MEC411-MECHANICS OF MATERIALS
CHAPTER 1
STRESS & STRAIN
INTRODUCTION
The study on strength of materials refer to various
methods of calculating the stresses and strains in structural
members (beams, columns, shafts etc.)
Knowledge of stresses and deflections allows for the safe
design of structures that are capable of supporting their
intended loads.
TYPE OF FORCES
STRESS
Stress (𝝈): Stress is the applied force acting perpendicular to the
surface area or system of forces that tends to deform a body.
Stress measures the intensity of the force per given area
o Normal stress (𝜎) results from the normal force 𝑁 and/or bending
moment 𝑀𝐵
o Shear stress (𝜏) results from shear stress 𝑉 and/or torsional
moment 𝑀𝑡
TYPE OF STRESS
NORMAL STRESS
When a force is applied to an elastic body, the body deforms. The
way in which the body deforms depends upon the type of force
applied to it.
Compression force makes the body shorter.
A tensile force makes the body longer
Tensile and compressive forces are called DIRECT FORCES
Stress is the force per unit area upon which it acts.
….. Unit is Pascal (Pa) or
( Simbol – Sigma)
Note: Most of engineering fields used kPa, MPa, GPa.
SHEAR STRESS
Shear stress – results when a force tends to make part of
the body or one side of a plane slide past the other.
When a pair of force cut a material
When a material is punched
When a beam has a transverse load
Shear stress is the force per unit area acting tangent to the
surface area or the cross sectional area of the material being
cut.
•Shear stress,
symbol is called Tau
The sign convention for shear force and stress is based on how it
shears the materials as shown below.
SINGLE SHEAR STRESS
DOUBLE SHEAR STRESS
EXAMPLE
The joint is fastened using 2 bolts as shown in the Figure.
Determine the required diameter of the bolts if allowable
shear stress for the bolts is
= 110 MPA.
allow
STRAIN
Normal strain (ε) is the deformation of a body which involved
elongation or contraction. It is a dimensionless quantity.
SHEAR STRAIN
Shear strain is a strain which involved a shear
deformation and changes in shape (angle) i.e.
body twist due to torsion or a distorted
cuboid as shown in Figure below. Strain
changes the angles of an object.
SHEAR MODULUS/MODULUS OF RIGIDITY
Shear modulus also known as Modulus of
rigidity is the measure of the rigidity of the
body, given by the ratio of shear stress to
shear strain.
STRESS & STRAIN DIAGRAM
• It is a tool for understanding material
behaviour under load. A stress strain
diagram help engineers to select the right
materials for specific loading conditions.
• It is a graph that represents how a material
behaves under an increasing load and used
by engineers when selecting materials for
specific designs.
A stress-strain diagram generally contains three regions:
•Elastic region: This portion is generally represented as a linear
relationship between stress and strain. If the load is released the
specimen will return to its original dimensions.
•Plastic region: In this portion, the specimen begins to yield. The
maximum strength of the specimen occurs in this zone. The
specimen endures some permanent deformation that remains
after the load is released.
•Rupture: The point at which a specimen breaks into two parts.
STRESS STRAIN DIAGRAM
Elastic limit
∙Upon reaching this point, if load is removed, the
specimen still return to original shape
Yielding
∙ A slight increase in stress above the elastic limit will
result in breakdown of the material and cause it to
deform permanently.
∙This behaviour is called yielding
∙ The point where the stress-strain diagram becomes
almost horizontal is called the yield point, and the
corresponding stress is known as the yield stress or
yield strength.
STRESS STRAIN DIAGRAM
Ultimate Stress
•The ultimate stress or ultimate strength, as it is often called, is the
highest stress on the stress-strain curve.
Necking
• At the ultimate stress, the cross-sectional area begins to decrease
• it is caused by slip planes formed within material
• As a result, “necking” begins to form
• Specimen breaks at FRACTURE STRESS
MODULUS OF ELASTICITY (E)
As seen in the Figure, the stress-strain diagram is a straight line from the origin
to a point called the proportional limit. This plot is a manifestation of Hooke’s
law
Stress is proportional to strain; that is,
σ= E Є
where E is material property
known as the modulus of
elasticity or Young’s modulus.
The units of E are the same as
the units of, Pa or psi. For
steel, E =29×106 psi, or 200
GPa, approximately.
Note that Hooke’s law does not apply
to the entire diagram;its validity ends
at the proportional limit. Beyond this
point, stress is no longer proportional
to strain.
MODULUS OF ELASTICITY (E)
•Elastic materials always spring back into shape when released. They
also obey HOOKE’s LAW.
•This is the law of spring which states that deformation is directly
proportional to the force. F/x = stiffness = kN/m
•The stiffness is different for different material and sizes of the material. We
may eliminate the size by using stress and strain instead of force and
deformation:
•The stiffness is now in terms of stress and strain only and this
constant is called the MODULUS of ELASTICITY (E)
DUCTILE MATERIALS
Ductility:
- Measure of the material property to deform before failure.
- Ductile materials can be pulled or drawn into pipes, wire,
and other structural shapes
- Examples of ductile material :
low carbon steel
aluminum
copper
brass
STRESS & STRAIN DIAGRAM: DUCTILE
MATERIAL
BRITTLE MATERIALS
Brittleness:
- Measure of the material’s inability to deform before failure.
- The opposite of ductility.
- Example of brittle materials :
glass
high carbon steel
ceramics
STRESS & STRAIN: BRITTLE MATERIAL
ELASTIC VS. PLASTIC
DOUBLE SHEAR
∙Consider a pin joint with a support on both ends as shown. This
is called CLEVIS and CLEVIS PIN
∙ By balance of force, the force in the two supports is F/2 each
∙The area sheared is twice the cross section of the pin
∙So it takes twice as much force to break the pin as for a case of
single shear
∙Double shear arrangements doubles the maximum force allowed
in the pin