# Benchmarking Geometrically Exact Beams: Clarification Needed on Material Density?

In an early paper on Simo, I came across an example titled "Cantilever 45-degree bend subject to fixed and follower end load". A force of $$600N$$ and $$300N$$ was applied at the end, resulting in a force-displacement curve. Given the physical parameters $$E=10^7$$ and $$G=0.5 \times 10^7$$ , why isn't the material density required for the calculation? Subsequent studies have also used this example without mentioning the beam's density. How should this problem be approached dynamically? I'm interested in calculating the equilibrium state.

The description of this example1 as follows:

EXAMPLE 7.5. Cantilever 45-degree bend subject to fixed and follower end load. This example has been considered by Bathe and Bolourchi [10] under fixed end load. The bend has a radius of 100 with a unit square cross-section. The material properties are $$E=10^7$$ and $$G=0.5 \times 10^7$$. These authors performed the analysis for conservative loading only using 8 three-dimensional degenerated beam elements. In the present calculation 8 linear elements are used. For comparison purposes with the results reported in [10] the bend is subject to a sequence of three load increments of magnitude 300,450 , and 600 . 1: Simo, Juan C., and Loc Vu-Quoc. "A three-dimensional finite-strain rod model. Part II: Computational aspects." Computer methods in applied mechanics and engineering 58.1 (1986): 79-116.

Can this example be mathematically expressed? I'd like to compute it myself using Matlab or Python. Any comments or suggestions would be greatly appreciated.

• Can this example be mathematically expressed? I'd like to compute it myself using Matlab or Python. Any comments or suggestions would be greatly appreciated!
– lumw
Aug 13, 2023 at 8:21
• Yes, of course. I have found Python quite suitable for calculating beam deflections and stresses. You can use matplotlib library for plotting deformed shapes or even to get colored results which look like from FEA (using contourf). If you want to do stuff in 3D, I would suggest meshio library for exporting into VTK file formats and exploring the results in ParaView. Aug 13, 2023 at 9:56
• Thank you, you're right. Python is a great tool. I'm thinking about how to lay out the expressions. Once the expressions are properly written, I should be able to compute it quickly. I'm quite proficient in Python.
– lumw
Aug 13, 2023 at 10:53

I'm not familiar with the finite strain approach to structural analysis, but I suspect they ignored density because density has no (direct) contribution to the strength or deflection of a structure.

There are some caveats:

• For some materials (e.g. concrete) density may have some correlation with stiffness, but density itself does not directly affect the displacement or strength of the structure.
• When combined with an acceleration (e.g. gravity) density may add a load to the structure, but I suspect the authors of that paper are ignoring self weight loads as well.

In linear elasticity, you can in some cases use superposition of multiple load cases results to get the results for the load combination. So focusing on one load makes more sense especially for benchmarking, where you might want to identify which cases you are still calculating incorrectly.

Here is an example applied to beam problems.

• Thank you. I'm trying to understand the related concepts. I appreciate the reference notes you provided.
– lumw
Aug 13, 2023 at 10:55