numerical

数字推理，主要是从一推数字中看出其规律，从而推敲出其所要求的答案。

虽然有很大部分的题目是属于数列这范畴的，但理论上，任何形式的问题，只要换上数字，也可名正言顺的收纳于数字推理这类试题中。

换言之，问题可以千变万化。

当然，归根究底，是我们能否看出个中玄机。

既然出题方法浩瀚如海，是不可能尽其详，穷其变的了，只能略举一二，使大家留意。

其中一些题目，只需略作更动，可以化成空间图形的问题，大家可小心的参考。

问题：以下的每一个数字都遵从某一规律，请从随后的数字中，选取有相同规律的作为答案。

例题N01：1579，19，357，468，13567，89（A）21（B）2389（C）6543（D）667（E）1323答案：B例题N01 的答案分析。

每组数字1579，19，357，468，13567，89 的每一个数位，都是从左至右，随小至大的序排列，选择中只有B 符合这一原则。

数字除了大小顺序的规律，还要留意次序。

无论是图形和数字，我们都应作这方面的处理，再举一些例子作参考。

例题N02：17340，69540，67354，16975，63540，169X3请问X 是以下那一个数字？（A）7（B）5（C）2（D）4（E）0提示：留意曾出现过的数字有无重复，数字的次序等。

例题N02 的答案分析。

答案：（A）7。

5，4，2，0 都不是。

在题目中曾出现的数字有1，6，9，7，3，5，4，0，每个数字只出现一次，而且是依这顺序决定先后。

例如，1 出现的话一定在最前，6 排在1 之后，但在其他任何数字之前。

从另一角度看，曾在 3 之前出现的有6 和7，6 是出现过的了，在9 之后的有5 和7，但5 还在3 之后，唯一选择是7 了。

再来一题。

例题N03：4253，3842，5384，2483，8425请问以下那一组数字与其他的不同？（A）4253（B）3842（C）5384（D）2483（E）8425于一些公开试中，不单止DS 所涉及的数学范围，无所不包，就是数字推理中涉及的数学范围，也浩翰如海。

专利名称：NUMERICAL-SIMULATION-RESULT DISPLAY PROGRAM, NUMERICAL-SIMULATION-RESULT DISPLAY METHOD, ANDNUMERICAL-SIMULATION-RESULT DISPLAYSYSTEM发明人：Hajime Yagi申请号：US12014553申请日：20080115公开号：US20080174611A1公开日：20080724专利内容由知识产权出版社提供专利附图：摘要：A numerical-simulation-result display system includes an image output apparatus, a three-dimensional-CAD-model-data input unit configured to read three-dimensional-CAD-model data of an object to be analyzed, a CAD-model-data converting unit configured to convert the read three-dimensional-CAD-model data into CAD graphic data capable of being displayed on the image output apparatus, a numerical-simulation-result-data input unit configured to read numerical-simulation-result data of the object, a simulation-result converting unit configured to convert the read numerical-simulation-result data into simulation-result graphic data capable of being displayed on the image output apparatus, and a display unit configured to display the CAD graphic data and the simulation-result graphic data on the image output apparatus in a superimposed manner.申请人：Hajime Yagi地址：Kanagawa JP国籍：JP更多信息请下载全文后查看。

根据下图回答1-3题：1 . 过去5年中，世界民航乘客的增长率是多少？A . 37%B . 47%C . 57%D . 67%E . 以上都不是答案：E2 . 过去5年中，欧洲民航乘客的增长超出美国多少？A . 40,000,000B . 60,000,000C . 400,000,000D . 600,000,000E . 无法判断答案：A3 . 5年前每名乘客在欧洲地区内的平均飞行距离为1,000公里。

现在这一数字已经下降了20%。

现在的欧洲航线飞行总距离变化的百分比是多少？A . 下了60%B . 下了30%C . 没有变化D . 增长了140%E . 增长了280%答案：D根据下图回答4-6题：4 . 1994年GREENCO每家潮汐能公司的平均收入大约是多少？A . 350,000欧元B . 375,000欧元C . 400,000欧元D . 425,000欧元E . 450,000欧元答案：B5 . 2000年GREENCO没加潮汐能公司实现的营业利润比1994年高多少欧元？A . 54,000欧元B . 108,000欧元C . 216,000欧元D . 432,000欧元E . 以上都不是答案：B6 . 1994年至2000年潮汐能部门的收入增长总量相应的年平均增长率是多少？A . 3.0%B . 3.3%C . 3.6%D . 3.9%E . 无法判断答案：A根据下图回答7-10题：7 . 效益率是指交易现金流量占总成本（售出商品成本及固定成本）的比率。

1996年IPG 的效益率是多少？A . 13.1%B . 13.3%C . 13.5%D . 13.7%E . 以上都不是答案：D8 . 1995年以后，该行业净销售额的年均增长率为10%。

到1997年IPG净销售额将会超过工业平均增长率的多少欧元？A . 64,000万B . 64,800万C . 65,600万D . 66,400万E . 以上都不是9 . 在哪两年之间销售利润增长率最高？A . 1995年-1996年B . 1996年-1997年C . 1997年-1998年D . 1998年-1999年E . 1999年-2000年答案：B10 . 如果1995年至1996年净销售额增长率与1994年到1995年相同，那么1994年的销售利润是多少欧元？A . 35,900万B . 36,900万C . 37,900万D . 38,900万E . 无法判断根据下图回答11-13题：11 . 如果明天日元对努尔特鲁姆的汇率下降幅度和今天相同，那么明天30,000日元可兑换多少努尔特鲁姆？A . 10,700B . 10,800C . 10,900D . 11,000E . 11,100答案：E12 .20，000欧元当日最高汇率比按收盘汇率可多兑换多少努尔特鲁姆？A . 29,300B . 37,300C . 39,200D . 72,800E . 85,800答案：C13 . 某商人用15,000英镑在今天汇率最高时兑换努尔特鲁姆。

Numerical Study of the Effect of Foundation Sizefor a Wide Range of SandsNobutaka Yamamoto1;Mark F.Randolph2;and Itai Einav3Abstract:This paper presents a numerical investigation of the effect of foundation size on the response of shallow circular foundations on siliceous and calcareous sands.The study is based on the predictive capabilities of the MIT-S1soil model for simulating both the compression and shear behaviors of natural sands over a wide range of densities,K0values and confining pressures.The paper highlights the variations in the deformation mechanisms for the siliceous and calcareous sands cases.The assessment of the bearing capacity factor, N␥,is examined,showing a dramatic decrease in the values with increasing foundation size for the case of footings on calcareous sands, eventually converging to a terminal N␥value.At this stage the sand resistance is insensitive to variations in initial density and foundation size because the sand tends to loose its initial characteristics due to grain crushing,leading the material rapidly toward ultimate conditions. In the silicious sand case,it is found that,eventually,for extremely large footing diameters,the deformation mechanism progresses toward a punching shear mechanism,rather than the classical rapture pattern accompanied by surface heave as employed in current bearing capacity equations.A dimensional transition between the failure mechanisms can clearly be defined,referred to as a“critical size”in the N␥–D relationship.DOI:10.1061/͑ASCE͒1090-0241͑2009͒135:1͑37͒CE Database subject headings:Sand;Calcareous soils;Silica;Shallow foundations;Size effect;Finite element method;Numerical analysis.IntroductionThe bearing capacity of foundations on granular materials has been studied extensively as one of the fundamental problems of geotechnical engineering.The most common method to estimate the bearing capacity is the classical Terzaghi equation that in-cludes three factors:the N c factor for cohesion,the N q for embed-ment depth,and the N␥for the self-weight component.These different factors are modified for the particular loading condition and material case in hand.The N␥factor is of primary importance for shallow foundations on sands but it is extremely sensitive to variations in the material properties.Early experimental studies of this factor in sand were mainly concerned with relatively small model foundations,tested in natural1g conditions.It was real-ized that N␥decreases with increasing footing width or diameter, and this is now widely recognized as the“foundation size effect”͑De Beer1963͒.De Beer reasoned that the foundation size ef-fect results from the fact that the strength criterion of sands should be nonlinear,with the friction angle decreasing with in-creasing stress level,rather than linear as in the conventional linear Mohr–Coulomb criterion.The nonlinear failure envelope arises from the stress dependency of dilation,which by itself arises from particle rearrangement and crushing͑Lee and Seed 1967;Bolton1986͒.There are several numerical approaches for assessing the in-fluence of N␥using nonassociated constitutive models͑i.e.,mod-els that incorporate a dilation angle that is not equal to the friction angle͒.Frydman and Burd͑1997͒and Erickson and Drescher ͑2002͒studied the effect of the dilation angle on N␥for strip and circular footings,respectively,using a nonassociated Mohr–Coulomb model.They found that the effect of dilation angle is negligible at low friction angles,but quite important for friction angles greater than35°,especially for the case of rough circular footings.However,these previous numerical studies are limited by the fact that the Mohr–Coulomb model cannot capture suffi-ciently well the stress and density state dependency of sand be-havior or the compressibility of sands.The intention of the work reported here was to perform a more comprehensive numerical study that accounts for many more be-havioral aspects of sand.For that purpose the MIT-S1constitutive model͑Pestana and Whittle1999͒was adopted as that model has sufficient complexity to simulate both compression and shear be-haviors of natural sands over a wide range of densities,K0values, and confining pressures using a single set of model parameters of a given sand type.In summary,this paper presents a numerical investigation of the foundation size effect in the case of shallow circular footings on siliceous and calcareous sands using the MIT-S1model.Fol-lowing a brief description on the MIT-S1model,the strength characteristics of siliceous and calcareous sands are discussed in the context of drained triaxial shear test results.The effects are1Engineer,Advanced Geomechanics,4Leura St.,Nedlands,WA, 6009,Australia;formerly,Ph.D.Student,Centre of Offshore Foundation Systems,Univ.of Western Australia,Crawley,WA6009,Australia. E-mail:nobutakay@.au2Professor,Centre for Offshore Foundation Systems,Univ.of Western Australia,35Stirling Highway,Crawley,WA6009,Australia.E-mail: randolph@.au3Senior Lecturer,School of Civil Engineering J05,Univ.of Sydney, Sydney,NSW2006,Australia.E-mail:I.Einav@.au Note.Discussion open until June1,2009.Separate discussions must be submitted for individual papers.The manuscript for this paper was submitted for review and possible publication on March15,2007;ap-proved on April30,2008.This paper is part of the Journal of Geotech-nical and Geoenvironmental Engineering,V ol.135,No.1,January1, 2009.©ASCE,ISSN1090-0241/2009/1-37–45/$25.00.then translated to explain the foundation problem for both typesof sand,followed by a discussion of the foundation size effect interms of the N␥factor.Modeling Sands Using the MIT-S1ModelFull details of the MIT-S1model can be found in Pestana andWhittle͑1999͒.According to Pestana et al.͑2002͒,the model iscapable of simulating many behavioral characteristics of sandbehavior,including nonlinearity of the compression curves andcritical state lines on e–ln pЈplots,the dilatancy behavior of sands,and the variation of peak friction angle as a functionof stress level and density.The model can capture a range ofdifferent characteristics of both compressible and incompressiblegranular materials through appropriate adjustment of the modelparameters.The MIT-S1model requires13input parameters to model thebehavior of freshly deposited sand͑which is the type of sand thispaper is concerned with͒.According to Pestana and Whittle ͑1999,2002͒these parameters can be determined from standard laboratory tests.This paper focuses on two different types of sands,Toyourasiliceous sand͑from Japan͒,and Goodwyn calcareous sand͑fromthe North West Shelf of Australia͒.The model parametersfor these sands were determined in Pestana et al.͑2002͒andYamamoto et al.͑2008͒.The model parameters for Dogs Bay calcareous sand and Goodwyn calcareous silt are also provided to enable a complete discussion on the foundation size effect.The physical properties of the sands and the silt are summarized in Table1,and the model input parameters are given in Table2.The particle size distributions for Toyoura siliceous sand͑Ishihara 1993͒,Dogs Bay calcareous sand͑Coop1990͒,Goodwyn calcar-eous sand͑Ismail2000͒,and Goodwyn calcareous silt͑Finnie 1993͒are shown in Fig.1.As may be seen,the Dogs Bay calcar-eous sand has larger particles than the Toyoura siliceous sand. Further,it is noted that the Goodwyn sand is relatively well graded with30%fines content.Compression BehaviorFig.2shows the MIT-S1predictions of the compression curves of both siliceous and calcareous sands.The initial densities and cur-vature of the compression curves vary significantly,but the model captures these variations well.Calcareous sands have higher ini-tial void ratios and greater reduction of volume than siliceous sands.The critical state lines of the sands are also significantly different,but again the model predicts them nicely.Shear BehaviorFig.3shows the MIT-S1predictions for drained isotropically consolidated shear tests on siliceous and calcareous sands with different initial densities but the same confining stress͑100kPa͒.Table1.Index Properties of SoilsPropertySiliceous CalcareousToyoura sand Goodwyn sand Dogs Bay sand a Goodwyn siltMineralogy Quartz,feldspar,magnetite Calcium carbonate͑94%͒Calcium carbonate͑98%͒Calcium carbonate͑94%͒Grain shape Subangular Skeletal grain Skeletal grain Skeletal grain Specific gravity,G s 2.65 2.72 2.75 2.77Mean particle size,D50͑mm͒0.16–0.200.1–0.20.20.03Coefficient of uniformity,C u 1.3–1.710–15 2.0645Maximum void ratio,e max0.98 2.32–1.97 2.21–1.83 2.40 Minimum void ratio,e min0.61–0.58 1.41–0.94 1.48–0.98 1.21a Properties of Dogs Bay sand were reassessed and different from the value provided by Pestana͑1994͒.Table2.MIT-S1Model Parameters for Various SoilsTest type Symbol Physical meaningSiliceous Calcareous b Toyoura sand a GW sand b DB sand b GW silt bCompression test␳c Compressibility at large stresses͑LCC regime͒0.3700.3500.3500.250 p refЈReference stress at unity void ratio for the H-LCC͑kPa͒5,5002,5004,0002,000␪First loading curve transition parameter0.2000.9000.4000.900 K0consolidation test K0NC K0in the LCC regime0.4900.4900.5100.450␮0ЈPoisson’s ratio0.2330.1500.2000.200␻Parameter for nonlinear Poisson’s ratio 1.00 2.00 1.00 2.00 Shear test␾cs Critical state friction angle͑°͒31.039.646.040.0␾mrЈPeak friction angle as a function of void ratio͑°͒28.560.080.072.0np Constant of peak friction angle 2.45 2.00 2.00 2.00m Geometry of bounding surface0.550.350.550.30␺Rate of evolution of anisotropy50.050.050.050.0Shear test with local measurement systems C b Small strain stiffness parameter750450750450␻s Small strain nonlinearity parameter 2.50 3.00 2.50 3.0a Pestana͑1994͒.b GW=Goodwyn;DB=Dogs Bay.In siliceous sand,denser samples exhibit a clear peak stress at relatively small strain levels,whereas no peak stress is found for looser samples.On the other hand,all calcareous samples show contractive behavior,although the experimental response from Finnie ͑1993͒for relative dense Goodwyn sand shows a slight peak at small strain levels.Sensitivity Study of the MIT-S1ParametersAs mentioned earlier,the MIT-S1model requires 13model pa-rameters to define the behavior of sand.Although the parameters should be specified precisely,the particular shallow footing prob-lem in this paper tends to be dominated by only a few parameters.Yamamoto ͑2006͒carefully investigated the effect of the different model parameters on the response of shallow circular footings on siliceous and calcareous sands.A summary of these sensitivityanalyses is given in Table 3.The compression parameters,p ref Јand ␪,and the shear parameter,m ,are the most significant,whereas the remaining parameters have less effect.It is found that the shear parameters ͑apart from m ͒have little effect on the results for calcareous sand,implying that the bearing response on calcar-eous sand is dominated more by the compression component than by shear.Hence,for the shallow footing problem the relatively large number of 13parameters was reduced to a more manageable study involving three significant parameters.Effects of Stress Level,Density,andCompressibility on the Strength Characteristics of SandsThe effects of stress level,density,and compressibility are of great importance for assessing the behavior of sands.The effects can be captured through a relationship between the peak friction angle,␾p Ј,the initial mean effective stress at failure,p 0Ј,and void ratio,e .Fig.4illustrates the relationship between ␾p Ј,p 0Јand e for theP e r c e n t a g e f i n e rFig.1.Particle size distributions forsandsV o i d R a t i o ,eMean Effective Stress,p'(kPa)Fig.2.Consolidation curves and critical state lines for siliceous andcalcareous sands510152025300100200300400500Linee 00.950.900.800.700.60CID testsToyoura siliceous sand p'0=100kPaD e v i a t o r i c s t r e s s ,q (k P a )Shear strain,H s (%)(a)0100200300400500D e v i a t o r i c s t r e s s ,q (k P a )Shear strain,H s (%)(b)Fig. 3.Triaxial drained shear tests results for siliceous and calcareous sands:͑a ͒siliceous sand;͑b ͒calcareous sandToyoura siliceous and Goodwyn calcareous sands.The peak fric-tion angles at lower stress levels for Toyoura siliceous sand are initially only weakly dependent on the increase in pressure,but this dependency then strengthens to a rapid reduction with in-creasing confining pressure.At higher stress levels,the peak fric-tion angles eventually converge to the critical state values͑i.e.,␾pЈ=␾csЈ͒at“critical stresses,”as suggested by Vesic and Clough ͑1968͒.It is noticed that the critical stress decreases as the densitydecreases.The peak friction angles for calcareous sands also de-pend on the combined influence of e and p0Ј.However,they re-duce rapidly with increasing p0Ј,even at low stress level.The critical stresses for calcareous sands are significantly lower than for siliceous sands.Three triaxial compression test results using Goodwyn sand are plotted in Fig.4,one for e0=1.1͓from Finnie ͑1993͔͒and two others for e0=1.4͓from Sharma͑2004͔͒.The MIT-S1predictions underestimate the peak friction angles for these data,which is consistent with the slight peak in deviator stress observed in triaxial tests͑Fig.3͒.The variation of peak friction angle raises questions on the applicability of conventional bearing capacity theories,which are based on constant friction angle with depth͑normalized by foun-dation size͒.For example,an analysis of a10m diameter foun-dation with practical settlement limits of5–10%of foundation diameter͑or width͒may be based on initial stresses of40kPa ͑multiplying half of the diameter,5m,by a soil effective unit weight of8kN/m3͒.However,when the same settlement level is applied to a100m diameter foundation,the corresponding stress level is simply ten times͑400kPa͒that for the10m diameter footing.At that stress level,the peak friction angles are no longer constant with depth.The peak friction angles for calcareous sands are obviously not constant at40kPa,thus for this sand the con-ventional bearing capacity formulas do notfit even for a moderate foundation size.Responses of Shallow Foundations on SandsThe following describes numerical results for the response of10 and100m diameter footings on siliceous and calcareous sands. Initial void ratios at the ground surface,e0,and effective unit weights,␥Ј,are0.8͑dense͒and8kN/m3for the siliceous sand, and1.3͑medium dense͒and7kN/m3for the calcareous sand.To carry out the100m diameter analyses,the effective unit weight has been taken ten times higher,avoiding the need to modify the finite-element meshes.Thus the increase in the foundation size is simulated simply by increasing the initial stress gradient. Pressure–Displacement CurvesFig.5͑a͒shows N␥and␦/D relationships for100m diameter smooth and rough footings on siliceous sand,with the10m di-ameter results also plotted for comparison.The bearing response of the large scale rough footing shows no peak value but rather increases continuously with increasing penetration depth.This is because the compression component of the material dominates the bearing response as the foundation size increases.For the 100m diameter smooth footing case,however,an ultimate bear-ing capacity is still observed although it needs much larger verti-cal displacement than for the small footing.This appears to be because the deformation mechanism for siliceous sand progres-sively shifts toward punching shear with increasing size of foun-dation.It is worth noting that the effect of roughness for larger foundations is much smaller.The bearing responses on calcareous sand with different foun-dation sizes show similar trends but the100m diameter founda-tion shows a more linear response͓Fig.5͑b͔͒,and the mobilized N␥for the100m case is smaller.Deformation MechanismsAs described earlier,a transformation in the mechanisms from small to large foundations may be seen,in particular for rough footings on siliceous sand.Fig.6͑a͒shows that at a penetration of 10%of the diameter the amount of surface heave reduces signifi-cantly with increasing diameter.However,for the smooth footing analysis͓Fig.6͑b͔͒,a classical rupture failure pattern with surface heave is still evident for the100m diameter calculations although more obvious downward deformations are exhibited at shallower penetration.The incremental displacement vectors for10and100m diam-eter footings on calcareous sand show almost identical defor-Initial mean effective stress,p'(kPa)Fig.4.Peak friction angle and initial state relationships for siliceous and calcareous sandsmation patterns at all penetration levels ͓Fig.6͑c ͔͒.The soil be-neath the footings compress almost in a one-dimensional vertical manner.Effect of Foundation Size on Bearing Capacity Factor,N ␥The following explores the effect of foundation size on the mo-bilized bearing resistance factor,N ␥.This effect of foundation size has been explained previously as due to the stress dependency of granular materials ͑De Beer 1963;Hettler and Gudehus 1988;Kusakabe et al.1991͒,or more precisely on the stress dependency of the peak friction angles.The numerical investigation using the MIT-S1model provides further explanations of this effect andalso allows a possible deduction of the dimensional transition between dilative and contractive responses of the soil.Siliceous SandFig.7summarizes the bearing response from analyses with dif-ferent footing sizes of fully smooth shallow circular footings on siliceous sand,by plotting the N ␥–␦/D relationships for e 0=0.8͑loose ͒,and N ␥-␦/D for e 0=0.65͑dense ͒,where ␦denotes the footing downward displacement.The effect of the foundation size has been recognized experimentally with the mobilized N ␥de-creasing with increasing diameter ͑e.g.,De Beer 1963͒,but with experimental evidence only over a relatively small diameter range.The numerical predictions using the MIT-S1model suggest that the foundation size effect exists for larger foundations as well.Moreover,as expected,a transition from dilative to contractive deformations can be seen as the foundation size increases.The smaller footings tend to show dilative behavior with clear peak stress,whereas the larger foundations present contractive re-sponse and exhibit lower mobilized N ␥values.This is also re-flected from the results of drained triaxial tests with different initial void ratios as shown in Fig.3.Fig.8shows N ␥–D rela-tionships for loose and dense siliceous sands.Two N ␥values are shown,one corresponding to the peak bearing resistance ͑if one exists ͒and the other corresponding to ␦/D =10%͑shown only if N ␥keeps increasing for ␦/D greater than 10%͒.The two020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement/Diameter,G /D (%)(a)020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement/Diameter,G /D (%)(b)Fig.5.Shallow foundation responses for siliceous and calcareous sands:͑a ͒siliceous sand;͑b ͒calcareous sandFig.6.Incremental displacement vectors after a penetration of 10%of the diameter:͑a ͒siliceous sand,rough,D =10m ͑left ͒,100m ͑right ͒;͑b ͒siliceous sand,smooth,D =10m ͑left ͒,100m ͑right ͒;and ͑c ͒calcareous sand,smooth,D =10m ͑left ͒,100m ͑right ͒N ␥values merge at about 20m diameter for loose ͑e 0=0.8͒samples and 60m diameter for dense ͑e 0=0.65͒samples ͑indi-cated by arrows ͒and this will be defined as the transition diam-eter point from dilative to contractive response.This diameter may be referred to as a “critical size,”D cr ,which basically fol-lows the same concept behind the definition of the ‘critical stress’by Vesic and Clough ͑1968͒,as described before.Kimura et al.͑1985͒suggested that the N ␥value reduces with reduction in density.Fig.8shows the great variation with density over a wide range of foundation size.The factor diminishes rap-idly with increasing foundation diameter for small diameters,but the effect reduces at larger diameters ͑noting the logarithmic scale of the plot ͒.Fig.8also compares the numerical results with centrifuge model tests for circular footings ͑D =1.5–3m ͒on Toyoura sili-ceous sand performed by Okamura et al.͑1997͒.Unfortunately,the finite-element results could not be obtained for small diam-eters owing to numerical instability for the high dilation rates associated with shearing at low stress levels.However,both re-sults show the reduction of N ␥with increasing diameter.Calcareous SandFig.9shows bearing responses of fully smooth shallow circular footings on calcareous sand with K 0=1for two representative densities ͑e 0=1.3for dense or e 0=1.9for loose ͒,applied over a wide range of diameters ͑1–100m ͒.It is noticed that the effect of foundation size and density are very strong for smaller diameters.Fig.10plots N ␥–D relationships for calcareous sand.Additional analyses to those in Fig.9were undertaken with identical soil parameters apart from taking K 0=0.49,and those results are shown in Fig.10alongside those for K 0=1.The rate of decrease of N ␥with increasing foundation size becomes gradually lower for larger foundation sizes and for loose samples the N ␥values become nearly constant for diameters of more than 30m.Physi-cal model results from Finnie and Randolph ͑1994͒are also shown,and though these show some decrease in N ␥with increas-ing foundation size,the rate of decrease is not as dramatic as for the numerical results.It may also be seen that the numerical re-sults give higher N ␥values,for a given void ratio,than those reported by Finnie and Randolph,in spite of giving lower peak friction angles for triaxial tests ͑see Figs.3and 4earlier ͒.Again,this emphasizes the importance of the soil compressibility in the bearing response.In Fig.9,none of the analyses exhibits a clear ultimate state.The calculation for a 1m diameter foundation on dense calcare-ous sand was terminated at about 15.5%normalized displace-ment,at which stage the incremental displacement vectors were as shown in Fig.11.These indicate a significant component of surface heave adjacent to the footing,as in a classical rupture failure pattern.It may be concluded that the critical foundation size for the dense calcareous sand may be estimated as about 1m.020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement /Diameter,G /D (%)(a)020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement /Diameter,G /D (%)(b)Fig.7.Effect of foundation size for shallow circular footings on siliceous sand:͑a ͒dense;͑b ͒loose020406080100120140160180200M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDiameter,D (m)Fig.8.N ␥and D relationships for siliceous sandN ␥–D Relationship for Various SandsThe investigation of the effect of foundation size has also been conducted with respect to two other types of calcareous soils,namely Dogs Bay calcareous sand and Goodwyn calcareous silt.The MIT-S1model parameters for the sand and silt are tabulated in Table 1.Effective unit weights were set to 7kN /m 3for the Dogs Bay sand and 6kN /m 3for the Goodwyn silt.The N ␥values from the analyses for Dogs Bay sand are shown in Fig.12.The computed factor is high for small diameters,even in comparison with those for siliceous sand shown in Fig.8.This appears related to the higher values of ␾mr Јand np ͑i.e.,higher friction angles ͒and higher p ref Ј͑i.e.,higher stiffness ͒.The com-puted results are still much lower than the experimental results from Klotz and Coop ͑2001͒,although these are taken from the end-bearing resistance of jacked piles,extrapolated back to thesurface.The values reduce strongly with increasing diameter,but still lie above the computed values for the overlapping diameter range of 2–3m.Thus,although the general trends are similar for the experimental and numerical results,it is difficult to demon-strate complete consistency.The calcareous silt analyses are based on extremely low p ref Јand ␪values and lead to very low N ␥values even for small foundation sizes ͑see Fig.13͒.The N ␥values for loose samples ͑e 0=2.7͒,in particular,are essentially independent of the founda-tion size.Physical model results ͑Finnie and Randolph 1994͒lie between the numerical predictions of loose and dense states.The experimental data also revealed that the N ␥values for calcareous silt are insensitive to the foundation size.The N ␥–D curves for all the above-presented materials are compared in Fig.13.For small diameters,Dogs Bay sand has the highest bearing capacity,whereas the Goodwyn silt gives the low-est,although it should be noted that results for Toyoura sand are050100150M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement/Diameter,G /D (%)(a)050100150M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDisplacement/Diameter,G /D (%)(b)Fig.9.Effect of foundation size for shallow circular footings on calcareous sand:͑a ͒dense;͑b ͒loose020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDiameter,D (m)Fig.10.N ␥and D relationships for calcareous sandFig.11.Incremental displacement vectors for 1m diameter footing on dense calcareous sand ͑␦/D =15.5%͒not available at smaller diameters.The different trends of the calcareous materials are evident and result from changes in the compression parameters,p ref Јand ␪,which primarily control the bearing response for calcareous materials ͑see Table 3͒.It is also found that the N ␥values for different calcareous materials and densities reduce with increasing diameter and merge to a some-what uniform N ␥͑in the range 5–10͒,independent of the density,foundation size,and material type.On the other hand,the N ␥values for large foundations on siliceous sand are significantly larger than those for calcareous soils ͑Fig.14͒.LimitationsThe principal limitation of the analyses conducted is that the finite-element results for smaller diameter foundations on sili-ceous sand could not be obtained due to calculation instability.One possible reason is the highly dilative response of silica sand at low effective stress levels,in conjunction with extremely large deformations at the edge of footings during loading.Due to the rupture type of failure pattern,neighboring element immediately inside and outside the footing show downward and upward defor-mations,respectively,which led to termination of the solution due to the extremely high displacement gradient.The smaller the foundation size ͑so low effective stresses ͒,the more significant this issue became.By contrast,the failure mechanism for calcar-eous sands gave downward deformations just beyond the edge of the footings.0200400600800M o b i l i z e d B e a r i n g R e s i s t a n c e ,NJ =2q b /J 'DDiameter,D (m)Fig.12.N ␥and D relationships for Dogs Bay calcareous sand020406080100M ob i l i z e d B e a r i n g R e s i s t a nc e ,N J =2q b /J 'DDiameter,D (m)Fig.13.N ␥and D relationships for Goodwyn calcareous silt020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDiameter,D (m)(a)020406080100M o b i l i z e d B e a r i n g R e s i s t a n c e ,N J =2q b /J 'DDiameter,D (m)(b)Fig.14.N ␥and D relationships for different types of soils:͑a ͒dense;͑b ͒loose。