The classical treatment of soil-structure interaction separates the problem into a kinematic part and an inertial part (Wolf, 1985). In the kinematic part, a massless foundation is subjected to the incoming waves and its motion, which differs from the free-field motion because the piles resist the deformation of the soil, becomes the input to the structure. In the inertial part, the structure is supported on frequency-dependent springs and dashpots, the impedance functions, that represent the stiffness of the foundation and the energy it radiates. Impedances for single piles were derived by Novak (1974) and are available as simple expressions for common soil profiles (Gazetas, 1991). Group effects arise from waves emitted by one pile reaching its neighbours and can be handled with interaction factors (Dobry and Gazetas, 1988). The kinematic response of piles and groups was characterised by Fan et al. (1991).

Nonlinearity near the pile is conventionally described by lateral load-transfer curves, the p-y curves developed from field loading tests in soft clay (Matlock, 1970) and sand (Reese et al., 1974) and adopted in offshore practice (API, 2000). For seismic loading the curves are attached to the free-field soil column, with dashpots to represent radiation, and the model is driven by the displacement history at each depth. Boulanger et al. (1999) evaluated this approach against centrifuge tests and found that it reproduced the main features of the measured response. It remains the reference against which a simpler model should be judged.

A macro-element replaces the foundation and the soil mobilised by it with a constitutive law written directly in terms of the resultant forces and the corresponding displacements at a reference point. The idea was introduced for strip footings on sand by Nova and Montrasio (1991), who borrowed the structure of hardening plasticity: a surface in force space bounding the admissible loads, a flow rule and a hardening law. Cremer et al. (2001) extended it to cyclic loading with uplift, and later formulations simplified the ingredients for dynamic analysis (Chatzigogos et al., 2009). For piles, the yield surface relates horizontal force and moment at the head, and couples them much more strongly than for a footing because lateral resistance comes from bending of the pile against the soil. A pile-head macro-element for bridge monoshafts was presented by Correia et al. (2012), and Li et al. (2016) developed a hypoplastic version for single piles in sand under three-dimensional loading.

The assessor assembles a soil profile with small-strain shear modulus and strength for each layer, the groundwater level, and the as-built pile details: diameter, length, spacing, reinforcement and the connection to the cap. For older bridges the last of these is often the least certain and the most consequential, since a pinned or weakly reinforced connection changes the head fixity on which both stiffness and capacity depend. Where liquefaction or lateral spreading is possible the procedure does not apply in its present form, and a different class of analysis is required.

The small-strain stiffness matrix of a single pile is obtained from closed-form expressions for the appropriate idealised profile, with the off-diagonal term that couples translation and rotation retained. Group stiffness is then built using dynamic interaction factors evaluated at the fundamental frequency of the bridge. Radiation damping coefficients are taken from the same sources. Below the fundamental frequency of the soil deposit little energy is radiated, and the dashpots are reduced or removed in that range (Gazetas, 1991).

Ultimate combinations of horizontal force and moment at the pile head are found from a mechanism analysis in the manner of Broms (1964), considering both failure of the soil along a short pile and formation of plastic hinges in a long one. Several ratios of moment to force are analysed and a smooth surface is fitted through the resulting points. For a group, axial capacities of the outer piles enter through the push-pull couple that resists overturning, so the vertical load becomes a third axis of the surface.

The transition from elastic response to failure is calibrated against a monotonic pushover of the pile or group on conventional p-y springs, run once for each foundation type. The macro-element hardening parameters are adjusted until its force-displacement and moment-rotation curves follow those of the pushover along two or three load paths. For unloading and reloading, the simplest defensible choice is a kinematic rule that reproduces the monotonic curve on first loading and gives closed loops thereafter. Gapping between pile and cohesive soil, which pinches the loops, can be represented by an additional parameter if the soil conditions suggest it.

Free-field motion at the surface is modified by kinematic interaction factors to give the foundation input motion. For slender piles in uniform soil and the frequencies that matter for most bridges the modification is small, and the free-field motion is commonly accepted in such cases; for large-diameter shafts and layered profiles with sharp stiffness contrasts it should be evaluated. The macro-element is inserted between the base of the pier and the node at which the input motion is applied, with the radiation dashpots in parallel with its elastic part so that they are not credited with damping once the near field has yielded. Nonlinear response history analysis is then carried out in the usual way.

The analysis returns histories of force, moment, displacement and rotation at the pile head. These are compared with acceptance limits for the foundation, and the peak head actions are applied to the pushover model of Step 4 to recover the bending moment distribution down the piles. Residual displacement and rotation of the element give a first estimate of permanent foundation movement, which for many bridges governs serviceability after the event.