Module 10 · Lesson 10.1

Axial load and moment together

Why the two cannot be checked separately, and what the interaction diagram is really showing.

Why this matters

A beam has one capacity: a moment of resistance. A column has a different moment capacity for every axial load it might be carrying — and the relationship is not monotonic. Compression helps up to a point and then hurts badly. A designer who checks axial load and moment as two separate sums will be wrong in both directions at once, and will not know which.

By the end of this lesson you should be able to

Explain the interaction physically, in terms of the compression zone
Construct a point on the interaction diagram from strain compatibility
Locate the balance point and say what it separates
Recognise which side of the diagram a given design sits on

What you should already know

Strain compatibility and the stress block (Module 4)
Ductility and the x/d limit (Module 4)
Equilibrium of a section (Module 4)

The same three principles, one more unknown

Nothing new is needed. A column section is analysed exactly as a beam section was in Module 4: material models, strain compatibility, equilibrium.

What changes is the bookkeeping. In a beam, equilibrium said the compression force equals the tension force, C = T, because there was no external axial load. In a column there is one, so:

C − T = N_Ed, and the moment of that force system about the section centroid is M_Ed.

Two equations, and a free choice of neutral-axis depth x. Every value of x satisfies both — it just corresponds to a different combination of N and M. Sweep x from very small to beyond the section depth, plot each resulting (N, M) pair, and you have traced the interaction diagram: the complete boundary of what the section can carry.

A column section is reinforced near BOTH faces, because the moment can act either way and because the bars are needed in compression as well as tension. That symmetry is why an interaction diagram for a column is symmetric about the moment axis.

Why compression helps, at first

This is the result that surprises people, and it is worth getting the physical picture rather than trusting the curve.

Take a section in pure bending. The neutral axis is shallow, the compression zone is small, and the lever arm between C and T is large — but C itself is limited by how little concrete is in compression.

Now add a modest axial compression. Equilibrium requires a larger compression force, so the neutral axis moves down and the compression zone deepens. The tension steel is still yielding, so T is unchanged. The couple that resists moment is now larger, because C has grown while the lever arm has shortened only slightly.

So the moment capacity rises.

Push further, and the neutral axis passes the point where the tension steel still yields. Now T falls too, the lever arm shortens sharply, and the moment capacity collapses towards zero at the squash load.

Predict first

A column section can resist 202 kNm of moment when carrying 500 kN of axial compression. What moment can it resist at 1000 kN?

Try it

Build the M–N interaction diagram

Change the section and watch the whole envelope move. The curve is computed by scanning the neutral-axis depth and applying equilibrium at each — it is not a drawn shape.

Depth h400 mm

Width b300 mm

Concrete f_ck30 N/mm²

Number of bars4 total

Bar diameter25 mm

Design point

Axial load N_Ed1,000 kN

Moment M_Ed150 kNm

Steel provided: 1963 mm² (4 H25)
Squash load: 2860 kN
Balance point: 864 kN, 226 kNm
M_Rd at this N_Ed: 216 kNm
Utilisation: 69%
Failure mode here: brittle — concrete crushes

Inside the envelope. At 1000 kN the section can carry 216 kNm, and 150 kNm is applied.

Things worth trying

Set N_Ed to 0 and raise it slowly. The capacity rises to the balance point before it falls — the opposite of what most people expect.
Add bars. The whole envelope grows, but the balance point barely moves: it is fixed by strain compatibility, not by how much steel is there.
Raise f_ck. The squash load climbs steeply while the peak moment grows only modestly — concrete grade buys axial capacity, not moment capacity.
Increase h and watch both axes grow. Depth is the only change that helps everywhere at once.

Four things are worth doing deliberately in that diagram, because each one isolates a different mechanism.

Sweep N_Ed from zero upward at fixed M_Ed. The design point rises through the envelope, and the horizontal distance to the curve — the spare moment capacity — grows at first. Only past the balance point does it start to shrink. Nothing else in concrete design behaves this way.

Add bars. The envelope grows everywhere, but the balance point hardly moves along the N axis. That is because the balance point is fixed by strain compatibility — the neutral-axis depth at which the steel reaches yield exactly as the concrete crushes — and that depth depends on the material strains, not on how much steel is present.

Raise f_ck. The squash load climbs steeply; the peak moment barely moves. Concrete grade buys axial capacity, and very little moment capacity. If a column is failing on moment, a stronger mix is close to useless.

Increase h. Both axes grow together. Depth is the only variable that improves everything at once, which is why it is the first thing to change when a column will not work.

Note

The balance point, and which side you want to be on

The balance point is where the concrete reaches its crushing strain at the same instant the tension steel reaches yield. It is the peak of the interaction diagram — the largest moment the section can ever carry.

It is also the boundary between two quite different failures:

Below it (lower axial load): the steel yields first. The column deflects visibly, cracks widen, and there is warning. This is the ductile side.
Above it (higher axial load): the concrete crushes while the steel is still elastic. Failure is sudden and explosive. This is the brittle side.

Most real building columns sit above the balance point, because axial loads in buildings are large. That is a fact worth sitting with: the typical column is on the brittle side of its own interaction diagram, and this is why column links, confinement and the minimum-eccentricity rule matter so much more than their small contribution to strength would suggest.

Worked example

Points on the interaction diagram of a 300 × 400 column

Given

Section 300 mm × 400 mm, C30/37, f_cd = 17.0 N/mm²
Reinforcement 1963 mm² total (4 H25), half near each face, 50 mm to the bar centres
Grade 500 steel, f_yd = 434.8 N/mm²

Find

The squash load, the balance point, and the moment capacity at several axial loads.

Assumptions

Rectangular stress block, ε_cu3 = 0.0035
Elastic–perfectly plastic steel
Moments taken about the SECTION CENTROID, not the tension steel

Practice

A 300 × 400 column has 1963 mm² of steel and C30/37 concrete (f_cd = 17.0, f_yd = 434.8 N/mm²). What is its squash load, in kN?

Practice

For the same section (d = 350 mm), at what neutral-axis depth does the balance point occur, in mm? Take ε_cu3 = 0.0035 and ε_yd = 0.00217.

Check yourself

Where on the interaction diagram is a column's failure most sudden?

Summary

C − T = N_Ed, and the moment of that system about the centroid is M_Ed
Sweeping the neutral-axis depth traces the whole interaction diagram
Below the balance point, adding axial load INCREASES moment capacity
Above it, moment capacity falls away towards zero at the squash load
The balance point separates ductile steel-controlled failure from brittle concrete crushing
Most building columns sit above the balance point, on the brittle side

Progress is kept in this browser only.

This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint