Design ESAL and AASHTO flexible pavement design: 16-Civ-A6 Highway Design, Construction, and Maintenance, May 2018, Q2

Question 2: Design ESAL and AASHTO flexible pavement design (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. A six-lane urban freeway carrying 21,000 veh/day in its opening year, with a vehicle mix, two growth regimes and a full set of material moduli:

Traffic and material data
Quantity	Value
First-year AADT (both directions)	21,000 veh/day
Passenger cars / 2-axle 4-tire / 2-axle 6-tire / multi-axle single units	83 % / 10 % / 5 % / 2 %
Growth, 2-axle 4-tire single units	5 %/yr for years 1–5, then 6 %/yr
Growth, all other classes	4 %/yr throughout
Design life	20 years
Subgrade resilient modulus $M_{R}$	15,000 psi
Base modulus $E_{B}$ / subbase modulus $E_{SB}$	27,000 psi / 20,000 psi
Asphalt concrete modulus $E_{AC}$	450,000 psi
Drainage	saturated 20 % of the time, drains in about 1 week
Serviceability	$p_{0}=4.5$, $p_{t}=2.5$

Find. (a) the cumulative 18-kip equivalent single-axle load applications in the design lane over 20 years, and (b) a layer-by-layer flexible pavement section whose structural number meets or exceeds the AASHTO requirement.

Approach. Convert each truck class to first-year ESALs with the Table IV-5 truck factors and a design-lane factor, grow each class with its own cumulative growth factor, then invert the 1993 AASHTO flexible design equation for the required structural number and distribute it over the layers by the standard layered-analysis procedure.

(a) Design ESAL (8 marks)

Select truck factors. The facility is an urban freeway that is not stated to be an Interstate, so the “Urban Systems — Other Freeways” column of Table IV-5 governs: $TF=0.015$ for two-axle four-tire single units, $0.13$ for two-axle six-tire single units and $0.74$ for single units with three or more axles (the heaviest single-unit row the table carries, which is the entry that applies to the six-axle-or-more class). Passenger cars have no row in the table and a truck factor of order 0.0004; they are neglected, which changes the total by well under one per cent.
Design-lane factor. AADT is a two-way total, so half of it travels in the design direction, and with three lanes per direction Garber & Hoel recommend 0.60–0.80 of the directional trucks in the design lane. Taking the mid-range 0.70,$$DF=D\times L=0.50\times 0.70=0.35$$
First-year ESALs by class. With $\text{ESAL}=AADT\times HVP\times DF\times TF\times 365$,

First-year ESAL applications in the design lane
Class	Share	Truck factor	First-year ESAL
Single unit, 2-axle 4-tire	10 %	0.015	4,024
Single unit, 2-axle 6-tire	5 %	0.13	17,438
Single unit, 6-axle or more	2 %	0.74	39,705
Passenger cars	83 %	negligible	—

Cumulative growth factors. For a single rate the appendix gives $GF=[(1+g)^{t}-1]/g$. All classes except the two-axle four-tire trucks grow at 4 % for 20 years:$$GF_{4\%,20}=\dfrac{(1.04)^{20}-1}{0.04}=29.778$$The two-axle four-tire class needs a two-stage treatment. Its first five years accumulate $GF_{5\%,5}=[(1.05)^{5}-1]/0.05=5.526$ years of first-year volume, and it enters year 6 at $(1.05)^{4}(1.06)=1.2884$ times the first-year volume, after which fifteen years at 6 % accumulate$$1.2884\times\dfrac{(1.06)^{15}-1}{0.06}=1.2884\times 23.276=29.989$$so $GF_{\text{two-stage}}=5.526+29.989=35.514$.
Cumulative ESALs. Multiplying each class by its own growth factor,$$W_{18}=4\,024(35.514)+17\,438(29.778)+39\,705(29.778)$$$$W_{18}=142\,913+519\,269+1\,182\,262=\boxed{1.84\times 10^{6}\ \text{ESAL}}$$Almost two thirds of the damage (64 %) comes from the 2 % of the traffic stream that is heavy single units — the usual and instructive result.

(b) Flexible pavement design (12 marks)

The question invites assumptions, so each is stated with its justification before the design proper begins.

Check: design assumptions. Reliability — Table 20.16 recommends 85–99.9 % for urban freeways; because “traffic detours are difficult and expensive” a high value is warranted, and $R=95\ \%$ ($Z_{R}=-1.645$) is adopted. Overall standard deviation — the usual flexible-pavement range is 0.40–0.50, but the question states that the materials were laboratory-tested and the traffic data are accurate, so the lower end is defensible: $S_{0}=0.35$. Drainage — water removed in about one week is “Fair” quality in Table 2.4 and 20 % saturation falls in the 5–25 % column, giving $m=1.00$ to $0.80$; the mid-range $m_{2}=m_{3}=0.90$ is used. Layer coefficients — from Figure 2.5 at 450,000 psi, $a_{1}=0.44$; from Figure 2.6 at 27,000 psi, $a_{2}=0.13$; from Figure 2.7 at 20,000 psi, $a_{3}=0.14$ (not needed below, because no structural subbase is required).

Assemble the design inputs. The serviceability loss is $\Delta PSI=p_{0}-p_{t}=4.5-2.5=2.0$, and the design equation reproduced on appendix pages 1 and 4 is$$\log_{10}W_{18}=Z_{R}S_{0}+9.36\log_{10}(SN+1)-0.20+\dfrac{\log_{10}\!\left(\frac{\Delta PSI}{4.2-1.5}\right)}{0.40+\frac{1094}{(SN+1)^{5.19}}}+2.32\log_{10}M_{R}-8.07$$As a check on the algebra, the worked example printed beside the nomograph ($W_{18}=5\times 10^{6}$, $R=95\ \%$, $S_{0}=0.35$, $M_{R}=5\,000$ psi, $\Delta PSI=1.9$) reproduces $SN=5.0$ from this equation.
Structural number required over the subgrade. Solving the equation for $SN$ with $W_{18}=1.84\times 10^{6}$ and $M_{R}=15\,000$ psi gives$$\boxed{SN_{\text{req}}=2.89}$$
Structural numbers required over each lower layer. The layered procedure repeats the same solution using the modulus of the material immediately beneath the layer being designed:$$SN_{1}=2.32\ \ (\text{over the base},\ E_{B}=27\,000\ \text{psi}),\qquad SN_{2}=2.60\ \ (\text{over the subbase},\ E_{SB}=20\,000\ \text{psi})$$
Asphalt concrete thickness. The surface course must by itself deliver the structural number required over the base:$$D_{1}\ge\dfrac{SN_{1}}{a_{1}}=\dfrac{2.32}{0.44}=5.28\ \text{in.}\quad\Longrightarrow\quad D_{1}=5.5\ \text{in.}\ (140\ \text{mm})$$which also satisfies the 3.0 in. minimum that the appendix minimum-thickness table sets for 500,001–2,000,000 ESAL. The structural number actually provided is $SN_{1}^{*}=0.44(5.5)=2.42$.
Granular base thickness. The base makes up the difference to the requirement over the subbase:$$D_{2}\ge\dfrac{SN_{2}-SN_{1}^{*}}{a_{2}m_{2}}=\dfrac{2.60-2.42}{0.13\times 0.90}=1.5\ \text{in.}$$That is far thinner than the appendix minimum of 6 in. of aggregate base for traffic between 500,001 and 2,000,000 ESAL, so construction practice governs and $D_{2}=6\ \text{in.}\ (150\ \text{mm})$.
Is a subbase required? With the minimum base in place the section already delivers$$SN^{*}=a_{1}D_{1}+a_{2}m_{2}D_{2}=0.44(5.5)+0.13(0.90)(6.0)=\boxed{3.12}$$against $SN_{\text{req}}=2.89$. The requirement is met with a surplus of $0.24$, so no structural subbase is needed: $D_{3}=0$. A 150 mm granular subbase may still be specified as a working platform, a drainage blanket and frost protection, and any such layer is a bonus on top of the structural design.
Confirm the capacity of the section specified. Substituting $SN=3.12$ back into the design equation returns an allowable $W_{18}=3.03\times 10^{6}$, i.e. about 1.6 times the design traffic — roughly 27 years of service at the assumed growth rates rather than 20.

Figure 2.1 — the design section: 140 mm of asphalt concrete over 150 mm of untreated granular base.

Question 2 — results
Quantity	Value
Design-lane factor	0.35 (0.50 directional × 0.70 lane)
Cumulative growth factor, 2-axle 4-tire	35.51
Cumulative growth factor, all other classes	29.78
Design ESAL, 20 years	1.84 × 10⁶
Design inputs	$R=95\ \%$, $Z_{R}=-1.645$, $S_{0}=0.35$, $\Delta PSI=2.0$
Structural number required	2.89
Asphalt concrete	5.5 in. (140 mm), $a_1=0.44$
Granular base	6.0 in. (150 mm), $a_2=0.13$, $m_2=0.90$
Granular subbase	not required structurally
Structural number provided	3.12 (allowable 3.03 × 10⁶ ESAL)

Topic. Traffic loading in equivalent single-axle loads, and the 1993 AASHTO structural-number method for flexible pavements.

Key relations. $\text{ESAL}=AADT\times HVP\times DF\times TF\times 365$; cumulative $=\text{ESAL}_{1}\left[(1+g)^{t}-1\right]/g$; structural number $SN=a_{1}D_{1}+a_{2}D_{2}m_{2}+a_{3}D_{3}m_{3}$; and the design equation quoted in step 1.

Why this works. The whole method rests on collapsing a mixed traffic stream onto a single damage scale. A truck factor is the number of 18-kip single-axle passes that would do the same damage as one pass of that vehicle, so multiplying volume by truck factor converts vehicles into a currency the pavement equation understands. Growth is handled as a geometric series, and when the rate changes part way through the analysis period the series must be split and the second stage started from the volume the first stage has already reached — that continuity factor of 1.2884 is where most errors occur. On the resistance side, the structural number is a weighted thickness in which each layer contributes in proportion to its stiffness through the layer coefficient and, for unbound layers, its drainage condition through the modifier $m$. The layered procedure works from the top down because each layer must protect the one below it: the asphalt alone has to satisfy the requirement computed with the base modulus, the asphalt plus base the requirement computed with the subbase modulus, and so on. Minimum thickness tables then override the arithmetic wherever it produces a layer too thin to place and compact.

Common pitfall. Applying one growth factor to the whole stream when the question deliberately gives one class a two-stage rate, and forgetting the design-lane factor entirely so the ESAL is nearly three times too large. A third trap is running the layered analysis with the subgrade modulus for every layer, which over-designs the asphalt and under-designs the granular layers.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., §20.3–20.5; AASHTO Guide for Design of Pavement Structures (1993), Part II Ch. 2 and Appendix D.

Question 2 of 7: Design ESAL and AASHTO flexible pavement design

Question 2: Design ESAL and AASHTO flexible pavement design (20 marks)

(a) Design ESAL (8 marks)

(b) Flexible pavement design (12 marks)