Every essay
Equilibrium
Nothing moves, so everything adds to nothing — and the free body decides what everything means.
Everything adds to nothing, and that is the whole of statics
A structure that stays put obeys two statements — the forces on it sum to zero, and so do the moments. Every number in the subject comes out of those two sentences.
The free body is a choice, and choosing it well is the whole skill
Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty.
Three forces must meet at a point, and a drawing can find it
A body held by exactly three forces has their lines of action concurrent. That is a theorem, it is enough to solve for direction and magnitude, and for a century it was done with a straightedge.
Counting the unknowns, and finding out whether statics can answer
Two equations per joint, one unknown per member, one per restraint. Subtract, and the sign of the answer says whether the structure is a mechanism, solvable, or beyond what equilibrium alone can settle.
Structural form
Trusses, cables and arches: the shapes that carry load by geometry rather than by bulk.
The triangle that cannot fold, and everything built out of it
A square of pinned bars is a mechanism. A triangle is not, and that single fact is the reason trusses exist and the reason they look the way they do.
The shape that carries itself, and the arch that is its reflection
Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is.
Depth is the cheapest strength there is
Doubling the depth of a truss halves its chord forces without adding a gram of material to the chords. Nothing else in structural design is that cheap, and almost every structure has already spent it.
Internal forces
What a cut reveals — shear, moment and axial force, and the diagrams that track them along a member.
What a cut reveals, and why it was there all along
Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing.
The diagram is an integral, and that is why it can be drawn by eye
Load, shear and moment are one function and its two integrals. Once that is seen, the diagrams stop being things to calculate and become things to sketch.
Where to put the supports, which is not at the ends
Moving the supports of a uniformly loaded beam inward by about a fifth of its length halves the worst bending moment. The load has not changed and nor has the beam.
Sections and stress
How a cross-section resists a moment, and why where the material sits matters more than how much there is.
The material far from the middle does nearly all the work
A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing.
Bending is a pair of forces, pushing and pulling
A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress.
The same steel in a different shape, and a factor of forty
Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.
Stability
Strong enough and still falling over: buckling, slenderness, and loads that make themselves worse.
Strong enough and still falls over
A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.
The ends decide the length that matters
Four columns of identical height and section, buckling at loads sixteen times apart. Nothing differs but what is holding the two ends.
The load that makes itself worse
A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load.
Deflection
Stiffness is not strength. What moves, how far, and why the answer goes as the fourth power of the span.
Stiffness is not strength, and usually it is the one that governs
A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all.
Span to the fourth, which is why spans are short
Doubling a span multiplies its deflection by sixteen. No other relationship in ordinary structural work is that steep, and it is the reason long spans are always a different kind of structure.
One support too many, and what it costs to know
Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question.