Deformations: Axial, Torsion, Bending and Thermal
NCEES FE Civil spec: Mechanics of Materials, C. Deformations (e.g., axial, torsion, bending, thermal). FE Reference Handbook: Mechanics of Materials chapter, deformations and beam deflection tables.
Deformation questions ask how much a member stretches, twists or deflects, rather than how highly it is stressed. The four cases on the FE Civil exam are axial elongation, angle of twist, beam deflection and thermal expansion, sometimes combined, as when a heated bar closes a gap and then pushes against a support.
What the NCEES spec covers
Section C covers axial, torsional, bending and thermal deformations. Beam deflection questions almost always use tabulated formulas rather than integration, so the skill is picking the right case and keeping units straight.
Key concepts
- Axial deformation is proportional to load and length and inversely proportional to stiffness AE. For stepped bars or varying loads, add the segments.
- Angle of twist in a circular shaft is proportional to torque and length and inversely proportional to GJ. The answer is in radians.
- Beam deflection scales with L³ for point loads and L⁴ for distributed loads, so span matters enormously.
- Superposition. For linear-elastic behavior, deflections from separate loads can be added.
- Thermal deformation of a free member causes no stress. If movement is blocked (fully, or after a gap closes), the blocked part of the expansion becomes elastic strain.
Core formulas
Where it is in the FE Reference Handbook
The Mechanics of Materials chapter has axial deformation, thermal deformation, torsion (angle of twist) and tables of slopes and deflections for simply supported and cantilevered beams. Those tables are among the most used pages in the handbook.
Worked example
A steel bar is 3 m long with A = 500 mm², E = 200 GPa and α = 12 × 10⁻⁶/°C. (a) Find its elongation under a 60 kN tensile load. (b) Separately, with no load, the bar sits with a 1.0 mm gap between its end and a rigid wall and is heated by 40°C. Find the stress in the bar.
(a) Axial elongation.
(b) Free thermal expansion.
That is more than the 1.0 mm gap, so the wall blocks 0.44 mm of expansion. The blocked amount becomes compressive strain:
The wall force is 29.3 × 500 = 14.7 kN. Without the gap, the fully restrained stress would be EαΔT = 96 MPa, so the gap relieves most of it.
Common mistakes
- Mixing GPa with MPa or m with mm. Using N, mm and MPa (N/mm²) throughout avoids most slips.
- Computing thermal stress for the full expansion when part of it is absorbed by a gap.
- Reporting angle of twist in degrees without converting from radians.
- Picking a cantilever deflection formula for a simple span, or the reverse.
- Using area moment of inertia in mm⁴ with E in GPa without matching units.
The shaft torsion calculator handles twist, and the beam calculator gives deflections for any load case.
Practice Problems
Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.
- A uniform linear-elastic bar has axial tensile load P = 100 kN, length L = 2 m, area A = 1,000 mm², and modulus E = 200 GPa. What is its elongation? See the worked solution or practice the Stress & Strain deck
- A simply supported steel beam has span L = 6 m, uniform load w = 12 kN/m, E = 200 GPa and I = 300 × 10^6 mm^4. What is the maximum deflection? See the worked solution or practice the Beam Analysis deck
- A 2 m cantilever with flexural rigidity E I = 4000 kN·m² carries a 10 kN point load at its free end. What is the deflection at the free end? See the worked solution or practice the Beam Analysis deck
- A hollow steel shaft 2 m long has outside diameter 60 mm and inside diameter 40 mm, carries a torque of 1.5 kN·m, and has G = 80 GPa. Its angle of twist is most nearly: See the worked solution or practice the Stress & Strain deck
- A material has coefficient of thermal expansion α = 12 × 10⁻⁶/°C and undergoes a temperature increase of 50°C. What is its free thermal strain? See the worked solution or practice the Stress & Strain deck
Related Calculators and Examples
- Beam calculator (deflection)
- Shaft torsion calculator
- Beam formulas reference
- Beam analysis practice deck
Frequently Asked Questions
What is the formula for axial deformation?
Elongation equals P L / (A E), where P is the axial load, L is the length, A is the cross-sectional area and E is the modulus of elasticity. Add segments separately if the load or section changes along the bar.
What is the maximum deflection of a simply supported beam with a uniform load?
It is 5 w L^4 / (384 E I) at midspan. Watch units: with w in N/mm, L in mm, E in MPa and I in mm^4 the result is in mm.
How do I find the angle of twist of a shaft?
Use phi = T L / (G J), with J = pi d^4 / 32 for a solid circular shaft. The result is in radians.