The one number a stronger steel does not change

The second of this site’s threads is that geometry beats material: move the same steel further from the neutral axis and the section gets stiffer for nothing, make the truss deeper and its chord forces fall without a gram being added. The argument works because shape is a variable a designer controls and material is not much of one.

That is normally read as an argument about economy. It is stronger than that. For a large class of structural problems, changing the material is not merely poor value — it does nothing whatever, because the property that decides the answer is the same for every version of the material available.

Why the slope is not for sale #

Strength is a statement about defects. A metal yields when dislocations move through it, and every strengthening mechanism metallurgy has — alloying, cold work, precipitation, grain refinement, quenching and tempering — works by making dislocations harder to move. There is a great deal of room in that: the yield stress of iron can be moved over a range of more than ten by processing, without changing what the material is.

Stiffness is a statement about bonds. The elastic modulus is, near enough, the second derivative of the interatomic potential at the equilibrium spacing, divided by that spacing. It is set by what the atoms are and how closely they are packed, and no amount of rearranging dislocations touches it. A steel with ten times the yield stress of pure iron has essentially the same modulus as pure iron, because the same atoms are the same distance apart.

That gives the two properties completely different characters. Strength is a processing variable. Stiffness is a periodic-table variable, and the table is short.

What that costs, in the places it decides #

Half the failure modes on this site do not contain the yield stress at all.

Deflection. The mid-span deflection of a simply supported beam under a uniform load is $5wL^4/384EI$. There is no strength anywhere in it. A beam that has moved too far to be usable is not helped by a stronger grade by one part in a thousand; the only routes are more depth, more material, a shorter span or a different support arrangement. Which is why the serviceability limit governs long-span members almost universally, and why the answer is always a change of shape.

Buckling. The Euler load is $\pi^2EI/L_e^2$, and the yield stress is not in it either. Two identical columns in mild steel and in a grade twice as strong buckle at the same load, provided they are slender enough to buckle at all. What the stronger grade buys is a higher squash load, which moves the crossover between the two failure modes to a higher slenderness and does nothing above it.

The crossover slenderness is $\pi\sqrt{E/f_y}$, which for $E = 210{,}000$ and $f_y = 275$ is 87, and for the same modulus and $f_y = 460$ is 67. Going to the stronger grade has therefore reduced the range of columns whose capacity it improves, from everything below a slenderness of 87 to everything below 67. Above that number the two grades are indistinguishable and the more expensive one is simply more expensive.

Vibration. A floor’s natural frequency goes as the square root of stiffness over mass. Stronger steel changes neither, and a floor that is unpleasant to walk on cannot be fixed by ordering a better grade. Worse, the usual response to a strength problem — a lighter section in a stronger grade — moves a vibration problem in the wrong direction, because it removes mass while leaving stiffness roughly alone. Long-span composite floors in high-strength steel are the standard modern case: every strength check passes comfortably and the floor is unusable at four hertz.

Anything decided by an imposed deformation. A settled support in a redundant structure generates moments proportional to $EI$, so a stiffer structure attracts more of them, and a stronger material attracts exactly the same ones while being no more able to shed them.

Which free body produced the number #

The modulus in every one of those expressions comes from one measurement, taken on the free body described in the essay on the tensile test: a bar cut anywhere along its gauge length carries only the applied axial force, and the slope of force-over-area against extension-over-length is what is being quoted.

It is worth being precise about which slope. A stress-strain curve is straight for a while and then is not, and the modulus is the slope of the straight part — the initial tangent. For steel the straight part is genuinely straight over 0.13% of strain and there is no ambiguity. For concrete there is no straight part at all: the curve is bending from the origin, and the number quoted as its modulus is a secant through some agreed point, usually a third of the compressive strength. That makes concrete’s modulus a construction of the same kind as the proof stress, with the same arbitrariness in it.

Two consequences follow that are easy to miss. Concrete’s quoted modulus varies with its strength — a stronger mix is stiffer, because both properties come from the same paste — so unlike steel, concrete does have a purchasable stiffness, though over a modest range and by a cube-root-ish relation rather than proportionally. And its modulus is not constant in time either, which is the subject of its own essay and is the largest single effect in the serviceability of concrete structures.

Where the model stops #

Stiffness per unit weight is a different question, and it has a different answer. Divide the modulus by the density and steel, aluminium and timber come out remarkably close: about 27, 26 and 22 in consistent units. The three materials are nearly interchangeable on specific stiffness and wildly different on absolute stiffness, and which of those matters depends on whether the structure is carrying its own weight or fitting into a depth.

That is the reason an aluminium beam of the same weight as a steel one can be made just as stiff, and the reason it has to be three times as deep to do it. The material choice becomes a geometry choice, which is the site’s second thread arriving from an unexpected direction.

The near-equality is not a coincidence and it is worth a sentence. Both the modulus and the density scale with how tightly and how closely the atoms are bound, so a heavy element tends to be a stiff one and the ratio varies far less than either term. That is why the specific-stiffness column of a materials table is nearly flat across the metals, and why the materials that break out of it — carbon fibre, and to a lesser extent timber along the grain — do so by being directional rather than by being better. A unidirectional composite is stiff along the fibres and unremarkable across them, and its advantage is available only to a structure whose load paths are known in advance and do not move.

The modulus is only a number for a material that is one thing. Reinforced concrete has two, and the standard treatment does not average them: it converts one material into an equivalent area of the other by the ratio of their moduli, which is the modular ratio and is the reason the neutral axis of a cracked section is nowhere near the centroid of its outline. The technique generalises to any composite — a timber flitch beam with a steel plate, a laminated beam with different grades in different laminations — and in every case what is being transformed by is a ratio of stiffnesses and never of strengths.

The modulus is not quite a constant even for one material. It falls with temperature, and it falls faster than the strength does — at 500°C a steel member has kept 78% of its yield stress and 60% of its modulus, which is why a member’s failure mode can change during a fire. It is also slightly different in tension and compression for some materials, and materially different along and across the grain for anything fibrous.

And the effective stiffness of a member is often not the material’s stiffness at all. A cracked reinforced concrete beam has lost most of its second moment of area to a material that has stopped working, and the number that governs its deflection is a property of the crack pattern rather than of the concrete.

The arithmetic of paying in the wrong currency #

It is worth putting numbers on how badly a strength purchase performs against a stiffness problem, because the answer is not “poorly” but “not at all, and the cost is the whole of it”.

Take a beam whose span has been fixed and whose deflection is 20% over its limit. The available moves are these. Increase the depth: deflection goes as the inverse cube of depth for a given shape, so 6% more depth suffices. Add material at the same depth: the second moment rises roughly in proportion to area for a flanged section, so 20% more steel does it. Shorten the span: deflection goes as the fourth power, so 5% off the span is enough. Change the support condition from simply supported to continuous: the coefficient falls from 5/384 to about 1/384 over an interior span, which is a factor of five in one step and costs nothing but a connection.

Increase the grade: nothing. The deflection is unchanged to the last decimal place, the beam still fails its serviceability check, and the only thing that has moved is the price. There is no partial credit and no diminishing return, because the quantity being purchased does not appear in the quantity being fixed.

The same arithmetic runs the other way and is just as sharp. A short stocky column governed by squashing gains its full 67% from a move from S275 to S460. The property is worth everything or nothing, and which of the two is decided by a slenderness the designer can read off before making the call.

The generalisation #

The pattern is that structural design has two kinds of variable, and they are not interchangeable. Some quantities can be bought: strength, toughness, corrosion resistance, and to a limited extent stiffness in composite and cementitious materials. Others can only be arranged: second moment of area, span, depth, restraint, redundancy, the route the load takes to the ground.

Almost everything interesting in this subject happens because a problem posed in one currency has to be paid in the other. A deflection problem cannot be paid in strength. A lateral-torsional buckling problem cannot be paid in strength either — it is decided by what is holding the compression flange, which is a matter of arrangement. Conversely, a fatigue problem cannot be paid in section size beyond a point, because the stress range falls in proportion while the detail category does not move at all.

Recognising which currency a problem is denominated in is most of the skill in choosing what to change, and the commonest error in the subject is reaching for strength because strength is the property with a number on the drawing.

A surprising place this turns up #

The constancy of the modulus is what makes Maxwell’s reciprocal theorem and the whole apparatus of elastic analysis useful in practice rather than merely true.

A redundant structure’s load distribution depends on the relative stiffnesses of its members. If stiffness were a purchasable property varying from member to member with the grade specified, then every indeterminate analysis would depend on a schedule of material grades that is usually decided after the analysis is done, and changing one member’s grade would redistribute the whole frame.

Because $E$ is the same everywhere in a steel structure, it cancels out of every relative-stiffness ratio. The distribution of forces in a redundant steel frame depends only on the geometry and the second moments of area — which is why an indeterminate structure can be analysed before anybody has decided what grade to build it from, and why the analysis does not have to be redone when they do.

Where the ladder goes next #

Later rungs on this anchor: the shear modulus and Poisson’s ratio, which are the two other elastic constants and are not independent of the first for an isotropic material. Anisotropy, where the single number becomes a matrix and timber, laminates and rolled plate each need a different amount of it. The tangent and secant moduli, and why buckling in the inelastic range is governed by the tangent one — which is the piece Engesser got wrong and then right. Specific stiffness as a selection criterion, and the material charts built on it. The modulus of concrete as a function of age, aggregate and stress level. And the strange corner where a structure’s stiffness is not the material’s at all: a cable net or a pneumatic structure whose stiffness comes from prestress and goes to zero when the prestress does.

Historically the constancy of the modulus took a long time to be recognised as remarkable. Young published his modulus in 1807 in a form nobody could use, and Navier put it into the shape now standard around 1826. Through the nineteenth century the great argument in the subject was about strength, because cast iron, wrought iron and steel differed enormously in it. Only when steel had displaced the others and its grades began to multiply did it become clear that the property everybody had been trying to improve and the property that governed half the failures were different properties, and that only one of them was moving.