Euler-Bernoulli Beam Finite Element - Deriving the Mass and Stiffness Matrices
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- Опубликовано: 16 июл 2024
- Download notes for THIS video HERE: bit.ly/3mrD4rW
Download notes for my other videos: bit.ly/37OH9lX
Derivation of the Euler-Bernoulli beam element for transverse vibrations of a slender beam.
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Thanks for some amazing fundamentals of vibration lecture!
Thank you!
Thanks a ton, this was so helpful..
If possible, please try to make one on FEM through Rayleigh-Ritz method.
@14:16
Thank you very much for amazing lecture, may we see also Stiffness matrix for Timoshenko beam, thank you
Yes, soon
Hello my friend! Nice work you have done here! REally appreciate it! I would like to ask you how with the differentiation of the w(x,t) the non transponse φ appears inside the intergration?
Not sure if I fully understand your question, but hopefully this covers it...
What appears in the integration is anything that is dependent on x. By assuming that we can write w(x,t) as a function of x times a function of t (typically called "Separation of Variables"), we can write w(x,t) = transpose(φ(x)) · q(t). Since we are dealing with vectors, the standard way of squaring a vector is to take the dot product of the vector and it's transpose - i.e. w^2 = Transpose(w) · w = w · Transpose(w). Any x-derivative is only applied to φ(x) while any time derivative is applied to q(t).
In the end, we remove from under the integral anything that is not a function of 'x' - so the q(t)'s are removed and the φ(x)'s (i.e. φ and it's transpose) remain under the integral.
So φ(x) and its transpose come from squaring of w(x,t). The reason it stays under the integral is because it depends on x.
Hope I haven't confused you.
Thank you my friend for your response. You have not confused me at all! The answer to my question exists in the line " Since we are dealing with vectors, the standard way of squaring a vector is to take the dot product of the vector and it's transpose - i.e. w^2 = Transpose(w) · w = w · Transpose(w)". Thank you again for your effort and good luck!
My pleasure. Glad I could help.
your lectures are superb. I have one question, I have beam which has two fixed supports and is tentioned from both side. I want to calculate the bending stiffness, which for that I need a 4x4 stiffness matrix, which I calculated the coefficient for that and it'll multiply to flexural rigidity and add it to another 4x4 matrix which will multiply to the tension of the beam. I do not have a mechanical background and I am a bit struggling understanding the nature of the second matrix is that the mass matrix??
The problem you are describing is an axially-loaded beam. This problem is particularly important in the analysis of rotor blades, turbo machinery and the buckling of columns. It is a well-known phenomenon that the axial load in a beam affects the transverse stiffness of the beam. Tensile loads increase transverse stiffness (much like a string under tension) while compressive loads decrease the transverse stiffness of the beam (this is why too much compressive load will cause a column to buckle - due to the fact that the flexural rigidity of the column decreases).
It sounds to me like both the matrices you are describing are stiffness matrices. The first is due to the natural flexural rigidity of the beam while the second deals with the stiffening (or de-stiffening) effect due to the axial load.
For linear problems (i.e. small deflections), these two matrices are added together to get the overall stiffness matrix. Since the 2nd matrix is multiplied by the axial load, it will disappear as the axial load -> 0 . Also for a negative axial load (compression), the 2nd matrix will subtract from the first thereby reducing the effective stiffness of them beam.
hello, its an impressive tutorial but I am lost at 14.23 how do you got those values? I tried substituting but its not giving right results. What u substituted for x? Thanks in advance
The idea here is that the products of the shape functions (the phi_i's) are being integrated from 0 to L (then multiplied by rho x A). I explained it in a little more detail in the video where I derived the matrices for the bar element (ruclips.net/video/HFkK6n7sKfc/видео.htmlm57s).
For each element in the mass matrix, you first need to multiply the corresponding shape functions and then integrate that product from 0 to L (multiplied by rho x A).
For example, in the case of the element location at (1,1), you'd find the product of phi_1 x phi_1 (i.e. phi_1 squared) and then integrate that between the limits of 0 and L. This can be done by hand, but you might find WolframAlpha to be a useful tool for this. Go to wolframalpha.com and paste in the following:
(rho * A) * integrate [1 - 3 (x/L)^2 +2 (x/L)^3] [1 - 3 (x/L)^2 +2 (x/L)^3] dx from x=0 to L
which will give you the (1,1) element as (rho x A x L) x 13/35 (which is exactly the same as 146/420 which I have in my video).
Similarly, for the (2,1) element (or the 1,2 element by symmetry), you would use the shape functions, phi_1 x phi_2. You can paste the following into WolframAlpha.com:
(rho * A) * integrate [1 - 3 (x/L)^2 +2 (x/L)^3] [x - 2 L (x/L)^2 + L (x/L)^3 ] dx from x=0 to L
which will give you (rho x A x L) x 11/210 (which is the same as 22/420 shown in the video).
In general, for the (i,j) element you will multiply shape functions phi_i and phi_j, then integrate the product from 0 to L (then multiply by rho x A).
WolframAlpha will also give you the form of the indefinite integral which will allow you to validate your math step-by-step.
For the elements of the stiffness matrix, you pretty much proceed in the same manner EXCEPT THAT you first take the 2nd derivative of each shape function before finding the product (and, of course, you replace the (rho x A) with (E x I)).
Hope I haven't confused you with this lengthy explanation!
great explanation.. thanks a lot
if we consider axial forces then how will we solve it?
what boundary conditions should i consider for displacement ?
If you have axial forces, then these are just placed in the force vector on the RIGHT-HAND_SIDE of the equation.
i.e. the equations of motion are [m]{x_ddot} + [k]{x} = {F}
Also, I am not sure I understand your second question regarding the boundary conditions for displacement.
Sorry i asked the wrong question.
1.What is the physical significance of stiffness and mass matrix?
2.How will we get mass and stiffness matrix if axial displacement is also taken into account?
Hi, its a good lecture.
Can u suggest any book for this ?
When I studied this FEM, my professor published his notes for us to use - there was no textbook. That said, there are so many texts out there on FEM, but you might have to search a little to find one that works for you. These books tend to be expensive, so I am hesitant to recommend one for you. Sometimes, the best book is the one that uses the same notation that you are used to reading (because it seems everyone uses different notation).
For a classical approach to FEM, I have always been a fan of Klaus-Jurgen Bathe's work. He is one of the "pioneers" of FEM and no-one understands it any better than he does. Bathe is currently a professor at MIT.
For a different, more modern take on FEM, I would point you to a previous professor of mine, Satya Atluri. Nobody know the current state-of-the-art of FEM any better. Novel, cutting edge stuff - like meshless methods!!
can u pls give ur profs published notes... by the way u taught very well
satya atluri is from india...andhra pradesh state..gudivada town.. which is near to my home town
Hello sir,
In the matrix given in example of pipes by k j bathe in his book some of the coefficients have -ve sign......I wanted you to see it once.....
It might be due to a difference in choice of coordinate systems. Can you perhaps link to a photo of the page showing both the matrix and the coordinate system?
Hello sir,
I have taken the snaps of the above reference.....
The link to that are:
drive.google.com/file/d/1-VcezJGGYufLEMzf6HHk6klTEOc1qah3/view?usp=drivesdk
drive.google.com/file/d/1-TGYO5is1dHgy5FmzXjdwsfLvUZUAb-9/view?usp=drivesdk
Hi ! Do you have any video on deriving the stiffness matrix for 4 nodes elements? I mean plane element mass matrix .
Are you referring to a plate finite element? I don't have a video on that...yet. I will be making one in the future.
@@Freeball99 Thank you! I am waiting for it!
@@Freeball99 yes I can't find such like video in youtube
video time - 10:40 >> substitute eqn 9 into eqn 8. How did you get the ao......a3 ?
This is just linear algebra. Might be easier to use matrix notation. If I substitute 9 into 8, I end up with 4 equations which can be written in matrix form as [B]{a} = {q}. We wish to write it in the form of {a} = inv([B]) {q}. This amounts to simply inverting the coefficient matrix and you're done. Alternatively, you can calculate each by hand since the algebra is relatively simple.
2d!!!
Hi, what is the software that you are using to write on top of screen? is it smart board? please let me know.
I'm using a 13 inch iPad Pro with an Apple Pencil. The app is called "Paper" by WeTransfer. Video is captured using Quicktime on my Mac and a USB connection to my iPad. Microphone is a Blue Yeti. .