Geotechnical centrifuge modeling is a technique for testing physical scale models of geotechnical engineering systems such as natural and man-made slopes and earth retaining structures and building or bridge foundations. [1]
The scale model is typically constructed in the laboratory and then loaded onto the end of the centrifuge, which is typically between 0.2 and 10 metres (0.7 and 32.8 ft) in radius. The purpose of spinning the models on the centrifuge is to increase the g-forces on the model so that stresses in the model are equal to stresses in the prototype. For example, the stress beneath a 0.1-metre-deep (0.3 ft) layer of model soil spun at a centrifugal acceleration of 50 g produces stresses equivalent to those beneath a 5-metre-deep (16 ft) prototype layer of soil in earth's gravity.
The idea to use centrifugal acceleration to simulate increased gravitational acceleration was first proposed by Phillips (1869). [2] Pokrovsky and Fedorov (1936) [3] in the Soviet Union and Bucky (1931) [4] in the United States were the first to implement the idea. Andrew N. Schofield (e.g. Schofield 1980) [5] played a key role in modern development of centrifuge modeling.
A geotechnical centrifuge is used to test models of geotechnical problems such as the strength, stiffness and capacity of foundations for bridges and buildings, settlement of embankments, [6] stability of slopes, earth retaining structures, [7] tunnel stability and seawalls. Other applications include explosive cratering, [8] contaminant migration in ground water, frost heave and sea ice. The centrifuge may be useful for scale modeling of any large-scale nonlinear problem for which gravity is a primary driving force.
Geotechnical materials such as soil and rock have non-linear mechanical properties that depend on the effective confining stress and stress history. The centrifuge applies an increased "gravitational" acceleration to physical models in order to produce identical self-weight stresses in the model and prototype. The one to one scaling of stress enhances the similarity of geotechnical models and makes it possible to obtain accurate data to help solve complex problems such as earthquake-induced liquefaction, soil-structure interaction and underground transport of pollutants such as dense non-aqueous phase liquids. Centrifuge model testing provides data to improve our understanding of basic mechanisms of deformation and failure and provides benchmarks useful for verification of numerical models.
Note that in this article, the asterisk on any quantity represents the scale factor for that quantity. For example, in , the subscript m represents "model" and the subscript p represents "prototype" and represents the scale factor for the quantity . [9]
The reason for spinning a model on a centrifuge is to enable small scale models to feel the same effective stresses as a full-scale prototype. This goal can be stated mathematically as
where the asterisk represents the scaling factor for the quantity, is the effective stress in the model and is the effective stress in the prototype.
In soil mechanics the vertical effective stress, for example, is typically calculated by
where is the total stress and is the pore pressure. For a uniform layer with no pore pressure, the total vertical stress at a depth may be calculated by:
where represents the density of the layer and represents gravity. In the conventional form of centrifuge modeling, [9] it is typical that the same materials are used in the model and prototype; therefore the densities are the same in model and prototype, i.e.,
Furthermore, in conventional centrifuge modeling all lengths are scaled by the same factor . To produce the same stress in the model as in the prototype, we thus require , which may be rewritten as
The above scaling law states that if lengths in the model are reduced by some factor, n, then gravitational accelerations must be increased by the same factor, n in order to preserve equal stresses in model and prototype.
For dynamic problems where gravity and accelerations are important, all accelerations must scale as gravity is scaled, i.e.
Since acceleration has units of , it is required that
Hence it is required that :, or
Frequency has units of inverse of time, velocity has units of length per time, so for dynamic problems we also obtain
For model tests involving both dynamics and diffusion, the conflict in time scale factors may be resolved by scaling the permeability of the soil [9]
(this section obviously needs work!)
scale factors for energy, force, pressure, acceleration, velocity, etc. Note that stress has units of pressure, or force per unit area. Thus we can show that
Substituting F = m∙a (Newton's law, force = mass ∙ acceleration) and r = m/L3 (from the definition of mass density).
Scale factors for many other quantities can be derived from the above relationships. The table below summarizes common scale factors for centrifuge testing.
Scale Factors for Centrifuge Model Tests (from Garnier et al., 2007 [9] ) (Table is suggested to be added here)
This section needs additional citations for verification .(January 2021) |
Large earthquakes are infrequent and unrepeatable but they can be devastating. All of these factors make it difficult to obtain the required data to study their effects by post earthquake field investigations. Instrumentation of full scale structures is expensive to maintain over the large periods of time that may elapse between major temblors, and the instrumentation may not be placed in the most scientifically useful locations. Even if engineers are lucky enough to obtain timely recordings of data from real failures, there is no guarantee that the instrumentation is providing repeatable data. In addition, scientifically educational failures from real earthquakes come at the expense of the safety of the public. Understandably, after a real earthquake, most of the interesting data is rapidly cleared away before engineers have an opportunity to adequately study the failure modes.
Centrifuge modeling is a valuable tool for studying the effects of ground shaking on critical structures without risking the safety of the public. The efficacy of alternative designs or seismic retrofitting techniques can compared in a repeatable scientific series of tests.
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Centrifuge tests can also be used to obtain experimental data to verify a design procedure or a computer model. The rapid development of computational power over recent decades has revolutionized engineering analysis. Many computer models have been developed to predict the behavior of geotechnical structures during earthquakes and other loads. Before a computer model can be used with confidence, it must be proven to be valid based on evidence. The meager and unrepeatable data provided by natural earthquakes, for example, is usually insufficient for this purpose. Verification of the validity of assumptions made by a computational algorithm is especially important in the area of geotechnical engineering due to the complexity of soil behavior. Soils exhibit highly non-linear behavior, their strength and stiffness depend on their stress history and on the water pressure in the pore fluid, all of which may evolve during the loading caused by an earthquake. The computer models which are intended to simulate these phenomena are very complex and require extensive verification. Experimental data from centrifuge tests is useful for verifying assumptions made by a computational algorithm. If the results show the computer model to be inaccurate, the centrifuge test data provides insight into the physical processes which in turn stimulates the development of better computer models.
Geotechnical engineering, also known as geotechnics, is the branch of civil engineering concerned with the engineering behavior of earth materials. It uses the principles of soil mechanics and rock mechanics to solve its engineering problems. It also relies on knowledge of geology, hydrology, geophysics, and other related sciences.
A centrifuge is a device that uses centrifugal force to subject a specimen to a specified constant force, for example to separate various components of a fluid. This is achieved by spinning the fluid at high speed within a container, thereby separating fluids of different densities or liquids from solids. It works by causing denser substances and particles to move outward in the radial direction. At the same time, objects that are less dense are displaced and moved to the centre. In a laboratory centrifuge that uses sample tubes, the radial acceleration causes denser particles to settle to the bottom of the tube, while low-density substances rise to the top. A centrifuge can be a very effective filter that separates contaminants from the main body of fluid.
In continuum mechanics, the Froude number is a dimensionless number defined as the ratio of the flow inertia to the external force field. The Froude number is based on the speed–length ratio which he defined as: where u is the local flow velocity, g is the local gravity field, and L is a characteristic length.
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.
Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids and particles but soil may also contain organic solids and other matter. Along with rock mechanics, soil mechanics provides the theoretical basis for analysis in geotechnical engineering, a subdiscipline of civil engineering, and engineering geology, a subdiscipline of geology. Soil mechanics is used to analyze the deformations of and flow of fluids within natural and man-made structures that are supported on or made of soil, or structures that are buried in soils. Example applications are building and bridge foundations, retaining walls, dams, and buried pipeline systems. Principles of soil mechanics are also used in related disciplines such as geophysical engineering, coastal engineering, agricultural engineering, hydrology and soil physics.
In fracture mechanics, the stress intensity factor is used to predict the stress state near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible and is known as plastic deformation.
Soil consolidation refers to the mechanical process by which soil changes volume gradually in response to a change in pressure. This happens because soil is a three-phase material, comprising soil grains and pore fluid, usually groundwater. When soil saturated with water is subjected to an increase in pressure, the high volumetric stiffness of water compared to the soil matrix means that the water initially absorbs all the change in pressure without changing volume, creating excess pore water pressure. As water diffuses away from regions of high pressure due to seepage, the soil matrix gradually takes up the pressure change and shrinks in volume. The theoretical framework of consolidation is therefore closely related to the concept of effective stress, and hydraulic conductivity. The early theoretical modern models were proposed one century ago, according to two different approaches, by Karl Terzaghi and Paul Fillunger. The Terzaghi’s model is currently the most utilized in engineering practice and is based on the diffusion equation.
In geotechnical engineering, bearing capacity is the capacity of soil to support the loads applied to the ground. The bearing capacity of soil is the maximum average contact pressure between the foundation and the soil which should not produce shear failure in the soil. Ultimate bearing capacity is the theoretical maximum pressure which can be supported without failure; allowable bearing capacity is the ultimate bearing capacity divided by a factor of safety. Sometimes, on soft soil sites, large settlements may occur under loaded foundations without actual shear failure occurring; in such cases, the allowable bearing capacity is based on the maximum allowable settlement. The allowable bearing pressure is the maximum pressure that can be applied to the soil without causing failure. The ultimate bearing capacity, on the other hand, is the maximum pressure that can be applied to the soil before it fails.
The lateral earth pressure is the pressure that soil exerts in the horizontal direction. It is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering structures such as retaining walls, basements, tunnels, deep foundations and braced excavations.
Shear strength is a term used in soil mechanics to describe the magnitude of the shear stress that a soil can sustain. The shear resistance of soil is a result of friction and interlocking of particles, and possibly cementation or bonding of particle contacts. Due to interlocking, particulate material may expand or contract in volume as it is subject to shear strains. If soil expands its volume, the density of particles will decrease and the strength will decrease; in this case, the peak strength would be followed by a reduction of shear stress. The stress-strain relationship levels off when the material stops expanding or contracting, and when interparticle bonds are broken. The theoretical state at which the shear stress and density remain constant while the shear strain increases may be called the critical state, steady state, or residual strength.
Critical state soil mechanics is the area of soil mechanics that encompasses the conceptual models representing the mechanical behavior of saturated remoulded soils based on the critical state concept. At the critical state, the relationship between forces applied in the soil (stress), and the resulting deformation resulting from this stress (strain) becomes constant. The soil will continue to deform, but the stress will no longer increase.
The Larson–Miller relation, also widely known as the Larson–Miller parameter and often abbreviated LMP, is a parametric relation used to extrapolate experimental data on creep and rupture life of engineering materials.
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.
Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of slopes of soil- and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. It is performed to assess the safe design of a human-made or natural slopes and the equilibrium conditions. Slope stability is the resistance of inclined surface to failure by sliding or collapsing. The main objectives of slope stability analysis are finding endangered areas, investigation of potential failure mechanisms, determination of the slope sensitivity to different triggering mechanisms, designing of optimal slopes with regard to safety, reliability and economics, and designing possible remedial measures, e.g. barriers and stabilization.
Preconsolidation pressure is the maximum effective vertical overburden stress that a particular soil sample has sustained in the past. This quantity is important in geotechnical engineering, particularly for finding the expected settlement of foundations and embankments. Alternative names for the preconsolidation pressure are preconsolidation stress, pre-compression stress, pre-compaction stress, and preload stress. A soil is called overconsolidated if the current effective stress acting on the soil is less than the historical maximum.
K-epsilon (k-ε) turbulence model is one of the most common models used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model that gives a general description of turbulence by means of two transport equations. The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.
Menter's Shear Stress Transport turbulence model, or SST, is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.