טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKrush Maor
SubjectLow Probability Response of Structures with Stochastic
Hetercgeneity
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Eli Altus
Full Thesis textFull thesis text - English Version


Abstract

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 (Smin<S(x)<Smax) was assumed. Extremal reaction forces for a singly indeterminate case (Rmin and Rmax) and their corresponding morphologies are found analytically, after finding a monotonic relation between R and S(x). These morphologies are step functions composed of Smin and Smax. 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 Rext 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.