Fundamental of Stress & Strains19 min read

A force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force is the newton, N. The three main types of mechanical force that can act on a body which change the shape of the body are as follows:

  1. Tensile Force
  2. Compressive Force
  3. Shear Force

Tensile Force :

Tension is a force that tends to stretch a material. A tensile force increases the length of the material on which it acts.  shown in given Figuretensile force

Example: A cable of a crane carrying the load

Compressive force :

Compression is a force that tends to squeeze or compress a material. A compressive force decreases the length of the material on which it acts.

compressive force

Example: A pillar supporting a bridge

Shear force :

Shear is a force that tends to slide one face of the material over an adjacent face. A shear force can cause a material to bend, slide or twist.

Example: a rivet holding two plates together is in shear if a tensile force is applied to the plates.shear force

Stress :

When a body is acted upon by some load or external force, the body changes its shape. During this deformation, the body resists the tendency of the load to deform the body. This internal resistance that the body offers to meet with the load is called Stress.

Stress is the ratio of the applied force F to cross-sectional area A of the material. The symbol used for tensile and compressive stress is σ (sigma).

Hence \boxed{\sigma = \frac{P}{A}}

where

  • P = External force or load in N (Newton) or kN.
  • A = Cross-sectional area in  m^2(square meters) or mm^2.
  • σ = Stress in  kN \setminus m^2         OR        kN \setminus mm^2

For tensile and compressive forces, the cross-sectional area is that which is at right angles to the direction of the force. For a shear force, the shear stress is equal to F/A, where the cross-sectional area A is that which is parallel to the direction of the force. The symbol used for shear stress is the Greek letter tau, τ.

Another unit of stress is Pascal, Pa. where 1 Pa= 1 N\setminus m^2

The various types of stresses may be classified into three categories as:

  • Simple Or Direct Stresses
  • Indirect Stresses
  • Combined Stresses

Simple stresses are generally called Direct stresses due to they developed under direct loading conditions. When tensile, compressive or shear force applied to a body the developed stress will be tensile stress, compressive stress or shear stress respectively.

Indirect Stresses are of two kinds (1. Bending Stress (2. Torsion Stress

Combined Stresses are any combination of Direct and indirect stresses.

Strain

Strain (e) is the deformation produced by stress. Various types of Strains are explained below:

Tensile Strain:

A uniform cross-sectioned piece of material is subjected to a uniform axial tensile stress. This tensile stress will its length from  l to  \left (l + \delta l \right ) and the increased length  \delta l is the actual deformation of the material. tensile strain

The fractional deformation or tensile strain is given by

Tensile Strain  \boxed{ e_{t} = \frac{\delta l }{l}}

Compressive Strain:

Under compressive stress a similar uniform cross-sectioned piece of material would be reduced in length from  l to  \left (l - \delta l \right ).

The fractional deformation or Compressive strain  is given by

Compressive Strain  \boxed{ e_{c} = \frac{\delta l }{l}}

Shear Strain:

shear strain

In the case of shearing load, a shear strain will be produced which is measured by the angle through which the material distorts.

In the fig. , a rectangular block LMNP of material fixed at the bottom face and a force F is subjected to distort through an angle  \phi and occupies a new position LM’N’P.

 

 

The Shear Strain  e_{s} is given by

Shear Strain  \boxed {e_{S} = \frac{NN'}{NP} =  \frac{MM'}{ML} =  \tan \phi }

 \Rightarrow e_{S} = \phi  ( radians). . . . . . . . . . . . since  \phi is very small.

Volumetric Strain:

The Ratio between change in volume and the original volume of the material called Volumetric Strain  e_{v}.

Volumetric Strain  \boxed{ e_{v} = \frac{\mbox{Change in Volume} }{ \mbox{ Original Volume}} = \frac{ \delta V}{V} }

Modulus Of Elasticity:

The strains which disappear with the removal of forces or loads are termed as Elastic strains, and such bodies said to be Elastic bodies. The bodies said to be Plastic bodies if strains exist after the removal of forces.  The stress developed by a limited value of load/force up to which the strain totally disappears after the removal of the force is called Elastic limit.

Stress is directly proportional to strain i.e.

 \Rightarrow   Stress     α      Strain

  \Rightarrow \frac{Stress}{Strain} = \mbox{A Constant}

This Constant is termed as Modulus of Elasticity.

Young’s Modulus:

The ratio between Tensile stress to the Tensile strain OR Compressive Stress to the Compressive Strain is called YOUNG’S MODULUS. It is the same as Modulus of elasticity and denoted by E.

 \Rightarrow E = \frac{\sigma}{e} = \frac{ \sigma_{t}}{e_{t}} =\frac{\sigma_{c}}{e_{c}}

Modulus of Rigidity:

The ratio of shear stress τ (tau) to the shear strain is Modulus of rigidity or Shear modulus of elasticity.. It is denoted by C.

 \Rightarrow          C = \frac{\tau }{e_{s}}

Bulk Modulus Of Elasticity:

The ratio of Normal stress on each face of a solid cube to the volumetric strain is called bulk modulus of elasticity or volume modulus of elasticity. It is denoted by K.

 \Rightarrow          K = \frac{\sigma_{n} }{e_{v}}

 

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