Точный метод решения задач на устойчивость с применением метода динамической жесткости

Стаття присвячена обговоренню застосування точного методу динамічної матриці жорсткості до проблем пружної стійкості інженерних конструкцій. В статті наведено вивід динамічної матриці жорсткості для структурних балочних компонентів з наступним керівництвом щодо автоматизації зборки глобальної динамічної матриці жорсткості для всієї конструкції з матриць для кожного компонента. Також обговорюються переваги метода динамічної матриці жорсткості у випадку параметричних розрахунків. Проаналізована проблема розрахунку власних значень трансцендентної матриці жорсткості. В деталях аналізується як метод прямого розрахунку власних значень, так і потужний метод Віттріка-Вільямса. В статті також наводиться загальна інструкція до програмування методу динамічної матриці жорсткості.The article is dedicated to the discussion on the exact dynamic stiffness matrix method applied to the problems of elastic stability of engineering structures. The detailed formulation of the member dynamic stiffness matrix for beams is presented along with the general guidelines on automatisation of the assembly of member dynamic stiffness matrices into the global matrix that corresponds to the whole structure. The advantage of the dynamic stiffness matrix in case of parametric studies is explained. The problem of computing the eigenvalues of transcendental matrix is addressed. The straightforward approach as well as a powerful WitrickWilliams algorithm are discussed in details. The general guidelines on programming the DS matrix method are given as well.Статья посвящена обсуждению применения точного метода динамической матрицы жесткости к проблемам упругой устойчивости инженерных конструкций. В статье представлен вывод динамической матрицы жесткости для структурных балочных компонентов с последующим руководством к автоматизации сборки глобальной динамической матрицы жесткости для всей конструкции из матриц для каждого компонента. Также обсуждаются преимущества метода динамической матрицы жесткости в случае параметрических расчетов. Проанализирована проблема расчета собственных значений трансцендентной матрицы жесткости. В деталях анализируются как метод прямого расчета собственных значений, так и мощный алгоритм Виттрика-Вильямса. В статье также приводится общее руководство к программированию метода динамической матрицы жесткости

ТОЧНЫЙ МЕТОД РЕШЕНИЯ ЗАДАЧ НА УСТОЙЧИВОСТЬ С ПРИМЕНЕНИЕМ МЕТОДА ДИНАМИЧЕСКОЙ ЖЕСТКОСТИ

The article is dedicated to the discussion on the exact dynamic stiffness matrix method applied to the problems of elastic stability of engineering structures. The detailed formulation of the member dynamic stiffness matrix for beams is presented along with the general guidelines on automatisation of the assembly of member dynamic stiffness matrices into the global matrix that corresponds to the whole structure. The advantage of the dynamic stiffness matrix in case of parametric studies is explained. The problem of computing the eigenvalues of transcendental matrix is addressed. The straightforward approach as well as a powerful Witrick-Williams algorithm are discussed in details. The general guidelines on programming the DS matrix method are given as well.Статья посвящена обсуждению применения точного метода динамической матрицы жесткости к проблемам упругой устойчивости инженерных конструкций. В статье представлен вывод динамической матрицы жесткости для структурных балочных компонентов с последующим руководством к автоматизации сборки глобальной динамической матрицы жесткости для всей конструкции из матриц для каждого компонента. Также обсуждаются преимущества метода динамической матрицы жесткости в случае параметрических расчетов. Проанализирована проблема расчета собственных значений трансцендентной матрицы жесткости. В деталях анализируются как метод прямого расчета собственных значений, так и мощный алгоритм Виттрика-Вильямса. В статье также приводится общее руководство к программированию метода динамической матрицы жесткости.Статья посвящена обсуждению применения точного метода динамической матрицы жесткости к проблемам упругой устойчивости инженерных конструкций. В статье представлен вывод динамической матрицы жесткости для структурных балочных компонентов с последующим руководством к автоматизации сборки глобальной динамической матрицы жесткости для всей конструкции из матриц для каждого компонента. Также обсуждаются преимущества метода динамической матрицы жесткости в случае параметрических расчетов. Проанализирована проблема расчета собственных значений трансцендентной матрицы жесткости. В деталях анализируются как метод прямого расчета собственных значений, так и мощный алгоритм Виттрика-Вильямса. В статье также приводится общее руководство к программированию метода динамической матрицы жесткости

Exact elastic stability analysis based on dynamic stiffness method

The article is dedicated to the discussion on the exact dynamic stiffness matrix method applied to the problems of elastic stability of engineering structures. The detailed formulation of the member dynamic stiffness matrix for beams is presented along with the general guidelines on automatisation of the assembly of member dynamic stiffness matrices into the global matrix that corresponds to the whole structure. The advantage of the dynamic stiffness matrix in case of parametric studies is explained. The problem of computing the eigenvalues of transcendental matrix is addressed. The straightforward approach as well as a powerful Witrick-Williams algorithm are discussed in details. The general guidelines on programming the DS matrix method are given as well

Response localization in disordered structures governed by the Sturm-Liouville differential equation. (Review)

The review is dedicated to the relatively new problem in structural engineering: localization of the response by structural irregularities. This review is aimed to outline all relevant discoveries in the response localization in mechanical problems (vibration, buckling) from the perspective of the common mathematical representation through Sturm-Liouville problem. Two possible approaches to analyze the influence of the disorder are discussed: exact dynamic stiffness formulation of the mistuned structure and the perturbation of the eigen solution of the tuned structure. Both approaches shown to lead to the same localization phenomena end exponential decay of the eigenvector from the source of disorder. In the section dedicated to the buckling mode localization the approach to analyze localization of the randomly disordered multi-span beam based on the Furstenberg’s theorem in presented. The examples of the localization phenomena in the real engineering structures are given.<br/

Stability of split structures: degeneracy breaking and the role of coupling

The present work is inspired by the need to understand the elastic stability of a class of structures that appear in a variety of seemingly unrelated fields. Here we consider several problems involving the stability of two or more slender structures coupled at the ends. In a sequence, we consider a bilayer beam, then a multilayer split beam, chains of elastically coupled rigid rods, a plate with a symmetric cut out, and finally several plate strips elastically coupled. We also study the instability of a biological structure known as the mitotic spindle. We report cooperative, competitive, and antisymmetric buckling of the bilayer split beam; and their dependence on the geometric parameters. Then we identify the mechanisms of elastic deformation, including additional strain induced by the misfit of two layers tied together at ends, that explains the observed behaviour. This is extended to buckling of a multilayer structure, i.e. a stack of thin elastic layers coupled at the ends. We also report rapid decay of the buckling amplitude of layers along the stacking direction, observed in simple experiments. We theoretically study a chain of elastically coupled rigid rods as the simplest model of this behaviour and report that coupled identical members, in the absence of any disorder, show spatially extended buckling modes, i.e. buckling amplitudes are periodically modulated. Analogies are drawn with a physically unrelated, yet mathematically close problem of wave propagation in periodic media. Introduction of irregularity leads to the spatial exponential decay of the amplitudes, i.e. localisation of buckling modes and thus associated Lyapunov exponents. We show that the strength of buckling localisation depends on the coupling-to-disorder ratio. Next, we study the instability of rectangular plates with one or more cut outs placed periodically. The first problem reveals two types of buckling modes – in-phase buckling and out-of-phase buckling of the two elastically coupled plate strips. Energy contributions from cylindrical bending and twist of the coupling region drive the structure from degeneracy to where the mode character changes. The second problem of multiple strips elastically connected reveals that the in-phase and out-of-phase modes become periodically modulated and the respective buckling loads appear in clusters. If the structure is perfectly ordered, the entire clusters of buckling loads are inverted in the degeneracy point via N-fold crossing. Infinitesimally small disorder triggers repulsion of eigenvalues and strong localisation occurs. We characterise this e↵ect comprehensively by calculating Lyapunov localisation factors and report regions of structural parameters for which high and moderate sensitivity to disorder is observed. Finally, mitotic spindles were studied using continuum modelling of the slender bio-structures also accounting for the interaction with the environment of the cell. Interesting buckling modes with spatial features such as coupled bending and torsion of filaments were observed

Structural stability of a Mitotic Spindle: parametric Finite element approach

Mitotic spindles are mechanical structures that play a critical role in cell division by generating forces to separate chromosomes. They are ordered assemblages of proteins that make up microtubules (MT) and microtubule connectors whose mechanical properties are responsible for their structural integrity under mitotic forces. We use a continuum mechanics approach to study the stability of equilibrium of a mitotic spindle as a whole. We create and apply a finite element (FE) parameterised model of interpolar MTs, astral MTs and MT connectors varying the number of MT filaments and the arrangement of their interconnections. The model is based on the experimental data on Fission Yeast spindles in late anaphase B and mitotic HeLa cells [1]–[3]. We account for the complex interactions between interpolar MTs, astral MTs, connectors and centrosomes through mechanical coupling. Comparing the results with experiments and Molecular dynamics-based simulations [1], we demonstrate the great potential of Structural mechanics methods to address the stability of spindles. Here we report how buckling states of the spindle get localised towards either of centrosomes due to the irregular placement of microtubules and irregularities in MT coupling. In certain cases, such behaviour may result in nuclear misplacement, asymmetric division or other abnormalities.[1] J. J. Ward, H. Roque, C. Antony, and F. Nédélec, “Mechanical design principles of a mitotic spindle,” eLife, vol. 3, p. e03398, 2014.[2] F. Pampaloni, G. Lattanzi, A. Jonas, T. Surrey, E. Frey, and E.-L. Florin, “Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistence length,” Proc. Natl. Acad. Sci., vol. 103, no. 27, pp. 10248–10253, 2006.[3] F. M. Nixon, C. Gutiérrez-Caballero, F. E. Hood, D. G. Booth, I. A. Prior, and S. J. Royle, “The mesh is a network of microtubule connectors that stabilizes individual kinetochore fibers of the mitotic spindle,” eLife, vol. 4, no. JUNE2015, pp. 1–21, 2015.<br/