WEBVTT 00:00:00.000 --> 00:00:01.000 Stress spreading beneath a circular load. 40 kN over radius 0.15 m; homogeneous elastic half-space. 00:00:01.000 --> 00:00:07.000 MATHEMATICS · define the relationship and the meaning of each symbol before calculating. 00:00:07.000 --> 00:00:14.000 P = 40 kN; a = 0.15 m; q = 565.884 kPa. z is depth below the circular load centre (m). Stress is positive in compression. 00:00:14.000 --> 00:00:23.000 INPUT · uniform radial contact pressure · read the axis units and the numerical values. 00:00:23.000 --> 00:00:31.500 40 kN over radius 0.15 m; homogeneous elastic half-space. 00:00:31.500 --> 00:00:34.500 At z = 0.15 m: σz = 365.8 kPa. Depth increases away from the loaded surface. 00:00:34.500 --> 00:00:37.500 At z = 0.30 m: σz = 161.0 kPa. Depth increases away from the loaded surface. 00:00:37.500 --> 00:00:40.500 At z = 0.60 m: σz = 49.2 kPa. Depth increases away from the loaded surface. 00:00:40.500 --> 00:00:43.000 At z = 1.20 m: σz = 13.0 kPa. Depth increases away from the loaded surface. 00:00:43.000 --> 00:00:52.000 OUTPUT · attenuation with depth · read the axis units and the numerical values. 00:00:52.000 --> 00:00:59.000 At z = 0.30 m, σz = 160.97 kPa. Load spreading reduces centreline stress with depth in this reference problem. 00:00:59.000 --> 00:01:05.000 CALCULATION · substitute the defined inputs; keep units through each intermediate step. 00:01:05.000 --> 00:01:08.000 Radius-depth ratio: a/z = 0.15/0.30 = 0.5 dimensionless 00:01:08.000 --> 00:01:11.000 Influence factor: 1 − (1 + 0.5²)^(−1.5) = 0.284458 dimensionless 00:01:11.000 --> 00:01:14.000 Contact pressure: 40,000 / (π × 0.15²) = 565.884 kPa 00:01:14.000 --> 00:01:17.000 Vertical stress: 565.884 × 0.284458 = 160.970 kPa, rounded 00:01:17.000 --> 00:01:26.400 SENSITIVITY · change one declared input · read the axis units and the numerical values. 00:01:26.400 --> 00:01:29.800 Depth 0.15 m: 365.814 kPa vertical stress. 40 kN uniform circular load; radius stays 0.15 m. 00:01:29.800 --> 00:01:33.200 Depth 0.30 m: 160.97 kPa vertical stress. 40 kN uniform circular load; radius stays 0.15 m. 00:01:33.200 --> 00:01:36.600 Depth 0.60 m: 49.1893 kPa vertical stress. 40 kN uniform circular load; radius stays 0.15 m. 00:01:36.600 --> 00:01:39.600 Depth 1.20 m: 13.0085 kPa vertical stress. 40 kN uniform circular load; radius stays 0.15 m. 00:01:39.600 --> 00:01:45.600 Check: Confirm that halving the load halves stress at every plotted point. Check the analytical centreline against the area-integrated kernel and verify the surface and far-field limits without evaluating a singular formula at z = 0. 00:01:45.600 --> 00:02:03.600 ONE ROW · Double the load and the response doubles here. Double the depth and it does not simply halve. 00:02:03.600 --> 00:02:13.600 PARAMETER MAP · 6161 evaluated cells. The colour limits stay fixed across every row. 00:02:13.600 --> 00:02:23.600 The load axis makes linearity visible while the depth axis exposes the nonlinear geometric attenuation. This centreline result integrates a uniformly loaded circular area. It is a 00:02:23.600 --> 00:02:30.975 useful reference check for a stress calculation before adding layer contrasts. The 3D height is compressive stress, not vertical displacement. 00:02:30.975 --> 00:02:42.975 VALUE LANDSCAPE · Height and colour encode Vertical stress (kPa). This is a graph over the two input axes, not automatically a physical pavement shape. 00:02:42.975 --> 00:02:51.392 The equation describes a homogeneous half-space. It cannot show stress redistribution caused by different layer stiffnesses or calculate fatigue life. 00:02:51.392 --> 00:02:56.808 Drawn layer depths are reference locations; this is not a multilayer solution.