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NCEES FE Civil Exam · 5 hr 20 min · 110 questions · Computer-based
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Mathematics & Statistics
~7-11% of the FE Civil exam
Key Focus: Derivatives, integrals, differential equations, matrix operations, and probability distributions. These appear across many other topics too.
Algebra & Trigonometry
Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
a, b, c = polynomial coefficients · x = roots · ± gives two solutions
Law of Sines
a/sin A = b/sin B = c/sin C
a, b, c = side lengths · A, B, C = angles opposite those sides
Law of Cosines
c² = a² + b² − 2ab cos C
c = side opposite angle C · a, b = other two sides · C = included angle
Pythagorean Identity
sin²θ + cos²θ = 1
θ (theta) = any angle · fundamental trigonometric identity
Double Angle
sin 2θ = 2 sin θ cos θ
θ (theta) = any angle · also: cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ
Logarithm Rules
log(ab) = log a + log b
log(a/b) = log a − log b · log(aⁿ) = n·log a · ln = natural log (base e ≈ 2.718)
Calculus
Power Rule (derivative)
d/dx [xⁿ] = n xⁿ⁻¹
x = variable · n = any real exponent · degree drops by 1
Chain Rule
d/dx [f(g(x))] = f′(g(x))·g′(x)
f, g = nested functions · f′, g′ = their derivatives · multiply outer by inner derivative
Power Rule (integral)
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
x = variable · n = exponent (n ≠ −1) · C = constant of integration
Fundamental Theorem
∫ₐᵇ f(x) dx = F(b) − F(a)
F = antiderivative of f · a, b = lower and upper limits of integration
Integration by Parts
∫ u dv = uv − ∫ v du
u, v = parts chosen to simplify · pick u to differentiate and dv to integrate
Taylor Series
f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
a = center of expansion · f⁽ⁿ⁾(a) = nth derivative at a · n! = n factorial · Σ = sum over all n
Probability & Statistics
Mean
μ = Σ xᵢ / n
μ (mu) = population mean · xᵢ = each data value · n = number of values · Σ = summation
Variance
σ² = Σ(xᵢ − μ)² / n
σ² (sigma squared) = variance (spread²) · xᵢ = each value · μ (mu) = mean · n = count
Standard Deviation
σ = √[Σ(xᵢ − μ)² / n]
σ (sigma) = standard deviation (same units as data) · measures spread around the mean μ
Normal Distribution
f(x) = e−(x−μ)²/2σ² / (σ√2π)
μ (mu) = mean (center) · σ (sigma) = standard deviation · π ≈ 3.14159 · e ≈ 2.71828
Binomial Probability
P(x) = C(n,x)·pˣ·(1−p)ⁿ⁻ˣ
n = trials · x = successes · p = probability per trial · C(n,x) = n!/(x!(n−x)!)
Bayes' Theorem
P(A|B) = P(B|A)·P(A) / P(B)
P(A|B) = probability of A given B · P(B|A) = likelihood · P(A), P(B) = prior probabilities
Statics
~7-11% of the FE Civil exam
Key Focus: Equilibrium equations, free body diagrams, truss analysis (method of joints & sections), centroids, and moments of inertia.
Equilibrium
Force Equilibrium
ΣFx = 0 · ΣFy = 0 · ΣFz = 0
Σ (sigma) = sum · Fx, Fy, Fz = force components along each axis · holds for any static body
Moment Equilibrium
ΣM = 0
M = moment · Σ = sum · can be taken about any convenient point
Moment of Force
M = F · d
M = moment (N·m or lb·ft) · F = force magnitude · d = perpendicular distance from line of action to pivot
Vector Cross Product
M = r × F
r = position vector from pivot to force application point · F = force vector · × = cross product
Dot Product
A · B = |A||B| cos θ
A, B = vectors · |A|, |B| = their magnitudes · θ (theta) = angle between A and B
Resultant Force
FR = √(Fx² + Fy²)
FR = resultant magnitude · Fx, Fy = x and y components · direction: θ = arctan(Fy / Fx)
Centroids & Moment of Inertia
Centroid (composite)
x̄ = Σ Aᵢ x̄ᵢ / Σ Aᵢ
x̄ = centroid location · Aᵢ = area of each sub-shape · x̄ᵢ = centroid x-coord of that sub-shape
Rectangle — Moment of Inertia
Iₓ = bh³/12
Iₓ = moment of inertia about centroidal horizontal axis · b = base width · h = height
Circle — Moment of Inertia
Iₓ = πr⁴/4
Iₓ = moment of inertia · r = radius · π ≈ 3.14159 · about any diameter
Parallel Axis Theorem
I = Iᶜ + Ad²
I = moment of inertia about offset axis · Iᶜ = centroidal MOI · A = area · d = distance between parallel axes
Friction
Friction Force
F = μN
F = friction force · μ (mu) = coefficient of friction (μs static, μk kinetic) · N = normal force
Angle of Friction
φ = arctan(μ)
φ (phi) = friction angle · μ (mu) = coefficient of friction · angle at which sliding impends
Mechanics of Materials
~8-12% of the FE Civil exam
Key Focus: Stress/strain, beam bending (flexure formula), shear stress, deflections, Mohr's circle, and combined loading.
Stress & Strain
Normal Stress
σ = P / A
σ (sigma) = normal stress (Pa or psi) · P = axial force · A = cross-sectional area · tension is positive
Shear Stress
τ = V / A
τ (tau) = average shear stress · V = shear force · A = cross-sectional area
Normal Strain
ε = δ / L = σ / E
ε (epsilon) = normal strain (dimensionless) · δ (delta) = total deformation · L = original length · E = Young's modulus
Shear Strain
γ = τ / G
γ (gamma) = shear strain (rad) · τ (tau) = shear stress · G = shear modulus
Hooke's Law
σ = Eε
σ (sigma) = stress · E = elastic (Young's) modulus · ε (epsilon) = strain · valid within the elastic limit
Poisson's Ratio
ν = −εlat / εax
ν (nu) = Poisson's ratio (≈ 0.25–0.33 for metals) · εlat = lateral (transverse) strain · εax = axial strain
Shear–Elastic Modulus
G = E / [2(1 + ν)]
G = shear modulus · E = elastic modulus · ν (nu) = Poisson's ratio · relates all three elastic constants
Thermal Deformation
δT = α · ΔT · L
δT (delta) = thermal elongation · α (alpha) = coeff. of thermal expansion (1/°C) · ΔT (delta T) = temperature change · L = original length
Beams — Bending & Shear
Flexure Formula
σ = M · c / I
σ (sigma) = bending stress · M = bending moment · c = distance from neutral axis to outermost fiber · I = moment of inertia
Section Modulus
σmax = M / S · S = I/c
σmax = peak bending stress · S = section modulus · M = bending moment · I = moment of inertia · c = distance to outermost fiber
Shear Stress in Beams
τ = VQ / (Ib)
τ (tau) = shear stress at a horizontal cut · V = shear force · Q = first moment of area above cut (∫y dA) · I = moment of inertia · b = width at cut
Beam Deflection — UDL
δmax = 5wL⁴ / 384EI
δ (delta) = max deflection at midspan · w = uniform load per unit length · L = span · E = elastic modulus · I = moment of inertia
Beam Deflection — Point
δmax = PL³ / 48EI
δ (delta) = max deflection at midspan · P = concentrated midspan load · L = span · E = elastic modulus · I = moment of inertia
Cantilever Deflection
δmax = PL³ / 3EI
δ (delta) = deflection at free end · P = concentrated load at free end · L = cantilever length · E = elastic modulus · I = moment of inertia
Torsion & Columns
Torsional Shear Stress
τ = Tc / J
τ (tau) = shear stress at outer surface · T = applied torque · c = outer radius · J = polar moment of inertia
Polar Moment (solid)
J = πc⁴ / 2
J = polar moment of inertia · c = outer radius · π ≈ 3.14159 · for hollow: J = π(co⁴ − ci⁴)/2
Euler Column Buckling
Pcr = π²EI / (KL)²
Pcr = critical buckling load · π ≈ 3.14159 · E = elastic modulus · I = moment of inertia · K = effective length factor (pin-pin = 1.0, fixed-free = 2.0) · L = column length
Slenderness Ratio
SR = KL / r · r = √(I/A)
SR = slenderness ratio · K = end-condition factor · L = length · r = radius of gyration · I = moment of inertia · A = cross-sectional area
Fluid Mechanics
~7-11% of the FE Civil exam
Key Focus: Fluid properties, hydrostatics, Bernoulli equation, continuity equation, pipe flow (Darcy-Weisbach), and dimensional analysis.
Fluid Properties
Density & Specific Weight
γ = ρg
γ (gamma) = specific weight (N/m³ or lb/ft³) · ρ (rho) = mass density (kg/m³) · g = 9.81 m/s² · ρwater = 1000 kg/m³ · γwater = 9.81 kN/m³
Specific Gravity
SG = ρ / ρwater
SG = specific gravity (dimensionless) · ρ (rho) = fluid density · ρwater = 1000 kg/m³ · SG > 1 means denser than water
Dynamic Viscosity
τ = μ (dv/dy)
τ (tau) = shear stress · μ (mu) = dynamic viscosity (Pa·s) · dv/dy = velocity gradient (rate of shearing strain)
Kinematic Viscosity
ν = μ / ρ
ν (nu) = kinematic viscosity (m²/s) · μ (mu) = dynamic viscosity · ρ (rho) = mass density
Hydrostatics
Hydrostatic Pressure
P = γh = ρgh
P = pressure at depth h · γ (gamma) = specific weight · ρ (rho) = density · g = gravity · h = depth below free surface
Force on Plane Surface
F = γ · h̄ · A
F = total hydrostatic force · γ (gamma) = specific weight · h̄ = depth to centroid of the submerged area · A = submerged area
Center of Pressure
ycp = ȳ + Iᶜ / (ȳ · A)
ycp = location of resultant force · ȳ = distance to centroid along inclined surface · Iᶜ = centroidal moment of inertia · A = area
Buoyancy (Archimedes)
FB = γ · Vsub
FB = buoyant force · γ (gamma) = specific weight of the fluid · Vsub = volume of fluid displaced by the object
Energy & Flow
Bernoulli Equation
P/γ + v²/2g + z = constant
P = pressure · γ (gamma) = specific weight · v = velocity · g = gravity · z = elevation · each term = head (length) · valid along a streamline in inviscid steady flow
Continuity (incompressible)
Q = A₁v₁ = A₂v₂
Q = volumetric flow rate (m³/s) · A = cross-sectional area · v = average velocity · conservation of mass for steady incompressible flow
Reynolds Number
Re = ρvD / μ = vD / ν
Re = Reynolds number (dimensionless) · ρ (rho) = density · v = velocity · D = pipe diameter · μ (mu) = dynamic viscosity · ν (nu) = kinematic viscosity · Re < 2000: laminar · Re > 4000: turbulent
Darcy-Weisbach
hL = f (L/D)(v²/2g)
hL = head loss · f = Darcy friction factor (Moody chart) · L = pipe length · D = diameter · v = velocity · g = gravity
Hazen-Williams
V = 0.849 · C · R0.63 · S0.54
V = velocity (m/s) · C = roughness coefficient (higher = smoother) · R = hydraulic radius · S = slope (head loss per unit length)
Flow Rate
Q = V · A
Q = volumetric flow rate · V = average flow velocity · A = cross-sectional flow area
Orifice Flow
Q = Cd · A · √(2gh)
Q = flow rate · Cd = discharge coefficient (≈ 0.61 sharp-edged) · A = orifice area · g = gravity · h = head driving flow
Energy Equation (full)
P₁/γ + v₁²/2g + z₁ + hp = P₂/γ + v₂²/2g + z₂ + hL
hp = pump head added · hL = head loss between sections · subscripts 1, 2 = upstream and downstream locations
Hydraulics & Hydrological Systems
~9-14% of the FE Civil exam — you know this well!
Key Focus: Manning's equation, open-channel flow, weirs, culverts, Rational Method, unit hydrograph, and flood routing. This is your strength — make sure you can apply these fast.
Open Channel Flow
Manning's Equation
Q = (1/n) · A · R2/3 · S1/2
Q = discharge (m³/s) · n = Manning's roughness coefficient · A = flow area · R = hydraulic radius (A/Pw) · S = channel slope · use 1.486/n for US customary
Hydraulic Radius
R = A / Pw
R = hydraulic radius · A = cross-sectional flow area · Pw = wetted perimeter (boundary in contact with water, not the free surface)
Froude Number
Fr = V / √(gD)
Fr = Froude number (dimensionless) · V = velocity · g = gravity · D = hydraulic depth (A/T) · T = top width · Fr < 1: subcritical · Fr = 1: critical · Fr > 1: supercritical
Critical Depth (rect.)
yc = (q²/g)1/3
yc = critical depth · q = unit discharge (Q/b) · b = channel width · g = gravity · occurs when Fr = 1
Specific Energy
E = y + V²/2g
E = specific energy · y = flow depth · V = velocity · g = gravity · V²/2g = velocity head
Chezy Equation
V = C √(RS)
V = velocity · C = Chezy resistance coefficient · R = hydraulic radius · S = channel slope
Weirs & Culverts
Sharp-Crested Weir
Q = (2/3) · Cd · L · √(2g) · H3/2
Q = flow rate · Cd = discharge coefficient (≈ 0.61) · L = weir crest length · g = gravity · H = head above weir crest
V-Notch Weir
Q = (8/15) · Cd · tan(θ/2) · √(2g) · H5/2
θ (theta) = total notch angle · Cd ≈ 0.61 · H = head above notch vertex · g = gravity
Broad-Crested Weir
Q ≈ 1.706 · L · H3/2
SI units (m, m³/s) · L = weir length · H = head above crest · 1.706 includes g and Cd
Orifice (submerged)
Q = Cd · A · √(2g · Δh)
Q = flow rate · Cd = discharge coefficient · A = orifice area · g = gravity · Δh (delta h) = head differential across orifice
Hydrology — Rainfall-Runoff
Rational Method
Q = CiA
Q = peak discharge (cfs) · C = runoff coefficient (0–1; higher = more impervious) · i = rainfall intensity (in/hr) · A = drainage area (acres)
SCS Curve Number
Q = (P − 0.2S)² / (P + 0.8S)
Q = runoff depth (in) · P = total precipitation (in) · S = max potential retention = 1000/CN − 10 · CN = curve number (0–100)
Time of Concentration
Tc = Tsheet + Tshallow + Tchannel
Tc = travel time from farthest point to outlet · three flow-regime segments summed
Kirpich Formula
Tc = 0.0078 · L0.77 · S−0.385
Tc = time of concentration (min) · L = longest flow path (ft) · S = average watershed slope (ft/ft)
Unit Hydrograph
Q(t) = Σ P(τ) · U(t−τ)
Q(t) = discharge at time t · P(τ) = rainfall excess in interval τ · U(t−τ) = unit hydrograph ordinate · Σ = convolution sum
Storage Routing
(I₁+I₂)/2 − (O₁+O₂)/2 = ΔS/Δt
I = inflow · O = outflow · ΔS (delta S) = change in storage · Δt (delta t) = time step
Groundwater
Darcy's Law
Q = K · i · A
Q = seepage flow · K = hydraulic conductivity (m/s) · i = hydraulic gradient (Δh/L) · A = cross-sectional area perpendicular to flow
Hydraulic Gradient
i = Δh / L
i = hydraulic gradient (dimensionless) · Δh (delta h) = head difference between two points · L = distance between those points
Thiem Well Equation
Q = πK(h₂² − h₁²) / ln(r₂/r₁)
Q = pumping rate · K = hydraulic conductivity · h₁, h₂ = water table heights at radii r₁, r₂ · π ≈ 3.14159 · unconfined aquifer, steady-state
Structural Analysis
~10-14% of the FE Civil exam
Key Focus: Determinate structures, influence lines, virtual work, indeterminate structures (force/displacement methods), and structural loads.
Determinacy & Reactions
Determinacy (truss)
i = m + r − 2j
i = degree of indeterminacy · m = members · r = reactions · j = joints · i = 0: determinate · i > 0: indeterminate · i < 0: unstable
Determinacy (frame)
i = 3m + r − 3j
i = degree of indeterminacy · m = members · r = reactions · j = joints · 3 DOF per joint for rigid frames
Virtual Work
δ = ∫ (m · M / EI) dx
δ (delta) = deflection at point of interest · m = moment from virtual unit load · M = real bending moment · E = elastic modulus · I = moment of inertia
Conjugate Beam
M/EI diagram → load on conjugate beam
Shear in conjugate beam = slope of elastic curve · Moment in conjugate beam = deflection · E = elastic modulus · I = moment of inertia
Common Beam Formulas
SS Beam — Uniform Load
Mmax = wL²/8 (midspan)
Mmax = maximum moment · w = uniform load per unit length · L = span · Reactions: R = wL/2 each end
SS Beam — Point Load
Mmax = PL/4 (midspan)
Mmax = maximum moment · P = concentrated load at midspan · L = span
Cantilever — Point Load
Mmax = PL (fixed end)
Mmax = moment at fixed support · P = point load at free end · L = cantilever length
Cantilever — Uniform Load
Mmax = wL²/2 (fixed end)
Mmax = moment at fixed support · w = uniform load per unit length · L = cantilever length
Fixed-Fixed — Uniform Load
Mend = wL²/12 · Mmid = wL²/24
Mend = moment at each fixed end · Mmid = moment at midspan · w = uniform load · L = span
Fixed-Pinned — Uniform Load
Mfixed = wL²/8
Mfixed = moment at fixed end · w = uniform load per unit length · L = span
Geotechnical Engineering
~10-15% of the FE Civil exam
Key Focus: Soil classification, compaction, effective stress, consolidation, shear strength (Mohr-Coulomb), bearing capacity, and slope stability.
Soil Phase Relationships
Void Ratio
e = Vv / Vs
e = void ratio (dimensionless) · Vv = volume of voids (air + water) · Vs = volume of solid particles
Porosity
n = Vv / V = e / (1+e)
n = porosity (decimal) · Vv = volume of voids · V = total volume · e = void ratio
Water Content
w = Ww / Ws × 100%
w = water content (%) · Ww = weight of water · Ws = weight of dry solids
Degree of Saturation
S = Vw / Vv
S = degree of saturation (0 = dry, 1.0 = fully saturated) · Vw = volume of water · Vv = volume of voids
Unit Weight
γ = W / V
γ (gamma) = total unit weight (kN/m³ or pcf) · W = total weight · V = total volume
Dry Unit Weight
γd = γ / (1 + w)
γd = dry unit weight · γ (gamma) = moist unit weight · w = water content expressed as decimal
Saturated Unit Weight
γsat = (Gs + e)γw / (1+e)
γsat = saturated unit weight · Gs = specific gravity of solids (≈ 2.65–2.72) · e = void ratio · γw = unit weight of water (9.81 kN/m³)
Submerged Unit Weight
γ′ = γsat − γw
γ′ (gamma prime) = buoyant/effective unit weight · γsat = saturated unit weight · γw = unit weight of water · use γ′ below the water table
Effective Stress & Consolidation
Effective Stress
σ′ = σ − u
σ′ (sigma prime) = effective stress carried by soil skeleton · σ (sigma) = total vertical stress · u = pore water pressure (= γw · hw)
Primary Consolidation
Sc = Cc·H/(1+e₀) · log(σ′f / σ′₀)
Sc = settlement · Cc = compression index · H = initial layer thickness · e₀ = initial void ratio · σ′f = final effective stress · σ′₀ = initial effective stress
Compression Index
Cc = 0.009(LL − 10)
Cc = compression index (slope of e–log σ′ curve) · LL = liquid limit (%) · approximate for normally consolidated clays
Time Factor
Tv = Cv · t / Hdr²
Tv = dimensionless time factor · Cv = coefficient of consolidation (m²/yr) · t = elapsed time · Hdr = drainage path (H/2 for double drainage)
Shear Strength & Bearing Capacity
Mohr-Coulomb
τf = c′ + σ′ tan φ′
τf (tau) = shear strength · c′ = effective cohesion · σ′ (sigma prime) = effective normal stress · φ′ (phi) = effective friction angle
Terzaghi Bearing Cap.
qu = cNc + qNq + 0.5γBNγ
qu = ultimate bearing capacity · c = cohesion · Nc, Nq, Nγ = bearing capacity factors · q = overburden pressure · γ (gamma) = soil unit weight · B = foundation width
Factor of Safety
FS = qu / qapplied
FS = factor of safety · qu = ultimate bearing capacity · qapplied = applied foundation pressure · typically FS ≥ 3
Active Earth Pressure
Ka = (1 − sin φ) / (1 + sin φ)
Ka = Rankine active pressure coefficient · φ (phi) = effective friction angle · lateral pressure: σh = Ka · σ′v
Passive Earth Pressure
Kp = (1 + sin φ) / (1 − sin φ)
Kp = Rankine passive pressure coefficient · φ (phi) = effective friction angle · note: Kp = 1/Ka
At-Rest Pressure
K₀ = 1 − sin φ
K₀ = at-rest coefficient (Jaky's formula) · φ (phi) = friction angle · for normally consolidated (NC) soils with no lateral strain
Transportation Engineering
~8-12% of the FE Civil exam
Key Focus: Horizontal/vertical curves, sight distance, traffic flow theory, pavement design (AASHTO), and level of service.
Horizontal Curves
Degree of Curve
D = 5729.58 / R
D = degree of curve (°/station) · R = radius (ft) · 1 station = 100 ft · arc definition
Tangent Length
T = R · tan(Δ/2)
T = tangent length (PC to PI) · R = radius · Δ (delta) = central (intersection) angle
Curve Length
L = π · R · Δ / 180
L = arc length · R = radius · Δ (delta) = central angle in degrees · π ≈ 3.14159
External Distance
E = R [1/cos(Δ/2) − 1]
E = external distance (PI to curve midpoint) · R = radius · Δ (delta) = central angle
Middle Ordinate
M = R [1 − cos(Δ/2)]
M = middle ordinate (chord midpoint to curve) · R = radius · Δ (delta) = central angle
Chord Length
C = 2R · sin(Δ/2)
C = long chord (PC to PT) · R = radius · Δ (delta) = central angle
Superelevation
e + f = V² / (127R)
e = superelevation (m/m) · f = side friction factor · V = design speed (km/h) · R = radius (m) · 127 = conversion constant
Vertical Curves
Parabolic Vert. Curve
y = g₁x + (g₂−g₁)x² / (2L)
y = elevation at x · g₁, g₂ = approach and departure grades (decimal) · L = curve length · x = distance from PVC
High / Low Point
x = g₁ · L / (g₁ − g₂)
x = distance from PVC to high/low point · g₁ = approach grade · g₂ = departure grade · L = curve length
Rate of Grade Change
r = (g₂ − g₁) / L
r = rate of grade change per station · g₁, g₂ = grades · L = curve length
Min Length (SSD)
L = A·S² / 2158 (crest, S < L)
L = min curve length (ft) · A = algebraic grade difference (%) · S = stopping sight distance (ft)
Traffic Flow
Flow-Speed-Density
q = k · v
q = flow rate (veh/hr) · k = density (veh/mi) · v = space mean speed · fundamental traffic flow equation
Headway
h = 3600 / q
h = average headway (seconds between successive vehicles) · q = flow rate (veh/hr) · 3600 s/hr
Spacing
s = 5280 / k
s = average vehicle spacing (ft) · k = density (veh/mi) · 5280 ft per mile
Stopping Sight Distance
SSD = 1.47Vt + V² / [30(f ± G)]
SSD = stopping sight distance (ft) · V = speed (mph) · t = perception-reaction time (2.5 s per AASHTO) · f = friction factor · G = grade (decimal, + uphill)
Environmental Engineering
~7-11% of the FE Civil exam
Key Focus: BOD, water/wastewater treatment, reactor kinetics, air quality basics, and solid waste concepts.
Water Quality
BOD Rate Equation
BODt = BODu (1 − e−k₁t)
BODt = oxygen demand at time t · BODu = ultimate BOD · k₁ = first-order decay rate (day⁻¹) · t = time (days) · e ≈ 2.718
DO Sag Equation
D = kd L₀/(kr−kd)·(e−kd t−e−kr t) + D₀e−kr t
D = dissolved oxygen deficit (mg/L) · kd = deoxygenation rate · kr = reaeration rate · L₀ = initial BOD · D₀ = initial DO deficit · t = time
Mass Balance
Σ Qin Cin = Σ Qout Cout + dM/dt
Q = flow rate · C = concentration · dM/dt = rate of mass accumulation · Σ = sum · at steady state dM/dt = 0
Stream Dilution
C = (Q₁C₁ + Q₂C₂) / (Q₁ + Q₂)
C = mixed concentration · Q₁, Q₂ = flow rates · C₁, C₂ = concentrations of the two streams
Reactor Kinetics
CSTR (1st-order, steady-state)
θ = (C₀ − C) / (k · C)
θ (theta) = hydraulic retention time · C₀ = influent concentration · C = effluent concentration · k = first-order rate constant
Plug Flow Reactor
C = C₀ · e−k · θ
C = effluent concentration · C₀ = influent concentration · k = first-order rate constant · θ (theta) = HRT (= V/Q) · e ≈ 2.718
Hydraulic Retention Time
θ = V / Q
θ (theta) = HRT · V = reactor volume (m³) · Q = volumetric flow rate (m³/day) · average time fluid spends in reactor
Engineering Economics
~5-8% of the FE Civil exam
Key Focus: Time value of money (P, F, A factors), NPV, IRR, benefit-cost analysis, depreciation, and breakeven analysis. Factor tables are in the FE Reference Handbook.
Time Value of Money Factors
Single Payment F/P
F = P(1+i)ⁿ
F = future value · P = present value · i = interest rate per period · n = number of periods · (1+i)ⁿ = compound interest factor
Single Payment P/F
P = F / (1+i)ⁿ
P = present value · F = future value · i = interest rate per period · n = number of periods · discounts future money back to today
Uniform Series F/A
F = A [(1+i)ⁿ − 1] / i
F = future value · A = uniform end-of-period payment · i = interest rate · n = number of periods
Uniform Series P/A
P = A [(1+i)ⁿ − 1] / [i(1+i)ⁿ]
P = present value · A = uniform annual payment · i = interest rate · n = number of periods
Capital Recovery A/P
A = P · i(1+i)ⁿ / [(1+i)ⁿ − 1]
A = equivalent uniform annual cost · P = present investment · i = interest rate · n = life in periods · this is the mortgage formula
Arithmetic Gradient P/G
P = G [(1+i)ⁿ−1−ni] / [i²(1+i)ⁿ]
P = present value · G = gradient amount (uniform increase each period) · i = interest rate · n = number of periods
Analysis Methods
Net Present Value
NPV = Σ Bt/(1+i)ᵗ − Σ Ct/(1+i)ᵗ
NPV = net present value · Bt = benefits in year t · Ct = costs in year t · i = discount rate · accept if NPV > 0
Benefit-Cost Ratio
B/C = PW(Benefits) / PW(Costs)
B/C = benefit-cost ratio · PW = present worth · accept if B/C ≥ 1.0 · used for public infrastructure projects
MARR
Accept if IRR ≥ MARR
MARR = Minimum Attractive Rate of Return · IRR = Internal Rate of Return (rate where NPV = 0) · project is acceptable when IRR exceeds MARR
Straight-Line Depreciation
d = (C − Sv) / n
d = annual depreciation · C = initial cost · Sv = salvage value at end of life · n = useful life (years)
Breakeven
Q* = FC / (SP − VC)
Q* = breakeven quantity · FC = fixed costs (do not change with output) · SP = selling price per unit · VC = variable cost per unit
Ethics & Professional Practice
~3-5% of the FE Civil exam
NSPE Code of Ethics — Fundamental Canons
1.Paramount
2.
3.
4.
5.
6.
Key Concepts to Know
Conflict of Interest
Disclose to all parties; do not accept gifts that influence judgment.
Whistleblowing
Engineers must report violations of law or ethical standards that endanger public safety.
PE Licensure
FE (EIT) → Work experience → PE exam. Must be licensed to offer engineering services to the public.
Confidentiality
Engineers shall not disclose confidential client info without consent, unless public safety is at risk.
FE Prep Resources
Curated books and tools to help you pass the FE Civil exam
These are books and resources I personally recommend for FE Civil prep. Links are affiliate links — if you purchase through them it costs you nothing extra and helps keep FEasyPass free. 📚
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NCEES FE Civil Practice Exam OFFICIAL
The real deal — 50-question interactive practice exam from NCEES themselves, matching the actual computer-based test format. Buy it through your MyNCEES account.
Get it from NCEES →
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FE Civil Practice Problems (PPI · Lindeburg)
466 exam-like, 3-minute practice problems with full step-by-step solutions using NCEES Handbook notation. The classic FE Civil problem book.
View on Amazon →
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FE Civil Review Manual (PPI · Lindeburg)
Comprehensive review covering every FE Civil topic with worked examples. Pairs chapter-by-chapter with the Practice Problems book above.
View on Amazon →
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TI-36X Pro Scientific Calculator
The most popular NCEES-approved calculator for the FE exam. Numeric solver, equation solver, and matrix tools — under $25.
View on Amazon →
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Casio fx-115ES PLUS Scientific Calculator
The other top NCEES-approved choice. A solid, budget-friendly alternative to the TI-36X Pro.
View on Amazon →
* As an Amazon Associate, FEasyPass earns from qualifying purchases. All recommendations are genuine — these are the resources I'd use to study for the FE Civil exam.
Full Practice Exam
Timed like the real FE Civil · 5 hr 20 min · All subjects · Step-by-step review after submission
Exam Settings
Real FE Civil: 110 questions · 5 hr 20 min · Open-book (digital handbook). Questions cover all 10 subject areas weighted by official NCEES specs.