# von Mises or Equivalent stress for Beam Elements

Are there analytical expressions for computing von Mises or equivalent stress at any point for Beam elements similar to the ones in continuum elements? I would like to design a structure with beam elements and ensure that the equivalent stress is within bounds (yield stress etc) for the structure to be safe.

### Re: von Mises for Beam Elements

Before using von Mises stress to design a structure with beam elements, I would check the structure  is not determined by buckling due to axial and/or bending forces. I usually never use von Mises on a beam structure unless I am sure it will fail by yielding. A common mistake that I have seen is that many mechanical engineers try to use the von Mises stress to analize by yielding a structural element that clearly would fail by buckling.

Is your beam element really a beam element? There are some software that use beam element to refer to beam as well as frame elements.

### Thanks for the response

Thanks for the response Alejandro. Yes, I am check buckling also along with stresses. But rather than enforcing multiple stress constraints (one for each stress - axial, bending shear, torsional shear), I want to enforce a single constraint on equialent stress. So back to original question, how do I compute this equivalent/von Mises stress?

### Re: von Mises for beam elements

If you know the analytical expressions for the axial, shear and torsional stresses at any point in the beam, then you can use

\sigma_\text{VM} = \sqrt{{1\over 2}\left[\left(\sigma_{xx} - \sigma_{yy}\right)^2 + \left(\sigma_{yy} - \sigma_{zz}\right)^2 + \left(\sigma_{zz} - \sigma_{xx}\right)^2 \right] + 3 \left(\tau^2_{xy} + \tau^2_{yz} + \tau^2_{zx}\right) }

### Re: von Mises for beam elements

So you are saying the definition of von mises stress used for continuum elements is valid for beam element too. Here, axial stress is \sigma_{xx}, shear in y-direction is \tau_{xy}, shear in z-direction is \tau_{xz}. All other components are zero? How do you resolve the torsional stress in terms of the stress components indicated in the expression?

### Re: von Mises for beam elements

The torsional stress must be included in one of these terms:

\tau^2_{xy} + \tau^2_{yz} + \tau^2_{zx}

### Re: von Mises for beam elements

Which one exactly? Implementation wise it won't matter but I am trying to understand the theoretical background. Any references I can look at?

### Re: von Mises stress for beams

It depends on how it is oriented the Cartesian coordinate in your problem. I think there is an example for a bike pedal arm that is subjected to combined bending and torsion, in this book:

http://www.amazon.com/gp/aw/d/0136123708/ref=mp_s_a_1_3?qid=1416374174&s...

or perhaps in this one:

http://www.amazon.com/gp/aw/d/0135077931/ref=mp_s_a_1_2?qid=1416374464&s...

### Re: von Mises stress for beams

Thanks for your help Alejandro! The first book has the example you mentioned.

### Using VABS will directly give

Using VABS will directly give you three stress defined in the framework of continuum mechanics. You will not have such confusion at all. Note unixial stress assumption is not valid for beams which are not isotropic homogeneous. You can try VABS in the cloud at https://cdmhub.org/tools/vabs. There is a detailed comparison between VABS and ABAQUS Beam Section Generate available at https://cdmhub.org/resources/912, showing VABS can reproduce 3D FEA accuracy at a much better efficiency that ABAQUS Beam Section Generate. 