Mechanics of materials carries 7 to 11 questions on the FE Civil exam and feeds directly into the structural and geotechnical items, so points here pay twice. The area rewards a small set of well-drilled moves: draw the shear and moment diagram, get the section property, apply the right stress formula, check the sign.
Every solution below shows the diagram or the stress element, not only the arithmetic, because the setup is where these problems are won or lost. Where a handbook table replaces a derivation, such as the standard beam deflection cases, the solution names the case to look up instead of integrating.
Exam weight: NCEES lists Mechanics of Materials at 7-11 questions of the 110-question FE Civil 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 Mechanics of Materials
NCEES specifies 7 to 11 mechanics of materials questions out of 110, placing it just behind statics among the mechanics areas. The core is stress and strain: axial members under load and temperature change, Hooke's law and Poisson's ratio, thin-walled pressure vessels, torsion of circular shafts, and bending stress from M c over I with transverse shear from V Q over I b.
Beyond that you get shear and moment diagrams, beam deflection from the handbook's standard cases, combined loading where axial and bending stresses superpose on the same fiber, principal stresses and Mohr's circle, and Euler buckling of columns with the effective length factor that depends on end conditions. Statically indeterminate axial members appear as compatibility problems.
5 Free Mechanics of Materials Practice Problems
Each mechanics of materials problem below comes from the FEExamClick FE Civil question bank, with worked solutions, 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 — B. Stresses and strains (e.g., diagrams, axial, torsion, bending, shear, thermal)
A free aluminum bar has \( \alpha = 23 \times 10^{-6}\,/{}^\circ\mathrm{C} \). If \( \Delta T = 35^\circ\mathrm{C} \), what is the thermal strain \( \varepsilon_t \)?
Answer: A) 0.00081
Step 1 – Identify given information:
\( \alpha = 23 \times 10^{-6} \;/{}^\circ\text{C} \) (aluminum), \( \Delta T = 35^\circ\text{C} \).
Step 2 – Thermal strain formula (FE Handbook):
$$\varepsilon_T = \alpha \Delta T$$
For a free (unrestrained) bar, no stress develops — only strain.
Step 3 – Solve:
$$\varepsilon_T = (23 \times 10^{-6})(35) = 805 \times 10^{-6} = 8.05 \times 10^{-4} \approx 0.00081$$
Step 4 – Answer: The thermal strain is \( 0.00081 \). Aluminum has a high coefficient of thermal expansion compared to steel, making thermal effects more significant.
Correct answer: (A)
Problem 2 — C. Deformations (e.g., axial, torsion, bending, thermal)
A hollow circular shaft has \( d_o = 70\,\mathrm{mm} \) and \( d_i = 50\,\mathrm{mm} \). With \( T = 1200\,\mathrm{N \cdot m} \), \( L = 1.5\,\mathrm{m} \), and \( G = 80\,\mathrm{GPa} \), what is the angle of twist \( \phi \) (degrees)?
Answer: A) 0.74
Step 1 – Identify given information:
\( T = 1{,}200 \;\text{N}\cdot\text{m} \), \( L = 1.5 \;\text{m} \), \( G = 80 \;\text{GPa} \), \( J = 1.74 \times 10^{-6} \;\text{m}^4 \).
Step 2 – Angle of twist formula (FE Handbook):
$$\phi = \frac{TL}{GJ}$$
Step 3 – Solve:
$$\phi = \frac{1{,}200 \times 1.5}{80 \times 10^9 \times 1.74 \times 10^{-6}} = \frac{1{,}800}{139{,}200} = 0.01293 \;\text{rad} = 0.74^\circ$$
Step 4 – Answer: The angle of twist is \( 0.74^\circ \). This is a small angle, consistent with the high shear modulus of steel.
Option (B) is what you get by averaging the two diameters into a single solid 60 mm shaft instead of subtracting the bore's polar moment. That shrinks \(J\) and so inflates the twist.
Correct answer: (A)
Problem 3 — B. Stresses and strains (e.g., diagrams, axial, torsion, bending, shear, thermal)
A fully restrained steel member develops a thermal stress magnitude of \( 100\,\mathrm{MPa} \). If \( E = 200\,\mathrm{GPa} \) and \( \alpha = 12 \times 10^{-6}\,/{}^\circ\mathrm{C} \), what temperature change \( \Delta T \) (°C) caused this stress?
Answer: D) 41.7
Step 1 – Identify given information:
\( \sigma = 100 \;\text{MPa} \), \( E = 200 \;\text{GPa} \), \( \alpha = 12 \times 10^{-6} \;/{}^\circ\text{C} \).
Step 2 – Solve for temperature change from thermal stress (FE Handbook):
$$\sigma = E \alpha \Delta T \quad \Rightarrow \quad \Delta T = \frac{\sigma}{E \alpha}$$
Step 3 – Solve:
$$\Delta T = \frac{100 \times 10^6}{200 \times 10^9 \times 12 \times 10^{-6}} = \frac{100 \times 10^6}{2.4 \times 10^6} = 41.7^\circ\text{C}$$
Step 4 – Answer: A temperature change of \( 41.7^\circ\text{C} \) causes \( 100 \;\text{MPa} \) of thermal stress in a fully restrained steel member.
Option (A) is what you get by plugging the volumetric expansion coefficient \(3\alpha = 36 \times 10^{-6}/^\circ\mathrm{C}\) into \(\sigma = E\alpha\Delta T\) instead of the linear coefficient, which triples the denominator.
Correct answer: (D)
Problem 4 — D. Combined stresses, principal stresses, and Mohr's circle
A plane stress state has \( \sigma_x = 60\,\mathrm{MPa} \), \( \sigma_y = 0\,\mathrm{MPa} \), and \( \tau_{xy} = 20\,\mathrm{MPa} \). What is the normal stress \( \sigma_\theta \) (MPa) on the plane whose outward normal is at \( \theta = 45^\circ \) from the +x-axis?
Answer: C) 50
Step 1 – Identify the given stress state:
| Parameter | Value |
|---|---|
| \( \sigma_x \) | \( 60 \;\text{MPa} \) |
| \( \sigma_y \) | \( 0 \;\text{MPa} \) |
| \( \tau_{xy} \) | \( 20 \;\text{MPa} \) |
| \( \theta \) | \( 45^\circ \) |
Step 2 – Write the stress transformation equation (from the FE Handbook):
$$\sigma_\theta = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos(2\theta) + \tau_{xy}\sin(2\theta)$$
Step 3 – Evaluate the trigonometric terms at \( 2\theta = 2(45^\circ) = 90^\circ \):
$$\cos(90^\circ) = 0, \qquad \sin(90^\circ) = 1$$
Step 4 – Substitute all values into the equation:
$$\sigma_\theta = \frac{60 + 0}{2} + \frac{60 - 0}{2} \cdot 0 + 20 \cdot 1$$
$$\sigma_\theta = 30 + 0 + 20 = 50 \;\text{MPa}$$
Correct answer: (C)
Problem 5 — A. Shear and moment diagrams
A simply supported beam has span \( L = 6\,\mathrm{m} \) with a uniform load \( w = 2\,\mathrm{kN/m} \) over the entire span and a point load \( P = 5\,\mathrm{kN} \) at \( x = 2\,\mathrm{m} \) from the left. What is the maximum absolute shear force in the beam (kN)?
Answer: C) 9.33
Step 1 – Identify given information:
\( L = 6 \;\text{m} \), \( w = 2 \;\text{kN/m} \) (full span), \( P = 5 \;\text{kN} \) at \( x = 2 \;\text{m} \).
Step 2 – Find reactions using moment equilibrium (FE Handbook):
Total load: \( W = wL + P = 12 + 5 = 17 \;\text{kN} \)
\( \sum M_A = 0 \): \( R_B(6) = 12(3) + 5(2) \)
Step 3 – Solve:
\( R_B(6) = 36 + 10 = 46 \), so \( R_B = 7.67 \;\text{kN} \)
$$R_A = 17 - 7.67 = 9.33 \;\text{kN}$$
Maximum \( |V| \) at left support: \( 9.33 \;\text{kN} \).
Step 4 – Answer: The maximum absolute shear is \( 9.33 \;\text{kN} \). The load is biased toward A, so \( R_A > R_B \).
Correct answer: (C)
Using the FE Reference Handbook for Mechanics of Materials
Two handbook resources decide your speed here: the table of simply supported and cantilever beam deflection cases, and the Mohr's circle and principal stress relations. Find both before exam day and note the sign convention the handbook uses for shear stress on the circle, because it differs from some textbooks and flips which principal plane you report.
Four Mistakes That Cost Points
- Using the wrong I for the axis that matters Bending about the strong axis uses I about that axis; buckling uses the smallest I of the section unless bracing prevents it. Reading the wrong column of the section table produces an answer that looks reasonable and is one of the offered choices.
- Putting c at the wrong distance In M c over I, c is measured from the neutral axis to the fiber you care about, which is only half the depth for a symmetric section. For a tee or an unsymmetric built-up shape the top and bottom values differ, and the larger one governs.
- Ignoring the effective length factor in buckling Euler's formula uses K L, not L. A fixed-pinned column is not a pinned-pinned column, and forgetting K changes the critical load by a factor of two or four. Identify the end conditions from the sketch before you touch the equation.
- Superposing stresses of different types Axial and bending stresses add algebraically because both act normal to the section. Shear and normal stresses do not add; they combine through Mohr's circle or the principal stress equations. Adding a torsional shear to a bending normal stress is a common and expensive slip.
Frequently Asked Questions
How many mechanics of materials questions are on the FE Civil exam?
NCEES specifies 7 to 11 questions out of 110. It sits just behind statics among the mechanics areas, and the skills carry over into the structural questions, where beam behavior, section properties and column capacity reappear in a design context.
Do I have to draw shear and moment diagrams on the exam?
Usually you only need one value, not the full diagram, but sketching it is still the fastest reliable route. Draw shear from the reactions, remember that moment is the area under the shear diagram, and maximum moment occurs where shear crosses zero. That single rule answers a surprising share of these questions.
Is Mohr's circle tested on the FE Civil exam?
Yes, principal stresses and maximum in-plane shear stress are part of the specification. You are rarely asked to draw the circle; you are asked for the numbers. The handbook gives the principal stress and maximum shear equations directly, so plugging in sigma x, sigma y and tau xy is often faster than sketching.
Do I need to memorize beam deflection formulas?
No. The NCEES FE Reference Handbook includes standard deflection and slope cases for simply supported and cantilever beams under point loads, uniform loads and end moments. Your job is recognizing which case the problem matches, and knowing that superposition lets you add two cases when the loading is a combination of them.
Keep Going
Done with mechanics of materials? Work through the other FE Civil knowledge areas with more free practice problems, review the full FE Civil exam topic breakdown, or plan your preparation with our FE exam study guide and study timeline. Or test yourself against the clock with the free FE Civil practice exam: 20 timed questions in the CBT format.