> For the complete documentation index, see [llms.txt](https://docs.openbrim.org/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.openbrim.org/technical-background/soil-structure-interaction/lateral-soil-models/hybrid-liquefied-sand-franke-and-rollins-2013.md).

# Hybrid Liquefied Sand (Franke and Rollins) \[Franke and Rollins, 2013]

The simplified hybrid *p-y* model of \[Franke and Rollins, 2013] represents a liquefied sand layer as the lower-bound envelope of two existing curves: the dilative, concave-up curve of [Liquefied Sand (Rollins)](/technical-background/soil-structure-interaction/lateral-soil-models/liquefied-sand-rollins-rollins-et-al-2005.md), which governs at small displacements, and the residual-strength curve of \[Wang and Reese, 1998], which uses the [Soft Clay (Matlock)](/technical-background/soil-structure-interaction/lateral-soil-models/soft-clay-matlock-matlock-1970.md) static curve with the residual shear strength of the liquefied soil as the undrained strength. The residual strength bounds the resistance at large displacements.

$$
\tag{1}
p = min(p\_{clay},p\_{liq})
$$

### Dilative branch (Rollins et al., 2005)

$$
\tag{2}
p\_{liq} = p\_dA(By)^{C}
$$

with $$A$$, $$B$$, $$C$$ and the diameter correction $$p\_d$$ as defined for [Liquefied Sand (Rollins)](/technical-background/soil-structure-interaction/lateral-soil-models/liquefied-sand-rollins-rollins-et-al-2005.md) ($$y$$ in mm, $$p$$ in kN/m, depth in m).

### Residual-strength branch (Wang and Reese, 1998)

$$
\tag{3}
p\_{clay} = 0.5p\_{u,clay}\Bigl(\frac{y}{y\_{50}}\Bigr)^{1/3} \quad for \quad y \leq 8y\_{50}, \qquad p\_{clay} = p\_{u,clay} \quad for \quad y > 8y\_{50}
$$

$$
\tag{4}
p\_{u,clay} = min\Bigl\[\Bigl(3+\frac{\gamma'}{s\_r}z+\frac{J}{b}z\Bigr)s\_rb, \ 9s\_rb\Bigr]
$$

$$
\tag{5}
y\_{50} = 2.5\varepsilon\_{50}b
$$

where,

$$s\_r$$ = residual shear strength of the liquefied soil (\[Franke and Rollins, 2013] recommend the 33rd-percentile values of Seed and Harder, 1990)\
$$γ'$$ = effective unit weight\
$$z$$ = depth below ground surface\
$$b$$ = pile diameter\
$$J$$ = 0.5\
$$ε\_{50}$$ = strain at one-half the maximum principal stress difference

\[Franke and Rollins, 2013] suggest estimating $$ε\_{50}$$ from the clean-sand corrected blow count to reflect the stiffer response of denser liquefiable soils,

$$
\tag{6}
0.0005 \leq \varepsilon\_{50} = 0.1537e^{-0.229(N\_1)\_{60-cs}} \leq 0.05
$$

and neglecting the residual strength ($$s\_r$$ = 0) for $$(N\_1)*{60-cs}$$ < 5 blows/ft until more case histories justify higher values. In OpenBrIM $$s\_r$$ and $$ε*{50}$$ are entered directly on the soil layer.

The envelope is evaluated on the displacement grid of the Rollins curve (0 to 6 in), so all three portions of the curve described by \[Franke and Rollins, 2013] can appear: the Rollins curve at small displacements, the residual-strength curve, and the Rollins curve again where its 15 kN/m cap is the lower of the two. The residual-strength branch is always the static Matlock curve; no cyclic degradation is applied to it.

{% hint style="info" %}
The journal article is not yet on file, so the model is implemented as it is reported in the secondary literature, which draws on \[Franke and Rollins, 2013] and \[Wang and Reese, 1998]. It will be checked against \[Franke and Rollins, 2013] once the article is obtained.
{% endhint %}

***

\[Franke and Rollins, 2013] Franke, K. W. and Rollins, K. M. (2013). Simplified hybrid p-y spring model for liquefied soils. *Journal of Geotechnical and Geoenvironmental Engineering*, 139(4):564–576.

\[Wang and Reese, 1998] Wang, S. T. and Reese, L. C. (1998). Design of pile foundations in liquefied soils. In *Geotechnical Earthquake Engineering and Soil Dynamics III*, GSP 75, ASCE, pages 1331–1343.

\[Rollins et al., 2005] Rollins, K. M., Gerber, T. M., Lane, J. D., and Ashford, S. A. (2005). Lateral resistance of a full-scale pile group in liquefied sand. *Journal of Geotechnical and Geoenvironmental Engineering*, 131(1):115–125.


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