Success on the PE Civil Transportation exam depends on quickly recognizing formulas in the NCEES Reference Handbook and matching them to the problem statement. This lesson serves as a final review guide, mapping critical equations across hydrology, drainage, traffic, geometry, and pavements.
1. Hydrology and Hydraulics
| Formula Name | USCS Form | SI (Metric) Form | Variables & Crucial Units |
|---|
| Rational Method | $Q = C i A$ | $Q = 0.00278 C i A$ | $Q$ [cfs or $m^3/s$], $C$ [dim], $i$ [in/hr or mm/hr], $A$ [ac or ha] |
| SCS Runoff Depth | $Q = \frac{(P-0.2S)^2}{P+0.8S}$ | Same | $Q$ [in], $P$ [in], $S$ [in] (Potential Max Retention) |
| SCS Retention | $S = \frac{1000}{CN} - 10$ | $S = \frac{25400}{CN} - 254$ | $S$ [in or mm], $CN$ [dim, 0-100] |
| Manning’s Velocity | $V = \frac{1.486}{n} R^{2/3} S^{1/2}$ | $V = \frac{1}{n} R^{2/3} S^{1/2}$ | $V$ [ft/s or m/s], $n$ [dim], $R = A/P_w$ [ft or m], $S$ [ft/ft] |
| Gutter Flow (Izzard) | $Q = \frac{0.56}{n} S_x^{5/3} S^{1/2} T^{8/3}$ | $Q = \frac{0.376}{n} S_x^{5/3} S^{1/2} T^{8/3}$ | $Q$ [cfs or $m^3/s$], $S_x$ [cross slope, ft/ft], $S$ [long slope, ft/ft], $T$ [spread, ft or m] |
| Broad-Crested Weir | $Q = C_w L H^{1.5}$ | Same | $Q$ [cfs], $C_w$ [roadway default $\approx 2.6$ to $3.0$], $L$ [width, ft], $H$ [head, ft] |
| Hydraulic Jump (Conjugate) | $\frac{y_2}{y_1} = 0.5 (\sqrt{1+8Fr_1^2}-1)$ | Same | $y_1$ [supercritical depth, ft], $y_2$ [subcritical depth, ft], $Fr_1$ [Froude number, dim] |
2. Traffic Engineering and Signals
| Formula Name | Equation Form | Variables & Crucial Units |
|---|
| Traffic Flow Relation | $q = u \cdot k$ | $q$ [flow, veh/h/lane], $u$ [speed, mph], $k$ [density, veh/mi/lane] |
| Greenshields Model | $u = u_f (1 - k/k_j)$ | $u_f$ [free-flow speed, mph], $k_j$ [jam density, veh/mi] |
| Greenshields Capacity | $q_{max} = \frac{u_f \cdot k_j}{4}$ | $q_{max}$ [capacity, veh/h/lane] |
| Yellow Change Interval | $Y = t + \frac{V}{2a + 2gG}$ | $t$ [reaction time, 1.0 s], $V$ [speed, ft/s], $a$ [deceleration, 10.0 $\text{ft/s}^2$], $G$ [grade, decimal] |
| Red Clearance Interval | $R_c = \frac{w+L}{V}$ | $w$ [intersection width, ft], $L$ [vehicle length, 20 ft], $V$ [speed, ft/s] |
| Webster’s Cycle Length | $C_{opt} = \frac{1.5L + 5}{1 - Y_{sum}}$ | $L$ [total lost time per cycle, s], $Y_{sum}$ [sum of critical flow ratios, dim] |
3. Roadway Geometry
| Formula Name | Equation Form | Variables & Crucial Units |
|---|
| Horizontal Radius | $R_{min} = \frac{V^2}{15(e + f)}$ | $V$ [design speed, mph], $e$ [superelevation, ft/ft], $f$ [side friction factor, dim] |
| Curve Stationing | $T = R \tan(\Delta/2)$ and $L = \frac{\pi R \Delta}{180}$ | $T$ [tangent length, ft], $L$ [curve length, ft], $\Delta$ [deflection angle, degrees] |
| Vertical Curve Profile | $y = y_{PVC} + g_1 x + \frac{r x^2}{2}$ | $y$ [elevation, ft], $g_1$ [initial grade, ft/ft], $r = \frac{g_2-g_1}{L}$ [ft/ft$^2$], $x$ [distance, ft] |
| High/Low Point Location | $x_{hp} = \frac{g_1 L}{g_1 - g_2}$ | $x_{hp}$ [distance from PVC, ft], $g_i$ [grades, percent], $L$ [curve length, ft] |
4. Pavement and Earthwork
| Formula Name | Equation Form | Variables & Crucial Units |
|---|
| Flexible Pavement SN | $SN = a_1 D_1 + a_2 D_2 m_2 + a_3 D_3 m_3$ | $a_i$ [layer coeff, dim], $D_i$ [thickness, inches], $m_i$ [drainage coeff, dim] |
| Earthwork Shrinkage | $V_C = V_B (1 - S_h)$ | $V_C$ [compacted volume], $V_B$ [bank volume], $S_h$ [shrinkage factor, decimal] |
| Earthwork Swell | $V_L = V_B (1 + S_w)$ | $V_L$ [loose volume], $V_B$ [bank volume], $S_w$ [swell factor, decimal] |
- Identify the Speed Units: In geometric radius $R_{min} = \frac{V^2}{15(e+f)}$, $V$ is in mph. In signal clearance $Y = t + \frac{V}{2a+2gG}$, $V$ is in ft/s. Double-check your units before squaring or dividing.
- Identify the Grade Form: In vertical curve high point formulas, $g_1$ and $g_2$ are entered in percent (e.g. $3.0$). In signal timing and hydraulic slope equations, grades are entered in decimal form (e.g. $0.03$).
- Verify the Area Units: Rational Method $Q = CiA$ uses area in acres. The SCS Peak flow equation $q_p = \frac{484AQ}{t_p}$ uses area in square miles.