The Finite Element Method | Part 9: Bending Moment and Shear Force in FEM

[@donaldkhanye9011]

Hello Dr. CEE, Thank you and thank you, these FEM videos are truly amazing. I try to follow along and yes, I do stumble more often than I would want to admit, but I dust myself and get up and press on. Your explanation and your inclusion of the seemingly little but key details in your explanation are golden and they really really help to keep going. I have now successfully completed this example, however I had to deal with the following which I am requesting confirmation/guidance from you Doctor. Stiffness Matrix Units: With previous topics, I did not have to worry much about the stiffness matrix units when assembling the (avengers, oh, scratch that) stiffness matrices because all the elements (entries) in the stiffness matrices had the same units. Now with the beam element, one has got a mix of EI/L, EI/L^2, and EI/L^3 units…. I had to very carefully match the corresponding units on each matrix entry even for the case where there are some Zero values entries on the stiffness matrices. Do you find the units to be more involved in the case of a beam element stiffness matrix as compared to the previous spring and truss elements stiffness matrices? Boundary Conditions: When considering the boundary conditions, M2 (along with M1 and M3) was set to zero, my first thoughts with regards to M2 were “…but wait Doctor…. 😊 isn’t there going to be a continuity moment over node #2?... I get that there is a roller but….” I then finally concluded that yes, there will be a continuity moment BUT the moment is NOT on the “roller support” it is on the “Local member (on the local element at node #2)” and this is why M2 which is the moment on the roller support is set to zero when considering boundary conditions at node #2. Is this the correct reasoning Doc? Your explanation of the FEM and the Statistics sign conventions to use to draw the SF and BM diagrams from the post processed local element results is to die for, thank you so much for this; and for throwing in the last nugget about differentiating or integrating the BM diagram function M(x) or SF diagram function V(x) to find the location of the maximum or minimum values. Your FEM lectures are unmatched and they are very informative. You are helping a lot with these and they are truly appreciated. Kind regards, DK

[@CivilEngineeringEssentials]

Hi there Engr. DK, I am happy you like those vids and they assist you in understanding how softwares "think" in general. I usually say: to be on the safe side, use "base" units. Here, I suggest to use Newtons for forces, meters for distances. And anything derived from that such as Stresses (Newton / meter ^2) if you do this, you are golden and have no unit problems. Let me put it that way: the unit difference in the beam stiffness matrix is just a matter of having L or not. Why? because L represents length in meters, and the load vector is either a force or a moment, you can see how L is playing a part here. It is a little deeper than this, but I hope that those little ideas help you grasp the unit concept. You are correct, in the force vector, the force entries are the "External forces" applied in the global axis sense + any force from distributed stuff. Those are not the internal forces. So even if M2 (external force) is 0, there will be local forces for m2 in both elements, and "here it is" the summation of those moments will equal the external moment, in that case 0 to achieve equilibrium. I am happy you liked the video, also, please tell me if you need any further explanation with regard to the points mentioned above. I am more than happy to provide more details. Regards, CEE

[@donaldkhanye9011]

@@CivilEngineeringEssentials Thank you, the explanation is good. I did (successfully) tried this yesterday with one of the recommended reading example 4.2 but then I used kilonewtons (kN) and meters (m) as my reference units. This allowed me to “temporarily - ditch” the units whist I perform matrices manipulations which in turn simplified things a lot for me during matrices manipulations. I therefore find your suggestion to use base units to be a more attractive and a consistent option. Furthermore, base units can also help to facilitate correct anticipation of the output units of everything else that is involved. Thanks for the idea. Your differentiation of “external forces” and “internal forces - pains ^_^”, with regards to M2 have put to bed all doubts I had with M2, thanks for that. I noticed in the “solution for example 4.2” - page 187 equations (4.3.15) that the author noted F2y and F4y as “positives”, aren’t these values supposed to be “negatives”? My calculations (I am using the metric version of the book) are giving me the same values but negative in sign. [these values are essentially the applied global nodal forces at node 2 and at node 4 and these can be read directly from the problem statement (the beam and loads layout figure 4-13 as facing the global negative y-direction). Your FEM lectures are helping me to understand and to appreciate how structures (theoretically) behave (or are assumed to behave) in general ^_^ it is a roller coaster ride, thanks for your work. Regards, DK

[@CivilEngineeringEssentials]

Just did a quick check on my version of the book Indeed, i think it should be negative as the external forces are negative. Please take my statement with a grain of salt as I quickly glimpsed over it without deeper consideration, as I am out of office ATM.

[@meditationandhealing5333]

My gratitude! Post processing was driving me crazy.

[@CivilEngineeringEssentials]

thnx a lot. I hope you do enjoy the series Regards, CEE

[@prakhars962]

This more of a matrix method not FEM. FEM involves shape functions, and you substitute the shape function in the weighted residual statement depending on different techniques.

[@CivilEngineeringEssentials]

A first course in the "finite element method" by daryl l logan. 👍 Still, I understand from where you are coming from. You are basically pointing to ritz methods and galerkin and min. Potential energy ajd so on. But that wont be for a first course in the method 🌹

[@prakhars962]

@@CivilEngineeringEssentials yes I have read the book. This is stiffness method, part of matrix method (chapter 2 of the book). anyways nice video.

[@CivilEngineeringEssentials]

I totally agree with you and understand ur point. Plz note that I did not cover ritz and galerkin extensively.