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Geotechnical

5% of exam

Soil classification, CBR/R-value and resilient-modulus correlations, phase relationships and soil properties, compaction and relative compaction, and earthwork cut/fill mass balance with shrinkage and swell.

5 concepts

A. Sampling, testing, evaluation, and stabilization

Subgrade Strength: CBR, R-Value & Resilient Modulus

Measure subgrade support with the CBR, R-value, and resilient modulus, convert among them and to soil class, and feed the right stiffness into pavement design.

Every pavement is only as strong as the dirt under it: the subgrade's stiffness sets how thick the pavement must be, and a soft, wet, or poorly characterized subgrade is where premature rutting and fatigue cracking begin. The PE Civil exam tests three measures of that support — the California Bearing Ratio (CBR), the R-value, and the resilient modulus (

M_R

) — plus the correlations that let you move between them when only one is reported. The handbook lists the bare test-method standards in

\S3.8

Material Test Methods, but the working definitions, the

M_R

correlations, and the typical-value tables and figure you actually use are gathered in

\S3.19

Pavements (printed pp. ~223–226). The unifying idea is that all three answer the same question, 'how stiffly does this soil push back under a wheel,' but at different load levels and with different test geometries, so you must know which one a design method wants and how to translate the others into it.

California Bearing Ratio (CBR)

The CBR rates a compacted soil's strength against a standard crushed-stone reference. A piston is pushed into the soil at a fixed rate and the resistance at

0.1\ \text{in}

penetration is divided by the standard value (

1000\ \text{psi}

) for that penetration, expressed as a percent. A CBR of

100

is as strong as the reference rock; a soft clay subgrade may be

2

to

5

, a good granular base

80

or more. Because it is cheap and familiar, CBR underlies many empirical pavement thickness charts, but it is a punching-shear index, not an elastic stiffness.

\text{CBR} = \dfrac{p_{\text{soil at }0.1''}}{p_{\text{standard at }0.1''}}\times 100\% = \dfrac{p_{\text{soil}}}{1000\ \text{psi}}\times 100\%

R-value (resistance value)

The Hveem stabilometer R-value (ASTM D 2844, AASHTO T 190) loads a confined, moisture-conditioned specimen vertically and measures the lateral pressure it transmits — a soil that behaves like a liquid transmits all the load sideways (

R \to 0

), a rigid material transmits none (

R \to 100

). The R-value scale runs

0

to

100

and feeds the historic state DOT and Asphalt Institute thickness methods. Like CBR it is an empirical strength index, but it captures a soil's response to confinement and saturation more directly, which is why western U.S. agencies adopted it.

R = 100 - \dfrac{100}{\dfrac{2.5}{D}\left(\dfrac{p_v}{p_h}-1\right)+1}

Resilient modulus — the mechanistic stiffness

The resilient modulus is the elastic stiffness the subgrade actually shows under a moving wheel: a repeated deviator stress is applied to a triaxial specimen and

M_R

is the recoverable (elastic) stress-to-strain ratio after the plastic deformation has shaken out over many cycles. It is the input that mechanistic-empirical and AASHTO pavement design demand because it represents the cyclic, non-failure loading a pavement really sees. For fine-grained soils

M_R

falls as deviator stress rises (stress-softening); for granular soils it rises with confining stress (stress-hardening), so a single number always implies a stress state.

M_R = \dfrac{\sigma_d}{\varepsilon_r} \qquad (\sigma_d = \text{repeated deviator stress},\ \varepsilon_r = \text{recoverable strain})

Correlations: CBR ↔ M_R

When only a CBR is in hand, the long-standing Heukelom–Klomp relation (handbook

\S3.19

Pavements) estimates the subgrade resilient modulus directly. It is valid only for fine-grained subgrade soils with

\text{CBR} \le 10

; above that, stiffer granular relations (and stress-dependent forms) apply, and the AASHTO 1993 guide gives separate base/subbase correlations keyed to the applied stress level. Treat any single correlation as an order-of-magnitude estimate, not a measurement — design guides accept it only when testing is impractical.

M_R\ (\text{psi}) = 1500\times \text{CBR} \qquad (\text{CBR}\le 10)

Correlations: R-value ↔ M_R and soil class

The Asphalt Institute relation converts an R-value to resilient modulus, with the recommended-value coefficients giving a convenient working form. The handbook's

\S3.19

Pavements also tabulates

M_R

against the soil support value, the AASHTO group index, and the Texas triaxial class in a single correlation figure, so a problem that hands you a group index or an R-value can still be driven to a modulus. Always check which correlation family the question intends — subgrade versus granular base — because the constants differ markedly.

M_R\ (\text{psi}) = 1000 + 555\times R\text{-value}

From support to pavement thickness

Whatever index you measure, the design step converts it to the stiffness or support term the method consumes: AASHTO flexible design uses

M_R

to set the subgrade contribution and, with layer coefficients, the required structural number; rigid design uses the modulus of subgrade reaction

k

(roughly

k \approx M_R/19.4

in consistent units, from plate-load correlations). A higher CBR, R-value, or

M_R

all mean a stiffer subgrade and therefore a thinner required pavement, all else equal. Knowing the direction of that trade is often enough to eliminate wrong multiple-choice answers.

\text{(flexible)}\ SN = a_1 D_1 + a_2 D_2 m_2 + a_3 D_3 m_3

Exam strategy

First identify which index the problem gives and which the design method wants, then pick the matching correlation — the

1500\times\text{CBR}

rule is subgrade-only and capped at CBR

10

, so do not apply it to a base course. When you flip to the handbook, go to

\S3.19

Pavements for the correlations and typical-value tables, not

\S3.8

(which only lists test-method standards). Carry units explicitly: CBR and R-value are dimensionless indices, while

M_R

and

k

have units (psi and pci). Remember the physical ordering — clays give low CBR/R/

M_R

, granular soils high — as a built-in sanity check. If a question asks merely whether a redesign gets thinner or thicker, reason from 'stiffer subgrade → less pavement' rather than grinding the full structural-number equation.

Key equations

California Bearing Ratio

\text{CBR} = \dfrac{p_{\text{soil}}}{p_{\text{standard}}}\times 100\%

Penetration resistance at 0.1 in vs. the 1000 psi standard-stone value; dimensionless percent.

R-value (stabilometer)

R = 100 - \dfrac{100}{\frac{2.5}{D}\left(\frac{p_v}{p_h}-1\right)+1}

Resilient modulus definition

M_R = \dfrac{\sigma_d}{\varepsilon_r}

CBR → M_R (subgrade)

M_R = 1500\times \text{CBR}\ (\text{psi})

Heukelom–Klomp; subgrade soils with CBR ≤ 10 only (handbook §3.19 Pavements).

R-value → M_R (subgrade)

M_R = 1000 + 555\times R\ (\text{psi})

Asphalt Institute recommended-value form; subgrade roadbed soils (handbook §3.19 Pavements).

CBR → M_R (granular base)

M_R = 740\times \text{CBR}\ (\text{psi}),\ \theta = 100\ \text{psi}

AASHTO 1993 base/subbase correlation at bulk stress θ = 100 psi (constants vary with θ).

Resilient modulus → k

k \approx \dfrac{M_R}{19.4}\ (\text{pci})

Approximate plate-load conversion to modulus of subgrade reaction for rigid pavement.

Worked examples

CBR to resilient modulus

Problem. A fine-grained subgrade clay tests at CBR $= 6$ . Estimate its resilient modulus for flexible pavement design.

Solution. The soil is fine-grained with CBR

= 6 \le 10

, so the Heukelom–Klomp subgrade relation applies.

M_R = 1500\times \text{CBR} = 1500(6) = 9{,}000\ \text{psi}

. Sanity check: subgrade clays typically fall in the

3{,}000

–

15{,}000\ \text{psi}

range;

9{,}000\ \text{psi}

for a moderate CBR of 6 sits squarely in that band. The CBR-10 cap was respected, so the correlation is valid.

M_R = 1500(6) = 9{,}000\ \text{psi}

R-value to resilient modulus

Problem. A subgrade reports an R-value of $20$ from the stabilometer. Find the design resilient modulus, and compare it to a CBR-based estimate if the soil's CBR is $5$ .

Solution. R-value path (Asphalt Institute recommended form):

M_R = 1000 + 555R = 1000 + 555(20) = 1000 + 11{,}100 = 12{,}100\ \text{psi}

Subgrade stiffness and thickness direction

Problem. A subgrade is reworked and its CBR rises from $4$ to $9$ . Using $M_R = 1500\times\text{CBR}$ , quantify the stiffness gain and state qualitatively what happens to the required pavement thickness.

Solution.

Common pitfalls

•Applying $M_R = 1500\times\text{CBR}$ to a base course or to CBR > 10. That relation is subgrade-only and capped at CBR 10; granular layers use the stress-dependent AASHTO forms.
•Treating CBR, R-value, and $M_R$ as the same scale. CBR and R-value are dimensionless indices; $M_R$ has units (psi) and is an elastic stiffness — convert, don't equate.
•Quoting a resilient modulus without a stress state. $M_R$ is stress-dependent (softening for clays, hardening for granular soils); a single value implies a specific deviator/confining stress.
•Confusing $M_R$ (resilient modulus) with $k$ (modulus of subgrade reaction). $M_R$ drives flexible design; $k$
•Reversing the strength–thickness logic. A higher CBR/R/ $M_R$ means a STIFFER subgrade and a THINNER required pavement, not thicker.
•Looking up the correlations in the wrong handbook section. The $M_R$ correlations and typical-value tables/figure live in $\S3.19$ Pavements; $\S3.8$ Material Test Methods only lists the ASTM/AASHTO test-method standards by name.
•Picking the wrong R-value coefficients. The handbook lists a range ( $A=772$ – $1155$ , $B=369$ – $555$ ); use the recommended $M_R=1000+555R$

References

NCEES PE Civil Reference Handbook 2.2 — §3.19 Pavements (printed pp. ~223–226): CBR/R-value/M_R correlations (1500×CBR Heukelom–Klomp; 1000+555R Asphalt Institute; A=772–1155, B=369–555; 740×CBR granular) and the typical-value tables and Subgrade Resilient Modulus figure — §3.8 Material Test Methods (p. 124) lists only the ASTM/AASHTO test-method standards by name; the correlations and definitions used here are in §3.19
FHWA-NHI-05-037, Geotechnical Aspects of Pavements — M_R correlations, typical CBR by USCS class
AASHTO Guide for Design of Pavement Structures (1993) — subgrade M_R and structural number framework

Sampling, Field Testing & Soil Stabilization

Plan borings and sampling, correct and use the SPT N-value, interpret Atterberg limits, and choose lime, cement, or mechanical stabilization for a problem subgrade.

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B. Soil properties

Phase Relationships & Soil Index Properties

Use the three-phase weight–volume diagram to link void ratio, porosity, water content, saturation, specific gravity, and the dry, total, and saturated unit weights.

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C. Compaction, excavation, embankment, and mass balance

Soil Classification, Compaction & Subgrade Strength

Classify a subgrade by AASHTO and USCS, read the group index, and tie the Proctor curve, maximum dry density, relative compaction, and the zero-air-voids line together.

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Earthwork Volumes & Mass Balance

Compute cut and fill by average end areas, convert bank–loose–compacted volumes with shrinkage and swell, and read a mass-haul diagram to balance earthwork and direct haul.

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Hveem resistance value, 0–100. $p_v$ vertical, $p_h$ horizontal (transmitted) pressure, $D$ = displacement.

Recoverable elastic stiffness under repeated load; $\sigma_d$ deviator stress, $\varepsilon_r$ recoverable strain (psi).

. CBR path (subgrade, CBR

= 5 \le 10

):

M_R = 1500(5) = 7{,}500\ \text{psi}

. The two correlations bracket the likely modulus (

7{,}500

–

12{,}100\ \text{psi}

); for design, prefer the index that matches the agency method, and treat the spread as correlation uncertainty. Sanity check: both are positive, in the thousands-of-psi range expected for a soft-to-moderate subgrade, and the lower CBR (5) sensibly yields the lower modulus.

M_R = 1000 + 555(20) = 12{,}100\ \text{psi}

Before:

M_R = 1500(4) = 6{,}000\ \text{psi}

. After:

M_R = 1500(9) = 13{,}500\ \text{psi}

. Ratio:

13{,}500/6{,}000 = 2.25

— the subgrade is

125\%

stiffer. In AASHTO flexible design, a higher subgrade

M_R

raises the allowable load repetitions for a given structure, so the required structural number — and thus pavement thickness — DECREASES. Sanity check: stiffer subgrade → less pavement needed; the modulus more than doubled, so a meaningful thickness reduction is expected. Both CBR values are ≤ 10, so the correlation holds throughout.

\dfrac{M_{R,\text{after}}}{M_{R,\text{before}}} = \dfrac{1500(9)}{1500(4)} = 2.25

(pci) drives rigid-slab design — they are related by correlation, not identical.

unless the problem specifies otherwise.