### Calculation of open-frame through beams according to the A.R. Rzhanitsyn’s theoryof compound rods

Pages 23-31

Through beams are widely used in the construction of large-span civil and industrial buildings, bridge engineering and mechanical engineering. They include open-frame girders and castellated beams. In order to determine their stress-strain state, software systems based on the finite element method are used or approximate calculations using simplified calculation patterns of Virendel beams are performed. Recently, many projects have been completed, in which A.R. Rzhanitsyn’s theory of compound rods is applied to calculate through structures.In this model, discrete links connecting upper and lower belts of the structure are re-placed by cross ties and shift connections continuously distributed along the length of the joint. Cross links hinder the convergence or separation of belts from one another. As a rule, pliability of cross links is neglected. This assumption, which substantially facilitates the calculation, is consistent with the hypothesis that there is no lateral strain in individual rods, calculated according to the theory. Therefore, whenever a compound rod is loaded, all its layers, and in this case – belts, are deformed according to the same curve pattern. In calculations, elastically compliant shift connections are replaced by the required distribution function of shear forces distributed along the length of the beam joint. Thus, the calculation of a through-beam is reduced to the solution of three ordinary differential equations of the second order, on the basis of which the following functions should be defined: beam deflection, bending moment and shear stress in the beam joint.The author discusses development of a numerical method of calculation of through beams based on the A.R. Rzhanitsyn’s theory of compound rods. To solve the system of differential equations, difference equations of the successive approximations method are involved to take account of finite discontinuities of the desired function, its first derivative and the right-hand side of the original differential equation. They demonstrate high accuracy if compared to well-known finite difference method equations.To illustrate the algorithm, the author provides sample calculations of open-frame girders and perforated beams having openings of different shapes. The results obtained by the authors are compared with a well-known analytical solution and a numerical solution based on the finite element method.

DOI: 10.22227/1997-0935.2013.9.23-31

- Biryukov V.V., Zabalueva T.R., Zakharov A.V. Proektirovanie bol'sheproletnykh mnogoetazhnykh sportivnykh zdaniy [Design of Long-span Multi-story Sports Buildings]. Arkhitektura i stroitel'stvo Rossii [Architecture and Construction of Russia]. 2011, no. 9, pp. 12—19.
- Shuller.V. Konstruktsii vysotnykh zdaniy [Structures of High-rise Buildings]. Moscow, Stroyizdat Publ., 1979, 248 p.
- Kartopol'tsev V.M., Balashov E.V. K voprosu issledovaniya napryazhenno-deformirovannogo sostoyaniya sovmestnoy raboty skvoznykh balok s zhelezobetonnoy plitoy na metallicheskom poddone [Towards Research into the Stress-Strain State of Combined Behaviour of Open-frame Beams and a Reinforced-concrete Slab Resting on the Metal Tray]. Vestnik TGASU [Bulletin of Tomsk State University of Architecture and Civil Engineering]. 2004, no. 1, pp. 169—178.
- Rabinovich I.M. Spravochnik inzhenera-proektirovshchika promsooruzheniy [Reference Book for Design Engineers of Industrial Buildings]. Tom 2 Raschetno-teoreticheskiy. [Vol. 2. Analysis and Theory]. Moscow – Leningrad, Gosstroyizdat Publ., 1934, 709 p.
- Drobachev V.M., Litvinov E.V. Analiticheskoe opredelenie napryazhenno-deformirovannogo sostoyaniya stenki-peremychki perforirovannoy balki [Analytical Methods of Identification of the Stress-strain State of a Partition Wall of a Castellated Beam]. Izv. vuzov. Stroitel'stvo [News of Institutions of Higher Education. Construction.] 2003, no. 5, pp. 128—133.
- Pritykin A.I. Progiby perforirovannykh balok-stenok s pryamougol'nymi vyrezami [Deflection of Castellated Deep Beam Having Rectangular Openings]. Izv. vuzov. Stroitel'stvo. [News of Institutions of Higher Education. Construction] 2009, no. 10, pp. 110—116.
- Pritykin A.I. Primenenie teorii sostavnykh sterzhney k opredeleniyu deformatsiy perforirovannykh balok [Application of the Theory of Compound Rods to Identification of Deformations of Castellated Beams]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2009, no. 4, pp. 177—181.
- Pimenov A.S., Kholopov I.S., Solov'ev A.V. Optimal'noe proektirovanie perforirovannykh balok [Optimal Design of Castellated Beams]. Vestnik transporta Povolzh'ya [News Bulletin of the Volga Region Transport]. 2009, no. 1, pp. 69—74.
- Bedi K.S., Pachpor P.D. Moment and Shear Analysis of Beam with Different Web Openings. International Journal of Engineering Research and Applications. November – December 2011, vol. 1, no. 4, pp. 1917—1921.
- Wakchaure M.R., Sagade A.V. Finite Element Analysis of Castellated Steel Beam. International Journal of Engineering and Innovative Technology. July 2012, vol. 2, no. 1, pp. 365—370.
- Chhapkhane N.K., Shashikant R.K. Analysis of Stress Distribution in Castellated Beam Using Element Method and Experimental Techniques. International Journal of Mechanical Engineering Applications Research. August - September 2012, vol. 3, no. 3, pp. 190—197.
- Kholoptsev V.V. Raschet sostavnykh mnogoproletnykh nerazreznykh balok [Analysis of Compound Multi-span Beams]. Stroitel'naya mekhanika i raschet sooruzheniy [Structural Mechanics and Analysis of Structures]. 1966, no. 3, pp. 26—29.
- Rzhanitsyn A.R. Sostavnye sterzhni i plastinki [Compound Rods and Plates]. Moscow, Stroyizdat Publ., 1986, 316 p.
- Gabbasov R.F., Filatov V.V. Chislennoe reshenie zadachi po raschetu sostavnykh sterzhney s peremennym koeffitsientom zhestkosti shva [Numerical Solution to the Problem of Analysis of Compound Rods Having Variable Seam Stiffness Coefficient]. ACADEMIA. Arkhitektura i stroitel'stvo [Academy. Architecture and Construction] 2007, no. 2, pp. 86—89.
- Gabbasov R.F., Gabbasov A.R., Filatov V.V. Chislennoe postroenie razryvnykh resheniy zadach stroitel'noy mekhaniki [Numerical Generation of Discontinuous Solutions to Problems of Structural Mechanics]. Moscow, ASV Publ., 2008, 280 p.