Equations and page numbers refer to Ashby, Materials Selection in Mechanical Design, 6th edition (Elsevier, 2025).
Columns
Euler load Fc = n²π²EI/L² (Eq 11.14, p.389), pinned ends n = 1. Shape factor φ = 12I/A² (Eq 11.3, p.380); tube 3r/πt, solid 3/π (Table 11.3), so the load ratio is r/t.
Real columns start slightly bent and carry residual stress, so they fail below Fc. The demo gives each column the EN 1993-1-1 equivalent bow e0 = α(λ̄ − 0.2)W/A (buckling curve a, α = 0.21, for the hot-finished tube; curve c, α = 0.49, for the solid bar). The bow grows as 1/(1 − F/Fc), and first yield, where axial plus bending stress reaches σy, then equals the code buckling resistance Nb = χAfy. After that the column collapses.
Sections: EN 10219 circular hollow section 60.3 × 2.9 mm; solid bar of equal area 25.8 mm. S355: E = 210 GPa, σy = 355 MPa.
Thin walls
Aluminium 6061-T6, E = 69 GPa, σy = 250 MPa, area 150 mm². The surface first yields at My = σyZ; the tube collapses when a full plastic hinge forms at Mp = (4/π)My, 27% higher (thin tube). The wall kinks at about 0.18 E t/r: the classical elastic value 0.6 E t/r cut by a typical 0.3 knock-down for real-world imperfections. Hinge and kink limits cross at r/t ≈ 39, φ ≈ 37, close to Ashby's φmax ≈ 2.3(E/σf)½ = 38 (Eq 11.15a, p.393).
Cell-wall solid ≈ 1500 kg/m³, 35 GPa, read from the cellulose bubble on Fig 13.34. Loads are given as a fraction of F₀, the load that breaks the solid bar. Wood crushes along the grain at about 39 MPa but breaks in bending at about 70 MPa (Sitka spruce, 12% moisture, USDA Wood Handbook), so the squeezed face creases at 0.55 of the breaking load; the beam then softens and breaks on the stretched face. The foam's walls yield in compression, so its collapse starts at 0.3 × 0.55 = 0.17 F₀.
Lattices
Octet truss (Fig 13.21): Maxwell M ≥ 0 (Eq 13.3, p.453), E ≈ ⅓(ρ/ρs)Es (Eq 13.27, p.473), collapse σ ≈ ⅓(ρ/ρs)σs (Eq 13.28) or strut buckling 0.2(ρ/ρs)²Es (Eq 13.29), whichever is lower.
Open (body-centred) lattice: M < 0, E ≈ (ρ/ρs)²Es (Eq 13.18), σpl ≈ 0.3(ρ/ρs)3/2σs (Eq 13.20).
Sheet gyroid and sheet diamond TPMS: E/Es = 0.90(ρ/ρs)1.58 and 0.713(ρ/ρs)1.21; σ/σs = 0.536(ρ/ρs)1.45 and 0.419(ρ/ρs)1.14. Fits to laser powder-bed Ti-6Al-4V scaffolds at ρ/ρs 0.33–0.47 (Naghavi et al., Bioengineering 9 (2022) 504, Table 7); below 0.33 they are extrapolated. Surfaces: gyroid sin x cos y + sin y cos z + sin z cos x; diamond (Schwarz D). The sheet is the region |f| < t, with t set so the solid fraction equals ρ/ρs.
Es = 115 GPa, σs ≈ 1000 MPa (as printed). As-built Ti-6Al-4V has little ductility, so strut lattices end with struts snapping along a 45° band and the load dropping; the block above the band slides down and the platen follows it. Sheet TPMS instead yield, then crush layer by layer along a long plateau (Naghavi et al. 2022).
Squash in the 3D view is drawn 8× larger than the elastic strain; the ratio between the two lattices is true.
Corrugation
φe = (2a+t)²/t² and φf = (2a+t)/t (Eqs 12.8–12.9, p.436) are ideal values for a section through the wave peaks; real gains are a little lower, and deep corrugations can also buckle locally at the crest. S280GD galvanized sheet, σy = 280 MPa. The surface first yields at My = σyZ; the sheet folds when the full plastic moment forms, 1.5 My for a flat strip, falling towards 1.27 My for a deep wave.
Microscope views
Grain and atom views are drawings of the mechanisms, not simulations. Strain is exaggerated so it can be seen. Atom views show one slip plane side-on; the real plane sits at about 45° to the load, where the shear stress on it is largest.
Mechanisms: slip on {110} planes in ferrite and dislocation pinning by carbon (steel); {111} slip and Mg–Si (β″) precipitates in 6061-T6; α′ martensite, partly melted powder and pores in printed Ti-6Al-4V; cellulose chains joined by hydrogen bonds in wood cell walls; zinc-coating cracking on galvanized sheet.