How far a wrong load reaches

Every load in this collection is a lie of some kind. A wheel is drawn as a point and is really a contact patch. A wall’s weight is drawn as a uniform line and is really a stack of bricks. A bearing reaction is drawn as an arrow and is really a pressure distribution over a plate, which is itself an idealisation of what happens between two surfaces that are neither flat.

The reason any of it works is Saint-Venant’s principle, and the reason the principle works is not a theorem about equilibrium. It is a statement about a length.

Which free body produced the number #

Two statically equivalent load systems differ by a system with no resultant force and no resultant moment. Subtract one from the other and what is left is self-equilibrating, and the whole principle is a claim about what such a system does.

So take the simplest self-equilibrating load there is: a normal traction on the boundary of a half-plane varying as $\cos(2\pi y/\lambda)$. It pushes as much as it pulls, over every wavelength, so its resultant is zero by construction.

The Airy stress function

$$\phi = \frac{p}{\alpha^2}(1 + \alpha x)\,e^{-\alpha x}\cos(\alpha y), \qquad \alpha = \frac{2\pi}{\lambda}$$

satisfies the biharmonic equation and both boundary conditions — the applied normal traction, and no shear on the loaded face. Differentiate it twice and

$$\sigma_{xx} = -p\,(1 + \alpha x)\,e^{-\alpha x}\cos(\alpha y)$$

The decay length is the load’s own wavelength, and nothing else appears. No modulus, no Poisson’s ratio, no thickness, no size of the loaded region except through $\lambda$ itself.

Which converts the principle into arithmetic #

A self-equilibrating traction across a depth $h$ must change sign at least twice — once to have no resultant force and once more to have no resultant moment. Its slowest Fourier component therefore has a wavelength of about $h$.

At $x = h$ that component is down to 1.4 per cent, and every faster component is down further. One depth is the answer, and it is an answer rather than a rule of thumb: it is $(1 + 2\pi)e^{-2\pi}$.

That single number is the licence under which every figure on this site is drawn. A point load is drawn as a point because a distance of one depth from it, the difference between the point and whatever the real contact was is under two per cent — and every stress the essay is about is quoted somewhere else.

The exact answer, which oscillates #

The half-plane is not a strip, and the strip has its own decay rates: the Papkovich–Fadle eigenvalues, the roots of $\sin 2\lambda \pm 2\lambda = 0$ for a strip with traction-free long edges.

They are complex. Solving the symmetric equation by Newton’s method in the complex plane gives $2.1062 + 1.1254i$, and the antisymmetric one gives $3.7488 + 1.3843i$.

The real part is the decay, and it agrees with the half-plane argument closely: measured from the centre of a strip of depth $h$, the first symmetric mode loses 99 per cent of itself in 1.09 depths against the half-plane’s 1.06. Two quite different routes, one number.

The imaginary part is the interesting one, because nothing in any statement of the principle mentions it. The residual stress changes sign as it dies away. It is not a decaying bump; it is a decaying oscillation, with a half-period of about 1.4 depths — so the stress a per cent past the end of the disturbed region is on the other side of the beam-theory answer, and a designer who has computed a peak near a discontinuity has not found the only place where the section is not doing what the model says.

It is worth saying plainly that the principle was a conjecture for more than a century. Saint-Venant stated it in 1855 as an observation about torsion and bending of prisms; Boussinesq gave it its general form thirty years later; and nobody proved anything until the middle of the twentieth century, when the honest statements turned out to be about energy rather than about stress. The theorem that exists says the strain energy stored beyond a distance falls exponentially, at a rate belonging to the body’s geometry — which is exactly what the eigenvalue above is, and which is why a counterexample has to be a body with a second length scale rather than a body with an unusual load.

The energy form also says what a body has to look like for the principle to fail, and one such body is a plate with a hole in it. A hole raises the stress by a factor that depends on its shape and not on its size, and the elevation dies away over the hole’s own dimension rather than the plate’s — the same wavelength argument, with the loaded region being the hole. Where the hole is small the disturbed region is small and nobody notices; where it is a fair fraction of the plate there is nowhere left for the disturbance to die in, which is the same failure the closing section of this essay reaches from the other end.

Three statically equivalent loads, and how long they stay different #

The principle is a statement about a difference, so the cleanest test of it is to build three loads with the same resultant and the same moment and watch them converge.

A point load at mid-depth starts at twenty times the mean stress. The same force spread over a fifth of the depth starts at five times. The same force split into two starts at nearly ten. All three are under a tenth of the mean by about three quarters of a depth, and under a hundredth of it by about 1.2.

That is the whole of it, and it is worth noticing how fast it is. The distinction between a point load and a patch — which decides everything about the bearing, the plate, the local crushing and the reinforcement — has ceased to be measurable within one depth of the place where it was decided.

The corollary is the one every essay on connections in this collection depends on: the region where it is not yet true is exactly the region a connection lives in. A joint is not a point precisely because it occupies the length over which Saint-Venant has not yet arrived, and a strut-and-tie model is the tool for that region for the same reason.

What plane sections is really assuming #

Plane sections stay plane is usually presented as a kinematic assumption about bending. It is better read as a consequence of what has just been derived.

Any deviation from a plane section is a strain distribution with no net force and no net moment — because the force and the moment are exactly what the plane part of the distribution carries. So a deviation from plane sections is a self-equilibrating system, and self-equilibrating systems decay over their own wavelength.

Plane sections is therefore not an assumption about materials or about bending. It is Saint-Venant’s principle, restated for the interior of a member. Which explains why it holds so well in the middle of a span and fails so completely at a support: nothing has changed about the material, and everything has changed about how far away the disturbance is.

The load that does not decay #

There is one self-equilibrating load in this collection that travels metres rather than depths, and it is the exception that says what kind of statement the principle is.

A bimoment on a thin-walled open section is four flange forces — two pushing, two pulling — arranged so that there is no axial force, no shear and no bending moment about either axis. Every resultant is zero. It is exactly the kind of load system Saint-Venant’s principle is about.

It decays over $\sqrt{EI_w/GJ}$, which for the 400 mm section here is 1.5 metres, or 3.8 depths. On a longer, thinner section — a channel purlin, a cold-formed zed — the same expression gives ten depths or more, and on a member only a few metres long the bimoment applied at one end has not decayed by the time it reaches the other.

The reason is visible in the expression. $\sqrt{EI_w/GJ}$ contains the ratio of two stiffnesses — the section’s resistance to warping against its resistance to twisting — and an open thin-walled section is enormously stiffer against the first than the second. Saint-Venant’s principle assumes a body with only one length scale in it. A thin-walled member has at least two, and the second is not a dimension of the cross-section at all.

A bimoment is a stress resultant with no resultant: it appears on no diagram, contributes to no equilibrium equation, and is the reason the principle that licenses this whole collection has an exception with a name.

The exception is not confined to thin-walled beams. Any body with a second length scale in it will produce one, and the cleanest example is a surface rather than a section.

That is the same shape of answer as the first two figures on this page and a completely different length. The disturbance is self-equilibrating, it decays exponentially, and it is gone within about three of something — but the something is $\sqrt{Rt}$ rather than a depth, and for this cylinder it is 170 mm against a radius of four metres. A rule of thumb quoted in section depths would have been wrong by a factor of twenty-three.

The shell also carries a ratio worth having beside the bimoment. The bending stress at the held edge is 1.82 times the membrane hoop stress the whole design is about, and that multiple is $\sqrt{3}/\sqrt{1-\nu^2}$: it contains no pressure, no radius and no thickness. A designer who sizes the wall for hoop tension and then welds it to a base has doubled the stress at the weld, at every pressure and every size, and the disturbance responsible is gone half a metre away.

The two-to-one spread, which is this principle wearing a hard hat #

Every design office rule about load dispersion is Saint-Venant’s principle with the arithmetic replaced by a slope.

A load on a bearing plate is taken to spread through the plate at 45 degrees; a load on a slab is taken to spread at two horizontal to one vertical; a patch load on a girder web is taken to be effective over a length that grows with the flange thickness and the fillet. None of those is derived anywhere. All of them are saying the same thing: the width over which a load is effective grows with distance from where it was applied, at a rate of order one.

The dispersion angle is that rate. A spread of 45 degrees means a load is distributed over a width of about $2x$ at a distance $x$, which is the same statement as “the disturbance has a wavelength of about $x$ and has therefore decayed” — and the reason the rules use a slope rather than an exponential is that a slope can be drawn on a detail and an exponential cannot.

What the slope loses is the fact that the convergence is quick. A designer who spreads a load at 45 degrees through 200 mm of concrete has, according to the arithmetic above, already lost more than nine tenths of the difference between a point load and a patch. The rule is conservative in the direction it usually needs to be, and its conservatism is not where it appears to be — it is not in the angle, it is in treating a continuous decay as though the load became uniform at some particular depth and not before.

The clearest case of a profession pricing that assumption is a gusset plate, because a gusset is the one piece of steel on this site that arrives with no cross-section at all.

Nothing in the arithmetic of this essay produced thirty degrees. What it produced was the statement that the width grows with distance at a rate of order one, and thirty degrees is one reading of “of order one” that somebody measured once and everybody has used since. The sensitivity is the part worth knowing, and it is easiest to see by moving the angle to the other rule in the same paragraph.

Ten degrees of angle is about a third of the answer, and there is no derivation anywhere that fixes the angle to better than that. The rules survive because the check they feed is rarely the one that governs — on this plate the Whitmore section buckles well before it yields, at 721 kN against a yield of 1564 in the 30° reading and 1120 against 2428 in the 45° one — and because a real spread converges so fast that the width is nearly right long before the rule says so. The same substitution of a slope for an exponential is what makes a patch load choose its own effective length on a girder web rather than take the one the detail imposes.

The principle decides how long a test specimen has to be #

Saint-Venant’s is usually met as a licence to simplify a calculation. Its most practical use is in the opposite direction: it says how much material an experiment needs before it is measuring anything.

A tensile coupon is gripped at both ends, and the grips apply a load distribution nothing like the uniform stress the test is about. So the gauge length has to sit far enough from both grips for the difference to have decayed — one specimen width is the usual rule, and it comes from the arithmetic above rather than from custom. A short coupon does not give a wrong number with a wide error bar; it gives a confident number about the grip.

The same reasoning fixes the length of a column test, the position of strain gauges on a beam specimen, and the distance from a weld at which a residual stress is quoted. In each of them the question is not what the load is but how far away the measurement has to be before the load’s shape stops mattering — and the answer is a section dimension, for the reason this page derives.

It is worth following that through to the end, because it says where every material property in this collection came from. A stress-strain curve is a measurement made at a gauge length chosen so that the grips have stopped being visible; the yield stress, the ultimate stress and the elongation are all read off it. So the one number a stronger steel does not change is a number whose very existence depends on the principle this essay is about, and a coupon too short to satisfy it does not measure a worse modulus — it measures the grip.

Why a refined mesh gives a worse answer #

The principle has a use that did not exist when it was stated and is now the commonest one: it says where a numerical model may be cut, and it explains a result that otherwise looks like a bug.

Apply a load to a single node of a finite element model and refine the mesh. The stress at that node does not converge. It grows, without limit, and it grows at a rate set by the element size: halve the elements and the peak rises again. A modeller watching that happen concludes the model is wrong, and the model is right — a point force on a continuum has infinite stress under it, so a mesh converging on the continuum solution is converging on infinity.

Now read the stress one depth away. It converged two refinements ago, and it has not moved since. Everything in this essay says it will not: the difference between the point load and any statically equivalent patch is a self-equilibrating system, and self-equilibrating systems are gone by then. The model is simultaneously divergent at the load and converged everywhere that matters, and the two facts have one cause.

The same argument sets where a sub-model may be cut out of a global one. Extract a region of interest, apply the parent model’s displacements or forces to its boundary, and mesh it finely: the answer inside is right provided the cut is at least one characteristic dimension away from anything being looked at, because the difference between the parent’s crude boundary distribution and the true one has decayed by then. Cut closer and the sub-model is a study of its own boundary conditions.

And it decides how large a model of a detail has to be. A bracket, a cope, a bolt group, a hole: the surrounding material has to extend a dimension or so past the feature in every direction before the artificial boundary stops being visible in the answer. Extending it further is free of information and not free of time, which is why the rule is a specific distance rather than “as much as fits”.

The general form is worth carrying into any analysis, numerical or not: a model is trustworthy at a distance from its own lies. Saint-Venant’s principle is what converts that sentence into a number, and the number is a section depth.

Where the model stops #

The half-plane solution is exact for a half-plane and an approximation for a strip. It gets the mechanism and the length scale right and does not know that the strip’s far face is free, which is what makes the exact eigenvalues complex.

The Fourier decomposition treats the end load as a boundary traction on an elastic half-space. A real end load is delivered through a plate, a weld or a bearing, whose own stiffness redistributes it before the member ever sees it — so the “point load” case here is the worst case rather than a case.

Everything is linear elastic. A local yield at the point of application redistributes the peak and shortens the decay, in the direction of safety, which is why the principle survives in structures whose materials do not obey any of the arithmetic above.

And the whole essay is about a plane problem. In three dimensions a self-equilibrating patch on a surface decays as the inverse cube of distance for a doublet and faster for higher-order systems, which is quicker than anything drawn here — so the plane result is the conservative one, which is a pleasant place for an approximation to land.

What the pictures cannot show #

The curves above are drawn against a distance measured in depths, and the region they are most interesting in — the first quarter of a depth — is where the model is least reliable, because that is where the loading detail the model has abstracted away actually is.

Nor can they show the thing that makes the principle useful in practice, which is not that stresses die away but that nobody has to model the end. The value of Saint-Venant’s principle is a permission to stop: to draw a bearing as an arrow, compute the member, and know that the arrow’s dishonesty is confined to a region that will be detailed separately by somebody applying a different set of rules.

The assumption the figure rests on #

The disturbance is assumed to be applied to a member that is otherwise uniform and continues far enough for it to decay in.

Both parts fail routinely. A beam with a bearing at one end and a splice a metre away has two disturbed regions overlapping, and their sum is not either of them. A short cantilever bracket is entirely disturbed region — there is nowhere in it that is a depth from both the load and the support — which is why brackets are designed by strut-and-tie rather than by section, and why the anchorage zone has an essay of its own.

The honest form of the principle is therefore conditional: stresses converge to the beam-theory answer at a distance of about one depth from a disturbance, provided there is a depth of member to converge in. A great many of the places where structures actually fail are places where there was not.

That is also the honest relation between this essay and the one the whole collection starts from. Everything a cut reveals about internal forces is exact, and it says nothing whatever about how those forces are distributed across the cut. Saint-Venant’s principle is the bridge between the two — and the length of the bridge is one depth, which is why a free body drawn a depth from anything is a calculation and one drawn at a bearing is a detail.