Structural Engineering & Statics
where the load goes
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Geometry beats material — page 14

Essays 313 to 329 of 329 on this thread, in the same order.
Ring frames close together are a continuous foundation. The corner-angle change along a 60.0 m box 4.0 m wide and 2.5 m deep, its top flange an orthotropic deck of 14 mm plate, its webs 14 mm and its bottom flange 20 mm, under 45 kN/m at 2.0 m from its centre line, in milliradians, with ring frames whose crossbeam has a second moment of 1.7 × 10⁹ mm⁴ every 4.0 m and every 12.0 m (solid), and each row smeared into a continuous foundation of the same stiffness per metre (dashed). At 4.0 m the two are indistinguishable, 1.12 and 1.12 at the worst point; at 12.0 m the box distorts between its frames, to 3.45 against 3.33. Without any frames the same deck distorts 145. Sections and stress

The deck whose frame is its crossbeams

A steel box girder resists distortion as a beam on an elastic foundation, the foundation being its walls' transverse bending. When the top flange is an orthotropic deck, that foundation is almost absent between the crossbeams and concentrated at them, so the box sits on a row of springs, not a bed. Smeared into a bed, the springs give the right answer to within a few per cent as long as they are closer together than the bed's own decay length — and the rule holds for every crossbeam stiffness on one curve. Past that spacing the frames act as separate diaphragms and the warping stress between them doubles.

 A stiff enough crown brace stops the arch snapping. The load parameter against the crown's deflection for a shallow pinned arch of rise 6.0 radii of gyration under a sinusoidal load, built with an antisymmetric imperfection of 0.10 radii of gyration, with a crown brace of κc = 0, 1, 2, 4; the dots mark the first peak of the load, where the arch snaps. κc = 0: peaks at 273; κc = 1: peaks at 299; κc = 2: peaks at 327; κc = 4: no peak, the load rising throughout. Stability

The prop at the crown of the arch

A shallow arch that turns sideways before it snaps can be braced at its quarter points, which holds the sideways mode and carries almost nothing. A single prop at the crown is cheaper and does something else. The crown does not move in the sideways mode, so the prop cannot stop it; it raises the load at which the arch turns aside only by taking part of the load itself. But behind a prop about three times as stiff as the rib's own first mode, the sideways branch stops falling, and the arch no longer snaps at all — at every rise, by the same prop.

 More hangers, each swinging through less. Each hanger's force range, plotted at its mid-point along the span, for a tied arch of 100 m span and 17 m rise on 32, 64, 96 crossed hangers of 3,000 mm² bar at 60°, 60 kN/m of dead load and 60 kN/m of live load run on from either end. 32 hangers: the worst 678 kN, the mean 528; 64 hangers: the worst 369 kN, the mean 273; 96 hangers: the worst 241 kN, the mean 184. Structural form

More hangers, and the same steel

A network arch's crossed hangers are the web of a truss, and a live load's shear passes through them, so every hanger swings through nearly all of its force. The obvious cure is more hangers, each crossing more of its neighbours, so that every section's shear is shared among more diagonals. It works hanger by hanger — three times the hangers, a third of the range — and not at all in total: the hangers' fatigue steel, sized for their ranges, is set by the shear the arch carries and barely changes with how many hangers it is divided among. What more hangers buy is elsewhere.

 A concrete rib creeps into snapping long before its prop matters. For the shallow pinned arch of rise 6.0 radii of gyration under a sinusoidal load, propped at its crown by a concrete column of stiffness κc = 3.5, the creep coefficient at which a load that stays on loses equilibrium, against that load as a share of the day's critical load: a steel rib on the creeping column (solid), which loses only the column's share, so that by a creep coefficient of 3 no load below 85 per cent snaps; and a concrete rib whose column creeps with it (dashed), whose lasting load acts as if (1 + φ) times larger, so that it snaps at φ = pcr/p − 1; dotted, a creep coefficient of 2. At half the critical load the concrete arch snaps at a creep coefficient of 1.00, at 70 per cent at 0.43; the steel arch on its column does not snap at either. Stability

The column that creeps under the arch

A shallow arch propped at its crown by a stiff enough column stops snapping — the sideways branch it would have fallen down turns upwards. If the column is concrete, it creeps under the force it holds, and after a decade it holds that force as a spring a third as stiff, well on the side where the arch snaps. But a load that arrives in seconds does not meet the crept column; it meets the column's instantaneous stiffness, from a rest length that creep has merely moved down, and the arch's branch still rises. Creep reaches only the load that stays on, and there it can take back only the column's own share — 13 per cent of it. A concrete rib is another matter: it creeps itself, and a lasting load at half its critical load snaps it before its creep coefficient reaches one.

 Whether the cross-frames' spacing matters depends on the crossbeams between them. For the 60.0 m orthotropic box 4.0 m wide and 2.5 m deep under 45 kN/m at 2.0 m from its centre line: the largest distortional warping stress against the spacing of cross-frames of two 1,900 mm² diagonals, with nothing between them (dashed), with open tee crossbeams every 4.0 m (light) and with ring frames every 4.0 m (solid); dotted, the ring frames alone, 4.1 N/mm². At 8 m, 3.8, 3.8 and 3.1 N/mm²; at 20 m, 21.8, 21.7 and 7.1; at 30 m, 60.8, 59.2 and 7.3. Sections and stress

The stiff frame among soft ones

A steel box girder with an orthotropic deck is held in shape by a frame at every crossbeam, and its designers usually add a stiffer frame — a pair of diagonals across the box — at every few crossbeams as well. Whether the soft frames between can be left to do the work depends on one length: how far a distortion reaches along a box held by the soft frames alone. Ring frames welded into the webs and bottom flange hold it to six and a half metres, and then cross-frames twenty metres apart barely change the distortion and add a peak of warping stress at each one, where a stiff frame grips a box that was distorting gently. Crossbeams that are only tees under the deck reach twenty-one metres, and then the box between the cross-frames distorts as if the crossbeams were not there.

 The worst direction is the long axis of an ellipse. For a bracket of six M20 grade 8.8 bolts in three rows of two at 75 mm, loaded 150 mm to the side of the group and 150 mm in front of the column face: the demand on the governing bolt, top right, per kilonewton of load, drawn as a radius in the direction the load acts — straight down at the bottom — under the elliptical interaction (solid), which in each quadrant of directions is an arc of one ellipse, and under the straight line (dashed), which is not. The ellipse's longest radius, at 16.0°, is its eigenvector: 0.0058 of the bolt's interaction used for every kilonewton, a capacity of 172.5 kN, found without trying a single direction. Connections

The worst direction is an eigenvector

A bracket whose load can come from any direction in the plane of its bolts has a capacity that is a closed curve, and once the load also pulls the bracket off its column the curve becomes a surface over direction and distance. Finding its lowest point looks like a search. Under the elliptical interaction of shear and tension it is not: inside each quarter-turn of directions every bolt's demand squared is a quadratic form in the load's direction, so the worst direction is that form's eigenvector, and the bracket's least capacity is a 2 × 2 eigenvalue per bolt. Under the straight line EN 1993-1-8 prints it is a search, because a straight line has no curvature to make an ellipse of — and the hand check that sets the worst shear anywhere against the worst tension anywhere is a tenth too low in the plane and a fifth too low once the bracket is pulled.

 A milliradian of slack costs the brace more than a milliradian of crookedness. For the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side by braces of 5,691 kN·m/rad — 2.0 times the 2,846 that just reaches 5,627 kN·m — erected with 5.0 mrad of initial twist: the moment in the middle brace when the girder reaches 5,627 kN·m, against each brace's slack (solid), and against the same amount added instead to the girder's initial twist, the braces tight (dashed). With tight braces and no extra twist, 34.1 kN·m; each milliradian of slack adds 8.0, each milliradian of initial twist 6.8. At 10.0 mrad, 114.0 against 102.4. Stability

The braces with slack in their bolts

A cross-frame bolted into clearance holes resists nothing until the girder it braces has twisted far enough to take up the clearance, and then resists fully. It is a two-way brace that starts late. The girder, erected a little crooked, twists freely into its braces while the load is still small, and from then on the braces hold a girder that is more crooked than the one they were fitted to. The slack acts as an imperfection, and a costly one: each milliradian of it adds more to the middle brace's force than a milliradian of extra crookedness would, and no stiffness takes it back out. On a 24 m plate girder braced at twice the stiffness it needs, three milliradians of slack — a pair of ordinary clearances over its depth — raise the middle brace's force by 70 per cent.

 The road is the harshest stretch of the beam's journey. The worst top fibre, at any corner of any section, of the 12 m pretensioned beam, 300 × 700 mm, with eight strands, its schedule drawn for the bed and for yard bearers 1.0 m in from each end, in N/mm² with tension negative: on the yard's bearers; on the haulier's trailer, 1.5 m and 2.5 m from its ends with a bounce of ±30 per cent; lifted by loops 1.0 m in from each end on slings at 60° with a snatch of ±20 per cent; slewed at 0.10 g; and set down on bearings sloping 1 per cent across it; dashed, the stored concrete's limit of −3.0. Yard bearers, −2.98; trailer, −4.88; lifted, −2.97; slewed, −3.13; set down, −1.79. Sections and stress

The beam hung from its loops

A pretensioned beam is cast in a bed, stored on bearers, hauled on a trailer and then lifted by a crane onto its bearings — and at every stage its top fibre, which the strands put in tension, has to stay inside the limit its debonding schedule was drawn for. The trailer, with a bouncing wheel and supports where the haulier put them, was the harshest stage the yard essays found. The last few metres, from the trailer to the bearings, are gentler: hung from loops at the yard's bearer positions the beam is on its bearers again, set down it is on the support its middle was designed for, and only a slew of the crane pushes it a little past its limit. Where the loops are cast decides how much margin the lift has, and the best place for them is not where the yard put its bearers but where the bed held the beam: at its ends.

 Lightened where it reverses, and the range follows. The stress range each post's corner sweeps over one crossing of a Vierendeel girder of six 2 m panels, 1.5 m deep, its chords all of one section (I = 117 × 10⁶ mm⁴, W = 780 × 10³ mm³), crossed by a 100 kN load: uniform posts, posts sized to their peaks, and posts sized to both their peaks and their ranges. Uniform, every interior corner ranges 76.1 to 78.3 MPa. Sized to peaks, the middle post's range is 150.8 MPa and its neighbours' 101.6, against the uniform girder's worst of 78.3. Sized to both, no corner exceeds 78.3 MPa. Structural form

The post sized by what it must survive

A Vierendeel girder's posts are usually all one section, and a crossing load gives them peaks that differ by a factor of two and ranges that barely differ at all. Size each post to its peak and the steel falls by a quarter — and the middle post, lightened to two-fifths of the chords, sweeps twice the stress range the whole girder used to, and lasts a seventh as long. Size each post to its range instead and the girder needs less steel than the uniform one, not more, because the posts that can be lightened are the ones at the ends.

 Two neutral axes in one cut. Stress against height through a girder with a 300 by 30.0 mm top flange, a 229 by 19.6 mm bottom flange and a 573 by 11.9 mm web, spliced at 700 kN·m. The girder's own neutral axis is 374 mm above its underside, and its flanges work at 133 and 201 N/mm². Its splice, each flange's covers sized to that flange's force at 250 N/mm², has its own centroid at 326 mm — 48 mm lower — and across the gap the plates bend about it: the top outer cover at 244 N/mm², the bottom outer at 269, the web plates from 179 at their top to 212 at their foot. Shaded: the covers' heights (darker) and the web plates'. Connections

The splice that finds its own axis

Splice a girder whose top flange is heavier than its bottom one and the splice plates, sized flange by flange, have a centroid 48 mm below the girder's. It looks as if the splice must then carry an axial force the girder does not, or be proportioned to keep the two axes together. It carries none: a cut through the gap carries what the girder carries, and the girder carries no force. Held at both ends, the mismatch makes less than a kilonewton. What the offset does instead is move the web's own force into plates that were not sized for it.

 Between the two counts, and the root's toughness says where. What a 20 mm S355 plate welded from one side to 0.5 of its thickness keeps of a double-sided butt throat, pulled straight across, against the toughness of the metal at its root: assessed as a crack on the strip-yield line (solid), by the toughness alone (dashed), and by the elastic and plastic counts (dotted, 0.25 and 0.41). Below 30 MPa√m the root tears before the elastic count's load; at 46 MPa√m the toughness alone would allow the whole plastic count, but the assessment gives 0.34, and it takes 66 MPa√m to reach 95 per cent of it. At 30, 50 and 100 MPa√m the root keeps 0.25, 0.35 and 0.41. Connections

The land that is already a crack

A partial-penetration weld made from one side leaves the rest of the joint unfused, and pulled across, that unfused land is an edge crack whose tip is the root. Counted as a crack it keeps neither the elastic quarter of a double-sided weld nor the plastic 0.41, but a share between them that the root's toughness chooses — and the toughness of a crack front is the toughness of its length, which for a weld is the whole weld. A long weld in a cold structure can tear below the elastic count, and the same joint drawn twice as large tears sooner, which no count of stresses can see.

 The hogging region cannot give its studs away. For a 457 × 191 × 67 steel beam under a 130 mm slab, continuous over two 12.0 m spans under 30 kN/m, 40 studs a span: the largest slip (solid, left scale) and the force in the slab's bars over the middle support (dashed, its value in the legend), against the density of studs beyond the point of contraflexure as a share of the density elsewhere, the span's total held at 40. At the same density everywhere the largest slip is 0.68 mm and the bars carry 344 kN; at half the density beyond contraflexure, 1.04 mm and 251 kN; at a quarter, 1.69 mm and 118 kN. The support's hogging moment falls from 448 to 398 kN·m as the bars stop working. Internal forces

The slip that peaks where the moment turns

A simply supported composite beam slips most at its supports, and its studs are crowded there. Make the same beam continuous over a middle support and the slip peaks twice: at the end support, and again, just as hard and the other way, at the point of contraflexure — not over the middle support, where the shear is largest but the slip falls to nothing. Studs graded toward the supports, as a simple span's would be, miss the second peak and buy a tenth; graded to the slip they buy more than a third. And the hogging region cannot give its studs away to the span, because its edge is where the second peak is.

 Which way the oval lies decides whether the pressure helps or harms. The limit moment of a steel tube of radius 600 mm and wall 2.5 mm (r/t = 240), whose ring buckles under 4.07 kPa, over its limit with no pressure (796 kN·m), against the angle between the bending plane and the diameter a non-uniform pressure pushes in on, for second harmonics of 0.10, 0.25 and 0.50 of that pressure; lines the closed form, dots the minimised energy. At 0.10, 0.93 aligned and 1.08 across; at 0.25, 0.82 aligned and 1.19 across; at 0.50, 0.65 aligned and 1.40 across; at 45° the oval does not touch the limit at all. Stability

The oval the pressure brings

A uniform pressure changes how stiff a tube's ring is against the oval that bending squashes it into, and nothing else: the tube still gives way at a flattening of two ninths of its radius. A pressure that is larger on one diameter than the other has an oval of its own, and that oval does work on the bending's directly. Lying in the bending plane it has done some of the flattening before the bending starts, and a pressure difference a quarter of the ring's own buckling pressure takes a fifth off the tube's bending capacity; lying across it, it adds a fifth. At forty-five degrees it does nothing at all. And the two ninths turn out to have been a property of a round tube.

 The more of the load that comes back, the less of it the row can carry for ever. The largest load 20 screws leaning at 45° in a row 56 mm apart carries for ever, as a share of its forward peak, against the share of it that comes back the other way each cycle (0 the same load repeated, 1 fully reversed), for each screw carrying none, half and all of its thread's first loss into the other way. Carrying 0 per cent: 0.96 repeated, 0.96 half reversed, 0.82 fully reversed; carrying 50 per cent: 0.96 repeated, 0.94 half reversed, 0.79 fully reversed; carrying 100 per cent: 0.96 repeated, 0.92 half reversed, 0.76 fully reversed. Connections

The row that is pushed as often as it is pulled

A row of leaning screws in a shear wall is loaded one way by the wind and then the other, and a screw that leans is pulled along its axis one way and pushed the other. Pulled, its axial force clamps the joint and friction helps it; pushed, it prises the joint open and the friction goes. So the row has a weaker side, and under a reversing load it settles exactly as it settles under a repeated one — the reversal does not start the ratchet again — but at a ceiling the pushed side sets, 0.79 of the pulled peak against 0.97. And it settles with a gap in the middle that every reversal crosses carrying nothing.

 The flange is bent hardest where it is thinnest, not where it curls most. The transverse bending stress at the root of the inner flange, where it meets the web, along a knee curved to 1,000 mm through a quarter turn, its flanges 600 mm apart on a 10 mm web, carrying 500 kN·m all round the curve; the dotted line is a yield stress of 355 N/mm². Uniform, 269 all round. Narrowing, 283 at the rafter end and 207 at the column end, largest where λ is; thinning, 191 at the rafter end and 397 at the column end, largest where λ is too. Tapered both ways, 205 at the rafter end and 325 at the column end: largest at the column end, where λ is 0.94 against 1.14 at the rafter end, because there the flange's own stress has nearly doubled. Sections and stress

The flange bent hardest where it curls least

A curved I-beam's flange curls towards its web and is bent across its own width at the root, by a stress that rises with the curling parameter λ. A portal knee's flange is cut from plate and usually tapers, and a taper changes λ along the curve. So the obvious guess is that the transverse stress peaks where λ is largest. For a flange that only narrows, or only thins, it does. Taper it both ways and the guess fails: λ is smallest at the narrow, thin end, and that is where the flange is bent hardest, because the same force is carried there on half the steel. The force the changing width hands to the web, meanwhile, is a few newtons a square millimetre.

 Where real roots sit on the warping curve. The elastic critical moment of a 3.0 m cantilever of 457 mm universal beam with its tip load at the shear centre, with its root holding the lateral slope and resisting warping through a spring, against the spring as a multiple of EIw/L on a logarithmic scale — from 521 kN·m free to warp to 1,266 built in — with four real roots placed on it. Top storey, column ends: 2.5 times EIw/L, 791 kN·m, 36% of the way; 15 mm plate, column runs on: 5.9 times EIw/L, 945 kN·m, 57% of the way; 40 mm plate, column runs on: 6.8 times EIw/L, 970 kN·m, 60% of the way; 15 mm plate, column boxed: 19.4 times EIw/L, 1,124 kN·m, 81% of the way. Stability

The root that belongs to the column

A cantilever's root must hold its flanges against warping at about fifty times the cantilever's own warping stiffness to count as built in, and doubles its buckling moment if it does. A flush end plate is the usual root, so it ought to be the plate's thickness that decides. It is not. The flanges warp by turning in plan in opposite directions, the plate passes that to the column as two opposite torques, and an open column twists between them more easily than the plate bends: three quarters of the root's flexibility is the column. An ordinary root is worth about six times the cantilever's stiffness, halfway up the curve, and a thicker plate barely moves it. Boxing the column does.

 A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54. Sections and stress

The bore with corners and the passage beside it

A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

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