טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentIsrael Idan
SubjectAnalysis of Thin Curved Beams with Stochastic Properties
DepartmentDepartment of Mechanical Engineering
Supervisors Professor Emeritus Eli Altus
Professor Josef Givli
Full Thesis textFull thesis text - English Version


Abstract

Today's manufacturing technologies are very advanced and allow producing parts with very complex shapes and from a variety of materials. However, the parts they produce have stochastic material and geometric imperfections.

It is important to analyze the relation between imperfections and stochastic mechanical responses (reaction forces, deflection, etc): a. to reduce the safety factors and obtain optimal designs, and b. to reduce manufacturing costs, by keeping highly accurate demands only where truly necessary.

In this research the imperfections are described by random fields. Three analytical models are introduced, one for material imperfections in the form of stochastic compliance (S=EI-1) and two models for geometric imperfections, each associated with different fabrication processes. The Two Point Correlation (TPC) functions of the imperfections, which are not necessarily statistically homogeneous, are derived analytically from these models.

In this research, analytical tools are developed for finding the mechanical responses of thin curved Euler-Bernoulli beams. When the responses are stochastic, due to imperfections, their statistical properties are found as functionals of the stochastic imperfections TPC.

Mathematical expressions with generalized meaning are found from various expressions of the responses of a 2D curved beam.

First, a curved beam with stochastic compliance which varies along the beam is analyzed. Using the Functional Perturbation Method (FPM) and the TPC of the compliance, an analytical approximation for the variance of the beam's reaction force is found. Afterwards, curved beams with geometric imperfections are analyzed. Again, by knowing the TPC of imperfections, and applying the FPM, analytical approximation for the variance of the beam's mechanical responses are found. The effect of correlation length of imperfections on the variance is investigated for both kinds of imperfections.

Monte Carlo Simulations are used as a tool for generating realizations of beams with pre-determined stochastic imperfections. The mechanical responses exact solutions are calculated for each realization. Thus, we obtain samples of responses which can be used for finding the responses variance and compare it with the analytical approximations.

The main conclusions from this research are: 1.The relative averaged errors of the analytical approximations, in most cases, are less than 5%. 2. The sensitivity of the moment and deflection to geometric imperfections is considerably higher than the reaction sensitivity. 3. The distribution of the maximal moment can be a priori predicted for large or small correlation length. 4. Correlation length of the imperfections has a great effect on the magnitude of the responses variance.