# Undergraduate researcher applies binning to study aleatory uncertainty in nonlinear buckling foundation models

Sophomore undergraduate, Katharin Jensen, has developed an easily understood illustration of the effect of aleatory uncertainty, which means natural point-to-point variability in systems. She has put statistical variability on the lengths of buckling elements in the following system:

Each of the buckling hinges has a lateral spring support (not shown above, but illustrated below) which resists axial loads up to a buckling limit. This system was chosen because it is typically the first example of a buckling system seen by undergraduates, and our goal for this work is to illustrate the profound effect of aleatory uncertainty (i.e., irreducible statistical variability in micromorphology such as link lengths) in a context that any undergraduate student of mechanics can understand. The trends shown here (especially when Katharin’s code is revised to allow damage in the form of broken lateral support springs) are similar to what we see in failure data for ceramics.

In the above plot, the red dashed line shows the force-displacement response of this 5-element system in the absence of any variability in buckling component lengths. The dark line illustrates that including variability in the buckling component lengths produces a reduction in the overall strength (peak force). This system is 100% elastic and recoverable, meaning that the force-displacement plot is retraced backward if the plate is pulled upward after any degree of compression.  In addition to illustrating the effect of geometric variability, this system therefore also debunks the myth that reaching a peak in a force-displacement plot indicates material damage — there is no damage in this plot!  This system is displacement controlled, and hence it is stable even after the peak.   The fact that the peak force with variability is smaller than that of the idealized perfect system also gives you a hint about why Euler buckling theory is not conservative: it fails to account for natural perturbations in the material that stimulate premature instability.

A two-element system drawn in 3D to clarify the equations being solved. Each buckling component has a horizontal supporting spring on a slider.

QUESTIONS: When the component geometry is allowed to vary statistically, how much change is expected in the force-displacement plot? Is there a dependence on the number of elements in the system? The 3D image (at left) shows two elements in parallel (each element is a spring and buckling component in series). The opening animation showed a five-element system.  The family of graphs below depicts the expected variation in force-displacement plots of a 10-element system. Each blue line is the force-displacement plot for a 10-element realization of statistically variable link lengths.

As the number of elements is increased, the simulations take a long time to run. Katharin was able, however, to use the binning scheme at https://csmbrannon.net/2015/07/20/publication-binning/ to reduce her calculation run time from 1 hour down to only 7 seconds!

For all the gory details of this work on probabilistic foundation modeling, please see the fifty-page report:
(written entirely by sophomore undergraduate investigator, Katharin Jensen)