> 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/massive-rock-liang-et-al-2009.md).

# Massive Rock (Liang) \[Liang et al., 2009]

\[Liang et al., 2009] derived a hyperbolic *p-y* criterion for drilled shafts in rock mass from theoretical failure mechanisms and a three-dimensional finite element parametric study, validated against six full-scale lateral load tests. The rock mass strength is described by the Hoek-Brown criterion with the Geological Strength Index; the criterion is intended for rock with enough closely spaced discontinuities to be treated as isotropic, not for response controlled by a single joint set.

$$
\tag{1}
p = \frac{y}{\frac{1}{K\_i}+\frac{y}{p\_u}}
$$

### Initial Tangent

$$
\tag{2}
K\_i = E\_m\Bigl(\frac{b}{b\_{ref}}\Bigr)e^{-2\nu}\Bigl(\frac{E\_pI\_p}{E\_mb^4}\Bigr)^{0.284}
$$

$$
\tag{3}
E\_m = \frac{E\_i}{100}e^{GSI/21.7}
$$

where,

$$b$$ = shaft diameter, $$b\_{ref}$$ = 0.3048 m (12 in; the reference rounds it to 0.305 m)\
$$E\_pI\_p$$ = flexural stiffness of the shaft (taken from the pile section in OpenBrIM)\
$$E\_m$$ = modulus of the rock mass, $$E\_i$$ = modulus of intact rock\
$$ν$$ = Poisson's ratio of the rock mass\
$$GSI$$ = geological strength index

### Rock Mass Strength

\[Hoek and Brown, 1980] with the constants of \[Marinos and Hoek, 2000] (disturbance factor 0):

$$
\tag{4}
\sigma'\_1 = \sigma'\_3+q\_u\Bigl(m\_b\frac{\sigma'\_3}{q\_u}+s\Bigr)^a, \qquad m\_b = m\_ie^{(GSI-100)/28}, \quad s = e^{(GSI-100)/9}, \quad a = 0.5
$$

The instantaneous Mohr-Coulomb parameters at the minor principal stress $$σ'\_3$$ (the effective overburden) follow Hoek (1990):

$$
\tag{5}
\sigma'\_n = \sigma'\_3+\frac{(\sigma'\_1-\sigma'\_3)^2}{2(\sigma'\_1-\sigma'\_3)+0.5m\_bq\_u}, \qquad \tau = (\sigma'\_n-\sigma'\_3)\sqrt{1+\frac{m\_bq\_u}{2(\sigma'\_1-\sigma'\_3)}}
$$

$$
\tag{6}
\phi' = 90° - \arcsin\Bigl(\frac{2\tau}{\sigma'\_1-\sigma'\_3}\Bigr), \qquad c' = \tau-\sigma'\_n\tan\phi'
$$

where $$q\_u$$ is the unconfined compressive strength of the intact rock and $$m\_i$$ its Hoek-Brown constant.

### Ultimate Resistance

$$p\_u$$ is the lesser of the wedge-failure value near the surface and the flow-around value at depth.

Failure at depth (Eqs. 13, 14, 21 of the reference), with $$c'$$ and $$ϕ'$$ evaluated at the depth of interest:

$$
\tag{7}
p\_u = \Bigl(\frac{\pi}{4}p\_L+\frac{2}{3}\tau\_{max}-p\_a\Bigr)b, \qquad p\_a = K\_a\sigma'\_v-2c'\sqrt{K\_a} \geq 0, \qquad K\_a = \tan^2\Bigl(45°-\frac{\phi'}{2}\Bigr)
$$

$$
\tag{8}
\tau\_{max} = 0.45\sqrt{q\_u} \ (MPa)
$$

where $$p\_L$$ is the limit normal pressure of the rock mass, Eq. (4) with $$σ'\_3$$ equal to the effective overburden $$σ'*v$$, and $$τ*{max}$$ the shaft-rock interface shear strength of Kulhawy and Phoon (1993).

Wedge failure near the surface (Eqs. 2–10 of the reference), with $$c'$$ and $$ϕ'$$ evaluated at a depth of $$H/3$$ because the side faces of the wedge are triangular:

$$
\tag{9}
p\_u = 2C\_1\cos\theta\sin\beta+C\_2\cos\beta+C\_3\sin\beta-2C\_4\sin\theta-C\_5
$$

$$
\tag{10}
C\_1 = H\frac{\tan\beta}{\cos\theta}\Bigl(c'+K\_0\sigma'\_{v0}\tan\phi'+\frac{H}{2}K\_0\gamma'\tan\phi'\Bigr)
$$

$$
\tag{11}
C\_2 = \frac{b\tan\beta(\sigma'*{v0}+H\gamma')+H\tan^2\beta\tan\theta(2\sigma'*{v0}+H\gamma')+c'(b+2H\tan\beta\tan\theta)+2C\_1\cos\beta\cos\theta}{\sin\beta-\tan\phi'\cos\beta}
$$

$$
\tag{12}
C\_3 = C\_2\tan\phi'+c'\Bigl(\frac{b}{\cos\beta}+2H\frac{\tan\beta\tan\theta}{\cos\beta}\Bigr)
$$

$$
\tag{13}
C\_4 = K\_0H\frac{\tan\beta}{\cos\theta}\Bigl(\sigma'\_{v0}+\frac{1}{2}\gamma' H\Bigr), \qquad C\_5 = \gamma' K\_a(H-z\_0)b \geq 0
$$

$$
\tag{14}
\beta = 45°+\frac{\phi'}{2}, \qquad \theta = \frac{\phi'}{2}, \qquad K\_0 = 1-\sin\phi', \qquad z\_0 = \frac{2c'}{\gamma'\sqrt{K\_a}}-\frac{\sigma'\_{v0}}{\gamma'}
$$

where,

$$H$$ = height of the failure wedge, the depth below the rock surface\
$$γ'$$ = effective unit weight of the rock mass\
$$σ'\_{v0}$$ = effective vertical stress applied on top of the rock mass (zero when the rock surface is the ground surface, as assumed in OpenBrIM)\
$$z\_0$$ = depth of the tensile crack

{% hint style="info" %}
The $$C\_2\cos\beta$$ term of Eq. (9) is retained, following Eq. 2 of \[Liang et al., 2009] and the force balance of the wedge (Eq. 23 of the reference). Some implementations of this model omit it.
{% endhint %}

***

\[Liang et al., 2009] Liang, R., Yang, K., and Nusairat, J. (2009). p-y criterion for rock mass. *Journal of Geotechnical and Geoenvironmental Engineering*, 135(1):26–36.

\[Hoek and Brown, 1980] Hoek, E. and Brown, E. T. (1980). Empirical strength criterion for rock masses. *Journal of the Geotechnical Engineering Division*, 106(9):1013–1035.

\[Marinos and Hoek, 2000] Marinos, P. and Hoek, E. (2000). GSI: a geologically friendly tool for rock mass strength estimation. In *Proc. GeoEng2000*, Melbourne, pages 1422–1440.


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