FE Civil formula sheet

Statistics: Distributions, Dispersion, and Confidence Intervals

Center and spread, the normal and standard-normal z-transform, confidence intervals for the mean, and least-squares regression with correlation — the FE statistics core.

Arithmetic mean

$$\bar{X}=\dfrac{1}{n}\sum_{i=1}^{n}X_i$$

Balance point of the data; same units as the data. Basis for variance and regression.

Median (sorted data)

$$\tilde{X}=\begin{cases}X_{(n+1)/2}, & n\text{ odd}\\[2pt]\tfrac{1}{2}\!\left(X_{n/2}+X_{n/2+1}\right), & n\text{ even}\end{cases}$$

Middle value of ascending-sorted data; resistant to outliers. The mode is the most frequent value.

Population variance

$$\sigma^2=\dfrac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2$$

Spread of a full population of size $N$ about true mean $\mu$. Units are (data units)$^2$.

Sample variance / standard deviation

$$s^2=\dfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2,\quad s=\sqrt{s^2}$$

Unbiased estimate from a sample; the $n-1$ divisor accounts for estimating $\mu$ by $\bar{X}$.

Standard-normal transform

$$z=\dfrac{x-\mu}{\sigma}$$

Number of standard deviations from the mean; converts any normal variable to the unit normal for table lookup.

Unit-normal table areas

$$F(-z)=1-F(z)=R(z)$$

$F(z)$ = area left of $z$, $R(z)$ = area right. Use symmetry for negative $z$; probabilities between values are area differences.

Confidence interval for the mean (σ known)

$$\bar{X}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$$

Brackets $\mu$ at confidence $1-\alpha$. $z_{\alpha/2}$: 1.645 (90%), 1.960 (95%), 2.576 (99%).

Confidence interval for the mean (σ unknown)

$$\bar{X}\pm t_{\alpha/2,\,n-1}\dfrac{s}{\sqrt{n}}$$

Use Student's $t$ with $n-1$ degrees of freedom when $\sigma$ is estimated by $s$ from a small sample; wider than the $z$ interval.

Least-squares slope and intercept

$$\hat{b}=\dfrac{S_{xy}}{S_{xx}}, \qquad \hat{a}=\bar{y}-\hat{b}\,\bar{x}$$

Best-fit line $\hat{y}=\hat{a}+\hat{b}x$ through the centroid $(\bar{x},\bar{y})$; minimizes squared vertical residuals.

Sums of squares

$$S_{xx}=\textstyle\sum x_i^2-\tfrac{1}{n}\bigl(\sum x_i\bigr)^2,\;\; S_{xy}=\textstyle\sum x_iy_i-\tfrac{1}{n}\bigl(\sum x_i\bigr)\bigl(\sum y_i\bigr)$$

Shortcut accumulators for regression; $S_{yy}$ has the analogous form in $y$.

Correlation and determination

$$R=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}, \qquad R^2=\text{fraction of }y\text{ variation explained}$$

$R\in[-1,1]$ measures linear association; $R^2\in[0,1]$ is the goodness of fit. Correlation is not causation.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics: Dispersion, Mean, Median, and Mode Values · NCEES FE Reference Handbook — Engineering Probability and Statistics: Normal Distribution, Unit Normal Distribution table, Confidence Intervals · NCEES FE Reference Handbook — Engineering Probability and Statistics: Linear Regression and Goodness of Fit (Least Squares) · Montgomery & Runger, Applied Statistics and Probability for Engineers

Codes of Ethics and Professional Practice

How engineering codes — the NCEES Model Rules and the ASCE code — order an engineer's duties, with the public held paramount, and how to resolve the conflicts the FE tests.

Obligation hierarchy (tie-breaker)

$$\text{public} \;\succ\; \text{employer / client} \;\succ\; \text{other licensees} \;\succ\; \text{self}$$

When duties conflict, the higher tier governs. The public's health, safety, and welfare is paramount over every other obligation.

Paramount-duty rule

$$\text{first duty} = \text{safeguard public health, safety \& welfare}$$

NCEES Model Rules, obligations to the public. The basis for the duty to report when overruled on a safety matter.

Sign-and-seal test

$$\text{may seal} \iff \text{competent in field} \;\wedge\; \text{prepared under your responsible charge}$$

Both conditions required. Stamping unfamiliar or unsupervised work violates the code and Model Law.

Conflict-of-interest rule

$$\text{actual or apparent conflict} \;\Rightarrow\; \text{full disclosure to all parties}$$

Disclosure (in writing where compensation is involved) cures the conflict; concealment is the violation. Triggered by mere appearance.

Single-source compensation

$$\text{pay from} > 1 \text{ party on a project} \Rightarrow \text{disclose} \wedge \text{written consent of all}$$

You may not be paid by more than one party for the same project work unless every interested party knows and agrees in writing.

No-gratuities rule

$$\text{gift / gratuity from contractor on your work} \Rightarrow \text{decline}$$

Licensees do not solicit or accept gratuities, directly or indirectly, from contractors or suppliers connected to client work.

Public-opinion validity

$$\text{permissible public opinion} \iff \text{adequate facts} \;\wedge\; \text{competent evaluation}$$

Testimony and public statements must be objective, fact-based, and within your competence; paid statements require disclosure of the sponsor.

Dilemma taxonomy

$$\text{ethical issue} \in \{\,\text{conceptual},\ \text{factual},\ \text{tradeoff}\,\}$$

Conceptual = disputed definitions; factual = contested data; tradeoff = competing goods (risk vs. benefit, safety vs. cost). No code gives a mechanical answer.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice · NCEES Model Rules, Section 240.15 — Rules of Professional Conduct · ASCE Code of Ethics

Time Value of Money and Equivalence

How interest moves money across time: compound vs simple interest, the six discrete factors, gradients, and converting between nominal and effective rates.

Single-payment compound amount (F/P)

$$F = P(1+i)^n$$

Future worth of a present sum. $i$ = rate per period, $n$ = number of periods (must match compounding).

Single-payment present worth (P/F)

$$P = F(1+i)^{-n}$$

Discounts a future lump sum to time zero.

Uniform-series present worth (P/A)

$$P = A\,\dfrac{(1+i)^n-1}{i(1+i)^n}$$

Present worth of an end-of-period annuity $A$; $P$ sits one period before the first payment.

Capital recovery (A/P)

$$A = P\,\dfrac{i(1+i)^n}{(1+i)^n-1}$$

Equal payment that recovers a present sum over $n$ periods — the loan-payment formula.

Uniform-series compound amount (F/A)

$$F = A\,\dfrac{(1+i)^n-1}{i}$$

Future worth accumulated by an end-of-period series.

Sinking fund (A/F)

$$A = F\,\dfrac{i}{(1+i)^n-1}$$

Deposit per period needed to accumulate a target future amount $F$.

Gradient present worth (P/G)

$$P = G\left[\dfrac{(1+i)^n-1}{i^2(1+i)^n}-\dfrac{n}{i(1+i)^n}\right]$$

Present worth of an arithmetic gradient; first increment $G$ occurs in period 2.

Gradient to uniform series (A/G)

$$A = G\left[\dfrac{1}{i}-\dfrac{n}{(1+i)^n-1}\right]$$

Converts an arithmetic gradient into an equivalent level annuity.

Effective annual rate

$$i_e = \left(1+\dfrac{r}{m}\right)^m-1$$

$r$ = nominal annual rate, $m$ = compounding periods per year. The continuous-compounding limit $i_e=e^r-1$ is beyond the handbook.

Where it lives: NCEES FE Reference Handbook — Engineering Economics · Newnan, Eschenbach & Lavelle, Engineering Economic Analysis

Equilibrium, Trusses, and Frames

Draw free-body diagrams, solve support reactions with ΣF=0 and ΣM=0, and analyze trusses by joints and sections, frames and machines, and zero-force members.

2-D equilibrium

$$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M_O=0$$

Three independent equations for a planar rigid body; solves up to three unknown reactions.

3-D equilibrium

$$\sum F_x=\sum F_y=\sum F_z=0,\qquad \sum M_x=\sum M_y=\sum M_z=0$$

Six equations for a spatial body; needed for 3-D supports and space trusses.

Method of joints

$$\sum F_H=0,\qquad \sum F_V=0$$

Applied at each pin; start where two or fewer members are unknown. Tension assumed positive.

Member force components

$$F_x=\frac{x}{L}F,\qquad F_y=\frac{y}{L}F$$

Resolve an axial member force along its geometry (run $x$, rise $y$, length $L=\sqrt{x^2+y^2}$).

Determinacy of a plane truss

$$m+r=2j$$

$m$ = members, $r$ = reactions, $j$ = joints. Equality → statically determinate (necessary condition).

Zero-force member rules

$$\text{2 non-collinear, unloaded}\Rightarrow 0;\ \ \text{3 with 2 collinear, unloaded}\Rightarrow \text{odd}=0$$

Identify members carrying no force by inspection before computing.

Newton's third law at a connection

$$\mathbf{F}_{AB}=-\mathbf{F}_{BA}$$

For frames/machines: force of member A on a pin equals minus the force of the pin on A.

Two-force member

$$\mathbf{F}_1=-\mathbf{F}_2,\ \text{collinear}$$

A member loaded only at two points carries a force directed along the line joining them (truss member).

Where it lives: NCEES FE Reference Handbook — Statics (Equilibrium, Method of Joints, Method of Sections, Zero-Force Members) · Hibbeler, Engineering Mechanics: Statics · Beer & Johnston, Vector Mechanics for Engineers: Statics

Kinematics, Newton's Second Law, and Work-Energy

Describe motion with the constant-acceleration and projectile equations, relate force to acceleration through F=ma, then short-circuit the algebra with work-energy and power.

Velocity and acceleration

$$v = \dfrac{ds}{dt}, \qquad a = \dfrac{dv}{dt} = v\dfrac{dv}{ds}$$

Definitions for rectilinear motion; the last form (from $a\,ds = v\,dv$) eliminates time. $s$ in m or ft, $v$ in m/s or ft/s, $a$ in m/s² or ft/s².

Constant-acceleration set

$$v = v_0 + a t, \quad s = s_0 + v_0 t + \tfrac{1}{2}a t^2, \quad v^2 = v_0^2 + 2a(s - s_0)$$

Valid only when $a$ is constant. Use the time-free third form when $t$ is neither given nor asked.

Projectile components

$$v_x = v_0\cos\theta, \quad v_y = v_0\sin\theta - g t, \quad y = (v_0\sin\theta)t - \tfrac{1}{2}g t^2$$

Horizontal velocity constant ($a_x = 0$); vertical motion under $a_y = -g$. Time links the two axes.

Normal and tangential acceleration

$$a_t = \dfrac{dv}{dt}, \qquad a_n = \dfrac{v^2}{\rho}, \qquad a = \sqrt{a_t^2 + a_n^2}$$

$\rho$ = radius of curvature (m or ft); $a_n$ points toward the center. For a circle $\rho = r$.

Circular motion

$$v = r\omega, \qquad a_t = r\alpha, \qquad a_n = r\omega^2$$

$\omega$ in rad/s, $\alpha$ in rad/s², $r$ in m or ft. Angles must be in radians.

Relative motion

$$\mathbf{v}_A = \mathbf{v}_B + \mathbf{v}_{A/B}, \qquad \mathbf{a}_A = \mathbf{a}_B + \mathbf{a}_{A/B}$$

Translating-frame vector addition; $\mathbf{v}_{A/B}$ is the velocity of $A$ as seen from $B$.

Newton's second law

$$\Sigma \mathbf{F} = m\mathbf{a}$$

One scalar equation per coordinate direction. $F$ in N or lbf, $m$ in kg or slug, $a$ in m/s² or ft/s².

Weight

$$W = mg$$

Convert between weight and mass; $g = 9.81\ \text{m/s}^2 = 32.2\ \text{ft/s}^2$. In USCS, $m = W/g$ in slugs.

Kinetic energy (particle)

$$T = \tfrac{1}{2}m v^2$$

Joules (N·m) in SI, ft-lbf in USCS. Always uses speed magnitude.

Work-energy theorem

$$U_{1\to2} = T_2 - T_1 = \tfrac{1}{2}m v_2^2 - \tfrac{1}{2}m v_1^2$$

Net work of all forces equals change in kinetic energy. Distance-based, time-free.

Work of common forces

$$U_{F} = F s\cos\theta, \quad U_{grav} = -W\Delta y, \quad U_{spring} = -\tfrac{1}{2}k(s_2^2 - s_1^2)$$

$\theta$ = angle between force and displacement; gravity work negative when rising; spring work negative when stretched or compressed from its free length.

Potential energy

$$V_g = mgh, \qquad V_e = \tfrac{1}{2}k s^2$$

Gravitational ($h$ above a datum) and elastic ($s$ = deflection from free length, $k$ in N/m or lbf/ft).

Energy conservation with nonconservative work

$$T_1 + V_1 + U_{1\to2} = T_2 + V_2$$

$U_{1\to2}$ is the work of nonconservative forces (negative for friction). Reduces to $T_1+V_1=T_2+V_2$ when only gravity/springs act.

Power and efficiency

$$P = \dfrac{dU}{dt} = \mathbf{F}\cdot\mathbf{v}, \qquad \varepsilon = \dfrac{P_{out}}{P_{in}}$$

$P$ in watts (J/s) or ft-lbf/s; $1\ \text{hp} = 746\ \text{W} = 550\ \text{ft-lbf/s}$.

Where it lives: NCEES FE Reference Handbook — Dynamics (Kinematics, Particle Kinetics, Work and Energy) · Hibbeler, Engineering Mechanics: Dynamics · Beer, Johnston, Vector Mechanics for Engineers: Dynamics

Axial, Torsion, Bending, and Shear Stress with Deformations

The four load cases — axial, torsion, bending, transverse shear — their stress formulas, Hooke's law, and the deformations (elongation, twist, thermal) each produces.

Axial normal stress

$$\sigma = \dfrac{P}{A}$$

Uniform stress on a cross section from axial load $P$ over area $A$; tension positive. Units: kip/in² = ksi.

Normal strain

$$\varepsilon = \dfrac{\delta}{L}$$

Fractional change in length; dimensionless (in/in).

Hooke's law (axial and shear)

$$\sigma = E\varepsilon, \qquad \tau = G\gamma$$

$E$ = modulus of elasticity, $G$ = shear modulus; valid in the linear-elastic range only.

Elastic-constant relation

$$G = \dfrac{E}{2(1+\nu)}$$

Links $E$, $G$, and Poisson's ratio $\nu = -\varepsilon_{lat}/\varepsilon_{long}$. Steel: $\nu\approx 0.30$.

Axial deformation

$$\delta = \dfrac{PL}{AE}$$

Elongation of a prismatic member; $AE$ is the axial stiffness. Sum segments for stepped members.

Thermal deformation

$$\delta_t = \alpha L\,(T - T_o)$$

Free-expansion length change; $\alpha$ = coefficient of thermal expansion. Stress-free if unrestrained.

Fully restrained thermal stress

$$\sigma = E\,\alpha\,\Delta T$$

Stress when expansion is completely prevented between rigid supports; compressive for heating.

Torsional shear stress

$$\tau = \dfrac{Tr}{J}$$

Linear from center; max at surface ($r$ = outer radius). $J$ = polar moment of inertia.

Polar moment of inertia (solid round)

$$J = \dfrac{\pi d^4}{32} = \dfrac{\pi r^4}{2}$$

Solid circular shaft; for a tube subtract the inner from the outer value.

Angle of twist

$$\phi = \dfrac{TL}{GJ}$$

Total twist in radians; $GJ$ is the torsional stiffness. Mirrors $\delta=PL/(AE)$.

Bending (flexure) stress

$$\sigma_x = -\dfrac{My}{I}, \quad \sigma_{max} = \dfrac{Mc}{I} = \dfrac{M}{S}$$

$I$ = second moment about the neutral axis, $c$ = distance to extreme fiber, $S=I/c$ = section modulus.

Transverse shear stress

$$\tau = \dfrac{VQ}{Ib}$$

$Q=A'\bar{y}'$ = first moment of the area beyond the layer; $b$ = width there. Max at neutral axis.

Rectangular shear shortcut

$$\tau_{max} = \dfrac{3V}{2A} \;(\text{rect}), \quad \tau_{max}=\dfrac{4V}{3A}\;(\text{circle})$$

Peak transverse shear at the neutral axis for common solid sections.

Where it lives: NCEES FE Reference Handbook — Mechanics of Materials · NCEES FE Reference Handbook — Statics · Hibbeler, Mechanics of Materials

Mix Design of Concrete and Asphalt

Proportion concrete by absolute volume around the water-cement ratio, and design asphalt by volumetrics to an optimum binder content — the two materials problems the FE Civil rewards most.

Water-cement ratio

$$w/c = \dfrac{W_{water}}{W_{cement}}$$

Mass of mixing water over mass of cementitious material (dimensionless). The primary determinant of concrete strength and durability; lower is stronger.

Abrams' strength law

$$f'_c = \dfrac{A}{B^{\,w/c}}$$

Empirical inverse-exponential drop of 28-day strength $f'_c$ with $w/c$; $A$, $B$ are material constants. Captures the handbook strength-vs-$w/c$ trend.

Absolute volume of an ingredient

$$V_i = \dfrac{W_i}{SG_i\,\gamma_w}$$

Solid volume occupied by mass $W_i$; $SG_i$ = specific gravity, $\gamma_w=62.4\ \text{lb/ft}^3$ ($1000\ \text{kg/m}^3$). Foundation of mix proportioning.

Volume balance (batch)

$$V_{cement}+V_{water}+V_{air}+V_{CA}+V_{FA}=V_{total}$$

Absolute volumes must fill the batch volume ($27\ \text{ft}^3$ per yd$^3$ or $1\ \text{m}^3$). Solve for the unknown ingredient (usually fine aggregate).

Air volume

$$V_{air}=(\text{air fraction})\,V_{total}$$

Entrained plus entrapped air as a real volume; e.g., $5\%$ of $27\ \text{ft}^3 = 1.35\ \text{ft}^3$. Must be included in the balance.

Concrete yield / unit weight

$$\gamma_{concrete}=\dfrac{\sum W_i}{V_{total}}$$

Total batch mass over total volume gives fresh unit weight (typ. $\approx 145\ \text{lb/ft}^3$ normalweight). Used to check the design and compute yield.

Asphalt air voids (VTM)

$$V_a = 100\!\left(1-\dfrac{G_{mb}}{G_{mm}}\right)$$

Percent air in the compacted mix; $G_{mb}$ = bulk SG of compacted specimen, $G_{mm}$ = maximum (rice) SG. Design target $4.0\%$.

Voids in mineral aggregate (VMA)

$$VMA = 100\!\left(1-\dfrac{G_{mb}\,P_s}{G_{sb}}\right)$$

Inter-granular void space (binder + air) as percent of total volume; $P_s=1-P_b$ = aggregate mass fraction, $G_{sb}$ = aggregate bulk SG. Has a minimum by NMAS.

Voids filled with asphalt (VFA)

$$VFA = 100\,\dfrac{VMA-V_a}{VMA}$$

Percent of VMA occupied by effective binder. Has both a minimum and maximum tied to traffic level.

Effective specific gravity of aggregate

$$G_{se}=\dfrac{100-P_b}{\dfrac{100}{G_{mm}}-\dfrac{P_b}{G_b}}$$

Aggregate SG including pores not filled by binder; $P_b$ = binder percent by total mass, $G_b$ = binder SG ($\approx1.02$). Lies between $G_{sb}$ and apparent SG.

Binder content by total mass

$$P_b = \dfrac{W_{binder}}{W_{mix}}\times100$$

Asphalt content as a percent of total mix mass (binder + aggregate). The design variable optimized to hit $4\%$ air voids.

Optimum binder content

$$P_{b,opt}=P_b\big|_{V_a=4.0\%}$$

Binder content giving exactly $4.0\%$ design air voids, provided VMA and VFA criteria are also satisfied there.

Where it lives: NCEES FE Reference Handbook — Materials Science/Structure of Matter: Concrete · ACI 211.1 — Standard Practice for Selecting Proportions for Normal, Heavyweight, and Mass Concrete · Asphalt Institute MS-2 — Asphalt Mix Design Methods · AASHTO M 323 / R 35 — Superpave Volumetric Mix Design

Energy Equation, Bernoulli, and Momentum

Continuity, the energy and Bernoulli equations in head form, energy and hydraulic grade lines, head loss, and the momentum equation for forces on bends and nozzles.

Continuity (incompressible)

$$Q=A_1v_1=A_2v_2$$

Volumetric flow rate constant. $Q$ in $\text{m}^3/\text{s}$ or cfs; velocity rises where area falls.

Mass flow rate

$$\dot m=\rho A v=\rho Q$$

Conserved in steady flow for any fluid; reduces to constant $Q$ when $\rho$ is constant.

Energy equation (head form)

$$\frac{P_1}{\gamma}+\frac{v_1^2}{2g}+z_1+h_p=\frac{P_2}{\gamma}+\frac{v_2^2}{2g}+z_2+h_t+h_f$$

Steady incompressible flow with pump head $h_p$, turbine head $h_t$, friction head $h_f$. Every term in length units.

Bernoulli equation

$$\frac{P_1}{\gamma}+\frac{v_1^2}{2g}+z_1=\frac{P_2}{\gamma}+\frac{v_2^2}{2g}+z_2$$

Frictionless, no-machine special case. Valid along a streamline for steady incompressible flow.

Hydraulic and energy grade lines

$$\text{HGL}=\frac{P}{\gamma}+z,\qquad \text{EGL}=\text{HGL}+\frac{v^2}{2g}$$

HGL = piezometric head; EGL = total head, one velocity head higher. EGL falls in the flow direction.

Pressure drop from head loss

$$P_1-P_2=\gamma h_f=\rho g h_f$$

For equal area and elevation, the pressure drop equals $\gamma$ times the friction head.

Darcy-Weisbach head loss

$$h_f=f\,\frac{L}{D}\,\frac{v^2}{2g}$$

Major friction loss; $f$ from the Moody diagram via $Re$ and $\varepsilon/D$. $L$, $D$ pipe length and diameter.

Minor (fitting) loss

$$h_{f}=C\,\frac{v^2}{2g}$$

Loss at valves, elbows, entrances, exits; $C$ (or $K$) given. Sudden exit to reservoir $C\approx1$.

Reynolds number

$$Re=\frac{\rho v D}{\mu}=\frac{v D}{\nu}$$

Laminar $Re<2100$, transitional $2100<Re<10^4$, fully turbulent $Re>10^4$. Sets $f$ and the flow regime.

Impulse-momentum principle

$$\textstyle\sum F=\rho Q\,(v_2-v_1)$$

Net external force = momentum flux out minus in. Apply per component.

Force on a pipe bend (x-component)

$$P_1A_1-P_2A_2\cos\alpha-F_x=\rho Q(v_2\cos\alpha-v_1)$$

$\alpha$ = bend angle; $F_x$ = anchoring force on fluid. Reaction on bend is $-F_x$.

Hydraulic power

$$\dot W=\gamma Q h=\rho g Q h$$

Fluid power for a pump/turbine head $h$; divide by efficiency for brake/input power.

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics (Continuity, Energy, and Bernoulli Equations) · NCEES FE Reference Handbook — Fluid Mechanics (Impulse-Momentum Principle; Pipe Bends) · NCEES FE Reference Handbook — Fluid Mechanics (Head Loss; Hydraulic and Energy Grade Lines)

Traverse, Bearings, and Coordinates

Bearings versus azimuths, the interior-angle check, latitudes and departures, traverse closure, the compass (Bowditch) rule, and coordinate computation — the spine of plane surveying.

Latitude and departure

$$\text{Lat}=L\cos\alpha,\qquad \text{Dep}=L\sin\alpha$$

Components of a course of length $L$ (ft) at angle $\alpha$ from the meridian. Lat positive N / negative S; Dep positive E / negative W.

Bearing-to-azimuth (by quadrant)

$$\text{Az}=\begin{cases}\beta & \text{NE}\\ 180^\circ-\beta & \text{SE}\\ 180^\circ+\beta & \text{SW}\\ 360^\circ-\beta & \text{NW}\end{cases}$$

Azimuth measured clockwise from north, $0^\circ$–$360^\circ$; $\beta$ = bearing angle ($0^\circ$–$90^\circ$).

Back azimuth

$$\text{Az}_{back}=\text{Az}\pm 180^\circ$$

Use $+180^\circ$ if the forward azimuth is below $180^\circ$, else $-180^\circ$. Back bearing reverses both letters, same angle.

Interior-angle check

$$\sum\theta_{\text{int}}=(n-2)\,180^\circ$$

Geometric sum of interior angles for an $n$-sided closed polygon traverse; basis of the angular misclosure check.

Angular misclosure correction

$$C_\theta=-\frac{\sum\theta_{\text{meas}}-(n-2)180^\circ}{n}$$

Equal correction applied to each measured angle when the misclosure is within tolerance.

Linear misclosure

$$e=\sqrt{\left(\textstyle\sum\text{Lat}\right)^2+\left(\textstyle\sum\text{Dep}\right)^2}$$

Resultant gap closing the traverse (ft). Ideally zero; the closures in Lat and Dep are its components.

Precision (relative accuracy)

$$\text{Precision}=\frac{\sum L}{e}=1\!:\!\left(\sum L/e\right)$$

Perimeter $\sum L$ divided by linear misclosure; reported as a ratio. Larger denominator = better survey.

Compass (Bowditch) rule

$$C_{\text{Lat},i}=-\left(\textstyle\sum\text{Lat}\right)\frac{L_i}{\sum L},\quad C_{\text{Dep},i}=-\left(\textstyle\sum\text{Dep}\right)\frac{L_i}{\sum L}$$

Distributes closure in proportion to course length. Adjusted Lat and Dep then each sum to zero.

Forward coordinate computation

$$N_{i+1}=N_i+\text{Lat}_i,\qquad E_{i+1}=E_i+\text{Dep}_i$$

Running sum of adjusted latitudes (northings) and departures (eastings). Returns to start if balanced.

Inverse (length and bearing from coordinates)

$$L=\sqrt{\Delta N^2+\Delta E^2},\qquad \tan\beta=\left|\frac{\Delta E}{\Delta N}\right|$$

Recovers a line from two coordinate pairs; quadrant of the bearing set by the signs of $\Delta N$ and $\Delta E$.

Where it lives: NCEES FE Reference Handbook — Civil Engineering (Latitudes and Departures) · NCEES FE Reference Handbook — Mathematics (Trigonometry) · Ghilani & Wolf, Elementary Surveying — Traverse computations and the compass rule

Hydrology: Rational Method, Rainfall, Runoff, and Flood Control

Convert rainfall into peak runoff with the rational and NRCS methods, read IDF curves through time of concentration, and size storage with return-period thinking.

Rational formula

$$Q = C\,i\,A$$

Peak discharge $Q$ (cfs) from runoff coefficient $C$ (dimensionless, 0-1), intensity $i$ (in/hr), and area $A$ (acres). Storm duration set to $t_c$.

Composite runoff coefficient

$$C_{comp} = \dfrac{\sum_j C_j A_j}{\sum_j A_j}$$

Area-weighted $C$ for a mixed-cover watershed; each subarea $A_j$ has coefficient $C_j$.

NRCS (SCS) runoff depth

$$Q = \dfrac{(P - 0.2S)^2}{P + 0.8S}$$

Runoff depth $Q$ (in) from storm depth $P$ (in); valid only when $P > 0.2S$. Below that no runoff occurs.

Maximum retention from curve number

$$S = \dfrac{1000}{CN} - 10$$

$S$ (in) is the potential maximum retention; $CN$ (30-98) encodes soil group, land use, and antecedent moisture.

Time of concentration (Kirpich — NOT in FE Handbook)

$$t_c = 0.0078\,\dfrac{L^{0.77}}{S^{0.385}}$$

The FE Reference Handbook gives NO $t_c$ formula, so the exam will supply whatever method/constant it wants you to use; this Kirpich form (empirical, $t_c$ in min, $L$ = flow length in ft, $S$ = slope in ft/ft) is shown only so the symbol $t_c$ is familiar. Do not assume handbook provenance — use whatever the problem provides.

Surface-water hydrologic budget

$$P + Q_{in} - Q_{out} + Q_g - E_s - T_s - I = \Delta S_s$$

Mass balance: precipitation and inflows minus outflows and losses equal change in surface storage.

Return period and annual probability

$$T = \dfrac{1}{p}$$

$T$ (years) is the reciprocal of the annual exceedance probability $p$. The 100-yr flood has $p = 0.01$.

Risk of exceedance in n years

$$R = 1 - \left(1 - \dfrac{1}{T}\right)^{n}$$

Probability a $T$-year event is exceeded at least once during $n$ years of project life.

Pan evaporation

$$E_L = P_c\,E_p$$

Lake evaporation $E_L$ from pan evaporation $E_p$ and pan coefficient $P_c$ (0.3-0.85, commonly 0.7).

Runoff volume from depth

$$V = Q\,(\text{in}) \times A \div 12$$

Volume (acre-ft) = runoff depth (in) times area (acres) divided by 12 (in/ft).

Where it lives: NCEES FE Reference Handbook — Civil Engineering: Hydrology · NCEES FE Reference Handbook — Civil Engineering: NRCS/SCS Rainfall-Runoff · USDA NRCS, National Engineering Handbook Part 630 (Hydrology)

Determinacy, Stability, and Indeterminate Structures

Count reactions, members, and joints to classify beams, trusses, and frames, then solve a single-degree indeterminate structure by the force method.

Truss classification

$$m + r \;\lessgtr\; 2j$$

$<2j$ unstable; $=2j$ determinate; $>2j$ indeterminate. $m$ = bars, $r$ = reactions, $j$ = joints.

Frame classification

$$3m + r \;\lessgtr\; 3j + c$$

$<$ unstable; $=$ determinate; $>$ indeterminate. $c$ = condition (release) equations.

Degree of static indeterminacy (frame)

$$\text{DSI} = (3m + r) - (3j + c)$$

Number of redundants; the count of extra equations (compatibility) needed beyond statics.

Degree of indeterminacy (truss)

$$\text{DSI} = (m + r) - 2j$$

Surplus of unknowns over the $2j$ joint-equilibrium equations.

Reaction components

$$r_{\text{roller}}=1,\ r_{\text{pin}}=2,\ r_{\text{fixed}}=3$$

Independent reaction components contributed by common 2-D supports.

Compatibility (force method)

$$\Delta_0 + R\,\delta_{11} = 0$$

$\Delta_0$ = deflection at the release under real load; $\delta_{11}$ = deflection per unit redundant $R$.

Propped-cantilever redundant

$$R_B = \dfrac{3wL}{8}$$

Roller reaction of a fixed-pinned beam under full-span UDL $w$; classic single-degree result.

Distribution factor

$$DF_i = \dfrac{(EI/L)_i}{\sum (EI/L)}$$

Fraction of joint unbalance taken by member $i$ in moment distribution; sums to 1 at a joint.

Fixed-end moment (UDL)

$$\text{FEM} = \dfrac{wL^2}{12}$$

Magnitude of the end moment of a fixed-fixed beam under uniform load; starting value for moment distribution.

Where it lives: NCEES FE Reference Handbook — Civil Engineering: Stability, Determinacy, and Classification of Structures · NCEES FE Reference Handbook — Civil Engineering: Elementary Statically Indeterminate Structures by Force Method · NCEES FE Reference Handbook — Mechanics of Materials · Hibbeler, Structural Analysis

Phase Relations and Effective Stress

The three-phase diagram, unit weights, and the principle of effective stress (σ' = σ − u) that governs every strength and settlement calculation in soil.

Void ratio and porosity

$$e = \dfrac{V_v}{V_s}, \qquad n = \dfrac{V_v}{V} = \dfrac{e}{1+e}$$

Two measures of void space; $e$ referenced to solids, $n$ to total volume.

Water content

$$w = \dfrac{W_w}{W_s}\times 100\%$$

Weight of water per weight of dry solids; from oven-drying.

Master saturation identity

$$S\,e = w\,G_s$$

Links saturation $S$, void ratio $e$, water content $w$, and specific gravity $G_s$. At full saturation $e = wG_s$.

Total (moist) unit weight

$$\gamma = \dfrac{W}{V} = \dfrac{(G_s + S e)\,\gamma_w}{1+e}$$

Includes pore water present. $\gamma_w = 9.81\,\text{kN/m}^3 = 62.4\,\text{lb/ft}^3$.

Dry unit weight

$$\gamma_d = \dfrac{\gamma}{1+w} = \dfrac{G_s\,\gamma_w}{1+e}$$

Solids weight per total volume; the target of compaction control.

Saturated unit weight

$$\gamma_{sat} = \dfrac{(G_s + e)\,\gamma_w}{1+e}$$

All voids full of water ($S=100\%$).

Submerged (effective) unit weight

$$\gamma' = \gamma_{sat} - \gamma_w$$

Buoyant unit weight below the water table; use to get $\sigma'$ directly.

Effective stress

$$\sigma' = \sigma - u$$

Terzaghi's principle; grain-skeleton stress governs strength and settlement.

Pore water pressure

$$u = \gamma_w\,h_w$$

Hydrostatic; $h_w$ = height of water above the point (zero above the water table).

Specific gravity of solids

$$G_s = \dfrac{W_s/V_s}{\gamma_w}$$

Grain density relative to water; typically 2.65–2.72.

Where it lives: NCEES FE Reference Handbook — Civil Engineering (Geotechnical, Phase Relationships) · ASTM D854 — Specific Gravity of Soil Solids · Terzaghi, Peck & Mesri, Soil Mechanics in Engineering Practice

Horizontal Curves and Superelevation

Simple-curve geometry (R, T, L, M, E), degree of curve, the e+f superelevation balance, minimum radius, and horizontal sight distance for the FE Civil exam.

Tangent distance

$$T = R\tan\dfrac{I}{2}$$

Distance from $PC$ (or $PT$) to the $PI$. $R$ in ft, $I$ = intersection/deflection angle in degrees.

Curve (arc) length

$$L = \dfrac{R\,I\,\pi}{180} = \dfrac{100\,I}{D}$$

Length of arc from $PC$ to $PT$ (ft). Uses the FULL central angle $I$, not $I/2$.

Long chord

$$LC = 2R\sin\dfrac{I}{2}$$

Straight-line distance from $PC$ to $PT$ (ft).

Middle ordinate

$$M = R\left(1-\cos\dfrac{I}{2}\right)$$

Offset from midpoint of long chord to midpoint of arc (ft).

External distance

$$E = R\left(\sec\dfrac{I}{2}-1\right)$$

Distance from $PI$ to midpoint of arc (ft); $\sec=1/\cos$.

Degree of curve (arc definition)

$$D = \dfrac{5729.58}{R}$$

$D$ (degrees) subtends a $100\,\text{ft}$ arc. $R$ in ft. Larger $D$ = sharper curve.

Superelevation balance

$$0.01e + f = \dfrac{V^2}{15\,R}$$

Handbook form with $e$ = superelevation in PERCENT, $f$ = side-friction factor, $V$ in mph, $R$ in ft. The equivalent reduced form $e+f=V^2/(15R)$ takes $e$ as a decimal. Constant $15$ embeds $g$ and unit conversion.

Minimum radius

$$R_{\min} = \dfrac{V^2}{15\,(e_{\max}+f_{\max})}$$

Sharpest allowable curve for a design speed $V$ (mph) given maximum $e$ and $f$ as DECIMALS (reduced form). With $e$ in percent use $15\,(0.01e_{\max}+f_{\max})$ in the denominator.

Spiral transition length

$$L_s = \dfrac{3.15\,V^3}{R\,C}$$

Length of spiral easement (ft); $V$ in mph, $R$ in ft, $C$ = rate of change of lateral acceleration (use $1\,\text{ft/s}^3$ unless stated).

Horizontal sightline offset

$$HSO = R\left[1-\cos\!\left(\dfrac{28.65\,S}{R}\right)\right]$$

Lateral clearance to a sight obstruction (ft); $S$ = sight distance along arc (ft), $R$ in ft. Argument is in degrees.

Curve stationing

$$\text{Sta }PT = \text{Sta }PC + L,\quad \text{Sta }PC = \text{Sta }PI - T$$

Stations advance along the alignment; $PT$ is reached via the arc $L$, never by adding $T$ to the $PI$.

Where it lives: NCEES FE Reference Handbook — Civil Engineering (Horizontal Curves) · AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book)

CPM Scheduling and Project Controls

Build an activity-on-node network, run the forward and backward passes for float and the critical path, then crash and track it with earned value (CPI/SPI).

Early finish (forward pass)

$$EF = ES + D$$

Earliest an activity can finish; $D$ = duration. In day-1 counting use $EF = ES + D - 1$.

Early start

$$ES = \max_{\text{predecessors}}(EF)$$

An activity cannot start until its latest-finishing predecessor is done.

Late start (backward pass)

$$LS = LF - D$$

Latest an activity can start without delaying the project.

Late finish

$$LF = \min_{\text{successors}}(LS)$$

Must finish before the earliest-starting successor's late start; last activity seeded at project duration.

Total float

$$TF = LS - ES = LF - EF$$

Slack before the project finish slips. $TF = 0$ defines the critical path.

Free float

$$FF = \min(ES_{\text{succ}}) - EF$$

Slack before any successor's early start slips; $FF \le TF$. The FE Reference Handbook 10.6 prints only total float (Float = LS − ES or LF − EF), not free float.

Crash cost slope

$$\text{slope} = \dfrac{C_{\text{crash}} - C_{\text{normal}}}{D_{\text{normal}} - D_{\text{crash}}}$$

Dollars per day to compress an activity; crash the cheapest critical activity first.

Cost variance

$$CV = EV - AC$$

Budgeted value earned minus actual cost; negative = over budget. $EV$=BCWP, $AC$=ACWP.

Schedule variance

$$SV = EV - PV$$

Earned minus planned value; negative = behind schedule. $PV$=BCWS (all in dollars).

Cost performance index

$$CPI = \dfrac{EV}{AC}$$

Dimensionless cost efficiency; $<1$ means each dollar buys less than a dollar of budgeted work.

Schedule performance index

$$SPI = \dfrac{EV}{PV}$$

Dimensionless schedule efficiency; $<1$ means work is being earned slower than planned.

Estimate at completion

$$EAC = \dfrac{BAC}{CPI}$$

Forecast final cost assuming current $CPI$ holds; $BAC$ = budget at completion.