The empirical methods of the previous posts ignored one obvious fact — that a pavement built of strong material should be able to be thinner than one built of weak material. The three methods in this post all address that gap by bringing modulus of elasticity into the calculation.
This post covers the triaxial method, its stiffness factor, the McLeod method and Burmister’s layered system analysis.
The Triaxial Method
Recall from the classification of design methods that the triaxial method is semi-empirical — built on stress–strain functions but using coefficients drawn from experience.
The Basic Thickness Formula
The pavement thickness Ts, consisting of material with modulus Es, is given by:
Ts = √[ (3 P X Y) / (2 π Es Δ) − a2 ]
| Symbol | Meaning | Unit or Value |
|---|---|---|
| P | Wheel load | kg |
| Δ | Design deflection | 0.25 cm |
| a | Radius of contact area | cm |
| X | Traffic coefficient | — |
| Y | Saturation coefficient | — |
| Es | Modulus of elasticity | kg/cm2 |
The design deflection of 0.25 cm is the key idea. The method does not ask “will the pavement fail?” but rather “how thick must it be so that it deflects no more than 2.5 mm under the design wheel?” Deflection is the design criterion.
Traffic Coefficient (X)
| Traffic Coefficient X | ADT (number) |
|---|---|
| 1/2 | 40 – 400 |
| 2/3 | 401 – 800 |
| 5/6 | 801 – 1200 |
| 1 | 1201 – 1800 |
| 7/6 | 1801 – 2700 |
| 8/6 | 2701 – 4000 |
| 9/6 | 4001 – 6000 |
| 10/6 | 6001 – 9000 |
| 11/6 | 9000 – 13,500 |
| 12/6 | 13,501 – 20,000 |
Note that the values are given as sixths, which makes the progression easy to see once you convert: 1/2 is 3/6, 2/3 is 4/6, 5/6 stays as it is, 1 is 6/6, and then 7/6, 8/6 and so on. The sequence simply climbs by one sixth per traffic band.
Rainfall or Saturation Coefficient (Y)
| Rainfall Coefficient Y | Average Annual Rainfall (cm) |
|---|---|
| 0.5 | 38 – 50 |
| 0.6 | 51 – 64 |
| 0.7 | 65 – 76 |
| 0.8 | 77 – 90 |
| 0.9 | 91 – 100 |
| 1.0 | 101 – 127 |
Both coefficients sit in the numerator of the thickness formula, so heavier traffic and higher rainfall both increase the required thickness — exactly as physical sense demands.
The Stiffness Factor
The basic formula treats the pavement as a single material. In reality, pavement and subgrade have different moduli, and that difference matters.
If pavement and subgrade are considered as a two layer system, a stiffness factor must be introduced to account for the different values of modulus of elasticity of the two layers.
Stiffness factor = (Es / Ep)1/3
where Es and Ep are the moduli of elasticity of the subgrade and pavement respectively.
The Modified Thickness
Tp = √[ (3 P X Y)/(2 π Es Δ) − a2 ] × (Es/Ep)1/3
What the Stiffness Factor Does
Look at the direction of the ratio — subgrade modulus on top, pavement modulus underneath.
Since a pavement is stiffer than the soil beneath it, Ep is greater than Es, so the ratio is less than 1. Multiplying by it therefore reduces the required thickness.
And the stiffer the pavement material relative to the soil, the smaller the ratio and the greater the reduction. This is precisely what the empirical methods could not do — reward the use of better material with a thinner section.
Equivalence Between Layers
The relation between pavement layers of thickness t1 and t2 having elastic moduli E1 and E2 is:
t1 / t2 = (E2 / E1)1/3
Note the cross-over — thickness 1 relates to modulus 2, and vice versa. This is an inverse relationship: a material of higher modulus needs less thickness to do the same job. It lets one material be substituted for another at an equivalent thickness.
The McLeod Method
From plate load tests, an empirical design equation was recommended:
T = K log10 (P / S)
| Symbol | Meaning |
|---|---|
| T | Required thickness of gravel base, cm |
| P | Gross wheel load, kg |
| S | Total subgrade support, kg — for the same contact area, deflection and number of repetitions of load P |
| K | Base course constant |
Two Points to Remember
- The base course constant K depends on the loaded area.
- The subgrade support S for highway pavement design is calculated from the support measured or calculated for a 30 cm diameter plate at 0.5 cm deflection and ten repetitions.
The formula’s logic is a ratio. If the wheel load P equals the subgrade support S, then log(1) = 0 and no thickness is needed — the soil can already carry the load. The more the load exceeds what the soil can support, the more thickness is required, and the logarithm makes that growth gradual rather than proportional.
Burmister’s Layered System Method
Donald M. Burmister developed the layered system analysis. It is the theoretical method of this chapter, based on modulus of elasticity of the different layers.
Four Assumptions
- Materials in each layer are isotropic, homogeneous and elastic.
- The pavement forms a stiffer layer, having a higher value of E than the subgrade.
- The surface layer is infinite in the horizontal direction but finite in the vertical direction — length and width infinite, height finite.
- Layers are in constant contact.
Listing assumptions explicitly is characteristic of a theoretical method, and it is honest — each one is an idealisation that real materials only approximately satisfy.
The Layered System Requirement
EB > ESB > ES
The Young’s modulus of the upper layers should be higher than that of the lower layers. Here EB is the base, ESB the sub-base and ES the subgrade.
This is the fundamental principle of layered pavement construction — strongest at the top, weakest at the bottom. It is exactly why the best material goes in the surface course, where stress is highest, and progressively cheaper material goes below as the load spreads.
The Reinforcing Effect
The vertical stress on the subgrade is reduced from 70 to 30 percent by introducing a pavement layer of thickness equal to the radius of the load (h = a), having elastic modulus 10 times higher than that of the subgrade soil, that is Ep/Es = 10.
This single result captures the whole value of Burmister’s approach. A stiff layer only one load-radius thick more than halves the stress reaching the soil — because the stiff layer bridges over the soft soil and spreads the load sideways rather than punching straight down.
Burmister’s approach therefore utilises the reinforcing action of the pavement layer.
The Deflection Factor
A deflection factor F2 is introduced in the two layered system, and it depends on:
- Ep / Es — the modulus ratio
- h / a — the thickness to load radius ratio
These are the same two ratios that appear in the reinforcing result above, which is no coincidence — they are the two quantities that fully describe a two layer system in this analysis.
The Three Methods Compared
| Triaxial | McLeod | Burmister | |
|---|---|---|---|
| Group | Semi-empirical | Empirical | Theoretical |
| Based on | Deflection criterion with modulus | Plate load tests | Elastic layer theory |
| Key formula | Ts = √[3PXY/(2πEsΔ) − a2] | T = K log10(P/S) | Deflection factor F2 |
| Accounts for material quality | Yes, via stiffness factor | Via base course constant K | Yes, via Ep/Es |
Formula Summary
| Quantity | Expression or Value |
|---|---|
| Triaxial thickness | Ts = √[3PXY/(2πEsΔ) − a2] |
| Design deflection | Δ = 0.25 cm |
| Stiffness factor | (Es/Ep)1/3 |
| Modified thickness | Tp = Ts × (Es/Ep)1/3 |
| Layer equivalence | t1/t2 = (E2/E1)1/3 |
| McLeod formula | T = K log10(P/S) |
| McLeod plate conditions | 30 cm diameter, 0.5 cm deflection, 10 repetitions |
| Layered system requirement | EB > ESB > ES |
| Burmister stress reduction | 70 % to 30 % when h = a and Ep/Es = 10 |
Quick Revision Notes
- Triaxial method design deflection Δ = 0.25 cm.
- X is the traffic coefficient (from ADT) and Y the saturation or rainfall coefficient (from average annual rainfall); both are in the numerator.
- Stiffness factor = (Es/Ep)1/3 — subgrade on top, pavement below, so it is less than 1 and reduces thickness.
- t1/t2 = (E2/E1)1/3 — note the cross-over of subscripts.
- McLeod: T = K log10(P/S); K depends on the loaded area; S from a 30 cm plate at 0.5 cm deflection and 10 repetitions.
- Burmister assumptions: materials isotropic, homogeneous and elastic; pavement stiffer than subgrade; surface layer infinite horizontally and finite vertically; layers in constant contact.
- EB > ESB > ES — upper layers must have higher modulus.
- Vertical stress on subgrade falls from 70 % to 30 % when h = a and Ep/Es = 10.
- Deflection factor F2 depends on Ep/Es and h/a.
Mistakes Students Commonly Make
- Inverting the stiffness factor. It is (Es/Ep)1/3 — subgrade modulus in the numerator.
- Forgetting the cross-over in the layer equivalence relation. t1/t2 = (E2/E1)1/3, not (E1/E2)1/3.
- Using a design deflection other than 0.25 cm.
- Writing the exponent as 1/2 rather than 1/3 in the stiffness factor.
- Confusing the McLeod plate conditions with the modulus of subgrade reaction test. McLeod uses a 30 cm plate at 0.5 cm deflection; K for rigid pavement uses a 75 cm plate at 0.125 cm.
- Reversing the Burmister modulus requirement. Upper layers must be stiffer, so EB is the largest.
- Misquoting the stress reduction. It falls from 70 to 30 percent, not the other way round.
Conclusion
These three methods share one improvement over the purely empirical approaches — they know the difference between good material and bad. The triaxial method designs to a deflection limit of 0.25 cm and then applies a stiffness factor that rewards a stiffer pavement with reduced thickness. McLeod relates thickness to the ratio of wheel load to subgrade support. And Burmister’s layered analysis shows quantitatively why layering works at all: a stiff layer just one load-radius thick can cut subgrade stress from 70 percent to 30 percent, which is the whole justification for building strong at the top and weak at the bottom.
Frequently Asked Questions
What is the design deflection in the triaxial method?
0.25 cm.
What are X and Y in the triaxial formula?
X is the traffic coefficient, read from a table of average daily traffic, and Y is the saturation or rainfall coefficient, read from a table of average annual rainfall.
What is the stiffness factor?
(Es/Ep)1/3, where Es and Ep are the moduli of elasticity of the subgrade and the pavement. It is introduced when pavement and subgrade are treated as a two layer system.
Why does the stiffness factor reduce the thickness?
Because the pavement is stiffer than the subgrade, so Ep exceeds Es and the ratio is less than one. A stiffer pavement material therefore requires less thickness.
What is the relation between equivalent layer thicknesses?
t1/t2 = (E2/E1)1/3, an inverse relation in which a higher modulus material needs less thickness.
What is the McLeod formula?
T = K log10(P/S), where T is the required gravel base thickness, P the gross wheel load, S the total subgrade support and K the base course constant.
How is subgrade support measured in the McLeod method?
From the support measured or calculated for a 30 cm diameter plate at 0.5 cm deflection and ten repetitions.
What are Burmister’s assumptions?
That materials in each layer are isotropic, homogeneous and elastic; that the pavement forms a stiffer layer with higher E than the subgrade; that the surface layer is infinite horizontally but finite vertically; and that the layers remain in constant contact.
What modulus relationship must a layered system satisfy?
EB > ESB > ES, meaning upper layers must have a higher Young’s modulus than the layers below them.
What stress reduction does Burmister’s analysis demonstrate?
That vertical stress on the subgrade falls from 70 percent to 30 percent when a pavement layer of thickness equal to the load radius is introduced, with elastic modulus ten times that of the subgrade.
What does the deflection factor depend on?
The modulus ratio Ep/Es and the thickness to load radius ratio h/a.