# Frontiers of Structural and Civil Engineering

 Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (2) : 387-410     https://doi.org/10.1007/s11709-019-0601-z
 RESEARCH ARTICLE
Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method
Faculty of Civil Engineering, University of Tabriz, Tabriz 51666-16471, Iran
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 Abstract The purpose of the present study was to develop a fuzzy finite element method, for uncertainty quantification of saturated soil properties on dynamic response of porous media, and also to discrete the coupled dynamic equations known as u-p hydro-mechanical equations. Input parameters included fuzzy numbers of Poisson’s ratio, Young’s modulus, and permeability coefficient as uncertain material of soil properties. Triangular membership functions were applied to obtain the intervals of input parameters in five membership grades, followed up by a minute examination of the effects of input parameters uncertainty on dynamic behavior of porous media. Calculations were for the optimized combinations of upper and lower bounds of input parameters to reveal soil response including displacement and pore water pressure via fuzzy numbers. Fuzzy analysis procedure was verified, and several numerical examples were analyzed by the developed method, including a dynamic analysis of elastic soil column and elastic foundation under ramp loading. Results indicated that the range of calculated displacements and pore pressure were dependent upon the number of fuzzy parameters and uncertainty of parameters within equations. Moreover, it was revealed that for the input variations looser sands were more sensitive than dense ones. Corresponding Authors: Farhoud KALATEH Just Accepted Date: 17 January 2020   Online First Date: 23 April 2020    Issue Date: 08 May 2020
 Cite this article: Farhoud KALATEH,Farideh HOSSEINEJAD. Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method[J]. Front. Struct. Civ. Eng., 2020, 14(2): 387-410. URL: http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0601-z http://journal.hep.com.cn/fsce/EN/Y2020/V14/I2/387
 Fig.1  Definition of triangular fuzzy number [a,b,c]. Fig.2  Structure chart for incremental form of Biot analysis with fuzzy finite element method. Fig.3  Communications between (a) deterministic and (b) fuzzy solvers. Fig.4  Soil column subjected to a surface step loading; the geometry and boundary condition [47]. Fig.5  Membership function of fuzzy input parameters in soil column. (a) Poisson’s ratio; (b) Young’s modulus (E); (c) permeability coefficient (k). Tab.1  Material properties of soil column Tab.2  Elastic constants for various soils (after AASHTO, 1995) Tab.3  Typical values of permeability coefficient for various soils Fig.6  Time history of pressure at (a) node 46, (b) node 36, (c) node 16, (d) node 6, and time history of displacement for (e) node 11, (f) node1 of soil column in deterministic solution. Fig.7  Time history of pore pressure at (a) node 36, (b) node 46, (c) node 6, (d) node 16, and time history of displacement at (e) node 1, (f) node 11, (g) node 36, (h) node 46 of soil column by three fuzzy input parameters in FFEM model for five membership grades. Fig.8  Fuzzy number of vertical displacement at (a) node 1, (b) node 6, (c) node 11, (d) node 36, and (e) node 46 of soil column within 200 s after loading. (Left side: results in five modes of analysis: E, $ν$, k fuzzy; E, $ν$ fuzzy; E fuzzy; $ν$ fuzzy; k fuzzy. Right side: results in three modes of analysis: E, $ν$ fuzzy; E fuzzy; $ν$ fuzzy. Fig.9  Results in five modes of analysis with E, v, k fuzzy; E, v fuzzy; E fuzzy; v fuzzy; k fuzzy and fuzzy number of pressure at (a) node 6, (b) node 11, (c) node 36, and (e) node 46 of soil column within 200 s after loading Fig.10  (a) Elastic foundation subject to a surface step loading; (b) geometry and boundary condition. Fig.11  (a) Young’s modulus (E); (b) Poisson’s ratio; (c) permeability coefficient (k). Tab.4  Elastic foundation material properties Fig.12  Pore pressure history of FFEM model in different membership grades (left) and membership grade of 0.9 (right) at different nodes of elastic foundation within three fuzzy input parameters: (a) node 35; (b) node 67, (c) node 99; (d) node 131. Fig.13  Displacement history of FFEM model in different membership grades (left) and membership grade of 0.9 (right), at different nodes of elastic foundation within three fuzzy input parameters: (a) node 35; (b) node 67; (c) node 131. Fig.14  Vertical displacement (a) node 35, (b) node 67, (c) node 99, (d) node 131, and pressure (e) node 35, (f) node 67, (g) node 99, (h) node 131 fuzzy number in five modes of analysis including: E, $ν$, k fuzzy; E, $ν$ fuzzy; E fuzzy; $ν$ fuzzy; k fuzzy for different nodes of elastic foundation within 100 s after loading. Fig.15  Variation ratio of (a) horizontal displacements, (b) vertical displacements, (c) pore water pressure in membership grade of 0.9 for foundation. Fig.16  Upper and lower bounds of (a) pressure, (b) vertical displacements, and (c) horizontal displacements of elastic foundation in membership grade of 0.9 by assuming E, $ν$, k as a fuzzy inputs within one second after loading (upper bound in left side and lower bound in right side). Fig.17  Upper and lower bounds of (a) pressure, (b) vertical displacements, and (c) horizontal displacements of elastic foundation in membership grade of 0.9 by assuming E, $ν$, k as a fuzzy inputs within 100 s after loading (upper bound in left side and lower bound in right side).