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# Craig's Soil Mechanics 7th Edition

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of superposition can be used in cases of a number of soil layers each having a different value of E (see Example 5.4). The above solutions for vertical displacement are used mainly to estimate the immediate settlement of foundations on saturated clays; such settlement occurs under undrained conditions, the appropriate value of Poisson’s ratio being 0.5. The value of the undrained modulus Eu is therefore required and the main difficulty in predicting immediate settlement is in the determination of this parameter. A value of Eu could be determined by means of the undrained triaxial test. However, such a value would be very sensitive to sampling disturbance and would be too low if the unconsolidated– undrained test were used. If the specimen were initially reconsolidated then a more realistic value of Eu would be obtained. Consolidation may be either isotropic under 1⁄2 to 2⁄3 of the in-situ effective overburden pressure, or under K0 conditions to simulate the actual in-situ effective stresses. If possible, however, the value of Eu should be deter- mined from the results of in-situ load tests or pressuremeter tests. It should be recognized, however, that the value obtained from load tests is sensitive to the time interval between excavation and testing, because there will be a gradual change from the undrained condition with time; the greater the time interval between excavation and testing the lower the value of Eu. The value of Eu can be obtained directly if settlement observations are taken during the initial loading of full-scale foundations. For particular clays, correlations can be established between Eu and the undrained shear strength parameter cu. It has been demonstrated that for certain soils, such as normally consolidated clays, there is a significant departure from linear stress–strain behaviour within the range of working stress, i.e. local yielding will occur within this range, and the immediate settlement will be underestimated. A method of correction for local yield has been given by D’Appolonia et al. [4]. Figure 5.14 Contact pressure under rigid area: (a) clay and (b) sand. Displacements from elastic theory 157 In principle the vertical displacement under fully drained conditions could be estimated using elastic theory if the value of the modulus for this condition (E0) and the value of Poisson’s ratio for the soil skeleton (�0) could be determined. Example 5.4 A foundation 4� 2m, carrying a uniform pressure of 150 kN/m2, is located at a depth of 1m in a layer of clay 5m thick for which the value of Eu is 40MN/m 2. The layer is underlain by a second clay layer 8m thick for which the value of Eu is 75MN/m2. A hard stratum lies below the second layer. Determine the average immediate settlement under the foundation. Now, D/B ¼ 0:5, and therefore from Figure 5.15, �0 ¼ 0:94. 1 Considering the upper clay layer, with Eu ¼ 40MN/m2: H=B ¼ 4=2 ¼ 2; L=B ¼ 2 ; �1 ¼ 0:60 Figure 5.15 Coefficients for vertical displacement. 158 Stresses and displacements Hence, from Equation 5.28 si1 ¼ 0:94� 0:60� 150� 2 40 ¼ 4:2mm 2 Considering the two layers combined, with Eu ¼ 75MN/m2: H=B ¼ 12=2 ¼ 6; L=B ¼ 2 ; �1 ¼ 0:85 si2 ¼ 0:94� 0:85� 150� 2 75 ¼ 3:2mm 3 Considering the upper layer, with Eu ¼ 75MN/m2: H=B ¼ 2; L=B ¼ 2 ; �1 ¼ 0:60 si3 ¼ 0:94� 0:60� 150� 2 75 ¼ 2:3mm Hence, using the principle of superposition, the settlement of the foundation is given by si ¼ si1 þ si2 � si3 ¼ 4:2þ 3:2� 2:3 ¼ 5mm PROBLEMS 5.1 Calculate the vertical stress in a soil mass at a depth of 5m vertically below a point load of 5000 kN acting near the surface. Plot the variation of vertical stress with radial distance (up to 10m) at a depth of 5m. 5.2 Three point loads, 10 000, 7500 and 9000 kN, act in line 5m apart near the surface of a soil mass. Calculate the vertical stress at a depth of 4m vertically below the centre (7500 kN) load. 5.3 Determine the vertical stress at a depth of 3m below the centre of a shallow foundation 2� 2m carrying a uniform pressure of 250 kN/m2. Plot the variation of vertical stress with depth (up to 10m) below the centre of the foundation. 5.4 A shallow foundation 25� 18m carries a uniform pressure of 175 kN/m2. De- termine the vertical stress at a point 12m below the mid-point of one of the longer sides (a) using influence factors, (b) by means of Newmark’s chart. 5.5 A line load of 150 kN/m acts 2m behind the back surface of an earth-retaining structure 4m high. Calculate the total thrust, and plot the distribution of pres- sure, on the structure due to the line load. 5.6 A foundation 4� 2m carries a uniform pressure of 200 kN/m2 at a depth of 1m in a layer of saturated clay 11m deep and underlain by a hard stratum. If Eu for the clay is 45MN/m2, determine the average value of immediate settlement under the foundation. Problems 159 REFERENCES 1 Britto, A.M. and Gunn, M.J. (1987) Critical State Soil Mechanics Via Finite Elements, Ellis Horwood, Chichester. 2 Burland, J.B. (1970) Discussion, in Proceedings of Conference on In-Situ Investigations in Soils and Rocks, British Geotechnical Society, London, p. 61. 3 Christian, J.T. and Carrier III, W.D. (1978) Janbu, Bjerrum and Kjaernsli’s chart reinter- preted, Canadian Geotechnical Journal, 15, 123, 436. 4 D’Appolonia, D.J., Poulos, H.G. and Ladd, C.C. (1971) Initial settlement of structures on clay, Journal of the ASCE, 97, No. SM10, 1359–77. 5 Fadum, R.E. (1948) Influence values for estimating stresses in elastic foundations, in Proceedings of the 2nd International Conference of SMFE, Rotterdam, Vol. 3, pp. 77–84. 6 Gibson, R.E. and Anderson, W.F. (1961) In-situ measurement of soil properties with the pressuremeter, Civil Engineering and Public Works Review, 56, 615–18. 7 Giroud, J.P. (1972) Settlement of rectangular foundation on soil layer, Journal of the ASCE, 98, No. SM1, 149–54. 8 Hill, R. (1950) Mathematical Theory of Plasticity, Oxford University Press, New York. 9 Hughes, J.M.O., Wroth, C.P. and Windle, D. (1977) Pressuremeter tests in sands, Geotech- nique, 27, 455–77. 10 Mair, R.J. and Wood, D.M. (1987) Pressuremeter Testing: Methods and interpretation, CIRIA/Butterworths, London. 11 Marsland, A. and Randolph, M.F. (1977) Comparisons of the results from pressuremeter tests and large in-situ plate tests in London clay, Geotechnique, 27, 217–43. 12 Newmark, N.M. (1942) Influence Charts for Computation of Stresses in Elastic Foundations, University of Illinois, Bulletin No. 338. 13 Palmer, A.C. (1972) Undrained plane strain expansion of a cylindrical cavity in clay: a simple interpretation of the pressuremeter test, Geotechnique, 22, 451–7. 14 Potts, D.M. and Zdravkovic, L. (1999) Finite Element Analysis in Geotechnical Engineering: Theory, Thomas Telford, London. 15 Potts, D.M. and Zdravkovic, L. (2001) Finite Element Analysis in Geotechnical Engineering: Application, Thomas Telford, London. 16 Poulos, H.G. and Davis, E.H. (1974) Elastic Solutions for Soil and Rock Mechanics, John Wiley & Sons, New York. 17 Scott, R.F. (1963) Principles of Soil Mechanics, Addison-Wesley, Reading, MA. 18 Timoshenko, S. and Goodier, J.N. (1970) Theory of Elasticity, 3rd edn, McGraw-Hill, New York. 19 Windle, D. and Wroth, C.P. (1977) The use of a self-boring pressuremeter to determine the undrained properties of clay, Ground Engineering, 10 (6), 37–46. 20 Wroth, C.P. (1984) The interpretation of in-situ soil tests, Geotechnique, 34, 447–89. 160 Stresses and displacements Chapter 6 Lateral earth pressure 6.1 INTRODUCTION This chapter deals with the magnitude and distribution of lateral pressure between a soil mass and an adjoining retaining structure. Conditions of plane strain are assumed, i.e. strains in the longitudinal direction of the structure are assumed