# Easy nonlinear model for large deformation of a beam

I have a beam subject to twisting and/or bending forces as well as linear compression force along its main axis. It is modelled as an isotropic beam, but if anisotropic is not too far off then that's okay too. The beam is capable of large deformations such that its maximum deformations are:

• 140 degrees in pure bending
• 140 degrees in pure twisting
• 70 degrees bending + 70 degrees twisting

What is an applicable nonlinear beam theory I can apply to this problem using equations rather than any software-based solutions?

I like using the basic undergrad Euler-Bernoulli beam theory, but the assumptions make it invalid in this case and I'm looking for something that is in the same vein as far as calculations goes and does not require significantly more advanced mathematics.

Ideally a theory that reduces the problem to a set of equations that can be solved without requiring multiple pages of tensor calculations that are hard to follow.

• I don't think you will get a satisfying answer to your question: non-linear AND no software. What material are we talking about here? Rubber? – Tim H Jan 26 '15 at 9:57
• It is PDMS. I mostly meant that I do not want to use a software where I plug in my forces, moments and modulus to get the results like ANSYS would. I intend to use matlab for solving equations as necessary. – imacube Jan 26 '15 at 10:02
• The words "easy" and "easiest" inject some element of opinion into this question. You could improve it by introducing some objective criteria of what's acceptable, for instance, you could restrict it to theory typically taught in undergraduate curricula, restricting graduate-level work. – Paul Gessler Jan 26 '15 at 12:39
• A quick search on scholar.google.com shows a bunch of articles dealing with PDMS beams... not sure what you'll have access to, but it looks like a large number of the PDFs are available w/o subscriptions to the sites. – andy mcevoy Jan 26 '15 at 15:20
• Just spit-balling here, but what about Timoshenko? Too simple? – Rick supports Monica Jan 26 '15 at 19:25

This may not fully answer your question but hopefully it will be a good start. I thought a distributed mass model would be a good approach for this so I did some searching and found this paper:

I also found this, which goes beyond what you need, including variable cross-sections and sheer stresses:

I think this second one is what you need, I included the first one because I can actually understand it whereas the second one is way beyond me. If you can simplify out the bits you don't need by substituting in suitable constants, it might be what you're looking for.

A similar question has been asked on the site with an answer posted here which shows the defining differential equations for large deformation of a beam.

The question was posed for uniform loading on a cantilever beam, but the solution can be extended to generalized loading and boundary conditions.

You have not mentioned what load you intend to apply for so much of bending / twisting deformation. Fiberglass (S-glass) has around 2 percent elongation and is used in Elastica/ pole vaulting/ Bow Archery/ automobile propeller shaft and similar large deformation applications, is a material choice that can be considered.

Since twisting is large,

• 1) Unsymmetrical section for eccentric placement of shear center for a channel beam
• 2) Flexible elastomer interleaving between composite plies or suitable resin system choice could be indicated.

The analysis requires consideration of in-plane/ bending forces occurring together. Interaction formulae involving EI/GJ are also used for structural dimensioning. With ANSYS you should use large deformation analysis.

If you have access to Matlab, the beam can be modeled as a collection of Finite Elements. Then the system can be represented with a Mass Matrix, Spring Constant Matrix, and a Damping Constant Matrix. The set of Forces also can be represented as a Force Vector. By solving the equations you have stress(sigma), strain(epsilon), and deflection (delta).

• This is very misleading, since it says nothing about the large-displacement aspect of the question. Also, the question seems to be about static analysis, in which case the mass and damping matrices are irrelevant. – alephzero Apr 28 '15 at 16:46
• The finite element method also deals with small deflections. Not large deflections. – Mark Jun 23 '15 at 15:50