Question 1 of 5: Load Application, Time-Dependent Response and Tension-Test Properties

Question 1: Load Application, Time-Dependent Response and Tension-Test Properties (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — static and dynamic loading, and the time-dependent response of materials (10 marks). A static load is applied slowly enough that the rate of loading has no influence on the response: the load is brought from zero to its full value over a period long compared with the natural period of the member, is then held essentially constant, and produces no measurable inertial force. The self-weight of a bridge deck, the weight of stored grain in a silo, the sustained portion of a snow load on a roof and the thrust of retained soil against a basement wall are all static. In the laboratory the standard tension test of ASTM E8 is a static test: the crosshead moves at a controlled low rate so that the specimen is in equilibrium at every instant of the test.

A dynamic load varies rapidly with time, so that acceleration of the mass of the structure generates inertial forces that must be carried in addition to the applied force. Dynamic loading covers impact (a vehicle striking a bridge pier, a pile hammer, a dropped weight in the Charpy test), cyclic or repeated loading (traffic axles crossing a pavement, wind-induced vortex shedding on a slender stack, machine vibration on a supporting slab) and blast or seismic ground motion. The engineering consequence is twofold: the internal force may greatly exceed the static value for the same nominal load, and many materials become stronger but less ductile as the strain rate rises, so a steel that is tough under static load may fracture in a brittle manner under impact at low temperature.

The time-dependent response of a material is the part of its deformation that is a function of the duration of loading rather than of the magnitude of loading alone. If a material were purely elastic, applying a constant stress would produce an instantaneous strain that then stayed constant for ever, and removing the stress would recover it completely and instantaneously. Real civil-engineering materials do not behave that way. Under a sustained stress they keep straining (creep); if instead the strain is held constant, the stress required to hold it decays with time (relaxation); and on unloading, part of the strain returns immediately, part returns slowly (delayed elasticity) and part never returns at all (permanent set). Any material whose behaviour depends on how long the load has acted, as well as on how large it is, is described as viscoelastic or time-dependent.

Creep is the continued increase of strain under a constant sustained stress. It is described by a creep curve with three stages: a primary stage in which the strain rate decreases, a secondary stage of nearly constant minimum strain rate, and a tertiary stage of accelerating strain that ends in creep rupture. In civil engineering the classic examples are concrete, which under a sustained service stress will eventually accumulate a creep strain one and a half to three times its initial elastic strain (this is why prestressing losses and long-term deflections of flat slabs must be computed); the sustained sag of a heavily loaded timber beam; the deformation of asphalt concrete under standing truck loads, which appears at the surface as rutting; and steel at elevated temperature, which is why creep governs the fire resistance of unprotected steelwork. Creep is stress-dependent and, in metals, becomes significant only above roughly 40 % of the absolute melting temperature.

Viscous flow is the continuous, entirely irrecoverable deformation of a material at a rate proportional to the applied shear stress, following the Newtonian relation $\tau = \eta\,\dot{\gamma}$, where $\eta$ is the viscosity and $\dot{\gamma}$ the shear strain rate. It differs from creep in that there is no elastic component to recover and no threshold stress: any shear stress, however small, produces flow. Asphalt binder is the standard civil-engineering example — at summer pavement temperatures a paving-grade bitumen behaves very nearly as a Newtonian liquid, which is exactly what the rotational-viscometer and dynamic-shear-rheometer grading tests measure, and it is this flow that lets a hot mix be compacted and that later permits rutting. Fresh concrete flowing down a chute or through a slump cone, wet clay squeezing from beneath a footing, and glass at annealing temperature are further examples. In a viscoelastic material such as asphalt concrete the two effects act together: the recoverable creep component and the permanent viscous component sum to the observed deformation, and separating them is the object of the repeated-load permanent-deformation test.

Part (b) — reading the two tension curves (10 marks). The eight quantities in the printed list reduce to six distinct answers per metal; item III on the examination paper carries only the words “in/in”, which is the wrapped tail of item II (“yield stress at an offset strain of 0.004 in/in”) and not a separate quantity. The parts are answered below in the printed order.

Given. The printed stress–strain chart for two metals tested in tension to fracture; stress in ksi on an axis running to 150 ksi with rules at 50 and 100 ksi, strain in in/in from 0.00 to 0.14 in steps of 0.02. Metal A is the solid curve, metal B the dashed curve.

Find. For each metal: the proportional limit, the yield stress at a 0.004 strain offset, the ultimate strength, the modulus of resilience and the toughness; then decide which metal is the more ductile and justify it.

Approach. Take the modulus of elasticity as the slope of the straight branch, use it to draw the 0.004-offset line and read its intersection with each curve, take the ultimate strength as the peak ordinate, compute the modulus of resilience as the triangular area under the elastic branch and the toughness as the whole area under the curve to fracture, and compare fracture strains for ductility.

1. I — Proportional limit: find where each curve leaves its straight line. The proportional limit is the largest stress for which stress and strain remain proportional, i.e. the last point on the initial straight segment. Reading the chart, metal A runs straight to $\sigma_{PL,A} = 50\ \text{ksi}$ at $\varepsilon = 0.0025$ and metal B runs straight to $\sigma_{PL,B} = 45\ \text{ksi}$ at $\varepsilon = 0.0045$. The slope of each straight segment is the modulus of elasticity, $E = \sigma_{PL}/\varepsilon_{PL}$: $$E_A = \frac{50}{0.0025} = 20\,000\ \text{ksi}, \qquad E_B = \frac{45}{0.0045} = 10\,000\ \text{ksi}.$$ These moduli are needed for every later part, so it is worth noting that $\boxed{\sigma_{PL,A} = 50\ \text{ksi},\ \sigma_{PL,B} = 45\ \text{ksi}}$ and that metal A is twice as stiff as metal B — values consistent with a structural steel and an aluminium alloy respectively.
2. II and III — Yield stress at a 0.004 in/in offset. The offset method draws a line of slope $E$ starting at the strain offset and takes the yield stress where that line cuts the curve: $\sigma = E(\varepsilon - 0.004)$. For metal A the construction line $\sigma = 20\,000(\varepsilon - 0.004)$ meets the curve at $\varepsilon = 0.0075$, giving $\boxed{\sigma_{y,A} \approx 70\ \text{ksi}}$. For metal B the line $\sigma = 10\,000(\varepsilon - 0.004)$ meets the curve at $\varepsilon = 0.0094$, giving $\boxed{\sigma_{y,B} \approx 54\ \text{ksi}}$. Both construction lines are drawn on Figure 1.1. Item III of the printed list supplies only the units of that offset, so no separate calculation is required for it.
3. IV — Ultimate strength: the peak of each curve. The ultimate (tensile) strength is the highest stress the specimen carries, read as the maximum ordinate before the curve terminates. Metal A peaks at the end of its curve at $\boxed{\sigma_{u,A} \approx 132\ \text{ksi}}$ and metal B at $\boxed{\sigma_{u,B} \approx 72\ \text{ksi}}$. In both cases the peak coincides with fracture on the plotted engineering-stress curve, so neither curve shows the pronounced post-necking drop of a very ductile metal; the reading is taken to the nearest ksi, which is about the precision the printed grid allows.
4. V — Modulus of resilience: the elastic strain energy per unit volume. Resilience is the area under the curve up to the proportional limit, which for a straight segment is a triangle: $$U_r = \tfrac{1}{2}\,\sigma_{PL}\,\varepsilon_{PL} = \frac{\sigma_{PL}^{2}}{2E}.$$ Substituting, $U_{r,A} = \tfrac{1}{2}(50)(0.0025) = 0.0625\ \text{ksi}$ and $U_{r,B} = \tfrac{1}{2}(45)(0.0045) = 0.101\ \text{ksi}$. Since 1 ksi of area equals $1000\ \text{in-lb/in}^3$, $\boxed{U_{r,A} = 62.5\ \text{in-lb/in}^3,\ U_{r,B} = 101.2\ \text{in-lb/in}^3}$. The softer metal B stores about 60 % more elastic energy per unit volume despite its lower strength, because resilience varies as $\sigma_{PL}^2/E$ and B's modulus is half A's.
5. VI — Toughness: the total area under the curve to fracture. Toughness is the strain energy absorbed per unit volume up to fracture, $U_T = \int_0^{\varepsilon_f}\sigma\,d\varepsilon$, evaluated here by the trapezoidal rule on the digitised curves. This gives $U_{T,A} = 8.20\ \text{ksi}$ and $U_{T,B} = 7.49\ \text{ksi}$, that is $\boxed{U_{T,A} \approx 8{,}200\ \text{in-lb/in}^3,\ U_{T,B} \approx 7{,}500\ \text{in-lb/in}^3}$. A quick check on the arithmetic: metal A averages roughly 104 ksi over a strain of 0.079, and $104 \times 0.079 = 8.2\ \text{ksi}$, which matches. The two areas are within about 9 % of each other — A wins on strength, B wins on elongation, and the products very nearly balance.
6. VII — Which metal is more ductile. Ductility is the amount of permanent strain a material can sustain before fracture, so the measure is the strain at which each curve ends: $\varepsilon_{f,A} = 0.079$ against $\varepsilon_{f,B} = 0.117$. Metal B therefore reaches $\boxed{48\ \%\ \text{more elongation than metal A}}$ — $0.117/0.079 = 1.48$ — and metal B is the more ductile of the two. The physical reason is visible in the shapes: metal A work-hardens steeply and fractures while still rising, whereas metal B flattens into a long low-slope plateau, which is the signature of extensive plastic flow before separation.

Final results.