Concept:

• The energy which is absorbed in body due to straining effect is known as strain energy (U). 

?\({\bf{U}} = \frac{1}{2} \times {\bf{\sigma }} ~\times~{\epsilon} = \frac{{{{\bf{\sigma }}^2}}}{{2{\bf{E}}}} \times {\bf{vol}}\)

• Whenever straining force is removed from strained body, body is capable of doing work. Hence resilience is also defined as capacity of a strained body for doing work on removal of straining force
• The strain energy stored in body will be maximum when body is stressed up to elastic limit
• Resilience is also defined as energy absorbed by a component within an elastic region. 
• The proof resilience is maximum quantity of strain energy stored in a body when strained up to elastic limit
• Modulus of resilience is defined as proof resilience of a material per unit volume
• Impact energy is defined as energy absorbed by component just before its fracture. It is also called as toughness. 

Stress = 60,000 / 400? = 47.746
Strain energy = ?²V/2E = 47.746×47.746×12,566,370 / (2×100000) = 143,236.54 N-mm = 143.2N-m.

Strain energy in gradual loading = ?²V/2E.

The forces are not concurrent so resultant force and couple both may be present. Thus best choice is that forces are not in equilibrium.

Resilience is ability of a material to absorb energy when elastically deformed and to return it when unloaded.

Explanation:

Resilience:

• The resilience of material is triangular area underneath elastic region of curve.  Resilience generally means ability to recover from (or to resist being affected by) some shock, insult, or disturbance.
• In physics and engineering, resilience is defined as capacity of a material to absorb energy when it is deformed elastically and n, upon unloading to have this energy recovered.
• In or words, it is maximum energy per volume that can be elastically stored. It is represented by area under curve in elastic region in Stress-Strain diagram. 
• This is usually measured by modulus of resilience, which is strain energy per unit volume required to stress material from, zero stress to yield stress.

Hardness:

• Hardness is resistance to plastic deformation (e.g., a local dent or scratch). Thus, it is a measure of plastic deformation, as is tensile strength, so y are well correlated.
• Hardness is resistance of a material to localized deformation. The term can apply to deformation from indentation, scratching, cutting or bending.
• There are a few different hardness tests: Rockwell, Brinell, Vickers, etc. They are popular because y are easy and non-destructive (except for small dent).

Ductility:

• The ability to deform before braking. It is opposite of brittleness. Ductility can be given eir as percent maximum elongation e_max or maximum area reduction. 

As stress = P/A ( 1 + ( 1 + 2AEh/PL)^1/2 )
Putting P = 100, h = 20, L = 1500, A = 150, E = 200,000 we will get stress = 60.23 N/mm².

We may do it taking average
Strain energy = Average force x displacement = (P/2) x PL³/3EI = P²L³/6EI.

The value of stress is calculated by equating strain energy equation and work done equation.

Concept:

In gradual loading, loading starts from zero and increases gradually till body is fully loaded, while in sudden loading, load is suddenly applied on  body.

\({σ _{gradual}} = \frac{P}{A}\)

\({σ _{sudden}} = \frac{{2P}}{A}\)

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

Calculation:

Given:

σ_gradual = Ïƒ, Ïƒ_sudden = 2σ.

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\({{\rm{U}}_{{\rm{gradual}}}} = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}{\rm{\;}},\)

\({{\rm{U}}_{{\rm{suddenly}}}} = \frac{{{{\left( {2{\rm{σ }}} \right)}^2}{\rm{V}}}}{{2{\rm{E}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\(\frac{{{{\rm{U}}_{{\rm{sudden}}}}}}{{{{\rm{U}}_{{\rm{gradual}}}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}} \times \frac{{2{\rm{E}}}}{{{{\rm{σ }}^2}{\rm{V}}}} = 4\)

The strain energy stored in a body due to suddenly applied load compared to when it is applied gradually is four times.