Tilted Piles Defy Their Bad Reputation by Stiffening Group Foundations Under Strong Shaking

Inclined piles, once banished from seismic design codes after a string of devastating earthquakes, are enjoying an unexpected scientific redemption. A new numerical study published in the Bulletin of Earthquake Engineering by Usama Zafar, Chandra S. Goit and Masato Saitoh of Saitama University shows that, when bundled into groups, tilted piles can actually make a foundation stiffer under vertical vibration than an identical group of perfectly vertical piles. The finding challenges decades of caution rooted in the damage observed at Loma Prieta in 1989, Northridge in 1994 and Kobe in 1995, and it hinges on a subtle interplay between geometry, soil nonlinearity and the way seismic waves travel from one pile to another.

The team set out to answer a question that has lingered in geotechnical engineering for years: does the loading intensity, meaning the strength of the imposed vibration, change the vertical dynamic impedance of inclined pile groups in cohesionless soil? Dynamic impedance, the frequency-dependent combination of stiffness and damping that connects a foundation to the ground beneath it, is the quantity that modern seismic codes such as Eurocode 8 and ASCE 7-16 require engineers to consider explicitly in advanced soil-structure interaction analyses. For vertical piles, that quantity is well documented. For inclined piles, and especially for groups of them, the picture has remained stubbornly incomplete.

The researchers built three-dimensional finite element models in PLAXIS 3D, replicating the dimensions of physical scale-model tests performed on single piles at inclinations of zero, five and ten degrees, and on a vertical three-by-three pile group. To capture the real behaviour of sand under cyclic loading, they employed two nonlinear constitutive descriptions: a Mohr-Coulomb model paired with discrete soil-pile interface elements that allow frictional slip, and the Hardening Soil model with small-strain stiffness, which reproduces the progressive degradation of soil stiffness as shear strain grows. A purely linear elastic model served as a benchmark to expose what nonlinear physics adds. The interface strength along the pile shaft was reduced by a factor of 0.5 to permit plastic slippage, a value calibrated against experimental impedance measurements at the lowest and highest frequencies of the study.

Validation was rigorous. The simulated static push-pull stiffnesses and the dynamic impedance functions of single piles at all three inclinations, and of the vertical pile group, matched published experimental data closely, whereas the linear elastic model systematically overestimated stiffness and underestimated damping. The validated framework was then extended to inclined three-by-three groups, for which no experimental data exist, under harmonic vertical excitation spanning 12 to 35 hertz at model scale, equivalent to roughly 1.2 to 3.5 hertz at prototype scale, a band that overlaps the natural frequencies of typical buildings and bridges. Two acceleration amplitudes, 0.5 and 5.0 metres per second squared, spanned the range from small-strain, nearly elastic response to progressively nonlinear soil behaviour.

The single-pile results confirmed a long-recognised but counterintuitive trend: tilting a pile makes it softer in the vertical direction. A vertical load applied to an inclined pile head decomposes into axial and lateral components, forcing the pile to bend as well as compress, and the resulting coupled deformation reduces the measured head stiffness. Both nonlinear models reproduced this degradation, and the effect intensified at higher excitation levels, where soil yielding near the shaft saps additional stiffness. In isolation, then, inclination looks like a liability, exactly as the post-earthquake reconnaissance reports of the 1990s concluded.

Groups of piles, however, tell a different story. When nine piles are connected by a rigid cap, their individual responses are no longer independent; each vibrating pile sends displacement waves through the soil that shake its neighbours, a phenomenon known as pile-to-pile interaction that typically softens the group and reduces its efficiency relative to the sum of its parts. The simulations revealed that inclination weakens this destructive interaction. As the piles splay outward, the spacing between their shafts increases with depth, so the stress and displacement fields generated by each pile overlap less with those of its neighbours. The result, across the considered configurations, was a pile-group head stiffness that rose with inclination angle, with the ten-degree group outperforming both the five-degree and vertical arrangements.

To dissect the mechanism, the team computed pile-to-pile interaction factors, the ratio of the induced displacement of a receiver pile to the motion of the loaded pile, for every pile in the array. Under the linear elastic model, the interaction factors decreased as inclination increased, most noticeably at ten degrees, but only when the stiffness of the receiver pile was included in the calculation. Without that stiffness contrast, the geometry alone produced almost no change, demonstrating that the reduction arises from wave scattering and modified stress transfer around stiff inclusions rather than from simple geometric separation. The magnitude of the reduction depended on excitation frequency, inclination angle and the position of the receiver pile within the group, with the closest neighbours showing the strongest effects.

Loading intensity left an unmistakable fingerprint. In the nonlinear models, the higher excitation level consistently produced lower interaction factors than the lower one, because soil yielding and interface slip around the loaded pile localise deformation near the pile tip and attenuate the displacement field before it can reach neighbouring piles. The Hardening Soil model attributed this to strain-dependent shear modulus degradation and increased hysteretic damping in the near-field soil, while the Mohr-Coulomb model attributed it to plastic yielding and frictional slip at the soil-pile contact. Crucially, the stiffness advantage of the inclined groups persisted even under the strong 5.0 metres per second squared excitation, although its magnitude varied with the constitutive representation. The study also showed that a superposition method built on these nonlinear interaction factors reproduced the direct group simulations closely, offering engineers a computationally efficient route to impedance functions without sacrificing nonlinear fidelity.

The authors are careful to frame the scope of their conclusions. The findings apply to a three-by-three floating pile group in dry, cohesionless soil under vertical excitation, with inclinations of five and ten degrees, at model scale interpreted through established similitude relationships. No experimental benchmarks exist yet for the inclined groups, so their impedance characteristics remain numerical predictions, and the two loading levels examined do not establish a continuous trend of loading-intensity effects. Nor does the study rehabilitate every battered pile: the historical failures involved inadequate pile-to-cap connections and kinematic distress under lateral shaking, issues that vertical impedance analysis does not address. Nevertheless, by demonstrating that inclination can enhance group efficiency under both weak and strong vertical vibration, the work provides a quantitative foundation for reconsidering batter piles in properly designed foundations, from bridge abutments to offshore platforms, and adds a striking twist to one of geotechnical engineering’s most contested debates.

Subject of Research: Vertical dynamic impedance and pile-to-pile interaction of inclined pile groups in cohesionless soil under varying loading intensity

Article Title: Influence of loading intensity on the vertical dynamic impedance of inclined pile groups in cohesionless soil

Article References: Zafar, U., Goit, C. S., & Saitoh, M. (2026). Influence of loading intensity on the vertical dynamic impedance of inclined pile groups in cohesionless soil. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02711-6

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02711-6

Keywords: inclined piles, pile groups, dynamic impedance, soil-structure interaction, seismic design, cohesionless soil, pile-to-pile interaction, finite element analysis, soil nonlinearity, loading intensity, foundation engineering, earthquake engineering