Structural Engineering & Statics
where the load goes
Connections

The width nobody drew

Every other member in this collection arrives with a cross-section. A gusset plate does not — it is a piece of steel with a brace bolted to it somewhere in the middle, and no geometry says how much of it is working. The profession's answer is a thirty-degree spread from a 1952 master's thesis, it invents three quarters of the area being checked, and the check it was written for is not the one that governs.

Assumes The connection is not a point, and every diagram on this site says it is, How far a wrong load reaches and The plate that ripples, and the width that is left.

18 min read 8 figures Drawing as calculationWhich failure arrives firstThe load must go somewhere

Every member in this collection has arrived with a cross-section. A beam has a depth and a width. A column has an area and a radius of gyration. Even a plate under a patch load has a thickness and a defined region to spread into.

A gusset plate has none of that. It is a piece of steel, cut to fit a corner, with a brace bolted or welded to it somewhere in the middle. Ask what area is carrying the brace force and the geometry does not answer, because there is no boundary anywhere that says where the member ends and the plate begins.

The width nobody drewA gusset plate with a brace bolted to it over 240 mm, and the width the profession has agreed to pretend is carrying the force. Everything else on this site arrives with a cross-section; a gusset does not, because it is a piece of steel with something attached somewhere in the middle of it and there is no geometry that says how much of it is working. The answer is the **Whitmore section**: assume the force spreads at 30° from the first fastener and take the width it has reached at the last, b_eff = w + 2L·tan30° = 367 mm. That is 4.08 times the width anything is actually attached to, and the rule comes from a 1952 master's thesis. It has since been checked against finite element work and holds to about ten per cent, which is fortunate, because moving the assumed angle by ten degrees moves the answer by 34%. On this plate the check that governs is not the stress the rule was written for: it is the Whitmore section buckles, at 721 kN against 1564.the brace force30°b_eff = 367 mm90 mm4.08× the width anything is actually attached togoverns on the Whitmore section buckles, at 721 kN
Fig. 1 A gusset with a brace connected over 240 mm, and the width the profession has agreed to pretend is carrying the force. It is four times the width anything is actually attached to.

The rule, and where it came from

The answer everybody uses is the Whitmore section: assume the force spreads at 30° from the first fastener to each side, and take the width it has reached at the last one.

beff=w+2Ltan⁡30°=90+2×240×0.577=367 mmb_{\text{eff}} = w + 2L\tan 30° = 90 + 2 \times 240 \times 0.577 = 367\ \text{mm}

against a connected width of 90 mm. The rule invents 4.08 times the width anything is attached to, and three quarters of the area being checked is a construction rather than a measurement.

Its provenance is worth stating plainly, because it changes how the number should be read. R. E. Whitmore’s 1952 master’s thesis at the University of Tennessee measured strains on photoelastic and aluminium models of a bridge gusset, found that a 30° spread reproduced the peak stresses he saw, and published. It is a fit, not a derivation, and there is no theory of gusset dispersion that produces thirty degrees.

Finite element work over the following seventy years has agreed with it to about ten per cent, which is why it survives — and it is worth knowing that a rule holding to ten per cent is doing well, because the sensitivity is not small.

assumed spread beffb_{\text{eff}} yield capacity
20° 265 mm 1,128 kN
30° 367 1,564
40° 493 2,099
45° 570 2,428

A ten-degree change in an angle nobody measured is worth 34% of the answer. That is the honest scale of what is being assumed, and it is larger than any of the material factors applied afterwards.

Which free body produced the number

There isn’t one, and that is the point of this essay.

Every other check on this site is made on a free body: cut the structure somewhere, put the internal forces on the cut, and require them to balance. The cut is chosen but the geometry of the cut is real — a section through a beam is a section through a beam, whatever the analyst thinks about it.

Here the cut is at the last row of fasteners, which is real, and the extent of the cut is chosen. Equilibrium is satisfied for any assumed width, because the total force across the cut is the brace force regardless; what the assumed width decides is the assumed distribution, and hence the peak stress.

So the Whitmore rule is not a statement about equilibrium. It is a statement about the shape of a stress distribution — a lower-bound-style assumption of the kind Saint-Venant’s principle licenses in general and does not quantify in any particular case.

Three ways to apply the same force, and one depth to forget the differenceThree end loads on a member 400 mm deep, all with the same resultant and the same moment: a point load, the same force spread over a fifth of the depth, and the same force split in two. What is plotted is the *difference* between each of them and the beam-theory answer — the self-equilibrating remainder — as a fraction of the mean stress. The point load starts at 20 times it and is under a tenth of it by 0.77 depths; all three are under one per cent by about 1.18. That distance is the licence every figure in this collection is drawn under, and the exact strip eigenvalue agrees with it: 2.106 + 1.125i, whose real part puts one per cent at 1.09 depths and whose imaginary part means the remainder changes sign on the way out, which no statement of the principle mentions.00.511.5200.20.40.60.81distance from the loaded end (depths)self-equilibrating remainder ÷ mean stressa point loadspread over h/5split in twoone depthsame resultant, same moment · one per cent left by 1.18 depths
Fig. 2 The principle the rule is standing on. A self-equilibrating disturbance dies out over a distance of the order of the depth it was applied over, so the detail of how a force is introduced stops mattering — and how quickly is exactly what the thirty degrees is a guess at.

The second invention, which is stacked on the first

A gusset in compression does not fail by reaching a stress. It buckles.

That requires an unbraced length, and a plate does not have one of those either. The convention is to take the average of three lines drawn from the ends and the middle of the Whitmore width to the nearest gusset edge, apply an effective length factor between 0.65 and 1.2 depending on whether the free edge is stiffened, and use the ordinary column curve on a strip of the assumed width.

That is two inventions stacked: an assumed width, and an assumed length over which that width is assumed to buckle. Neither has a derivation, and the second is applied to a strip that only exists because of the first.

For the plate here — 12 mm thick, with 300 mm to the nearest edge, unstiffened at K=1.2K = 1.2 — the radius of gyration is t/12=3.46t/\sqrt{12} = 3.46 mm, the slenderness is 104, and the critical stress is 164 MPa against a yield of 355.

Nbuckle=721 kNagainstNyield=1,564 kNN_{\text{buckle}} = 721\ \text{kN} \qquad\text{against}\qquad N_{\text{yield}} = 1{,}564\ \text{kN}

The plate loses 54% of its capacity to a stability check based on two assumed dimensions.

What actually governs, which is fourth on the list

Run all five limit states on the same plate.

limit state capacity
1 the Whitmore section buckles 721 kN
2 a block tears out 1,150
3 the bolts bear out 1,242
4 the Whitmore section yields 1,564
5 the net section tears 1,595

The check the rule was written for — the Whitmore stress against yield — is fourth of five, and gives an answer more than twice the governing one. A designer who checks the Whitmore stress and stops has satisfied the least demanding structural criterion on the plate.

That ordering is not universal, and its sensitivity is the practical content:

Stiffen the free edge and KK falls to 0.65. The buckling capacity rises to 1,246 kN and block shear takes over as the governing mode at 1,150. A stiffener costs a rib of plate and buys 73%.

Bring the edge closer — 150 mm instead of 300 — and the slenderness halves to 52, the buckling capacity rises to 1,289 kN, and block shear governs again. A more compact gusset is a stronger one in compression, which is the opposite of the instinct that a bigger plate is safer.

Thicken the plate from 12 mm to 20 and the buckling capacity goes from 46% of yield to 76%, because thickness enters the area linearly and the radius of gyration linearly as well. Thickness helps the stability check twice and the stress check once — the only variable on the plate that does.

A 12 mm plate, and the width it can beThe elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 555 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 700 mm only 65 per cent of it is still working.1002003004005006007000200400600plate width (mm)slender beyond 555 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width
Fig. 3 The stability problem the invented strip is standing in for. A real gusset is a plate with two free edges and two attached ones, and its buckling is a plate problem — the column strip is a device for making it tractable.

Why the honest description is uncomfortable

Set out plainly: the design of a gusset plate rests on an assumed dispersion angle from a 1952 thesis, an assumed buckling length from an averaging convention, and an effective length factor chosen from a short table by judgement — and the answer is used to size a member that holds a bracing system together.

That is not a criticism of the practice, because there is no better available and the practice works. It is an argument for knowing which parts of a calculation are physics and which are convention, and this calculation is unusually rich in the second.

The place it went wrong publicly is instructive. The I-35W bridge collapse in Minneapolis in 2007 was initiated by a gusset plate half the thickness the original design should have specified — a drawing office error from 1964 — and the investigation found that the plates had never been checked at all in the original design, because gusset capacity was assumed to exceed member capacity by convention. The convention was reasonable for the plates people were drawing. It contained no mechanism for noticing when one of them was wrong.

What the joint does to the beamEnd moment as a fraction of the fixed-end value wL²/12, against the joint's rotational stiffness, for a beam of EI/L = 14000. At the rigid boundary of 112000 kN·m/rad the joint delivers 80% of it and at the pinned boundary 20%. Everything between the two lines is a redistribution nobody chose and every analysis assumed away.02000040000600008000010000012000014000016000000.20.40.60.81joint rotational stiffness, kN·m/radend moment ÷ wL²/126.04%30%62.16%rigid boundarysemi-rigidfixed ended
Fig. 4 The wider point the gusset is a case of. A connection is not a point, its capacity is a set of limit states rather than a number, and the one that governs is rarely the one the member’s own design suggests.

What the plate is actually doing, which is spreading a force

Strip away the conventions and the gusset’s job is one this collection has met before: take a concentrated force and deliver it to a member that is wider than the place it arrived.

That is an anchorage zone with the prestress removed, a base plate turned through ninety degrees, and a patch load on a web with the web replaced by a plate. All four are dispersion problems, and all four are handled by assuming a spread angle and computing a stress on the resulting area.

The angles differ — 30° here, 45° for a patch load through a flange, a bell-shaped field for an anchorage — and the differences are historical rather than principled. Each was fitted to the geometry it belongs to, and none of them is the answer to a general question about how a force spreads in a plate.

What is general is the accompanying transverse tension. A force that spreads apart pulls apart, which is the bursting force behind an anchorage plate and the splitting that a strut-and-tie model of any dispersion region contains. The gusset has it too — a tension across the spreading force, in the plane of the plate — and it is the one action in this list that the Whitmore check does not compute at all, because the check is about a longitudinal stress on a transverse section.

It is usually satisfied by inspection, since a plate is continuous across the spread and there is a great deal of it. It stops being satisfied by inspection when the plate is narrow, notched, or has a bolt line running down the spread — which is exactly where gusset failures have been observed to start.

The force spreads, and the spreading needs a tieThe end block behind an anchorage of 1200 kN on a 200 mm plate, in a section 700 mm deep. Half the force enters at the quarter point of the plate and leaves at the quarter point of the section, so a strut between the two rises 125 mm and needs a transverse tie to turn it. Placing the tie 0.5 depths from the face makes that tie force 214 kN — and at exactly half a depth this reproduces Guyon's 0.25P(1 − a/h) to the digit, which makes that famous coefficient a lever arm somebody chose rather than a property of concrete. The bearing stress under the plate is 20.0 N/mm² against 5.7 once the force has spread.1200 kN214 kN of tiestruts at 20°0.5 huniform from here onGuyon: 0.25 P (1 − a/h) = 214 kN — the same number, from the same lever armbearing 20.0 N/mm² · once spread 5.7 N/mm²
Fig. 5 The same dispersion problem where the transverse tension is the whole subject. A force spreading apart pulls apart, and the tie carrying that pull is the reinforcement an anchorage zone exists to place.

Where this model stops

The plate is treated as a strip. It is a two-dimensional element with a complex boundary, and its real buckling mode involves the free edge waving rather than a strip bowing. Finite element studies show the strip model to be conservative in most configurations and unconservative in some — particularly where the free edge is long and unstiffened.

One brace at a time. A real gusset usually carries two or three braces plus a beam and a column, and the Whitmore sections for the different members overlap. There is no accepted way to handle that, and practice is to check each independently and hope, which is a lower-bound argument that has never been made rigorously.

The connection is assumed concentric. Where the brace’s line of action does not pass through the working point, there is a moment on the plate as well, and the assumed strip is carrying bending it was not sized for.

And the seismic case is different. A gusset in a braced frame designed to yield is expected to accommodate the brace’s post-buckling rotation, which is a ductility requirement rather than a strength one — and the detail that satisfies it, a clear length of plate between the brace end and the beam, is in direct tension with the compact geometry the buckling check wants.

Block shear: the metal between the holesTwo bolts in a 12 mm plate end connection. The shaded block tears out along a shear plane 120 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 387.24 kN, of which the shear plane carries 61.36%.pullshear plane, 120 mmtension plane, 40 mmcapacity 387.24 kN0.6 fu Anv = 269.35 kN · 0.6 fy Agv = 237.6 kN · fu Ant = 149.64 kNthe yield value governs the shear plane
Fig. 6 The limit state that takes over once the buckling one is fixed. A block of metal comes out bounded by two surfaces with two different strengths on them, and no stress on any assumed section predicts it.
A bolt group under an eccentric loadA 2 by 2 bolt group carrying 400 kN at 120 mm from its centroid, with the resultant force on each bolt drawn to scale, by the elastic vector method. The load is shared equally and the torque is not, so the worst bolt carries 310.48 kN against 100 kN of direct shear alone — 3.1 times as much.400 kNe = 120the dashed ring is the group's centroidworst bolt 310.48 kNFour bolts · direct shear 100 kN eachelastic vector method
Fig. 7 And the check underneath both of them, which is real rather than assumed. Where the bolts are is a fact, what each one carries follows from compatibility, and nothing in that calculation depends on a judgement about a distribution.
How much of a flange works is decided by the span, not by the flangeThe working fraction of a flange overhang against its width as a fraction of the span, with the code rule and the exact elastic ceiling on the same axes. At b/L = 0.075 — the 1.5 m overhang on the 20 m span — the model gives 0.955 of the width and the code's min(L/8, b) gives 1.000, while the exact ceiling of L/2π per side is 3.183 m, wider than the flange itself, so it does not bind until b/L reaches 1/2π = 0.159. Doubling the flange at a fixed span moves the curve down, not the effective width up: at b/L = 0.050 the fraction is 0.979 and at 0.199 it is 0.759, so 4 times the flange buys 3.06 times the working width. Past b/L ≈ 0.22 the one-term shape rises above the exact ceiling and is optimistic; the ceiling governs there, and the curve is drawn no further than the model is good for.00.050.10.150.20.2500.20.40.60.81b ÷ L, the overhang as a fraction of the spanb_eff ÷ b1.5 m on 20 m: 0.955 of the flange worksthe modelenergy minimummin(L/8, b)the code ruleL / 2π per sidethe exact ceiling
Fig. 8 The other invented width on this site, for comparison. An effective flange width is the same kind of device — a field converted into an area so that a stress can be computed — and it carries the same kind of sensitivity to the convention.

What the picture cannot show

The figure draws a trapezium of stressed metal with clean edges, which is the one thing the Whitmore section certainly is not. The real stress field spreads continuously, peaks somewhere near the middle of the last bolt row, and decays without a boundary anywhere. The trapezium is a bookkeeping device for turning a field into an area, and the 30° is what makes the average over that area come out close to the peak of the field.

That also explains why the rule is robust despite being a guess. It is not predicting a boundary; it is calibrating an average against a peak. Any assumption that gets the ratio approximately right will work, and thirty degrees is one that does.

How a plate would be designed if the rule did not exist

It is worth asking what the alternative looks like, because the answer explains why a fitted rule from 1952 is still in use.

A finite element model gives the real stress field, and it gives it in an afternoon on any modern machine. It is unarguably more accurate than a 30° spread. It is also unusable as a design procedure for a routine connection: the mesh has to be built, the boundary conditions decided, the buckling done as a separate eigenvalue analysis, and the result checked against a criterion that a stress field does not directly supply. Nobody designs the two hundred gussets in an ordinary braced frame that way, and the ones that matter are checked that way afterwards.

A strut-and-tie model gives a lower bound with no invented sections at all: choose a load path through the plate, size a strut and a tie for it, and detail accordingly. That is the honest treatment, and it is the standard one in concrete. In steel it is rarely used, largely because a plate has no reinforcement to be the tie — the tie is the plate, and sizing it returns to a question about width.

So the Whitmore rule persists because it is a routine procedure for a routine member, in a discipline where the alternative is either bespoke analysis or a method that does not transfer from a different material. That is the ordinary reason engineering conventions outlive their derivations, and it is not a bad reason.

What it does argue for is knowing where the convention is thin. A gusset that is unusually large, unusually thin, unusually shaped, carrying more than one brace, or expected to yield in an earthquake is outside the geometry the rule was fitted on — and those are exactly the cases where a model is worth building.

The generalisation

The habit worth carrying is a question to ask when a check is being made: is the section being checked a real boundary or an assumed one?

A beam’s section is real; the material stops there. A net section through a bolt line is real; the holes are where they are. A block shear surface is real; it is a path through material that exists.

A Whitmore section is not. Nor is an effective width in a wide flange, nor a punching perimeter at a fixed distance from a column, nor the tributary area of a column in a two-way slab. Each of those is a convention that converts a field into an area so that a stress can be computed, and each carries a sensitivity to the convention that is usually larger than the material factors applied on top.

Those two kinds of section are treated identically in a calculation and they should not be read identically. The first has an error that comes from the material data. The second has an error that comes from someone’s judgement about a distribution, and it is bigger.

Where this goes next

100 kNe = 150the dashed ring is the group's centroidworst bolt 50.37 kNSix bolts · direct shear 16.67 kN eachelastic vector method The connection is not a point, and every diagram on this site says it is Every free body drawn here has joined its members at points. Real structures fail at the joints far more often than in the members, and the reason is that a joint is exactly the region the theory behind every other page explicitly excludes. 00.511.5200.20.40.60.81distance from the loaded end (depths)self-equilibrating remainder ÷ mean stressa point loadspread over h/5split in twoone depthsame resultant, same moment · one per cent left by 1.18 depths How far a wrong load reaches Every figure in this collection applies a load as a point, a line or a uniform pressure, and no real load is any of those. The licence is Saint-Venant's, it is usually quoted as a principle, and it is really a statement about a wavelength. pullshear plane, 180 mmtension plane, 40 mmcapacity 421.7 kN0.6 fu Anv = 322.5 kN · 0.6 fy Agv = 297 kN · fu Ant = 124.7 kNthe yield value governs the shear plane The metal between the holes, which comes out as a block A bolted end connection can fail without a single bolt breaking and without the plate reaching its tensile strength anywhere. A block of metal simply comes out, bounded by two surfaces with two different strengths on them. 1002003004005006007000200400600plate width (mm)slender beyond 370 mmyieldcritical stress — inverse square in the widthwhat the plate actually delivers, over its full width The plate that ripples, and the width that is left A wide thin plate in compression buckles at a stress that has nothing to do with the strength of the material. It then goes on carrying load — the middle drops out, and the edges work harder.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

BearingBlock shearBraceConnection designDispersionEffective lengthGussetLimit stateLoad pathNet sectionPlate bucklingSaint-Venant's principleStiffenerStressWhitmore section