The section stronger for being cut

What is left after the first fibre yields put a number on the reserve in each shape of section, and at the top of its list it placed the diamond: a square on its corner, a shape factor of exactly 2, the worst possible way of arranging an area for bending and therefore the largest reserve. It noted the ordering was backwards. The worse a section is at first yield, the more it holds in reserve afterwards, because a poor section is one whose area was near the middle in the first place and so had a lot left to give.

Push on that ordering a little and something stranger comes out. If the diamond’s elastic modulus is so poor because a thin tip stands far out and reaches yield first, what happens if the tip is simply cut off? That is removing material from the place every other essay on this subject has said is the most valuable, as far from the middle as it can get. The tip’s lever arm is the largest in the section. The expected answer is that the section gets weaker, and that expected answer is wrong for one definition of weakness and right for another, and the difference between the two is worth more than the 5 per cent.

A few lines of algebra and a surprising maximum #

Take the diamond with half-depth $h$ and half-width $b$, and for arithmetic take both as one. Its half-width at height $y$ is $1 - y$. Cut flat at a height $c$ from the centre, keeping $|y| \le c$. Then

$$I(c) = 4\int_0^c (1 - y)\,y^2\,dy = 4\left(\frac{c^3}{3} - \frac{c^4}{4}\right),$$

and since the furthest fibre is now at $c$, the elastic modulus is

$$Z(c) = \frac{I(c)}{c} = 4\left(\frac{c^2}{3} - \frac{c^3}{4}\right).$$

The whole section has $Z(1) = 1/3$. Differentiate: $dZ/dc = 4(2c/3 - 3c^2/4)$, which is zero at $c = 8/9$. So the largest elastic modulus does not belong to the whole section. It belongs to the section with a ninth of each tip cut away, and there

$$\frac{Z(8/9)}{Z(1)} = \frac{256}{243} = 1.0535.$$

The derivative at $c = 1$ is negative: reducing $c$ from the whole section raises the modulus at once, without any threshold to cross. Note also how little is taken. The two removed tips are small triangles whose area goes with the square of what was cut, so a ninth of the depth takes $1/81$ of the area, or 1.2 per cent of the steel, for 5.35 per cent more modulus.

So two of the three lines behave as anyone would expect. The area goes down, gently, and then more quickly. The plastic modulus goes down too, since a fully plastic section has every fibre at yield, so every piece of material removed takes its full stress times its full lever arm with it. Only the elastic line goes up, peaks at the ninth, and returns to the whole section’s value somewhere near three quarters of the depth. This one line is what the result is, and the reason for it is in what the modulus is measuring.

For scale, take a bar with a half-diagonal of 100 mm at 355 MPa. The whole section first yields at 118 kN·m and has a plastic moment of 237. Cut to 88.9 mm, it first yields at 125 kN·m and has a plastic moment of 229. A designer working to first yield would take the cut section. One working to a plastic moment would take the whole one.

A modulus is a ratio, and a ratio has a denominator #

The moment a section takes at first yield is $f_y I/c$. Two things happen when a tip goes. The second moment falls, as a part of it has gone. And the distance to the furthest fibre also falls, and that is a separate gain, because the stress limit is applied at the furthest fibre whatever it happens to be, and now the furthest fibre is nearer in. The first effect depends on how much material was at the tip, and the second does not. For a pointed tip, the first is very small at the start, as there’s almost no material up there, while the second is immediate. Thus a small cut has to win.

Stated in general, with $w(c)$ the section’s whole width at the height being cut:

$$\frac{dZ}{dc} = \frac{2w(c)\,c^3 - I}{c^2}.$$

The modulus grows as the section is cut for as long as $w(c)\,c^3 < I/2$. That is a useful test, and one that applies to any symmetric shape. It says when a fibre at the edge is costing the section more, by standing out, than it adds in bending stiffness, and it gives the answer in terms of how wide the section is up there.

The test is a statement about shape, and every shape on it has a width that’s not zero at some point. A diamond’s and a circle’s tip has zero width, so each starts at a test value of nothing, and each must therefore gain from the first cut. A rectangle’s edge is full width, so its test value starts at 1.5, and any cut at all simply loses. An I-section is the same, even more so. The whole logic of the same steel in a different shape is about putting a wide flange at the edge, so that its value of $w c^3/I$ is large.

What turning a square on its corner costs #

The numbers look different if the whole comparison is started one step earlier, with a square bar lying flat. A square of side $s$ has $Z = s^3/6$ and $S = s^3/4$. Turn it on its corner, with the same area and a half-diagonal of $s/\sqrt{2}$, and it has $Z = s^3/8.49$ and $S = s^3/4.24$.

So turning the square costs 29 per cent of its elastic modulus, and only 6 per cent of its plastic modulus. That single comparison holds most of the argument. A designer working to first yield would never stand a square on its corner, since a third of the strength appears to go with that rotation. In plastic terms it barely matters, as the same material is in the same bands of the section, and the fully plastic moment reads the first moment of each half, which turns out to be nearly unchanged. In other words, the elastic check is penalising the rotated square almost entirely for having a thin tip stand out, and the cut gets back about an eighth of that penalty by removing the tip, at a price of a little of the plastic moment. Both shape factors are right: 1.5 flat and 2 on the corner, and the difference between them is almost entirely the difference in the denominators.

Which shapes come to a point #

The gain is a modest one everywhere, and the size of it depends on how sharply the section comes to its point. A hexagon on its corner gains nearly as much as a diamond, since its tip has the same 45-degree-or-so shape, and only the base of the tip differs. A circle has only 0.7 per cent to offer, since it arrives at its top nearly flat and has a substantial width just below the tip. That’s the reason nobody bothers to machine a flat on a round bar to gain strength. A rectangle gains nothing, and any section with a wide flange at its edge also gains nothing.

Most real sections are in that last group, for good reason. A designer starts with an I-section because it is already at the far end of what this test says, and there are no thin tips to cut. It is only with shapes chosen for some other reason, such as a square bar rotated to shed water or to fit into a corner, a timber beam cut from a log so that its round corners stand off, or the trapezoid of a concrete haunch, that the question ever comes up. So the result is not a design tool. What it does is make a claim about the meaning of the modulus.

The whole section was stronger all along #

This figure settles the paradox. Read along any vertical line: at every curvature, the whole section carries more moment than the cut one. It has to, since the whole section contains every fibre of the cut section, at the same strain, with the same stress, plus some more fibres at the tips, each of which is pulling in the same direction. There is no curvature at which removing those fibres helps. In a ductile material, cutting off the tips does not make the bar stronger in any sense that a beam would notice.

So what is the first-yield moment? It is the height on the whole section’s curve at which a very small part of the bar, its tip, has reached yield. On a curve like that one, the tip’s yielding has almost no visible effect. There’s a slight loss of slope, with the tip having so little area, and the line carries on. The moment 0.333 marks nothing structural, and it marks only the instant a thin wedge at the outer edge reached its limit. The cut section’s 0.351 is a higher number because it refers to a different fibre, reached a little later. At the moment the cut section’s new tip yields, the whole section is carrying 0.374, and that’s more than the cut section has at that point. Thus the elastic modulus has been ranking the two sections by how soon an arbitrary event happens, and in this case that isn’t how much they can carry.

This is what the section that yields from the outside in describes from the other direction. A section does not fail at first yield. A check on $f_y Z$ is a convenient and safe statement that the worst fibre is still elastic, which is useful for several purposes, and the name “section strength” for it is a piece of shorthand. Here the shorthand has produced an ordering that the structure doesn’t have. The same comparison has a twin in the section that cannot reach its own strength, where the question is the reverse one: whether a section’s thin plates let it get as far as its modulus claims at all.

Where a first-yield number does mean something #

There is, however, a set of cases in which the first fibre really is the end of the story, and in those the cut gives a real gain.

A brittle material. Cast iron, glass and plain concrete don’t redistribute. When the outer fibre breaks, a crack runs inward from it, and the section is gone. For those materials the elastic modulus is the strength, near enough, and a chamfered tip is stronger by what the algebra says. The case isn’t quite as clean as the algebra, though: the bigger one is the weaker one argues that strength in a brittle material falls with the volume under high stress, and a sharp tip has very little volume at its peak stress, so a whole tip may be stronger than its nominal stress suggests. The two effects pull against each other, and it’s a matter of testing whether the tip’s small highly stressed volume or the cut section’s shorter lever is the more important. Chamfers on precast concrete and glass come from the requirements of handling, and they’ve never been there for the 5 per cent.

A fatigue range. Fatigue reads stress range at a detail, and a fatigue crack begins at a single point. In a welded or machined part, what matters is the stress at the furthest fibre, and if that fibre is a sharp corner, then a flat on it has a double benefit. A sharp corner has its own stress concentration and is the usual site of a crack, and a chamfered edge removes that as well. The detail decides and the steel does not gives the general rule, and here the fibre itself has been moved.

An elastic limit used as a service limit. Some structures are required to stay elastic for reasons other than strength, such as precision instruments, optical benches or springs, where the first permanent set is itself the failure. A spring engineer who cares about the elastic range will prefer the cut section. In this group the modulus is the correct objective, though the stiffness has fallen by more than 6 per cent, since $I(8/9) = 0.312$ against $1/3$.

That last note is worth keeping. At the point of the best modulus the section has lost more stiffness than it has gained in modulus, and so it deflects more. The modulus has gone up, the second moment has gone down, and those two are usually thought of as moving together.

Per unit of steel #

Divide by the area, and the elastic efficiency keeps improving past the ninth: its peak per unit of area is further along, since area falls with the square of what’s cut and the modulus only starts falling after the ninth. This is another way of making the same point, and it’s closer to the question a designer asks. If a section’s weight were being paid for at first yield, the cut would pay twice. It also raises the question of what shape would be best if the cut could go anywhere, rather than only a flat across a chosen shape. The answer to that for a fixed area and a first-yield criterion is a wide-flange form with no thin tips, which is where the same steel in a different shape arrived from the other side. So the diamond’s cut is really one small step toward the I-section.

There is also a pleasing symmetry in the axis that moves when the section yields and two strengths, depending which way up, both about sections asymmetric about the neutral axis. A tee has its furthest fibre on one side only, at the end of its stem, where the width is small, so its test value at the stem’s tip is low. By this argument, trimming a tee’s stem raises its elastic modulus on the stem side at first, and it did so in cast iron beams, whose makers arrived by trial at a short stem with a heavy bottom flange. That case has its own neutral axis to move, so the algebra here doesn’t carry over directly, and the result is left as a pointer rather than a number.

What the elastic–plastic bar leaves out #

The material is elastic and perfectly plastic. A hardening steel has a curve that goes on climbing past the plateau, which favours the whole section further, since its tips have a large strain to work with.

Shear is ignored. The cut changes the shear stress at the new flat from nothing to something, as the width is no longer zero there, but at the flat the stress is very low in any case.

The stress at a tip is taken as what plain bending gives. In a real rolled or cast diamond the tip is rounded, and in an elastic analysis a sharp corner on the far fibre has a stress field that’s a little different from beam theory near the tip. The small numbers involved mean that a rounded tip already has part of this benefit.

Lateral buckling and local stability are ignored. A diamond is a stocky solid and it has no such troubles. That is one reason the case can be stated so cleanly.

The moment acts about one axis. A section loaded on the diagonal of its diagonal, about the other axis, has the corners in different places. A real bar bent in two planes sees the tips at some angle, and a flat in the wrong place does nothing.

Still open: the cut that moves with the moment #

A flat cut along a whole bar treats all sections alike, though the moment does not. A bar with a sharply varying moment has a highly stressed tip only near its peak, and it could keep the tip everywhere else, where the second moment is worth more than the modulus. Whether a tip that tapers out along the length, cut at the peak and left whole near the supports, can gain the elastic number without giving up the stiffness, and how that compares with the classic varying-depth shape of a fully stressed beam, is a question about how a cut’s benefit is traded against its loss along a span. It would need a calculation for the deflection of a section that changes along the length, which this single cross-section has no place for.