Modulus Reduction Curves Research Articles

A non-linear stress–strain relationship for soil and rock

While many employ a hyperbolic stress–strain relationship for soils, it is known that such a relationship is accurate over the small-strain range as encountered in earthquake and soil dynamics problems. For large strains, a relationship with different input parameters is needed, as is required for finite-element analyses of large-deformation behaviour. These separate characterisations do not carry over into the other application. Here, a proposed power relationship is presented that was developed to characterise the triaxial test stress–strain behaviour of cohesionless material from lubricated or ‘frictionless’ cap and base tests (144 tests) covering a range in the natural variation in particle size, particle shape and surface roughness, over low to high confining pressure. This relationship covers the strain range from 10−6 to soil failure. To date, it has been successfully used in laterally loaded pile response characterisation (the strain wedge model) and shallow-foundation load–settlement–bearing capacity response. Most recently, it has been extended to assess the behaviour of rock-like material (caliche). The relationship and its comparison with the hyperbolic relationship for large strain and the shear modulus reduction curve for seismic behaviour are presented.

Using shear modulus to predict the bearing capacity of strip foundations on sand

This paper reports results of a study to predict load-settlement behaviour of strip foundations on a dry uncemented sand using only soil stiffness data, under both working loads and extreme loads. This involved predicting the strength of a foundation system without using any strength parameters. A non-linear stepwise load-settlement prediction algorithm was utilised, which used the small-strain shear modulus of layers below the footing, a suitable stiffness modulus reduction curve and elasticity theory. The algorithm was adapted to predict the ultimate bearing capacity, as well as to construct astress-settlement curve and determine the bearing pressure experienced beneath a foundation at any particular settlement. The method was evaluated using two different sets of modulus reduction curves, for small-scale model foundations on loose, medium dense and dense sand. Model tests were conducted at 1g. Particle Image Velocimetry (PIV) was used to observe strains within the soil during loading. These strains were used to calculate the shear modulus of discrete layers throughout loading and back-estimate the layer small-strain shear moduli. Additionally, the mode of shear failure for each test at ultimate bearing capacity was observed using PIV. Modulus reduction curves that initially degrade shear modulus rapidly beyond the elastic threshold strain (0.001%) gave superior predictions under the given conditions. Using such curves, ultimate bearing capacity predictions produced errors of 0.4% to 10.8%. Using the same input parameters, stresssettlement predictions were of satisfactory accuracy over the entire settlement range.

Effects of consolidation conditions on the dynamic shear modulus of saturated coral sand over a wide strain range

The utilization of coral sand as a construction material for ports and other infrastructure systems in coral reefs is prevalent, wherein the soil elements within slopes and embankments typically experience an anisotropic stress state. It is imperative to accurately characterize the degradation behavior of the stratigraphic material's stiffness across a wide range of strains. In this study, the maximum and strain-dependent shear modulus (G0 and G) characteristics of saturated coral sand under isotropic and anisotropic consolidations were investigated by performing multi-stage stress-controlled undrained cyclic torsional shear tests with cyclic hollow cylinder shear apparatus. The experimental results indicated that G0 and G generally rose with increasing initial effective consolidation pressure but decreased with increasing inclination of initial major principal stress from the vertical (αc). In contrast, the effect of consolidation stress ratio (kc) on G0 and G was nonmonotonic, and that of kc and αc on the strain-dependent shear modulus reduction (G/G0) curve was slight. By defining anisotropic consolidation indexes δ0 and δr used for quantifying the effect of consolidation-induced anisotropy on the G0 and the G/G0, respectively. A unified formula of G0 obtained by multiplying Hardin's model by δ0, as well as a unique formula of G/G0 obtained by correcting γr in the Davidenkov skeleton curve expression as a linear function of δr were established under various consolidation conditions, respectively. The proposed G formula over a wide strain range is well validated by the experimental data for siliceous sandy soils from the literature.

Undrained response of fibre reinforced expansive soil subjected to cyclic loading

The evaluation of dynamic soil characteristics is a crucial task in understanding the geotechnical engineering problems associated with dynamic loads. Most of the investigations concentrated on non-expansive materials, and there is a dearth of information on the dynamic performance of fibre reinforced expansive soil. This research paper presents the experimental soundings on unreinforced and reinforced expansive soil with randomly distributed polypropylene fibre with varying confining pressures under cyclic loading. Furthermore, to understand the micro structural behavior of the fibre reinforced soil, samples are examined under SEM analysis after being subjected to cyclic loading. Quantitative analysis of soil micro pores was done through image processing. In addition, the study also included the presentation of maximum shear modulus, a modulus reduction curve and a pore pressure variation curve. Experimental results indicate that, the addition of fibres caused an improvement in the dynamic shear modulus and damping ratio with a rise in confining pressure. Inclusion of fibres reduces the permanent strain. The change of dynamic shear modulus and damping ratio over shear strain amplitude and the dynamic shear stress–strain curves were analyzed under different confining pressures. From the SEM analysis, fibre reinforced soil has a fabric with large packets of soil particles. Moreover, the presence of fibres enhanced pore distribution and diminished considerable amount of voids. The results of image processing shown the area porosity of unreinforced soil is 25.3% and that for 1% of fibre reinforced soil is 6.9%. This shows the enhancement of the interface effect of soil reinforced with fibre, which strengthens the soil and integrity.

Resonant Column Testing Procedure for Microbial-Induced Carbonate-Precipitated Sands

Abstract One of the fundamental inputs to site response analysis is the characterization of the dynamic properties of the soil, namely the shear modulus and material damping ratio. Because of soil’s nonlinear behavior, these properties change with induced shear strains, and modulus reduction and damping (MRD) curves have been proposed to capture that cyclic shear strain dependency. Microbial-induced carbonate precipitation (MICP) is a natural cementation process that induces the precipitation of calcium carbonate, bonding soil particles together. Laboratory experiments have demonstrated that MICP can improve the mechanical behavior of soils and mitigate their liquefaction potential. However, MRD curves have not been developed for MICP-treated soils, which hinders further evaluations of their suitability as a ground improvement technique for geotechnical earthquake engineering applications. To the best of our knowledge, this paper provides the first empirical study measuring the shear strain–dependent dynamic properties of MICP-treated sands for different cementation levels (i.e., lightly to heavily cemented). Outcomes from this work include empirical models of the maximum shear modulus and minimum shear strain damping ratio of MICP-treated clean sands and the corresponding mean MRD curves. The experimental program includes MICP-treated specimens of Ottawa 20-30 sand tested with a resonant column (RC) device incorporating a modified RC porous disk that (1) minimizes the disturbance between treatment and installation of MICP-treated samples prior to being tested in the RC device, (2) prevents slippage between the specimen and the modified RC porous disk during RC shearing, and (3) enables repeatability of the MICP-treated sample preparation. We find that the level of cementation influences the MRD curves of MICP-treated sands. Linear elastic and volumetric shear strain thresholds for MICP-treated sands are smaller than those for untreated sands, whereas the initial shear modulus for treated soils is larger than its counterpart for untreated sands.

The PoliTO–UniRoma1 database of cyclic and dynamic laboratory tests: assessment of empirical predictive models

The soil nonlinear hysteretic behaviour is usually described, in the moderate strain range, through the shear modulus reduction and material damping ratio (MRD) curves. In common practice, in absence of specific laboratory tests, the curves are estimated by employing empirical regression models. Such predictive models, typically calibrated on large experimental datasets, correlate the soil response to its physical properties. This research fits within this context, presenting a comprehensive database of cyclic and dynamic laboratory tests conducted on natural Italian soils. The database, publicly available as supplementary data of the paper, contains the results of the tests conducted by the geotechnical laboratories of the Politecnico di Torino (Turin, Italy) and the Sapienza Università di Roma (Rome, Italy) over the past 30 years. The experimental data are employed to assess the performance of some widely used empirical models in predicting the MRD curves of natural uncemented fine-grained soils, emphasizing the importance of using an independent dataset for conducting a reliable statistical analysis. The results show that the use of many soil parameters as proxies for predicting the soil response does not necessarily lead to an improvement in the performance of the model. Therefore, according to Occam’s razor principle, simple models are to be preferred.

Resonant column and cyclic torsional shear tests on Sutlej river sand subjected to the seismicity of Himalayan and Shivalik hill ranges: A case study

The study of site-specific dynamic properties of soil holds vital importance in many geotechnical engineering problems for the region holding high active seismicity. The Sutlej River basin, a valley and high-risk seismic zone in the north-eastern part of the Punjab state, has been considered for the study. The site-specific shear modulus reduction curves and material damping ratio curves for a range of shear strain amplitudes have been developed under varying experimental conditions of confining pressure, relative density, and loading frequency using the resonant column (RC) tests and cyclic torsional shear (CTS) tests. The study addresses the usage of two testing approaches, i.e., resonant column test (RC) and cyclic torsional shear test (CTS), for the estimation of shear modulus and damping ratio values at varying shear strain amplitudes. The regression analysis has been performed for the two testing techniques viz. RC and CTS to study the sensitivity of shear modulus values and material damping ratio values towards varying experimental conditions. The shear modulus and damping ratio values determined from cyclic torsional shear tests were observed to have great dependency on the frequency of loading. The knowledge of site-specific dynamic properties will support practical interpretation for the geotechnical engineering problems in the region.

Dynamic characterization of tire derived aggregates using cyclic simple shear and bender element tests

The relatively recent use of tire-derived aggregates (TDA) in civil engineering applications makes it necessary to thoroughly understand and characterize its behaviour under static and dynamic loads. Only a limited number of experimental studies have investigated the dynamic behaviour of TDA with large aggregate sizes. This paper aims to investigate the dynamic and cyclic behaviours of TDA and factors that could affect the characterization of TDA using cyclic simple shear tests. A comprehensive series of cyclic direct simple shear tests were performed in this study on TDA under vertical stress and shear strain amplitude ranges of 15–400 kPa and 0.01–10%, respectively. Small-strain shear moduli (Gmax) of TDA were also determined from shear wave velocities measured at vertical stresses of 15–800 kPa using bender elements. The results were used to study the effects of cyclic loading frequency, drainage condition, fabric anisotropy, shear strain amplitude, and vertical stress on the dynamic behavior of TDA. Shear modulus and damping ratio of TDA ranged from 85 to 4290 kPa and 7–25%, respectively. While the shear modulus of TDA was found to be strongly influenced by the loading frequency, drainage condition, anisotropy, shear strain amplitude and vertical stress, these testing parameters had little or reduced effects on the damping ratios of TDA specimens. Finally, shear modulus reduction curves were obtained by combining the results of cyclic simple shear and bender element tests.

A new FounDyn module in OpenFAST to consider foundation dynamics of monopile supported wind turbines using a site-specific soil reaction framework

A FounDyn module is created in OpenFAST to consider foundation dynamics, which is an appealing supplement to the current version of OpenFAST. The FounDyn module receives the motions from the SubDyn module and sends the forces back to the SubDyn module. In FounDyn, the soil-monopile interaction is captured using a site-specific soil reaction framework. The soil reaction framework possesses the same configuration of the semi-analytical 1D model to consider effects of the large plie diameter and the small pile aspect ratio but uses new site-specific soil reaction models. The soil reaction models are nonlinear and hysteretic, which match the desired modulus reduction curve by identifying three parameters in a hyperbolic function and a linear function using genetic algorithm (GA) and manual parameter tuning, and the desired damping curve by applying the Ishihara-Yoshida rule that controls the unloading-reloading curves iteratively through three parameters. The FounDyn module is verified by the well-confined OC3 project in terms of modal frequencies, tower top displacement and shear force and moment at the mudline, and reasonable agreements are achieved between them. A series of emergency shutdown analyses of the NREL 5 MW wind turbine are performed using OpenFAST plus FounDyn. The results show that the misalignment of wind and earthquake affects the tower bending moments largely. The earthquake excitation is found to be the design driving load, prevailing over the wind excitation for the design of wind turbine supporting structures.

New p-y model for seismic loading prediction of pile foundations in non-liquefiable and liquefiable soils considering modulus reduction and damping curves

In this study, a new p-y model is proposed for the seismic loading prediction of pile foundations using the Beam on Nonlinear Winkler Foundation (BNWF) method. It matches the desired modulus reduction curve by identifying three parameters in a hyperbolic function and a linear function using a genetic algorithm (GA), and the desired damping curve by applying the Ishihara-Yoshida rule that controls the unloading–reloading curves iteratively through the three parameters. The rate effect is integrated into the proposed PySimple5 model for clay by exerting influence on the ultimate capacity and maximum material damping through a power function, while the pore pressure effect is reflected in the proposed Pyliq5 model for sand by relating the ultimate capacity to the mean effective stress. For a single pile in non-liquefiable soil, the predicted superstructure acceleration and pile bending moment by PySimple5 agree well with those from centrifuge tests for different soil shear strain levels, while the equivalent linear and PySimple1 models underestimate them for soil shear strain levels higher than 1%. For a pile in liquefiable soil, PyLiq5 shows a reasonable agreement with the centrifuge tests in terms of the superstructure acceleration and pile bending moment by considering the pore pressure effect.

Significance of Determination Methods on Shear Modulus Measurements of Fujian Sand in Cyclic Triaxial Testing

It has long been known that the hysteresis loops of sand under cyclic loading gradually become asymmetric with the increase of strain amplitudes, but a symmetrical hysteresis loop is widely assumed in current practice. Despite several methods which have been proposed recently to consider the hysteresis loop irregularities, previous research has lacked a quantitative study on the effects of determination methods on the shear modulus G and modulus reduction curve G/Gmax. The primary objective of the current study is to evaluate the uncertainties associated with the shear modulus measurements introduced by four determination methods. Reconstituted sand specimens prepared at three relative densities are tested using strain-controlled cyclic triaxial tests, at various effective confining pressures. The results in terms of G and G/Gmax with increasing shear strain are presented, following by the difference quantification in the calculated G/Gmax caused by the determination methods, the Gmax definition and the cycle number. The results show that the calculated G/Gmax may differ significantly for the same hysteresis loop, with a maximum percentage change of 40~50%. The aggravated influence at low confining pressure highlights that careful consideration of the asymmetrical hysteresis loop at large strains is warranted.

PredictingVpand constrained modulus reduction curve based onVsand shear modulus reduction curve in accordance with poroelastic theory

The compression wave velocity (Vp) of sediments plays a key role in seismic wave amplification of vertical motion and is required in site response analysis. However, such information is usually lacking during field exploration (e.g. the surface wave method) because only shear wave velocity (Vs) is obtained. This aim of this study is to predict Vpbased on Vsempirically and theoretically, particularly focusing on saturated conditions. The empirical approach is to establish the Vpcorrelation dependency on Poisson's ratio and Vs, and the theoretical approach is based on poroelastic theory that accounts for the interaction between fluid and soil skeleton. The engineering geological database for the Taiwan strong motion instrumentation program and the Kiban Kyoshin network database in Japan are adopted to establish an empirical model and validate poroelastic theory. The validated poroelastic approach is used to develop a constrained modulus reduction curve dependency on the porosity, Vs, Poisson's ratio and degree of saturation with a shear modulus reduction curve. The proposed approach can be used to develop generic Vpprofiles and constrained modulus reduction curves for the site response to vertical motion given a site-specific Vsprofile.

A homogenization approach to estimate the shear modulus of spatially variable soil materials

Engineers often use seismic site response analyses to evaluate the propagation of seismic waves through a soil deposit. These analyses have traditionally relied on dynamic properties (e.g., the maximum shear modulus, Gmax, and shear modulus reduction curves G/Gmax) measured on homogeneous soil specimens. However, soil properties are not uniform, and it is common to see variabilities at different scales. This study uses physical models with controlled variability to evaluate the Gmax and G/Gmax in heterogeneous soil samples to reflect the effects of soil variability. We used a 3D soil printer to create heterogeneous soil models of 20x20x20 cm that match target random field simulations generated by spatially varying the liquid limit on clayey-type soils. The models were used to sample cylindrical specimens, then tested in a torsional shear rheometer to measure Gmax and G/Gmax. The results show that Gmax and the G/Gmax curves are affected by the internal specimen variability, particularly by the coefficient of variation (COV) in the underlain random fields. Using the experimental results, we also show that the Hill [1] approach for estimating properties of composite materials provides a good estimation for the experimentally measured G/Gmax curves in a heterogeneous soil media, especially in specimens with low to medium variability. Finally, we evaluated the performance of the homogenization bounding theory of Hashin and Shtrikman [2], also formulated for composite materials, on estimating Gmax. We find that the Gmax of specimens with low and medium variability can be reasonably estimated using the upper bound. In contrast, samples with high variability are better estimated using the lower bound.

Deep, In Situ Nonlinear Dynamic Testing of Soil with Controlled Blasting: Instrumentation, Calibration, and Application to a Plastic Silt Deposit

Abstract This study presents a novel test method to obtain dynamic properties of soil in situ using controlled blasting. Experimental and calibration protocols have been developed to conduct such investigations for any soil at any depth to obtain the coupled response between nonlinear shear modulus and pore pressure generation in soil. Controlled blasting using buried explosives with different charge weights at various depths were used as an active seismic source and the soil response measured. Micro electro mechanical system (MEMS) accelerometers, geophones, pore pressure transducers, and the MEMS-based inclinometers were calibrated to develop unique excitation voltage-to-unit relationships. Procedures to determine the post-installation sensor locations and orientations were developed to quantify the appropriate body wave velocity and shear strain in the soil. The conversion of calculated strains to a constant-volume direct simple shear-equivalent shear strain facilitated direct comparison to previously reported shear modulus reduction curves. Although this manuscript primarily focuses on instrumentation and calibration protocols, an example of the results of a test blast program is provided to demonstrate observation of the in situ, coupled, nonlinear fluid-mechanical response of an instrumented plastic silt deposit at a depth of 9 to 11.5 m below ground surface.

Small-strain behaviour of unsaturated silty clay: experiments and model interpretation

The experimental data dealing with the so-called small strain stiffness of soils are indispensable in developing and calibrating advanced numerical models. A literature review revealed a lack of such data for unsaturated soils. In this study, the laboratory measurements of very small and small strain stiffness are carried out in a double-walled triaxial cell using bender elements and LVDT transducers. The tested soil is a reconstituted silty clay, representing a typical soil encountered in engineering practice. A so-called effective stress for unsaturated soil is used in interpreting and numerically simulating the effects of suction magnitude, net stress magnitude and suction history on the initial elastic shear modulus, and on its decrease with increasing strain. The increase of both net stress and suction leads to increase in stiffness. This is observed by both elastic shear modulus and shear modulus reduction curve measurements. The “overconsolidation” by suction leads to the increase of stiffness, too. The experimental data are used for calibrating constitutive parameters and for assessing the capabilities of an advanced hypoplastic model for unsaturated soils. The model predictions are consistent with the laboratory measurements and the model can capture the increase of stiffness with respect to the increase of suction and net stress reasonably well. Moreover, the model is able to predict the increase of stiffness due to suction “overconsolidation”.

Dynamic Characterization of Tire Derived Aggregates

Despite the recent efforts in characterizing and investigating the dynamic properties of tire-derived aggregates (TDA) and its use in civil engineering applications, the size of TDA used in most of the previous work was significantly smaller than the TDA size range mentioned in ASTM D6270 standards. There are two types of TDA, Type A and Type B, in which the maximum aggregate size of Types A and B are 200 and 450 mm, respectively. Thus, the availability of experimental data and shear modulus reduction curves for the dynamic behavior of TDA of larger sizes are deemed necessary for practical engineers to assess and investigate the behavior of such material under dynamic loads. This paper provides the results of large-scale undrained cyclic triaxial tests performed on Type A TDA at large strain amplitudes. The tests were performed under confining pressures ranging from 25 to 200 kPa and shear strain levels ranging from 0.1% to 10%. Furthermore, each test consisted of successive stages in which the number of cycles and axial strain levels varied. The shear modulus of Type A TDA has a range of 245–1,796 kPa, with a decreasing trend with increasing shear strain amplitudes. Furthermore, the dampening ratios were found to range from 11% to 23.5%. Finally, the shear modulus reduction curves were obtained through curve fitting techniques and nonlinear regression.

Deterministic ground motion simulations with shallow crust nonlinearity at Garner Valley in Southern California

AbstractWe present deterministic ground motion simulations that account for the cyclic multiaxial response of sediments in the shallow crust. We use the Garner Valley in Southern California as a test case. The multiaxial constitutive model is based on the bounding surface plasticity theory in terms of total stress and is implemented in a high‐performance computing finite‐element parallel code. A major advantage of this model is the small number of free parameters that need to be calibrated given a shear modulus reduction curve and the ultimate soil strength. This, in turn, makes the model suitable for regional‐scale simulations, where geotechnical data in the shallow crust are scarce. In this paper, we first describe a series of numerical experiments designed to verify the model implementation. This is followed by a series of idealized large‐scale simulations in a 35 26 4.5 km domain that encompasses the Garner Valley downhole array site, which is an instrumented and well‐characterized site in Southern California. Material properties were extracted from the Southern California Earthquake Center Community velocity model, CVM‐S4.26, considering its optional geotechnical layer, while the modulus reduction curves and soil strength were selected empirically to constrain the nonlinear soil model parameters. Our nonlinear simulations suggest that peak ground displacements within the valley increase relative to the linear case, while peak ground accelerations can increase or decrease, depending on the frequency content of the excitation. The comparisons of our simulations against hybrid three‐dimensional–one‐dimensional site response analyses suggest the inadequacy of the latter to capture the complexity of fully three‐dimensional simulations.

Integration of viscoplastic effects in a one-dimensional constitutive model for ground response analysis

During an earthquake, strain-rate effects affect both the stiffness and damping behaviour of soils, yet existing constitutive models for ground response analysis are typically formulated within a rate-independent framework. In this paper, a one-dimensional (1D) viscoplastic stress–strain model is presented to introduce strain rate effects in ground response analysis. Its constitutive equations are based on a model that uses a cubic spline fit of the modulus reduction curve and a coordinate transformation technique to match any input modulus reduction and damping curve. A viscous stress component is added to model the effect of strain rate on the mechanical behaviour of soils using a single input parameter. The model is able to reproduce the linear increase in shear strength with the logarithm of shear strain rate and allows the introduction of viscous effects in 1D ground response analysis with control over damping and modulus reduction behaviour. The model is implemented in a software for ground response analysis and used to predict the results of a centrifuge test modeling 1D wave propagation. The results show that the model accurately predicts the amplification and attenuation of shear waves, in a context where strain rates impact the response of the model significantly.

Dynamic soil properties and seismic ground response analysis for North Indian seismic belt subjected to the great Himalayan earthquakes

The paper highlights the importance of using site-specific shear modulus reduction (G/Gmax versus shear strain, γ) curves and damping ratio (D versus shear strain, γ) curves for ground response analysis. In order to develop comprehensive G/Gmax–γ and D–γ curves (i.e. over a wide range of strain level), two types of apparatus, viz. resonant column and cyclic simple shear, have been used. The case study considered the geological deposits from the river beds of Yamuna River originating from the Himalayan seismic zone of North India. The tests results have been analysed to develop G/Gmax–γ and D–γ curves and compared with standard curves. It has been observed that upper and lower boundaries for the standard curves are remarkably different for the geological deposit under consideration. In order to assess the impact of using standard curves rather than site-specific curves, ground response analysis has been carried out at five sites along the Yamuna River using two types of curves (standard and site-specific developed in this study). The study showed that the amplification of shear waves at these sites based on the experimentally derived curves is much higher as compared to the standard curves. The proposed curves better represent dynamic behaviour of the soil deposits of the region and will provide a realistic response as far as practically possible, for the structures constructed in the states of Haryana and Delhi and nearby areas. It is anticipated that the data presented in this paper will have wide application and usage.

Effect of Nonlinear Soil Model on Seismic Response of Slopes Composed of Granular Soil

A series of two‐dimensional finite element analyses are performed to simulate the seismic response of slope composed of granular soil. Four sets of input parameters for the nonlinear soil model are used to fit the reference the shear modulus reduction and damping curves, thereby to evaluate the influence of the nonlinear soil model. The first set is fitted to the shear modulus reduction curve. The second and third sets are fitted simultaneously to both shear modulus reduction and damping curves. The final set applied the shear strength adjustment to adequately capture the nonlinear soil response at large strains. The accuracy of each set of parameters are evaluated through comparison with centrifuge model test measurements. It is observed that the nonlinear soil model has a marginal influence on the acceleration response. On the contrary, the vertical settlement is highly influenced by the nonlinear soil model. The discrepancy is shown to increase with an increase in the intensity of the input ground motion. It is demonstrated that the adjustment for the shear strength is important in performing seismic analyses of slopes, which is most often ignored in practice. Based on the results, practical guidelines on how to select the parameters for the nonlinear soil model are provided.