Mechanics of
STATICS
K N PRAJAPTI
O
R
ORBITAL
PUBLICATION
BRIEF
CHAPTER ONE
STATICS OF PARTICLES
CHAPTER TWO
EQUILIBRIUM OF RIGID BODIES
CHAPTER THREE
CENTROID AND CENTRE OF GRAVITY
CHAPTER FOUR
ANALYSIS OF STRUCTURE
CHAPTER FIVE
FORCES IN BEAMS AND CABLE
CHAPTER
FRICTION
SIX
CHAPTER SEVEN
MOMENT OF INERTIA
CHAPTER EIGHT
METHOD OF VIRTUAL WORK
CHAPTER NINE
STRESS AND STRAIN
CHAPTER TEN
AXIAL LOADING
CHAPTER
TORSION
ELEVEN
CHAPTER TWELVE
BENDING STRESS IN ELEMENTS
CHAPTER THIRTEEN
TRANSVERSE SHEAR STRESS IN ELEMENTS
CHAPTER FOURTEEN
COMBINED LOADING IN ELEMENTS
CHAPTER FIFTEEN
STRESS TRANSFORMATION
CHAPTER FIFTEEN
STRAIN TRANSFORMATION
CHAPTER SIXTEEN
DEFLECTION OF BEAMS AND SHAFTS
CHAPTER SEVENTEEN
BUCKLING OF COLUMNS
CHAPTER EIGHTEEN
POWER SCREW
CHAPTER NINETEEN
THREADED JOINTS
CHAPTER TWENTY
PERMENENT JOINTS
CHAPTER
SPRING
TWENTY–ONE
CHAPTER TWENTY–TWO
STATICALLY INDETERMINATE STRUCTURES
CHAPTER TWENTY–THREE
FLUID PROPERTIES
CHAPTER TWENTY–FOUR
PRESSURE AND ITS MEASUREMENT
CONTENTS
CHAPTER
ONE
STATICS OF PARTICLES
INTRODUCTION
Mechanics can be defined as that science which describes and predicts the
conditions of rest or motion of bodies under the action of forces. It is divided into
three parts: mechanics of rigid bodies, mechanics of deformable bodies, and
mechanics of fluids.
The mechanics of rigid bodies is subdivided into statics and dynamics, the
former dealing with bodies at rest, the latter with bodies in motion. In this part of
the study of mechanics, bodies are assumed to be perfectly rigid. Actual structures
and machines, however, are never absolutely rigid and deform under the loads to
which they are subjected. But these deformations are usually small and do not
appreciably affect the conditions of equilibrium or motion of the structure under
consideration. They are important, though, as far as the resistance of the structure
to failure is concerned and are studied in mechanics of materials, which is a part of
the mechanics of deformable bodies. The third division of mechanics, the mechanics
of fluids, is subdivided into the study of incompressible fluids and of compressible
fluids. An important subdivision of the study of incompressible fluids is hydraulics,
which deals with problems involving water.
FUNDAMENTAL CONCEPTS AND PRINCIPLES
The basic concepts used in mechanics are space, time, mass, and force. These
concepts cannot be truly defined; they should be accepted on the basis of our
intuition and experience and used as a mental frame of reference for our study of
mechanics.
The concept of space is associated with the notion of the position of a point P.
The position of P can be defined by three lengths measured from a certain reference
point, or origin, in three given directions. These lengths are known as the coordinates
of P.
To define an event, it is not sufficient to indicate its position in 3 space. The
time of the event should also be given.
The concept of mass is used to characterize and compare bodies on the basis
of certain fundamental mechanical experiments. Two bodies of the same mass, for
example, will be attracted by the earth in the same manner; they will also offer the
same resistance to a change in translational motion.
A force represents the action of one body on another. It can be exerted by
actual contact or at a distance, as in the case of gravitational forces and magnetic
forces. A force is characterized by its point of application, its magnitude, and its
direction; a force is represented by a vector.
In Newtonian mechanics, space, time, and mass are absolute concepts,
independent of each other. (This is not true in relativistic mechanics, where the time
of an event depends upon its position, and where the mass of a body varies with its
velocity.) On the other hand, the concept of force is not independent of the other
three. Indeed, one of the fundamental principles of Newtonian mechanics listed
below indicates that the resultant force acting on a body is related to the mass of the
body and to the manner in which its velocity varies with time.
A particle has a mass, but a size that can be neglected. For example, the size of
the earth is insignificant compared to the size of its orbit, and therefore the earth can
be modeled as a particle when studying its orbital motion. A rigid body can be
considered as a combination of a large number of particles in which all the particles
remain at a fixed distance from one another, both before and after applying a load.
This model is important because the body’s shape does not change when a load is
applied, and so we do not have to consider the type of material from which the body
is made. In most cases the actual deformations occurring in structures, machines,
mechanisms, and the like are relatively small, and the rigid-body assumption is
suitable for analysis. A concentrated force represents the effect of a loading which is
assumed to act at a point on a body.
Newton’s Laws of Motion
First Law
If the resultant force acting on a particle is zero, the particle will remain at rest (if
originally at rest) or will move with constant speed in a straight line (if originally in
motion).
Second Law
If the resultant force acting on a particle is not zero, the particle will have an
acceleration proportional to the magnitude of the resultant and in the direction of
this resultant force.
Third Law
The forces of action and reaction between bodies in contact have the same
magnitude, same line of action, and opposite.
Gravity Principle
This states that two particles of mass M and m are mutually attracted with equal and
opposite forces F and -F of magnitude F given by the formula
F=G
Mm
(N)
r2
where r = distance between the two particles, G = universal constant called the
constant of gravitation.
A particular case of great importance is that of the attraction of the earth on a particle
located on its surface. The force F exerted by the earth on the particle is then defined
as the weight W of the particle. Taking M equal to the mass of the earth, m equal to
the mass of the particle, and r equal to the radius R of the earth, and introducing the
constant,
F=G
M
R2
(N)
The value of R in formula depends upon the elevation of the point considered; it also
depends upon its latitude, since the earth is not truly spherical. The value of g
therefore varies with the position of the point considered. As long as the point
actually remains on the surface of the earth, it is sufficiently accurate in most
engineering computations to assume that g equals 9.81 m/s2 or 32.2 ft/s2.
VECTOR AND SACLAR
Many physical quantities in engineering mechanics are measured using either
scalars or vectors. A scalar is any positive or negative physical quantity that can be
completely specified by its magnitude. Examples of scalar quantities include length,
mass, and time. A vector is any physical quantity that requires both a magnitude and
a direction for its complete description. Examples of vectors encountered in statics
are force, position, and moment. A vector is shown graphically by an arrow. The
length of the arrow represents the magnitude of the vector, and the angle 𝜃 between
the vector and a fixed axis defines the direction of its line of action. The head or tip
of the arrow indicates the sense of direction of the vector.
Vectors are represented by arrows in the illustrations and will be distinguished
from scalar quantities in this text through the use of boldface type (P).
A vector used to represent a force acting on a given particle has a well-defined
point of application, namely, the particle itself. Such a vector is said to be a fixed, or
bound, vector and cannot be moved without modifying the conditions of the
problem. Other physical quantities, however, such as couples are represented by
vectors which may be freely moved in space; these vectors are called free vectors.
Still other physical quantities, such as forces acting on a rigid body, are represented
by vectors which can be moved, or slid, along their lines of action; they are known
as sliding vectors. Two vectors which have the same magnitude and the same
direction are said to be equal, whether or not they also have the same point of
application equal vectors may be denoted by the same letter. The negative vector of
a given vector P is defined as a vector having the same magnitude as P and a direction
opposite to that of P the negative of the vector P is denoted by -P. The vectors P and
-P are commonly referred to as equal and opposite vectors.
FORCE
Push or pull of an object is considered a force. Push and pull come from the
objects interacting with one another. Terms like stretch and squeeze can also be used
to denote force. In Physics, force is defined as:
The push or pull on an object with mass causes it to change its velocity.
Force is an external agent capable of changing a body’s state of rest or motion.
It has a magnitude and a direction. The direction towards which the force is applied
is known as the direction of the force, and the application of force is the point where
force is applied. The Force can be measured using a spring balance. The SI unit of
force is Newton(N).
The force is an important factor in the field of Mechanics, which may be broadly
defined as an agent which produces or tends to produce, destroys or tends to destroy
motion. e.g., a horse applies force to pull a cart and to set it in motion. Force is also
required to work on a bicycle pump. In this case, the force is supplied by the
muscular power of our arms and shoulders. A force may produce the following
effects in a body, on which it acts:
1. It may change the motion of a body. i.e. if a body is at rest, the force may set it in
motion. And if the body is already in motion, the force may accelerate it.
2. It may retard the motion of a body.
3. It may retard the forces, already acting on a body, thus bringing it to rest or in
equilibrium.
4. It may give rise to the internal stresses in the body, on which it acts.
In order to determine the effects of a force, acting on a body, we must know the
following characteristics of a force:
1. Magnitude of the force (i.e., 100 N, 50 N, 20 kN, 5 kN, etc.)
2. The direction of the line, along which the force acts (i.e., along OX, OY, at 30°
North of East etc.). It is also known as line of action of the force.
3. Nature of the force (i.e., whether the force is push or pull). This is denoted by
placing an arrow head on the line of action of the force.
4. The point at which (or through which) the force acts on the body.
When two or more forces act on a body, they are called to form a system of forces.
Following systems of forces are important from the subject point of view:
1. Coplanar forces. The forces, whose lines of action lie on the same plane, are known
as coplanar forces.
2. Collinear forces. The forces, whose lines of action lie on the same line, are known
as collinear forces.
3. Concurrent forces. The forces, which meet at one point, are known as concurrent
forces. The concurrent forces may or may not be collinear.
4. Coplanar concurrent forces. The forces, which meet at one point and their lines
of action also lie on the same plane, are known as coplanar concurrent forces.
5. Coplanar non-concurrent forces. The forces, which do not meet at one point, but
their lines of action lie on the same plane, are known as coplanar non-concurrent
forces.
6. Non-coplanar concurrent forces. The forces, which meet at one point, but their
lines of action do not lie on the same plane, are known as non-coplanar concurrent
forces.
7. Non-coplanar non-concurrent forces. The forces, which do not meet at one point
and their lines of action do not lie on the same plane, are called non-coplanar nonconcurrent forces.
RESOLUTION OF FORCE INTO COMPONENT
Two or more forces acting on a particle may be replaced by a single force which
has the same effect on the particle. Conversely, a single force F acting on a particle
may be replaced by two or more forces which, together, have the same effect on the
particle. These forces are called the components of the original force F, and the
process of substituting them for F is called resolving the force F into components
Clearly, for each force F there exist an infinite number of possible sets of components.
Sets of two components P and Q are the most important as far as practical
applications are concerned. But, even then, the number of ways in which a given
force F may be resolved into two components is unlimited.
1. One of the Two Components, P, Is Known. The second component, Q, is obtained
by applying the triangle rule and joining the tip of P to the tip of F the magnitude
and direction of Q are determined graphically or by trigonometry. Once Q has been
determined, both components P and Q should be applied at A.
2. The Line of Action of Each Component Is Known. The magnitude and sense of the
components are obtained by applying the parallelogram law and drawing lines,
through the tip of F, parallel to the given lines of action. This process leads to two
well-defined components, P and Q, which can be determined graphically or
computed trigonometrically by applying the law of sines.
RECTANGULAR COMPONENTS OF FORCE
The force F has been resolved into a component Fx along the x axis and a
component Fy along the y axis. The parallelogram drawn to obtain the two
components is a rectangle, and Fx and Fy are called rectangular components.
The x and y axes are usually chosen horizontal and vertical, respectively, they may,
however, be chosen in any two perpendicular directions. In determining the
rectangular components of a force, the student should think of the construction lines
as being parallel to the x and y axes, rather than perpendicular to these axes. This
practice will help avoid mistakes in determining oblique components. Two vectors
of unit magnitude, directed respectively along the positive x and y axes, will be
introduced at this point. These vectors are called unit vectors and are denoted by i
and j, respectively. The rectangular components Fx and Fy of a force F may be
obtained by multiplying respectively the unit vectors i and j by appropriate scalars.
Fx = Fx i
and
Fy = Fy i
F = Fx i + F y j
While the scalars Fx and Fy may be positive or negative, depending upon the
sense of Fx and of Fy, their absolute values are respectively equal to the magnitudes
of the component forces Fx and Fy. The scalars Fx and Fy are called the scalar
components of the force F, while the actual component forces Fx and Fy should be
referred to as the vector components of F. However, when there exists no possibility
of confusion, the vector as well as the scalar components of F may be referred to
simply as the components of F. We note that the scalar component Fx is positive when
the vector component Fx has the same sense as the unit vector i (i.e., the same sense
as the positive x axis) and is negative when Fx has the opposite sense. A similar
conclusion may be drawn regarding the sign of the scalar component Fy.
Denoting by F the magnitude of the force F and by 𝜃 the angle between F and
the x axis, measured counterclockwise from the positive x axis, we may express the
scalar components of F as follows:
Fx = F cos θ and
Fy = F sin θ
The relations obtained hold for any value of the angle 𝜃 from 0° to 360° and
that they define the signs as well as the absolute values of the scalar components F x
and Fy. When a force F is defined by its rectangular components Fx and Fy, the angle
defining its direction can be obtained by writing
tan θ =
Fy
Fx
When three or more forces are to be added, no practical trigonometric solution can
be obtained from the force polygon which defines the resultant of the forces. In this
case, an analytic solution of the problem can be obtained by resolving each force
into two rectangular components. Consider, for instance, three forces P, Q, and S
acting on a particle A.
R=P+Q+R
Resolving each force into its rectangular components,
Rx i + Ry j = Px i + Py j + Qx i + Qy j + Sx i + Sy j
Rx i + Ry j = (Px + Qx + Sx )i + (Py + Qy + Sy ) j
from which it follows that
Rx = Px + Qx + Sx
Ry = Py + Qy + Sy
The scalar components Rx and Ry of the resultant R of several forces acting on a
particle are obtained by adding algebraically the corresponding scalar components
of the given forces.
EQUILIBRIUM OF PARTICLES
A particle which is acted upon by two forces will be in equilibrium if the two
forces have the same magnitude and the same line of action but opposite sense. The
resultant of the two forces is then zero. In such a case, the net effect of the given
forces is zero, and the particle is said to be in equilibrium. We thus have the
following definition: When the resultant of all the forces acting on a particle is zero,
the particle is in equilibrium.
Sir Isaac Newton formulated three fundamental laws upon which the science
of mechanics is based. The first of these laws can be stated as follows: If the resultant
force acting on a particle is zero, the particle will remain at rest (if originally at rest)
or will move with constant speed in a straight line (if originally in motion).
As an example, consider the 75-kg crate shown in the space diagram of Figure.
This crate was lying between two buildings, and it is now being lifted onto a truck,
which will remove it. The crate is supported by a vertical cable, which is joined at A
to two ropes which pass over pulleys attached to the buildings at B and C. It is desired
to determine the tension in each of the ropes AB and AC. In order to solve this
problem, a free-body diagram showing a particle in equilibrium must be drawn.
Since we are interested in the rope tensions, the free-body diagram should include
at least one of these tensions or, if possible, both tensions. Point A is seen to be a good
free body for this problem. The free-body diagram of point A is shown in Figure b.
It shows point A and the forces exerted on A by the vertical cable and the two ropes.
The force exerted by the cable is directed downward, and its magnitude is equal to
the weight W of the crate.
W = mg = (75 kg)(9.81 m/s2 )=736 N
and indicate this value in the free-body diagram. The forces exerted by the two ropes
are not known. Since they are respectively equal in magnitude to the tensions in rope
AB and rope AC, we denote them by TAB and TAC and draw them away from A in the
directions shown in the space diagram. No other detail is included in the free-body
diagram. Since point A is in equilibrium, the three forces acting on it must form a
closed triangle when drawn in tip-to-tail fashion. This force triangle has been drawn
in Figure c. The values TAB and TAC of the tension in the ropes may be found
graphically if the triangle is drawn to scale, or they may be found by trigonometry.
If the latter method of solution is chosen, we use the law of sines and write
TAB
TAC
736
=
=
sin 60°
sin 40°
sin 80°
When a particle is in equilibrium under three forces, the problem can be
solved by drawing a force triangle. When a particle is in equilibrium under more
than three forces, the problem can be solved graphically by drawing a force polygon.
If an analytic solution is desired, the equations of equilibrium. These equations can
be solved for no more than two unknowns; similarly, the force triangle used in the
case of equilibrium under three forces can be solved for two unknowns. The more
common types of problems are those in which the two unknowns represent (1) the
two components (or the magnitude and direction) of a single force, (2) the
magnitudes of two forces, each of known direction. Problems involving the
determination of the maximum or minimum value of the magnitude of a force are
also encountered.
FORCES IN SPACE
Consider a force F acting at the origin O of the system of rectangular
coordinates x, y, z. To define the direction of F, we draw the vertical plane OBAC
containing F. This plane passes through the vertical y axis; its orientation is defined
by the angle f it forms with the xy plane. The direction of F within the plane is defined
by the angle 𝜃𝑦 that F forms with the y axis. The force F may be resolved into a
vertical component Fy and a horizontal component Fh; this operation, shown in
Figure b, is carried out in plane OBAC according to the rules developed in the first
part of the chapter. The corresponding scalar components are
Fy = F cos θy
Fh = F sin θy
But Fh may be resolved into two rectangular components Fx and Fz along the x and z
axes, respectively.
Fx = Fh cos φ = F sin θy cos φ
Fz = Fh sin φ = F sin θy sin φ
The given force F has thus been resolved into three rectangular vector components
Fx, Fy, Fz, which are directed along the three coordinate axes. Applying the
Pythagorean theorem to the triangles OAB and OCD of Figure,
F2 = (OA)2 = (OB)2 + (BA)2 = F2y + F2h
F2h = (OC)2 = (OD)2 + (DC)2 = F2x + F2z
The relationship existing between the force F and its three components Fx, Fy, Fz is
more easily visualized if a “box” having Fx, Fy, Fz for edges. The force F is then
represented by the diagonal OA of this box. Figure b shows the right triangle OAB
used to derive the first of the formulas: Fy = F cos θy . In Figures a and c, two other
right triangles have also been drawn: OAD and OAE. These triangles are seen to
occupy in the box positions comparable with that of triangle OAB. Denoting by θx
and θz , respectively, the angles that F forms with the x and z axes, we can derive two
formulas similar to Fy = F cos θy .
Fx = F cos θ𝑥
F𝑦 = F sin θy
F𝑧 = F sin θz
The three angles θ𝑥 , θ𝑦 , θ𝑧 define the direction of the force F; they are more
commonly used for this purpose than the angles uy and f introduced at the beginning
of this section. The cosines of θ𝑥 , θ𝑦 , θ𝑧 are known as the direction cosines of the
force F. Introducing the unit vectors i, j, and k, directed respectively along the x, y,
and z axes.
𝑭 = Fx 𝒊 + Fy 𝒋 + Fz 𝒌
The angle a force F forms with an axis should be measured from the positive side of
the axis and will always be between 0 and 180°. An angle ux smaller than 90°
(acute) indicates that F (assumed attached to O) is on the same side of the yz plane
as the positive x axis; cos θ𝑥 and Fx will then be positive. An angle θ𝑥 larger than 90°
(obtuse) indicates that F is on the other side of the yz plane; cos θ𝑥 and Fx will then
be negative. In Example 1 the angles ux and uy are acute, while θ𝑧 is obtuse;
consequently, Fx and Fy are positive, while Fz is negative.
F = F (cos θx i+ cos θy j+ cos θz k)
which shows that the force F can be expressed as the product of the scalar F and the
vector.
λ = cos θx i + cos θy j + cos θz k
Clearly, the vector 𝝀 is a vector whose magnitude is equal to 1 and whose direction
is the same as that of F. The vector 𝝀 is referred to as the unit vector along the line of
action of F. It follows from that the components of the unit vector 𝝀 are respectively
equal to the direction cosines of the line of action of F:
cos θx = Ix
cos θy = Iy
cos θz = Iz
The three angles θ𝑥 , θ𝑦 , θ𝑧 are not independent. Recalling that the sum of the squares
of the components of a vector is equal to the square of its magnitude.
I2x + I2y + I2z = 1
cos2 θx + cos2 θy + cos2 θz = 1
When the components Fx, Fy, Fz of a force F are given, the magnitude F of the force is
obtained from equation. The relations can then be solved for the direction cosines,
cos θx =
Fx
F
cos θy =
Fy
F
cos θz =
Fz
F
In many applications, the direction of a force F is defined by the coordinates of two
points, M(x1, y1, z1) and N(x2, y2, z2), located on its line of action. Consider the vector
⃗⃗⃗⃗⃗⃗⃗
𝑀𝑁 joining M and N and of the same sense as F. Denoting its scalar components by
dx, dy, dz, respectively.
⃗⃗⃗⃗⃗⃗⃗
MN = dx i + dy j + dz k
The unit vector L along the line of action of F (i.e., along the line MN) may be
⃗⃗⃗⃗⃗⃗⃗ by its magnitude MN. Substituting for ⃗⃗⃗⃗⃗⃗⃗
obtained by dividing the vector MN
MN from
and observing that MN is equal to the distance d from M to N,
L=
⃗⃗⃗⃗⃗⃗⃗
MN 1
= (d i + dy j + dz k)
MN d x
Recalling that F is equal to the product of F and L,
F = FL =
F
(d i + dy j + dz k)
d x
from which it follows that the scalar components of F are, respectively.
Fx =
Fdx
d
Fy =
Fdy
d
Fz =
Fdz
d
The relations considerably simplify the determination of the components of a force
F of given magnitude F when the line of action of F is defined by two points M and
N. Subtracting the coordinates of M from those of N, we first determine the
components of the vector ⃗⃗⃗⃗⃗⃗⃗
MN and the distance d from M to N:
dx = x2 - x1
dy = y2 - y1
dz = z2 - z1
d = √d2x + d2y + d2z
Substituting for F and for dx, dy, dz, and d into the relations, we obtain the
components Fx, Fy, Fz of the force.