1 · Overview

LEAPS computes the stress, strain and displacement fields inside a pavement idealised as a stack of homogeneous, isotropic, linear-elastic layers resting on a semi-infinite elastic halfspace. Each layer is defined by a thickness h, an elastic modulus E and a Poisson's ratio ν; the bottom layer is infinitely deep. Surface loads are uniform circular contact pressures (tyres). This is the classical layered elastic analysis (LEA) idealisation used throughout mechanistic-empirical pavement design.

The tool is meant for intuition, teaching and rapid mechanistic checks — not as a replacement for a validated design package. Its core assumptions are:

• Each layer is homogeneous, isotropic and linearly elastic (no stress dependence, no visco-elasticity, no temperature gradient).
• Layers extend infinitely in the horizontal plane; the lowest layer is a halfspace.
• Loads are static, vertical and applied as uniform pressure over circular contact areas.
• Full continuity, frictionless slip or a linear shear-spring is enforced at each interface.

2 · Multilayer elastic theory

For an axisymmetric vertical load, the elasticity equations admit a solution through a single biharmonic Love stress function φ(r, z) that satisfies ∇⁴φ = 0. Taking the zeroth-order Hankel transform in the radial coordinate maps the biharmonic equation to an ordinary differential equation in the transform variable m, whose general solution in layer i is

The four constants Aᵢ … Dᵢ per layer are fixed by boundary and interface conditions. Stresses and displacements are recovered from φ̄ by differentiation in Hankel space and an inverse Hankel transform (an integral over m weighted by Bessel functions J₀, J₁).

Boundary conditions

• Surface (z = 0): prescribed normal pressure from the load transform, zero shear.
• Each interface: continuity of vertical stress and vertical displacement always; the treatment of shear stress and radial displacement depends on the bond (see §4).
• Depth (z → ∞): the growing exponentials in the halfspace must vanish, so C_N = D_N = 0.

These assemble into a 4N − 2 linear system per transform parameter m for an N-layer system, solved independently for every quadrature abscissa.

3 · Numerical method

Each response is an infinite integral of an oscillatory, slowly-decaying integrand. LEAF-JS evaluates it with:

• Panel-wise Gauss–Legendre quadrature between consecutive zeros of the relevant Bessel function, so each panel integrates a smooth half-wave.
• Wynn ε-acceleration of the sequence of panel partial sums, which converts the slowly-converging alternating tail into a rapidly-convergent estimate.
• Decay-scale refinement for deep evaluation points, where the integrand's envelope decays fast and panels are re-spaced accordingly.
• Closed-form surface tails: at z = 0 the integrals reduce to Weber–Schafheitlin forms evaluated with complete elliptic integrals K, E (arithmetic–geometric mean). Surface deflection of a halfspace is therefore returned to machine precision rather than by truncated quadrature.

The solver tolerance (Solver panel) sets the convergence target of the accelerated tail; the contour resolution sets the size of the sampling grid used only for the stress-bulb visualisation, not for the reported key responses.

4 · Interfaces

Between adjacent layers, vertical stress and vertical displacement are always continuous. The shear behaviour is one of three models:

The spring stiffness k spans the two limits continuously: k → ∞ recovers the fully-bonded solution, k → 0 recovers the frictionless one. Both limits are checked in the validation suite (§9).

5 · Loads & gears

Each wheel applies a uniform pressure p over a circular contact patch. The contact radius follows from the wheel load F and pressure:

Multiple wheels are superposed: LEAPS solves each wheel independently in its own axisymmetric frame, then rotates every stress/strain tensor into the global frame before summation, so shear coupling between offset wheels is captured exactly. The Gear generator builds standard layouts, all parameterised by wheel load, pressure, dual spacing Sd (transverse gap between paired tyres) and tandem spacing St (longitudinal gap between axles):

Individual wheels can be repositioned and re-loaded in the wheel table after a gear is built, so irregular configurations are fully supported.

6 · Conventions & units

• Stresses are tension-positive. Compression under a wheel is therefore negative.
• z is measured downward from the surface; deflection w is positive downward.
• The section is a vertical cut in the x–z plane at the plan y-coordinate set by the draggable section line.
• Internally everything is mm · N · MPa; the UI converts to the SI or US display system you select.

7 · Response quantities

At every evaluation point LEAPS reports the full state:

• Stress tensor — σx, σy, σz and the shear components τxy, τxz, τyz.
• Principal stresses σ1 ≥ σ2 ≥ σ3, the maximum shear τmax = (σ1 − σ3)/2, and the von Mises stress.
• Strain tensor — εx, εy, εz (via the 3-D elastic compliance of the local layer).
• Displacement — ux, uy and the vertical deflection w = uz.

The Key responses rail highlights the classic design drivers: maximum surface deflection, horizontal tensile strain at the bottom of the asphalt (fatigue), vertical compressive strain at the top of the subgrade (rutting), tensile stress at the bottom of bound layers, and peak interface shear.

8 · Distress models

The Performance tab converts the two critical mechanistic strains into allowable load repetitions using the Asphalt Institute transfer functions. Fatigue cracking is driven by the horizontal tensile strain εt at the bottom of the asphalt and the asphalt modulus E (in psi):

Subgrade rutting is driven by the vertical compressive strain εv at the top of the subgrade:

The governing life is the smaller of the two. N_f and N_r are allowable repetitions of the analysed load, not calendar life — no seasonal, reliability or shift factors are applied.

9 · Validation suite

LEAF-JS ships with a 69-check regression suite (run under Node during development; a subset runs in your browser at every startup and drives the status-bar badge). The checks cover:

The startup self-check compares the engine against the Boussinesq closed forms; a green badge means those passed on your machine, in your browser, this session.

10 · Using LEAPS

Workflow

• Structure — pick a material per layer and edit thickness, modulus and Poisson's ratio inline. The menu renames, reorders, duplicates or deletes a layer; drag the handle to restack. The bottom layer is always the infinite subgrade.
• Interfaces — set the bond below each layer; enter k for shear springs.
• Loads — build a standard gear, then fine-tune any wheel's position and load in the wheel table.
• Evaluation points — double-click the section to drop a point; it snaps to interfaces and wheel axes. Drag to move, select and press Delete to remove.
• Results — key responses appear in the right rail; the bottom dock holds depth profiles, surface responses, the per-layer response table, the point table and the performance estimates.

Plan view

The plan inset shows the gear from above with contact footprints, axle and tandem connectors, detected spacings and a grid. Drag the dashed line to move the analysis section along the y-axis; every response updates for the new cut.

Keyboard & interaction

Exports

The right rail exports the evaluation points and the contour grid to CSV, the whole project to a .leaps.json file (re-openable), the section to PNG, and a printable report (opens in a new tab — use Print → Save as PDF). Work autosaves to your browser between sessions.

11 · References

• Burmister, D. M. (1945). The general theory of stresses and displacements in layered soil systems. Journal of Applied Physics, 16.
• Huang, Y. H. (2004). Pavement Analysis and Design (2nd ed.). Pearson.
• The Asphalt Institute (1982). Research and Development of the Asphalt Institute's Thickness Design Manual (MS-1), Research Report 82-2.
• ARA / NCHRP (2004). Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures (MEPDG), NCHRP 1-37A.
• FAA (2021). Airport Pavement Design and Evaluation, AC 150/5320-6G, and the LEAF layered elastic engine used in FAARFIELD.
• Watson, G. N. (1944). A Treatise on the Theory of Bessel Functions. Cambridge University Press (Weber–Schafheitlin integrals).

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