The bar that was bent before it was loaded

A crane hook is a beam that was bent before anybody hung anything on it. Its throat — the section directly across from the load line — is a piece of steel in bending, and everything the trade knows about beams ought to apply to it. It does not, and the failure is not a small one: the section at the throat carries roughly half as much again as the standard bending formula reports, at the exact fibre where hooks break.

What makes it worth an essay is that the failure has nothing to do with the assumption everyone expects to blame.

The assumption survives, and is not enough #

The bending formula is usually taught with one hypothesis attached. Plane sections stay plane: a flat cut face remains flat after the beam bends, merely rotating about some axis in the section. Everything else — the linear strain, the linear stress, the second moment of area, the section modulus — is arithmetic laid on top of that one geometrical claim, and where the claim fails the whole edifice fails with it.

In a curved bar the faces stay flat. Two plane sections a small angle apart rotate relative to one another, remaining plane throughout, exactly as in a straight beam. Photoelastic work and finite-element analysis both confirm it, and the exact elasticity solution quoted further down assumes nothing about plane sections at all and produces the same answer.

The formula is wrong anyway, and the missing hypothesis was never stated because in a straight beam it is invisible.

The fibres of a curved bar are different lengths before anything is applied. A fibre at radius $r$ subtending an angle $\theta$ has length $r\theta$. The fibre at the inner face is shorter than the fibre at the outer face, in the ratio $r_i / r_o$ — for the hook above, 50 to 120, which is a factor of 2.4. In a straight beam every fibre has the same length and the point never arises.

Strain is a change in length divided by the original length. Equal rotation of two plane faces gives every fibre the same change in length. Dividing that same change by lengths that differ by a factor of 2.4 gives strains that differ by a factor of 2.4 in the other direction. Strain goes as $1/r$, not as $r$ — and since the material is still perfectly elastic and perfectly ordinary, so does stress.

The hyperbola, and where its zero went #

Two things have happened, and they push in the same direction.

The stress is a hyperbola rather than a straight line, so it is steeper near the inner face and flatter near the outer. And the axis of zero stress has moved toward the centre of curvature, which lengthens the lever arm to the outer fibre and shortens it to the inner one. Both effects raise the inner stress and lower the outer. The straight formula gets the peak wrong by 38% here, in the unsafe direction, and it gets the outer fibre wrong by 32% in the safe one.

The neutral axis is worth deriving, because it is the number the whole theory turns on. The face carries no net force, so

$$\int_A \sigma \, dA = 0 .$$

With $\sigma \propto (R_n - r)/r$ that condition becomes $\int (R_n - r)/r \, dA = 0$, and rearranging gives

$$R_n = \frac{A}{\displaystyle\int_A \frac{dA}{r}} .$$

That is a harmonic mean of the radii weighted by area, where the centroid $R$ is an arithmetic one. A harmonic mean is always the smaller, so the neutral axis is always inside the centroid, in every curved bar there has ever been. It never coincides with the centroid and never sits outside it.

Everything then divides by the gap between them,

$$e = R - R_n, \qquad \sigma = \frac{M\,(R_n - r)}{A\,e\,r} ,$$

and $e$ is a difference of two nearly equal numbers. For this hook it is 4.65 mm out of a 70 mm depth. For a mildly curved beam it is a fraction of a millimetre, which is why hand calculations of curved bars were notorious: five-figure accuracy in $R_n$ buys three figures in $e$ and two in the answer. The solver behind these figures computes $\int dA/r$ in closed form and then checks it against a 20,000-point quadrature before anything is divided by it.

What the straight formula was actually saying #

Side by side, the structure of the error is clear. $\sigma = My/I$ is not a general consequence of plane sections; it is the special case of $\sigma = M(R_n - r)/(Aer)$ in the limit where the fibres are equal. Take $R \to \infty$ with the depth held and $R_n \to R$, $e \to 0$, and the hyperbola straightens into the familiar line. The straight formula is the outer limit of the curved one, and the second moment of area is what $A\,e\,R$ collapses to when it gets there — the same relationship that a pair of forces has to the stress block, a resultant description exactly right in one geometry and quietly conditional everywhere else.

Checked against a theory that assumes nothing #

The argument so far still contains an assumption — that the plane faces rotate rigidly, which is Winkler’s hypothesis and which was asserted rather than proved. There is a way to check it that does not use it.

Golovin, in 1881, solved the curved bar in pure bending as a problem in plane-stress elasticity: an Airy stress function in $r$ alone, with zero radial traction on both curved faces and a moment on the ends. Nothing in it says anything about plane sections. The two answers can be drawn on top of each other.

Four parts in a thousand, on a bar curved about as sharply as anything gets made. Winkler’s theory is not an approximation that happens to be good; it is very nearly the elasticity, and the plane-sections assumption it rests on turns out to be true rather than merely convenient — a rarer outcome than it sounds. It also means the whole discrepancy with $My/I$ is attributable to one thing. Not to shear, not to the ends, not to any subtlety of the elasticity: entirely to fibres that started at different lengths.

Which free body produced the number #

The number quoted at the top of this page is not the bending stress. It is larger, and the reason is a free body rather than a formula.

Cut the hook through its throat and take everything below the cut — the lower half of the hook, the sling, and the load. The load $P$ hangs on a line that passes through the centre of curvature, and the cut face is off to one side of that line. So the face has to supply a direct tension $N = P$ to balance the load vertically, and a moment $M = N \cdot R$ to balance it rotationally, where $R$ is measured from the load line to the section’s own centroid. The two are not alternatives, and neither is optional. Choosing the free body is the whole skill; getting this one wrong loses a tenth of the answer silently.

Ten per cent is not the interesting part. The interesting part is that both terms peak at the inner fibre and neither has a compensating term there. Curvature raises the inner stress, the direct tension raises it again, and the outer fibre — compression falling from a predicted 219.4 to an actual 166.6, then to 140.6 once the tension is added — is doing progressively less. A hook is a section in which all the demand has collected at one edge.

Why a hook is that shape #

The trapezoid follows directly, and it is one of the cleaner cases of a shape that exists because of an equation rather than because of a process.

Widening the section at the inner fibre and narrowing it at the outer does three separate things at once, and only the first is obvious.

It puts area where the stress is. More material at the fibre carrying 249 N/mm² and less at the fibre carrying 141.

It moves the centroid inward, which shortens the lever arm. The trapezoidal hook’s centroid sits at 79.70 mm against 85.00 mm for a rectangle between the same two radii. Since $M = N \cdot R$, the moment the section has to carry falls by 6% — the shape has reduced its own load, which is not something a straight beam’s shape can do.

It moves the neutral axis inward too, to 35.8% of the depth from the inner face rather than the 42.8% the rectangle manages, shortening the inner lever arm further.

The three together are worth measuring. The trapezoid has an area of 1,925 mm² and peaks at 248.7 N/mm². A rectangle of identical area between the same two radii — 27.5 mm wide — peaks at 288.3. The hook shape is 13.7% better for the same steel, in a component where the steel is the whole cost. The same argument about shape rather than material reaches a factor of forty in a straight beam; here it reaches fourteen per cent, and fourteen per cent is enough to have fixed the geometry of a machine component for a century and a half.

Where the correction matters, and where it does not #

The obvious next question is when to bother, and the answer has a shape worth seeing rather than a threshold worth memorising.

Three per cent at ten depths of radius; forty-five per cent at one. The curvature correction is not a refinement to be applied everywhere but a different answer in one region, and the region is narrow: hooks, chain links, C-frames, press frames, clamps, ring segments, the eye of a lifting shackle. Anything a designer would describe as a curved beam — an arched roof rib, a bowed girder — sits at R/h of twenty or more and is straight for this purpose to within the accuracy of anything else in the calculation. That is a different question from the one the thrust line asks of an arch, which is curved so that it need not bend at all. A hook is curved for reasons of use and bends because of it.

The same peak, arriving from somewhere else #

An engineer meeting a stress several times what the load over the area would suggest has usually met a stress concentration. This is not one, and the distinction is worth being exact about.

A stress concentration decays over a few radii and is governed by Saint-Venant’s principle; a curved bar’s elevation does not decay at all, because it is not a disturbance. Every section of the curved region carries the hyperbola. So the two stack: a hook with a hole through it, or a sharp fillet at the throat, multiplies an already elevated field by the geometric factor of the notch, on top of the factor the curvature had already supplied. And the two behave differently once the material yields.

A stress concentration in a ductile material is largely a fatigue problem rather than a static one, because a small yielded zone at the notch redistributes and the section carries on. A curved bar’s elevated inner fibre yields across the whole width of the section, so the redistribution is a change of state rather than a local accommodation — and the section still has substantial reserve, since the outer half of it was never near capacity. Static failure of a hook is well past first yield and is preceded by a visible opening of the throat, which is why hooks are gauged rather than tested. Under repeated lifting the reserve is worth nothing. Fatigue counts cycles at the elastic stress, and the elastic stress at the inner fibre is the 248.7 rather than the 161.7, so a hook designed with the straight formula sees an endurance calculation performed on a number 35% below the truth. That, rather than a static rupture, is how the error usually presents itself.

Where the model stops #

The section is prismatic and the curvature is constant. A real hook varies both along its length. The formula is applied section by section, and the resulting field is not in equilibrium with itself between sections — a small error nobody has ever measured on a hook.

The material is linear. Everything above divides by $e$ and multiplies by $M$, which requires a modulus the same at both faces. Past first yield the inner fibre softens, the neutral axis moves again, and the elastic answer overestimates the stress and underestimates the capacity.

Radial stress has been ignored. A curved bar in bending develops a radial stress as the fibre forces try to straighten. For a solid section it is small; for a curved I-section it governs, and the shear flow that no diagram shows has a radial cousin here that is worse.

The load line passes through the centre of curvature. That is what makes $M = N \cdot R$. A hook loaded off that line, or a sling at an angle, has a different moment and a different free body, and the difference can go either way.

Nothing here is a stress concentration. The fillet at the root of the throat, the machining marks, the hole for the safety catch: each multiplies the number this page computes, and none of them is in it.

What the pictures cannot show #

Every figure on this page is a section — a stress plotted against radius across one cut face. The hook itself appears in two of them as an outline, and the outline is doing no work.

That hides the thing an engineer would most like to see: how the peak moves around the hook as the load line swings. The throat is the worst section for a load hanging straight down and is not the worst section for a sling pulling at 45°, and finding the governing section means repeating this calculation at every angle rather than reading it off a picture.

The figures also cannot show the assumption that makes them possible: that the curved bar is in pure bending plus direct tension, with the shear on the cut face ignored. A cut through the throat of a hook carrying a vertical load has no shear on it, which is exactly why the throat is where the calculation is done — the section was chosen to make the assumption true. Choosing where to cut has quietly done half the work before any arithmetic began.

And the hyperbola is drawn as a smooth curve to the very inner fibre, where the real material has a surface, a finish and a residual stress field from forging. The last half-millimetre of every figure here is a mathematical extrapolation into a region the theory does not describe.

The ladder from here #

Later rungs on this anchor: the closed-form $\int dA/r$ for the sections that have one, and what to do about the ones that do not. Deflection of curved members, where Castigliano’s theorem earns its place because the geometry defeats direct integration. The closed ring and the chain link, which are indeterminate and where the curvature correction meets a redundant structure sharing load by stiffness. Radial stress and the curved I-section, where a flange tries to pull itself off the web. Curved bars past yield and the plastic hinge in a hook. Initial curvature as an imperfection rather than a shape, which is where this argument meets column buckling. The curved bar as a two-dimensional elasticity problem, and what Golovin’s solution says that Winkler’s cannot. And the analogy nobody expects — a surface that carries by being curved is the same geometry used the opposite way round, where curvature removes bending instead of complicating it.

Winkler published the theory in 1858, and it has survived unchanged, which is unusual. The reason it survived is visible in the third figure on this page: an approximate theory that agrees with the exact elasticity to four parts in a thousand does not get improved on. It gets taught, forgotten, and rediscovered every time somebody works out why a hook broke at a load the arithmetic said it could hold.