A section made of two materials, one of them pretended away

A flitch beam is a timber joist with a steel plate bolted to the side of it. The plate is 6 mm thick and the joist is 150 mm wide, so the steel is one twenty-sixth of the section’s area, and a reasonable first guess would be that it carries about that share of the load. It carries 43% of it, and the reason is a ratio of moduli rather than anything about strength.

The operation is usually taught as a trick, and it is not one. It is two sentences of theory written down in the only form that lets a section calculation proceed.

Which free body produced the number #

Cut the flitch beam and take one side. Across the cut is a bending moment, and the material has to supply it as a distribution of direct stress. Two conditions decide that distribution and there are no others.

Compatibility. Plane sections stay plane, so the strain at a height $y$ from the neutral axis is $\varepsilon = \kappa y$ — one number, the curvature, for the whole face. The steel and the timber are bolted together and cannot slide, so at any height they have the same strain. That is the whole of the compatibility statement and it is where everything follows from.

Constitution. Each material turns its strain into a stress by its own modulus: $\sigma_i = E_i \kappa y$. Equal strain, different moduli, so stress in the ratio of the moduli.

Now the force on a strip of the stiffer material, of width $b$ and height $dy$, is $E_s \kappa y \, b \, dy$. The force on a strip of the softer material of width $n b$, where $n = E_s/E_t$, is $E_t \kappa y \, (nb) \, dy = E_s \kappa y\, b\, dy$ — the same force, at the same height, and therefore the same first moment about any axis. So replacing the stiff material by $n$ times as much of the soft one changes nothing that any equilibrium equation can see, and the result is a section of one material to which every ordinary formula applies.

The transform is exact. What comes out of it has to be converted back — a stress computed on the transformed section is a stress in the reference material, and the real material’s stress is $n$ times it — but nothing was approximated on the way.

The check that there is no choice in it #

Transforming to the steel instead — dividing the timber’s width by 19.09 rather than multiplying the steel’s — gives a different fictitious section, a different second moment and different numbers throughout, and the stresses that come out of it are identical to the last figure.

That is the closure worth having. The reference material is a choice of units, like deciding whether to work in millimetres or metres, and a result that depended on it would be a result that was wrong.

What the transform manipulates is the second moment computed strip by strip. Each strip contributes its area times the square of its distance from the axis; widening a strip multiplies its contribution in exact proportion; and widening the stiff material by the modular ratio is precisely the operation that makes one integral do for two materials. Nothing else in the calculation has to know that a second material was ever present.

Where the neutral axis goes #

The transform also settles a question a single-material section never has to ask: where is the neutral axis?

For one material it is the centroid of the area, and everybody knows it. For two it is the centroid of the transformed area — which is to say the point about which $\sum E_i A_i y_i$ vanishes rather than $\sum A_i y_i$. The two coincide only when the materials are arranged symmetrically about the same line, which the flitch beam is and most sections are not.

The machinery the transform hands its output to is entirely ordinary. The parallel-axis theorem assembles a section’s second moment from its pieces and their distances and does not care whether the widths it is given are real widths or transformed ones, which is the entire reason the transform is worth making rather than merely correct.

For the flitch beam the answer is 150 mm from the bottom either way, because both materials run the full depth. For a composite floor beam — a steel section with a concrete slab on top — the transformed neutral axis is usually up in the slab, so the whole steel section is in tension. And for a reinforced concrete section it is somewhere that has to be solved for, because part of the section has been deleted.

The section that is missing a piece #

A reinforced concrete section in bending is the same operation with one extra rule: the concrete below the neutral axis is a material with no tensile strength and is deleted. Because the axis’s position decides what is deleted and what is deleted decides the axis’s position, the calculation is no longer a centroid — it is the root of a quadratic, and the section’s second moment falls by a large fraction as soon as it cracks.

The modular ratio is doing exactly the same job in both cases. In the flitch beam it makes a plate act like 114 mm of timber; in the concrete section it makes 1,200 mm² of bar act like 9,000 mm² of concrete. And the consequence is the same in both: the reinforcement carries a share of the moment out of all proportion to its area, because stress follows strain times modulus.

The consequence that generalises #

Read that figure the other way round and it is a general statement about structures rather than about sections: stiffness attracts load, and it attracts it in proportion to stiffness rather than to strength or to size.

The same sentence explains why the stiffest path takes the load in a frame with parallel routes, why a stiff member’s stress rises when it is stiffened, and why a repair made by adding steel to a member attracts the very force it was added to relieve. The transformed section is that principle written for the inside of one member instead of across a structure, and it is the cleanest place to see it because the arithmetic is a single ratio.

The corollary is the one people find uncomfortable. A stiff material in a section cannot decline to carry load. It has no choice: the strain is imposed by the section’s curvature, and the stress follows. A designer who adds a steel plate to a timber beam “for a bit of extra safety” has added a member that will reach its own limit first.

What happens when a modulus is not a constant #

The whole transform assumes each material has a modulus. Concrete does not: under sustained load its strain grows at constant stress, which is the same thing as its effective modulus falling, and the usual accounting is $E_{c,eff} = E_c/(1+\phi)$ with $\phi$ around 2 at the end of a structure’s life. The modular ratio is therefore not one number, and a section has two different transformed sections at two different times.

So a composite section’s modular ratio is about 7 for short-term loading and about 21 for long-term. Three consequences follow and none of them is small.

The neutral axis moves. As the concrete softens, the transformed slab narrows and the axis descends toward the steel.

The steel takes more. With the concrete’s share falling, the steel picks up the difference at constant total moment — so a composite beam’s steel stress increases over decades with no change of loading.

And the deflection grows, which is the mechanism the camber question turns on.

The same effect appears in reverse in reinforced concrete: the compression zone creeps, the steel in it is stressed further, and a heavily reinforced column can shed a substantial fraction of its concrete stress into the bars over its lifetime with no external change at all.

The harder limit is the one the transform cannot be argued past. It is a linear operation and it needs a modulus to exist; past the elastic limit of either material there is no modulus, the strain distribution is still linear and the stress distribution is not, and the section has to be integrated fibre by fibre instead.

What it is worth, and what it is not #

The flitch beam is a good case for one more reason: it makes the two things a designer wants easy to separate.

Stiffness gained: 1.70 times. The transformed second moment is 595 × 10⁶ mm⁴ against 351 for the same shape with the moduli ignored — so the plate nearly doubles the beam’s stiffness while adding 4% of its area, and if the beam was sized by deflection then that is the whole benefit.

Strength gained: rather less than that, and limited by the timber. The plate reaches 96 N/mm² while the timber is at 5.04, and the timber’s own limit is around 8; so the section reaches the timber’s limit at a moment of about 32 kNm, at which point the steel is at 154 — a quarter of what steel can do.

The steel is not being used, and no amount of adding more of it changes the situation, because adding steel raises the modular contribution and lowers the timber’s stress in the same breath.

That is the general shape of every two-material section that is not designed as one. The stiff material is over-stressed relative to its share and under-used relative to its capacity, and the only arrangement that fixes it is the one composite construction uses: put each material where its own strength is wanted, with the concrete in compression and the steel in tension, so that the modular ratio works with the geometry rather than against it.

Where the ratio comes from #

One number has been carrying the whole page and it is worth asking what kind of number it is.

The modular ratio is a ratio of moduli, and a modulus is the one number a stronger steel does not change. Two steels of 275 and 690 N/mm² have $n = 1$ between them; a section made of both has one linear stress distribution across it, and the high-strength half is not attracting anything.

Compare that with the pairs that do produce a large ratio:

Every large entry in that table is a pair in which one material is stiff and one is not, and the stiff one is being asked to share a strain that suits the other. That is the situation the transform exists to describe, and it is also the situation a designer should be suspicious of — because it is exactly the arrangement in which one material reaches its limit while the other is barely working.

Transform either way, and check that they agree #

The choice of which material to transform into is free, and making it both ways is the cheapest available check on the arithmetic.

Transform the timber into steel and its width is divided by $n = E_s/E_t$; transform the steel into timber and its width is multiplied by the same $n$. The two transformed sections look nothing alike — one is a sliver and the other is a slab — and they must agree on everything physical.

The neutral axis is invariant, and it has to be: it is the height at which $\int E\,y\,dA$ vanishes, and multiplying the whole integrand by a constant does not move a zero. So a discrepancy between the two neutral axes is an arithmetic error and nothing else.

The strains are invariant, being a curvature times a distance, with no modulus in them.

And the stresses come out identical, provided each is read with its own factor: a stress computed on a transformed area belongs to the material it was transformed into, and recovering the real stress in the other material means multiplying by $n$ again. That last step is the one that gets dropped, and dropping it produces a stress in the transformed material that nothing in the beam is made of.

The second transform costs a minute. It catches a wrong $n$, a wrong direction of division, and a first moment taken about the wrong axis — which between them are most of the ways this calculation goes wrong.

The shear flow has to be transformed too #

There is a trap in taking the transformed section into a shear calculation, and it catches people who have got the bending right.

The interface between the two materials has to carry a shear flow, and the flow is $q = VQ/I$ — with both $Q$ and $I$ taken from the transformed section. That is not optional: the flow is the rate of change of the force in the material above the cut, and that force was computed on the transformed geometry, so its first moment has to be as well. Using the real geometry’s $Q$ against the transformed $I$ mixes two sections and produces a number belonging to neither.

But the stress at the interface is $q$ divided by the actual width, not the transformed one. The flow is a real force per unit length crossing a real surface, and the surface has the width the material has. So the calculation uses transformed properties for the flow and true geometry for the stress — a mixture that looks like an inconsistency and is exactly right.

The practical consequence is on the connection. A flitch beam’s bolts, a composite beam’s studs, a laminated member’s glue line: each is sized on a flow computed from a section that does not exist, delivered across a surface that does. And the flow at the interface of a two-material section is generally larger than intuition suggests, because the interface sits where the stiff material meets the soft one — which is where the force being redistributed per unit length is greatest.

Where the model stops #

Both materials must be linear. Past yield in either, the transform is dead: the section has to be integrated fibre by fibre with each material’s own constitutive law. The ultimate-strength calculation for a reinforced concrete section is that integration, and it looks nothing like a transformed section.

They must not slip. The whole of the compatibility statement is that the two materials share a strain, which requires them to be connected — and a connection that is not rigid gives a partly composite section whose behaviour is between two transforms rather than at one.

Shrinkage and temperature are outside it. Two materials that want to change length by different amounts generate stress with no load applied. The transform handles the resulting curvature perfectly well once the free strains are supplied, but nothing in the section’s geometry supplies them.

And a “modulus” is an idealisation for both of the materials most often transformed. Timber’s modulus depends on moisture, load duration and grain direction; concrete’s depends on age, aggregate and how fast the load arrived. Quoting a modular ratio to three figures, as this page has done throughout, is a statement about the arithmetic and not about a beam.

What the pictures cannot show #

The transformed section is drawn as a shape, and there is no such shape. Nothing 114 mm wide exists anywhere in the flitch beam; the drawing is a computational device given a width and a hatching so that it can be looked at, and a reader who imagines it as a piece of timber has imagined a piece of timber carrying 96 N/mm².

The stress diagram beside it is drawn as two blocks with a step between them, and the step is at the interface. There is no step in the strain — that is the whole point — and the discontinuity is entirely a property of the constitutive law rather than of anything happening at the boundary. Drawing the strain instead would give a single straight line and would show nothing.

And the share figure plots two percentages against a plate thickness as though a designer might choose 7.3 mm. Plate comes in sizes, the bolts through it come in sizes, and the practical choice is among four or five geometries — so the curve’s job is to show the shape of the relation rather than to be read off.

The ladder from here #

Later rungs on this anchor: the age-adjusted effective modulus and the ageing coefficient, which is how creep under a changing stress is handled rather than under a constant one. The transformed section with three or more materials, and the reinforced concrete column where steel, cracked concrete and uncracked concrete are all present. Shrinkage-induced curvature in an asymmetrically reinforced section, which produces deflection with no load at all. The cracked-uncracked transition and the effective second moment used to interpolate across it. Prestressed sections, where the transform has to be made about a section that includes the tendon and the tendon has an initial strain. Sandwich sections, where the modular ratio is so large that the core’s contribution to bending is negligible while its contribution to shear is everything. And the transformed section under axial load and moment together, where the neutral axis leaves the section entirely.

The idea’s origin is worth noting because it is older than reinforced concrete. Composite beams of timber and iron were being calculated by this method in the 1850s, and the arithmetic reached concrete only when somebody noticed that a material with no tension is a material whose transformed width is zero.