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Three-ways to derive the Euler-Bernoulli-Saint Venant Beam Theory

Wenbin Yu's picture

After having taught graduate structural mechanics for several years, I am finally
able to write down my lecture notes (attached) for teaching the beam theory. In
the notes, we formulated the complete classical beam model
(extension/torsion/bending in two directions), which is also called
Euler-Bernoulli-Saint beam theory, in three ways: Newtonian method, variational
method, and variational asymptotic method, using 3D elasticity theory as the
starting point. Many self-contradictions of the various assumptions used in both
Newtonian method and variational method are clearly pointed out. The
variational asymptotic method does not rely on any ad hoc assumptions, ending
in a self-consistent theory. Many confusion points (at least for myself when I was an undergraduate student) about the beam theory have been clarified. Deriving
Timoshenko model was left as a homework problem. It can be considered as a theoretical
tutorial for VABS, a tool for blade modeling extensively used in helicopter and
wind turbine industries.


comments/critiques are welcome.

PDF icon EulerBernoulli3D.pdf252.11 KB


Pu Zhang's picture

I read this lecture note carefully and I really enjoy it. The Euler beam is a classical model one can find in any text books about Mechanics of Materials. However, this note tells readers a long but fascinating story and shows different approaches for deriving the beam model together.

In addition, I feel it is also necessary to compare the Euler beam and Euler elastica in the last page. The latter one is quite useful but not introduced by most of the textbooks. 

Wenbin Yu's picture

Dear Pu,


Thanks a lot for your nice comments. I hope you noticed all the contraditing assumptions one has to use to derive classical beam models using ad hoc approaches which were what we have been taught. Can you point out a good reference for Euler elastica so that I can add it to the notes?



stan's picture

Thank you for the useful notes Wenbin Yu!

I wish to ask you how do you use the Variational Asymptotic Method for modelling
beams with large displacements and large rotations? I suppose that using the
nonlinear strain-displacement relations instead of the linear ones in equation
(70) will not be enough. Do you use co-rotational formulation or some other

Thank you in advance!

S. Stoykov

Wenbin Yu's picture

Dr. Stoykov,

Thanks a lot for your nice comments. For large displacements and large rotations, as long as the warping functions are small comparing to the beam displacements/rotations, it is possible to decouple the original nonlinear 3D problem into a nonlinear 1D nonlinear beam problem and 2D linear cross-sectional analysis. That has been done in VABS. If the warping functions are not small, other techniques such as co-rotational formulation should be used. We are currently doing some research for US Army for this aspect.



 I have been reviewing your notes as I am beginning my graduate study in the field of Aerospace Engineering. I am finding that they are very clear and helpful. Accordingly, I was hoping you could maybe provide the solutions to the problems at the end of the notes. I am trying to build a strong understanding of classical beam theory and modeling. I think that the solutions to your problems would be quite useful. Thanks for your time!



Wenbin Yu's picture

Thanks a lot for your comments. We at Purdue recently launched a website called, a web-based collaborative platform. Kind of plus the capabilities to run software in a web broswer, and many other nice features. I also posted this document on that site. The discussions will be followed there and the solutions will be posted there too. 

I have created a profile on your website and I found the document under Teaching Materials. However, I could not find a discussion about the paper. Did you mean that the solutions and discussion will appear in the near future? If so, how soon do you think they will be posted? Thanks again for the help!

Wenbin Yu's picture

Dear Preston,

Thanks for pointing this out. Please feedback this and your other comments through the ticket system of cdmHUB so that our support personnel can quickly take care of it. 


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