3.6.3 · D5Spacecraft Structures & Systems Engineering
Question bank — Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E
Before we start, a glossary so no symbol is used unearned:
- = the applied pulling (or pushing) force, in newtons (N).
- = the cross-sectional area of the bar — the size of the face you would see if you sliced straight through it — in m².
- = the original (unloaded) length of the bar, in m; = the change in length after loading, in m.
- (stress) , force divided by cross-section area — "how crowded the pull is", in pascals (Pa = N/m²).
- (strain) , stretch divided by original length — a pure ratio, no units.
- (Young's modulus) , the slope of the straight part of the stress–strain graph — "how stubborn the material is", in Pa.
- (yield stress) = the stress ceiling where the material stops springing back and starts deforming permanently. Below it the material is elastic (returns to shape); above it, plastic (stays bent). See Yield Strength and Plastic Deformation.
The two figures below are the visual backbone the traps keep pointing at: Figure 1 shows what "slope " versus "ceiling " means on the real curve; Figure 2 shows why the deflection formula puts and on the bottom.
True or false — justify
TF1 — "A thick bar and a thin wire of the same aluminium have the same stiffness ."
True. is a material property; it does not care about the part's shape. The part stiffness (force per stretch) differs, but is identical.
TF2 — "Doubling the cross-sectional area doubles the stress for a fixed force."
False. Stress is , so doubling halves the stress — the same force is shared over more material.
TF3 — "Strain is measured in metres."
False. Strain is , length over length, so units cancel — it is dimensionless (sometimes quoted as microstrain, ).
TF4 — "A high Young's modulus guarantees the material is strong."
False. is the slope (stiffness); strength is the stress ceiling (Figure 1) at which it fails. A stiff material like cast iron can be brittle and fail early. See Yield Strength and Plastic Deformation.
TF5 — "Below the yield point, stress and strain are proportional."
True. That linear region (left of in Figure 1) is where Hooke's Law lives; the constant of proportionality is .
TF6 — "Stretching a bar by 2 mm always means the same strain."
False. Strain depends on original length: 2 mm on a 2 m strut is , but 2 mm on a 20 mm sample is — a hundred times more.
TF7 — "If two struts carry the same force but one is longer, the longer one stretches more."
True (same , , ). From , stretch grows in direct proportion to length .
TF8 — "A material with lower stretches more under the same stress."
True. Strain , so smaller gives larger strain — this is why aluminium ( GPa) strains ~2.9× more than steel ( GPa) at equal stress.
TF9 — "Stress inside a bar depends on which imaginary cut you choose."
False (for a uniform axial bar). By Newton's 3rd law every cut must carry the same internal force , so is the same across every cross-section.
Spot the error
SE1 — "The band stretched 3 mm, so its strain is 3 mm."
Error: strain is a ratio, , never a raw length. You must divide the 3 mm by the original length before it means anything.
SE2 — "Stress is 70, so I'll plug 70 straight into ."
Error: units. "70" is likely MPa; the formula needs SI (Pa and m). Convert Pa first, or the answer is off by .
SE3 — "Steel is stiffer than aluminium, so steel can carry more stress before breaking."
Error: conflates stiffness with strength. Stiffness () sets how much it bends, not when it fails. A soft aluminium alloy can out-strength some steels; check the actual .
SE4 — "I found , so I'll just use anyway."
Error: is Hooke's Law, valid only in the elastic region (left of the ceiling in Figure 1). Past yield the material deforms permanently and this formula overshoots badly. See Stress-Strain Curve.
SE5 — "Strain has no units, so I can add it to a length."
Error: dimensionless does not mean 'a length'. You may only recover a length by multiplying strain by (), not by adding.
SE6 — "The strut is stiff ( high), so it can never fail."
Error: stiffness only limits deflection; a stiff-but-brittle strut can snap suddenly once stress hits its (possibly low) ceiling . Stiffness and margin against failure are separate questions — the second needs a Factor of Safety.
SE7 — "This composite has GPa, so it stretches more than aluminium at GPa."
Error: reversed logic. Higher means less stretch for the same stress, since shrinks as grows (steeper slope in Figure 1).
Why questions
WHY1 — Why do we divide force by area instead of just using force?
Because failure and stretching depend on how concentrated the force is, not its raw size. A 1000 N pull crowds a hair but is nothing to a girder — dividing by strips out the part size.
WHY2 — Why do we divide stretch by original length?
To describe the material, not the particular part. The same fractional stretch is the meaningful quantity; a fixed 1 mm means very different things on a short vs long bar.
WHY3 — Why does have the same units as stress (Pa)?
Because and is dimensionless — dividing pascals by a pure number leaves pascals.
WHY4 — Why is called a slope and strength a ceiling?
On the stress–strain graph (Figure 1), is the gradient of the initial straight line (rise in per rise in ); strength is the height where the line stops being straight. A slope and a height are different things.
WHY5 — Why does the deflection formula put and on the bottom?
More area or more stiffness both resist stretching (parallel springs in Figure 2), so both reduce — they belong in the denominator. Longer length gives strain more distance to accumulate over, so is on top.
WHY6 — Why can two materials fail at the same stress yet stretch by wildly different amounts first?
Because failure stress (the ceiling) and (the slope up to it) are independent. Spring steel and a stiff polymer might share a but have very different , so very different strain at failure.
WHY7 — Why must we check before trusting Hooke's Law?
Because is only the linear part of the real curve (Figure 1). Above the line bends over into permanent deformation and the simple proportionality breaks. See Yield Strength and Plastic Deformation.
Edge cases
EC1 — What is the stress if the applied force is zero?
. No load, no internal crowding — and by Hooke's Law strain is zero too, so the bar sits at its natural length.
EC2 — What happens to stress as area shrinks toward zero (a sharp notch or thinning)?
blows up toward infinity as — this is why cracks and notches are dangerous: they locally starve the area and spike the stress.
EC3 — Can strain be negative, and what does that mean physically?
Yes. Under compression the bar shortens, , so (our sign convention). The stress is then compressive (); Hooke's Law still holds with matching sign in the elastic range.
EC4 — What is strain for a perfectly rigid (infinitely stiff) material?
As , for any finite stress — it carries load without stretching at all. No real material is truly rigid, but very high approaches it.
EC5 — Does axial stretching change the cross-section, and does our notice?
Physically yes — pulling lengthwise thins the bar sideways (Poisson's Ratio: stretch one way, shrink the perpendicular ways). But engineering stress uses the original , so it ignores this small change; only true stress accounts for the shrinking area.
EC6 — What is the stress in a strut heated with both ends clamped, carrying no external force?
It can be large and compressive even with : the material wants to expand but can't, so the clamps push back. This Thermal Stress shows stress does not require an applied external pull.
EC7 — In the limit of a very long strut, what dominates its total stretch?
The length itself: grows linearly with , so a long tie-rod stretches proportionally more even though its strain (per-metre) stays the same. This drives strut sizing in Spacecraft Load Paths and Struts.
Recall One-sentence summary of the traps
Nearly every trap comes from confusing a ratio (stress, strain, ) with a raw amount (force, stretch), or a slope () with a ceiling () — anchor each symbol to "per unit area", "per unit length", or "slope of the graph" and the traps dissolve.