A Middle Way

This book focuses on a method for exploring, explaining, and understanding the behavior of large many-body systems. It describes an approach to non-equilibrium behavior that focuses on structures (represented by correlation functions) that characterize mesoscale properties of the systems. In other words, rather than a fully bottom-up approach, starting with the components at the atomic or molecular scale, the “hydrodynamic approach” aims to describe and account for continuum behaviors by largely ignoring details at the “fundamental” level. This methodological approach has its origins in Einstein’s work on Brownian motion. He gave what may be the first instance of “upscaling” to determine an effective (continuum) value for a material parameter—the viscosity. His method is of a kind with much work in the science of materials. This connection and the wide-ranging interdisciplinary nature of these methods are stressed. Einstein also provided the first expression of a fundamental theorem of statistical mechanics called the Fluctuation-Dissipation theorem. This theorem provides the primary justification for the hydrodynamic, mesoscale methodology. Philosophical consequences include an argument to the effect that mesoscale parameters can be the natural variables for characterizing many-body systems. Further, the book offers a new argument for why continuum theories (fluid mechanics and equations for the bending of beams) are still justified despite completely ignoring the fact that fluids and materials have lower scale structure. The book argues for a middle way between continuum theories and atomic theories. A proper understanding of those connections can be had when mesoscales are taken seriously.

CALCULATION OF THE STABILITY OF A PLANE BENDING OF BEAMS WITH A VARIABLE CROSS-SECTIONAL WIDTH

The article proposes a technique for calculating the side buckling of beams of variable rectangular section based on the energy method. This method is considered on the example of a two-section cantilever beam of variable width under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized eigenvalue problem.

Problems of beam bending solution on the basis of variation criterion of critical energy levels

Introduction. The article is devoted to the development of variational formulations of structural mechanics problems using the example of the problems of bending beams. The existing variational approaches, the nonlinear theory of bending of beams, as well as the classical methods of resistance of materials, are not able to explain a number of issues related to the discrepancy between the results of theory and experiments, for example, in problems of pure and transverse bending of beams. To solve these issues, variational formulations and the criterion of critical levels of the internal potential energy of deformation, developed by the authors, are used. Materials and methods. For the internal potential energy of a deformed body, the stationarity condition at critical levels is written, which makes it possible to obtain equations of state that describe the self-stress of the structure. It is shown that a mathematical model of the state of a structure at critical levels of potential energy of deformation leads to an eigenvalue problem. The quantities characterizing the formulation of problems when formulating in generalized efforts and generalized displacements are discussed. Results. Using the examples of problems of pure bending and direct transverse bending of simple beams by a concentrated force, the formulation of the problem and the method of its solution are shown. The diagrams of deflections and bending moments are given, and the magnitudes of the amplitude values in the middle of the span are given. It is shown that for simple beams in problems of pure bending and transverse bending, the maximum values of the moments are achieved in the middle of the beam span, as in the experiment. Conclusion. The results are discussed and compared with the data obtained in the theory of flexible rods. It is noted that the dangerous section in two approaches having different physical nature is located in the middle of the beam span. The boundaries of discrepancy between the results for displacements, moments of internal forces and stresses are shown. It is noted that the results obtained according to the linear theory of strength of materials lead to a significant margin of safety. The prospects for the development of the theory of critical levels of internal potential energy of deformation, and the possibility of applying the technique to problems of structural mechanics are discussed.

Effects of Shear Lag in Steel Box Girders of a Crane Runway

The term of shear lag is related to the discrepancies between the approximate theory of the bending of beams and their real behaviour. It refers to the increases of the bending stresses near the flange-to-web junctions and the corresponding decreases in the flange stresses away from these junctions. In the case of wide flanges of plated structures, shear lag caused by shear strains, which are neglected in the conventional theory, may be taken into account by a reduced flange width concentrated along the webs of the steel girders. In EN 1993-1-5, the concept of taking shear lag into account is based on effective width of the flange which is defined in order to have the same total normal force in the gross flange subjected to the real transverse stress distribution as the effective flange subjected to a uniform stress equal to the maximum stress of the real transverse distribution. Some aspects concerning the shear lag phenomenon and a working example for a box girder of a heavy crane runway to illustrate the determination of the shear lag effect are also presented.

Local stresses in the keel when the ship is docked

При постановке судна в док на днищевые перекрытия со стороны килевой дорожки действуют значительные реактивные усилия, вызывающие местные деформации и напряжения в киле, стрингерах и флорах. С днищевых связей усилия передаются в основном на поперечные переборки и в меньшей степени на бортовые перекрытия, что вызывает общий изгиб корпуса судна. Расчеты общей прочности при постановке судов и кораблей в док показывают, что напряжения от общего изгиба корпуса незначительны. Дополнительные локальные напряжения от местного изгиба продольных днищевых связей, и в первую очередь в киле, при использовании балочных моделей либо не учитываются совсем, либо определяются достаточно условно. Альтернативой является использование метода конечных элементов (МКЭ) при достаточно подробном пространственном моделировании связей судна, дока и опорного устройства, что весьма затратно. В данной работе предлагается достаточно простая методика оценки локальных напряжений в киле при постановке судна в сухой док. Методика основана на использовании теории изгиба балок на сплошном упругом основании. Приводится пример расчета баржи-площадки. Показано, что напряжения в киле баржи вблизи поперечных переборок могут достигать недопустимо больших значений. Полученные результаты подтверждаются расчетом по МКЭ трехмерной модели баржи. When the ship is docked, significant reactive forces act on the bottom slabs from the keel track side, causing local deformations and stresses in the keel, stringers and floras. From the bottom braces, forces are transmitted mainly to the transverse bulkheads and, to a lesser extent, to the side floors, which causes a general bending of the ship's hull. Calculations of the total strength when ships and ships are docked show that the stresses from the general bending of the hull are insignificant. Additional local stresses from local bending of longitudinal bottom ties, and primarily in the keel, when using beam models, are either not taken into account at all, or are determined rather conditionally. An alternative is to use the finite element method (FEM) with a sufficiently detailed spatial modeling of the ship, dock and support device connections, which is very costly. This paper proposes a fairly simple method for assessing local stresses in the keel when the ship is in dry dock. The technique is based on the use of the theory of bending of beams on a solid elastic foundation. An example of calculation of the platform barge is given. It is shown that stresses in the keel of a barge near transverse bulkheads can reach unacceptably high values. The results obtained are confirmed by FEM calculations of a three-dimensional model of the barge.