Principal Stresslines

Hi Panagiotis and Sawako,

first of all congratulations to this amazing add on for Grasshopper, I have been trying it out for a couple of days and I am quite impressed.

I have had a close look at the slab example and at the stresslines. When you take a point in the middle of the slab and plot the S1 Stress through it you can see the following (SC_01.jpg) The vector display is the X Component of the Quad Results Planes. It seems to me that when the Field does a sharp turn (90 degree between one step and the next one) it starts plotting the other direction of stress (S2). Could you maybe explain what is happening there, I really would like to understand that.

Many thanks,

Philipp

Attachments:

SC_01.jpg, 351 KB

Replies to This Discussion

Permalink Reply by Panagiotis Michalatos on November 29, 2012 at 12:11pm

Hi Philipp,

The reason for this is that the stressline tracing algorithm always selects the most likely direction to continue along. So S1 and S2 lines are seeded at the same points and start with different directions but there is no guarantee that they will always follow the 1st or 2nd stress direction.The order of principal stresses is rather arbitrary as they are the eigenvalues /vectors of the stress matrix. the first has a smaller value than the second but its absolute value may be larger if it is negative. In general when generating patterns on such tensor fields [the principal curvature directions being another example] you need an algorithm that is invariant to 90 degrees rotations otherwise patterns seem erratic. Depending on what you need to visualize the separation of the stresslines into groups could be different [for example you could separate portions of stresslines in tension and compression etc..]

Permalink Reply by Philipp Ostermaier on November 30, 2012 at 8:48am

Hi Panagiotis,

first of all thank you for your fast reply. It is definitely a bit clearer to me now. I was thinking that the 2 directions are strictly separate, which does not seem to be the case, but at the same time just seems logical to me that a force would always seek the path of least resistance, so rather than making a 90 degree turn follow a more similar direction. The thought of separating stresslines into groups of tension and compression ist interesting from a design perspective. I wondered how tension and compression forces relate to the S1 and S2 lines, so what I did is pluging the outputs of P1 and P2 into the respective vector display for S1 and S2 and coloring the vectors blue for compression (negative values) and red for tension (positive values). So when you look at the upper side of the surface S1 (SC_02), Tensors along S1 show compression towards the middle and towards the supports Tension. However the Principal Stress Display of the Mesh Visualisation Component for the upper side shows it the other way round, again Red/ Tension and Cyan/ Compression as it says in your manual. Did I miss out on something ? When I look at the lower side (SC_03) I find it more or less matching up (I am just decerning between negative and positive values) so that might make the difference in the size of the compression area. So, does this mean that the S1 and S2 lines are related to the upper and lower side of the surface ? One beeing predominantely in compression(upper side) and one being stressed(lower side) ? That would also explain to me why S1 and S2 swap when you change the side of the surface. I am sorry, many questions... If you have time to explain, would be great. Also, maybe you have a book or article in mind which would explain those things more in depth....

Many thanks,

Philipp

Attachments:

SC_02.JPG, 360 KB
SC_03.JPG, 354 KB

Permalink Reply by Panagiotis Michalatos on December 4, 2012 at 5:51pm

Hi Philipp,

The slab in the millipede example has two heavy loads applied in the middle of its long sides. This means that if you see its displacement near the middle the dominant effect is not the bending between the supports but the even stronger bending perpendicular to this axis [as the two lateral loads pull its sides down]. So although the first principal stress vector along the long axis of the slab on the top plane has turned from tension into compression there is tension which is in absolute value stronger than this compression along the second direction.
The color mesh diagram DOES NOT visualize the tension/compression along the 1st or second direction, instead it looks at both of them and decides the color depending on which of the two is larger in absolute value [so it visualizes the dominant effect regardless of the direction along which that happens]

In Reality the stresses within the thickness of the slab are much more complicated [they are three dimensional tensors with three eigenvectors per point] and they twist in space. What you see in the millipede diagram are the normal stresses within a layer [including the effect of bending etc...] however as you can see in the two attached images of a cantilevering slab the situation is more complex with tension and compression lines crisscrossing as seen from the side [which is the well known cantilever stress diagram] and forming three dimensional loops within the bulk of the slab.

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