Structural Engineering & Statics
where the load goes
Theme

Geometry beats material — page 13

Essays 289 to 312 of 329 on this thread, in the same order.
Below a limit the row settles; above it, it ratchets. The end screw's slip at the top of each cycle, against the number of cycles, for 20 leaning screws 56 mm apart with 40 mm of member depth per row loaded again and again from new to four levels, each screw, reloaded, reaching the slip it had been to with 85% of the load it had there. At 0.94 it settles at 2.50 mm after 3 cycles; at 0.96 it settles at 2.83 mm after 7 cycles; at 0.97 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 21; at 0.99 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 8. The row's first peak is 263 kN; every level drawn is below it. Connections

The row that remembers its overload

A row of twenty leaning screws reaches 98% of twenty screws' peak, once. Loaded again, a screw that has been past its peak comes back along a softer path and reaches the slip it had been to with less load than it carried there. So a row loaded again and again neither simply evens out nor simply ratchets down: below 0.965 of its first peak it settles after a few cycles with the middle screws catching up, and above it the end screws are pushed back onto their curve every cycle, lose a little more each time, and the row fails at a load it carried the first time. One large overload costs it less than many smaller ones near the peak.

 Squeezed while it bends, the hinge rides down its plastic moment. Moment against curvature for an I-section of 3,000 mm² and 200 mm depth in elastic–perfectly plastic steel, the moment over its unloaded plastic moment and the curvature over its unloaded first-yield curvature: at a constant 10% of its squash load, at a constant 60%, and with the compression rising from the first to the second as it bends (dashed: the plastic moment at the compression the rising path has reached). At constant loads it settles at 0.96 and 0.44. The rising path carries 0.86 at twice the first-yield curvature, 0.70 at 6.0 times it and 0.43 at 12.0: it forms its hinge early, at the lower load, and then loses moment as fast as the plastic moment falls with the load. Materials

The hinge squeezed while it turns

The leeward column of a frame overturning in an earthquake gains compression as its base hinge rotates. Followed fibre by fibre, with every fibre's plastic strain remembered, the hinge does neither of the things the constant-load curves suggest. It forms early, at the low load it started with, and then rides down the section's plastic moment as the load rises: an I-section whose compression goes from a tenth to six-tenths of its squash load holds 0.86 of its unloaded plastic moment at twice its first-yield curvature and 0.43 at twelve times it. The steel never softens, and the hinge still has a falling branch, set by statics. The relieved column on the other side gains what this one loses, less the curvature of the interaction curve.

 The film over an angle's corner. Contours of Prandtl's stress function over the corner of an equal angle 100 × 100 × 10 mm with a 12.0 mm root radius and 6.0 mm toe radii: left as rolled, right with sharp corners. Along each leg the contours run parallel to its faces; at the corner they bulge into the metal the fillet adds, round the outside of the L. The rolled angle's J is 6.80 cm⁴ and the sharp one's 6.18 cm⁴. The dot marks the steepest slope: on the fillet in the rolled angle, in the re-entrant corner in the sharp one, where it grows without limit as the grid is refined. Sections and stress

The corner with one fillet

A rolled I-section's web meets its flanges in two T junctions, four fillets, and its torsion constant is 7 to 12 per cent above Σbt³/3 because of them. An angle meets itself in one L, and a channel's web meets its flanges in two. Solved over the real outline, an L junction's fillet adds about two-thirds of what a T junction's does with the same plates. The sum then errs on either side: low by 8% for an I-section with a channel's plates, high by 0.8% for the channel itself, and anywhere from 5% low to 2% high for angles, depending on which length of leg is counted. The fillet's peak stress is still the inscribed circle's, to a few per cent.

 A brace on the sideways mode gives back the snap. The load parameter against the crown's deflection, both in radii of gyration, for a shallow pinned arch whose rise is 8.0 radii of gyration, built 0.1 of a radius of gyration out of shape in its sideways mode, unbraced and with a brace on the sideways mode of 0.88, 1.75, 3.50 times the rib's own antisymmetric bending stiffness. Unbraced it peaks at 410; with 0.88 it peaks at 658; with 1.75 it peaks at 794; with 3.50 it peaks at 840. The perfect arch's limit point, which a brace can only return it to, is 844; unbraced, it turns sideways at 461. Stability

The brace that holds a mode

A shallow arch rising eight radii of gyration turns sideways at a load of 461 when its symmetric snap would not come until 844. A member that only stops the quarter points moving in opposite senses — a brace on the sideways mode, not on any one point — gives the whole difference back once it is 1.75 times as stiff as the rib's own resistance to that mode, and nothing more after that. The stiffness needed grows as the square of the rise. And the brace is not the light member the argument suggests: an arch built a tenth of a radius of gyration out of shape puts an eighth of its whole load through it at the peak.

 The shear term cuts the beam's stiffness, not its share. For 20 storeys of two 6.0 m walls 0.35 m thick either side of a 2.4 m opening, coupling beams 0.35 m wide, 60 kN at every floor, against the coupling beams' depth: each beam's stiffness with its shear deformation counted over its stiffness in bending alone, the degree of coupling likewise, and the largest beam's shear likewise. At a depth of 0.6 m (span over depth 4.0) the stiffness is 0.85 of the bending figure, the coupling 0.977 and the shear 0.961; at 2 m (1.2), 0.33, 0.973 and 0.937. Where the shear term is large the beam is stiff enough that its stiffness has stopped deciding anything. Deflection

The coupling beam asked for shear

The coupling beams of a pair of shear walls are the stubbiest members in a building, and the shear term in their deflection is largest exactly there: a beam two metres deep over a 2.4 m opening is a third as stiff as bending alone says. The walls hardly notice. The pair's degree of coupling moves by under 3%, the beams' shears by under 7%, because a beam that stiff has stopped deciding how much of the overturning the walls carry as a couple. What decides it is the shear the beams can give. A 600 mm beam limited to 2 N/mm² starts to yield at 1.9 times the storey force; past the last yield the pair is two cantilevers again, and the beams supply rotation instead.

 The series sum is exact for matched flanges and errs by their mismatch. The pair's stiffness over the series of its two flanges as tee stubs on rigid bases, each with half the bolt's length, against the end plate's lever arm from web to bolt, for end plates 14, 18, 25 mm thick on a column flange 14 mm thick with its bolt 30 mm from its web, tips 35 mm beyond the bolt. Where the end plate matches the column flange — 14 mm thick at 30 mm — the ratio is exactly one: two identical flanges back to back meet on their plane of symmetry, which is the rigid base. A 14 mm end plate is 0.95 of the series with its bolt at 45 mm and 0.85 at 65; an 18 mm end plate is 0.97 of the series with its bolt at 45 mm and 0.86 at 65; a 25 mm end plate is 1.00 of the series with its bolt at 45 mm and 0.93 at 65. Connections

The flanges that lean on each other

A bolted end-plate joint's stiffness is built from springs in series: the end plate as a tee stub on a rigid base, the column flange as another, the bolt shared between them. The rigid base is not a fiction. It is exactly the plane of symmetry between two identical flanges bolted back to back, so for matched flanges the series sum is exact. A real end plate and column flange are not matched, and each tip bears on a tip that moves differently from its own. Solved as a pair, the error is the mismatch: 3% for an 18 mm end plate on a 14 mm column flange with similar lever arms, 12–15% when the end plate's bolt is twice as far from its web.

 The deck's width over the pier is set by its bars. The local effective width, as a fraction of the overhang, along a deck flange overhanging 3.0 m each side of its girder, continuous over two 20 m spans under a uniform load, drawn where the moment is at least three tenths of its largest: uncracked (dashed), and over the pier, where the deck is in tension, cracked with 1.5% of steel along the girder and 1.5% across it and with half that transverse steel. EN 1994's widths are drawn as the short level lines, Le/8 over the pier and in the spans. Over the pier the uncracked deck works over 0.325 of its overhang, the cracked one with equal bars over 0.328, and with half the transverse steel over 0.281; in the spans, uncracked, the deck works over 0.81. The rule gives 0.417 over the pier whichever way the deck is reinforced. Sections and stress

The deck that cracks over the pier

Over the pier of a continuous composite bridge the concrete deck is in tension and cracks, and what is left of it is a grid of bars and struts. Shear lag does not stop at the cracks: the bars still receive their force through the concrete between them. Whether the cracked deck works over a narrower width than the whole one depends on one ratio, the deck's shear stiffness over its axial stiffness — and once the concrete has cracked, both of those come from the reinforcement, so the answer is set by how the bars are arranged.

 A haunch moves the worst stress range to its toe. The stress range up the height of the first interior post of a Vierendeel girder of six 2 m panels, 1.5 m deep, every member of the same section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load, over one crossing: the plain post (dashed), and the post with 250 mm haunches 2.00 times as deep at both corners (solid); with the constant-amplitude limits 52.3 MPa for detail category 71 (dotted). Plain, the range is 78.3 MPa at the corner. Haunched, it is 43.3 at the corner and jumps to 61.2 at the toe, 250 mm along the post — the largest anywhere in it, 61.2. Structural form

The corner that counts its crossings

As a load crosses a Vierendeel girder, the moment at a post's corner changes sign when the load passes that post. A static check reads the peak, and the peak says the posts beside the supports work twice as hard as the middle one. Fatigue reads the range, and the range says every interior post works the same. A haunch at the corner lowers the corner's range and moves the worst of it to the haunch's toe — so a deeper haunch soon buys nothing, a longer one keeps paying, and a toe two detail categories worse than the corner cancels the haunch outright.

 The single side's strongest direction is not the double side's. The resultant a millimetre of throat can carry in S355 against the direction of the load: a double-sided butt throat, a fillet throat, and a 20 mm plate's single-sided throat at penetrations of 0.5 and 0.7, counted plastically. The double-sided butt is strongest at 22.7°, 478 N/mm². The single side at 0.5 rises from 183 straight across to 314 straight along and is strongest along the weld; at 0.7 it rises from 291 to 357 at 40° and falls to 314, and from 38° to 88° it carries more than the fillet. Connections

The root that has to stretch for it

A partial-penetration weld made from one side carries its pull off the plates' line, and counted elastically it keeps a quarter of a double-sided weld at half penetration. Let the throat yield and it keeps 0.41 — but the same 0.41 at every direction up to the one where the double-sided weld is strongest, so that weld's peak advantage is never won back. What the yielding does instead is move the single side's own strongest direction out to 40° or beyond, where a deep single-sided throat outcarries a fillet. Every per cent of it is paid for with strain at the root, which is the tip of an unfused land.

 The same damper behind a lever is best at a larger motion. The damping of one storey with 2% of its own damping and a brace as stiff as the storey, with an α = 0.3 damper sized to its brace's best at the design amplitude with no amplifier, against the storey's amplitude, with the same damper behind stroke amplifiers of ratio 1, 2 and 3. A ratio f makes the damper act on the storey as one f to the power 1 + α larger — 2.46 and 4.17 times — so it locks more, and its band moves up by about f to the power (1 + α)/(1 − α): its best is near 3.8 and 7.5 times the design amplitude. At a hundredth of the design amplitude all three give 1.4%; at the design amplitude 22.7%, 6.4% and 2.3%. Dynamics

The lever that moves the damper and not the band

A damper whose force goes as a low power of its speed is sized for the earthquake and locks in the wind. A toggle or scissor linkage multiplies its stroke, and it seemed it might move the band of amplitudes the damper serves down towards the wind. It cannot. Behind a lever of ratio f the same damper acts on the storey as one f to the power 1 + α larger, so its band moves up, not down; re-sized to the design amplitude, it gives the storey exactly the curve it had without the lever, and what has changed is the damper — a quarter of the size, a third of the force, three times the stroke. What would widen the band is a ratio that grows with the motion, and a toggle has one — until its links lie straight and it locks.

 The mechanism outcarries the frame that is nearly one. The load on the joint of two bars of 200 MN axial stiffness across 8.0 m against how far it moves down, followed to large displacement: the flat pair with its bars tensioned to 0, 50 and 200 kN with nothing applied, and the same bars raised 3 degrees with no tension, a rise of 210 mm. The raised frame is stiffer at first and snaps through at 11.1 kN, 89 mm down. The flat pair at 200 kN starts at 100 N/mm, carries 11.1 kN at 89 mm and 50 kN at 210 mm, and goes on stiffening; at 50 kN of tension, 50 kN at 242 mm; with none, at 252 mm. Equilibrium

The mechanism that stands up anyway

Two bars in a straight line between two pins are a mechanism by the count, by the rank and by every measure of how nearly a frame folds. Tension them and they carry load: 200 kN of tension gives the joint the stiffness of the same bars raised 1.8 degrees, and where the bars raised 3 degrees snap through at 11 kN, the tensioned flat pair carries 50 kN and goes on stiffening. The stiffness is a lack of fit of four millimetres, which 83 degrees of warmth takes away; compressed, the same pair buckles to rest crooked on one side or the other; and no tension can reach a mechanism that swings rather than shivers.

 The wire's strength buys the wall. The thinnest wall, as a multiple of the bore, on which a wound tube can carry the ceiling, against the wire's strength over the liner's. Wire as strong as the liner needs 3.01; 1.2 times as strong, 2.30; 1.5 times, 1.69; from about 2.00 times, any wall. Dotted, 2.22: the thinnest wall on which shrunk-on layers of the liner's own steel reach the ceiling, where σy·ln k is twice the first-yield pressure — thinner than the same steel wound, because a layer can carry compression and a wire cannot. Materials

The wire that cannot push

A tube's liner wound with wire under tension is a compound tube of a hundred layers, each shrunk on by its own winding. It reaches the same ceiling, twice the first-yield pressure, and no further, because the bore's range belongs to the whole wall and its floor to the liner's steel. What the wire's strength decides is where the ceiling can be reached at all. A wire cannot be compressed, so a jacket of the liner's own steel needs a wall three times the bore, where shrunk-on layers of the same steel need 2.22; wire half as strong again needs 1.69; wire twice as strong reaches it on any wall. Every turn is wound harder than it will rest, by what all the turns after it take away.

 Rigid enough long before it is a plate. A box 4.0 m wide and 2.5 m deep on a 60 m span, 45 kN/m at 2.0 m off its axis, with three diaphragms, each diaphragm of racking stiffness K, against K over the girder's own stiffness against a point distortional load, 15.0 × 10⁹ N·mm/rad: the largest warping stress (solid) and the largest distortion, which drives the walls' transverse bending (dashed), each over its value with rigid diaphragms. At ten times the girder's stiffness the warping stress is 1.05 of the rigid value and the distortion 8.37 times it; at a hundred, 0.89 and 1.50; at a thousand, 0.99 and 1.05. Marked: a cross-frame of two 1,000 mm² diagonals (26 times) and a 10 mm plate (524 times). Sections and stress

The diaphragm that is a few angles

A box girder's diaphragm holds its section square against the distortion an eccentric load drives. It does not have to be rigid; it has to be stiff against the girder's own resistance to a point racking load, which is small — twice the section's frame stiffness times its decay length. A cross-frame of two 1,000 mm² diagonals is twenty-six times that and holds the warping stress at the rigid value, while a 10 mm plate is five hundred times it. The cross-frame lets the section rack three times as far as a plate, which leaves the walls bending at 5 N/mm² instead of 1.5, against 275 with no diaphragm at all. And its diagonals carry nearly a rigid diaphragm's force whatever their stiffness, so it is sized for strength.

 Caught from both sides without a mode shape. The elastic critical load factor of a one-bay frame of six storeys, 3.5 m high and 6 m wide, columns of 40,000 kN·m² and beams of 60,000, 400 kN on every floor, 14.96 (level line), and the bounds the traces of its flexibility give with no assumed shape, against how many traces are taken: from below, 5.12, 10.82, 13.11, 14.07 for the first four; from above, 22.90, 19.23, 17.39, 16.39. Dotted, the storey estimate of EN 1993-1-1, 12.86, and Rayleigh's quotient on the first-order sway, 15.88. Stability

The trace that adds up every storey

A column's buckling load can be bounded from below by the traces of its flexibility, with no shape guessed. A sway frame has a flexibility too — one number per pair of floors — and its traces bound the frame's critical load factor from below and from above, closing in as more are taken: on a six-storey frame whose factor is 14.96, the first four give 14.07 to 16.39. But the first trace is Dunkerley's sum over the storeys, counting each as though its sway mode were the frame's, and it bounds a well-balanced frame worst of all: one over the number of storeys when every storey is alike, which is exactly when Eurocode 3's storey-by-storey estimate is exact.

 Shorter after every cycle, at the same load. The shortening of a column hinge at the end of each cycle, in yield strains, for an I-section of 3,000 mm² and 200 mm depth and the rectangle of the same area, each carrying a constant 30% of its squash load while its curvature cycles between ±3.0 times its first-yield curvature, in elastic–perfectly plastic steel. The I-section shortens by 8.14 yield strains every cycle, 97.1 after 12; the rectangle by 1.60, 19.4. Neither settles: the load has not changed, and the column is shorter after every cycle. Materials

The column that shortens on every cycle

A column hinge bent back and forth by an earthquake carries its share of the building's weight the whole time. In elastic–perfectly plastic steel an I-section at three tenths of its squash load, cycled to three times its first-yield curvature each way, comes back from every cycle shorter by eight yield strains, and goes on doing so: it never settles. The flange that yields in compression shortens, and the other flange, carrying its part of the column's load, never reaches yield in tension to pull it back. A rectangle of the same area shortens a fifth as fast, and the overturning that swings the column's load up and down hardly matters. The weight and the curvature make the ratchet between them.

 The knee moves out as the braces multiply. The critical load of a pin-ended column, over the Euler load of the segments between its braces, against the braces' stiffness in units of that load over the segment's length, for one, two and four braces. Each curve rises to the segment's Euler load and stays there; the knee where it arrives is at 2.00, 3.00, 3.62. With no braces at all the column buckles at 1/(n + 1)² of the segment's load: 0.25, 0.11, 0.04. Stability

The braces that have to hold each other

One brace at mid-height must have a stiffness of twice the column's load over the length it halves. Spread the same column over four braces and each one must be stiffer, not softer: 3.62 times the load over its segment, rising towards four with every brace added, because the shape the braces have to stop is a zigzag in which every brace point is pushed by both its neighbours. Braces that tie the column to itself instead of to the ground need one, however many there are. A row with surplus stiffness can carry one brace at two thirds of the ideal, but not one that is missing, and the force each brace has to resist grows with the count.

 A crack starts at the corner and runs along the grain. The end of a timber beam 400 mm deep and 140 mm wide, notched on its underside so that 200 mm stands over the support, with the notch's corner 50 mm from the line of the reaction, drawn to scale. A crack starts at the corner and runs along the grain towards mid-span. With a shear strength of 3.5 N/mm² the un-notched end would carry 130.7 kN; Eurocode 5's factor kᵥ of 0.40 on the reduced depth leaves 25.8 kN, 20% of it. Sections and stress

The notch that costs more than its depth

Cut a timber beam's underside away at the support, so that half its depth sits on the bearing, and a stress check on what is left says the end has lost half its shear strength. The beam says otherwise. A crack starts at the notch's corner and runs along the grain, and whether it runs is a question of energy, not stress — so the answer has a size in it. A 400 mm glulam beam notched to half its depth keeps a quarter of its shear resistance, an 800 mm one keeps 18 per cent, and the deeper the beam the less a notch leaves. Sloping the notch buys much of it back, and moving the corner away from the support costs it again.

 A screen damps a tank on a tower and barely touches a standpipe. The rms drift under broadband ground shaking, as a share of the same structure with its water frozen to the wall, against the damping a screen gives the sloshing: a steel standpipe 10 m across and 30 m tall with a 12 mm shell, full (solid), and the water tower of the earlier essay, a 12 m tank 8 m deep on a tower of 2.5 s (dashed). Plain water, at half a per cent: 0.82 and 1.01. A screen giving 10%: 0.76 and 0.40. The standpipe's gain comes almost entirely from leaving the lens out of the shaft's motion, which plain water already does. Dynamics

The tank that is its own tower

A standpipe holds its water in its own shaft, so the water fills a height several times its width. Only a lens of it near the surface sloshes — 0.455ρπR³, 178 t of the 2,199 t in a steel standpipe 10 m across and 30 m tall — and that lens stops growing once the water is deeper than a radius, so its period is set by the width alone: 3.3 s, against the shaft's 0.38. The lens is 8% of the water but a quarter of the shaft's moving mass, because it sits where the shaft moves most. It is far too slow to be tuned as a damper, and a screen can do little with it; what it does instead is stay still while the shaft sways, which takes nearly a fifth off the drift a frozen-water calculation predicts.

 The pair holds its clamp longer than the stub it is checked by. The tension a bolt at which the clamp lets go, as a share of the 110 kN preload, against the end plate's lever arm from web to bolt, for an end plate 18 mm thick on a column flange 14 mm thick at 30 mm: the pair (solid), the end plate's stub (dashed) and the column flange's (dotted). With the end plate's bolt at 30 mm the pair and the stubs nearly agree. At 45 mm the pair lets go at 88.2 kN and the end plate's stub already at 81.1, the pair holding 9 per cent more; at 60 mm, 82.0 against 71.2, 15 per cent more. Connections

The clamp the pair holds longer

A preloaded bolt clamps an end plate to a column flange, and the joint stays closed until the tension relieves the clamp. The joint diagram says that happens at the preload over one minus the bolt's share — 122 kN for a 110 kN preload — but prying adds to the tension at the bolt line and the clamp lets go at 88. A component model checks each flange as a stub on a rigid base and lets the joint go when its weaker stub does, at 81 kN. The real pair, its tips leaning on a column flange that gives way, pries less than the end plate alone would, and holds its clamp nine per cent longer — fifteen when its lever arms are two to one.

 In a core, the depth has to tune the tank, and only shallow water is slow. The sloshing period of rectangular tanks 6, 10, 15 m long against the depth of their water (logarithmic), with building periods of 4, 5, 6 and 8 s (dotted). Deep, each tank is as fast as it can be: 2.77 s, 3.58 s, 4.38 s. To be tuned to a 5 s building the 10 m tank needs 1.80 m of water; to 4 s, 3.50 m; to 8 s, 0.65 m. A building faster than 3.58 s cannot be reached by the 10 m tank at any depth. Dynamics

The tank that must be shallow to be slow

A deep tank's sloshing period is set by its width alone, so a tank in a tall building's core ought to be tuned by choosing how wide to make it. It cannot be. A deep tank fast enough to fit is too fast for the building: 3.6 s for a core 10 m across, where a 5 s building would need a tank 19.5 m long and an 8 s one 50 m. The depth has to tune it, and only shallow water is slow — 1.73 m for 5 s, 0.63 m for 8. Shallow water is the efficient kind, with four fifths of it sloshing. But at the accelerations the tank exists to reduce, the wave it makes is a quarter of its depth on a 5 s building and seven tenths on an 8 s one, and the linear tuning it was designed by has stopped describing it.

 The pressure on the wall jumps where the bin meets the hopper. The normal pressure on the wall of a bin 8 m across holding 30 m of a solid weighing 9 kN/m³, wall friction 0.45 and K = 0.45, over a conical hopper of half-angle 20° with wall friction 0.40, the solid's internal friction 35°, down the bin and then down the hopper: in the bin, K times Janssen's vertical stress (solid) and that times the 1.35 discharge factor (dotted); in the hopper, filling (light) and emptying (dark). At the foot of the bin the wall carries 38.1 kPa, 51.4 with the discharge factor. On the hopper's top edge it carries 75.8 kPa filling and 95.7 emptying — 1.99 and 2.51 times the bin wall's. Half way down the hopper, 46.3 and 36.7 kPa. Equilibrium

The pressure that switches at the hopper

Below a silo's bin a conical hopper funnels the grain to its outlet, and the wall pressure there does not keep growing with depth as a liquid's would. It peaks at the hopper's top edge. Filling, the edge carries twice the bin wall's pressure just above it; emptying, the grain converges on the outlet, its largest stress turns from vertical towards horizontal, and the edge carries two and a half times. Half a metre above the apex the stress is 2.6 kPa whether the bin holds 5 m of grain or 60 — which is why a silo empties at the same rate full or nearly empty.

 Past yield, the notch trades stress for strain. The root of a notch of elastic factor 3.0 in a mild steel whose cyclic curve reaches 0.2 per cent strain at 252 N/mm², under a nominal stress of 200 N/mm²: the cyclic stress–strain curve (solid), the elastic line the root would follow if it stayed elastic (thin), and Neuber's hyperbola σε = (KₜS)²/E (dashed), which the curve crosses at the root's state. Elastic, the root would carry 600 N/mm² at 0.30 per cent. By Neuber it carries 293 at 0.61 per cent; by Glinka's energy rule 272 at 0.44; by the linear strain rule 246 at 0.30. Materials

The notch that trades stress for strain

A notch with an elastic factor of three triples the stress at its root only while the root stays elastic, which in a mild steel is up to a nominal stress of about 84 N/mm². Past that the root's stress and strain part company: at a nominal 200 the root carries 293 N/mm², not 600, and strains 0.61 per cent, twice what the elastic factor implies. Neuber's rule and Glinka's bracket the truth a factor apart, and in the low-cycle range that factor is a factor of two in life. The trade has two gifts in it: one overload leaves the root in compression, and a mean stress mostly yields itself away.

 Braces that pull one way never brace the girder to its moment. The critical moment of the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side, against each brace's stiffness (logarithmic), when every brace resists twist both ways, when the three alternate in the direction they resist, when two face one way and one the other, and when all three face the same way; dotted, the 5,627 kN·m the girder must reach. Two-way braces of 2,846 kN·m/rad reach it. At that stiffness the alternating row reaches 3,087, the two-and-one row 1,734 and the row facing one way 947, the girder's unbraced moment; with rigid braces, 4,063, 1,891 and 947. Stability

The braces that pull only one way

A brace made of a single tie, or a cross-frame whose diagonals can only pull, resists a girder twisting one way and not the other. A buckling girder has no preference between the two directions, so it takes the one its braces leave free. Three such braces all facing the same way leave a 24 m girder exactly as weak as if they were not there. Facing alternately, they reach 3,087 kN·m at the stiffness that takes two-way braces to 5,627, and no stiffness takes them past 4,063. Opposing the outer pair does not help; one one-way brace anywhere in the row is enough to cap it. Every brace has to resist both ways, and made of ties that costs twice the steel.

 Across the gap the splice plates bend as one section. The top half of the splice of a 612 mm deep I-beam with 229 by 19.6 mm flanges, spliced at 700 kN·m by a 15 mm outer cover on each flange, two 8 mm inner covers beside the web under it and two 8 mm web plates 480 mm deep, to scale, cut at the gap where only the plates cross (the beam's own flange and web dashed beyond), and the stress through it. Bent as one section, the outer cover works at 206 N/mm² at its centre and the inner covers at 186, in proportion to their distances from the neutral axis; the hand design's area rule gives both 193; the plate splice's lever rule gives the outer 123 and the inner 350. Connections

The splice that bends as one section

A plate spliced with two unequal covers shares its load by lever arm about the line the load is held on, and the thinner cover carries more. A beam's flange splice looks like the same joint, and the rule that worked for the plate puts the thin inner covers at the edge of yield. They are working at half that. A beam's halves are not held on any line; they turn, and every splice plate across the gap turns with them, so the plates bend as one section and share the beam's moment by their distances from its neutral axis. The outer cover works hardest, the inner covers less, and the web plates less than the hand design hands them.

All themes