WEBVTT

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From falling mass to stress field. a = 0.15 m; E = 200 MPa; ν = 0.35; 0 ≤ t ≤ 30 ms.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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Pmax = 40 kN; pulse duration T = 30 ms; circular radius a = 0.15 m. E = 200 MPa; ν = 0.35. Compression and downward movement are positive. The spatial response is quasi-static; pavement inertia is not included.

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INPUT · prescribed load history · read the axis units and the numerical values.

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a = 0.15 m; E = 200 MPa; ν = 0.35; 0 ≤ t ≤ 30 ms.

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Mass drops onto a buffer and loading plate; impact mechanism shown schematically.

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Prescribed 30 ms half-sine pulse, slowed for inspection. Stress and displacement scale with P(t).

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Peak: q = 565.9 kPa; w₀ = 744.8 μm; σz at 0.30 m = 161.0 kPa.

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OUTPUT · calculated sensor displacement · read the axis units and the numerical values.

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At the centre, q = 565.88 kPa gives w₀ = 744.85 μm. Further sensors move less under this model.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Loaded area: π × 0.15² = 0.0706858 m²

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Peak pressure: 40,000 / 0.0706858 = 565884.242 Pa

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Centre displacement: 2 × (1 − 0.35²) × q × 0.15 / 200,000,000 = 744.845 μm

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Stress at 0.30 m: q × [1 − (1 + 0.15²/0.30²)^(−1.5)] = 160.970 kPa

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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E = 100 MPa: 1489.69 μm centre displacement. Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.

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E = 200 MPa: 744.845 μm centre displacement. Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.

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E = 400 MPa: 372.423 μm centre displacement. Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.

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Check: Check zero-load response, inverse modulus scaling and agreement between the integrated stress field and the analytical centreline expression. Keep the separate illustrative sensor record distinct from these computed readings.

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ONE ROW · The sensors sit on the surface. The load does not stop there.

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PARAMETER MAP · 6161 evaluated cells. The colour limits stay fixed across every row.

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Start at the centreline, then compare neighbouring radial positions at the same depth. Each cell integrates the Boussinesq kernel over the uniformly loaded circular plate: P = 40

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kN and a = 0.15 m. Colour and height encode the same compressive stress. The surface in the right-hand view is a graph of stress over position and depth; it is not a deformed

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pavement shape.

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VALUE LANDSCAPE · Height and colour encode Vertical stress (kPa). This is a graph over the two input axes, not automatically a physical pavement shape.

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Homogeneous half-space, quasi-static pressure, no layer interfaces or inertia. The spatial quadrature is approximate. Stiffness affects displacement but does not enter this prescribed-load stress field.

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Uniform pressure, homogeneous half-space, quasi-static response; no inertia or layer interfaces.