"Вестник МГСУ"

The proposed paper covers the operator-related formulation of the eigenvalue problem of analysis of a three-dimensional structure that has piecewise-constant physical and geometrical parameters alongside the so-called basic direction within the framework of a discrete-continual approach (a discrete-continual finite element method, a discrete-continual variation method).
Generally, discrete-continual formulations represent contemporary mathematical models that become available for computer implementation. They make it possible for a researcher to consider the boundary effects whenever particular components of the solution represent rapidly varying functions. Another feature of discrete-continual methods is the absence of any limitations imposed on lengths of structures. The three-dimensional problem of elasticity is used as the design model of a structure. In accordance with the so-called method of extended domain, the domain in question is embordered by an extended one of an arbitrary shape. At the stage of numerical implementation, relative key features of discrete-continual methods include convenient mathematical formulas, effective computational patterns and algorithms, simple data processing, etc. The authors present their formulation of the problem in question for an isotropic medium with allowance for supports restrained by elastic elements while standard boundary conditions are also taken into consideration.

The paper covers the analytical construction of fundamental functions of ordinary differential equations with constant coefficients and their wavelet approximations specific to problems of the structural mechanics. The definition of the fundamental function of an ordinary linear differential equation (operator) with constant coefficients is presented. A correct universal method of analytical construction of the fundamental function in the context of problems of structural analysis is described as well. Several basic elements of the multi-resolution wavelet analysis (basic definitions, wavelet transformations, the Haar wavelet etc.) are considered. Fast algorithms of analysis and synthesis (direct and inverse wavelet transformations) for the Haar basis and a corresponding algorithm of averaging are proposed. It is noteworthy that the algorithms of analysis and synthesis are the relevant constituents of all wavelet-based methods of structural analysis. Moreover, the effectiveness of these algorithms determines the global efficiency of respective methods. A few examples of fundamental functions of ordinary linear differential equations (the problem of analysis of beam, the problem of analysis of the beam resting on the elastic foundation) are presented.

The paper covers the operator-related formulation of the eigenvalue problem of analysis of the wall beam with piecewise-constant physical and geometrical parameters alongside the so-called basic direction within the framework of the discrete-continual approach (discrete-continual finite element method, discrete-continual variation-difference method). Generally, discrete-continual formulations are contemporary mathematical models which are currently becoming available for computer-based implementation. They allow investigators to consider the boundary effects whenever solution components represent rapidly varying functions. Another feature of discrete-continual methods is the absence of limitations imposed on lengths of structures. Two-dimensional model of elasticity is used as a design model of a structure. In accordance with the so-called extended domain method, the domain is limited by the boundary of arbitrary shape. Corresponding key features at the stage of numerical implementation of discrete-continual methods include convenient mathematical formulas, effective computational patterns and algorithms, simple data processing techniques, etc. The definition of an expression for an operator of the problem under consideration, if resolved in the isotropic medium, is presented; the allowance for supports restrained by elastic members is provided; standard boundary conditions are taken into account

The Wavelet analysis, that replaces the conventional Fourier analysis, is an exciting new
problem-solving tool employed by mathematicians, scientists and engineers. Recent decades have
witnessed intensive research in the theory of wavelets and their applications. Wavelets are mathematical
functions that divide the data into different frequency components, and examine each
component with a resolution adjusted to its scale. Therefore, the solution to the boundary problem
of structural mechanics within multilevel wavelet-based methods has local and global components.
The researcher may assess the infl uence of various factors. High-quality design models and reasonable
design changes can be made.
The Haar wavelet, known since 1910, is the simplest possible wavelet. Corresponding
computational algorithms are quite fast and effective. The problem of computing the convolution
of functions in the Haar basis, considered in this paper, arises, in particular, within the waveletbased
discrete-continual boundary element method of structural analysis. The authors present
their concept of convolution of functions within the Haar basis (one-dimensional case), share
their useful ideas concerning Haar functions, and derive a relevant convolution formula of Haar
functions.

Earthquakes can be very strong and can lead to significant damages. Effect of earthquakes depend on seismic action characteristics (intensity, spectral composition, etc.), foundation soil properties in region of construction, design and construction quality. In seismically dangerous regions structural calculations the current design standards suppose the use of the coefficient K1, which takes account the non-linear work of construction material and the allowable damages of structures. Our research shows that a stiffening core fails in case of intensive earthquake if the walls are designed according to current design standards. Thus, plastic deformations do not occur and develop in the supporting elements at the beginning of the process, so the lowering coefficient K1 should be disregarded. As stiffening core is projected with account for the reduction factor K1, the existing reinforcement is not enough for standing the emerging stress and its failure happens followed by a redistribution of the stress to frame columns. The columns are also projected with account for the reduction factor K1 and are not able to take such an increase stress beyond design. There is destruction of column frame and complete collapse of the building. So seismic resistance of bearing structures is reduced several times. The approach to estimating K1 must be responsible, based on the latest scientific research, which sometimes could not be done according to the acting design standards.

An earthquake is a rapid highly nonlinear process. In effective normative documents there is a coefficient K1, which takes into account limit damage of building structures, i.e. non-linear work of building materials and structures during seismic load. Its value depends on the building constructive layout. However, because of the development of construction and new constructive solutions this coefficient should be defined according to design-basis justification. The article considers the five-storey building calculation on seismic impact by linear-spectral and direct dynamic methods. Our research shows that the coefficient K1 for this building is 0.4, which was calculated using nonlinear dynamic method. According to effective normative documents K1 is 0.25…0.3 for buildings of this type. Thus we get a lack of seismic stability of bearing structures by 1.5…2 times. In order to ensure the seismic safety of buildings and facilities, especially of unique objects, the coefficient K1 should be determined by calculations with sufficient scientific justification, particularly with the use of non-linear dynamic methods.

The authors highlight three problems of the age of information technologies and proposes the strategy for their resolution in relation to the computer-aided design of civil engineering structures.
The authors express their concerns in respect of globalization of software programmes designated for the analysis of civil engineering structures and employed outside of Russia. The problem of the poor quality of the input data has reached Russia. Lately, the rate of accidents of buildings and structures has been growing not only in Russia. Control over efficiency of design projects is hardly performed. This attitude should be changed. Development and introduction of CAD along with the application the efficient methods of projection of behaviour of building structures are in demand. Computer-aided calculations have the function of a logical nucleus, and they need proper control. The system approach to computer-aided calculations and technologies designated for the projection of accidents is formulated by the authors.
Two tasks of the system approach and fundamentals of the strategy for its implementation are formulated. The study of cases of negative results of computer-aided design of engineering structures was performed and multi-component design patterns were developed. Conclusions concerning the results of researches aimed at regular and wide-scale implementation of the strategy fundamentals are formulated.
Organizational and innovative actions concerning the projected behaviour of civil engineering structures proposed in the strategy are to facilitate:
safety and reliability improvement of buildings and structures;
saving of building materials and resources;
improvement of labour efficiency of designers;
modernization and improvement of accuracy of projected behaviour of buildings and building standards;
closer ties between civil and building engineering researchers and construction companies;
development of competitive environment to boost competition in the market of structural design companies and in the market of developers.