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

Question 7 of 7: AASHTO-93 Rigid Pavement Thickness Design

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 7: AASHTO-93 Rigid Pavement Thickness 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 jointed plain concrete pavement on a granular subbase, with a full set of AASHTO-93 design inputs: traffic and its growth, reliability and its standard deviation, the serviceability window, the concrete properties, the foundation stiffness and a qualitative drainage description.

Given data and the design coefficients they imply
Input	As given	Value used in the AASHTO equation
Analysis period, n	2 years	2
First-year traffic	150 000 ESAL	150 000
Traffic growth rate, g	4 %	0.04
Reliability, R	95 %	Z_R = −1.645
Overall standard deviation, S₀	0.4	0.40
Serviceability, p_i to p_t	4.5 to 2.5	ΔPSI = 2.0
Concrete compressive strength	35 MPa	E_c = 4.06 × 10⁶ psi
Modulus of rupture, S′_c	4.0 MPa	580 psi
Modulus of subgrade reaction, k	1.0 MPa (read as MPa/m)	3.68 pci
Drainage	fair, saturated 10 % of the time	C_d = 0.95
Load transfer	dowelled joints, untied asphalt shoulders	J = 3.2

Find. The required thickness of the concrete slab, and the thickness to be adopted for construction.

Figure 7.1 — Cross-section of the designed pavement. The asphalt shoulder is untied, which is what fixes the AASHTO load-transfer coefficient at J = 3.2.

Approach. Accumulate the two-year traffic with the growth-factor series to get the design ESAL, translate every material and site input into the units the 1993 AASHTO rigid equation expects, and then solve that equation for the slab thickness D by trial, since D cannot be isolated algebraically.

Design traffic. Traffic grows geometrically, so the total repetitions over the analysis period are the first-year figure times the growth factor $$\text{GF}=\frac{(1+g)^n-1}{g}=\frac{(1.04)^2-1}{0.04}=2.040$$ $$W_{18}=150\,000\times 2.040=\boxed{306\,000\ \text{ESAL}}$$ The stated 150 000 ESAL is taken to be the design-lane figure already; treating it as a two-way total and applying the usual directional and lane factors (D_D = 0.5, D_L = 0.9) would give 137 700 ESAL instead, and that sensitivity is revisited in step 7.
Concrete properties in AASHTO units. The 1993 equation is written in US customary units, so $$S'_c=4.0\ \text{MPa}\times 145.04=580\ \text{psi}$$ $$E_c=57\,000\sqrt{f'_c}=57\,000\sqrt{35\times 145.04}=57\,000\sqrt{5\,076} =4.06\times 10^6\ \text{psi}$$
Foundation stiffness. Reading the given “1.0 MPa” as a modulus of subgrade reaction of 1.0 MPa/m, $$k=\frac{1.0\ \text{MPa/m}}{0.2714\ \text{MPa/m per pci}}=3.68\ \text{pci}$$
Reliability, serviceability and the coefficients. At 95 % reliability the standard normal deviate is $Z_R=-1.645$, and with $S_0=0.40$ the reliability term is $Z_RS_0=-0.658$. The serviceability loss is $\Delta\text{PSI}=4.5-2.5=2.0$. For fair drainage with saturation reached 10 % of the time, AASHTO Table 2.5 gives $C_d$ between 1.00 and 0.90, and the mid-range value $C_d=0.95$ is adopted. For a dowelled jointed plain pavement whose shoulder is asphalt and therefore not tied to the slab, Table 2.6 gives $J=3.2$.
Set up the AASHTO-93 rigid design equation. $$\log_{10}W_{18}=Z_RS_0+7.35\log_{10}(D+1)-0.06 +\frac{\log_{10}\!\left(\dfrac{\Delta \text{PSI}}{4.5-1.5}\right)} {1+\dfrac{1.624\times 10^7}{(D+1)^{8.46}}}$$ $$+\,(4.22-0.32p_t)\log_{10} \!\left[\frac{S'_c\,C_d\left(D^{0.75}-1.132\right)} {215.63\,J\left(D^{0.75}-\dfrac{18.42}{(E_c/k)^{0.25}}\right)}\right]$$ with $D$ in inches. The required $\log_{10}W_{18}$ is $\log_{10}(306\,000)=5.486$.
Solve for D by trial. Substituting successive trial thicknesses: $$\begin{aligned} D=7.0\ \text{in}:&\quad \log_{10}W_{18}=5.215\ \ (163\,900\ \text{ESAL, too thin})\\ D=7.5\ \text{in}:&\quad \log_{10}W_{18}=5.408\ \ (255\,700\ \text{ESAL, still short})\\ D=8.0\ \text{in}:&\quad \log_{10}W_{18}=5.592\ \ (391\,200\ \text{ESAL, ample}) \end{aligned}$$ Interpolating between the last two and refining gives $$D=7.71\ \text{in}=\boxed{196\ \text{mm required}}$$
Adopt a construction thickness and test the result. Concrete pavement is specified in convenient increments, so round up to $$D=\boxed{200\ \text{mm}\ (8\ \text{in})}$$ which also satisfies the practical minimum for a dowelled highway slab. Two sensitivities confirm that this choice is robust. If the 150 000 ESAL is instead a two-way figure, the design-lane traffic drops to 137 700 and the requirement falls to 173 mm — still below the adopted 200 mm. If the quoted subgrade stiffness is unrepresentative and a more usual granular-subbase effective value of about 54 MPa/m (200 pci) is used, the requirement falls to 164 mm. In every reading of the data, 200 mm governs.

Check — two features of the given data worth flagging on the answer sheet. First, a design life of 2 years is far shorter than the 20 to 40 years a concrete pavement is normally analysed over; the arithmetic is carried out exactly as printed, but the same inputs over a 20-year period would accumulate 4.47 million ESAL and require about 290 mm. Second, “modulus of subgrade reaction = 1.0 MPa” is dimensionally incomplete, and read literally as 1.0 MPa/m it is roughly an order of magnitude below anything measured on a real prepared subgrade. Both are reported here as assumptions, with the sensitivity of the answer quantified above, which is what the paper’s own Note 1 asks a candidate to do.

It is worth seeing why the very low k barely moves the answer. The foundation enters the design equation only through the term 18.42/(E_c/k)^0.25, and a fourth root is an exceedingly weak function — increasing k from 3.7 to 200 pci, a factor of more than fifty, changes that term from 0.57 to 1.54 and the required thickness by only about 30 mm. That insensitivity is a genuine and often misunderstood property of rigid pavements: the slab, not the soil beneath it, carries the load by bending, which is exactly the opposite of the situation in a flexible pavement, where subgrade stiffness dominates the design.

Quantity	Value
Growth factor over 2 years at 4 %	2.040
Design traffic, W₁₈	306 000 ESAL
Concrete elastic modulus, E_c	4.06 × 10⁶ psi (28 000 MPa)
Modulus of rupture, S′_c	580 psi (4.0 MPa)
Effective k	3.68 pci (1.0 MPa/m as given)
Drainage coefficient, C_d	0.95
Load transfer coefficient, J	3.2
Required slab thickness	D = 7.71 in = 196 mm
Adopted slab thickness	D = 200 mm (8 in)

Concept & Theory

Topic: Empirical thickness design of jointed plain concrete pavement by the 1993 AASHTO Guide.

Key relations: Growth factor GF = ((1+g)ⁿ−1)/g; W₁₈ = (first-year ESAL) × GF × D_D × D_L; Z_R = −1.645 at 95 % reliability; E_c = 57 000√f′_c (psi); k in pci = (k in MPa/m)/0.2714; the AASHTO-93 rigid equation solved for D by trial; C_d from Table 2.5 and J from Table 2.6.

Why this works: The 1993 equation is a regression fitted to the AASHO Road Test, in which the observed loss of serviceability of instrumented slabs was correlated with axle repetitions, slab thickness, concrete strength and foundation support. Each term has a recognisable role: 7.35 log(D+1) is the direct thickness effect; the serviceability term scales the answer to how much ride quality the owner is willing to lose; the final logarithmic term is a stress ratio, comparing the flexural capacity of the slab (S′_c) against the Westergaard edge stress it must resist, which is why J, C_d, E_c and k all appear inside it; and Z_RS₀ shifts the whole design to a conservative percentile of the scatter observed in the test.

Common pitfall: Entering the compressive strength where the equation wants the modulus of rupture — the two differ by roughly a factor of nine and the error is silent because the equation still returns a number. Almost as common is mixing unit systems: S′_c must be in psi, E_c in psi and k in pci, and a metric k left in MPa/m understates the foundation by a factor of about 3.7. Finally, forgetting that AASHTO’s J penalises an untied shoulder (3.2 rather than 2.7 for a tied concrete shoulder) costs about 17 mm of slab here.

Source: AASHTO, Guide for Design of Pavement Structures (1993), Part II, Ch. 3 (rigid pavement design), Tables 2.5 and 2.6; N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Ch. 20; Transportation Association of Canada, Pavement Asset Design and Management Guide (Canadian practice for rigid pavement design inputs).

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