Application of Mapping and Superimposing Rule for Solution of Parabolic PDE in Porous Medium under Cyclic Loading
This paper presents an analytical method to solve
governing consolidation parabolic partial differential equation (PDE)
for inelastic porous Medium (soil) with consideration of variation of
equation coefficient under cyclic loading. Since under cyclic loads,
soil skeleton parameters change, this would introduce variable
coefficient of parabolic PDE. Classical theory would not rationalize
consolidation phenomenon in such condition. In this research, a
method based on time space mapping to a virtual time space along
with superimposing rule is employed to solve consolidation of
inelastic soils in cyclic condition. Changes of consolidation
coefficient applied in solution by modification of loading and
unloading duration by introducing virtual time. Mapping function is
calculated based on consolidation partial differential equation results.
Based on superimposing rule a set of continuous static loads in
specified times used instead of cyclic load. A set of laboratory
consolidation tests under cyclic load along with numerical
calculations were performed in order to verify the presented method.
Numerical solution and laboratory tests results showed accuracy of
presented method.
[1] Wilson N. E, Elgohary M. M, "Consolidation of soils under cyclic
loading," Canadian Geotechnical Journal. 1974:11(3): 420-433.
[2] Baligh M. M, Levadoux J "Consolidation theory for cyclic loading,"
Journal of the Geotechnical Engineering Division, ASCE. 1978:104(4):
415-431.
[3] Rahal M. A, Vueza A. R, "Analysis of settlement and pore pressure
induced by cyclic loading of silo," Journal of Geotechnical and
Geoenvironmental Engineering, December 1998: 1208-1210.
[4] Vueza A. R, Rahal M. A, "Cyclic loading for the measuring of soil
consolidation parameters," In: Proceedings of Settlement. ASCE,
Reston, 1994: 1: 760-774.
[5] Favaretti M, Mazzucato A, "Settlements of a silo subjected to cyclic
loading," In: Proceedings of Settlement. ASCE, Reston, 1994: 1: 775-
785.
[6] Favaretti M, Soranzo M, "A simplified consolidation theory in cyclic
loading condition," Int: Symposium on Compression and Solidation of
Clayey Soils, A. A. Balkema, Rotterdam, The Netherlands, 1995: 405-
409.
[7] Zhuang Ying-chun, Xie Kange-he, "Study on one-dimensional
consolidation of soils under cyclic loading with varied compressibility,"
Journal of Zhejiang University Science 2005; (6A):141-147.
[8] Toufigh M. M, Ouria A, "Nonlinear consolidation of inelastic soils
under rapidly cyclic loading," Proceeding of Iranian NCCE1, Iran,
(Persian) 2004: 174-182.
[9] Toufigh M. M, Shafiei B, "Finite element consolidation model for
subsidence problem based on Biot's three dimensional theory," Indian
Geotechnical Journal, 1996: 26(3):286-296.
[10] Toufigh, M.M, Ouria A, "Laboratory investigation of land subsidence
under cyclic loading in Kerman province, Iran," Proc of Seventh
International Symposium on Land Subsidence, China, 2005:(1): 335-
343.
[11] Toufigh M. M, Shafiei B, "Prediction of future land subsidence in
Kerman, Iran, due to groundwater withdrawal," International Journal of
Rock Mechanics and Mining Sciences & Geomechanics Abstracts,
1996: 33(8): A344.
[1] Wilson N. E, Elgohary M. M, "Consolidation of soils under cyclic
loading," Canadian Geotechnical Journal. 1974:11(3): 420-433.
[2] Baligh M. M, Levadoux J "Consolidation theory for cyclic loading,"
Journal of the Geotechnical Engineering Division, ASCE. 1978:104(4):
415-431.
[3] Rahal M. A, Vueza A. R, "Analysis of settlement and pore pressure
induced by cyclic loading of silo," Journal of Geotechnical and
Geoenvironmental Engineering, December 1998: 1208-1210.
[4] Vueza A. R, Rahal M. A, "Cyclic loading for the measuring of soil
consolidation parameters," In: Proceedings of Settlement. ASCE,
Reston, 1994: 1: 760-774.
[5] Favaretti M, Mazzucato A, "Settlements of a silo subjected to cyclic
loading," In: Proceedings of Settlement. ASCE, Reston, 1994: 1: 775-
785.
[6] Favaretti M, Soranzo M, "A simplified consolidation theory in cyclic
loading condition," Int: Symposium on Compression and Solidation of
Clayey Soils, A. A. Balkema, Rotterdam, The Netherlands, 1995: 405-
409.
[7] Zhuang Ying-chun, Xie Kange-he, "Study on one-dimensional
consolidation of soils under cyclic loading with varied compressibility,"
Journal of Zhejiang University Science 2005; (6A):141-147.
[8] Toufigh M. M, Ouria A, "Nonlinear consolidation of inelastic soils
under rapidly cyclic loading," Proceeding of Iranian NCCE1, Iran,
(Persian) 2004: 174-182.
[9] Toufigh M. M, Shafiei B, "Finite element consolidation model for
subsidence problem based on Biot's three dimensional theory," Indian
Geotechnical Journal, 1996: 26(3):286-296.
[10] Toufigh, M.M, Ouria A, "Laboratory investigation of land subsidence
under cyclic loading in Kerman province, Iran," Proc of Seventh
International Symposium on Land Subsidence, China, 2005:(1): 335-
343.
[11] Toufigh M. M, Shafiei B, "Prediction of future land subsidence in
Kerman, Iran, due to groundwater withdrawal," International Journal of
Rock Mechanics and Mining Sciences & Geomechanics Abstracts,
1996: 33(8): A344.
@article{"International Journal of Architectural, Civil and Construction Sciences:61777", author = "Mohammad M. Toufigh and Ahad Ouria", title = "Application of Mapping and Superimposing Rule for Solution of Parabolic PDE in Porous Medium under Cyclic Loading", abstract = "This paper presents an analytical method to solve
governing consolidation parabolic partial differential equation (PDE)
for inelastic porous Medium (soil) with consideration of variation of
equation coefficient under cyclic loading. Since under cyclic loads,
soil skeleton parameters change, this would introduce variable
coefficient of parabolic PDE. Classical theory would not rationalize
consolidation phenomenon in such condition. In this research, a
method based on time space mapping to a virtual time space along
with superimposing rule is employed to solve consolidation of
inelastic soils in cyclic condition. Changes of consolidation
coefficient applied in solution by modification of loading and
unloading duration by introducing virtual time. Mapping function is
calculated based on consolidation partial differential equation results.
Based on superimposing rule a set of continuous static loads in
specified times used instead of cyclic load. A set of laboratory
consolidation tests under cyclic load along with numerical
calculations were performed in order to verify the presented method.
Numerical solution and laboratory tests results showed accuracy of
presented method.", keywords = "Mapping, Consolidation, Inelastic porous medium,
Cyclic loading, Superimposing rule.", volume = "2", number = "5", pages = "108-6", }