The section that is checked is not the one chosen

A secondary beam framing into a girder cannot sit on top of it if the floor is to be level, so its top flange is cut away over the length the girder occupies. The cut is called a cope, it is a couple of hundred millimetres long and fifty deep, and it is drawn on the fabrication ticket rather than on the design calculations.

What is left at the end of the beam is a different section.

A third of the material, three quarters of the resistance #

The arithmetic is the same one that makes depth the cheapest strength there is, running backwards.

The cope removes a flange 190 × 14.5 mm and 35.5 mm of web with it — 3,075 mm² of a 9,362 mm² section, 33 per cent. But that flange is the material furthest from the neutral axis, and its contribution to the second moment is $A\bar{y}^2$ with $\bar y$ nearly half the depth. Removing it moves the neutral axis down, halves the effective depth on one side, and leaves a tee whose section modulus is 27 per cent of the original — more than twice as much resistance lost as material.

A cope is the largest single reduction in capacity that anybody makes to a structural member without noticing, and it is made by a fabricator working from a detail rather than by an engineer working from a calculation. The gap between the two percentages is the whole of why depth is bought and area is spent: every strip of a section counts by the square of its distance, so the strips at the extremes are the expensive ones to remove.

The saving grace is where it happens. The cope is at the end of the beam, where the bending moment on a simply supported member is nothing, so a section with a quarter of the modulus is being asked for a moment that is a fraction of a per cent of mid-span’s. The cut is enormous and it is in the right place, which is why the arrangement survives at all and why the check is unfamiliar rather than routine.

Which free body produced the number #

Cut the beam just beyond the end of the cope and take the coped length as the free body. On it: the reaction $R$ at the bolt group, and the internal forces on the cut — a shear, an axial force and a moment.

That moment is $R$ times the distance from the bolt line to the cut, which is the cope length plus the setback: 130 mm here, giving 23.4 kNm. It is small compared with anything the beam carries at mid-span, and the section carrying it is small too.

The important feature of the free body is what is not on it. There is no bending moment applied at the connection — the beam is designed as simply supported, the connection is nominally a pin, and the design calculations contain no moment at the end of this beam at all. The moment exists because the shear has to travel from the bolt line to the point where the full section resumes, and no analysis of the beam as a line element can see it.

The free edge #

The unfamiliar check is the third one, and it is about a boundary condition rather than about a stress.

A plate supported along both its longitudinal edges buckles at $k\pi^2D/(b^2t)$ with $k = 4$. A plate with one edge free — an outstand — has $k = 0.425$. The ratio is 9.4, and the cope has converted the top of the web from the first into the second.

Treated as a bare outstand of the coped web’s own height, the plate buckles at 41 N/mm² against an applied 61. It fails, and by a large margin.

Two models, a factor of 139 apart #

The bare-outstand answer is not what anybody uses, and the reason is that it ignores something real: the coped web is not free-standing. Beyond the end of the cope the web is full depth and continuous, and it holds the coped region along its whole vertical edge.

Cheng and Yura’s 1988 expression is calibrated on tests of coped beams and keeps that restraint:

$$F_{cr} = \frac{0.62\,\pi E t_w^2\,f\,k}{h_o\,c}, \qquad k = 2.2\left(\frac{h_o}{c}\right)^{1.65},\ f = \frac{2c}{d}$$

For the cope drawn it gives 5,740 N/mm² — far above yield, so buckling never governs and the flexural check at a utilisation of 0.17 is the answer.

Forty-one against five thousand seven hundred and forty. A factor of 139 between two models of the same thing, one derived and one fitted, and no smooth transition between them.

Which model is right is a question about how much of the uncoped web reaches into the cope, and the honest answer is that it depends on the cope length: a short cope is held from both ends and a long one is not. The $(h_o/c)^{1.65}$ in Cheng and Yura’s $k$ is that dependence, fitted — which is why the expression has an exponent nobody can derive.

What the check is really about #

Three checks have now been made on the same 130 mm of steel, and it is worth stepping back to say what kind of calculation this is.

It is not a member calculation. A member calculation takes a section, a length and a load, and asks whether the section can carry the internal forces the analysis found. Here the analysis found a shear and nothing else — no moment, no axial force, nothing at the connection at all, because the connection was modelled as a pin.

Everything checked on this page is a force the analysis does not contain, generated by the geometry of a detail that exists only on the fabrication drawing. That is the defining property of the connection region and the reason it needs its own methods: beam theory has stopped applying, the internal forces are not what the diagram says, and the section under consideration is one nobody selected.

The corner #

The last feature is the one a fatigue engineer looks at first.

Where the cope meets the full-depth web there is a re-entrant corner, and a re-entrant corner in a plate carrying stress is a stress raiser. With a radius $r$ and a notch depth $d_c$, the concentration factor is roughly

$$K_t \approx 1 + 2\sqrt{d_c/r},$$

which for a 50 mm cope with a 10 mm radius is 5.5.

The concentration is not a strength problem: the corner yields locally and the stress redistributes, exactly as a hole in a plate does. It is a fatigue problem, and a beam in a bridge or a crane runway with a square-cut cope is a detail category well down the table.

The cope length is the variable, and nobody owns it #

Of the three dimensions in the problem — cope depth, cope length and setback — the depth is fixed by the girder the beam frames into, the setback is a standard 10 mm, and the length is chosen by whoever draws the detail.

It matters twice over and in opposite directions.

A longer cope raises the moment, linearly: the lever arm is the cope plus the setback, so doubling the cope from 120 to 240 mm nearly doubles the flexural stress in the tee.

A longer cope lowers the buckling resistance, and faster: Cheng and Yura’s $k$ goes as $(h_o/c)^{1.65}$ and $f$ as $c/d$, so the net dependence is close to $c^{-0.65}$ on top of the moment’s $c^{+1}$.

The correct cope length is the shortest one that clears the girder, and the way to get it is to dimension it on the drawing rather than to let it be the residue of four clearances. That is a detailing instruction and not a calculation, which is exactly why it is the one most often missed.

The shear check, which turns out to govern #

With the calibrated buckling stress in hand the three utilisations are 0.17 for flexure of the tee, 0.25 for shear on the reduced web, and 0.17 for buckling. Shear governs, which is not what the drama of the previous sections suggests and is worth recording honestly.

The reason is that the cope removes a flange and the flange was carrying almost none of the shear. The shear area is the web, the cope shortens it from 428 mm to 392, and 8 per cent is the whole of what the cut cost the shear capacity — against 73 per cent of the section modulus and 33 per cent of the area. Meanwhile the flexural demand is small in absolute terms — 23.4 kNm on a beam that carries several hundred at mid-span.

So the honest summary is that a well-proportioned cope on an ordinary beam is comfortable in all three checks, and the calculation exists for the cases where it is not: a deep cope, a long cope, a thin web, a high reaction, or all four on a beam somebody chose for deflection and is barely stressing.

Where the model stops #

One flange is coped. Where a beam frames between two girders at the same level, both flanges are cut and what is left is a bare web plate — no flanges at all, a section modulus a fraction of the tee’s, and free edges top and bottom. Double-coped ends are much worse and are checked by a different set of expressions again.

The tee is treated as a beam. It is 130 mm long and about 400 mm deep, which is a span-to-depth ratio of a third: plane sections do not stay plane in anything like that, and the flexural check is a member calculation applied to a region that is entirely disturbed. A strut-and-tie model would be the honest treatment and nobody uses one.

The bolt group is a point. The reaction is taken as acting at the bolt line, and it does not — it is distributed over the bolts, so the lever arm is a little shorter than assumed and the moment a little smaller.

The stress concentration is elastic and two-dimensional. The real corner is a three-dimensional notch in a plate with a flame-cut surface and its own residual stresses, and the fatigue category assigned to it is a test result rather than a $K_t$.

And the drawing shows the cope as designed. What arrives is a cope cut to a tolerance, sometimes over-cut at the corner, sometimes with the radius omitted, and occasionally deeper than the drawing because the girder turned out to be a different serial size. Of everything on this page, the dimension least under anybody’s control is the one the answer depends on most.

What it would take to avoid the cope entirely #

The connection exists because the two beams have to be at the same level, and there are three other ways to arrange that.

Frame the secondary beam onto a seat. The beam sits on a bracket bolted to the girder web, its bottom flange bearing on the seat, and nothing is cut. The cost is a bracket and the loss of the flush soffit.

Use a shallower secondary beam. If the secondary is shallow enough to fit between the girder’s flanges, no cope is needed. The cost is a heavier beam, since the depth was doing useful work.

Make the girder deeper than it needs to be. The cope depth is the girder’s top flange plus its fillet plus a clearance; a girder with a thinner flange needs a shallower cope. This is the one nobody does, because the girder is sized by its own bending and the flange thickness is not free.

The ladder from here #

Later rungs on this anchor: the double-coped end, where both flanges go and the web plate is on its own. Reinforced copes — a doubler plate, a horizontal stiffener along the cut, or a longitudinal plate welded to the free edge — and which of the three checks each of them fixes. The lateral-torsional buckling of the coped tee, which is a fourth mode nobody has mentioned and is the one Cheng and Yura’s tests were actually measuring. Extended shear tabs, where the connection is deliberately made flexible and the eccentricity is designed in rather than ignored. The fatigue detailing of copes in bridges, where the radius, the surface finish and the grinding are specified because they have to be. And the general point this is a case of: a fabrication detail can be the governing check on a member, and it is drawn by somebody who has not seen the calculation.