When we replace supports distributed reaction by point reaction in thin beams, is that considered an application of Saint-Venant principle?

Question

I read about Saint-Venant principle which is about the effect of replacing forces by equivalent forces system that effect on the same small region of a rigid body, all examples I come across are on replacing axial force acting on a column by equivalent distributed force,so my question is about replacing distributed force on small region by point force in thin beams, is that considered an application of Saint-Venant principle?

edited: thanks to everyone tried to help me, I get many unsewrs in short time, may be because i have defficulties with English language those unswers were not clear to me, also i didn't explane my question well finally i find this page which contain three exemples the second one (table exemple) is what i was asking about ... thank you again https://www.fembestpractices.com/2020/11/saint-venants-principle-interpretations.html

Answer 1

I would like to add something to the already existing answer here.

Yes, it would be considered as an application of Saint Venant's principle. Since the law just mentions about replacing a point force with an equivalent distributed force over the area and vice versa, it can be an externally applied force or it can be a reaction force, so it doesn't matter.

However, it should be remembered that Saint Venant's Principle comes with its own sacrificies. For example, if you are concerned about the stresses near the region of application of force (or the reaction force), then you have to apply the force in the manner how it is actually applied in the real world. What I mean is that if the force is applied at a small area as compare to the overall size of the beam in reality, then it can idealized as a point force. But if you replace this point force with an equivalent system of forces over the whole area, then the stresses near the force application region will be different than what it should be. Usually, after a distance of height or width (which ever is bigger) of the cross section over which a point load is applied, the stresses render to become uniform and redistributes (ofcourse, given that there is no blockage like a hole or a discontinuity during this distance).

If you are already familiar with Finite Element Analysis (FEA or FEM), then you would know that a point load basically generates singularity at the point of application of load and the stresses near this point load application cannot be trusted at all. After a few elements (usually 3) around this point load, the stresses can be trusted. But if you are actually interested in observing if the region near the application of point load would be sensitive to failure or not, then you can replace this point force over an extremely small area. In this way, the singularity effect will dispel and then you can safely trust the stresses near the region of application of load (without caring about after how many elements should these stresses be trusted).

P.S: Since analytical equations cannot solve for the amount of stresses near the application of load (if it is at a point or at a very small region) mathematically, that is why we use FEA/FEM to predict the stresses near this load application.

Answer 2

Yes, I think you can say so if you reverse its concept exactly the way it is concerned. The explanation (of the concept) below is quite straightforward and precise:

Saint-Venant's Principle simply states that the stress measured at any point on an axially loaded cross section is uniform given that the measured location is far enough away from the point of load application or any discontinuity in the member’s cross section. In other words, when we calculate stress by conventional methods, i.e.,

σ = P / A

we have assumed that we are reasonably far away from the point of application or any discontinuity such that the normal stress is uniform.

In reality, when a point load is applied to a surface, the stress is concentrated at the point of application and eventually evens out as the distance from the point is increased. This increase in stress, also known as a stress riser, also occurs during abrupt changes in the material’s cross section.

ADD:

Three different load types are applied at the rightmost boundary:

• A constant axial stress of 100 MPa
• A symmetric parabolic stress distribution with peak amplitude 150 MPa
• A centered point load with the same resultant as the two previous load cases

As seen in the plots below, the stress distribution at the hole is not affected by how the load is applied. The key here is, of course, that the hole is far enough from the load.

Reading: https://www.comsol.com/blogs/applying-and-interpreting-saint-venants-principle/

Answer 3

Yes it is correct. This is not a dire t answer but can shed some light on the question.

We do a lot of simlar substitution of point load for distributed loads and vice versa.

we do the same exchenge between point loads to one moment e g, in bolted hangar roof beam to column or bolted brackets supperting a beam.

in the elbow bolt group we find the CG of the group and multiply the point load of each bolt force by that arm and sum to one moment. the post is then solved for that moment.

Answer 4

This is an added effort in response to your question after your latest update.

In the situation above, if a < d (beam depth), practically we can replace the uniform load with a concentrated load, and say it is justified by the Saint-Venant Principle.

However, it will run into problems with the actual behaviors from the beam theory if we proceed to check the beam internal forces, for example, the internal reactions at the mid-span:

For the beam with the uniform load:

$M = \dfrac{wa}{2}*\dfrac{L}{2} - w*\dfrac{a}{2}*\dfrac{a}{4} = \dfrac{waL}{4} - \dfrac{wa^2}{4}$

$V = \dfrac{wa}{2} - w*\dfrac{a}{2} = \dfrac{wa}{2} - \dfrac{wa}{2} = 0$

For the beam with the concentrated load:

$M = \dfrac{wa}{2}*\dfrac{L}{2} = \dfrac{waL}{4}$, and

$V = \dfrac{wa}{2}$

Conclusion: By comparing the results, we can see the danger of using the analogy of the Saint-Venant Principle, it can lead to erroneous results; except in very limited cases, or we can tolerate the mistakes from an approximation, or the calculations do not concern member internal forces/stresses, such as in the stability analysis, for which only the external forces/reactions are of interests.

However, Saint-Venant is not to blame for the limitation in its applicability, it's us to be blamed for not truly understand what engineering phenomenon it encompasses.

Hope this helps to erase your confusion over this matter.