will cancel.
0
U
 =

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Huang
10.3 PRINCIPLE OF VIRTUAL WORK FORASYSTEM OF
382
The method of virtual work is particularly effective for solving equilibrium problems that involve a system
of several connected rigid bodies.
CONNECTED RIGID BODIES

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10.3 PRINCIPLE OF VIRTUAL WORK FORASYSTEM OF
383
CONNECTED RIGID BODIES

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10.4 CONSERVATIVE FORCES
392
When a force does work that depends only upon the initial and final positions of the force, and it is
independent of the path it travels, then the force is referred to as a conservative force.
• Weight
If the block moves from A to B, through the vertical displacement h, the work is
• Spring Force
The work of Fs when the block is displaced from s1 to s2 is
• Friction
In contrast to a conservative force, consider the force of friction exerted on a
sliding body by a fixed surface.
The work done by the frictional force depends on the path; the longer the path, the greater the work.
Consequently, frictional forces are nonconservative, and most of the work done by them is dissipated
from the body in the form of heat.
0
h
U Wdy Wh
= − = −

2
1
2 2
2 1
1 1
2 2
s
s
U ksds ks ks
 
= = − −
 
 


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10.5 POTENTIAL ENERGY
Gravitational and Elastic Potential Energy
393
A conservative force can give the body the capacity to do work and this capacity,
measured as potential energy, depends on the location or “position” of the body
measured relative to a fixed reference position or datum.
• Gravitational Potential Energy
Measuring y as positive upward, the gravitational potential energy of the body’s
weight W is therefore
• Elastic Potential Energy
When a spring is either elongated or compressed by an amount s from its
unstretched position (the datum), the energy stored in the spring is called
elastic potential energy, determined from
g
V Wy
=
2
1
2
e
V ks
=

Shieh-Kung
Huang
10.5 POTENTIAL ENERGY
Potential Function
394
In the general case, if a body is subjected to both gravitational and elastic forces, the potential energy or
potential function V of the body can be expressed as the algebraic sum
In particular, if a system of frictionless connected rigid bodies has a single degree of freedom, such that its
vertical distance from the datum is defined by the coordinate y, then the potential function for the system
can be expressed as .
If the block moves from y1 to y2, then the work of W and Fs is
g e
V V V
= +
( )
V V y
=
( ) ( )
1 2 1 2
U V y V y
− = −
( ) ( ) ( ) 2 2
1 2 1 2 1 2 1 2
1 1
2 2
U V y V y W y y ky ky
− = − = − − + −

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Huang
10.6 POTENTIAL-ENERGY CRITERION FOR EQUILIBRIUM
395
If the system is in equilibrium and undergoes a virtual displacement q, rather than an actual displacement
dq, then the equation is
Hence, when a frictionless connected system of rigid bodies is in equilibrium, the first derivative of its
potential function is zero.
0 0 0 0
dU dV U V
dV dV
U V V q
dq dq
 
   
= −  = −
 
=  =  = =  =
 
 
0
dV
W ky
dy
= − + =

Shieh-Kung
Huang
10.7 STABILITY OF EQUILIBRIUM CONFIGURATION
396
The potential function V of a system can also be used to investigate the stability of the equilibrium
configuration, which is classified as stable, neutral, or unstable.
• Stable Equilibrium
A system is said to be in stable equilibrium if a system has a tendency to return to its original position
when a small displacement is given to the system.
• Neutral Equilibrium
A system is said to be in neutral equilibrium if the system still remains in equilibrium when the system
is given a small displacement away from its original position.
• Unstable Equilibrium
A system is said to be in unstable equilibrium if it has a tendency to be displaced farther away from its
original equilibrium position when it is given a small displacement.

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Shieh-Kung
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405
Thanks for your attention!
See you next semester!
Of course, don’t forget the final exam!

Engineering Mechanics I (Statics) (Part 2/2)

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Engineering Mechanics I (Statics) (Part 2/2)