Structural Engineering & Statics
where the load goes
Concept

Load-sharing — where it appears

The division of a load between parallel paths in proportion to their stiffnesses, which holds exactly when the paths share a displacement. It is what makes a stiff element attract force whether or not that is where the strength is, and it is the reason a strengthening scheme that adds stiffness somewhere unexpected can make matters worse.

Named by 17 essays across 7 fields — each of them below, with the objects they name alongside it.

Two beams tied together, and the deeper one takes 89% of the load. Two simply supported beams of 6 m, one twice as deep as the other, tied together at midspan so that they have to move as one. A load of 100 kN stands on the tie. Point stiffness is 48EI/L³, so the deeper beam is 8 times as stiff — depth cubed, nothing else — and the load divides in that ratio: 11.1 kN into the shallow beam and 88.9 kN into the deep one, 11% against 89%. Both midspan points move 5.00 mm, which is the whole of the argument: the geometry of the load never entered it. The deflection is drawn 78 times full size — the real sag is 5.00 mm on a 6 m span, about 1 in 1200.

The stiffest path takes the load

When two members share a force the split can be argued about. When they share a displacement it cannot — stiffness settles it, and nothing about the load or the plan drawing gets a vote. The consequence is that stiffening a lightly loaded member raises its stress, and the way to unload something is to soften it.

internal-forces · Load-sharing A cable alone goes to a kink, and a kink is not a road. A point load of 1000 at mid-span of a 900 m suspended deck. The upper shape is the cable with no girder at all: two straight lines meeting under the load, because a cable takes the funicular shape of whatever is on it and the funicular of a point load is a kink — 0.0083 radians of it here. The lower shape is the same cable with the girder present, peaking at 1.125 against the bare cable's 1.873. The girder is not carrying the load — it takes only 17% of it — it is spreading it, over a characteristic length of √(EI/H) = 183 m, and what reaches the cable is spread over that length rather than arriving at a point.

The deck is not there to carry the load

A cable takes the shape of whatever is on it, which is exactly the problem — under a point load its shape is a kink, and a kink is not a road. The stiffening girder exists to spread the load until what reaches the cable is something the cable's own shape is right for.

structures · Stiffening girder A frame with no plane to be drawn in. The tetrahedron, solved: three equations at every free joint, one axial force in every member. Members drawn heavy carry more; tension and compression are separated by the sign that came back from the solve rather than by inspection. The count is m + r = 12 against 3j = 12, which makes it exactly determinate, and joint equilibrium closes to 1.8e-15.

Three equations at every joint

A plane truss is determinate when m + r = 2j. A space frame needs 3j, and that one changed digit is why a cube of twelve bars is six mechanisms short while looking perfectly solid — and why every three-dimensional frame ever built is made of triangles in several planes at once.

structures · Space frame Two centres, and the distance between them is a torque. A storey 30 by 18 m with its walls drawn heavy, pushed in one direction by 1000 kN. The force acts through the centre of mass and the storey turns about the centre of rigidity — the stiffness-weighted centroid of the walls, at x = 15.0 m — and the distance between the two is an eccentricity of 0.00 m before the 5% that has to be assumed anyway. The table below the plan splits each wall's force into its direct share and its torsional one. Torsion relieves the walls near the centre of rigidity and loads the far ones, so the wall in trouble is not the wall carrying the most: west wall is asked for 8% more than its direct share, and the walls at right angles to the push carry 19 kN each with nothing applied along them at all.

The corner that moves most

A lateral force is shared out in proportion to stiffness only if it passes through the centre of rigidity, which is not the centre of the plan and not the centre of mass. The distance between the two is a torque, and the wall that pays for it is the one furthest away and carrying least.

structures · Plan torsion Whether the floor shares the load out by stiffness or by area. The share of a uniform storey force taken by each of three equally stiff walls, against the stiffness of the floor plate that spans between them. A plate far stiffer than the walls translates almost rigidly, every wall deflects the same and the share is the ratio of stiffnesses — 60% to the middle wall here, the same as everyone else. A plate far softer than the walls behaves as a continuous beam over them and the middle wall takes 33%. Neither end is the tributary-area answer of 50%, which assumes a plate that is both soft and discontinuous over the wall, and which nothing here ever reaches.

The floor is a beam lying down

A floor plate spans horizontally between the walls that resist a lateral load, carries a distributed inertia load, and has chords, a web and a span-to-depth ratio like any other beam. Its stiffness decides whether the walls share the load by their stiffness or by the area of floor nearest them — and the familiar tributary answer turns out to be neither limit.

structures · Diaphragm How much of a deflection belongs to the beam. The share of the total deflection that is the beam's own bending, against the stiffness of what it sits on. A 8 m beam on two supports under a uniform load: on rigid supports every millimetre is the beam's, and the share falls away as the supports soften until almost none of it is. The beam drawn beside this figure sits at 51% — so 49% of what it does is happening somewhere a beam calculation never looks. The two flexibilities are in series, which means the softer one governs and stiffening the other buys nothing.

The deflection that belongs to the support

A beam calculation answers a question about a beam sitting on things that do not move. Real ones sit on bearings, on other beams and on columns that shorten, and every one of those is a spring in series with the member — so a deflection is the sum of two things and only one of them is a property of the beam.

deflection · Support flexibility The end bolts do the work and the middle ones very nearly nothing. A lap of 10 bolts at 75 mm pitch transferring 800 kN between two plates, with the force each bolt actually carries drawn above it and the flat line a division by the bolt count would have given drawn behind. The end bolts carry 1.16 of their nominal share and the middle ones 0.89. The reason is not in the bolts: at the leading end the first plate is carrying everything and the second nothing, so the two strain at different rates and the slip between them is largest there. In the middle they strain alike, there is almost no slip, and a bolt with no slip across it transfers almost no force. The mean over the worst is 0.861, and the end bolt has to slip 1.16 mm before the rest catch up.

The joint that has to be as good as the member

A splice exists because members come in lengths and structures do not. It has to deliver the same force, at the same stiffness, in the same distribution across the section, through a discontinuity — and each of those three requirements is met by a different feature of the detail, with the third one usually left to look after itself.

connections · Splice Two drawings of one deck, and they are not the same structure. A 112 m viaduct on five supports, articulated two ways. Above, the fixed point is at the left abutment: the far end has to be given 45 mm of movement, and the friction of every sliding bearing runs one way, so the fixed support takes 660 kN before any wind or braking is applied. Below, the fixed point is at the middle pier: the largest joint halves to 22 mm and the friction now cancels across the fixed point, leaving 0 kN. The movement arrows are drawn at 900 times the scale of the deck, because a 45 mm movement on a 112 m span is thinner than the line the deck is drawn with. Nothing about the deck, the loads or the ground has changed between the two.

Where the structure is allowed to move

One drawing decides how big every movement joint on a bridge is and where every horizontal force goes, it takes an afternoon, and it appears on no calculation sheet. Move the fixed point from an abutment to the middle pier and the largest joint halves and the horizontal force on that support drops from the whole of the friction to none of it.

structures · Articulation Effective length is a property of the storey. The effective length factor of the one column that resists sway, against the total gravity load on the storey as a multiple of its own. At the left-hand end it carries the storey alone and its K is 1.99 — the 2.0 every chart gives a column fixed at the base and free to sway at the top, reproduced here by a route that never mentions a chart. Then columns are added that have pinned bases and therefore no lateral stiffness whatever. They contribute load and nothing else, so they cannot buckle on their own and they lower the load at which everything buckles together. K rises as the square root of the load ratio, exactly, and at the storey drawn — three leaning columns carrying 69% of the gravity load — it is 3.57. That is off the end of every published alignment chart, and the leaning columns themselves, which a designer would take at K = 1.0 for pinned ends, are at 2.54.

The column that leans on its neighbours

A column with a pinned base and a pinned top has no lateral stiffness at all and cannot stand up alone, and yet thousands of them do. What holds them is the rest of the storey, and what it costs is paid by whichever columns do have stiffness — whose effective length rises as the square root of the load being leaned on them, straight off the end of every chart.

stability · Storey buckling The group is not weaker; it is very much softer. A 3 × 3 pile cap on the left, with each pile's share of 9.0 MN and 4.5 MNm in meganewtons — N/n plus M·y/Σy², the same three terms in the same order as a bolt group under an eccentric load and a section under biaxial bending. The corner piles take 1.25 times the average and a pile added at the centroid would change that by nothing at all, because it adds to neither second moment. On the right is the effect a bolt group cannot have: the piles share ground, so the stress bulbs overlap and the group settles 3.9 times as much as a single pile at the same load per pile, rising to 14.2 for 144 of them. The capacity check everyone makes — block failure against the sum of the piles — comes out at 4.54 here and does not govern at all. The check nobody tabulates is the one that does.

Nine piles, and four times the settlement

A pile cap divides its load between its piles by the same three terms a bolt group uses and a section under biaxial bending uses. What a bolt group does not have is neighbours it shares ground with — and the group effect that matters is not the strength check everybody makes, but a stiffness effect nobody tabulates.

structures · Pile group Two identical pipes, and one carries three times the other. Load per metre on a buried conduit against the depth of cover, in trench widths, with the weight of the prism of soil directly above it drawn between them. A conduit laid in a narrow trench is stiffer than nothing and softer than the sides: the backfill settles relative to the undisturbed ground, the friction on the trench walls acts upward, and the conduit gets 64% of the prism. Lay the same conduit on the ground and build an embankment over it and it is now stiffer than the fill beside it, the interior prism settles less, the friction acts downward, and it gets 172% — a factor of 2.71 between two pipes with nothing different but which way the ground moved. The equation is Janssen's, the same one a silo wall obeys, with a trench for a silo; both curves start on the prism line, because with no depth there is no shear to redistribute anything. This is why a flexible pipe is buried rather than a rigid one: making the conduit weaker moves it down the page.

The pipe decides what the soil weighs

A buried conduit is not loaded by the soil above it. It is loaded by whatever share of that soil the relative movement leaves it — and which way the shear on the sides of the prism acts depends on whether the conduit settles more or less than the ground beside it. Two identical pipes under identical fill, one carrying two thirds of the prism and one carrying nearly twice it.

equilibrium · Soil arching The column under the first interior support is given a quarter more. Two equal spans of 8.0 m carrying 5.0 kN/m, continuous over three supports. The tributary rule gives each interior support one span's worth of load, 40.0 kN, and each end support half of that. The continuous beam gives 15.0 kN, 50.0 kN, 15.0 kN — ratios of 0.75, 1.25, 0.75 to what the areas say. The reactions still add to the whole load, because they must; what has moved is which support gets it. The end supports are relieved because the span next to them hogs over the first interior support, lifting their end of it.

The column given more than its rectangle

The tributary rule draws a rectangle round each column and hands it whatever stands inside. A floor is continuous over its columns, and a continuous beam does not give each support the load above it — the first interior one takes a quarter more and the end ones a quarter less. On a grid the two directions multiply, and two columns on the same floor differ by a factor of nearly three.

equilibrium · Tributary area A load on one span moves the other. The change in level along two 100 m spans of stressed ribbon at a sag of 2.00 per cent, carrying 35 kN/m, when 20 kN/m of live load is added to the left-hand span, with a rigid pier (dashed) and with a pier of 200,000 kN/m (solid). With a rigid pier the loaded span sags a further 195 mm and the other does not move. With a pier of 200,000 kN/m the pier top leans 25 mm towards the load, the loaded span sags 358 mm and the unloaded span rises 191 mm; the pier carries 4,953 kN of the 9,440 kN a rigid pier would.

The pier that carries only the difference

A stressed ribbon continued over a pier pulls on it from both sides, and under its own weight the two pulls cancel. Load one span and they do not: the pier is asked for the difference, nearly ten meganewtons for a crowd on one of two hundred-metre spans. Whether it gives it depends on its stiffness against a number the ribbon supplies itself, and whatever the pier does not carry, the ribbon turns into movement — a third of a metre of extra sag in the loaded span and a fifth of a metre of rise in the empty one.

structures · Stressed ribbon Fully stressed, with a choice of diagonals. Two trusses sized so that every member is at the allowable stress under the full load, each member drawn as wide as its area, tension and compression in two colours. Above, an eight-panel Pratt truss as deep as a panel is long, with a counter-diagonal in each of its six interior panels, loaded by 10 at every bottom joint, found by resizing and reanalysing until nothing changes; below, the same truss without its counters. The counters in the two middle panels have shrunk to nothing (dotted) and the four nearer the supports have stayed, working in compression beside the diagonals in tension. The truss that kept them needs 1,200 units of steel against the Pratt's 1,220, and deflects 24.0 at mid-span against 26.0.

The truss whose forces follow its sections

In a determinate truss each member's force is fixed before its section is chosen, so sizing for strength and sizing for stiffness can be done in either order. Put a counter-diagonal in every panel and they cannot. The fully stressed design becomes an iteration that starves some counters to nothing and keeps others, lands on a different truss from every start, and — whichever it lands on — weighs the same and deflects the same. Then enlarge one group of members to stiffen it, and a vertical nobody touched is overloaded by 58 per cent.

deflection · Truss deflection More props, heavier slabs — unless they go in after the weight. The heaviest load any slab carries during construction, as a multiple of its own weight, against the number of levels of props under the slab being cast. Every level a shore that carried the wet concrete and moves up with the formwork: 2.00, 2.25, 2.37, 2.44 w for one to four levels. One level of shores and the rest backprops, put in snug after the formwork is struck: 2.00, 1.50, 1.33, 1.25 w. The same number of props, differently timed: a shore enters the load path before the weight it carries and passes it down, and a backprop enters after and shares only what arrives later.

Loaded twice over before it is a month old

A concrete floor in a building going up a storey a week carries, in its first weeks, the wet weight of the floors being cast above it, handed down through the props. By Grundy and Kabaila's arithmetic that is twice its own weight on a single level of shores, and more with more levels — 2.25 on two, 2.37 on three — because every prop that carried wet concrete passes its load down the stack. The same props put in after the formwork is struck carry nothing until something new arrives, and the peak falls to 1.5, 1.33, 1.25. The heaviest day of a floor's life is decided by when its props went in, not how many there are.

materials · Maturity One quantity, three names. The middle braced bay's share of the storey force (solid) and one end bay's (dashed), for a 48 by 12 m plate, 200 mm of concrete, spanning two 24 m bays between three braced bays of 100,000 kN/m each, under 40 kN/m, against a factor on the plate's stiffness as built, on a logarithmic scale. The share moves continuously from 61 per cent, a continuous plate on unyielding supports, to 33 per cent, the rigid plate. Left of 0.02 (shaded) ASCE 7 calls the plate flexible and assigns tributary shares, 50 per cent to the middle; the plate there actually gives it 59 per cent at the boundary. Right of 1.54 (shaded) EN 1998-1 calls it rigid and assigns stiffness shares, 33 per cent; at that boundary the plate gives 37. The plate as built, factor 1, gives 38 per cent to the middle and 31 to each end.

Rigid by one code and not by the other

A floor plate between shear walls shares its storey force out in proportions that move continuously with its stiffness, from a continuous beam's to a rigid body's. The codes put it in one of two boxes and use a different model in each. ASCE 7 and EN 1998-1 draw their boxes' edges a hundred times apart in plate stiffness, each at a point where its own licensed model already misstates the middle wall by about a tenth — and an ordinary 200 mm slab sits in the gap between them.

structures · Diaphragm One span hung, the other still bare tendon. The level along two 100 m spans at a sag of 2.00 per cent, 35 kN/m finished and 3.0 kN/m of bare tendon, the tendons at 1100 N/mm² when finished, with the deck hung on the left span only (solid) and as finished (dashed), on a pier of 200,000 kN/m. The hung span sags 2.04 m and pulls 21,442 kN; the bare tendon beside it sags 0.20 m and already pulls 18,363 kN, because it was cut 455 mm shorter than the span so as to carry its share when finished. The pier is asked for the difference, 3,080 kN, and leans 15 mm towards the hung span — against 4,953 kN under 20 kN/m of crowd on one finished span.

The tendon that pulls before the deck arrives

A stressed ribbon is hung one span at a time, and the obvious fear is the stage at which one span carries its deck and the next does not: the pier between them asked for a whole span's thrust. It is not asked for that, because tendons cut to the finished length are already stretched across the empty span and pulling. For two 100 m spans at a fiftieth, the rigid pier's stage force is 4,109 kN against a crowd's 9,440 — unless the tendons were sized generously, when the stage overtakes the crowd.

structures · Stressed ribbon

Named alongside it

The objects these essays reach for when they reach for this one.

StiffnessLoad pathFree bodyTributary areaDeflectionDiaphragmElastic foundationGeometric stiffnessLateral systemServiceabilityShear wallAccidental eccentricity

All concepts