Beam bending and pressure vessels, explained
The two closed-form results behind most everyday structures: how a beam bends and how much it deflects, why depth beats width, the four textbook load cases, and why a pressurised cylinder is pulled apart twice as hard around its girth as along its length.
A beam is a fight between tension and compression
Load a shelf in the middle and it curves. The top face gets shorter and is squeezed; the bottom face gets longer and is stretched. Somewhere in between is a layer that is neither — the neutral axis. The further a fibre of material sits from that axis, the harder it is strained, and the more it resists the bending.
That gives the bending stress formula every engineer knows: σ = M·c/I. M is the bending moment at that point, c the distance from the neutral axis to the outer surface, and I the second moment of area, which measures how much material sits far from the axis.
Why depth beats width
For a solid rectangle, I = b·h³/12. Width b counts once, but depth h counts three times over. Double the width of a beam and it gets twice as stiff; double its depth and it gets eight times as stiff.
That is why floor joists stand on edge, why a ruler is floppy flat and rigid on its side, and why an I-beam puts its material into flanges far from the middle. A plank twice as wide as it is deep is four times stiffer turned on edge — same wood, same weight.
Strong enough is not stiff enough
Two separate questions decide a beam. Strength asks whether the peak stress M·c/I stays below what the material can take. Stiffness asks how far the beam deflects, and that depends on E·I, the material's modulus times the section's second moment.
A beam can pass one and fail the other. A floor can be far too strong to break and still bounce underfoot, which is why building rules commonly cap floor deflection at around span/360. With a low-modulus material such as a polymer, deflection usually governs long before strength does.
The four textbook cases
A simply supported beam with a central point load P deflects P·L³/(48·E·I), with a peak moment of P·L/4. Spread the load evenly as w per unit length and the deflection becomes 5·w·L⁴/(384·E·I), the moment w·L²/8.
A cantilever, fixed at one end, with a point load at its tip deflects P·L³/(3·E·I), with a moment of P·L at the wall; under an even load it deflects w·L⁴/(8·E·I) with a moment of w·L²/2. For the same load, a cantilever tip moves sixteen times as far as the middle of a simply supported beam of the same span.
Notice the powers of L. Double the span under the same point load and the deflection rises eightfold; under the same load per metre, sixteenfold. Span is the most expensive thing you can add to a beam.
Pressure vessels: hoop stress is twice axial
Pressurise a thin-walled cylinder of radius r and wall thickness t and the wall is stretched two ways. Around the circumference the hoop stress is p·r/t; along the length the axial stress is p·r/(2·t), exactly half. Cut the cylinder lengthways and the pressure pushes on a strip as long as the vessel, held by only two thin walls; cut it across and the pressure pushes on a disc held by the whole circumference.
So cylinders fail along their length: a boiled sausage splits lengthways, and so does an over-pressurised pipe. A sphere carries p·r/(2·t) in every direction, which is why it needs only half the wall of a cylinder of the same radius, and why the most demanding tanks are spheres or have domed ends. The wall needed for an allowable stress follows directly: t = SF·p·r/σ for a cylinder, half that for a sphere.
Safety factors, and what these formulas leave out
A safety factor divides the strength by a margin that covers uncertain loads, scatter in the material and the simplifications in the model. It is not a licence to ignore what the model does not see.
These formulas assume a slender, linear-elastic beam with small deflections, and they neglect shear deformation, which matters for short, deep beams and for sandwich panels with soft cores. They say nothing about buckling, about the stress concentrations at holes, notches and joints, or about the ends and nozzles of a pressure vessel. Thin-wall vessel theory also assumes t is under about a tenth of r; thicker walls need the Lamé equations. Use them for sizing, then check the details the formulas cannot see.
Same total load, four ways to hold it
The sag is drawn to scale between the cases. Spreading the load evenly cuts the deflection to five-eighths; holding the beam at one end instead of both multiplies it by six (even load) or sixteen (tip load). Exact deflections for your own beam come from the calculator.
Hoop stress is twice axial
Pressurise a cylinder and it is pulled apart around its circumference twice as hard as along its length. That 2:1 ratio is why hydrogen tanks are filament-wound with mostly circumferential and helical fibre, over a liner that does the gas sealing — the composite carries the load, the liner keeps the hydrogen in.
Check yourself
1. A plank twice as wide as it is deep is turned from flat onto its edge. How much stiffer is it in bending?
2. Why does a boiled sausage split along its length rather than around it?
3. Double the span of a simply supported beam carrying the same central point load. Its deflection becomes…
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