The stress that was there before the load

Cut a rolled steel I-section into narrow strips along its length and the strips spring apart. Some are longer than they were when they were part of the section, some are shorter, and none of them is the length the section thought it was. The section was holding them in place, and the forces it was using to do it were there before anybody applied a load.

This is a stress field with no external cause, in equilibrium with nothing. It sums to zero axial force and zero moment across the section — it has to, since no force and no moment are being applied — and that is precisely why no static calculation on this site can see it. Every free body drawn anywhere in the preceding forty-six essays would be in equilibrium with or without it.

Where it comes from #

The mechanism is differential cooling and it is unavoidable rather than a defect.

A section leaves the rolls at around 1000°C and cools in air. The parts with the most exposed surface per unit of material — the tips of the flanges, the middle of the web — cool fastest. They contract first, while the rest is still hot and soft enough to accommodate the contraction without resisting it. Then the last regions to cool, the junctions where the flange meets the web, try to contract in their turn, and by now they are surrounded by material that has already set and will not let them.

The junctions therefore end up stretched, in tension, and the regions that cooled first end up compressed by the reaction. The magnitude depends on the section’s proportions and on how it was cooled, and for a hot-rolled section it is typically 20 to 30% of the yield stress. For a welded section it is very much worse: a weld is a small region of molten metal cooling inside a large cold one, and the tension along a weld line reaches yield.

The same mechanism produces residual stress in almost everything. A quenched bar has compression at the surface and tension inside. A cold-formed section has been bent past yield and carries the stresses that unloading leaves behind. Flame-cut plate has a strip of yield-level tension along the cut edge. Concrete shrinks against its own reinforcement. None of these appear in any analysis, and all of them are present in the structures those analyses describe.

What it costs, measured #

The cost is not capacity. A section with residual stress reaches exactly the same plastic moment as one without: once everything has yielded, the section has no memory of where it started, because yielding erases it. Residual stress is a statement about the path, not the destination.

What it costs is the point at which the path stops being straight.

Computed on this I-section: first yield arrives at 92% of the plastic moment with no residual stress, at 88% with a hot-rolled pattern of ±83 N/mm², and at 80% with a welded pattern of ±138. The plastic moment is 72.6 kNm in all three cases.

Those numbers look small, and for a beam they are. The reason residual stress matters is not the beam. It is the column.

Why it is the knee in the column curve #

Euler’s load is the load at which a perfectly straight elastic column becomes indifferent to being bent, and it depends on $EI$ alone. The squash load is $A f_y$. Plot both against slenderness and they cross, and the lower of the two is the capacity — a pair of straight-ish lines with a corner.

Real columns do not follow either line near the corner. They fall below both, by as much as a third, over a band of slenderness that happens to be where most real columns live.

The mechanism is that a column carrying axial compression has residual compression already in its flange tips. Those tips therefore reach yield at an average stress well below $f_y$ — for a pattern of 30% of yield, at about 70% of it. Once they yield, they contribute nothing to further stiffness: their tangent modulus is zero. The section resisting the next increment of bending is not the whole section but the elastic core, and its second moment of area is much smaller.

So the effective $EI$ collapses progressively as the yielded zone spreads, and the buckling load is governed by the tangent stiffness of the remaining core rather than by the elastic stiffness of the whole. The column buckles at a load Euler’s formula never contemplated, because Euler’s formula was written for a section whose stiffness is a constant.

And the geometry of the pattern makes it worse than it needs to be. The flange tips are the material furthest from the minor axis — the axis a column usually buckles about — so the first material to be lost is the material contributing most to the resistance. A residual pattern that put the compression in the web instead would cost far less.

The same pattern, twice as large #

A welded section is not a rolled one with slightly more of the same. The mechanism is more violent — a weld pool at 1500°C solidifying inside plate at ambient — and the tension along the weld line reaches the yield stress, because there is nothing to stop it: the metal contracts, the surroundings hold it, and it stops contracting only when it starts yielding.

The rounding is the whole effect in one word. Residual stress does not lower a capacity; it removes the corner from a curve, and everything that is decided by stiffness rather than by strength is decided in the region the corner used to be.

Which free body produced the number #

The free body is the cross-section, cut, with the stresses on the face — the same one as in the moment-curvature essay, and the residual field enters it in a specific way that is worth naming.

A locked-in stress is a locked-in strain. A fibre carrying 80 N/mm² of compression with nothing applied is a fibre already sitting at a strain of $-80/210000$ on its own stress-strain curve, and everything that happens subsequently starts from there. So the calculation adds a pre-strain of $\sigma_r/E$ to each fibre and is otherwise unchanged: plane sections still stay plane, the axial force on the face still has to equal the applied one, and each fibre’s stress still comes from its own total strain.

The check that the pattern is admissible is that it must be self-equilibrating, and this is not an assumption but a requirement: $\int \sigma_r\,dA = 0$ and $\int \sigma_r\,y\,dA = 0$, because the section is carrying no force and no moment before it is loaded. The cosine pattern used here satisfies both by construction — a cosine over the full depth has zero mean and, being symmetric about mid-depth, zero first moment. Any pattern that failed either test would be describing a section that was accelerating.

What the picture cannot show #

The figures on this page draw a stress that varies over the depth of a section, and the real field varies in three directions, over the length of a member, and between one piece of steel and the next.

It varies along the length. A section cools at its ends differently from its middle, and a member that has been flame-cut, drilled or welded has a local pattern superimposed wherever that happened. The single distribution drawn here is a representative cross-section and there is no such thing as the cross-section.

It varies between nominally identical pieces. Two beams of the same designation from the same mill differ, and beams from different mills differ more. This is one of the reasons column tests scatter, and the reason design curves are lower-bound envelopes rather than fits.

And it is not visible in any way at all. The one honest thing to say about the pictures here is that nothing corresponding to them can be seen by looking at a piece of steel, measured without destroying the piece, or inferred from any load test that stays elastic. Sectioning — cutting the member into strips and measuring how much each one springs — is the classical method, and it works by destroying the thing being measured, which is the clearest possible statement of how thoroughly this stress field hides from statics.

Where the model stops #

The pattern used here is a smooth idealisation of a jagged reality. Measured residual stress in a rolled section is closer to a set of straight-line segments with sharp changes at the junctions, and the actual distribution varies between mills, between section sizes and along the length of one piece. The cosine is a reasonable stand-in for the shape of the effect and should not be read as a measurement.

Residual stress is three-dimensional and this is one component of it. The longitudinal stress is the one that matters for a column and it is the one drawn. There are transverse and through-thickness components too, and in a thick welded joint they are what turn a ductile steel into a brittle one, because triaxial tension suppresses yielding without suppressing fracture.

It can be removed, at a price. Stress relief — heating the whole piece to around 600°C and cooling it slowly — works, and is routine for pressure vessels and for thick welded fabrications where the alternative is unacceptable. It is expensive, it distorts, and for ordinary structural steelwork nobody does it, so the column curves are calibrated on sections that have it.

A pattern can be put in on purpose, and then it is an asset. Shot peening drives the surface into compression to resist fatigue cracking, which starts at surfaces in tension. Autofrettage pressurises a thick cylinder past yield once so that unloading leaves the bore in compression. Prestressed concrete is the same idea at the scale of a structure. The same phenomenon that costs a column a third of its capacity is, applied on purpose, one of the most valuable techniques in the subject.

The generalisation #

The pattern is that a self-equilibrating stress field is invisible to statics and consequential to everything else.

This site has met the shape before from the other direction. Settle one support of a redundant beam and a complete set of bending moments appears, in equilibrium with no load at all. That field is self-equilibrating in exactly the same sense: cut anywhere and the sums cancel. The difference is only that the settlement’s field lives at the scale of a structure and this one lives at the scale of a cross-section.

Both share the consequence that makes them dangerous: they are strain-generated rather than force-generated, so they do not go away when the load does and they do not scale with it. And both share the consequence that makes them survivable: ductility erases them. Yield the section and the memory is gone. That is why residual stress costs a beam nothing at collapse and a column a great deal — a beam is allowed to yield and a column, which fails by instability before yielding is complete, is not.

A surprising place this turns up #

The most economically significant consequence of residual stress is that structural steel sections are hot-rolled rather than welded from plate wherever there is a choice.

A welded plate girder can be given exactly the flange and web the designer wants, at exactly the depth wanted, with the material distributed exactly where it is useful. A rolled section comes from a catalogue. On every geometric argument the welded one should win, and for deep girders it does.

For columns it usually does not, and the reason is on this page: welding puts yield-level tension along the weld lines and yield-level compression in the flange tips, roughly twice the rolled pattern. Welded columns therefore sit on a lower column curve than rolled ones of the same dimensions — this is explicit in the design curves, which are selected by fabrication method as well as by axis — and the geometric advantage of getting the material where it is wanted is spent paying for the stress that putting it there introduced.

Where the ladder goes next #

Later rungs on this anchor: the tangent-modulus and reduced-modulus theories of inelastic buckling, the argument between Engesser and Considère about which is right, and Shanley’s resolution in 1947 that neither is and the tangent modulus is the lower bound that matters. Measured residual stress patterns and the sectioning method used to obtain them. The multiple column curves and how fabrication method, axis and thickness select between them. Residual stress in cold-formed sections, which have the opposite sign at the corners and are strengthened by it. Welding sequence as a design variable. And stress relief, both thermal and mechanical, with what each actually removes.

Historically the effect was suspected long before it was measured. Nineteenth-century column tests scattered far more than the theory allowed and the scatter was blamed on end conditions and initial curvature, both of which are real and neither of which is sufficient. The systematic measurement of residual stress in rolled sections belongs to the 1950s and 1960s, principally at Lehigh, and the multiple column curves in modern codes are the direct result: a body of work whose conclusion was that a column’s capacity depends on how it was made, which is not a quantity structural theory had any place for.