The section calculation with no formula in it

There is a shelf of section formulas on this site. The transformed section for two elastic materials. The cracked section for concrete that has given up in tension. The plastic moment for a fully yielded steel shape. The stress block for concrete at its crushing strain. Each was derived once, for one arrangement of material, and each stops at the boundary of the case it was derived for.

They are all the same computation.

Slice the section into strips. Assume a curvature. Give every strip the strain that curvature and a trial neutral axis put it at. Look up each strip’s stress in whatever law belongs to its material. Sum the forces; if they do not balance the applied axial force, move the neutral axis and do it again. When they balance, take moments.

That loop is twenty lines and it does not know what kind of section it has been handed. It has never heard of cracking, yielding, confinement or prestress. Hand it an elastic section with two materials and it reproduces the transformed-section answer. Hand it a concrete section with tension excluded and it finds the cracked neutral axis. Hand it a steel section at large curvature and it finds the plastic moment. It is not an alternative to those formulas; they are what it does under particular conditions.

Which free body produced the number #

The cut face, taken as a set of strips. Two equations govern it, and they are the two equations every section calculation on this site has used:

$$\sum_i \sigma_i A_i = N \qquad\qquad \sum_i \sigma_i A_i y_i = M$$

with the strain in each strip fixed by the assumption that plane sections stay plane:

$$\varepsilon_i = \kappa (y_i - y_{na})$$

Two unknowns are hiding in there. The curvature $\kappa$ is chosen — it is the independent variable, and the answer comes out as a moment for each curvature. The neutral axis position $y_{na}$ is not chosen: it is whatever makes the first equation come out right, and there is no way to write it down in advance.

That is the whole reason the method is a loop rather than a formula. Axial equilibrium is one equation in one unknown and it is not linear, because the relationship between strain and stress in each strip is not. Bisection settles it in forty iterations to any precision anybody wants, and forty iterations of a twenty-line sum is nothing.

For a 300 by 450 section with 1,200 mm² of steel and a modular ratio of 7.5, the closed form puts the cracked neutral axis at 137.03 mm. The strip loop, told nothing about cracking and given only “concrete takes no tension”, puts it at 137.03 mm.

How many strips, which turns out not to be the question #

The obvious worry about a discretisation is how fine it has to be, and the obvious answer is “as fine as the machine allows”. Both are wrong here.

Two strips give 1.4% error. Eight give 0.088%. Two thousand give nothing worth having over eight, and the error between those points does not fall monotonically — it can be worse at sixteen than at eight.

The reason is worth understanding because it generalises. The error does not depend on the strip count; it depends on where the neutral axis falls relative to a strip boundary. A strip that straddles the neutral axis has part of it in compression and part in tension, and the method gives the whole strip one stress — the stress at its centre. If the neutral axis happens to land on a boundary, that error is zero at any strip count. If it lands at a strip centre, the error is at its worst.

So the convergence plot is not a smooth curve with a slope to be read off; it is a noisy descent whose envelope falls as $1/n^2$ and whose individual points bounce around inside it. The useful reading is the envelope, and the practical rule is that eight to sixteen strips across a compression zone is plenty for anything, which is far fewer than instinct suggests.

There is a cheap fix that removes the noise entirely, and it is the same one every quadrature scheme uses: put a strip boundary at every known discontinuity — the neutral axis after the first iteration, each material change, each layer of reinforcement — and integrate exactly within each region. It converts the noisy descent into an exact answer at very few strips.

What it buys, which is not accuracy #

The strip loop is not more accurate than the closed forms. On every problem a closed form covers, the two agree to whatever precision the strips are taken to, and the closed form gets there instantly.

What it buys is generality without new derivations, and that is a different currency. Four examples, none of which has a closed form worth writing down:

A confined core inside an unconfined cover. Confinement raises the peak stress and multiplies the crushing strain, but only inside the hoops. So a column section has two concrete laws in it, on regions with a boundary that is not a straight line, and the cover spalls off at a strain the core is nowhere near. As a formula this is unpleasant. As strips it is a conditional inside the loop.

A prestressed section. The tendon has a strain in it before anything is applied, so its strain is the section’s strain plus a constant. One extra term, in one strip.

A composite section part way through construction. The steel carries the wet concrete alone and the composite section carries everything after. The strips have different strain origins depending on when they arrived, which as algebra means tracking two neutral axes and as strips means a per-strip offset.

A section that has been loaded and unloaded. Residual stresses and previous yielding mean each strip starts on a different point of its own hysteresis path. There is no closed form at all for this, and there does not need to be.

The output that a formula cannot give at all #

There is something the fibre model produces as a by-product which no closed form does, and it is the reason the method is worth having even where a formula exists.

Run the loop over a range of curvatures rather than at one, and what comes out is the whole moment-curvature relationship — not a capacity but a curve, with the first-yield point on it, the plateau, the descending branch, and the curvature at which the extreme fibre crushes.

That curve is what rotation capacity is read from, what a plastic hinge’s length is estimated from, and what a nonlinear frame analysis needs at every section it integrates. A capacity formula answers one question about the section; the curve answers the questions that come after the capacity has been reached, and those are the ones a collapse calculation is made of.

It also makes visible something a capacity hides. Two sections with identical moment capacities can have completely different curvatures at failure, and the difference is the whole of the difference between a ductile member and a brittle one. The number that is checked is the same; the number that decides what happens is not checked at all.

Two ways to drive the loop, and only one of them survives the peak #

The description above fixes the curvature and solves for the neutral axis. There is an obvious alternative — fix the moment and solve for the curvature — and it is the one most people reach for first, because a moment is what a structure hands a section.

It fails, and it fails in exactly the place the analysis is being run for.

Past the peak of the moment-curvature curve there are two curvatures at every moment, and past the end of the curve there are none. A solver hunting for the curvature that produces a given moment will find the wrong root, oscillate between the two, or diverge — and it will do all three within a few per cent of the capacity, which is the region a collapse calculation lives in. Fixing the curvature instead makes the moment a single-valued output at every step, and the descending branch is traced as easily as the rising one.

That is a general property of nonlinear analysis rather than a quirk of sections: drive the calculation with the quantity that stays monotonic. The same choice appears one level up, where an arch or a shallow frame that snaps through has two loads at some displacements, and the analysis has to be displacement-controlled for the same reason.

What the axial force does to all of this #

Everything so far has had $N = 0$ on the right of the first equation. Put a real number there and the loop is unchanged — the neutral axis simply settles somewhere else — but two things about the answer change, and both are worth naming.

The first is that the moment capacity becomes a function of the axial force, which is the interaction curve every column is designed on. Running the loop at a series of axial forces and recording the peak moment at each traces that curve out directly, including the balance point and the reversal in slope near it. There is no closed form for the interaction curve of a real reinforced section, and there does not need to be.

The second is that the section may have no tension zone at all, in which case there is no neutral axis inside the section and the bisection has to be allowed to look outside it. That is not a numerical detail: it is the whole difference between a member that cracks and one that does not, and a bisection bounded to the section’s own depth will silently report the wrong answer for every section in that state.

The same idea, one level up #

The pattern here — replace an integration that has no closed form by a sum over pieces, and iterate the one unknown that equilibrium fixes — is not confined to sections.

It is exactly what the stiffness method does to a structure: replace the continuum by elements, write equilibrium at the nodes, and solve. It is what a numerical integration does to a deflection. It is what the yield-line search does to a slab. In each case the invention is bookkeeping rather than mechanics, and in each case what it replaced was a shelf of special cases.

The fibre model is the smallest instance of it, which is why it is a good one to learn from. There is one integral, one unknown, and one loop, and the whole apparatus fits on a screen.

Every strip can have a different material #

The feature that makes the method worth having is not that it integrates a curve. It is that each strip carries its own stress-strain law, and nothing in the two equations cares whether they are all the same.

That is what a closed form cannot do at any price, and it is where most of the method’s real work happens. The simplest case has only two laws in it and is old enough to have its own closed form, which makes it the one place the claim can be checked against something.

That figure is also the honest answer to the charge that the method assumes nothing. It assumes three things, and they are visible in it: plane sections stay plane, which is what makes one strain profile serve both materials; each strip is at a single stress, which is what lets a width be multiplied; and the law measured on a specimen is the law the member obeys. The strip loop removes the algebra and none of the modelling — the transformed section is that same pair of assumptions solved by hand for the one case where the hand can reach.

The share of the work that follows from them is not intuitive, and it is worth seeing swept rather than stated once.

Which is also the warning that comes with the method. A stiff element attracts the load it was added to relieve, and a strip loop will report that faithfully whether or not anybody expected it — the arithmetic has no opinion about which material was supposed to be doing the work.

A confined section. The core is confined by hoops and the cover is not, so the strips inside the links follow one curve — higher peak, much longer tail — and the strips outside them follow another that ends at 0.0035 and stops. The section’s moment-curvature curve then has a step in it where the cover leaves, which is a real feature of a real column and appears in no formula anywhere.

A section in a fire. The temperature varies through the depth, so every strip is at a different temperature, and every temperature implies a different reduced yield stress and a different reduced modulus. Feed each strip its own pair and the same two equations return the section’s capacity — and, as a by-product, the position of the neutral axis, which has migrated toward the cool face because the hot strips have lost their stiffness. The hour that is really a temperature supplies the retention factors; this supplies everything downstream of them, and there is no closed form for any of it.

A member that has corroded. Reduced bar areas at some levels and not others, a changed curve for a pitted bar, cover that has spalled and is no longer there. Three separate changes, each expressible as a different strip, none of them expressible as a modified $Z$.

And a material known only as points. A heritage cast iron, a fibre composite, a timber with different curves in tension and compression: the method takes a table of measured values as readily as an equation, because it never differentiates the curve and never inverts it. It only ever asks what stress goes with this strain, which is the one question a set of measurements can answer directly.

Where the model stops #

Three assumptions, and they are precisely the ones the closed forms make, which is the point of the third refutation above.

Plane sections stay plane. Every strip’s strain comes from one linear profile. Near a support, a load or a hole that is not true, and no amount of refinement in the strip direction fixes it, because the error is in the strain assumption rather than in the integration.

Each strip is at one stress. True in the limit and the source of the discretisation error discussed above. It also quietly assumes the stress varies only across the depth, which is wrong for a section under biaxial bending — that needs a two-dimensional mesh and a neutral line whose angle is a second unknown.

The material’s law is the member’s law. The stress-strain curve came from a specimen. Applying it strip by strip assumes a region of the member behaves like a small specimen, which is the assumption the size effect says is not generally safe, and which is why the method is trustworthy for a confined compression zone and not for a plain concrete one.

There is a fourth, less often stated: the method finds an equilibrium state, not necessarily a stable one. On the descending branch of a moment-curvature curve the section is shedding load as it deforms, and whether the member can follow that path depends on what is around it — which is a question about the structure and not about the section.

What the picture cannot show #

The convergence plot shows an error against a strip count, which makes the discretisation look like the accuracy question. In practice it is nowhere near the largest one.

The concrete strength is known to about 15%. The steel’s yield stress is a characteristic value with a real distribution behind it. The modular ratio depends on a creep coefficient quoted to one significant figure. Against any of those, the difference between eight strips and eight hundred is invisible — which is the honest reading of the figure, and is why the practical answer to “how many strips” is “far fewer than instinct suggests, and it is not the number to worry about”.

The generalisation #

The habit worth carrying is a question to ask of any closed form: what did it assume, and what does the calculation look like if that assumption is removed?

For section formulas the answer is almost always the same. Every one of them is a case where the integral over the section happened to be doable — a rectangle with a linear stress, a rectangle with a constant stress, two rectangles with a step between them. The formula is not a theory of the section; it is a theory of that integral being tractable.

Remove the tractability and nothing about the mechanics changes at all. The two equations are the same equations, the plane-sections assumption is the same assumption, and what has gone is only the possibility of writing the answer down. That is worth knowing before reaching for a formula, because it says exactly which of its features are physics and which are convenience — and it is nearly always fewer of the first than the printed version suggests.