Sand (O'Neill) [Murchison and O'Neill, 1983]

[Murchison and O'Neill, 1983] evaluated four p-y criteria for sand against a database of fourteen full-scale lateral load tests and recommended an "extended hyperbolic" model: the continuous hyperbolic-tangent curve of Parker and Reese with the simplified ultimate resistance expressions of Bogard and Matlock, an empirical depth factor, and a pile shape factor. The same curve was later adopted by API RP 2A (see Sand (API)). The model documented here is the report's curve with its pile shape factor, and the ultimate resistance in the simplified form used by FB-MultiPier's Sand (O'Neill) model (Help Manual, Sec. 13.7.1), which estimates the earth pressure coefficient at rest from Jaky's relationship.

To develop the p-y curve, the effective unit weight, the internal friction angle and the initial modulus of subgrade reaction are required.

Ultimate Resistance

At a given depth, $p_u$ is the lesser of the wedge-failure value at shallow depths and the horizontal flow-failure value at greater depths (Eqns 13.7.D–F of the FB-MultiPier Help Manual):

$$\tag{1} p_u = min(p_{us},p_{ud})$$

$$\tag{2} p_{us} = \gamma' z\Bigl[b(K_p-K_a)+zK_p\tan{\phi}\tan{\beta}\Bigr]$$

$$\tag{3} p_{ud} = \gamma' bz\Bigl(K_p^{3}+2K_0K_p^{2}\tan{\phi}+\tan{\phi}-K_a\Bigr)$$

$$\tag{4} K_0 = 1-\sin{\phi}, \quad K_a = \tan^2{(\frac{\pi}{4}-\frac{\phi}{2})}, \quad and \quad K_p = \tan^2{(\frac{\pi}{4}+\frac{\phi}{2})}$$

where,

$b$ = pile diameter (average diameter between the surface and depth $z$ for tapered piles) $z$ = depth below ground surface $γ'$ = effective unit weight of sand $ϕ$ = internal friction angle $β$ = π/4 + $ϕ$/2 $K_0$ = coefficient of earth pressure at rest, from Jaky's relationship (Eqn 13.7.G) $K_a$ = coefficient of Rankine's active earth pressure $K_p$ = coefficient of Rankine's passive earth pressure $p_{us}$ = horizontal ultimate resistance per unit length at shallow depths (wedge failure) $p_{ud}$ = horizontal ultimate resistance per unit length at greater depths (flow-around failure)

Eqs. (2)–(3) are the simplified wedge and flow expressions of the FB-MultiPier implementation. The report itself (Eqs. 3.50–3.54) groups the Reese et al. (1974) expressions into the Bogard and Matlock coefficients $C_1$, $C_2$, $C_3$ with $K_0$ = 0.4, the form used by Sand (API); for a 36 in pile at 10 ft in sand with $ϕ$ = 30° the two forms differ by about 40 % in $p_u$.

p-y Curve

The p-y curve is Eq. 7.3 of the report (Eqn 13.7.A of the FB-MultiPier Help Manual):

$$\tag{5} p = \eta A p_u\tanh{\Bigl[\frac{kz}{Ap_u}y\Bigr]}$$

where,

$p$ = horizontal resistance per unit length $y$ = horizontal displacement $k$ = initial modulus of subgrade reaction, a function of $ϕ$ (Fig. 3.18 of the report) $η$ = pile shape factor: 1.0 for circular and prismatic piles, 1.5 for tapered piles and H-piles (Eq. 7.3) $A$ = empirical adjustment factor fitting the theoretical $p_u$ to the measured behavior:

$$\tag{6} A = 0.9 \quad (cyclic \ loading)$$

$$\tag{7} A = 3 - 0.8\frac{z}{b} \geq 0.9 \quad (static \ loading)$$

In OpenBrIM the shape factor is taken from the pile section type: $η$ = 1.5 for H-shaped piles and 1.0 otherwise. The report notes that the model was derived from tests in nearly uniform, uncemented medium-to-fine sands and that the results are more sensitive to the friction angle profile than to the exact form of the curve.

The ultimate resistance follows the FB-MultiPier implementation (Eqs. 2–4, Jaky's $K_0$) rather than the Bogard and Matlock coefficients of the original report with $K_0$ = 0.4; the curves generated by OpenBrIM reproduce FB-MultiPier's Sand (O'Neill) output. The report's own form is available as Sand (API).

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[Murchison and O'Neill, 1983] Murchison, J. M. (1983). An Evaluation of p-y Relationships in Sands. M.S. Thesis, University of Houston (report to the American Petroleum Institute, PRAC 82-41-1, O'Neill, M. W. and Murchison, J. M., 1983).

[Murchison and O'Neill, 1984] Murchison, J. M. and O'Neill, M. W. (1984). Evaluation of p-y relationships in cohesionless soils. In Analysis and Design of Pile Foundations, ASCE, pages 174–191.