The hole that multiplies the stress by three

Every stress on this site has been an average over a cut. A tensile stress is a force divided by an area, a bending stress is a moment divided by a section modulus, and both are statements about a whole face rather than about any point on it. That is a sound way to work as long as the stress really is nearly uniform over the region it is being averaged across, which is what Saint-Venant’s principle licenses.

Put a hole in the plate and it stops being true near the hole. The interesting question is by how much, over what distance, and whether it matters — and all three have exact answers.

The factor does not depend on the size of the hole #

This is the first surprise and it is a strong one. A hole ten millimetres across and a hole a metre across, in plates large enough that the edges are far away, both multiply the applied stress by exactly three.

The reason is dimensional. The solution has one length in it, the hole radius $a$, and one other length, the distance from the centre $r$ — so every dimensionless quantity in the answer can depend only on the ratio $a/r$. At the hole’s edge that ratio is 1 whatever $a$ is, and the concentration factor is therefore a pure number.

The consequence for design is worth stating plainly: a bigger hole is not a worse hole, it is a bigger hole. What a large hole costs is net section — there is less material left to carry the load — and the peak stress at its edge is three times the net stress rather than something worse. The two effects are separate and both are computable, and the intuition that a big hole is dangerous because it concentrates stress more is simply false.

The compression nobody mentions #

At the crown of the hole — the point on the loading axis — the hoop stress is $-\sigma$. A plate in pure tension has a region of compression in it, and the compression is a third of the peak tension in magnitude.

The mechanism is Poisson’s: the plate is being stretched along the load axis and is therefore contracting across it, and the material immediately above the hole has nothing beneath it to resist the contraction. It closes in over the hole, which puts it into compression along the direction it is closing.

That is a real stress and it does real things. A plate with a hole, loaded in tension and unloaded repeatedly, has a location on it that cycles into compression on every tensile cycle. It is also why the same hole in a plate loaded in compression has tension at its crown, which is where a fatigue crack in a compression member starts — a crack growing under a nominally compressive load, at a location that is in tension for a reason that has nothing to do with the sign of the applied stress.

How far it reaches #

The decay panel answers a question this site has asked before without a number on it. Saint-Venant’s principle says that the details of how a load is applied stop mattering a short distance away, and “a short distance” is usually given as “of the order of the dimension of the region”.

Here it can be made exact. The hoop stress on the transverse axis is

$$\sigma_{\theta\theta}(r) = \frac{\sigma}{2}\left(2 + \frac{a^2}{r^2} + 3\frac{a^4}{r^4}\right)$$

which is 3σ at $r = a$ and falls quickly. It is within 5% of the applied stress at 3.52 radii and within 1% at 7.27. So a hole disturbs the plate over a region a few times its own size, and everything outside that is undisturbed to a precision better than the material properties are known to.

That is Saint-Venant’s principle with a number on it rather than a gesture, and the number is useful: it says that two holes more than about seven radii apart do not interact, which is why a bolt group’s holes can be treated independently and why a perforated plate’s behaviour changes character when the perforations get close.

What the concentration costs a real member #

Putting the two effects together for a bolted tension member gives the number a designer actually needs.

A plate 200 mm wide and 10 mm thick, in steel with a yield stress of 355, has a gross capacity of 710 kN. Drill a 22 mm hole through it and the net area falls by 11%, so the net-section capacity is 632 kN. That is the whole of the static penalty.

The elastic peak stress at the hole, at that load, is three times the net stress — which is 3 × 355 = 1065 N/mm², a stress the material cannot reach. What that means is that the member has yielded locally long before it gets there, at about a third of its capacity, and has spent the remaining two-thirds redistributing. By the time the net section is fully plastic the stress across it is uniform and the factor of three has vanished from the arithmetic entirely.

Which free body produced the number #

The free body is the whole plate, and this is the one calculation in this field that is not solved by cutting anything.

Kirsch’s solution is obtained by writing the two-dimensional elasticity problem in polar coordinates, choosing an Airy stress function whose form is dictated by the boundary conditions — uniform stress at infinity, no traction on the hole’s surface — and solving for its coefficients. Equilibrium is satisfied identically by the construction of the stress function; what has to be imposed is compatibility, which is the statement that the strains derived from the stresses correspond to some actual displacement field.

That is a different method from anything else on this site and it is worth naming the difference. Every other number here comes from cutting a free body and insisting the sums cancel. This one comes from writing down a field that satisfies equilibrium everywhere by construction and then finding the member of that family that fits the boundaries. It is the elasticity-theory route rather than the statics route, and it is what a problem with no statically determinate cut requires.

The checks are on the boundaries. At $r = a$ the radial stress and the shear both come out zero at every angle, which is what “the hole’s surface carries nothing” means. Far from the hole the field tends to the uniform state — $\sigma_{xx} \to \sigma$, everything else to zero. And the whole field satisfies equilibrium exactly, so a cut taken anywhere returns the applied load: integrating the hoop stress across the net section gives the applied force, with the peak of 3σ and the deficit near the hole exactly compensating.

Where the model stops #

The plate is elastic, and a ductile plate is not for long. At an applied stress of a third of yield the peak has reached yield, and the material there stops taking more. What happens next is redistribution: the yielded region sheds stress to its neighbours, the peak flattens, and by the time the net section is fully plastic the stress across it is uniform. A ductile member’s static strength is not reduced by a stress concentration at all — it is reduced only by the loss of net area. That is the same mechanism as a section reaching its plastic moment, and it has the same precondition: enough strain capacity to hold a stress while the material beside it catches up.

That is the single most important practical consequence on this page, and it is a debt to ductility. A bolt hole costs a tension member its net area and nothing more, and the factor of three simply does not appear in the calculation. In a brittle material it appears in full, which is why cast iron members were never drilled if it could be avoided.

The plate is infinite, and a real one is not. The solution assumes the hole is far from any edge. A hole close to an edge, or a line of holes close together, has a higher factor, and the handbooks that tabulate those factors exist because the closed form does not extend to them.

And nothing here has a fatigue crack in it. Everything on this page is a first-application calculation. Under repeated load the local yielding that saves the member statically is precisely what damages it cyclically, and the detail’s fatigue category is set by the concentration that the static check was right to ignore.

And the concentration says nothing about a crack. Sharpen the hole into an ellipse and the factor rises: 5 for a 2:1 ellipse, 7 for 3:1, 11 for 5:1. Sharpen it to a crack, with a tip radius approaching zero, and the factor for a 5 mm flaw with a 0.01 mm tip radius is 46 — and as the radius goes to zero the factor goes to infinity. That divergence is not a large number, it is a signal that the whole concept has failed, and it is why fracture mechanics had to be invented: a quantity that goes to infinity has to be replaced by the coefficient of its divergence.

What the picture cannot show #

The tint is a scalar and the stress is a tensor. What is drawn is the hoop stress $\sigma_{\theta\theta}$, which is the component that decides the peak and is the right one to plot. There are two others at every point — a radial stress and a shear — and the direction of the principal stress rotates as the hole is walked around. A crack starting at the edge of a hole runs perpendicular to the hoop direction, which is why it starts radially and then curves.

The plate is drawn with an edge and solved without one. The solution assumes the hole sits in an infinite plate, and the rectangle around it in the figure is a viewing window rather than a boundary. A real plate whose edges are within a few radii has a higher factor, and the closed form does not extend to it.

And the field is elastic, which the member is not for most of its life. Everything drawn here is the state at low load. At working stress the peak has yielded, the picture is wrong at the hole and right everywhere else, and by ultimate load the redistribution has flattened it entirely. The most consequential thing about this figure is the range of load over which it applies, and that range is not on it.

The generalisation #

The pattern is that a stress concentration matters exactly to the extent that the material cannot relieve it.

Three cases, and they cover almost everything. Under static load in a ductile material, it does not matter: the peak yields, redistributes, and the member’s strength is set by the net section. Under static load in a brittle material, it matters in full: there is no redistribution, and the member fails when the peak reaches the strength. Under cyclic load in any material, it matters more than in either static case, because fatigue cracks start where the local stress range is largest and a crack does not care that its neighbourhood has yielded — indeed the local yielding is what starts it.

That third case is the one that catches people, because a member designed and checked statically with a concentration correctly ignored is a member whose fatigue life is decided by the concentration that was correctly ignored.

A surprising place this turns up #

The most consequential stress concentrations in structural engineering are not holes. They are the ones nobody drew.

A hole appears on a drawing and can be looked up. What does not appear is the toe of a fillet weld, which is a sharp re-entrant corner with a small and uncontrolled radius; the end of a cover plate, where a load path stops abruptly; the corner of a cope cut into a beam’s flange; the root of a thread; the change of section at a bolt shank. Each of those is a stress raiser of a factor between two and five, none of them is drawn as a feature, and every one of them is where a fatigue crack starts.

This is the reason fatigue design proceeds by detail category rather than by calculating stress concentrations. The concentration at a weld toe cannot be computed, because it depends on the weld’s profile, which depends on the welder. So the profession gave up computing it and started classifying: a table of details, each with an experimentally determined strength, and the concentration folded into the classification rather than into the stress.

That is a rare and instructive retreat. The exact solution on this page is a century and a quarter old and completely reliable, and for the geometry that actually decides structural life it is unusable, because the geometry is not known well enough to solve.

Where the ladder goes next #

Later rungs on this anchor: stress concentration factors for the configurations that have no closed form — holes near edges, rows of holes, filleted shoulders, keyways — and how such a handbook is built. The elliptical hole solution in full, and Inglis’s 1913 paper that produced it. Redistribution after local yielding, computed rather than asserted, and the load at which the net section goes fully plastic. Notch sensitivity, which is the empirical observation that fatigue strength is reduced by less than the elastic factor, and why. Shear lag, which is the same phenomenon in a wide flange whose load enters through a narrow web. And the three-dimensional problem, where a spherical cavity concentrates by about 2 rather than 3 and a plate’s through-hole concentrates differently at its surface than at mid-thickness.

Historically Kirsch published in 1898 and Inglis’s elliptical generalisation followed in 1913, and it was Inglis’s result — with its factor going to infinity for a sharp tip — that gave Griffith the problem he solved eight years later. The sequence is unusually clean for this subject: an exact solution, a limit in which it becomes meaningless, and a new theory built precisely to handle the case where the old one diverges.