# Evaluating stiffness with varying MOE through cross section

1. Oct 17, 2011

### ClydeF

I am replacing an exisitng solid rod with a new rod with a rod made from a different material and a different cross section. The new material has a MOE that varies with skin depth which is making it hard for me to develop an expression for stiffness (EI).

The new rod has a hollow rectangular cross section with base and height b and h and wall thickness t. The MOE is given by E = M - dN where M (Max MOE) and N (MOE slope) are constants and d is skin depth. M and N are specific to bamboo species. (Think bamboo bike frame) The rectangular rod is composed of four strips. The MOE at the outermost fibers is always M (shown in red below). The figure below illustrates the rod.

If E were constant the stiffness would be given by EI = E * 1/12 * ( (bh3) - ( (b-2t)(h-2t)3 ) )

I have successfully simulated the stiffness as a discrete summation of thin walled rectangles using MatLab. However, this is part of a larger simulation and the discrete summation is really slowing down the overall simulation.

Can someone please point me in the right direction to develop a closed form expression for EI for the rod described above? I've spent some time trying to integrate the MOE expression with the second moment of inertia expression but I think I'm not correctly accounting for the MOE.

If the problem is not clear please let me know and I'll draw up some better graphics.

Thank you !!

-Tim