Numerical analysis of the influence of porosity and void size on soil stiffness using random fields

Transcription

1 Numerical analysis of te influence of porosity and oid size on soil stiffness using random fields D.V. Griffits, Jumpol Paiboon, Jinsong Huang Colorado Scool of Mines, Golden, CO, USA Gordon A. Fenton Dalousie Uniersity, Halifax, NS, Canada ABSTRACT: Te purpose of te study is to inestigate te influence of porosity and oid size on equialent geotecnical engineering properties. Te paper uses random field teory to generate models of geomaterials containing spatially random oids wit controlled porosity and oid size. Analyses ten use finite elements and Monte-Carlo simulations, were eac simulation leads to an effectie alue of te property under inestigation. Te effectie alue is defined as te property tat would ae led to te same response if te geomaterial ad been omogeneous. Altoug te metodology can be applied to a ariety of soil properties, te paper concentrates on elastic stiffness. A noel finite element model is deeloped inoling tied freedoms tat allows analysis of an ideal element test and ence direct ealuation of te effectie Young s modulus and Poisson s ratio. A foundation problem of a footing on an elastic foundation containing oids is also described. 1 INTRODUCTION Te motiation for tis work came from a study of foundations resting on subsurface materials containing oids of ariable porosity and size (Griffits et al. 211). Te geology of te region consisted of karst wic is a special type of landscape and subsurface caracterized by te dissolution of soluble rocks, including limestone and dolomite. Gien limited site inestigation data and te spatial unpredictability of te underground oid locations, a numerical study was conducted to determine te effectie elastic properties of materials wit randomly distributed oids. Te goal of te study was to generate carts giing guidance on te statistical distribution (mean, standard deiation and PDF) of effectie Young s modulus and Poisson s ratio as a function of porosity and oid size. Te effectie alue is defined as te property tat would ae led to te same response if te geomaterial ad been omogeneous. Te oid size component of te study was intended to inestigate weter two deposits wit te same porosity but quite different oid size distributions would perform differently. Haing establised te statistical distributions of effectie properties as mentioned aboe, probabilistic statements relating to foundation performance under design loads can ten be made, e.g. te probability of excessie settlement for a foundation resting on subsurface materials wit gien porosity and oid size. 2 PRVIOUS WORK Te macro-scale response of a omogeneous material wic as a eterogeneous micro-structure as been studied by a number of inestigators. Te goal is to obtain te effectie or equialent properties at te macro-scale. An important objectie of micromecanics is to link mecanical relations at different lengt scales ranging from finer to coarser lengt scales. Under te assumption tat te material beaior at te micro-scale is known and is representatie of te entire eterogeneous material, a representatie olume element (RV) can be employed in te omogenization process. Te concept of RV was first introduced by Hill (1963) since wen tere ae been many numerical simulations deeloped and applied to determine RV size (e.g. Kulatilake 1985, Ning et al. 28 and smaieli et al. 21). Seeral teoretical models ae been proposed for dealing wit scale effects ranging from micro to macro leels. Te Differential Metod (Roscoe 1952) as a long and interesting istory, and as been one of te most effectie and widely used metods. Te Composite Speres Model (Hasin 1962) considered only a single inclusion and led to simple closed-form expressions. Te Self Consistent Metod (Budiansky 1965, Hill 1965) and te Generalized Self Consistent Metod, formalized by Cristensen and Lo (1979) inoled embedding an inclusion pase directly into an infinite medium. Cristensen and Lo (1979) explained tat te final D.V. Griffits et al. / Numerical analysis of porosity and oids size 1

2 form of tis metod can sole te sperical inclusion problem. Finally, te Mori-Tanaka (M-T) metod as described by Beneniste (1987) as attracted a lot of interest but is quite different to te oter metods. Te M-T metod inoles complex manipulations of te field ariables along wit special concepts of strain and stress. Altoug tere are many analytical models for estimating te effectie elastic properties of a material containing oids, tey are often limited to oids wit simple sapes. Numerical metods suc as te Finite lement Metod (FM) or te Boundary lement Metod (BM) ae been used to alidate some of te teoretical approaces. Two major ariables can be inestigated in a realistic representation of a defectie material; namely te olume and size of te oids or inclusions. Isida and Igawa (1991) considered seeral kinds of periodic arrays of oles, wile Day et al. (1992) considered a material containing circular oles witin a triangular or exagonal matrix. Models wit random circular oles, occasionally oerlapping eac oter were also analyzed. Hu et al. (2) deeloped a numerical model based on BM to estimate te effectie elastic properties, e.g. Young s modulus, bulk modulus and sear modulus. Te main objectie was to inestigate te influence of te sortest distance between oles of random size and olume based on a normal distribution function. Cosmi (24) introduced a new numerical model called te Cell Metod (CM) to inestigate te effect of randomly located oids. Te model consisted of a omogeneous matrix of cells wic contains randomly located oids. Li et al. (29) deeloped an FM model to calculate te elastic properties of porous materials wit randomly distributed oids. Altoug tere are many teoretical and numerical approaces, te results are rater unsatisfactory because of te uncertainties in te caracterization of te geometry canging from place to place orizontally and ertically. In tis paper, we will use te random finite element metod (RFM) first deeloped by Griffits and Fenton (1993) and Fenton and Griffits (1993). In tis metod, conentional finite element analysis of a plane strain elastic solid using 8 nodes square elements (e.g. Smit and Griffits 24) is combined wit random field generation (e.g. Fenton and Vanmarcke 199, Fenton and Griffits 28) and Monte-Carlo simulations to deelop output statistics of quantities suc as te effectie Young s modulus and Poisson s ratio. Not only te olume of inclusions can be considered by RFM, but also te size of te inclusions troug control of te spatial correlation lengt. 3 FINIT LMNT MODL Te finite element mes for tis study considers a square plane strain block of material of unit side lengt modeled by node square elements of side lengt x = y =.2 as sown in Figure 1. Te boundary conditions allow ertical moement only of nodes on te left side, orizontal moement only of nodes on te bottom side, wit te bottomleft corner node fixed. Te ertical components of all nodal freedoms on te top loaded side were tied, as were te orizontal components of all nodal freedoms on te rigt side. Tied freedoms are forced to moe by te same amount in te analysis by giing tem all te same freedom number during stiffness assembly. Te tied freedom approac offers an elegant way of modeling a eterogeneous medium as an ideal element of material. Oter metods employing stress or strain control may also be used to gie similar outcomes. Te unit ertical force sown as Q in te figure is applied to te tied ertical freedom on te top of te square. Te boundary conditions ensure tat no matter wat degree of eterogeneity is introduced, suc as, for example, te darker regions in Figure 1 indicating oids, te mes will always deform as an ideal element wit te top surface remaining orizontal and te rigt side remaining ertical. From tese ertical and orizontal moements under te application of a unit ertical pressure, te effectie Young s modulus and Poisson s ratio can be easily back-figured from linear elastic teory. Figure 1. Tied freedom model wit oids, under a unit ertical pressure. 4 RANDOM VOIDS MODL Input to te analysis consists of te target mean porosity n and spatial correlation lengt θ, wit te latter offering some degree of control oer te oid size. A random ariable Z is assigned to eac ele- D.V. Griffits et al. / Numerical analysis of porosity and oids size 2

3 ment of te mes based on a standard normal random field wit spatial correlation lengtθ. Te random field generation properly accounts for local aeraging, wic is to say tat te point ariance of te random field is reduced as a function of te ratio x θ. In tis study x θ 1, so te ariance reduction due to local aeraging sould be less tan 1%. Once te standard normal random field alues ae been assigned to te mes, cumulatie distribution tables (suitably digitized in te software) are ten used to estimate te alue of te standard normal ariable z n 2 for wic ( z n ) ( ) Φ 2 Φ = n 2 (1) as sown in Figure 2. Tereafter, any element assigned a random field alue in te range Z > z n 2 is treated as intact material wit Young s modulus and Poisson s ratio of = 1 and υ =.3, respectiely, wile any element were Z z n 2 is treated as a oid element wit an assigned Young s modulus of =.1 (1 times smaller tan te surrounding intact material). points are reasonably correlated (i.e. ρ >.14 ). By canging te alue of θ in te parametric studies, te degree to wic oid elements wit random alues in te range Z z n 2 tend to be clustered togeter can be influenced. A small alue of θ will imply fewer adjacent elements meeting te criterion at a gien location, and ence smaller and more frequent oids wile a large alue of θ will imply larger and less frequent oids as sown in Figure 3. Figure 3. Typical simulations sowing generation of oids at low and ig spatial correlation lengts θ (n=.2 in bot cases). 6 MONT-CARLO SIMULATIONS Figure 2. Target porosity area in standard normal random field. 5 VOID SIZ It is clear tat two rock or soil samples wit te same porosity could ae quite different oid sizes. For example one material could include numerous small olume oids, wile te oter could include less frequent larger olume oids. Te random field spatial correlation lengt θ offers some quantitatie control of oid size. As in preious work by te autors, te Marko spatial correlation as been used as sown in q.2, were te correlation coefficient ρ between two points separated by an absolute distance τ is gien ( τ θ ) ρ = exp 2 (2) Te function models an exponentially decaying correlation wic ranges from 1 for points ery close togeter to for points a long way apart. Loosely speaking θ is te distance witin wic A Monte-Carlo process means tat analyses are repeated numerous times until te statistical properties of te output parameters start to stabilize. In tis work, eac Monte-Carlo simulation inoles te generation of a random field and oid distribution as explained preiously. Tis is followed by an elastic analysis of te block suc as tat sown in Figure 1. Te primary outputs from eac elastic analysis are te ertical and orizontal deformations of te block, and, respectiely. Altoug eac simulation is based on te same θ and n, te spatial location of te oids will be different eac time, tus in some cases te oids may be just below te top of te block leading to a relatiely ig wile in oters, te oids may be buried in te middle of te block leading to a relatiely low. Following eac simulation, te computed displacements and are conerted into effectie alues of Young s modulus and Poisson s ratio from te plane strain elastic stress-strain relations to gie = υ + (3) D.V. Griffits et al. / Numerical analysis of porosity and oids size 3

4 2 1 υ = (4) ac Young s modulus alue is ten normalized as by diiding by te intact Young s modulus Discussion of te Poisson s ratio response is left for future work. Figure 6. σ / s. n for.2 Θ.8. Figure 4. Te accuracy of te estimated finite element statistics depend on te number of realizations for n =.3 and θ =.1. 7 RSULTS OF RFM In tis study, 1 simulations was sown to be enoug to gie statistically repeatable results as sown in Figure 4, so following eac set of Monte- Carlo simulations, te mean and standard deiation of te normalized Young s modulus was computed for a range of parametric ariations of n and Θ, wit results sown in Figures 5 and 6 respectiely. In te current work we ae expressed te spatial correlation lengt in dimensionless form θ Θ = (5) L were L is te widt of te loaded element in Fig- L = 1 ure 1 ( ) It can be noted from Figure 5 tat te mean Young s modulus µ obiously drops towards zero wit increasing porosity n and tat Θ does not ae muc influence on µ. Figure 6 sows tat Θ as more influence on te standard deiation of Young s modulus σ and also displays a maximum at n.3. 8 COMPARISON WITH OTHR SOLUTIONS Te current results are now compared wit existing teoretical and numerical approaces. From Figure 7, it can be obsered tat te current metod tends to gie lower alues of te effectie Young s modulus tan te teoretical metods described preiously. Figure 8 sows a comparison between te current metod and numerical results of Isida and Igawa (1991) and Day et al. (1992). Once more, te current metod tends to gie lower alues of te effectie Young s modulus, especially for lower alues of n. Figure 7. Comparison of te effectie Young s modulus obtained from RFM and different teoretical models. Figure 5. µ / s. n for.2 Θ.8. D.V. Griffits et al. / Numerical analysis of porosity and oids size 4

5 Figure 8. Comparison of te effectie Young s modulus obtained from RFM and different numerical models. Te lower alues generated by te RFM results suggests greater conseratism in design, but te reasons for tis trend are under continued inestigation. 9 FOUNDATION STTLMNT A similar tecnique using RFM as also been applied to study te influence of oids on foundation settlement using te mes sown in Figure 9. Te foundation sub-soil was modelled using 5 x 3 square 8-node finite elements of unit side lengt. Te strip footing at te ground surface was represented using tied ertical and orizontal freedoms coering ten elements wit a widt of 1 m. For a uniform foundation wit no oids, a comparison was made wit results of Sudret and Der Kiuregian (2) and te computed ertical displacement of.43 m was in ery good agreement. It may be noted tat te ertical displacement of a rigid footing is approximately te same as te aerage settlement of te centre and edge of a flexible footing carrying te same total load (Dais and Taylor 1962, Poulos and Dais 1974). From te omogeneous alidation example, te constant of proportionality relating te inerse of te effectie Young s modulus to ertical footing displacement gien by C = (6) was found to be 3 C = m /MN. Tis constant was ten used to compute te effectie Young s modulus during 1 Monte-Carlo simulations based on te ertical footing displacement, using C i =, i = 1,2,,1 (7) i Figure 9. Tied freedom analysis in foundation settlement on material wit no oids and material containing oids wit n=.2 and Θ =.5 and 1. (Q =.2 MPa, = 5 MPa, ν =.3 and B = 1 m). Te results of Monte-Carlo simulations for a range of parametric ariations of n and Θ are presented in Figure 1. In te settlement analyses, we ae expressed te spatial correlation lengt Θ in dimensionless form by diiding te actual correlation lengt θ by te footing widt B = 1 m. ac computed alue of from q. (4) is also expressed in dimensionless form by diiding by te intact alue = 5 MPa. Figure 1 sows reasonable agreement between te effectie Young s modulus from te settlement analysis and element tests for a range of porosities. Furter work is needed in tis area to assess te scaling effects. D.V. Griffits et al. / Numerical analysis of porosity and oids size 5

6 Sample calculation: 1) For input alues n =.2 and Θ =.5 2) From Monte-Carlo simulations, µ =.454, σ.74 = 3) Probability of design failure P[ / <.3] = Φ = 1.88%.74 were Φ [.] is te standard cumulatie distribution function. Figure 1. Comparison of results from settlement analysis and element test wit µ / s. n for Θ = PROBABILISTIC INTRPRTATION In order to make probabilistic interpretations from a Monte-Carlo analysis, we can eiter count te number of simulations tat exceed an allowable design alue as a proportion of te total number of simulations, or we can fit a probability density function to te data as in Figure 11. Te istogram sown in te figure indicates te frequency distribution of effectie Young s modulus alues following a suite of 1 Monte-Carlo simulations. Te smoot line is a fitted normal distribution based on te computed mean and standard deiation alues ( µ =.454, σ =.74 ) CONCLUSION RFM sows promise as a powerful alternatie approac for modeling te mecanical influence of inclusions and oids in geomaterials. Te oids are not restricted to being simple sapes as in some of te teoretical metods, and te user can control te oid olume and size troug spatial correlation, tus two rocks wit te same porosity could ae quite different oid sizes. Te RFM as been used in tis study to inestigate te influence of porosity and oid size on te effectie stiffness of geomaterials containing random oids. A noel tied freedom approac as been used to model an idealized element test leading to predictions of te effectie Young s modulus as a function of porosity and oid size. Some preliminary studies of foundation settlement on sub-soils containing oids were also presented using a similar RFM metodology and it was demonstrated ow tese results could be used to delier probabilistic conclusion regarding foundation settlement. 5 /o (design) =.3 RFRNCS frequency /o Figure 11. Histogram of Young s modulus alues following a suite of Monte-Carlo analyses togeter wit a fitted normal distribution. n =.2, Θ =.5. Let us assume tat te design of a footing as failed if / <.3. Tus for te particular parametric combination of n =.2 and Θ =.5 we migt wis to estimate P[ / <.3]. Beneniste Y., A new approac to te application of Mori-Tanaka teory in composite materials, Mec Mater, 6: Budiansky Y., On te elastic moduli of some eterogeneous material, J Mec Pys Solids, 13: Cristensen R.M. and Lo K.H., Solutions for effectie sear properties in tree pase spere and cylinder models, J Mec Pys Solids, 27: Cosmi F., 24. Two-dimension estimate of effectie properties of solid wit random oids, Teoretical and applied fracture mecanics, 42: Dais.H. and Taylor H., Te moement of bridge approaces and abutment on soft foundation soils, Proc. 1 st Biennial Conf. Aust, Road Res. Board, p.74. Day A.R. Snyder K.A. Garboczi.J. and Torpe M.F., Te elastic moduli of a seet containing circular oles, J Mec Pys Solids, 4: smaieli K. Hadjigeorgiou J. and Grenon M., 21. stimating geometrical and mecanical RV based on syntetic rock mass models at Brunswick Mine, International Journal of Rock Mecanics and Mining Sciences, 47, D.V. Griffits et al. / Numerical analysis of porosity and oids size 6

7 Fenton G.A. and Griffits D.V., Statistics of block conductiity troug a simple bounded stocastic medium, Water Resour Res, ol.29, no.6, pp Fenton G.A. and Griffits D.V., 28. Risk Assessment in Geotecnical ngineering, Jon Wiley & Sons, Hoboken, NJ. Fenton G.A. and Vanmarcke.H., 199. Simulation of random fields ia local aerage subdiision, J ng. Mec., ol.116, no.8, pp Griffits D.V. and Fenton G.A., 27. Probabilistic Metods in Geotecnical ngineering, Springer, Wien, Newyork, CISM Courses and Lectures No. 491, International Centre for Mecanical Sciences. Griffits D.V. Dotson D. and Huang J., 211. Probabilistic finite element analysis of a raft foundation supported by drilled safts in karst, To appear in GeoRisk 211, Atlanta. Hasin Z., Te elastic moduli of eterogeneous materials, J Appl Mec, 29: Hill R., lastic properties of reinforced solids: some teoretical principles, J. Mec. Pys. Solids, 11, Hill R., A self-consistent mecanics of composite materials, J. Mec. Pys. Solids, 13: Hu N. Wang B. Tan GW. Yao ZH. and Yuan WF., 2. ffectie elastic properties of 2D solids wit circular oids: numerical simulations, Composites Science and Tecnology, 6: Isida M. and Igawa H., Analysis of zig-zag array of circular oles in an infinite solid under uniaxial tension, Int J Solids Struc, 27: Kulatilake P. H. S. W., stimating elastic constants and strengt of discontinuous rock, J. Geotec. ngrg., 111, Li B. Wang B. and Reid S.R., 21. ffectie elastic properties of randomly distributed oid models for porous materials, International Journal of Mecanical Sciences, 52: Ning Y. Xu W.Y. Zeng, W.T. Meng G.T. and Si A.C., 28. Study of random simulation of columnar jointed rock mass and its representatie elementary olume scale. Cinese Journal of Rock Mecanics and ngineering, 27(6), Poulos H.G. and Dais.H., lastic solution for soil and rock mecanics. Jon Wiley and sons, New York, London, Sydney, Toronto. Roscoe R., Te iscosity of suspensions of rigid speres, J. appl. Pys, 3, 267. Smit I.M. and Griffits D.V., 24. Programming te finite element metod, Jon Wiley and sons, Cicester, New York, 4 t edition. Sudret B. and Der Kiuregian A., 2. Stocastic finite elements and reliability, a state-of-te-art report, Tecnical Report n UCB/SMM-2/8, Uniersity of California, Berkeley. ACKNOWLDGMNT Te autors wis to acknowledge te support of NSF grant CMMI on GOALI: Probabilistic Geomecanical Analysis in te xploitation of Unconentional Resources" and KGHM Cuprum, Wrocław, Poland troug te Framework 7 U project on Industrial Risk Reduction (IRIS). D.V. Griffits et al. / Numerical analysis of porosity and oids size 7