PEP cheat sheet

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a); discriminant b² − 4ac < 0 means complex conjugate roots.

Ideal gas law: PV = nRT with R = 8.314 J/(mol·K) = 0.08206 L·atm/(mol·K); STP (0 °C, 101.325 kPa) molar volume ≈ 22.41 L/mol.

First law of thermodynamics: ΔU = Q − W for a closed system; sign convention must be stated explicitly before use.

Arrhenius equation: k = A·e^(−Ea/RT); a plot of ln k versus 1/T has slope −Ea/R.

Log rules: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a; ln x = 2.303·log x.

Derivative rules: d/dx(xⁿ) = n·xⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x; product and chain rules apply to rate expressions.

Statistics: standard deviation s = √[Σ(xᵢ − x̄)²/(n − 1)]; standard error of the mean = s/√n; 95% interval ≈ mean ± 1.96·SE for large n.

Newton's second law in engineering form: ΣF = ma = d(mv)/dt; momentum is conserved when net external force is zero.

Normal stress σ = F/A; axial deformation δ = FL/(AE); shear stress τ = VQ/(It) for transverse loading.

Hooke's law in 1D: σ = Eε; Poisson's ratio ν = −ε_lateral/ε_axial, typically 0.25–0.35 for metals.

Beam bending: σ = Mc/I with I = bh³/12 for a rectangle; maximum bending moment for a simply supported beam under central point load P is PL/4.

Euler buckling: P_cr = π²EI/(KL)²; effective length factor K = 1.0 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2.0 fixed-free.

Bernoulli (steady, incompressible, frictionless along a streamline): P₁/γ + z₁ + v₁²/(2g) = constant.

Continuity for incompressible flow: A₁v₁ = A₂v₂; Reynolds number Re = ρvD/μ, laminar pipe flow below about 2100.

Thermal resistance of a plane wall: R = L/(kA); resistances in series add, giving q = ΔT/ΣR.

Second law limits: Carnot efficiency η = 1 − T_C/T_H with absolute temperatures; no cyclic device exceeds it.

Bearing designation suffix codes (e.g., ZZ metal shield, 2RS rubber seal, C3 internal clearance) change lubrication, sealing, and fit selection, not the bore size.

Gear ratio = N_driven/N_driver = ω_driver/ω_driven; for a simple gear train speed ratio is the product of driven teeth over driving teeth.

Four-bar Grashof condition: s + l ≤ p + q gives at least one fully rotating link; equality is a change-point linkage.

Stress concentration factor K_t multiplies nominal stress at geometric discontinuities; fatigue notch factor K_f ≤ K_t.

S-N fatigue curve: below the endurance limit (ferrous alloys) life is conventionally treated as infinite; nonferrous alloys lack a true limit.

Shaft critical speed is proportional to √(EI/(mL⁴)); operation within roughly ±20% of a critical speed is avoided.

Hydraulic circuit rule: flow determines actuator speed, pressure determines force; a relief valve sets maximum system pressure.

Measurement uncertainty for a result y = f(x₁, x₂) is combined as u_y² = Σ(∂f/∂xᵢ)²u_xᵢ² for independent inputs.

Factor of safety: FS = failure strength/allowable stress, or FS = allowable load/applied load; it must be defined consistently in a calculation.

Design for static ductile failure uses yield strength (von Mises or maximum shear); brittle materials use ultimate strength with a stress-concentration factor.

Von Mises for plane stress: σ' = √(σx² − σxσy + σy² + 3τxy²).

Tolerance stack-up: worst-case = sum of tolerances; statistical (RSS) = √(Σ toleranceᵢ²), which assumes independent, centered distributions.

Clearance fit, interference fit, and transition fit are determined by comparing hole and shaft tolerance zones, not nominal sizes alone.

A design matrix identifies functional requirements mapped to design parameters; an uncoupled or decoupled matrix is preferred over a coupled one.

Reliability in series: R_system = R₁·R₂·…·R_n; parallel (redundant) systems give R = 1 − Π(1 − Rᵢ).

Safety factor on fatigue: use modified Goodman, σ_a/S_e + σ_m/S_ut = 1/FS, for combined mean and alternating stress.

Standard vs code: a standard is a consensus technical document; a code is a standard adopted into law by a jurisdiction.

ANSI, ASME, ASTM, ISO, IEC, and NFPA documents are voluntary consensus standards unless referenced by regulation or contract.

ASME Boiler and Pressure Vessel Code Section VIII governs pressure vessel design; Section IX covers welding qualification.

Load and resistance factor design applies factored loads (e.g., 1.2D + 1.6L) and resistance factors φ < 1; allowable stress design uses service loads with FS.

NFPA 70 (National Electrical Code) is commonly adopted by state or local law; adoption edition varies by jurisdiction.

ISO 9001 is a quality management system standard and does not itself specify product technical requirements.

Units: 1 in = 25.4 mm exactly; 1 psi = 6.894757 kPa; 1 ft·lbf = 1.355818 J.

Hierarchy of documents in a contract: specification and drawings govern over general standards only where the contract says so; verify precedence clauses.

NCEES Model Rules establish licensure requirements: accredited degree, FE and PE exams, and qualifying experience.

Only a licensed PE may seal and sign engineering work for public use; sealing work outside one's competence is a violation.

Conflict of interest: disclose in writing and withdraw from decisions where a personal financial interest exists.

Confidentiality obligations persist after employment ends; proprietary client data is not reused without authorization.

Safety hierarchy: eliminate the hazard, then substitute, then engineer controls, then administrative controls, then PPE.

Whistleblower duty: report a condition that endangers public health and safety to the employer first, then to the authority having jurisdiction.

Sustainable development: hold paramount public health, safety, and welfare, and consider environmental and resource impacts of the design.

Continuing professional competency (PDH) requirements vary by jurisdiction; record retention and audit are common conditions of renewal.

Unit check: carry units through every step; a dimensional mismatch signals an error before the numeric answer is chosen.

Order-of-magnitude estimate first: round inputs to one significant figure to eliminate implausible answer choices.

Sign conventions: state the assumed positive direction (tension, clockwise, heat in) before writing equilibrium or energy equations.

Absolute temperature: gas-law and Carnot calculations require kelvin; using °C is the most common single error.

Answer-choice discipline: a choice that differs by a factor of 10, 100, or π usually reflects a missed unit conversion or radius/diameter confusion.

Time budget: allocate roughly 6 minutes per multiple-choice item; mark and return rather than stalling on one problem.

Reference lookup: know which handbook table you need before opening it; the index is faster than scanning.

Sanity check: verify that the computed stress, flow, or efficiency lies within a physically plausible range before finalizing.

Carbon steel classification: AISI/SAE 10xx plain carbon, 41xx chromium-molybdenum, 43xx nickel-chromium-molybdenum; the last two digits give nominal carbon points (e.g., 1045 ≈ 0.45% C).

Heat treatment: annealing softens and relieves stress; normalizing refines grain; quenching plus tempering trades hardness for toughness.

Cast iron: grey iron is brittle with high damping; ductile (nodular) iron has graphite nodules and appreciable elongation.

Aluminum alloys: 6061-T6 is weldable and machinable; 7075-T6 has higher strength with lower corrosion resistance.

Welding: a fillet weld throat is 0.707 times the leg size; weld strength is checked on the throat area.

Casting defects include shrinkage porosity, hot tearing, and gas porosity; riser design controls shrinkage feeding.

Surface finish: Ra is the arithmetic average roughness; lower Ra generally raises fatigue life by reducing crack initiation.

Corrosion: galvanic coupling requires dissimilar metals, an electrolyte, and electrical contact; the more anodic metal corrodes.

Present worth: P = F/(1 + i)ⁿ; annuity: P = A·[(1 + i)ⁿ − 1]/[i(1 + i)ⁿ].

Capital recovery factor converts P to A: A = P·[i(1 + i)ⁿ]/[(1 + i)ⁿ − 1].

Net present value > 0 indicates a project exceeds the minimum attractive rate of return; IRR is the i where NPV = 0.

Depreciation: straight-line D = (cost − salvage)/n; MACRS uses declining-balance with a mid-year convention.

Break-even quantity Q = fixed cost/(unit price − unit variable cost).

Benefit-cost ratio > 1 supports a public project; incremental analysis is required when comparing mutually exclusive alternatives.

Inflation: real rate ≈ (1 + nominal)/(1 + inflation) − 1, which approximates nominal − inflation for small values.

Sunk costs are excluded from incremental economic analysis; only future differential cash flows matter.

Normal distribution: approximately 68.3% of values fall within ±1σ, 95.4% within ±2σ, 99.7% within ±3σ.

Binomial probability: P(x) = C(n,x)·pˣ·(1 − p)^(n−x) for n independent trials with constant success probability p.

Poisson distribution: P(x) = e^(−λ)·λˣ/x!, used for counts of rare events in a fixed interval.

Control charts: X-bar and R charts monitor process mean and variation; a point beyond 3σ limits signals special-cause variation.

Process capability: C_p = (USL − LSL)/(6σ); C_pk accounts for centering: min[(USL − μ), (μ − LSL)]/(3σ).

Hypothesis testing: reject the null when the p-value is below the chosen significance level α; a Type I error is a false rejection.

Regression: least-squares slope b = Σ[(xᵢ − x̄)(yᵢ − ȳ)]/Σ(xᵢ − x̄)²; r² is the fraction of variance explained.

Sampling: increasing sample size reduces standard error as 1/√n; it does not remove bias.

Ohm's law V = IR; power P = VI = I²R = V²/R for resistive elements.

Three-phase power: P = √3·V_L·I_L·cos θ for a balanced load, with V_L the line-to-line voltage.

Kirchhoff's laws: sum of currents at a node is zero; sum of voltages around a loop is zero.

Impedance: Z_R = R, Z_L = jωL, Z_C = 1/(jωC); magnitude |Z| and phase angle determine current.

Thevenin equivalent: open-circuit voltage in series with the resistance seen with sources deactivated.

First-order step response: time constant τ = RC or L/R; the response reaches about 63.2% of final value in one τ.

PID control: proportional acts on present error, integral removes steady-state offset, derivative adds damping and amplifies noise.

Motor torque: T = (P·60)/(2πN) with P in watts and N in rpm; horsepower = 745.7 W.

Hydrostatic pressure: P = ρgh; gauge pressure excludes atmospheric pressure, absolute pressure includes it.

Darcy-Weisbach head loss: h_f = f·(L/D)·(v²/2g); friction factor f depends on Reynolds number and relative roughness.

Pump power: P = ρgQH/η, with Q in m³/s, H in m, and η the pump efficiency.

Heat conduction through a composite wall: q = ΔT/Σ(L/kA); convection q = hAΔT.

Sensible heat: Q = m·c_p·ΔT; latent heat changes phase at constant temperature.

Log-mean temperature difference: ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) for heat exchangers.

Water quality: BOD measures oxygen demand of biodegradable organics; COD includes non-biodegradable oxidation.

Hydraulic detention time: t = V/Q; overflow rate is Q/A and governs particle settling in clarifiers.