M-N interaction

Theory behind axial force-moment interaction diagrams for reinforced concrete columns and beam-columns, including biaxial bending methods.

Introduction

Concrete columns and beam-columns resist combined axial force and bending moment simultaneously. The interaction diagram defines the complete envelope of ( $N$ , $M$ ) combinations that the section can resist at ultimate. Any design point falling inside this envelope is adequate; any point outside indicates failure.

Understanding the interaction diagram is essential for column design because the moment capacity depends strongly on the axial load level — moderate compression increases moment capacity (up to the balanced point), while high compression reduces it.

Background

The interaction diagram concept was developed in the 1960s as researchers recognised that treating axial force and bending independently was unconservative for reinforced concrete. The key insight is that concrete is strong in compression but weak in tension, so axial compression pre-compresses the tension zone and delays cracking, initially increasing the moment capacity.

Mathematical formulation

Uniaxial interaction

For a given neutral axis depth $c$ , the section is at a unique ( $N$ , $M$ ) state. By sweeping $c$ from zero (pure tension) to infinity (pure compression), the full interaction curve is traced.

At each value of $c$ :

The strain distribution is linear with $\varepsilon_{top} = \varepsilon_{cu}$ (ultimate concrete strain, typically 0.003):

\varepsilon(y) = \varepsilon_{cu} \cdot \frac{c - y}{c}

Concrete force:

C_c = \alpha_2 \cdot f'_c \cdot \gamma \cdot c \cdot b

Where $\alpha_2$ and $\gamma$ are stress block parameters (code-dependent).

Steel forces at each bar layer:

F_{si} = A_{si} \cdot \sigma_s(\varepsilon_i)

Axial capacity: $N_u = C_c + \sum F_{si}$
Moment capacity: $M_u = C_c \cdot \bar{y}_c + \sum F_{si} \cdot (y_i - y_{ref})$

Special points

Point	Condition	Physical meaning
Pure compression ( $N_{u0}$ )	$c \to \infty$ , all concrete and steel in compression	Maximum axial load, zero moment
Balanced ( $N_b$ , $M_b$ )	$\varepsilon_s = \varepsilon_y$ at the tension steel	Simultaneous crushing and yielding
Pure bending ( $M_u$ )	$N = 0$ , force equilibrium with zero net axial force	Maximum moment, zero axial load
Pure tension ( $N_{t}$ )	All steel in tension, concrete ignored	Maximum tensile capacity

Strength reduction factor

The $\phi$ factor varies with the axial load level:

Code	Compression-controlled	Transition	Tension-controlled
AS 3600	$\phi = 0.65$	Linear interpolation	$\phi = 0.85$
ACI 318	$\phi = 0.65$	Linear interpolation	$\phi = 0.90$
EN 1992	$\phi = 1.0$ (uses partial factors on materials instead)	—	—

Biaxial bending

When both $M_x$ and $M_y$ are present, the uniaxial interaction diagram is insufficient. The capacity becomes a 3D surface in ( $N$ , $M_x$ , $M_y$ ) space.

Rigorous method

ACS generates the full 3D surface by sweeping the neutral axis orientation angle $\theta$ from 0 $^\circ$ to 360 $^\circ$ at each value of $c$ . For each ( $c$ , $\theta$ ) combination, the equilibrium equations give a unique ( $N$ , $M_x$ , $M_y$ ) point on the surface.

The design point is then checked by interpolation: if it lies inside the surface, the section is adequate.

Bresler reciprocal method

A simplified approach per AS 3600 Cl. 10.6.4:

\frac{1}{N_u} = \frac{1}{N_{ux}} + \frac{1}{N_{uy}} - \frac{1}{N_{u0}}

Where:

$N_u$ = biaxial capacity at ( $M_x$ , $M_y$ )
$N_{ux}$ = uniaxial capacity at $M_x$ alone (from $N$ - $M_x$ curve)
$N_{uy}$ = uniaxial capacity at $M_y$ alone (from $N$ - $M_y$ curve)
$N_{u0}$ = squash load (pure compression)

This method is accurate for sections with symmetric reinforcement but can be unconservative for highly asymmetric layouts.

Bresler load contour method

An alternative per ACI 318-19:

\left(\frac{M_x}{M_{ux}}\right)^\alpha + \left(\frac{M_y}{M_{uy}}\right)^\alpha \leq 1.0

Where:

$M_{ux}$ , $M_{uy}$ = uniaxial moment capacities at the applied axial load $N^*$
$\alpha$ = contour exponent (typically 1.0 to 2.0; 1.5 is common for rectangular sections)

Design standard treatment

Aspect	AS 3600:2018	ACI 318-19	EN 1992-1-1
Uniaxial method	Rectangular stress block	Rectangular stress block	Parabolic-rectangular or rectangular
$\varepsilon_{cu}$	0.003	0.003	0.0035
Biaxial simplified	Bresler reciprocal (Cl. 10.6.4)	Bresler contour (R6.3.3)	Cl. 5.8.9 (contour method)
$\phi$ approach	Variable $\phi$	Variable $\phi$	Partial material factors ( $\gamma_c = 1.5$ , $\gamma_s = 1.15$ )

Implementation in ACS

ACS generates interaction diagrams by:

Sweeping the neutral axis depth $c$ over 100 evenly-spaced values, from near-zero (pure tension) up to 1.5 times the section depth measured in the bending frame
At each depth, solving for force equilibrium and computing ( $N$ , $M$ )
Applying the code-specific $\phi$ factor at each point
For biaxial analysis, repeating at multiple angles (default 24 slices for the 3D surface)

The computation uses the user-selected stress-strain model. The rectangular stress block matches simplified code calculations; Hognestad or Mander models provide more realistic post-peak behaviour for advanced analysis.

Navigating the interaction surface

The 3D N– $M_x$ – $M_y$ surface on the Interaction tab is an interactive viewport. Because it is a solid rather than a flat drawing, its controls differ from the 2D canvases in one respect — left-drag rotates the surface rather than panning it:

Action	Control
Rotate	Left-drag
Move in / out	Scroll wheel
Pan	Middle-drag, or Ctrl (⌘) + left-drag
Reset the camera	The house icon in the bottom-left corner

Rotating is usually what you want here: examining the balanced point, the pure-bending intercept, or closely-spaced design points near the boundary generally means viewing the envelope from a different angle rather than sliding it across the screen.

Middle-drag pans on this surface. It previously moved the camera in and out, which duplicated the scroll wheel and disagreed with every other canvas in the platform; the scroll wheel still does that job. See Canvas navigation for the controls shared across all surfaces.

Where the sweep stops

Step 1’s range covers the states that matter for bending, but not the top of the diagram. As $c$ grows the strain field flattens toward a uniform $\varepsilon_{cu}$ , and the axial force keeps climbing until every bar has passed its compression yield strain — which for AS 3600 grade-500 reinforcement takes

c \;\ge\; \frac{\varepsilon_{cu}}{\varepsilon_{cu} - \varepsilon_{y}} \, d \;=\; \frac{0.003}{0.003 - 0.0025} \, d \;=\; 6d

several times deeper than the geometric range above. ACS therefore continues the sweep past step 1 with geometrically-growing steps until the axial force stops increasing, so each curve (and, for the 3D surface, each of the 24 angle slices) ends at this uniform-strain limit rather than at an arbitrary depth.

This matters most for the biaxial surface. The uniform-strain limit does not depend on the bending angle $\theta$ — a uniform strain field does not know which axis you are bending about — so every one of the 24 angle slices terminates at the same point, and the top of the surface closes cleanly. Stopping instead at a fixed multiple of the rotated section depth $w|\sin\theta| + h|\cos\theta|$ would end each slice at a different physical state, distorting the surface near its apex.

The same rule handles a descending-branch stress model such as Hognestad: there the axial force reaches a peak and then falls as the whole section is driven onto the falling branch, and “stop when the axial force stops increasing” ends the sweep at that peak.

The pure-axial squash point $N_{u0}$ still sits above the top of the swept curve, and the step between them is real rather than an artefact of the sweep. Two code provisions separate them: the squash load uses the Cl. 10.6.2.2 stress factor $\alpha_1$ while the swept curve integrates the Cl. 8.1.3 bending stress block $\alpha_2 < \alpha_1$ , and AS 3600 Table 2.2.2 assigns the pure-axial point a different $\phi$ (0.65) from the compression-controlled surface ( $\phi_o$ ).

:::caution[Result change — 2026-07] The convergence-tail extension described above landed with an update that also invalidated previously cached interaction diagrams. Sections above the old geometric-sweep rim — the region that was previously truncated at a fixed multiple of the rotated section depth — now show correct values at the top of the curve. Existing cached results were automatically invalidated (CalculationVersions.ConcreteInteraction advanced from 2.11 to 2.12); any saved design computed before this update will recompute the first time it is opened. :::

Why the 3D surface steps at $M_y = 0$ and $M_x = 0$

The AS 3600:2018 Cl. 8.1.3 Note 2 (and the matching Cl. 10.6.2.5 Note 3) requires a further reduction to the stress block:

for any other sections where the width reduces from the neutral axis towards the compression face, $\alpha_2$ shall be reduced by 10 percent

ACS applies this per bending direction — that is, once per angle slice of the 3D surface, assessed on the section’s compression half in that slice’s own rotated frame. The consequence for a plain rectangular column is worth understanding before reading the surface:

Bent about a principal axis ( $\theta = 0^\circ$ , $90^\circ$ , $180^\circ$ , $270^\circ$ ), the compression zone is a rectangle of constant width. The Note does not apply, and $\alpha_2$ is used in full.
Bent about any skew axis, the compression zone is a triangle narrowing to a corner — its width does reduce toward the compression face. The Note applies, and $\alpha_2$ is reduced by 10%.

So the concrete contribution drops by 10% the moment the bending direction leaves a principal axis, and the 3D surface carries a visible step along the $M_y = 0$ and $M_x = 0$ meridians. This is intentional and follows the Note; it is not a meshing or interpolation artefact. The skew directions are the reduced — that is, conservative — ones; the principal axes are not penalised.

Two related behaviours:

The factor is fixed for the whole depth sweep of a slice, classified about the gross centroid rather than re-assessed at every neutral-axis depth. Re-assessing per depth would let the factor flip partway up a non-convex section’s sweep, which would put a step into the concrete force and break the monotonic ordering that the constant- $N$ contours and the design check rely on.
Circular sections take precedence and are reduced by 5% instead (Cl. 8.1.3 Note 2’s circular provision), applied at every angle. A circular column therefore shows no step — its surface is smooth in $\theta$ .

Prestressed sections

When the section contains prestressing tendons, they participate in the neutral-axis-depth sweep via full strain compatibility (AS 3600:2018 Cl. 8.1). At each value of $c$ and bending angle $\theta$ , the tendon strain is computed from the same fibre equilibrium used for the M- $\kappa$ curve:

\varepsilon_p(c) = \varepsilon_{pe} + \varepsilon_{decomp} - \varepsilon_{cu} \cdot \frac{c - y_p}{c}

Where $y_p$ is the tendon distance from the extreme compression fibre and $\varepsilon_{pe} + \varepsilon_{decomp}$ is the initial tensile strain from effective prestress and decompression (see the moment-curvature theory page for the full definition).

The tendon force is $F_p = A_p \cdot \sigma_p(\varepsilon_p)$ clamped at the yield force $A_p f_{py}$ . Because the tendon displaces concrete, the displaced concrete force $0.85 f'_c \cdot A_p$ (or code equivalent) is subtracted from the concrete resultant — identical to the netting used for reinforcing bars.

At the end-caps:

Squash load ( $N_{u0}$ , $c \to \infty$ ): all concrete and steel in compression; tendons contribute $A_p \cdot \min(f_{py},\, \sigma_{pe})$ toward compression.
Pure tension ( $N_t$ , $c = 0$ ): concrete ignored; tendons contribute $A_p \cdot f_{py}$ toward tension.

For reinforced (non-prestressed) sections the effectivePrestressMpa parameter defaults to zero, so the interaction diagram path is byte-identical to the pre-PT implementation.

Limitations and assumptions

Plane sections remain plane
No second-order (P- $\delta$ ) effects at the section level — these are member-level effects that must be handled separately
Slenderness effects are not included (the interaction diagram represents the cross-section capacity, not the member capacity)
No time-dependent redistribution in the interaction diagram