\(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-curve

General

The \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-curve of a cross-section is described by the moment \(M\), the axial-force between the sub-cross-sections \(N\), the curvature \(\kappa\) and the strain-difference \(\varepsilon_\mathrm{\Delta}\). It assumes the cross-section has two sub-cross-sections that move independently from each other.

The \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-curve composes in general of three types of points:

1. \(M\)-\(N\): where \(\kappa = 0\) and the rest is unequal zero

2. \(M\)-\(\kappa\): where \(N = 0\) and the rest is unequal zero

3. \(M\)-\(N\)-\(\kappa\): where all of the three parameters are unequal zero

Due to this definitions these points are computed independently from each other. Their computation is described hereafter.

\(M\)-\(N\)-curve

\(M\)-\(N\) points assume zero curvature and therefore an uniform axial-force on each of both sub-cross-sections, but with reverse sign.

computation of \(M\)-\(N\)-points

The strain-difference \(\varepsilon_\mathrm{\Delta}\) is then computed by adding the strain that occurs by the axial-force on each of the sub-cross-sections. In case each sub-cross-section consists of a number of sub-sections an equilibrium must be found.

An axial-force \(N\) is computed for each strain-point \(\varepsilon\) of the materials denoted to the given sub-(cross-)sections. After the axial-force \(N\) is computed for the first sub-cross-section from the given strain \(\varepsilon\) the same axial-force is applied to the other sub-cross-section, but with reversed sign.

The moment is computed by the multiplying the computed axial-force of each sub-section with the corresponding lever-arm. The lever-arm is in each case the distance between the top of the cross-section and the centroid of the axial-force on this sub-section.

\(M\)-\(\kappa\)-curve

A \(M\)-\(\kappa\)-point assumes that no axial-force \(N\) is transferred between the sub-cross-sections. Therefore, each sub-cross-section adds its individual resisting moment to the overall resisting moment. No composite-effect is considered.

computation of \(M\)-\(\kappa\)-points

To compute the full \(M\)-\(\kappa\)-curve first the \(M\)-\(\kappa\)-curves of the sub-cross-sections are computed as described here. Afterwards for each computed curvature \(\kappa\) the moment \(M\) of the other sub-cross-section is computed. Adding the corresponding moments of the sub-cross-sections results in the moment of this point.

The strain-difference \(\varepsilon_\mathrm{\Delta}\) is computed by evaluation of the strain of both sub-cross-sections at the same vertical position.

\(M\)-\(N\)-\(\kappa\)-curve

The \(M\)-\(N\)-curve and the \(M\)-\(\kappa\)-curve are edge-cases whereas the \(M\)-\(N\)-\(\kappa\)-curve describes most of the points. All values (\(M\), \(N\), \(\kappa\) and \(\varepsilon_\mathrm{\Delta}\) are unequal zero.

computation of \(M\)-\(N\)-\(\kappa\)-points

In this case for each axial-force \(N\) computed for the M-N-curve the corresponding material-point is used to compute first a curvature \(\kappa_\mathrm{fail}\) leading to a failure of the sub-cross-section considering also this axial-force \(N\).

Then, all material-points are collected that are include between \(\kappa_\mathrm{fail}\) and zero curvature also considering the axial-force.

This procedure is repeated for each material-point of both sub-cross-sections. In each case the computed curvature \(\kappa\) and the reversed axial-force \(N\) are applied to the other sub-cross-section. The resulting moments are then summed up forming an individual \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-point.

Summary

The M-N-curve, M-\kappa-curve, M-N-\kappa-curve form together the \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-curve. This curve (or more precise ‘surface’) describes the relationship between moment \(M\), axial-force \(N\), curvature \(\kappa\) and strain-difference \(\varepsilon_\mathrm{\Delta}\) for the given cross-section.

This relationship allows to consider the behaviour of the cross-section under considering the effect of slip, shear-connectors and axial-force in the sub-cross-sections.