Structural Engineering & Statics
where the load goes
Depth

Series — page 4

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The building is a filter with one very sharp tooth. What a component mounted on the roof of a eight-storey building feels, divided by what the same component would have felt on the ground. A rigid item sees the floor's peak acceleration rather than the ground's, which is already 2.29 times as much. An item whose own period matches the building's sees 6.1 times as much, at 0.94 s — a resonance inside a resonance, and the two amplifications multiply rather than add. Far above the building's period the ratio falls back toward one, because a component much softer than its support simply follows the ground. The shape of this curve is a property of the building and not of the earthquake, which is why equipment is qualified against a floor spectrum rather than a ground one.

Floor spectrum

  1. 1 The spectrum a floor hands on
  2. 2 The pipe held at every floor
2 essays · dynamics Which floor frequencies a 2 Hz pace punishes. The response factor of a floor of 12 tonnes modal mass and 3.0% damping, against its own natural frequency, under a walker at 2 steps per second. The peaks are at 2, 4, 6, 8 Hz — the harmonics of the pace — and they fall away sharply: four peaks, each smaller than the one below it, because the harmonics of walking get smaller. A floor at 2 Hz reaches R = 31 and one at 8 Hz reaches 6. Between them the floor is quiet.

Floor vibration

  1. 1 The floor that is strong and unusable
  2. 2 The crowd that is also the structure
2 essays · dynamics The cable and the arch are the same curve. The shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is.

Funicular

  1. 1 The shape that carries itself, and the arch that is its reflection
  2. 2 The polygon that finds the shape
2 essays · structures The ductility number depends on the ruler. Elongation after fracture against the gauge length it was measured over, in units of √S₀. A tensile specimen extends uniformly until the ultimate load and then localises into a neck, so the total is a strain (16 per cent here) plus a length (7.2 mm), and dividing a length by the gauge length is what makes the curve fall. At the two standard gauges the same steel reports 27.5 per cent over 5.65√S₀ and 21.8 over 11.3√S₀ — a ratio of 1.264, and 42 per cent of the first number is a property of the specimen rather than of the material.

Gauge length

  1. 1 The ductility that depends on the ruler
  2. 2 The curve was rising the whole time
2 essays · materials The strength was bought in a furnace and the welder gives it back. Proof stress as delivered and beside a weld, for four aluminium alloys. The heat-treated alloys lose half of it: the strength of a 6xxx extrusion is in precipitates formed by an ageing treatment, and the arc dissolves them for 32 mm either side of the weld, permanently. The work-hardened tempers lose nearly as much, because the heat undoes exactly the work. The annealed ones lose nothing at all, because there is nothing left in them to anneal. The consequence is the crossover: 5083-H22 is 1.04 times 6061-T6 as delivered and 0.92 times it once welded, so the stronger alloy is the weaker member. The 240 mm member drawn, with two longitudinal welds, keeps 87% of its parent capacity — a weld along a member softens a strip and leaves a section, and the same weld across it softens the whole of one.

Heat-affected zone

  1. 1 The strength the welder gives back
  2. 2 The soft zone that takes all the stretch
2 essays · materials 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.

Indeterminacy

  1. 1 One support too many, and what it costs to know
  2. 2 The support that moved
2 essays · deflection Why the curve sags, and why the two axes are not the same column. The same column curve with the sag computed rather than drawn. A hot-rolled section carries a residual compression of 30% of yield at its flange tips before anything is applied, so the tips yield first and what is left resisting a change of shape is the elastic core. About the major axis the stiffness follows the core's width; about the minor axis it follows its cube. The worst loss is 27% at λ = 74 about the minor axis against 23% about the major, and the whole effect lives between λ = 75 and λ = 89 — outside that band nothing has yielded, or everything has. No imperfection appears anywhere in this figure.

Inelastic buckling

  1. 1 The column that had yielded before it was loaded
  2. 2 The column with no plateau
2 essays · stability Influence line for the bending moment at x = 3. The bending moment at one fixed station, plotted against the position of a unit load walking across the span. The beam was re-solved at 301 load positions. The worst position is x = 3.00, giving 2.100.

Influence line

  1. 1 The worst place to stand
  2. 2 The train that is worse than its heaviest axle
2 essays · internal-forces The same beam, cut at x = 5. A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.

Internal forces

  1. 1 What a cut reveals, and why it was there all along
  2. 2 The moment that will not lie flat
2 essays · internal-forces A soap film over a hole, and its volume is the torsion constant. The stress function for Saint-Venant torsion of a square of 100 mm, relaxed on a 97 by 97 grid until it stopped moving — 385 sweeps. Its contours are the lines the shear stress runs along, its slope is the magnitude of that stress, and twice its volume is the torsion constant: 1.4053e+7 mm⁴ against a closed form of 1.4058e+7, an error of -0.035 per cent. The steepest slope is 67.515 at the boundary, at the point on it nearest the centre — which is why the peak shear in a solid section is at the middle of the longest side and never at a corner, where the film comes down to zero from two directions and its slope vanishes.

Membrane analogy

  1. 1 Two volumes, and both of them are torques
  2. 2 The corner a soap film cannot finish
2 essays · sections The same load, two diagrams, both in equilibrium. One span of a pair of 7 m spans under 5 kN/m, drawn twice. The elastic solution puts 31 kNm over the support and 17 in the span. Reducing the support moment by 30% and taking what statics then gives leaves 21 and 21: the section the beam needs falls from 31 kNm to 21, a saving of 30%. Both curves are in equilibrium with the same load — the mid-span ordinate plus half the support moment is the free moment 31 kNm for either — and the second is legitimate for that reason alone. What it costs is 1.0 milliradians of rotation at the support, which the section has to be able to deliver.

Moment redistribution

  1. 1 The moment that was moved on purpose
  2. 2 The moment that was shed has to land
2 essays · internal-forces The web down the middle carries no torsion at all. A 2-cell box 3.0 m wide and 1.5 m deep under 4000 kNm of torque. Statics gives one equation, T = 2 Σ q_i A_i, and there are 2 unknown flows — so the section is torsionally redundant and the missing statements are that every cell twists by the same amount. Solving that system gives 444.4 and 444.4 N/mm in the cells, so the internal web carries 0.00 N/mm — 0.00 per cent of the outer wall's. J is 2.3679e+12 mm⁴ against 2.3679e+12 for the same outline with no internal web at all, a ratio of 1.000000. The same walls slit open would give 7.413e+10, so closure is worth 32 times and the internal web is worth what the ratio says.

Multicell torsion

  1. 1 Two cells, one equation, and a web with nothing in it
  2. 2 Three actions on one web
2 essays · sections The moment a joint makes out of geometry. Bending stress in the chord divided by the axial stress already in it, against how far the diagonal's working line misses the node. A truss analysed as pin-jointed is drawn with its members as lines through their own centroids meeting at a point, because that is what makes the joint a pin with nothing but forces in it. A fabricated joint is not obliged to oblige, and the resultant then has a moment about the node — 16 kNm at the 60 mm drawn. It divides between the members meeting there in proportion to their rotational stiffnesses, 4EI/L or 3EI/L, which is the moment-distribution rule applied to an unbalanced moment that came from geometry rather than from load. The chord takes 48 per cent of it and carries 25 per cent of its axial stress again in bending. Ten per cent arrives at 28 mm, which on a chord of any depth is a detailing decision rather than a mistake.

Noding eccentricity

  1. 1 The joint that is not where it was drawn
  2. 2 The joint whose lines may not meet
2 essays · connections A couple applied to the core, and two columns to make it. A 20-storey core with one outrigger at 59% of its height. The arm is stiff in bending and the perimeter columns are stiff in tension and compression, so between them they resist the core's rotation at that level — a couple of 35283 kNm here, carried as a 294 kN pair in the columns at 120 m centres. The compatibility is one equation: the core's rotation at that level, less what the couple takes back out of it, equals the rotation the arm and its columns allow. The top drift falls from 360 mm to 74, which is 80% of it, and the base moment from 73500 to 38217 kNm. The deflected shape is drawn hugely exaggerated: the real top drift is about one five-hundredth of the height.

Outrigger

  1. 1 The arm that makes the columns work
  2. 2 What the second arm is worth
2 essays · structures The bearing is one length and the web is loaded over another. A load applied over a stiff bearing of 100 mm on the flange of a girder with a 1200 × 8 mm web. The flange bends under it and the yield lines that form spread the load along the web over 559 mm — 5.6 times the bearing, and 82% of the yield resistance is that spread rather than the bearing. The effective length is not a decision anybody made: it is what the flange's own bending stiffness against the web's own strength works out to.

Patch loading

  1. 1 The load that chooses its own length
  2. 2 A column nine hundred millimetres long
2 essays · stability Loaded straight down, and moving sideways. An equal angle with a moment applied about the horizontal axis. Its principal axes lie at 45.0° to the drawn ones, so the neutral axis runs at -30.6° rather than horizontally, and the section moves 59% as far sideways as it moves down. The product of inertia that causes it is -1.066 × 10⁶ mm⁴, and it is zero for every section drawn in this field until now.

Principal axes

  1. 1 Loaded straight down, and it moves sideways
  2. 2 The axis a column buckles about
2 essays · sections One point, every plane through it, one circle. A point carrying 140 N/mm² across one face, 0 across the other and 45 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 83.2 centred at 70.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 153.2 and -13.2, on planes 16.4° from the face the 140 acts on; the largest shear on any plane is 83.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 160.2.

Principal stress

  1. 1 The worst stress is not where the worst bending is
  2. 2 The circle nobody draws
2 essays · sections A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658.

Punching shear

  1. 1 A check made on a perimeter, not on a section
  2. 2 Turn the column, and the slab passes
2 essays · internal-forces 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.

Reciprocity

  1. 1 The theorem that swaps the question round
  2. 2 An influence line is a deflected shape
2 essays · deflection Restraint is a fraction, and the length decides how far up it reaches. A 20 m wall 3.0 m high cast against a base that has already hardened — a length-to-height ratio of 6.67. The base holds the bottom of the wall at R = 0.50 and the top of it at 0.304, decaying as 0.609 to the power of the height in wall heights. The free contraction is 380 microstrain, of which 84 per cent is the wall cooling from its own hydration peak and the rest is drying; the concrete's own strain capacity is 50. Everything to the right of the dashed line cracks, which here is the bottom 3.00 m of it. Nothing has been loaded.

Restraint cracking

  1. 1 The steel decides how many, not how much
  2. 2 The bay that is cast last
2 essays · internal-forces

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