M.Sc Student | Krush Maor |
---|---|

Subject | Low Probability Response of Structures with Stochastic Hetercgeneity |

Department | Department of Mechanical Engineering |

Supervisor | Professor Emeritus Eli Altus |

Full Thesis text |

Estimating the
statistical behavior of a mechanical structure is crucial for the evaluation of
its reliability.** **Probabilistic behavior originates from stochastic
material properties, geometry or loads.

When the size of the structure’s sub-elements (grains) is negligible, a homogeneous effective property can be used to calculate the response. In these cases the statistical scatter of the response is very small. However, when the sub-element’s size is not negligible the statistical nature of the response may be crucial, as in failure predictions.

In many studies, the mean and variance of such beams are discussed and regarded as the most important statistical information for design (Altus 2001). However, since most structures are designed against failure the statistical data that concerns the low probability events is also important.

In this
research, the low probability behavior of a statically indeterminate, heterogeneous
beam is studied. The beam’s compliance is longitudinally stochastic. The target
function is the reaction force at the support *R*, which is a stochastic
functional of the compliance (Altus 2001).

A stochastic
bounded compliance field (*S*_{min}<*S*(*x*)*<S*_{max})
was assumed. Extremal reaction forces for a singly indeterminate case (*R*_{min}
and *R*_{max}) and their corresponding morphologies are found
analytically, after finding a monotonic relation between *R* and *S*(*x*).
These morphologies are step functions composed of *S*_{min} and *S*_{max}.
For a polynomial load distribution the maximal possible number of “steps” is
also obtained.

The approach is
then generalized to a multiply-supported beam (**R**). Extremal reaction
forces and corresponding morphologies are found for each support. The
components of **R** are confined within an envelope whose shape is obtained.

Based on the
above boundaries, the statistical behavior of *R* is studied. For a
discrete *N* elements beam, the full CDF of *R* could be expressed as
an implicit,* N*-dimensional integral.

A Fréchet
approximation of* R* is used to evaluate the CDF using a Monte-Carlo simulation.
This CDF is compared to the exact *R* CDF. It is shown that the
morphologies which yield *R*_{ext} are a much better basis for
approximating the CDF at the low probability region, compared to ⟨*S*⟩.

The maximal stress is often an important design parameter against failure. Knowing the probabilities of high stresses is crucial for structural reliability analysis. Therefore, in the last part, the reaction force’s statistics are used to find the probability of having a maximal stress at a certain level.