Department:
AERONAUTICAL ENGINEERING
Course:
UCK203E-STRENGTH OF
MATERIALS I
Week: 2
Lecturer: Assist. Prof. Dr. SAMUEL MOVEH
E-mail: samoveh@gelisim.edu.tr
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Text Books-:
1. Strength of materials by G. H. Ryder, Mc Millan India Ltd.,
2. Elements of Strength of Materials by S.P. Timoshenko and D.H. Young, East
West Press Pvt. Ltd.,
Ref. Books:1. Introduction to solid mechanics by H. Shames, Prentice Hall India, New Delhi
2. Engineering mechanics of solid by E. P. Popov, Prentice Hall India, New Delhi
3. Engineering Physical Metallurgy, by Y. Lakhtin, MIR pub, Moscow
Concept of Modeling and Basic Principles
Why do we need models?
Model (Definition):
A representation of a
system that allows for
investigation of the
properties of the
system and, in some
cases, prediction of
future outcomes.
Models provide a framework for conceptualizing our
ideas about the behavior of a particular system
Models allow us to find structure in complex systems
and to investigate how different factors interact
Models can play an important role in informing
policies:
i. By providing understanding of underlying causes for a
complex phenomenon
ii. By predicting the future
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PRINCIPLE AND ELEMENTS OF MODELLING
Design should be traceable to the
requirements model.
Always consider the architecture
of the system to be built
Understand the problem you’re
trying to
Understand the basic design
principles and concepts
Elements of Models
 Decisions variables
 Constraints
 Objective function
 Parameter
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SOME EXAMPLES OF AIRCRAFT MODELLINGS
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MODEL CLASSIFICATIONS
 Some models are replicas of the physical properties
(relative shape, form, and weight) of the object they
represent.
 Others are physical models but do not have the same
physical appearance as the object of their
representation.
 A third type of model deals with symbols and numerical
relationships and expressions.
physical models
Each of these fits within an overall classification of four
main categories: physical models, schematic models,
verbal models, and mathematical models.
schematic models
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TYPES OF MATHEMATICAL MODELS
1. DESCRIPTIVE MODELS
Descriptive models are used to merely describe something mathematically. Common statistical models in this
category include the mean, median, mode, range, and standard deviation
2. OPTIMIZATION MODELS
Optimization models are used to find an optimal solution. These models share certain common characteristics.
Knowledge of these characteristics enables us to recognize problems that can be solved using linear programming.
3. DETERMINISTIC MODELS
Deterministic models are those for which the value of their variables is known with certainty.
4. SPECIFIC MODELS.
5. GENERAL MODELS
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Introduction to Stress
 Stress is the force applied to a material, divided by the material's cross-sectional area.
TYPES OF STRESSES :
 Only two basic stresses exists :
(1) Normal stress and
(2) Shear stress.
 Other stresses either are similar to these basic stresses or are a
combination of this e.g. bending stress is a combination tensile,
compressive and shear stresses. Torsional stress, as encountered
in twisting of a shaft is a shearing stress. Let us define the
normal stresses and shear stresses in the following sections.
Area
σ
σ
 Normal stresses : We have defined stress as force per unit area. If
the stresses are normal to the areas concerned, then these are
termed as normal stresses. The normal stresses are generally
denoted by a Greek letter (σ)
 This is also known as uniaxial state of stress, because the
stresses acts only in one direction
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 However, such a state rarely exists, therefore we have biaxial
Tensile or compressive Stresses:
and triaxial state of stresses where either the two mutually
 The normal stresses can be either tensile or compressive
perpendicular normal stresses acts or three mutually
whether the stresses acts out of the area or into the area
perpendicular normal stresses acts as shown in the figures
below :
Units: SI Unit: N/m2 (Pa), kPa, MPa,
Gpa
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Shear Stresses:
Let us consider now the situation, where
the cross – sectional area of a block of
material is subject to a distribution of
forces which are parallel, rather than
normal, to the area concerned.
Such forces are associated with a
shearing of the material, and are
referred to as shear forces. The resulting
stress is known as shear stress.
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Introduction to Strain
 When a single force or a system force acts on a body, it undergoes some deformation. This deformation
per unit length is known as strain.
Poisson’s Ratio;
 The ratio lateral strain to longitudinal strain produced by a single stress
is known as Poisson‟s ratio. Symbol used for poisson‟s ratio is nu or 1/
m.
Shear Strain
Elasticity;
 The property of material by virtue of which it returns to its  The shear strain or “slide‟ is expressed by angle φ and it
can be defined as the change in the right angle. It is
original shape and size upon removal of load is known as
measured in radians and is dimensionless in nature.
elasticity.
 Mathematically strain may be defined as
deformation per unit length.
 So, Strain=Elongation/Original length
Hooks Law
 states that within elastic limit stress is proportional to strain.
𝑆𝑇𝑅𝐸𝑆𝑆
E = 𝑆𝑇𝑅𝐴𝐼𝑁
E is Young‟s Modulus OR Modulus of Elasticity is defined as the ratio of stress to strain within elastic limit.
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Modulus of Rigidity
 For elastic materials it is found that shear stress is
proportional to the shear strain within elastic limit. The ratio
is called modulus rigidity. It is denoted by the symbol G or C.
Bulk modulus (K):
 It is defined as the ratio of uniform stress intensity to the volumetric strain. It is
denoted by the symbol K
 A stress-strain curve is a graphical way to show the reaction of a material
when a load is applied. It shows a comparison between stress and strain.
 The stress-strain diagram provides a graphical measurement of the strength
and elasticity of the material. Also, the behaviour of the materials can be
studied with the help of the stress-strain diagram, which makes it easy to
understand the application of these materials.
(i) Proportional Limit
It is the region in the stress-strain curve that obeys Hooke’s Law. In this limit,
the stress-strain ratio gives us a proportionality constant known as Young’s
modulus. The point OA in the graph represents the proportional limit.
(ii) Elastic Limit
It is the point in the graph up to which
the material returns to its original
position when the load acting on it is
completely removed. Beyond this limit,
the material doesn’t return to its
original position, and a plastic
deformation starts to appear in it.
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(iii) Yield Point
The yield point is defined as the point at which the material
starts to deform plastically. After the yield point is passed,
permanent plastic deformation occurs. There are two yield
points (i) upper yield point (ii) lower yield point.
Elastic Moduli of Materials
The following table lists Young’s modulus, shear
modulus and bulk modulus for common
materials.
(iv) Ultimate Stress Point
It is a point that represents the maximum stress that a
material can endure before failure. Beyond this point,
failure occurs.
(v) Fracture or Breaking Point
It is the point in the stress-strain curve at which the failure
of the material takes place.
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Exercise 1:
 Find the modulus of elasticity of a rod , which tapers uniformly from
30mm to 15mm diameter in a length of 350mm. The rod is subjected to
an axial load of 5.5KN and extension f the rod is ‘0.025mm
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