show that the instantaneous stress produced by suddenly applied load is twice that due to same load being applied gradually.
Answers
Answer:
Basically there are two things to keep in mind
Energy can be neither created nor destroyed. It has to be conserved.
Equilibrium is necessary for a body to be stationary.
For gradually applied loads, with the slowly attained stresses and strains, there is energy dissipation in the material as heat. Hence we don't care about conserving energy of the system (it's conserved automatically through heat dissipation) and only solve equations to get “equilibrium”.
For sudden loads, the material has no time to dissipate energy, so we have to solve equations such that first, “energy” is conserved and then, slowly, “equilibrium” is reached. Hence the difference in results in the two approaches.
The “2″ only comes from the (1/2) in area of triangle, which is valid for linear elastic materials.
Example:
If you load a spring gradually, the spring starts to deform until internal force in spring can balance the load. Hence
W = K*x or,. … x = W/K.
2. If you load it suddenly, then now energy has to be conserved.
Work done by load = W*x
Internal energy stored in spring = area under the force displacement curve of the spring = (1/2) * (x) * (K*x)
On equating, x= 2W/K (twice, as expected !)
But at this stage, equilibrium is not attained, do the body cannot be stationary (it has to move). Hence it oscillates until both energy balance and equilibrium is attained.
By the time the spring stops oscillating, it would have dissipated energy equal to “(1/2) * (K*x) *(x)” hence
New energy conservation equation becomes
W*x = (1/2) * (x) * (Kx) + (1/2) * (Kx) * (x)
First half of RHS = energy stored in spring
Second half of RHS = energy dissipation by heat
On solving, x= W/K
Which does not violate equilibrium. Hence the system is now stable.
Hope I made sense.
Explanation: