Structural Engineering & Statics
where the load goes
The collection

Every essay — page 18

Essays 409 to 432 of 728, in the same order.
Nine-tenths of a tightening torque stretches nothing. Where the torque applied to the nut of an M20 grade 10.9 bolt goes, against the coefficient of friction in its thread and under its nut, taken as equal. The bottom band is the thread's lead — the only part of the work that stretches the bolt — the middle band is friction in the thread and the top band is friction under the nut. At μ = 0.14 the lead takes 11% of the torque, the thread 39% and the nut face 50%. At μ = 0.06 the lead's share is 22% and at 0.24 it is 6%, so a coefficient nobody measured decides how much of a specified torque arrives in the bolt as preload. Connections

The torque that goes into the thread

A preload specified as a torque is a preload specified through two coefficients of friction that nobody measures. Nine-tenths of the torque on a bolt is spent turning against its own thread and the face of its nut, so a change in the grease moves the clamping force by half — and the one method that escapes it does so by yielding the bolt on purpose.

7 figures · Slip-critical The compression is under the flange, not at the edge of the plate. A 500 × 400 mm base plate, 20 mm thick, under a 260 mm column carrying 300 kN and 120 kN·m, with holding-down bolts 60 mm from the tension edge. A plate this thick can deliver bearing only 43 mm past the compressed flange, so the compression sits under that flange, 124 mm from the column's centre, and the lever arm to the bolts is 314 mm: the bolts carry 264 kN and the flange zone 564 kN. The dashed block is the rigid-plate answer, 54 mm deep at the plate's far edge with its resultant 223 mm out, which needs only 128 kN in the bolts. Both satisfy equilibrium; only one is a plate that can carry the pressure where it is drawn. Connections

The compression that stays under the flange

Put a moment on a base plate and a pressure-block calculation pushes the compression out to the plate's edge, where the lever arm to the holding-down bolts is longest. A plate that is not rigid cannot put it there. The compression stays within a few tens of millimetres of the compressed flange, the lever arm shrinks, and the bolts carry twice what the pressure block said.

7 figures · Base plate Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 200 mm, carrying a reaction of 300 kN through three bolts, as utilisations, for the end as coped and with three repairs: an 8 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.46, shear 0.41, tear-out 0.57 and local buckling 0.46. The doubler thickens the web, which is most of what a coped tee is, and lowers all four; the edge plate gives the tee back a flange and lowers only flexure and buckling, leaving shear and tear-out exactly where they were. What governs: as coped, tear-out at 0.57; doubler, tear-out at 0.30; edge stiffener, tear-out at 0.57; both, tear-out at 0.30. Connections

The repair that fixes the wrong check

A coped beam end that fails its checks is usually repaired by welding a plate along the edge the cope left, because the cope removed a flange and the plate puts one back. On ordinary proportions that repair restores the check that was not failing. The check that governs is carried by the web, and only a plate on the web reaches it.

7 figures · Coped beam The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing. Dynamics

A wall that is allowed to lift

A block that rocks inherits everything that decides whether it survives — a restoring moment set by its weight and shape that falls as it leans, and a loss of energy set by its proportions at each landing. Put a tendon through a wall and yielding bars across its base and both become design quantities, and the ratio between them decides whether the wall comes home.

8 figures · Rocking Two separate checks, and the block that is neither. Amplification of the steady displacement at a machine 2.0 m above the base of a 150 tonne block 5.0 m across and 2.0 m deep, against forcing frequency. Checked separately, the block sways at 11.2 Hz with 29 per cent of critical damping and rocks at 15.0 Hz with 13 per cent, and the rocking check's peak is 3.99. Because its mass sits above its base the two are one system, with modes at 9.9 Hz and 20.9 Hz — one below both checks and one above — and the block's own peak is 2.49 at 9.7 Hz. Dynamics

The frequency below both checks

A machine block is checked for sway and for rocking as if they were two oscillators, each with its own spring, its own radiation damping and its own natural frequency. A block whose mass sits above its base is one oscillator with two modes, and neither of them is a sway or a rocking — one sits below both checks, one above both, and the lower one is where the machine resonates.

8 figures · Vibration isolation However stiff the ties, a free line stops short. The first three frequencies of stays 180, 150 and 120 m long joined by cross-ties, against the stiffness of each tie from 1.0 kN/m to 1000 MN/m, with the line free at its ends and, dashed, anchored to the deck at both ends with the same stiffness. With the line free the first frequency rises from 0.73 Hz and levels off at 0.82 Hz, however stiff the ties are made — short of the 1.01 Hz of the 120 m stay, which a free line cannot pass. Anchored, the first frequency reaches 1.09 Hz. At the 6.0 MN/m marked, the free line gives 0.82 Hz and the anchored line 1.09 Hz. Dynamics

The line of ties that stops short

Cross-ties are the one intervention on a stay cable that changes its frequencies rather than damping them, and the usual account says they lift the stays out of the range that excites them. A line of ties that is not anchored to anything cannot lift the first frequency above the shortest stay's own, however stiff the ties are made. What lifts it is carrying the line to the deck.

8 figures · Cable dynamics One collision, four contact laws, one impulse. Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The elastic linear spring peaks at 19.7 MN and lasts 58 ms; the Hertz law peaks at 22.0 MN over 61 ms. Set to a restitution of 0.65, the spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN, a tension two faces in contact cannot carry, while the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 instead. The areas under the curves are the impulses: 600 kN·s for the spring and dashpot, which is what momentum requires at 0.65, 646 for the damped Hertz law, and 727 for both elastic laws. Dynamics

The force that belongs to the model

When two buildings meet, momentum decides what each of them feels, and no contact law can change it. What the contact law decides is the force, and the force it reports is the contact stiffness somebody assumed, raised to a power. Four laws and a hundredfold change of stiffness move the buildings by a few per cent and the force by a factor of eight.

7 figures · Pounding The rates a seated crowd makes worse. The stand's steady rms acceleration under 40 people jumping, against the rate they jump at, for a stand of 30 t modal mass at 6 Hz with 2 per cent damping, empty and with 15 t of spectators sitting on it. Empty, the worst rate is 3.00 Hz, where the second harmonic of the jump lands on the stand's own frequency, and the stand reaches 2.54 m/s², 25.9 per cent of gravity. Occupied, the worst anywhere from 1.5 to 3.5 Hz is 0.49 m/s² at 3.50 Hz. But from 1.64 to 2.32 Hz, shaded, the occupied stand responds more than the empty one — 1.6 times as much at 2.00 Hz. Dynamics

The crowd that is also the structure

A crowd jumping to music loads a grandstand with harmonics several times those of walking, and nothing about their size is measured: they follow from a pulse that has to average one body weight. But the people who jump arrive with the people who sit, and the seated crowd is mass, stiffness and damping bolted to the stand. It lowers the worst case by a factor of five and makes the most common jumping rates worse.

7 figures · Floor vibration The frequencies at which a moving deck makes a stay grow. Stability of the first mode of a 120 m stay at 3,500 kN inclined at 25°, whose own frequency is 1.01 Hz, when the deck at its anchorage moves at a frequency Ω against the stay's ω, plotted against the swing in tension the movement produces. Inside each shaded region a small disturbance grows. Each region's tip is at a swing of four times the damping ratio — 0.40%, 2.0% and 4.0% for damping of 0.10%, 0.50% and 1.0% — and it widens as the swing grows, to a deck between 1.970 and 2.030 times the stay's frequency at a 6.0% swing with 0.10% damping. With no damping, dashed, the region reaches down to no swing at all. A deck moving ±5, ±10 and ±20 mm vertically swings the tension by 0.70%, 1.4% and 2.8%. At ±10 mm and exactly twice the stay's frequency, marked, the stay grows with damping of 0.10% and settles with 0.50% and 1.0%. Dynamics

The stay shaken along its own length

A deck that moves at a stay's anchorage pushes nothing across the stay. It stretches the stay along its own line and lets it go again, so the tension swings, and at twice the stay's frequency that swing drives the stay with no sideways force at all. Whether the swing grows is one comparison — a quarter of the tension swing against the damping ratio — and it is a comparison the capped damper wins.

7 figures · Cable dynamics A link that keeps two buildings apart has made them one. The largest closing movement between a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, and each building's largest displacement, against the size of a viscous damper joining them across the gap, from 0.01 MN·s/m to 541.3 MN·s/m. With no link they close by 515 mm, and the stiffer building moves 220 mm and the softer 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm. The least that keeps the 50 mm gap is 4.8 MN·s/m, where the stiffer building moves 269 mm and the softer 280 mm. At the largest link the two move together, 281 mm and 281 mm. Dynamics

The damper that ends up as a joint

A damper across the gap between two buildings acts on exactly the motion the gap is sized for, and it can be sized for two different things. The size that takes the most energy out of the pair still lets the buildings collide. The size that keeps them apart has nearly stopped moving: it has joined them into one building, and the stiffer of the two pays for it in drift.

7 figures · Pounding A filler sets the force, until it runs out of thickness. Contact force against the movement since first touch for one collision between 500 t and 300 t on a 40 mm filler crushing at 4 MPa over 5 m², 60 per cent of it crushable, at closing speeds of 1.00, 1.94 and 2.85 m/s. The filler loads elastically, then crushes at 20.0 MN for as long as it has thickness to give. At 1.00 m/s it crushes 4 mm and the force never passes 20.0 MN; at 1.94 m/s it crushes 17 mm and the force never passes 20.0 MN; at 2.85 m/s it crushes all 24 mm it can and the faces meet through it, peaking at 23.7 MN. Dashed, the bare contact at 2.85 m/s peaks at 35.0 MN. Dynamics

The force a filler can promise

A crushable filler in the gap between two buildings replaces a contact stiffness nobody knows with a crush strength somebody chose, and the force of a collision becomes that strength — for as long as the filler has thickness left to crush. Energy decides how much thickness that has to be, and in a gap sized for buildings that were never meant to meet it is not much.

6 figures · Pounding After a yield, the prestress left does not remember the prestress put in. The prestress left in the tendon of a 2.0 × 8.0 m rocking wall of 400 kN once it has rocked to each rotation and come back upright, for a tendon yielding at 1,370 kN and stiffening by 20 kN a millimetre as the base opens, prestressed to 300 kN, 600 kN and 900 kN. Each keeps all its prestress until it yields — at 53.5 mrad, 38.5 mrad and 23.5 mrad — and then loses 20 kN for every further milliradian, so past 53.5 mrad the three lines are one: the yield force less the stiffness times the stretch. Every one of them is slack at upright past 68.5 mrad. With bars of 400 kN the wall re-centres only while 400 kN remains, dashed, and it keeps that only up to 48.5 mrad, whatever it was prestressed to. Dynamics

The tendon that forgets its prestress

A rocking wall comes home because a tendon pulls it, and the tendon must not yield — yet the rotations that test the wall are exactly the ones that stretch it. Past its yield, the force a tendon keeps is its yield force less its stiffness times the stretch, whatever it was prestressed to, so the guarantee a design needs is a limit on rotation, and more prestress only reaches that limit sooner.

7 figures · Rocking Set into the ground, the block is quieter only while its sides radiate. Displacement at the top per kilonewton of machine force for a 150 tonne block 3.2 m across and 3.2 m deep, on a logarithmic scale: on the surface, dashed, and with 1.60 m of it set into the ground, first with the side soil radiating as the side-layer model gives it and then with no radiation from the sides at all, dashed. On the surface it peaks at 191.8 µm per kN at 5.5 Hz. Embedded and radiating, it peaks at 11.0 µm per kN at 8.2 Hz. With the same embedment and silent sides the resonance moves up to 8.4 Hz and peaks at 168.7 µm per kN, because the lower mode's damping falls from 3.2 per cent to 1.6 per cent. At 5.5 Hz the three give 191.1, 8.0 and 9.5 µm per kN. Dynamics

The damping that comes through the sides

A machine block set into the ground is held at its sides as well as its base, and the side soil does two things at once. It lifts the height at which the ground's resistance acts toward the block's centre of mass, which weakens the coupling between sway and rocking without ever removing it. And it radiates, which is what actually flattens the resonance — so the quiet an embedded block promises rests on the stiffness of backfill nobody measured.

7 figures · Vibration isolation The whole deflected shape, found from the members' changes of length. A Pratt truss of eight panels at a depth of 1, carrying 15 kN at each top joint, at EA = 1, drawn as built and in its deflected shape, with every movement magnified the same number of times. The shape comes from Williot's construction and Mohr's correction: every joint's movement from the members' extensions alone, less the rigid rotation the supports forbid. The mid-span joint L4 moves down 2019.41 units, and the dots at every joint are a stiffness solution sharing none of that arithmetic, agreeing to 1e-14 of the largest movement. One construction gives all sixteen joints at once. Deflection

The drawing that is right except for a rotation

Williot's construction finds every joint of a truss from its members' changes of length alone, in one drawing — and puts the roller six thousand units off its support. The error is one rigid rotation, Mohr's diagram takes it away, and a drawing started from the member symmetry holds still never makes it.

7 figures · Truss deflection Soft in shear, the wall gives up moment to the span. The bending moment along a 300 × 600 rectangle fixed at its left end and propped at its right, under a uniform load, per unit load and span, at span-to-depth ratios of 4, 2, 1, and for the same beam treated as rigid in shear, dashed. Rigid in shear the wall carries 0.125 wL² and the prop 0.375 of the load. Counting the shear it deforms by, the wall moment falls to 0.119 wL² at 4, 0.105 wL² at 2, 0.070 wL² at 1, and the prop's share rises to 0.381, 0.395, 0.430. The load has not changed and the beam is no weaker: its forces have moved, because in a redundant beam they come from how it deforms. Deflection

The shear that moves the moments

In a statically determinate beam, shear deformation adds movement and changes no force. In a redundant one the forces come from how the beam deforms, so a stubby beam soft in shear carries less moment at its wall and more in its span — and the carry-over factor, the one half every hand method passes along, falls to nothing at φ = 2 and then changes sign.

7 figures · Shear deflection A fixed-ended member's end moments are the edge stresses of a column. A prismatic member, fixed at both ends under a point load 0.333 of the span from its left end, drawn three ways. At the top, the member. In the middle, the column the analogy puts in its place: a strip as long as the member and as wide at each point as 1/EI there, with its centroid, the elastic centre, 0.500 of the span from the left. At the bottom, the simply supported moment diagram, dashed, which is the load on that column; the straight line, which is the stress that load produces, P/A + M(x − x̄)/I; and the member's own moment diagram, which is the difference. The end moments are the column's edge stresses: 0.148 PL at the left and 0.074 at the right, which are Pab²/L² and Pa²b/L², and the largest sagging moment is 0.099 PL. Deflection

The fixed-end moment is a column stress

A fixed end forbids a change of slope and a deviation, so on a member fixed at both ends Mohr's two theorems both have the answer nothing. Written with 1/EI as a width, those two conditions are the P/A + My/I of a short column under an eccentric load — the end moments are its edge stresses, and a haunched member's are read from where its elastic centre has moved.

6 figures · Moment-area The largest cycle in the record is not between two neighbouring reversals. A 26-second stress record at the mid-span detail of a 20 m road bridge, as seven vehicles cross it and the deck rings at 4 Hz after each: a heavy lorry at 1.0 s, a car at 5.2 s, a van at 8.0 s, an ordinary lorry at 12.0 s, a second lorry meeting it at 12.3 s, a car at 17.0 s, the heaviest vehicle of the record at 20.0 s. With a reversal threshold of 0.5 N/mm² the record has 185 reversals, and rainflow pairs them into 92 cycles. The four largest loops are marked, each by a line joining the two reversals that close it: 74.3 N/mm² between 20.6 s and 21.2 s; 45.1 N/mm² between 1.5 s and 2.2 s; 38.5 N/mm² between 12.6 s and 13.5 s; 16.5 N/mm² between 8.5 s and 8.9 s. The largest is the record's whole range, from its lowest point to its highest. Counted between successive reversals instead, the largest range anywhere in the record is 38.1 N/mm², because the deck's ringing puts reversals on every rise and every fall. Materials

Which reversal closes the loop

A strain-gauge record is a wiggle, not a list of cycles, and its damage depends on how its reversals are paired. Rainflow pairs each one with the reversal that closes its hysteresis loop. Counting the ranges between neighbours breaks every lorry's cycle into pieces, and on one bridge record it gives a detail 424 years where rainflow gives it 34.

5 figures · Fatigue A room's fire has a peak and an end, and the furnace has neither. Gas temperature in a compartment with 200 MJ/m² of fire load per square metre of its enclosing surface, linings of thermal inertia 1160 J/m²s½K and a medium growth rate, for opening factors of 0.02, 0.04, 0.08, 0.14 m½, against the standard furnace curve, dashed. At 0.02 the fire peaks at 841 °C after 120 minutes and is back at ambient by 435; at 0.04 the fire peaks at 944 °C after 60 minutes and is back at ambient by 171; at 0.08 the fire peaks at 1048 °C after 30 minutes and is back at ambient by 92; at 0.14 the fire peaks at 900 °C after 20 minutes and is back at ambient by 37, having used its fuel before the growth limit, so the fuel rather than the air decided it. A larger opening lets more air in: the fire burns hotter and gets through its fuel sooner. The furnace curve is still rising when every one of them is out. Materials

The fire that goes out

The furnace curve rises for ever. A room's fire has a peak and an end, set by its window, its walls and its fuel, and a bigger window makes it hotter and shorter — which is worse for bare steel and better for protected steel. The protected member is hottest half an hour after the fire has started to die.

6 figures · Fire Half of the creep is the member drying out. The creep coefficient of a concrete of mean strength 38 N/mm² loaded at 28 days, in a member of notional size 150 mm, against time since loading: exposed to air at 50 per cent relative humidity, and the same member sealed so that it cannot dry. After a year the exposed member's coefficient is 1.93 and the sealed member's 0.81; after 50 years they are 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member's final creep, is what drying adds. The sealed member's coefficient has no size in it; the drying part is where the member comes in. Materials

The creep that belongs to the member

A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether.

5 figures · Creep The joint yields first and the span takes the rest. The end and span moments of a beam of span 9 m and flexural rigidity 61,700 kN·m², with a plastic moment of 522 kN·m, on joints of rotational stiffness 41,133 kN·m/rad, 6.00 EI/L, and resistance 261 kN·m, as a uniform load rises to collapse. While elastic the joints carry 0.75 of the fixed-end moment. The joints reach their resistance first, at 51.6 kN/m, and from there every further increment of load goes to the span, which reaches 522 kN·m at 77.3 kN/m. The collapse load is 8(Mp + Mj)/L² = 77.3 kN/m, between the 51.6 at which the beam would collapse on pins and the 103.1 it would reach on rigid full-strength ends, and it contains the joint's strength and not its stiffness. Connections

The joint that has to keep turning

A beam on partial-strength joints collapses at 8(Mp + Mj)/L² however stiff the joints are. Stiffness decides only which yields first, and a joint that yields first has to go on rotating at full moment until the span catches up. A weak joint on a stiff connection has the most turning to do — more than a rigid full-strength one.

5 figures · Joint classification Made from one side, the throat misses the load and its root takes the difference. Two plates 20 mm thick joined by a partial-penetration butt weld with 10 mm of throat, pulled apart by a force on their mid-thickness, with the stress across the throat drawn beside each joint as a multiple of the mean. Above, welded from one side: the throat runs from the top face down to 10 mm, so the force passes 5.0 mm below its centroid. The moment adds tension at the root and removes it at the face: the root carries 4.00 times the mean stress and the face −2.00, which is compression. The unfused 10 mm below the root is a notch, and it is on the tension side. Below, the same 10 mm of throat split between the two faces: the force passes through its centroid and every part of it carries the mean stress. Connections

The throat that misses the load

A partial-penetration butt weld's throat faces the load squarely, so a millimetre of it carries more than a fillet's, and in every grade what limits it is a cap on normal stress rather than the directional criterion. Made from one side, the throat sits off the plates' line of action. At half penetration it keeps a quarter of that capacity, and the stress the eccentricity adds lands on the root, which is a crack.

5 figures · Weld strength A plate welded to a loaded beam carries only the load that comes after it. The bending stress across the depth of a 457 mm beam coped 200 mm long and 50 mm deep, carrying a reaction of 300 kN, at the end of the cope, repaired with a doubler 8 mm thick on the web. With 60 per cent of the reaction already on the beam when the plate is welded, the original steel carries that share on its coped section and the rest on the repaired one. At the web's free edge it reaches 137 N/mm², against 164 as coped and 97 if the plate had been there from the start. The plate carries only the later load, 39 N/mm² at its outer edge. The repair has removed 40 per cent of the stress it would have removed on an unloaded beam. Connections

The plate that arrives after the load

A repair plate welded to a beam already in use carries only the load that arrives after its weld has cooled. The elastic checks remember that order — a doubler welded under 60 per cent of the reaction removes 40 per cent of the stress it would remove from an unloaded beam — while tear-out, a plastic mechanism, gets the whole of the repair. Propping the beam is what hands the plate the load already there.

5 figures · Coped beam Four forces pair off, and the line joining the pairs carries both resultants. A beam 8 m long held by the vertical link at the left end, the strut at 6 m, the horizontal link at the right end, under a load of 10 kN at 3 m inclined at −60.0°. Pairing the load with line A: the two cross at P, (0.00, 5.20) m, and lines B and C cross at Q, (6.00, 0.00). The resultant of the first pair passes through P and that of the second through Q, and since they balance each other both lie on PQ, dashed. The force polygon on the right is the load, then A, B and C, closing where it began, with PQ's direction as the diagonal that splits it into two triangles. The forces are A 4.33 kN along its line, B 6.12 kN along its line, C 0.67 kN against its line, as the three equations of equilibrium give them. Equilibrium

The line that pairs four forces

Three forces in equilibrium meet at a point; four need not. But they pair off. The resultant of two passes through the point where their lines cross, the resultant of the other two through theirs, and the two resultants must share the line joining those points. Culmann's line turns a four-force body into two triangles — and the method of sections into a drawing.

5 figures · Graphic statics The pole, the strings, and the resultant of any number of forces. Five downward loads on a span of 10 m — 30 kN at 1.5 m, 20 kN at 3.5 m, 45 kN at 5.0 m, 25 kN at 7.0 m, 35 kN at 8.5 m — adding to 155.0 kN. On the right, the loads laid end to end down one line, with a pole 60.0 kN to the left of it and a ray drawn to every division between them. On the left, the funicular polygon: each segment parallel to the ray of the loads it has passed, so the shape is the one a string carrying these loads would hang in. The first and last strings are extended until they cross, at 5.24 m, and that crossing is where the 155.0 kN resultant acts — the same station the moment sum Σ P x / Σ P gives, 5.24 m, reached with no pole in it at all. Equilibrium

The pole decides the drawing, not the answer

Five forces will not pair off the way four do. They need a point that is nowhere on the structure — chosen freely, by whoever is holding the pencil — and the string of lines it generates. Every choice draws a different polygon and finds the same resultant, and the shape it draws turns out to be the beam's bending moment diagram.

6 figures · Graphic statics