Two ways to fail, and the curve between them

Almost nothing in a real structure carries only one kind of action. A column in a frame that resists sway by bending its corners carries axial force from the floors above and moment from the beams framing into it. An arch rib carries thrust and, wherever the line of thrust departs from the arch’s own centreline, bending as well. A truss chord with a load applied between its panel points carries both.

So a section has two capacities and needs a rule for how they share. The rule almost universally applied is that they share in proportion: $N/N_{pl} + M/M_{pl} \le 1$, a straight line between the two intercepts. It is easy to remember, obviously safe-looking, and it is wrong by a great deal.

Why there is a curve at all #

The exact curve is easier to compute than the approximate rule is to justify, and it needs no iteration whatever.

Take the section fully plastic: every fibre at $+f_y$ or $-f_y$, depending on which side of some horizontal axis it lies. Slide that axis from the top face to the bottom. At each position, the axial force is the yield stress times the difference between the compressed and tensioned areas, and the moment is the yield stress times the difference between their first moments. Both fall out of areas; no stress-strain law is involved, no curvature is imposed, and nothing has to be solved.

With the axis at mid-depth of a symmetric section the areas balance, the axial force is zero, and the moment is the plastic moment. Slide it to the top face and everything is in tension: the axial force is the full squash load and the moment is zero. Every intermediate position is a point on the curve, and sweeping the axis traces the whole thing.

For a rectangle the arithmetic closes in one line. With the axis a distance $a$ from mid-depth, the unbalanced area is $2ab$ so $n = N/N_{pl} = 2a/d$, and the lost moment is that of a central block of depth $2a$, which is $f_y b a^2$. So

$$\frac{M}{M_p} = 1 - \left(\frac{2a}{d}\right)^2 = 1 - n^2$$

The interaction is a parabola, and at $n = 0.5$ it gives $M/M_p = 0.75$ where the straight line gives 0.50. The figure’s 25.0% is exactly $1 - n^2 - (1 - n)$ at $n = 0.5$, computed by a routine that has never seen the closed form.

Why the two sections differ so much #

The rectangle’s bulge is enormous and the I-section’s is small, and the reason is the same one that runs through the whole shape-factor argument.

Axial force is carried by area, wherever it is. Moment is carried by area far from the middle. A section that has arranged nearly all of its material in two flanges has almost nothing near the neutral axis to give to the axial force, so the axial force has to eat into the flanges, and eating into the flanges costs moment quickly. A rectangle has a third of its area within a sixth of the depth of the axis, doing very little for the moment, and the axial force can be carried there almost for free.

So the efficient section for bending is the inefficient one for combined action, which is the reverse of the pattern everywhere else on this site and is worth stating clearly: the I-section wins on bending, wins on axial force, and loses on the interaction between them.

Which free body produced the number, and about which point #

This site’s standing obligation is to name the free body that produced a number. For an interaction curve that obligation has a second half which is not optional: a moment is not defined until the point it is taken about is named, and here the choice changes the answer.

The free body is the fully plastic cross-section with the stress block on the cut face. The axial force is unambiguous — it is the integral of the stresses and does not care where anything is measured from. The moment is the integral of the stresses against distance from some datum, and there are two candidates.

For every section drawn so far the two coincide, because both sit at mid-depth. For a tee they are 38.5 mm apart: the elastic centroid, where the first moment of area vanishes, is 149 mm up; the equal-area axis, where the areas above and below balance, is 187.5 mm up.

That is the whole of it, and the check that makes it trustworthy is that for the symmetric sections the two references give identical curves to the last digit, as they must. A difference that appears only for the section where the two points differ, and vanishes for the sections where they do not, is a difference caused by the reference point and by nothing else.

What the axial force does to the section itself #

The interaction curve is a boundary, and boundaries say nothing about what is happening inside them. What is happening is that the neutral axis moves, and it moves a long way.

For pure bending on a symmetric section the axis is at mid-depth, because that is where equal areas above and below produce equal and opposite resultants. Add compression and the compressed zone has to grow, so the axis descends. At the squash load it has left the section entirely, and every fibre is in compression.

This is the same migration that a cracked reinforced concrete section undergoes for a different reason. There the axis moves because part of the material has stopped working; here it moves because the resultant it has to produce is no longer zero. Both are the same statement — the neutral axis sits wherever the axial force on the section comes out right — and neither is a definition.

It is worth noticing what has not changed between the figures. The strain diagram is a straight line in all of them, at every load, with and without axial force. Everything interesting is happening between that line and the stress diagram, and the axial force enters only by changing where the line crosses zero.

Where the model stops #

All of this is a section calculation and a column is a member. The interaction drawn here is between axial force and moment at a cross-section. A slender column has a third failure mode neither axis contains: it can buckle, and it can buckle at an axial force well below the squash load with a moment well below the plastic moment. The design interaction for a real column is therefore a much lower and differently shaped surface, and this one is its upper limit.

And the moment on a column is not the moment applied to it. A column that has deflected carries its axial load off the axis, which adds a moment, which increases the deflection. The amplification is $1/(1 - N/N_{cr})$ and it runs away long before the buckling load. A member checked against this curve using the first-order moment is being checked against the wrong point on it.

The curve assumes full plasticity, which assumes rotation. Reaching any point on it requires the section to yield throughout, which requires the curvature demands of the moment-curvature essay to be met, which requires ductility nobody has checked and plates that do not buckle first.

Bending is drawn about one axis. A real column bent about both has a three-dimensional interaction surface, and the section of it in any plane containing the axial axis is a curve of this kind. The rule usually applied to combine them is again linear, and again conservative, and again by an amount nobody computes.

The generalisation #

The pattern is that a linear interaction rule is the honest answer to a question nobody wants to solve, and its error is always in the same direction.

Two capacities that draw on the same material do not generally share it in proportion, because they draw on different parts of it. Whenever the two demands can be satisfied by different material within one member, the true interaction bulges outward and a straight line is conservative. Whenever they compete for the same material — as axial force and moment do in a section that is all flange — the bulge closes up and the straight line is nearly right.

That gives a usable rule of thumb with no calculation in it: the more efficient the section is at the individual actions, the closer the linear interaction is to the truth. A rectangle, which is bad at bending, has a huge bulge. An I-section, which is good at it, has almost none. The reserve the linear rule ignores is exactly the reserve a good section did not have in the first place.

What the picture cannot show #

An interaction diagram is a picture of a boundary, and three things it does not contain matter more than the shape of the boundary does.

It has no length in it, so it cannot show buckling. Every point on every curve here describes a cross-section that has yielded throughout. A slender member fails long before any of its sections do, by going sideways, and the axis it goes sideways about may not be the axis the moment is applied about. The interaction surface for a real column is this curve pulled down towards the origin by an amount that depends on slenderness, and the amount is not a property of the section at all.

It has no sequence in it. The curve says which combinations of $N$ and $M$ the section can hold, and says nothing about how it got there. A section taken to a point on the boundary by increasing $N$ at constant $M$ has a different residual state from one taken to the same point the other way round, because plasticity is path-dependent — and the difference matters as soon as the load is applied more than once.

And the two axes are not measured in the same way. The axial force on a column is usually known to within a few per cent, since it is mostly the weight of what is above. The moment is not: it depends on the stiffness of every beam framing in, on the fixity of the bases, and on how far the frame has already leaned. Reading a design point off this diagram gives an accuracy on one axis that the other cannot support.

A surprising place this turns up #

The parabola $M/M_p = 1 - n^2$ has an immediate consequence that is easy to miss: for small axial loads, the moment capacity is essentially untouched.

At $n = 0.1$ the rectangle keeps 99% of its plastic moment. At $n = 0.2$ it keeps 96%. The axial force is free until it is not, and then it becomes expensive quickly — the curve is flat at the top and steep at the bottom, because it is a parabola in $n$ rather than a line.

The flatness is not a rectangle’s peculiarity. Every curve on this page leaves the moment axis horizontally, and for a reason that needs no algebra: at zero axial force the plastic neutral axis is at the balance point, and moving it a little transfers a little area from one side to the other — which changes the axial force in proportion to that area, and changes the moment in proportion to the area times its distance from the axis, which is zero at the axis. The moment is stationary there. Any interaction curve of this kind is flat where it meets the moment axis, whatever the section.

Codes recognise this with a threshold below which axial force may be ignored entirely in a beam’s design, and the threshold looks arbitrary until it is read off this curve. It is the axial ratio at which the moment capacity has fallen by some agreed small amount, and its existence is a statement about the shape of a parabola rather than a concession. A beam carrying an incidental tie force is genuinely a beam, and the arithmetic says so.

Where the ladder goes next #

Later rungs on this anchor: the interaction including shear, which is a third axis and reduces the yield stress available for bending by a von Mises factor. The buckling interaction for real columns, and where the amplification factor enters it. Biaxial bending and the interaction surface in three dimensions. The interaction curve for a reinforced concrete section, which is not convex at all and has a balance point where the failure mode changes from steel-governed to concrete-governed — the one place in the subject where more axial load genuinely does increase moment capacity, and for a reason that has nothing to do with a reference axis. Interaction in connections, where bolt groups carry tension and shear together. And the yield surface as a mathematical object, of which every curve here is a section.

Historically the linear rule predates the exact one and has outlived several attempts to replace it. Exact plastic interaction curves for standard sections were computed in the 1950s as part of the same programme that produced plastic design, and were immediately simplified back into piecewise-linear approximations for use — not out of conservatism, but because a designer working by hand needed an expression that could be inverted to find a section, and a parabola in two variables cannot be. That constraint disappeared some decades ago and the rule has not.