PROBLEMS ON VOLUMETRIC STRAIN,LATERAL STRAIN,LONGITUDINAL STRAIN AND POISSON’S RATIO

    

      

      1)    Determine the changes in length, width and thickness of a steel bar which is 4m long, 30 mm wide and 20mm thick and is subjected to an axial pull 30KN in the direction of length. E=2x10⁵N/mm² and poison’s ratio=0.3.Also determine the volumetric strain, change in volume and final volume.

      

            Step 1: Data

          Length: 4m = 4000 mm

            Width = 30mm

Thickness = 20mm

Load = 30KN

E=2x10⁵N/mm²

poison’s ratio= 0.3

volumetric strain, change in volume and final volume = ??

Step 2: Calculation of area of the material

A= b X d

A = 30 X 20

A = 600mm²

Step 3: Calculation of stress:

Stress = Load/area of cross section

Stress = 30 X 1000 /600

Stress= 50 N/mm²

Step 4: Calculation of longitudinal strain:

E = σ/e

2x10⁵ = 50/e

Longitudinal strain = 0.25X10^-3

Step 5: Calculation of lateral strain:

ϻ = lateral strain/ longitudinal strain

0.3 = lateral strain / 0.25X10^-3 

lateral strain = 0.075X10^-3 

Step 6: Calculation of change in volume

e_v = (dl/l)(1-2ϻ)

e_v=(0.25X10^-3)(1-2(0.3))

e_v= 0.1X10^-3    

Step 7 Calculation of change in volume:

e_v = dv/v

Volume = lbd

             = (4000) (30) (20)

             = 24x10⁵ mm³

0.1X10^-3 = dv/24x10⁵ 

dv= 240 mm³

VOLUMETRIC STRAIN

Volumetric strain: Volumetric strain of a deformed body is defined as the ratio of the change in volume of the body to the deformation to its original volume. If V is the original volum and dV the change in volume occurred due to the deformation, the volumetric strain ev induced is given by

                                                                      

  e_v=dV/V 

Consider a uniform rectangular bar of length l, breadth b and depth d as shown in figure. Its volume V is given by,

This means that volumetric strain of a deformed body is the sum of the linear strains in three mutually perpendicular directions.

LATERAL STRAIN,LONGITUDINAL STRAIN AND POISSON'S RATIO

Longitudinal strain : Whenever the bar is subjected to the axial load ,there will be increase in the length of the bar along the direction of loading. Therefore the longitudinal strain is defined as ratio of increase in the length of the bar in the direction of applied load to that of the original length (gauge length).

i.e, e = dL/L

where

e= longitudinal strain

dl= increase in length

L = gauge or original length

Lateral strain: Whenever the bar is subjected to the axial load ,there will be decrease in the dimensions of the bar in the perpendicular direction of loading. Therefore lateral strain is defined as ratio of decrease in the length of the bar in the perpendicular direction of applied load to that of the original length (gauge length).

i.e, e = dB/B or dD/D

where

e= lateral strain

dd= decrease in depth

D= gauge or original depth

db= decrease in breadth

B = gauge or original breadth

Poisson’s ratio:  The ratio of lateral strain to that of the longitudinal strain is termed as poisson’s ratio and it is represented by ϻ or 1/m.

i.e, ϻ or 1/m =  lateral strain/longitudinal strain

Value of the Poisson’s ratio for most materials lies between 0.25 and 0.33.