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Normal soil stiffness

Both static and dynamic normal soil stiffness may be derived from soil characteristics and pipe submerged weigh and diameter through different models such as DnV Guidelines 14, Dunlap, Verley and Lund and Audibert. These stiffness are calculated in the GUI.

The formulas to be used to compute the static or dynamic stiffness are given in the table below. Usually the formula computes a penetration of the pipe in the soil. The stiffness is then the ratio between the pipe weight and this penetration.

Notations:

• \(K_s\) : Static stiffness

• \(K_d\) : Dynamic stiffness

• \(Z_p\) : Penetration of the pipe in the soil

• \(W_p\) : Weight of the pipe

• \(D\) : Diameter of the pipe

• \(K\) : Soil shear coefficient

• \(g\) : Submerged weight of soil

• \(C_u\) : Undrained shear strength

• \(f\) : Friction angle

• \(\nu\) : Poissons coefficient

• \(e\) : Void ratio

• \(I_p\) : Plasticity index

\[ \newcommand{\arraystretch}{2} \begin{array}{|c|c|c|} \hline \bold{Model} & \bold{Static~Stiffness} & \bold{Dynamic~Stiffness} \\ \hline DNV~Guideline~14 & \begin{array}{l} \scriptsize W_{p} = B(\frac{1}{2}\gamma B N_{\gamma}+C_u N_c+\gamma Z_p N_q) \\ \scriptsize B = \begin{cases} 2\sqrt{Z_p(D-z_p)} &\text{if}~Z_p \leqslant \frac{D}{2} \\[0.2cm] D &\text{if}~Z_p>\frac{D}{2} \end{cases} \\ \scriptsize K_s = \displaystyle \frac{W_p}{Z_p} \end{array} & \begin{array}{l} \scriptsize K_d = \displaystyle \frac{0.88}{1-\nu}G \\ \scriptsize G_{clay}=1300 \displaystyle \frac{(2.97-e)^2}{1+e}(OCR^k)\sqrt{\sigma_s} \\ \scriptsize G_{sand}=1955 \displaystyle \frac{(2.97-e)^2}{1+e}\sqrt{\sigma_s} \\ \scriptsize \sigma_s=0.75\gamma \beta \quad OCR=1 \end{array} \\ \hline Dunlap & \begin{array}{l} \scriptsize W_p=7.32 D C_u \displaystyle \Big(\frac{Z_p}{D}\Big)^{0.64} \\ \scriptsize K_s=\displaystyle \frac{W_p}{Z_p} \end{array} & \begin{array}{l} \scriptsize W_p=8.332 D C_u \displaystyle \Big(\frac{Z_p}{D}\Big)^{0.615} \\ \scriptsize K_s=\displaystyle \frac{W_p}{Z_p} \end{array} \\[0.7cm] \hline Verley~and~Lund & \begin{array}{l} \scriptsize \displaystyle \frac{Z_p}{D}=0.0071(SG^{0.3})^{3.2}+0.062(SG^{0.3})^{0.7} \\ \scriptsize \text{With}~G=\displaystyle \frac{C_u}{D \gamma}, \quad S=\frac{W_p}{DC_u} \\ \scriptsize K_s=\displaystyle \frac{W_p}{Z_p} \end{array} & \text{No Dynamic Formula} \\[1cm] \hline Audibert & \begin{array}{l} \scriptsize W_p= \displaystyle \frac{BK(\pi+2)C_uZ_p}{0.15\times0.1B+0.85Z_p} \\[0.5cm] \scriptsize B = \begin{cases} 2\sqrt{Z_p(D-z_p)} &\text{if}~Z_p \leqslant \frac{D}{2}\\[0.2cm] D &\text{if}~Z_p>\frac{D}{2} \end{cases} \end{array} & \text{No Dynamic Formula} \\[1cm] \hline \end{array} \]

\(N_c\), \(N_q\) and \(N_\gamma\) correspond to bearing capacity factors, the formula to be used is the following:

For sandy soils : \(\begin{cases} N_q &= e^{arctan\phi}tan^2 \Big(\frac{\pi}{4}+\frac{\phi}{2}\Big) \\ N_{\gamma} &= 1.8(N_q-1)tan\phi \\ N_c &= (N_q-1)cotan\phi \end{cases}\)

For clayey soils : \(\begin{cases} N_q &= 1 \\ N_{\gamma} &= 0 \\ N_c &= 5.14 \end{cases}\)