Essential FE Exam Equations
While the FE Reference Handbook provides all the formulas you need during the exam, knowing the most important equations by heart saves valuable time and helps you recognize which formula to apply. This guide highlights the equations that appear most frequently on the FE exam and explains when to use each one.
Mathematics
- Quadratic Formula: x = [-b plus or minus sqrt(b squared minus 4ac)] / (2a). Use for solving second-degree polynomial equations.
- Chain Rule: dy/dx = (dy/du)(du/dx). Use for differentiating composite functions.
- Integration by Parts: Integral of u dv = uv minus integral of v du. Use when integrating products of functions.
- Taylor Series: f(x) = sum of [f(n)(a)/n!](x-a)^n. Use for approximating functions near a point.
Statics
- Equilibrium: Sum of F_x = 0, Sum of F_y = 0, Sum of M = 0. The foundation of every statics problem.
- Moment: M = F times d (force times perpendicular distance). Use for calculating moments about a point.
- Centroid: x_bar = Sum(A_i times x_i) / Sum(A_i). Use for finding centroids of composite areas.
- Parallel Axis Theorem: I = I_c + A times d squared. Use for transferring moments of inertia to different axes.
Mechanics of Materials
- Hooke's Law: sigma = E times epsilon. Relates stress to strain in the elastic range.
- Axial Deformation: delta = PL / (AE). Deformation of a bar under axial load.
- Flexure Formula: sigma = My / I. Bending stress in a beam at distance y from neutral axis.
- Shear Formula: tau = VQ / (Ib). Shear stress in a beam at a given cross-section location.
- Torsion Formula: tau = Tr / J. Shear stress in a circular shaft under torsion.
- Euler Buckling: P_cr = pi squared EI / (KL) squared. Critical buckling load for a column.
Fluid Mechanics
- Hydrostatic Pressure: P = gamma times h + P_0. Pressure at a depth h in a fluid.
- Bernoulli's Equation: P/(gamma) + V squared/(2g) + z = constant. Energy conservation along a streamline.
- Reynolds Number: Re = rho V D / mu = VD / nu. Determines laminar vs. turbulent flow regime.
- Darcy-Weisbach: h_f = f (L/D) (V squared / 2g). Head loss due to friction in pipe flow.
- Manning's Equation: V = (1/n) R^(2/3) S^(1/2). Open channel flow velocity.
Thermodynamics
- First Law (Closed System): Q minus W = delta U. Energy balance for a closed system.
- First Law (Open System): Q_dot minus W_dot = m_dot times (delta h + delta KE + delta PE). Steady-state energy balance.
- Ideal Gas Law: PV = nRT. Relates pressure, volume, and temperature for ideal gases.
- Carnot Efficiency: eta = 1 minus T_cold / T_hot. Maximum efficiency of a heat engine.
- COP (Refrigeration): COP = Q_cold / W_net. Coefficient of performance for a refrigerator.
Engineering Economics
- Future Value: F = P(1 + i)^n. Compound interest formula.
- Present Worth of Annuity: P = A[(1+i)^n minus 1] / [i(1+i)^n]. Present value of a uniform series.
- Straight-Line Depreciation: D = (Cost minus Salvage) / Life. Annual depreciation amount.
Electrical Circuits
- Ohm's Law: V = IR. Voltage equals current times resistance.
- Power: P = VI = I squared R = V squared / R. Electrical power in a resistor.
- Series Resistance: R_total = R_1 + R_2 + ... Sum of individual resistances.
- Parallel Resistance: 1/R_total = 1/R_1 + 1/R_2 + ... Reciprocal sum for parallel circuits.
How to Use This List
Do not try to memorize every formula listed here. Instead, use this list as a study guide to ensure you understand when to apply each equation and can find it quickly in the FE Reference Handbook. During your study sessions, practice looking up each formula in the handbook so you know exactly where to find it on exam day. The goal is to minimize lookup time while maximizing your problem-solving speed.