Physics · Mechanics · Grade 9

Elasticity and deformation

How much load can a bridge cable carry, and when does it snap?

The situation

The deck of a suspension bridge hangs from steel cables. Each cable holds tens of tonnes and stretches a few centimetres under this load — when the load is removed, it returns to its original length. But if the load exceeds the limit, the cable stretches permanently and then breaks. How does an engineer choose a cable, and how much load can be hung on it? To find out, a steel sample is pulled apart in a testing machine in the laboratory.

Topic

Elasticity and deformation. Bridge cable and tensile test of a steel sample: Hooke's law and Young's modulus

Key ideas

Hooke's law
Young modulus
Breaking stress

Formulas

F_{e l} = k x, E_{p} = \frac{k x^{2}}{2}

Hooke's law for a spring and the potential energy of a deformed spring.

σ = \frac{F}{S}, ε = \frac{Δ L}{L_{0}}, σ = E ε

Mechanical stress, strain and Hooke's law for a rod ( $E$ — Young's modulus).

\frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}}, k = k_{1} + k_{2}

The stiffness of springs connected in series and in parallel.

ε_{⟂} = - ν ε, τ = G γ, G = \frac{E}{2 (1 + ν)}

Poisson's ratio, shear and the shear modulus.

w = \frac{σ ε}{2} = \frac{E ε^{2}}{2}, δ = \frac{F l^{3}}{48 E I}, φ = \frac{32 M L}{π G d^{4}}

Elastic energy density; formulas for bending (a beam loaded at the middle) and torsion (a wire).

Theory

Where does the elastic force come from?

In a solid, atoms are bound by electromagnetic forces. When a body is stretched or compressed, the distances between atoms change and they tend to return to their former positions — the resulting elastic force is directed against the deformation. The tension in a string, the spring force and the normal reaction force of a support are all elastic forces: the table, the string and the hook deform imperceptibly. Such constraints limit the motion of a body, and the reaction force adjusts to the load: in the balance experiment $N = m g - k x$ . The gravitational force $m g$ is the Earth's attraction (universal gravitation and the gravitational field — see the “Law of universal gravitation” experiment), while weight is the force a body exerts on its support or suspension.

Hooke's law, stiffness and Young's modulus

For small deformations the force is proportional to the deformation: $F = k x$ for a spring, $σ = E ε$ for a rod, that is $k = E S / L_{0}$ . The stiffness $k$ depends on the shape of the body and on the material; Young's modulus $E$ depends only on the material: steel ≈ 200 GPa, copper ≈ 120 GPa, aluminium ≈ 70 GPa, rubber — a few MPa. Elasticity is the property of regaining shape once the load is removed, stiffness is resistance to deformation. For springs in series the same force acts on each and the extensions add: $1 / k = 1 / k_{1} + 1 / k_{2}$ . In parallel the extension is the same and the forces add: $k = k_{1} + k_{2}$ . The deformation energy is $E_{p} = k x^{2} / 2$ , and per unit volume $w = E ε^{2} / 2$ .

The stress–strain diagram

At the start of the graph is the Hooke region (a straight line) up to the proportional limit $σ_{p r}$ . If the load is removed before the elastic limit $σ_{e l}$ , the sample returns to its original length. Then comes yielding: the force hardly changes while the sample “flows” (copper has no clear plateau — a conventional $σ_{0.2}$ is used, at which the residual strain is 0.2 %). The metal then work-hardens, the stress reaches the peak of the diagram — the ultimate tensile strength $σ_{m}$ — a neck forms in the sample and it breaks. If the load is removed in the plastic region, the unloading line is parallel to the Hooke line — a permanent deformation remains. The area under the diagram is the work done per unit volume. Ductile materials (steel, copper) stretch a lot before breaking, while brittle materials (cast iron, glass) break with almost no stretching.

Types of deformation

Tension (a cable, a chain), compression (a column, a foundation), shear (a rivet, a bolt: layers slide relative to each other, $τ = G γ$ ), torsion (a shaft, an axle: cross-sections are twisted relative to each other — this is non-uniform shear), bending (a beam, a bridge span: the upper layer is compressed, the lower is stretched, with a neutral layer in the middle). A stretched sample becomes thinner: $ε_{⟂} = - ν ε$ ; Poisson's ratio is ≈ 0.29 for steel, ≈ 0.34 for copper and ≈ 0.5 for rubber (the volume hardly changes). For an isotropic material $G = E / 2 (1 + ν)$ .

Elastic hysteresis

If you stretch rubber and then release it, the unloading curve runs below the loading curve — a loop forms. The area of the loop is the energy per unit volume that is converted to heat in each cycle. That is why car tyres heat up on a long journey. Rubber withstands a very large (hundreds of per cent) elastic deformation before breaking, while metals work in the elastic region with almost no hysteresis.

Laboratory and engineering

Young's modulus can be found from tension ( $E = F L_{0} / (S Δ L)$ ) and from bending ( $δ = F l^{3} / 48 E I$ , $I = b h^{3} / 12$ ), and the shear modulus from torsion ( $φ = 32 M L / π G d^{4}$ ). An engineer keeps the stress in a structure several times below the ultimate strength: the factor of safety is $n = σ_{m} / σ$ . Real bridge cables are twisted from high-strength steel wires with a strength of ≈ 1.6–1.9 GPa. Solids expand when heated ( $Δ L = α L_{0} Δ T$ ) — which is why expansion joints are left in bridges (the “Thermal expansion” experiment). Model: the material curves are fitted to typical values, and the temperature of the sample is constant.

Did you know?

Robert Hooke published the law of elasticity in 1676 as a coded anagram and revealed it in 1678: the Latin “ut tensio, sic vis” — “as the extension, so the force”.
The Young's modulus of rubber (a few MPa) is more than ten thousand times smaller than that of steel (≈ 200 GPa) — which is why rubber stretches noticeably even under the force of a hand, while a steel wire does not.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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