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

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.

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[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.