Exercises: Structures and Materials

Work these with a calculator and $g_0 = 9.81\ \text{m/s}^2$ unless told otherwise. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each one cold before you peek. For the "implement it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does. Carry units through every step and sanity-check every number (against $g$, against a material's strength, or against sea-level pressure).

Part A — Warm-ups: loads, stress, and strain (⭐)

23.1 † A $1{,}500\ \text{kg}$ spacecraft rides an axial load factor of $5\ \text{g}$ during ascent. Compute the axial force on its payload adapter, in kN, and express it as an equivalent hanging mass in tonnes.

23.2 Convert a sound-pressure level of $145\ \text{dB}$ (a loud spot inside a fairing) to an RMS pressure in pascals, using $\text{SPL} = 20\log_{10}(p/p_{\text{ref}})$ with $p_{\text{ref}} = 2\times10^{-5}\ \text{Pa}$.

23.3 † A tension rod has a cross-sectional area of $20\ \text{mm}^2$ and carries $8\ \text{kN}$. Compute the stress in MPa. Is an aluminum rod (yield ~350 MPa) safe under this load, ignoring safety factors?

23.4 A $2.0\ \text{m}$ strut stretches by $1.2\ \text{mm}$ under load. Compute the strain (dimensionless, and in microstrain).

23.5 † A material at a strain of $6\times10^{-4}$ has Young's modulus $E = 72\ \text{GPa}$ (aluminum). Use Hooke's law $\sigma = E\varepsilon$ to find the stress in MPa.

23.6 A pressurized tank has radius $r = 1.5\ \text{m}$, wall thickness $t = 3\ \text{mm}$, and internal pressure $p = 4\ \text{bar}$. Compute the hoop stress $\sigma_{\text{hoop}} = pr/t$ in MPa.

Part B — Materials and specific properties (⭐⭐)

23.7 † Compute the specific strength (UTS/$\rho$) of titanium Ti-6Al-4V (UTS $950\ \text{MPa}$, $\rho = 4.43\ \text{g/cm}^3$) and of a 2xxx aluminum ($470\ \text{MPa}$, $2.7\ \text{g/cm}^3$). Which is higher, and by what percentage? Why, then, is aluminum the more common choice?

23.8 Compute the specific stiffness $E/\rho$ (in SI, $\text{m}^2/\text{s}^2$) for stainless steel ($E = 193\ \text{GPa}$, $\rho = 7{,}900\ \text{kg/m}^3$) and aluminum ($70\ \text{GPa}$, $2{,}700\ \text{kg/m}^3$). Comment on how close they are and what that implies for a buckling-limited tank.

23.9 † The mass of a buckling-limited cylinder wall scales as $\rho/E$ (the inverse of specific stiffness). Using the values in 23.8, estimate the ratio of a steel wall's mass to an aluminum wall's mass for the same buckling resistance. Does steel "triple" the mass?

23.10 In two sentences, explain why carbon-fiber composite reaches far higher specific stiffness than any metal, and name one drawback that keeps metals in wide use.

Part C — Implement it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it. (Reference code is in code/exercise-solutions.py.)

23.11 † Write hoop_stress(p, r, t) returning $pr/t$ in pascals. Trace hoop_stress(3.5e5, 1.83, 0.004) and report it in MPa.

23.12 Write margin_of_safety(sigma_allow, sigma_limit, fos) returning $\sigma_{\text{allow}}/(\text{FoS}\cdot\sigma_{\text{limit}}) - 1$. Trace it for margin_of_safety(570, 360, 1.4).

23.13 † Write kg_per_kg_structure(dv, isp) returning the stage mass added per kilogram of structure at fixed delta-v — i.e. the mass ratio $e^{\Delta v/v_e}$ with $v_e = I_{sp}g_0$ (use $g_0 = 9.80665$). Trace it for kg_per_kg_structure(6000, 348).

Part D — Find the error (⭐⭐)

23.14 † A student computes the hoop stress for $r = 1.83\ \text{m}$, $t = 0.004\ \text{m}$, $p = 3.4\times10^{5}\ \text{Pa}$ and writes "$\sigma = pr\,t = 3.4\times10^{5}\times1.83\times0.004 = 2{,}489\ \text{Pa}$." Identify the mistake and give the correct hoop stress in MPa.

23.15 A subsystem lead reports proudly that a bracket has "a margin of safety of 2.5 — extremely safe." The chief engineer sends it back to be made lighter. Explain why, and clearly distinguish a factor of safety (a requirement) from a margin of safety (the reserve above it).

Part E — Design it (⭐⭐ / ⭐⭐⭐)

23.16 † A strut sees a worst-case (limit) stress of $300\ \text{MPa}$ and must meet a factor of safety of $1.4$ on ultimate. (a) What ultimate strength must the material have, at minimum? (b) If you pick 7075-T6 aluminum (UTS $570\ \text{MPa}$), compute the margin of safety. (c) Is the part lean or overweight, and what would you do about it?

23.17 (Mission project) Build the mass budget for your MDR track (A comsat · B lunar lander · C Mars orbiter · D asteroid probe): a subsystem table with first-pass masses (use §23.5's representative shares as a sanity check), a dry-mass sum, a $25$–$30\%$ mass margin, a note tying your wet mass to your delta-v budget (Chapter 3) via the mass ratio, and the launch-vehicle payload cap it must fit under. Save it; you will size the vehicle from it in Chapter 29.

23.18 † A cryogenic tank must hold $p = 5\ \text{bar}$ at radius $r = 2.25\ \text{m}$. (a) For an aluminum whose allowable hoop stress (safety factor already applied) is $300\ \text{MPa}$, find the minimum wall thickness from $\sigma_{\text{hoop}} = pr/t$. (b) Compute the wall's mass per metre of length ($\rho_{\text{Al}} = 2{,}700\ \text{kg/m}^3$; treat the wall as a thin cylinder, cross-sectional material area $2\pi r t$). (c) In one or two sentences, explain — invoking $E/\rho$ — why switching this wall to steel would not triple its mass if the design were buckling-limited.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

23.19 † Why can't you just use a fat factor of safety of 3 on a rocket "to be safe," the way a building does? Answer in terms of the rocket equation and mass, and say what aerospace does instead to stay safe at FoS $\approx 1.4$.

23.20 Why can't you just build the whole spacecraft out of the strongest material (say titanium everywhere) so nothing ever breaks? Give two reasons rooted in cost, manufacturability, and specific (not raw) strength.

23.21 † Estimate, to an order of magnitude, how many kilograms at lift-off one extra kilogram of payload "costs" on a two-stage rocket in which each stage has a mass ratio of about $3$. What does this say about where in the stack a saved kilogram is worth the most?

Part G — Interleaved & synthesis (Chapters 1–5, 22) (⭐⭐/⭐⭐⭐)

23.22 † (Ch. 3) A stage must keep a mass ratio of $4.5$ to deliver its delta-v. How much extra propellant does $3\ \text{kg}$ of added structure require, at fixed delta-v? (Use "extra propellant $= (R-1)\times\Delta m_{\text{dry}}$.")

23.23 (Ch. 5) The max-Q dynamic-pressure force and the $q\alpha$ bending load of Chapter 5 are launch loads. Classify each among the four load types of §23.1, and name the structure (primary or secondary) that must carry the $q\alpha$ bending.

23.24 † (Ch. 22) Reusability (Chapter 22) adds landing legs, grid fins, and reinforcement to a booster. In mass-budget terms, why is reuse a structural "tax" that the rocket equation punishes — and why is it worth paying anyway (theme #5)?

23.25 (Ch. 2, synthesis) Using Newton's second law $F = ma$, explain why the peak axial launch load usually occurs near a stage's burnout rather than at lift-off, and connect this to why engines often throttle down near the end of the burn.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: a correct set-up with explicit units, a sanity check (against $g$, a material strength, or the rocket equation), and a clear statement of which trade-off — mass vs. strength, mass vs. cost, or performance vs. reusability — is being made.