Abstract
The flexure function \( W(x,y) \) of a ribbed rectangular plate is determined by the Kantorovich-Vlasov method on the basis of the sum of series \( \sum\nolimits_{i = 1}^{n} {W_{i} (y)X_{i} (x)} \) with functions \( X_{i} (x) \) satisfying specified kinematic boundary conditions, while functions \( W_{i} (y) \) are determined as the solutions of differential equations of the fourth order. The problem is complicated by the absence of the target selection of functions \( X_{i} (x) \) required to attain the desired accuracy. A special beam function \( X_{1} (x) \) defined by the initial parameters method was proposed to be used in the first term of series. This function is defined at a single beam with the boundary conditions equal to those of a ribbed plate with respect to the selected coordinate. This allows one to reasonably increase in the accuracy of the specified variation method when one series term is used. The practical calculations of a real object—shaped sheet as per Russian GOST—conducted using the proposed function are provided. Flexure tables were obtained for the shaped sheets provided in the specified standard. The analysis of resulting flexures shows that the use of the sheet profile with more reinforcement ribs is more cost effective as the sheet flexure decreases faster than the sheet weight increases. For example, if one rib is added to S-15-1000-06 profile sheet, sheet flexure in the initial range of 8–12 ribs decreases by no less than 8% with the sheet weight increase by 5%.
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References
Gots AN (2013) Numerical methods of calculation in power plant engineering. Study guide. Part 2. Publishing House of the Vladimir State University, Vladimir
Introduction to Elasticity/Rayleigh-Ritz method (2018) https://en.wikiversity.org/wiki/Introduction_to_Elasticity/Rayleigh-Ritz_method
Ivanov VN (2004) Variational principles and problem-solving techniques related to the elasticity theory. RUDN Publishing House, Moscow, Study guide
Mama BO, Nwoji CU, Onah HN et al (2017) Bubnov-Galerkin method for the elastic stress analysis of rectangular plates under uniaxial parabolic distributed edge loads—Dept of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria. https://doi.org/10.21817/ijet/2017/v9i6/170906060
Suryaninov NG, Kozolup GN (2009) Kantorovich-Vlasov method applied in the ribbed plate bending problem. Proc Odessa Polytech Univ 1(33)–2(34):205–209
Orobey VF, Purich VN (2013) Accuracy of the variation Kantorovich-Vlasov method. Proc Odessa Polytech Univ 2(41):19–24
Aleksandrov AV, Potapov VD, Derzhavin BP (2003) Strength of materials. Textbook for institutions of higher education. Vysshaya Shkola, Moscow
GOST RF 24045-2016. Bent steel sheet profiles with stair landings and railings for construction. Specifications (2018). Standartinform, Moscow
BS EN 506 (2008) Roofing products of metal sheet. Specification for self-supporting products of copper or zinc sheet
BS EN 14782 (2006) Self-supporting metal sheet for roofing, external cladding and internal lining-Product specification and requirements
Belyaev NM (2014) Strength of materials. Study guide. Alyans, Moscow
Makarov EG (2007) Engineering calculations in Mathcad-14. Piter Publishing House, St.-Petersburg
Kiryanov DV (2007) Mathcad-14. BHV-Peterburg, St.-Petersburg
Zwillinger D (1997) Handbook of differential equations. Academic Press, Boston
Polyanin AD, Zaitsev VF (2003) Handbook of exact solutions for ordinary differential equations. Chapman & Hall/CRC Press, Boca Raton
Arnold VI (1992) Ordinary differential equations. Springer, Berlin
Petrovskii IG (1973) Ordinary differential equations. Dover Publications, New York
Acknowledgements
The authors wish to thank Vladimir F. Mikhaylets for his helpful discussions and consultations.
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Konev, S.V., Fainshtein, A.S., Teftelev, I.E. (2020). Application of Kantorovich-Vlasov Method for Shaped Plate Bending Problem. In: Radionov, A., Kravchenko, O., Guzeev, V., Rozhdestvenskiy, Y. (eds) Proceedings of the 5th International Conference on Industrial Engineering (ICIE 2019). ICIE 2019. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-22041-9_11
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DOI: https://doi.org/10.1007/978-3-030-22041-9_11
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