Mechanical Engineering - Strength of Materials - Discussion

Strain energy stored in the body = Work done by the load in deforming the body.
Strain energy stored in the body = Area of the load-extension curve.
Strain energy stored in the body = P. x.
U = P. x.

As we know that maximum strain energy stored in the body U will be provided by the following expression as mentioned here.
U=(σ^2/2E)*V.
U=(σ^2/2E)*A*L.
P*x=(σ^2/2E)*A*L.

Now we will secure the value of extension x in terms of Stress, Length of the body and Young's modulus of the body by using the concept of Hook's Law.
According to Hook's Law.

Within elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above the law as mentioned here.

Stress = E. Strain
Where E is Young's Modulus of elasticity of the material.

σ = E. ε.
σ = E. (x/L).
x = σ. L/ E.

Let use the value of the extension or deformation "x" in the above equation and we will have
P*(σ/E)*L=[(σ^2)/(2*E)]*A*L.
P=σ*A/2.
σ=2(P/A).

Therefore, we can say here that maximum stress induced in the body due to suddenly applied load will be twice the stress induced in the body with the same value of load applied gradually.

When we loaded suddenly load 'w' then we consider 'w' load in stress while we loaded gradually load "w' then we consider average value of 'w' which will be w/2 in the stress calculation.

So we know that,
Stress = load/area.

So let p1 is stress when suddenly load and p2 is stress when load is gradually.

So p1 = w/area.

And p2 = (w/2)/area.

So p1/p2 = [w/aria]/[ (w/2) /aria].

So p1/p2 = 2.

p1 = 2p2.

Thats proves that suddenly loaded stress is double to the gradually loaded stress.

In case of impact loading the dynamic force experienced by the body is given by P(dyn.) = W*[1+sqrt(1+2h/D)].

Here P = dynamic load experienced by the body.
W = static load which is impacting.
D = stact deflection due to above load.
h = height from which load is acting.

In this case h=0, So dynamic load experienced by the body is twice the load applied. As load to area is stress so stresses induced will be twice.

Please, define "gradually" and "suddenly".

Only if the load is related to weight loading, (dropping a weight vs supporting a weight) , only then the rate of the loading is actually a factor that affect the stress induced to the body. If the load is related to a spring (releasing a compressed spring that is already in contact to a body) then the rate of the loading does not play any role.

As in gradually applied load, load varies from 0 to a.

The net load applied is w1 = w/2.

So, the stress is p = w/(2a).

For suddenly applied load stress p1 = w/a.

p1/p = 2, p1 = 2p.

So, stress due to suddenly applied is twice of gradually applied load.

In case of impact loading:

σ or δ(impact)=(I.F.)*σ or δ(static) I.F.=Impact Factor.

I.F.=1+(1+2h/δ).

h= height of load drop, δ= static change in dimension.

Here in this case h=0.

So, I.F.= 2.

So, stress and strain will be double.

@Aman.

How can you say that the average load is w/2?

In case w is split into 4 loads (w1+w2+w3+w4) applied gradually then average would come w/4.

In this case answer would come 4 times. Your method is completely wrong.

When the load applied gradually then stress induce = W/A.

When the load applied suddenly then stress induce = 2 * W/A.

By comparing.

(stress) suddenly = 2 * (stress) gradually.

When the load is applied slowly due to viscous dissipation by the atmosphere only half of the energy is stored as compared to the case when the load is applied suddenly.

It is experimentally proved that the stress developed by impact load is twice than developed by gradually applied load.