Physics question: Why is the spacing between bands on a silo smaller at the bottom? Thanks for the help!?
Due to pressure, the closer the bands are from one another add more strength to the bottom where there is more pressure. As the product in the silo builds up there is more pressure on the bottom of the silo.
The silo is banded to reinforce the cylinder and keep it from buckling under the pressure of the grain. You can derive how pressure changes as a function of height along the silo very easily by P=F/A using Newton's 2nd law F=mg, so P=mg/A. Differentiate P with respect to mass to yield dP=(g/A)*dm. Since density of the grain (p)=m/V, solve for m=V*p, so dm=p*dV=p*A*dh <because V=Adh is the unit volume element). Substitute the expression for dm into the differential equation dP=(g/A)*dm to get dP=(g/A)*pAdh - the A's cancel leaving dP=pgdh. Solving this first order linear o.d.e. yields pressure as a function of height along the silo - solve it directly by integration since variables are already seperated - result is P(h)=pgh. From this result you can see how pressure varies linearly with height along the silo, in fact it's a linear increase as height increases. If you're interested in the pressure at a plane that cuts the silo transversely - which we are - you'll have to integrate or add the sum of all pressures above this plane to arrive at the final pressure at the plane of interest. Clearly, since P increases with h, these sums are not linear from plane to plane - meaning more lateral reinforcement is required as you move to lower and lower planes to offset the downward pressure of the grain - i.e., more bands needed at the bottom. You could figure out the spacing of the bands by knowing the density of the grain stored in the silo and the material properties of the silo construction (steel, aluminum, wood, etc...), assuming a silo that's fillled with grain the band spacing will scale downward with lower material strength, or Young's modulus. I'll leave that derivation for you to have fun with since I don't know the material properties or whether the silo is filled with corn, rice, wheat, barley, or cous-cous.
that is to provide one direction of materials when they are going to be discharged, and to have one point of discharge or withdrawal
There is more pressure at the bottom, so this calls for more strength.
