Nominal Stress, True Stress and Factor of Safety

Direct stress is the value obtained by dividing the load by original cross-sectional area. That is the reason why the value of stress started dropping after neck is formed in mild steel (or any ductile material). But actually as material is stressed its cross-sectional area changes. We should divide load by the actual cross-sectional area to get true stress in the material. To distinguish between the two values, the terms nominal stress and true stress is introduced.

Because we consider nominal stress, after neck formation started (after ultimate stress), stress-strain curve started sloping down and the breaking took place at lower stress (nominal). If we consider true stress, it is increasing continuously as strain increases as shown in Fig. 1.

Fig. 1 Nominal Stress - Strain Curve and True Stress - Strain Curve for Mild Steel

Factor of Safety

In practice, it is not possible to design a mechanical component or structural component permitting stressing up to ultimate stress for the following reasons.

1. Reliability of material may not be 100 per cent. There may be small spots of flaws.
2. The resulting deformation may obstruct the functional performance of the component.
3. The loads taken by designer are only estimated loads. Occasionally there can be overloading. Unexpected impact and temperature loadings may act in the lifetime of the member.
4. There are certain ideal conditions assumed in the analysis (like boundary conditions).

Actually ideal conditions will not be available and, therefore, the calculated stresses will not be 100 per cent real stresses. Hence, the maximum stress to which any member is designed is much less than the ultimate stress and this stress is called Working Stress. The ratio of ultimate stress to working stress is called factor of safety. Thus

In case of elastic materials, since excessive deformation create problems in the performance of the member, working stress is taken as a factor of yield stress or that of a 0.2 proof stress (if yield point does not exist). Factor of safety for various materials depends up on their reliability. The following values are commonly taken in practice.

1. For steel – 1.85
2. For concrete – 3
3. For timber – 4 to 6

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Stress-Strain Relationship of Mild Steel

The stress-strain relation of any material is obtained by conducting tension test in the laboratories on standard specimen. Different materials behave differently and their behaviour in tension and in compression differ slightly.

Behaviour in Tension

Fig.1 shows a typical tensile test specimen of mild steel. Its ends are gripped into universal testing machine. Extensometer is fitted to test specimen which measures extension over the length L1, shown in Fig. 2. The length over which extension is measured is called Gauge Length. The load is applied gradually and at regular interval of loads extension is measured. After certain load, extension increases at faster rate and the capacity of extensometer to measure extension comes to an end and hence, it is removed before this stage is reached and extension is measured from scale on the universal testing machine. Load is increased gradually till the specimen breaks.

Fig. 1 Tension Test Specimen

Fig. 2 Tension Test Specimen after Breaking

Load divided by original cross sectional area is called as nominal stress or simply as stress. Strain is obtained by dividing extensometer readings by gauge length of extensometer (L1) and by dividing scale readings by grip to grip length of the specimen (L2). The uniaxial tension test is carried out on tensile testing machine and the following steps are performed to conduct this test.

• The ends of the specimen are secured in the grips of the testing machine.
• There is a unit for applying a load to the specimen with a hydraulic or mechanical drive.
• There must be some recording device by which you should be able to measure the final output in the form of Load or stress. So the testing machines are often equipped with the pendulum type lever, pressure gauge and hydraulic capsule and the stress vs strain diagram is plotted. Fig.3 shows stress vs strain diagram for the typical mild steel specimen.

Fig. 3 Stress-Strain Curve of Mild Steel

The following salient points are observed on stress-strain curve.

a) Limit of Proportionality (A)

It is the limiting value of the stress up to which stress is proportional to strain.

b) Elastic Limit

This is the limiting value of stress up to which if the material is stressed and then released (unloaded) strain disappears completely and the original length is regained. This point is slightly beyond the limit of proportionality.

c) Upper Yield Point (B)

This is the stress at which, the load starts reducing and the extension increases. This phenomenon is called yielding of material. At this stage strain is about 0.125 per cent and stress is about 250 N/mm².

d) Lower Yield Point (C)

At this stage the stress remains same but strain increases for some time.

e) Ultimate Stress (D)

This is the maximum stress the material can resist. This stress is about 370 - 400 N/mm². At this stage cross sectional area at a particular section starts reducing very fast. This is called neck formation. After this stage load resisted and hence the stress developed starts reducing.

f) Breaking Point (E)

The stress at which finally the specimen fails is called breaking point. At this strain is 20 to 25 per cent. If unloading is made within elastic limit the original length is regained i.e., the stress-strain curve follows down the loading curve shown in Fig. 3. If unloading is made after loading the specimen beyond elastic limit, it follows a straight line parallel to the original straight portion as shown by line FF′ in Fig. 3. Thus if it is loaded beyond elastic limit and then unloaded a permanent strain (OF) is left in the specimen. This is called permanent set.

Stress-Strain Relation in Aluminium and High Strength Steel

In these elastic materials there is no clear cut yield point. The necking takes place at ultimate stress and eventually the breaking point is lower than the ultimate point. The typical stress-strain diagram is shown in Fig. 4. The stress at which if unloading is made there will be 0.2 per cent permanent set is known as 0.2 per cent proof stress and this point is treated as yield point for all practical purposes.

Fig. 4 Stress-Strain Relation in Aluminium and High Strength Steel

Stress-Strain Relation in Brittle Material

The typical stress-strain relation in a brittle material like cast iron, is shown in Fig. 5. In these material, there is no appreciable change in rate of strain. There is no yield point and no necking takes place. Ultimate point and breaking point are one and the same. The strain at failure is very small.

Fig. 5 Stress-Strain Relation for Brittle Material

Percentage Elongation and Percentage Reduction in Area

Percentage elongation and percentage reduction in area are the two terms used to measure the ductility of material.

a) Percentage Elongation

It is defined as the ratio of the final extension at rupture to original length expressed, as percentage. Thus,

where

     L – Original length

     L′– Length at rupture

The code specify that original length is to be five times the diameter and the portion considered must include neck (whenever it occurs). Usually marking are made on tension rod at every ‘2.5d’ distance (d - diameter of rod) and after failure the portion in which necking takes place is considered. In case of ductile material percentage elongation is 20 to 25.

(b) Percentage Reduction in Area

It is defined as the ratio of maximum changes in the cross sectional area to original cross-sectional area, expressed as percentage. Thus,

where

     A – Original cross-sectional area

     A′ – Minimum cross-sectional area

In case of ductile material, A′ is calculated after measuring the diameter at the neck. For this, the two broken pieces of the specimen are to be kept joining each other properly. For steel, the percentage reduction in area is 60 to 70.

Behaviour of Materials under Compression

As there is chance to bucking (laterally bending) of long specimen, for compression tests short specimens are used. Hence, this test involves measurement of smaller changes in length. It results into lesser accuracy. However precise measurements have shown the following results.

a) In case of ductile materials stress-strain curve follows exactly same path as in tensile test up to and even slightly beyond yield point. For larger values the curves diverge. There will not be necking in case of compression tests.

b) For most brittle materials ultimate compressive stress in compression is much larger than in tension. It is because of flows and cracks present in brittle materials which weaken the material in tension but will not affect the strength in compression.

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Relationship between Elastic Constants

 Modulus of elasticity, modulus of rigidity and bulk modulus are the three elastic constants.

1) Modulus of Elasticity (Young’s Modulus) ‘E’

It is defined as the ratio of linear stress to linear strain within elastic limit.

2) Modulus of Rigidity (G or N)

It is defined as the ratio of shearing stress to shearing strain within elastic limit and is usually denoted by letter 'G' or 'N'. Thus

where

     G = Modulus of rigidity

     q = Shearing stress

     ϕ = Shearing strain

3) Bulk Modulus (K)

When a body is subjected to identical stresses p in three mutually perpendicular directions, as shown in Fig. 1, the body undergoes uniform changes in three directions without undergoing distortion of shape.

Fig. 1 Stresses p acting in three mutually perpendicular directions

The ratio of change in volume to original volume has been defined as volumetric strain (e_v). Then the bulk modulus, 'K' is defined as

where

      p = identical pressure in three mutually perpendicular directions

  , Volumetric strain

      Δ v = Change in volume

      v = Original volume

Thus bulk modulus may be defined as the ratio of identical pressure ‘p’ acting in three mutually perpendicular directions to corresponding volumetric strain.

Fig. 1 shows a body subjected to identical compressive pressure ‘p’ in three mutually perpendicular directions. Since hydrostatic pressure i.e. the pressure exerted by a liquid on a body within it, has this nature of stress, such a pressure ‘p’ is called as hydrostatic pressure.

Relationship between Modulus of Elasticity and Modulus of Rigidity

Consider a square element ABCD of sides ‘a’ subjected to pure shear ‘q’ as shown in Fig. 2. AEC’D shown is the deformed shape due to shear ‘q’. Drop perpendicular BF to diagonal DE. Let ′ϕ′ be the shear strain and G is the modulus of rigidity.

Fig. 2

              

Since angle of deformation is very small we can assume ∠BEF = 45°, hence EF = BE cos 45°

                    

                      

Now, we know that the above pure shear gives rise to axial tensile stress ‘q’ in the diagonal direction of DB and axial compression q at right angles to it. These two stresses cause tensile strain along the diagonal DB.

Relationship between Modulus of Elasticity and Bulk Modulus

Consider a cubic element subjected to stresses p in the three mutually perpendicular direction x, y, z as shown in Fig. 3.

Fig. 3

Now the stress p in x direction causes tensile strain p/E in x direction while the stress p in y and z direction cause compressive strains μp/E in x direction.

Relationship between E,G and K

                                 We know that       E = 2G(1 + μ)                  (1)

                                                                     E = 3K(1 – 2 μ) (2)

By eliminating ′μ′ between the above two equations we can get the relationship between E, G and K, free from the term μ.

     

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Volumetric Strain (εv)

When a member is subjected to stresses, it undergoes deformation in all directions. Hence, there will be change in volume. The ratio of the change in volume to original volume is called volumetric strain.

Thus,

Where,

           εv = Volumetric strain 

          δV = Change in volume

           V = Original volume

It can be shown that volumetric strain is sum of strains in three mutually perpendicular directions.

For example consider a bar of length L, breadth b and depth d as shown in Fig. 1.

Fig. 1 Rectangular Bar

Now,

Volume, V = L b d

Since volume is function of L, b and d, by using product rule (The derivative of the product of two differentiable functions is equal to the addition of the first function multiplied by the derivative of the second, and the second function multiplied by the derivative of the first function.) we may write as;

Consider a circular rod of length ‘L’ and diameter ‘d’ as shown in Fig. 2.

Fig. 2 Circular Rod

Volume of the bar

                                                                                         (Since V is function of d and L)

Dividing the equation by V

In general for any shape volumetric strain may be taken as sum of strains in three mutually perpendicular directions.

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Hooke’s Law and Poisson’s Ratio

Hooke’s Law

Robert Hooke, an English mathematician conducted several experiments and concluded that stress is proportional to strain up to elastic limit. This is called Hooke’s law. Thus Hooke’s law states that ‘stress is proportional to strain up to elastic limit.’

σ ∝ ϵ

where ′σ′ is stress and ′ϵ′ is strain

Hence,

σ = E ϵ

Where ‘E’ is the constant of proportionality of the material, known as Modulus of Elasticity or Young’s modulus, named after the English scientist Thomas Young (1773–1829). 

The present day sophisticated experiments have shown that for mild steel the Hooke’s law holds good up to the proportionality limit which is very close to the elastic limit. For other materials, Hooke’s law does not hold good. However, in the range of working stresses, assuming Hooke’s law to hold good, the relationship does not deviate considerably from actual behaviour. Accepting Hooke’s law to hold good, simplifies the analysis and design procedure considerably. Hence Hooke’s law is widely accepted. The analysis procedure accepting Hooke’s law is known as Linear Analysis and the design procedure is known as the working stress method.

Poisson’s Ratio (μ)

When a material undergoes changes in length, it undergoes changes of opposite nature in lateral directions. For example, if a rectangular bar is subjected to direct tension in its axial direction it elongates and at the same time its sides contract as shown in Fig.1.

Fig. 1 Changes in Axial and Lateral Directions due to Tensile Force

If we define the ratio of change in axial direction to original length as linear strain and change in lateral direction to the original lateral dimension as lateral strain, it is found that within elastic limit there is a constant ratio between lateral strain and linear strain. This constant ratio is called Poisson’s ratio. 

Thus,

It is denoted by 1/m or μ.

For most of metals its value is between 0.25 to 0.33. Its value for steel is 0.3 and for concrete 0.15.

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Basic Terminologies in Mechanics

1) Mass (m)

The quantity of the matter possessed by a body is called mass. The mass of a body will not change unless the body is damaged and part of it is physically separated. If the body is taken out in a space craft, the mass will not change but its weight may change due to the change in gravitational force. The body may even become weightless when gravitational force vanishes but the mass remain the same.

2) Weight (w)

Weight of a body is the force with which the body is attracted towards the centre of the earth. The weight of the body is equal to the product of mass and the acceleration due to gravity. This quantity of a body varies from place to place on the surface of the earth.

Mathematically,

w=mg

Where ‘w’ is the weight of the body, ‘m’ is the mass of the body and ‘g’ is the acceleration due to gravity.

Table 1 Difference between Mass and Weight

Mass	Weight
Mass is the total quantity of matter contained in a body.	Weight of a body is the force with which the body is attracted towards the centre of the earth.
Mass is a scalar quantity, because it has only magnitude and no direction.	Weight is a vector quantity, because it has both magnitude and direction.
Mass of a body remains the same at all places. Mass of a body will be the same whether the body is taken to the centre of the earth or to the moon.	Weight of body varies from place to place due to variation of ‘g’ (i.e., acceleration due to gravity.
Mass resists motion in a body.	Weight produces motion in a body.
Mass of a body can never be zero.	Weight of a body can be zero.
Using an ordinary balance (beam balance), the mass can be determined.	Using a spring balance, the weight of the body can be measured.
The SI unit of the mass is the kilogram (kg).	The SI unit of the weight is Newton (N).

3) Time

The time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. To overcome difficulties due to irregularities in the earth’s rotation, the unit of time is taken as second, which is defined as the duration of 9192631770 period of radiation of the cesium-133 atom.

4) Space

The geometric region in which study of body is involved is called space. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements. The reference point is called the origin and the set of measurements as coordinates. If the coordinates involved are only in mutually perpendicular directions, they are known as cartesian coordination. If the coordinates involve angles as well as the distances, it is termed as Polar Coordinate System.

5) Length

It is a concept to measure linear distances. Meter is the unit of length. However depending upon the sizes involved micro, milli or kilo meter units are used for measurements. A meter is defined as length of the standard bar of platinum-iridium kept at the International Bureau of weights and measures. To overcome the difficulties of accessibility and reproduction now meter is defined as 1690763.73 wavelength of krypton-86 atom.

5) Continuum

A body consists of several matters. It is a well known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as conglomeration of such discrete particles. The body is assumed to be a continuous distribution of matter. In other words the body is treated as continuum.

6) Particle

A particle may be defined as an object which has only mass and no size. Theoretically speaking, such a body cannot exist. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as a particle, without sacrificing accuracy.

For example:

• A bomber aeroplane is a particle for a gunner operating from the ground.
• A ship in mid sea is a particle in the study of its relative motion from a control tower.
• In the study of movement of the earth in celestial sphere, earth is treated as a particle.

7) Rigid Body

A body is said to be rigid, if the relative positions of any two particles do not change under the action of the forces acting on it i.e., the distances between different points of the body remain constant. No body is perfectly rigid. Rigid body is ideal body.

Fig. 1 Rigid Body due to the action force F

8) Deformable Body

When a body deforms due to a force or a torque it is said deformable body. Material generates stresses against deformation. All bodies are more or less elastic.

Fig. 2 Deformable Body due to the action force F

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Strain (ε)

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. Strain is a dimensionless unit since it is the ratio of two lengths. But it also a common practice to state it as the ratio of two length units like m/m or mm/mm etc. Strain is represented by 'ε' (Greek lowercase alphabet Epsilon).

No material is perfectly rigid. Under the action of forces it undergoes changes in shape and size. All materials including steel, cast iron, brass, concrete etc. undergo deformation when loaded. But the deformations are very small and hence we cannot see them with naked eye. There are instruments like extensometer and electric strain gauges which can measure this extension. Strain may be of linear strain or lateral strain.

The bars extend under tensile force and shorten under compressive forces along axial direction. The change in length per unit length is known as linear strain/longitudinal strain. Thus, 

When there is a changes in longitudinal direction takes place change in lateral direction also take place. The nature of these changes in lateral direction are exactly opposite to that of changes in longitudinal direction i.e., if extension is taking place in longitudinal direction, the shortening of lateral dimension takes place and if shortening is taking place in longitudinal direction extension takes place in lateral directions. The lateral strain may be defined as changes in the lateral dimension per unit lateral dimension. Thus,

Consider a square bar of length ‘L’ and breadth ‘b’. The linear dimension (length) changes by ‘Δ’ due to the application of tensile or compressive force. The lateral dimension (breadth) changes by b’ due to the application of tensile or compressive force as shown in the figure.

Fig.1 Deformation of a square bar due to axial tensile/compressive force

Shear Strain (ϕ)

This type of strain is produced when the deforming force causes change in the shape of the body. The distortion produced by shear stress on an element or rectangular block is shown in the figure. The shear strain is expressed by angle ‘ϕ’ and it can be defined as the change in the right angle. It is measured in radians and is dimensionless in nature.

Shearing stress has a tendency to distort the element to position AB′C′D from the original position ABCD as shown in figure. This deformation is expressed in terms of angular displacement and is called shear strain. 

Fig.2 Deformation of a rectangular body fixed at bottom due to shear force

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Stress

Stress is the internal resistance offered by the body to the external load applied to it per unit cross sectional area. Stresses are normal or tangential to the plane to which they act and are tensile, compressive or shearing in nature.

When a member is subjected to loads it develops resisting forces. To find the resisting forces developed a section plane may be passed through the member and equilibrium of any one part may be considered. Each part is in equilibrium under the action of applied forces and internal resisting forces. The resisting forces may be conveniently split into normal and parallel to the section plane. The resisting force parallel to the plane is called Shearing resistance. The intensity of resisting force normal to the sectional plane is called Normal resistance.

Forces Acting on Rectangular Rod

Consider a rectangular rod subjected to axial pull P. Let us imagine that the same rectangular bar is assumed to be cut into two halves at section XX. The each portion of this rectangular bar is in equilibrium under the action of load P and the internal forces acting at the section XX has been shown in figure. The symbol ‘σ’ is used to represent stress.

Where A is the area of the X –X section

Here we are using an assumption that the total force or total load carried by the rectangular bar is uniformly distributed over its cross section. But the stress distributions may be far from uniform, with local regions of high stress known as stress concentrations. If the force carried by a component is not uniformly distributed over its cross sectional area, A, we must consider a small area, ‘δA’ which carries a small load ‘δP’, of the total force ‘P', Then definition of stress is

Unit of Stress

The basic units of stress in S.I units i.e. (International system) are N/m^2 (or Pa). When Newton is taken as unit of force and millimeter as unit of area, unit of stress will be N/mm^2. The other derived units used in practice are kN/mm^2, N/m^2 or kN/m^2. A stress of one N/m2 is known as Pascal and is represented by Pa.

Hence, 1 MPa = 1 MN/m^2 = 1 × 10^6 N/(1000 mm^2) = 1 N/mm^2.

Thus one Mega Pascal is equal to 1 N/mm^2.

Types of Stress

The two basic stresses exists are Normal stress and Shear stress. Other stresses either are similar to these basic stresses or as a combination of this.

Example : Bending stress is a combination tensile, compressive and shear stresses.

                  Torsional stress, as encountered in twisting of a shaft is a shearing stress.

1) Normal stress

If the stresses are normal to the areas concerned, then these are termed as normal stress. The normal stress is generally denoted by a Greek letter (σ). Stress is said to be normal stress when the direction of the deforming force is perpendicular to the cross-sectional area of the body. Normal stress can be further classified into three types based on the dimension of force. This is also known as uniaxial state of stress, because the stresses acts only in one direction however, such a state rarely exists, therefore we have biaxial and triaxial state of stresses where either the two mutually perpendicular normal stresses acts or three mutually perpendicular normal stresses acts as shown in the figures below.

Uniaxial State of Stress

Biaxial State of Stress

Triaxial State of Stress

The normal stresses can be either tensile or compressive whether the stresses acts out of the area or into the area.

a) Tensile Stress

Consider a bar subjected to force P as shown in figure. To maintain the equilibrium the end forces applied must be the same, say P. If the deforming force or applied force results in the increase in the object’s length then the resulting stress is termed as tensile stress.For example when a rod or wire is stretched by pulling it with equal and opposite forces (outwards) at both ends. 

Tensile Force

b) Compressive Stress 

If the deforming force or applied force results in the decrease in the object’s length then the resulting stress is termed as compressive stress. For example: When a rod or wire is compressed/squeezed by pushing it with equal and opposite forces (inwards) at both ends.

Compressive Force

Sign convections for Normal stress

Tensile stress is taken as +ve

Compressive stress is taken as –ve

2) Shear Stress

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.

Shear Force

Bearing Stress

When one object presses against another, it is referred to a bearing stress (They are in fact the compressive stress).

Bending Stress 

Bending stress is the stress that results from the application of a bending moment to a material, causing it to deform. This results in the development of a combination of tensile and compressive stresses through the cross-section of the material and creates a stress gradient that causes the material to bend. 

Bending Stress in Beam

Torsional stress 

Torsional shear stress or Torsional stress may be defined as that shear stress which acts on a transverse cross-section that is caused by the action of a twist.

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