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

Question 6 of 7

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 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 printed on page 1 confirms 20 marks per question, split as: Q1 20; Q2 20; Q3 (a) 15 and (b) 5; Q4 (a) 8 and (b) 12; Q5 (a) 8 and (b) 12; Q6 20; Q7 (a) 8 and (b) 12. All seven printed questions are worked below, 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 required but not given may be assumed and that assumptions should be recorded with the answer — several questions need that licence, and every assumption is flagged where it is made.

Reference texts. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed. (sight distance, vertical and horizontal alignment, traffic stream models, earthwork); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (Canadian design-domain values for stopping sight distance, perception-reaction time and deceleration); AASHTO, A Policy on Geometric Design of Highways and Streets (the tabulated metric stopping sight distances); AASHTO, Guide for Design of Pavement Structures (1993) (rigid pavement thickness, reliability, drainage and load-transfer coefficients); Asphalt Institute, Mix Design Methods MS-2 (gradation charts, the 0.45 power chart, aggregate blending); M.S. Mamlouk and J.P. Zaniewski, Materials for Civil and Construction Engineers, 4th ed. (aggregate moisture states, sieve analysis); Transportation Association of Canada, Pavement Asset Design and Management Guide (Canadian pavement design practice).

Check — assumptions carried through this paper. Four inputs the exam does not supply are assumed under its own Note 2 (“any data required, but not given, can be assumed”), and each is restated at the point of use: (i) Question 2 needs a stopping-sight-distance basis — a 2.5 s perception-reaction time and a 3.4 m/s² deceleration, the TAC and AASHTO design values, giving the tabulated 185 m at 100 km/h; (ii) Question 2 also needs to know whether the 600 m radius is to the road centreline — it is taken as the centreline, and Step 5 shows the alternative reading changes the answer by 0.02 m; (iii) Question 6 does not say whether the transverse joints are dowelled — dowels are assumed, giving a load-transfer coefficient J = 3.2, with the undowelled case quantified in a callout; (iv) Question 6 gives a drainage description rather than a coefficient, so C_d = 1.00 is read from the AASHTO table, again with the alternative quantified.

Question 6 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 full set of AASHTO 1993 rigid-pavement design inputs, in metric units, for a jointed plain concrete pavement with flexible (asphalt) shoulders:

Given data — AASHTO 1993 rigid pavement design
Input	Symbol	As given (metric)	Converted (US customary)
Design traffic	$W_{18}$</td><td>$5 \times 10^6$ ESALs</td><td>$5 \times 10^6$
Modulus of subgrade reaction	$k$	30 MPa/m	110.5 pci
Subgrade resilient modulus	$M_R$	150 MPa	21 756 psi
Design reliability	$R$</td><td>95%</td><td>$Z_R = -1.645$
Overall standard deviation	$S_0$	0.50	0.50
Modulus of rupture of concrete	$S'_c$	2.5 MPa	362.6 psi
Modulus of elasticity of concrete	$E_c$</td><td>30 000 MPa</td><td>$4.351 \times 10^6$ psi
Initial serviceability	$p_i$	4.5	4.5
Terminal serviceability	$p_t$	2.0	2.0
Drainage (water drains in two days, saturated < 5% of the time)	$C_d$	1.00 (fair drainage, 1–5% column)
Load transfer (JPCP, asphalt shoulders, dowelled joints assumed)	$J$	3.2

Find. The required thickness $D$ of the concrete slab, and a construction thickness to adopt.

Jointed plain concrete pavement with asphalt shoulders: the design inputs and the resulting slab thickness.

Approach. The AASHTO 1993 rigid design equation is an empirical regression fitted to the AASHO Road Test in US customary units, so every input is first converted to those units; the drainage and load-transfer coefficients are then read from the AASHTO tables using the verbal descriptions given, and the equation is solved iteratively for $D$, which appears on both sides.

Convert the material and support properties into the units the equation requires. Using $1$ MPa $= 145.038$ psi and $1$ MPa/m $= 3.6839$ pci, $$k = 30 \times 3.6839 = 110.5\text{ pci}, \qquad S'_c = 2.5 \times 145.038 = 362.6\text{ psi}$$ $$E_c = 30\,000 \times 145.038 = 4.351 \times 10^6\text{ psi}$$ The subgrade resilient modulus of 150 MPa (21 756 psi) is not an input to the rigid equation; it would be used to estimate $k$ if $k$ were not supplied, and it is supplied here directly.
Select the drainage coefficient from the verbal description. The AASHTO drainage-quality classes are defined by the time taken to remove water: two hours is excellent, one day good, one week fair, one month poor. Water that drains “in two days” falls between good and fair, and the conservative reading is fair. The pavement is saturated less than 5% of the time, which places it in the 1–5% column, for which fair drainage gives $C_d$ between 1.00 and 1.10. Take $$C_d = 1.00$$ and treat 1.10 as the optimistic bound, quantified in the callout below.
Select the load-transfer coefficient. AASHTO tabulates $J$ by shoulder type and by whether load-transfer devices are present. For a jointed plain concrete pavement with asphalt (untied, flexible) shoulders, $J = 3.2$ with dowels at the transverse joints and $J = 3.8$ to $4.4$ without. At $5 \times 10^6$ ESALs a dowelled joint is standard practice, so $$J = 3.2$$
Compute the serviceability loss and the reliability term. $$\Delta PSI = p_i - p_t = 4.5 - 2.0 = 2.5, \qquad Z_R S_0 = (-1.645)(0.50) = -0.8225$$
Write the design equation and evaluate the terms that do not depend on $D$. The AASHTO 1993 rigid equation is $$\log_{10}W_{18} = Z_RS_0 + 7.35\log_{10}(D+1) - 0.06 + \frac{\log_{10}\left[\Delta PSI/(4.5-1.5)\right]}{1 + \dfrac{1.624 \times 10^7}{(D+1)^{8.46}}} + (4.22 - 0.32p_t)\log_{10}\!\left[\frac{S'_cC_d\left(D^{0.75}-1.132\right)} {215.63\,J\left(D^{0.75} - \dfrac{18.42}{(E_c/k)^{0.25}}\right)}\right]$$ with $D$ in inches. The support ratio is $$\frac{E_c}{k} = \frac{4.351 \times 10^6}{110.5} = 39\,370, \qquad \left(\frac{E_c}{k}\right)^{0.25} = 14.086, \qquad \frac{18.42}{14.086} = 1.3077$$ and the traffic requirement is $\log_{10}(5 \times 10^6) = 6.6990$.
Solve iteratively for the slab thickness. Because $D$ appears in three separate terms the equation is solved by trial. Trying $D = 13$ in returns $\log_{10}W_{18} = 6.510$, short of the 6.699 required; $D = 14$ in returns 6.727, slightly more than required. Interpolating and refining, $$D = 13.86\text{ in} = \boxed{352\text{ mm}}$$ at which the right-hand side reproduces 6.699 exactly.
Adopt a construction thickness. Slabs are specified in round increments, and the sensitivity study below shows the plausible range of inputs spans roughly 335 mm to 416 mm. Adopt $$D_{adopted} = \boxed{355\text{ mm}}$$ which is the next 5 mm increment above the computed value and corresponds to the familiar 14-inch slab. If the joints cannot be confirmed as dowelled, the design must instead be built at 390 mm or more.

Check — how sensitive is 352 mm to the judgement calls? Four inputs are either assumed or unusual, and each was re-solved to bound the answer. (i) Load transfer: undowelled joints, $J = 3.8$ to 4.4, give 386 mm to 416 mm — by far the largest single influence, and the reason the dowel assumption is stated explicitly. (ii) Drainage: taking the optimistic $C_d = 1.10$ instead of 1.00 gives 335 mm, a saving of only 17 mm. (iii) The standard error: the paper specifies $S_0 = 0.50$, which is the value AASHTO recommends for flexible pavements; the rigid range is 0.30 to 0.40, and $S_0 = 0.35$ would give 324 mm. The printed 0.50 is used, and it is conservative. (iv) The modulus of rupture: 2.5 MPa is low for paving concrete, where 4.0 to 4.5 MPa at 28 days is normal; a 4.0 MPa concrete would need only 274 mm. The slab is thick chiefly because the concrete specified is weak in flexure, and the single most cost-effective change available to the designer is to specify a higher flexural strength rather than more thickness.

Final results — Question 6
Quantity	Symbol	Value
Modulus of subgrade reaction, converted	$k$	110.5 pci
Modulus of rupture, converted	$S'_c$	362.6 psi
Concrete modulus, converted	$E_c$</td><td>$4.351 \times 10^6$ psi
Drainage coefficient (fair, 1–5% saturated)	$C_d$	1.00
Load transfer coefficient (asphalt shoulders, dowelled)	$J$	3.2
Reliability term	$Z_RS_0$</td><td>$-0.8225$
Serviceability loss	$\Delta PSI$	2.5
Computed slab thickness	$D$	13.86 in = 352 mm
Adopted slab thickness	—	355 mm

Topic. AASHTO 1993 rigid pavement thickness design for a jointed plain concrete pavement.

Key relations. The design equation quoted in Step 5; the reliability term $Z_RS_0$ with $Z_R = -1.645$ at 95%; $\Delta PSI = p_i - p_t$; the support ratio $(E_c/k)^{0.25}$ entering only through $18.42/(E_c/k)^{0.25}$; the unit conversions 1 MPa = 145.038 psi and 1 MPa/m = 3.6839 pci; and the tabulated coefficients $C_d$ for drainage and $J$ for load transfer.

Why this works. The equation is not derived from mechanics; it is a regression fitted to slab performance observed at the AASHO Road Test and extended by theory. Read structurally, it says that the traffic a slab can carry before its serviceability falls from $p_i$ to $p_t$ rises steeply with thickness (the $7.35\log(D+1)$ term), is scaled by the ratio of the concrete’s flexural strength to the stress the load induces (the final logarithmic term), and is penalised by uncertainty through $Z_RS_0$. Notice how weakly the foundation enters: $k$ appears only inside $18.42/(E_c/k)^{0.25}$, so multiplying $k$ by a factor of ten changes that term by well under a factor of two. Slab thickness in rigid design is governed by concrete flexural strength and by joint load transfer, not by subgrade stiffness — which is the opposite of the flexible-pavement intuition.

Common pitfall. The equation is valid only in US customary units; substituting MPa and MPa/m directly is the single most common failure, and it produces an answer that looks plausible. Second, $Z_R$ is negative for reliability above 50%; a positive 1.645 would make the pavement thinner as reliability increased. Third, the subgrade resilient modulus is not an input when $k$ is given — converting it with $k = M_R/19.4$ would return 1121 pci here, an order of magnitude above the stated value, because that relation returns a composite $k$ on top of a subbase. Fourth, $\Delta PSI$ is divided by $(4.5 - 1.5) = 3.0$ in the numerator of the serviceability term regardless of the actual $p_i$; that 4.5 and 1.5 are fixed constants of the regression, not the design values.

Source. AASHTO, Guide for Design of Pavement Structures (1993), Part II Chapter 3 (rigid pavement design), including Table 2.5 for drainage coefficients and Table 2.6 for load transfer; Transportation Association of Canada, Pavement Asset Design and Management Guide, for current Canadian practice; N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Chapter 20.

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