3.3 Mechanics of Materials

The Analytical Core of FE Civil

Mechanics of Materials carries 6–9 questions and underpins every Structural Engineering problem. The skill tested is matching a loading (axial, torsion, bending, shear, combined) to the correct stress equation and the right section property.

Stress, Strain, and Hooke's Law

Normal stress is force per unit area, σ = P/A. Normal strain is ε = δ/L. In the elastic range they follow Hooke's law:

$\sigma = E\varepsilon$

where E is the modulus of elasticity (≈ 29,000 ksi / 200 GPa for steel). Poisson's ratio ν relates lateral to axial strain (ν ≈ 0.3 for steel). The shear modulus is G = E / [2(1 + ν)].

Axial Loading

Elastic axial deformation of a prismatic bar:

$\delta = \frac{PL}{AE}$

Thermal elongation is δ = αLΔT. If the bar is restrained, a thermal stress σ = EαΔT develops.

Torsion

For a circular shaft, the torsional shear stress is:

$\tau = \frac{Tr}{J}, \qquad \phi = \frac{TL}{JG}$

The polar moment of inertia J = πr⁴/2 = πd⁴/32 (solid). Stress is maximum at the outer surface and zero at the center.

Shear and Moment Diagrams

The load, shear, and moment are linked by calculus:

Maximum bending moment occurs where shear crosses zero. For a simply supported beam of span L with a central point load P, M_max = PL/4; with a uniform load w, M_max = wL²/8.

Bending and Transverse Shear Stress

Flexure stress varies linearly through the depth:

$\sigma = \frac{Mc}{I}$

maximum at the extreme fiber (distance c from the neutral axis). Transverse shear stress is τ = VQ/(Ib), maximum at the neutral axis where Q (first moment of area) is largest.

Beam Deflection

Standard cases are tabulated in the NCEES FE Reference Handbook — look them up, do not derive:

• Simply supported, central point load P: δ_max = PL³/(48EI)
• Simply supported, uniform load w: δ_max = 5wL⁴/(384EI)
• Cantilever, end point load P: δ_max = PL³/(3EI)

Combined Stress and Mohr's Circle

When axial, bending, and torsion act together, superpose normal stresses and use Mohr's circle to find principal stresses and maximum shear:

$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

The maximum in-plane shear stress equals the circle's radius, τ_max = √[((σx − σy)/2)² + τxy²].

Columns and Buckling

Long, slender columns fail by elastic (Euler) buckling, not crushing:

$P_{cr} = \frac{\pi^2 EI}{(KL)^2}$

The effective-length factor K depends on end conditions: K = 1.0 (pinned–pinned), 0.5 (fixed–fixed), 0.7 (fixed–pinned), 2.0 (fixed–free cantilever). Buckling controls when the slenderness ratio KL/r is large. These K values are tabulated in the handbook — confirm the support condition before selecting K.