Upgrading an unpaved road: organise the decision evidence

Where did the missing gravel go?

A uniform forty-millimetre allowance hides the places where the surface has lost much more.

The official catalogue lists a TRH24 draft standard dated October 2024. Check that document and its applicability rather than relying on an undated summary. (Department of Transport, n.d.)

Put a volume against the missing thickness

Governing relationship: V = length × width × lost thickness.

L = 1,000 m; b = 5 m; h is prescribed loss in metres.

Each 1 mm loss represents 5 m³ across this area; no wear-rate law is inferred.

The film first plots converted thickness (m), then follows the calculation and plots in-place volume (m³). A 40 mm loss gives 200 m³. This is in-place geometry; loose haul volume additionally requires density and compaction factors.

Geometric scenario

Material loss and replacement volume

Loss is prescribed; density, compaction and haul factors are not included.

Download example data (CSV)

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

Understand the model

A thickness loss over an area gives an in-place material volume. Predicting when loss occurs requires a separate deterioration model.

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

L; b; h

Road length, width and prescribed thickness loss (m).

V

In-place volume corresponding to that loss (m³).

Values used in this film

• L = 1,000 m; b = 5 m; h is prescribed loss in metres.
• Each 1 mm loss represents 5 m³ across this area; no wear-rate law is inferred.

The mathematical relationships

1. Equation 1

   Source and scope
   $$V=Lbh$$

   Multiply length, width and thickness in metres to obtain cubic metres.

2. Equation 2

   Source and scope
   $$\frac{dV}{dh}=Lb=5000\ \mathrm{m^2}$$

   The derivative is the area exposed to loss: every additional metre of loss changes volume by Lb cubic metres. Here 1 mm corresponds to 5 m³.

Work through the example

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

1. Area

   1000 × 5 = 5000

   m²

2. Thickness conversion

   40 / 1000 = 0.04

   m

3. In-place volume

   5000 × 0.04 = 200

   m³

4. Sensitivity

   dV/dh = 5000; 1 mm corresponds to 5

   m³ per mm

Read the output

A 40 mm loss gives 200 m³. This is in-place geometry; loose haul volume additionally requires density and compaction factors.

Check your understanding

Check metre conversions and reconstruct a nonuniform case section by section. Keep in-place, compacted and loose quantities labelled separately. Do not multiply by a bulking factor without its source and material-state definition.

Assumptions and limits

Loss is prescribed; density, compaction and haul factors are not included.

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

Timestamped film transcript 25 short entries · select a time to revisit the film

1. 0:00

   Material loss and replacement volume. 1,000 m × 5 m; prescribed loss increases from 0 to 40 mm.

2. 0:01

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

3. 0:07

   L = 1,000 m; b = 5 m; h is prescribed loss in metres. Each 1 mm loss represents 5 m³ across this area; no wear-rate law is inferred.

4. 0:14

   INPUT · loss scenarios · read the axis units and the numerical values.

5. 0:23

   1,000 m × 5 m; prescribed loss increases from 0 to 40 mm.

6. 0:37

   40 mm = 0.040 m; 1,000 × 5 × 0.040 = 200 m³. Surface movement is exaggerated.

7. 0:43

   OUTPUT · missing material · read the axis units and the numerical values.

8. 0:52

   A 40 mm loss gives 200 m³. This is in-place geometry; loose haul volume additionally requires density and compaction factors.

9. 0:59

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

10. 1:05

    Area: 1000 × 5 = 5000 m²

11. 1:08

    Thickness conversion: 40 / 1000 = 0.04 m

12. 1:11

    In-place volume: 5000 × 0.04 = 200 m³

13. 1:14

    Sensitivity: dV/dh = 5000; 1 mm corresponds to 5 m³ per mm

14. 1:17

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

15. 1:26

    Loss 35 mm: 175 m³ in-place volume. Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.

16. 1:29

    Loss 40 mm: 200 m³ in-place volume. Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.

17. 1:33

    Loss 45 mm: 225 m³ in-place volume. Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.

18. 1:36

    Check: Check metre conversions and reconstruct a nonuniform case section by section. Keep in-place, compacted and loose quantities labelled separately. Do not multiply by a bulking factor without its source and material-state definition.

19. 1:42

    ONE ROW · A uniform forty-millimetre allowance hides the places where the surface has lost much more.

20. 2:00

    PARAMETER MAP · 12261 evaluated cells. The colour limits stay fixed across every row.

21. 2:10

    This 1 km by 5 m synthetic loss field varies from 25 to 55 mm. Integrate thickness over area after converting millimetres to metres: V = ∫∫h/1000 db dx. Four complete longitudinal

22. 2:22

    sine cycles cancel the signed fluctuation, leaving exactly 200 m³. Equal total volume therefore coexists with substantial local variation.

23. 2:29

    VALUE LANDSCAPE · Height and colour encode Prescribed thickness loss (mm). This is a graph over the two input axes, not automatically a physical pavement shape.

24. 2:41

    Prescribed loss geometry, not an erosion model or a measured change survey. Bulk density, compaction and loose supply volume are separate quantities.

25. 2:49

    Loss is prescribed; density, compaction and haul factors are not included.

Download the timed captions (VTT)

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

Compare alternatives consistently

Our suggested decision table states the objective, baseline condition, service requirements, evidence gaps and alternatives. Compare each option using the same analysis period and clearly declared assumptions. Keep unresolved environmental, operational and engineering questions visible.

A decision matrix without invented precision

For a synthetic road, compare maintaining the current service, targeted improvements and a broader upgrade. Describe what evidence would distinguish the alternatives. Do not assign numerical scores where the evidence only supports qualitative judgement.

Separate service needs from treatment preference

This article does not prescribe an upgrading warrant or replace the applicable road authority process. Verify current source status before applying detailed requirements.

Compare options on a shared brief

An original options table can begin with a common objective: the service the road is expected to provide. State the baseline and what each alternative changes, then list the evidence needed to compare them.

Separate a missing estimate from a low cost and an unknown effect from a negligible effect. Assigning zero to an unknown entry can make a poorly understood option appear attractive.

Keep the same analysis period and cost scope across alternatives. If the team changes one of those assumptions, update every option or explain why a comparison is no longer like-for-like. This article supplies a structure for discussion; the detailed upgrading criteria must come from the applicable authority and source edition.

An evidence gap is not a disadvantage score

Suppose three alternatives remain in discussion: maintaining the current service, targeted works, and a broader upgrade. Their review registers contain two, four and five unresolved questions respectively. The counts describe work still needed. They do not rank the alternatives.

Case review

One volume can hide very different repair locations

Does the 200 m³ total tell us where material is needed?

Local minimum · 25 Thickness loss (mm)

The discriminating check

Check the change-survey basis, accessible width, in-place density and the service objective before comparing options.

Evidence supports the comparison

With credible geometry, carry quantities into comparable maintenance and upgrading scenarios.

Evidence is missing or incompatible

With an assumed uniform loss or uncertain density, show the sensitivity rather than claiming a measured supply quantity.

What you can conclude. In-place loss, loose supply volume and the preferred treatment are three different conclusions.

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

A broader intervention may have more open questions because its scope is larger. A familiar option may have fewer because more evidence already exists. Treating the smaller count as the better option would confuse knowledge about an alternative with the value of the alternative.

Comparison dimension	Question to make explicit
Service objective	What must the road provide?
Baseline	What happens under the agreed reference case?
Technical suitability	Can the option meet the objective?
Appraisal basis	Are period, prices and cost scope comparable?
Unresolved evidence	What could change the comparison?

The official catalogue lists TRH24 as a draft dated October 2024. Confirm its status and the authority's applicable requirements before using detailed upgrading criteria. This article provides an original comparison structure, not a warrant for upgrading a particular road. (Department of Transport, n.d.)

A defensible recommendation explains why an option is preferred against the agreed objective and how uncertainty affects that preference. Where evidence is insufficient, identify the investigation that would matter to the decision. Assigning a numerical score to every empty cell creates precision in the spreadsheet without creating knowledge about the road.

Calculation laboratory

How does a thickness estimate become a material quantity?

Calculation step	Substitution or result	Units and interpretation
Area	1000 × 5 = 5000	m²
Thickness conversion	40 / 1000 = 0.04	m
In-place volume	5000 × 0.04 = 200	m³
Sensitivity	dV/dh = 5000; 1 mm corresponds to 5	m³ per mm

The quantity is an area integral of thickness loss. The rectangular expression V = Lbh is its uniform-width, uniform-loss special case. If measured loss varies by section, calculate each section’s area times its representative loss and sum the volumes. An unweighted mean of measurements with unequal represented areas can bias the total.

The ±5 mm cases create a ±25 m³ change around 200 m³, or ±12.5%. This is a sensitivity interval arising from a stated thickness range, not a statistical confidence interval. Width and length uncertainty can contribute too. For small changes, δV/V is approximately δL/L + δb/b + δh/h; direct recomputation is preferable when changes are large or inputs covary.

Haul quantities require a further material-state conversion. Conserving dry mass gives loose volume equal to in-place volume multiplied by the in-place-to-loose dry-density ratio. The required densities must come from applicable evidence; no universal bulking factor is assumed here. Nor does a calculated missing volume predict when the next loss will occur. An upgrading decision must compare service requirements, materials, drainage, maintenance and costs with the relevant design context. Geometry quantifies one part of the evidence, not the preferred alternative. (CSRA, 1996)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
Loss 35 mm	175.000000 m³ in-place volume	Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.
Loss 40 mm	200.000000 m³ in-place volume	Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.
Loss 45 mm	225.000000 m³ in-place volume	Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.

CALCULATION LABORATORY

How does a thickness estimate become a material quantity?

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

Sensitivity case1. Loss 35 mm2. Loss 40 mm3. Loss 45 mm

Loss 35 mm: 175 m³ in-place volume. Uniform prescribed loss over 1000 m by 5 m; no wear law assumed.

Reproduce and challenge the result

Check metre conversions and reconstruct a nonuniform case section by section. Keep in-place, compacted and loose quantities labelled separately. Do not multiply by a bulking factor without its source and material-state definition.

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.

Keep the local loss in the volume calculation

This 1 km by 5 m synthetic loss field varies from 25 to 55 mm. Integrate thickness over area after converting millimetres to metres: V = ∫∫h/1000 db dx. Four complete longitudinal sine cycles cancel the signed fluctuation, leaving exactly 200 m³. Equal total volume therefore coexists with substantial local variation.

Reproduce the experiment. h(x,b) = 40 + 15 sin(2πx/250)sin(πb/5) mm. The downloadable table contains 12,261 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 · 12,261 cells

Where did the missing gravel go?

A uniform forty-millimetre allowance hides the places where the surface has lost much more.

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

Prescribed loss geometry, not an erosion model or a measured change survey. Bulk density, compaction and loose supply volume are separate quantities.

References and method notes

1. Department of Transport (n.d.). Roads: Engineering Standard Guidelines and Manuals. Official catalogue; accessed 5 September 2026. Original source (Accessed: 5 September 2026).TMH22 V8b Draft; TRH12; TRH15; TRH24 catalogue entries
2. CSRA (1996). TRH4: Structural Design of Flexible Pavements for Interurban and Rural Roads. 1996 edition. Original source (Accessed: 5 September 2026).Design framework; confirm project applicability

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