How Does St. Venant's Principle Apply to a Cut Annular Ring?

[jrm2002]

St. Venant's principle!

Can anyone give me an intuitive example explaining St. Venant's principle?

 

[Astronuc]

Two ways to state St. Venant's principle are:

The localized effects caused by any load acting on the body will dissipate or smooth out within regions that are sufficiently away from the location of the load.

best.umd.edu/publications/2005TRB_STMfinal.pdf

statically equivalent systems of forces produce the same stresses and strains within a body except in the immediate region where the loads are applied.

from http://me.queensu.ca/courses/mech422/Notes422.pdf

Basically the point being made is that loads must be applied in a finite area (area of contact) and they will produce local stresses and strains. However, as a far as the rest of the structure is concerned, the local effects will not be experienced, and the load could be treated as a point load.

 

[jrm2002]

Yes, I understand that but I want an intuitive example!
Thankx anyway.

 

[FredGarvin]

Another way to look at it is that the local deformations caused by a force can be looked at on the basis of the entire body, not at the point of application. The example I was always shown was a beam. A cantilever beam's support reactions, etc...are not effected by the local effects of a force at the opposite end. The catch phrase is "several characteristic dimensions away." So you can do the overall analysis of the beam's deflections and stresses, etc...but you should go back afterwards and look at the immediate area of where that force was applied because there will be some very localized things that are happening there.

 

[mbaraka]

Yes, I understand that but I want an intuitive example!
Thankx anyway.

I am presently working on distributing some high seismic brace forces to the spillway piers of a dam. I am doing a finite element model of the piers and am making use of St. Venant's Principle in justifying a partial model of the dam that only encompasses the spillway piers. If you like, in a couple of weeks I can share some of the results.

Mike

 

[M_Schaefer]

Suppose you were analyzing the stress in a cantilever beam. The real beam is 20 inches long and has a distributed load of 25 lb/inch over the last 4 inches of the beam, near the free end. To simplify the analysis you use a statically equivalent point load, 100 lbs at a point 2 inches from the free end.

Stresses caclulated at the fixed end would be nearly identical for your simplified case compared to the real beam, because it is "far" from the idealized point load. Stresses calculated at a point 3 inches from the free end would be significanlty different for the simplified case compared to the real beam, because it is "close" to the idealized point load.

Where does the analysis transition from "far" to "close"? Generally it depends on the dimensions of the beam. Two or 3 times the beam height would be a distance (lengthwise) that I may define as the "St. Venant's" distance, where the results would be vaid. In the above example, if the beam were 1 inch thick, I might assume that the results of the simplified analysis were valid except for the 6" of the beam near the free end (the actual distributed load was along 4 inches plus 2 x beam height = 6 inches) or something like that. There's not hard rule about this.

 

[L.Richter]

@M.Schaefer~ How would St. Venant's principle apply to an annular ring/washer that has been cut by a saw producing a distance between the ends and producing two opposing forces?