PE Civil — Transportation Formula Sheet

Project Management
9% of the exam

Quantity take-off and cost estimating, CPM schedules with activity sequencing and float, and engineering-economic comparison of alternatives (present worth, EUAC, lifecycle, benefit-cost).

Engineering Economics & Benefit-Cost

Move cash flows to one date with the interest-factor family, compare alternatives by present worth and EUAC, and justify a project with the benefit-cost ratio.

Single payment (F/P, P/F): $F = P(1+i)^n, \qquad P = F(1+i)^{-n}$; Grow ( $F/P$ ) or discount ( $P/F$ ) a lump sum $n$ periods at rate $i$ . $P$ , $F$ in dollars, $i$ per period, $n$ periods.
Uniform series present worth (P/A): $(P/A,i,n) = \dfrac{(1+i)^n - 1}{i\,(1+i)^n}$; Present worth of a level annual amount $A$ over $n$ periods. Inverse is capital recovery $(A/P)$ .
Capital recovery factor (A/P): $(A/P,i,n) = \dfrac{i\,(1+i)^n}{(1+i)^n - 1}$; Converts a present cost $P$ into a level annual amount; the core of every EUAC.
Sinking-fund factor (A/F): $(A/F,i,n) = \dfrac{i}{(1+i)^n - 1}$; Converts a future amount (e.g., salvage) into a level annual amount; credited in EUAC.
Uniform gradient present worth (P/G): $(P/G,i,n) = \dfrac{(1+i)^n - 1}{i^2 (1+i)^n} - \dfrac{n}{i\,(1+i)^n}$; Present worth of a series that increases by a constant $G$ each period (first cash flow at end of period 2). $G$ in dollars/period.
Net present worth: $\text{PW} = -P_0 + \sum_{t=1}^{n}\dfrac{A_t}{(1+i)^t} + \dfrac{S}{(1+i)^n}$; Discount all cash flows to time zero; accept if $\text{PW} \ge 0$ , choose the largest among net-benefit alternatives over equal lives.
Equivalent uniform annual cost: $\text{EUAC} = P\,(A/P,i,n) + A_{\text{O\&M}} - S\,(A/F,i,n)$; Annualized total cost; compare alternatives of any lives by lowest EUAC. $S$ = salvage at end of life $n$ .
Benefit-cost ratio: $\dfrac{B}{C} = \dfrac{\text{PW or AW of benefits}}{\text{PW or AW of costs}} \ge 1$; A single project is justified when $\ge 1$ ; for mutually exclusive alternatives use incremental $\Delta B/\Delta C$ .
Rate of return condition: $\text{PW}(i^*) = 0, \quad \text{accept if } i^* \ge \text{MARR}$; $i^*$ zeroes present worth; compare to the minimum attractive rate of return.
Capitalized cost (perpetual): $P = \dfrac{A}{i}$; Present worth of a perpetual annual cost $A$ (infinite life), e.g., permanent corridor maintenance.
Effective interest rate: $i_e = \left(1 + \dfrac{r}{m}\right)^{m} - 1$; Convert nominal annual rate $r$ with $m$ compounding periods/yr to an effective annual rate before using annual factors (handbook $\S1.7.2$ ).
Inflation-adjusted rate: $d = i + f + i\,f$; Combined rate when cash flows are in actual (then-current) dollars and inflation $f$ applies (handbook $\S1.7.4$ ).

Where it lives: NCEES PE Civil Reference Handbook — §1.7 Engineering Economics · NCEES PE Civil Reference Handbook — §1.7.9 Benefit-Cost Analysis · Newnan, Lavelle & Eschenbach, Engineering Economic Analysis · AASHTO, User and Non-User Benefit Analysis for Highways (the 'Red Book')

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Traffic Capacity & LOS
9% of the exam

Uninterrupted and interrupted flow: peak-hour factor, flow rate, density and speed relationships, freeway/segment level of service, and signalized-intersection and roundabout capacity.

Flow, Speed, Density & the Peak-Hour Factor

The fundamental q = k·v relation, the Greenshields speed-density line, jam density and capacity, and the peak-hour factor that scales an hourly volume to a 15-minute flow rate.

Fundamental relation: $q = k\,u_s$; Flow $q$ (veh/h) equals density $k$ (veh/mi) times space-mean speed $u_s$ (mph). The backbone of all uninterrupted-flow analysis.
Density from a detector: $k = \dfrac{q}{u_s}$; Density inferred from measured flow and space-mean speed; vehicles per mile per lane (or per roadway).
Equivalent hourly flow: $q = \dfrac{3600\,n}{T}$; Converts $n$ vehicles counted in $T$ seconds to an equivalent hourly rate (vph). Same identity scales a 15-min count by 4.
Space-mean speed: $u_s = \dfrac{nL}{\sum t_i}$; Harmonic-type average: section length $L$ over mean travel time. The only speed valid in $q=k\,u_s$ .
Greenshields speed-density: $u_s = u_f\left(1 - \dfrac{k}{k_j}\right)$; Linear model; $u_f$ = free-flow speed (mph), $k_j$ = jam density (veh/mi). Speed is zero at jam.
Greenshields flow-density: $q = u_f\,k\left(1 - \dfrac{k}{k_j}\right)$; Parabola obtained by substituting the speed-density line into $q=k\,u_s$ ; peak gives capacity.
Greenshields capacity: $q_{max} = \dfrac{u_f\,k_j}{4}$; Maximum flow of the Greenshields stream (vph), occurring at $k_{cap}=k_j/2$ and $u_{cap}=u_f/2$ .
Optimum density and speed: $k_{cap} = \dfrac{k_j}{2}, \qquad u_{cap} = \dfrac{u_f}{2}$; Density and speed at capacity. Capacity is reached at half the free-flow speed, not at maximum speed.
Peak-hour factor: $PHF = \dfrac{V}{4\,V_{15}}$; Hourly volume $V$ over four times the peak 15-min count $V_{15}$ . Range 0.25–1.0; uniform hour = 1.0.
Peak flow rate: $v = \dfrac{V}{PHF}$; Equivalent hourly flow rate during the peak 15 minutes; the value used in capacity and LOS work.

Where it lives: NCEES PE Civil Reference Handbook — §5.1.1 Uninterrupted Flow (§5.1.1.2 Space Mean Speed, §5.1.1.4 Greenshields Maximum Flow Rate, $q_{max}=u_f k_j/4$ ) · NCEES PE Civil Reference Handbook — §5.1.2 Street Segment Interrupted Flow (§5.1.2.1 Speed-Density Model, §5.1.2.2 Flow-Density Model) · NCEES PE Civil Reference Handbook — §5.1.3 Traffic Analysis (§5.1.3.3 Peak-Hour Factor) · Garber & Hoel, Traffic and Highway Engineering, 5th ed.

Read the Traffic Capacity & LOS chapter

Traffic Analysis & Safety
6% of the exam

Volume and speed studies, trip generation and modal split, traffic forecasting, crash rates and severity, collision analysis, and Highway Safety Manual crash modification factors and predicted crashes.

Crash Rates & Collision Analysis

Normalize crashes by exposure (MEV at intersections, HMVM on segments), weight by severity with EPDO, and screen high-crash locations against a critical rate.

Intersection crash rate (per MEV): $R_{\text{MEV}} = \dfrac{C \times 10^{6}}{365\,T\,V}$; $C$ = crashes in study period, $T$ = years, $V$ = total entering AADT (vpd). Result in crashes per million entering vehicles.
Segment crash rate (per HMVM): $R_{\text{HMVM}} = \dfrac{C \times 10^{8}}{365\,T\,(\text{AADT})\,L}$; $L$ = segment length (mi). Result in crashes per hundred million vehicle-miles of travel.
Exposure — intersection (MEV): $M = \dfrac{365\,T\,V}{10^{6}}$; Million entering vehicles over the period; reused in the critical-rate term.
Exposure — segment (VMT): $\text{VMT} = 365\,T\,(\text{AADT})\,L$; Vehicle-miles of travel over the period; divide by $10^{8}$ for HMVM.
Equivalent property-damage-only index: $\text{EPDO} = w_F N_F + w_I N_I + w_P N_P$; $N$ = counts by severity (fatal/injury/PDO), $w$ = agency severity weights relative to PDO $=1$ .
Critical crash rate: $R_c = R_a + K\sqrt{\dfrac{R_a}{M}} + \dfrac{1}{2M}$; $R_a$ = average rate of similar sites, $M$ = site exposure (MEV or HMVM), $K=1.645$ at 95%. Flag if $R_{\text{site}}>R_c$ .
Crash frequency: $f = \dfrac{C}{T}$; Crashes per year — raw, un-normalized; use only to compare identical-exposure sites.
Severity index: $SI = \dfrac{\text{EPDO}}{C}$; Average equivalent-PDO weight per crash; higher means more severe crash mix.

Where it lives: AASHTO Highway Safety Manual, 1st Edition (HSM-1) — Part B, network screening and crash-rate/critical-rate methods · NCEES PE Civil Reference Handbook — §5.1.4 Accident Analysis (names accident rates and collision diagrams) · FHWA, Highway Safety Improvement Program (HSIP) Manual — site screening and EPDO weighting

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Roadside & Cross-Section Design
10% of the exam

Forgiving-roadside clear zone and recoverable slopes, barrier length-of-need and crash cushions, cross-section elements (lanes, shoulders, side slopes), and nonmotorized/ADA design.

Forgiving Roadside & Clear Zone

The forgiving-roadside philosophy, the clear-zone width from design speed, traffic volume and slope, recoverable vs non-recoverable slopes, and the horizontal-curve adjustment.

Clear-zone distance (chart): $L_{C} = f(V_{design},\,\text{ADT},\,\text{slope})$; Lateral recovery distance from the edge of the traveled way, read from the AASHTO RSDG-4 clear-zone chart on design speed, design-year ADT, and roadside slope (ft).
Horizontal-curve adjustment: $L_{C,\text{curve}} = K_{cz}\,L_{C}$; Outside of a horizontal curve only. $K_{cz}$ = curve correction factor (>1, from radius and design speed, ~1.1-1.5).
Recoverable foreslope: $\text{slope} \le 1V{:}4H$; Counted within the clear zone; driver can typically recover or stop.
Non-recoverable foreslope: $1V{:}4H < \text{slope} \le 1V{:}3H$; Steeper than $1V{:}4H$ but no steeper than $1V{:}3H$ (i.e. horizontal run $3 \le H < 4$ per unit vertical): traversable but the vehicle reaches the toe; not counted as recovery — provide a clear runout area at the toe.
Critical foreslope: $\text{slope} > 1V{:}3H$; Rollover risk; within the clear zone, flatten it or evaluate a barrier with the RSDG embankment warrant (fill height vs slope).
Slope ratio (H:V): $\text{slope ratio} = \dfrac{\Delta H}{\Delta V}$; A $1V{:}6H$ slope drops $1$ vertical per $6$ horizontal; horizontal projection of a slope of height $h$ is $(\text{ratio})\times h$ . Note the two readings of the same slope: use the $H{:}V$ projection multiplier (here $6$ ) to compute the horizontal width $\Delta H$ , but use the $V{:}H$ steepness value (here $1/6 \approx 0.167$ ) when classifying recoverable vs critical — do not interchange $6$ and $0.167$ .
Slope length: $L_{s} = \sqrt{(\Delta H)^2 + (\Delta V)^2}$; Actual surface length of a foreslope, vs its horizontal projection $\Delta H$ used in clear-zone measurement (ft).
Treatment hierarchy: $\text{remove} \to \text{redesign} \to \text{relocate} \to \text{breakaway} \to \text{shield} \to \text{delineate}$; RSDG-4 order of roadside hazard treatment; shielding is a late resort because a barrier is itself a hazard.

Where it lives: AASHTO Roadside Design Guide (RSDG-4), Ch. 3 — Roadside Topography and Drainage Features (clear zone, slopes, channels) · AASHTO Green Book — A Policy on Geometric Design of Highways and Streets (GDHS-7), roadside and cross-section design · FHWA — Clear Zone and Horizontal Clearance guidance

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Horizontal Design
11% of the exam

Circular-curve elements (T, L, M, E, LC, degree of curve) and PC/PT stationing, superelevation rate and side friction, superelevation runoff, minimum radius, sight-distance offset, and compound/reverse curves.

Circular Curve Geometry & Stationing

The radius, tangent, length, middle ordinate, external distance, chord, and degree of curve of a simple circular curve, and how they fix PC and PT stations.

Tangent length: $T = R\tan\dfrac{\Delta}{2}$; Distance from PI to PC (equals PI to PT). $R$ = radius (ft), $\Delta$ = central angle, $T$ in ft.
Curve (arc) length: $L = \dfrac{\pi}{180}R\Delta = \dfrac{100\,\Delta}{D}$; Length measured along the arc, PC to PT. Second form uses arc-definition degree of curve $D$ (deg).
Long chord: $C = 2R\sin\dfrac{\Delta}{2}$; Straight-line distance PC to PT (ft).
Middle ordinate: $M = R\!\left(1-\cos\dfrac{\Delta}{2}\right)$; Offset from midpoint of long chord to midpoint of arc (ft); the sightline-clearance ordinate.
External distance: $E = R\!\left(\sec\dfrac{\Delta}{2}-1\right)$; Distance from PI to midpoint of arc along the bisector (ft).
Degree of curve (arc): $R = \dfrac{5{,}729.58}{D}$; Arc definition: $D$ = central angle subtended by 100 ft of arc (deg); $R$ in ft.
Degree of curve (chord): $R = \dfrac{50}{\sin(D/2)}$; Chord definition: $D$ = central angle subtended by a 100 ft chord (deg). Railroad practice.
Deflection angle to a point: $\delta = \dfrac{90}{\pi}\cdot\dfrac{\ell}{R}$; Deflection from back tangent to a point at arc distance $\ell$ from PC (deg); total to PT is $\Delta/2$ .
PC and PT stations: $\text{Sta}_{\text{PC}} = \text{Sta}_{\text{PI}} - T, \quad \text{Sta}_{\text{PT}} = \text{Sta}_{\text{PC}} + L$; Stationing rule. The PT follows the PC by the ARC length, not the tangent.

Where it lives: NCEES PE Civil Reference Handbook — §5.2.1 Basic Curve Elements · AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book, GDHS-7) · Hickerson, Route Location and Design, 5th ed.

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Vertical Design
11% of the exam

Equal-tangent parabolic curves: elevations and offsets, high/low points, PVI/PVC/PVT stationing, and crest/sag curve lengths from stopping and passing sight distance and the rate of vertical curvature K.

Equal-Tangent Parabolic Curves

How the symmetrical parabola joins two grades — PVC/PVI/PVT stationing, the tangent-offset elevation model, and locating the high or low point.

Rate of change of grade: $r = \frac{g_2 - g_1}{L}$; Constant for a parabola. $g_1,g_2$ = grades (decimal or %), $L$ = curve length (ft or stations); the handbook's $A=|g_2-g_1|$ is the magnitude of the grade change.
Curve elevation (tangent-offset form): $Y = Y_{PVC} + g_1\,x + \frac{g_2 - g_1}{200\,L}\,x^2$; Elevation at horizontal distance $x$ (ft) from the PVC. $g_2 - g_1$ in percent, $g_1$ in decimal, $L$ , $x$ in ft. Use $g_1$ as decimal here (e.g., $+0.03$ ).
Tangent offset: $y = \frac{g_2 - g_1}{200\,L}\,x^2$; Vertical distance from the back tangent to the curve. $g_2 - g_1$ in percent (its sign sets the direction: negative = crest, offset below the tangent; positive = sag, offset above), $x,L$ in ft. The reported offset is the magnitude $|y|$ ; the sign only flags crest vs sag.
PVC and PVT stations: $\text{Sta}_{PVC}=\text{Sta}_{PVI}-\tfrac{L}{2},\quad \text{Sta}_{PVT}=\text{Sta}_{PVI}+\tfrac{L}{2}$; Symmetrical curve: half the length each side of the PVI. Convert $L/2$ (ft) to station form ( $1\ \text{sta}=100\ \text{ft}$ ).
PVC elevation from PVI: $Y_{PVC}=Y_{PVI}-g_1\,\tfrac{L}{2}$; Back off the back-tangent grade over the half-length. Likewise $Y_{PVT}=Y_{PVI}+g_2\,L/2$ .
Middle ordinate (external) at PVI: $E=\frac{|A|\,L}{800}$; Maximum tangent offset, occurring at the PVI, reported as a positive magnitude. $A$ in percent, $L$ in ft, $E$ in ft. The sign of $g_2 - g_1$ (crest vs sag) sets which side of the tangent the offset falls on, not the size of $E$ .
High / low point station: $x_m=-\frac{g_1}{2a}=\frac{g_1\,L}{g_1-g_2}$; Distance from PVC to the turning point. Valid only if $0\le x_m\le L$ ; otherwise no interior high/low point. $g_1$ and $g_2$ in the same units.
Rate of vertical curvature: $K=\frac{L}{|A|}$; Horizontal length per 1% change of grade (ft/%). The single parameter that links curve length to sight-distance design.
Grade on the curve: $g(x)=g_1+\frac{g_2-g_1}{L}\,x$; Instantaneous grade (%) at distance $x$ from PVC; equals zero at the high/low point.

Where it lives: NCEES PE Civil Reference Handbook — §5.3.1 Symmetrical Vertical Curve Formula · AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book, GDHS-7) — Vertical Alignment

Read the Vertical Design chapter

Intersection Geometry
10% of the exam

Intersection sight distance and sight triangles, at-grade and roundabout layout geometry, and interchange/ramp design including acceleration and deceleration lane lengths and gap acceptance.

Intersection Sight Distance

Build the departure and approach sight triangles, size intersection sight distance from the gap-acceptance time as ISD = 1.47·V·t_g, and keep the triangle clear of obstructions.

Intersection sight distance (departure): $\text{ISD} = 1.47\,V_{\text{major}}\,t_g$; Major-road leg of the sight triangle, ft. $V_{\text{major}}$ = major-road design speed (mph), $t_g$ = time gap (s), $1.47$ = ft/s per mph. AASHTO Green Book Case B/F.
Time gap with multilane adjustment: $t_g = t_{g,\text{base}} + 0.5\,(n_L - 1)$; Passenger-car gap for crossing/left turns; $n_L$ = number of through lanes crossed. Base $7.5$ s (left turn) or $6.5$ s (right turn/cross) onto a two-lane road; $0.7$ s/lane for trucks.
Allowable speed from a fixed sight triangle: $V_{\text{major}} = \dfrac{\text{ISD}_{\text{avail}}}{1.47\,t_g}$; Inverts the ISD relation when the available major-road leg is fixed by the site, mph.
Approach-triangle leg: $d = 1.47\,V\,t_a$; Leg length for uncontrolled (Case A) / yield (Case C) approaches; $t_a$ = travel time (≈ 6.5 s base, reduced at low speed), ft.
Stopping sight distance (lower-bound check): $\text{SSD} = 1.47\,V\,t_r + \dfrac{V^2}{30\left(\tfrac{a}{32.2}\pm G\right)}$; Major-road SSD that ISD must not fall below; $t_r=2.5$ s, $a=11.2$ ft/s², $G$ = grade (decimal). From AASHTO sight-distance criteria.
Sight-line eye and object heights: $h_{\text{eye}} = 3.5\ \text{ft},\quad h_{\text{obj}} = 3.5\ \text{ft}$; AASHTO design heights defining the clear plane within the sight triangle for intersection design.
Minor-road (decision-point) leg: $a = 14.5\ \text{ft}$; Nominal set-back of the stopped driver's eye from the edge of the major-road traveled way (passenger car, Case B departure triangle).

Where it lives: AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book, GDHS-7) — Intersection Sight Distance (Cases A–F) · NCEES PE Civil Reference Handbook — §5.2 Horizontal Design · AASHTO Green Book — Stopping Sight Distance criteria

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Traffic Signals
7% of the exam

Signal timing kinematics: yellow change and all-red clearance intervals, pedestrian crossing time, cycle length and green splits, lost time and capacity, left-turn phasing, and signal warrants.

Yellow & All-Red Clearance Intervals

Time the ITE yellow change and all-red clearance intervals, then locate the dilemma zone, keeping the grade sign and the mph-to-ft/s conversion the way examinees keep dropping them.

ITE yellow change interval: $Y = t + \dfrac{v}{2a + 2gG}$; Yellow time in s. $t$ = reaction time (1.0 s default), $v$ = approach speed in ft/s, $a$ = deceleration (10 ft/s² default), $g = 32.2\,\text{ft/s}^2$ , $G$ = grade as signed decimal (downgrade negative).
All-red clearance interval: $R = \dfrac{w + L}{v}$; Red clearance in s. $w$ = width from near stop line to far edge of conflict (ft), $L$ = vehicle length (20 ft default), $v$ = approach speed in ft/s.
mph to ft/s conversion: $v\,[\text{ft/s}] = 1.47\,V\,[\text{mph}]$; Exact factor 5280/3600 = 1.4667 ≈ 1.47. Apply before any kinematic term.
Cannot-stop distance: $X_c = 1.47\,V\,t_{stop} + \dfrac{(1.47\,V)^2}{2a_2}$; Minimum safe stopping distance (ft). $V$ in mph, $t_{stop}$ = stopping perception time (s), $a_2$ = stopping deceleration (ft/s²).
Cannot-clear distance: $X_0 = 1.47\,V\,Y - W$; Max distance from which a vehicle at constant speed clears during the yellow (ft). $W$ = intersection width plus vehicle length (ft), $Y$ = yellow (s).
Dilemma-zone existence: $X_c > X_0 \;\Rightarrow\; \text{zone width} = X_c - X_0$; A dilemma zone forms only when the cannot-stop boundary lies upstream of the cannot-clear boundary.
Yellow that eliminates the zone: $Y_{min} = \dfrac{X_c + W}{1.47\,V}$; Set the constant-speed X₀ (a₁ = 0) equal to X_c and solve; the smallest yellow that closes the dilemma zone at speed V (mph). With a₁ > 0, solve the handbook's full X₀ instead.
Total change-plus-clearance: $CP = Y + R$; Sum of yellow and all-red. Only its unused tail (the clearance lost time) is lost; effective green g = G + Y + R − t_L.

Where it lives: NCEES PE Civil Reference Handbook — §5.4.3 Interval Timing · NCEES PE Civil Reference Handbook — §5.4.1 Dilemma Zones · ITE Traffic Engineering Handbook, 6th ed. — Yellow Change and Red Clearance Intervals

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Traffic Control Design
6% of the exam

Permanent signs and pavement markings, sign placement and legibility, and temporary traffic control: work-zone taper lengths, buffer space, and channelizing-device spacing.

Work-Zone Taper Lengths & TTC

How to size a merging taper from the MUTCD L = WS²/60 and L = WS rules, set the shifting/shoulder ratios, space channelizing devices, and lay out the four temporary-traffic-control areas.

Merging taper, low speed: $L = \dfrac{W S^{2}}{60}$; Minimum merging-taper length (ft) for $S \le 40\,\text{mph}$ . $W$ = lateral offset (ft), $S$ = speed (mph).
Merging taper, high speed: $L = W S$; Minimum merging-taper length (ft) for $S \ge 45\,\text{mph}$ .
Shifting taper: $L_{\text{shift}} \ge 0.5\,L$; Lateral-shift taper (no lane drop) is at least half the merging length.
Shoulder taper: $L_{\text{shoulder}} \ge \tfrac{1}{3}\,L$; Closing a shoulder needs about one-third of the merging length.
One-lane two-way taper: $50\ \text{ft} \le L_{\text{1L2W}} \le 100\ \text{ft}$; Short blunt taper where one lane serves both directions alternately.
Device spacing (taper): $s_{\text{taper}} \le S\ (\text{ft})$; Channelizing-device spacing in a taper, in feet, not to exceed the speed in mph.
Device spacing (tangent): $s_{\text{tangent}} \le 2S\ (\text{ft})$; Along tangent activity-area sections spacing may roughly double.
Device count in a taper: $N = \dfrac{L}{s} + 1$; Number of channelizing devices over a taper of length $L$ at spacing $s$ (add one for the upstream device).
Longitudinal buffer space: $L_{\text{buffer}} \approx \text{SSD}(S)$; Empty recovery cushion ahead of the work space, sized to stopping sight distance.

Where it lives: Manual on Uniform Traffic Control Devices (MUTCD), Part 6 — Temporary Traffic Control (taper lengths, channelizing-device spacing, buffer space) · AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book, GDHS-7) — stopping sight distance used to size the longitudinal buffer · NCEES PE Civil Reference Handbook — Construction §2.6.2 Work Zone and Public Safety (context only; contains no taper method)

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Geotechnical
5% of the exam

Soil classification, CBR/R-value and resilient-modulus correlations, phase relationships and soil properties, compaction and relative compaction, and earthwork cut/fill mass balance with shrinkage and swell.

Subgrade Strength: CBR, R-Value & Resilient Modulus

Measure subgrade support with the CBR, R-value, and resilient modulus, convert among them and to soil class, and feed the right stiffness into pavement design.

California Bearing Ratio: $\text{CBR} = \dfrac{p_{\text{soil}}}{p_{\text{standard}}}\times 100\%$; Penetration resistance at 0.1 in vs. the 1000 psi standard-stone value; dimensionless percent.
R-value (stabilometer): $R = 100 - \dfrac{100}{\frac{2.5}{D}\left(\frac{p_v}{p_h}-1\right)+1}$; Hveem resistance value, 0–100. $p_v$ vertical, $p_h$ horizontal (transmitted) pressure, $D$ = displacement.
Resilient modulus definition: $M_R = \dfrac{\sigma_d}{\varepsilon_r}$; Recoverable elastic stiffness under repeated load; $\sigma_d$ deviator stress, $\varepsilon_r$ recoverable strain (psi).
CBR → M_R (subgrade): $M_R = 1500\times \text{CBR}\ (\text{psi})$; Heukelom–Klomp; subgrade soils with CBR ≤ 10 only (handbook §3.19 Pavements).
R-value → M_R (subgrade): $M_R = 1000 + 555\times R\ (\text{psi})$; Asphalt Institute recommended-value form; subgrade roadbed soils (handbook §3.19 Pavements).
CBR → M_R (granular base): $M_R = 740\times \text{CBR}\ (\text{psi}),\ \theta = 100\ \text{psi}$; AASHTO 1993 base/subbase correlation at bulk stress θ = 100 psi (constants vary with θ).
Resilient modulus → k: $k \approx \dfrac{M_R}{19.4}\ (\text{pci})$; Approximate plate-load conversion to modulus of subgrade reaction for rigid pavement.

Where it lives: NCEES PE Civil Reference Handbook 2.2 — §3.19 Pavements (printed pp. ~223–226): CBR/R-value/M_R correlations (1500×CBR Heukelom–Klomp; 1000+555R Asphalt Institute; A=772–1155, B=369–555; 740×CBR granular) and the typical-value tables and Subgrade Resilient Modulus figure · FHWA-NHI-05-037, Geotechnical Aspects of Pavements · AASHTO Guide for Design of Pavement Structures (1993)

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Pavement
5% of the exam

Traffic characterization and ESALs, flexible-pavement AASHTO structural number, rigid-pavement slab design parameters, and pavement evaluation and rehabilitation procedures.

ESALs & Traffic Characterization

Convert a mixed truck stream into 18-kip equivalent single-axle loads with the fourth-power law, then accumulate design-lane ESALs over the design period with growth and lane factors.

Load-equivalency factor (fourth-power law): $F_E = \left(\dfrac{W_x}{18}\right)^{4}$; Approximate ESALs per pass of a single axle carrying $W_x$ kips; exact values come from AASHTO single/tandem/tridem tables (functions of $SN$ and $p_t$ ). $W_x$ in kips.
Truck factor: $TF = \sum_i F_{Ei}$; Average ESALs applied by one truck of a class; the sum of load-equivalency factors over all of that truck's axle groups.
Growth factor (cumulative): $G_{rn} = \dfrac{(1+r)^{n}-1}{r}$; Sums the geometric growth series over $n$ years at annual rate $r$ (decimal). Reduces to $n$ when $r=0$ ; never use $n$ alone when $r>0$ .
ESAL accumulation (handbook p. 274): $\text{ESAL}_i = f_d\,G_{rn}\,\text{AADT}_i\,N_i\,F_{Ei}\,(365)$; $f_d$ design-lane factor, $\text{AADT}_i$ first-year daily volume in axle category $i$ , $N_i$ axles per vehicle, $F_{Ei}$ load-equivalency factor; $365$ annualizes.
Design-lane ESAL (handbook §5.1.6): $\text{ESAL}_{DL} = D_D\,D_L\,\text{ESAL}$; $D_D$ directional distribution (≈0.5), $D_L$ lane distribution ratio; ESAL is the cumulative two-directional 18-kip total over the analysis period.
First-year daily ESALs from truck factor: $\text{ESAL}_{\text{day}} = \text{AADT}\times T \times TF$; $T$ = decimal fraction of trucks in the stream; multiply by $365$ then $G_{rn}$ to accumulate over the design period.
Cumulative design ESALs (assembled): $W_{18} = \text{AADT}\,T\,TF\,(365)\,G_{rn}\,D_D\,D_L$; The full chain from a traffic count to the design-lane $W_{18}$ used in AASHTO-93 / MEPDG.

Where it lives: NCEES PE Civil Reference Handbook — §5.1.6 Design Traffic · NCEES PE Civil Reference Handbook — §5.1.6–5.1.8 and the ESAL accumulation equation (p. 274) · AASHTO Guide for Design of Pavement Structures (1993) — Appendix D, Load Equivalency Factors

Read the Pavement chapter

Drainage
11% of the exam

Highway hydrology (rational method, time of concentration, runoff) and hydraulics: gutter and inlet capacity, storm-drain pipe flow, open-channel flow, culvert headwater, energy dissipation, and detention.

The Rational Method & Time of Concentration

Size storm drainage from peak runoff with Q = CiA, build a composite runoff coefficient, and find the intensity from an IDF curve at the time of concentration.

Rational formula: $Q = CiA$; Peak discharge $Q$ (cfs) from runoff coefficient $C$ (dimensionless), intensity $i$ (in/hr), area $A$ (acres). In SI, $Q\,(\text{m}^3/\text{s}) = 0.00278\,CiA$ with $i$ in mm/hr, $A$ in hectares.
Composite runoff coefficient: $C_w = \dfrac{\sum_{i} C_i A_i}{\sum_{i} A_i}$; Area-weighted $C$ for a mixed-cover catchment; form it before applying $Q=CiA$ .
Time of concentration: $t_c = \sum t_{\text{segment}} = t_{\text{sheet}} + t_{\text{shallow}} + t_{\text{channel/pipe}}$; Travel time from the hydraulically most distant point to the outlet; sets the IDF duration.
Sheet-flow travel time (NRCS TR-55): $t_{\text{sheet}} = \dfrac{0.007\,(nL)^{0.8}}{P_2^{\,0.5}\,S^{0.4}}$; TR-55 form, result in hours; $n$ overland roughness, $L$ length (ft, $\le\!300$ ft), $P_2$ 2-yr 24-hr rainfall (in), $S$ slope (ft/ft). The handbook $\S6.5.4$ uses an intensity-based FHWA form ( $K_u=0.933$ USCS, minutes) instead.
Shallow concentrated velocity: $V = k\sqrt{S}$; $k = 16.1\ \text{ft/s}$ unpaved, $20.3\ \text{ft/s}$ paved; then $t = L/V$ .
IDF intensity (fitted form): $i = \dfrac{a}{(t_c + b)^{\,c}}$; Regional constants $a,b,c$ for a given return period; enter with duration $=t_c$ (min).
Customary unit identity: $1\ \text{acre}\cdot\text{in/hr} = 1.008\ \text{cfs} \approx 1\ \text{cfs}$; Why $Q=CiA$ returns cfs directly with $i$ in in/hr and $A$ in acres.

Where it lives: NCEES PE Civil Reference Handbook — §6.5.2 Runoff Analysis (Rational Formula Method) · NCEES PE Civil Reference Handbook — §6.5.4 Time of Concentration (FHWA intensity-based sheet-flow form) and §6.5.3 Point Precipitation (IDF) · NRCS TR-55, Urban Hydrology for Small Watersheds · FHWA HEC-22, Urban Drainage Design Manual (3rd ed.)

Read the Drainage chapter

PE Civil: Transportation formula sheet

Project Management
9% of the exam

Engineering Economics & Benefit-Cost

Traffic Capacity & LOS
9% of the exam

Flow, Speed, Density & the Peak-Hour Factor

Traffic Analysis & Safety
6% of the exam

Crash Rates & Collision Analysis

Roadside & Cross-Section Design
10% of the exam

Forgiving Roadside & Clear Zone

Horizontal Design
11% of the exam

Circular Curve Geometry & Stationing

Vertical Design
11% of the exam

Equal-Tangent Parabolic Curves

Intersection Geometry
10% of the exam

Intersection Sight Distance

Traffic Signals
7% of the exam

Yellow & All-Red Clearance Intervals

Traffic Control Design
6% of the exam

Work-Zone Taper Lengths & TTC

Geotechnical
5% of the exam

Subgrade Strength: CBR, R-Value & Resilient Modulus

Pavement
5% of the exam

ESALs & Traffic Characterization

Drainage
11% of the exam

The Rational Method & Time of Concentration

Now use them on real questions

Now use them on real questions

Now use them on real questions

Project Management9% of the exam

Engineering Economics & Benefit-Cost

Traffic Capacity & LOS9% of the exam

Flow, Speed, Density & the Peak-Hour Factor

Traffic Analysis & Safety6% of the exam

Crash Rates & Collision Analysis

Roadside & Cross-Section Design10% of the exam

Forgiving Roadside & Clear Zone

Horizontal Design11% of the exam

Circular Curve Geometry & Stationing

Vertical Design11% of the exam

Equal-Tangent Parabolic Curves

Intersection Geometry10% of the exam

Intersection Sight Distance

Traffic Signals7% of the exam

Yellow & All-Red Clearance Intervals

Traffic Control Design6% of the exam

Work-Zone Taper Lengths & TTC

Geotechnical5% of the exam

Subgrade Strength: CBR, R-Value & Resilient Modulus

Pavement5% of the exam

ESALs & Traffic Characterization

Drainage11% of the exam

The Rational Method & Time of Concentration

Now use them on real questions

Now use them on real questions

Now use them on real questions

Project Management
9% of the exam

Traffic Capacity & LOS
9% of the exam

Traffic Analysis & Safety
6% of the exam

Roadside & Cross-Section Design
10% of the exam

Horizontal Design
11% of the exam

Vertical Design
11% of the exam

Intersection Geometry
10% of the exam

Traffic Signals
7% of the exam

Traffic Control Design
6% of the exam

Geotechnical
5% of the exam

Pavement
5% of the exam

Drainage
11% of the exam