Structural Engineering & Statics
where the load goes
The collection

Every essay — page 24

Essays 553 to 567 of 567, in the same order.
What a 60-degree seam leaves of the wall. The combinations of axial stress σa and hoop stress σh that a tube of 6082-T6 (parent ultimate 310 N/mm², softened zone 185) carries, with its seam at 60° to the axis (solid), inside the parent wall's von Mises ellipse (dashed). Where the solid line falls inside the ellipse the seam's zone fails first. axial tension: 60 per cent of the parent; closed pressure: 78 per cent of the parent; hoop alone: 76 per cent of the parent. The seam is weakest, at 60 per cent, along the loadings whose unstretched line lies on it. Materials

The seam that pressure protects

A soft weld zone pulled at 55 degrees to the load gets no help from the metal around it, because 55 degrees is the line a plate under uniaxial tension does not stretch. A spiral-welded tube winds its seam at about that angle. Pressurise the tube and the line that does not stretch moves to the tube's own axis, so the spiral seam is suddenly well protected and the longitudinal seam is the exposed one. The soft zone's weakness belongs to the loading as much as to the weld: a seam is in danger only where the field around it leaves its line unstretched.

6 figures · Heat-affected zone An inclined web takes the shear centre off its line. Where the shear centre of a trapezoidal box 1,500 mm deep, 3,000 mm across the top and 2,000 mm across the bottom (15 mm top plate, 12 mm elsewhere) lies as an interior web from the top flange's middle is leaned so that its foot moves along the bottom flange (solid, dots at each position), against where it would lie with a vertical web at the inclined web's mid-depth position (dashed); the cross marks the box with a vertical web at the middle, 791 mm up. A vertical web moves it sideways only, within 2 mm of that height. The inclined web lifts it as it leans: with its foot 500 mm left of the middle it is −66 mm across and 800 mm up; 500 mm right, 66 across and 800 up. Sections and stress

The web that leans and lifts the centre

In a rectangular box an interior web pulls the shear centre sideways toward itself and leaves its height alone, because a vertical web cannot carry horizontal shear. Most steel box girders are trapezoids, and their webs lean. Leaning the outer webs puts the shear centre below the centroid instead of above it; leaning an interior web lifts the shear centre as well as pulling it, by up to 35 mm in a 1.5 m box; and a vee of interior webs meeting at the bottom flange drops it by as much as 200. A web that leans carries horizontal shear, and whatever carries horizontal shear decides where the shear centre is in height.

7 figures · Shear centre Two formulas that agree until the shear is soft. The buckling load over the Euler load, by Engesser (dashed) and Haringx (solid), against the shear stiffness over the Euler load on a logarithmic scale. Marked: double lacing, 23.48: Engesser 0.959, Haringx 0.961; battens, 0.30: Engesser 0.232, Haringx 0.419; rubber bearing, 6.6e-4: Engesser 0.001, Haringx 0.025. They agree within 5 per cent while the shear stiffness is more than about 4 times the Euler load, and for a shear stiffness a thousandth of it Haringx's is 31 times Engesser's. Stability

The bearing Engesser says has buckled

A member that is flexible in shear buckles below its Euler load, and there are two classical formulas for how far below. For a laced column they agree to a fraction of a per cent. For a battened column one gives nearly twice the other. For a laminated rubber bearing, Engesser's says it buckles at a fourteenth of the load it carries every day and Haringx's says it is safe by a factor of nearly three. They are not two theories. Put the axial load into Engesser's shear stiffness and it becomes Haringx's, exactly — so the whole disagreement is about which shear stiffness was measured.

6 figures · Built-up column The slab carries what the collector does not, beside the bay. The plate's shear flow just beside the same brace line, along it: the rule's uniform 40 kN/m (dashed); with a 150 mm concrete slab (solid), concentrated beside the bay — 132 kN/m on average over the bay's 6 m, 3.3 times the rule's, with peaks at the bay's ends that grow as the mesh is refined, because a stiff support ending in a plate is a stress singularity — and little elsewhere; with a steel deck of shear stiffness 10,000 kN/m (dotted), close to uniform. Each curve adds up to the line's 960 kN. Structural form

The collector the slab does not need

A floor delivers its storey force to a short braced bay through a collector, a member along the brace line that gathers the floor's shear and carries it to the bay. The rule that sizes it assumes the floor hands over its shear uniformly along the whole line. A concrete slab does not: it is so stiff in shear that it sends most of its load straight into the bay, the collector carries about a quarter of the rule's force, and the slab beside the bay works at more than three times the rule's shear. A steel deck is the other way round. The rule is right for one floor, and safe for the other only while the slab stays uncracked.

6 figures · Diaphragm Preload makes the base rigid, until the bed lets go. Moment against rotation at the base of a 500 × 400 × 20 mm base plate under a 260 mm column, with two M24 holding-down bolts in each of two rows 60 mm from its ends: with no preload (dashed), 58,609 kN·m per radian, 9.8 times the column's EI/L; with bonded anchors whose free length is 136 mm (dotted), 13.9 EI/L; preloaded to 150 kN a bolt (solid), 45.6 EI/L until the bed under the plate lets go at its tension edge at about 57 kN·m, then softening toward the unpreloaded base. Faint: the slope of EN 1993-1-8's rigid boundary, 30 EI/L. Connections

The base that is rigid until the bed lets go

A column base drawn as fixed is a spring about a tenth as stiff as the rule for "rigid" requires, and most of its softness is the holding-down bolts stretching. Shortening the bolts helps less than it seems: even 60 mm of free length leaves the base below two thirds of the rigid boundary. Preloading them works completely — the base becomes half as stiff again as the boundary asks — but only while the grout under the plate stays in compression. The preload is spent at a definite moment, the same moment the column's own weight would buy for free, and the shorter the bolt the faster creep of the grout takes the preload away.

6 figures · Base plate Where the pile and the ground settle alike. For a pile 24 m long and 600 mm across, carrying 800 kN, in ground whose top 18 m settles — by 100 mm at the surface, less with depth — over a toe that resists by moving: on the left, how far the ground (dashed) and the pile (solid) settle down the pile's length; on the right, the force in the pile. Above 15.9 m the ground settles more than the pile and hangs on it; below, the pile settles more and the ground holds it up. The force peaks there, at 1,407 kN against the 800 kN applied, and the toe, settling 10.6 mm, mobilises 623 kN; the head settles 13.6 mm. Internal forces

The plane where the pile and the ground agree

Downdrag is usually found by equilibrium: the ground hangs on the pile above a neutral plane, holds it up below, and the plane is where those balance the load and the toe's resistance. But the toe's resistance is not a number the designer can supply. It is whatever the toe's own settlement mobilises, and the toe settles because the ground drags the pile down. Solved by compatibility instead — the plane where the pile and the ground settle by the same amount — the drag, the neutral plane, the toe's force and the pile's own settlement all turn out to grow with how far the ground goes, and each kilonewton put on the head costs the pile three quarters of one, not half.

6 figures · Downdrag Where a bending stem sheds its compaction pressure. The share of the rigid build-up's pressure shed at each depth by a cantilever stem 3 m high, bending stiffness 67,500 kN·m² a metre (a concrete stem about 300 mm thick), backfilled in 250 mm lifts of soil at 20 kN/m³ and 30°, each compacted by a roller of 60 kN a metre: nothing at the base, which does not move; little in the top lift, which nothing is placed after; most, 20 per cent, at 0.9 m down, where the pressure was locked early and the stem went on moving away from it. Equilibrium

The stem that bends away from the roller

A roller compacting backfill against a wall leaves a lateral pressure behind it that the soil's weight alone would never make, and for a wall that cannot move that residual is Ingold's envelope. A cantilever stem moves as the fill goes up, and a residual locked in against it relaxes as the stem bends away. But not everywhere: the base never moves, so it keeps its pressure, and the last lifts have nothing placed after them, so they keep theirs. A 300 mm stem sheds an eighth of the rigid wall's base moment and the softest stem a third — never more, because the two ends of the stem are where the relief cannot reach.

6 figures · Lateral pressure A point at mid-span is worth twice its total spread out. The mid-span twist of an open section 8 m long on forks, carrying 12 kN/m at 75 mm from its shear centre, against the stiffness of one rotational spring at mid-span, logarithmic; dashed, the twist with the uniform deck of 5 kN·m per metre per radian, whose whole restraint is 40 kN·m per radian. A mid-span spring of 19.5 kN·m per radian — 49 per cent of the deck's total — twists the beam as little as the deck does; the deck's whole total at mid-span leaves 3.13°. Deflection

The restraint that works where the twist is

A deck fastened along a beam's top flange resists its twist a little everywhere. A secondary beam framing in at mid-span resists it a lot at one point. Given the same total stiffness, the point does better — half the deck's total at mid-span holds the beam as well as the whole deck — because it works where the beam twists most. But a point restraint has a price the deck never charged: it takes the torque through one connection, up to three fifths of the whole applied torque, and if it is stiff it moves the worst twist out to the quarter-points rather than removing it.

6 figures · Twist serviceability A sine's requirement grows with the deck, and a random deck's with its square. The damping a 120 m stay at 3,500 kN, 1.01 Hz needs to hold still, against the rms of the deck's movement at the anchorage, with the deck tuned to twice the stay. A sine of that rms: √2·σ·(dε/dx)/4, a straight line. Against it, a random deck with 1.0% damping of its own: the swing's variance over sixteen times the deck's damping ratio, a parabola for a typical history and, dotted, twice that for the mean square, with dots from integrating the stay and the deck together over five 2,000-second histories each. With the stay's own 0.10% the sine grows above an rms of 2 mm and the random deck above 9.1 mm — 6.4 mm for its mean square. At 7.5 mm the integration asks for 0.06% against an average of 0.07%, at 15 mm the integration asks for 0.25% against an average of 0.27%, at 23 mm the integration asks for 0.49% against an average of 0.61%, past the range the average claims and at 30 mm the integration asks for 0.71% against an average of 1.1%, past the range the average claims. Dynamics

The deck that forgets its own rhythm

A deck at twice a stay's frequency grows the stay whenever a quarter of the tension swing beats the damping. But a deck pushed by gusts is not a sine: its amplitude comes and goes and its phase drifts, and it holds a rhythm for seconds where the stay needs minutes to answer one. Against that deck the requirement is not a quarter of the swing but its square over sixteen times the deck's own damping — a sixth of the sine's at service, and a number set by how much the deck remembers rather than by how far it moves.

7 figures · Cable dynamics Where in a slab's first month its heaviest load lands. The load on the sixth floor of a frame of 250 mm slabs on a 7.5 m grid cast a floor every 7 days at 20 °C, on one level of shores and one of backprops, as a multiple of its own weight, against its age. Rigid and snug, no creep: a peak of 1.51 w at 14 days; steel springs and creep: a peak of 1.71 w at 21 days; the same, 1.0 mm of slack: a peak of 1.78 w at 7 days. Snug and rigid, the slab carries its heaviest load twice, a week apart, as the struck slab and then as the one below it. With creep the second is the heavier, because in the intervening week the slab above has sunk onto the backprops; with slack, the first. Materials

The backprop that takes load nobody gave it

The arithmetic of propped construction says a backprop put in snug carries nothing until the next floor is cast, and then only a share of it. Count the props as the springs they are and that barely changes, because a steel prop is about seven times stiffer than the slab per square metre. Two other things change it. The young slab just struck creeps onto its backprops within a day and hands a quarter of a slab's weight down to the one below, which then carries 1.71 times its weight rather than 1.5. And a millimetre of slack at installation — about half the slab's own deflection — gives the struck slab back everything a backprop was meant to save.

6 figures · Maturity The film over a bored shaft has an island that floats. Contours of Prandtl's stress function — a soap film blown over the section — for a 50 mm shaft with a 10 mm hole 15.0 mm off its axis, at the same pressure; the shear stress is the film's slope. Left, the bored shaft: the film is pinned to the outline and to a flat island over the hole, shaded, which floats at 55 per cent of the solid shaft's peak height, where the film's pull round its edge balances the pressure on it. The shaft keeps 93 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.70 times the solid shaft's surface stress under the same torque. Right, the same shaft slit along its length from the hole to the surface: the island is pinned at the outline's height, the film sags into it, and the shaft keeps 67 per cent, its largest stress 2.81 times. Sections and stress

The island the film floats

Bore a hole along a shaft, off centre for a lubrication passage or a cable, and the soap film that solves its torsion has a second edge: a flat island over the hole whose height nobody knows in advance. Let it float where the film's pull balances the pressure on it, and the shaft keeps nearly all its stiffness and doubles the stress beside the hole, as a hole in any shear field does. Pin it at the outline's height instead and the same shaft has been slit to its hole, and loses a third of its stiffness. The island's height is the whole difference between a bore and a cut.

5 figures · Membrane analogy Two theories, one plate, and a column beside it. The buckling stress of a long plate of an austenitic stainless steel of proof stress 230 N/mm² and modulus 200,000 (Ramberg–Osgood exponent 6), simply supported on its edges and compressed along them, as a share of the proof stress, against its slenderness λ̄p, the square root of the proof stress over the elastic plate's buckling stress. Flow theory: 2.30 at λ̄p = 0.6 and 0.86 at 1. Deformation theory: 0.99 and 0.71. A column of the same material at the same slenderness, by its tangent modulus: 0.76 and 0.58. Dashed, the sharp envelope, the lesser of the proof stress and the elastic stress. Flow theory runs far above the proof stress on stocky plates — 5.29 at 0.4 — which no plate test has reached. Stability

The plate the wrong theory gets right

A column of stainless steel buckles by its tangent modulus, the one stiffness its rounded curve has left. A plate needs three — along the load, across it, and in twist — and the two classical theories of plasticity give it different ones. Flow theory, the one whose physics is right, keeps most of the plate's elastic stiffness and puts a stocky stainless plate's buckling stress at twice its proof stress. Deformation theory, whose physics is wrong, softens every direction and agrees with the tests. Two thirds of the gap is the twist alone, and either way a rounded curve costs a plate much less than it costs a column.

6 figures · Inelastic buckling The pier reads the difference between the two spans' temperatures. Contours of the pier force between two 100 m spans at a sag of 2.00 per cent with the deck hung on one, the tendons at 1,100 N/mm² when finished, on a pier of 200,000 kN/m, every half meganewton, over the warming of the hung span's tendons (across) and of the bare tendon's (up) since they were cut. With neither warmer the pier carries 3,080 kN; warming the bare tendon 30 °C alone gives 4,126 kN, the hung span alone 2,342 kN, and both together 3,392 kN. The contours are straight and parallel: the pier gains 35 kN for each degree the bare tendon warms and loses 25 for each degree the hung span does, so a warming both share moves it only 30 per cent as far as the same warming on the bare tendon alone, and what the pier feels is mostly how much warmer one span's strand is than the other's. Structural form

The strand in the sun beside the strand in the shade

Halfway through hanging a stressed ribbon, one span is a sagging cable full of segments and the next is bare tendon stretched nearly straight, and the pier between them carries the difference of their pulls. A warm day takes tension off both — and at the flat sag a ribbon is laid at, the hung span is so nearly all stretch that it loses three quarters of what the bare one does. What moves the pier is not the weather but the shade: strand in the sun beside strand under concrete, 35 kN on the pier for every degree between them. No sun slackens the bare tendon. The hour does change something else, which is the level the joints are cast at.

6 figures · Stressed ribbon One ratio decides how much of a row works. The share of n single screws' peaks a row reaches, for rows of 10, 20 and 30 8 mm screws at 45° to the joint, 56 mm apart along the grain, in two 100 mm softwood members of modulus 11,500 N/mm² with 40, 80 and 160 mm of member depth each, and threads peaking at 0.5, 1, 1.5 and 2.5 mm, against one ratio: the slip difference the members' stretch puts between the end screw and the middle one at the row's peak, PL(1/EA₁ + 1/EA₂)/8, over the width of a single screw's curve above nine tenths of its peak. Thirty-six rows fall on one curve: every row with the ratio below 0.3 reaches at least 1.00 of its screws' sum; at a ratio of 1 they reach about 0.96; and the most stretched, at 3.5, 0.72. The colour is the thread's peak slip; a brittler thread has a narrower plateau and lands further right for the same row. Connections

Twenty screws that peak together

A leaning screw reaches its peak at two millimetres of slip and then lets go, so a row of twenty along the grain looks like the long bolted joint again — the end screws past their peak and falling while the middle ones have barely started. Follow the row slip by slip and it is not: in a member of ordinary depth twenty leaning screws carry 98 per cent of twenty times one, against the 74 per cent the effective-number rule allows. What decides it is one ratio — the slip the timber's stretch spreads along the row against the width of a screw's peak — and the rule is right only for a brittle thread in a thin member.

6 figures · Dowel yield What leaves the interior panel lands in the span. A frame of two 8 m bays and two 4 m storeys with fixed bases, 457 mm beams and 254 mm columns, with 20 kN/m on the left bay's beams only, against the stiffness of every beam-to-column joint as a multiple of the beam's EI/L, on a logarithmic scale. The shear across the floor interior joint's column-web panel (solid): 154 kN with rigid joints, 146 at EN 1993-1-8's rigid limit of 25 EI/L, 104 at 3.0 EI/L, 23 MN·m/rad, 63 at EI/L. The loaded beam's largest sagging moment (dashed), in kN·m: 71, 76, 102 and 126, against 160 for a simply supported span. The faint lines are EN 1993-1-8's boundaries for a frame that sways: nominally pinned below 0.5 EI/L, rigid above 25. Internal forces

The panel a soft joint empties into the span

Under a load on one bay, an interior column's web panel carries the difference between the two beams' moments. Make the beam-to-column joints semi-rigid and the panel is relieved — by 5 per cent at the stiffness EN 1993-1-8 still calls rigid, by a third at three times the beam's EI/L — because a softer joint takes less moment. The moment it does not take lands at mid-span, where the beam was not designed for it, and in sway it lands in the columns. And soften only some of a frame's joints, and the sway goes looking for the ones still stiff: their panels carry up to a quarter more than they would in an all-rigid frame.

6 figures · Corner moment