The material that has a direction

Steel has one modulus and one yield stress. That is not a simplification made for teaching; it is very nearly exactly true, and it is the reason a steel section can be described by a handful of geometric properties and a grade.

Timber has three moduli, three strengths in compression, three in tension and three in shear, and the ratios between them run to thirty. What follows from that is not a correction to the arithmetic of a steel beam. It is a different arithmetic, with a different check governing and a limit state that steel design does not contain.

Which free body produced the number #

Hankinson’s formula is

$$f_\alpha = \frac{f_0\,f_{90}}{f_0\sin^2\alpha + f_{90}\cos^2\alpha}$$

and it is worth being honest about what it is: an interpolation that fits, not a failure theory. Nothing derives it, and its exponent is 2 for compression and closer to 1.5 or 2.5 for other actions depending on whose tests are being fitted.

What makes it worth using is the shape. At forty-five degrees, where $\sin^2 = \cos^2 = \tfrac12$, it collapses to

$$f_{45} = \frac{2f_0f_{90}}{f_0 + f_{90}}$$

the harmonic mean of the two strengths — which for 21 and 2.5 N/mm² is 4.47. A harmonic mean is dominated by the smaller of its arguments in a way an arithmetic mean is not, and that is the entire content of the formula: in a material with a weak direction, having any component in the weak direction is most of the way to being in the weak direction.

That is the same arithmetic as a biaxially bent section, where a five-degree tilt costs ninety per cent of the stress. Both are a small angle multiplying a large ratio.

Three moduli, and a logarithm to draw them on #

The stiffnesses are as lopsided as the strengths and are less often discussed.

The number that does the most damage quietly is $E/G$. For steel it is 2.6; for this timber it is 15.9.

Shear deflection is the term every beam calculation drops, and its size goes as $E/G$ times the square of the depth over the span. Six times the ratio means six times the term — so the correction that is worth one per cent on a rolled steel beam is worth six or more on a timber joist of the same proportions, and on a short deep one it is worth a great deal more than that.

Which means a timber beam’s deflection cannot be computed from its second moment alone, and the codes that govern it say so explicitly, with a shear term written into the serviceability check rather than mentioned in a footnote.

The check with no steel equivalent #

Bearing across the grain is where a timber structure is usually decided, and it has no counterpart in steel design at all.

A support reaction arrives perpendicular to the member. In steel that is a stress against $f_y$ and is never critical. In timber it is a stress against $f_{c,90}$, which is 2.5 N/mm² — a twelfth of the strength along the grain — and it is applied over whatever bearing length the detail happens to provide.

The failure is not dramatic. The member crushes locally at the support, sinks a few millimetres into whatever it is bearing on, and stops — because crushing across the grain is a ductile mechanism with a long plateau, unlike almost everything else timber does. A “failed” bearing is a floor that has dropped at one end and is otherwise intact, which is why the limit state is written as a serviceability one in some codes and a strength one in others.

Shear governs a beam that steel would not #

The crossover between bending and shear is a ratio of two strengths and nothing else.

The bending utilisation of a uniformly loaded rectangular beam is $0.75wL^2/(bd^2f_m)$; the shear utilisation is $0.75wL/(bd\,f_v)$. Setting them equal cancels the load, the width, and one power of the span, leaving

$$\frac{L}{d} = \frac{f_m}{f_v}$$

For this timber that is 6.0, and for grade 355 steel it is 1.73. So a timber joist shallower than six times its depth is governed by shear parallel to the grain — an ordinary lintel, a floor joist over a corridor, a beam trimming a stair — and a steel beam would have to be shorter than twice its own depth before the same thing happened, which is not a beam.

And shear parallel to the grain is the least forgiving failure timber has. It is a split running along the member, on the plane where the material is weakest, at a stress that arrives suddenly rather than after any visible warning. It is the reason notches in timber beams are treated with far more care than notches in steel ones: a notch at a support introduces a crack tip on exactly the plane the shear is trying to open.

What the direction actually is #

Everything above treats “the grain” as a single direction, and timber has three.

There is the longitudinal direction along the fibres, the radial direction out from the centre of the tree, and the tangential direction round the growth rings. Longitudinal is very stiff and strong; the other two are weak and are not the same as each other — tangential shrinkage is about twice radial, which is why a plank cups as it dries and why the direction of the rings across a section decides which way it will move.

Structural design ignores the distinction and works with “parallel” and “perpendicular”, which is a decision rather than an approximation: the two perpendicular directions differ by a factor of two, and the design value is the worse of them.

The distinction reappears the moment moisture is involved. Shrinkage across the grain is of the order of a per cent for every four per cent change in moisture content, and along the grain it is negligible. A timber frame therefore moves in one direction only, by amounts that accumulate through every horizontal member in a wall, and the detailing rule that follows — keep the number of horizontal members in a load path small — is a consequence of anisotropy with no strength calculation anywhere in it.

The section that is chosen for it #

Because the properties are so unequal, timber’s structural geometry is not steel’s with different numbers in it. Three consequences show up in what gets built.

Sections are rectangular and deep. There is no timber I-section, because a flange would have to be connected to a web across the grain and the connection would be the weak link — so the whole of the section-shape argument that produces the I-beam is unavailable, and depth is bought directly.

Members are laminated. Glued laminated timber is a way of removing the defects that dominate the scatter: cut a plank into laminations, discard or finger-joint the worst, glue them back up, and the resulting member has a strength closer to the mean of many pieces than to the minimum of one. It is a statistical improvement rather than a material one, and it is why glulam design values are markedly higher than sawn timber’s for the same species.

And the plane of lamination is the plane of shear. A glulam beam’s glue lines run along its length, on exactly the plane the horizontal shear is trying to open — so the shear check on a laminated beam is a check on the material and on the manufacture at once, and a delamination and a shear split are the same failure with different causes.

Where the same argument appears without timber #

Anisotropy is not a timber problem. It is a property of any material made of aligned things, and there are three others in this collection wearing different names.

A corrugated web has an effective modulus along the girder of a thousandth of its modulus across it — an anisotropy ratio of a thousand, larger than any natural material’s, and produced entirely by geometry.

A reinforced concrete slab spanning one way is far stiffer and stronger in that direction than across it, which is why two-way spanning is a design decision rather than a description.

And a fibre composite is a manufactured version of the same thing, with the ratios chosen rather than inherited: a unidirectional laminate has an $E$ ratio of twenty or more, and the whole design method is about laying up plies so that the resulting orthotropy matches the load.

Which suggests the useful way to hold the idea. Anisotropy is not a defect of a natural material; it is the efficient answer whenever the loads have a direction. Timber is a cantilever optimised by evolution for wind, and every property in the bar chart above is a consequence of a tree not needing to be strong sideways.

What the anisotropy is worth, which is a great deal #

It would be easy to read all of the above as a list of difficulties, and it would be the wrong reading.

Measured along the grain, timber’s strength-to-weight ratio is comparable with structural steel’s and its stiffness-to-weight ratio is better than concrete’s. A material with a density of 420 kg/m³ carrying 21 N/mm² in compression is doing something no isotropic material of that weight does, and the reason it can is precisely that it has spent nothing on the two directions it does not need.

Anisotropy is what the efficiency is made of. An isotropic material with timber’s density and timber’s longitudinal properties does not exist, because making a material strong in three directions costs three times as much material as making it strong in one.

That reframes the design problem. The checks in this essay — the Hankinson collapse, the bearing limit, the early shear crossover — are not defects to be worked around; they are the bill for an efficiency that has already been taken. A designer’s job with an anisotropic material is to arrange the structure so that the loads arrive along the direction that was paid for, and every timber detailing rule worth knowing is a version of that instruction.

Cross-laminated timber, which is the anisotropy cancelled #

The most interesting recent thing done with timber is to arrange it so that the direction stops mattering.

Cross-laminated timber is boards glued in alternating layers at right angles, and the effect is a panel with two comparable strong directions instead of one. What it buys is not strength — the crossing layers contribute almost nothing to bending in the other direction, so a five-layer panel is roughly three layers’ worth of material in each direction — but isotropy, which is what makes a panel rather than a beam possible.

The price is paid in shear, and it has a name: rolling shear. The crossing layers carry the panel’s shear across their own grain, on the plane where timber is weakest of all, and the strength there is a fraction of the already-low parallel value. So a cross-laminated panel is governed by shear at span-to-depth ratios where a solid timber beam would not be — the same crossover this essay computed, moved further along the axis by a material property that only exists because the layers cross.

Where the model stops #

Hankinson’s formula is a fit. It has no derivation, its exponent depends on the action being interpolated, and it is applied to compression, to bearing and to bolted connections with different exponents in different codes.

The properties here are characteristic values for one grade. Timber’s scatter is far wider than steel’s — a coefficient of variation of 20 to 30 per cent on strength is ordinary — and the design values used are low percentiles of a distribution rather than measurements of a material.

Nothing here is about duration or moisture, and both are large. Timber’s strength under a permanent load is around 60 per cent of its short-term strength, and its stiffness under one creeps by a factor approaching two; both depend on the service moisture content, and neither has an equivalent in steel design.

And the member is assumed defect-free. Real timber has knots, sloping grain around them, and drying checks — and the strength-reducing effect of a knot is not that it removes area but that it locally rotates the grain direction, which by Hankinson’s formula is a much larger effect.

What the pictures cannot show #

The polar curve is drawn as a smooth function of angle, which assumes the grain has a single well-defined direction at the point of interest. Around a knot it does not: the grain sweeps around the knot at every angle from zero to ninety within a few tens of millimetres, so the strength varies over a region rather than following the curve.

Nor can the bar chart show what a designer really works with, which is a species and a grade rather than six numbers. The properties in the chart are inferred from a visual or machine grading of a piece of sawn timber, and the inference — not the measurement — is what the whole design rests on.

The assumption the figure rests on #

Every number here assumes the load is applied at a known angle to a known grain direction.

That is the assumption a connection breaks. A bolt in a timber member bears against the wood at whatever angle the force happens to arrive, and the force in a truss diagonal arrives at the angle of the diagonal — so the bearing strength at a joint is a Hankinson value rather than a material property, and it changes with the geometry of the truss rather than with the timber.

Which is why timber connections, rather than timber members, are what timber engineering is mostly about. A member’s capacity is a grade and a section; a joint’s capacity is a set of angles, and the arithmetic on this page says that those angles are worth more than the grade.