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This paper presents the results of a series of triaxial tests with dry sand at confining pressures varying from 1.5 kPa to 100 kPa at relative densities of 0.20, 0.59, and 0.84. The results, which are in reasonable accordance with an equation given by Bolton, show that the friction angle is strongly dependent on the stress level and on the basis of the test results, a nonlinear Mohr failure criterion has been proposed. This yield criterion has been implemented in a finite element program and an analysis of the bearing capacity of a circular shaped model foundation, diameter 100 mm, has been conducted. Comparisons have been made with results from 1g model scale tests with a foundation of similar size and a good agreement between numerical results and test results has been found.

In conventional design of shallow foundations, earth retaining structures, slopes, and the friction angle of sand are regarded as a constant, being primarily dependent on the relative density of the material. Experimental work (e.g., [

One of the main effects in geotechnical structures always stems from body forces and therefore the stress level in simple laboratory experiments is always much lower than the stress level in real life. For this reason very often, one has to accept large discrepancies between results from these laboratory tests and the equivalent conventional geotechnical calculations. As examples of this Hansen [

When the friction angle is constant and the response is considered linearly elastic perfectly plastic, the material model is termed Mohr-Coulomb, which is the one used in standard geotechnical designs. For this reason a lot of experience has been gathered regarding the parameters used in the model (Young’s modulus

The Mohr-Coulomb model predicts a linear relation between the normal stress acting on a slip surface and strength. Experiments, however, have shown that this is not true. When carried out at low stress levels in the laboratory model experiments predict a far higher friction angle compared to the one obtained at realistic stress levels.

To allow for the nonlinear dependency of the strength on the stress level, physical modelling in connection with centrifuge tests has become widespread over the past thirty to forty years. Also over these years, there has been a great increase in the computational power available, due to bigger and cheaper computers combined with very efficient modelling methods such as the finite element method, and it seems obvious that everyday design of soil structures could now be based on a more advanced model than the Mohr-Coulomb model. In comparison rock mass structures are now routinely being designed using the nonlinear Hoek-Brown criterion [

The main aims of the work described in the present paper are threefold. Firstly, it is to determine a simple failure envelope of the type of sand used for simple model tests in the geotechnical laboratory of Esbjerg Institute of Technology in Denmark. Secondly, the model is implemented in an elastoplastic finite element code in order to carry out bearing capacity calculations. Thirdly, these bearing capacity calculations are compared with model tests carried out in the laboratory.

Another purpose of the paper is to advocate the need for a slightly more complicated failure criterion in practical calculations, compared to conventional use of the constant Mohr-Coulomb friction angle. With the modern day availability of inhouse and commercial numerical computations codes, it should now be tractable to carry out routine geotechnical design with more realistic material models. The purpose of this paper is not to propose an all-encompassing constitutive model as it is for example seen in [

As a validation of the model, results from numerical simulations of the bearing capacity of a circular model footing are compared with the results of simple model tests carried out in the geotechnical laboratory of Esbjerg Institute of Technology in Denmark. The footing has a diameter of 10 cm and tests are carried out on sand at different relative densities. The numerical simulations are based on a nonlinear Mohr failure envelope of which the parameters have been determined through triaxial tests.

The sand used in the experiments is Esbjerg sand, which is an alluvial, medium grained sized quartz sand of subangular shape with the characteristics given in Table

Properties of Esbjerg sand.

Parameter | Value |
---|---|

_{10} | 0.25 |

_{60} | 0.58 |

_{u} = D_{60}/D_{10} | 2.32 |

_{50} | 0.50 |

Specific density | 2.621 |

Maximum void ratio e_{max} | 0.733 |

Minimum void ratio e_{min} | 0.449 |

Relative density in tests | 0.20, 0.59, 0.84 |

Dry unit weight in tests (kN/m^{3}) | 15.64 16.74 17.54 |

The triaxial tests were carried out with dry sand in a triaxial testing apparatus. The confining pressure was applied by lowering the air pressure inside the test specimen. The diameter and height of the test specimens were 70 mm. A schematic drawing of the load test set up is shown in Figure

Schematic drawing and photograph of the triaxial test setup.

The deviatoric stress was applied using a hydraulic cylinder operated with a hand pump, and the force was recorded by means of an electronic load cell which was placed inside the cell to obtain as accurate values as possible. The confining pressure was provided by a vacuum pump connected to the specimen through the loading plates at either end of the specimen and the amount of vacuum was controlled by a valve operated by hand and recorded using a pressure transducer. The vertical displacements were measured by means of two displacement transducers mounted outside the cell and the displacements were taken as average values of these two transducer readings. The volumetric changes of the specimens were recorded by a displacement transducer which could register the movements of the water table in a burette connected to the water filled cell. The diameter of both loading plates is 90 mm, and the thickness is 50 mm. The membrane, made from 0.30 mm rubber, was fastened to the base plate and sealed with an O-ring and held by vacuum to the inner surface of a cylindrical split mould. Both the bases made from aluminium and the nylon cap were provided with 2 layers of lubricated 0.30 mm latex sheets.

The specimens were prepared by preweighing the specific amount of sand to obtain the desired relative density and the sand was placed through a funnel in an appropriate number of steps. Between each of the steps the sand was compacted by tamping to obtain the desired density. For the loose samples (relative density,

After the three displacement transducers and the hydraulic cylinder had been put in position, the vacuum was removed and all transducers were zeroed. Hereafter, the specimen was isotropically consolidated by application of vacuum to the desired level of the effective confining pressure and after a few minutes when it was clear from the transducer readings, that there were no further volume changes, all transducers were zeroed again and the load was applied, so as to produce a rate of displacement of approximately 5mm pr minute. For the two lowest values of the confining pressure the confining was provided solely by the water pressure.

The additional radial stress due to the stiffness of the membrane has been taken account by the equation [

Corrections to the deviatoric stress due to the increase in the cross-sectional area of the sample during testing have been made by computing the deviatoric stress from the equation:

Triaxial tests were carried out for relative densities of 0.20, 0.59, and 0.84 and for each relative density, the following values of the initial confining stress were applied: 1.5 kPa, 5.3 kPa, 20 kPa, 50 kPa, and 100 kPa. The results are shown in Figures

Test results for Esbjerg sand, *estimated value.

Confining pressure _{3} [kPa] | Relative density | Peak angle of friction _{peak} [deg.] | Angle of dilation _{max} [deg.] | Modified peak angle of friction _{mod} [deg.] |
---|---|---|---|---|

1.5 | 0.20 | 44.3 | 7.0* | 37.2 |

5.3 | 0.20 | 39.0 | 5.2 | 33.6 |

20 | 0.20 | 35.2 | 3.5 | 30.8 |

50 | 0.20 | 33.0 | 2.7 | 29.2 |

100 | 0.20 | 31.8 | 1.5 | 28.1 |

1.5 | 0.59 | 47.6 | 17.0* | 42.0 |

5.3 | 0.59 | 44.8 | 13.4 | 39.3 |

20 | 0.59 | 42.4 | 10.8 | 37.2 |

50 | 0.59 | 40.0 | 11.0 | 35.7 |

100 | 0.59 | 38.9 | 9.4 | 34.6 |

1.5 | 0.84 | 53.3 | 23.0* | 47.0 |

5.3 | 0.84 | 48.6 | 18.3 | 43.0 |

20 | 0.84 | 46.1 | 14.8 | 40.1 |

50 | 0.84 | 42.4 | 15.5 | 38.7 |

100 | 0.84 | 41.3 | 13.7 | 37.6 |

Test results for Esbjerg sand—

Test results for Esbjerg sand—

Test results for Esbjerg sand—

In this equation, which is derived from the linear Mohr-Coulomb failure criterion,

The angle of dilation

It is a well-known fact that dilation plays an important role in the study of the strength of soil [

The test results have been compared with values found with the Bolton equation [

Friction angles for Esbjerg Sand.

The tests have shown that the angle of friction varies significantly with the confining pressure and that the greater the confining stress, the smaller the angle of friction. This variation is greater for smaller values of the confining stress. The values of the friction angle according to Bolton’s equation are, in general, smaller than the test values. These differences are greater for smaller values of the relative density and the confining pressure and this tendency has been confirmed by the tests carried out by Ponce and Bell, while the tests by Fukushima and Tatsouka have shown no variation of the friction angle with confining pressures below 50 kPa. From Figures

The linear Mohr-Coulomb failure envelope is given by the equation:

As discussed above the linear Mohr-Coulomb envelope is a poor fit to the test results at small stress levels where the dependency of

Several authors have suggested yield functions that take the

Parameters in nonlinear yield function.

Relative density | _{0} | _{c0} | |
---|---|---|---|

0.20 | 2.6613 | 12.0619 | 0.9082 |

0.59 | 3.4409 | 18.8313 | 1.0887 |

0.84 | 3.9355 | 19.7073 | 1.4330 |

Failure envelopes and test results,

The failure criterion of (

As pointed out by Siddiquee et al. [_{50}

To verify the validity of the proposed nonlinear yield function, 1g model tests in axisymmetric conditions were carried out. The sand used for the tests was the Esbjerg sand described above, and the tests were conducted at relative densities

The load was applied to the footing by a hydraulic jack mounted on a steel beam, which was fastened to the concrete floor in the lab using 28 mm threaded steel bars. During the test, the load was recorded using a load transducer, HBM S9, and the vertical displacements were recorded by a displacement transducer HBM WA/50 mm. Both load and displacement transducers were calibrated before the tests. The load was raised continuously, and the rate of displacement was app. 5 mm/minute. The test values were recorded by means of a datalogger Spider 8 from HBM. A photo of the load test setup is shown in Figure

Load test arrangement.

Bulging was observed both for the medium dense and the dense sand. The bulge never reached the edge of the testing container, indicating that the size was adequately large, in order to have only minor influence on the test results. Especially for the dense sand, heave of the surface was pronounced, meaning that the outermost slip line, separating the soil body at yield from the soil at rest, was easily identified.

The ultimate bearing capacity of a rough, shallow, circular foundation resting on a cohesionless material is traditionally found from the following equation, given in several textbooks:

In the tests in this project, the footings are initially resting on the surface of the sand; that is the contribution from the overburden pressure to the ultimate load is zero at the beginning of the test, but as the load increased, the footing sinks into the ground introducing a vertical effective stress at the foundation level, which cannot be ignored, as it accounts for a significant part of the bearing capacity of the foundation.

The bearing capacity factors _{f}

Definition of failure loads.

For shallow foundations, three failure modes have been described by Vesić [

When the footing fails in general shear, there is no doubt as to the magnitude of the failure load, as the loaddisplacement curve displays a pronounced peak, as shown by the red curve in Figure

Failure loads from tests.

Test no. | Relative density | Displacement at failure [mm] | Load at failure in test [kPa] | Modified failure load [kPa] |
---|---|---|---|---|

A1 | 0.59 | 12.6 | 51 | 51 |

A2 | 0.59 | 9.8 | 50 | 50 |

A3 | 0.59 | 8.7 | 67 | 67 |

A4 | 0.59 | 14.8 | 60 | 60 |

A5 | 0.56 | 26.7 | 44 | 50 |

A6 | 0.61 | 9.0 | 62 | 56 |

A7 | 0.58 | 8.0 | 56 | 58 |

A8 | 0.58 | 10.0 | 59 | 62 |

A9 | 0.58 | 9.5 | 59 | 62 |

A10 | 0.61 | 9.5 | 61 | 56 |

B1 | 0.75 | 4.5 | 110 | 158 |

B2 | 0.80 | 5.6 | 132 | 155 |

B3 | 0.84 | 6.3 | 116 | 116 |

B4 | 0.84 | 7.5 | 158 | 158 |

B5 | 0.80 | 6.7 | 120 | 141 |

B6 | 0.84 | 7.3 | 144 | 144 |

B7 | 0.80 | 6.5 | 138 | 161 |

B8 | 0.82 | 6.5 | 136 | 147 |

B9 | 0.75 | 8.0 | 104 | 152 |

B10 | 0.80 | 6.5 | 126 | 147 |

Summary of results from tests and FEM calculations.

Relative density | Diameter of footing [mm] | Failure load from FEM analysis [kPa] | Failure load from tests [kPa] | Standard deviation of failure loads [kPa] |
---|---|---|---|---|

0.59 | 100 | 71 | 57 | 5.7 |

0.84 | 100 | 151 | 148 | 13.0 |

For tests where the relative density is not equal to the reference values of 0.59 or 0.84, a modification of the measured value has been made. This modification is necessary in order to be able to calculate the average failure load for

It can be seen from (

Results from linear Mohr-Coulomb analysis.

Friction angle [°] | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 |
---|---|---|---|---|---|---|---|---|

Failure load [kPa] | 37 | 46 | 57 | 71 | 89.0 | 111 | 141 | 174 |

By comparing the values of the failure loads in Table

As an example, the modification of the result for test A7 is demonstrated.

Triaxial tests carried out on medium and dense Esbjerg sand at low stress levels show that the triaxial angle at peak depends strongly on the stress level. For confining pressures above 20 kPa, there is a reasonable match with results obtained from the equation proposed by Bolton. Because the friction angle is dependent on the confining pressure, the linear Mohr-Coulomb yield criterion is ill-suited for the determination of the failure load of 1g model scale footings. Therefore, on the basis of the results from the triaxial tests, a nonlinear Mohr failure criterion has been proposed and implemented in a finite element program. To overcome numerical difficulties due to the nonassociative behaviour of sand the associative flow rule is used, but with yield parameters modified with the equation given by Davis. Results from the finite element analysis of the bearing capacity of 100 mm diameter footings show a good agreement with results obtained from simple model-scale footing tests on medium and dense sand.

The authors wish to thank the Lida and Oskar Nielsen Foundation for a substantial, financial support. Also the guidance given by professor Poul V. Lade in connection with the triaxial tests is highly appreciated.