16-Civ-A6 Highway Design, Construction, and Maintenance · December 2018

Question 5 of 5: Design ESAL for a new urban freeway with staged traffic growth

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

Notes on this paper

Paper format. National Examinations, December 2018 — 16-Civ-A6 Highway Design, Construction, and Maintenance. Three-hour, closed-book paper with a ten-page appendix. Five questions are printed and four solutions constitute a complete paper; all questions carry equal value (25 % each) and the marks for sub-questions are shown in brackets. Note 1 invites the candidate to state any interpretation assumed, and Note 2 permits any required data that is not given to be assumed. All five questions are worked here, because the set is a study resource rather than an examination attempt.

Reference texts. AASHTO, Guide for Design of Pavement Structures, 1993 — Part II Ch. 2 (flexible pavements) and Ch. 3 (rigid pavements); Figures 2.5–2.7, 3.1, 3.3, 3.6, 3.7 and Tables 2.4–2.6 are reproduced on appendix pages 1–8. Garber, N.J. and Hoel, L.A., Traffic and Highway Engineering, 5th ed. — Ch. 3 (driver characteristics, stopping sight distance), Ch. 15 (geometric design), Ch. 20 (flexible and rigid pavement design). AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book), 7th ed. — Ch. 3. Transportation Association of Canada, Geometric Design Guide for Canadian Roads — Ch. 1.2 (design controls), Ch. 2.1 (sight distance), Ch. 3.2 (horizontal alignment and spirals), Ch. 4 (roadside design and clear zones). AASHTO, Roadside Design Guide, 4th ed. — Ch. 3 (clear zone). Huang, Y.H., Pavement Analysis and Design, 2nd ed. — Ch. 11 and 12.

Check — chart readings and assumed data. Three of the five questions are solved from nomographs and tables reproduced on the appendix pages. Each chart reading used here was taken from the printed appendix chart and is quoted in the step where it is used, so that a reader working from a different print can substitute their own reading. Two items are genuinely not supplied by the paper and are assumed under Note 2: the percentage of time the pavement structure is near saturation in Question 1 (needed for the drainage coefficients $m_2$, $m_3$), and AASHTO Figure 3.4, the rigid-foundation correction needed in Question 3(b), which the ten-page appendix does not include. Both are flagged where they arise and the sensitivity of the answer to each is quantified.

Question 5: Design ESAL for a new urban freeway with staged traffic growth (25 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. A four-lane urban freeway opening at 17,500 veh/day two-way, with a four-class vehicle mix and three different growth histories, over a 25-year design life.

Traffic composition, truck factors and growth
Vehicle class	Share of AADT	Truck factor $TF$	Growth
Passenger cars	78 %	not tabulated (taken as 0)	2 % for 2 yr, then 6 %
Single unit, two-axle, four-tire	12 %	0.015	3 % for 5 yr, then 4 %
Single unit, two-axle, six-tire	7 %	0.13	3 % throughout
Single unit, six-axle or more	3 %	0.74	3 % throughout

Find. The cumulative 18-kip equivalent single-axle load applications in the design lane over 25 years.

Figure 5 — how the cumulative design ESAL divides between the three truck classes; passenger cars contribute nothing.

Approach. Compute a first-year design-lane ESAL for each vehicle class from the appendix relation $ESAL = AADT\times HVP\times DF\times TF\times TDY$, form a cumulative growth factor for each class over the 25-year life — using a two-stage factor with a continuity term wherever the growth rate steps — and sum the products.

Select the truck factors. Table IV-5 is entered in its urban half. The road is a freeway but is not described as an Interstate, so the "Other Freeways" column applies, giving $$TF_{\text{2-axle, 4-tire}} = 0.015 \qquad TF_{\text{2-axle, 6-tire}} = 0.13 \qquad TF_{\text{3-axle or more}} = 0.74$$ The paper's "six-axle or more single unit" class maps to the table's "3-axle or more" single-unit row, which is the heaviest single-unit row tabulated. Passenger cars do not appear in Table IV-5 at all; their truck factor is of order $10^{-4}$ and AASHTO practice is to omit them, so they are carried through the calculation at $TF = 0$ and contribute nothing. That the 78 % of the traffic which is cars is structurally irrelevant is itself one of the lessons of the question.
Establish the design lane factor. The AADT quoted is two-way, so a directional split of 0.50 applies. A four-lane freeway carries two lanes in each direction, for which the AASHTO lane-distribution factor is 0.80–1.00; taking the customary 0.90, $$DF = 0.50 \times 0.90 = 0.45$$
Compute the first-year design-lane ESAL for each class. With $TDY = 365$ days, $$ESAL_{\text{4-tire}} = 17\,500(0.12)(0.45)(0.015)(365) = 5{,}174$$ $$ESAL_{\text{6-tire}} = 17\,500(0.07)(0.45)(0.13)(365) = 26{,}157$$ $$ESAL_{\text{6-axle+}} = 17\,500(0.03)(0.45)(0.74)(365) = 63{,}811$$ Already the ordering is instructive: the 3 % of the fleet that is heavy single-unit trucks does twelve times the damage of the 12 % that is light four-tire trucks.
Form the growth factor for the classes with a single rate. For a constant rate the cumulative factor over $t$ years is the ordinary annuity sum, and for the two classes growing at 3 % throughout $$GF = \frac{(1+g)^{t} - 1}{g} = \frac{1.03^{25} - 1}{0.03} = \frac{2.09378 - 1}{0.03} = 36.459$$
Form the two-stage growth factor, with its continuity term. When the rate steps after $t_1$ years, the second stage must start from the volume the first stage has already reached, not from the opening-year volume. Summing year by year, the first stage contributes $[(1+g_1)^{t_1}-1]/g_1$ and the second stage begins at the year-$t_1$ volume $(1+g_1)^{t_1-1}$, grows once by $(1+g_2)$ to reach year $t_1+1$, and then runs as its own annuity: $$GF = \frac{(1+g_1)^{t_1} - 1}{g_1} + (1+g_1)^{t_1-1}(1+g_2)\,\frac{(1+g_2)^{t_2} - 1}{g_2}$$ For the two-axle four-tire trucks, $g_1 = 3\ \%$ for $t_1 = 5$ years then $g_2 = 4\ \%$ for the remaining $t_2 = 20$ years: $$GF_{\text{4-tire}} = \frac{1.03^{5}-1}{0.03} + 1.03^{4}(1.04)\frac{1.04^{20}-1}{0.04} = 5.3091 + 34.856 = 40.165$$ The middle factor $1.03^{4}(1.04) = 1.1706$ is the continuity term. Dropping it — that is, restarting the second stage at the opening-year volume — would give 35.1 and understate this class by 13 %. As a check on the result, a flat 4 % for all 25 years would give 41.646, and the two-stage value is 96.4 % of that, which is the right side of it and the right order.
Compute the passenger-car factor for completeness. With $g_1 = 2\ \%$ for two years then $6\ \%$ for 23, $$GF_{\text{cars}} = \frac{1.02^{2}-1}{0.02} + 1.02(1.06)\frac{1.06^{23}-1}{0.06} = 2.020 + 50.812 = 52.832$$ Since $TF = 0$ for cars, this factor contributes nothing to the design ESAL; it is reported only to show the class was considered.

Assemble the design ESAL. Multiplying each class's first-year ESAL by its own growth factor:

Cumulative design-lane ESAL by vehicle class over 25 years
Class	First-year ESAL	Growth factor	Cumulative ESAL
Passenger cars	0	52.832	0
Single unit, two-axle, four-tire	5,174	40.165	207,810
Single unit, two-axle, six-tire	26,157	36.459	953,658
Single unit, six-axle or more	63,811	36.459	2,326,507
Total	—	—	3,487,975

$$\boxed{W_{18} = 3.49\times10^{6}\ \text{18-kip ESAL applications in the design lane}}$$

Sanity-check the answer against the structure it will produce. Three and a half million ESALs sits in the appendix's "2,000,001–7,000,000" minimum-thickness band, calling for at least 3.5 in. of asphalt concrete over a 6 in. aggregate base — a plausible new urban freeway. The heaviest 3 % of the traffic contributes 67 % of the total damage, and the 78 % that is passenger cars contributes essentially none. That concentration is the practical message: the accuracy of a pavement design rests almost entirely on the truck classification count, and refining the passenger-car growth forecast — the one growth rate the question makes most elaborate — changes nothing at all.

Question 5 — final results
Quantity	Value
Design lane factor	$DF = 0.50 \times 0.90 = 0.45$
Truck factors (urban, other freeways)	0.015 / 0.13 / 0.74
First-year design-lane ESAL, four-tire / six-tire / six-axle+	5,174 / 26,157 / 63,811
Growth factor, two-axle four-tire (3 % then 4 %)	40.165
Growth factor, classes at a steady 3 %	36.459
Growth factor, passenger cars (2 % then 6 %)	52.832
Cumulative ESAL by class	207,810 / 953,658 / 2,326,507
Design ESAL over 25 years	$3.49\times10^{6}$
Share carried by the heaviest 3 % of vehicles	67 %

Topic. Converting mixed traffic into equivalent single-axle loads with staged growth.

Key relations. $ESAL = AADT \times HVP \times DF \times TF \times TDY$; single-stage growth factor $[(1+g)^{t}-1]/g$; two-stage growth factor with the continuity term $(1+g_1)^{t_1-1}(1+g_2)$; truck factors from Table IV-5 by functional class.

Why this works. Pavement damage does not scale with traffic volume; it scales roughly with the fourth power of axle load, which is exactly what a truck factor encapsulates — the number of 18-kip single-axle passes that one vehicle of a given class is worth. Once each class carries its own truck factor, the mixed fleet collapses to a single damage count that the design equations can use. The growth factor is simply the sum of a geometric series, and the continuity term is what keeps that series unbroken when the rate changes: the second stage is a new geometric series, but its first term is the last volume the first series reached, not the volume in year one.

Common pitfall. Applying the two-stage formula without the continuity term, which restarts the second stage at the opening-year volume and understates the class by low double-digit percentages. The second common error is forgetting that AADT is two-way, so the 0.50 directional split must be applied in addition to the lane-distribution factor; omitting it doubles the design ESAL.

Source. AASHTO, Guide for Design of Pavement Structures (1993), Part I Ch. 2 and Appendix D; Asphalt Institute MS-1, Ch. IV (Table IV-5 truck factors); Garber and Hoel, 5th ed., Ch. 20.

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