Timoshenko bending theory implemented in MAST

[JohnDN90]

Reviewing the stiffness matrix for Timoshenko beam's in Cook's finite element textbook (4th edition), when kappa-->inf (or k=1/kappa=0 in the Cook textbook), the bending stiffness matrix reduces to that of Euler-Bernoulli beam theory.

I tried this in MAST, using MAST::BERNOULLI for Euler-Bernoulli theory and using MAST::TIMOSHENKO and kappa=4.503e+15 to simulate Euler-Bernoulli theory with a Timoshenko formulation.

However, when I do this, the Jacobians between the two formulations do not match. I end up with some very large values on the diagonal for the Timoshenko case. Alternatively, I checked the case where kappa=0, and I get some zero's on the diagonal.

Could you shed any light on why Jacobian_bernoulli does not equal Jacobian_timoshenko as kappa-->inf?

[manavbhatia]

I would not expect the Jacobians to match since the two use very different discretizations. The formulation of the Timoshenko beam has the shear energy terms that depend linearly on the shear correction factor. So, if you put a large value of kappa, you get that diagonal block (from quadrature of shear energy) to be scaled by the same value. Likewise when you scale it with 0.

Most likely Cook is talking about the two formulations becoming theoretically equivalent such that they give the same displacements. But, there is no way that the two stiffness matrices will be same.

[JohnDN90]

Hmm, he doesn't show the derivation, but the stiffness matrix he presents does reduce to to the Euler-Bernoulli bending stiffness when kappa->inf.

Do you happen to remember which reference you used when you implemented Timoshenko bending in MAST? Perhaps there are additional terms that Cook neglects in his book.

Edit: Here, on page 227 is the same Timoshenko bending stiffness that Cook presents (other than the extension terms, I think they may be incorrect in that link).

[manavbhatia]

I used the standard formulation for Timoshenko beams. You can find it in books by Bathe or by Hughes. It is also detailed in my paper:

https://arc.aiaa.org/doi/abs/10.2514/1.C034301

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On Nov 6, 2019, at 2:09 PM, David John Neiferd notifications@github.com wrote:

 Hmm, he doesn't show the derivation, but the stiffness matrix he presents does reduce to to the Euler-Bernoulli bending stiffness when kappa->inf.

Do you happen to remember which reference you used when you implemented Timoshenko bending in MAST? Perhaps there are additional terms that Cook neglects in his book.

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[JohnDN90]

Thanks, I found the formulation in Section 5.4.1 of Bathe's book here. I see what you're doing now, although it is different than the stiffness matrix I'm typically used to seeing for Timoshenko beams. I'll have to look into both formulas more so I can get an understanding of the difference.

[manavbhatia]

Where does Cook talk about the equivalence of the formulations and stiffness matrices? I have the book and can look it up if you give me the page number.

[JohnDN90]

Pages 26-27 is where Cook discusses Timoshenko beams, only briefly. Although, the same stiffness matrix can be found in multiple journal articles, he references a few in the book.

Edit: He discusses it on pages 170 and 171 as well. Reading through the Bathe textbook, it looks like the formulation you used is prone to shear locking if the shear contribution is fully integrated. I see that you used reduced integration in MAST however. In the Cook textbook, page 171, he states "As the element becomes more and more slender, phi_y approaches zero, and [k] reduces to the familiar stiffness matrix of a beam that has only bending deformation, without exhibiting shear locking or ill-conditioning of [k]." So perhaps this formulation is even less prone to shear locking than the other formulation, but this is only speculation on my part as of now. I need to look into it more.

[manavbhatia]

I just reviewed it. He seems to be using the Hermite shape functions to compute the shear terms. These are cubic shape functions. We have used Lagrange shape functions (of specified order) to compute the bending and shear terms in the stiffness matrix. So, two different discretiztions.

The Bernoulli beam is a 4th order differential equation and the Timoshenko beam is a set of DEs with orders less than or equal to 2. Equations 2.3.7 in Cook’s book seem so be adding the shear terms as corrections to the Bernoulli beam expressions. I have not seen the papers where is borrowing this formulation from, but there appears to be some manipulation of the DEs.

[JohnDN90]

Comparing the two stiffness matrices for a single element cantilever beam, Cook's stiffness matrix matches Nastran and the analytical result available on Wikipedia.

The stiffness matrix from Bathe, the one implmeneted in MAST, is a lot stiffer, resulting in about 11.5% smaller displacements.

[JohnDN90]

I just reviewed it. He seems to be using the Hermite shape functions to compute the shear terms. These are cubic shape functions. We have used Lagrange shape functions (of specified order) to compute the bending and shear terms in the stiffness matrix. So, two different discretiztions. The Bernoulli beam is a 4th order differential equation and the Timoshenko beam is a set of DEs with orders less than or equal to 2. Equations 2.3.7 in Cook’s book seem so be adding the shear terms as corrections to the Bernoulli beam expressions. I have not seen the papers where is borrowing this formulation from, but there appears to be some manipulation of the DEs.

Ah ok, thanks for clarifying. I might take a crack at implementing the Cook textbook method in MAST tomorrow, allowing it to be selected using "MAST:TIMOSHENKO2" or something along those lines.

[manavbhatia]

Comparing the two stiffness matrices for a single element cantilever beam, Cook's stiffness matrix matches Nastran and the analytical result available on Wikipedia.

For thin sections the Timoshenko beam with equal order Lagrange shape functions for both v and theta will be stiffer due to shear locking, which is a numerical artifact. This goes away with reduced order integration. For thick sections shear licking is not an issue.

We have implemented both Bernoulli and Timoshenko beams in Mast. As you have identified, the latter uses Lagrange shape functions and can lock up unless reduced integration is used, which should be turned on by default.

Since cook’s formulation uses a different approach, it may be able to circumvent the locking problem.

The stiffness matrix from Bathe, the one implmeneted in MAST, is a lot stiffer, resulting in about 11.5% smaller displacements