Engineering mechanics. Statics of Rigid Bodies

1. INTRODUCTION TO STATICS
Dr. Ahmet KIRLI

2. Course Description: The course covers the following topics; statics of
particles: forces in plane, forces in space, equilibrium, moment of a force,
moment of a couple, equivalent systems of forces on rigid bodies, equilibrium
in two dimensions, equilibrium in three dimensions, distributed forces:
centroids and center of gravity, analysis of structures: trusses, frames and
machines, internal forces in beams and cables, friction, moments of inertia of
areas, moments of inertia of masses, method of virtual work.
Course Objectives:
1) To provide definition of force and moment vectors and give necessary
vector algebra
2) To explain the concept of equilibrium of particles and rigid bodies in plane
and 3D space
3) To give information about support types and to give ability to calculate
support reactions
4) To explain the equilibrium of structures and internal forces in trusses, and
frames
5) To give information about distributed loads
6) To provide information on moment of inertia
7) To explain virtual work concept.

3. Chapter 1 INTRODUCTION
Mechanics - the physical science which describes or predicts the
conditions of rest or motion of bodies under the action of forces.
A.Rigid bodies
1.Statics
2.Dynamics
B.Deformable bodies
C.Fluid Mechanics
1. Compressible - gas
2. Incompressible - liquids
Mechanics is the foundation of most engineering sciences
mechanics is an applied science.
The purpose of mechanics is to explain and predict physical phenomena
and thus to lay the foundations for engineering applications.

4. Mechanics Of Rigid Bodies,
It is the analysis of the actions of forces,
1. Statics : the study of systems that are in a state of constant motion, either at
rest or moving at a constant velocity
and
2. Dynamics: the study of systems in which acceleration is present.

5. What is Rigid Body?
Rigid body - a body is considered rigid when the relative movement between its parts are
negligible.
Assumptions
In Statics we will assume the bodies to be perfectly rigid, no deformation.
This is never true in the real world, everything deforms a little when a load is applied.
These deformations are small and will not significantly affect the conditions of equilibrium
or motion, so we will neglect the deformations.

6. Statics: are they in rest or in constant velocity?
or what should be the force to be able to identify this system as static?
Strength od Materials: What should be the cross section area?
Or what should be the material properties?
Dynamics: steady state response?
Transient response
Frcition dynamics?
Vibration?

7. 1- space - the geometric region occupied by bodies whose positions are described
by linear and angular measurements relative to a coordinate system.
2- time - the measure of the succession of events
3- mass - the measure oaf the inertia of a body,
It is the resistance to a change of motion.
4- force - the action of one body on another
Basic concepts: space, time, mass, force
Coordinate Systems:
Cartesian: x, y, z
Cylindrical: r, θ,z
Spherical: r, θ,ϕ

8. Newton developed the fundamentals of mechanics.
The concepts above, space, time, and mass are absolute, independent of each other in Newtonian
Mechanics.
In physics, classical mechanics (also known as Newtonian mechanics) is one of two major sub-fields
of mechanics. The other sub-field is quantum mechanics.
Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the
influence of a system of forces.
Newtonian mechanics, along with Lagrangian mechanics and Hamiltonian mechanics, as the three main
formalisms of classical mechanics.
Classical mechanics describes the motion of macroscopic objects
such as spacecraft, planets, stars and galaxies.
classical mechanics are sub-fields, including those that describe the behavior of solids, liquids and gases.
Classical mechanics provides extremely accurate results when studying large objects and speeds not
approaching the speed of light.
When the objects being examined are sufficiently small, it becomes necessary to introduce the other
major sub-field of mechanics: quantum mechanics.

9. Hamiltonian Mechanics - impulse and momentum
Langrangian Mechanics - energy
Newtonian Mechanics - forces
Newton's laws of motion
2nd
Law - the acceleration of a particle is proportional to the resultant force acting on it and is
in the direction of this force.
(1)
F = ma
3rd
Law - the forces of action and reaction between interacting bodies are equal in magnitude,
opposite in direction, and act along the same line of action (Collinear).
1st
Law - A particle remains at rest or continues to move in a straight line with a constant
speed if there is no unbalanced force acting on it (resultant force = 0).

10. System of Units
Base units are units of length, mass and time.
Length Mass Time
SI Units Meter (m) Kilogram (kg) Second (s)
English Units Foot (ft) Slug (slug) Second (s)
Force: Newton (N)
1 N = (1 kg)(1 m/ s²)
1 Newton is the force required to give a mass of 1 kg an acceleration of 1 m/ s².
Weight is a force. The weight of 1 kg Mass is:
W = mg
W = (1 kg)(9.81 m/ s²)
W = 9.81 N
Pounds in EU

11. Conversion from one System of Units to Another:
1 ft = 0.3048 m
1 lb = 4.448 N
1 slug = 1 lb s² /ft = 14.59 kg
What is the mass of an object that weighs 19.62 Newtons?
This is the difference between Mass and Weight.
F = mg
19.62 N = m (9.81 m/ s²)
m = 19.62 N /(9.81 m/ s²)= 2 kg
Example:

12. Scalars and Vectors
Vectors – A mathematical quantity possessing magnitude and direction.
Scalar – A mathematical quantity possessing magnitude only.
Name some vectors: forces, velocity, displacement
Name some scalars: Area, volume, mass energy
Representation of vector
Bold R – Word Processors  Book uses this.
Arrow R

– Long Hand, Word Processors
Underline R – Long Hand, Typewriter, Word Processors
Magnitude of a Vector
R Book uses italics for all scalars

13. Types of Vectors
1). Fixed (or bound) vectors – a vector for which a unique point of
application is specified and thus cannot be moved without
modifying
the conditions of the problem.
2). Free vector – a vector whose action is not confined to or associated
with a unique line in space. (couple)
3). Sliding vector – a vector for which a unique line in space (line of
action) must be maintained.
For 2 vectors to be equal they must have the same:
1). Magnitude P P
2). Direction
They do not need to have the same point of application.
A negative vector of a given vector has same magnitude but opposite
direction.
P -P
P and –P are equal and opposite P + (-P) = 0

14. Vector Operations
Product of a scalar and a vector
P + P + P = 4P (the number 4 is a scalar)
This is a vector in the same direction as P but 4 times as long.
(+n)P = vector same direction as P, n times as long
(-n)P = vector opposite direction as P, n times as long
Vector Addition
The sum of 2 vectors can be obtained by attaching the 2 vectors to the
same point and constructing a parallelogram – Parallelogram
law.
R
Q
R = P + Q
P
R = resultant vector
Note: The magnitude of P + Q is not usually equal to Q
P  .
Addition of vectors is communative: P + Q = Q + P

15. Triangle Rule
P
Q R R Q
P
Let’s add 3 vectors!
Parallelogram Law
Q Q
Q
P P P
S S S
R1 = Q + P R = R1 + S = Q + P + S
Triangle Rule
Q Q P P S
R
1
Q R
1
P R
S S
R = R1 + S = Q + P + S
R1 = Q + P
1
R 1
R

R


16. Polygon Rule – Successive applications of triangle rule.
Q
Q P
S
P R
S
Note: P +Q + S = (P + Q) + S = P + (Q + S) vector addition is associative
Vector Subtraction – the addition of the corresponding negative vector
P – Q = P + (-Q)
Resolution of vector into components
A single vector can be represented by 2 or more vectors. These vectors
are components of the original vector. Finding these is called
resolving the vector into its components.
There is an infinite number of ways to resolve one vector.
etc.
P
R
-Q
P
P1
P2 P1
P2
P

17. When would #2 happen? When you are given a
coordinate system!
What are the x and y components of P if P = 1000 N, = 30o
Px = P cos 30o
= 866 N Py = P sin 30o
= 500 N
Note: Given Px and Py, what is P?
P2
= Px
2
+ Py
2
=8662
+ 5002
= 1000 N
P

x
y

18. EXAMPLE
1). Given: The fixed structure shown below.
P = 500 N
T = 200 N
Find: Combine P and T into a single force R



4
.
48
75
cos
5
3
75
sin
5
tan






AD
BD
Law of cosines:
N
R
R
c
ab
b
a
c
5
.
396
)
4
.
48
cos(
)
500
)(
200
(
2
500
200
)
cos(
2
2
2
2
2
2
2








Law of sines:

4
.
48
sin
5
.
396
sin
200



2
.
22


3m
T
P
A
B
C D
5
m
 75o
R
P = 500
T = 200


