FWD deflection bowls: read the shape before choosing a treatment

The bowl is a response, not a diagnosis

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

A falling weight deflectometer applies a load and measures the pavement response at defined offsets. The resulting bowl is a spatial record: the values belong together, but each also belongs to a sensor position, a test load and a set of test conditions. Reading only the central value discards information. Reading the shape without its measurement context gives the picture more authority than the record supports. (FHWA, 2017)

Horak's work places deflection-bowl parameters within relative structural benchmarking, alongside visual survey information. That is a useful starting point for investigation. It does not turn every bowl shape into a unique identification of a damaged layer. (Horak, 2008)

For a reader new to the subject, imagine observing how a layered system moves under a known action. For a technical reviewer, the question becomes more specific: which measured responses, under which conventions, support the interpretation being proposed?

From a forty-kilonewton pulse to a bowl

Governing relationship: q(t) = P(t)/(πa²); P(t) = 40 sin(πt/30) kN.

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.

The film first plots applied force (kn), then follows the calculation and plots peak displacement (μm). At the centre, q = 565.88 kPa gives w₀ = 744.85 μm. Further sensors move less under this model.

Elastic mechanics

From falling mass to stress field

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

Download example data (CSV)

Model, calculations and transcriptEquations, worked steps, source notes and a timed reading guide

Understand the model

A prescribed force creates contact pressure. In the reference half-space, that pressure produces a displacement bowl and a stress field.

Work through the inputs and equations below, then compare the calculated output with the film. All numerical inputs in this study are synthetic.

Inputs and symbol key

P(t); Pmax

Applied force and its peak (N in SI substitution).

T; t

Pulse duration and elapsed pulse time (s, or consistently converted ms).

a; q

Loading radius (m); uniform contact pressure (Pa).

E; ν

Young’s modulus (Pa); dimensionless Poisson ratio.

w₀

Downward centre displacement (m; reported in μm).

z; σz

Depth below the load centre (m); vertical compressive stress (Pa).

Values used in this film

• 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.

The mathematical relationships

1. Equation 1

   Source and scope
   $$\begin{gathered}P(t)=P_{max}\sin(\pi t/T)\\q(t)=P(t)/(\pi a^2)\end{gathered}$$

   Convert force to newtons and divide by the circular plate area. The half-sine specifies the force during the 30 ms pulse.

2. Equation 2

   Source and scope
   $$w_0=2(1-\nu^2)qa/E$$

   At fixed pressure and geometry, displacement varies inversely with modulus E. Convert the resulting metres to micrometres for the plotted bowl.

3. Equation 3

   Source and scope
   $$\sigma_z(0,z)=q[1-(1+a^2/z^2)^{-3/2}]$$

   This expression gives vertical stress directly beneath the load centre at depth z. Its units are pressure; it is not a strain or damage calculation.

Work through the example

Use the relationships above with consistent units. These substitutions are our worked example, not values quoted from the source.

1. Loaded area

   π × 0.15² = 0.0706858

   m²

2. Peak pressure

   40,000 / 0.0706858 = 565884.242

   Pa

3. Centre displacement

   2 × (1 − 0.35²) × q × 0.15 / 200,000,000 = 744.845

   μm

4. Stress at 0.30 m

   q × [1 − (1 + 0.15²/0.30²)^(−1.5)] = 160.970

   kPa

Read the output

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

Check your understanding

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.

Assumptions and limits

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

Return to the full references and method notes to check the cited edition and its applicability.

Timestamped film transcript 28 short entries · select a time to revisit the film

1. 0:00

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

2. 0:01

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

3. 0:07

   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.

4. 0:14

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

5. 0:23

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

6. 0:24

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

7. 0:27

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

8. 0:49

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

9. 0:56

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

10. 1:05

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

11. 1:12

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

12. 1:18

    Loaded area: π × 0.15² = 0.0706858 m²

13. 1:21

    Peak pressure: 40,000 / 0.0706858 = 565884.242 Pa

14. 1:24

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

15. 1:27

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

16. 1:30

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

17. 1:39

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

18. 1:43

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

19. 1:46

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

20. 1:49

    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.

21. 1:55

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

22. 2:13

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

23. 2:23

    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

24. 2:34

    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

25. 2:46

    pavement shape.

26. 2:51

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

27. 3:03

    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.

28. 3:12

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

Download the timed captions (VTT)

The transcript records the on-screen wording. Formula provenance and interpretation are explained in the source note above.

Begin with a table that fixes the geometry

The following synthetic readings illustrate a single test. Deflection is in micrometres; sensor offset is in millimetres. Mixing those units would corrupt an otherwise correct subtraction.

Offset from load centre (mm)	Deflection (μm)
0	600
300	350
600	220
900	130
1,200	80

Case review

A bowl gives you more than its deepest point

Does the response change only near the plate, or across the measured bowl?

0 · 600 Deflection (μm)

The discriminating check

Check applied load, sensor offsets, temperature context and repeat drops before comparing bowls.

Evidence supports the comparison

With comparable tests, contrast near-plate and outer response and formulate competing explanations to investigate.

Evidence is missing or incompatible

With inconsistent loading or uncertain geometry, resolve the test record before fitting stiffness parameters.

What you can conclude. A bowl-shape difference is evidence to interpret alongside layer and material information, not a unique diagnosis.

Original worked case. Read the source editions and method notes.

The difference between the first two readings is 600 − 350 = 250 μm. The next two adjacent differences are 350 − 220 = 130 μm and 220 − 130 = 90 μm. The central reading is also 0.600 mm, because 1,000 μm equals 1 mm. These are transparent operations on the table, not backcalculated moduli or remaining-life estimates.

A published interpretation may attach specific names to differences measured at specific offsets. The name and sensor convention must travel together. If a device has a different sensor arrangement, selecting the nearest column and keeping the same parameter name is not a neutral substitution. Follow the selected method and document any permitted transformation.

What changes when the test context changes?

Suppose a second record has a central reading of 660 μm. Relative to 600 μm, that is a 10% increase: (660 − 600) / 600 × 100. Before calling it a structural change, establish whether the two tests have comparable load conditions, positions and relevant environmental context. The arithmetic establishes a difference between records; comparability determines whether it supports the proposed interpretation. (FHWA, 2017)

The same discipline applies to plotted bowls. A graph can make two tests look nearly identical by changing its vertical scale. It can make a modest difference look dramatic by truncating the axis. Keep a common scale when a visual comparison is intended, state the units, and leave access to the underlying readings.

Distinguish measured response, a parameter and an inferred property

The measured deflection is an observation. A difference such as 250 μm is a derived parameter. A statement about a layer's stiffness or condition is an interpretation that requires a method and applicable assumptions. These three levels should not be collapsed into a single coloured label.

A useful calculation record identifies the source readings, offset convention, load treatment, calculation version and rounding. An interpretation record adds the selected benchmark, applicability and corroborating evidence. That separation makes an explanation readable without depriving a specialist of the information needed to audit it.

Build competing explanations before selecting a response

An elevated response can prompt questions about material behaviour, moisture, construction history, layer geometry or measurement conditions. An investigation should identify which evidence would distinguish the relevant explanations for the site. A visual survey may locate distress; construction records may explain a change in structure; additional investigation may be needed where the record remains ambiguous.

The important step is to name the uncertainty. “This section has a higher central reading” is a measurement comparison. “This layer has failed” is a much stronger claim. If the evidence only supports the first, write the first and identify the investigation needed to evaluate the second. Horak's benchmarking context supports using bowl information with wider pavement evidence. (Horak, 2008)

Turn the result into a useful investigation brief

A defensible brief from this example would retain the five readings, their offsets and the test record; identify the differences to be assessed under the selected method; and state what evidence is needed before a structural interpretation is accepted. That creates a path from measurement to judgement that another engineer can follow. It is more useful than a bowl coloured with unexplained severity bands.

The mechanics behind the film

The sequence begins with the mass, buffer and loading plate. That mechanism sketch explains how an FWD applies an impulse; it does not infer a peak force from an animated drop height. For the calculation, we prescribe a half-sine load with 40 kN peak force and 30 ms duration. The peak load is within the example testing levels described in FHWA guidance. The film slows the pulse to make its phases visible. (FHWA, 2017)

The pavement reference model is a homogeneous, isotropic, linear-elastic half-space with Young's modulus E = 200 MPa and Poisson's ratio ν = 0.35. A flexible circular area of radius a = 0.15 m carries uniform pressure. This is an explicit benchmark for understanding load spreading; it is not a rigid-plate contact solution or a multilayer pavement solver.

P(t) = Pmax sin(πt/T), 0 ≤ t ≤ T; otherwise P(t) = 0
q(t) = P(t) / (πa²)
dσz = [3 q(t) z³ / (2π R⁵)] dA
σz(x,z,t) = integral over the circular loaded area of dσz
w(r,t) = [(1 − ν²) q(t)/(πE)] integral over the disk of dA/ρ

Here R is the three-dimensional distance from an elemental load to a point below the surface; ρ is the horizontal distance to a surface observation. Compression and downward displacement are positive. The cross-section is calculated by area integration of the Boussinesq vertical-stress kernel. The 3D surface uses the corresponding circular-load displacement integral, evaluated with complete elliptic integrals. Both views therefore respond to the same pressure history. (Rocscience, n.d.) (University of Cambridge, 2005)

Under the load centre the integrals simplify, giving useful independent checks:

w₀ = 2(1 − ν²)qa / E
σz(0,z) = q[1 − (1 + a²/z²)^(−3/2)]

At peak load, q = 565.88 kPa and w₀ = 744.85 μm. At z = 0.30 m, the vertical compressive stress is 160.97 kPa. The colour scale shows stress in kPa, while the surface displacement is exaggerated 400 times. These are different quantities with separate scales. The synthetic five-sensor readings discussed earlier are a separate interpretation example; the film is not fitted to that table.

What changes in a dynamic layered analysis?

This film evaluates a sequence of elastic equilibrium states as the prescribed load rises and falls. The two spatial views are synchronised; stress-wave travel time, inertia and damping inside the pavement are not solved. The buffer sketch is not coupled to an impact model. No delayed wavefront or permanent rut is drawn because the selected model does not calculate one.

A layered dynamic model requires layer thicknesses, material constitutive laws, densities, damping or viscoelastic parameters, interface conditions, plate contact and a measured load history. It also needs mesh and time-step checks, boundary treatment and comparison against suitable observations. Those inputs determine what a simulation can credibly predict. In this benchmark, doubling E halves displacement while leaving the stress distribution under the prescribed surface pressure unchanged. That is a consequence of this homogeneous elastic problem, not a general statement about a multilayer pavement. The distinction helps a reviewer understand precisely which inference the animation supports.

Calculation laboratory

Which part of the response changes when stiffness changes?

Calculation step	Substitution or result	Units and interpretation
Loaded area	π × 0.15² = 0.0706858	m²
Peak pressure	40,000 / 0.0706858 = 565884.242	Pa
Centre displacement	2 × (1 − 0.35²) × q × 0.15 / 200,000,000 = 744.845	μm
Stress at 0.30 m	q × [1 − (1 + 0.15²/0.30²)^(−1.5)] = 160.970	kPa

The pressure is force divided by area, so converting kilonewtons to newtons must precede substitution into SI equations. The displacement then has units Pa × m / Pa = m. Multiply by one million only when reporting micrometres. These checks catch a visually plausible but thousandfold erroneous animation.

At fixed load and geometry, centre displacement is inversely proportional to E: doubling E halves movement. In this homogeneous elastic half-space solution, the centreline vertical stress does not depend on E. That distinction matters: the stress heatmap should not brighten merely because the displacement grows under a softer modulus. Increasing the applied load, by contrast, scales both stress and displacement. The sensitivity experiment isolates stiffness so that the two mechanisms are not confused.

The falling mass and buffer explain the loading arrangement; the calculation prescribes a half-sine force history rather than deriving impact force from the mass, drop height and buffer constitutive law. The time-scaled response is quasi-static and excludes pavement inertia. The spatial surface field and integrated stress field are calculated, but layer interfaces and wave propagation are absent. Use this reference to understand the mechanics before interpreting measured bowls with the test context and a suitable model. (University of Cambridge, 2005) (FHWA, 2017)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
E = 100 MPa	1,489.690267 μm centre displacement	Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.
E = 200 MPa	744.845134 μm centre displacement	Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.
E = 400 MPa	372.422567 μm centre displacement	Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.

CALCULATION LABORATORY

Which part of the response changes when stiffness changes?

Compare the cases on a fixed scale. Select a case to inspect its input change and calculated output.

Sensitivity case1. E = 100 MPa2. E = 200 MPa3. E = 400 MPa

E = 100 MPa: 1,489.690267 μm centre displacement. Fixed 40 kN load, 0.15 m radius and Poisson ratio 0.35.

Reproduce and challenge the result

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.

These are original synthetic calculations. The sensitivity cases vary the stated input while retaining the other assumptions; they are not field observations or treatment thresholds. The film’s calculation and sensitivity chapters use the same numerical cases as this table.

Trace the stress away from the load

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 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 pavement shape.

Reproduce the experiment. σz = ∫A 3qz³/[2π((x−ξ)²+η²+z²)^(5/2)] dA. The downloadable table contains 6,161 evaluated cells, with the two input coordinates and output value retained on every row. These are parameter combinations, not that many independent field observations. The equation extends the worked example above; it is original teaching arithmetic, not a new standard or calibrated prediction.

Computed parameter atlas · 6,161 cells

Follow the load below the plate

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

Download every calculated cell (CSV) · Open full-size explorer · Equation source and tests (ZIP)

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.

References and method notes

1. Horak (2008). Benchmarking the structural condition of flexible pavements with deflection bowl parameters. Journal of the South African Institution of Civil Engineering, 50(2), pp. 2–9. Original source (Accessed: 5 September 2026).Relative structural benchmarking with deflection bowls
2. FHWA (2017). Using Falling Weight Deflectometer Data With Mechanistic-Empirical Design and Analysis, Volume III. FHWA-HRT-16-011, Chapter 2. Original source (Accessed: 5 September 2026).Deflection testing guidelines
3. Rocscience (n.d.). Settle3: Settlement and Consolidation Theory Manual. Theory manual. Original source (Accessed: 5 September 2026).Section 2.3.1: Boussinesq stress kernel and integration over a loaded area
4. University of Cambridge (2005). Engineering Tripos Part IIA: 3D5 Geotechnical Engineering. Examination paper and formula information, 2005. Original source (Accessed: 5 September 2026).Data sheet Sections 3.1 and 3.3, printed pp. 6 and 8: circular-load stress and surface settlement

Use the source edition and project requirements that apply to your work. Educational examples are not site-specific design advice.