The shape that came out of the shop

A welded plate girder is drawn straight. It does not come out straight, and the amount by which it does not is computable from the welding procedure rather than from the workmanship.

The mechanism has no mystery in it. A weld is a volume of metal placed molten and then cooled by six or seven hundred degrees while the plate around it refuses to let it contract. It cannot contract, so it yields in tension, and when everything is cold what is left is a locked-in tensile force along the weld balanced by compression in the plate either side. That force is not a stress state to be noted and forgotten; it is a force, with a magnitude and a line of action, and everything a force does it does.

How large the force is #

Okerblom’s model is the one used here, and it is a free body rather than a correlation.

The arc delivers a heat input $Q$ per unit length of weld, of which a fraction $\eta$ enters the plate. That heat raises a region above the temperature at which the steel has no strength left to resist — around 580 °C for structural grades — and everything inside that region ends up at yield in tension when cold, because it was free to be compressed while hot and is not free to expand back. So the shrinkage area follows from a heat balance,

$$A_{sh} = \frac{\eta Q}{\rho c\, \Delta T_y},$$

and the force is that area at yield:

$$N = f_y A_{sh}.$$

For 1.2 kJ/mm of heat, an arc efficiency of 0.8 and ordinary steel, $A_{sh}$ is 439 mm² per weld. Two of them at 355 N/mm² is 312 kN.

That number is worth sitting with. It is larger than the axial force in most of the members the girder will be connected to, it was applied by nobody, it appears on no drawing, and it is present in every welded assembly in the world.

An eccentric axial force is a curvature #

The force acts at the weld. On a plate girder that is at the web-to-flange junction, which for an asymmetric section — a heavier bottom flange, a girder designed for composite action — is a couple of hundred millimetres from the centroid.

An axial force $N$ applied at an eccentricity $e$ along the whole length of a member is a uniform moment $Ne$, and a uniform moment is a uniform curvature:

$$\kappa = \frac{Ne}{EI}, \qquad \delta = \frac{\kappa L^2}{8}.$$

For the girder drawn — 12 m long, $I = 4.5\times10^8$ mm⁴, welds 210 mm off the centroid — that is a radius of curvature of 1,443 m and a midspan bow of 12.5 mm. The straightness tolerance is L/1000, which is 12.0. The girder is out of tolerance before it has been touched, and the calculation that says so uses nothing but the welding procedure.

It is also 1.19 mm shorter than drawn, which is the half of the effect nobody photographs and the half that shows up as a member that will not fit.

Which free body produced the number #

Take a transverse cut through the girder, well away from its ends, after it has cooled and before anything has been applied to it.

Crossing the cut are two things: a tension in the shrinkage zone at the weld, and a compression distributed over the rest of the section. The two must sum to zero force, because nothing external is pushing on the girder — it is lying in the shop. They must also sum to whatever moment the section has, and that moment is not zero: the tension is concentrated at the weld and the compression is spread, so their resultant couple is $Ne$ about the centroid.

The free body is in equilibrium with no external force and a non-zero internal moment, which is exactly the condition that produces curvature and nothing else. That is why the bow is a circular arc rather than a kink: the moment is the same at every section because the shrinkage force is the same at every section.

It also says what happens if the girder is straightened. Straightening applies an external moment opposing $Ne$; the internal state is unchanged; and the moment goes back the moment the jack comes off, unless the straightening was itself plastic. Mechanical straightening works by adding a second plastic strain, which is why it is done with a press or with heat and not with a clamp.

What the bow is worth as an imperfection #

A ten-millimetre bow in a twelve-metre member sounds cosmetic. It is not cosmetic because an initial bow is precisely what the column curve was fitted to.

A compression member with an initial deflection $e_0$ has that deflection amplified by the axial force it carries,

$$e_{total} = \frac{e_0}{1 - N/N_{cr}},$$

and the amplified bow produces a bending stress $N e_{total}/Z$ that adds directly to the axial one. For the girder drawn, at 120 N/mm² of applied compression against a critical stress of 432, the amplification is 1.385, the total bow is 17.3 mm, and the extra bending stress is 17 N/mm².

Seventeen is not alarming on its own. What is alarming is where it sits: on top of an imperfection allowance the buckling curve already contains. The column curves in every code were fitted to test data on members with the fabrication imperfections of their day; a member whose welding has produced an additional bow of the same order has consumed the allowance twice, and no check adds the two.

Symmetry is the whole remedy #

The force cannot be avoided. Its line of action can be moved, and moving it onto the centroid removes the moment exactly.

A doubly symmetric section welded on both sides of the web at both flanges has four shrinkage forces whose resultant passes through the centroid by symmetry. The curvature is zero — not small, zero — and the member comes out straight and slightly short. The same girder with one flange welded first and the other a day later comes out bowed, straightens partially when the second is done, and finishes somewhere in between depending on how much of the first bow has been locked in by the second’s restraint.

That is why welding sequence is a design output rather than a shop preference on anything sensitive, and why the sequence is specified on drawings for box girders, orthotropic decks and heavy moment connections. The instruction is always the same shape: balance about the centroid, and work outward from the middle.

The other lever is heat input. The force is linear in $Q$, so halving the heat halves the force and halves the bow. That points at process selection — a lower-heat process, more passes of smaller size, or a mechanised run at higher speed — and it points against the instinct to lay a bigger weld for comfort, which buys strength linearly and buys distortion linearly too.

The scaling, which is the part worth carrying #

Three exponents come out of the arithmetic and they are more useful than any of the numbers.

Linear in heat input. The shrinkage force is $f_y \eta Q/(\rho c \Delta T_y)$, so halving the heat halves the bow. Every process decision that lowers heat input — a faster travel speed, more passes of smaller size, a lower-heat process — buys distortion back proportionally.

Linear in eccentricity, through zero. There is no threshold and no dead band. A section symmetric about the axis in question has no bow at all, and one with the welds a hundred millimetres out has half the bow of one with them at two hundred.

Quadratic in length. $\delta = \kappa L^2/8$ with $\kappa$ fixed by the section, so the bow of a given girder is set by its length and not by anything the fabricator can do at the weld. Doubling a girder’s length quadruples its bow and only doubles its tolerance, which is why long members are the ones that fail the check and why the same section detail is acceptable at 6 m and not at 18.

That last one is the practical rule. A welding procedure that produces an acceptable member at one length produces an unacceptable one at twice it, and nothing in the procedure says so.

The three distortions, and why only one of them is here #

Welding distortion comes in three kinds and this page is about one of them, which is worth saying so the other two are not mistaken for it.

Longitudinal shrinkage is the effect computed above: a force along the weld, an eccentric moment, a bow in the plane containing the weld and the centroid. It is the one that has a clean free body and a closed form.

Transverse shrinkage pulls the plates together across the weld. It closes a gap, shortens a girder across its width, and — as the through-thickness essay describes — is the one that threatens to tear a plate. Its magnitude is empirical rather than derivable.

Angular distortion rotates the plates about the weld line, because a fillet is deposited on one side and shrinks more at the top of the throat than at the root. It is what makes a T-joint’s flange curl up and a butt weld’s plates fold, and it is the most visible and the least computable of the three.

All three happen at once. Only the first can be predicted from a heat balance, and the others are handled by procedure trials, which is a fair description of the state of the subject.

What it does to a built-up column #

The case where all of this matters most is the one where the member is in compression and the welds are at the extremes of the section.

A built-up column — two channels laced together, or four angles battened — is welded at intervals rather than continuously, but each weld is a shrinkage force applied a long way from the centroid, and their sum is a moment about whichever axis the welds are unbalanced on. Built-up columns are also the members whose buckling calculation is most sensitive to shear flexibility and to imperfection, so the additional bow is applied to the section least able to absorb it.

This is the reason the column curves distinguish welded from rolled sections, and put welded ones on a lower curve. It is usually explained by residual stresses — the compression left in the flange tips brings forward first yield, which is true — but the geometric half is the same phenomenon and is not usually separated from it. The curve is fitted to whatever the tests contained, and what they contained was members that were both residually stressed and bowed.

The residual stress and the bow are one object #

It is worth being precise about the relationship between this page and the one about residual stress, because the two are usually treated as separate subjects and are the same one.

A residual stress field is self-equilibrating: its resultant force is zero and its resultant moment is zero, taken over the whole section. That is the definition. A welded girder’s shrinkage field satisfies the first condition and not the second — not because the physics is different, but because the section is asymmetric, so the tension at the weld and the compression spread over the rest have a couple between them.

Straighten the girder and the couple is removed by an external moment; the stresses are still there. Weld the section symmetrically and the couple was never there; the stresses still are. The stresses and the curvature are two readings of one field, and which of the two appears depends entirely on the section’s symmetry.

That is why a rolled section, whose residual stresses come from uneven cooling of a symmetric shape, is stressed and straight, while a welded asymmetric section is stressed and bent — and why both are on lower column curves than a hypothetical member with neither.

The measurement that settles it #

Everything here is predicted rather than observed, and the observation is easy and rarely made.

Lay a string line along the member and measure the offset at midspan: that is $\delta$. Measure the length against the cutting list: that is the shortening. Both take a minute and both are compared against numbers the welding procedure predicts before the member is made.

The comparison is worth making because it separates two explanations that get confused. A member that is bowed and the right length was distorted by something other than shrinkage — a restraint during welding, an uneven support in the shop. A member that is bowed and short in proportion — 12.5 mm of bow against 1.19 mm of shortening, a ratio of about ten for the girder drawn — is showing exactly the signature the model predicts, and the ratio is a property of the section rather than of the shop.

Where it is designed for, and where it is discovered #

There are three attitudes to this in practice and they map onto three kinds of structure.

Designed for. Box girders, orthotropic decks and heavy welded columns have a welding sequence specified on the drawings, a heat input limit in the procedure, and often a pre-set built into the assembly jig. The distortion is treated as a load case with a remedy, and the remedy is chosen before anything is cut.

Absorbed. Ordinary building steelwork specifies a straightness tolerance and leaves the fabricator to meet it however they like — usually by sequence, sometimes by straightening afterwards. This works, and it works because the tolerance is generous relative to what an ordinary symmetric section does.

Discovered. The failure mode is a member that cannot be erected: it does not line up with its holes, or it is visibly bowed and someone objects. At that point the options are to straighten it, to accept it with a check, or to remake it, and all three are expensive in a way that a five-minute calculation at detailing was not.

The distinction between the second and third is not the quality of the fabricator. It is whether the section is symmetric about the axis the welds are on, which is a decision made by the person who drew it.

Where the model stops #

The heat balance is a one-dimensional idealisation. A real weld pool has a three-dimensional temperature field and the shrinkage zone is not a rectangle. The model gets the magnitude right to within tens of per cent and the scaling — linear in heat input, linear in eccentricity, quadratic in length — exactly.

The plate is assumed elastic outside the shrinkage zone. On a thin flange the compression balancing the shrinkage force can be a large fraction of yield, and a section that yields locally redistributes the force rather than carrying it where the model put it.

Nothing here treats sequence. The forces are added as though every weld were made simultaneously. Real sequences lock some of the earlier distortion in, and a full treatment is an incremental analysis rather than a sum.

And the tolerance is not the criterion. L/1000 is a fabrication limit, not a structural one; a member outside it is not thereby unsafe and a member inside it is not thereby unaffected. The structural question is what the bow does to the buckling check, and the answer to that is continuous.

Where the ladder goes #

Later rungs on this anchor: transverse and angular distortion, and why neither has a closed form. Welding sequence as an incremental problem, and the software that solves it. Residual stress patterns in welded and rolled sections, and their separate contribution to the column curves. Pre-setting and mechanical straightening, which are plastic operations on a member that is already plastically strained. Distortion in orthotropic decks, where the plate is thin and the weld length per square metre is enormous. Heat input control and process selection as a structural decision. And the general form of the argument: a fabrication process that applies a force nobody specified to a member nobody checked for it.