# Calculating the von mises stress in a 3D beam when given stresses and moments on both ends

I want to calculate the von mises stress in a given structural member. I want to generate a von-mises plot from this data. The forces are according to the local coordinate system of the beam. Also, I am using linear simplification for the beam.

I have an FEA output file that has provided the following:

Node 1 Location: -5.200901, -4.278615, 4.095733

Fx(axial force), Fy(shear force), Fz(shear force), Mx (Axial torsion), My (shear moment), Mz (shear moment)

Node 2 Location: -6.272305, 2.599551, 2.654122

Fx(axial force), Fy(shear force), Fz(shear force), Mx (Axial torsion), My (shear moment), Mz (shear moment)

Cross sectional area of beam (A): 0.001764 The beam has a square cross section of width and depth of 0.015 and a thickness of 0.003

Length of beam (L): 7.10882

Given structural properties of the isotropic metal

Also, is it possible to interpolate the values in between and figure out the plotting in between?

• Reading at your question I feel that there are ambiguities on the setup. Are you using a linear simplification for the beam? Please update your question with a graph of the nodes (just the locations). Jun 11 at 8:58
• @NMech Thank you for your feedback, I have updated the question Jun 11 at 9:06

If you only have those two nodes for the whole beam then it won't be possible, even if external loads are not applied between the nodes. The reason is the bending moments that follow (usually) a non linear relationship with x, so you won't be able to accurately interpolate between the nodes.

## Interpolation if more node data are available

If you had more nodes along the span of the beam, then it would be possible under certain conditions.

For example, if you have concentrated loads on the structure that will create discontinuities on the diagrams, and you won't be able to interpolate accurately. So you'd need to have concentrated loads only at the nodes (and preferably at the end nodes).

If the loads are distruted, then things are easier, however then the shape of the bending moment diagram will have a power of x shape.

In the simple case of having concentrated loads at the end of the structural members, regarding the interpolation of forces and moments then you could assume that the forces $$N_x, N_y, N_z$$ (what you call $$F_X, F_y, F_z$$) and $$M_x, M_y, M_z,$$ are continuous and interpolate the forces.

## how to calculate the von mises stress at each point coordinates (y,z) of a crosssection at distance x

First, you will also need to provide also

• the dimension of the beam cross-section (breadth, height)
• and also the elastic modulus (and depending on the implementation poisson's ratio) of the material.

Then (in the simplest form) you could use the equation which will allow you to calculate for each cross-section:

• $$\sigma_{xx}$$ : affected mainly by $$N_x, M_y, M_z$$
• $$\tau_{xz}$$ : affected mainly by $$Q_z, M_x$$ ($$Q_z$$ is $$F_z$$)
• $$\tau_{xy}$$ : affected mainly by $$Q_y, M_x$$ ($$Q_y$$ is $$F_y$$)

you should be able to get formulas for each point on the cross-section.

The next step is to calculate the von Mises stress (At each point on the cross-section) and plot it.

• Thank you for your reply. Just wanted to clarify what you mean with Qz, and Qy, what are these quantities? I understand the sigma is the stress, Nx is the normal force, and tou is the torque. And the member is a square hollow section, with sides of 0.015 and a wall thickness of 0.003 Jun 11 at 9:34
• As I was reading it, I realized that I missed something and had to rewrite the post. Jun 11 at 11:22
• I'm not entirely satisfied but I feel a bit under the weather today. Maybe it was the jab yesterday. Jun 11 at 12:31
• No problem, thanks for your detailed reply. I Still couldn't figure out what was Q and T. Also, what I have is an entire finite element model. I was trying to see if I can generate the von-Mises plot for individual elements using the given data. Jun 14 at 8:15
• $Q_y$ is what you called $F_y$. $T$ I don't think I've used it. Jun 14 at 8:52

Von Misses Stress is a distinct state of material measure at a certain location of an element. It is not a measure of stress variations between two points. The equations below demonstrate how to calculate the Von Misses Stress for an element under various states/types of stresses.

https://en.wikipedia.org/wiki/Von_Mises_yield_criterion#:~:text=Summary%20%20%20%20State%20of%20stress%20,2%20%7Bdisplays%20...%20%202%20more%20rows%20