Strain Energy In Axial Loaded bars

The Strain Energy in axially loaded bars can be expressed as

U = 1/2 (Stress)x(Strain)x volume of the bar 

U = 1/2 (s).(e).V  = 1/2 (s).(s/E).V 

The stresses in Axial loaded bars due to different cases will be given as below

1)The stress induced in the bar due to gradually applied load s = P/A

2)The stress induced in the bar due to suddenly applied load s = 2P/A

3)The stress induced in the bar due to Impact load s

Example:

1.Calculate the strain energy stored in a bar 50mm diameter & 2.5 m long subjected to tensile load of 100 kN take E= 200 GPa if 

a) load is gradually applied

b) load is suddenly applied​

Solution:

a) If load is gradually applied, the strain energy 

U = 1/2 (Stress)x(Strain)x volume of the bar 

    = 1/2 (s).(e).V  = 1/2 (s).(s/E).V ---(1)

The stress induced in the bar due to gradually applied load s = P/A 

Load on the bar P  = 100 kN = 100x1000N

Diameter of bar =50mm

Length of bar = 2.5m 

Area of the bar  A  = p d2/4 

Substitute the values in equation (1),

The strain energy U=  31.81 x 10 3 N - mm  

b)  If load is suddenly applied, the strain energy 

U = 1/2 (Stress)x(Strain)x volume of the bar 

    = 1/2 (s).(e).V  = 1/2 (s).(s/E).V ---(2)

But the stress induced in the bar due to gradually applied load s = 2P/A 

{Twice of gradually applied load}

If this stress is used , 

The strain energy due suddenly applied  U = Four times of strain energy of gradually applied load

U =  4 x 31.81 x 10 3 N - mm 

    =127.28 x 10 3N -mm

Assignment:

1. A square steel rod 50 mm size & 3 m long is subjected to a pull of 100 kN suddenly applied to it Calculate , i) Strain energy, ii) Elongation and

iii) Modulus of resilience , Take E = 210 Gpa.

2. A bar 1.5 m long10 mm diameter hangs vertically , it has a collar fixed at lower end .a load of 120 N falls on the collar from the height of 30 mm. Calculate strain energy absorbed. Take Young's modulus for material is 210 Gpa.

3. A steel bar 20 mm diameter is 7 m long & has collar attached to it .A load of 800N falls on it from a height of 60 mm. Find - i) Stress ii) Change in length. iii) Strain energy and iv) modulus of resilience. Take Young's modulus E is 210 Gpa.

4. A bar 30 mm x 30 mm is 4.5 m long is subjected to impact load of 300 N which falls on collar from a height of 400 mm. 

Find i)Stress ii)Strain energy iii) Change in length.iv) modulus of resilience.