When half the section has given up

Every treatment of bending starts by putting the neutral axis through the centroid of the section, and most of them present it as a definition. It is not a definition. It is a result, and it follows from one line of statics: for a member in pure bending the axial force on the section must be zero, so the stresses — which are proportional to distance from the axis — must integrate to nothing, so the first moment of the area about the axis must vanish. The place where the first moment of an area vanishes is that area’s centroid.

Every word of that argument survives if the material is not uniform, and if part of the section has stopped working. What changes is which area the first moment is taken of.

The axis has moved 113 millimetres and no material has been removed. What was removed is the contribution of the material below the axis, which was never doing much and is now doing nothing at all.

Why the axis moves, in one equation #

Below the neutral axis the concrete is in tension and has cracked, so it carries nothing. Above it the concrete is in compression and carries a stress rising linearly from zero at the axis. The steel, at depth $d$, carries a tensile stress — and it is a different material, with a modulus about eight times larger.

The trick that makes this a one-material problem is the modular ratio, $n = E_s/E_c$. A strain $\varepsilon$ in steel produces $n$ times the stress it would produce in concrete, so an area $A_s$ of steel behaves exactly like an area $nA_s$ of concrete placed at the same depth. Replace the bars by that fictitious area and the section is homogeneous again, and every result from elementary bending theory applies.

Now impose the zero-axial-force condition on the effective area. With the axis at depth $x$:

$$\underbrace{b \cdot x \cdot \frac{x}{2}}_{\text{first moment of the compression block}} = \underbrace{n A_s (d - x)}_{\text{first moment of the transformed steel}}$$

which is a quadratic in $x$ with one positive root. Everything else — the second moment of the cracked section, the two stresses, the lever arm — follows from where that root lands.

What cracking actually costs #

The intuitive fear is that cracking costs strength. It does not, or very little: the concrete in tension was carrying a stress of at most a few newtons per square millimetre over an area with a small lever arm, and losing it changes the moment capacity by a few per cent.

What cracking costs is stiffness, and it costs a great deal of it. The second moment of the cracked section in the figure is 1139 × 10⁶ mm⁴ against 3422 × 10⁶ for the uncracked one — a loss of 67%. A member that cracks deflects three times as far under the same load, and does so at a load far below anything resembling failure.

This is why serviceability rather than strength governs so much reinforced concrete design, and why the deflection calculation for a concrete beam is a genuinely harder problem than the strength calculation. The beam is not uncracked and it is not fully cracked; it is cracked where the moment is large and uncracked between the cracks, with the concrete between them still carrying tension by bond — an effect called tension stiffening that has no closed form and is handled by interpolating between the two second moments.

Which free body produced the number #

The section is 300 wide and 500 deep overall, with 1200 mm² of steel at an effective depth of 450, a modular ratio of 7.5, and an applied moment of 150 kNm.

Solving the quadratic: with $p = nA_s/b = 7.5 \times 1200/300 = 30$, the root is $x = p(\sqrt{1 + 2d/p} - 1) = 30(\sqrt{1 + 30} - 1) = 30 \times 4.568 = 137.0$ mm.

The cracked second moment is $bx^3/3 + nA_s(d-x)^2 = 300 \times 137^3/3 + 9000 \times 313^2 = 257 \times 10^6 + 882 \times 10^6 = 1139 \times 10^6$ mm⁴.

The free body is the cut section itself, with the stress block on one face. The concrete carries a triangular compression whose resultant is $C = \frac{1}{2}\sigma_c b x = \frac{1}{2} \times 18.05 \times 300 \times 137 = 371$ kN, acting at $x/3$ from the top. The steel carries $T = \sigma_s A_s = 309.2 \times 1200 = 371$ kN. The two are equal — which is the zero-axial-force condition, satisfied rather than assumed — and they are separated by a lever arm of $z = d - x/3 = 404$ mm.

Their moment is $C z = 371 \times 0.404 = 150$ kNm, which is the applied moment. That closure is the check the figure prints: if the neutral axis were in the wrong place, $C$ and $T$ would not be equal, and if the stresses were wrong the couple would not come back to the moment that was applied.

The assumption the figure rests on is that the concrete’s stress-strain relation is linear right up to the top fibre. At 18 N/mm² on an ordinary concrete that is defensible, and at working loads it is roughly what happens. At the ultimate limit state it is false — the real distribution is a curve flattening near the top, conventionally idealised as a rectangle — and the whole calculation changes character, though the reasoning that locates the axis does not.

The lever arm barely moves, and that is the useful part #

One number in the figure is remarkably stable, and it is the one design leans on.

The lever arm between the compression resultant and the tension resultant is $z = d - x/3$. Doubling the steel moved the axis from 137 to 180 — a 31% change — and moved the lever arm from 404 to 390, which is 3.5%. The lever arm is insensitive to almost everything, because $x$ enters it divided by three and subtracted from a much larger number.

That insensitivity is why the working design rule for a reinforced section is as crude as it is and as reliable as it is: assume a lever arm of about $0.9d$, and the required steel follows from $A_s = M/(f_s z)$ without solving any quadratic at all. The approximation is not lazy; it is an exploitation of the fact that the quantity being approximated hardly varies.

Two points do not establish an insensitivity, so here is the third, taken the other way from the opening figure.

A factor of four in the steel buys a 76 per cent change in the position of the neutral axis and a six per cent change in the arm the design uses. What does move sharply is the stress in the bar — 601 N/mm² in the lightly reinforced case against 309 in the opening one, for exactly the same applied moment — because the couple is very nearly fixed and the force in it is therefore inversely proportional to the area carrying it. That is the same insensitivity read from its useful end: the arm is what design assumes, and the stress is what design solves for.

The same reasoning, three other places #

The move — take the first moment of whatever is effective, and put the axis where it vanishes — is not about concrete. It is about any section whose effective area is not its geometric area.

A composite steel-and-concrete beam. A steel section with a concrete slab on top, connected so they act together. The slab is transformed by the modular ratio into an equivalent width of steel, the axis is found by the same first-moment condition, and it usually lands inside the slab — meaning the entire steel section is in tension and the whole of it is working at its best lever arm, which is why composite construction is efficient. Whether the two act together at all depends on the connection carrying the shear flow between them, which is a separate calculation and the one that decides whether the transformed section exists.

A plated girder with a hole in the web. Cut a service penetration through a web and the effective area changes; the axis moves toward the remaining material. The same is true of a plate that has locally buckled, where the middle of the plate stops carrying and only an effective width near the edges still works — the section’s properties are then computed on the effective area, by exactly this arithmetic.

A section with residual stresses that have yielded locally. Once part of a steel section has yielded, its incremental stiffness is zero and it contributes nothing to further bending, so the axis for the next increment of moment is found from the still-elastic part. This is exactly how the plastic hinge develops: the axis migrates as yield spreads, ending up at the equal-area axis rather than the centroid, and the two coincide only for a symmetric section.

Plane sections staying plane is doing the real work here. The strain distribution is linear, and that is a kinematic statement independent of material. Everything else in this essay is the consequence of feeding that linear strain into different stress-strain relations for different parts of the section.

That figure is the long-term case, and it is the reason a concrete deflection calculation is quoted as two numbers. Stiffness rather than strength is what time attacks, and it attacks it through a material constant rather than through anything a load case would show.

One beam, two cracked sections, and a circularity #

A continuous beam has moments of both signs along it, and a cracked section is not the same object in the two.

Take an ordinary floor beam cast monolithically with its slab — a T-beam. In the sagging regions the flange is at the top, in compression, and the compression zone is very wide: the neutral axis sits high, often inside the flange, the lever arm is nearly the full depth, and the cracked second moment is large. In the hogging regions over the supports the flange is in tension and has cracked, so what is left is the narrow rib with some top steel in it. The compression zone is now the rib’s width rather than the flange’s, the axis drops a long way, and the cracked second moment can be half or a third of the sagging value.

So the member’s stiffness varies along its length by a factor of two or more, with the variation switching where the moment changes sign — and the position of that point is an output of the analysis.

Which produces a circularity worth naming. The distribution of moments in a continuous beam depends on the relative stiffnesses of its parts. Those stiffnesses depend on whether each part is cracked and in which direction. Whether it is cracked, and in which direction, depends on the moment. Nothing in the chain can be started without one of the others.

Practice resolves it by refusing to enter the loop: the analysis is run on gross, uncracked section properties throughout, uniform along the member, and the resulting moments are used. That is not accurate and it is defensible — a uniform error in stiffness cancels out of a relative-stiffness calculation entirely, and a non-uniform correction applied inconsistently would be worse than none. It is also why redistribution is permitted so readily in concrete: the real member has already redistributed toward the answer the uniform analysis did not give, before anything yielded.

Where the model stops #

Cracking is not uniform along the member. The section analysed is a section at one place. A real beam is cracked at intervals of a couple of hundred millimetres, and between the cracks the concrete is still gripping the bar and carrying tension. The member’s stiffness is somewhere between the cracked and uncracked values, closer to cracked at high load and closer to uncracked at low, and interpolating between them is what the deflection calculation actually does.

Two things about that figure are worth carrying. The first is how far across the beam sits: everything on this page about a cracked section is very nearly the whole answer at service load, and the 80 per cent stiffness loss the opening figures reported is not an upper bound being approached slowly. The second is that the interpolation is not a constant. The coefficient that positions the beam between the bounds halves under sustained or repeated load, because it is the bond between bar and concrete that drags the intact concrete into the calculation, and that bond deteriorates. A beam measured on the day it is loaded and the same beam a year later are at different points between the same two lines.

The modular ratio is not a constant. Concrete creeps. Under sustained load its effective modulus falls by a factor of two or three over years, which raises $n$, which moves the axis, which changes every stress in the section. The long-term analysis is the same arithmetic with a different $n$, and the difference between short-term and long-term deflection is often larger than the difference between cracked and uncracked.

Shrinkage does the same thing without any load. Concrete shrinks and the steel does not, so the bars restrain the shrinkage, and the restraint puts the concrete into tension and the steel into compression with no external action whatever. It is a self-equilibrating stress field of exactly the kind an imposed deformation produces in a redundant structure, and it can crack a member that has never been loaded.

One axis, one direction. The analysis finds a horizontal neutral axis, which is correct only when the bending is about a principal axis of the effective section. Bend the same section about a skew axis, or reinforce it asymmetrically, and the axis is no longer horizontal — the first-moment condition becomes two conditions, and the axis takes up whatever inclination satisfies both. A section under biaxial bending has a neutral axis that is not parallel to either applied moment, which is one of the reliably counter-intuitive results in the subject.

Bond. All of it assumes the steel and concrete strain together. Where the bond fails — at a lap that is too short, at a bar that is too smooth, in a member that has been overloaded — the transformed-section idea has no basis at all, because the transformation was a statement that they share a strain.

The figures have a limitation worth naming. The cracked zone is drawn hatched, as though the crack were a clean horizontal boundary at a known depth. Real cracking is a set of discrete, roughly vertical cracks with irregular tips, and “the depth of the crack” is a statistical property of a pattern rather than a line. What the hatching honestly represents is the part of the section the analysis has stopped counting, which is an accounting boundary and not a physical one.

The generalisation #

The pattern generalises to any composite: find the effective area, transform everything into one material by the ratio of moduli, and apply single-material theory to the result. It works for timber flitch beams with a steel plate between two joists, for glass-reinforced plastics, for laminated timber with different grades in different laminations, and for steel sections with a cover plate of a different grade.

There is one composite in the list that behaves differently from the rest, and it is worth separating out. In a flitch beam or a composite deck the two materials are both elastic and both stay elastic, so the transformation is a fixed geometric fact. In reinforced concrete one of the two materials has changed state — it has cracked — and the effective area therefore depends on the load. That makes the section nonlinear in a way the others are not: the same beam has one set of section properties below its cracking moment and another above it, and the transition is not gradual.

The deeper point is about what a neutral axis is. It is tempting to think of it as a physical feature of the beam — a plane where nothing is happening. It is better understood as a bookkeeping consequence: the location at which the section’s own resultant axial force comes to zero, given the current effective area and the current stress-strain relation of each part. Change any of those and it moves, without anything moving in the beam.

The history is short and instructive. Reinforced concrete was patented and built decades before it was understood: Monier’s pots and tanks date from the 1860s, Hennebique’s frames from the 1890s, and the transformed-section analysis was worked out afterwards to explain what was already standing. Koenen proposed putting the axis at mid-depth in 1886, which is wrong for a cracked section, and the correct treatment — cracked section, modular ratio, first-moment condition — settled around the turn of the century, in the hands of Considère, Ritter and Mörsch. For fifteen years the material was designed by rules of thumb calibrated on tests, which worked, and which is a reminder that the analysis is a description of the thing rather than a permission for it.

The ladder from here #

Later rungs on this anchor: the ultimate limit state, where the linear stress block is replaced by a rectangular one and the axis moves again. Tension stiffening and the interpolation between cracked and uncracked. Creep and the long-term modular ratio. The prestressed section, where the axis is manipulated deliberately by putting stress in backwards before the load arrives. The doubly reinforced section, and what compression steel does to the axis. And crack width itself, which is the quantity the whole serviceability argument is really about, and which depends on bar diameter and spacing rather than on anything in the stress block.