This paper proposes a numerical framework to simulate in-situ free fall cone penetration tests conducted on soft and sensitive marine clay. First, the free fall cone penetration tests are carried out at an offshore site in the northern part of the Gulf of Finland, Baltic Sea. The numerical simulation employs the Generalized Interpolation Material Point Method to replicate the process of indentation of the cone penetrometer into the clay. The clay is modelled using an advanced constitutive model that considers the effect of strain rate and strain softening associated with the dynamic penetration process. The simulation uses a friction contact model to represent the interface between the cone penetrometer and the clay. The numerical simulation accurately replicates the penetration process associated with the tests. @en

The paper presents experimental results of investigations of offshore deposits from the Gulf of Finland in the Baltic Sea. Firstly, it briefly reviews the sedimentary history of the studied site. Then, it presents results from marine geological investigations, including seismoacoustic reflection sub-bottom profiles and long sediment cores collected by piston coring. Finally, the paper presents the physical and mechanical properties of the intact soil samples through geotechnical laboratory tests. The data show the distribution of water content, bulk density, undrained shear strength, sensitivity, and organic content of the samples at various depths up to approximately 4 meters. The undrained shear strengths, obtained by fall cone tests, are scattered and the variations are ±35% for the studied specimens. A quantitative correlation between geotechnical data and local sedimentary units and depositional history was not possible in this study. Further efforts are needed to reduce the uncertainties and measurement errors in determining the undrained shear strengths in geotechnical laboratory.@en

This paper shows a numerical replication of a laboratory-scale free fall cone penetrometer test of marine clay. The numerical simulation involves large deformations and considers the destructuration of clay, strain rate effects, and non-linear material behaviour. The numerical simulation well replicates the laboratory experiment captured on a high-speed camera. The penetration process is replicated accurately in time, and the depth of the penetration corresponds to that obtained in an experiment. The simulation results indicate that the numerical framework implemented in Uintah software, consisting of an advanced soil model and the Generalized Interpolation Material Point Method, is well-suited for replication of the dynamic penetration process in soft and sensitive marine clay.@en

The paper shows a numerical simulation of a fall cone test done on a soft and sensitive marine clay sample collected from the Gulf of Finland in the Baltic Sea. The simulation involves large deformations and takes into account the destructuration of clay, strain rate effects, and non-linear material behaviour. First, the analysed fall cone test was captured with a high-speed camera (1000 frames/sec). MATLAB image processing tools helped to quantify the cone displacement, velocity, and acceleration during the indentation process. Further, we simulate the process of indentation of the cone into the soils using the Generalized Interpolation Material Point Method (GIMP). In the simulation, the undrained shear strength of soil (𝑠𝑢) is taken as a function of strain rate to consider the effect of strain rate on cone indentation, as well as a function of destructuration. The presented numerical simulation replicates well the shown experiment. The simulation results indicate that the present material framework, along with GIMP, can be used to simulate the dynamic penetration process on soft and sensitive clay.@en

In situ free-fall cone penetrometers (FF-CPTs) allow for rapid investigations of undrained shear strength of soft shallow water deposits. Often samples are also studied in geotechnical laboratory by fall cone tests. This study presents in situ FF-CPT and laboratory fall cone testing results of an offshore site in the northern part of the Gulf of Finland, Baltic Sea. Site investigations included seismoacoustic surveys and sediment coring. Based on seismoacoustic reflection profiles and long sediment cores, the local deposits were classified and divided into glaciolacustrine, post-glacial lacustrine and brackish-water stratigraphic units. FFCPT measurements provided in situ undrained shear strength and dynamic cone resistance of the seabed geological units. Laboratory fall cone tests were performed on the vertical core samples at every 0.5m. Comparisons were established between undrained shear strength from FF-CPT tests and from fall cone tests.@en

The present study examines the stability of an embankment of cohesive-frictional material placed over a purely cohesive stratum which is reinforced by means of uniformly spaced vertical granular cylindrical columns. By using the kinematic yield homogenization concept, the macroscopic yield strength envelope associated with the reinforced clay layer has been obtained numerically for different volumetric fractions of the granular column to the composite unit cell by using the semidefinite programming (SDP) technique. An upper bound approximation of the macroscopic yield criterion, using the least square method, is expressed as a summation of a number of convex ellipsoidal envelopes so that the solution for computing the stability factors for the embankment on granular column (in the presence of surcharge pressure) can be obtained later with the usage of the second order cone programming (SOCP) while performing the upper bound finite elements limit analysis. The methodology avoids the usage of any stress variable and the complete procedure has been clearly described and the associated code has been written in MATLAB. The study will be not only useful for designing embankments constructed over soft clay reinforced with granular columns but also for solving different geomechanics stability problems by using the homogenization based kinematic limit analysis approach.

@en

The present study examines the stability of an embankment of cohesive-frictional material placed over a purely cohesive stratum which is reinforced by means of uniformly spaced vertical granular cylindrical columns. By using the kinematic yield homogenization concept, the macroscopic yield strength envelope associated with the reinforced clay layer has been obtained numerically for different volumetric fractions of the granular column to the composite unit cell by using the semidefinite programming (SDP) technique. An upper bound approximation of the macroscopic yield criterion, using the least square method, is expressed as a summation of a number of convex ellipsoidal envelopes so that the solution for computing the stability factors for the embankment on granular column (in the presence of surcharge pressure) can be obtained later with the usage of the second order cone programming (SOCP) while performing the upper bound finite elements limit analysis. The methodology avoids the usage of any stress variable and the complete procedure has been clearly described and the associated code has been written in MATLAB. The study will be not only useful for designing embankments constructed over soft clay reinforced with granular columns but also for solving different geomechanics stability problems by using the homogenization based kinematic limit analysis approach.

@en

Bearing capacity of rectangular foundations, with varying aspect ratios (1 ≤ L/B ≤ 5) and embedded at shallow to medium depths (D/B ≤ 5), has been evaluated by using the three-dimensional cell-based smoothed finite elements quasi-kinematic limit analysis. The results have also been obtained for circular and strip foundations by performing exclusively the axisymmetric and plane strain analyses. The associated optimization problem has been solved by employing the semidefinite programming (SDP). Shape factors (sc, sq and sγ) and depth factors (dc, dq and dγ) are given as a function of soil internal friction angle (ϕ). The shape factors have been expressed as a function of L/B, on the other hand, the depth factors have been given as a function of D/B and L/B. A thorough comparison of the results has been made with the different solutions available in literature. The variations of (i) power dissipation function, (ii) maximum plastic shear strain rates, and (iii) the nodal velocities patterns, have also been examined to interpret the failure mechanism.@en

Finite element-based upper bound and lower bound limit analysis are found to be an excellent tool for solving various stability structural problems with less computational cost. The accuracy and efficiency of the obtained solution is highly dependent on the intensity of elements in the high stress or stain gradient region. Therefore, an adaptive meshing procedure is required to reduce the gap between upper and lower bound solutions. The adaptive meshing procedure presented in this paper is based on two basic concepts (a) a posterior error indicator and (b) an unstructured mesh generator. The posterior error estimator is based on intensity of maximum shear strain rate (γmax) has been used. Open-source software TRIANGLE and TETGEN have been used to generate unstructured meshes for 2D and 3D problems, respectively. A plane strain and a 3D foundation problem have been analyzed and steps involve in calculation have been clearly mentioned in the paper. The present analysis method is found to be efficient and accurate.@en

While performing the upper bound limit analysis, using an associated flow rule, the expressions for the power dissipation function corresponding to different forms of yield criteria for clays exhibiting shear strength anisotropy and transversely asymmetric properties (implying different strengths in compression and tension), have been derived in terms of velocities, strain rates and shear strength parameters without involving stress variables by using the concept of duality in an optimization theory. The optimization has been performed in an efficient manner by using the second order cone programming (SOCP). The problems of strip and square footings have been dealt with. The lower bound solutions have also been established for the plane strain problems, and the two bound solutions have been found to match very closely with each other. The results also match well with the different solutions reported in literature. The variations of the power dissipation function and nodal velocity patterns in the problem domain have also been studied. The proposed formulation(s) will be useful for solving different two and three-dimensional geomechanics problems in clays to account for shear strength anisotropy as well as transversely asymmetric properties.@en

By using the lower bound finite elements limit analysis, this paper presents a methodology to evaluate the bearing capacity of piles in cohesive soils. The non-homogeneous nature of soil has been assumed for which the cohesion increases linearly with depth. An axisymmetric formulation, comprising of a planar domain with a mesh of three noded triangular elements has been adopted. Here, Mohr–Coulomb (MC) yield criterion has been employed. The yield criterion is expressed in terms of three quadratic constraints so that the problem can be solved using second-order cone programming (SOCP). Bearing capacity factors have been generated as a function of (i) height to radius ratio of the pile, (ii) rate of increase of cohesion and (ii) roughness of base and shaft of pile. The results computed from the analysis have been compared thoroughly with that reported in literatures.@en

A three-dimensional kinematic limit analysis approach based on the radial point interpolation method (RPIM) has been used to compute collapse loads for rectangular foundations. The analysis is based on the Mohr-Coulomb yield criterion and the associated flow rule. It is understood that the internal plastic power dissipation function and flow rule constraints can be expressed entirely in terms of plastic strain rates without involving stresses. The optimization problem has been solved on basis of the semidefinite programming (SDP) by using highly efficient primal-dual interior point solver MOSEK in MATLAB. The results have been presented in terms of the variation of the shape factors with changes in the aspect ratio (L/B) of the footing for different values of soil internal friction angle (ϕ). Computations have revealed that the shape factors, sc and sq, due to effects of cohesion and surcharge increase continuously with (1) decrease in L/B and (2) increase in ϕ. On the other hand, the shape factor sγ, due to the effect of soil unit weight, increases very marginally with an increase in L/B up to (1) ϕ = 25° for a rough footing and (2) ϕ = 35° for a smooth footing. Thereafter, for greater values of ϕ, the variation of sγ with L/B has been found to be quite similar to that of the factors sc and sq. The variations of (1) nodal velocity patterns, (2) plastic power dissipation, and (3) maximum plastic shear strain rates have also been examined to interpret the associated failure mechanism.@en

By employing different smoothed finite element (S-FE) methods, the kinematic limit analysis approach has been presented by using three noded triangular elements to solve plane strain and plane stress stability problems on basis of the Mohr-Coulomb yield criterion. By using the concept of duality in conic programming, the expressions for the power dissipation function and the associated constraints have been written entirely in terms of kinematic variables. The optimization formulation is framed as a second order cone programming problem. Based on the absolute maximum magnitude of the shear strain rate, an algorithm to update the mesh has been provided. Amongst the different approaches, the node-based S-FE formulation has been found to be very accurate as well as efficient to solve large scale stability problems. The selective edge-based and the edge-based with a bubble node approaches also generate quite accurate solutions though require a little more computational time.@en

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1) three-noded constant strain finite elements and (2) six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements’ interfaces.@en

The writers are thankful to the discussers for their constructive discussion on the work. Rather than keeping 𝛼=0.5 for all geological strength index (GSI) values, the discussers presented a method to account for the actual value of 𝛼 in the generalized Hoek-Brown (GHB) criterion by using the reduced values of the material input parameters to solve the problem by using the modified Hoek-Brown (MHB) criterion; for the definitions of GSI and 𝛼, refer to the original paper, and the MHB and GHB criteria are defined exactly the same way as introduced by the discussers. In the original paper, it was pointed out that due to an assumption of using 𝛼=0.5, a significant overestimation of the collapse load occurs for GSI≤30. The discussers provided a detailed investigation on exploring how the choice of 𝛼 affects the solutions for values of GSI≤30. Figs. 2–4 plotted by the discussers provide the difference between the MHB and GHB criteria for different values of GSI, stress range, and 𝑚𝑖. These figures explain the reason for the difference between the obtained solution and that reported in the literature by Merifield et al. (2006) and Clausen (2013). The main contribution of the discussers is that a new method has been proposed to take into account the true value of 𝛼 by choosing the reduced values of the input strength parameters (𝜎𝑐⁢𝑖, 𝑚𝑏, and 𝑠) to be used in the MHB criterion based on the least-square optimization so that the yield envelope of the MHB criterion fits closely with the GHB criterion. The method introduced by the discussers is simple and straightforward and can be easily implemented. By using two worked examples, the discussers indicated that the reduced material input parameters can significantly reduce the strength overestimation of the MHB criterion for lower values of GSI, although no comparison of the final solution with the GHB was provided in the discussion. The major disadvantage, however, associated with the recommended approach of the discussers is the choice of appropriate range of 𝜎1 in order to compute the reduced values of the input strength parameters to be used in the MHB criterion. It is, however, going to be a difficult exercise to select the correct range of 𝜎1. If the selected range of 𝜎1 is kept very large, it is not going to be an easy task to find the perfect match between the two yield envelopes using the GHB and the MHB criteria.@en

The lower-bound finite elements limit analysis in combination with semidefinite programming (SDP) has been used to solve stability problems involving a modified Hoek-Brown (HB) yield criterion with the exponent 𝛼=0.5. In order to demonstrate the applicability of the proposed computational approach, bearing capacities of strip and circular foundations on rock mass have been determined. The results have been given in the form of nondimensional bearing capacity factors as a function of different material input parameters. By comparing the results obtained from the present analysis with that reported in literature, it is noted that the proposed approach remains quite accurate and is highly efficient to deal with any large-scale optimization problem. The method does not require any kind of the smoothing of the yield surface and it can even be implemented to deal with any three-dimensional problem as well.@en

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs wave propagating outwards from the domain interior. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non-tangential angle-of-incidence and all non-zero frequencies. In this thesis different modeling procedure of unbounded domain and their merits and demerits ha been discussed. Detailed description of PML theory has been presented. A seven-step process to implement a PML element in a finite element model is proposed with a special reference to Opensees. Opensees is an object-oriented, open source framework developed by University of California, Berkeley which gives an advanced computational tool to simulate nonlinear soil and structure response and provides facility to add new material and elements. In this thesis, a new material code named PML material and a new element code named PML element has been written in C++ in Opensees compatible format using polymorphism and inheritance property of C++. A new material class named as soil has also been added to Opensees framework which can take care of Kelvin-Voigt damping (most common type of damping for soil). A shallow foundation model is considered for verifying coding results. Different boundary conditions are considered for modeling of unbounded domain and results are compared to perfectly matched layer modeling procedure. Numerical experiments were carried out to study effects of length of linear part, thickness of PML layer and different parameters involved in modeling PML layer. On the basis of these results, a set of guidelines has been presented for good modeling practices using perfectly matched layer. The effect of different ranges of frequency on PML modeling has been studied and found that PML modeling of unbounded domain is not significantly affected by the variation of frequency. Finally, the seismic response of a 2-D soil-pile model using PML model has been studied.@en