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The figure library — page 5

Every picture here is generated from code. This page lists the generator families, how many essays call each one, and how many separate drawings each has produced. Page 5 of 7, ring-buckling to stiffened-deck.

The cheapest way out of being round. A ring under uniform external pressure, drawn in its first four buckling modes with the pressure each one needs underneath it, in N/mm². The pressure has no direction: it stays normal to the wall wherever the wall goes, so it does work on any change of shape that reduces the enclosed area, and the ring buckles into whichever shape is cheapest. Bare, that is the oval — n = 2 at 3EI/R³ — and the modes rise as n² − 1, so three lobes cost 2.67 times as much. Nothing in the drawing prefers any orientation, which is the point — a column has an axis to buckle about and a ring has none.

`ring-buckling`

A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 1.00 m wide and 4.00 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 14.0° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done.

`rocking-block`

Two faces, a couple, and a core that does none of it. A sandwich section 61.4 mm deep: two 0.7 mm faces separated by 60 mm of core. The bending is carried as a couple between the faces — 41 N/mm² of tension in one and compression in the other, over a lever arm of 60.7 mm — and the core carries a shear stress of 0.047 N/mm² and nothing else. The parallel-axis term is 98.8% of the section's second moment; the faces' own bending about their own centroids is 0.004% of it, and the core's is 1.2%. Separated by nothing at all the same two faces would be 2.3e+4 times less stiff.

`sandwich`

20 kN applied at once and held, on a structure of 0.500 s period. Displacement against time for a single-degree-of-freedom structure of natural period 0.500 s and 2.0% damping, under 20 kN applied at once and held. The static deflection under the same peak force is 12.67 mm and the peak response is 24.56 mm — a factor of 1.94.

`sdof-response`

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.

`second-moment`

Where the class limits come from. The width-to-thickness ratio at which two kinds of plate reaches its own elastic critical stress at the yield stress, for three steel grades. A flange outstand (buckling coefficient 0.43) derives to 18.6, 15.2, 13.3 at 235, 355, 460 N/mm², against quoted limits of 14.0, 11.4, 10.0; A web, in bending (buckling coefficient 4) derives to 56.8, 46.2, 40.6 at 235, 355, 460 N/mm², against quoted limits of 42.0, 34.2, 30.0. The derived number is the larger every time, and by the same factor at every grade — flange outstand 1.33, web, in bending 1.35 — because both the derivation and the quoted limit go as one over the root of the yield stress. A constant ratio is what a fixed knockdown looks like: the derivation is for a perfect plate and the quoted limit is for a rolled one, carrying residual stress and not quite flat.

`section-class`

`section-compare`

A truss cut through panel 3. The truss severed through one panel, with everything to the right removed and the three cut member forces drawn on the exposed faces. Taking moments about the marked joint removes two of the three unknowns, so one equation gives the third: -52.94.

`section-cut`

The damping a crowd leaves behind, and the number of people that uses it up. Total damping ratio against pedestrians, for a structure with 0.60% of its own and a modal mass of 120 tonnes at 0.5 Hz. Each person walking laterally puts a force in phase with the deck's velocity into it — about 300 N·s/m per person — so the crowd subtracts from the damping, and at 30.16 people what is left is nothing. Below the threshold a disturbance dies away. Above it, the structure feeds itself and the motion grows out of nothing at all — which is what makes this a different mechanism from resonance rather than a severe case of it.

`self-excitation`

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.

`shear-centre`

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.

`shear-flow`

The coefficient is a slope, and that is why it can exceed one. The crack magnified: two rough faces, drawn as a sawtooth at 54° to the plane. Sliding one over the other cannot happen without lifting it, so a shear displacement forces a separation in fixed proportion — the tangent of that angle, which is the number written down as a coefficient of friction and here is 1.40. The bars crossing the plane are stretched by the separation and clamp the faces back together; they are not carrying the shear, they are supplying the normal force that lets the roughness carry it. A bar that is not anchored on both sides supplies nothing.

`shear-friction`

An angle bolted through one leg. A 100 × 75 × 10 angle connected through its 100 mm leg with three bolts at 75 mm pitch. The centroid sits 19.77 mm from the connected face over a connection 150 mm long, so U = 1 − 19.77/150 = 0.87 and 13.18% of the net area is not working.

`shear-lag`

Load, shear and moment — a simple span. The applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.

`shear-moment`

Flattening the truss saves stirrups and crushes the web. Two capacities against the angle of the cracks, for a web 300 mm wide with a lever arm of 495 mm. The rising line is the stirrups: a cut along the crack severs z·cot θ/s of them, so flattening the crack from 45° to cot θ = 2.5 takes the 225 kN they carry to 563 — 2.5 times as much from the same steel. The falling line is the concrete strut, whose stress is V(cot θ + tan θ)/b_w z and therefore least at 45°. They cross at cot θ = 2.44, and 550 kN is the most this section will carry however it is reinforced.

`shear-truss`

The hoops change their mind at an angle no proportion chose. The two membrane forces of a spherical dome of radius 30 m under 3 kN/m² of surface, taken from the crown to a base at 60°. The free body for the meridional force is the cap above a cone of half-angle φ, and vertical equilibrium of it gives N_φ = −wR/(1 + cos φ) directly: -45.0 kN/m at the crown falling to -60.0 at the base, compression everywhere. Equilibrium normal to the surface then gives the hoop force, which starts at -45.0 kN/m and reaches 15.0 — it changes sign, and the angle at which it does was found here by bisecting N_θ rather than quoted: 51.827292°. Setting N_θ = 0 gives cos²φ + cos φ − 1 = 0, so cos φ is (√5 − 1)/2, the reciprocal of the golden ratio — an identity this site's solver gate checks against the bisection to nine decimals rather than asserting, because it is too pretty to be believed on sight. Below that parallel the hoops are in tension, which masonry has none of, and that is where every old dome is cracked.

`shell-membrane`

What the shape of a load in time is worth, for two load shapes. The peak displacement as a multiple of the static deflection, against the load's duration divided by the structure's natural period, for two load shapes: a load that rises linearly, then stays; a rectangular pulse, then nothing. The lines are closed forms and eight dots are the peak of a complete time integration of an oscillator of 0.400 s period under that load, agreeing with the line to within 0.19% everywhere.

`shock-spectrum`

Three minima, and only two of them get a check. Elastic buckling stress against half-wavelength for a 200 × 65 × 15 × 1.5 mm lipped channel in uniform compression. The local minimum is at 200 mm and 41 N/mm²; the distortional at 689 mm and 287; the global curve falls away to the right and reaches 489 at the 1.5 m member. The distortional branch is a strut on an elastic foundation — the flange and lip rotating about the web junction, restrained by the web's own bending at 627 N·mm per radian per millimetre — so its minimum is at π(EC_w/k_φ)^¼ and its value is (2√(EC_wk_φ) + GJ)/I₀, the same closed form a continuously braced strut has. The elastic stresses are in the order local, distortional, global, and the mode that governs the strength is not the lowest of them, because they have very different amounts of post-buckling reserve.

`signature-curve`

The ground is a structure, and it has a period. The transfer function of 38 m of soil at 75 m/s over bedrock at 800 m/s: how much the surface moves for a given motion in the rock, at every frequency. It is a column fixed at the bottom and free at the top, so its resonances are the odd harmonics — the peaks stand at 1, 3, 5, 7 times the first, which is a fixed-free column and nothing else. The fundamental is at 0.493 Hz, a period of 2.03 s, and it is 4H/v_s exactly. The peak amplification is 7.5 against the bound 1/(α + πξ/2) = 7.5, which agrees to 0.2% — and the α in it is the impedance ratio, 0.0554 here. That is the term that keeps the answer finite: assume rigid bedrock and α is zero, the bound becomes 13 and the model is predicting an amplification set by damping alone. What limits the surface motion is that the energy can leave downward.

`site-amplification`

A strength that is a property of the specimen. Nominal strength against size for geometrically similar specimens of one material. On the left the specimen is too small for a crack to run and the strength is a plateau — a plastic limit, and the regime laboratory specimens sit in. On the right a crack releases more energy than it consumes as soon as it starts and the strength falls as the inverse square root of size, which is the regime real structures sit in. The turn happens at D₀ = 120 mm. A 100 mm specimen reads 3.10 N/mm² and a 1500 mm member of the same material carries 1.14: the test overestimates the structure by a factor of 2.71.

`size-effect`

`slab-action`

A preloaded joint, before and after it slips. Two preloaded bolts at 172 kN each, on one friction face at μ = 0.5. The joint carries 172 kN by friction with the bolts in tension and not in shear at all; past that it slips into bearing and carries 250 kN with the bolts now in shear. Two different mechanisms, one joint.

`slip-curve`

A broad tank sloshes and a tall one does not. The liquid's division into the part that moves with the wall and the part that sloshes, against the tank's proportion. The convective masses come from the potential-flow solution and the impulsive mass is whatever is left, so the two sum to the liquid's mass exactly at every proportion rather than approximately over part of the range. A tall tank at H/R = 3 is 84% impulsive and behaves almost like a solid; a shallow one at H/R = 0.5 is 72% convective and most of its contents never notice the earthquake. This tank sits at H/R = 1.33, which is 65% impulsive.

`sloshing`

`sn-curve`

The snow that left the roof is standing against the wall. A roof in section with a 1.0 m obstruction at its downwind end, drawn with the vertical scale exaggerated 4 times because a metre of snow on twenty metres of roof is thinner than the line it would be drawn with. The balanced layer is 0.48 kN/m² everywhere; the wedge against the wall holds the snow that 30% of the 20 m upwind gave up, so its area is fixed by conservation of mass rather than chosen. That makes it 0.85 m deep and 3.4 m long, with a peak load of 2.60 kN/m² — 5.4 times the balanced value, and still only 27% of the snow on the roof. An intensity several times the design load, made of a quantity nobody would notice had moved.

`snow-drift`

Three legs, three equations, one answer. A rigid top on three legs carrying 100 kN at (0.4, 0.25) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 60.4, 3.1, 36.5 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.

`space-equilibrium`

A frame with no plane to be drawn in. The tetrahedron, solved: three equations at every free joint, one axial force in every member. Members drawn heavy carry more; tension and compression are separated by the sign that came back from the solve rather than by inspection. The count is m + r = 12 against 3j = 12, which makes it exactly determinate, and joint equilibrium closes to 1.8e-15.

`space-truss`

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.

`span-fourth`

`stay-system`

A cable alone goes to a kink, and a kink is not a road. A point load of 1000 at mid-span of a 900 m suspended deck. The upper shape is the cable with no girder at all: two straight lines meeting under the load, because a cable takes the funicular shape of whatever is on it and the funicular of a point load is a kink — 0.0083 radians of it here. The lower shape is the same cable with the girder present, peaking at 1.125 against the bare cable's 1.873. The girder is not carrying the load — it takes only 17% of it — it is spreading it, over a characteristic length of √(EI/H) = 183 m, and what reaches the cable is spread over that length rather than arriving at a point.

`stiffened-deck`