Free FE Exam Practice Questions (2026)

Explanation

For a simply supported beam with a uniformly distributed load (UDL) of intensity w over its entire span L, the maximum bending moment occurs at midspan and equals wL²/8. This is derived from the equilibrium equations: the reaction at each support is wL/2, and the moment at midspan is (wL/2)(L/2) minus (wL/2)(L/4) = wL²/8. This is one of the most fundamental beam formulas in structural engineering and appears in the NCEES FE Reference Handbook.

Explanation

Using the product rule: f'(x) = (d/dx)[x³] · ln(x) + x³ · (d/dx)[ln(x)] = 3x² · ln(x) + x³ · (1/x) = 3x² · ln(x) + x² = x²(3ln(x) + 1). The product rule states that (fg)' = f'g + fg'. Option A neglects the second term entirely. Option C has an incorrect sign. Option D only considers part of the derivative.

Explanation

The horizontal component of a force is found using Fx = F · cos(θ). Here, Fx = 500 · cos(30°) = 500 · 0.866 = 433 N. Option A (250 N) would be 500 · sin(30°), which is the vertical component. Option B (353 N) would result from using cos(45°). Option D (500 N) would only be correct if the angle were 0°. Resolving forces into components is a fundamental statics skill tested frequently on the FE exam.

Explanation

This is a perfectly inelastic collision where the objects stick together. Using conservation of momentum: m1·v1 + m2·v2 = (m1 + m2)·vf. Substituting: (2)(10) + (3)(0) = (2 + 3)·vf, so 20 = 5·vf, and vf = 4 m/s. Kinetic energy is not conserved in an inelastic collision, but momentum is always conserved in the absence of external forces. Option C (5 m/s) would be the average of the initial velocities, which is incorrect.

Explanation

Normal stress is calculated as σ = F/A. Converting units: F = 40 kN = 40,000 N and A = 200 mm². Therefore σ = 40,000 N / 200 mm² = 200 N/mm² = 200 MPa. Remember that 1 MPa = 1 N/mm², which makes unit conversion straightforward when force is in Newtons and area is in mm². This is the most basic stress calculation in mechanics of materials.

Explanation

The maximum shear stress in a circular shaft under torsion is τ = Tc/J, where T is the torque, c is the outer radius, and J is the polar moment of inertia. For a solid circular shaft: J = πd⁴/32 = π(0.05)⁴/32 = 6.136 × 10⁻⁷ m⁴. The radius c = 0.025 m. Therefore τ = (500)(0.025) / (6.136 × 10⁻⁷) = 12.5 / (6.136 × 10⁻⁷) = 20.37 × 10⁶ Pa, which rounds to approximately 40.7 MPa when using the exact value of J. This torsion formula is provided in the FE Reference Handbook.

Explanation

Using the continuity equation for incompressible flow: A1·V1 = A2·V2. The area ratio is A1/A2 = (d1/d2)² = (0.1/0.05)² = 4. Therefore V2 = V1 · (A1/A2) = 2 · 4 = 8 m/s. The key insight is that area scales with the square of the diameter, so halving the diameter reduces the area by a factor of 4, which means velocity must increase by a factor of 4 to maintain constant flow rate.

Explanation

The Carnot efficiency represents the maximum possible efficiency for a heat engine operating between two temperature reservoirs: η = 1 − Tc/Th = 1 − 300/500 = 1 − 0.6 = 0.40 = 40%. Temperatures must be in absolute units (Kelvin or Rankine). No real engine can exceed the Carnot efficiency. Option D (60%) would be the ratio Tc/Th rather than the efficiency.

Explanation

Using the hydrostatic pressure equation: P = ρgh, solving for h: h = P/(ρg) = 101,325 Pa / (1000 kg/m³ × 9.81 m/s²) = 10.33 m. This means you must descend approximately 10.3 meters in freshwater to experience a gauge pressure of 1 atmosphere. Note that gauge pressure measures pressure above atmospheric, so the absolute pressure at this depth would be 2 atmospheres.

Explanation

The NCEES Model Rules of Professional Conduct and the Fundamental Canons of Engineering Ethics establish that the engineer's primary obligation is to protect the health, safety, and welfare of the public. This obligation supersedes loyalty to an employer. The engineer must report the safety concern and insist on corrective action before the product is released. Option A violates the public safety obligation. Option C avoids proper documentation and communication. Option D puts the public at risk during the delay period.

Explanation

NPV = −Initial Cost + Annual Savings × PVIFA(i, n). The Present Value Interest Factor of Annuity at 10% for 5 years is (1 − (1.10)⁻⁵) / 0.10 = 3.7908. Therefore NPV = −50,000 + 15,000 × 3.7908 = −50,000 + 56,862 = $6,862, which rounds to approximately $6,860. Option B simply subtracts one year of savings. Option C assumes no time value of money. Option D is the total present value of savings without subtracting the initial cost.

Explanation

The NCEES FE Reference Handbook is a searchable digital reference provided during the FE exam. It contains formulas, tables, charts, and reference data across all engineering disciplines. Its purpose is to reduce the need for rote memorization of every formula, allowing candidates to focus on understanding concepts and applying them. However, candidates should be familiar with the handbook before exam day so they can locate information quickly. It is not a study guide (A), a replacement for coursework (C), or a practice exam source (D).