Water Resources HubFE section 9 of 16 · free theory
Materials
Reading stress–strain curves, how steel, concrete, timber, and asphalt behave, what the standard material tests actually measure, and concrete fundamentals — water–cement ratio and curing.
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Reading a stress–strain curve
The tension test is the single most informative experiment in this section, and the exam expects you to read its curve fluently. The initial straight-line slope is the modulus of elasticity E. Where the curve stops being straight, the material yields; the peak is the ultimate (tensile) strength; the strain at fracture measures ductility.
E = σ / ε (initial linear slope)
| E | modulus of elasticity — stiffness, not strength |
% elongation = (Lf − L0) / L0 × 100
| L0, Lf | original and final gauge lengths — ductility measure |
% reduction of area = (A0 − Af) / A0 × 100
Landmarks on the curve, in order: proportional limit, yield strength (often taken at 0.2% offset strain for steels without a sharp yield point), ultimate tensile strength (the peak — note the specimen keeps stretching after it as necking begins), and fracture. Toughness is the area under the whole curve — a strong-but-brittle material can be less tough than a weaker, ductile one.
Worked example Reducing a tension test
Given:
- Steel bar, original diameter 12.7 mm, gauge length 200 mm.
- In the elastic range: at P = 30 kN the elongation is 0.237 mm.
- Yielding begins near P = 34 kN; maximum load 50 kN.
- After fracture: gauge length 238 mm, necked diameter 9.5 mm.
Solution:
- Original area: A0 = π(12.7)²/4 = 126.7 mm².
- Modulus: σ = 30,000/126.7 = 236.8 MPa at ε = 0.237/200 = 0.001185, so E = 236.8/0.001185 = 200,000 MPa = 200 GPa.
- Yield strength ≈ 34,000/126.7 = 268 MPa; ultimate strength = 50,000/126.7 = 395 MPa.
- Ductility: % elongation = (238 − 200)/200 × 100 = 19.0%. Final area Af = π(9.5)²/4 = 70.9 mm², so % reduction of area = (126.7 − 70.9)/126.7 × 100 = 44.0%.
Answer: E ≈ 200 GPa, yield ≈ 268 MPa, ultimate ≈ 395 MPa, 19% elongation — a textbook ductile structural steel.
Steel, concrete, timber, asphalt — what distinguishes them
The exam does not ask for deep materials science, but it does expect one or two behavioural facts per material:
- Structural steel — ductile, strong in both tension and compression, E ≈ 200 GPa, G ≈ 75–79 GPa. Yields before it fractures, which is why steel structures give warning.
- Concrete — strong in compression, roughly one-tenth as strong in tension, so it is almost always paired with steel reinforcement. Stiffness is lower and more variable than steel (E ≈ 25–35 GPa for normal-weight concrete). It creeps under sustained load and shrinks as it dries.
- Timber — strongly anisotropic: much stronger and stiffer parallel to the grain than across it. Strength drops as moisture content rises, so design values assume a service moisture condition.
- Asphalt — viscoelastic and temperature-sensitive: stiff and brittle when cold, soft and rut-prone when hot. Pavement questions lean on this temperature dependence.
The one-line version the exam rewards: steel is ductile and symmetric in tension/compression; concrete needs help in tension; timber cares about grain direction and moisture; asphalt cares about temperature.
Materials testing concepts
Each standard test answers a different question, and the exam checks that you know which is which:
- Tension test — E, yield strength, ultimate strength, ductility (% elongation, % reduction of area).
- Hardness tests (Brinell, Rockwell) — resistance to surface indentation. Hardness correlates roughly with tensile strength for steels, but it says nothing about toughness.
- Impact test (Charpy) — energy absorbed in sudden fracture of a notched specimen. This is the notch-toughness check: it catches brittle behaviour that a slow tension test can miss, especially at low temperatures.
- Fatigue — failure under repeated loading at stresses well below the static strength. Steel shows an endurance limit; aluminium generally does not.
Concrete fundamentals: water–cement ratio and curing
Two ideas dominate every FE-level concrete question. First, the water–cement ratio (w/c): for a given cement content, less water means higher strength and lower permeability — but the mix still needs enough water to hydrate the cement and stay workable. Raising w/c trades strength away for workability. Second, curing: fresh concrete must be kept moist and warm enough for hydration to continue; drying out early stops strength gain. That is why standard test cylinders are moist-cured and tested at 28 days.
w/c = (mass of water) / (mass of cement)
| w/c | water–cement ratio by mass — lower means stronger, less permeable concrete |
Typical structural w/c values sit around 0.40–0.50. The 28-day compressive strength (f′c) is the standard reference strength because most of the strength gain is complete by then under proper curing.
Worked example Batch water from the w/c ratio
Given:
- Concrete batch with 350 kg of cement per cubic metre.
- Specified w/c = 0.45.
Solution:
- Water required: 0.45 × 350 kg = 157.5 kg per m³ (about 158 litres).
- If the w/c drifted to 0.60, water would rise to 0.60 × 350 = 210 kg per m³ — an extra 52.5 kg of water doing nothing but leaving capillary pores behind when it evaporates.
- Those pores are why the higher-w/c batch ends up weaker and more permeable, even with identical cement content and curing.
Answer: 157.5 kg of water per m³ at w/c = 0.45; pushing to w/c = 0.60 adds 52.5 kg of water and costs strength.