Geotechnical centrifuge modeling

Model of a port structure loaded on the UC Davis centrifuge

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 ${\displaystyle x^{*}={\frac {x_{m}}{x_{p}}}}$, the subscript m represents "model" and the subscript p represents "prototype" and ${\displaystyle x^{*}\,}$ represents the scale factor for the quantity ${\displaystyle x\,}$.[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

${\displaystyle \sigma '^{*}={\frac {\sigma '_{m}}{\sigma '_{p}}}=1}$

where the asterisk represents the scaling factor for the quantity, ${\displaystyle \sigma '_{m}}$ is the effective stress in the model and ${\displaystyle \sigma '_{p}}$ is the effective stress in the prototype.

In soil mechanics the vertical effective stress, ${\displaystyle \sigma '}$ for example, is typically calculated by

${\displaystyle \sigma '=\sigma ^{t}-u\,}$

where ${\displaystyle \sigma ^{t}}$ is the total stress and ${\displaystyle u}$ is the pore pressure. For a uniform layer with no pore pressure, the total vertical stress at a depth ${\displaystyle H}$ may be calculated by:

${\displaystyle \sigma ^{t}=\rho gH\,}$

where ${\displaystyle \rho }$ represents the density of the layer and ${\displaystyle g}$ 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.,

${\displaystyle \rho ^{*}=1\,}$

Furthermore, in conventional centrifuge modeling all lengths are scaled by the same factor ${\displaystyle L^{*}}$. To produce the same stress in the model as in the prototype, we thus require ${\displaystyle \rho ^{*}g^{*}H^{*}=(1)g^{*}L^{*}=1\,}$, which may be rewritten as

${\displaystyle g^{*}={\frac {1}{L^{*}}}}$

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.

${\displaystyle a^{*}=g^{*}={\frac {1}{L^{*}}}}$

Since acceleration has units of ${\displaystyle {\frac {L}{T^{2}}}}$, it is required that

${\displaystyle a^{*}={\frac {L^{*}}{T^{*2}}}}$

Hence it is required that :${\displaystyle {\frac {1}{L^{*}}}={\frac {L^{*}}{T^{*2}}}}$, or

${\displaystyle T^{*}=L^{*}\,}$

Frequency has units of inverse of time, velocity has units of length per time, so for dynamic problems we also obtain

${\displaystyle f^{*}={\frac {1}{L{*}}}}$

${\displaystyle v^{*}={\frac {L^{*}}{T{*}}}=1}$

${\displaystyle T^{*}=L^{*2}\,}$

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)