Shear Stress in Tall Cross-Sections

The correct terminology is "Deep Beam Theory". The difference is due to shear lag which, is not significant in shallow beams.

Hi,

It is meaning torsional moment of inertia?

So likely Ib if you consider bending and It if you consider torsion.?

How do you define your coordinates? z seems to be in one direction of the cross section?

"only gives good results if the hight of the cross-section divided by the length of the beam is 1/10"

this is valid in bending and torsion but usually you can go down to 1/4.

At 1/4 (in bending so that the main deformation is in the stiff plane) the maximum equivalent (von Mises) stress from shear equals the tension/compression-stress at the edges of the beam.

The problem arises if the "beam" is shorter than 1/4. In this situation the matching of the beam to the (assumed rigid) frame will be most important.

To avoid stress concentrations from fast changing cross-sections an optimised shape is a necessity.

Look at trees transition from earth to tree there is an optimised shape, same where heavily loaded branches exit. Typical 2 times the diameter lengthwise and one diameter crosswise an optimum shape near an ellipse or hyperbola is matching the two different cross-sections. A simple radius (same optimum size) will give 6% stress rise above these elliptic or hyperbolic best fit curves that give a stress intensity factor of 1.01.

You can think about the stress lines as stream-lines in a laminar flowing fluid. Any sharp transition in cross section will cause trouble: too high stress in beams, vortices in tubes.

So if you make a container of liquid with a hole in the bottom of the shape to be tested and observe the free flow out of the sharp-edged hole then the shape of the flow is an optimum shape also in bending and torsion.

Coming back to your original question: " how does a shear strain over the beam cross-section give a strain that ruins the linear normal strain", it is not ruining the normal strain but in bending it is too low to be important and in bending and torsion the stress intensity factors are a result of a more or less important notch. "Notch" is a sharp change in cross section that cannot be followed by the stress lines.

Make a test for yourself and glue a stick of chalk or similar brittle stuff to a plane with a good glue, then break by torsion or bending and look at the fractured surfaces. Much to learn from these where the cracks started and how these propagated.

More to learn about the working of additional notches and to use these as notch-modifiers.

I use flexures in bending of very thin thickness (20... 40 ... any thickness) and length to widths of 1/1 ... 4/1, so shorter than the above-mentioned 4/1 ratio below which shear from bending is more important than lengthwise tensile or compressive stress.

RHABE

(Examples in http://www.precision-engineering.de/EB.FLEX.html)

Precision flexures and flexure-mechanisms

In the case of deep beams, plane sections do not remain plane as assumed in beam theory. Check out some of these links for a better understanding.