Dynamic soil property correlations

groundhog.soildynamics.soilproperties.dampingratio_sandgravel_seed(cyclic_shear_strain, **kwargs)

Damping ratios for sand are compiled from a dataset comprising several sands and gravels. Average values and upper and lower bounds are provided. The comparison of the trends proposed for sand with the datapoints measured on gravel suggests that the trend is applicable for gravels too.

Parameters:

cyclic_shear_strain – Cyclic shear strain (\(\gamma_{cyc}\)) [\(pct\)] - Suggested range: 0.0001 <= cyclic_shear_strain <= 1.0

Returns:

Dictionary with the following keys:

• ’D LE [pct]’: Low estimate damping ratio (\(D_{LE}\)) [\(pct\)]

• ’D BE [pct]’: Average or best estimate damping ratio (\(D_{BE}\)) [\(pct\)]

• ’D HE [pct]’: High estimate damping ratio (\(D_{HE}\)) [\(pct\)]

Proposed trends and measurement data on gravels

Reference - Seed, H. B., Wong, R. T., Idriss, I. M., & Tokimatsu, K. (1986). Moduli and damping factors for dynamic analyses of cohesionless soils. Journal of geotechnical engineering, 112(11), 1016-1032.

groundhog.soildynamics.soilproperties.gmax_shearwavevelocity(Vs, gamma, g=9.81, **kwargs)

Calculates the small-strain shear modulus (shear strain < 1e-4%) from the shear wave velocity and the bulk unit weight if the soil based on elastic theory.

Often, the result of an in-situ or laboratory test will provide the shear wave velocity, which is then converted to the small-strain shear modulus using this function.

Parameters:

• Vs – Shear wave velocity (\(V_s\)) [\(m/s\)] - Suggested range: 0.0 <= Vs <= 600.0

• gamma – Bulk unit weight (\(\gamma\)) [\(kN/m3\)] - Suggested range: 12.0 <= gamma <= 22.0

• g – Acceleration due to gravity (\(g\)) [\(m/s2\)] - Suggested range: 9.7 <= g <= 10.2 (optional, default= 9.81)

\[ \begin{align}\begin{aligned}G_{max} = \rho \cdot V_s^2\\\rho = \gamma / g\end{aligned}\end{align} \]

Returns:

Dictionary with the following keys:

• ’rho [kg/m3]’: Density of the material (\(\rho\)) [\(kg/m3\)]

• ’Gmax [kPa]’: Small-strain shear modulus (\(G_{max}\)) [\(kPa\)]

Reference - Robertson, P.K. and Cabal, K.L. (2015). Guide to Cone Penetration Testing for Geotechnical Engineering. 6th edition. Gregg Drilling & Testing, Inc.

groundhog.soildynamics.soilproperties.modulusreduction_darendeli(mean_effective_stress, pi, ocr, N, frequency, soiltype, min_strain=0.0001, max_strain=1.0, no_points=250, custom_coefficients=None, **kwargs)

Darendeli (2001) proposed a comprehensive framework for estimating the modulus reduction curve and damping curve for sand, fine sand, silt and clay based on extensive laboratory testing. The framework is initially based on the work by Hardin and Drnevich but extends the formulation to include the effect of soil type, plasticity, overconsolidation ratio, stress ratio, loading frequency and number of cycles applied.

The author used a Bayesian approach to calibrate the parameters of the parametric equations and also formulated expressions to estimate the standard deviation on the estimates. Parameters for individual soil types can be used as well as parameters calibrated to the entire credible dataset (using 'all' for soiltype).

The formulation is based on Masing damping but takes into account the damping at small strains, which is not zero but difficult to estimate from resonant column or cyclic DSS tests. The Masing damping at large strains is also adjusted to be in line with experimental observations (Masing damping overestimates the damping ratio at large strains).

Parameters:

• mean_effective_stress – Mean effective stress at the depth under consideration (\(\sigma_0^{\prime}\)) [\(kPa\)] - Suggested range: 0.0 <= mean_effective_stress <= 1000.0

• pi – Plasticity index (difference between liquid limit and plastic limit) (\(PI\)) [\(pct\)] - Suggested range: 0.0 <= plasticity_index <= 60.0

• ocr – Overconsolidation ratio of the soil (\(OCR\)) [\(-\)] - Suggested range: 1.0 <= OCR <= 20.0

• N – Number of cycles (\(N\)) [\(-\)] - Suggested range: N >= 1.0

• frequency – Loading frequency (\(f\)) [\(Hz\)] - Suggested range: 0.05 <= frequency <= 20.0

• soiltype – Soil type used for calculating modulus reduction and damping curves - Options: (‘sand’, ‘fine sand’, ‘silt’, ‘clay’, ‘all’)

• min_strain – Minimum value for the strain (\(\gamma_{min}\)) [\(pct\)] (optional, default= 0.0001)

• max_strain – Maximum value for the strain (\(\gamma_{max}\)) [\(pct\)] (optional, default= 1.0)

• no_points – Number of points used for the strain curve calculation (:math:``) [\(-\)] - Suggested range: no_points >= 10.0 (optional, default= 250)

• custom_coefficients – Dictionary with custom calibration coefficients (:math:``) [\(-\)] (optional, default= None)- Elementtype: float, order: ascending, unique: True, empty entries allowed: False

\[ \begin{align}\begin{aligned}\frac{G}{G_{max}} = \frac{1}{1 + \left( \frac{\gamma}{\gamma_r} \right)^a}\\\gamma_r = \left( \phi_1 + \phi_2 \cdot PI \cdot OCR^{\phi_3} \right) \cdot \sigma_0^{\prime \phi_4}\\a = \phi_5\\D_{\text{adjusted}} = b \cdot \left( \frac{G}{G_{max}} \right)^{0.1} \cdot D_{\text{Masing}} + D_{\text{min}}\\D_{\text{min}} = \left( \phi_6 + \phi_7 \cdot PI \cdot OCR^{\phi_8} \right) \cdot \sigma_0^{\prime \phi_9} \cdot \left[1 + \phi_{10} \cdot \ln(f) \right]\\b = \phi_{11} + \phi_{12} \cdot \ln(N)\\\sigma_{\text{NG}} = \exp(\phi_{13}) + \sqrt{\frac{0.25}{\exp(\phi_{14})} - \frac{\left(G / G_{max} - 0.5 \right)^2}{\exp(\phi_{14})}}\\\sigma_D = \exp (\phi_{15} ) + \exp (\phi_{16} ) \cdot \sqrt{D}\\\rho_{i,j} = \exp \left( \frac{-1}{\exp ( \phi_{17} )} \right) \cdot \exp \left( \frac{- | \ln \gamma_i - \ln \gamma_j |}{\exp ( \phi_{18} )} \right)\end{aligned}\end{align} \]

Returns:

Dictionary with the following keys:

• ’strains [pct]’: List of strains for the modulus reduction curve (\(\gamma\)) [\(pct\)]

• ’G/Gmax [-]’: Modulus ratio for given strains (\(G / G_{max}\)) [\(-\)]

• ’D [pct]’: Damping ratios for given strains (\(D\)) [\(pct\)]

• ’sigma_ND [-]’: Standard deviation for the modulus reduction curve (\(\sigma_{ND}\)) [\(-\)]

• ’sigma_D [pct]’: Standard deviation for the damping curve (\(\sigma_D\)) [\(pct\)]

Reference - Darendeli, M. B. (2001). Development of a new family of normalized modulus reduction and material damping curves. The university of Texas at Austin.

groundhog.soildynamics.soilproperties.modulusreduction_plasticity_ishibashi(strain, pi, sigma_m_eff, multiplier_1=0.000102, exponent_1=0.492, multiplier_2=0.000556, exponent_2=0.4, multiplier_3=-0.0145, exponent_3=1.3, **kwargs)

Calculates the modulus reduction curve (G/Gmax) as a function of shear strain. The curve depends on the plasticity of the material (plasticity index) and the mean effective stress at the depth of interest.

The curve for cohesionless soils can be established by using a plasticity index of 0. At low plasticity, the effect of confining pressure on the modulus reduction curve is more pronounced.

Also calculates the damping ratio of plastic and non-plastic soils based on a fit to empirical data.

Parameters:

• strain – Strain amplitude (\(\gamma\)) [\(pct\)] - Suggested range: 0.0 <= strain <= 10.0

• PI – Plasticity index (\(PI\)) [\(pct\)] - Suggested range: 0.0 <= PI <= 200.0

• sigma_m_eff – Mean effective pressure (\(\sigma_m^{\prime}\)) [\(kPa\)] - Suggested range: 0.0 <= sigma_m_eff <= 400.0

• multiplier_1 – Multiplier in equation for K (:math:``) [\(-\)] (optional, default= 0.000102)

• exponent_1 – Exponent in equation for K (:math:``) [\(-\)] (optional, default= 0.492)

• multiplier_2 – First multiplier in equation for m (:math:``) [\(-\)] (optional, default= 0.000556)

• exponent_2 – First exponent in equation for m (:math:``) [\(-\)] (optional, default= 0.4)

• multiplier_3 – Second multiplier in equation for m (:math:``) [\(-\)] (optional, default= -0.0145)

• exponent_3 – Second exponent in equation for m (:math:``) [\(-\)] (optional, default= 1.3)

\[ \begin{align}\begin{aligned}\frac{G}{G_{max}} = K \left( \gamma, \text{PI} \right) \left( \sigma_m^{\prime} \right)^{m \left( \gamma, \text{PI} \right) - m_0}\\K \left( \gamma, \text{PI} \right) = 0.5 \left[ 1 + \tanh \left[ \ln \left( \frac{0.000102 + n ( \text{PI} )}{\gamma} \right)^{0.492} \right] \right]\\m \left( \gamma, \text{PI} \right) - m_0 = 0.272 \left[ 1 - \tanh \left[ \ln \left( \frac{0.000556}{\gamma} \right)^{0.4} \right] \right] \exp \left( -0.0145 \text{PI}^{1.3} \right)\\\begin{split}n ( \text{PI} ) = \begin{cases} 0.0 & \quad \text{for PI } = 0 \\ 3.37 \times 10^{-6} \text{PI}^{1.404} & \quad \text{for } 0 < \text{PI} \leq 15 \\ 7.0 \times 10^{-7} \text{PI}^{1.976} & \quad \text{for } 15 < \text{PI} \leq 70 \\ 2.7 \times 10^{-5} \text{PI}^{1.115} & \quad \text{for } \text{PI} > 70 \end{cases}\end{split}\\\xi = 0.333 \frac{1 + \exp(-0.0145 PI^{1.3})}{2} \left[ 0.586 \left( \frac{G}{G_{max}} \right)^2 - 1.547 \frac{G}{G_{max}} + 1 \right]\end{aligned}\end{align} \]

Returns:

Dictionary with the following keys:

• ’G/Gmax [-]’: Modulus reduction ratio (\(G / G_{max}\)) [\(-\)]

• ’K [-]’: Factor K in the equation (\(K ( \gamma, \text{PI} )\)) [\(-\)]

• ’m [-]’: Exponent m in the equation (\(m \left( \gamma, \text{PI} \right) - m_0\)) [\(-\)]

• ’n [-]’: Factor n in equations (\(n ( \text{PI} )\)) [\(-\)]

• ’dampingratio [pct]’: Damping ratio (\(\xi\)) [\(pct\)]

Reference - Ishibashi, I., & Zhang, X. (1993). Unified dynamic shear moduli and damping ratios of sand and clay. Soils and foundations, 33(1), 182-191.