Stress analytical solution for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform distributed pressure

Abstract

The stress state around circular openings, such as boreholes, shafts, and tunnels, is usually needed to be evaluated. Solutions for stresses, strains and ultimate bearing capacities of pressurized hollow cylinder are common cases. Stress analytical method for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform pressure on the outer surface and uniform radial pressure on the inner surface is given. The power series method of complex function is used. The stress analytical solution is obtained with the assumption that two layers of a cylinder are fully contacted. The distributions of normal and tangential contact stress along the interface, tangential stress on the inner boundary and stresses in the radial direction at θ=0°, 45° and 90°, are obtained. An example indicates that, when the elastic modulus of the inner layer of a double-layered thick-walled cylinder is smaller than that of the outer layer, the tangential stress is smaller than that in the corresponding point for a traditional cylinder composed of homogeneous materials. In that way, stress concentration at the inner surface can be alleviated and the stress distribution is more uniform. This is a capable way to enhance the elastic ultimate bearing capacity of thick-walled cylinder.

Keywords

thick-walled cylinder / stress analytical solution / complex variable function / non-uniform distributed pressure / stress concentration / combination of different elastic moduli

References

[1] ObaraY, IshigurobY. Measurements of induced stress and strength in the near-field around a tunnel and associated estimation of the Mohr-Coulomb parameters for rock mass strength [J]. Int J Rock Mech Min Sci, 2004, 41(5): 761-769

[2] MartinC D, ReadR S, MartinoJ B. Observations of brittle failure around a circular test tunnel [J]. Int J Rock Mech Min Sci, 1997, 34(7): 1065-1073

[3] ObertL, DuvallW I. Rock mechanics and the design of structures in rock [M]. New York: Wiley, 1967542-545

[4] MarchettiM E, CapraJ C. Analysis of circular tunnels in soft ground [J]. Tunn Tunn Int, 1986, 18(10): 45-48

[5] LooT T. On axisymmetric deformation of a finite hollow cylinder [J]. Chin J Theor Appl Mech, 1979, 15(3): 219-228

[6] HouY, HeF-bao. Elasticity solution of axisymmetric problem of a finite hollowed cylindrical [J]. Appl Math Mech, 1991, 12(10): 903-912

[7] LinX-song. An application of the extended kantorovich method to the three-dimensional axial symmetry stress analysis for thick-walled cylinder with finite length [J]. Eng Mech, 1997, 14(3): 70-77

[8] LinX-song. A kantorovich-type stress variational method of higher approximation for the finite thick walled cylinders under the axisymmericalloan [J]. Eng Mech, 1999, 16(6): 119-132

[9] ChenX, TanC P, HaberfieldC M. Solutions for the deformations and stability of elastoplastic hollow cylinders subjected to boundary pressures [J]. Int J Numer Anal Meth Geomech, 1999, 23(8): 779-800

[10] ShiZ-f, ZhangT-t, XiangH-jun. Exact solutions of heterogeneous elastic hollow cylinders [J]. Compos Struct, 2007, 79(7): 140-147

[11] Sudara Raja LyengarK T, YoganandaC V. Long circular cylinderical laminated shells subjected to axisymmetric external loads [J]. ZAMM, 1964, 44: 270-272

[12] ChandrashekharaK, BhimaraddiA. Elasticity solution for a long circular sandwich cylindrical shell subjected to axisymmetric load [J]. Int J Solids Struct, 1982, 18(7): 611-618

[13] HorganC O, ChanA M. Pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials [J]. J Elast, 1999, 55(1): 43-59

[14] GreenA E. Three-dimensional stress systems in isotropic plates [J]. Philosophical Transactions of the Royal Society of London. Series A, Math Phys Sci, 1948, 240: 561-597

[15] SternbergE, SadowskyM A. Three dimensional solution of the stress concentration around a circular hole in a plate [J]. J Appl Mech, 1949, 16(1): 27-36

[16] OuyangC, XuL-lin. On 3-dimensional non-axisymmetrical deformation analysis for finite hollow circular cylinder [J]. Chin J Theor Appl Mech, 1986, 18(2): 115-122

[17] YiannopoulosA Ch. Simplified solution for stresses in thick-wall cylinders for various loading conditions [J]. Comput Struct, 1996, 60(4): 571-578

[18] PanE, RoyA K. A simple plane-strain solution for functionally graded multilayered isotropic cylinders [J]. Struct Eng Mech, 2006, 24(6): 727-740

[19] JabbariM, SohrabpourS, EslamiM R. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads [J]. J Appl Mech (Transactions ASME), 2003, 70(1): 111-118

[20] LiuZ-b, ShanR, LiuW, NiL-yong. Solution of a hollow thick-wall cylinder subject to quadric function pressures and its limit when l→∞ [J]. Science in China, Series E: Technol Sci, 2004, 47(2): 229-236

[21] LiuWen. Mathematical model and analytical solution for cylinder subject to uneven pressures [J]. Chin J Mech Eng (English Edition), 2006, 19(4): 574-578

[22] LiuW, ShanRui. Mathematic model and analytic solution for a cylinder subject to exponential function [J]. Chin J Mech Eng (English Edition), 2009, 22(4): 587-593

[23] MuskhelishviliNi. Some basic problems of the mathematical theory of elasticity [M]. Groningen: Noordhoof, 1963199-210

[24] LuA-z, ZhangL-qingComplex function method on mechanical analysis of underground tunnel [M], 2007, Beijing, Science Press: 18-30

[25] LuA-z, ZhangL-q, ZhangNing. Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay [J]. Int J Rock Mech Min Sci, 2011, 48(3): 514-519