16-Civ-B7 Transportation Planning and Engineering · May 2013

Question 4 of 7: Sag-Curve Comfort Length and the CBR Test

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 98-Civ-B7 Highway Engineering, National Examinations May 2013 — a three-hour open-book examination; any non-communicating calculator is permitted. The cover page states that a total of five solutions is required, that only the first five as they appear in the answer book will be marked, and that all questions are of equal value. The grading scheme on the last page confirms 20 marks per question: Q1 (a) and (b) 10 marks each; Q2 (a) through (e) 4 marks each; Q3 (a) to (j) 2 marks each; Q4 (a) and (b) 10 marks each; Q5 (a) and (b) 10 marks each; Q6 (a) through (e) 4 marks each; Q7 20 marks. All seven printed questions are worked here, because this set is a study resource rather than a timed attempt; on exam day a candidate submits only the first five, in order. The paper also states that any data not given but required may be assumed, and that assumptions should be recorded with the answer — several questions below need that licence, and each assumption is flagged where it is made.

Reference texts. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed. (geometric design, sight distance, vertical curves, earthwork, pavement design); AASHTO, Guide for Design of Pavement Structures (1993) (rigid and flexible thickness design, reliability, drainage and load-transfer coefficients); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (Canadian design-domain values for sight distance and vertical curvature); Asphalt Institute, Mix Design Methods MS-2, 7th ed. (mixture volumetrics, VMA, VFA, absorbed binder); B.M. Das, Principles of Geotechnical Engineering, 9th ed. (compaction, Proctor testing, zero-air-voids line, CBR); M.S. Mamlouk and J.P. Zaniewski, Materials for Civil and Construction Engineers, 4th ed. (concrete and asphalt materials); A.M. Neville, Properties of Concrete, 5th ed., and CSA A23.1 (air entrainment, curing, joints in concrete pavement).

Check — assumptions carried through this paper. Three items are not supplied by the exam and are assumed under the paper’s own Note 2 (“any data, not given but required, can be assumed”), each stated again at the point of use: (i) Question 5 gives the mass of the Proctor mould but not its volume, so the ASTM D698 / AASHTO T99 standard 101.6 mm mould volume of 944 cm³ is used; (ii) Question 6 does not name a design speed, so the available stopping sight distance is computed from the Canadian/AASHTO eye and object heights of 1.08 m and 0.60 m; (iii) Question 7 lists the modulus of subgrade reaction as “1.0 MPa”, which is dimensionally incomplete — it is read as 1.0 MPa/m and the sensitivity of the answer to that reading is reported with the result.

Question 4: Sag-Curve Comfort Length and the CBR Test (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Part (a) supplies a sag between a falling 4 % grade and a rising 3 % grade with a running speed and a permissible centripetal acceleration; part (b) supplies seven load-penetration readings from a bearing-ratio test, written as products so that the imperial test values convert to SI on the page (the extraction prints the heading as “CDR Test”, an optical mis-read of CBR).

Given data
Item	Value
(a) Approach and departure grades	g₁ = −4 %, g₂ = +3 %
(a) Running speed	V = 80 km/h = 22.222 m/s
(a) Permissible centripetal acceleration	a = 0.3 m/s²
(b) Penetrations (mm)	1.270, 1.905, 2.540, 5.080, 7.620, 10.160, 12.700
(b) Stresses (MPa)	0.3450, 0.5175, 0.8625, 2.7600, 4.8300, 5.9340, 6.1410
(b) Standard stress at 2.54 mm	1000 × 0.0069 = 6.90 MPa

Find. The minimum sag-curve length that keeps the vertical acceleration within the stated comfort limit; then the corrected origin of the load-penetration curve, the corrected stress at 2.54 mm penetration, and the CBR of the sample.

Figure 4.1 — The sag curve of part (a). The comfort criterion limits the vertical acceleration a driver feels as the vehicle follows the equivalent arc of radius R = 100L/A.

Approach. For part (a), a vehicle traversing a parabolic sag experiences an essentially constant vertical acceleration equal to v²/R, where R is the equivalent radius 100L/A of the parabola, so setting that acceleration to the permissible value fixes L directly. For part (b), the concave-upward toe of the load-penetration curve is a seating artefact, so the tangent at the steepest point of the curve is projected back to zero stress to define a new origin, and every penetration is then measured from that origin.

(a) Algebraic change of grade. The sag turns the profile from falling to rising, so $$A=|g_2-g_1|=|+3-(-4)|=7\ \text{percent}$$ Working in decimals, the curvature of the parabola is $A/(100L)$ per metre.
(a) Equivalent radius of the parabola. A parabolic vertical curve of length $L$ and grade change $A$ (in percent) has an essentially constant curvature, so it behaves like a circular arc of radius $$R=\frac{100\,L}{A}$$ This is the standard small-grade approximation, and it is what allows a comfort criterion written for circular motion to be applied to a parabola.
(a) Impose the comfort limit. The vertical acceleration a driver feels is the centripetal acceleration $a=v^2/R$; setting it to the permissible value and solving for $L$, $$\frac{v^2}{R}=a\ \Longrightarrow\ L=\frac{A\,v^2}{100\,a}$$ Substituting $v=80/3.6=22.222$ m/s and $a=0.3$ m/s², $$L=\frac{7\times(22.222)^2}{100\times 0.3}=\frac{7\times 493.83}{30} =\boxed{115.2\ \text{m}}$$
(a) Cross-check against the tabulated form. The familiar design formula $L=AV^2/395$, with $V$ in km/h, is exactly this expression with a = 0.3 m/s² and the constant rounded from 388.8 up to 395: $$L=\frac{7\times 80^2}{395}=113.4\ \text{m}$$ The two agree to within 1.6 %, which confirms the derivation. Adopt L = 115.2 m, say 120 m in practice.
(a) A caution the comfort criterion invites. Comfort almost never governs a real sag curve. Applying the headlight-sight-distance criterion $L=AS^2/(120+3.5S)$ with a stopping sight distance of about 130 m at 80 km/h gives $$L=\frac{7\times 130^2}{120+3.5(130)}=\frac{118\,300}{575}=205.7\ \text{m}$$ which is roughly 1.8 times the comfort length. The question asks only for the comfort value, and 115.2 m is the answer to give, but a design submitted at that length would not light the road far enough ahead at night.
(b)(i) Convert and plot the readings. Each stress is the tabulated integer multiplied by 0.0069 (pounds per square inch converted to MPa) and each penetration is the tabulated fraction of an inch multiplied by 25.4, giving the seven points listed in the data table above and plotted below. The curve rises slowly at first, steepens through the middle, and then flattens — the classic seating-plus-bearing shape.

Figure 4.2 — Measured load-penetration curve (blue), the tangent at the steepest point projected back to zero stress (red), and the two corrected readings taken from the shifted origin (green).

(b)(ii) Correct the concave-upward toe. Successive secant slopes of the plotted curve are 0.272, 0.543, 0.747, 0.815, 0.435 and 0.082 MPa/mm, so the steepest portion — the point of inflection — lies between 5.080 and 7.620 mm, where the slope is $$m=\frac{4.830-2.760}{7.620-5.080}=\frac{2.070}{2.540}=0.8150\ \text{MPa/mm}$$ Projecting that tangent back to zero stress gives the corrected origin: $$x_0=5.080-\frac{2.760}{0.8150}=5.080-3.387=\boxed{1.693\ \text{mm}}$$ All penetrations are now measured from 1.693 mm rather than from zero.
(b)(iii) Corrected stress at 2.54 mm penetration. A corrected penetration of 2.540 mm corresponds to a measured penetration of $1.693+2.540=4.233$ mm. Interpolating linearly on the measured curve between (2.540 mm, 0.8625 MPa) and (5.080 mm, 2.760 MPa), $$\sigma_{2.54}=0.8625+\frac{4.233-2.540}{2.540}\,(2.760-0.8625) =0.8625+0.6667(1.8975)=\boxed{2.128\ \text{MPa}}$$ The same construction at a corrected 5.080 mm (measured 6.773 mm) gives $\sigma_{5.08}=4.140$ MPa, which part (iv) uses as a check.
(b)(iv) Bearing ratio. The CBR is the corrected stress expressed as a percentage of the standard stress for the same penetration: $$\text{CBR}_{2.54}=\frac{\sigma_{2.54}}{\sigma_{\text{std}}}\times 100 =\frac{2.128}{6.90}\times 100=\boxed{30.8\ \%}$$ Repeating at 5.08 mm against the conventional standard of 1500 psi = 10.35 MPa gives $4.140/10.35\times 100=40.0$ %, which is higher than the 2.54 mm value. Under ASTM D1883 that is the signal to repeat the test, and if the repeat confirms it the 5.08 mm value governs. The question supplies only the 2.54 mm standard, so CBR = 30.8 % is the answer to report, with the 40 % check noted as a qualification.

A CBR near 31 % describes a good granular base-course material — well above the 2–8 % of a plastic subgrade clay and below the 80 %-plus of a dense crushed base. Combined with the fact that the corrected origin sits at 1.693 mm, nearly two thirds of the first standard penetration, the test tells a consistent story: the specimen surface was irregular or the top layer slightly loose, so a meaningful amount of plunger travel was consumed before the material began to carry load in earnest. Ignoring that correction would have given a stress at 2.540 mm of only 0.8625 MPa and a CBR of 12.5 %, under-rating the material by a factor of two and a half.

Quantity	Value
(a) Grade change, A	7 %
(a) Minimum sag length from comfort	L = 115.2 m (say 120 m)
(a) Comparison, L = AV²/395	113.4 m
(a) Headlight criterion, for context	205.7 m — governs in design
(b)(ii) Corrected origin	1.693 mm (tangent slope 0.8150 MPa/mm)
(b)(iii) Corrected stress at 2.54 mm	2.128 MPa
Corrected stress at 5.08 mm	4.140 MPa
(b)(iv) CBR at 2.54 mm penetration	30.8 %
CBR at 5.08 mm (check)	40.0 % — repeat test indicated

Concept & Theory

Topic: Comfort control of sag vertical curves, and origin correction in the California Bearing Ratio test.

Key relations: R = 100L/A for a parabolic vertical curve; a = v²/R, hence L = Av²/(100a) and the tabulated L = AV²/395 for a = 0.3 m/s²; sag headlight criterion L = AS²/(120+3.5S); CBR = (corrected stress / standard stress) × 100, with standard stresses 6.9 MPa at 2.54 mm and 10.35 MPa at 5.08 mm.

Why this works: On a sag the vehicle and the road curve the same way, so gravity and the centripetal acceleration add rather than oppose, and the occupant feels an apparent weight increase of a/g. Because a parabola of small grade change has almost constant curvature, the whole problem collapses to circular motion at radius 100L/A, and the permissible acceleration converts straight into a minimum length. In the CBR test, the plunger must first bed into an imperfectly trimmed surface, which produces a load-penetration curve that is concave upward at small penetrations; the true material response is the steep, straight portion beyond that, so extrapolating its tangent to zero load recovers the penetration at which real bearing began.

Common pitfall: In part (a), taking A as the difference of the absolute grade values (4 − 3 = 1 %) instead of the algebraic difference (3 − (−4) = 7 %) — the sign convention is the whole content of the step, and it changes the answer sevenfold. In part (b), reading the stress at the measured 2.54 mm instead of at 2.54 mm past the corrected origin, which here would report a CBR of 12.5 % rather than 30.8 %.

Source: N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Ch. 15 (vertical curves, comfort and headlight criteria); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (sag curve K values); B.M. Das, Principles of Geotechnical Engineering, 9th ed., Ch. 6 (California Bearing Ratio); ASTM D1883.

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