Stresses and Strains

NCEES FE Civil spec: Mechanics of Materials, B. Stresses and strains (e.g., diagrams, axial, torsion, bending, shear, thermal). FE Reference Handbook: Mechanics of Materials chapter, stress, strain, torsion and beam stresses.

Stress is internal force per unit area, and strain is deformation per unit length. This section asks you to compute stress for each basic loading (axial, bending, transverse shear, torsion and thermal) and to relate stress to strain through material properties. These formulas account for a large share of the mechanics of materials questions on the FE Civil exam.

What the NCEES spec covers

Section B covers stress-strain diagrams and their key points, Hooke's law, Poisson's ratio, axial stress, torsional shear stress in circular shafts, bending stress, transverse shear stress in beams, and thermal stress and strain.

Key concepts

Stress-strain diagram. The initial straight line has slope E. Key points are the proportional limit, the yield strength (often by the 0.2% offset method), the ultimate strength and fracture.
Poisson effect. A bar stretched axially contracts laterally. Poisson's ratio is the ratio of lateral to axial strain, about 0.3 for steel.
Bending stress varies linearly from zero at the neutral axis to a maximum at the extreme fiber.
Transverse shear stress is zero at the top and bottom and largest at the neutral axis. For a rectangle the maximum is 1.5 times the average.
Torsion in a circular shaft gives shear stress that grows linearly with radius.
Thermal stress only develops when thermal expansion is restrained.

Core formulas

σ = \frac{P}{A} ε = \frac{δ}{L} σ = E ε ν = - \frac{ε _{lat}}{ε _{axial}} G = \frac{E}{2 ( 1 + ν )}

σ_{b} = \frac{M c}{I} = \frac{M}{S} τ = \frac{V Q}{I b} τ_{m a x, rect} = \frac{3 V}{2 A}

τ = \frac{T ρ}{J} J_{solid} = \frac{π d ^{4}}{32} ε_{T} = α Δ T σ_{T} = E α Δ T (fully restrained)

Where it is in the FE Reference Handbook

All of these are in the Mechanics of Materials chapter: uniaxial stress-strain, Hooke's law and elastic constants, thermal deformations, torsion, and beam bending and shear stress. Typical material properties (E, G, ν, α) are tabulated in the same chapter.

Worked example

A rectangular timber beam is 150 mm wide and 300 mm deep. At one section it carries a bending moment of 40 kN·m and a shear force of 60 kN. Find the maximum bending stress and the maximum transverse shear stress.

Step 1: section properties.

I = \frac{150 ( 300 ) ^{3}}{12} = 337.5 \times 1 0^{6} mm^{4} c = 150 mm A = 45, 000 mm^{2}

Step 2: bending stress.

σ_{b} = \frac{M c}{I} = \frac{( 40 \times 1 0 ^{6} ) ( 150 )}{337.5 \times 1 0 ^{6}} = 17.8 MPa

This is tension on one face and compression on the other.

Step 3: shear stress at the neutral axis.

τ_{m a x} = \frac{3 V}{2 A} = \frac{3 ( 60 , 000 )}{2 ( 45 , 000 )} = 2.0 MPa

The same result follows from VQ/(Ib) with $Q = 150 (150) (75) = 1.6875 \times 1 0^{6} mm^{3}$ . Shear stress is much lower than bending stress here, which is typical for all but short, heavily loaded beams.

Common mistakes

Using the full depth h for c instead of the distance from the neutral axis to the extreme fiber.
Mixing units: N·mm with mm gives MPa. Convert kN·m to N·mm (multiply by 10⁶).
Using b in VQ/(Ib) as the full flange width when the cut is in the web.
Calculating thermal stress for a member that is free to expand.
Using the radius in J = πd⁴/32, or the diameter for ρ in Tρ/J.

See the bending stress example and the VQ/Ib shear stress example.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A 50 kN axial load acts uniformly over a 500 mm² cross-sectional area. What is the normal stress magnitude? See the worked solution or practice the Stress & Strain deck
A linear-elastic material has normal stress 200 MPa and elastic modulus E = 200 GPa. What is its axial strain? See the worked solution or practice the Stress & Strain deck
A rectangular beam 200 mm wide and 400 mm deep carries a bending moment of 50 kN·m. What is the maximum bending stress? See the worked solution or practice the Beam Analysis deck
A rectangular beam 200 mm wide × 300 mm deep carries a shear force of 60 kN. What is the maximum transverse shear stress? See the worked solution or practice the Beam Analysis deck
A solid circular shaft 40 mm in diameter carries a torque of 1.2 kN·m. The maximum shear stress is most nearly: See the worked solution or practice the Stress & Strain deck
A steel member is held between rigid supports so it cannot expand, then heated by 40 °C. Taking α = 12 × 10⁻⁶ per °C and E = 200 GPa, the magnitude of the induced axial stress is most nearly: See the worked solution or practice the Stress & Strain deck

Related Calculators and Examples

Frequently Asked Questions

What is the flexure formula?

Bending stress equals M c / I, where M is the bending moment, c is the distance from the neutral axis to the point of interest and I is the moment of inertia of the cross section. The maximum is at the extreme fiber.

Where is the maximum shear stress in a rectangular beam?

At the neutral axis. For a rectangular section it equals 1.5 times the average shear stress, or 3V / (2A).

When does thermal stress occur?

Only when thermal expansion or contraction is restrained. A fully restrained bar develops stress E alpha delta T; a bar free to move develops none.

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