4 · Interfaces

Between adjacent layers, vertical stress and vertical displacement are always continuous. The shear behavior is set by a dimensionless slip value, the same input WinJULEA takes, where 0 is fully bonded and 1 is frictionless. Both endpoints are exact. Anything between is Goodman's shear spring:

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).

Turning a dimensionless slip into a stiffness necessarily introduces a length. The literature, BISAR's shear-spring compliance, Uzan's friction parameter, scales it on the loaded radius, which is why a dimensionless friction parameter is documented as depending on the diameter of the load circle. LEAPS uses

with aref the reference contact radius, and prints the resulting stiffness beside the control. An intermediate value is a defensible partial bond, but it is not guaranteed to reproduce another program's intermediate value digit for digit, because each program normalizes its own way. Enter k directly when the number matters.

5 · Loads & gears

The load model selector in the toolbar chooses how each load is applied. Every model carries the same total force F, so switching between them is a direct experiment in Saint-Venant's principle: near the contact the three answers differ by a factor of two or more, and by a couple of contact radii down they agree to a fraction of a per cent.

A singular point is reported as such rather than given the finite number a quadrature happens to return there. That the concentrated idealizations have no answer at the load is not a defect in the solver, it is the reason design methods spread a wheel over a contact area in the first place.

Load, pressure and area are one equation

A circular contact is described by three numbers and they are not independent:

Any two fix the third, and which two you have depends on where the number came from: a legal axle load and a placarded inflation pressure, or a measured footprint and a scale reading. The Solve for switch names the one LEAPS computes. That quantity becomes a read-only readout rather than an input, so there is no state in which three numbers can sit on screen that do not satisfy the relation, and no question about which of them the solver believed. Contact radius is offered alongside area because a footprint is quoted both ways.

The same switch governs the gear generator, so a gear built from load and pressure and a gear built from load and area are the same gear.

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 parameterized by wheel load, pressure, dual spacing Sd (transverse gap between paired tires) 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.
• Shear strains are engineering strains, γ = τ/G.
• Internally everything is mm · N · MPa, and SI display is that same system, so every SI conversion factor is exactly one. The English system is the column set WinJULEA prints: inches, pounds and psi, with a modulus in psi rather than ksi and a displacement in inches rather than mils, so a result reads straight across from the other program.
• LEAPS asks which system you want once, before anything is typed. Every number in the workspace is in it, and an analysis read in the wrong system is not wrong by a factor anyone notices in a plot. The choice is remembered and stays changeable from the toolbar.

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. Those cards carry a tint of their own so they read as conclusions rather than as more of the table, and the one that governs is outlined.

Reading the results table

The table is transposed the way a layered-elastic program prints it: one column per evaluation point, one row per quantity. Every cell also carries a bar scaled to the largest magnitude in its own row and tinted by sign, and the largest cell of each row is marked. A row of numbers in scientific notation is hard to scan and easy to misread by a decade; the bar is there so the shape of a response along the section can be seen before any single value is read. Because each row is scaled to itself, bars are comparable across a row and never across rows.

A response that is zero by symmetry comes back from the quadrature as roundoff rather than as zero: on a wheel axis the shear is 2e-18 where the answer is exactly nought. Anything twelve orders below the largest value in its own row is printed as 0, in the table and in the exported analysis alike. Twelve, not six: the engine agrees with an independent solver to about five significant figures, so a number that small is not a small response, it is the last bits of the double. The project file keeps the raw value.

Greek letters, subscripts and superscripts are typeset rather than spelled: the table, the field picker, the colorbar and the canvas all draw from one notation table, so τ_xz is the same glyph everywhere it appears and the CSV twin is a plain-text transliteration of the same entry. Real equations are set with KaTeX, each carrying a plain-text twin that shows until it loads.

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 analyzed load, not calendar life, no seasonal, reliability or shift factors are applied.

9 · Validation suite

LEAF-JS ships with a 112-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:

Two quadrature defects were found by cross-checking the engine against an independent implementation, and both were converged and wrong, stable at every tolerance, so no amount of tightening would have shown them. The panel mesh near m = 0 now resolves the interface reflections rather than the evaluation depth, and the first panel is graded geometrically toward zero. A surface deflection that came out 0.8 % high on a three-layer section now agrees with the reference to 4e−5. The lesson is recorded here because it generalizes: a single solver cannot check its own quadrature.

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 between two rows inserts a layer there, the deletes one, the handle restacks, and the menu renames, duplicates and reorders. The bottom layer is always the infinite subgrade and its thickness is ignored.
• Interfaces: bonded, partial or free, one click apart, with the shear stiffness a partial value implies printed beside the slider. Every interface also carries a chip on the section itself.
• Load model: circular imprint, point or line, in the toolbar. The total load does not change when you switch.
• Loads: type any two of load, pressure and area; build a standard gear from them; then fine-tune any wheel's position and magnitude in the load 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 opens on the results table: one column per evaluation point, one row per quantity, in the order a layered-elastic program prints them, position, the six stress components, the six strains, the three displacements, then the principal stresses and principal strains. The quantity and unit columns stay frozen while you scroll sideways, and the six groups fold. The other dock tabs hold depth profiles, surface responses, the per-layer summary and the performance estimates.

Two views of one model

The Section view is the cut itself: the x-z plane at the section line, drawn to scale with a ruler along the bottom for x, the depth rail at the left for z, and a gnomon in the corner carrying all three axes. The third one is the reason the gnomon is there: y runs into the page, the drawing is one cut along it, and the gnomon prints which cut. Contour lines carry their own values in a gap broken in the line, the way a contour map engraves them, so a bulb can be read without walking back to the color bar.

The 3D view is the same model in the box it sits in, cut open at the section line. The far half is solid and carries the contour on the exposed face; the near half is drawn as the glass it was cut out of, so the wheels standing on it stay where they are. It answers the question a flat cut cannot: where is the section, and what else is up there? A dual tandem has four wheels and a plane through two of them; a line load at the default azimuth runs along y and crosses the section at a single point, which in the flat view is a knife edge and in this one is a line on the ground. Drag to orbit, scroll to zoom, shift-drag to pan.

Two things about that camera are deliberate. It is orthographic, so nothing foreshortens and a length is a length wherever in the box it sits: the figure can be measured. And the orbit is held within one quadrant, because past about 80 degrees the cut face is edge on, the contour disappears, and the reader is looking along the plane whose values they came to read.

Editing on the section

The section drawing is an input, not an illustration. Three gestures edit the structure directly, and each one changes the same state the panel does, so the two are never out of step:

• All three gestures are for the Section view. In 3D the canvas is a camera: a screen position out there is a whole line of world positions, and guessing one would move a layer nobody meant to touch.
• Drag an interface line to set the thickness of the layer above it. The thickness is drawn beside the cursor while you drag, and it lands on a round number: the step follows the zoom, so the same gesture is coarse on a whole section and fine on a close-up, and it never produces a thickness nobody would type.
• Click an interface chip to cycle bonded, partial, free. The chip prints the state it is in and sits clear of the colorbar.
• Click a layer to select it; the panel row highlights with it, which is how a thick section stays navigable.

Before you run

Two things guard a run. The Build button lights when anything the gear is built from has been edited, because those values seed the loads rather than driving them and nothing moves until it is pressed. And the badge beside Run is a preflight checklist: a tick for every input that is complete, a cross for anything that needs review, and the count of the two. Pressing Run with a failing check opens the list instead of solving. A layered-elastic solver will return plausible numbers for a section nobody meant to type, and a wrong answer that looks right is worse than no answer.

The design study

Every other panel answers one question: what does this section do? A design asks the other one. How thick does the asphalt have to be. How much does bonding buy. What happens when the subgrade is soft. Answering that a section at a time means editing a number, pressing Run, writing the answer down, and doing it eight more times, which is enough friction that nobody does it eight times.

So the Design study tab sweeps one quantity across a range: any layer's thickness or modulus, any interface's slip, the wheel load, or the contact pressure. Every step is a full layered-elastic solve of a section that differs from the one on screen in exactly that quantity, read at the same critical points and through the same transfer functions the panels use, so a point on the curve and the panel beside it agree by construction rather than by coincidence.

Two things make it a design tool rather than a plotting one. The current design is marked on the curve, so the study always says where you are standing. And for a life, a target draws a line and the crossing is solved for and printed in words: 1 × 10⁶ repetitions at Dense-graded HMA thickness = 98.8 mm. The crossing is interpolated in log N between the two steps that straddle it, which is the space a transfer function is a straight line in.

A load or pressure sweep holds the other of the two and lets the contact area follow, which is what happens on a truck: inflation pressure is a property of the tire and the load is a property of the trip. The subgrade's thickness is deliberately not on the list, because it is a half-space and its thickness is ignored: a sweep of it would draw a flat line and teach the wrong thing. Every step of the study exports to CSV with all five critical responses and the governing mechanism.

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

Export full analysis writes one file carrying the whole run: the structure, every interface and its slip value, every load with its derived radius or line intensity, the solver settings, and every response at every evaluation point, in the units on screen, with the sign convention named in the header. Values are rounded to six significant figures, because the solver agrees with an independent implementation to about five and the digits past that belong to the double rather than to the pavement. The project .leaps.json keeps full precision and re-opens the run exactly; the contour field exports to CSV, the design study exports to CSV, and the section to PNG. 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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