AM  Vol.4 No.4 , April 2013
Exact Solution of Terzaghi’s Consolidation Equation and Extension to Two/Three-Dimensional Cases
ABSTRACT

The differential equation by Terzaghi and Fr?hlich, better known as Terzaghi’s one-dimensional consolidation equation, simulates the visco-elastic behaviour of soils depending on the loads applied as it happens, for example, when foundations are laid and start carrying the weight of the structure. Its application is traditionally based on Taylor’s solution that approximates experimental results by introducing non-dimensional variables that, however, contradict the actual behaviour of soils. The proposal of this research is an exact solution consisting in a non-linear equation that can be considered correct as it meets both mathematical and experimental requirements. The solution proposed is extended to include differential equations relating to two/three dimensional consolidation by adopting a transversally isotropic model more consistent with the inner structure of soils.


Cite this paper
R. Francesco, "Exact Solution of Terzaghi’s Consolidation Equation and Extension to Two/Three-Dimensional Cases," Applied Mathematics, Vol. 4 No. 4, 2013, pp. 713-717. doi: 10.4236/am.2013.44099.
References
[1]   K. Terzaghi, “Die Berechnung der Durchlassigkeitsziffer des Tones aus Dem Verlauf der Hidrodynamichen Span-nungserscheinungen Akademie der Wissenschaften in Wien,” Mathematish-Naturwissen-Schaftiliche Klasse, Vol. 132, 1923, pp. 125-138.

[2]   K. Terzaghi and O. K. Fröhlich, “Theorie der Setzung von Tonschichte,” Franz Deuticke, Leipzig/Wien, 1936.

[3]   T. W. Lambe and R. W. Whitman, “Soil Mechanics,” John Wiley & Sons, New York, 1969.

[4]   K. Terzaghi, “Principles of Soil Mechanics: I—Phenomena of Cohesion of Clays,” Engineering News-Record, Vol. 95, No. 19, 1925, pp. 742-746.

[5]   A. Casagrande, “Structure of Clay and Its Importance in Foundation Engineering,” Journal of Boston Society of Civil Engineers, Vol. 19, No. 14, 1932, pp. 168-209.

[6]   G. Gilboy, “Improved Soil Testing Methods,” Engineering News Record, 21 May 1936.

[7]   P. C. Rutledge, “Relation of Undisturbed Sampling in Laboratory Testing,” Tr. Am. Soc. C.E., No. 109, 1944.

[8]   C. Viggiani, “Foundations,” Hevelius Editore, Benevento, 1999.

[9]   R. Nova, “Fundamental of Soil Mechanics,” McGraw-Hill, Milano, 2002.

[10]   R. Lancellotta, “Geotechnical,” 3rd Edition, Zanichelli Editore, Bologna, 2004.

[11]   M. A. Biot, “General Theory of Three-Dimensional Consolidation,” Journal of Applied Physics, Vol. 12, No. 2, 1941, pp. 155-164.

[12]   M. A. Biot and F. M. Clingan, “Consolidation Settlement of Soil with an Impervious to Surface,” Journal of Applied Physics, Vol. 12, No. 7, 1941, pp. 578-581.

[13]   J. Mandel, “Consolidation des Couches d’Argille,” Proc. IV ICSMFE, London, 1957.

[14]   J. De Jong, “Application of Stress Function to Consolidation Problems,” Proc. IV ICSMFE, London, 1957.

[15]   R. E. Gibson and J. McNamee, “The Consolidation Settlement of a Load Uniformly Distributed over a Rectangular Area,” Proc. IV ICSMFE, London, 1957.

[16]   J. McNamee and R. E. Gibson, “Plane Strain and Axially Symmetric Problems of the Consolidation of a Semi Infinite Clay Stratum,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 13, No. 2, 1960, pp. 210-217.

[17]   J. Mandel, “Tassements Produits par la Consolidation d’une Couche d’Argille de Grande Epaisseur,” Proc. V ICSMFE, Paris, 1961.

[18]   C. W. Cryer, “A Comparison of the 3-Dimensional Theories of Biot and Terzaghi,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 16, 1963, pp. 401-412.

[19]   R. L. Schiffman and A. Fungaroli, “Consolidation Due to Tangential Loads,” Proc. VI ICSMFE, Montreal, 1965.

[20]   R. E. Gibson, J. K. L. Schiffman and S. L. Pu, “Plain Strain and Axially Symmetric Consolidation of a Clay Layer of Limited Thickness,” University of Illinois, Mate Report, 1967, pp. 67-74.

[21]   R. L. Schiffman, A. T. F. Chen and J. C. Jordan, “An Analisys of Consolidation Theories,” Proceedings of the American Society of Civil Engineers, Vol. 98, No. SM1, 1969, pp. 285-312.

[22]   P. A. Vermeer and A. Verruijt, “An Accuracy Condition for Consolidation by Finite Element,” International Journal for Numerical Analytical Methods in Geomechanics, Vol. 5, No. 1, 1981, pp. 1-14. doi:10.1002/nag.1610050103

[23]   T. J. R. Hughes, “The Finite Element Method,” Prentice-Hall, Upper Saddle River, 1987.

[24]   D. W. Taylor, “Fundamental of Soil Mechanics,” Jonh Wiley & Sons, New York, 1948.

 
 
Top