Abstract
We provide analytical formulas and design charts for computing displacements, strains and stresses in transversely isotropic half-spaces subject to linearly distributed vertical pressures. The domain integrals extended to the loaded region, resulting from the solution associated with a vertical point load, are first transformed into boundary integrals. For polygonal domains, the boundary integrals are further reduced to algebraic sums depending upon the loading function and the position vertices of the loaded region. A detailed account of this transformation is reported for all integrals in order to highlight the singularities to be coped with and how they can be circumvented. Design charts for the vertical stress are reported in order to validate the proposed formulation. Finally, the values of displacements and stresses underneath a foundation of arbitrary shape comparatively show the influence of modelling half-spaces as isotropic or transversely isotropic media.
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Anyaegbunam AJ (2014) Complete stress and displacements in a cross-anisotropic half-space caused by a surface vertical point load. Int J Geomech 14(2):171–181
Argatov I, Sabina F (2012) Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer. Int J Eng Sci 50:132–143
Atkinson J (1975) Anisotropic elastic deformation in laboratory tests on undisturbed London clay. Géotechnique 25:357–374
Barden L (1963) Stresses and displacements in a cross-anisotropic soil. Géotechnique 13:198–210
Chen WT (1966) On some problems in transversely isotropic elastic materials. J Appl Mech 33(2):347–355
Ding H, Chen W, Zhang L (2006) Elasticity of transversely isotropic materials. Springer, Dordrecht
Ding HJ, Xu BH (1988) General solutions of axisymmetric problems in transversely isotropic body. Appl Math Mech 9:135–142 (in Chinese)
D’Urso MG (2012) New expressions of the gravitational potential and its derivates for the prism. In: Sneeuw N, Novák P, Crespi M, Sansò F (eds) VII Hotine-Marussi International Symposium on Mathematical Geodesy. Springer, Berlin, pp 251–256
D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geodesy 87(3):239–252
D’Urso MG (2014) Analytical computation of gravity effects for polyhedral bodies. J Geodesy 88:13–29
D’Urso MG (2014) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120(4):349–372
D’Urso MG (2015) Some Remarks on the Computation of the Gravitational Potential of Masses with Linearly Varying Density. In: Sneeuw N, Novák P, Crespi M, Sansò F (eds) VIII Hotine-Marussi Symposium. Rome
D’Urso MG (2015) The gravity anomaly of a 2D polygonal body having density contrast given by polynomial functions. Surv Geophys 36(3):391–425
D’Urso MG, Marmo F (2009) Vertical stresses due to linearly distributed pressures over polygonal domains. In: ComGeo I, First International Symposium on Computational Geomechanics. Juan les Pins, France, pp 283–289
D’Urso MG, Marmo F (2013) On a generalized Love’s problem. Comput Geosci 61:144–151
D’Urso MG, Marmo F (2015) Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains. Zeitschrift für Angewandte Mathematik und Mechanik 95(1):91–110
D’Urso MG, Russo P (2002) A new algorithm for point-in polygon test. Surv Rev 36(284):410–422
Elliott HA (1948) Three-dimensional stress distributions in hexagonal aeolotropic crystals. Math Proc Cambr Philos Soc 44:522–533
Eskandari M, Shodja HM (2010) Green’s functions of an exponentially graded transversely isotropic half-space. Int J Solids Struct 47:1537–1545
Eubanks RA, Sternberg E (1954) On the axisymmetric problem of elasticity theory for a medium with transverse isotropy. J Ration Mech Anal 47:1537–1545
Gerrard C, Wardle L (1973) Solutions for point loads and generalized circular loads applied to a cross anisotropic halfspace. Tech. Rep. 13, CSIRO (Commonwealth Scientific and Industrial Research Organization) Australia, Division of Applied Geomechanics, Sydney, Australia
Hu H (1953) On the three-dimensional problems of the theory of elasticity of a transversely isotropic body. Acta Phys Sin 9(2):130–148
Kalantari M, Khojasteh A, Mohammadnezhad H, Rahimian M, Pak R (2015) An inextensible membrane at the interface of a transversely isotropic bi-material full-space. Int J Eng Sci 91:34–48
Koning H (1957) Stress distribution in a homogenous, anisotropic, elastic semi-infinite solid. In: 4th international conference on soil mechanics and foundation engineering. Butterworths, London, pp 335–338
Kuzkin V, Kachanov M (2015) Contact of rough surfaces: conductance-stiffness connection for contacting transversely isotropic half-spaces. Int J Eng Sci 97:1–5
Lekhniskii SG (1981) Theory of elasticity of an anisotropic body. MIR Publishers, Moscow
Liao J, Wang C (1998) Elastic solutions for a transversely isotropic half-space subjected to a point load. Int. J. Numer. Anal. Meth. Geomech. 22:425–447
Lin W, Kuo CH, Keer LM (1991) Analysis of a transversely isotropic half space under normal and tangential loadings. ASME J Tribol 113:335–338
Lodge AS (1955) The transformation to isotropic form of the equilibrium equations for a class of anisotropic elastic solids. Q J Mech Appl Mech 8:211–225
Marmo F, Rosati L (2016) A general approach to the solution of Boussinesq’s problem for polynomial pressures acting over polygonal domains. J Elast 122:75–112
Marmo F, Sessa S, Rosati L (2016) Analytical solution of the Cerruti problem under linearly distributed horizontal pressures over polygonal domains. J Elast 124:27–56
Michell JH (1900) Some elementary distributions of stress in three-dimensions. Proc Lond Math Soc 32:23–35
Nowacki W (1954) The stress function in three-dimensional problems concerning an elastic body characterized by transverse isotropy. Bull Polish Acad Sci 4(2):21–25
Okumura IA (1987) Generalization of Elliott’s solution to transversely isotropic solids and its application. Proc Jpn Soc Civ Eng 386:185–195
Pan E, Chen W (2015) Static Green’s function in anisotropic media. Cambridge Press, Cambridge
Pan Y, Chou T (1976) Point force solution for an infinite transversely isotropic solid. J Appl Mech 43(4):608–612
Pan Y, Chou T (1979) Green’s function solutions for semi-infinite transversely isotropic materials. Int J Eng Sci 17:545–551
Rosati L, Marmo F (2014) A closed form expression of the thermo-mechanical fields induced by a uniform heat source acting over an isotropic half-space. Int J Heat Mass Transf 75:272–283
Selvadurai APS, Nikopour H (2012) Transverse elasticity properties of a unidirectionally reinforced composite with a random fibre arrangement. Compos Struct 94:1973–1981
Sessa S, D’Urso MG (2013) Employment of Bayesian networks for risk assessment of excavation processes in dense urban areas. In: 11th International Conference of Structural Safety and Reliability. ICOSSAR, New York, pp 3163–3169
Shield RT (1951) Notes on problems in hexagonal aeolotropic materials. Math Proc Camb Philos Soc 47:401–409
Tang KT (2006) Mathematical methods for engineers and scientists. Springer, Berlin
Ting TCT (1996) Anisotropic elasticity: theory and applications. Oxford University Press, Oxford
Toraldo F (2015) Elastic solutions for transversely isotropic half-spaces subject to vertical pressures. Ph.D. thesis, University of Naples, Federico II, Naples (in Italian)
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Marmo, F., Toraldo, F. & Rosati, L. Analytical formulas and design charts for transversely isotropic half-spaces subject to linearly distributed pressures. Meccanica 51, 2909–2928 (2016). https://doi.org/10.1007/s11012-016-0443-x
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DOI: https://doi.org/10.1007/s11012-016-0443-x