FMfiURE chumom OF A SATURA'FED REMGULDED CLAN Thesis fat the MM cf Fifi. $. MlCHEGAN STATE UNNERfiiW Abraham Kwan-Xuen Lek 1963 This is to certify that the thesis entitled "Failure Conditions of 8 Saturated Remoulded Clay" presented by Ab raham Kwan-Yuen Loh has been accepted towards fulfillment of the requirements for M.S . degree in Civil Engineering @4554 it, Major professor pm [cf/h my 0-169 LIBRARY Michigan Sm: University ABSTRACT FAILURE CONDITIONS OF A SATURATED REMOULDED CLAY by Abraham Kwan-Yuen Loh An experimental study of the failure conditions under various states of applied stress has been made on a saturated remoulded clay. The purpose of the study was to examine the behavior of a clay under different stress states and the effect of stress changes on the shear strength. In standard triaxial tests, the states of uniaxial or biaxial com- pression, superimposed on a hydrostatic stress state were examined. The effect of the intermediate principal stress on the yielding of clays was studied by using a hollow cylindrical sample, subjected to com- pression under various intermediate stress states. All tests were of the consolidated-undrained type. A Glacial Lake clay from Sault Ste. Marie, Michigan was used for the experiments. It is shown that the Mohr effective stress failure envelopes are independent of the loading paths for each type of test, and that the intermediate principal stress has only a minor effect on the strength parameters of the clay. It is also shown that the stress-strain and pore-pressure change characteristics are dependent on the stress paths for each type of test. Abraham Kwan- Yuen Loh On the assumption that the material is isotropic and homogeneous, the test results have been used to compute a failure surface in the effective principal stress space. Results are compared with the generalized Coulomb's criterion. Good agreement is obtained, in spite of some minor deviations. Much experimental work remains to be carried out before the correctness of Coulomb's criterion can be established. FAILURE CONDITIONS OF A SATURATED REMOULDED CLAY BY ABRAHAM KWAN- YUEN LOH A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1961 ii ACKNOWLEDGEMENTS The author is indebted to Dr. T. H. Wu, Department of Civil Engineering, without whose guidance and encouragement this thesis would not have been possible; he also wishes to express his sincere gratitude to Dr. L. E. Malvern, Department of Applied Mechanics, for his constructive comments and assistance, and to Dr. 0. B. Andersland, Department of Civil Engineering, for his valuable sug- gestions. To his colleagues A. G. Douglas and R. D. Goughnour, the author wishes to express his sincere thanks for their criticism and assistance in the laboratory. CONTENTS 2. l. Shearing Resistance of a Cohesive Soil 2. 2. Mohr's Theory of Rupture 2. 3. Yield Surface in Principal Stress Space EXPERIMENTAL PROGRAM ......................... 3. 1. Materials Used 3. 2. Sample Preparation 3. 3. The Shear Tests APPARATUS ....................................... The Removable Top Cap and Pressure Cylinder The Loading Ram The Base and the Base Adaptor The Sealing Cap and the Sample End-piece Rubber Membranes The Formers for Preparing Hollow Cylindrical Specimens RESULTS .......................................... 5. 1. Deviator Stress 5. 2. Pore-pressure 5. 3. Effective Principal Stresses iii 14 14 14 15 19 19 19 20 20 20 21 22 22 23 25 iv Page 5. 4. Axial Strain at Failure 27 5. 5. Failure Conditions 28 5. 6. Mohr Failure Envelopes 29 5. 7. Yield Surface 30 6. CONCLUSIONS ..................................... 33 BIBLIOGRAPHY ......................................... 58 APPENDIX ........ . ................ . ................... . 60 Al. Testing Procedure 61 1. Sample Preparation 61 2. Consolidation 62 3. Standard Triaxial Tests 62 4. Hollow Cylinder Triaxial Tests 63 A2. Data 65 UCl Tests 65 UC2 Tests 67 U11 Tests 68 UIZ Tests 70 U13 Tests 72 UEl Tests 74 UE2 Test 76 A3. Sample Calculation s 1. UC1 Test UC2 Test U11 Test UI2 Test UI3 Test UE1 Test UE2 Test Page 77 77 78 79 80 81 82 83 10. 11A. 11B. 12. 13A. 13B. 14A. 14B. 15A. LIST OF TAB LES Index properties of Michigan clay .................. Deviator stress at failure ......................... Pore-pressure at failure ......................... Effective principal stresses at failure .............. Axial strain at failure ........................... Details of failure conditions ....................... Stress conditions at failure ....................... Comparison of stress conditions at failure for Michigan clay and Weald clay ................... Summary of effective strength parameters from consolidated-undrained triaxial tests on Michigan Clay .......................................... Coordinates in effective principal stress space of experimental stress points on the plane o"+cr"+cr"=1 1 2 3 UC1 Test. 0- = 3 kg. /Cm. 2 ....................... c 2 UC1 Test, 0C: 4 kg. /cm. ....................... UC2 Test, (re: 3 kg. /cm. 2 ....................... 2 U11 Test, ch = 3 kg./cm. ....................... 2 UN Test, O‘C = 4 kg./Cm. ....................... U12 Test, O'C = 3 kg. /cm. 2 ....................... UIZ Test, ac = 4 kg. /cm. 2 ....................... UI3 Test, (re 2 3 kg. /cm. 2 ....................... vi Page 14 23 25 26 27 53 54 55 56 57 65 66 67 68 69 7O 71 72 Table Page 15B. UI3 Test, O'C = 4 kg. /cm. 2 ....................... 73 16A. UE1 Test, cc 2 3 kg. /cm. 2 ..... . ........ . ....... . 74 16B. UE1 Test, o-C = 4 kg./cm ....................... 75 17. UE2 Test, O'C = 4kg./cm .................. 76 Figure 4B. 4C. 10. ll. 12. 13. 14. LIST OF FIGURES Mohr's Diagram and Inclination of Failure Planes . . . . Mohr's Envelope for Coulomb's Criterion ..... . . . . . Yield Surface in Principal Stress Space ............ Right Section of Yield Surface by the Plane 0" +0" +0" =1 ............................... 1 2 3 The Stress System in the Standard Triaxial Test ..... Failure Envelopes and the Stress-plane for Axially- symmetrical Stres ses .......................... Hollow Cylinder Triaxial Cell ..................... The Formers for Preparing Hollow Cylindrical Specimens .................. . .................. Results from UC1 Tests .......................... Results from UIl Tests . .............. . .......... Results from UI2 Tests .......................... Results from UI3 Te,sts ........................... Results from UE1 Tests .......................... Results from UC2 Test ............................ Results from UE2 Test ............................ Summary of Deviator Stress vs. Strain Curves for Tests at O'C = 3 kg. /cm. 2 ....................... viii Page 35 35 36 36 37 37 38 39 40 41 42 43 44 45 45 46 ix Figure Page 15A. Summary of Pore-pressure vs. Strain Curves for 2 Tests at 0C = 3 kg. /cm. ...................... 47 15B. Summary of Pore-pressure vs. Strain Curves for 2 Tests at (TC 2 4 kg. /cm. ...................... 48 16A. Mohr's Effective Stress Failure Envelopes and Stress Paths ......................................... 49 16B. Niohr‘s Effective Stress Failure Envelopes and Stress Paths .................................. 50 17A. Mohr‘s Failure Envelope from UC1. UC2 and UE1 Tests ........................................ 51 17B. 1VIohr's Failure Envelope from U11, U12 and U13 Tests ......................................... 51 18. Failure Surface in Effective Principal Stress Space . . 52 NOTATION vertical intercept of a line drawn through top points of Mohr's circles pore-pressure coefficient at failure cohesion apparent cohesion in terms of effective stress effective cohesion depending on the water content at failure only arbitrary constant deviator stress = 0' . ~ or , major minor coordinate of apex of Coulomb's failure surface in principal stress space ratio of external radius to internal radius (Rb/R1) proportionality factor flow coefficient = tanz‘(450 + (19/2) 1—(c? +& +&)1— 31 2 3X0 C internal pressure external pressure radius of element radial distance on octahedral plane of principal stress space to failure surface in compression radial distance on octahedral plane of principal stress space to failure surface in extension xi Ri internal radius R external radius 0 s shearing resistance u pore-pressure w water content Au pore-pressure change 6 axial strain n angle between space diagonal and abscissa-axis in a stress-plane for axially- s ymmetrical stre s se 3 c|> angle of internal friction cl) angle of shearing resistance in terms of effective stress (be effective cohesion, depending on the water content at failure only q) slope angle of a line drawn through top points of Mohr's circles 0 normal compressive stress 0' effective normal compressive stress 0C consolidation pressure 01, 0 2, 0' 3 axial, circumferential and radial stresses respectively 0'1, 5'2, 0' 3 effective, axial, circumferential and radial stresses respectively (Ti, 0' '2, 0'3 principal stresses 0"1', 0' '2' , 0'3' stress space coordinates of projection onto an octahedral plane 1' shear stress 9 angle of inclination of shear plane 1. INTRODUC TION The shear strength of clays is commonly defined by the Coulomb-Mohr failure criterion. The correctness of the Coulomb- Mohr criterion for a clay under various stress states has not been thoroughly investigated. It is, therefore, desirable to obtain exper- imental data on the failure conditions under various states of applied stress. BISHOP and ELDIN (1953) and KIRKPATRICK (1957) found good agreement between values of the effective angles of shearing resis- tance of sand obtained from compression and extension tests, under drained conditions. KIRKPATRICK, in addition, studied the effect of the intermediate principal stress on the yielding of sands by using thick-walled cylindrical samples. He found that the values of the effective angle of shearing resistance so obtained were only slightly higher than those obtained from the conventional compression and extension tests. He also obtained an experimental failure surface in principal stress space which shows good agreement with the Coulomb- Mohr theory. In contrast, HABIB (1953) and HAYTHORNTHWAITE (1960a) found that the values of the effective angle of shearing resistance of sand obtained in extension tests were considerably smaller than those obtained in compression tests, under drained conditions. HAYTHORNTHWAITE (1960b) also studied the yield conditions of a remoulded silt by subjecting hollow cylindrical samples to combined torsion and compression, under drained conditions. He obtained an experimental failure surface in principal stress space which shows substantial departure from the Coulomb-Mohr theory. HENKEL (1959, 1960) and PARRY (1960) reported good agreement between values of the effective angle of shearing resistance obtained from compression and extension tests on normally-consolidated remoulded clays, under both drained and undrained conditions. Exper- imental information on the condition of failure for clays under inter- mediate stress states is not available. In view of the conflicting evidence presented by the afore- mentioned investigators and the need for experimental data on the condition of failure for clays under various intermediate stress states, a series of consolidated-undrained triaxial tests has been carried out on a saturated remoulded clay. The general objectives were: to examine the behavior of clays under different stress states; to study the effect of the intermediate principal stress on the shear strength of clays; and to obtain an experimental failure surface in principal stress space, in order that the experimental evidence can be compared with the Coulomb-Mohr predictions. 2. THEORETICAL CONSIDERATIONS 2.1. Shearing Resistance of a Cohesive Soil The shearing resistance of a cohesive soil, in general, has been defined by Coulomb as follows: 5 = c + 0 tan <1) (1) where s is the shearing resistance c is the cohesion 0‘ is the normal compressive stress ct) is the angle of internal friction The parameters c and <1> in the preceding equation are assumed to be constant and are independent of the state of stress preceding failure. The rational use of Coulomb's Law requires an understanding of the many complex. factors affecting the shear strength of clays. It is of utmost importance to realize that c and <1) are not constant soil proper- ties but are merely empirical coefficients, which may vary for a given soil depending on a number of variables (stress history, drainage conditions, water content or void ratio, structure, speed of shear and others). In the conventional triaxial test, the values of c and <1) obtained depend on the stress history of the soil as well as the applied stress conditions. Terzaghi (1936) concluded that the stress conditions for failure in soils as well as the volume changes, depend solely on the intensity of effective stresses. The stress conditions for failure in soils therefore depend exclusively on the effective stress which is given by C} = 0' - u (2) where 0 is the total stress 0' is the effective stress and u is the pore-pressure. In terms of effective stresses, Coulomb's failure criterion for the shearing resistance of a saturated clay may be expressed by the following: 5 = E + t} tan a) (3) where 5 denotes the shearing resistance 6 denotes the apparent cohesion 0' denotes the effective normal stress on the failure plane considered and (i) denotes the angle of shearing resistance. Hvorslev (1937, 1960) expressed the shear strength of a cohesive soil in terms of the fundamental soil properties as follows: s=ce+€rtan¢e (4) where s is the shearing resistance Ce is the effective cohesion, depending on the water content at failure only. 0' is the effective normal stress on the failure plane and Cb is the effective angle of friction. The effective angle of friction (be is a property of the material, whereas the angle of shearing resistance c); is merely the rate of increase of overall shear strength with pressure. In the case of a saturated normally-consolidated clay, for instance, the shear strength, in terms of effective stresses, may be expressed as follows: 5 = 0' tan <1.) The apparent cohesion E is equal to zero. The term 0' tan c); in- cludes both the effective cohesion and the effective friction and may be expressed in terms of Hvorslev's parameters as follows: 0' tan - = c + 0' tan . e 4.8 2. 2. Mohr's Theory of Rupture The state of stress at a point may be represented graphically by a plot known as the Mohr diagram. This diagram is of the utmost value in the study of stress conditions. Mohr's rupture theory may be expressed by the statement that there exists for a material a boundary called Mohr Envelope such that a Mohr circle within the envelope represents a stable condition, whereas a circle tangent to the envelope represents failure on the plane denoted by the point of tangency. The Mohr envelope or line of rupture is a property of the material, whereas the Mohr circle depends only on the stresses caused by the loading, and it is independent of what the material may be. The hypothesis formulated by Mohr. demonstrated in Fig. 1 can be summarized as follows: 1. The line of rupture is independent of the means by which it is obtained. 2. The line of rupture is independent of the intermediate principal stress. 3. The angle between the line of rupture and a vertical is equal to the angle between planes of rupture in the material at failure. Mohr's representation of the stress condition for failure is extremely useful in solving problems in soil mechanics in that Coulomb's criterion can be readily expressed by means of the Mohr's diagram. In Fig. 2 is shown Mohr's graphical representation of the principal stress components acting on a soil element. Assuming that the Mohr failure envelope is the straight line 5 = c + 0 tan 4), the geometrical relationship in Mohr's diagram yields equations (5), (6), (7) and (8) in the following paragraph. The Mohr's diagram shown in Fig. 2 shows that the normal and shear stresses on any plane can be expressed as 0' 0 2 1+ 3 1 : e - e:____ _ - '9 0 01cos +0351n 2 +Z(0'1 0'3)c052 (5) 01-0-3 'r=(01-0'3) $1n8c059=——2——sm 26 (6) Mohr's envelope for Coulomb's failure criterion is s = c + 0 tan Cb. Hence, 01- 0'3 = 2c cos d) + (01+ 03) sm¢ (7) which can be written as 0' tan2 (450 + 43/2) + 2c tan (450 + 43/2) “1 z 3 =03N¢+ 2cVN¢ (8) where N4) = tan2 (450 + ¢/2). 2. 3. Yield Surface in Principal Stress Space Coulomb's failure criterion can be generalized to extend its validity to three-dimensional representation of stress conditions in stress fields (Drucker, 1953; Shield, 1955). This interpretation of Coulomb's criterion leads to a yield surface for three-dimensional stress fields (Hill, 1950). In principal stress space, the yield surface is a right hexagonal pyramid equally inclined to the 0', 0' , 0" axes, with its apex on the line 123 0i=0' '2=0'3. The hexagon is irregular since the yield stress in tension differs from that in compression (Shield, 1955). The yield surface in principal stress space is shown in Fig. 3, bounded by 6 planes, the equations of which are as follows: ._ .1, W 01— 0'3N¢+2c N4) 0'3 0'N ¢+2c1’N Cb criNcP + ZCRINCP a. e 01:0'2N¢+ZCJ§; F (9) o'é-T-O’ 0"2N¢+Zc‘j_1\§ .1. +2.); J where as before N4) : tanz (45° + ¢/2) The apex of the yield surface is determined by the equation I: l: l: 01 0'2 03 k where ZC‘J’N; kz—fi— =—C COtCP (10) 4° Fig. 4A shows the right section of the yield surface by the octahedral plane oi + 0"Z + 0"?) = 0. Right sections by other planes 0"1 + 0"2 + 0' '3: constant are similar in shape but are of varying size, depending on the distance along the space diagonal at which they are taken. It is shown in the following paragraphs that the ratio RC = OC : Re = OE in Fig. 4A is independent of c, when this parameter c is a constant. Fig. 4B shows the stress system and the stress-plane that can be investigated in the standard triaxial test. Fig. 4C shows the Mohr effective stress failure envelopes and the effective stress-plane for axially-symmetrical stresses. The right section of the yield surface by the octahedral plane FBADG, whose equation is &l + &Z + 63 = 0'a is similar in shape to the one shown in Fig. 4A. The space diagonal in Fig. 4C defines the isotropic stress condition where 0'1 = 0'2 = 33. If the all-round stress at point A is 0a the stresses at points B (compres- sion) and D (extension) can be obtained as follows: At point B (compression), RC = AB, and Rc = (0'l - 03) V273. 01 0a -R cosn=0a-Re'\/273 e _ + . .—. 3 0a V172 Re SlnT) 0'a + Re V176 and _ - _ Re —(0'3 - 0'1) “(2273. On the right section by the plane (fl + 5'2 + 53 = 0 through the q I! origin O (Fig. 4C), (Ta = 0. We have then in a compression test _1=RC'\/TZ7§. s3 = sci/77?. Rc =61 - s3)V27§ and in an extension test 10 6-1 = .Re 3/2/3. s3 = +Re V176, Re = (63 - 61) V273 where 01, 0'3 are the effective stress components R is the radial distance in compression c onthe lanec-r +0' +0 =0 p 1 2 3 and R is the radial distance in extension on the plane 01 + 0'2 + 03 = 0. Using equation (7), we obtain R : 2c cos 6V? (11) c 3 - sm (P _ 2ccos¢V6 — 12 Re 3 + sin 4) ( ) On dividing, we have R . c : 3 + sm 4) (13) Re 3 - sin (I) It is to be noted that this ratio Rc/Re is independent of c, when c is a constant. It can be seen from equation (13) or from equation (7) that when <1) = 0, Coulomb's criterion becomes Tresca's yield condition (maximum shear stress at yield equal to a constant, Hill, 1950) where Rc/Re = l. The corresponding right section by the plane 0' + 0' + 0" = 0 is then a regular hexagon with the yield stress in 1 2 3 tension equal to that in compression. 11 To study the validity of Coulomb's criterion in three-dimensional stress fields, the experimental stress points at failure can be projected onto a chosen octahedral plane. These points may then be compared with Coulomb's failure surface. The projection of any stress point at failure on this octahedral plane is equal to the point where the straight line joining the stress point and the apex 0i = 0'2 = 0'3 = k of Coulomb's yield surface intersects the octahedral plane. Any straight line joining a stress point in stress space and the apex of Coulomb's yield surface can be expressed by the following: _ ll _ H _ H °'1 “1 z 0'2 “2 z “3 “3 (14) 0'-k 0'-k 0-k 1 2 3 where 01, 02, 0'3 are the experimental failure stresses and o‘i, 0"2', 0'3 are the stress space coordinates of the projection of the experimental point onto the plane 0'1' + 0"2' + 03 = C (15) From (14) we have -0"1'((rZ - k) + 03(01 -k) + 1((0'2 - 0'1) = 0 (16) -cr‘1'(0'3 - k) + 03(01 - k) + k(0'3 - 01) = 0 (17) By solving simultaneously (15), (16) and (17) we obtain 0.” _—_ -k(201- 02- 0'3) + C(Ul-k) (18) 1 0'1 + 02 + 0'3 - 3k Similarly, we also obtain 12 - - - + C - k 0‘“ : k(2cr2 0'3 01) (0'2 ) (18) 2 01 + 0'2 + 0'3 — 3k -k(2cr3- 0'1- 02) + C(03- k) 0'3' z 0' + 0' + 0 - 3k (18) 1 2 3 In the case of a normally-consolidated clay, in terms of effective stresses, c = 0 and the apex of Coulomb's yield surface is at the origin. For points on the Coulomb's surface, we have then from (9) and (18) in a compression test P (19) and in an extension test P (20) for a normally-consolidated clay. 13 Equations (19) and (20) yield C(N¢-1) R. "“‘i' “3W = n72“ M ¢ C(N¢-l) R. =-M=T2'N—.‘+T-M for a normally-consolidated clay, where M is a proportionality factor. On dividing, we have as before (13) :9_3+sin¢ R 3—sincp e 14 3. EXPERIMENTAL PROGRAM 3.1. Materials Used The clay used in the testing program was obtained from Sault Ste. Marie, Michigan. Details of the geotechnical properties of the Glacial Lake clay in its natural state have been published (Wu, 1958). The index properties are given in Table 1 below. Table 1. Index properties of Michigan clay L. L. P. L. P. 1. Clay fraction Activity % % % % 55 23 32 60 0. 53 3. 2. Sample Preparation Bulk samples of air-dried clay were thoroughly remoulded and mixed with distilled water to a consistency corresponding to a water content of 40-45%. Hollow cylindrical samples were prepared by injecting the remoulded clay into a mould provided by the formers, described in Chapter 4 "Apparatus, " by means of a jack. Cylindrical samples for the standard triaxial cell were prepared by the conventional procedure described by Bishop and Henkel (1957). The remoulded clay samples were normally-consolidated under all-round pressures of 3 or 4 kg. /cm.2 before the shear tests were carried out. Filter paper side drains were used to accelerate the 15 consolidation process. Volume changes were measured throughout the consolidation process. Water contents were measured before consolidation and after completion of the shear tests. 3. 3. The Shear Tests Consolidated-undrained triaxial tests were performed on the remoulded clay. Descriptions of the stress changes applied to the samples are given below, together with the abbreviations used to describe the tests: UC1 01 > 02 = 03 = constant. Axial stress increased, radial stress constant. UC2 0' > 0 = 0' , 0 = constant. Axial stress constant. 1 2 3 1 . radial stress decreased. U11 0' > 0' > 0 Axial stress increased first, 2 1 3 . . followed by 1ncreasrng all three, the axial, radial and circumferential stresses. UI2 0 > 0 > 0' , 0' , 0' =constant. A11 stresses increased first; 1 2 3 2 3 . . . radial and Circumferential stresses then maintained constant, followed by axial stress increase. UI3 0'2 > 01 > 0'3, 0'1 = constant. Axial stress increased first; and then maintained con- stant, followed by increasing radial and circumferential stresses. UE1 0'1 < 02 = 0'3 = constant. Axial stress decreased, radial stress constant. UE2 0 <0' :0 0' constant. Radial stress increased, axial 1 2 3' 1 stres s constant. 16 0'1, 0'2, 03 represent the axial, circumferential and radial stresses respectively. UE1, UE2 and UC2 tests were carried our on the standard tri- axial cell, whereas UCl, U1 1, U12 and UI3 tests were carried out on the hollow-cylinder triaxial cell. Pore-pressures were measured in all the tests. UC1, U11, U12 and U13 tests were carried out with controlled stress increments, allowing 10 minutes for equilibrium to be attained for each increment. Usually, thirty to thirty-five increments were required to reach failure UC2, UE1 and UE2 tests were carried out at controlled rate of strain of 2-3% per hour. The specimens usuallyreached failure in about 5 hours after the initial loading. Area corrections have been applied in the evaluation of all test data (Bishop and Henkel, 1957). Corrections for membrane and filter strip restraint have not been applied to the results. All tests were run against a back pressure of l to 1. 5 kg. /cm.2, maintained for at least 10 hours before testing, in order to eliminate the possibility of developing negative pore-pressures, and to force into solution any free or entrapped air, which may exist in the sample or in the testing apparatus. In the evaluation of the test results, the elastic theory of thick- walled cylinders has been used. According to this theory (Timoshenko, 1940), plane sections perpendicular to the longitudinal axis are assumed 17 to remain plane and consequently the longitudinal strain is independent of the radius. It can be shown that for a thick—walled cylinder sub- jected to internal and external pressures, the radial and circumferential stresses can be expressed as follows: 2 I>- K p (p.-p ) Ii 1 o 1 o o “z z _'_2_"' + “—T' ‘7:— ‘2" K -1 r K - 1 2 p.- sz (p.- p ) R 1 1 o o “3 “—2—— ' —'z——' 2 (22’ K - 1 r K - 1 where 0'2 is the normal circumferential stress 0'3 is the normal radial stress r is the radius of the elment Ri is the internal radius RO is the external radius pi is the internal pressure pO is the external pressure K = R /R.. o 1 and the conventional positive sign convention for tension is adopted. From (21) and (22), we have atr=R o (Z3) 18 and at r = R 2 2 \ Pi(K +1) - 2K p0 K -1 P (24) “3 : ’pi J Assuming a linear stress distribution across the cylinder wall for both 0'2 and 03, from (23) and (24), the average circumferential and radial stresses can be expressed as follows: 2 _ 3 K +1 (25) 0‘3av : -(pi + Pol/Z Equation (25) was used to calculate the stresses. 19 4. APPARATUS UE1, UE2 and UC2 tests were carried out on the standard tri- axial cell, whereas UCl, U11, U12 and U13 tests were carried out on the hollow cylinder triaxial cell described below. Details of the hollow cylinder triaxial cell are shown in Fig. 5. The construction of the apparatus was mainly governed by the necessity to have a complete pressure seal between the bore and the outside chamber. An air release valve at Y and rubber O-rings at P in the sample end-piece F and at Q in the base adaptor S have been provided for this purpose. The principal features to be noted are: The Removable Top Cap and Pressure leinder A transparent plexiglass cylinder M, fitted between rubber gaskets and held against the top cap C and the base J by 4 steel tie bars, is used. The top cap is provided with an oil filler valve at B and an air release valve at D. The Loading Ram A 1/2” diameter stainless steel ram is used. It is fitted inside the bronze bushing in the top cap C in order to minimize friction and leakage. 20 The Base and the Base Adaptor The base J is provided with connections for the outside chamber fluid pressure at L, the bore chamber fluid pressure at K, and for pore-pressure measurements at W and drainage at H in the Pedestal X. The base adaptor S was made removable to allow for the seal of the inner membrane at Q. It is held tight against the pedestal X by a tightening bolt and nut at R, provided with a rubber O-ring at T. The SealinLCap and the Sample End-piece Seal between the bore and the outside of the sample G is provided by the air release valve Y in the sealing cap O and the rubber O-ring at P in the sample end-piece F. The sealing cap 0 is screwed onto the threaded sample end-piece F and the outside membrane is sealed off by O-rings at U and V. Rubber Membranes The sample G is enclosed between two rubber membranes, forming a hollow cylinder of 3 inches in inside diameter, 4 inches in outside diameter snd 5 inches in height. The membranes are 0. 013" thick, supplied by Testlab Corporation in Chicago, Illinois. The outside and the bore chamber pressures are each controlled by a Norwegian Constant Pressure Cell (Andresen and Simons, 1960). The pressure difference between the outside and the bore chambers is measured by a mercury manometer. Pore-pressures are measured 21 by connecting the sample to a manometer and a Bourdon pressure gauge, through a Null Indicator (Andresen and Simons, 1960). The Formers for Preparing Hollow Cylindrical Specimens Hollow cylindrical specimens were prepared by using a mould formed by an inner plexiglass former and an outer split aluminum jacket shown in Fig. 6. The remoulded clay was placed in a 6" hollow steel cylinder and injected into the mould, by means of a jack connected to the steel cylinder. The sample end-piece was then placed on the sample after the trimming of the top of the sample had been carried out. The inside membrane was secured to the sample end-piece by a rubber band. The inner former was then withdrawn and the sealing cap was screwed tight onto the threaded sample end- piece. The outer former was removed and the sample was ready for testing after placing O-rings on the outside membrane at U and V (Fig. 5). The clay specimen, prepared in this way, is likely to be slightly over-consolidated due to the injecting pressure, but it is believed that the magnitude of the over-consolidation pressure would be of insigni- ficant importance. It was found that if the consistency of the clay was such that its water content was below about 40%, difficulties were encountered during the injection process. 22 5. RESULTS Applied stresses and pore—pressures were recorded for each test against axial strain. Complete details of each test are shown in Figs. 7-13 and in Tables 11-17 in Appendix 2, "Data, " together with sample calculations for each type of test. In this paper failure is defined as the peak point on the deviator stress-strain curve. 5.1. Deviator Stress Values of maximum deviator stress (designated as D) are shown plotted against axial strain in Figs. 7-13. For comparison, typical test results from each type of test have been plotted on the same diagram in Fig. 14. In all cases, with the exception of the UE1 and UE2 tests, the deviator stress increases rapidly with strain in the beginning of the test, and remains virtually constant at failure. In the case of the UE1 and UE2 tests, similar behavior of the deviator stress is observed, except that the deviator stress decreases after a maximum has been reached. The average maximum deviator stress, expressed in terms of the consolidation pressure 0C, for each type of test is summarized in Table 2 below. 23 Table 2. Deviator stress at failure Test Df/o'c Df/Df, UC1 UC1 0.82 1. 00 UC2 0. 72 0. 88 U11 0. 91 1.11 U12 0.83 1.01 U13 0. 99 1. 21 UE1 0. 62 0.76 UE2 0. 52 0. 64 It is interesting to note that the so—called "undrained shear strength" Df/Z varies with the type of test and appears to be depen- dent upon the loading path. It can be seen that the difference in undrained shear strength, expressed in terms of the consolidation pressure, obtained from the various types of tests, is quite remark- able. The only close agreement is cited between the UC1 and U12 tests. The differences can only be explained by the fact that the pore-pressure behaves differently in each type of test. It appears that a Tresca—type condition, D = constant, cannot be used as a failure criterion. 5. 2. Pore-pressure Values of pore-pressure change (designated as Au) are shown plotted against axial strain in Figs. 7-13. For comparison, typical 24 test results from each type of test have been plotted on the same diagram in Figs. 15A and 15B. The UC1, U11 and U12 tests show an initial rapid rate of pore- pressure increase, reducing to a negligible rate of change at failure. The UC2 test shows a small initial rapid increase, followed by a rapid decrease in pore-pressure. The rate of decrease then becomes negligible during the remainder of the test. The decrease continues after failure has been reached. The UE1 and UE2 tests show a small initial rapid pore-pressure decrease, followed by an increase in pore- pressure which is considerably less than that for the UC1, U11 and U12 tests at failure. The rate of pore-pressure change at failure is again negligible. The UI3 test shows an initial rapid rate of pore- pressure increase, reducing to a negligible rate of change at failure. The average pore-pressure at failure, expressed in terms of the consolidation pressure 0C, and the pore-pressure parameter Af (Skempton, 1954) for each type of test is summarized in Table 3 below. It can be seen from Table 3 that A is dependent on the stress f path for each type of test and varies with the type of test. 25 Table 3. Pore-pressure at failure Test (Au)f = (Au)f 0C D UC1 +0. 64 +0. 78 UC2 -0. 33 -0. 46 U11 +0. 63 +0. 69 U12 +0. 63 +0. 76 UI3 +0. 40 +0. 41 UE1 +0. 08 +0. 13 UE2 +0. 77 +1. 48 5. 3. Effective Principal Stresses Values of the effective principal stresses (designated as 0' , are shown plotted against axial strain in Figs. 7—13. In the UC1 and UC2 tests, the axial stress is the major principal stress while both the intermediate and minor principal stresses are equal to the radial stress. In the UE1 and UE2 tests, however, the axial stress becomes the minor principal stress while both the major and intermediate principal stresses are now equal to the radial stress. In all the remaining tests, where the intermediate principal stress is different from the major and the minor principal stresses the radial stress 0'3 is always the minor principal stress. Whether the axial or circumferential stress is the major or the intermediate principal stress depends on the method of applying the stresses. In the U11 and U13 tests, 26 a change of the circumferential stress 02 from the intermediate to the major principal stress takes place during the test. In the U12 test, however, a change of the circumferential stress 0'2 from the major to the intermediate principal stress takes place during the test. In all cases, all the three principal stresses remain virtually constant at failure. The average effective principal stresses at failure, expressed in terms of the consolidation pressure 0C and the effective principal stress ratio are summarized in Table 4 below. Table 4. Effective principal stresses at failure (0 major + 0 ma'or Test (Tl/ac UZ/Uc 0p3/O'c 0' minor) 0' minor UC1 1.16 0. 34 0.34 1.49 3. 47 UC2 1.12 0.36 0. 36 1.49 3.11 UIl 1.12 1.31 0.40 1.71 3.32 U12 1.15 0.61 0.33 1.48 3.54 U13 0. 88 1. 43 0. 44 1. 87 3. 25 UE1 0. 26 0. 88 0. 88 1.14 3. 39 UE2 0. 43 0. 94 0. 94 1. 37 2.18 It is interesting to note from Table 4, that with the exception of the UE2 test, the effective principal stress ratio at failure changes very little with the type of test. For a normally-consolidated clay, Coulomb's criterion, in terms of effective stresses, can be expressed as 27 Hq ' H «‘9' 2 where as before 2 o - N = tan (45 + 0/2) ¢ and (1,0, denote the effective major and minor principal stresses 1 J respectively. The fact that (ii/(.1 = constant as shown implies that J Coulomb's criterion is valid for clays under various intermediate stress states. 5. 4. Axial Strain at Failure Although the axial strain at failure cannot be defined with great accuracy, it appears that within experimental error, the axial strain at failure varies with the stress path for each type of test. The average axial strain at failure for each type of test is summarized in Table 5 below. Table 5. Axial strain at failure Test Axial strain at failure % UC1 0' > 0 = 0 = constant 13. 4 1 2 3 > = = 4. UC2 0'1 0'2 03, 0'1 constant 1 6 > > . UIl 0'2 01 0'3 2 7 > > Z . 4 U12 0'1 0'2 0'3, 0'2, 03 constant 7 > > : UI3 0'2 0'1 03, 0'1 constant 1. 3 UE1 0'1 < 0‘ = 03 = constant 10. 7 UE < : : . 2 01 0'2 0'3, 01 constant 7 0 28 The interesting feature to be noted is the wide range of axial strains at failure, varying from 1. 3% to 14. 6%. The wide range of variation of axial strains is to be expected since the axial strains are not the major principal strains in some of the tests. In the UCI, UC2 and U12 tests where the axial stress 01 is the major principal stress at failure, a variation of 7. 4% to 14. 6% is noted. Although a comparison of axial strains would be inconsistent for the other tests where the axial stress 01 is not the major principal stress at failure, variation between U11 and U13 tests, UE1 and UE2 tests is also observed. The results obtained appear to be in agreement with the formulations by Levy—Iviises (Hill, 1950) that the principal axes of stress and strain coincide. 5. 5 Failure Conditions Details pertinent to the condition at failure for each type of test have been summarized and are shown in Table 6. The subscript f has been used to designate the principal stresses, pore—pressures, maximum deviator stress, axial strain or water content at failure. The mode of failure, observed from visual inspection of the specimen after failure, of each type of test has been described and is also given in Table 6. Table 7 summarizes the stress conditions at failure for the various types of tests. For ease of comparison the stresses at failure have been expressed in terms of the consolidation pressure. 29 The test results for the stress conditions at failure obtained for Michigan clay in this investigation have been compared with those obtained on Weald clay by Parry (1960) and are presented in Table 8. Although the nature of Michigan clay is expected to be different from Weald clay, it is interesting to note that the trend is remarkably similar. It is to be noted that for both clays, the undrained shear strength Df/Z and the pore-pressure change characteristics are dependent on the stress paths for each type of test and vary with the type of test, and that, for all practical purposes, the effective principal stress ratio at failure is independent of the stress path, irrespective of the type of test. 5. 6. Mohr Failure Envelopes The Mohr effective stress failure envelopes and the corresponding loading paths for different types of tests are shown in Figs. 16A and 16B. Instead of plotting the Mohr failure circles, the top point of each failure circle has been plotted. If a and (,0 are the vertical intercept and the slope angle for the straight line through such points, it can be shown that sin ()3 = tan ((1, and 6 = a/cos <1). Fig. 17A shows the Mohr effective stress failure envelope obtained from UC1, UC2 and UE1 tests. Fig. 17B shows the Mohr effective stress failure envelope obtained from U11, U12 and U13 tests. 30 It can be seen that the values ofc); from the different types of tests do not differ by more than 10. A value of 330 is obtained from U11, U12 and U13 tests where the intermediate principal stress is different from the major and the minor principal stresses, whereas a value of 320 is obtained from UC1, UC2 and UE1 tests where the intermediate principal stress is equal to either the major or the minor principal stresses. In a stress system where the intermediate principal stress is different from the major and the minor principal stresses, the Coulomb-Mohr theory underestimates the shear strength of a clay. The error, however, is slight and has very little practical significance. The effective strength parameters obtained from the different types of tests are summarized in Table 9. 5. 7. Yield Surface Fig. 18 shows the experimental stress points representing failure projected onto the plane 0'1' + 0‘2' + 0'3' = 1. Coulomb's criterion, based on the effective strength parameters of E = 0 and 4.: = 320 obtained from compression and extension tests on the standard triaxial cell, is shown as dotted lines in the same figure. The representation of experimental stress points projected onto the plane 0'" + 0" + 0'" =1 is readily obtained by using equation (18) as l 2 3 follow 5: 31 For E = 0, k = 0 and with C = 1 equation (18) becomes The projected effective principal stress space coordinates of the experimental stress points from each type of test are shown in Table 10. The representation of Coulomb's criterion projected onto the plane 0']: + 0'): + 0'3' = 1 is readily obtained by using equation (19) and (20) as follow 5: forEannd$=32° N¢=3.25. With C =1, we have in a compression test 0"=CN¢=—-2'—Zi=062 1 N+2 2+3.25 ' ' ¢ c 1 n _—_ H : : —— = 0. c'2 U3 N+2 3.25+2 191 4> and in an extension test C 1 1' : :—-————— : “1 ZN¢+1 2x3.25+1 0"“ CN _ _ o _ 3.25 H _ II _— _———— = ,4 4 “2 “3 2N¢+1 2x3.25+1 0 3 It can be seen that the experimental data are in good agreement with Coulomb's prediction. The test data also show minor departure from Coulomb's criterion. Since only a limited number of triaxial tests have been carried out in the present investigation, a considerably greater volume of experimental evidence is needed before any strong assertions can be made. 32 33 6. CONCLUSIONS The conclusions summarized below reflect the findings of this investigation and are limited to the soil used, the methods of sample preparation and the testing procedure employed: 1. The undrained shear strength of a clay is dependent on the loading path for each type of test and varies with the type of test. The variation may be explained by the different pore-pressure behavior under different applied stress changes. 2. The pore-pressure change characteristics are dependent on the loading path for each type of test and vary with the type of test. 3. The effective principal stress ratio at failure, for all practical purposes, is independent of the loading path, irrespective of the type of test. 4. The axial strain at failure appears to vary with the loading path for each type of test. Test results appear to be in agreement with the concept that the principal axes of stress and strain are coincident. 5. The Mohr effective stress failure envelopes are independent of the loading paths for each type of test. In a stress system where the intermediate principal stress is different from the major and the minor principal stresses, the Coulomb-Mohr theory underestimates the shear strength of a clay. 34 6. The validity of Coulomb's criterion in three-dimensional stress fields has been studied. Test results show that the experi- mental failure surface is, in general, in good agreement with Coulomb's prediction, although minor deviations are noted. 7. In this investigation, it has been tacitly assumed that the elastic theory is valid in the calculation of stresses at failure. The fact that the behavior of clays is not truly elastic has not been con- sidered. In the evaluation of the stresses and strains at failure, an error or inaccuracy is likely to be introduced by assuming that the material is fully elastic. It is believed that the elasto-plastic and/or the plastic solution would be more satisfactory in the understanding of the behavior of clays. 8. So far it has not been possible to relate the stresses with the strains measured in the various types of tests. For a complete understanding of the deformation behavior of clays, it appears neces- sary that stress and strain should be related. In the future, it is hoped to extend the range of tests to include stress states such as torsion, combined torsion and compression, in order that some of the limitations on the validity of the test data can be eliminated. SHEAR STRESS ’C 26 . l n:- n ~..,_ .. .' N LINE OF RUPTURE -\ l " 5 I \ \ . 7/ a ’ J p I NORMAL STRESS 0' , \\ " fl \ 1 .12 _ v Mot—112's CIRCLE OF STRESS 2 \ 1 FAILURE PLANE Fig. l. Mohr's Diagram and Inclination of Failure Plane FAILURE ENVELOPE ’1' \ 0" I 0‘ 53—. r// I 0" = 65059 + Ggsm’e ’i: =(0,-C73)smecose I 0', -T(I 2. \ 35 Fig. 2. Mohr's Envelope for Coulomb's Criterion “6-“; l"'.t'"“lfl'l.w-ff *~v<-'-- - ~_--..'. Fig. 4A. Right Section of Yield Surface by the Plane 0"1 + 0"2+0"3= I to: O‘,r-O,N¢+2c./N}. I , l /‘ 62 0;,=G,N¢+2c~/N-¢ . _. _ \ / \ / l / I / / 0‘3'=G;N,+2cJ_N-¢ J\ / / I 0‘;=O';N¢+ 2cm; I / / : I G / ’ 433(1/ I u G ' / | / / 05:@"£.i 2.5.53 0,’= 0$N¢+2cm CI , o PURE TENSION ’- CI x PURE COMPRESSION Fig. 3. Yield Surface in Principal Stress Space II (I), w \ I I I I \ 61:02 PLANE C «62:03 PLANE // \/ \ / \ Re // \ \O-fi 03 PLANE \ O / \ / R,‘ 62 E' 63 0 36 Stress-plane that can be investigated in the standard triaxial / apparatus ‘ / ‘ SPA CE DIAGDNAL o / so, 502-2 Jigs / 02 Fig. 4B. The Stress System in the Standard Triaxial Test C, II F RIGHT SECTION SPACE DIAGGNAL B SLOPE HE. ””” I Re I o -- ---"r, A \EXTENSION MOHR ' | 0 ' I 4L l l a L 0 G 5 61: 563 Fig. 4C. Failure Envelopes and the Stress Plane for Axially- Symmetrical Stresses ‘ U M —— - e o /,/$ \P F . . _______,,.II—«~—- ””6 W \ 'g.‘.-.'H-—-- 3" I.D.——-—-u.5.:', H . ,, ‘ 4” O. D. 4 /// ‘\\\ .. //" \ ' . / L h . Q R S '.' /,/‘K 1“ / ' l .- ' V 1 \\ o' a: .__._. _ __ “‘\\ \ \ \ \\ V ., / \\ \x a?) x J ‘f , Z/ //:/’ /T // it! M. .31 l .| C a. 1 7 m wugx ” M .. 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Nm.H wmwd Nod Humd Himd wH .H o.m o o b . . Ix Imus x o o o Eo\ muH o o m H o H b b b b b .- . MOSHE .m HHOGHE m b I I I o pooh. .94 .m. x-Q nowmfib + “oh-men .3 m Hmb+N H VWHQ m N H b deHHdH um mGoHqucoo mmonum .N. oHnH-mH. 55 H . No.0 24. Nm.~ 34 $5 :5 $6 $5 $5 2:328” b a 2 .o S .o S .N om 4 2. .o E .o S .o 5 .o 3. .o .umcooumfmb 6 4005450me .N mo .o- g .o 0 2 .N 2. 4 E .o S .o a. .o 3 .0 £040 05328 u C .n 35 $5 £0 84 $5 3.0 3.5 $0 $4 .umcooumbumb .m ConmvHQEOU .H 432 953?: 0.2 .520 383 E .o S .o M: .N S 4 mm .o E .o i .o 4.0 .o 3 .o mm: mo .o no .0 am .m E 4 3 .o S .o mm .o mm .o 8 .o 4m: 2. .o mo 4 3 ..o. 5 4 2. .o No .0 4.4. .o 3 4 mm .o 20 3 .o 2 4 E .m a. 4 mm .o on .o mm .o 3 .o 2 4 NS 3 .o 3 .o mm ..o. E 4 3 .o 3 .o 2. .o 3 4 N4 4 :0 mm .o- 2 4 2 .m a. 4 N» .o 3 .o om .o om .o 2 4 ~00 4.0 .o mm 4 S. .m a. 4 mm .o 3 .o 4% .o 3. .o 2 4 400 w x .3 x 0b 0b m MOSHE b HHOHHE “0+ 0.0 Hmw+Nw+vaW who: W :0” “mum. :4 .4. x IQ] I. m mm. 4. nomebv Mommabv Q n AH >30 3003 van >30 GmeoHHZ new mndHHmm ad 9536480 mmmuum Ho GOmHHMnHEOO .w 3nt Table 9. Summary of effective strength parameters from consolidated undrained triaxial tests on Michigan clay Test C (i9 UC1, UC2 0 32° UE1 0 32° U11 0 33° U12 0 32° U13 0 33° UC1, UC2, UE1 0 32° U11, U12, U13 0 33° 56 57 wow .0 wow .0 vaH .0 NM .0 on .m 4:. .m. on .H o .HV NMHD mmw .o mmHu .o omH .0 PM .w 00 .m 00 .m no .H o .HV HMO. mmv .o mmHu .o hNH .0 Ho .m mm .N mm .N m0 .0 o .m mmH .o mom .0 Ham .0 cm .oH Hp .H 0H» .m 00 .m o .v MHD. Hi: .0 mmm .0 cm .0 0m. .w PM .H wHu .Hu Hm .N o .m HoH .o Hom .0 Ham .0 mHu .w om. .H Hum .N mm .Hu 0 .Hu NmH .0 00m .0 Hum .o 9v .0 mwm .0 mo .H 3» .m o .m NHD cmH .0 HvoN .0 mm .o o .0 mmm .o mm .H 3. .m o .m mHuH .0 34v .0 mow. .0 mm .NH on .H uvm .m ow .HV 0 .v HHD oHuH .o wvé mwm .o Hung. mo.H H05. wmcd o.m mmvH .o moH .0 H05 mm.m wo.H mo.H hm .m 04m NOD med mde Humod 0H .N. Hm.H Hm.H meJV 0% HUD Hum: .o wwH .0 Hmod mmd mo.H mo.H mm.m o.m mmmwmg rafnfne Assuming meow? N.§\.E was? Néoxwx .483? N .Hmw HNm. HH.my 0 umOH. m N H Awmb+HNb+mev Hmw Hmw ..HHb b H :b H :b H :b Humb+mb+wb meHnH 05 co muGHom mmouum 3308:0me Ho 000mm mmmuum HmQHUGHHm 0300030 CH mmumcHUuooU .OH 0308 58 BIBLIOGRAPHY ANDRESEN, A. and SIMONS, N. E., 1960. Norwegian triaxial equipment and technique, Norwegian Geotechnical Institute, Publication No. 35. BISHOP, A. W. and ELDIN, A. K. GAMAL, 1953. The effect of stress history on the relation between 4) and porosity in sand. Proc. 3rd Int. Conf. Soil Mech. , 1 : 100. BISHOP, A. W. and HENKEL, D. J., 1957. The measurement of soil properties in the triaxial test, Edward Arnold, London. DRUCKER, D. C. , 1953. Limit analysis of two- and three- dimensional soil mechanic problems. J. Mech. Phys. Solids, 1 : 4 : 217. HABIB, P. , 1953. Influence de la variation de la contrainte prin- cipale moyenne sur la résistance au cisaillement des sols. Proc. 3rd Int. Conf. Soil Mech. , l : 131. HAYTHORNTHWAITE, R. M. , 1960a. Mechanics of the triaxial test for soils, Proc. A. S. C. E. , Vol. 86, SMS. HAYTHORNTHWAITE, R. M. , 1960b. Stress and strain in soils, Plasticity, Proc. 2nd Symposium on Naval Structural Mechanics, Pergamon Press. HENKEL, D. J. , 1959. The relationships between the strength, pore-pressure, and volume-change characteristics of saturated clays, Geotechnique, 9 : 3 : 119. HENKEL, D. J. , 1960. The relationships between the effective stresses and water content in saturated clays, Geotechnique, 10 : Z : 41. HILL, R. , 1950. The mathematical theory of plasticity, Oxford Press, London. HVORSLEV, M. J. , 1960. Physical components of the shear strength of saturated clays, Proc. A. S. C. E. Research Conf. on Shear Strength of Cohesive Soils. 59 KIRKPATRICK, W. M. , 1953. The condition of failure for sands, Proc. 3rd Int. Conf. Soil Mech., 1b : 172. PARRY, R. H. G. , 1960. Triaxial compression and extension tests on remoulded saturated clay, Geotechnique, 10 : 4 : 166. RUTLEDGE, P. C. , 1940. Theories of failure of materials applied to the shearing resistance of soils, Proc. Purdue Conf. Soil Mech. and Applications, 1940. SHIELD, R. T., 1955. On Coulomb's law of failure in soils, J. Mech. Phys. Solids, 4: 10. SKEMPTON, A. W. , 1954. The pore-pressure coefficients A and B, Geotechnique, 4 : 4 z 143. TAYLOR, D. W., 1948. Fundamentals of soil mechanics. John Wiley & Sons. New York. TERZAGHI. K. . 1936. The shearing resistance of saturated soils and the angle between the planes of shear, Proc. lst Int. Conf. Soil Mech. Vol. 1:D-7, pp. 54-56. TIMOSHENKO, S., 1940. Strength of materials, Macmillan, London. WU, T. H., 1958. Geotechnical properties of Glacial Lake Clays, Proc. A. S. C. E. , V01. 84, 5M3. APPENDIX 60 61 A1. TESTING PROCEDURE 1. Sample Preparation Bulk samples of clay were thoroughly remoulded and mixed with distilled water and left in the moist room for a few days before use. The remoulded clay was prepared at a water content of 40-45%. Hollow cylindrical samples were prepared by injecting the clay into a mould formed by two formers, described previously under ”APPARATUS. " Solid cylindrical samples in the standard triaxial cell were prepared by the conventional procedure described by Bishop & Henkel (1957). 1.4" in diameter and 3" in height specimens were used in the UC2 tests whereas 1. 4" in diameter and 2. 5" in height specimens were used in the UE1 and UE2 tests. 2. Consolidation The remoulded clay sample was normally-consolidated under an all-around pressure of 3 or 4 kg./cm.2 The consolidation process was accelerated by using filter paper stripe side drains on solid cylindrical samples, and by using a continuous jacket of filter paper side drains on the inside of hollow cylindrical samples. It was found that 24-30 hours were required for 95-100% consolidation. After consolidation, a back pressure of 1 to l. 5 kg. /cm.2 was applied to the sample through the basal porous stone, in order to eliminate the possibility of developing negative pore-pressures and to dissolve any free or entrapped air which 62 may exist in the sample or in the testing apparatus. The back pressure was maintained for at least 10 hours before the shear test was carried out. The all-around pressure in the cell was raised the same amount simultaneously. 3. Standard Triaxial Tests UC2, UE1 and UE2 tests were carried out in the standard triaxial cell at controlled rate of strain of 2-3% per hour (0. 04 - 0. 06 in. per hourL In the UC2 and UE2 tests, the cell pressure was adjusted so that the axial stress was kept constant and equal to the initial cell pressure (0') b the followin e uation 30 Y g q _() N6 1 U3- O.3ona l-arfa where 0'3 is the cell pressure at any time during the test N is the proving ring factor expressed in load per division 6 is the proving ring deflection in div. from zero load ar is the area of the ram and a is the area of the sample at any time. The weight of the ram has been neglected in all the calculations. In order to determine rapidly the value of 0' required to keep the axial 3 stress 01 constant at any stage. a series of curves relating all the 63 variables was prepared before starting the test. Some modifications on the triaxial cell are necessary for carrying out the UE1 and UE2 tests. The loading cap is provided with a threaded hole at the center, while the loading ram is provided with a bayonet catch at the top and threads at the bottom. The bored threaded hole in the loading cap was used as a guide during the con- solidation process. When the extension test is to be started, the ram is screwed into the loading cap and pushed through a slotted plate mounted onto the proving ring, and engaged by rotating the ram through 90 degrees. All the tests were carried out beyond the point of failure. The mode of failure was noted in all the tests. Water contents were measured after completion of the tests. 4. Hollow Cylinder Triaxial Tests U11, U12 and U13 tests were carried out in the hollow cylinder triaxial cell with controlled stress increments. The cell was placed on a balance scale provided with an adjustable yoke, controlled by a hand wheel. Dead loads were applied to the sample by balancing the yoke against the loading ram. Axial dead loads of Z. 5 and 5. 0 kilograms were applied to the specimens normally-consolidated under all-around pressures of 3 and 4 kg. /cm.2 respectively. A period of 10 minutes was allowed for equilibrium to be attained for each increment. The 64 pressure difference between the bore and the outside chambers, meas- ured by a mercury manometer, was applied at the rate of O. 05 kg./cm. every 10 minutes. In the U12 tests, the bore chamber pressure was maintained constant by a Norwegian constant pressure cell while the outside chamber pressure was increased by a lever—screw until the desired amount of pressure difference was reached. The outside chamber pressure was then maintained constant by a Norwegian constant pres- sure cell throughout the test. The specimen was then axially-loaded by increments to failure. In the U11 tests, the specimen was first axially-loaded by incre- ments, followed by increasing the outside chamber pressure until failure has been reached, while maintaining both the axial load and the bore chamber pressure constant. In the U13 tests, the axial stress was applied first and kept con- stant by taking weights off the scale when the outside chamber pressure was increased by the following axial dead load decrease = (po- pi) T} (4)2(2. 54:).Z : 81 _ (pO pi) The bore chamber pressure was kept constant throughout the test. The tests were stopped as soon as failure was reached. The mode of failure was noted in all the tests. Water contents were measured after completion of the tests. I. lil iiil‘lllllII‘II . ({l‘l‘llllll A2. Test: DATA Table 11A 0C: 3kg. /Cm.2 { i = 42. 5% w = Z9. 3% %z 61 2 3 (0' — 0' ) (&1+&3)/2 Au 0.000 .87 2.87 2.87 .00 .87 .00 0.185 .33 2.74 2.74 .59 .04 .13 0.304 .48 2.70 2.70 .78 .08 .17 0.490 .84 2.49 2.49 .35 .17 .38 0.85 .93 2.18 2.18 .77 .05 .71 1.00 .90 2.05 2.05 .85 .98 .82 1.48 .89 1.89 1.89 .00 .89 .98 2.28 .80 1.85 1.85 .15 .73 .22 3.02 .72 1.50 1.50 .22 .81 .37 4.05 .83 1.35 1.35 .28 .49 .52 5.24 .58 1.25 1.25 .33 .42 .82 8.55 .53 1.15 1.15 .38 .34 .72 7.90 .47 1.05 1.05 .42 .28 .82 9.55 .51 1.05 1.05 .48 .28 .82 10.75 .52 1.03 1.03 .49 .28 .84 13.30 .53 1.03 1.03 .50 .28 .84 Table 11B 2 wi = 41% Test: UC1 0C=4 kg./cm. {wf : 28.2% 1:: 0‘1 0'2 (73 D = (01- 0'3 (81+ (13)/2 Au 0.00 3.96 .96 3.96 0. 00 3.96 .00 0.50 4.79 .55 3.55 1.24 4.17 .41 0. 86 5. 28 . 06 3. 06 2. 22 4.17 . 90 1.13 5.14 .68 2.68 2.46 3.91 .28 l. 54 5. 08 . 38 2. 38 2. 70 3. 73 . 58 2.00 5.05 . 20 2. 20 2.85 3. 63 .76 2. 85 4.96 .98 1.98 2.98 3. 47 .98 3.95 4.78 .76 1.76 3.02 3. 27 . 20 5. 07 4. 65 . 57 1. 57 3. 08 3.11 . 39 6. 30 4.58 .46 1.46 3.12 3.02 .50 7.45 4.53 .38 1.38 3.15 2.96 .58 8.70 4.54 .36 1. 36 3.18 2.95 .60 10.42 4.51 .31 1.31 3.20 2.91 .65 13.40 4.54 .31 1.31 3.23 2.93 .65 ii 11 il‘t!‘[‘{'l‘ In“! Table 12 2 Wi = 41. 5% Test: UCZ 0' —3 kg./cm. {Wf : 27. 6% %z 1 2 83 D=(&1-&3) (&l+c}3)/2 Au 0. 00 2. 57 .57 2. 57 0. 00 2. 57 0. 00 0.109 2.94 .43 2. 43 0.51 2. 89 0.09 0. 292 2.94 .93 1.93 1. 05 2. 44 -0. 01 0.802 3.12 .72 1.72 1.40 2.42 -0. 23 1. 385 3.17 .59 1. 59 1. 58 2. 38 -0. 30 1. 82 3.15 .55 1. 55 1. 80 2. 35 -0. 30 2.55 3.13 .43 1. 43 1. 70 2. 28 -0. 30 3. 28 3.11 . 35 1. 35 1. 78 2. 23 -0. 30 4.00 3.11 .31 1. 31 1. 80 2. 21 -0. 31 5.10 3.09 . 22 1. 22 1. 87 2.18 -0. 35 8.20 3.14 .21 1.21 1.95 2.18 -0.42 . 7.30 3.18 .11 1.11 2.07 2.15 —0. 54 8. 39 3. 27 .09 1. 09 2.18 2.18 -0. 84 9.12 3.29 .11 1.11 2.18 2.20 —0. 89 10. 20 3. 33 .14 1.14 2.19 2. 24 -0. 77 12.40 3. 38 . 11 1.11 2. 25 2. 24 -0. 88 13.10 3. 34 . 08 1. 08 2. 28 2. 21 —0. 89 13.85 3. 38 .10 1.10 2. 28 2. 25 -0. 95 14.80 3. 37 .08 1. 08 2. 30 2. 23 -0. 98 Table 13A w, = 41.05% Test‘ U11 0' =3 kg /cm 2 1 ° 1 ° ° wf = Z9. 53% 6 - — D: 0' - Au 2 1 62 U3 - - max "/0 (CT '0 . ) max min 2 0. 00 .73 .73 2.73 0. 00 .73 .00 0.109 . 31 .78 2.34 0.97 .83’ .43 0. 217 .47 .39 2.06 1.41 .77 .81 0.391 .42 .57 1.80 1.77 .69 .12 1.39 . 17 .46 1.30 2.16 .39 .62 1. 825 . 96 . 27 1.10 2.17 .19 . 82 2.06 .91 .26 1.05 2.21 .16 .87 2.24 .85 .21 1.00 2.21 .11 .92 2. 41 . 83 .19 0. 98 2. 21 . 09 . 94 2.5+ .98 .71 1.05 2.66 .38 .97 68 Table 13B 2 wi = 45. 50% Test: UIl 8C=4 kg./cm. {wf : 28. 6% E _ _ - _ - Z 0’1 °2 °3 - D =— (Umax min) A“ ‘70 (Gmax- 0.min) 2 0 00 3 83 3 83 3.83 0 00 3 83 0 00 0. 202 3. 99 3. 81 3. 81 0.18 3. 90 0.02 0. 302 4. 45 3. 74 3. 74 0. 71 4.10 0. 09 0. 470 4. 97 3. 58 3. 58 1. 41 4. 27 0. 27 0. 582 5. 33 4. 83 3. 25 2.12 4. 29 0. 78 0. 781 5. 32 5. 09 2. 74 2. 58 4. 03 1. 34 0.940 5. 30 5. 31 2. 49 2. 82 3. 90 1. 84 1.12 5. 24 5. 38 2. 33 3. 05 3. 79 1. 83 150 5 08 5 38 2.07 3 29 3 58 211 2 02 5 01 5 42 1.90 3 52 3 48 2 31 Table 14A w. = 41.4% Test: U12 0‘ = 3 kg./cm.2 { 1 _ c wf — 29. 5% 6 ‘ ' ' — ' ' A %z cr1 0‘2 03 (6'. D-_ ' (Umaxflr u max min) 2 0. 00 2.63 .63 2.63 0.00 2.63 .00 0. 00 2. 67 . 78 2. 57 0. 21 2. 68 . 09 O. 096 2. 83 . 30 2. 31 0. 99 2. 81 . 45 0. 214 3.15 . 21 2. 22 0. 99 2. 72 . 54 0.30 3.40 . 14 2.15 1.25 2.78 .61 0.45 3.70 .96 1.97 1.73 2.84 .79 0.60 3.78 .80 1.81 1.97 2.80 .95 0.90 3.83 64 1.65 2.18 2.74 .11 1.30 3.89 .53 1.54 2.35 2.72 .22 1.67 3.85 .43 1.44 2.41 2.65 . 32 2.78 3.76 .28 1. 29 2.47 2.53 .47 4. 70 3. 58 . 08 1. 09 2. 49 2. 34 . 67 7.05 3. 49 . 98 0. 99 2. 50 2. 24 . 77 70 Testz'UIZ Table 14B w. 11 1h 5" 01 c.Q 2 1 = 4 . . o-C kg /Cm {w 28. 7% %z 01 02 3 (0' D_ :3 . ) (cg—max min) Au xnax nn 2 0.00 3.86 .86 .86 0.00 3.86 0.00 0.077 3.88 .48 .32 1.16 3.90 0.67 0.273 4.01 .45 .29 1.16 3.87 0.70 0.48 4.43 .20 .04 1.39 3.74 0.95 0.70 4.80 .90 .74 2.06 3.77 1.25 0.985 4.88 .65 .49 2.39 3.69 1.50 1.16 4.90 .51 .35 2.55 3.63 1.64 1.59 4.84 .30 .14 2.70 3.49 1.85 2.71 4.62 .95 .79 2.83 3.22 2.20 4.15 4.51 .72 .56 2.95 3.04 2.43 6.70 4.43 .57 .41 3.02 2.92 2.58 7.00 4.55 .54 .36 3.19 2.96 2.61 Table 15A 2 W' = Test: 0' = 3 kg./cm. 1 _ c w — f e — — D= +8 Au z 1 0.2 0F3 (0' _5_ max % max 2 0. 00 . 86 . 86 2. 86 0. 00 . 86 . 00 0.41 .00 .83 2.83 0.17 .92 .03 0.481 . 28 .78 2.78 0. 50 .03 .08 0.542 .41 .75 2.75 0.66 .08 .11 0. 650 . 65 . 66 2. 66 0. 99 .16 . 20 0.866 .87 .55 2.55 1.32 .21 .31 0.910 .79 .55 2. 33 1.46 .06 .56 1.00 .26 .50 1.95 1.55 .73 .09 1.06 .91 .89 1.67 2.22 .78 .44 1.08 .76 .99 1.58 2.41 .79 .59 1.28 .51 .48 1.37 3.11 .93 .84 Table 15B ll 2 wi Test: U13 0 = 4kg./cm. C W f E 0 0 0 D z 1 2 3 - (0' ‘70 max 0. 00 3. 65 3. 65 3. 65 0 0.044 3.96 3. 62 3.62 0 0. 088 4. 11 3. 60 3. 60 0. 0.132 4. 26 3. 58 3. 58 0 0.198 4. 41 3. 55 3. 55 0 0. 220 4. 54 3. 52 3. 52 1. 0. 308 4. 79 3. 43 3. 43 1. 0. 396 4. 99 3. 29 3. 29 1. 0. 460 5. 09 3. 22 3. 22 1. 0. 5277 5.17 3.13 3.13 2. 0. 615 5. 27 3. 06 3. 06 2. 0. 79 5. 18 2. 80 2. 80 2. 0. 855 4. 84 3. 24 2. 54 2. 0. 91 4.14 4. 62 2. 04 2. 0.945 4. 04 4. 78 1.96 2. 1.05 3.92 4.92 1.87 3. 1.14 3.82 5.08 1.79 3. 1. 25 3.74 5. 26 1.74 3. 1.30 3. 69 5.46 1.71 3. 1.71 3.64 5.41 1.66 3. :‘OOPPPPPPPPPP g... Q PPPPPPPWPPPPPPPPPPPP p—a P591919!" Table 16A w, = 41. 8% Test: UE1 0' = 3 kg. /cm.‘2 { 1 _ c wf —- 27. 02% :7: {.1 2 '3 D: 75-3- 61) (41+831/2 Au 0.00 2.85 2.85 2.85 0.00 2.85 0.00 0.218 2.15 3.03 3.03 0.88 2.59 -0.18 0.436 1.96 2.98 2.98 1.02 2.47 -0.13 0.655 1.82 2.88 2.88 1.06 2.35 -0.03 0.874 1.70 2.80 2.80 1.10 2.25 +0.05 1.31 1.54 2.68 2.68 1.14 2.11 +0.17 2.18 1.37 2.58 2.58 1.21 1.98 +0.27 3.06 1.20 2.48 2.48 1.28 1.84 +0.37 3.93 1.12 2.46 2.46 1.34 1.79 +0.39 4.36 1.05 2.43 2.43 1.38 1.74 +0.42 5.24 0.90 2.39 2.39 1.49 1.65 +0.46 6.10 0.88 2.42 2.42 1.54 1.65 +0.43 7.42 0.79 2.48 2.48 1.69 1.64 +0.37 8.30 0.74 2.48 2.48 1.74 1.61 +0.37 9.17 0.72 2.48 2.48 1.76 1.60 +0.37 10.05 0.74 2.53 2.53 1.79 1.64 +0.32 10.90 0.75 2.58 2.58 1.83 1.67 +0.27 11.80 0.78 2.58 2.58 1.80 1.68 +0.27 12.67 0.79 2.59 2.59 1.80 1.69 +0.26 14.40 0.86 2.60 2.60 1.74 1.73 +0.25 74 Table 16B w, 42% Test: UE1 0' = 4 kg./Cm.2 { 1 _ c wf — 26.1% ez (T c7 0 D=(0‘ —c-r) (8+8 )/2 Au ,% 1 2 3 3 1 1 3 10.00 .81 3.81 .81 .00 .81 0.00 0.218 .86 4.05 .05 .19 .46 -0.24 0.435 .61 3.98 .98 .37 .30 -0.17 0.870 .31 3.81 .81 .50 .06 0.00 1.31 .10 3.68 .68 .58 .89 +0.13 2.39 .80 3.50 .50 .70 .65 +0.31 3.48 .61 3.43 .43 .82 .52 +0.38 4.00 .55 3.42 .42 .87 .49 +0.39 5.22 .34 3.40 .40 .06 .37 +0.41 6.30 .19 3.43 .43 .24 .31 +0.38 7.82 .08 3.48 .48 .40 .28 +0.33 8.80 .07 3.52 .52 .45 .30 +0.29 10.43 .07 3.60 .60 .53 .34 +0.21 11.30 .06 3.62 .62 .56 .34 +0.19 12.20 .08 3.62 .62 .54 .35 +0.19 13.05 .14 3.62 .62 .48 .38 +0.19 14.80 .26 3.62 .62 .36 .42 +0.19 Table 17 w. = 42. 1% Test: UE2 0' = 4kg./Cm.z { 1 _ c wf - 26. 1% 0:02 51 62 {:3 13:03:3— 81) (81+é3)/2 Au 0 00 3 84 3.84 3 84 0 00 .84 .00 0 19 4 43 5.24 5 24 0 81 .84 .40 0.58 3.94 5.38 5.38 1.39 .87 .93 1 18 3 35 4.91 4 91 1 58 .13 .58 1 55 2 98 4.58 4 58 1 80 .78 .91 2 71 2 39 4.09 4 09 1 70 .24 .50 4.25 1.84 3.89 3.89 1.85 .77 .00 5 81 1 88 3.88 3 88 1 98 .87 .08 6. 65 1. 62 3. 66 3. 66 2. 04 . 64 . 08 8.98 1.70 3.78 3.78 2.08 .73 .08 7 35 1 72 3.78 3 78 2 04 .74 .08 8 15 1 88 3.81 3 81 1 93 .85 .13 9 70 1 73 3.88 3 88 1 93 .70 .08 12.00 1.79 3.53 3.53 1.74 .88 .01 13.80 1.80 3.38 3.38 1.58 .59 .01 76 A3. SAMPLE CALCULATIONS 2 l. UC1 Test: 0‘ = 3 kg./cm. ———— c Before consolidation: Length of sample = 5" Area of sample = 35. 4 sq. cm. Volume of sample = 450 c. c. After consolidation: Volume of sample = 341. 8 c. c. 3 Length of sample = 5 V341. 8/450 : 4. 6" 2/3 Area of sample 2 (341. 8/450) x 35.4 = 29. 5 sq. cm. . 4 Axial compression 2 0. 046" E % = 04066 x 100 = 1. 0% z . Axial load = 55 kg. Cell pressure 0'3 = 4 kg. /cm. 2 2 Pore-pressure u =1. 95 kg./cm. D . t t D _ _ 55 (4. 6-0. 046) ev1a or s ress — 01-03 — 29. 5 x 4.6 2 = 1. 85 kg./cm. 2 81:83 +13 =4.0 +1.85 =5.85 kg./cm. - 2 01:01- 11 = 5.85 -1.95 = 3.90 kg./cm. - 2 0' :0 -u =4.00 —1.95 = 2.05 kg./cm. 77 78 2 2. UC2 Test: O'C = 3 kg./Cm. Before consolidation 2 L23" A=10cm. V=76c.c. After consolidation L = 2. 75" A = 8. 44 sq. cm. V = 58 c. c. . . 0.11 Ax1a1compressmn = 0.11" 62% = 2 75 x 100 = 4. 0% Load dial 6 = 95 div. 2 Cell pressure 0 2. 43 kg. /cm. 3 2 Pore-pressure u 1.12 kg. /cm. Proving ring factor N = 0. 33 1b. /div. __ ' (2.75-0.11) (:1— 0‘3 —95x0.33x0.4545 2.75X 8.44 Deviator stres s D 2 1. 8 kg./cm. 2 0'1=03+D=2.43+1.8=4.23kg./cm. 6—1=81—u=4.23..1.12::3.111