An average stiffness is not a safe stiffness

Euler’s load is $\pi^2EI/L^2$, and every symbol in it is a single number. A column with a splice, a service hole, a change of section at a floor, a haunch, a cast-in plate or a corroded length has no single $EI$, and the question of what to put in the formula is answered on drawings every day by taking an average.

That answer is wrong, and it is wrong in the direction that matters.

Which free body produced the number #

Not a free body — an eigenvalue, and it is worth being clear about the difference because it is the reason averaging fails.

A column’s critical load is where the elastic stiffness $\mathbf{K}$ and the geometric stiffness $\mathbf{K}_g$ balance: $\det(\mathbf{K} - P\mathbf{K}_g) = 0$. Both matrices are integrals over the length — $\mathbf{K}$ weights $EI$ by the square of the curvature of the mode, and $\mathbf{K}_g$ weights the axial force by the square of the slope. So

$$P_{cr} = \frac{\int EI\,(v'')^2\,dx}{\int (v')^2\,dx}$$

which is guessing the shape written as a quotient. The numerator is a weighted average of $EI$, and the weight is $(v'')^2$.

For a pinned column the mode is a half sine, so $v'' \propto \sin(\pi x/L)$ and the weight is $\sin^2$ — zero at both ends, maximum at the middle, and with 61% of its total in the middle third. Stiffness in the middle third counts for nearly two thirds of the answer, and stiffness in the outer sixth at each end counts for about four per cent.

An arithmetic average weights every part of the column equally. That is the right weighting for exactly no buckling problem, and for a pinned column it is wrong by the ratio of $\sin^2$ to a constant.

Two wrong answers, and where the right one sits #

Take a pinned column whose middle third is at half the stiffness of the rest.

By average. The mean stiffness is $\tfrac{2}{3}(1) + \tfrac{1}{3}(0.5) = 0.833$, so the average answer is 0.833 of the uniform critical load.

By weakest. Use 0.5 throughout and the answer is 0.5.

Solved. 0.61.

The average over-predicts by 36%, on the unsafe side; the weakest under-predicts by 18%, on the safe side. Neither is close, and the reason they are not close is the weighting: the weak band is exactly where the weighting is heaviest, so the answer is dragged toward the weak value rather than sitting near the middle of the two.

The general position is that the true answer always lies between the two hand estimates, which is genuinely useful: it means a designer who computes both has a bracket, and one who computes only the average has an unsafe number with no indication of by how much.

Position, which matters more than amount #

Move the weak band rather than changing it, and the effect is startling.

A band of a sixth of the length at 40% stiffness costs 4.6% of the critical load when it sits against a pin, and 32.6% when it sits at mid-height — a factor of seven for the same material removed from the same column. Nothing about the amount of weakening changed; only where it was.

That has direct consequences on a drawing.

A splice belongs near a point of contraflexure. In a pinned column that is the end; in a fixed-ended one it is the quarter points. The traditional rule of splicing “just above floor level” is right for a sway frame, where the column’s mode has its inflection near mid-height and its greatest curvature at the floors — which is the opposite conclusion, reached by the same reasoning applied to a different mode.

A service hole belongs where the curvature is small, and “small” means somewhere the designer has to identify from the mode rather than from the moment diagram — the ends decide the length is what fixes the mode, and a column’s restraint conditions therefore decide where it may be weakened.

And corrosion at a column base is worse than it looks in a fixed-base frame and better than it looks in a pinned one. Same corrosion, same section loss, two different answers, because the mode’s curvature at the base is different.

The strongest column, which is not the uniform one #

If a weak middle is so costly, a strong middle should be worth having — and the question of how much is one of the oldest in the subject.

Lagrange asked it in 1770 and got it wrong. Clausen posed it properly in 1851 for a column of similar cross-sections — where the second moment goes as the square of the area, which is the case for a family of shapes scaled up and down. Keller settled it in 1960: the best distribution of a given volume beats the uniform column by exactly 4/3.

That is a large margin for a result nobody uses. A search over twenty-four segments, normalising the total area and hill-climbing on the eigenvalue, reaches 1.32 here — the shortfall from 4/3 being the discretisation rather than the search — and the profile it finds is a spindle: thickest at mid-height, tapering smoothly to nothing at both pins.

Two reasons nobody builds it, and both are worth stating because they are the interesting part.

The optimum is flat near its peak and sharp near its ends. Most of the 33% is available from a mild taper, and the last few per cent demand a section that goes to zero area at the pins — where it has to carry the whole axial reaction. The optimum is unbuildable at exactly the point where it earns its last gain.

And it is optimal for one thing only. The spindle is the strongest column of that volume against buckling about that axis at that end condition. It is a poor column for an accidental moment, a poor column in fire, and a much worse column than the uniform one if the ends turn out to be more restrained than assumed — because then the mode has curvature near the ends, where there is nothing left.

The built-up column, which is the same argument in disguise #

A column made of two chords laced or battened together is a column with a stiffness that varies not along its length but with the mode it is asked about, and the machinery is the same.

About the axis through both chords the section is solid and behaves as its second moment says. About the axis between them the two chords have to shear relative to one another for the column to bend, and the lacing has to carry that shear — so the effective stiffness is the full second moment reduced by a shear flexibility, and it falls as the lacing gets weaker. The column made of two columns is that reduction, and a battened column with light battens can be far weaker than the sum of its parts suggests.

What makes it the same argument is where the shear is. The shear in the mode goes as $v'$, which for a pinned column is largest at the ends — so lacing at the ends is what matters, and lacing at mid-height carries almost nothing. That is the mirror image of the bending result, and it is why the two rules for a laced column sound contradictory: put the material at mid-height and put the lacing at the ends.

The column that stops, and the one that changes floor #

Two cases in real buildings are stepped columns and are not usually recognised as such.

A column that changes section at a floor is a stepped column whose steps are at the storey heights. If the floors provide full lateral restraint then each storey is its own column and the steps are irrelevant; if they do not — a mezzanine, a plant deck, a floor with a large void beside the column — then the buckling length spans two storeys and the step is inside it. The practical failure is to check each storey separately when the mode does not stop at the floor, which is the column that leans on its neighbours seen one column at a time.

A column supporting a transfer structure has a step of a completely different kind: the load steps, not the section. Above the transfer the column carries a fraction of what it carries below, and the geometric stiffness integral is dominated by the heavily loaded part. The column that stops is about the load path; the stability consequence is that the effective length is decided by the lower part almost alone.

The general point in both is that the mode does not respect the drawing’s divisions. A column is not the piece between two floor levels; it is whatever length has to bend for the structure to fail, and that is a property of the restraints rather than of the setting-out.

Doing it properly, which is cheap #

None of the above is an argument for a hand method, because the eigenvalue is not expensive.

Discretise the member into twenty or thirty beam elements, assemble $\mathbf{K}$ from each element’s own $EI$ and $\mathbf{K}_g$ from its own axial force, apply the end restraints and find the smallest $P$ at which the pencil becomes singular. That is a few dozen lines and a few milliseconds, it reproduces $\pi^2EI/L^2$ to seven figures for the uniform case, and it handles a step, a taper, a varying axial force, an intermediate spring and a partly fixed end without any of them being a special case.

The reason to know the weighting anyway is not to avoid the computation. It is to know what to expect from it, and to notice when it is wrong. An eigenvalue solver that reports a critical load above the average estimate has either been given a stiffness distribution that is heavy in the middle or has an error in it, and being able to say which without re-running is the difference between using a tool and trusting one.

That is also the argument for the two hand bounds. They are worthless as design values and valuable as a bracket: the answer must be between the weakest-segment value and the average value, and a computation outside that range is a computation to look at again.

Where the model stops #

The load was constant along the column. A column carrying its own weight has an axial force that varies, and the geometric stiffness integral has to carry that variation — too tall for nothing but itself is that problem, and its answer involves Bessel functions rather than a sine.

The ends were pinned. Everything about the weighting changes with the mode. A fixed-fixed column has curvature at the ends and at mid-height and a zero at the quarter points, so a weak band at the quarter point costs nothing and one at the end costs a great deal — precisely inverted from the pinned case.

The behaviour was elastic. A stocky column yields before it buckles and none of this applies; a column of intermediate slenderness does both, and the effective stiffness in the partly yielded region is the tangent modulus rather than $E$. The column that had yielded before it was loaded is what residual stresses do to that, and it makes the “stiffness” a function of the load — so the eigenvalue problem becomes non-linear and the averaging question gets worse rather than better.

Only the first mode was found. A stepped column can have a second critical load very close to its first — a column with two similar weak bands, or one with an intermediate restraint near the load at which the mode number changes — and two nearly coincident eigenvalues make a structure far more sensitive to imperfection than either alone. Two ways of buckling at once is what that costs, and the check is to find two eigenvalues rather than one.

And the column was assumed straight. A real one is not, and its response is a growth rather than a bifurcation. The interesting consequence is that a column with a weak band develops its imperfection growth there, so the weak band is also where the second-order moment is largest, and the two effects compound.

The generalisation #

The idea worth carrying is that an eigenvalue is a weighted average, and the weight is the mode.

That single sentence covers a surprising amount. A structure’s fundamental frequency is a weighted average of its stiffness with the mode’s curvature as the weight and of its mass with the mode’s displacement — which is why most of the mass moves together and why a heavy plant room at the top of a building matters far more than the same mass in the basement. A buckling load is a weighted average of $EI$. A modal damping ratio is a weighted average of the local damping. In every case the weighting is not uniform, and in every case the naive average is what a designer reaches for.

The practical form of the rule is a question: where does the mode put its energy? Answer that first and the weighting follows, and with it the answer to where a weakening hurts, where a stiffening helps and where neither does anything at all. A structure is not equally sensitive everywhere, and the map of its sensitivity is a picture of its mode.

There is a corollary about design that follows immediately and is worth stating on its own. Material added where the mode has no energy is material wasted, and material removed there is free. That is the licence behind every haunched frame, every tapered mast, every castellated beam and every voided slab — each of them is a structure that has noticed where its own governing mode does nothing and has taken the material out of exactly that place. The stepped column is the same observation with the sign reversed: a step put in for a reason unconnected with stability lands wherever it lands, and if it lands in the middle it has cost a third of the column.