How much energy is produced during thermal expansion?

[R.Mesri]

We say the expansion coefficient of steel is 12 × 10 ^-6 m/m°C, Young's modulus 210 × 10⁹ N/m² and yield stress 250 mega Pascal. With 100 meters length and 0.1 m² section area , If we increase its temperature 10 °C, how much energy will be produced by expansion force in Joules?

with young's modulus the answer is:
E= 12 × 10 ^-6 × 100 m × 10 °C × 210 × 10 ⁹ × 0.1 m = 252 × 10 ⁶ Joules

with yield stress the answer is :
E= 12 × 10 ^-6 × 100 m × 10 °C × 250 × 10 ⁶ × 0.1 m = 0.3 × 10 ⁶ Joules

I have been confused that which one could be right?

 

[nasu]

It may be that neither is right.
The energy density depends on the square of the strain. I don't think you have it in the first formula,

For the second, why would you even use the yield stress? Do you know that it reaches the yield limit by increasing the temperature by 10 degree?

 

[R.Mesri]

In piping and mechanical engineering the force of a heated steel bar or pipe calculates by Young's modulus multiply by expansion coefficient. On the other hand the yield strength is the maximum force can be applied to a metal without deforming it after that point the plastic deformation begins. I would like to know how much real energy could be get from an expansion. When I use yield strength, the resulting amount looks very small.