DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY

Vestnik MGSU 8/2012
  • Nemchinov Vladimir Valentinovich - Moscow State University of Civil Engineering (MGSU) Candidate of Technical Sciences, Professor, Department of Applied Mechanics and Mathematics, Mytischi Branch 8 (495) 583-73-81, Moscow State University of Civil Engineering (MGSU), 50 Olimpiyskiy prospekt, Mytischi, Moscow Region, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 104 - 111

Numerical modeling of dynamic problems of the theory of elasticity remains a relevant task.
A complex network of waves that propagate within solid bodies, including longitudinal, transverse,
conical and surface Rayleigh waves, etc., prevents the separation of wave fronts for modeling purposes.
Therefore, it is required to apply the so-called "pass-through analysis".
The method applied to resolve dynamic problems of the two-dimensional theory of elasticity
employs finite elements to approximate computational domains of complex shapes, whereby the
software calculates the speed and voltage in the medium at each step. Preset boundary conditions
are satisfied precisely.
The resulting method is classified as explicit bilayer difference schemes that form special
relationships at the boundary points.
The method is based on an implicit bilayer time-difference scheme based on a system of
dynamic equations of the theory of elasticity of the first order, which is converted into an explicit
scheme with the help of a Taylor series in time, while basic relations are resolved with the help of
the Galerkin method. The author demonstrates that the speed and voltage are calculated with the
same accuracy as the one provided by the classical finite element method, whereby determination
of stresses has to act as a numerically differentiating displacement.
The author identifies the relations needed to calculate both the internal points of the computational
domain and the boundary points. The author has also analyzed the accuracy and convergence
of the resulting method having completed a numerical simulation of the well-known problem
of diffraction of a longitudinal wave speed in a circular aperture. The problem has an analytical
solution.

DOI: 10.22227/1997-0935.2012.8.104 - 111

References
  1. Baron M.L., Matthews. Difraktsiya volny davleniya otnositel’no tsilindricheskoy polosti v uprugoy srede [Diffraction of a Pressure Wave with Respect to a Cylindrical Cavity in an Elastic Medium]. Prikladnaya mekhanika [Applied Mechanics]. A series, no. 3, 1961, pp. 31—38.
  2. Klifton R.Dzh. Raznostnyy metod v ploskikh zadachakh dinamicheskoy uprugosti [Difference Method for Plane Problems of Dynamic Elasticity]. Mekhanika [Mechanics]. 1968, no. 1 (107), pp. 103—122.
  3. Musaev V.K. Primenenie metoda konechnykh elementov k resheniyu ploskoy nestatsionarnoy dinamicheskoy zadachi teorii uprugosti [Application of the Finite Element Method to Solve a Transient Dynamic Plane Elasticity Problem]. Mekhanika tverdogo tela [Mechanics of Solids]. 1980, no. 1, p. 167.
  4. Musaev V.K. Vozdeystvie prodol’noy stupenchatoy volny na podkreplennoe krugloe otverstie v uprugoy srede [Impact of the Longitudinal Steo-shaped Wave on a Supported Circular Hole in an Elastic Medium]. All-Union Conference “Modern Problems of Structural Mechanics and Strength of Aircrafts.” Collected abstracts. Moscow Institute of Aviation, 1983, p. 51.
  5. Sabodash P.F, Cherednichenko R.A. Rasprostranenie uprugikh voln v polose, sostavlennoy iz dvukh raznorodnykh materialov [Propagation of Elastic Waves in a Band Composed of Two Dissimilar Materials]. Collected works on “Selected Problems of Applied Mechanics” dedicated to the 60th Anniversary of Academician V.N. Chelomey. Moscow, VINITI, pp. 617—624.
  6. Clifnon R.J. A Difference Method for Plane Problems in Dynamic Elasticity. Quart. Appl. Mfth. 1967, vol. 25, no. 1, pp. 97—116.

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Solid models of rectangular section columns within the framework of analysis of building structures using the method of finite elements

Vestnik MGSU 9/2012
  • Agapov Vladimir Pavlovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Department of Applied Mechanics and Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoye shosse, Moscow, 129337, Russian Federation; +7 (495) 583-47-52; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Vasilev Aleksey Viktorovich - Rodnik Limited Liability Company design engineer 8 (482) 2-761-004, Rodnik Limited Liability Company, 22 Kominterna st., Tver, 170000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 55 - 59

The theory of the strength of materials has produced a substantial influence on the development
and practical implementation of computer methods of the strength analysis of beams and
beam systems. Beams are modeled through the employment of one-dimensional elements within
the overwhelming majority of the finite element method software programmes; the stiffness matrix
is derived on the basis of the hypothesis of flat sections, and end forces concentrate in the centres
of the gravity of cross sections. This approach makes it possible to develop effective algorithms,
although it has several drawbacks. They include an incorrect transmission of forces from beams
to plates and massive elements of structures, difficulties in taking account of the warping effect of
the beam, and the complexity of taking account of physical and geometrical nonlinearities. Some
authors suggest using the three-dimensional theory with account for the flat sections hypothesis. It
encompasses the patterns of rotations of sections in the analysis of structures, although the problems
of warping and shear deformations remain.
The authors propose a new approach to rectangular column modeling by means of the finite
element analysis of building structures. Each column is presented as a set of three-dimensional
8-node elements with arbitrary discretization alongside the cross section and the height of the column.
The inner nodes of the finite element mesh are excluded sequentially layer by layer, thus,
reducing the stiffness matrix and other characteristics of the column with reference to its top and
bottom cross sections. The finite element method has been adapted to PRINS software programme.
The comparative analysis of the two structures has been completed with the help of this software.
The structures exposed to the structural analysis included slabs and columns. In one case,
columns were modeled with the help of one-dimensional elements, and in the another case, the
proposed elements were used. The comparison of the results demonstrates that the employment
of the proposed elements makes it possible to avoid problems associated with the transmission of
the force in a particular point.

DOI: 10.22227/1997-0935.2012.9.55 - 59

References
  1. Filin A.P. Matritsy v statike sterzhnevykh sistem [Matrices in the Statics of Framework Structures]. Ìoscow-Leningrad, Izd-vo literatury po stroitel’stvu publ. [Publishing House of Civil Engineering Literature]. 1966, 438 p.
  2. Rabotnov Yu.N. Soprotivlenie materialov [Strength of Materials]. Moscow, Fizmatgiz Publ., 1962, 456 p.
  3. Feodos’ev V.I. Soprotivlenie materialov [Strength of Materials]. Moscow, Nauka Publ., 1986, 512 p.
  4. Aleksandrov A.V., Lashchennikov B.Ya., Shaposhnikov N.N., Smirnov V.A. Metody rascheta sterzhnevykh sistem, plastin i obolochek s primeneniem EVM [Computer Methods of Analysis of Framework Structures, Plates and Shells]. Moscow, 1976.
  5. Kornoukhov N.V. Prochnost’ i ustoychivost’ sterzhnevykh sistem [Strength and Stability of Framework Structures]. Moscow, Stroyizdat Publ., 1949, 376 p.
  6. Zienkiewicz O.C., Taylor R.L. The Finite Element Method for Solid and Structural Mechanics. McGraw-Hill, 2005, 631 p.
  7. Bathe K.J. Finite Element Procedures. Prentice Hall, Inc., 1996, 1037 p.
  8. Ayoub À., Filippou F.C. Mixed Formulation of Nonlinear Steel-concrete Composite Beam Element. J. Structural Engineering. ASCE, 2000.
  9. Hjelmstad K.D., Taciroglu E. Mixed Variational Methods for Finite Element Analysis of Geometrically Non-linear, Inelastic Bernoulli-Euler Beams. Communications in Numerical Methods in Engineering. 2003.
  10. Agapov V.P. Issledovanie prochnosti prostranstvennykh konstruktsiy v lineynoy i nelineynoy postanovkakh s ispol’zovaniem vychislitel’nogo kompleksa «PRINS» [Strength Analysis of Three-dimensional Linear and Non-linear Structures Using PRINS Software Programme]. Collection of works “Threedimensional Constructions of Buildings and Structures: Research, Analysis, Design and Application”. no. 11, Moscow, 2008, pp. 57—67.

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