Shear center, tension axis and cross-sectional stiffness/flexibility

Most often a genreral beam theory requires a 6×6 cross-sectional stiffness/flexibility (s/f). Though it is great to be able to give this completely-general, coupled s/f matrix, it is sometimes difficult to figure out how to specify the s/f matrix if one has a simple (isotropic/quasi-isotropic) cross-section. This post tries to address this issue. Consider a cross-section as shown below.
Figure A: Beam cross-section showing the shear center (s) and the tension center (t)

For a simple cross-section, one can always find two points on the cross-section, viz., shear center (s) and the tension center (t). By defining the shear forces and torque at the shear center while defining the axial force and the bending moments at the tension center, we have the relation between the beam generalized forces and the corresponding beam generalized strains as:



The equations can be written as:

and, the corresponding inverse relation:

Figure B: Forces and moments at the beam reference line

Now to solve the beam equations one needs to transform all the variables to the beam reference axis. The transformations are:

where:


The above constitutive law can now be written for the beam reference axis as:

Adding pictures/illustrations/graphs etc

To add pictures,

1. Create the picture
2. Convert to JPG
3. Use Picasa to view and modify the picture
4. From Picasa use Hello to post the picture to Blogger using Bloggerbot
5. Edit the post to add other stuff

All the three software, viz., Picasa, Hello and of course Blogger are free and are maintained by Google. For the picture below you will note that it shows a 320 size JPG file in the post but one can click on it to see a much bigger 640 size. You can use Hello to specify these two sizes.


Trial Photo

Equations formatted using LaTeX on this blog

This is to check if I can insert equations into my log. For example if I wanted to add the Energy-Mass equation, I can add:
<EMBED TYPE="application/x-techexplorer"
TEXDATA="$$E = m c^2$$"
HEIGHT=20 WIDTH=100%></EMBED>
which translates to:

As one can see it does do a LaTeX conversion. The only hassel is that I have to specify the HEIGHT. Otherwise it seems like an ideal way to communicate in LaTeX on the Web.

Note: You do need to download the techexplorer plugin from Integre Technical Publishing Co.

Here I go ...

Just checking if this thing works!