Structural Engineering & Statics
where the load goes
Theme

One support too many — page 8

Essays 169 to 186 of 186 on this thread, in the same order.
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 loaded end stops; the unloaded one takes over. The moments at the four floor beam ends of the frame of two 8 m bays and two 4 m storeys, its beam ends joined to the columns by end plates of 3.0 EI/L initial stiffness and 120 kN·m resistance, under λ times 20 kN/m on the left bay and 20 kN at each floor, against λ: on the softening joints (solid) and on joints that keep their initial stiffness (dashed). At λ = 1 the loaded beam's inner end carries 78 kN·m and the unloaded beam's inner end, across the same column, 13. The loaded end reaches its resistance at λ = 2.69; at λ = 3 the four ends carry 89, 120, 81, 88 kN·m, against 97, 235, 39, 68 on stiff joints. The unloaded inner end has gone from 13 to 81, 6.2 times as much for three times the load. Internal forces

The joint that softens under its own load

A bolted end plate is stiff until its moment passes two thirds of its resistance and then softens until, at its resistance, it is a third as stiff. Put such joints at every beam end of a frame under a pattern load and a push, and the most loaded joint softens first and hands its share across the column to the joint on the other side, which carries six times as much for three times the load. The redistribution grows as the frame approaches its strength. Every joint ends at its own stiffness, so the single secant stiffness an elastic analysis is allowed asks the loaded joint for half as much again as it can carry, and gives the exterior panel three quarters of what it gets.

 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.

 Allowed to yield, the damaged truss is hardly lighter. The steel in an eight-panel Pratt truss with a counter-diagonal in each interior panel, loaded by 10 at every bottom joint, sized to survive the loss of any one interior diagonal, against the share of the full load it must carry with the diagonal gone: checked elastically (solid) and by the lower-bound theorem, every member allowed to yield (dashed); the intact design's 1,200 as a level line. At a third, 1,221 elastic and 1,206.7 plastic; at two thirds, 1,304 elastic and 1,296.7 plastic; at the full load, 1,555 elastic and 1,545.0 plastic; at four thirds, 2,074 elastic and 2,060.0 plastic. The plastic design saves 7.4 units at two thirds of the load out of the 104 that robustness costs. Deflection

The section that yielding cannot reach

A truss that loses a member in an accident is allowed to yield while it finds another way to stand, so a counter-braced truss sized by the lower-bound theorem ought to need far less steel than one sized elastically. It needs 1,296.7 units against 1,304 — seven of the 104 that robustness costs. One diagonal gone, the damaged panel is determinate at its own section, and no redistribution can change a force a cut decides; yielding can reach only the verticals. What does move the price is a counter that buckles: good for half its tension in compression, it adds 120 units, more than robustness did.

 The order of building decides which supports keep the secondary moment. The secondary moment at each interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built and stressed from one end, from both ends towards a closure bay, from the middle outwards, and all at once. From one end: 161, 175, 160, 219 kN·m; from both ends: 161, 172, 172, 161 kN·m; from the middle out: 215, 120, 120, 215 kN·m; all at once: 344, 258, 258, 344 kN·m. Internal forces

The beam built towards its own middle

A continuous prestressed beam built one bay at a time from one end keeps about half the secondary moment a beam stressed at once would carry, and most of it at the end where it was finished. Built from both ends towards a closure bay, it is symmetric, and the closure's supports keep only a little more than the rest: 172 kN·m against 161. Built from the middle outwards, the first interior supports — the ones the load hogs hardest — keep 215, 63 per cent of the at-once value, for any length of beam. The order of construction is a design variable. Creep then takes back most of the choice.

 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.

 Closing late beats jacking again on the same date. For the 60 m concrete rib rising 6 m under 60 kN/m, struck at 28 days, on ground that spreads 8 mm at once and as much again as it consolidates, 90 per cent in 10 years: the springing moment after a century, against a date in the arch's first years — left three-hinged and closed on that date (solid), or fixed at striking and jacked back to no moment on that date (dashed); dotted, fixed and jacked at striking only, −173. At three months, −124 against −140; at a year, −84 against −117; at ten years, −15 against −76. The difference is the memory of the years before: the jacked arch's early restraint goes on relaxing after the second jacking, and the open arch had none. Deflection

The arch that waits with its hinges open

A concrete arch fixed on the day it is struck has to restrain every millimetre its rib creeps and shrinks and its ground settles for the rest of its life, and jacking it again a year later cannot make it forget the year before. Struck as a three-hinged arch and left that way, it takes the fastest part of every movement for nothing, because a determinate arch has no moment to spend on a change of length. Closed at a year it ends the century with half the springing moment of the arch fixed at striking, and a third less than the arch jacked again on the same date. The price is a crown that has dropped 78 mm while it waited, and a rib that is no longer the shape of its own thrust.

 A jack set for the first day is wrong by the last. The secondary moment over the first interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built from both ends towards a closure bay, against the creep coefficient: not jacked; jacked at closure by the whole of what the order took, 183 kN·m there; jacked by 0.60 of it, so as to reach the at-once moment at a creep coefficient of 2.0; and by 0.89, so that the shortfall on the day and the overshoot at the end are equal; dotted, stressed at once, 344. On the day and at 2.0: not jacked, 161 and 302; jacked by the whole, 344 and 372; for the end, 271 and 344; balanced, 324 and 365. Internal forces

The jack that creep carries past the mark

A continuous beam built from both ends towards a closure bay keeps less secondary moment at its supports than the same beam stressed all at once, and creep gives back three-quarters of the difference over the years. Bridge builders do not always wait: before the closure is cast, the supports can be lowered on jacks so that the beam starts with the moments it would have had. Jacked by the whole difference, it does start there — and creep, which goes on returning the staged deficit while it relaxes the jacked correction, carries it past them, by a seventh of what was jacked. Jacked by 0.89 of the difference, it starts a ninth short and ends a ninth over. And it takes a jack at every interior support: with fewer, the moment goes to the wrong places.

 The flatter the ribbon, the more the casting temperature matters. For two 100 m spans of a stressed ribbon under 35 kN/m on a pier of 200,000 kN/m, one span's joints cast 20 °C warmer than the other's: the force left on the pier as a share of the force a crowd of 20 kN/m on one span puts there, against the ribbon's sag as a percentage of its span. At a sag of 1 per cent the casting leaves 82 per cent of a crowd (1,902 kN against 2,316); at 2 per cent, 14 per cent; at 3, 3 per cent; at 4, 1 per cent. Structural form

The span cast in the afternoon

A stressed ribbon's joints are cast on whatever day the programme reaches them, and a span cast on a warm afternoon sets around strand that has lengthened in the sun. It cools that night as a concrete band eleven times stiffer than the strand, and the band does not stretch back as the strand would have. On a ribbon with a sag of 2 per cent, a span cast 20 °C warmer than its neighbour ends 72 mm higher than it was set and leaves 713 kN on the pier between them — a seventh of what a crowd on one span puts there. On a flatter ribbon the same afternoon costs more than the crowd. The difference between the two spans is the strand's flexibility against the band's, and only the elastic part of it differs; the part that comes from the sag is the same for both.

 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.

 Where the knees begin to alternate. The shakedown factor (solid) and collapse factor (thin) of a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, on pinned bases, with 160 kN of gravity that comes and goes and a 1,000 kN roof carried at its knees, against the peak of a wind that reverses, first-order (dashed) and amplified. First-order, the shakedown and collapse factors are one line until the knees alternate, from a wind of 110 kN; amplified, from 90 kN. At 60 kN: shakedown 0.727 first-order and 0.692 amplified, collapse 0.727 and 0.692. At 120 kN: shakedown 0.533 first-order and 0.488 amplified, collapse 0.571 and 0.539. At 200 kN: shakedown 0.348 first-order and 0.326 amplified, collapse 0.400 and 0.366. Materials

The roof that makes the wind worse every storm

A portal frame's shakedown check is made on its elastic moments, and the elastic moments a heavy roof sways are larger than first-order analysis says: each storm's wind is amplified by 1/(1 − N/Ncr), on its way out and on its way back. On pinned bases under a 1,000 kN roof that takes the wind at which the knees start to alternate from 110 kN down to 90, and the shakedown load down by 4.8 per cent where gravity leads and 8.4 where the wind does. Fixed bases are worth more under a heavy roof than without one, but by half again, not twice.

 The middle reaction is a dimension, read off in millimetres. The middle reaction under dead load of a Pratt truss continuous over two 24.0 m spans, 3.0 m deep, carrying 60 kN of dead load at each interior bottom joint, against the height of its middle bearing above the level drawn, cut to the continuous list (solid) and as two simple spans (dashed). At the drawn level the continuous list lands with 572 kN, the analysis's value, and the simple-span list with 480 kN, the two spans' own share. Every millimetre the bearing stands high adds 2.6 kN: 10 mm high, 598 kN; 10 mm low, 545; the simple-span list's 92 kN shortfall is the bearing 35 mm low. Deflection

The pier's share is cut into the members

A truss continuous over a middle support has one reaction that statics cannot find, and the lengths its members are cut to decide it. Cambered to land level under its dead load on three bearings, a two-span truss puts 572 kN on its pier; cambered span by span, as two simple trusses, the same members on the same bearings put 480 kN there and carry the dead load with no moment over the pier at all. The list chooses the moment diagram. And a millimetre of shop tolerance on every member moves the pier's share as much as a bearing set fourteen millimetres out of level, so a reaction the design depends on has to be set by a jack rather than by a drawing.

 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.

 The column line's sum agrees with every analysis, because it cannot see the beams. The roof sway of a 4-bay, 10-storey concrete frame, 8 m bays and 3.5 m storeys, 600 mm columns and 400 × 700 mm beams, 40 kN/m on every beam and 40 kN of wind at every floor, against the beams' stiffness as a share of their gross value, from the whole frame (line) and from the windward column line's integral of the moments that analysis gives it (dots). At the gross stiffness, 15.7 mm; at 0.5, 25.4; at 0.35, the usual allowance for cracked beams, 33.3; at 0.2, 52.0. The windward column's base moment moves from 126 to 235 kN·m, and the line's integral follows the sway exactly at every one. Deflection

The check that cannot see the beams

The unit load that finds a frame's sway may be carried on any structure that can carry it, and in a frame of four bays and ten storeys it can be carried down one line of columns alone. Then the roof sway is that line's real moments against the unit load's, a few dozen products by hand — exact for the frame as drawn, for its beams cracked to a third of their stiffness, for its joints made springs and for its bases made springs, the last only once the bases' rotation is given its own term. It is exact because the unit load never enters a beam. Which makes it a check on the analysis's arithmetic that cannot say anything about the analysis's beams.

 The first hinge comes latest where the beam's end and its span yield together. The load factor at which the first plastic hinge forms in the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, against the initial stiffness of its full-strength end plates (1.20 times the beam's plastic moment), softening as EN 1993-1-8 has it and kept linear. With softening, 3.39 at 100 EI/L, 3.78 at 25 and 3.91 at 3; kept linear, 3.32, 3.51 and 3.91. Stiff joints hinge first at the inner end and flexible ones in the span; the first hinge comes latest between, 4.52 at 10.0 EI/L with softening and 4.31 at 5.0 kept linear. Rigid joints give 3.25, and the beam collapses at 5.06 for every joint. Internal forces

The joint stronger than its beam

A full-strength joint never reaches its own resistance: the beam beside it hinges first. But an end plate softens from two thirds of its resistance, and a joint only a little stronger than its beam is already on the soft part of its curve before the beam yields. The softening sheds moment from the beam's end into its span, so it delays the first hinge — by 8 per cent for a joint at 1.2 times the beam's strength, 18 for one at exactly the beam's, and not at all from 1.5 on. It never moves the collapse. And too much shedding hands the first hinge to the span, so the first hinge comes latest at the stiffness where the end and the span yield together.

 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.

 Where the shelter is decides whether the leading row works harder. The share of the cap's shear the leading row and the last row of three rows carry, as a multiple of the average pile's, against the thickness of soft ground over stiffer, in characteristic lengths, with the row factors applied at every depth (solid) or in the soft layer only (dashed). With the factors at every depth the leading row carries 1.24 and the last 0.83 at 2.0 lengths; in the soft layer only, 1.07 and 0.96. In uniform ground the usual reading gives 1.35 and 0.75. Internal forces

The shelter that stops at the interface

Piles in a group shelter one another, and practice counts the shelter as a factor on each row's soil, calibrated on groups in uniform ground. Put the same group in soft ground over stiff and the calibration has to be read one of two ways — the shelter at every depth, or only where the ground is soft — and the two readings disagree about which row works hardest. The first gives the leading pile 17 per cent more moment at the cap; the second gives the last pile 24 per cent more in the ground.

 A floor that shears lowers the storey's buckling load before the core does. The buckling factor of a storey of eleven lines of leaning columns, ten bays, each column carrying 3,000 kN over 3.6 m and built a three-hundredth out of plumb, held by a floor to a core sized for a buckling factor of 4.0 were the floor rigid, against the floor's shear stiffness per bay as a multiple of a column's sway softening P/h, logarithmic, with the core at one end, in the middle, or split between both ends; dotted, the rigid floor's 4.0. At 100 times the factor is 1.52, 2.80 and 3.02; at 1,000 times, 3.49, 3.84 and 3.88; a concrete slab is several thousand times. Stability

The floor the columns lean on

A storey of pin-ended columns, every one a little out of plumb, leans on its floor, and the floor leans on a core. Count the floor as rigid and the core needs the sum of the columns' demands. A floor that shears in its own plane is a brace in series with the core, and the far columns lean on it before their push arrives. With a floor a hundred times stiffer per bay than a column's own sway softening, the storey's buckling factor falls from 4.0 to 1.5 and the core is asked for twice as much.

All themes