Soilmix

groundhog.excavations.soilmix.bendingstiffness_soilmix_method1(moment_inertia_reinforcement, modulus_soilmix, height_soilmix, reinforcement_offset, height_reinforcement, flange_thickness, connection_thickness, flange_width, modulus_reinforcement=210000000.0, participating_width=nan, tensile_strength_soilmix=nan, **kwargs)[source]

Calculates the bending stiffness of a soilmix wall (for use in geotechnical retaining wall calculations) using the combined stiffness of reinforcements and the soilmix material itself. The stiffness is calculated as the average of the cracked and uncracked stiffness of the material according to §5.4.2.3 if EN 1992-1-1. The formulae contain simplications that result in a small error of 1 to 3%. Note that all dimensions are in meters and units of force are in kN.

Parameters:

  • moment_inertia_reinforcement – Moment of inertia of the reinforcement (\(I_a\)) [m4] - Suggested range: moment_inertia_reinforcement >= 0.0

  • modulus_soilmix – Youngs modulus of the soilmix (\(E_{sm}\)) [kPa] - Suggested range: modulus_soilmix >= 0.0

  • height_soilmix – Height of the soilmix material (\(h_{sm}\)) [m] - Suggested range: height_soilmix >= 0.0

  • reinforcement_offset – Offset of reinforcement elements (center to center) (\(a\)) [m] - Suggested range: reinforcement_offset >= 0.0

  • height_reinforcement – Longest dimension of the reinforcement (\(h_a\)) [m] - Suggested range: height_reinforcement >= 0.0

  • flange_thickness – Thickness of the flanges (\(t_f\)) [m] - Suggested range: flange_thickness >= 0.0

  • connection_thickness – Thickness of the connection between the flanges (\(t_w\)) [m] - Suggested range: connection_thickness >= 0.0

  • flange_width – Width of the flanges (\(b_f\)) [m] - Suggested range: flange_width >= 0.0

  • modulus_reinforcement – Youngs modulus of the reinforcement (steel by default) (\(E_a\)) [kPa] - Suggested range: 0.0 <= modulus_reinforcement <= 300000000.0 (optional, default= 210000000.0)

  • participating_width – Participating width for bending. For most cases, this is equal to the center-to-center distance of reinforcement (default when np.nan is used) (\(b_{c1}\)) [m] - Suggested range: participating_width >= 0.0 (optional, default= np.nan)

  • tensile_strength_soilmix – Tensile strength of soilmix to calculate cracking moment (\(f_{sm,t}\)) [kPa] - Suggested range: tensile_strength_soilmix >= 0.0 (optional, default= np.nan)

\[ \begin{align}\begin{aligned}EI\text{-eff} = \frac{EI\text{-1} + EI\text{-2}}{2}\\EI\text{-1}=E_a I_a + E_{sm}(I_{sm} - I_a)=E_{sm} \left[ (n-1) I_a + I_{sm}\right] \quad \text{where } n=E_a/E_{sm}\\I_{sm} = \frac{b_{cl} h_{sm}^3}{12}\\c_1=c_2=\frac{h_{sm}-h_a}{2}\\d=h_{eq}-c_2-\frac{t_f}{2} \quad{where } h_{eq} \text{ can be taken as } h_{sm}\\c_{1,b}=c_1+\frac{t_f}{2}\\h_w=h_a-2 t_f\\\xi_e=-(2n-1) \rho + \sqrt{(2n-1)^2 \rho^2 + 2 \left[ (n-1) \frac{h_{sm}}{d} + 1 \right] \rho}\\\xi_e=\frac{x_e}{d}\\\rho=\frac{A_f}{d b_{cl}}\\A_f=t_f b_f\\I_2=\frac{b_{cl} x_e^3}{3} + (n-1) A_f (x_e - c_{1,b})^2 + n A_f (d-x_e)^2 + n t_w \left( \frac{(x_e - c_1 - t_f)^3}{3} + \frac{(h_w - x_e + c_1 + t_f)^3}{3} \right)\\EI\text{-2} = E_{sm} I_2\\EI\text{-eff/m} = EI\text{-eff} = \frac{EI\text{-1} + EI\text{-2}}{2a}\end{aligned}\end{align} \]
Returns:

Dictionary with the following keys:

  • ’n [-]’: Ratio of reinforcement and soilmix moduli (\(n\)) [-]

  • ’Ism [m4]’: Moment of inertia of soilmix (\(I_{sm}\)) [m4]

  • ’EI-1 [kNm2]’: Bending stiffness of the uncracked section (\(EI\text{-1}\)) [kNm2]

  • ’Mcr [kNm]’: Cracking moment (if tensile strength is specified) (\(M_{cr}\)) [kNm]

  • ’c1 [m]’: Net cover above reinforcement (\(c_1\)) [m]

  • ’c2 [m]’: Net cover below reinforcement (\(c_2\)) [m]

  • ’d [m]’: Useful height of lower flange (\(d\)) [m]

  • ’c1b [m]’: Gross cover on upper flange (\(c_{1b}\)) [m]

  • ’hw [m]’: Height of the body of the reinforcement (\(h_w\)) [m]

  • ’Af [m2]’: Flange area (\(A_f\)) [m2]

  • ’rho [-]’: - (\(\rho\)) [-]

  • ’xi,e [-]’: - (\(\xi_e\)) [-]

  • ’xe [m]’: Effective height of the section (\(x_e\)) [m]

  • ’I2 [m4]’: Moment of inertia of the cracked section (\(I_2\)) [m4]

  • ’EI-2 [kNm2]’: Bending stiffness of the cracked section (\(EI\text{-2}\)) [kNm2]

  • ’EI [kNm2]’: Bending stiffness of the combined section (\(EI\)) [kNm2]

  • ’EI-eff/m [kNm2]’: Bending stiffness of the combined section per unit length (\(EI\text{-eff}\)) [kNm2]

../_images/IPE_conventions.png

Naming conventions for reinforcement geometry

Reference - Denies, N. and Huybrechts, N. (2016). Handboek soimix-wanden - Ontwerp en uitvoering, SBRCURnet

groundhog.excavations.soilmix.bendingstiffness_soilmix_method2(bendingstiffness_reinforcement, modulus_soilmix, height_soilmix, reinforcement_offset, participating_width=nan, **kwargs)[source]

Calculates the bending stiffness of a soilmix wall with internal reinforcements according to a simplified method. The effective bending stiffness is calculated as the sum of the stiffness of the reinforcement and the stiffness of the soilmix zone under compression. This method is usually on the conservative side, with bending stiffnesses 10 to 20% lower than the ones calculated with the more accurate method 1.

Parameters:

  • bendingstiffness_reinforcement – Bending stiffness of the reinforcement (\(E_a I_a\)) [m4] - Suggested range: bendingstiffness_reinforcement >= 0.0

  • modulus_soilmix – Youngs modulus of the soilmix material (\(E_{sm}\)) [kPa] - Suggested range: modulus_soilmix >= 0.0

  • height_soilmix – Height of the soilmix material (\(h_{sm}\)) [m] - Suggested range: height_soilmix >= 0.0

  • reinforcement_offset – Center-to-center offset of reinforcement elements (\(a\)) [m] - Suggested range: reinforcement_offset >= 0.0

  • participating_width – Participating width for bending. For most cases, this is equal to the center-to-center distance of reinforcement (default when np.nan is used) (\(b_{c1}\)) [m] - Suggested range: participating_width >= 0.0 (optional, default= np.nan)

\[ \begin{align}\begin{aligned}EI\text{-eff} = E_a I_a + E_{sm} \left[ \frac{b_{c1} \cdot \left( \frac{h_{sm}}{2} \right)^3}{3} \right]\\EI\text{-eff} / m^{\prime} = EI\text{-eff} / a=\frac{E_a I_a}{a} + E_{sm} \left[ \frac{\left( \frac{h_{sm}}{2} \right)^3}{3} \right]\end{aligned}\end{align} \]
Returns:

Dictionary with the following keys:

  • ’EaIa [kNm2]’: Bending stiffness of the reinforcement (\(E_a I_a\)) [kNm2]

  • ’EI_sm [kNm2]’: Bending stiffness of the soilmix (\(EI_{sm}\)) [kNm2]

  • ’EI-eff [kNm2]’: Effective bending stiffness of the combined section (\(EI\text{-eff}\)) [kNm2]

  • ’EI-eff/m [kNm2/m]’: Effective bending stiffness per unit wall length (\(EI\text{-eff/m}\)) [kNm2/m]

Reference - Denies, N. and Huybrechts, N. (2016). Handboek soimix-wanden - Ontwerp en uitvoering, SBRCURnet