Physics · strength of materials
Every material has its own “character”: steel stretches, glass suddenly snaps, rubber extends by several hundred percent. Pull samples in a tensile testing machine, design a truss bridge and drive a truck across it, compare beams of the same mass, buckle a column, flip a hanging chain into an arch and see why airplane windows have rounded corners.
A testing machine pulls a sample and draws its “signature” — the stress–strain curve.
A rod is a very stiff spring: k = EA/L. How much will a one-meter steel rod 1 cm thick stretch when you hang a car from it?
Typical values — specific grades can differ by tens of percent.
| Material | E [GPa] | Rm [MPa] | elongation |
|---|---|---|---|
| structural steel | 200–210 | 400–550 | 20–25% |
| aluminum 6061-T6 | 69 | ~310 | ~12% |
| annealed copper | 110–130 | 200–250 | ~45% |
| window glass | 70 | 30–100 | ≈ 0 |
| concrete | 25–40 | 2–4 (compression 20–50) | ≈ 0 |
| pine along the grain | 9–12 | 80–100 | ~1% |
| rubber | 0.001–0.1 | 15–30 | 500–700% |
| aramid (Kevlar) | 70–130 | 3 000–3 600 | 2.5–4% |
| nylon line | 2–4 | 400–900 | 20–30% |
Build a bridge from steel bars and drive a truck across it. The computer calculates the force in every member.
The bars meet at hinges — as if on pins. Push the frames sideways. Then brace the square with a single bar: tap one corner, then the opposite one.
Nothing but triangles, with diagonals alternately in compression and tension. Few members — quick to assemble. Patented by James Warren and Willoughby Monzani.
Diagonals sloping toward the middle are in tension, while the shorter posts are in compression. Good for steel: the long members are in no danger of buckling.
A mirror image of the Pratt: the diagonals are in compression and the verticals in tension. It suited the era of timber diagonals and iron vertical rods.
Each bar is a spring with stiffness EA/L. The program assembles them into a big system of node equilibrium equations (the stiffness method) and solves it by Gaussian elimination. When the system has no solution, the structure is a mechanism.
A member in compression usually fails before one in tension — by buckling (Euler's critical load falls as 1/L²). That's why it pays to thicken long compression members. More in the “Buckling” tab, and about triangles in Geometry.
The same amount of steel — yet the stiffness differs more than two hundredfold. It all depends on the shape of the cross-section.
Deflection under the same force at the same position. Bar: stiffness (logarithmic scale).
Try it at home: lay a ruler flat across two books and press it with a finger — it bends easily. Stand it on edge — it barely moves. Only the shape of the cross-section changed.
Concrete handles compression superbly, but tension ten times worse. In a beam, the bottom is in tension… so that's where the steel goes.
Steel and concrete expand with heat almost identically, and concrete protects steel from rust. Cracks in a reinforced concrete beam are normal — what matters is that they stay fine.
Steel tendons are tensioned before the concrete is poured, then released. They squeeze the concrete from below “in advance,” so a load first merely reduces that squeeze. That's how long bridge girders and floor slabs are made.
A slender column in compression doesn't crack — it suddenly kicks out sideways. And an arch can work in compression alone.
A hanging chain is only in tension. Turn its shape upside down and you get an arch that is only in compression. Tap a link to hang a weight; drag the handles at the ends.
The stones of an arch are wedges. Each presses on its neighbor, and the keystone at the top locks the whole thing. Roman aqueducts and bridges still stand today, because stone handles compression superbly — and an arch needs no tension.
A concrete dome about 43 m across, with no reinforcement. The Romans used ever lighter aggregate toward the top and left an opening (the oculus) at the crown.
A cathedral's high vault pushes the walls outward. Flying buttresses carry that thrust to outer piers — which let the walls become walls of stained glass.
Antoni Gaudí designed vaults from hanging strings with little bags of shot, then looked at them in a mirror. As early as the 17th century, Robert Hooke wrote: as hangs the flexible line, so but inverted will stand the rigid arch.
A hanging chain forms a catenary (a hyperbolic cosine), not a parabola — although for a small sag they are very similar. You get a parabola when the weight is spread evenly horizontally, like the deck of a suspension bridge. More about functions: Functions and Growth.
Scale an animal up ten times: it will weigh a thousand times more, but its bones will only be a hundred times thicker in cross-section.
Model: an animal at scale ×1 has about a threefold safety margin in its bones when running (the order of magnitude known from studies of mammals). Muscle force grows with cross-section (k²), weight with volume (k³). Galileo described this as early as 1638 and drew two bones: the larger one must be disproportionately thicker.
Wood along the grain has almost the same E/ρ ratio as steel and aluminum, and as a beam it clearly beats them. Across the grain it is tens of times weaker — it is an anisotropic material.
Bone is a composite: collagen fibers (ductile) and hydroxyapatite crystals (hard). Bamboo is a tube with its fibers on the outside — where bending stresses are greatest. A honeycomb gives enormous stiffness for minimal mass.
Engineers learn from mistakes — including tragic ones. That's why airplane windows are oval today, and large halls must be inspected before winter.
We describe these in simplified form, following the findings of investigators and experts. We remember the victims of these disasters.
The slender, flexible deck hanging over the strait began to twist in a wind of about 65 km/h, until it collapsed. It wasn't ordinary resonance but flutter: the wind itself fed the oscillations. Nobody died. Since then, bridges have been tested in wind tunnels. More: Vibrations and Resonance.
The welded hulls of ships mass-produced during the war cracked in cold waters — several broke clean in half. The steel grades of the time turned brittle at low temperature, and cracks started at the sharp corners of hatches. Research (by Constance Tipper, among others) led to impact testing of steel at low temperatures.
The first jet airliner. After two crashes, a fuselage was tested in a water tank, pressurized again and again. A fatigue crack started at the corner of a cutout in the skin, where the stress was many times higher than average. Since then, airplane windows have had rounded shapes.
In Kansas City, two suspended walkways full of people collapsed — 114 people died. During construction, the hanging arrangement was changed: two shorter rods instead of one long one. That “minor” change doubled the force on the upper walkway's connection (diagram alongside).
On January 28, 2006, in Chorzów, Poland, during a racing-pigeon show, the hall's roof collapsed — 65 people died and more than 140 were injured. A thick layer of snow and ice lay on the roof. According to the investigation and the courts, the cause was a combination of design and construction errors and neglected maintenance.
After this disaster, Polish building law introduced mandatory inspections of large halls twice a year — before and after winter.
In memory of the victims.
On August 14, 2018, a section of the viaduct collapsed and 43 people died. Experts pointed to corrosion of the steel tendons hidden inside the concrete stays — hard to spot from the outside. The disaster accelerated the spread of continuous sensor monitoring of bridges.
Bending a paper clip once won't break it. Bending it many times will. The bigger the bend, the fewer cycles it survives.
A structure is designed to carry several times more than we expect: in building typically 1.5–2 times (depending on loads and material), in elevators and cables — more. This margin covers variation in material quality, imprecise loads, corrosion and human error. But it does not protect against skipped inspections.
Today engineers divide a structure into thousands of small elements and compute the stresses in each one by computer — exactly how the color map above is made (only on a much smaller scale).
When you pull a rod, its atoms move apart, and the bonds between them act like tiny springs. To compare thick and thin rods, long and short, engineers calculate stress (force per unit of cross-sectional area) and strain (relative change in length):
So a rod is a spring with stiffness k = EA/L. A steel rod Ø1 cm and 1 m long has k ≈ 15.7 kN/mm: a car of about 1.5 t will stretch it by less than a millimeter. Above the yield strength, the metal flows and stays permanently elongated; the highest point of the curve is the tensile strength Rm, after which a neck forms and the sample breaks.
Truss members are pin-jointed, so they can only be in tension or compression. At every node the forces must balance. For n nodes that's 2n equations; the computer assembles them into a system K·u = F and solves it by Gaussian elimination. A pin-jointed triangle can't change shape without changing the lengths of its sides — a square can. A plane truss is rigid and statically determinate when the number of members is m = 2n − 3 (and the members are well placed).
A bent beam shortens at the top (compression) and lengthens at the bottom (tension); in the middle lies the neutral axis. Stress grows linearly with distance from it, and stiffness comes from the cross-section's moment of inertia I — material far from the axis does the most work. Hence the I-beam: two thick flanges far from the axis and a thin web.
A slender rod in compression stays straight up to a certain force, then suddenly bows sideways. Leonhard Euler derived this force in 1744:
An arch shaped like an inverted chain contains only compression — which is why stone and brick are enough.
When scaled up ×k, weight grows as k³ and cross-section as k², so stress grows as k. Near a hole, stress locally rises Kt times (for a round hole in a wide plate Kt = 3, for an elliptical one 1 + 2a/b), and sharp corners are the worst. Repeated loading, even below the yield strength, creates microcracks in such spots that grow with every cycle — that's metal fatigue.
Simplifications: materials are homogeneous; the tensile curves are smooth approximations of typical values. The truss has ideal pin joints and members made of steel tubes (E = 200 GPa, Re = 355 MPa); the truck is a single point load; we ignore the bridge's own weight and apply no safety factors. Beams: linear Euler–Bernoulli theory (small deflections), without self-weight or lateral-torsional buckling. Reinforced concrete: a simplified model that cracks at 3 MPa with capacity from the steel. Buckling: Euler's formula for an elastic rod with a small initial curvature. The chain is a simple simulation of links. Stress map: linear FEM (plane stress, about 3–4 thousand elements on a quarter of the plate). The paper-clip fatigue curve is illustrative. Deflections in the drawings are heavily exaggerated — the scale is always shown.