Determination of isolation layer thickness for undersea mine based on differential cubature solution to irregular Mindlin plate

Abstract

The differential cubature solution to the problem of a Mindlin plate lying on the Winkler foundation with two simply supported edges and two clamped edges was derived. Discrete numerical technology and shape functions were used to ensure that the solution is suitable to irregular shaped plates. Then, the mechanical model and the solution were employed to model the protection layer that isolates the mining stopes from sea water in Sanshandao gold mine, which is the first subsea mine of China. Furthermore, thickness optimizations for the protection layers above each stope were conducted based on the maximum principle stress criterion, and the linear relations between the best protection layer thickness and the stope area under different safety factors were regressed to guide the isolation design. The method presented in this work provides a practical way to quickly design the isolation layer thickness in subsea mining.

Keywords

subsea mine / irregular Mindlin plate / differential cubature method / isolation layer / protection layer / thickness optimization

References

[1] LiX, LiD, LiuZ, ZhaoG, WangW. Determination of the minimum thickness of crown pillar for safe exploitation of a subsea gold mine based on numerical modeling [J]. International Journal of Rock Mechanics & Mining Sciences, 2013, 57(1): 42-56

[2] NilsenB, DahldT S. Stability and rock cover of hard rock subsea tunnels. Tunnelling & Underground Space Technology, 1994, 9(2): 151-158

[3] NilsenB. The optimum rock cover for subsea tunnels. Rock mechanics contributions and challenges [C]. Proceeding of the 31st US Symposium on Rock Mechanics. Golden, BALKEMA A A, Rotterdam Colorado: ASME, 19901005-1012

[4] LiS C, LiS C, XuB S, WangH P, DingW T. Study on determination method for minimum rock cover of subsea tunnel [J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(11): 2289-2295

[5] OuD Y, MakC M, KongP R. Free flexural vibration analysis of stiffened plates with general elastic boundary supports [J]. World Journal of Modelling and Simulation, 2012, 8(2): 96-102

[6] LiW L, ZhangX, DuJ, LiuZ. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports [J]. Journal of Sound and Vibration, 2009, 321(1): 254-269

[7] Civalek. Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods [J]. Applied Mathematical Modelling, 2007, 31(3): 606-624

[8] CataniaG, SorrentinoS. Spectral modeling of vibrating plates with general shape and general boundary conditions. Journal of Vibration and Control, 2012, 18(18): 1607-1623

[9] CheungY K, ThamL G, LiW Y. Free vibration and static analysis of general plate by spline finite strip [J]. Computational Mechanics, 1988, 3(3): 187-197

[10] ShiJ, ChenW, WangC. Free vibration analysis of arbitrary shaped plates by boundary knot method [J]. Acta Mechanica Solida Sinica, 2009, 22(4): 328-336

[11] KangS W, AtluriS N. Free vibration analysis of arbitrarily shaped polygonal plates with simply supported edges using a sub-domain method [J]. Journal of Sound and Vibration, 2009, 327(3): 271-284

[12] SaadatpourM M, AzhariM. The Galerkin method for static analysis of simply supported plates of general shape [J]. Computers & Structures, 1998, 69(1): 1-9

[13] KhovH, LiW L, GibsonR F. An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions [J]. Composite Structures, 2009, 90(4): 474-481

[14] QuintanaM V, NallimL G. A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports [J]. International Journal of Mechanical Sciences, 2013, 69: 1-9

[15] LiewK M, HanJ B, XiaoZ M, DuH. Differential quadrature method for Mindlin plates on Winkler foundations [J]. International Journal of Mechanical Sciences, 1996, 38(4): 405-421

[16] ReissnerE. The effect of transverse shear deformation on the bending of elastic plates [J]. Journal of Applied Mechanics, 1945, 12(2): 69-77

[17] ReissnerE. On bending of elastic plates [J]. Quarterly of Applied Mathematics, 1947, 5(1): 55-68

[18] MindlinR D. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates [J]. Journal of Applied Mechanics, 1951, 18: 31-38

[19] TsaiC C, WuE, MY. Analytical particular solutions of augmented polyharmonic spline associated with Mindlin plate model [J]. Numerical Methods for Partial Differential Equations, 2012, 28(6): 1778-1793

[20] SetoodehA R, MalekzadehP, VosoughiA R. Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory [J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2012, 226(7): 1896-1906

[21] HouG L, QiG W, XuY N, ChenA. The separable Hamiltonian system and complete biorthogonal expansion method of Mindlin plate bending problems [J]. Science China Physics: Mechanics and Astronomy, 2013, 56(5): 974-980

[22] KobayashiH, SonodaK. Rectangular Mindlin plates on elastic foundations [J]. International Journal of Mechanical Sciences, 1989, 31(9): 679-692

[23] LiuF L, LiewK M. Differential cubature method for static solutions of arbitrarily shaped thick plates [J]. International Journal of Solids and Structures, 1998, 35(28): 3655-3674

[24] CivanF. Solving multivariable mathematical models by the quadrature and cubature methods [J]. Numerical Methods for Partial Differential Equations, 1994, 10(5): 545-567

[25] TimoshenkoS, Woinowsky-KriegerSTheory of plates and shells [M], 1959New YorkMcGraw-Hill

[26] IyengarK, TS R, ChandrshekharaK, SebastianV K. On the analysis of thick rectangular plates [J]. Ingenieur-Archive, 1974, 43(5): 317-330

[27] PengK, LiX B, WangZ W. Hydrochemical characteristics of groundwater movement and evolution in the Xinli deposit of the Sanshandao gold mine using FCM and PCA methods [J]. Environmental Earth Sciences, 2015, 73(12): 7873-7888

[28] PengK, LiX B, WanC C, PengS Q, ZhaoG Y. Safe mining technology of undersea metal mine [J]. Transactions of Nonferrous Metals Society of China, 2012, 22(22): 740-746

[29] KimS M, McculloughB F. Dynamic response of plate on viscous Winkler foundation to moving loads of varying amplitude [J]. Engineering Structures, 2003, 25(9): 1179-1188

[30] GhavanlooE, DaneshmandF, RafieiM. Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation [J]. Physica E: Lowdimensional Systems and Nanostructures, 2010, 42(9): 2218-2224

[31] HardenC W, HutchinsonT C. Beam-on-nonlinear-Winklerfoundation modeling of shallow, rocking-dominated footings [J]. Earthquake Spectra, 2009, 25(2): 277-300

[32] DalogluA T, VallabhanC, VG. Values of k for slab on winkler foundation [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2000, 126(5): 463-471

[33] ChenY M, LiX BSubsea large metal deposits safe and efficient mining technology [M], 2013BeijingPress of Metallurgy Industry