Structural Engineering & Statics
where the load goes
The collection

Every essay — page 2

Essays 25 to 48 of 469, in the same order.
Every strip counts by the square of its distance. A rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs. Sections and stress

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.

7 figures · Second moment Bending is a push and a pull. A section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis. Sections and stress

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.

8 figures · Bending stress The same material, four ways. Four cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest. Sections and 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.

7 figures · Section shape Where plane sections stop staying plane. Strain across a cut face at four span-to-depth ratios, with the straight line the theory assumes drawn faintly behind. For a slender beam the two coincide; for a beam as deep as its span the real distribution is nothing like a straight line, and beam theory has no claim on it. Sections and stress

Plane sections stay plane, and what the assumption costs

Beam theory rests on one sentence about geometry. It is very nearly true for a slender member, wrong for a deep one, and everything in the subject that fails does so where it stops holding.

7 figures · Plane sections Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 0.41 against a mean of 0.19 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero. Sections and stress

The shear nobody draws

A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains.

7 figures · Shear flow The shear centre of a channel. A channel of 80 by 200, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 31.7 outside the web to leave the section untwisted — a point in the air, outside the material entirely. Sections and stress

The point that is not in the section

A channel loaded down its web twists. To stop it, the load must be applied through a point outside the steel entirely — in the air beside the section, where nothing can be attached.

7 figures · Shear centre The neutral axis is wherever the first moment vanishes. A 300 by 500 section with 1200 mm² of steel at a depth of 450, carrying 150 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 137.0 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 18.0 N/mm² at the top fibre and the steel carries 309 N/mm²; the resulting couple is 371 kN on a lever arm of 404 mm, which multiplies back to the 150 kNm applied. The uncracked section would have had 3422×10⁶ mm⁴ against the cracked 1139×10⁶ — a loss of 67% of the stiffness. Sections and stress

When half the section has given up

Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely.

7 figures · Bending stress Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 82.77 against a mean of 38.46 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero. At the junction between flange and web the flow is 820.0 per unit length, so connectors of 20000 each have to be spaced no further apart than 24 — which is what turns two pieces into one section. Sections and stress

The section made of pieces

Two plates bolted together are twice as strong as one plate. Two plates made to act as one section are eight times as stiff. The difference is entirely in the joint, and the joint has a calculable spacing.

7 figures · Second moment The column curve. Failure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it. Stability

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.

8 figures · Buckling The ends decide the length that matters. Four columns of identical height and section, buckling under four sets of end conditions. The effective length factor is the fraction of the column that behaves like a pin-ended one, and the buckling load goes as its inverse square. Stability

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.

7 figures · Effective length The load that makes itself worse. The amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached. Stability

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.

7 figures · Second-order The length at which a beam stops being a beam. Elastic critical moment against the distance between lateral restraints, with the section's plastic capacity drawn across it. The two cross at 3803 — beyond that length the beam buckles sideways before it reaches the strength its cross-section has, and the capacity is set by the restraints rather than by the steel. Stability

The beam that fails sideways

A deep narrow beam bending in its strong plane can, at a moment well below its capacity, swing out of that plane and twist. The failure has nothing to do with how much it can carry and everything to do with what is holding it.

7 figures · Lateral-torsional A 8 mm plate, and the width it can be. The elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 370 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 700 mm only 47 per cent of it is still working. Stability

The plate that ripples, and the width that is left

A wide thin plate in compression buckles at a stress that has nothing to do with the strength of the material. It then goes on carrying load — the middle drops out, and the edges work harder.

7 figures · Plate buckling A column that was never straight. Load against lateral deflection at mid-height, for a column starting with an initial bow of 0.002. There is no critical value to reach: the deflection grows from the first increment, slowly at first and then without bound as the ratio approaches 1.00 — which is the Euler load, and which the column therefore never attains. The perfect column, drawn for comparison, sits on the vertical axis until it arrives there and then has no answer at all. Stability

The column that was never straight

Euler's load is the load at which a perfectly straight column becomes indifferent to being bent. No column is perfectly straight, so no column ever reaches it — and the load it never reaches can still be measured.

7 figures · Buckling A brace is a stiffness requirement, not a strength one. Critical load against brace stiffness for a pinned column braced at mid-height. The curve climbs from the unbraced Euler load of 9.87EI/L² and flattens at 39.48EI/L², which is the Euler load of the braced segment — past that the column buckles in a shape the brace does not obstruct, and further stiffness buys nothing. The knee is at about 159EI/L³. A stiffness of 60EI/L³ is marked, reaching 21.75EI/L². Stability

The brace that need not be strong

A brace holding a column at mid-height carries almost no force. What it has to be is stiff — and the stiffness required is exact, large, and reached at a knee past which more buys nothing at all.

7 figures · Effective length Which limit arrives first. Utilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry. Deflection

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.

8 figures · Serviceability Deflection goes as the fourth power of the span. Deflection against span for a constant load intensity and section, with two slower relationships drawn faintly behind it for comparison: the load itself, which grows in proportion to the span, and the bending moment, which grows as its square. Doubling the span multiplies the deflection by sixteen, while the moment only quadruples. Deflection

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.

7 figures · Span scaling Where the peak moment goes when a restraint is added. The same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics. Deflection

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.

7 figures · Indeterminacy The deflection at x = 4, by virtual work. Three diagrams: the moment from the real load, the moment from a unit load placed where the answer is wanted, and their product. The area under the third, divided by EI, is the deflection — 213.33 here. No standard case was consulted, so the method works for any load pattern at all. Deflection

One deflection, without solving everything

To find how far one point of a structure moves, put an imaginary force of one unit there, multiply two moment diagrams together, and integrate. The answer arrives without ever solving for the deflected shape.

8 figures · Virtual work Maxwell's reciprocal theorem. A load at one point and the deflection it causes at another, against the same load moved to the second point and the deflection read at the first. Both integrals return 93.3335, and neither calculation was told about the other. The two deflected shapes are entirely different; the two readings are identical. Deflection

The theorem that swaps the question round

Push here and measure there; push there and measure here. The two readings are identical, for every elastic structure, whatever its shape — and that fact turns an influence line into something a model can be asked for directly.

8 figures · Reciprocity 3 continuous spans against 3 simple ones. The bending moment in a continuous beam whose support 1 has settled by 0.01. Three curves: the moment the load causes, the moment the settlement causes on its own — dashed, peaking at 73.5, and in equilibrium with no applied load at all — and their sum, which is what the beam carries, peaking at 73.5 against 24.5 without the settlement. The settlement field is proportional to EI: a stiffer beam is punished harder for the same movement, which is the opposite of every intuition load-carrying gives. Deflection

The support that moved

A redundant structure knows things statics cannot see. Settle one support by ten millimetres and a complete set of bending moments appears — in equilibrium with no load at all, and larger for a stiffer beam.

8 figures · Indeterminacy The water that will not run off. Water depth against sag at the middle of the bay, for a roof bay starting with a reversed construction camber of 0.01. There is no critical value to reach: the deflection grows from the first increment, slowly at first and then without bound as the ratio approaches 1.00 — which is the depth at which the bay fills faster than it stiffens, and which the roof bay therefore never attains. The perfect roof bay, drawn for comparison, sits on the vertical axis until it arrives there and then has no answer at all. Deflection

The water that will not run off

A flat roof deflects, the deflection makes room for water, the water deepens the deflection. It is the same equation as a buckling column, with rain in place of the axial load.

8 figures · Serviceability Three materials pulled until they stop. Three stress-strain curves — mild steel, high-strength steel, aluminium alloy — plotted to a strain of 2.0%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. The 0.2% offset construction is drawn on the high-strength steel: a line of slope E from a strain of 0.002, cutting the curve at 460 N/mm². Materials

The stress at which nothing in particular happens

One material in six has a yield point that a specimen actually does something at. For all the others the yield stress is a construction — a line drawn at an arbitrary offset — and every strength calculation for those materials depends on it.

7 figures · Proof stress Four materials pulled until they stop. Four stress-strain curves — mild steel, aluminium alloy, concrete, timber, along the grain — plotted to a strain of 0.6%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. No offset construction is drawn. Materials

The one number a stronger steel does not change

Geometry beats material almost everywhere on this site. Stiffness is the exception in the other direction — it cannot be bought at all, because every steel ever made has the same elastic modulus.

8 figures · Elastic modulus