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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Basic functions and bilateral estimatesin the stability problems of elastic non-uniformly compressed rods expressed in terms of bending moments with additional conditions

Vestnik MGSU 2/2014
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 39-46

The method of two-sided evaluations is extended to the problems of stability of an elastic non-uniformly compressed rod, the variation formulations of which may be presented in terms of internal bending moments with uniform integral conditions. The problems are considered, in which one rod end is fixed and the other rod end is either restraint or pivoted, or embedded into a support which may be shifted in a transversal direction.For the substantiation of the lower evaluations determination, a sequence of functionals is constructed, the minimum values of which are the lower evaluations for the minimum critical value of the loading parameter of the rod, and the calculation process is reduced to the determination of the maximum eigenvalues of modular matrices. The matrix elements are expressed in terms of integrals of basic functions depending on the type of fixation of the rod ends. The basic functions, with the accuracy up to a linear polynomial, are the same as the bending moments arising with the bifurcation of the equilibrium of a rod with a constant cross-section compressed by longitudinal forces at the rod ends. The calculation of the upper evaluation is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the elements of the modular matrices. It is noted that the obtained upper bound evaluation is not worse thanthe evaluation obtained by the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2014.2.39-46

References
  1. Kupavtsev V.V. Variatsionnye formulirovki zadach ustoychivosti uprugikh sterzhney cherez izgibayushchie momenty [Variational Formulations of the Problems of Elastic Rods Stability Using Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, vol. 3, no. 4, pp. 285—289.
  2. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh sistem [Fundamentals of the Stability Analysis of the Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Kupavtsev V.V. Dvustoronnie otsenki v zadachakh ustoychivosti uprugikh sterzhney, vyrazhennykh cherez izgibayushchie momenty [Bilateral Estimates in Elastic Rod Stability Problems Formulated through Bending Moments]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 2, pp. 47—54.
  4. Rektoris K. Variatsionnye metody v matematicheskoy fizike i tekhnike [Variational Methods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  5. Doraiswamy Srikrishna, Narayanan Krishna R., Srinivasa Arun R. Finding Minimum Energy Configurations for Constrained Beam Buckling Problems Using the Viterbi Algorithm. International Journal of Solids and Structures. 2012, vol. 49, no. 2, pp. 289—297. DOI: 10.1016/j.ijsolstr.2011.10.003.
  6. Panteleev S.A. Dvustoronnie otsenki v zadachakh ob ustoychivosti szhatykh uprugikh blokov [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestiya RAN. MTT [News of the Russian Academy of Sciences. Mechanics of Solids]. 2010, no. 1, pp. 51—63.
  7. Santos H.A., Gao D.Y. Canonical Dual Finite Element Method for Solving Post-buckling Problems of a Large Deformation Elastic Beam. International Journal of Non-Linear Mechanics. 2012, vol. 47, no. 2, pp. 240—247. DOI: 10.1016/j.ijnonlinmec.2011.05.012.
  8. Selamet Serdar, Garlock Maria E. Predicting the Maximum Compressive Beam Axial Force During Fire Considering Local Buckling. Journal of Constructional Steel Research. 2012, vol. 71, pp. 189—201. DOI: 10.1016/j.jcsr.2011.09.014.
  9. Tamrazyan A.G. Dinamicheskaya ustoychivost' szhatogo zhelezobetonnogo elementa kak vyazkouprugogo sterzhnya [Dynamic Stability of the Compressed Reinforced Concrete Element as Viscoelastic Bar]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, vol. 2, no. 1, pp. 193—196.
  10. Manchenko M.M. Ustoychivost' i kinematicheskie uravneniya dvizheniya dinamicheski szhatogo sterzhnya [Dynamically Loaded Bar Stability and Kinematic Equations of Motion]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 71—76.

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BASIC FUNCTIONS FOR THE METHOD OF TWO-SIDED EVALUATIONS IN THE PROBLEMS OF STABILITY OF ELASTICNON-UNIFORMLY COMPRESSED RODS

Vestnik MGSU 6/2013
  • Kupavtsev Vladimir Vladimirovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associated Professor, Department of Theoretical Mechanics and Aerodynamics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Мoscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 63-70

The author considers the method of two-sided evaluations in the problems of stability of a one-span elastic non-uniformly compressed rod under various conditions of fixation of its ends.The required minimum critical value of the loading parameter for the rod is the minimum value of the functional equal to the ratio of the norms of Hilbert space elements squared. Using the inequalities following from the problem of the best approximation of a Hilbert space element through the basic functions, it is possible to construct two sequences of functionals, the minimum values of which are the lower evaluations and the upper ones. The basic functions here are the orthonormal forms of the stability loss for a rod with constant cross-section, compressed by longitudinal forces at the ends, which are fixed just so like the ends of the non-uniformly compressed rod.Having used the Riesz theorem about the representation of a bounded linear functional in the Hilbert space, the author obtains the additional functions from the domain of definition of the initial functional, which correspond to the basic functions. Using these additional functions, the calculation of the lower bounds is reduced to the determination of the maximum eigenvalues of the matrices represented in the form of second order modular matrices with the elements expressed in the form of integrals of basic and additional functions. The calculation of the upper bound value is reduced to the determination of the maximum eigenvalue of the matrix, which almost coincides with one of the modular matrices. It is noted that the obtained upper bound evaluations are not worse than the evaluations obtained through the Ritz method with the use of the same basic functions.

DOI: 10.22227/1997-0935.2013.6.63-70

References
  1. Kupavtsev V.V. K dvustoronnim ocenkam kriticheskih nagruzok neodnorodno szhatyh uprugih sterzhnej. [On Bilateral Evaluations of Critical Loading Values in Respect of Non-uniformly Compressed Elastic Rods]. Izvestija vuzov. Stroitel’stvo I arhitektura. [News of Institutions of Higher Education. Construction and Architecture]. 1984, no. 8, pp. 24—29.
  2. Alfutov N.A. Osnovy rascheta na ustojchivost’ uprugih sistem. [Fundamentals of Stability Analysis of Elastic Systems]. Moscow, Mashinostroenie Publ., 1991, 336 p.
  3. Rektoris K. Variatsionnye metody v matematicheskoy fizike I tekhnike. [Variational Metods in Mathematical Physics and Engineering]. Moscow, Mir Publ., 1985, 589 p.
  4. Panteleev S.A. Dvustoronie otsenki v zadache ob ustojchivosti szhatyh uprugih blokov. [Bilateral Assessments in the Stability Problem of Compressed Elastic Blocks]. Izvestyja RAN. MTT. [News of Russian Academy of Sciences. Mechanics of Solids]. 2010, no. 1, pp. 51—63.
  5. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Bar of Variable Cross Section with Different Variants of Ends Fastening (Informftion 1)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 10, pp. 19—25.
  6. Bogdanovich A.U., Kuznetsov I.L. Prodol’noe szhatie tonkostennogo sterzhnja peremennogo sechenija pri razlichnyh variantah zakreplenija torcov [Longitudinal Compression of a Thin-Walled Core of Variable Cross Section with Different Variants of Ends Fastening (Informftion 2)]. Izvestija vuzov. Stroitel’stvo [News of Institutions of Higher Education. Construction]. 2005, no. 11-12, pp. 10—16.
  7. Nicot Francois, Challamel Noel, Lerbet Jean, Prunier Frorent, Darve Felix. Some in-sights into structure instability and the second-order work criterion. International Journal of Solids and Structures. 2012. Vol. 49, no. 1. pp. 132—142.
  8. Aristizabal-Ocha J. Dario. Matrix method for stability and second rigid connections. Engineering Structures. 2012. Vol. 34. pp. 289—302.
  9. TemisYu.M.,Fedorov I.M. Sravnenie metodov analiza ustojchivosti sterzhnej peremennogo sechenija pri nekonservativnom nagruzhenii [Comparing the Methods for Analysing the Stability of Rods of a Variable Cross-section under Non-conservative Loading]. Problems of strength and plasticity [Proceeding sof Nizhni Novgorod University]. 2006, no. 68, pp. 95—106.
  10. Le Grognec Philippe, Nguyen Quang-Hay, Hjiaj Mohammed. Exat buckling solution for two-layer Timoshenko beams with interlayer. International Journal of Solids and Structures. 2012. Vol. 49, ¹ 1. pp. 143—150.
  11. Chepurnenko A.S., Andreev V.I., Yazyev B.M. Energeticheskiy metod pri raschete na ustoychivost’ szhatykh sterzhney s uchetom polzuchesti. [Energy Method of Analysis of Stability of Compressed Rods with Regard for Creeping]. Vestnik MGSU. [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 1, pp.101—108.

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Calculation model of non-linear dynamic deformation of composite multiphase rods

Vestnik MGSU 5/2014
  • Mishchenko Andrey Viktorovich - Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NGASU) Candidate of Technical Sciences, Associate Professor, Department of Structural Mechanics, Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NGASU), 113 Leningradskaya str., Novosibirsk, 630008, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 35-43

The method of formulating non-linear physical equations for multiphase rods is suggested in the article. Composite multiphase rods possess various structures, include shear, polar, radial and axial inhomogeneity. The Timoshenko’s hypothesis with the large rotation angles is used. The method is based on the approximation of longitudinal normal stress low by basic functions expansions regarding the linear viscosity low. The shear stresses are calculated with the equilibrium equation using the subsidiary function of the longitudinal shift force. The system of differential equations connecting the internal forces and temperature with abstract deformations are offered by the basic functions. The application of power functions with arbitrary index allows presenting the compact form equations. The functional coefficients in this system are the highest order rigidity characteristics. The whole multiphase cross-section rigidity characteristics are offered the sums of the rigidity characteristics of the same phases individually. The obtained system allows formulating the well-known particular cases. Among them: hard plasticity and linear elastic deformation, different module deformation and quadratic Gerstner’s low elastic deformation. The reform of differential equations system to the quasilinear is suggested. This system contains the secant variable rigidity characteristics depending on abstract deformations. This system includes the sum of the same uniform blocks of different order. The rods phases defined the various set of uniform blocks phase materials. The integration of dynamic, kinematic and physical equations taking into account initial and edge condition defines the full dynamical multiphase rods problem. The quasilinear physical equations allow getting the variable flexibility matrix of multiphase rod and rods system.

DOI: 10.22227/1997-0935.2014.5.35-43

References
  1. Andreev V.I., Potekhin I.A. O sposobe sozdaniya optimal'nykh konstruktsiy na osnove resheniya obratnykh zadach teorii uprugosti neodnorodnykh tel [On the Method of Creating Optimal Structures Based on Solving Inverse Problems of Inhomogeneous Bodies Elasticity Theory]. Vestnik otdeleniya stroitel'nykh nauk [Bulletin of Civil Engineering Scientific Department]. 2007, no. 11, pp. 48—52.
  2. Andreev V.I. Optimization of Thick-walled Shells Based on Solutions of Inverse Problems of the Elastic Theory for Inhomogeneous Bodies. Computer Aided Optimum Design in Engineering. 2012, pp. 189—202. DOI: 10.2495/OP120171.
  3. Al'tenbakh Kh. Osnovnye napravleniya teorii mnogosloynykh tonkostennykh konstruktsiy [Main Directions of Multi Layered Thin-walled Structures Theory]. Mekhanika kompozitnykh materialov [Mechanics of Composite Materials]. 1998, vol. 34, no. 3, pp. 333—348.
  4. Teters G.A., Kregers A.F. Mnogotselevoe optimal'noe proektirovanie kompozitnykh konstruktsiy. Obzor [Multi-purpose Optimal Design of Composite Structures. Review]. Mekhanika kompozitnykh materialov [Mechanics of Composite Materials]. 1996, vol. 32, no. 3, pp. 363—376.
  5. Mishchenko A.V., Nemirovskiy Yu.V. Optimizatsiya sloistykh sterzhney pri var'irovanii geometricheskikh funktsiy naruzhnykh i vnutrennikh sloev [Optimization of Layered Rods by Means of Geometrical Functions Variation of External and Internal Layers]. Izvestiya vuzov. Stroitel'stvo [News of Higher Educational Institutions. Construction]. 2005, no. 3, pp. 19—24.
  6. Mishchenko A.V. Primenenie szhato-izognutykh sterzhney so smeshchennymi tsentrami secheniy v ramnykh konstruktsiyakh [Using Compressed-Bending Bars with Displaced Cross Section Centers in Frame Structures]. Izvestiya vuzov. Stroitel'stvo [News of Higher Educational Institutions. Construction]. 2007, no. 6, pp. 4—11.
  7. Vasil’ev V.V. Klassicheskaya teoriya plastin — istoriya i sovremennyy analiz [Classical Plates Theory — History and Contemporary Analysis]. Izvestiya RAN. Mekhanika tverdogo tela [Journal of Russian Academy of Sciences. Mechanics of Solids]. 1998, no. 3, pp. 45—58.
  8. Piskunov V.G. Iteratsionnaya analiticheskaya teoriya v mekhanike sloistykh kompozitnykh sistem [Iterative Analytical Theory in the Mechanic of Layered Composite Systems]. Mekhanika kompozitnikh materialov [Mechanics of Composite Materials]. 2003, vol. 39, no. 1, pp. 3—24.
  9. Baxter S.C., Horgan C.O. End Effects for Anti-plane Shear Deformations of Sandwich Structures. Journal of Elasticity. 1995, vol. 40, no. 2, pp. 123—164. DOI: 10.1007/BF00042458.
  10. Cowper G.R. The Shear Coefficient in Timoshenko’s Beam Theory. Trans. of ASME. 1966, vol. E88, no. 2, pp. 335—340. DOI: 10.1115/1.3625046.
  11. Foraboschi P. Analytical Solution of Two-Layer Beam Taking into Account Nonlinear Interlayer Slip. J. Eng. Mech. 2009, vol. 135, no. 10, pp. 1129—1147. DOI: 10.1061/(ASCE)EM.1943-7889.0000043.
  12. Karama M., Afaq K.S., Mistou S. Mechanical Behaviour of Laminated Composite Beam by New Multi-layered Laminated Composite Structures Model with Transverse Shear Stress Continuity. International Journal of Solids and Structures. 2003, vol. 40, no. 6, pp. 1525—1546.
  13. Ambarsumian S.A. Eshche odna utochnennaya teoriya anizotropnykh obolochek [Another Refined Theory of Anisotropic Shells]. Mekhanika polimerov [Mechanics of Polymers]. 1970, no. 5, pp. 884—896.
  14. Nemirovskiy Yu.V. Raschet i ratsional'noe proektirovanie derevyannykh sterzhnevykh elementov [Calculation and Rational Design of Wood Rods]. Sovremennye problemy sovershenstvovaniya i razvitiya metallicheskikh, derevyannykh, plastmassovykh konstruktsiy v stroitel'stve i na transporte: sbornik nauchnykh trudov III Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii [Contemporary Problems of Improving and Development of Steel, Wood and Plastic Structures in Construction and Transport: Collection of Scientific Works of the 3rd International Scientific and Technical Conference]. Samara, SamGASU Publ., 2005, pp. 247—251.
  15. Mishchenko A.V., Nemirovskiy Yu.V. Raschet i proektirovanie derevyannykh sterzhnevykh sistem s uchetom fizicheskoy nelineynosti [Analysis and Rational Designing of Wood Rod Systems with Regard for Physical Nonlinearity]. Stroitel'naya mekhanika i raschet sooruzheniy [Structural Mechanics and Analysis of Building]. 2007, no. 6, pp. 46—52.

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