The smallest of six failures

A bolt in a steel plate has a small number of ways to fail and they are all checks on parts: the bolt shears, the plate tears, the plate splits to the edge, the metal between the holes comes out as a block. Each is a strength of one component compared with a force.

A dowel through timber is not like that. The steel and the wood deform together, and the shape they take is what decides the capacity — the dowel can stay straight while the timber crushes against it, or rotate rigidly while both members crush, or bend into one plastic hinge, or into two. Each of those is a mechanism rather than a component, and the capacity of the joint is the smallest load that any of them can be sustained at.

That is exactly the structure of a plastic collapse calculation, and it produces the same theorem: every mechanism gives an upper bound, and the true capacity is the lowest.

Which free body produced the number #

Take the simplest mechanism first, because it fixes the units for everything else.

Mode a: the dowel stays perfectly straight and the timber in the first member crushes uniformly against it. The free body is the dowel, with a uniform bearing pressure over the projected area $t_1 d$ on one side and whatever the other member supplies on the other. The pressure the timber can take is the embedment strength $f_h$ — a property of the wood measured by pushing a dowel into it, not a compressive strength of the material — so

$$F_a = f_{h,1}\, t_1\, d$$

Mode f is the other extreme. The dowel bends into two plastic hinges, one in each member, and the timber crushes over the short lengths between each hinge and the shear plane. Both the hinge positions and the crushed lengths are unknowns, and minimising the load over them gives

$$F_f = 1.15\sqrt{\frac{2\beta}{1+\beta}}\,\sqrt{2 M_y f_{h,1} d}$$

with no member thickness in it at all. That is the result worth the essay. Past a certain embedment the timber stops being what decides, and the capacity depends only on the dowel’s plastic moment, the wood’s embedment strength and the dowel diameter.

The four mechanisms in between — rigid rotation, one hinge in each member in turn — are algebraically messier and are roots of quadratics rather than products, but they are the same calculation: assume a shape, write equilibrium of the dowel with the timber at its crushing pressure wherever it is in contact, and solve.

Where the crossover is #

Plot every mode against member thickness and the shape of the answer appears at once.

The crushing modes are straight lines through the origin, because a crushed length is proportional to a thickness. The one-hinge modes are roots and rise more slowly. The two-hinge mode is horizontal. They cross at about six and a half diameters for the 12 mm dowel drawn here, and past the crossover the joint’s capacity is a constant.

That is why every timber connection table in existence is indexed by dowel diameter and by member thickness in diameters, and why the thickness column stops. A 12 mm dowel in 100 mm members and the same dowel in 200 mm members are the same connection.

Below the crossover the picture inverts, and the governing mode says something about which material is being wasted. In a thin member the timber crushes before the steel can bend, so a stronger dowel buys nothing; in a thick one the steel bends before the timber crushes, so a denser timber buys nothing. The efficient joint is at the crossover, where both materials reach their limits together, and it is the only place in the whole design where they do.

Why the joint is what sizes a timber structure #

It is worth saying plainly why a connection essay is doing so much work in a subject about members, because timber is the field where the ratio is most extreme.

A softwood joist is a perfectly respectable beam: it has a bending strength of twenty-odd newtons per square millimetre and there is plenty of it. What it does not have is any way of transferring that force into anything else. A steel member is welded or bolted through a section whose strength is comparable with the member’s; a timber member is doweled through holes in a material whose embedment strength is a fraction of its bending strength and whose splitting resistance is a fraction of that again.

The arithmetic is stark. A 12 mm dowel in this joint carries five kilonewtons. A 200 by 50 joist in the same timber carries about sixty in bending. So a moment connection able to develop the member would need a group of dowels with a lever arm — twenty or more of them — and the spacing rules will not allow twenty dowels in a 200 mm depth without splitting the wood.

That is why timber structures are pin-jointed. Not because the joints cannot be made continuous in principle, but because a connection that develops a timber member’s own capacity is larger than the member. Every design decision downstream follows from it: trusses rather than frames, bracing rather than moment resistance, and glued rather than mechanical connections wherever continuity is genuinely wanted.

The two material properties, and neither is a strength #

The arithmetic uses exactly two material numbers, and both are unusual.

The embedment strength $f_h$ is not a compressive strength. It is measured by loading a dowel into a hole in a block and recording the pressure at a defined deformation, and it is substantially higher than the wood’s compressive strength — because the timber under the dowel is confined by the timber around it, and confined wood crushes at a higher stress than a free prism. It scales with density and falls with dowel diameter, and the code’s regression is $f_{h,0} = 0.082(1 - 0.01d)\rho_k$, which for a 12 mm dowel in a 350 kg/m³ softwood is 25.3 N/mm².

The yield moment $M_y$ is not the dowel’s plastic moment either. It is a fitted quantity, $0.3 f_u d^{2.6}$, and the exponent is the tell: a plastic modulus goes as $d^3$, and the fit comes out at 2.6 because the “hinge” in a dowel embedded in a soft material is not a hinge in a beam sense — it is a curved length with a stress gradient along it, and the effective moment falls short of the full plastic value by an amount that varies with size.

Both are regressions on tests. That is a different epistemic status from most of the numbers on this site, and it is worth being honest about: the derivation is exact and the inputs are fitted, which is a common shape in connection design and always worth noticing.

The same joint rotated is a weaker joint #

Timber is strongly anisotropic, and the embedment strength is one of the properties that shows it most plainly. Across the grain the wood has no fibres running along the bearing direction and the dowel crushes it far more easily.

The correction is Hankinson’s, with a diameter-dependent factor:

$$f_{h,\alpha} = \frac{f_{h,0}}{k_{90}\sin^2\alpha + \cos^2\alpha}, \qquad k_{90} = 1.35 + 0.015 d$$

so across the grain the strength is $f_{h,0}/k_{90}$, which for a 12 mm dowel is a factor of 1.53 lower.

A connection detail that is right at one angle is not right at another, and unlike almost everything else in structural engineering the difference is not a second-order effect. A truss node with members arriving at four angles has four different embedment strengths in the same block of wood.

Ductility, and why the mode matters as much as the number #

The governing mode is not only the capacity; it is also the character of the failure, and for a timber joint that matters more than for most.

A crushing mode is ductile. The timber under the dowel yields, the hole elongates, the joint deforms visibly and goes on carrying its load. A joint that fails this way gives warning, redistributes to its neighbours, and behaves acceptably under an earthquake.

A two-hinge mode is more ductile still, because the steel is doing the yielding.

What is neither is splitting, and it is not in Johansen’s list at all. A dowel bearing on timber pushes the fibres apart as well as crushing them, and if the edge distance or the spacing is too small the member splits along the grain instead — a brittle failure at a load that can be well below any of the six. That is why timber connection rules are so heavily populated with minimum spacings and edge distances, and why the arithmetic in this essay is conditional on them being met. The model assumes the failure it describes is the one that happens, and the detailing rules are what make that assumption true.

Double shear, which is not twice single shear #

Put the dowel through three members instead of two — an inner member between two outer ones, which is the commonest timber joint there is — and the list changes rather than doubling.

There are two shear planes, so the load is shared between them, and each plane has its own mechanisms. But the shapes available are not the same: the inner member is loaded from both sides, and the dowel’s symmetry means the hinge that would have formed in the middle of a single-shear joint cannot form at all. What is left is four mechanisms rather than six — the outer members crushing, the inner member crushing, one hinge in each outer member, and two hinges per shear plane.

The consequence is a good one and it is not the one a count of shear planes predicts. Double shear removes the rigid rotation mode, which is the one that has governed every single-shear joint on this page.

So the second plane is not what is bought. What is bought is the deletion of a mechanism, and the gain is 37 per cent rather than 100 because the mechanism that replaces it was always close behind.

That is the same argument as the reason a spliced member with cover plates on both faces is preferred to one with a plate on a single face, and it appears here as a whole failure mode disappearing rather than as an eccentricity being removed.

The joint is also the softest thing in the structure #

Everything above is a capacity, and for a timber structure the capacity is often not what decides. The same dowel that governs the strength governs the stiffness, and by a wider margin.

A dowelled connection has a slip modulus — a force per unit of relative movement between the two members — and the code’s regression for it is $K_{ser} = \rho_m^{1.5} d / 23$ newtons per millimetre, per fastener per shear plane. For the 12 mm dowel in 350 kg/m³ timber used throughout this essay that is 3.4 kN/mm, which means the joint has slipped about 1.5 mm by the time it reaches its 5 kN capacity.

Set that beside the member. Take a 3 m tie of 200 by 50 in the same timber, carrying 30 kN through six dowels at each end. Its own elastic extension is $PL/EA = 30{,}000 \times 3000 / (11{,}000 \times 10{,}000) = 0.82$ mm. The slip at each connection is $5/3.4 = 1.47$ mm, and there are two of them.

The joints contribute 2.94 mm and the member 0.82 — three and a half times as much movement in the connections as in the thing they connect.

Three consequences follow, and they are why a timber structure behaves so unlike a steel one of the same layout.

A timber truss’s deflection is a joint calculation. Analysing it as pin-jointed bars and summing $PL/EA$ gets the answer wrong by a factor rather than a percentage, and the discrepancy is in the direction that matters — the real truss sags considerably more. Every node is two slips.

The load distribution in a redundant timber structure is not what a member analysis says. Load follows stiffness, and if most of the flexibility is at the joints then the distribution is decided by how many fasteners each path has rather than by the sections.

And the stiffness itself is load-dependent. The slip modulus above is a service value; timber codes use a reduced $K_u = \tfrac{2}{3}K_{ser}$ for ultimate-limit-state analysis, because the connection softens as it approaches the mechanisms this essay is about. So the structure’s stiffness is one number for deflection and another for stability, from the same joint.

Which is the same finding as the shear-lag stiffness loss and the slip in a bearing joint, except that here it is not a correction. In timber the connection is not a softening of the load path; it is the load path’s flexibility, and the members are the stiff part.

Where the model stops #

The rope effect is left out and it is not small. A dowel with a nut and washer on it, or a screw with a head, resists being pulled through as it bends — so as the joint deforms, the dowel goes into tension and its axial force clamps the two members together, adding a friction contribution. That “rope effect” can add a quarter to the capacity of a bolted joint and much more to a screwed one, and the code adds it as a percentage of the yield term with a cap. It is a genuine mechanism and it is entirely absent from the six free bodies here.

Multiple fasteners are not $n$ times one. A row of dowels along the grain splits the wood more readily than one does, so the effective number of fasteners in a row is less than the actual number — typically $n^{0.9}$ for bolts. That is the timber version of the long-joint effect, arriving through a splitting mechanism rather than an elastic one.

And the model is a plasticity model for a material that is not plastic. Wood crushing under a dowel does have a plateau, which is what makes the model work, but it does not have an indefinite one — it softens eventually. Every mechanism above assumes the crushing pressure holds while the shape develops, and for a very unsymmetrical joint the far end can be softening while the near end is still developing.

Johansen’s list, and why it stayed #

The model dates from 1949 and it is essentially unaltered. That is rare in connection design, where most rules have been replaced twice, and the reason is worth naming: it is a mechanism model rather than a strength model, so it survives every recalibration of its inputs.

When better embedment tests changed $f_h$, the six expressions did not change — only the numbers going into them. When the yield-moment fit changed, likewise. A rule of the form “capacity equals a coefficient times a strength times an area” cannot do that, because the coefficient is doing the work of the mechanism and has to be refitted whenever anything moves.

It is the same durability that the truss analogy has in concrete design and for the same reason. A model that says what shape the failure takes can absorb new numbers indefinitely; one that says only how strong something is cannot.

Both of the model’s inputs are curves this collection has drawn elsewhere and neither is used here as it is drawn. The embedment strength is the plateau of a bearing curve, which is the same character a bolt in a steel plate has — a pressure that holds while a hole elongates — and what differs between the two materials is by how much and for how long. The yield moment is a point on a moment–curvature relation, and the fitted exponent of 2.6 rather than 3 measures how much of the real hinge that relation does not describe.

There is one more property of the model worth recording, because it is the reason it can be used at all in a spreadsheet. Every one of the six expressions is explicit — a product or a root, never an iteration — so a designer evaluates six formulae and takes a minimum, and a piece of software does the same in microseconds across a whole structure. A mechanism model that needed a numerical minimisation over hinge positions would have stayed in the literature. Johansen did the minimisation once, symbolically, and what is tabulated is the answer.

The generalisation #

The habit worth taking away is about how to write a capacity down.

Where two materials deform together, the capacity is a minimum over the shapes they can take, and not a minimum over their strengths. That is true of a dowel in timber, of a shear stud in concrete, of an anchor bolt in a base, of a bolt bearing in a plate thin enough to bend, and of every fastener whose failure involves both sides deforming.

The practical consequence is that such a capacity cannot be found by checking components. It has to be found by enumerating mechanisms, and the enumeration has to be complete — because a mechanism that was not written down does not get a lower bound put under it by any amount of care with the ones that were. That is the same warning the upper-bound theorem carries, arriving in a joint a few centimetres across.