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