Endochronic Constitutive Model for Sands and Its Application to Geotechnical Problems

Abstract

The introductions of large digital computers in the field of engineering have rendered possible the solution of a wide variety of problems without the need to violate the equilibrium and compatibility. The major requirement for such analysis is a good constitutive model that represents the stress strain behaviour of the materials in an accurate way. Nowadays for most of the geotechnical engineering applications the elastoplastic models like Mohr Coulomb model are widely used. All the existing constitutive models which represent the plastic behaviour of soil are developed from the fundamentals of classical theory of plasticity. The classical theory of plasticity is always associated with the concept of yield surface and potential surface to represent the plastic behaviour. The definition of yield surface depends on the location of the yield point. But in practical sense it is very difficult to find out the exact yield point for a material. The expression for yield and potential surfaces are simply mathematical expressions formulated for computational efficiency. Experimentally it is very difficult to find out the yield surface in the case of three dimensional stress spaces. The classical theory of plasticity is developed based on the mechanical process. It is believed that a theory which violates the thermodynamic principle is not able to represent the material behaviour accurately. the initial stage and combined to give the final state of stress. It was proved that the equation proposed by Wu and Wang (1983) can be used to represent the triaxial behaviour of sand very well. The dilation and densification behaviour can be predicted very well with the endochronic constitutive equations. The principal aim of this work is to implement the endochronic constitutive equation in the FLAC3D model library like any other constitutive model and validate it with the triaxial test data. After implementation and validation, the application of the particular constitutive model is extended to some practical geotechnical engineering problems like the stresses and displacements around an underground opening such as tunnels, surface settlement due to shallow tunneling, stress distribution below the footing, settlement analysis of footing on various foundation beds such as sand, clay and sand overlying clay bed, lateral displacement of the secant pile wall due to excavation and the force developed in the horizontal support etc. All the three problems validate the model with the analytical, experimental and field data respectively. The equation proposed by Wu and Wang (1983) is used for the present study. In order to validate the equation proposed by Wu and Wang (1983), MatLab programming is used. The hydrostatic, deviatoric and volumetric behaviour is obtained separately using the concerned equations. The equation is coded in the MatLab and analysis is done for a triaxial element test. Both drained and undrained analyses were done in order to study the behaviour. The pore pressure developed is captured separately with the equation proposed by Geoffrey et al (1975). The results obtained from the analysis of the MatLab were compared with that of the experimental results. The analysis shows that the equation captures the least plastic behaviour well for the triaxial element test. The dilation and densification behaviour obtained using the respective equation shows that it matches well with the experimental results. A parametric study is also conducted in MatLab to see how the parameters affect the stress strain and volumetric behaviour of the sand. The parametric study conducted with the MatLab shows that most of the parameters involved in the equation affect the plastic part of the stress strain curve rather than the initial elastic part. User defined constitutive model was written in visual C++ and compiled as DLL (Dynamic Link Library) files that will be loaded whenever it is needed in FLAC3D. In visual C++, header and source files were written by incorporating the constitutive equation proposed by Wu and Wang (1983), defining the variables and other functions etc, and a dynamic link library is created, which is then integrated into the 3D finite difference code FLAC3D using the CPPUDM module to simulate the stress strain behaviour of the materials. CPPUDM module is an additional option in FLAC3D to implement the user defined constitutive models. The visual C++ code was written in the form of incremental stress strain relationship. The model acts like any other constitutive model in the FLAC3D model library and can be loaded whenever it is required. For the validation of the model in FLAC3D, the data for the MatLab simulation were used. Both drained and undrained tests were simulated with the model. The results obtained from the analysis shows that by suitably selecting the parameters the model can simulate the stress strain behaviour of sand very well. The volumetric and deviatoric behaviour were observed and is matching well with the experimental data. In the case of the undrained test the pore pressure generation is well captured by the equation proposed by Geoffrey et al (1975). In urban areas the construction of shallow tunnels results in excessive settlements of the ground surface and thereby causes damage to the existing above ground structures. In order to minimize the settlements and to reduce the impact due to that, a prior analysis of the displacements and stresses around the opening is very important. Nowadays numerical analysis is widely used for the analysis of such structures. The most important requirement of such analysis is a constitutive model that can represent the unloading behaviour around the tunnel opening of sand very well. Here the endochronic constitutive model implemented in the FLAC3D model library is used to evaluate the stresses and displacements around the tunnel. In the analysis the tunnel is simulated as a cylindrical hole in an infinite medium with the in situ stress. The stresses at the springing line was observed and compared with the analytical solution. The results show that the results are matching well with the analytical results. The comparison of the results with that obtained using the Mohr Coulomb model is also done to see how the model differs from a widely used plastic model. By slightly adjusting the parameters the results obtained from both the models are in well agreement. The strain softening effect which is predominant around an underground opening due to the loosening of soil mass is well captured by the endochronic model compared to the Mohr Coulomb model. The settlement analysis shows that the model is almost in close agreement with the closed form solution proposed by Oteo & Sagaseta (1982) and the results obtained with the Mohr Coulomb model. The settlement trough formed for various shapes is wider and deeper than the Mohr Coulomb model. The vertical stress distribution around the opening of the tunnel is studied with varying the shape of the openings using the proposed constitutive model. The results obtained were compared with that of the Mohr Coulomb model. The slightly higher values in the case of endochronic model are basically due to its plastic nature. The displacement and stresses in the axial direction (along the excavation) is observed with the model. In the case of shallow tunnel the surface get influenced by the loosening of the soil mass which necessitates the use of the support system. The study shows that the model can be used for the simulation of underground opening like tunnel and will capture the behaviour well. Footings are structures used to support the buildings constructed above the ground. The settlement analysis of footings is very important when we consider the stability of the structures supported by it. The vertical stress distribution below the footing is studied using the endochronic constitutive model and compared with the analytical solution proposed by Boussinesq (1885). In the elastic range the model shows matching results with the Boussinesq’s solution. The settlement analysis of footing on various foundation beds such as sand, clay and sand overlying the clay bed were studied using the endochronic constitutive model implemented in the FLAC3D model library. The experimental data conducted in our lab (Sireesh (2006)) was used for the study. The results show that with the chosen parameters the results obtained with the endochronic model are in good agreement with the experimental data. The Mohr Coulomb model over predicts the results. This shows higher modulus value for the Mohr Coulomb model. By conducting the parametric study it was seen that by reducing the value of modulus for the Mohr Coulomb model, the results are in good agreement with the experimental value. The displacement and stress contours obtained for the two models were compared. By analyzing the displacement contours it is seen that the Mohr Coulomb model shows uniform settlement. In the case of endochronic model uniform settlement is observed for about 5% settlement that is in the elastic range. After a certain strain level the displacement contours are tilted more towards one side showing the rotational failure. Here the endochronic model captures the anisotropic behaviour associated with the materials like sand at higher strain level. This result is a concrete evidence that the model can capture the realistic behaviour very well compared to any other model. Even though the model developed is for sand its application can be extended to clay also. The size and shape of the footing is varied to study its effect on the pressure settlement curve. The analysis is done with square shape of 150mm side and circular shape of 150mm diameter. As there is not much variation in the area of influence, the shape has little influence on the pressure settlement curve. As the size of the footing increases the settlement increases for a given pressure. A parametric study is conducted by varying the modulus value used. The study shows that as the modulus value increases, the settlement reduces for a given bearing pressure. The endochronic model analysed with the lower modulus value shows that the model predicts the perfectly plastic behaviour, here the settlement increases for low value of bearing pressure. The application of endochronic model for the simulation of complex geotechnical engineering problems like footings is highly explored in the study. Nowadays most of the infrastructure facilities are concentrated towards the underground space. The excavation and construction of such structures in the urban areas results in damage to the existing above ground structures if the construction is done in close proximity to the structures. In the present study a staged excavation of an underground construction for the Bangalore metro project is simulated with the endochronic constitutive model. In the Bangalore metro project the excavation for the underground station is done at the cricket stadium site. At the site there are two major buildings such as the six storied Hindustan Aeronautical Limited building and 100 years old BSNL masonry building. To minimize the impact on these structures were a major concern during the construction of the work. A robust support system consists of secant pile walls, soldier piles and horizontal struts are installed at the site. The OSV method known as the Onsite Visualization and monitoring is conducted to minimize the damage to the existing structures and the accidents at the construction site. Sensors are connected to LEDs which show change in color when the displacements and forces cross the triggered value. The field instrumentation is done with inclinometers, tilt meters and strain gauges connected to the sensors to observe the lateral deformation of the secant pile wall, tilt of the HAL building and the forces developed in the horizontal struts. The monitoring of field data is done for a period of five months from July to November. From the analysis of the field observed data it is clear that the support system provided were strong enough to resist the forces due to unloading. The lateral deformation of the secant pile wall and the forces developed in the strut were numerically analysed using the endochronic constitutive model and the results were compared with the field monitored data. The results show that the model captures the behaviour very close to the field data for a slightly higher modulus than that reported in the geotechnical report (BMRC report). This may be due to the fact that the value of modulus calculated experimentally might have some error. The analysis with the Mohr Coulomb model shows that the model over predicts the results very close to the surface of the excavation. This indicates that the influence of load is more on that particular depth for the Mohr Coulomb model. But the stiffness of the lateral support system is uniform throughout the depth; the endochronic model predicts the result more accurately than the Mohr Coulomb model. The strut forces developed in the horizontal support system is observed using the two models. The strut forces in the field is affected by so many factors such as the temperature variation, stages of excavation and other live loads acting on the site, so an exact comparison with the field data is quite difficult. The analysis shows that even though it is difficult to simulate the exact three dimensional nature of the problem in the present study the endochronic constitutive model captures the behaviour very well. The results obtained shows that the endochronic constitutive model implemented in the FLAC3D model library provides a very promising solution like any other constitutive model. As the theory is based on the irreversible law of thermodynamics and the formulation of the constitutive equation are based on the internal energy concept it can represent the material behaviour in accordance with the laws of continuum mechanics. The anisotropic behaviour of soil at higher strain level is well represented through the footing problem. The endochronic constitutive model is a very simple one to simulate the stress strain behaviour of the materials without the concept of yield surface; the parameters used in the equation can be obtained directly from a single triaxial stress strain plot. This study highlights the importance of a model without the concept of yield surface to capture the stress strain behaviour of any materials. Since the model is of completely plastic nature it has its own uniqueness in capturing the material behaviour due to loading and unloading.