The wide side goes inside

A bar that was bent before it was loaded has its neutral axis inside its centroid, and the stress across it is a hyperbola rather than a straight line. That rung establishes the correction. This one is about what to do with it.

The hero is a hook of trapezoidal section, 50 to 120 mm radius, 40 mm wide at the inside and 15 at the outside, carrying 100 kN. Cutting the throat leaves a face with a direct tension of 100 kN and a moment of 7.970 kN·m, and the stress reaches 497.5 N/mm² at the inner fibre — where the straight-beam calculation says 323.4, and where hooks break.

The question this rung answers is why that section is that shape.

Two effects, and only one of them is new #

Hold the area at 1,925 mm², the depth at 70 mm, the radii at 50 and 120, and the moment at 4 kN·m. Change only the distribution of the area across the depth.

Now turn the first section round, keeping every quantity in the comparison fixed and changing only which end of the depth the material sits at.

Read the two columns separately, because they behave completely differently.

The correction factor runs 1.378, 1.386, 1.415 across the three. That is a spread of under three per cent, and it is the quantity the rung below this one is about.

The straight-beam stress runs 162.3, 178.1, 220.3 — a spread of 36 per cent, and it is ordinary section-modulus arithmetic that has nothing to do with curvature at all.

So the shaping is mostly a straight-beam decision. Putting area near the inner fibre raises the section modulus about the inner fibre, exactly as it would in a straight beam loaded one way round, and that is worth most of the 39 per cent. The curvature correction adds a further couple of per cent in the same direction, because a section with its area concentrated near the inside has a smaller $e$ — the gap between centroid and neutral axis relative to the depth.

Both point the same way, which is why the shaping is unambiguous. In a straight beam a section shaped for one direction of bending is worse for the other; here there is only one direction, because the load hangs on one side of the throat and always will.

Which free body produced the moment #

The 7.970 kN·m in the hero is not read off a bending moment diagram, and it is worth deriving because the lever arm is the quantity the section shape moves.

Take the hook and cut it across the throat. Keep everything below the cut — the load, the shank of the hook it hangs on, and the lower part of the curve. One force enters that body: the 100 kN hanging on a line that passes through the centre of curvature, because that is where a load in the bight of a hook sits.

Resolve that force onto the cut face, at the section’s own centroid. It becomes a direct tension of 100 kN and a couple of $100 \times R$, where $R$ is the perpendicular distance from the load line to the centroid — 79.70 mm for the wide-in section.

So $R$ is a property of the section, not of the hook. It is the radius to the centroid, and moving material across the depth moves it. That is the third penalty of the previous section, and it is invisible in any calculation that treats the moment as given.

Two further things fall out of that free body and both are easy to get wrong.

The moment is taken about the centroid and the stress is zero somewhere else. Those are not in conflict: the moment is a statement about the resultant of the stresses, and the position where the stress vanishes is a consequence of the strain distribution. On a straight beam in pure bending the two coincide, which is why the distinction never arises there.

And the direct tension does not shift the neutral axis in the way an axial load usually does. It adds a uniform stress to a hyperbola, so the position of zero total stress moves outward — but the position used in the bending calculation, $r = R_n$, is a property of the geometry alone and does not move at all. Confusing the two is the commonest slip in a hand calculation of a hook.

What turning it round does to the hook #

That figure carries a third effect the profile comparison could not show, and it is the largest of the three.

The moment on a hook is $N \cdot R$, where $R$ is the distance from the load line to the section’s own centroid. Move the centroid outward and the lever arm grows with it — so a section shaped wide-side-out is not merely worse at resisting the moment, it is being asked for a larger one.

Three penalties, then, all from the same decision: a smaller section modulus at the governing fibre, a slightly larger curvature correction, and a longer lever arm. They compound to a factor of 1.52, and the only one of them a straight-beam intuition would have predicted is the first.

And the outer fibre goes the other way, which is what makes it a choice rather than an oversight. The wide-in section reaches 281.3 N/mm² of compression outside against the wide-out section’s 231.9. A designer looking at the outer fibre would turn the section the wrong way round and be right about the number they were looking at.

Shaping is a redistribution and it is paid for #

Nothing in the comparison of the three sections added material. The area was 1,925 mm² throughout, and the 39 per cent came from where it was put.

That makes the shaping a redistribution, and redistributions are always paid for somewhere. Here the payment is explicit and it is at the outer fibre: 281.3 N/mm² of compression on the wide-in section against 231.9 on the wide-out one. The section that is 39 per cent better inside is 21 per cent worse outside.

The trade is worth taking because the inside governs, for three reasons that are worth separating.

The stresses are larger there. 223.6 against 167.3 on the wide-in section, before the direct tension is added.

The direct tension adds to the inside and subtracts from the outside. 51.9 N/mm² each way, so the gap widens by 104 N/mm² on a hook carrying 100 kN.

And the inside is in tension. A compressive peak on a stocky forged section is not a failure mode; a tensile one at a forged surface, cycled thousands of times, is the failure mode. The two fibres are not equivalent even at equal stress.

So the shaping rule has a condition attached that is easy to lose: put the material where the tension is. A curved member loaded the other way — an arch rib, a portal haunch hogging rather than sagging — has its tension at the outer fibre and wants the trapezoid the other way round, and it is the same essay with the sign changed.

A hook is unusual in never being loaded the other way, which is why its section can be optimised so hard. Most curved members in structures see both signs and get a symmetric section, and the shaping argument then buys nothing at all.

The tension nobody adds #

The free body is the reason it cannot be left out. Cut the throat of the hook and take everything below the cut. The load hangs on a line through the centre of curvature, so what crosses the cut is a force of 100 kN along that line — which is to say a direct tension of 100 kN plus a moment of that force times its distance to the centroid.

Both are the same force resolved about the same point, and taking one without the other is not a conservative simplification, it is an incomplete free body.

They also land on the same fibre. The bending puts tension inside; the direct force puts tension everywhere. So the inner fibre gets both, the outer fibre gets their difference, and the shaping argument of the previous section is reinforced: every effect on this section is worst at the inside.

Which is the whole of the design rule. Put the steel where all three things are worst, accept that the outer fibre is then the less efficient one, and shape the section so that the fibre that governs is the one carrying the most material.

Why $e$ is the difficult number #

The correction is governed by $e = R - R_n$, the gap between the centroid and the neutral axis, and every stress on this page divides by it.

For the wide-in trapezoid $e$ is 4.65 mm on a section 70 mm deep, and $R_n$ itself is computed from $A / \int \mathrm{d}A/r$ — an area divided by an integral, both of which are large numbers whose ratio is close to $R$. Subtracting one from the other is a difference of two nearly equal quantities, which is the classic way to lose precision, and the loss gets worse exactly as the bar gets straighter.

At $R/h = 30$ the gap is 0.26 per cent of the depth. Computing $R_n$ to four significant figures and subtracting leaves $e$ with no significant figures at all, and the stress that divides by it is arbitrary.

Which is why the correction is not simply applied everywhere. The arithmetic of a curved beam is unstable in precisely the regime where the answer is closest to the straight beam’s, so the sensible procedure is to use it where it matters and use $My/I$ where it does not — a rule about numerical conditioning masquerading as a rule about mechanics.

Two defences exist and this site’s generator uses the second.

Compute $e$ from a series in $h/R$ rather than by subtraction, which keeps the cancellation out of the arithmetic and is what handbooks tabulate.

Or compute $R_n$ by quadrature and check the closed form against it, which is what happens before anything on this page is divided by $e$. The check is not decoration: a closed form for $\int \mathrm{d}A/r$ over a trapezoid contains a logarithm of a ratio of radii, and that logarithm is the other place the same cancellation hides.

It is the same hazard the section’s two areas raise from the other side — a section property that is a small residue of large terms is a property whose accuracy is a computational question rather than a modelling one.

Where the correction stops mattering #

That convergence is the honest limit of this essay’s argument. The shaping is worth 39 per cent at $R/h = 1.14$ and almost nothing at $R/h = 10$, not because the correction has gone away — it has, but it was never the large term — but because the reason to shape a section for one fibre disappears when the bending is not so overwhelmingly one-sided.

The straight-beam stress is identical, which is the cleanest statement of the split this essay is built on. Curvature changes the correction and nothing else; the section shape changes the base number and the correction together; and the base number is the larger of the two.

What the section costs to make #

Three practical consequences follow from a section shaped this hard, and they are the reason the shaping is not carried further than it is.

A trapezoid is a forging, not a rolled section. The taper across the depth has to be produced by dies, which is economic for a component made in thousands and not for a one-off. A curved member fabricated for a single structure is made from plate or from rolled sections, and its shaping is limited to what can be cut and welded.

The inner fibre is the hardest surface to finish. It is on the inside of a curve, it is where the forging flash and the die parting line tend to land, and it carries the peak of every stress on this page. A surface defect there is a fatigue crack starter at the one location the essay has spent its length identifying.

And the inspection criterion is not a stress at all. A hook in service is checked by measuring the throat opening — the distance across the mouth, against a stamped original — because a hook that has yielded at the inner fibre opens permanently and does so long before it breaks. That measurement is available to somebody with a rule and no calculation, and it is the sensible criterion precisely because the elastic peak this essay computes is not what the hook eventually fails at.

Which puts the arithmetic in its place. It says where to put the material and how much margin the shape delivers. It does not say when to take the hook out of service, and the quantity that does is a length anybody can measure.

The same shaping argument elsewhere on this site #

A section shaped for one fibre because only one fibre governs is a recurring move, and it is worth collecting the instances, because the condition that licenses it is the same every time.

A reinforced concrete beam is the purest case: the steel goes where the tension is and the concrete carries the compression, and the section is asymmetric because the two materials are. Where a bar may stop is the same argument along the length rather than across the depth.

A crane rail’s head is thickened because wear and contact stress are on one surface. A hook’s throat is thickened because tension is on one fibre. Neither section would survive being turned over.

A plate girder with unequal flanges is the version that fails most often, because a girder that is asymmetric for its sagging region is wrong in its hogging one, and the section that is checked is not always the one chosen is what that looks like when the asymmetry is produced by a cut rather than by a choice.

And a channel or an angle is asymmetric for reasons of assembly rather than of stress, which is why the shear centre is somewhere the load is not and why those sections twist under a load applied where anybody would apply it.

The rule the collection suggests is narrow and worth stating as such. Shape a section for one fibre only when the sign of the bending cannot change, and check that nothing about erection, reversal or a second load case can turn it over. A hook satisfies that condition absolutely. Very little else does.

What to carry away #

The wide face goes inside, and it is worth 39 per cent. Same area, same depth, same curvature, same moment.

Most of that is ordinary section-modulus arithmetic. The curvature correction spans three per cent across the sections compared; the straight-beam stress spans thirty-six.

And the hook adds a fourth effect the profile cannot show. Turning the section round moves the centroid outward, which lengthens the lever arm and increases the moment being resisted — so the total penalty is 52 per cent rather than 39.

Every term is worst at the inner fibre. Bending, curvature and direct tension all peak there, which is why there is no trade-off in the shaping.

Where the model stops #

Everything here is elastic. A hook that is overloaded yields at the inner fibre and redistributes towards a plastic hyperbola, and its collapse load is well above what the elastic peak suggests — which is why hooks in service are inspected for permanent opening of the throat rather than for cracks.

No stress concentration is modelled. The throat of a real hook is a fillet, not a corner, and the concentration there sits on top of every number on this page — the same problem a weld group’s geometry has at the point that governs.

The section is prismatic and a real hook’s is not. A forged hook’s section varies continuously round the curve, and the throat is chosen as the critical one on the argument that both the moment and the curvature are worst there — which is true and is not the same as proving no other section governs.

And fatigue is the actual design case. A lifting hook is loaded and unloaded thousands of times, at a stress range this essay computes the peak of but says nothing about, and the inner fibre is a forged surface whose finish decides the category.

The ladder from here #

Later rungs on this anchor: the closed-form $\int \mathrm{d}A/r$ for the sections that have one, and numerical integration for those that do not. Deflection of curved members, where Castigliano earns his place because the geometry defeats direct integration. Radial stress in a curved I-section, where the flange tries to pull itself off the web and the web needs a thickness nothing about bending would have asked for. The plastic hinge in a hook, and the throat opening that is the real inspection criterion. The closed ring and the chain link, which are indeterminate and where the stiffest path takes the load meets the curvature correction. And initial curvature read as an imperfection rather than a shape, which is where this argument runs into column buckling.

Hooks were trapezoidal for a century before anyone could say by how much it helped. The shape came from forging practice and from watching where hooks broke, and Winkler’s theory of 1858 arrived to explain a section that blacksmiths had already converged on. That is the ordinary order of events for a component made in large numbers: the geometry is optimised by failure long before it is optimised by arithmetic, and the arithmetic’s contribution is to say how much margin the optimisation left.