16-Civ-B7 Transportation Planning and Engineering · December 2015

Question 7 of 8: Economic Comparison of Two Low-Volume Pavement Sections

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

Notes on this paper

Paper format. 98-Civ-B7 Highway Engineering, National Examinations, December 2015. Three hours, open book, any non-communicating calculator. Eight questions of equal value (20 marks each); five solutions constitute a complete paper and only the first five in the answer book are marked. Note 1 invites the candidate to state any assumption made about an ambiguous input, and Note 2 permits any datum that is required but not given to be assumed. All eight questions are solved here, because the set is a study resource rather than a timed attempt.

Reference texts. Garber & Hoel, Traffic and Highway Engineering, 5th ed. (geometric design, sight distance, pavement design); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (superelevation and spiral tables — the paper's Table 2.1.2.5 is TAC page 2.1.2.12); AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book) for runoff distribution and relative-gradient limits; AASHTO, Guide for Design of Pavement Structures (1993) for the flexible pavement equation and layer/drainage coefficients; Asphalt Institute MS-2, Asphalt Mix Design Methods and Mamlouk & Zaniewski, Materials for Civil and Construction Engineers, for mixture volumetrics and binder grading.

Check: assumptions carried through this paper. Under the paper's own Note 2 the following values are assumed and stated where used: the AASHTO maximum relative gradient (0.50 % at 80 km/h) and the 70 % / 30 % split of superelevation runoff either side of the PC for two lanes rotated (Question 2); a truck factor of 0.52 for all trucks on a rural Interstate and a lane-distribution factor of 0.70 for three lanes in one direction (Question 6); and a downhill 2 % ramp grade in Question 4, since the freeway is elevated above the local street. Each is flagged again at the point of use with the sensitivity of the answer to it.

Question 7: Economic Comparison of Two Low-Volume Pavement Sections (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.

Load, subgrade and material data
Item	Value
Wheel load, tyre pressure	11 300 lb per tyre, $q = 100$ psi
Subgrade modulus, allowable vertical stress	$E_2 = 5\,000$ psi; 15 psi
Option 1 — crushed rock	$E_1 = 50\,000$ psi, 120 lb/ft³, $5.00 per ton
Option 2 — asphalt concrete	$E_1 = 250\,000$ psi, 145 lb/ft³, $32.00 per ton
Conversion	1 ton = 2 000 lb; seal coat cost negligible

Find. The thickness of each candidate surfacing that limits the vertical stress on the subgrade to 15 psi, and hence which option is the cheaper per unit area of road.

The two candidate sections as Burmister two-layer systems. Both must attenuate the 100 psi contact pressure to 15 psi at the top of the subgrade; the stiffer layer does it in less thickness.

Approach. Reduce the wheel load to an equivalent circular contact area, enter the Burmister two-layer chart supplied with the paper at the required stress ratio to obtain $a/h_1$ for each modulus ratio, convert to a thickness, and price the two sections per square foot of pavement.

Radius of the loaded area. A tyre print carrying $P$ at a contact pressure $q$ has radius $$a = \sqrt{\frac{P}{\pi q}} = \sqrt{\frac{11\,300}{\pi(100)}} = \sqrt{35.97} = \boxed{6.00\ \text{in}}$$ The axle carries two such prints far enough apart that each may be analysed on its own.
Required stress ratio. The subgrade may see no more than 15 psi under a 100 psi contact pressure, so the chart is entered at $$\frac{\sigma_c}{q} = \frac{15}{100} = 0.15$$
Read the chart for each modulus ratio. The two candidate ratios are $E_1/E_2 = 50\,000/5\,000 = 10$ and $250\,000/5\,000 = 50$. At $\sigma_c/q = 0.15$ the chart gives $$\left(\frac{a}{h_1}\right)_{rock} \approx 0.73, \qquad \left(\frac{a}{h_1}\right)_{AC} \approx 1.25$$ Both readings can be confirmed independently by the equivalent-thickness argument: a one-layer Boussinesq system reaches $\sigma_z/q = 0.15$ at a depth $z = 2.956\,a$, and a two-layer system reproduces that stress when $h_1(E_1/E_2)^{1/3} = 2.956\,a$, so $$\frac{a}{h_1} = \frac{(E_1/E_2)^{1/3}}{2.956} \ \Rightarrow\ 0.729\ (\text{ratio }10), \qquad 1.246\ (\text{ratio }50)$$ which matches the plotted curves to within the width of a pencil line.
Thicknesses. Inverting the ratios with $a = 6.00$ in, $$h_{rock} = \frac{6.00}{0.729} = \boxed{8.23\ \text{in}\ (209\ \text{mm})}, \qquad h_{AC} = \frac{6.00}{1.246} = \boxed{4.81\ \text{in}\ (122\ \text{mm})}$$ The asphalt does the same job in 58 % of the thickness, the ratio being $(50/10)^{1/3} = 1.71$.
Cost of the crushed-rock section. Per square foot of pavement the mass of rock is $(8.23/12)(120) = 82.3$ lb, that is 0.04115 tons, so $$C_{rock} = 0.04115(5.00) = \$0.206\ \text{per ft}^{2}$$
Cost of the asphalt section. Likewise $(4.81/12)(145) = 58.1$ lb, or 0.02906 tons, so $$C_{AC} = 0.02906(32.00) = \$0.930\ \text{per ft}^{2}$$
Compare, and state the margin. The asphalt option costs $0.930/0.206 = 4.5$ times as much as the crushed rock. On a single 3.5 m lane one kilometre long (37 674 ft²) the two sections price out at $7 751 and $35 052 respectively: $$\boxed{\text{crushed rock with a seal coat is the economical choice}}$$ The break-even point is instructive. The two sections cost the same when $$\frac{h_{rock}}{h_{AC}} = \frac{(32.00)(145)}{(5.00)(120)} = 7.73$$ whereas the structural requirement only asks for a ratio of 1.71. Crushed rock would have to be more than four times thicker than the mechanics demand before asphalt became competitive, so the conclusion is robust against any plausible error in reading the chart.
Engineering judgement beyond the arithmetic. The road carries a heavy axle only once a month, so fatigue of a bound layer is not the control — subgrade rutting is, which is precisely what the 15 psi criterion expresses. That is the situation in which a thick granular section sealed against water is the right answer; a 122 mm asphalt pavement would be structurally adequate and economically indefensible. The seal coat still matters: the 50 000 psi modulus assumed for the crushed rock only holds while the layer stays unsaturated, which is exactly why the question specifies an impermeable surface.

Question 7 — final results
Quantity	Crushed rock	Asphalt concrete
Modulus ratio $E_1/E_2$	10	50
Chart entry $a/h_1$ at $\sigma_c/q = 0.15$	0.729	1.246
Required thickness	8.23 in (209 mm)	4.81 in (122 mm)
Mass per square foot	82.3 lb (0.0412 ton)	58.1 lb (0.0291 ton)
Cost per square foot	$0.206	$0.930
Cost per 3.5 m lane-kilometre	$7 751	$35 052
Radius of the loaded area	$a = 6.00$ in
Break-even thickness ratio	7.73, against a structural ratio of 1.71
Recommendation	crushed rock with a bituminous seal coat, 4.5 times cheaper

Topic. Burmister two-layer elastic theory used as a subgrade-protection criterion, and the cost comparison that follows from it.

Key relations. Contact radius $a = \sqrt{P/(\pi q)}$; one-layer Boussinesq stress on the axis $\sigma_z/q = 1 - z^{3}/(a^{2}+z^{2})^{3/2}$; two-layer chart $\sigma_c/q = f(a/h_1,\ E_1/E_2)$; equivalent thickness $h_e = h_1 (E_1/E_2)^{1/3}$; cost per unit area $= (h/12)\,\gamma\,(\text{unit price})/2000$.

Why this works. A stiff layer over a soft one spreads the wheel load over a wider area of subgrade than the same thickness of soft material would, and Burmister's solution quantifies exactly how much. The equivalent-thickness idea makes the result memorable: a layer of modulus $E_1$ behaves like a thickness $h_1(E_1/E_2)^{1/3}$ of subgrade material, so stiffness buys thickness only as the cube root. That cube root is the whole economics of the problem — multiplying the modulus by five saves only 42 % of the thickness, while the unit price has gone up more than sixfold.

Common pitfall. Using the full axle load rather than the wheel load when computing the contact radius, which inflates $a$ by $\sqrt{2}$ and every thickness with it. The second pitfall is to compare the two options by thickness alone and conclude that the thinner section must be cheaper; thickness and cost move in opposite directions here, and only the cost per unit area settles it.

Source. Huang, Pavement Analysis and Design, §2.2 (Burmister two-layer systems); Garber & Hoel, Traffic and Highway Engineering, 5th ed., §19.3; Mamlouk & Zaniewski, Materials for Civil and Construction Engineers, Chapter 9.

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