Stress-displacement Relationship Research Articles

Frictional pull-out work capacity of macro-SSF in concrete composites and evaluation of matrix deterioration due to kinematic fiber-end-slip after full-debonding point

ABSTRACT This paper presents a parametric study on the effect of frictional pull-out work capacity on the performance of fiber/matrix regime during the kinematic pull-out process. This study involves an experimental program of pull-out test of straight steel fiber embedded in concrete composites, numerical study based on finite element modeling, and theoretical approaches based on mathematical model. Various parameters were considered such as fiber geometry, Poisson’s ratio, elasticity modulus, and aspect ratio. The experimental results were calibrated with the results of numerical study, and the theoretical approaches were controlled using deterioration coefficient based on a given curve of uniaxial stress–displacement relationship plotted experimentally. The results showed that the capacity of pull-out work decreases 62.6% when the fiber diameter decreases 6 times. Decreasing the fiber diameter also improves the resistance of the matrix against deterioration. While decreasing the fiber embedded length by 1.8 times, the frictional pull-out work decreases by 56.3% and boosts relatively the resistance of matrix against crumbling, where the frictional shear bond strength improved by 91%. Likewise, decreasing the fiber aspect ratio by 1.8 times decreases the done frictional pull-out work by 56.3% and boosts the fiber duct resistance against damage by improving the frictional shear bond strength.

Unveiling the reinforcement benefits of innovative textured geogrids

The smooth surface texture of the commercially available geogrids limits the shear strength mobilization at the interfaces. This study presents the design, manufacturing, and interface performance evaluation of innovative textured geogrids. Geogrids with square, triangular, and hexagonal apertures with and without inherent surface texture were manufactured through additive manufacturing (3D printing) technique, using PLA (Poly Lactic Acid) filament. The texture includes elevated pins of 3 mm height at the junctions and inherent diamond pattern of 1 mm height on the ribs. The individual and combined effects of surface texture and aperture shape on the stress–displacement relationship, dilation angle, and the thickness of shear zone are quantified using large-scale direct shear tests and Particle Image Velocimetry (PIV) analysis. Results showed that the textured geogrid with hexagonal aperture has exhibited the maximum interface coefficient of 0.96 with sand followed by the geogrids with triangular and square apertures. Irrespective of the aperture shape, provision of the surface texture resulted in an overall increase of interface shear strength by more than 13%. Further, PIV analysis revealed that the shear zone is 25% thicker for textured geogrids of different aperture shapes, suggesting higher interlocking and passive resistance offered by their textured surfaces.

A damage rock model considering shear failure by modified logistic growth theory

Localized rock failures, like cracks or shear bands, demand specific attention in modeling for solids and structures. This is due to the uncertainty of conventional continuum-based mechanical models when localized inelastic deformation has emerged. In such scenarios, as macroscopic inelastic reactions are primarily influenced by deformation and microstructural alterations within the localized area, internal variables that signify these microstructural changes should be established within this zone. Thus, localized deformation characteristics of rocks are studied here by the preset angle shear experiment. A method based on shear displacement and shear stress differences is proposed to identify the compaction, yielding, and residual points for enhancing the model's effectiveness and minimizing subjective influences. Next, a mechanical model for the localized shear band is depicted as an elasto-plastic model outlining the stress-displacement relation across both sides of the shear band. Incorporating damage theory and an elasto-plastic model, a proposed damage model is introduced to replicate shear stress-displacement responses and localized damage evolution in intact rocks experiencing shear failure. Subsequently, a novel nonlinear mathematical model based on modified logistic growth theory is proposed for depicting the shear band's damage evolution pattern. Thereafter, an innovative damage model is proposed to effectively encompass diverse rock material behaviors, including elasticity, plasticity, and softening behaviors. Ultimately, the effects of the preset angles, temperature, normal stresses and the residual shear strength are carefully discussed. This discovery enhances rock research in the proposed damage model, particularly regarding shear failure mode.

Simulation of soil-structure interaction using inelasticity-separated finite element method

The soil–structure interaction (SSI) effect is nonnegligible during the nonlinear seismic response analysis of large-scale or underground structures. However, the computation of nonlinear response of a structure considering SSI effect usually is a time-consuming process because the numerical model should involve the structure itself, a wide-range soil domain and the soil–structure interface, especially for large-scale structure. This study aims to incorporate the inelasticity-separated finite element method (IS-FEM), which is an algorithm recently developed for efficient structural nonlinear analysis, to improve the efficiency of the soil-structure interaction system. To this end, an inelasticity-separated contact element model for modeling the nonlinear behavior of soil and structure interface is developed in this study. To derive the inelasticity-separated governing equation of this model, a reference elastic stiffness is defined so that the normal and shear deformation of any point pair in the presented contact element is decomposed into reference linear and reference nonlinear parts according to this reference elastic stiffness. As a result, the problem of inconsistency between the requirement of IS-FEM for constant initial linear elastic stiffness and the existence of inflection point for the stress versus relative displacement relationship of contact behavior can be solved. Then, the Goodman element formulation for depicting the nonlinear contact behavior is introduced and an interpolation scheme to establish the reference nonlinear deformation field in an element is incorporated. By combining the proposed model with existing inelasticity-separated elements, a global analytical model of the soil–structure interaction system can be established within the framework of IS-FEM. Because the nonlinearity usually occur in some local regions of the global model and the use of the IS-FEM only require performing these operations for a small scale matrix representing local nonlinearity, the computational efficiency of nonlinear structural response analysis considering SSI effect can be improved significantly. The applicability and efficiency of the proposed method is finally verified by two numerical examples.

Softening-based interface model and nonlinear load-settlement response analysis of piles in saturated and unsaturated multi-layered soils

This work presents a simplified method for the nonlinear analysis of the load–displacement response of piles in multi-layered soils. As a starting step, a new interface model based on the disturbed state concept (DSC) is put forth to simulate the interface shear stress-displacement relationship by considering the nonlinear hardening–softening behaviour. In the new model, input parameters can be conveniently calibrated using conventional interface shear tests or on-site tests. The good agreement between predictions and experimental data from interface direct shear tests validated the performance of the proposed DSC model. The DSC model performed better in terms of predictions when compared to the hyperbolic one. Next, the soil-structure interface model and bearing capacity theory are coupled to provide a theoretical framework for the analysis of pile load-transfer in saturated and unsaturated multi-layered soils, where the DSC model is employed to represent base resistance as well as skin friction. This work also discusses the profile of steady-state in-situ matric suction, soil–water characteristic curve, and pore-water pressure of unsaturated soils. The proposed method has the advantage of being used in practice as it is simple to obtain input parameters from laboratory tests, as well as Standard Penetration or Cone Penetration Tests. The proposed framework is finally applied to the analysis of five well-documentedcase studies. The proposed approach and the static load test results from the field measurements are found to be in satisfactory agreement, indicating that the proposed method performs well. The proposed method is suggested to be utilised for preliminary analysis, planning a suitable programme of loading tests, as well as optimizing the pile design by back analysis ofthe load test results.

Terramechanics Models for Tracked Vehicle Terrain Interaction Analysis A Review

Efficient manoeuvrability of off-road tracked vehicles such as military tanks and rovers is essential in ensuring the success of military/extra-terrestrial operations. To achieve this, in-depth research on vehicle-terrain interaction is crucial. This manuscript deals with reviewing the ways to study terramechanics viz. theoretical, empirical, and field tests, and proposing the merits and demerits of each method. Under the theoretical approach, empirical, numerical, and semi-empirical methods are discussed. Under the empirical approach, the method based onthe vehicle cone index for tracked and wheeled vehicles is discussed. Under the numerical approach, advantages and disadvantages of Finite Element Method (FEM) and Discrete Element Method (DEM) are discussed. Semi-empirical method, based upon a combination of the best features of numerical and empirical approaches discusses terrain response to normal repetitive loads and shear repetitive loads for tracked as well as wheeled vehicles. Pressure sinkage relationship for terrains at various loading conditions and shear stress displacement relationship for different terrains obtained through penetration and shear tests are discussed to determine the vehicle’s mobility parameters under a semi-empirical approach. Further, the Super element model, multi-body simulation model, and ride and cornering vibration model are discussed under computer simulation models. A detailed review of various models customized towards tracked vehicle-terrain interaction discussed in this manuscript helps the authors set up a laboratory for terramechanics at DIAT. Preliminary analysis along with conceptual design of the experimental setup is also discussed. In a nutshell, this paper attempts to summarize the research that has been carried out in the field of tracked vehicle-terrain inter action comprising of VCI, MMP, FEM, DEM, Super element model, Multibody technique, and Semi-empirical methods helping the authors to establish a laboratory of terramechanics for their M. Tech. program on Armament and Combat Vehicles at DIAT Pune.

A constitutive model for the gravelly soil-structure interface considering particle breakage and fabric change

The isolated foundation with cushions is gaining popularity as a viable option for building bridges in active seismic zones due to its exceptional seismic isolation performance. However, the efficacy of this foundation relies heavily on the interactions within the cushion layer and between the caisson foundation and the cushion layer. The cushion layer, composed of 0.5 m thick crushed aggregate layer and 2.5 m thick pebble gravel, is susceptible to particle breakage and fabric change, which can significantly impact the overall behavior of the foundation. To better characterize the mechanical behavior of the cushion layer and the interface between aggregates and the structure, a new elastoplastic constitutive model is proposed in this study. The current model is built based on an existing constitutive model, Saberi’s model (Model-S). Two parameters are introduced, including a particle breakage coefficient, B10, calculated by total plastic work, and a fabric-dilatancy variable, z, to determine the interfacial dilatancy variation. The current model consists of 13 parameters in total. Comparison evident that the current interfacial constitutive model can better simulate the change of the physical state and asymmetrical response in the stress-displacement relationship due to symmetrical loading in different directions.

Mechanical properties and soil failure process of interface between grouser of tracked mining vehicle and deep-sea sediment

Tracked mining vehicles (TMVs) are key equipment in deep-sea mining systems and can easily slip when walking on soft sediments on the seabed. Thus, it is important to investigate the effect of track parameters and grounding-specific pressure on the shear interaction between mining vehicles and sediment. The sediment was simulated using a mixture of bentonite and water, and a shear test of the track with a single grouser and sediment was conducted. A numerical model of the track–sediment shear test was established, and the shear strength and failure mode of the sediment soil under the track action were examined. The results demonstrate that the peak shear strength of the track–sediment interface increases with an increase in the grounding-specific pressure, gradually increases with an increase in grouser height, and tends to stabilize at a given level of grouser height. Owing to the presence of the grouser, the deep-sea sediment transitions from the elastic state to the plastic state faster than the typical clay during the shear test. The variation law of peak shear strength at the track-sediment interface with the height of track grouser under different grounding-specific pressures was analyzed, and a modified shear stress-displacement relationship model was proposed. Wedge failure is the predominant method of failure for a single-grouser shear sediment soil at various heights. The primary factor affecting the horizontal disturbance range is the grouser height, and the range increases as the grouser height increases; the primary element affecting the vertical disturbance range is the grounding-specific pressure, and the range increases as the grounding-specific pressure increases. The relationship of the grouser height and cohesiveness with the interfacial shear strength was examined and a new shear stress–displacement relationship model that can accurately capture the shear mechanical behavior between the track and sediment under various grouser heights was proposed. In conclusion, the study findings could serve as a guide for the engineering design of a TMV walking system.

Bond-slip model of anchor-soil interface and analysis of bearing capacity of anchor in frozen soil

It has been recognized that temperature and moisture have obvious influence on the bonding characteristics of anchor-frozen soil interface. Following the principle and method used to explore the interface performance of microelement anchor-soil, this research starts by performing the pullout tests of anchor-soil under different temperature and moisture content conditions. Then, we analyze the characteristics of interface bond stress-displacement and compare the effects of temperature and moisture content on interface mechanical properties. The test results show that the failure mode of anchor changes slightly when soil temperature and moisture content vary, wherein the shear failure which shows brittle characteristics at the anchor-frozen soil interface is dominant. Additionally, we have found that the effect of temperature on the peak bond stress is more significant than moisture. Under different soil temperatures and moisture content, the shape of bond stress-displacement curve of anchor-frozen soil interface is similar. Furthermore, the bond stress-displacement relationship of anchor-frozen soil interface can be fitted by a composite function which is composed of exponential and ideal elastic-plastic model function. Thus, this research recommends to propose a bond-slip model of anchor-frozen soil interface considering temperature effect. Combined with the load transfer method, the force and deformation equation of anchor in frozen soil is derived. Through example analysis, it is discovered that the pullout bearing capacity of anchor increases with the decrease of soil temperature. Moreover, the initial shear stiffness of anchor-soil interface is positively related to the bearing capacity of anchor. At the end, the influence of interfacial residual strength on the bearing capacity of anchor in the later stage of pullout is superior to that in the earlier stage.

Nonlinear Dynamical Analysis of Composite Boring Bar with Nano-carbon Materials

Preventing resonance in boring bar vibration is the key to achieve high quality and efficiency. In this paper, using the three-dimensional nonlinear dynamic model of boring process, the cutter bar is simplified as a rotating non planar bending axis, starting from the constitutive relationship and stress displacement relationship of composite materials. Based on Hamilton principle, the nonlinear motion differential equation of cutting process is established. The nonlinear partial differential equations of bending vibration are discretized by Galerkin method. Using the multi-scale method, the frequency response functions of the system in the primary resonance and super resonance are obtained. Using this model, the effects of the length diameter ratio of nano-carbon materials and the content of nano-carbon materials on the vibration frequency response of the boring bar are studied. The results show that the steady vibration amplitude of the cutter bar can be reduced by adding nano-carbon materials to the fiber reinforced polymer composite (FRP) beam.

The effects of tensile strain rate on the dynamic tensile behavior of ultra high toughness cementitious composite

In this study, dogbone-shaped block specimens with cross section aspect ratio of 1.25 were prepared and tested at different strain rates ranging from 10−5 to 10−1 s−1 to investigate the dynamic tensile behavior of ultra high toughness cementitious composite (UHTCC) under different strain rates. The dynamic tensile responses were characterized in terms of the tensile strength, strain capacity, elastic modulus, specific energy absorption (SEA) and crack pattern. The results indicated that, with the increase of strain rate, the ultimate tensile strength increased by 160% but the strain capacity decreased by 80%. As the strain rate increased from 10−5 to 10−2 s−1, the elastic modulus and SEA increased by 40% and 50%, respectively. Reduction in the width of cracks in UHTCC was observed with the increase of strain rate while the number of cracks kept almost constant. Compared with thin planar specimens, the rate of increase in the ultimate tensile strength of UHTCC block specimens caused by strain rate increase was higher whereas that in the strain capacity was lower. Mesoscale numerical model with randomly distributed polyvinyl alcohol (PVA) fibers was built to analyze relationship between the cross section stress and displacement of specimens with different thicknesses. The results showed that the stress in the cross section and their corresponding relative displacement of block specimen were smaller than those of thin planar specimen under the same loading status. In addition, the cross section stress-displacement relationship of thin planar specimen was more sensitive to the strain rate than that of block specimen, which may be the reason for the weaker strain hardening behavior of thin planar specimens under dynamic tensile loading.

Approximating Constant-Q Seismic Wave Propagations in Acoustic and Elastic Media Using a Cole–Cole Model

ABSTRACT Seismic wave propagation in the Earth’s interior inevitably encounters attenuation and dispersion effects, which usually can be represented by a constant-Q model. However, solving the constant-Q wave equations formulated by fractional Laplacians is computationally intensive. Alternatively, the Cole–Cole model provides an optimal description of seismic attenuation. Because of the fractional time derivatives of both stress and strain in the expression, this method exhibits good adaptability and flexibility. In this article, we investigate the performance of the Cole–Cole model to approximate constant-Q behaviors with different fractional orders in acoustic and elastic media. The phase velocity and quality factor are compared to determine an optimal fractional order. After that the Cole–Cole model can be easily represented by the other three parameters (i.e., relaxed modulus, minimum angular frequency, and minimum quality factor), which are similar to the constant-Q theory (reference modulus, reference angular frequency, and frequency-independent quality factor). The first-order viscoacoustic and viscoelastic wave equations are derived to implement seismic wavefield simulations by combining conservation equation and stress–displacement relation. Furthermore, a time-domain algorithm is developed to solve the wave equations based on the Grunwald–Letnikov approximation and finite-difference scheme. Numerical modeling results calculated by our proposed method have good consistencies with the reference solutions from the constant-Q theory, suggesting that a small fractional order can well approximate the constant-Q outputs in a broad frequency band.

Discrete element method investigation of shear behaviour of 3D geogrid–sand interface

Interface shear characteristics have an important impact on the stability of geosynthetically reinforced soil structures. The shear characteristics of three-dimensional (3D) geogrid–sand interfaces were investigated using large-scale direct shear tests and the discrete element method (DEM). Geogrids were manufactured by 3D printing. The effect of mesh pattern and transverse-rib thickness on the stress–displacement relationship, strength parameters, coordination number and porosity distribution were evaluated. The results showed that the mesh pattern and transverse-rib thickness have an impact on the interface shear characteristics. The peak and residual interface shear strength of the modified geogrid mesh pattern exceeded that of biaxial geogrids. The average coordination numbers of the modified geogrid mesh pattern were greater than those of biaxial geogrids. The variability of particle compactness, as characterised by the porosity distribution, shows how the modified mesh pattern increases the interface shear strength. The interface shear strength of the geogrid–sand interface was improved by thickening the transverse ribs of the modified geogrid mesh pattern.

Study on the cohesive shear characteristics and intrinsic modelling of the root–tailing soil interface of Amorpha fruticosa

To study the soil consolidation effect of shrub plant roots on tailings soil and to explore the frictional characteristics of plant roots on tailings soil, three experimental conditions of the root–soil interface were established by using a modified indoor direct shear instrument with binders such as liquid sodium silicate and cyanoacrylate to conduct direct shear frictional tests at the root–soil interface using the roots of the typical slope protection plant Amorpha fruticosa. The Gompertz improved curve model was established by using the relationship between shear stress and shear displacement and the trend of the root–soil interface parameter index. The results were compared between the improved Gompertz curve model and the Clough–Duncan hyperbolic model, and a two-factor coupled improved Gompertz interfacial intrinsic structure model with normal stress and cohesive strength factor was established. The results showed that the interface shear stress and shear displacement showed strain hardening characteristics at different normal pressures for cohesive strength ratios of 1.5 and 1.7 at the root–tailing soil interface. At a cohesive strength ratio of 1.6, strain-softening was observed from 100 to 300 kPa and strain hardening was observed at 400 kPa. The improved Gompertz curve model predicts the shear stress and shear displacement curves at the root–soil interface with different cohesive strengths more reasonably than the Clough–Duncan hyperbolic model, and the maximum accuracy can be improved by nearly 40%. The two-factor coupled improved Gompertz curve model can fit the shear stress versus shear displacement relationship at the A. fruticosa root–tailing soil interface.

Bonding Performance of CFRP-Steel Interface after Continuous High Temperature

The bonded shear performance of CFRP-reinforced steel structures following continuous high-temperature natural cooling was investigated experimentally, and 30 CFRP-steel specimens were subjected to single-shear tensile testing. In analyzing the shear stress-displacement relationship in each working condition, the equation for the coupling effect of the heating temperature and duration on the ultimate shear stress was proposed. The results show that the ultimate shear stress at the CFRP-steel interface tends to decrease with the increase of heating temperature, and the ultimate shear stress at the heating temperature of 300°C is 32.37% of that at 25°C. The high-temperature duration has an obvious influence on the bonding performance of the CFRP-steel interface, and the CFRP-steel specimens fail when the heating temperature is 300°C and lasts for 120 min.

Influence of Lode parameter on damage and fracture behavior of polyethylene materials

AbstractIn this study, a series of tests on notched polyethylene (PE) plate specimens providing clues to damage and fracture behavior for a wide range of Lode parameters was carried out. Two series of numerical simulations of each test were performed, the first one without taking damage evolution into consideration and the second one taking damage evolution into consideration. Good correlation of experimental and numerical results in terms of the engineering stress–displacement relation has been achieved. The results show that the average Lode parameter increases with the increasing notch radius of the plate specimens. In addition, the damaged and undamaged constitutive equations were determined from the first and second series of finite element (FE) simulation, respectively. The critical damage parameter calculated from the damaged and undamaged constitutive equations was found to decrease when the average Lode parameter is increased. Furthermore, fracture occurs at the center of the minimum cross section, where the maximum damage parameter and plastic strain occur. The fracture strain was found to increase with the increase in the average Lode parameter.

Effects of fracture network distribution on excavation-induced coupled responses of pore pressure perturbation and rock mass deformation

We perform a numerical study of the spatio-temporal evolution of rock damages and fracture displacements around underground excavation in fractured rocks. We model fracture networks based on a power law model and explore different values of the length exponent and fracture intensity. An elasto-brittle constitutive relation is employed to characterise the deformation/failure of intact rocks, while a non-linear stress–displacement relation is used to mimic the normal/shear displacement of pre-existing fractures. Fluid flow through fractured porous media is simulated based on Darcy’s law. Both direct coupling (via poroelasticity) and indirect coupling (via changes of rock mass properties) are considered, with excavation-induced stress alteration, pore pressure variation, damage evolution, and fracture deformation realistically captured. It is observed that, fracture systems dominated by large fractures in general accommodate significant sliding along pre-existing fractures but minor damage in intact rocks, while fracture systems governed by small fractures can have profound damage and slippage simultaneously. Excavation-induced stress variation near the tunnel can instantly cause pore pressure fluctuations in low-permeability rock matrix, promoting fracture shear and rock damage. In addition, we investigate the microseismicity generated during and after the excavation as well as analyse the mechanisms of excavation and drainage-induced disturbances. Our results have important implications for underground excavation activities involved in many geoengineering applications.

Assessment of frictional-bond strength at the interface of single SSF in cementitious composite and prediction of accompanying pressure of surrounding matrix

ABSTRACT This paper deals with the study of bond in fiber-reinforced concrete composites and the effect of fiber/concrete interfacial friction on the fiber bond capacity. The paper presents the results of an experimental campaign including direct pull-out test of straight steel fiber-SSF embedded in concrete matrix, aimed at investigating the effect of some parameters, such as bonded length, yielding strength of steel fiber, fiber/concrete roughness, surrounding pressure of concrete matrix, and Poisson’s ratio of concrete. Two analytical models based on the experimental results were proposed. Groups of single fiber embedded in concrete were prepared using different values of bonded length. The results showed that the frictional-bond strength, during fiber-slip stage, can be considered as constant despite changing the fiber embedment length. Furthermore, the SSF/concrete system can be considered as an elastic system, whereas the stress–displacement relation is semilinear. Finally, the results showed good improvements in the efficiency of frictional-bond strength when using fresh concrete with higher self-compacting, greater roughness at the interface of SSF/hardened-concrete, and fiber with lower yield strength. While no significant effect of Poisson’s ratio on the frictional-bond strength was noticed.

Numerical Investigations of Damage Behaviour at the Weld/Base Metal Interface

In the present paper, the numerical modelling to predict the interface damage of weld defect in a steel pipeline was studied. This work focused on determination of the maximum operating pressure and the characterisation of mechanical behaviour at a weld-base metal interface. The operating pressure can fluctuate leading to the phenomenon of fatigue and consequently to the failure of pipeline. Experimental investigations were carried out using non-destructive test (NDT) in order to locate and determine size of defects. A bilinear interface decohesion model is used to simulate the damage behaviour considering a stress-relative displacement laws. Numerical simulations based on the finite element method were used to study the influence of size defects and young's moduli ratio on the operating pressure as well as interfacial damage between the weld and base metal. The obtained results showed that the interface damage depending on shape and material properties of defects has an impact on pipeline safety and integrity.

Failure characterization of Longmaxi shale under direct shear mode loadings

Insight into the shear failure of shale is significant for the analysis of hydraulic fracturing and wellbore stability in a shale gas reservoir. In this work, a series of direct shear tests is conducted on cubic Longmaxi shale samples with various bedding orientations, i.e., α = 0°, 30°, 60° and 90°, where the bedding orientation α is defined as the angle between the bedding and shearing direction. The anisotropies in the shear stress-displacement relation, failure pattern and shear strength are then analyzed and highlighted. Stress fluctuation is widely observed near the peak shear stress for most samples with α = 30° and 60°, and the shear displacement at the peak shear stress reaches its maximum and minimum values at α = 60° and α = 0°, respectively, for a given normal stress. The failure surface is formed by shear sliding along a weak bedding plane if α = 0°, resulting in a smooth failure surface with a much lower fractal dimension value; when α = 30°, 60° and 90°, some inclined cracks form across the failure surface during shearing, resulting in much rougher fracture surfaces. The shear strength and cohesion reach their minimum values at α = 0°, while the smallest internal friction angle is observed at α = 90°. Finally, by considering the anisotropies induced by the material microstructure and stress state, a new anisotropic shear failure criterion is presented and found to achieve good agreement with the test results. With the new criterion, the bedding angle corresponding to the maximum shear strength is found to vary with the applied normal stress.