My main scientific interest lies in the area of nonlinear problems of thin shell theory, especially in studying shell behavior by large deflections.
The study of equilibrium states of shells with large deflections is of a practical interest for two reasons. First, the information concerning the initial post-buckling behavior is frequently not sufficient for analysis of stability of thin shells due to the acute imperfection sensitivity of such systems. It means that small imperfections both in geometry and loading can result in strong reduction of the classical critical load obtained for a perfect shell. Koiter’s perturbation method and other standard methods of bifurcation theory fundamentally furnish the possibility of taking into account only small imperfections. This results in a critical load value which is close to a classical one. However, the ranges of admissible loads of real structures are restricted by significantly smaller values and conform to non-close and non-imminent post-buckling equilibrium states of shells, which are characterized by large displacements. Thus post-buckling path in the load-displacement diagram with large deflections could provide additional required information on post-buckling behavior of shells in order to be able to estimate their stability. Moreover, the methods of local or standard bifurcation theory based on using the eigenfunctions of appropriate linear buckling problem are not applicable to many practically important problems for shells, such as circular cylinders under axial compression or spheres under external pressure, because of a large number of closely spaced eigenvalues. Furthermore, experiments, as well as numerical and analytical solutions show that shell equilibrium configurations with large deflections are quite different from classical buckling modes. They consist of regions of inextensional strain (for example, an inverted part of sphere) surrounded by narrow zones of extensive bending and membrane deformations (inner boundary layers). Investigation of such configurations calls for development of new sophisticated methods of analysis.
Second reason is that structures with thin-walled elements whose application involves large displacements, rotations and deformations are recently used intensively, for example, metallic bladders for aircraft cryogenic fluid tanks and expulsion systems, elements of computer keyboards and shock absorbers. The typical problem in this case is to design a structure with given load-displacement diagram and to estimate stress and strain states of appropriate equilibrium shapes.
A new asymptotic method is proposed to obtain solutions of nonlinear theory of thin shells. The method is based on a new definition of the small parameter that is selected to be proportional to the ratio of the shell thickness to the deflection amplitude of structure. This parameter is actually small by large deflections compared to the shell thickness, for instance, if the shell is in the post-buckling stage. Furthermore, this parameter is equal to the well-known one in the shell theory when deflections are of the order of curvature radius of a structure.
The singular perturbation method is applied in combination with the variational method to the general Reissner’s equations describing axially symmetric large deflections of thin isotropic, orthotropic and composite shells of revolution with varying material and geometrical parameters in meridian direction. The obtained asymptotic nonlinear boundary value problem is significantly simpler in comparison to the original one. The asymptotic model has the following advantages: the number of the geometrical and stiffness parameters of shell is effectively reduced, and singularities are eliminated without loss of the accuracy of the solution. The explicit asymptotic formulae describing both shell behavior with change of load and stress states of structure have been obtained. It is important to note that the accuracy of obtained asymptotic results coincides with the accuracy of the original boundary value problem for a thin shell which itself is based on the asymptotic limit of three-dimensional equations of the theory of elasticity with respect to the same small parameter.
To obtain the solution in the entire range of deflection amplitude, we can construct a linear or nonlinear one by small deflections and then join both asymptotic expansions by small and large deflections. This approach is illustrated for the case of spherical shell under concentrated load in the following figure:
Here curve 5 corresponds to the asymptotic solution by large deflections, line 2 corresponds to the linear solution, curve 3 is the next step of the nonlinear asymptotic approximation of solution by small deflections, curves 4 and 5 represent result of joining both asymptotic expansions. The numerical solution is shown as a dashed line. The experimental results, which have been obtained by testing the clamped semi-spherical rubber shell, are marked in this figure by circles and crosses.
As it is well known, the problem of correct determination of the suitable load for carrying shell structures can’t be effectively resolved without taking into account influences of various perturbation factors on their stability. The main attention of both researchers and practical engineers was devoted to the internal perturbations, which take place in the shells. These are initial geometrical and other imperfections, which appear during the manufacture of the structure. Their influence is significant, but admissible load calculated without taking into account external perturbations, which appear during the exploitation process, can’t guarantee the sufficient reliability of structures. The asymptotic analysis of shell behavior in pre-buckling and post-buckling ranges allowed us to select the level of basic load, which separates areas of high and low sensitivity of shell concerning the internal and external perturbations. This load could be recommended as an admissible one. Corresponding formula and curve 1 are shown in the figure below for the case of isotropic spherical shell under external pressure.
The area between two dashed curves corresponds to the known experimental results. The similar formulae have been obtained for orthotropic spherical shells as well as for isotropic axially compressed cylindrical shells.
Suggested asymptotic method allows also to predict the critical basic load if the imperfections or external perturbations are determinate.
In the following figure the experimental and theoretical results of investigation of stability of cylindrical shell under uniform external pressure are shown. To obtain an asymptotic solution, minimum of information about the structure has been used: the amplitude of initial imperfection (it was a local dimple specially created on the shell surface) and the value of additional deflection amplitude by small (not dangerous) load (the circle point in the figure). The linear pre-buckling solution was jointed with asymptotic solution by large deflections.
The unique experimental studies of elastic rubber spherical shell under uniform external pressure and concentrated load were conducted together with my postgraduate student Dr. S. Duginetz. The load-displacement diagrams have been obtained for deflections, whose value reached radius of the shell. Equilibrium configurations were also studied.
I had an exiting collaboration with a brilliant experimenter Dr. V. Krasovski. We investigated the pre-buckling and post-buckling behavior of isotropic cylindrical shells under axial load and additional local perturbations. Previous experimental investigations of buckling of compressed high-quality shells, conducted using a high-speed video camera, have shown that the loss of stability begins with formation of one or several small local dents, which initiate continuous formation process of stable periodic post-buckling configuration. Stable local post-buckling configurations were also obtained in special experiments with high-quality shells under external local perturbations of their surface in the certain loading ranges. This implies that the local character of buckling is an essential feature of real shells.
These experimental studies were fundamental in creating asymptotical models of thin shells.
Research in Reliability Analysis of complex systems[top]
In my work with Dyadem Int. I am involved in designing and optimizing algorithms to compute reliability of complex systems using Monte Carlo Simulation, Markov Chain, Decomposition and Binary Decision Diagram methods.
Two papers are published and presented at the most prestigious "Reliability and Maintenance Symposium" (RAMS)
Truncation approach in decomposition method for system reliability analysis, RAMS 2009
The Esary Proschan (EP) method and other approximations, based on minimal cut sets (MCSs) calculation, are commonly used in reliability system analysis. However, it is time consuming and even impossible to calculate all MCSs of large systems. Cut off by order or by probability is applied to find the most important MCSs. In this paper we discuss the efficiency and accuracy of this approach by comparing results of calculations, obtained for large industrial benchmarks using both EP and pivotal decomposition method (DM).
DM (and similar binary decision diagram method) allows obtaining exact solutions for large fault trees, but it is still time consuming. We have shown that for large fault trees it is efficient to apply approximate DM in combination with truncation approach. We suggest an algorithm, which is mathematically sound and accurate enough in the entire range of probability. It is also applied efficiently to calculation of different system parameters: total downtime, frequency, expected number of failures and mean time to failure (MTTF). The efficiency and accuracy of the approach is demonstrated by numerous calculations of most complex industrial benchmarks. For large industrial examples it is 100-1000 times faster than exact decomposition method.
An Improved Monte Carlo Method in Fault Tree Analysis, RAMS 2010
The Monte Carlo (MC) method is one of the most general ones in system reliability analysis, because it reflects the statistical nature of the problem. It is not restricted by type of failure models of system components, allows to capture the dynamic relationship between events and estimate the accuracy of obtained results by calculating standard error. However, it is rarely used in Fault Tree (FT) software, because a huge number of trials are required to reach a tolerable precision if the value of system probability is relatively small. Regrettably, this is the most important practical case, because nowadays highly reliable systems are ubiquitous.
In the present paper we study several enhancements of the raw simulation method: variance reduction, parallel computing, and improvements based on simple preliminary information about FT structure. They are efficiently developed both for static and dynamic FTs. The effectiveness and accuracy of the improved MC method is confirmed by numerous calculations of complex industrial benchmarks.