# Lateral load

How Rischio calculates a single pile under a lateral load and a moment at its head: a beam on each stratum's p–y curve, its deflection and its moments.

A single pile under a lateral load $Q_g$ and a moment $M_g$ at its head deflects, bends and pushes on the soil. Rischio solves the pile as a beam on springs for its deflection and its moments. Units, symbols and assumptions are set out in [Method and assumptions](https://rischio.io/docs/deep-foundation/theory/).

## The loads and the head

-   $Q_g$ acts at the head, and $M_g$ is positive when it bends the pile the way $Q_g$ does: $Q_g$ at a height $e$ above the head is $M_g = Q_g e$.
-   The head is **free**, and turns, or **fixed**, held by a cap that doesn’t let it turn (Das and Sivakugan’s restrained head). A fixed head takes the moment the cap gives it, so it has no $M_g$ of its own.
-   The vertical load may be zero, a pile under a lateral load only, such as a sign or a sound wall. Then there is no axial check and no settlement.

## The pile

The pile’s stiffness is $E_pI_p$. $I_p$ comes from the section: $\pi D^4/64$ for a circle, $\pi[D^4 - (D - 2t)^4]/64$ for a steel pipe of wall $t$, $B^4/12$ for a square, $0.0547 D^4$ for an octagon, and $L t^3/12$ for a barrette bent across its thickness. An H-pile is a section of Das and Sivakugan’s Table 12.1a, bent about its strong axis (the load against the flanges, which face the soil with their width) or its weak one.

The yield moment of a steel pile is $M_y = F_y S$, at first yield (Eq. 12.116), with $F_y$ = 345 MPa for an H-pile (ASTM A572 Grade 50) and 310 MPa for a pipe (ASTM A252 Grade 3) unless another is given (FHWA NHI-16-009, §6.3 and Tables 8-1 and 8-2). A concrete or timber pile’s $M_y$ depends on its reinforcement and is given; without it, neither the bending check nor Broms’s long pile is computed.

## The beam on springs

$$
E_pI_p \frac{d^4x}{dz^4} + p = 0
$$

with $x$ the deflection and $p$ the soil’s reaction per unit length (Eq. 12.99). Rischio solves it by finite elements, 5 cm long, with each spring at its node and the toe free; a fixed head has no rotation.

### p–y curves

Each spring follows the p–y curve that FHWA GEC 9 (App. A) gives for its stratum, the curves practice designs with:

-   **sand:** Reese, Cox and Koop (1974), an initial line $p = k z y$ with $k$ the stratum’s $n_h$, a parabola to $y_m = D/60$, a line to $y_u = 3D/80$ and then its ultimate $\bar{A} p_s$;
-   **soft clay**, below 48 kPa: Matlock (1970), $p/p_u = 0.5 (y/y_{50})^{1/3}$ to $8 y_{50}$, $y_{50} = 2.5\,\varepsilon_{50} D$, with $J = 0.5$ below 24 kPa and 0.25 above;
-   **stiff clay with free water**, the water at or above the ground over it: Reese, Cox and Koop (1975);
-   **stiff clay without free water**: Welch and Reese (1972), $p/p_u = 0.5 (y/y_{50})^{1/4}$ to $16 y_{50}$. GEC 9 prefers it whenever the water isn’t at the ground, “even below groundwater”.

Each curve is static or cyclic; a cyclic load softens it, and on a stiff clay without free water it takes the number of cycles. $\varepsilon_{50}$ is the stratum’s own, or by $c_u$ from FHWA GEC 12, Table 7-21 (0.02 for a soft clay to 0.004 for a hard one). Where a curve takes $\gamma z$, Rischio uses the effective vertical stress there, which is the same in one stratum and holds through strata and the water table. The coefficients GEC 9 gives only as charts (Figs. A-6, A-10 and A-11) are read from them. Each spring’s stiffness is its secant $p/y$, and the solution iterates until the deflections settle.

API’s sand curve (RP 2A), the most widely used, isn’t in GEC 9 or in any source Rischio has yet, and isn’t offered.

## The checks

Each has a verdict once the designer sets its limit (under AASHTO, the code’s φ and FHWA’s 25 mm). Beside each, as a reference:

-   **Deflection at the head**, $x_0 \le$ 25 mm: 1 in., the most common of the limits 22 DOTs use (FHWA GEC 9, App. E). The structural engineer sets the one that applies.
-   **Bending of the pile**, $M_y/M_{max} \ge 1$.

## References

-   Matlock, H. (1970). Correlations for design of laterally loaded piles in soft clay. *Offshore Technology Conference*, Paper 1204.
-   Reese, L. C., Cox, W. R., & Koop, F. D. (1974). Analysis of laterally loaded piles in sand. *Offshore Technology Conference*, Paper 2080.
-   Reese, L. C., Cox, W. R., & Koop, F. D. (1975). Field testing and analysis of laterally loaded piles in stiff clay. *Offshore Technology Conference*, Paper 2312.
-   Welch, R. C., & Reese, L. C. (1972). *Laterally Loaded Behavior of Drilled Shafts*. Research Report 3-5-65-89, Center for Highway Research, University of Texas at Austin.
-   Parkes, J., Castelli, R., Zelenko, B., O’Connor, M., Montesi, A., & Godfrey, J. (2018). *Design, Analysis, and Testing of Laterally Loaded Deep Foundations that Support Transportation Facilities*. FHWA GEC 9, FHWA-HIF-18-031.

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Source: https://rischio.io/docs/deep-foundation/theory/lateral/