Moment-curvature analysis

Theory behind nonlinear moment-curvature (M-kappa) analysis for reinforced and prestressed concrete sections.

Introduction

Moment-curvature (M- $\kappa$ ) analysis traces the full nonlinear response of a reinforced concrete cross-section from initial loading through cracking, yielding, and ultimate failure. Unlike the simplified rectangular stress block approach used for routine design, M- $\kappa$ analysis uses the actual nonlinear stress-strain relationship of the concrete and steel to compute the moment capacity at every level of curvature.

This analysis reveals information that simplified methods cannot provide: cracking moment, yield moment, ultimate curvature, ductility ratio, and energy absorption capacity. These quantities are essential for seismic design, progressive collapse assessment, and understanding member behaviour under extreme loading.

Background

The concept of moment-curvature analysis for reinforced concrete was formalised by Hognestad, Hanson, and McHenry in the 1950s and extended by Park and Paulay in their foundational text Reinforced Concrete Structures (1975). The method is rooted in the Euler-Bernoulli beam hypothesis: plane sections remain plane after bending.

Under this assumption, the strain at any fibre is linearly proportional to its distance from the neutral axis:

\varepsilon(y) = \kappa \cdot (y - y_{NA})

Where:

$\varepsilon(y)$ = strain at depth $y$
$\kappa$ = curvature (1/mm)
$y_{NA}$ = neutral axis depth

Mathematical formulation

Strain distribution

For a given curvature $\kappa$ and neutral axis depth $c$ , the strain at any fibre location $y$ measured from the extreme compression fibre is:

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

Where $\varepsilon_{top}$ is the strain at the extreme compression fibre. The curvature is:

\kappa = \frac{\varepsilon_{top}}{c}

Concrete stress

The concrete stress at each fibre depends on the strain and the selected stress-strain model. For the Hognestad model:

\sigma_c(\varepsilon) = \begin{cases} f'_c \left[ \frac{2\varepsilon}{\varepsilon_0} - \left(\frac{\varepsilon}{\varepsilon_0}\right)^2 \right] & \text{for } \varepsilon \leq \varepsilon_0 \\ f'_c \left[ 1 - 0.15 \cdot \frac{\varepsilon - \varepsilon_0}{\varepsilon_u - \varepsilon_0} \right] & \text{for } \varepsilon_0 < \varepsilon \leq \varepsilon_u \end{cases}

Where:

$f'_c$ = compressive strength (MPa)
$\varepsilon_0$ = strain at peak stress (typically $2f'_c / E_c$ )
$\varepsilon_u$ = ultimate compressive strain (typically 0.003—0.004)

Concrete in tension is assumed to carry no stress after cracking (tension stiffening can be optionally included).

Reinforcement stress

Steel stress follows the elastic-perfectly-plastic model:

\sigma_s(\varepsilon) = \begin{cases} E_s \cdot \varepsilon & \text{for } |\varepsilon| \leq \varepsilon_y \\ f_y \cdot \text{sign}(\varepsilon) & \text{for } |\varepsilon| > \varepsilon_y \end{cases}

Where $\varepsilon_y = f_y / E_s$ .

Equilibrium

For a given curvature $\kappa$ , the neutral axis depth $c$ is found by enforcing axial force equilibrium:

\sum F = \int_A \sigma_c \, dA + \sum_i A_{si} \cdot \sigma_{si} - N^* = 0

Where:

The integral is over the concrete compression zone
$A_{si}$ and $\sigma_{si}$ are the area and stress of each reinforcement bar
$N^*$ is the applied axial force (zero for pure bending)

This equation is solved iteratively using Newton-Raphson or bisection method.

Moment

Once equilibrium is satisfied, the moment is computed by summing moments about the centroid:

M = \int_A \sigma_c \cdot (y - y_{ref}) \, dA + \sum_i A_{si} \cdot \sigma_{si} \cdot (y_i - y_{ref})

Algorithm

The M- $\kappa$ analysis proceeds as follows:

Start with a small curvature increment $\Delta\kappa$
For each curvature step: a. Assume a trial neutral axis depth $c$ b. Compute the strain at every concrete fibre and bar c. Evaluate stresses using the stress-strain models d. Check axial equilibrium; iterate on $c$ until $\sum F = N^*$ e. Compute the moment $M$
Record the $(M, \kappa)$ pair
Increment curvature and repeat until the concrete crushes ( $\varepsilon_{top} \geq \varepsilon_u$ ) or steel ruptures

Key points on the M- $\kappa$ curve

Point	Definition	Engineering significance
Cracking ( $M_{cr}$ , $\kappa_{cr}$ )	Concrete tensile stress reaches $f_{ct}$	Onset of visible cracking; stiffness drops
Yield ( $M_y$ , $\kappa_y$ )	Tension steel reaches $f_y$	Beyond this point, deformation increases rapidly
Ultimate ( $M_u$ , $\kappa_u$ )	Concrete crushes or steel ruptures	Maximum capacity; section fails

Ductility ratio

\mu_\kappa = \frac{\kappa_u}{\kappa_y}

A ductility ratio $\mu_\kappa > 3$ is generally considered ductile. Higher values indicate more deformation capacity before failure, which is desirable for seismic and progressive collapse resistance.

Energy absorption

The area under the M- $\kappa$ curve represents the energy absorption capacity of the section:

U = \int_0^{\kappa_u} M(\kappa) \, d\kappa

This quantity is relevant for impact and blast loading scenarios.

Implementation in ACS

ACS implements M- $\kappa$ analysis using a fibre-based discretisation of the cross-section. The number of fibres (default 100) determines the resolution of the concrete stress integration. More fibres give more accurate results at the cost of computation time.

The analysis:

Uses the user-selected stress-strain model for concrete
Supports any cross-section shape (not limited to rectangles)
Handles voids by excluding void regions from the fibre mesh
Accounts for prestressing tendons with initial strain offset
Can generate curves at multiple axial load levels simultaneously

Prestressed sections

When the section contains prestressing tendons, each tendon enters the fibre equilibrium with an initial tensile strain that is present before any applied loading. For bonded tendons the initial strain at each curvature step is:

\varepsilon_p = \varepsilon_{pe} + \varepsilon_{decomp} - \varepsilon_c

Where:

$\varepsilon_{pe} = \sigma_{p.ef} / E_p$ — effective prestrain after all losses
$\varepsilon_{decomp} = \sigma_{cp} / E_c$ — elastic decompression strain locked into the section at the tendon centroid (the concrete strain that existed before any applied moment)
$\varepsilon_c$ — concrete strain at the tendon location at the current curvature step

For unbonded tendons the tendon strain is held constant at the effective prestrain: $\varepsilon_p = \varepsilon_{pe}$ . The tendon cannot develop additional strain compatible with local concrete deformations.

This $\sigma_{p.ef}$ hold is specific to the moment-curvature (service-through-ultimate) sweep. The ultimate flexural check instead gives an unbonded tendon the code’s simplified $\sigma_{pu}$ (AS 3600 Cl. 8.1.8 / ACI 318-19 Cl. 20.3.2.4.1), which is an ultimate-state quantity and would be wrong at the service end of this curve.

The cracking point ( $M_{cr}$ , $\kappa_{cr}$ ) on the M- $\kappa$ curve uses the same $M_{cr,t}$ formula as the PT cracking moment panel, so both panels always display an identical cracking moment for the same section and prestress state.

The M- $\kappa$ interaction surface extends this to biaxial bending by sweeping the bending angle from 0 $^\circ$ to 360 $^\circ$ and generating a 3D surface of $(M_x, M_y, N)$ triplets.

Limitations and assumptions

Plane sections remain plane (Euler-Bernoulli hypothesis)
No bond-slip between steel and concrete
Uniaxial stress state in concrete (no confinement modelling unless Mander model is selected)
Monotonic loading only (no cyclic or hysteretic behaviour)
Shear deformations not included (pure flexural response)

Moment-curvature analysis

Introduction

Background

Mathematical formulation

Strain distribution

Concrete stress

Reinforcement stress

Equilibrium

Moment

Algorithm

Key points on the M-κ\kappaκ curve

Ductility ratio

Energy absorption

Implementation in ACS

Prestressed sections

Limitations and assumptions

Further reading

Key points on the M- $\kappa$ curve