Structural Engineering & Statics
where the load goes
Concept

Detailing — where it appears

Deciding the geometry a calculation assumed, which is where most of the decisions a structure's behaviour depends on are actually taken. A fatigue category, a bend radius, a bursting bar's position and a weld's orientation are all detailing choices, and each is worth a factor rather than a percentage.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

The same beam, the same load, and one of them has to lift it. Two lower-bound models of one beam, differing only in which chord the 600 kN is applied to. The chord forces are identical in both — the moment diagram does not know where the load arrived — and so are the struts. What is not identical is the vertical at the load: nothing in the top-loaded model, and the whole 600 kN in the hung one, because a bottom-chord panel point touches no strut and the load has nowhere to go but up. That tie is 1200 mm² of steel against the 96 mm² the shear calculation asks for over the same length — a factor of 12.5, in the same place, and additional to it.

The load that has to be lifted

Every shear calculation on this site assumes the load arrives on the top of the beam and walks down a diagonal to the support. Hang the same load from the soffit and the diagonal cannot start — the load has to be carried up to the compression zone first, by a tie no sectional calculation contains.

internal-forces · Indirect support The gap is a sum of five things and only one of them is computed. What a 30 mm movement joint is asked to accommodate, by three combination rules. The top bar is every term at its extreme, added: 33.2 mm, which assumes the hottest day, the fullest floor, the whole of the shrinkage and the worst-placed wall arrive together. The chance of that is about 1.5%. The bottom bar treats them as independent and asks for 16.0 mm. The middle bar is the rule used for actions and almost never for movements — one term at its full value and the rest at their coincidence factors — and gives 25.5 mm. The segments across the top bar are the terms themselves, and the ordering is the finding: the largest is tolerance at 10.0 mm, which is not a structural quantity at all, and the smallest is deflection at 3.2 mm — the only one anybody computes carefully, and 10% of the total.

The gap nobody computed

A movement joint is sized by adding up everything the structure will do to it, and the deflection calculation — the only term anybody computes carefully — is usually the smallest one in the list. The largest is a construction tolerance, which is not a structural quantity at all, and the sum of the extremes is nearly twice what treating them as independent would ask for.

deflection · Movement budget The load path has a kink in it, and the kink is a plate thick. Two 10 mm plates lapped over 60 mm and pulled with 60 kN. The two load paths are offset by the thickness of a plate, so the joint carries a moment nobody applied: P × 10.0 mm / 2. Taken at face value that gives a peak stress 4.00 times the mean. The joint rotates under load and the moment falls to 86 per cent of it, leaving 3.57 times — a saving of 11 per cent and not, on a plate this thick, a rescue. The bolt is bent as well as sheared: 382 N/mm² of bending against 191 of shear.

The joint that is crooked by construction

Lap two plates and fasten them and the two load paths are offset by the thickness of a plate. The joint carries a moment nobody applied, the peak stress is four times the mean, and the rotation that is supposed to straighten it out saves eleven per cent — because the rescue works for thin sheet with a long lap and a bolted structural joint is neither.

connections · Single lap The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 120 mm to frame into a girder. What is left is a tee with a section modulus of 3.836e+5 mm³ against the whole section's 1.438e+6 — 27 per cent. The moment at the end of the cope is the reaction on a lever arm of 130 mm: 23.4 kNm, giving 61 N/mm² and a flexural utilisation of 0.17. The web now has a free edge along the cope, so its buckling coefficient collapses from 4 to 0.425 — a factor of 9.4 — and the re-entrant corner has a stress concentration of 5.5 on a 10 mm radius.

The section that is checked is not the one chosen

A beam framing into a girder has its top flange cut away so the two can sit at the same level. What is left is a tee with a quarter of the section modulus, a web with a free edge, and a re-entrant corner — and the beam was selected on a table entry that describes none of it.

connections · Coped beam The moment a joint makes out of geometry. Bending stress in the chord divided by the axial stress already in it, against how far the diagonal's working line misses the node. A truss analysed as pin-jointed is drawn with its members as lines through their own centroids meeting at a point, because that is what makes the joint a pin with nothing but forces in it. A fabricated joint is not obliged to oblige, and the resultant then has a moment about the node — 16 kNm at the 60 mm drawn. It divides between the members meeting there in proportion to their rotational stiffnesses, 4EI/L or 3EI/L, which is the moment-distribution rule applied to an unbalanced moment that came from geometry rather than from load. The chord takes 48 per cent of it and carries 25 per cent of its axial stress again in bending. Ten per cent arrives at 28 mm, which on a chord of any depth is a detailing decision rather than a mistake.

The joint that is not where it was drawn

A truss is drawn as lines through the centroids of its members meeting at points. A fabricated joint is under no obligation to oblige, and when the working lines miss each other the resultant has a moment about the node that has to go somewhere.

connections · Noding eccentricity Where a bar may stop, and how far past there it goes anyway. The tension the bottom steel must carry along a 9 m beam, and the resistance of the bars actually present, drawn as a staircase. The demand is the moment diagram divided by the lever arm and then SHIFTED 270 mm toward midspan, because the truss inside the beam delivers its shear diagonally — the two constructions agree to 0.36 per cent, which is the second-order term and nothing else. Each curtailed layer then runs a further 1150 mm to develop, so the outer layer stops at 245 mm rather than the 1665 mm the moment diagram allows. The tail is 1420 mm at each end — 16 per cent of the span — and it is what turns a 20 per cent saving into 2.2.

Where a bar may stop

The moment diagram falls away from midspan, so the steel midspan needs is not needed everywhere, and curtailing it saves real money. Then two things get in the way, and between them they take nine tenths of what the moment diagram promised.

sections · Curtailment Where two buildings touch, and what is there when they do. Closure between an 8-storey building 24 m tall and a 4-storey one 15 m tall, drawn against height, with the 50 mm gap between them. The two swaying out of phase close on each other more the higher up they are, so they first touch at 7.6 m and are in contact above it. The dots are the taller building's floors: its storeys are 3.00 m and the other's are 3.75, so two of them arrive part way up a column rather than at a slab. A blow at mid-height of a column asks it for a shear of half the impact and an end moment of Fh/8, neither of which is a demand any part of the design contains.

The floor that arrives at a column

The gap between two buildings is computed from their roof displacements, which is where each of them moves most. It is not where they touch, and it is not what is there when they do — a slab edge meeting a column part way up its height is a different event from two slabs meeting, and it is the one that appears in the photographs.

dynamics · Pounding The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything.

The plate that is only as big as it is thick

A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all.

connections · Base plate Force and capacity round a weld group, which do not vary together. Utilisation round the c shape group under 100 kN at 150 mm, walked from one end of the weld to the other, with the force scaled onto the same axis for comparison. The capacity is 1460 N/mm where the force runs along the weld and 1789 where it runs across it — √1.5 more — so the utilisation is not simply the force in different units. The worst point is at 47 degrees to the weld, where the capacity is 1612 N/mm, and the group carries 182 kN against the 165 the along-the-weld value would allow.

The weld that is stronger where it is pulled

A fillet weld pulled across its axis is √1.5 stronger than the same weld pulled along it. On a group under an eccentric load the direction of the pull varies from point to point, so the capacity does too — and the gain that follows is worth twenty per cent at one eccentricity and nothing at all at another.

connections · Weld strength 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.

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.

connections · Coped beam 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.

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.

dynamics · Pounding 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.

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.

connections · Weld strength A tubular K joint with a 12.6 mm gap. A 168.3 × 8.0 mm chord with two 114.3 × 6.3 mm braces at 45°, their toes 12.6 mm apart on the chord's crown. The braces' working lines, carried down through the chord, meet 3.0 mm below the chord's axis — an eccentricity of 0.02 chord diameters, inside the band of −93 to +42 mm in which the joint rules let its moment be neglected. Nobody chose that number. It follows from the gap, the brace diameter and the angle, and the gap was chosen so both toes could be welded. The working lines would meet on the chord's axis at a gap of 6.7 mm, which is a joint no rule permits.

The joint whose lines may not meet

In a welded tubular truss the noding eccentricity is not a fabrication error. It is fixed by the brace diameter, the brace angle and the gap between the toes, and the gap is chosen for the welder. For a third of ordinary proportions the one gap at which the working lines would meet lies in a band no fabrication rule permits — so the concurrent joint the truss was analysed with is the one joint that cannot be built.

connections · Noding eccentricity The area rule's even stress, and the stress the cut finds. The tensile stress through the depth of the two covers where they cross the gap of a 20 mm main plate 200 mm wide, spliced with a 15 mm cover on one face and an 8 mm cover on the other, carrying 900 kN; the main plate's centre line, on which the load acts, is at zero. Shared by area every fibre would work at 196 N/mm² (dashed). The free body puts the covers' resultant on the load line instead, and the stress varies across them: 107 to 179 N/mm² in the 15 mm cover and 275 to 313 in the 8 mm one — the thin cover's outer face at 1.60 times the area rule's stress.

The thinner cover carries more

A plate spliced with a cover on each face is the standard cure for a lap joint's eccentricity, and the covers are usually sized as if they shared the load in proportion to their areas. A free body cut across the splice says otherwise. With nothing but the two covers crossing the cut, their forces must balance about the line the load acts on, so they share by lever arm and not by area — and the thinner cover, which sits nearer that line, carries more than half the load on less than half the steel.

connections · Single lap A crack at the cope's corner, and the hole its tip needs. A 457 × 190 beam end coped 150 mm long and 80 mm deep, reacting 350 kN of which 70 per cent comes and goes with traffic, to scale. A fatigue crack has grown 30 mm down the web from the corner where the cope's two cuts meet, the most stressed point of the coped section. At that depth the live load's range of stress intensity at its tip is 15.6 MPa√m, and a hole drilled with the tip inside it stops the crack only if its radius is at least (ΔK/10.5√fy)², 6.2 mm: a 12 mm hole. The crack and the hole together leave 341 mm of the 377 mm coped section below them.

The hole at the tip of the crack

The commonest reason to repair a coped beam end is a fatigue crack at the corner of the cope, and the commonest repair needs no plate at all: a hole drilled so that the crack's tip is inside it, turning a crack into a smooth notch that the traffic can no longer drive. The hole has to be big enough, and how big is set by the crack — its radius grows in step with the crack's length. A 30 mm crack at a stringer's cope needs a 12 mm hole, a 60 mm crack a 24 mm one. The repair stops working at about 100 mm, where the hole outgrows a site drill and the section left beside it runs out of capacity at nearly the same depth. Most of the crack's life is spent far shorter than that, which makes the repair a matter of how early the crack is found.

connections · Coped beam The room left between two planes of braces. The surface of a 168.3 mm chord unrolled flat — along the chord across, round it up — with the footprints where 114.3 mm braces at 45 degrees are welded to it: two from a K joint in one plane (solid), two from a K joint in a plane 90 degrees round the chord (dashed), both nodes at the same section, each with a 12.6 mm gap between its own toes. The narrowest gap between a brace of one plane and a brace of the other is 6.5 mm, against the 12.6 mm two welds need: too little to weld. Each footprint spans 85.6 degrees of the chord's circumference.

Two planes of braces and one circumference

A tower leg or a triangular girder's chord receives K joints from two planes at once, and each can be checked as a planar joint: its gap, its eccentricity, its band. What the planar checks cannot see is that the two planes' braces are welded to the same tube. A 114 mm brace takes up 86 degrees of a 168 mm chord's circumference whatever its gap, so a square tower's two planes leave 6.5 mm between them where the welds need 12.6, and a triangular girder's braces land on each other. And two eccentricities at the edge of their band give the chord √2 or √3 times the moment the band was written for.

connections · Noding eccentricity Across the gap the splice plates bend as one section. The top half of the splice of a 612 mm deep I-beam with 229 by 19.6 mm flanges, spliced at 700 kN·m by a 15 mm outer cover on each flange, two 8 mm inner covers beside the web under it and two 8 mm web plates 480 mm deep, to scale, cut at the gap where only the plates cross (the beam's own flange and web dashed beyond), and the stress through it. Bent as one section, the outer cover works at 206 N/mm² at its centre and the inner covers at 186, in proportion to their distances from the neutral axis; the hand design's area rule gives both 193; the plate splice's lever rule gives the outer 123 and the inner 350.

The splice that bends as one section

A plate spliced with two unequal covers shares its load by lever arm about the line the load is held on, and the thinner cover carries more. A beam's flange splice looks like the same joint, and the rule that worked for the plate puts the thin inner covers at the edge of yield. They are working at half that. A beam's halves are not held on any line; they turn, and every splice plate across the gap turns with them, so the plates bend as one section and share the beam's moment by their distances from its neutral axis. The outer cover works hardest, the inner covers less, and the web plates less than the hand design hands them.

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

The splice that finds its own axis

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

connections · Single lap

Named alongside it

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

EccentricityLoad pathBolt groupFree bodyLever armFatigueNet sectionSecond momentSecondary momentStress concentrationWorking lineBeam-column

All concepts