Theory and practice of the fall cone test

Koumoto, T. & Houlsby, G. T. (). GeÂotechnique 5, No. 8, 7±7 and practice of the fall cone test T. KOUMOTO and G. T. HOULSBY{ The fall cone is considered as a more reliable method for determining the liquid limit than the Casagrande method, and is standardised in many countries as the preferred liquid limit test method. In this paper the theory and practice of the fall cone test are described. First the penetration mechanism of a fall cone into clay is analysed, introducing the concept of dynamic strength to the static results. Next the applicability of dynamic analysis to the fall cone test is examined with regard to Hansbo's cone factor, K, for various cone angles. The theoretical K value for the 68 cone with a semi-rough surface is found to agree better with experimental results than is the case for the 38 cone. It is proposed that the liquid limit be rede ned internationally as the water content at which a 68, 6g fall cone penetrates mm. Finally the applicability of the fall cone test as a device to relate the strength of a clay with the index properties is examined. KEYWORDS: clays; laboratory tests; plasticity; shear strength; soil classi cation; standards. Pour deâterminer la limite liquide, la meâthode du coãne tombant est consideâreâe comme plus able que la meâthode de Casagrande. Elle est la norme dans de nombreux pays pour tester la limite liquide. Dans cet exposeâ, nous deâcrivons les aspects theâoriques et les aspects pratiques des essais au coãne tombant. D'abord nous analysons le meâcanisme de peâneâtration d'un coãne dans de l'argile, appliquant le concept de force dynamique aux reâsultats statiques. Ensuite, nous examinons l'applicabiliteâ de l'analyse dynamique au test du coãne tombant aá la lumieáre du facteur K du coãne d'hansbo pour divers angles du coãne. Nous avons trouveâ que la valeur theâorique K pour le coãne aá 68 avec surface semi rugueuse se rapproche davantage des reâsultats expeârimentaux que celle du coãne aá 38. Nous proposons de redeâ nir la limite liquide sur le plan international comme la teneur en eau aá laquelle un coãne de 68 et de 6 g peâneátre de mm. En n, nous examinons l'applicabiliteâ de l'essai au coãne tombant comme dispositif permettant d'eâtablir la relation entre la reâsistance de l'argile et les proprieâteâs indexeâes. INTRODUCTION The fall cone test was originally developed as a method for estimating the strength of remoulded cohesive soils in Scandinavia, and has become widely used as a standard method for determining the liquid limit of clays. In the past, several empirical estimates have been made of the strength at the liquid limit. Houlsby (98) made a dynamic analysis of the fall cone test, in which the single most important factor affecting strength at the liquid limit was found to be the cone roughness. He also observed that the predicted strengths were rather higher than those obtained from experimental estimates, and was unable to resolve these differences entirely. Fujikawa & Koumoto (98) made a static analysis of the fall cone test in which the theoretical values of cone factor, K (Hansbo, 957), were rather lower than experimental values. Koumoto (989) obtained improved agreement between theory and experiment by introducing the concept of dynamic strength into Houlsby's analysis, and reported that the calculated values of K agreed well with the fall cone test results. The purpose here is to draw these results together and resolve the differences between theory and experiment. An analysis of the fall cone test is presented, in which all the factors that are believed to affect the test are taken into account. Reasonable agreement between experimental and theoretical results is obtained. surface becomes distorted. The effects of the change of position of the soil surface are taken into account approximately. A study is made of the maximum value of the load on the cone for a given depth of penetration (see Fig. ). The penetration is assumed to take place suf ciently rapidly for undrained conditions to apply. It is assumed that the plastic deformations of the soil are suf ciently large for the soil to be modelled as a rigid±plastic cohesive material (with a Tresca yield surface). Since fall cone tests are usually carried out on remoulded material, no account is taken of the possibility of sensitivity: that is, there is no loss of undrained strength with shear strain. The geometry of the cone is described simply by two variables: the cone apex angle, â, and the depth of the penetration, h s (Fig. ). The vertical force exerted by the cone on the soil is Q, and the properties of the soil are given by the undrained shear strength, s u, and the bulk unit weight, ã. The surface properties of the cone are speci ed by an adhesion value a u ( < a u < s u ), which speci es the maximum allowable shear stress on the cone surface. The position of the cone is referred to a cylindrical set of coordinates (r, z) with the z-axis vertically downwards. A simple dimensional analysis shows that the load on the cone must be given by an expression of the form Q ãh s s u h ˆ f, â, á () s s u STATIC ANALYSIS As a preliminary to the analysis of the fall cone test, in which the cone is released and falls under its own weight, an analysis is carried out of a quasi-static penetration of the soil surface by the cone. In the following static analysis, the cone is assumed to penetrate a body of soil with an initially horizontal surface. As the cone penetrates, the soil is displaced so that the Manuscript received 4 December ; revised manuscript accepted 9 May. Discussion on this paper closes April, for further details see inside back cover. Saga University, Japan. { University of Oxford, United Kingdom. 7 where á is the ratio of adhesion to undrained shear strength, a u /s u. In the plastic analysis of cohesive materials with selfweight, if the soil surface is horizontal, then the additional resistance due to the self-weight is simply equal to the weight of the soil displaced. Equation () therefore becomes Q s u h ˆ f â, á ð tan = â= 3 s ãh s s u () Since the contribution of the self-weight of a typical soil to the cone resistance will be small (typically 5% for a 38, 8g code, and even less for 68 cones (Houlsby, 98), the effects of soil self-weight are ignored in the following calculations. The dimensionless static penetration resistance of a cone is

7 KOUMOTO AND HOULSBY C δ F E O A θ B r h s β D z Fig.. Schematic diagram of fall cone test therefore a function of apex angle of cone and the cone surface roughness. The load, Q, may alternatively be expressed in the terminology of bearing capacity theory as Q ˆ N c s u A ˆ N c s u ð tan â= h s (3) where N c is the cone bearing capacity factor. From equations () and (3): f â, á ˆ F ˆ ðn c tan â= (4) The problem can therefore be re-expressed in terms determination of the factor N c as a function of â and á. The in uence of soil displacement As the cone penetrates the soil, the clay is displaced so that the soil surface is no longer at. Plasticity calculations for the factor N c are much simpli ed if the soil heave is ignored, but this results in an underestimate of the cone resistance. A full analysis accounting for the heave is extremely complex (Lockett, 963), and an approximate method is used here instead. A section through the deformed soil surface is assumed to be a straight line from the cone surface down to a point on the original clay surface at the outer extremity of the plastically deforming region. The inclination of this line can be determined as a function of the extent of the plastically deforming region, by making use of the fact that the volume of heave is equal to the volume of the cone (Houlsby, 98). An iterative calculation is necessary since the extent of the plastic region depends in turn on the inclination of the surface. Bearing capacity factors in which heave is taken into account will be denoted by N ch. (a) (b) Analysis to determine N c and N ch The soil is assumed to be rigid±plastic, and deforms according to Tresca's yield condition with a maximum shear stress s u. The calculation is by the method of characteristics, making use of the Harr±von Karman assumption to allow the hoop stress to be determined. Details of the method used are given by Houlsby (98) and Houlsby & Wroth (98). The calculated slip-line nets for semi-rough cones of angle 38, 68 and 98 are shown in Fig.. In this gure the slip lines on the left side of the cone axis are for the case when heave is ignored, and those to the right are for the case when heave is taken into account. The bearing capacity is calculated by summing the appropriate components of the normal and shear tractions calculated on the cone surface. The values of N c, N ch (c) Fig.. Slip line elds for cones of angle (a) 38, (b) 68 and (c) 98 and ë ˆ N ch =N c calculated for different values of â and á are summarised in Table. Figure 3 shows N c and N ch values for smooth and fully rough cones as a function of cone angle, â. In Fig. 3 the N ch values for â ˆ 88 are calculated by multiplying the N c values by the coef cient

THEORY AND PRACTICE OF THE FALL CONE TEST 73 5 Table. Calculated values of Nch, ä and ë for range of values of â and á á â ˆ 38 â ˆ 68 â ˆ 98 â ˆ 8 â ˆ 58 Nch ä:deg ë Nch ä: deg ë Nch ä: deg ë Nch ä:deg ë Nch ä:deg ë 4 99 7 87 83 5 66 6 86 6 9 97 6 943 9 4 398 7 9 5 9 488 6 3 6 84 98 5 933 6 95 6 55 55 98 7 35 7 8 4 8 5 4 5 45 4 6 99 6 6 98 6 59 9 4 98 6 94 9 49 94 7 55 7 37 8 8 3 9 43 5 7 457 5 77 98 6 8 8 98 7 7 8 98 9 7 65 6 6 34 8 3 5 4 6 7 97 5 5 7 8 8 54 7 33 8 67 9 7 8 6 339 8 8 3 39 8 8 85 5 99 7 55 7 66 95 7 67 7 88 83 7 93 5 8 338 8 3 9 38 9 66 4 49 96 7 977 7 6 9 7 885 7 8 78 8 5 4 338 8 8 5 369 N c, N ch 5 3 6 9 5 8 β: deg c f ˆ c c N c Smooth cone Rough cone Fig. 3. Calculated bearing capacity factors for smooth and rough cones where c ˆ OB=OA in Fig. (see Appendix for the derivation of this factor). Both the N ch and N c values for rough cones are (as expected) greater than those for smooth cones, and N ch values are greater than N c values for any â value. The effect of heave Figure 4 shows the relation between the angle of the heaved surface, ä, and the cone angle, â. As can be seen, ä is zero for both â ˆ 8 and â ˆ 88, and the maximum values are :58 at â ˆ 738 in the case of a smooth cone and 7 58 at â ˆ 798 in the case of a rough cone. The effects of heave on the cone bearing capacity factor are shown in Fig. 5, in which ë ˆ N ch =N c is plotted against cone angle. The values of ë for smooth and rough cones are almost the same for â values up to about 88. Above that value, the effect for the smooth cone is somewhat greater than that for the rough cone. This can be explained by the shapes of the slip-line elds shown in Fig.. The extent of the slip-line eld for smooth cones increases continuously with an increase in the value of â, while for a rough cone the extent of the deforming region remains about constant for values of â greater than about 98. δ: deg 5 5. 5 at 73 7. 5 at 79 Rough cone (α ) 3 6 9 5 8 β: deg Smooth cone (α ) Fig. 4. Relationship between angle of heaved surface, ä, and cone angle N ch

74 KOUMOTO AND HOULSBY λ ( N ch /N c ). 6. 5. 4. 3. Smooth cone (α ) Rough cone (α ) values are found to lie in between the theoretical values for smooth and rough cones. The coef cient F ˆ 7: for the regression curve in Fig. 6 corresponds to a value 7: N ch ˆ ð tan 68= ˆ 6 :8 This corresponds closely to the value obtained for á ˆ :5 (see Table ), so that the roughness factor for these 68 cones appears to be about 5. Cones manufactured by different processes may of course have other roughnesses.. Values of cone surface roughness Figure 6 shows the results of tests of the static cone penetration resistance in remoulded clays for â ˆ 68. The physical properties of the clays are given in Table (which also includes information to be used later). As expected, the experimental Q/s u : mm 3 6 9 5 8 Fig. 5. Variation of ë 3 5 5 5 Experiments Clays LL IP A 43 M 68 33 B 38 93 Rough cone (α ) β: deg N ch =N c with cone angle 5 5 5 h s : mm Smooth cone (α ) Q/s u 7. h s (α. 5) α a u /s u Fig. 6. Comparison between measured and theoretical penetration of static cone DYNAMIC ANALYSIS In the above analysis the penetration of the cone was assumed to be quasi-static. To apply these results to the fall cone, the dynamics of the cone are taken into account. Houlsby (98) carried out such an analysis. In the following we introduce the possibility that the undrained strength of the clay may be a function of the strain rate, and at the high rates of strain in the fall cone test this may be signi cant. We use s ud to denote the undrained strength under dynamic conditions. The dynamic resistance of cone penetrated to a depth z is therefore Fs ud z, and the dynamic analysis of the fall cone test involves solution of the equation d z dt ˆ g Fs ud m z (5) where m is the mass of the fall cone assembly, g is gravitational acceleration, z is the penetration, and t is the time from the beginning of penetration. Noting that the results of the static penetration give Q ˆ mg ˆ Fs u h s, equation (5) may be rewritten as! d z dt ˆ í dí dz ˆ g s udz s u h (6) s where v is the velocity of the cone. Equation (6) may be integrated to give v! u í ˆ t gz s udz 3s u h (7) s if v ˆ atz p ˆ. The cone p therefore comes to rest (v ˆ ) at z ˆ h ˆ h s 3s u =s ud ˆ h s 3æ, where æ ˆ su =s ud (typically æ will be a factor less than p ). The dynamic cone penetration depth, h, is therefore 3æ times the static cone penetration depth, h s. Table. Physical properties of clays Sample r s :kg=m 3 Consistency limits Casagrande method Fall cone method w L :% w P :% I P :% w L :% w P :% I P :% Yagusa clay Y 65 66 7 8 6 37 7 69 6 43 Ariake clay A 58 5 68 8 5 69 Ariake clay 6 3 9 5 79 9 7 48 79 Ariake clay A3 58 4 54 86 39 53 86 Clay mixture{ M 67 74 36 38 76 34 4 Clay mixture{ M 65 78 4 6 37 4 68 35 33 Clay mixture} M3 66 8 5 9 75 49 6 Clay mixture} 65 9 5 35 6 54 9 3 69 Bentonite B 65 334 43 3 9 9 95 39 56 Bentonite B 74 4 35 6 365 4 35 9 3 Fall cone method uses 68, 6g cone (wl is water content at h ˆ :5 mm, w P is water content at h ˆ :5 mm) { M mixture is 8 6% A3, 8 4% sand { M mixture is 5% A, 5% B } M3 mixture is 5% A3, 5% B } mixture is 66 3% B, 33 3% Yagusa clay

THEORY AND PRACTICE OF THE FALL CONE TEST 75 The fall cone factor, K standard triaxial test. Extrapolating to higher strain rates, the Combining the above results, the undrained shear strength (at above expression gives s u =s u(%=h) of 6 at _ã ˆ : 3 low strain rates) may be expressed as a function of the fall cone 6 %=h (for the 38 cone) and 64 at _ã ˆ :5 3 6 %=h (for penetration, h, as 68 and 98 cones). If standard triaxial tests are adopted as the s u ˆ 3Qæ Fh ˆ 3Qæ ðn ch tan â= h ˆ KQ benchmark for comparison of undrained strength values, æ can h (8) be estimated as follows: where K is the fall cone factor as de ned by Hansbo (957). æ :9 :6 :74 for the 38 cone (a) The above equation results in 3æ æ :9 K ˆ ðn ch tan :64 :73 for 68 and 98 cones (b) â= Koumoto (989) carried out quasi-static penetration tests (at a rate of mm/s) and standard fall cone tests on similar clay samples. Accounting for the estimated rate of strain in his quasi-static tests would lead to expected values of æ of 74 for his tests with 38 cones, and 76 for the tests with 68 and 98 cones. In fact, from the measured ratios of the static cone penetration depth, h s, to the fall cone penetration depth, h, the experimentally deduced value of æ was 7 for all three cone angles. The indication is that strain rate effects are in fact slightly higher (by 4±7% than those indicated by the extrapolation in Fig. 7. Considering that an extrapolation was required to strain rates about two orders of magnitude higher than in any of the triaxial tests, the agreement is remarkably good. Determination of æ To obtain the K values, æ must be known, and to make a reasonable estimate of this factor the rate of shear strain during the fall cone test is in turn needed. The average shear strain rate during penetration, _ã, is estimated by the following equation (see Appendix ): :67 3 6 ä deg _ã %=h p (9) ä deg =45 h mm where ä is the angle of the heaved surface of the clay (in degrees). For the range of penetration depth measured in practice in the fall cone test, a typical strain rate value would be : 3 6 %=h for the 38 cone (:89 3 6 %=h to:5 3 6 %=h for h in the range 5±5 mm) and about :5 3 6 %=h for a 68 or 98 cone (:94 3 6 %=h to3:37 3 6 %=h for h in the range 5±5 mm). These values are approximate only. A relationship between the normalised strength and the rate of shear strain is summarised in Fig. 7, from results published by Berre & Bjerrum (973), Vaid & Campanella (977) and Lefebvre & Leboeuf (987). The line shown on the gure is given by s u s u %=h ˆ : : log _ã () where _ã is in %=h. This line ts the data quite well, and corresponds to the commonly used rule of thumb that the undrained strength increases by about % for every tenfold increase in strain rate. By interpolation, s u =s u(%=h) is 9 at _ã ˆ 79%=h, which corresponds approximately to the strain rate for a typical s u /s u(%/h). 5. 5 Standard triaxial test. 9 at 79%/h Fall cone test (6 and 9 cone). 64 at. 5 6 %/h. 6 at. 6 %/h (3 cone) COMPARISON OF THEORETICAL AND EXPERIMENTAL VALUES OF K The fall cone factors, K, calculated using the above values of æ for different values of cone angle are shown in Table 3, and compared in Fig. 8 with experimentally determined values of K (Hansbo, 957; Karlsson, 96; Wood, 98, 985). Note that, as implicitly accepted in the later papers, it appears that Karlsson's gures were reported as too low by a factor of. In Fig. 8 the values of K calculated for smooth and rough cones are shown for comparison. From Fig. 8, the theoretical K values show good agreement with the experiments for â. 458, and especially for â ˆ 68. Leroueil & Le Bihan (996) report that the ratio of penetration of the 68, 6g to that of the 38, 8g cone is on average 5, and this implies that the K value for the 38 cone is exactly K. 8. 6. 4.. 8. 6 α.. 5. Experiments Hansbo l Karlsson Wood 98 Wood 985. 5 y.. log x Triaxial test Berre & Bjerrum Vaid & Campanella Lefebre & LeBoeuf. 4. 3 3 4 5 6 7 8. γ: %/h Fig. 7. Variation of normalised shear strength with strain rate 5 3 45 6 75 9 β: deg Fig. 8. Variation of fall cone factor, K, with cone angle: theory and experiments Table 3. Calculated values of K for different values of â â 38 458 68 758 98 Smooth á ˆ : 835 4 6 Semi-rough á ˆ :5 33 58 35 7 97 Rough á ˆ : 3 495 5 5 9

76 KOUMOTO AND HOULSBY three times that for the 68cone. This is entirely consistent with the experimental data in Fig. 8. The theoretical values, however, give a ratio of :33=:35 4:4. The discrepancy between theory and measured values lies almost entirely with the 38 cone. Furthermore, the difference between the theoretical K values for smooth and rough 38 cones is comparatively large, so that the cone surface roughness of a 38 cone needs to be known reasonably accurately. Given that (a) there is better agreement between theory and experiment and (b) the 68 cone is less sensitive to the surface roughness, the 68 cone is expected to provide a more reliable indication of the strength of a clay in the fall cone test. The difference between experiment and theory for the 38 cone has not been resolved completely. Contributing factors probably include the following: (a) (b) (c) The buoyant effect of the self-weight of the clay. This can account, however, for only about a 5% decrease in the apparent K value. The shape of the deformed free surface. When the effect of soil heave is (approximately) introduced, the effect is to decrease the K value by about % from the values obtained ignoring heave. It may be that accounting for the shape of the free surface more rigorously would further decrease the value of K. The strain rates have been estimated simply from a very approximate global estimate of the rates. It may be that, if a more precise estimate of the strain rates could be made (accounting for variation with both time and position), then a lower value of K would be obtained. It is unlikely, however, that this would make a very large difference. Undrained shear strength at the liquid limit Table 4 shows national standards adopted for the fall cone liquid limit test in various countries, together with some published methods that are not adopted as national standards. The theoretical K values given above, and the implied undrained shear strengths at the liquid limit s ul, are also tabulated. Unfortunately, the theoretical s ul value varies from 38 kpa to 4 5 kpa. Wroth & Wood (978) suggest a strength of s ul ˆ :7 kpa from the synthesis of several sources of data, and indicate a possible range of 7± 65 kpa. Theoretical s ul values are all within between this range except for the Indian and Chinese standards (see Table 4). THE FALL CONE LIQUID LIMIT The relationship between w and s u Typical results for undrained shear strength, s u, obtained by vane tests for remoulded clays are shown in Fig. 9 where s u has been made dimensionless by dividing by atmospheric pressure w: % 45 4 35 3 5 5 5 Clays Y A M B Hor Lon Gos She LL 69 9 68 38 3 73 8 97 IP 43 69 87 33 69 93 4 48 5 65. 4... 4... 4 s u /p a Fig. 9. Log±linear relationship between undrained strength and water content Table 4. Standards for the fall cone liquid limit test in different countries Country and/or publication Cone details Container details Penetration Penetration at liquid Fall cone factor, Undrained strength at time: s limit, hl: mm K (theory) liquid limit, Mass: g Length: mm Diameter: mm Depth: mm ˆ KQ=h : sul L : kpa (theory) Apex angle, â: deg Sweden, Norway 6 6. 6 or 5 3 or 5 35 83 Canada 6 6 5 >5 5 35 83 Japan 6 6 > >6 >3 5 5 35 38 India 3 48 3 48??? 5 4 4 84 UK 3 8 35 >55 4 5 33 66 New Zealand 3 8 35 55 >4 5 33 66 France (Le aive, 97) 3 8 35??? 7 33 3 68 China 3 76 5 5 3 5 7 33 3 5 USSR{ 3 76 5?? 5 595 4 5 Georgia. Institute of Technology{ 3 75???? 595 4 46 { The USSR and Georgia Institute of Technology cones are not fall cones; instead the cone is lowered slowly until the weight is carried by the soil. The K factors are calculated on the basis of ignoring dynamics and rate effects.

THEORY AND PRACTICE OF THE FALL CONE TEST 77 Table 5. Coef cients in equation () Clay a b r Y 7 6 997 A 6 5 56 994 39 4 4 99 A3 9 54 36 98 M 44 34 7 M 39 97 34 999 M3 38 4 336 99 38 8 4 998 B 3 45 535 97 B 3 7 56 995 P a. The physical properties of these clays are shown in Table 5. As can be seen in Fig. 9, the w log s u relationship is not linear, but curves over the whole range of water content, which varies from higher than the liquid limit to near the plastic limit. This has already been observed by Karlsson (96) and Youssef et al. (965). Fujikawa & Koumoto (98) and Koumoto (989, 99) suggest that the log w log s u relationship is more nearly linear for a wide range of water content. Consequently, the w s u relation may be expressed as w ˆ as u = p a ) b () where a and b are coef cients that vary with the type of the clay. The values of a and b obtained by regression analysis are shown together with the regression coef cient, r, in Table 5. Note that the values of r are extremely high. In Fig. the linear equations obtained above are shown by broken lines. s u : kpa 4 4 s u /p a (83/p a )/h. 4.. 4 h: mm Clays LL IP Y 69 43 B 7 79 69 35 3 Fig.. Relationship between s u values obtained by vane tests and h w: % 4 4 Clays Y B LL 69 7 35 IP 43 79 69 3 The relationship between w and h The relationship between w and cone penetration, h, can be obtained by combining equations (8) and () to give KQ w ˆ a bˆ p a h Ah B (3) where A ˆ a(kq= p a ) b and B ˆ b. For the 68, 6g cone K is equal to 35, and A ˆ a( 83) b. Figure shows the relationship between s u values obtained by vane tests and h. Equation (8) is shown by a solid line, and is in very good agreement with experimental results, so that this equation could be used to estimate s u with reasonable accuracy from measurements of h. Equation (3) shows that the w h relationship is linear on a double logarithmic scale. In Fig. the linear log w log h equations obtained by regression analysis for each clay are shown by broken lines (for each clay, the correlation coef cient w: % 4 w a(s u /p a ) b Clays Y A M B Hor Lon Gos She LL 69 9 68 38 3 73 8 97 IP 43 69 87 33 69 93 4 48 5 65 w Ah B 4 4 h: mm Fig.. Log±log relationship between penetration depth and water content r is greater than 9). A series of fall cone tests at different water contents of a clay can be used to give the coef cients A and B, and hence also the coef cients a and b in equation (). The value of h L at the liquid limit Details of fall cone tests, as adopted in various countries as the liquid limit test, are summarised in Table 4. For the 68, 6g cone, the penetration at liquid limit h L ˆ mm is usually adopted. However, if the Casagrande method is taken as de ning the liquid limit, then for clays of rather high liquid limit (w L ˆ 7±35%) h L increases with w L, and varies between mm and 4 mm, as shown in Fig. 3. Kumapley & Boakye 5 4 h L : mm h L,average. mm 5. 4... 4... 4 3 4 5 s u /p a Fig.. Log±log relationship between undrained strength and water content Fig. 3. Variation of h L with w L w L : %

78 KOUMOTO AND HOULSBY (98) reported that for a 68, 6g cone, h L varied between mm and mm for soils with w L < 9%. The Japanese Geotechnical Society () has adopted h L ˆ :5 mm as the Japanese Industrial Standard. The intention is that fall cone tests should be applicable for all clays with w L < 6%. A series of tests has been carried out on a number of Japanese clays. The water contents at h L ˆ :5 mm on the log w log h plots are taken as the liquid limit and are shown in Table. As seen in this table, the difference between the Casagrande liquid limits and the fall cone liquid limit (at h L ˆ :5 mm) is less than 4% for all clays with w L < 6%. The clays tested with higher liquid limits are all Bentonite clays. h P : mm h P,average. 3 mm 4 6 Strength at the fall cone liquid limit Using a 68, 6g cone, the undrained shear strength at liquid limit s ul is (using the theoretical results) 83 kpa for h L ˆ mm and 38 kpa for h L ˆ :5 mm. Fig. 4 shows the variation of undrained shear strength at liquid limit with the plasticity index. In the gure the above s ul values are shown as a dotted broken line and a broken line respectively. The use of h L ˆ 5 mm as the penetration at liquid limit is seen to give a good estimate of s ul. The strength of 38 kpa is smaller than the ±3 kpa suggested by Casagrande (958), 5± kpa by Karlsson (96), 7 kpa by Wroth & Wood (978) and 6 kpa by Whyte(98), but is in the range of the values of 7± 65 kpa suggested by Wroth & Wood (978). THE FALL CONE TO MEASURE THE PLASTIC LIMIT The value of h P at the plastic limit At present the liquid limit and the plastic limit of soils are determined by two completely different methods. Wood & Wroth (978) tried to determine the plastic limit indirectly from the fall cone tests. If fall cone tests were to be available for the plastic limit as well as the liquid limit, then the two tests might become easier, more useful and more meaningful mechanically. If the fall cone test were to be used to determine the plastic limit, one of the merits of plotting the data on the log w log h diagram would be to allow extrapolation to the penetration at the plastic limit. Fig. 5 shows the relationship between the extrapolated h P values at the plastic limit on the log w log h diagram and the plastic limit itself. From Fig. 5, the average h P value is 3 mm for the 6g, 68 cone. If a ratio of was assumed between the strengths at the liquid and plastic limits (as suggested by Skempton & Northey, (953), and numerous subsequent authors), then a penetration of about 5 mm (one tenth of h L ˆ :5 mm) would be expected. It is suggested therefore that the plastic limit for clays could be de ned by extrapolation of the log w log h diagram to the point at which the penetration is one tenth of that at the liquid limit. This would establish a rm mechanical basis for the ratio 3. 5 s ul.83 kpa (h L mm) Fig. 5. Variation of h P with w P of between the strengths of the two limits. The clear disadvantage of this approach is that it involves extrapolation. This could be circumvented by either of two means: (a) the use of a heavier cone for a new plastic limit test, which would involve rather larger penetrations; or (b) the de nition of a new index value at, say, a strength that is only a factor of higher than that at the liquid limit. Undrained strength at fall cone plastic limit By rede ning the plastic limit as, for instance, the water content at h P ˆ :5 mm for a 68, 6g cone, the value of s up given as times the s ul value: that is, 38 kpa from equation (8). Fig. 6 shows the undrained shear strength at this newly rede ned plastic limit. However, this requires extrapolation on the plots already employed (Figs ±Fig. ) as the very small penetration is rarely attained. In Fig. 6, the average s up value for the h P,average is 8 kpa, rather less than the value implied by the analysis given above. INDEX PROPERTIES It is usual to express the form of the critical-state line for a clay in the form v ˆ à ë ln p9= p a where v is the speci c volume, and this implies a linear relationship between the water content and the logarithm of pressure, and implicitly therefore also with the logarithm of the undrained strength. Butter eld (979), however, suggested that the form ln v ˆ ln à ë ln p9= p a provided a better t to the data. This form is particularly appropriate for remoulded plastic clays with a wide range of 5 w P : % s up /P a 38 (h P. 5 mm) s ul : kpa. 5. 5 s ul.38 kpa (h L. 5 mm) s up : Pa 5 s up /P a 8 (h P,average. 3 mm) 5 5 5 3 35 4 5 5 5 3 35 4 I P I P Fig. 4. Variation of s ul with I P Fig. 6. Variation of s up with I P

water contents. A minor alteration to Butter eld's expression is the use of ln e ˆ ln à ë ln p9= p a where e is the voids ratio, and this form proves to be convenient in the following. This expression can also be written e ˆ à p9= p a ë Since the critical-state line is given by q ˆ Mp9, and undrained shear strength is half the deviator stress at failure, s u ˆ q=, then we can obtain ë s ë e ˆ à u M p a For saturated clays e ˆ wg s =, where G s is the speci c gravity of the soil particles and w is in %. Substituting also the expression used above for the undrained strength in terms of fall cone penetration, we obtain w ˆ à KQ ë h ë p a M G s Comparing this expression with equation (3) gives a ˆ à ë, b ˆ ë, A ˆ à KQ G s M G s p a M ë and B ˆ ë Thus the tting parameters a, b, A and B are closely related to the engineering properties of the soil, and in particular to the compressibility. THEORY AND PRACTICE OF THE FALL CONE TEST 79 Intercept a measured 8 7 6 5 4 3 Koumoto Skempton & Northey 3 4 5 6 7 8 Intercept a predicted comparison of measured (shown in Table 5) and predicted values of a by the above expression. As can be seen, measured and predicted values are again in very good agreement. : w a(s u /Pa) b a w L (. 36) b b I PN /4. 6 Fig. 8. Comparison of measured and predicted a values Determination of the value of a and b From equation (), w L and w P are given as w L ˆ as ul b ˆ a:38 b and w P ˆ as up b ˆ a:38 b (where the units are in kpa). The form of these expressions suggests that it would be convenient to de ne a modi ed plasticity index, I PN, as I PN ˆ ln w L ln w P (cf. I P ˆ w L w P ). Substituting the above gives I PN ˆ b ln or b ˆ I PN =4:6. The value of a is given as a(kpa) ˆ :38 b w L. Figure 7 shows a comparison of measured values of b (as given in Table 5) and those predicted from the measured ë values, and shows that these are consistent. Fig. 8 shows a. 8 Undrained strength from fall cone tests Combining equations (8), () and (3) gives w ˆ :38 IPN=4: 6 w L s IPN=4: 6 u (4) or s u ˆ :38w 4: 6=I PN L w 4: 6=I PN (5) The undrained shear strength, s u, of remoulded clay should be generally expressed as a function of type and condition of clay. Equation (5) is consistent with this notion, since it is expressed as a function of w L and I PN (representing types of clay), and w (representing the condition of the clay). Figure 9 shows a comparison of measured values of s u with those predicted by equation (5). Measured values of s u are in very good agreement with the predicted ones. Exponent b measured. 6. 4 Koumoto Skempton & Northey : s u : kpa, measured 4 4 Clays LL IP Y 69 43 B 7 69 69 35 3 :. w a(s u /Pa) b b I PN /4. 6. 4.. 4. 6. 8 Exponent b predicted Fig. 7. Comparison of measured and predicted b values... 4 4 4 s u : kpa, predicted Fig. 9. Comparison of predicted and measured values of s u

7 KOUMOTO AND HOULSBY The relationship between I LN and s u The relationship between w and s u is sometimes plotted on the liquidity index diagram of I L against ln s u. Fig. shows the relation between I L and ln s u for remoulded clays. As can be seen, the I L ±lns u relation for different clays gives different curves, as Wasti & Bezirci (985) also report for a variety of soils. Consistent with the above discussions of the variation of strength with water content, it would be appropriate to de ne a new liquidity index, I LN,as I LN ˆ ln w ln w P = ln w L ln w P ˆ ln w ln w P =I PN (6) and substituting equations (4) gives I LN ˆ ln s u ln s up = ln s ul ln s up ˆ :7 :7 ln s u (7) or s u ˆ exp :7 I LN =:7Š (8) Figure shows the relationship between I LN and ln s u for remoulded clays. In Fig. the theoretical equation (7) is shown as a solid line. As can be seen, the experiments are in fairly good agreement with theory. Equation (8) can be used to predict the undrained strength I L. 4.. 8. 6. 4. Clays LL IP Y 69 43 A M B 69 7 79 68 33 69 35 3.. 4 4 4 s u : Pa Fig.. Log±linear relationship between undrained strength and liquidity index I LN. 6. 4.. 8. 6. 4. I LN. 7. 7ln Su/kPa Clays LL IP Y 69 43 A M B 69 7 79 68 33 69 35 3.. 4 4 4 Su: kpa Fig.. Log±log relationship between undrained strength and newly de ned liquidity index for a remoulded clay from the value of the newly de ned liquidity index, I LN. STANDARDISATION OF THE FALL CONE TEST It is unfortunate that so many different cones have been adopted worldwide (see Table 4). Such a proliferation is clearly unhelpful if the fall cone is to be used as a standard test. For any one country to change its standards is a process fraught with dif culties, some of which are not purely related to technical matters. On the basis of the work reported here, we feel compelled, however, to present the scienti c case for certain changes. The views expressed are our own and represent no interest of outside bodies. There are two widely used cones: the 68, 6g cone and the 38, 8g cone. Of these we recommend the 68, 6g cone on the basis that: (a) the theoretical understanding of this cone is in much better accord with the experimental results than is the case for the 38 cone. This agreement gives added con dence to the use of the results. (b) the results of the 68 cone tests are less sensitive to cone roughness than are those from the 38 cone. This sensitivity is thought to be a major contributor to the variability of 38 one test results. (c) the blunter tip of the 68 cone is, presumably, less prone to wear. Against these arguments it must be acknowledged that the larger penetration depths for the 38 cone mean that the depth can be determined with (proportionally) more accuracy. The authors consider, however, that on balance the 68 cone is superior, and would strongly recommend that those countries (including of course the UK) currently using the 38 cone should consider transferring to the 68 cone as standard. The second issue to be resolved is the depth of penetration to be adopted for the liquid limit. There seems little doubt that the fall cone is a more reliable device for measuring the properties of clays than is the Casagrande apparatus. The process of benchmarking the fall cone against the Casagrande apparatus is therefore scienti cally an indefensible procedure, and can only be justi ed on historical grounds. It is clear that the penetration depth at the (Casagrande) liquid limit increases marginally with liquid limit. Those countries concerned mainly with low to medium plasticity clays have typically adopted a penetration depth of mm. Those in which the test is also to be applied to very high plasticity clays prefer a larger value (e.g. 5 mm in Japan). Such differences are highly undesirable. On the grounds of (a) simplicity and (b) historical precedent we strongly recommend the adoption of mm as standard: The liquid limit should be internationally rede ned as the water content at which a 68, 6g fall cone penetrates mm. Consistent with the well-accepted observation that the strength at the plastic limit is about times that at the liquid limit, then it might seem logical to recommend: that the plastic limit should be internationally rede ned as the water content at which a 68, 6g fall cone would penetrate mm. Such a recommendation poses a practical problem in that, since measurement of a penetration of mm is clearly unrealistic, then serious consideration needs to be given to (a) the need for extrapolation to the plastic limit from higher water contents, and the limits that need to be set for this, and (b) whether the use of a heavier cone should be adopted for the plastic limit test or (c) whether it would be more appropriate to introduce a new limit corresponding to a higher water content. Options (a) and (c) have the advantage that they avoid the problem of compacting a homogeneous sample of clay at near the plastic limit into a suitable container: the friability of samples near the plastic limit makes this rather dif cult. Further work is clearly required, but nevertheless we recommend that the plastic limit should be rede ned in terms of a fall cone test, probably involving a heavier weight and a smaller penetration than for

the liquid limit. For instance, a 5 mm penetration of a 5g cone would be a possible combination. CONCLUSIONS The fall cone test is a simple and quick method for determining the undrained shear strength, which can be interpreted in terms of simple mechanics. Factors affecting the fall cone penetration are the angle of the cone tip, the cone surface roughness, and the rate of shear strain during penetration. The theoretical fall cone factor K ˆ 3 calculated here for the 68 cone agrees well with published experimental values, while the K factor for the 38 cone is slightly greater than the value derived from experimental data. The 68 cone is recommended as more suitable than the 38 cone to determine strength, because it is less sensitive to the cone roughness. The theoretical undrained shear strengths at the liquid limit are consistent with those determined experimentally. The relationship between water content, w, and undrained shear strength, s u, for highly plastic remoulded clay is approximately linear on a double logarithmic scale over the whole range of water content from above the liquid limit to near the plastic limit. Consequently the relationship between w and cone penetration depth, h, is also linear on the double logarithmic plot. The fall cone liquid limit, w L, can be determined from the log w log h line as the water content at h ˆ mm for a 68, 6g cone. The plastic limit could be de ned in a similar way, but a suitable combination of cone mass and penetration remains to be determined. From the h value for a test on a 68, 6g cone the s u value for remoulded clay can be estimated by the relation s u (kpa) ˆ 83=h (h in mm). The implied s u values at the liquid and plastic limits are 83 kpa and 83 kpa respectively. The intercept A and the exponent B of the log w log h line are related to the parameters à and ë, which de ne the position of the critical state line. By rede ning the plasticity index as I PN ˆ ln w L ln w P and the liquidity index as I LN ˆ (ln w ln w P )=I PN, the I LN ln s u relationship is given by the equation I LN ˆ :7 :7 ln s u. ACKNOWLEDGEMENTS The research described in this paper was supported partly by the Japanese Ministry of Education and conducted at the Department of Engineering Science, the University of Oxford. The rst author is grateful to Saga University, which granted him the leave for his research. APPENDIX : DETERMINATION OF THE COEFFICIENT f Examining the right hand side of Fig., the volumes shown in section as the two shaded triangles may be equated (making use of Pappas's theorem): 3 å ˆ 3 [ b 3 å a Š 3 a b (9a) Fig.. Limiting case of heave solution as cone angle approaches 88 THEORY AND PRACTICE OF THE FALL CONE TEST 7 or ˆ a b a b (9b) When å is a very small angle, the in uence of the slightly sloping surface on the geometry of the failure mechanism will become in nitesimal, so that c (shown on the left-hand side of Fig. ), will be equal to the ratio b=a. Substituting this into equation (9a) leads to the solution a ˆ c c () c (The above solution is strictly applicable only for a. :5, but this always proves to be the case for the problems examined here.) The coef cient f is simply the ratio of the area of the cone in contact with the soil when heave is taken into account to that when heave is not taken into account: thus f ˆ a. APPENDIX : ESTIMATION OF THE RATE OF SHEAR STRAIN DURING FALL CONE PENETRATION Examining the right-hand side of Fig., the shear strain of the soil, ã, may be estimated from the change of the angle è ˆ ð=4 in the undeformed state to è ä in the deformed state. Since ã represents the change of an angle that is originally ð=, an estimate of the shear strain is therefore ã ˆ ä. This is clearly no better than an approximate estimate of the order of magnitude of the strain undergone by the soil, since the actual strain pattern will be rather complex. An estimate of the rate of shear strain, _ã, is therefore _ã ˆ ä=t, where t is time for the cone penetration time, which is given by Houlsby (98) as s s h s h t ˆ :44 ˆ :44 p g g 3 Thus: s _ã ˆ ä p g 3 (a) :44 h Since it is convenient to specify ä in degrees, h in millimetres and _ã in percent per hour, equation (a) becomes s p ä(deg) 3 (ð=8) 9:8 3 _ã(%=h) ˆ 3 36 3 :44 h(mm) 3 3 ˆ :67 3 6 ä(deg) p h(mm) (b) NOTATION a u adhesion on cone surface F non-dimensional cone resistance factor g gravitational acceleration h dynamic penetration depth of cone h s static penetration depth of cone m mass of cone N c cone bearing capacity factor (not accounting for heave around cone) N ch cone bearing capacity factor (accounting for heave around cone) P a atmospheric pressure ( kpa) Q cone resistance r, z coordinates s mean in-plane stress s u undrained shear strength s ud dynamic undrained shear strength t time w water content v velocity of cone á a u =s u â cone apex angle ã unit weight of saturated clay ä inclination of heaved surface å (ð â)= æ s u =s ud è direction angle of s slip line to the r-axis ë N ch =N c ã average shear strain in deforming soil around cone _ã average shear strain rate in deforming soil around cone

7 KOUMOTO AND HOULSBY REFERENCES Berre, T. & Bjerrum, L. (973). Shear strength of normally consolidated clays. Proc. 8th ICSMFE, Moscow (), 39±49. Butter eld, R. (979). A natural compression law for soils (an advance on e±log p9). GeÂotechnique 9, No. 4, 469±48. Casagrande, A. (958). Notes on the design of the liquid limit device. GeÂotechnique 8, No., 84±9. Fujikawa, T. & Koumoto, T. (98). Three-dimensional analysis of the penetration of fall cone. J. Jpn Soc. Irrigation, Drainage and Reclamation Engng 83, 38±43 (in Japanese). Hansbo, S. (957). A new approach to the determination of the shear strength of clays by the fall-cone test. Proc. Roy. SGI 4, 7±48. Houlsby, G. T. (98). Theoretical analysis of the fall cone test. GeÂotechnique 3, No., ±8. Houlsby, G. T. & Wroth, C. P. (98). Direct solution of plasticity problems in soils by the method of characteristics. Proc. 4th Int. Conf. Numer. Methods Geomech., Edmonton 3, 59±7. Japanese Geotechnical Society (). The Japanese Geotechnical Society Standards: Test method for liquid limit of soils by the fall cone, JGS 4- (in Japanese). Karlsson, R. (96). Suggested improvements in the liquid limit test, with reference to ow properties of remoulded clays. Proc. 5th Int. Conf. Soil Mech. Found. Engng, Paris, 7±84. Koumoto, T. (989). Dynamic analysis of the fall cone test. J. Jpn Soc. Irrigation, Drainage and Reclamation Engng 44, 5±56 (in Japanese). Koumoto, T. (99). Determination of both liquid and plastic limits of clay by the fall cone test. J. Jpn Soc. Irrigation, Drainage and Reclamation Engng 46, 95± (in Japanese). Kumapley, N. K. & Boakye, S. Y. (98). The use of cone penetrometers for the determination of the liquid limits of soils of ow plasticity. Proc. 7th Regional Conf. for Africa on Soil Mech. Found. Engng, 67±7. Lefebvre, G. & LeBoeuf, D. (987). Rate effects and cyclic loading of sensitive clays. J. Geotech. Engng, ASCE 3, No. 5, 476±489. Le aive, E. (97) Les limites d'atterberg et le penetrometer aá cone. Bulletin de Liaison des Laboratoires des Ponts et ChausseÂes, France, No. 5, pp. 3±3. Leroueil, S. & Le Bihan, J.-P. (996) Liquid Limits and Fall Cones, Canadian Geotechnical Journal, 33, No. 5, October, 793±798. Lockett, F. J. (963). Indentation of a rigid/plastic material by a conical indenter. J. Mech. Phys. Soils, 345±355. Skempton, A. W. & Northey, R. D. (953). The sensitivity of clays. GeÂotechnique 3, No., 3±53. Vaid, Y. P. & Campanella, R. G. (977). Time-dependent behavior of undisturbed clay. J. Geotech. Engng Div., ASCE 3, No. GT7, 693±79. Wasti, Y. & Bezirci, M. H. (985). Determination of the consistency limits of soils by the fall cone test. Can. Geotech. J. 3, 4±46. Whyte, I. L. (98). Soil plasticity and strength: a new approach using extrusion. Ground Engng 5, No., 6±4. Wood, D. & Wroth, C. P. (978). The use of the cone penetrometer to determine the plastic limit of soils. Ground Engng, No. 3, 37. Wood, D. M. (98). Cone penetrometer and liquid limit. GeÂotechnique 3, No., 5±57. Wood, D. M. (985). Some fall cone tests. GeÂotechnique 35, No., 64±68. Wroth, C. P. and Wood, D. M. (978). The correlation of index properties with some basic engineering properties of soils. Can. Geotech. J. 5, No., 37±45. Youssef, M. S., El Ramli, A. H. and El Demery, M. (965). Relationships between shear strength, consolidation, liquid limit and plastic limit for remoulded clays. Proc. 6th Int. Conf. Soil Mech. Found. Engng, Montreal, 6±9.