Total Mechanical Engineering

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Simple Stresses And Strains.

        1. Equilibrium:-

             ”Any system of forces which keeps the body at rest is said to be in equilibrium" or "when the condition of the body is unaffected even though it is acted upon by number of forces, it is said to be equilibrium".

    2. Concept of stresses:-

            "Stress is defined as the ratio if applied force per unit cross-sectional area".

       σ = P/A

where,

σ is the stress applied
F is the force applied
                 A is the area of force application

• Stress Units:-

        Stress can be expressed using multiple units. Refer to the table given below for Stress units.

System of units	Stress units
Fundamental units	Kg.m-1.s-2
SI (derived units)	N.m2
SI (derived units)	Pa or pascal
SI (mm)(derived units)	M.Pa or N/(mm)2
US unit (ft)	lbf/ft2
US unit (inch)	Psi (lbf/inch2)

• Types of Stress:-

A stress acts on a body may be normal stress or shear stress.

 

1.               Normal Stress:

 Normal stress is a stress that acts perpendicular to the area. The formula for the normal stress is given by

 

The normal stress is again subdivided into two parts.

 

I.                        Tensile Stress:-

 The stress-induced in a body when it is subjected to two equal and opposite pulls as shown 

in the figure given below is called tensile stress.

§  

§  Due to the tensile stress there is an increase in the length of the body and decrease in the cross section area of the body.

§  Tensile stress is a type of normal stress, so it acts at 90 degree to the area.

§   The strain which is induced due to tensile stress is called tensile strain. It is equals to the ratio of increase in the length to the original length.

I.                        Compressive Stress:-

        The stress which induced in a body when it is subjected to two equal and opposite pushes as shown in the figure given below is called compressive stress.

§ Due to the compressive stress, there is a decrease in the length and increase in the cross section area of the body.

§ Compressive stress is also a type of normal stress and so it also acts at 90 degree to the area.

§ The strain which is induced due to compressive stress is called compressive strain. It is equals to the ratio of decrease in the length to the original length.

2.              Shear Stress:-

         Shear stress induced in a body when it is subjected to two equal and opposite forces that acts tangential to the area.

§  The strain produced due to the shear stress is called shear strain.

§  The shear stress is denoted by the symbol τ (tau). It is a Greek letter.

§  It is defined as ratio of shear resistance to the shear area.

§  The formula for the shear stress is given below.

 

       3.    Concept of strain:-

            Strain is defined a ratio of change in dimension to original dimension of a body when it is deformed. It is a dimensionless quantity as it is a ratio between two quantities of same dimension.

                             ϵ=δlL

                                    where,

ϵ is the strain due to stress applied

δl is the change in length

L is the original length of the material.

• Strain Units:-

              Not that strain has no unit.

• Types of Strain:-

• Longitudinal Strain:-

           When the body of elastic materials is subjected to load, then there is change in length in the direction of applied force. This change in length per original length is called as longitudinal strain. It is denoted by e or  έ. Tensile strain and compressive strain are linear strain.

    

• Volumetric Strain:-

               When the body of elastic materials is subjected to the external forces on its faces, then there will be change in its volume. This changed in volume per original volume is called as volumetric strain.

• Shear Strain:-

                Shear strain is defined as the change in right angle.    

$$        

4. Hooke's Law

• Statement:-

        According to Hooke’s law, when a material is loaded within elastic limit, the stress induced in the material is directly proportional to the strain produced. It means that the ratio of stress with the corresponding strain gives us a constant within elastic limit. The constant is known as Modulus of Elasticity or Modulus of Rigidity or Elastic Modulii.

    In the Stress strain curve the proportionality limit indicates the Hooke’s law. The curve shows linearity within elastic limit. The ratio of stress and corresponding strain in the stress strain curve gives us Young’s Modulus

Mathematically,

• Young’s Modulus or Modulus of Rigidity or Elastic Modulii.

        It is defined as the ratio of stress to the strain within elastic limit. It is denoted by the letter ‘E’.

•  It represents the elastic property of a material.
•  The unit of Young’s modulus is N/m². or N/mm². The unit of young’s modulus is same as that of units of stress.

     Where,

•   σ = Stress
•   e = Strain
•   E = Proportionality constant called as Young’s Modulus.