FE Civil Formula Cheat Sheet

The FE Civil exam is open-book — NCEES gives you the digital FE Reference Handbook during the test. The goal isn't memorizing every formula, it's recognizing which one applies fast. Here are the highest-yield formulas per subject to drill until they're automatic.

Math & Stats

Quadratic Formula

x = (−b ± √(b²−4ac)) / 2a

Roots of any quadratic ax² + bx + c = 0.

Power Rule (derivative)

d/dx [xⁿ] = n xⁿ⁻¹

Degree drops by 1 · n = any real exponent.

Standard Deviation

σ = √[Σ(xᵢ − μ)² / n]

Measures spread of data around the mean μ.

Statics

Force Equilibrium

ΣFx = 0 · ΣFy = 0

Sum of force components along each axis is zero for any static body.

Moment Equilibrium

ΣM = 0

Sum of moments about any point is zero.

Parallel Axis Theorem

I = I꜀ + Ad²

I꜀ = centroidal moment of inertia · A = area · d = distance between axes.

Dynamics

Newton's Second Law

ΣF = ma

Net force equals mass times acceleration — the foundation of kinetics.

Projectile Time of Flight

t = 2v₀sin(θ) / g

Time in the air when launch and landing elevations are equal.

Mechanics of Materials

Normal Stress

σ = P / A

Axial force P over cross-sectional area A.

Flexure Formula

σ = Mc / I

Bending stress from moment M, distance to outer fiber c, moment of inertia I.

Hooke's Law

σ = Eε

Stress equals elastic modulus times strain, within the elastic limit.

Materials

Water-Cement Ratio

w/c = W_water / W_cement

Lower w/c generally means higher strength, lower permeability.

ACI Required Avg. Strength

f'cr = f'c + 1.34s

Mix design target strength · s = standard deviation of past tests.

Fluid Mechanics

Bernoulli Equation

P/γ + v²/2g + z = const.

Valid along a streamline in steady, inviscid flow.

Continuity

Q = A₁v₁ = A₂v₂

Conservation of mass for steady incompressible flow.

Reynolds Number

Re = vD / ν

Re < 2000: laminar · Re > 4000: turbulent.

Hydraulics & Hydrology

Manning's Equation

Q = (1/n)·A·R^(2/3)·S^(1/2)

Open-channel flow · n = roughness · R = hydraulic radius · S = slope.

Rational Method

Q = CiA

Peak discharge · C = runoff coefficient · i = rainfall intensity · A = drainage area.

Darcy's Law

Q = K · i · A

Groundwater seepage flow · K = hydraulic conductivity · i = hydraulic gradient.

Structural Analysis

SS Beam — Uniform Load

M_max = wL² / 8

Maximum moment at midspan for a simply supported beam.

SS Beam — Point Load

M_max = PL / 4

Maximum moment at midspan under a concentrated load.

Truss Determinacy

i = m + r − 2j

i = 0: determinate · m = members · r = reactions · j = joints.

Geotechnical Engineering

Effective Stress

σ′ = σ − u

Stress carried by the soil skeleton · u = pore water pressure.

Mohr-Coulomb

τf = c′ + σ′ tan φ′

Shear strength from effective cohesion and friction angle.

Terzaghi Bearing Capacity

qu = cNc + qNq + 0.5γBNγ

Ultimate bearing capacity of a shallow foundation.

Transportation Engineering

Degree of Curve

D = 5729.58 / R

Arc definition, R in feet, 1 station = 100 ft.

Stopping Sight Distance

SSD = 1.47Vt + V² / [30(f ± G)]

V = speed (mph) · t = perception-reaction time (2.5s per AASHTO).

Flow-Speed-Density

q = k · v

The fundamental traffic flow equation.

Construction Engineering

Total Float

TF = LS − ES = LF − EF

TF = 0 marks the critical path in a CPM schedule.

Cost Performance Index

CPI = EV / AC

CPI > 1.0 = under budget · CPI < 1.0 = over budget.

Crew Productivity

Output = Crew Size × Hours × Rate

Estimates duration for a work item given a known production rate.

Environmental Engineering

BOD Rate Equation

BODt = BODu(1 − e^(−k₁t))

Oxygen demand as a function of time.

Hydraulic Retention Time

θ = V / Q

Average time fluid spends in a treatment reactor.

Engineering Economics

Single Payment F/P

F = P(1+i)ⁿ

Future value from present value, compound interest.

Capital Recovery A/P

A = P·i(1+i)ⁿ / [(1+i)ⁿ−1]

Equivalent uniform annual cost — the mortgage formula.

Ethics & Professional Practice

Fundamental Canon #1

Safety, health & welfare of the public

Paramount above all other duties in the NSPE Code of Ethics.

Licensure Path

FE (EIT) → experience → PE

Must be licensed to offer engineering services to the public.

Surveying

Latitude / Departure

Lat = L·cos(brg) · Dep = L·sin(brg)

North-south and east-west components of a traverse line.

Linear Misclosure

e = √[(ΣLat)² + (ΣDep)²]

Error of closure for a traverse loop.

This is a starting point, not the whole handbook — the real FE Reference Handbook runs well over 200 pages. The point of drilling these isn't recall under pressure with no reference; it's building enough familiarity that you can find and apply the right formula in under a minute during the actual exam.

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