Formula quick-reference index

γ = ρg

γ	specific weight — weight per unit volume
ρ	mass density — mass per unit volume
g	gravitational acceleration, 32.2 ft/s² (9.81 m/s²)

SG = γ_substance / γ_water

SG	specific gravity, dimensionless

ν = μ / ρ

μ	dynamic (absolute) viscosity
ν	kinematic viscosity

p = γh

p	gauge pressure at depth h
h	vertical depth below the free surface

F_R = γ h_c A

F_R	magnitude of hydrostatic force on a plane surface
h_c	vertical depth of the surface's centroid
A	area of the surface

↑ Topic 2: Buoyancy & Flotation

y_p = y_c + I_xx,cy_cA

y_p	distance from the free surface to the centre of pressure, measured along the incline
y_c	distance from the free surface to the centroid, measured along the incline
I_xx,c	second moment of area about the centroidal axis (rectangle: bh³/12, with h along the incline)

Topic 2: Buoyancy & Flotation

F_b = γ_fluid · V_displaced

F_b	buoyant force, acting vertically upward through the centroid of the displaced volume
V_displaced	volume of fluid displaced = submerged volume of the body only

↑ Topic 2: Buoyancy & Flotation

Floating equilibrium: W = F_b

W	total weight of the floating body

↑ Topic 2: Buoyancy & Flotation

GM = MB − GB stable if GM > 0

G	centre of gravity of the body
B	centre of buoyancy (centroid of submerged volume)
M	metacentre — where the tilted buoyant-force line crosses the body's centreline
MB = I₀/V_sub	distance from B to M; I₀ is the second moment of the waterplane area about its centroidal axis

Topic 3: Continuity, Energy & Momentum

Q = A₁V₁ = A₂V₂

Q	volumetric flow rate (discharge)
A	cross-sectional area normal to the flow
V	mean velocity through the section

ρ₁A₁V₁ = ρ₂A₂V₂

Used when density changes (gases, compressible flow). For water, ρ cancels and you get Q = AV.

p₁/γ + V₁²/2g + z₁ + h_A = p₂/γ + V₂²/2g + z₂ + h_T + h_L

p/γ	pressure head
V²/2g	velocity head
z	elevation head (same datum for both points!)
h_A	head added by a pump
h_T	head removed by a turbine
h_L	total head loss between 1 and 2

ΣF = ρQ(β₂V₂ − β₁V₁)

ΣF	vector sum of forces on the fluid (pressure + weight + reaction), in the chosen direction
β	momentum correction factor, ≈ 1.0 for turbulent flow — the exam usually lets you drop it

Topic 4: Pipe Flow

h_f = f LD V²2g

h_f	friction head loss
f	Darcy friction factor (dimensionless — not the Fanning factor, which is f/4)
L, D	pipe length and diameter
V	mean velocity

Re = VDν laminar: f = 64Re

Re	Reynolds number; laminar below ≈ 2,300, turbulent above ≈ 4,000
ν	kinematic viscosity of the fluid

Turbulent f: Haaland approximation 1/√f ≈ −1.8 log[(ε/3.7D)^1.11 + 6.9/Re]

ε	absolute roughness of the pipe wall (commercial steel ≈ 0.00015 ft)
Use the Moody chart or Colebrook equation equivalently; Haaland is calculator-friendly.

V = 1.318 · C · R^0.63 · S^0.54 (English units, V in ft/s)

C	Hazen-Williams roughness coefficient (≈ 150 new PVC, 130 new steel, 100 old cast iron)
R	hydraulic radius = A/P (D/4 for a full circular pipe)
S	slope of the energy grade line = h_f/L

h_m = K V²2g

K	minor loss coefficient (entrance ≈ 0.5, exit = 1.0, valves/fittings from tables)
Equivalently, fittings can be converted to an equivalent length of straight pipe.

Series pipes: Q is the same everywhere; head losses add. Parallel pipes: head loss is the same in each branch; discharges add.

Topic 5: Pumps & System Curves

H_sys = H_static + KQ²

H_sys	total head the system demands at flow Q
H_static	static head: elevation lift + (p_discharge − p_suction)/γ
KQ²	friction + minor losses, which scale with the square of flow

Operating point: the (Q, H) where the pump curve crosses the system curve.

Q₂Q₁ = N₂N₁ H₂H₁ = (N₂N₁)² P₂P₁ = (N₂N₁)³

Q	discharge
H	head
P	power
N	rotational speed (or impeller diameter D)

Water power: P = γQHη English shortcut: bhp = Q(gpm) × H(ft)3960 × η

η	pump efficiency as a decimal
The 3960 shortcut already includes unit conversions for water — do not multiply by γ again.

NPSH_A = p_atm/γ − p_v/γ ± h_s − h_L,suction > NPSH_R

NPSH_A	net positive suction head available from the system
NPSH_R	net positive suction head required by the pump (from its curve)
p_v	vapour pressure of the liquid at its temperature
h_s	static suction head: + if the supply sits above the pump, − if the pump must lift (suction lift)
h_L,suction	head loss in the suction piping

Topic 6: Open-Channel Flow

Want this topic as a printable one-pager? See the Open Channel Flow Cheat Sheet — every formula below plus worked examples.

V = (k/n) · R^2/3 · S^1/2 Q = AV

V	mean velocity
Q	discharge
k	unit constant: 1.486 for English units (ft/s), 1.0 for SI
n	Manning's roughness coefficient (concrete ≈ 0.013, earth ≈ 0.022, natural channel ≈ 0.03–0.05)
R	hydraulic radius = A/P, where P is the wetted perimeter
S	longitudinal slope of the channel (energy slope for uniform flow)

Rectangular: A = by, P = b + 2y Trapezoidal: A = (b + zy)y, P = b + 2y√(1+z²) Full circular: R = D/4

b	bottom width
y	flow depth
z	side slope, horizontal:vertical
Do NOT use R = D/4 for a partly full pipe — recompute A and P for the actual depth.

Fr = V / √(g·D_h) with D_h = A/T

Fr	Froude number: < 1 subcritical, = 1 critical, > 1 supercritical
D_h	hydraulic depth = area / top width T (for a rectangle, D_h = y)

Rectangular channels: y_c = (q²/g)^1/3 and E_min = 3y_c/2

y_c	critical depth
q = Q/b	discharge per unit width
E = y + V²/2g	specific energy — minimised at critical depth

Topic 7: Hydrology & Runoff

Q_p = C · i · A

Q_p	peak runoff rate (cfs when i is in in/hr and A in acres — the units work out)
C	runoff coefficient (0–1); use an area-weighted composite for mixed land use
i	rainfall intensity for a duration equal to the time of concentration, t_c
A	drainage area

Composite C = Σ(C_jA_j) / ΣA_j

t_c

time of concentration — travel time from the hydraulically most distant point

S = 1000/CN − 10 (inches) I_a = 0.2S

CN	curve number, 0–100 (higher = more runoff); from TR-55 tables by soil group and cover
S	potential maximum retention after runoff begins
I_a	initial abstraction — interception, depression storage, early infiltration

Q = (P − I_a)²(P − I_a) + S for P > I_a; Q = 0 otherwise

Q	direct runoff depth (inches)
P	storm rainfall depth (inches)

t_p = ΔD/2 + 0.6 t_c q_p = 484 A / t_p

t_p	time to peak (hr)
q_p	peak of the unit hydrograph (cfs)
ΔD	unit storm duration (hr)
A	area in square miles

Topic 8: Groundwater Flow

Q = K · i · A with i = dh/dl

Q	discharge through the porous medium
K	hydraulic conductivity (permeability)
i	hydraulic gradient — head loss per unit length, dimensionless
A	bulk cross-sectional area normal to flow (solids + voids)

v = Q/A (Darcy flux) v_s = v/n (seepage velocity)

v	discharge per unit bulk area — not the actual pore-water speed
v_s	true average velocity through the pores
n	porosity

Confined (constant thickness b): q = T · (dh/dl), T = K·b

T	transmissivity — the aquifer's headline property for confined flow
b	saturated thickness of the confined aquifer

Unconfined (Dupuit): q = (K/2L)·(h₁² − h₂²) per unit width

h	saturated thickness (water-table height above the impermeable base)
Assumes nearly horizontal flow — the Dupuit approximation; fine for gentle gradients.

Confined: Q = 2πT(h₂ − h₁)ln(r₂/r₁)

h₁, h₂	hydraulic heads at radial distances r₁, r₂ from the well
T = Kb	transmissivity

Unconfined: Q = πK(h₂² − h₁²)ln(r₂/r₁)

Heads h are water-table heights above the aquifer base — and they enter squared.

Topic 9: Water Treatment

Coagulation/flocculation: rapid mix then gentle stirring; design by detention time t = V/Q and the Camp number Gt

Typical: rapid mix 30–60 s; flocculation 20–40 min; Gt ≈ 10⁴–10⁵.

Sedimentation: surface overflow rate = QA weir loading = QL_weir

Surface overflow rate (gpd/ft²)	the controlling design parameter — particles settle if their settling velocity exceeds it
Weir loading (gpd/ft)	checked separately so settled sludge is not scoured over the weirs

Filtration: filtration rate = Q/A (gpm/ft²); backwash reverses the flow to clean the media

Rapid sand filters run ≈ 2–4 gpm/ft²; head loss grows as the bed clogs, triggering backwash.

Disinfection: CT = C × T

C	disinfectant residual concentration (mg/L)
T	contact time (min) — use T₁₀, the time 90% of the water exceeds, for credit
CT (mg·min/L)	compared against regulatory tables for the target pathogen and disinfectant

Topic 10: Wastewater Treatment

BOD exerted at time t: y = L₀(1 − e^−kt) (base e) or y = L₀(1 − 10^−Kt) (base 10)

y	oxygen consumed by time t
L₀	ultimate BOD
k, K	deoxygenation rate constants — k = 2.303K, so check which base the question uses

BOD₅ ≈ 0.68 × L₀ (for the standard k = 0.23 day⁻¹, base e, at 20°C)

Handy when a question gives one and asks for the other.

F/M = Q · S₀V · X

Q	influent flow
S₀	influent BOD₅
V	aeration tank volume
X	MLSS — mixed liquor suspended solids
Units: day⁻¹; conventional plants run ≈ 0.2–0.5 day⁻¹.

MCRT = mass of solids in the systemmass of solids wasted per day = V·XQ_wX_w + Q_eX_e

Q_w, X_w	waste sludge flow and concentration
Q_e, X_e	effluent flow and suspended solids (often negligible)
MCRT in days; conventional ≈ 5–15 days, extended aeration 20–30+.

Surface overflow rate = QA Solids loading rate = (Q + Q_r) · XA

Q_r	return sludge flow
Design SOR ≈ 400–700 gpd/ft²; solids loading ≈ 20–30 lb/day/ft² for conventional plants.

↑ Topic 11: Water Hammer & Surge

SVI = (settled volume in 30 min, mL/L) × 1000 / MLSS (mg/L) (mL/g)

SVI < 100 good settling; > 150 suggests bulking.

Topic 11: Water Hammer & Surge

a = √(K/ρ)√[1 + (K/E)(D/e)]

a	pressure-wave celerity
K	bulk modulus of the fluid (water ≈ 2.2 GPa)
E, D, e	pipe material modulus, diameter, wall thickness
Rigid-pipe limit: a₀ = √(K/ρ) ≈ 1,480 m/s for water.

↑ Topic 11: Water Hammer & Surge

Δp = ρ · a · ΔV ΔH = a · ΔVg

Δp, ΔH	Joukowsky pressure / head rise for a rapid (instantaneous) velocity change
ΔV	sudden change in flow velocity

↑ Topic 11: Water Hammer & Surge

Critical closure time: t_c = 2La

Closure faster than 2L/a is "rapid" — the full Joukowsky rise applies; slower closures give a reduced surge.

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Topic 12: Pipe Networks

↑ Topic 12: Pipe Networks

Node: ΣQ_in = ΣQ_out + q_demand Loop: Σh_L = 0

The two conditions every network solution must satisfy: continuity at each junction, energy around each closed loop.

↑ Topic 12: Pipe Networks

Hardy Cross: ΔQ = −Σh_Ln · Σ|h_L/Q|

ΔQ	flow correction applied to every pipe in the loop (iterate until Σh_L ≈ 0)
n	head-loss exponent: 1.852 for Hazen-Williams, ≈ 2 for Darcy-Weisbach

↑ Topic 12: Pipe Networks

Series: r_eq = r₁ + r₂ + … Parallel: 1√r_eq = 1√r₁ + 1√r₂ + …

r	pipe resistance in h_L = rQ² (Darcy-Weisbach form)

Topic 13: Culverts & Spillways

↑ Topic 13: Culverts & Spillways

Outlet control: HW = TW + (K_e + 1)V²2g + h_f

HW	headwater depth above the culvert invert
TW	tailwater depth above the outlet invert
K_e	entrance loss coefficient; the "+1" accounts for the exit loss
Inlet control: capacity depends on entrance geometry and HW only — barrel roughness does not matter.

↑ Topic 13: Culverts & Spillways

Q = C · L · H^1.5

Weir equation (broad-crested, sharp-crested, ogee spillway, roadway overtopping)
C	weir coefficient (depends on crest shape; SI vs US units)
L	crest length perpendicular to flow
H	head above the crest

Topic 14: Stormwater Management & BMPs

↑ Topic 14: Stormwater Management & BMPs

S = 12(I_p − Q_o) · T_b

S	detention storage (triangular-hydrograph approximation)
I_p	peak inflow rate
Q_o	allowable (constant) outflow rate
T_b	base time of the inflow hydrograph

↑ Topic 14: Stormwater Management & BMPs

Orifice: Q = C_d · A · √(2gH)

H	head measured to the orifice centreline
C_d	discharge coefficient (≈ 0.6 for a sharp-edged orifice)

↑ Topic 14: Stormwater Management & BMPs

Water-quality volume: WQV = (first-flush depth) × (drainage area)

Detention controls peak flow; retention/BMPs treat the first flush, which carries most pollutants.

Topic 15: Water Distribution Systems

↑ Topic 15: Water Distribution Systems

Q_max-day = PF_md · ADD Q_peak-hour = PF_ph · ADD

ADD	average-day demand = population × per-capita use
PF	peaking factor (max-day ≈ 1.5–2.5, peak-hour ≈ 2.5–4 of ADD)

↑ Topic 15: Water Distribution Systems

V_total = V_eq + V_fire + V_emerg

Storage = equalization (diurnal balancing) + fire storage + emergency reserve.

↑ Topic 15: Water Distribution Systems

p = γ(HGL − z)

Available pressure at a node = hydraulic grade line minus ground elevation; minimum pressures (often ≈ 35–40 psi) govern tank height and pressure zones.

Topic 16: Wastewater Collection Systems

↑ Topic 16: Wastewater Collection Systems

V = 1n · R^2/3 · S^1/2 Q = AV (SI; 1.486/n for US units)

Manning's applied to partial-full circular sewers — use the hydraulic-element ratios (theta-based geometry) for d/D < 1.

↑ Topic 16: Wastewater Collection Systems

V_min ≈ 0.6 m/s (2 ft/s) design d/D ≤ 0.8

Self-cleansing minimum velocity prevents solids deposition; maximum velocity (≈ 3 m/s) limits scour. Force mains flowing full under pressure use Darcy-Weisbach/Hazen-Williams, not Manning's geometry.

Topic 17: Sedimentation & Erosion

↑ Topic 17: Sedimentation & Erosion

A = R · K · LS · C · P (USLE)

A	average annual soil loss (tons/acre/year)
R, K	rainfall erosivity, soil erodibility
LS	topographic (slope length-steepness) factor
C, P	cover-management and support-practice factors — the management levers
USLE estimates sheet and rill erosion only, not gully or channel erosion.

↑ Topic 17: Sedimentation & Erosion

Overflow rate v_o = QA_s Detention time t_d = VQ

Particles with settling velocity > v_o are removed in an ideal settling basin.

Topic 18: Flood Frequency & Reservoirs

↑ Topic 18: Flood Frequency & Reservoirs

P = 1T Risk_n = 1 − (1 − 1T)ⁿ

T	return period (years) — an average, not a schedule
Risk_n	probability of at least one exceedance in n years

↑ Topic 18: Flood Frequency & Reservoirs

Log-Pearson III: log Q_T = (mean of logs) + K · (std. dev. of logs)

K	frequency factor, a function of the skew coefficient and T — the standard for US flood-frequency analysis.

↑ Topic 18: Flood Frequency & Reservoirs

I − O = ΔSΔt (level-pool routing: storage attenuates the flood peak)

Rippl mass curve: required storage = maximum cumulative (demand − inflow) deficit.

Mathematics

d/dx xⁿ = nxⁿ⁻¹ d/dx e^x = e^x d/dx ln x = 1/x

∫_a^b f(x) dx = F(b) − F(a) f̄ = 1/(b − a) ∫_a^b f(x) dx

dy/dx + P(x)y = Q(x), μ = e^{∫P(x) dx}, d/dx(yμ) = Qμ

x = D_x/D, y = D_y/D (Cramer's rule, D ≠ 0)

A · B = |A||B| cos θ |A × B| = |A||B| sin θ

(a + bi)(c + di) = (ac − bd) + (ad + bc)i e^iθ = cos θ + i sin θ

x_n+1 = x_n − f(x_n)/f′(x_n) (Newton) ∫_a^b f ≈ (h/3)(f₀ + 4Σf_odd + 2Σf_even + f_n), n even (Simpson's)

Probability & Statistics

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P(not A) = 1 − P(A)

_nC_k = n!/(k!(n − k)!) _nP_k = n!/(n − k)!

P(X = k) = _nC_k p^k(1 − p)^n−k (binomial; μ = np, σ² = np(1 − p))

z = (x − μ)/σ ≈68% within ±1σ, ≈95% within ±2σ, ≈99.7% within ±3σ

s² = Σ(x_i − x̄)²/(n − 1) (sample variance)

x̄ ± z* · s/√n (confidence interval; z* = 1.96 for 95%)

Computational Tools

=IF(test, value_if_true, value_if_false) $A1 / A$1 / $A$1 lock column / row / both

absolute error = |measured − true| relative error = absolute error / |true|

sums/differences: δS = √(δa² + δb²) (absolute errors, RSS)

products/quotients: δQ/Q = √((δa/a)² + (δb/b)²) (relative errors, RSS)

z = xⁿ ⇒ δz/z = n · δx/x

central difference (f(x+h) − f(x−h))/(2h) beats forward difference (f(x+h) − f(x))/h

Ethics & Professional Practice

Public safety, health & welfare > employer/client > profession > colleagues

Paramount duty

engineers shall hold paramount the safety, health, and welfare of the public

Seal & sign only work under your responsible charge

Responsible charge

direct personal control and supervision of the work being sealed

Conflicts of interest: disclose to all parties, proceed only with informed consent

Dual compensation

no payment from more than one party on the same project without full disclosure and agreement of all

Valid contract = offer + acceptance + consideration

Consideration	something of value exchanged by each party
Negligence	failure to exercise the care of a reasonably prudent, competent engineer

Engineering Economics

(F/P, i, n) = (1 + i)ⁿ (P/F, i, n) = (1 + i)⁻ⁿ

P / F	present / future worth
i	effective rate per period
n	number of periods

(F/A, i, n) = [(1 + i)ⁿ − 1] / i (A/F, i, n) = i / [(1 + i)ⁿ − 1]

A	uniform end-of-period series

(P/A, i, n) = [(1 + i)ⁿ − 1] / [i(1 + i)ⁿ] (A/P, i, n) = [i(1 + i)ⁿ] / [(1 + i)ⁿ − 1]

Note

(A/P, i, n) = (A/F, i, n) + i; each factor is the reciprocal of its partner

(P/G, i, n) = (1/i){[(1 + i)ⁿ − 1]/[i(1 + i)ⁿ] − n/(1 + i)ⁿ}

G	arithmetic gradient (starts at 0 in period 1)
(A/G, i, n)	= (P/G, i, n) × (A/P, i, n)

i_eff = (1 + r/m)^m − 1 NPV = −C₀ + Σ C_t/(1 + i)^t B/C = PW(benefits) / PW(costs)

r	nominal annual rate; m = compounding periods per year
IRR	the rate i* with NPV(i*) = 0; accept if IRR > MARR

Straight-line: D = (Cost − Salvage)/n BV_t = Cost − t·D

i_f	inflated rate: i_f = (1 + i)(1 + f) − 1
Q*	breakeven: Q* = fixed costs / (unit price − unit variable cost)

Statics

F_R = ΣF F_x = F cos θ, F_y = F sin θ

F_R	resultant force — vector sum of the system
θ	angle from the positive x-axis

M_O = r × F |M| = F d

M_O	moment about point O
d	perpendicular distance from O to the line of action

ΣF = 0 ΣM_O = 0

Equilibrium

3 scalar equations in 2D, 6 in 3D; supports: roller 1, pin 2, fixed 3 reactions (2D)

x̄ = Σx̃_iA_i / ΣA_i ŷ = Σŷ_iA_i / ΣA_i

Composite centroid

weighted average of piece centroids; holes enter as negative areas

Rectangle: bh³/12 Triangle: bh³/36 Circle: πr⁴/4

Centroidal I

h is the dimension perpendicular to the axis

I = Ī + Ad² J_O = I_x + I_y

Parallel-axis theorem	d = distance between the two parallel axes
J_O	polar moment of area

F_R = ∫w dx (area under w) triangular: w_maxL/2 at L/3 from the max end

Distributed load

resultant acts at the centroid of the intensity diagram

F ≤ μ_sN (impending: =) T₂ = T₁e^μβ

Dry friction	μ_s = tan θ at impending slip on an incline
Belt friction	T₂ = tight side, β in radians

FE Civil: Dynamics

v = v₀ + at s = s₀ + v₀t + ½at² v² = v₀² + 2a(s − s₀)

v₀, s₀

initial velocity and position (constant acceleration)

R = v₀² sin 2θ/g h_max = v₀² sin²θ/(2g) T = 2v₀ sin θ/g

R	projectile range (same launch/landing elevation, no air resistance)
h_max	maximum height above launch level
T	time of flight

a_t = dv/dt a_n = v²/ρ a = √(a_t² + a_n²)

a_t	tangential acceleration — rate of change of speed
a_n	normal acceleration toward the centre of curvature (ρ = radius of curvature)

ΣF = ma F_friction = μN

m = W/g	mass — convert weight to slugs in US units
μ	friction coefficient (μ_s impending slip, μ_k sliding)

T₁ + ΣU_1–2 = T₂ T = ½mv²

U_weight = ±mgΔh	positive for a drop, negative for a rise
U_spring = ½k(s₁² − s₂²)	work of a spring between stretches s₁ and s₂
U_friction = −F_fd	friction always does negative work

mv₁ + Σ∫F dt = mv₂ e = (v_B2 − v_A2) / (v_A1 − v_B1)

e	coefficient of restitution: 1 elastic, 0 perfectly plastic

f_n = √(k/m)/(2π) τ = 2π√(m/k) f_d = f_n√(1 − ζ²)

f_n	undamped natural frequency (Hz)
τ	period of one oscillation (s)
ζ	damping ratio; f_d is the damped natural frequency

Mechanics of Materials

σ = P/A ε = δ/L δ = PL/(AE)

σ, ε	axial stress and strain
E	modulus of elasticity

τ = Tr/J φ = TL/(GJ) J = πd⁴/32 (solid)

τ	torsional shear stress (max at r = c)
φ	angle of twist, radians
J	polar moment of inertia

σ = My/I τ = VQ/(It)

M, V	bending moment and shear force at the section
I	second moment of area about the neutral axis (rectangle: bh³/12)

σ_1,2 = (σ_x+σ_y)/2 ± √[((σ_x−σ_y)/2)²+τ_xy²]

σ₁, σ₂

principal stresses

P_cr = π²EI/(KL)²

P_cr	Euler buckling load
K	1.0 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2.0 fixed–free

Materials

↑ Materials

E = σ/ε (initial linear slope of the stress–strain curve)

E	modulus of elasticity — stiffness, not strength

↑ Materials

% elongation = (L_f−L₀)/L₀ × 100 % reduction of area = (A₀−A_f)/A₀ × 100

ductility measures from the tension test

↑ Materials

w/c = (mass of water)/(mass of cement)

w/c

water–cement ratio by mass — lower means stronger, less permeable concrete

Structural Analysis

m + r = 2j (planar truss determinacy; degree = (m + r) − 2j)

m, r, j

members, reaction components, joints

dV/dx = −w dM/dx = V

w	distributed load, positive downward

Simply supported: centre P → M_max = PL/4; uniform w → M_max = wL²/8. Cantilever: tip P → M_max = PL; uniform w → M_max = wL²/2.

Simply supported, span L, unit load at x from A: η_A = (L − x)/L, η_B = x/L; moment at C (a from A, b from B): peak η = ab/L at C

Portal method: inflection at mid-height of columns and mid-span of beams; interior column shear = 2 × exterior column shear

Cantilever tip P: Δ = PL³/(3EI). Simply supported: centre P → Δ = PL³/(48EI); uniform w → Δ = 5wL⁴/(384EI).

Structural Design

LRFD: 1.4D; 1.2D + 1.6L + 0.5(L_r/S/R); 1.2D + 1.0W + L + 0.5(L_r/S/R); 1.2D + 1.0E + L; 0.9D + 1.0W; 0.9D + 1.0E / ASD: D; D + L; D + (L_r/S/R); D + 0.75L + 0.75(L_r/S/R); D + (0.6W or 0.7E); 0.6D + 0.6W; 0.6D + 0.7E

Steel tension: yielding P_n = F_yA_g (φ = 0.90); rupture P_n = F_uA_e (φ = 0.75); A_e = U·A_n

Steel column: F_e = π²E/(KL/r)²; KL/r ≤ 4.71√(E/F_y) → F_cr = 0.658^(F_y/F_e)F_y; else F_cr = 0.877F_e; P_n = F_crA_g (φ = 0.90)

Steel beam (compact, L_b ≤ L_p): M_n = Z_xF_y (φ = 0.90). Shear: V_n = 0.6F_yA_wC_v (φ = 0.90).

RC flexure: a = A_sf_y/(0.85f′_cb); M_n = A_sf_y(d − a/2) (φ = 0.90 tension-controlled)

RC shear (SI, f′_c in MPa): V_c = 0.17λ√(f′_c)bd; V_s = A_vf_ytd/s; φ(V_c + V_s) ≥ V_u (φ = 0.75)

One-way slab min thickness: simply supported l/20; one end continuous l/24; both ends continuous l/28; cantilever l/10

Timber: F′_b = F_b·C_D·C_M·C_t·C_L·C_F…; f_b = M/S ≤ F′_b; rectangular shear f_v = 1.5V/A ≤ F′_v

Geotechnical Engineering

n = e / (1 + e) S = w G_s / e

e	void ratio
n	porosity
S	degree of saturation (0–1)
w	water content
G_s	specific gravity of solids

γ_d = G_sγ_w / (1 + e) γ_sat = (G_s + e)γ_w / (1 + e) γ = γ_d(1 + w)

γ_w

unit weight of water: 9.81 kN/m³ (62.4 pcf)

PI = LL − PL C_u = D₆₀/D₁₀ C_c = D₃₀² / (D₁₀ D₆₀)

PI	plasticity index
C_u, C_c	uniformity and curvature coefficients (well-graded sand: C_u ≥ 6, 1 ≤ C_c ≤ 3)

σ′ = σ − u i_c = γ_sub / γ_w

σ′	effective stress — controls strength and settlement
i_c	critical hydraulic gradient (quick condition)

q = k i A v_s = v / n q = k h (N_f / N_d)

k	hydraulic conductivity
v_s	seepage velocity (true pore-water speed)
N_f, N_d	flow-net channels and equipotential drops (q per unit length)

s_c = H₀1 + e₀ C_c log₁₀σ′_v0 + Δσσ′_v0 T_v = c_v t / H_dr²

s_c	primary consolidation settlement (normally consolidated clay)
T_v	time factor (U = 50% at T_v = 0.197)

τ = c + σ′ tan φ s_u = q_u / 2

τ	Mohr–Coulomb shear strength (effective stress)
s_u	undrained shear strength (UU: φ = 0)

K_a = tan²(45° − φ/2) K_p = tan²(45° + φ/2) σ′_h = K σ′_v

K_a, K_p

Rankine active / passive coefficients (φ in degrees)

q_ult = s_c c N_c + γD_fN_q + s_γ ½ γB N_γ

s_c, s_γ

shape factors: strip 1.0/1.0, square 1.3/0.8, circular 1.3/0.6

FS = c + γz cos²β tan φγz sin β cos β

FS	infinite-slope factor of safety (resisting / driving)

Transportation Engineering

SSD = 1.47 V t + V² / [30 (f ± G)]

SSD	stopping sight distance, ft
V	design speed, mph
t	perception–reaction time, 2.5 s for design
f	coefficient of friction (0.35 typical)
G	grade as a decimal: + upgrade, − downgrade

T = R tan(Δ/2) L = R Δ (Δ in radians) E = R [sec(Δ/2) − 1] M = R [1 − cos(Δ/2)]

T, L, E, M	tangent length, curve length, external distance, middle ordinate, ft
R	curve radius, ft; Δ deflection (central) angle

R_min = V² / [15 (e + f)] D = 5,729.58 / R

e	superelevation rate (decimal)
f	side-friction factor
D	degree of curve (arc definition), degrees per 100-ft arc

y(x) = y_PVC + g₁x + (g₂ − g₁) x² / (2L) x_hp = g₁L / (g₁ − g₂) K = L / A

g₁, g₂	grades as decimals, signed
x_hp	distance from PVC to high/low point, ft
A	\|g₁ − g₂\| in percent; K = ft per 1% grade change

Crest (SSD): L = AS²/2158 (L ≤ S) or L = 2S − 2158/A (L > S) Sag: L = AS²/(400 + 3.5S) (L ≤ S) or L = 2S − (400 + 3.5S)/A (L > S)

S	sight distance (SSD), ft

q = k v q_max = v_f k_j / 4 c = s (g / C)

q, k, v	flow (veh/h), density (veh/mi), space-mean speed (mph)
v_f, k_j	free-flow speed, jam density; capacity at k = k_j/2
c	signal lane-group capacity; s ≈ 1,900 veh/h/ln, g effective green, C cycle length

LEF ≈ (P / 18)⁴ SN = a₁D₁ + a₂D₂m₂ + a₃D₃m₃

LEF	load equivalency factor vs an 18-kip single axle; P in kips
SN	structural number; a_i layer coefficient, D_i thickness (in), m_i drainage coefficient

Construction Engineering

Forward: ES = max(EF of predecessors), EF = ES + duration Backward: LF = min(LS of successors), LS = LF − duration

ES, EF	early start / early finish
LS, LF	late start / late finish

Total float = LS − ES = LF − EF Free float = min(ES of successors) − EF

Critical path

longest-duration path; zero total float; its length is the project duration

Production (LCY/h) = (60 × capacity (LCY) × efficiency) / cycle time (min) Trucks required = truck cycle time / loader cycle time (round up)

Efficiency

50-minute hour → 0.833 unless stated otherwise

V = L (A₁ + A₂) / 2 V = L (A₁ + 4A_m + A₂) / 6

V	earthwork volume, ft³ (divide by 27 for CY)
L	distance between end sections, ft; A₁, A₂ end areas, A_m mid-section area, ft²

Volume (CY) = Volume (ft³) / 27 LCY = BCY × (1 + swell)

BCY / LCY / CCY

bank (in place), loose (excavated), compacted cubic yards

Surveying

HI = elev_BM + BS elev_TP = HI − FS ΣBS − ΣFS = last elevation − first elevation

HI	height of instrument
BS	backsight — rod reading on known elevation (adds)
FS	foresight — rod reading on wanted elevation (subtracts)

C_t = α(T − T₀)L C_p = (P − P₀)L / (AE) C_s = −w²L_s³ / (24P²)

C_t, C_p, C_s

temperature, pull, and sag corrections; α = 6.45×10⁻⁶/°F, T₀ = 68°F

NE: Az = bearing SE: Az = 180° − bearing SW: Az = 180° + bearing NW: Az = 360° − bearing

Back azimuth

Az ± 180° (add if Az < 180°, subtract if Az > 180°)

Lat = D cos(Az) Dep = D sin(Az) e = √[(ΣLat)² + (ΣDep)²] precision = 1 : (perimeter/e)

e	linear misclosure, ft

Corr_Lat,i = −ΣLat × (D_i / perimeter) Corr_Dep,i = −ΣDep × (D_i / perimeter)

Compass rule

corrections oppose the error sums; longer legs absorb more

A = ½|Σ(E_iN_i+1 − E_i+1N_i)|

A	enclosed area, ft²; 1 acre = 43,560 ft²

Environmental Engineering

Steady-state mass balance: 0 = Σ(Q·C)_in − Σ(Q·C)_out − k·V·C (first-order loss)

Accumulation = 0 at steady state, but the reaction term k·V·C stays.

Completely mixed reactor, first-order decay: C = C₀1 + k·t_d (t_d = V/Q)

Not the plug-flow form C = C₀e^−kt — a mixed tank removes less for the same detention time.

Streeter–Phelps: D(t) = k₁L₀k₂ − k₁(e^−k₁t − e^−k₂t) + D₀e^−k₂t

D(t)	oxygen deficit = saturation DO − actual DO
k₁, k₂	deoxygenation / reaeration constants, day⁻¹, base e

t_c = 1k₂ − k₁ · ln[k₂k₁(1 − D₀(k₂ − k₁)k₁L₀)] DO_min = DO_sat − D(t_c)

Natural log, not log₁₀; report DO_min, not the deficit.

k_T = k₂₀ θ^T−20 (θ ≈ 1.047 for k₁, 1.024 for k₂) k = 2.303 K

Temperature correction and the base-10 → base-e conversion — multiply, never divide.