Mechanics of Materials

Whichever of the three the question leaves blank is the one you solve for.

Both lengths must be in the SAME unit before you divide — that is the entire difficulty of this formula, and it catches somebody every year.

Four knowns, one unknown, every single time. Read the fraction as an argument rather than a spelling.

Stress that acts ALONG a member is normal stress.

An isotropic material — one with the same properties in every direction, which covers nearly every metal you will meet — does not get to choose its three elastic constants independently.

Read it as a declaration of ignorance rather than of strength.

Twist a shaft and every fibre in it resists — but not equally. The ones far from the centre line get strained the most and fight back the hardest, which means the section's resistance depends on how its metal is spread about the axis.

Now put a torque on that section.

A shaft has two ways to be wrong. It can break — that is the stress check you just did — or it can simply wind up too far, so that the thing on the far end arrives late and out of position.

Motors are sold in kilowatts. Shafts, keys and couplings feel none of it — they feel torque.

Everything so far has been analysis: here is a shaft, how hard is it working.

A shaft that survives its torque still has to hand that torque to something — a gear, a sprocket, a coupling hub.

Stand a plank on edge and it carries a floor. Lay the same plank flat and it carries itself, badly.

Every formula in this chapter measures from the neutral axis, and in elastic bending the neutral axis passes through the centroid of the cross-section.

Real sections are made of parts, and a part's own I is quoted about its own centre. Move it away from the assembly's neutral axis and it becomes far more useful than it was.

Bending stress is largest at the fibre furthest from the neutral axis, so in practice that is the only fibre anybody checks.

Before a section can be checked, the demand has to be known.

Everything so far has been ingredients.

Bending is not the only thing happening inside a loaded beam.

A beam can be perfectly safe and completely unacceptable. Plaster cracks, doors bind, floors bounce — and none of that is a strength problem.

Take away one support and everything gets worse at once. A cantilever is built in rigidly at one end and free at the other — a balcony, a canopy, a diving board, the arm of a crane.

Every compression member has two ways to fail, and they are not the same failure. A stubby one crushes — it simply runs out of strength.

Read aloud: P-crit equals pi squared E I over K L, all squared.

Put pressure inside a thin-walled cylinder and the wall is pulled two ways at once.

The hoop is always double. The reason is worth thirty seconds. Cut the cylinder lengthwise and the pressure inside is resisted by TWO strips of wall, one each side of the cut.

A cylinder has a bad direction. Around the girth it carries pd/2t, along the axis only half of that, so half the wall's strength along the axis is never called on.

The thin-wall formula rests on one quiet assumption: that the hoop stress is the same all the way through the wall. Equilibrium only fixes the AVERAGE, and for a thin wall the average is all there is.

So far the wall has been a given and the stress the answer. A vessel designer works the other way round.

Push a post off-centre and it is doing two things at once: carrying an axial load and resisting a moment.

All three in MPa, tension positive.

A yield strength comes from a tensile test: one bar, one stress, pulled until it gives. A real part is never that polite.

Nominal stress is a comfortable fiction: load over area, spread evenly. Real parts have holes, grooves, keyways and shoulders, and at every one of them the stress lines crowd together and the local stress leaps.

Cut a thread on a bar and you have taken metal away.

A bolt does not hold things together by being strong. It holds them together by being STRETCHED, and the tension you put in at assembly — the preload — is the whole point of the exercise.

Here is the most misunderstood fact in machine design. Hang a working load on a properly preloaded bolted joint, and the bolt does NOT pick up that load.

A bolt in a lap joint is not being pulled along its axis any more; the plates are trying to slide past each other and the bolt is in the way.

A bolt is a screw whose job is to stay put. A power screw is one whose job is to move: a jack lifting, a vise clamping, a press ram advancing.

A fillet weld's cross-section is a right-angled triangle with both legs lying on the plates. It does not fail along a leg.

The last lesson checked a weld somebody else had drawn.

Whether a weld cracks is settled by two things: how fast the joint cools, and how much the steel wants to harden when it does.

A machinist sets the spindle to 250 rpm and ruins a carbide insert on a 20 mm bar, or scraps it instantly on a 200 mm one.

Three questions get asked of every roughing pass: how much metal comes off, how long it takes, and what it costs the insert.

Two of the fastest questions you can ask a metal.

A tensile machine divides load by the ORIGINAL cross-section, because that is the only area it was ever told.

Almost nothing in a plant is loaded once. A stress that cycles between a maximum and a minimum is described by two numbers, and both matter.

Plot alternating stress against life on log-log paper and the S-N curve becomes a straight line.

And a is the crack length, in metres: the HALF-length for a centre crack, the full depth for an edge crack. Reading a centre crack's full width as a is the classic inspection-report error.

That is first yield, and it is not collapse.

Strength design does not add loads up and hope.