Chapter 23 — Key Takeaways (Structures and Materials)

A one-page reference. Reread this before an exam, or before you size a structure, pick a material, or close a mass budget.

The equations that matter

Equation Use it to find Notes
$F = m\,n\,g_0$ force from a load factor $n$ (in g's) quasi-static launch load; $g_0 = 9.81\ \text{m/s}^2$
$\text{SPL} = 20\log_{10}(p/p_{\text{ref}})$ acoustic pressure from dB $p_{\text{ref}} = 2\times10^{-5}\ \text{Pa}$; 140 dB $\to$ 200 Pa
$\sigma = F/A$ stress (Pa, usually MPa) internal force per unit area
$\varepsilon = \Delta L/L$ strain (dimensionless) Hooke's law: $\sigma = E\varepsilon$
$\sigma_{\text{hoop}} = pr/t$ hoop stress in a thin cylinder longitudinal is half: $pr/2t$; sphere $= pr/2t$ both ways
$\text{FoS} = \sigma_{\text{allow}}/\sigma_{\text{limit}}$ factor of safety (a requirement to meet) aerospace: ~1.25 yield, 1.4 ultimate
$\text{MS} = \dfrac{\sigma_{\text{allow}}}{\text{FoS}\cdot\sigma_{\text{limit}}} - 1$ margin of safety (reserve above FoS) $\ge 0$ passes; a small positive MS is a lean part
$\sigma/\rho,\ E/\rho$ specific strength, specific stiffness the real material figures of merit (per kilogram)
$m_0/m_f = e^{\Delta v/v_e}$ mass ratio $R$ 1 kg of structure costs $R$ kg of stage: $(R-1)$ kg is propellant
$\sigma_{\text{cr}} \propto E\,(t/r)$ buckling stress of a thin cylinder depends on stiffness & geometry, not strength

What every symbol means (and its units)

Symbol Name Units Notes
$\sigma$ stress Pa (MPa) tension, compression, or shear
$\varepsilon$ strain — $\Delta L/L$; often microstrain ($10^{-6}$)
$E$ Young's modulus (stiffness) Pa (GPa) steel ~193, Ti ~114, Al ~72, CFRP ~70 GPa
$\rho$ density kg/m³ (g/cm³) Al 2.7, CFRP 1.6, Ti 4.43, steel 7.9 g/cm³
$p, r, t$ pressure, radius, wall thickness Pa, m, m tank sizing
$n$ load factor g's axial 3–6 g, lateral 1–2 g
FoS, MS factor / margin of safety — requirement vs. reserve above it
$R$ mass ratio — $= m_0/m_f = e^{\Delta v/v_e}$

Launch loads at a glance

Load Typical magnitude Threatens Peaks
Quasi-static acceleration 3–6 g axial, 1–2 g lateral primary structure near stage burnout
Random / sine vibration 5–15 $\text{g}_{\text{rms}}$ (20–2000 Hz) fatigue, brackets, joints lift-off, transonic
Acoustic 140–145 dB in fairing (~180 dB near engines) light, large-area parts (arrays, dishes) lift-off, transonic
Pyroshock 100s–1000s of g, milliseconds brittle parts, electronics separation events

The materials, ranked by specific properties (room temperature)

Material $\rho$ (g/cm³) $\sigma/\rho$ $E/\rho$ Pick it for
Aluminum (2xxx / Al-Li) 2.7 ~174 ~27 the default: cheap, weldable, heritage
CFRP composite 1.6 ~375† ~44† best specific strength/stiffness; fairings, panels
Titanium (Ti-6Al-4V) 4.43 ~214 ~26 hot, strong, propellant-compatible parts
Stainless steel (301/304L) 7.9 ~71–165 ~24 cryo-strengthening + heat tolerance + cost (Starship)

† composite values are layup-dependent (directional). Key fact: $E/\rho$ is nearly identical (~2.5×10⁷ m²/s²) for all metals, so a buckling-limited steel tank weighs about the same as an aluminum one.

Decision aid — "which idea / when"

You want… Use
force on a component $F = m\,n\,g_0$ (load factor)
does a part survive its load $\text{MS} = \sigma_{\text{allow}}/(\text{FoS}\cdot\sigma_{\text{limit}}) - 1 \ge 0$
tank wall thickness $t = pr/\sigma_{\text{allow}}$ (cylinder) or $pr/2\sigma_{\text{allow}}$ (sphere)
which material (mass-limited) highest $\sigma/\rho$ (strength case) or $E/\rho$ (buckling/stiffness case)
cost of adding structure $\Delta m_0 = R\,\Delta m_{\text{dry}}$; extra propellant $= (R-1)\Delta m_{\text{dry}}$
is a thin tube strength- or buckling-limited compare applied $\sigma$ to yield and to $\sigma_{\text{cr}}\propto E(t/r)$

Common pitfalls

Pitfall Reality
"Choose the strongest material." Choose the highest specific strength/stiffness — per kilogram, not raw.
"A big factor of safety is good." On a rocket it is wasted mass; target FoS met with MS near zero.
"Deceleration/force = safety." FoS is a requirement (≥1.4); MS is the reserve above it. Don't confuse them.
"Steel is too heavy for a rocket." $E/\rho$ ≈ aluminum's, and it strengthens at cryo — buckling-limited mass barely changes.
"1 kg of structure = 1 kg off payload." It costs $R$ kg of stage (structure + propellant); compounds per stage.
"A stronger material fixes buckling." Buckling depends on $E$ and geometry, not strength — use stiffness/pressure/shape.

Numbers worth memorizing

  • $g_0 = 9.81\ \text{m/s}^2$; aluminum $\rho \approx 2.7$, steel $\approx 7.9\ \text{g/cm}^3$.
  • Aerospace factors of safety: ~1.25 yield, 1.4 ultimate — the lowest in engineering.
  • $\sigma_{\text{hoop}} = pr/t$ (twice the longitudinal); sphere $= pr/2t$.
  • $E/\rho \approx 2.5\times10^{7}\ \text{m}^2/\text{s}^2$ for Al, Ti, and steel alike.
  • "1 kg of structure costs several ($R$) kilograms at lift-off"; mass margin ~25–30% early → a few % at launch.

Mission / project additions this chapter

  • MDR: started the spacecraft mass budget (major increment) — a subsystem table, a dry-mass sum, a 25–30% margin, and the wet mass tied to the delta-v budget (Ch. 3) and the launch-vehicle cap (Ch. 30). Feeds vehicle sizing in Ch. 29.
  • astrotools: created structures.py — dry_mass(subsystems), apply_margin(mass, frac), mass_budget(subsystems, margin_frac), margin_of_safety(sigma_allow, sigma_limit, fos). Signatures are canonical; Ch. 25/29 call them.