Mechanical Design and Analysis is 10-15 questions on the FE Mechanical exam, tied for the largest knowledge area. It is also the most distinctly mechanical part of the paper: nothing equivalent appears on FE Civil.
Every question here is downstream of something else. The stress comes from mechanics of materials, the material limit comes from material properties, and the loading comes from statics. Study it last and it takes half the time.
Exam weight: NCEES lists Mechanical Design and Analysis at 10-15 questions (9-14%) of the 110-question FE Mechanical exam. Work each problem below on paper first, then reveal the worked solution — reading a solution you have not attempted builds recognition, not recall.
What NCEES Tests in Mechanical Design and Analysis
The specification covers stress analysis of machine elements, failure theories and failure prevention, deformation and stiffness, pressure vessels and piping, bearings, power screws, power transmission, joining methods, manufacturability, quality and reliability, and components such as springs, gears, clutches and brakes.
Failure theories are the backbone: maximum shear stress and distortion energy for ductile materials, and fatigue analysis using the endurance limit with stress concentration and surface factors. Expect a thin-walled pressure vessel hoop and longitudinal stress calculation, a bolted joint sized against a load, and a gear or belt drive ratio.
5 Free Mechanical Design and Analysis Practice Problems
Each problem below comes from the PECivilClick FE Mechanical question bank and matches the style, difficulty and format of the real exam. Attempt each one under a three-minute limit — roughly the pace the exam demands.
Problem 1 — Stress analysis of machine elements
A solid circular shaft is subjected to a torque of 500 N·m. If the shaft diameter is 40 mm, what is the maximum shear stress in the shaft?
A) 79.6 MPa
B) 39.8 MPa
C) 59.7 MPa
D) 19.9 MPa
Answer: B) 39.8 MPa
For a solid circular shaft, the maximum shear stress is:
\(\tau_{max} = \frac{Tc}{J} = \frac{16T}{\pi d^3}\)
Given:
- \(T = 500\) N·m
- \(d = 40\) mm \(= 0.040\) m
\(\tau_{max} = \frac{16 \times 500}{\pi \times (0.040)^3} = \frac{8000}{\pi \times 6.4 \times 10^{-5}}\)
\(\tau_{max} = \frac{8000}{2.01 \times 10^{-4}} = 39.8 \times 10^6 \text{ Pa}\)
\(\boxed{\tau_{max} = 39.8 \text{ MPa}}\)
Problem 2 — Stress analysis of machine elements
A rotating shaft experiences both bending moment M = 800 N·m and torque T = 600 N·m. If the shaft diameter is 50 mm, what is the equivalent stress using the maximum shear stress theory?
A) 32.6 MPa
B) 81.5 MPa
C) 65.2 MPa
D) 40.7 MPa
Answer: B) 81.5 MPa
For combined bending and torsion, first get the surface stresses:
\(\sigma = \frac{32M}{\pi d^3}\) (bending), \(\tau = \frac{16T}{\pi d^3}\) (torsion)
Given:
- \(M = 800\) N·m
- \(T = 600\) N·m
- \(d = 50\) mm \(= 0.050\) m, so \(\pi d^3 = 3.927 \times 10^{-4} \text{ m}^3\)
\(\sigma = \frac{32(800)}{3.927 \times 10^{-4}} = 65.2 \text{ MPa}\), \(\tau = \frac{16(600)}{3.927 \times 10^{-4}} = 24.4 \text{ MPa}\)
Maximum shear stress in the shaft:
\(\tau_{max} = \sqrt{(\sigma/2)^2 + \tau^2} = \frac{16}{\pi d^3}\sqrt{M^2 + T^2}\)
\(\sqrt{M^2 + T^2} = \sqrt{800^2 + 600^2} = \sqrt{1{,}000{,}000} = 1000\) N·m
\(\tau_{max} = \frac{16(1{,}000)}{3.927 \times 10^{-4}} = 40.7 \text{ MPa}\)
The maximum shear stress (Tresca) theory compares \(\sigma_1 - \sigma_3\) against \(S_y\), so the equivalent stress is twice the maximum shear stress:
\(\sigma_1 = \frac{\sigma}{2} + \tau_{max} = 73.3 \text{ MPa}\), \(\sigma_3 = \frac{\sigma}{2} - \tau_{max} = -8.1 \text{ MPa}\)
\(\sigma_{eq} = \sigma_1 - \sigma_3 = 2\tau_{max} = \frac{32}{\pi d^3}\sqrt{M^2 + T^2} = 81.5 \text{ MPa}\)
\(\boxed{\sigma_{eq} \approx 81.5 \text{ MPa}}\)
Distractors: (D) 40.7 MPa is \(\tau_{max}\) itself, a shear stress rather than an equivalent normal stress; (C) 65.2 MPa is the bending stress alone, ignoring the torque; (A) 32.6 MPa is \(\sigma/2\), the center of Mohr's circle.
Problem 3 — Stress analysis of machine elements
A hollow shaft with outer diameter 80 mm and inner diameter 70 mm transmits power at 1800 rpm. If the allowable shear stress is 60 MPa, what is the maximum power the shaft can transmit?
A) 585 kW
B) 285 kW
C) 385 kW
D) 471 kW
Answer: D) 471 kW
For a hollow shaft, the polar moment of inertia is:
\(J = \frac{\pi}{32}(d_o^4 - d_i^4)\)
Given:
- \(d_o = 80\) mm \(= 0.080\) m
- \(d_i = 70\) mm \(= 0.070\) m
- \(\tau_{allow} = 60\) MPa
- \(n = 1800\) rpm
\(J = \frac{\pi}{32}(0.080^4 - 0.070^4) = \frac{\pi}{32}(1.695 \times 10^{-5}) = 1.664 \times 10^{-6} \text{ m}^4\)
Maximum torque from \(\tau = \frac{Tc}{J}\):
\(T = \frac{\tau J}{c} = \frac{(60 \times 10^6)(1.664 \times 10^{-6})}{0.040} = 2496 \text{ N·m}\)
Angular velocity:
\(\omega = \frac{2\pi n}{60} = \frac{2\pi(1800)}{60} = 188.5 \text{ rad/s}\)
Power:
\(P = T\omega = 2496 \times 188.5 = 470{,}500 \text{ W}\)
\(\boxed{P \approx 471 \text{ kW}}\)
Problem 4 — Stress analysis of machine elements
A stepped shaft has a stress concentration factor Kt = 1.8 at a fillet. If the nominal bending stress is 50 MPa, what is the maximum stress at the fillet?
A) 130 MPa
B) 90 MPa
C) 50 MPa
D) 27.8 MPa
Answer: B) 90 MPa
The maximum stress at a stress concentration is:
\(\sigma_{max} = K_t \times \sigma_{nom}\)
Given:
- \(K_t = 1.8\)
- \(\sigma_{nom} = 50\) MPa
\(\sigma_{max} = 1.8 \times 50 = 90 \text{ MPa}\)
\(\boxed{\sigma_{max} = 90 \text{ MPa}}\)
Stress concentration factors account for geometric discontinuities that cause local stress increases beyond the nominal stress.
Problem 5 — Failure theories and analysis
Which failure theory is most appropriate for predicting yielding in ductile materials under combined loading?
A) Coulomb-Mohr theory
B) Modified Mohr theory
C) Von Mises (distortion energy) theory
D) Maximum normal stress theory
Answer: C) Von Mises (distortion energy) theory
The Von Mises (distortion energy) theory is the most accurate and widely used theory for predicting yielding in ductile materials under combined loading conditions.
It states that yielding occurs when the distortion energy per unit volume equals the distortion energy at yield in a simple tension test.
- Maximum normal stress theory → brittle materials
- Coulomb-Mohr theory → brittle materials with unequal tensile/compressive strengths
- Modified Mohr theory → brittle materials
\(\boxed{\text{Answer: Von Mises (distortion energy) theory}}\)
Using the FE Reference Handbook for Mechanical Design and Analysis
This section is the widest in the handbook and the easiest to get lost in, so navigate by component rather than by concept. If the question names a spring, a bearing, a bolt or a gear, search that word directly. Two relationships worth locating before exam day are the thin-walled pressure vessel stresses and the distortion energy expression for combined loading, because they appear more than any other formulas in this area.
Four Mistakes That Cost Points
- Swapping hoop and longitudinal stress. In a thin-walled cylinder hoop stress is twice the longitudinal stress. Getting them the wrong way round produces a factor-of-two error, and both values sit on the answer list.
- Applying a ductile failure theory to a brittle material. Maximum shear stress and distortion energy describe ductile yielding. Brittle materials call for a maximum normal stress or a Mohr-based criterion, and the exam tests whether you notice which material you were given.
- Using ultimate strength directly as the fatigue limit. The endurance limit is a fraction of ultimate strength, then reduced further by surface finish, size, loading and stress concentration factors. Skipping the modifying factors gives an unconservative answer that is deliberately offered.
- Ignoring stress concentration at a discontinuity. A hole, fillet or keyway raises local stress by the concentration factor, and it matters most in fatigue. Computing nominal stress and stopping there is the most common fatigue error.
Frequently Asked Questions
How many mechanical design questions are on the FE Mechanical exam?
NCEES specifies 10-15 questions from Mechanical Design and Analysis out of 110, roughly 9-14 percent. It is tied with Dynamics, Fluid Mechanics and Thermodynamics for the heaviest area on the exam.
Which failure theory should I use?
Read the material first. For ductile materials use maximum shear stress for a conservative result or distortion energy for a more accurate one; the question usually names which it wants. For brittle materials use a maximum normal stress criterion. Choosing the theory is often the entire question.
How much fatigue analysis is tested?
Enough to require the endurance limit and its modifying factors, not enough to require a full cumulative-damage analysis. Know how to get from ultimate strength to a corrected endurance limit, and how stress concentration enters that chain.
Should I study this area first or last?
Last. It builds directly on statics, mechanics of materials and material properties, so studying it before those means relearning their content inside harder problems. Once the three feeder areas are solid, this area is mostly about knowing which criterion applies.
Keep Going
These topics feed into each other on the exam:
- FE Mechanical Mechanics of Materials practice problems — 9-14 questions on the exam
- FE Mechanical Material Properties and Processing practice problems — 7-11 questions on the exam
- FE Mechanical Statics practice problems — 9-14 questions on the exam
Browse every knowledge area from the free FE Mechanical practice problem hub, see what the full bank covers on the FE Mechanical exam prep page, or plan your schedule with the FE study timeline.