Integraph

Moment-curvature analysis

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

Introduction

Moment-curvature (M-κ\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-κ\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:

ε(y)=κ⋅(y−yNA)\varepsilon(y) = \kappa \cdot (y - y_{NA})

Where:

  • ε(y)\varepsilon(y) = strain at depth yy
  • κ\kappa = curvature (1/mm)
  • yNAy_{NA} = neutral axis depth

Mathematical formulation

Strain distribution

For a given curvature κ\kappa and neutral axis depth cc, the strain at any fibre location yy measured from the extreme compression fibre is:

ε(y)=εtop⋅c−yc\varepsilon(y) = \varepsilon_{top} \cdot \frac{c - y}{c}

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

κ=εtopc\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:

σc(ε)={fc′[2εε0−(εε0)2]for ε≤ε0fc′[1−0.15⋅ε−ε0εu−ε0]for ε0<ε≤εu\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:

  • fc′f'_c = compressive strength (MPa)
  • ε0\varepsilon_0 = strain at peak stress (typically 2fc′/Ec2f'_c / E_c)
  • εu\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:

σs(ε)={Es⋅εfor ∣ε∣≤εyfy⋅sign(ε)for ∣ε∣>εy\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 εy=fy/Es\varepsilon_y = f_y / E_s.

Equilibrium

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

∑F=∫Aσc dA+∑iAsi⋅σsi−N∗=0\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
  • AsiA_{si} and σsi\sigma_{si} are the area and stress of each reinforcement bar
  • N∗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=∫Aσc⋅(y−yref) dA+∑iAsi⋅σsi⋅(yi−yref)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-κ\kappa analysis proceeds as follows:

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

Key points on the M-κ\kappa curve

PointDefinitionEngineering significance
Cracking (McrM_{cr}, κcr\kappa_{cr})Concrete tensile stress reaches fctf_{ct}Onset of visible cracking; stiffness drops
Yield (MyM_y, κy\kappa_y)Tension steel reaches fyf_yBeyond this point, deformation increases rapidly
Ultimate (MuM_u, κu\kappa_u)Concrete crushes or steel rupturesMaximum capacity; section fails

Ductility ratio

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

A ductility ratio μκ>3\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-κ\kappa curve represents the energy absorption capacity of the section:

U=∫0κuM(κ) dκU = \int_0^{\kappa_u} M(\kappa) \, d\kappa

This quantity is relevant for impact and blast loading scenarios.

Implementation in ACS

ACS implements M-κ\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

The M-κ\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 (Mx,My,N)(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)

Further reading

  • Park, R. and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, 1975. Chapters 3—4 cover M-κ\kappa analysis in detail.
  • Hognestad, E., Hanson, N.W., and McHenry, D., “Concrete Stress Distribution in Ultimate Strength Design,” ACI Journal, 1955.
  • Priestley, M.J.N., Seible, F., and Calvi, G.M., Seismic Design and Retrofit of Bridges, John Wiley & Sons, 1996. Chapter 4 for seismic applications of M-κ\kappa.