Lecture - 4
Review
In previous lectures we have discussed about engineering design, design cycle, design process
etc. The importance of other factors such as ergonomics, aesthetics, environmental and cost
was also discussed. We then move to machine design which deals with the designing any
machine member for its strength and rigidity. The design equation was also developed which
can be written as:
Maximum induced stress at any point in a loaded machine member <= Strength (S)
4.1
STRENGTH
The strength is the property of the material and is obtained experimentally in laboratory.
Property is the term used when it does not depend on the place, shape, size and other conditions.
For instance, if a person is capable of supporting 100 kg of load on his head, his capability is
always with him whether he is carrying load or not. Similarly, the strength of a material exists
with material whatever may be loading conditions. As the same person may have different
strength when tested for different conditions such as, running, climbing, etc. Likewise, the
material exhibit different strength when tested under different conditions such as tension,
compression, torsion etc.
4.2
TENSILE TEST DATA (STRESS – STRAIN CURVE)
The material properties are adequately described by the stress-strain curve for a material on
precise test specimen (IS 1608-1972) and controlled loading and operating conditions. Figure
4.2.1 shows a typical tensile test specimen of gauge length, l o and original cross-section area
Ao and the stress-strain curve for a mild steel is shown in Figure 4.2.2.
The cross-section of specimen can be circular, square and rectangular and the gauge length is
taken as
l o  5.65 Ao
(4.2.1a)
and for circular section l o  5  diameter of specimen
(4.2.1b)
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Figure 4.2.1: Tensile test specimen
True curve
F’
Stress
U
F
Engineering curve
Y
E
P
Plastic range
Strain hardening
Necking and failure
Elastic range
Strain
Figure 2.2: Stress-strain curve under tensile loading of a mild steel specimen
We may obtain the various properties from a tensile test such as proportional limit, elastic limit,
yield point, yield strength, ultimate point, ultimate strength, fracture point, modulus of
elasticity, percentage elongation, reduction in area and Poisson ratio.
Point P: Proportional Limit
The proportional limit (point P) is the point on the stress-strain curve up to which the stressstrain curve follows a straight line. It is the upper limit for which Hooke’s law (stress is directly
proportional to strain within the elastic limit; in actual practice stress is proportional to strain
within proportional limit) is valid.
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The slope of straight line represents the modulus of elasticity (E). Typical value of E for streel
is 210 × 103 𝑀𝑃𝑎. This value give the initial slope of stress-strain curve close to 90 deg
(tan−1 210000).
Point E: Elastic Limit
It is the point up to which the material regains its original shape after the load is removed. It
may also be defined as the point corresponding to the maximum stress that can be induced in
a material without causing permanent deformation. It represents the boundary between the
elastic and the plastic behavior of a material. It may be noted that the curve from point P to E
is not a straight line and, therefore, the Hooke’s law, which states that stress is proportional to
the strain, does not apply from P to E. It is important to observe that a single tensile test cannot
determine elastic limit, as the exact determination of the elastic limit needs a repeated loading
and unloading of the specimen beyond the proportional limit with incremental step up in the
load. Every time, the length is checked after removing the load. The elastic limit is marked
when first change in the gauge length is observed after the removal of the load.
Point Y: Yield Strength
This is the point when the material deforms for the first time without any increase in loading.
The curve is horizontal at this point and is represented by Y. The stress corresponding to yield
point (Y) is known as yield strength (𝑆𝑦𝑡 ). S stands for strength, subscript y for yield and
subscript t for tension.
Point U: Ultimate strength or Tensile Strength
Beyond point Y, the material begins to strain-harden and recovers some of its elastic property.
As the deformation (plastic) increases, the metal becomes stronger and thus, greater loading is
required for carrying out further deformation. However, this increase in the actual strength of
the material is also accompanied by a reduction in the cross-sectional area due to elongation.
Eventually, a stage is reached when the loss of strength (i.e. the load carrying capacity) due to
reduction in cross-sectional area of the specimen under test dominates the gains due to strainhardening of the material. Obviously, the load supported by the specimen is maximum at this
point. The load reaches this maximum value at point U and the stress at this point is called the
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ultimate or tensile strength( S ut ). It is characterized by the beginning of necking. This point
represents the maximum tensile stress or ultimate strength.
Point F: Fracture Strength
Beyond U, the cross section of the specimen goes on decreasing rapidly with drop in the load
and this continues until the specimen fractures at point F. The stress at point F is called the
fracture strength ( S F ). The fact that the nominal strength at F is less than the ultimate strength
at U is somewhat misleading. In fact, the cross section of the specimen between U and F goes
on decreasing and if tensile load is divided by the reduced cross sectional area, the actual stress
developed in the specimen would be higher. The stress-strain curve obtained in the case when
instantaneous area is used for stress computation is called true stress-strain curve as shown by
dotted line UF’ in the Figure 4.2.2. However, the solid line represents the engineering stressstrain diagram. The fracture strength has little design significance because no component is
designed to be load till fracture point.
You may refer lecture notes --- of stress and strain published in strength of material section for
more details of true stress-strain and loading and unloading behavior of the material.
4.3
MORE ON TESILE TEST
For most of the machine design applications, yield strength acts as a basis of design (design
criterion). This is because, a small amount of permanent deformation may result in
nonfunctional of the machine. Hence, measurement of the yield strength of materials is highly
important. Different materials yield in a different way. Structural steel or low carbon steels
(soft steels) are the only materials which exhibit pronounced yield point (Figure 4.2.2). The
material properties are affected by alloying, heat treatment, and he manufacturing process used.
For example, we note from stress-strain curves of pure iron and of three different grades of
steel (Figure 4.2.3).
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Stress
Quenched, tempered alloy steel
Stress
High strength, low alloy steel
Proof stress
Carbon steel
Pure iron
A
0.002
O H
Strain
Strain
Figure 4.2.3: Stress-strain curve for different
Figure 4.2.4: Proof stress
ferrous alloys
It is noted from the curves that there is no actual dip in the curves as observed for soft steels.
The yield strength, for the materials which do not exhibit a well-defined yield phenomenon or
yield point, is measured as the stress required to produce a small amount of permanent
deformation. It implies that as the load is removed, the specimen will not regain its original
shape and a permanent strain or elongation will set in it. The term yield strength for such
material is also referred to as proof stress or offset yield strength.
It is the stress at which the permanent elongation or the total elongation is equal to the specified
value and in general it is taken as 0.2%. To mark the yield point, strain equal to 0.002 is marked
(OA) on the abscissa of the stress-strain curve and a line parallel to the initial slope is drawn
to intersect the curve at Y. This is shown in Figure 4.2.4. The additional information that can
be obtained from the stress-strain curve is given in the following sections
Behavior of Material: Ductile or Brittle
The classification of material such as ductile and brittle is done on the basis of their behavior
under the application of load. A ductile material has the ability to undergo appreciable plastic
deformation when loaded beyond the elastic limit. A completely brittle material fractures at
the elastic limit although some brittle materials like white cast iron show a little plasticity
before fracture. A material is accepted as ductile if it shows more than 5 percent elongation at
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fracture. Ductility is the most desirable property for the operations like bending, drawing,
forming etc.
The ductility and brittleness of a material may also be affected due to manufacturing process
e.g. the casting of a material is less ductile than the cold/hot working of the same material. In
general, the tendency of a material to be brittle increases with decrease in temperature;
increases with rate of loading; and change in state of stress from uniaxial to tri-axial tension.
Tensile Fracture of Specimen
When a ductile material is fractured under tension, the necking (local reduction in cross-section
due to sudden flow of material) occurs and leads to a most popularly known cup and cone
fracture as shown in Figure 4.2.5(a). Unlike ductile material, brittle material does not exhibit
any necking before fracture. The typical brittle fracture of a brittle material under tension is
shown in Figure 4.2.5(b).
Figure 4.2.5: Fractured tensile test specimens
4.3
COMPRESSION TEST DATA
It is a usual assumption that materials are equally strong in tension and compression. This is
specifically true for a wide variety of steels. Generally, the strength of the material in
compression is higher than that observed in tension.
The proportional limit, the elastic limit and the ultimate strength of the material in compression
can be obtained from the compression test. The exact determination of the ultimate strength of
a ductile material in compression is difficult. The brittle materials shatter at the compressive
fracture whereas ductile materials do not and hence a particular degree of distortion is
considered as an indicative of ultimate point, i.e. material failure.
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The initial regions of stress strain diagrams in compression and tension are more or less similar
for ductile materials such as steel, aluminum and copper. However, after yielding compressed
ductile material budges outward and flattened out with increase in load. This increases the
resistance to compression. Since the actual cross sectional area of a specimen tested in
compression is larger than the initial area, the true stress in compression test is smaller than the
nominal stress.
4.4
TORSION TEST
Torsional properties evaluate the behavior of materials for engineering application involving
shear stress. Torsion test can determine the proportional limit, the elastic limit, the shear yield
strength (𝑺𝒚𝒔 ), shear ultimate strength (𝑺𝒖𝒔 ) and modulus of rigidity (G). All the definitions
are analogous to the corresponding properties defined for tension test. The modulus of rigidity
is the ratio of shear stress within the proportional limit to the corresponding shear (angular)
strain in radians. In most of the cases, the properties of a material in shear are not available, so
we can use the following empirical relation to obtain the shear data for practical purposes to
be used in the design of machine elements:
S us  0.8S ut
for steel
S us  0.75 S ut
nonferrous ductile material
S ys  0.577S yt
according to maximum distortion energy theory
The shear fracture is quite different from those observed in tension or compression. In case of
shear test, no reduction of area or elongation is observed in specimen, and the fracture has
distinct texture characteristics of the twisting effect. The typical fractured specimens of cast
iron and mild steel are shown in Figure 4.4.1.
Mild steel
cast iron
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Figure 4.4.1: Fractured torsion test specimens
4.5
FAILURE CRITERION: 𝑺𝒚 or 𝑺𝒖
A machine element is said to be failed when it becomes unable to perform the desired or
intended function. A machine element is always connected to other elements in a machine for
performing the intended function. It may not be possible for any machine to perform its
function if the deformation in any of the element is beyond an acceptable limit. The excess
deformation may lead to jamming of the machine operation. Hence, to avoid this, deformation
or deflection is considered as one of the failure criterion. Likewise, other failure criterion may
be the value of stress in the component. We have two salient stress points for any material on
the basis of which failure criteria for stress are chosen i.e. failure due to yielding or fracture.
Again, both of these criteria imply that a component is assumed to be failed if either some
permanent deformation in the component has set or the component is completely fractured.
The permanent deformation will be induced if the member or component is stressed up to yield
point stress i.e. yield strength and the fracture takes place when the component is stresses till
ultimate point stress i.e. ultimate or tensile strength. Yielding and fracture of material are the
most common mechanical failure criteria. A machine element of ductile material does not
remain useful after yielding (after some permanent deformation is setup in the element);
therefore, yielding is taken as failure criterion for ductile material and ultimate strength for
brittle material.
The wear is used as failure criterion in case of components rubbing against each other with
appreciable relative velocity as in case of bearings, clutches, brakes etc. Wear can also be set
as a failure criterion when two components are having sliding or rolling motion as in the case
of gears, bearings, bushes, piston-cylinders etc.
4.6
FACTOR OF SAFETY AND DESIGN STRESS
The factor of safety is a measure of the uncertainties associated with load calculation, failure
criteria adopted, failure theory selected (to be discussed in Chapter 5), material properties etc.
In general, following factors should be considered for the determination of the factor of safety:

Effect of failure (damage associated with failure)

Uncertainties of load
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
Uncertainties in material data

Uncertainties in modeling and analysis

Operating temperature variation

Degree of economy desired

Atmospheric conditions

Manufacturing aspects etc.
Load acting on an element to be designed and the other service conditions are not predicted
accurately during the design stage and hence uncertainties are always associated. It is not only
the load and the service conditions but the mechanical properties of the materials, obtained
from the experiments are also not exactly same. In practical application, we cannot load a
component to reach its yield strength or ultimate strength. This is because experiments are
always performed under controlled conditions using standard specimen on standard machine.
All these parameters are not identical when a component of the same material is put in use. Its
size and shape may be different, the loading may also be different and even manufacturing
operation also affects the failure stress. To account for all these uncertainties, the strength of
material used for design calculation must be reduced by a factor called factor of safety. The
resultant value of stress i.e. ratio of material strength and factor of safety is called design stress.
Design stress (  d ) is also known as allowable stress, limiting stress or permissible stress.
d 
Strength ( S )
Factor of Safety (n)
As discussed for ductile material yielding is considered as failure criterion and fracture for
brittle material, therefore,
d 
Sy
d 
Su
n
n
for ductile material
for brittle material
or
n
Strength ( S y or Su )
Design Stress ( d )
where S y , S u are yield strength and ultimate strength.
The resistance to sliding is less in case of ductile material than resistance to fracture that is why
the ductile material first slides to yield substantial deformation and then fractures. The
resistance to fracture is less than resistance to sliding for brittle material which makes fracture
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before any sliding, therefore, the deformation in the brittle material is very less before fracture.
As a thumb rule, a material is said to be brittle if the percentage elongation at fracture is less
than 5.
Factor of safety is the ratio of strength and design stress. This factor can also be applied to
other parameter as load, speed etc. as under:
Factor of safety 
Maximum load or Critical load or Load at Failure
Applied Load or Calculated load
Factor of safety 
Maximum speed or Safe Speed
Operating speed
In the design of machine element under static load we can use factor of safety with load as well
as with strength. If correct and precise calculation of load is ensured, factor of safety is used
only on strength to compute the design stress. It may be taken as a thumb rule that factor of
safety on load is used to calculate maximum load when the component or machine is manually
operated or load application is manual. For example, a man may use jack to lift a heavy duty
vehicle which was otherwise designed to a lift light duty vehicle. There is no mean to measure
load applied on a spanner to tighten even small size nut and may be overloaded. In the design
of such components, factor of safety on load should also be used and maximum load is
computed as
Maximumload  (applied load )  factor of safety
Here, we are using factor of safety to take into account the possible overloading of the
component and hence may also be called as overloading factor.
Design load  (applied load )  Over loading factor
Now, the design equation can be written as
Maximum induced stress at any point in a loaded machine member <= Design Stress (𝜎𝑑 𝑜𝑟 𝜏𝑑 )
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