Where the steel is, not how much of it

Bending is a pair of forces. A section carrying a moment has a compression resultant somewhere in its upper part and an equal tension resultant somewhere in its lower part, and the moment is the product of one of them with the distance between the two.

$$M = C z = T z$$

Two quantities, and they are bought on entirely different terms. The force is an area of material at a stress, ordered by the tonne and priced accordingly. The distance is a dimension on a drawing, and it costs nothing at all.

Which free body produced the number #

Cut the beam and take everything on one side. Two conditions have to hold on the cut face: the axial forces sum to zero, and their moment equals the applied moment.

The first condition is what fixes the neutral axis. It says the compression in the concrete equals the tension in the steel, and since the compression grows with the depth of the compression zone, there is exactly one depth at which the two balance. Nothing about the applied moment enters that calculation for a cracked elastic section — the neutral axis is a property of the geometry and the modular ratio alone.

The second condition then gives the moment, and it does so by multiplying the resultant found in the first by the distance between the two resultants. That distance is the internal lever arm $z$, and everything in this essay is about it.

On the section drawn, $z = 462$ mm against an effective depth of 540 — a ratio of 0.856, which is where the familiar rule of thumb $z \approx 0.9d$ comes from. It is not a constant, and its variation is the interesting part.

Two integrals, and no third #

A concrete compression zone has a curved stress distribution whose shape depends on the material’s own law, and every design office replaces it with a rectangle. That substitution is exact for the only two things the moment equation asks about.

The equilibrium equations need the area under the stress distribution — the total compression — and its centroid — where that compression acts. Any two distributions with the same two integrals are indistinguishable to a bending calculation, and the rectangle was constructed from the curve’s own integrals precisely so that they match.

Deliberately the wrong shape is the essay about that construction. The consequence here is worth stating positively: the lever arm is not a property of the stress block’s shape. It is a property of two integrals, and the only way to change it is to change where the material is.

What adding steel actually does #

Follow the two conditions through and the answer to “what does more reinforcement buy” is not what the linear intuition says.

Adding steel increases $T$ at a given stress. Equilibrium then requires a larger $C$, which requires a deeper compression zone, which lowers the compression resultant and therefore shortens the lever arm. So the capacity $T z$ grows less than in proportion to the steel area, and the shortfall accelerates.

Push it far enough and the lever arm collapses toward zero while the steel is still elastic: the compression zone reaches most of the depth, the concrete crushes before the steel yields, and the section fails without warning. That is the over-reinforced case, and every design standard forbids it by putting a limit on the neutral axis depth rather than on the steel area — a limit on the lever arm, expressed in the variable that decides it.

The mirror image is what makes the depth so much better a variable. Increasing $d$ increases the lever arm in proportion and does nothing to the force at all.

Where the effective depth is actually decided #

If capacity is proportional to $d$, then the design of a reinforced beam is largely the design of $d$, and it is worth listing who decides it. It is not the structural engineer.

$$d = h - \text{cover} - \phi_{\text{link}} - \phi_{\text{bar}}/2$$

Cover is a durability requirement, set by the exposure class and the fire period. The link diameter is set by the shear design, or by a minimum. The bar diameter is set by what is available and by crack width rules. On a 600 mm deep beam with 25 mm cover, an 8 mm link and 25 mm bars, $d = 554$. Change the cover to 50 for an aggressive exposure and $d = 529$ — 4.5% of the capacity, gone, for a reason that has nothing to do with structure.

Put the bars in two layers, because they will not fit in one, and the centroid of the steel drops another 30 mm or so: another 5%. Ten per cent of a beam’s bending capacity can be decided by a cover requirement and a bar spacing rule between them, and neither appears in any equation a structural calculation contains.

The bar has to be anchored to be worth anything #

A tension resultant is only there if the bar can develop the force. That is a bond question, and it is the one place where the amount of steel and its position interact.

The development length grows with the bar diameter, so a section reinforced with a few large bars needs longer anchorages than one with many small ones at the same area — and the large bars also sit further from the face, lowering $d$. Both effects push the same way, which is why heavily reinforced members are detailed with more, smaller bars than an area calculation alone would suggest.

The distribution matters as much as the length. At a code’s own development length the elastic bond is only 44% used, because the slip is concentrated at the loaded end and dies away over a decay length. The uniform-bond assumption behind every tabulated anchorage is a statement about what happens after the bond has yielded along the whole length, which is a claim about ductility rather than about strength.

Two section moduli, and only one of them is a strength #

The lever arm has an elastic counterpart, and for an asymmetric section there are two of them.

$Z = I/y$ is a lever arm in disguise: it is the second moment divided by the distance to the fibre being checked, and a section with material concentrated near one face has a large modulus for that face and a small one for the other. Which one governs is decided by the sign of the moment rather than by anything about the section — so a member that can be bent either way has the smaller of the two as its capacity, and the extra material on one side is dead weight.

That is why a tee is an efficient shape for a slab and a poor one for a beam that can hog, and it is the same trade two strengths depending which way up is about.

Prestress, where the lever arm is a profile #

The same variable appears in prestressed concrete under a different name and with much more freedom.

A tendon’s eccentricity is a lever arm the designer draws, and unlike a reinforcing bar it can vary along the member. It is largest where the moment is largest and zero where the moment is zero, which is the profile in the figure — and the limits it has to stay within are four inequalities and a wedge, two at transfer and two in service.

The freedom has a price. A tendon low at the ends would crack the top of a beam carrying nothing but itself, which is a stage that exists on a casting bed and nowhere in the finished structure.

What a stiff material does to the share #

There is one further way in which position beats quantity, and it is about which material carries what.

Because plane sections stay plane, the strain at a height is fixed and the stress follows the modulus. A stiff material at a large distance therefore takes a share of the moment out of all proportion to its area — the modular ratio times the lever arm ratio, and nothing else.

The practical corollary is uncomfortable: a stiff repair attracts the very load it was added to relieve. Bond a steel plate to the soffit of a cracked beam and it picks up load in proportion to its modulus, which is exactly what was wanted; add a stiff element beside a soft one anywhere in a structure and the same thing happens whether it was wanted or not.

The number that says how much is left #

There is a single ratio that carries most of what a designer needs to know about a bending section, and it is the lever arm written as a fraction of the effective depth.

At $z/d$ close to 0.95 the section is very lightly reinforced: the compression zone is shallow, the concrete is barely working, and adding steel is almost perfectly efficient because the lever arm hardly moves. This is the state of a slab, and it is why slabs are designed with a rule of thumb rather than an iteration.

At $z/d$ around 0.85 — the section in the figure at the top — the section is doing what it was designed to do. Adding steel still helps, at a diminishing rate.

Below about 0.82 the neutral axis has descended past the limit standards impose, and the section is refused rather than being allowed to be inefficient. The refusal is about ductility: a section whose concrete crushes before its steel yields gives no warning, and the depth limit is how that is prevented. It is also, incidentally, the point past which adding steel has stopped being good value.

So a single dimensionless number tells a designer where a section sits between three quite different regimes, and it is a ratio of two lengths with no material in it. That is characteristic of this subject and worth noticing: the quantities that generalise are ratios of lengths, and the ones that do not are stresses.

The corresponding statement for a steel section is the shape factor, which is the ratio of the plastic lever arm to the elastic one and is 1.5 for a rectangle, about 1.15 for an I-section, and 1.7 for a solid circle. Same question, same kind of answer, and again a pure geometric ratio.

Where the model stops #

The lever arm is not constant along the member. Everything here is a cross-section calculation at one station, and $z$ changes as the moment changes and as bars are curtailed. Near a support it is not even a bending quantity — the chord carries half the shear as tension, and the force in the bar is the moment diagram shifted rather than divided.

Plane sections is doing all the work. The whole construction depends on the strain varying linearly across the depth, which is the assumption underneath and which fails in deep members, near openings and next to concentrated loads.

The steel may not reach its yield stress. Everything above assumed the tension resultant was the area times the design strength, which requires the steel to have strained past yield — and whether it has depends on where the neutral axis ended up, which is the quantity being computed. The check closes on itself and, in the region where it does not close, what is left after the first fibre yields is the correct question.

The neutral axis moves as the material yields. The elastic result above and the plastic one are different calculations with different lever arms, and the ratio between the two moments is the shape factor — 1.50 for a rectangle, and reached only at several times the curvature at first yield.

What the picture cannot show #

A section drawing shows bars as circles at positions, and the positions are nominal. What is built is decided by chairs, spacers, the sequence of fixing, and whether the top steel was walked on before the pour — and effective depth is measured to the real position rather than the drawn one.

The consequence is that $d$ is the least reliably delivered quantity in a reinforced concrete member and the one its capacity is most directly proportional to. Bars in the top of a slab pushed down 20 mm by a boot lose 10% of a 200 mm slab’s hogging capacity, in the region where the hogging moment is largest.

Nor does the drawing show the fixing tolerance running the other way, which it also does. This is a quantity with a scatter and a mean, applied to a capacity that is linear in it, and the scatter has a bigger effect than any material variability in the section.

The generalisation #

The habit worth carrying is to look for the distance in every capacity, because there is nearly always one and it is nearly always the cheap variable.

A section’s bending capacity is a force times a lever arm. A truss’s chord force is a moment over a depth. A shear wall’s overturning resistance is a weight times a width. A bolt group’s moment capacity is a force times a distance to the group’s centroid. In every one of them the force is a material quantity that costs money and the distance is a geometric quantity that costs a decision.

The reason it is under-used is the same in every case: the force appears in a specification and the distance appears on a drawing, and only one of those is checked by the calculation. Where the material sits beats how much of it there is, and the whole of the difficulty is that nobody is asked the question in those terms.